20214	----	[Transcriber's Notes] Conventional mathematical notation requires specialized fonts and typesetting conventions. I have adopted modern computer programming notation using only ASCII characters. The square root of 9 is thus rendered as square_root(9) and the square of 9 is square(9). 10 divided by 5 is (10/5) and 10 multiplied by 5 is (10 * 5 ). The DOC file and TXT files otherwise closely approximate the original text. There are two versions of the HTML files, one closely approximating the original, and a second with images of the slide rule settings for each example. By the time I finished engineering school in 1963, the slide rule was a well worn tool of my trade. I did not use an electronic calculator for another ten years. Consider that my predecessors had little else to use--think Boulder Dam (with all its electrical, mechanical and construction calculations). Rather than dealing with elaborate rules for positioning the decimal point, I was taught to first "scale" the factors and deal with the decimal position separately. For example: 1230 * .000093 = 1.23E3 * 9.3E-5 1.23E3 means multiply 1.23 by 10 to the power 3. 9.3E-5 means multiply 9.3 by 0.1 to the power 5 or 10 to the power -5. The computation is thus 1.23 * 9.3 * 1E3 * 1E-5 The exponents are simply added. 1.23 * 9.3 * 1E-2 = 11.4 * 1E-2 = .114 When taking roots, divide the exponent by the root. The square root of 1E6 is 1E3 The cube root of 1E12 is 1E4. When taking powers, multiply the exponent by the power. The cube of 1E5 is 1E15. [End Transcriber's Notes] INSTRUCTIONS for using a SLIDE RULE SAVE TIME! DO THE FOLLOWING INSTANTLY WITHOUT PAPER AND PENCIL MULTIPLICATION DIVISION RECIPROCAL VALUES SQUARES & CUBES EXTRACTION OF SQUARE ROOT EXTRACTION OF CUBE ROOT DIAMETER OR AREA OF CIRCLE [Illustration: Two images of a slide rule.] INSTRUCTIONS FOR USING A SLIDE RULE The slide rule is a device for easily and quickly multiplying, dividing and extracting square root and cube root. It will also perform any combination of these processes. On this account, it is found extremely useful by students and teachers in schools and colleges, by engineers, architects, draftsmen, surveyors, chemists, and many others. Accountants and clerks find it very helpful when approximate calculations must be made rapidly. The operation of a slide rule is extremely easy, and it is well worth while for anyone who is called upon to do much numerical calculation to learn to use one. It is the purpose of this manual to explain the operation in such a way that a person who has never before used a slide rule may teach himself to do so. DESCRIPTION OF SLIDE RULE The slide rule consists of three parts (see figure 1). B is the body of the rule and carries three scales marked A, D and K. S is the slider which moves relative to the body and also carries three scales marked B, CI and C. R is the runner or indicator and is marked in the center with a hair-line. The scales A and B are identical and are used in problems involving square root. Scales C and D are also identical and are used for multiplication and division. Scale K is for finding cube root. Scale CI, or C-inverse, is like scale C except that it is laid off from right to left instead of from left to right. It is useful in problems involving reciprocals. MULTIPLICATION We will start with a very simple example: Example 1: 2 * 3 = 6 To prove this on the slide rule, move the slider so that the 1 at the left-hand end of the C scale is directly over the large 2 on the D scale (see figure 1). Then move the runner till the hair-line is over 3 on the C scale. Read the answer, 6, on the D scale under the hair-line. Now, let us consider a more complicated example: Example 2: 2.12 * 3.16 = 6.70 As before, set the 1 at the left-hand end of the C scale, which we will call the left-hand index of the C scale, over 2.12 on the D scale (See figure 2). The hair-line of the runner is now placed over 3.16 on the C scale and the answer, 6.70, read on the D scale. METHOD OF MAKING SETTINGS In order to understand just why 2.12 is set where it is (figure 2), notice that the interval from 2 to 3 is divided into 10 large or major divisions, each of which is, of course, equal to one-tenth (0.1) of the amount represented by the whole interval. The major divisions are in turn divided into 5 small or minor divisions, each of which is one-fifth or two-tenths (0.2) of the major division, that is 0.02 of the whole interval. Therefore, the index is set above 2 + 1 major division + 1 minor division = 2 + 0.1 + 0.02 = 2.12. In the same way we find 3.16 on the C scale. While we are on this subject, notice that in the interval from 1 to 2 the major divisions are marked with the small figures 1 to 9 and the minor divisions are 0.1 of the major divisions. In the intervals from 2 to 3 and 3 to 4 the minor divisions are 0.2 of the major divisions, and for the rest of the D (or C) scale, the minor divisions are 0.5 of the major divisions. Reading the setting from a slide rule is very much like reading measurements from a ruler. Imagine that the divisions between 2 and 3 on the D scale (figure 2) are those of a ruler divided into tenths of a foot, and each tenth of a foot divided in 5 parts 0.02 of a foot long. Then the distance from one on the left-hand end of the D scale (not shown in figure 2) to one on the left-hand end of the C scale would he 2.12 feet. Of course, a foot rule is divided into parts of uniform length, while those on a slide rule get smaller toward the right-hand end, but this example may help to give an idea of the method of making and reading settings. Now consider another example. Example 3a: 2.12 * 7.35 = 15.6 If we set the left-hand index of the C scale over 2.12 as in the last example, we find that 7.35 on the C scale falls out beyond the body of the rule. In a case like this, simply use the right-hand index of the C scale. If we set this over 2.12 on the D scale and move the runner to 7.35 on the C scale we read the result 15.6 on the D scale under the hair-line. Now, the question immediately arises, why did we call the result 15.6 and not 1.56? The answer is that the slide rule takes no account of decimal points. Thus, the settings would be identical for all of the following products: Example 3: a-- 2.12 * 7.35 = 15.6 b-- 21.2 * 7.35 = 156.0 c-- 212 * 73.5 = 15600. d-- 2.12 * .0735 = .156 e-- .00212 * 735 = .0156 The most convenient way to locate the decimal point is to make a mental multiplication using only the first digits in the given factors. Then place the decimal point in the slide rule result so that its value is nearest that of the mental multiplication. Thus, in example 3a above, we can multiply 2 by 7 in our heads and see immediately that the decimal point must be placed in the slide rule result 156 so that it becomes 15.6 which is nearest to 14. In example 3b (20 * 7 = 140), so we must place the decimal point to give 156. The reader can readily verify the other examples in the same way. Since the product of a number by a second number is the same as the product of the second by the first, it makes no difference which of the two numbers is set first on the slide rule. Thus, an alternative way of working example 2 would be to set the left-hand index of the C scale over 3.16 on the D scale and move the runner to 2.12 on the C scale and read the answer under the hair-line on the D scale. The A and B scales are made up of two identical halves each of which is very similar to the C and D scales. Multiplication can also be carried out on either half of the A and B scales exactly as it is done on the C and D scales. However, since the A and B scales are only half as long as the C and D scales, the accuracy is not as good. It is sometimes convenient to multiply on the A and B scales in more complicated problems as we shall see later on. A group of examples follow which cover all the possible combination of settings which can arise in the multiplication of two numbers. Example 4: 20 * 3 = 60 5: 85 * 2 = 170 6: 45 * 35 = 1575 7: 151 * 42 = 6342 8: 6.5 * 15 = 97.5 9: .34 * .08 = .0272 10: 75 * 26 = 1950 11: .00054 * 1.4 = .000756 12: 11.1 * 2.7 = 29.97 13: 1.01 * 54 = 54.5 14: 3.14 * 25 = 78.5 DIVISION Since multiplication and division are inverse processes, division on a slide rule is done by making the same settings as for multiplication, but in reverse order. Suppose we have the example: Example 15: (6.70 / 2.12) = 3.16 Set indicator over the dividend 6.70 on the D scale. Move the slider until the divisor 2.12 on the C scale is under the hair-line. Then read the result on the D scale under the left-hand index of the C scale. As in multiplication, the decimal point must be placed by a separate process. Make all the digits except the first in both dividend and divisor equal zero and mentally divide the resulting numbers. Place the decimal point in the slide rule result so that it is nearest to the mental result. In example 15, we mentally divide 6 by 2. Then we place the decimal point in the slide rule result 316 so that it is 3.16 which is nearest to 3. A group of examples for practice in division follow: Example 16: 34 / 2 = 17 17: 49 / 7 = 7 18: 132 / 12 = 11 19: 480 / 16 = 30 20: 1.05 / 35 = .03 21: 4.32 / 12 = .36 22: 5.23 / 6.15 = .85 23: 17.1 / 4.5 = 3.8 24: 1895 / 6.06 = 313 25: 45 / .017 = 2647 THE CI SCALE If we divide one (1) by any number the answer is called the reciprocal of the number. Thus, one-half is the reciprocal of two, one-quarter is the reciprocal of four. If we take any number, say 14, and multiply it by the reciprocal of another number, say 2, we get: Example 26: 14 * (1/2) = 7 which is the same as 14 divided by two. This process can be carried out directly on the slide rule by use of the CI scale. Numbers on the CI scale are reciprocals of those on the C scale. Thus we see that 2 on the CI scale comes directly over 0.5 or 1/2 on the C scale. Similarly 4 on the CI scale comes over 0.25 or 1/4 on the C scale, and so on. To do example 26 by use of the CI scale, proceed exactly as if you were going to multiply in the usual manner except that you use the CI scale instead of the C scale. First set the left-hand index of the C scale over 14 on the D scale. Then move the indicator to 2 on the CI scale. Read the result, 7, on the D scale under the hair-line. This is really another way of dividing. THE READER IS ADVISED TO WORK EXAMPLES 16 TO 25 OVER AGAIN BY USE OF THE CI SCALE. SQUARING AND SQUARE ROOT If we take a number and multiply it by itself we call the result the square of the number. The process is called squaring the number. If we find the number which, when multiplied by itself is equal to a given number, the former number is called the square root of the given number. The process is called extracting the square root of the number. Both these processes may be carried out on the A and D scales of a slide rule. For example: Example 27: 4 * 4 = square( 4 ) = 16 Set indicator over 4 on D scale. Read 16 on A scale under hair-line. Example 28: square( 25.4 ) = 646.0 The decimal point must be placed by mental survey. We know that square( 25.4 ) must be a little larger than square( 25 ) = 625 so that it must be 646.0. To extract a square root, we set the indicator over the number on the A scale and read the result under the hair-line on the D scale. When we examine the A scale we see that there are two places where any given number may be set, so we must have some way of deciding in a given case which half of the A scale to use. The rule is as follows: (a) If the number is greater than one. For an odd number of digits to the left of the decimal point, use the left-hand half of the A scale. For an even number of digits to the left of the decimal point, use the right-hand half of the A scale. (b) If the number is less than one. For an odd number of zeros to the right of the decimal point before the first digit not a zero, use the left-hand half of the A scale. For none or any even number of zeros to the right of the decimal point before the first digit not a zero, use the right-hand half of the A scale. Example 29: square_root( 157 ) = 12.5 Since we have an odd number of digits set indicator over 157 on left-hand half of A scale. Read 12.5 on the D scale under hair-line. To check the decimal point think of the perfect square nearest to 157. It is 12 * 12 = 144, so that square_root(157) must be a little more than 12 or 12.5. Example 30: square_root( .0037 ) = .0608 In this number we have an even number of zeros to the right of the decimal point, so we must set the indicator over 37 on the right-hand half of the A scale. Read 608 under the hair-line on D scale. To place the decimal point write: square_root( .0037 ) = square_root( 37/10000 ) = 1/100 square_root( 37 ) The nearest perfect square to 37 is 6 * 6 = 36, so the answer should be a little more than 0.06 or .0608. All of what has been said about use of the A and D scales for squaring and extracting square root applies equally well to the B and C scales since they are identical to the A and D scales respectively. A number of examples follow for squaring and the extraction of square root. Example 31: square( 2 ) = 4 32: square( 15 ) = 225 33: square( 26 ) = 676 34: square( 19.65 ) = 386 35: square_root( 64 ) = 8 36: square_root( 6.4 ) = 2.53 37: square_root( 498 ) = 22.5 38: square_root( 2500 ) = 50 39: square_root( .16 ) = .04 40: square_root( .03 ) = .173 CUBING AND CUBE ROOT If we take a number and multiply it by itself, and then multiply the result by the original number we get what is called the cube of the original number. This process is called cubing the number. The reverse process of finding the number which, when multiplied by itself and then by itself again, is equal to the given number, is called extracting the cube root of the given number. Thus, since 5 * 5 * 5 = 125, 125 is the cube of 5 and 5 is the cube root of 125. To find the cube of any number on the slide rule set the indicator over the number on the D scale and read the answer on the K scale under the hair-line. To find the cube root of any number set the indicator over the number on the K scale and read the answer on the D scale under the hair-line. Just as on the A scale, where there were two places where you could set a given number, on the K scale there are three places where a number may be set. To tell which of the three to use, we must make use of the following rule. (a) If the number is greater than one. For 1, 4, 7, 10, etc., digits to the left of the decimal point, use the left-hand third of the K scale. For 2, 5, 8, 11, etc., digits to the left of the decimal point, use the middle third of the K scale. For 3, 6, 9, 12, etc., digits to the left of the decimal point use the right-hand third of the K scale. (b) If the number is less than one. We now tell which scale to use by counting the number of zeros to the right of the decimal point before the first digit not zero. If there are 2, 5, 8, 11, etc., zeros, use the left-hand third of the K scale. If there are 1, 4, 7, 10, etc., zeros, then use the middle third of the K scale. If there are no zeros or 3, 6, 9, 12, etc., zeros, then use the right-hand third of the K scale. For example: Example 41: cube_root( 185 ) = 5.70 Since there are 3 digits in the given number, we set the indicator on 185 in the right-hand third of the K scale, and read the result 570 on the D scale. We can place the decimal point by thinking of the nearest perfect cube, which is 125. Therefore, the decimal point must be placed so as to give 5.70, which is nearest to 5, the cube root of 125. Example 42: cube_root( .034 ) = .324 Since there is one zero between the decimal point and the first digit not zero, we must set the indicator over 34 on the middle third of the K scale. We read the result 324 on the D scale. The decimal point may be placed as follows: cube_root( .034 ) = cube_root( 34/1000 ) = 1/10 cube_root( 34 ) The nearest perfect cube to 34 is 27, so our answer must be close to one-tenth of the cube root of 27 or nearly 0.3. Therefore, we must place the decimal point to give 0.324. A group of examples for practice in extraction of cube root follows: Example 43: cube_root( 64 ) = 4 44: cube_root( 8 ) = 2 45: cube_root( 343 ) = 7 46: cube_root( .000715 ) = .0894 47: cube_root( .00715 ) = .193 48: cube_root( .0715 ) = .415 49: cube_root( .516 ) = .803 50: cube_root( 27.8 ) = 3.03 51: cube_root( 5.49 ) = 1.76 52: cube_root( 87.1 ) = 4.43 THE 1.5 AND 2/3 POWER If the indicator is set over a given number on the A scale, the number under the hair-line on the K scale is the 1.5 power of the given number. If the indicator is set over a given number on the K scale, the number under the hair-line on the A scale is the 2/3 power of the given number. COMBINATIONS OF PROCESSES A slide rule is especially useful where some combination of processes is necessary, like multiplying 3 numbers together and dividing by a third. Operations of this sort may be performed in such a way that the final answer is obtained immediately without finding intermediate results. 1. Multiplying several numbers together. For example, suppose it is desired to multiply 4 * 8 * 6. Place the right-hand index of the C scale over 4 on the D scale and set the indicator over 8 on the C scale. Now, leaving the indicator where it is, move the slider till the right-hand index is under the hairline. Now, leaving the slider where it is, move the indicator until it is over 6 on the C scale, and read the result, 192, on the D scale. This may be continued indefinitely, and so as many numbers as desired may be multiplied together. Example 53: 2.32 * 154 * .0375 * .56 = 7.54 2. Multiplication and division. Suppose we wish to do the following example: Example 54: (4 * 15) / 2.5 = 24 First divide 4 by 2.5. Set indicator over 4 on the D scale and move the slider until 2.5 is under the hair-line. The result of this division, 1.6, appears under the left-hand index of the C scale. We do not need to write it down, however, but we can immediately move the indicator to 15 on the C scale and read the final result 24 on the D scale under the hair-line. Let us consider a more complicated problem of the same type: Example 55: (30/7.5) * (2/4) * (4.5/5) * (1.5/3) = .9 First set indicator over 30 on the D scale and move slider until 7.5 on the C scale comes under the hairline. The intermediate result, 4, appears under the right-hand index of the C scale. We do not need to write it down but merely note it by moving the indicator until the hair-line is over the right-hand index of the C scale. Now we want to multiply this result by 2, the next factor in the numerator. Since two is out beyond the body of the rule, transfer the slider till the other (left-hand) index of the C scale is under the hair-line, and then move the indicator to 2 on the C scale. Thus, successive division and multiplication is continued until all the factors have been used. The order in which the factors are taken does not affect the result. With a little practice you will learn to take them in the order which will require the fewest settings. The following examples are for practice: Example 56: (6/3.5) * (4/5) * (3.5/2.4) * (2.8/7) = .8 Example 57: 352 * (273/254) * (760/768) = 374 An alternative method of doing these examples is to proceed exactly as though you were multiplying all the factors together, except that whenever you come to a number in the denominator you use the CI scale instead of the C scale. The reader is advised to practice both methods and use whichever one he likes best. 3. The area of a circle. The area of a circle is found by multiplying 3.1416=PI by the square of the radius or by one-quarter the square of the diameter Formula: A = PI * square( R ) A = PI * ( square( D ) / 4 ) Example 58: The radius of a circle is 0.25 inches; find its area. Area = PI * square(0.25) = 0.196 square inches. Set left-hand index of C scale over 0.25 on D scale. square(0.25) now appears above the left-hand index of the B scale. This can be multiplied by PI by moving the indicator to PI on the B scale and reading the answer .196 on the A scale. This is an example where it is convenient to multiply with the A and B scales. Example 59: The diameter of a circle is 8.1 feet. What is its area? Area = (PI / 4) * square(8.1) = .7854 * square(8.1) = 51.7 sq. inches. Set right-hand index of the C scale over 8.1 on the D scale. Move the indicator till hair-line is over .7854 (the special long mark near 8) at the right hand of the B scale. Read the answer under the hair-line on the A scale. Another way of finding the area of a circle is to set 7854 on the B scale to one of the indices of the A scale, and read the area from the B scale directly above the given diameter on the D scale. 4. The circumference of a circle. Set the index of the B scale to the diameter and read the answer on the A scale opposite PI on the B scale Formula: C = PI * D C = 2 * PI * R Example 60: The diameter of a circle is 1.54 inches, what is its circumference? Set the left-hand index of the B scale to 1.54 on the A scale. Read the circumference 4.85 inches above PI on the B scale. EXAMPLES FOR PRACTICE 61: What is the area of a circle 32-1/2 inches in diameter? Answer 830 sq. inches 62: What is the area of a circle 24 inches in diameter? Answer 452 sq. inches 63: What is the circumference of a circle whose diameter is 95 feet? Answer 298 ft. 64: What is the circumference of a circle whose diameter is 3.65 inches? Answer 11.5 inches 5. Ratio and Proportion. Example 65: 3 : 7 : : 4 : X or (3/7) = (4/x) Find X Set 3 on C scale over 7 on D scale. Read X on D scale under 4 on C scale. In fact, any number on the C scale is to the number directly under it on the D scale as 3 is to 7. PRACTICAL PROBLEMS SOLVED BY SLIDE RULE 1. Discount. A firm buys a typewriter with a list price of $150, subject to a discount of 20% and 10%. How much does it pay? A discount of 20% means 0.8 of the list price, and 10% more means 0.8 X 0.9 X 150 = 108. To do this on the slide rule, put the index of the C scale opposite 8 on the D scale and move the indicator to 9 on the C scale. Then move the slider till the right-hand index of the C scale is under the hairline. Now, move the indicator to 150 on the C scale and read the answer $108 on the D scale. Notice that in this, as in many practical problems, there is no question about where the decimal point should go. 2. Sales Tax. A man buys an article worth $12 and he must pay a sales tax of 1.5%. How much does he pay? A tax of 1.5% means he must pay 1.015 * 12.00. Set index of C scale at 1.015 on D scale. Move indicator to 12 on C scale and read the answer $12.18 on the D scale. A longer but more accurate way is to multiply 12 * .015 and add the result to $12. 3. Unit Price. A motorist buys 17 gallons of gas at 19.5 cents per gallon. How much does he pay? Set index of C scale at 17 on D scale and move indicator to 19.5 on C scale and read the answer $3.32 on the D scale. 4. Gasoline Mileage. An automobile goes 175 miles on 12 gallons of gas. What is the average gasoline consumption? Set indicator over 175 on D scale and move slider till 12 is under hair-line. Read the answer 14.6 miles per gallon on the D scale under the left-hand index of the C scale. 5. Average Speed. A motorist makes a trip of 256 miles in 7.5 hours. What is his average speed? Set indicator over 256 on D scale. Move slider till 7.5 on the C scale is under the hair-line. Read the answer 34.2 miles per hour under the right-hand index of the C scale. 6. Decimal Parts of an Inch. What is 5/16 of an inch expressed as decimal fraction? Set 16 on C scale over 5 on D scale and read the result .3125 inches on the D scale under the left-hand index of the C scale. 7. Physics. A certain quantity of gas occupies 1200 cubic centimeters at a temperature of 15 degrees C and 740 millimeters pressure. What volume does it occupy at 0 degrees C and 760 millimeters pressure? Volume = 1200 X (740/760) * (273/288) = 1100 cubic cm. Set 760 on C scale over 12 on D scale. Move indicator to 740 on C scale. Move slider till 288 on C scale is under hair-line. Move indicator to 273 on C scale. Read answer, 1110, under hair-line on D scale. 8. Chemistry. How many grams of hydrogen are formed when 80 grams of zinc react with sufficient hydrochloric acid to dissolve the metal? (80 / X ) = ( 65.4 / 2.01) Set 65.4 on C scale over 2.01 on D scale. Read X = 2.46 grams under 80 on C scale. In conclusion, we want to impress upon those to whom the slide rule is a new method of doing their mathematical calculations, and also the experienced operator of a slide rule, that if they will form a habit of, and apply themselves to, using a slide rule at work, study, or during recreations, they will be well rewarded in the saving of time and energy. ALWAYS HAVE YOUR SLIDE RULE AND INSTRUCTION BOOK WITH YOU, the same as you would a fountain pen or pencil. The present day wonders of the twentieth century prove that there is no end to what an individual can accomplish--the same applies to the slide rule. You will find after practice that you will be able to do many specialized problems that are not outlined in this instruction book. It depends entirely upon your ability to do what we advocate and to be slide-rule conscious in all your mathematical problems. CONVERSION FACTORS 1. Length 1 mile = 5280 feet =1760 yards 1 inch = 2.54 centimeters 1 meter = 39.37 inches 2. Weight (or Mass) 1 pound = 16 ounces = 0.4536 kilograms 1 kilogram = 2.2 pounds 1 long ton = 2240 pounds 1 short ton=2000 pounds 3. Volume 1 liquid quart = 0.945 litres 1 litre = 1.06 liquid quarts 1 U. S. gallon = 4 quarts = 231 cubic inches 4. Angular Measure 3.14 radians = PI radians = 180 degrees 1 radian = 57.30 degrees 5. Pressure 760 millimeters of mercury = 14.7 pounds per square inch 6. Power 1 horse power = 550 foot pounds per second = 746 watts 7. Miscellaneous 60 miles per hour = 88 feet per second 980 centimeters per second per second = 32.2 feet per second per second = acceleration of gravity. 1 cubic foot of water weighs 62.4 pounds 1 gallon of water weighs 8.34 pounds Printed in U. S. A. INSTRUCTIONS FOR USING A SLIDE RULE COPYRIGHTED BY W. STANLEY & CO. Commercial Trust Building, Philadelphia, Pa. 
201	----	Flatland: A Romance of Many Dimensions Edwin A. Abbott (1838-1926. English scholar, theologian, and writer.) ----------------------------------------------------------------- | "O day and night, but this is wondrous strange" | | ______ | | / / /| ------ / /| /| / /-. | | /---- / /__| / / /__| / | / / / | | / /___ / | / /___ / | / |/ /__.-' | | | | No Dimensions One Dimension | | . A ROMANCE OF MANY DIMENSIONS ----- | | POINTLAND LINELAND | | | | Two Dimensions Three Dimensions | | ___ __ | | | | /__/| | | |___| |__|/ | | FLATLAND SPACELAND | | "Fie, fie, how franticly I square my talk!" | ----------------------------------------------------------------- With Illustrations by the Author, A SQUARE (Edwin A. Abbott) To The Inhabitants of SPACE IN GENERAL And H. C. IN PARTICULAR This Work is Dedicated By a Humble Native of Flatland In the Hope that Even as he was Initiated into the Mysteries Of THREE Dimensions Having been previously conversant With ONLY TWO So the Citizens of that Celestial Region May aspire yet higher and higher To the Secrets of FOUR FIVE OR EVEN SIX Dimensions Thereby contributing To the Enlargement of THE IMAGINATION And the possible Development Of that most rare and excellent Gift of MODESTY Among the Superior Races Of SOLID HUMANITY Preface to the Second and Revised Edition, 1884. By the Editor If my poor Flatland friend retained the vigour of mind which he enjoyed when he began to compose these Memoirs, I should not now need to represent him in this preface, in which he desires, firstly, to return his thanks to his readers and critics in Spaceland, whose appreciation has, with unexpected celerity, required a second edition of his work; secondly, to apologize for certain errors and misprints (for which, however, he is not entirely responsible); and, thirdly, to explain one or two misconceptions. But he is not the Square he once was. Years of imprisonment, and the still heavier burden of general incredulity and mockery, have combined with the natural decay of old age to erase from his mind many of the thoughts and notions, and much also of the terminology, which he acquired during his short stay in Spaceland. He has, therefore, requested me to reply in his behalf to two special objections, one of an intellectual, the other of a moral nature. The first objection is, that a Flatlander, seeing a Line, sees something that must be THICK to the eye as well as LONG to the eye (otherwise it would not be visible, if it had not some thickness); and consequently he ought (it is argued) to acknowledge that his countrymen are not only long and broad, but also (though doubtless in a very slight degree) THICK or HIGH. This objection is plausible, and, to Spacelanders, almost irresistible, so that, I confess, when I first heard it, I knew not what to reply. But my poor old friend's answer appears to me completely to meet it. "I admit," said he--when I mentioned to him this objection--"I admit the truth of your critic's facts, but I deny his conclusions. It is true that we have really in Flatland a Third unrecognized Dimension called 'height', just as it is also true that you have really in Spaceland a Fourth unrecognized Dimension, called by no name at present, but which I will call 'extra-height'. But we can no more take cognizance of our 'height' than you can of your 'extra-height'. Even I--who have been in Spaceland, and have had the privilege of understanding for twenty-four hours the meaning of 'height'--even I cannot now comprehend it, nor realize it by the sense of sight or by any process of reason; I can but apprehend it by faith. "The reason is obvious. Dimension implies direction, implies measurement, implies the more and the less. Now, all our lines are EQUALLY and INFINITESIMALLY thick (or high, whichever you like); consequently, there is nothing in them to lead our minds to the conception of that Dimension. No 'delicate micrometer'--as has been suggested by one too hasty Spaceland critic--would in the least avail us; for we should not know WHAT TO MEASURE, NOR IN WHAT DIRECTION. When we see a Line, we see something that is long and BRIGHT; BRIGHTNESS, as well as length, is necessary to the existence of a Line; if the brightness vanishes, the Line is extinguished. Hence, all my Flatland friends--when I talk to them about the unrecognized Dimension which is somehow visible in a Line--say, 'Ah, you mean BRIGHTNESS': and when I reply, 'No, I mean a real Dimension', they at once retort, 'Then measure it, or tell us in what direction it extends'; and this silences me, for I can do neither. Only yesterday, when the Chief Circle (in other words our High Priest) came to inspect the State Prison and paid me his seventh annual visit, and when for the seventh time he put me the question, 'Was I any better?' I tried to prove to him that he was 'high', as well as long and broad, although he did not know it. But what was his reply? 'You say I am "high"; measure my "high-ness" and I will believe you.' What could I do? How could I meet his challenge? I was crushed; and he left the room triumphant. "Does this still seem strange to you? Then put yourself in a similar position. Suppose a person of the Fourth Dimension, condescending to visit you, were to say, 'Whenever you open your eyes, you see a Plane (which is of Two Dimensions) and you INFER a Solid (which is of Three); but in reality you also see (though you do not recognize) a Fourth Dimension, which is not colour nor brightness nor anything of the kind, but a true Dimension, although I cannot point out to you its direction, nor can you possibly measure it.' What would you say to such a visitor? Would not you have him locked up? Well, that is my fate: and it is as natural for us Flatlanders to lock up a Square for preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube for preaching the Fourth. Alas, how strong a family likeness runs through blind and persecuting humanity in all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes--we are all liable to the same errors, all alike the Slaves of our respective Dimensional prejudices, as one of your Spaceland poets has said-- 'One touch of Nature makes all worlds akin'." [Note: The Author desires me to add, that the misconception of some of his critics on this matter has induced him to insert in his dialogue with the Sphere, certain remarks which have a bearing on the point in question, and which he had previously omitted as being tedious and unnecessary.] On this point the defence of the Square seems to me to be impregnable. I wish I could say that his answer to the second (or moral) objection was equally clear and cogent. It has been objected that he is a woman-hater; and as this objection has been vehemently urged by those whom Nature's decree has constituted the somewhat larger half of the Spaceland race, I should like to remove it, so far as I can honestly do so. But the Square is so unaccustomed to the use of the moral terminology of Spaceland that I should be doing him an injustice if I were literally to transcribe his defence against this charge. Acting, therefore, as his interpreter and summarizer, I gather that in the course of an imprisonment of seven years he has himself modified his own personal views, both as regards Women and as regards the Isosceles or Lower Classes. Personally, he now inclines to the opinion of the Sphere that the Straight Lines are in many important respects superior to the Circles. But, writing as a Historian, he has identified himself (perhaps too closely) with the views generally adopted by Flatland, and (as he has been informed) even by Spaceland, Historians; in whose pages (until very recent times) the destinies of Women and of the masses of mankind have seldom been deemed worthy of mention and never of careful consideration. In a still more obscure passage he now desires to disavow the Circular or aristocratic tendencies with which some critics have naturally credited him. While doing justice to the intellectual power with which a few Circles have for many generations maintained their supremacy over immense multitudes of their countrymen, he believes that the facts of Flatland, speaking for themselves without comment on his part, declare that Revolutions cannot always be suppressed by slaughter, and that Nature, in sentencing the Circles to infecundity, has condemned them to ultimate failure--"and herein," he says, "I see a fulfilment of the great Law of all worlds, that while the wisdom of Man thinks it is working one thing, the wisdom of Nature constrains it to work another, and quite a different and far better thing." For the rest, he begs his readers not to suppose that every minute detail in the daily life of Flatland must needs correspond to some other detail in Spaceland; and yet he hopes that, taken as a whole, his work may prove suggestive as well as amusing, to those Spacelanders of moderate and modest minds who--speaking of that which is of the highest importance, but lies beyond experience--decline to say on the one hand, "This can never be," and on the other hand, "It must needs be precisely thus, and we know all about it." CONTENTS: PART I: THIS WORLD Section 1. Of the Nature of Flatland 2. Of the Climate and Houses in Flatland 3. Concerning the Inhabitants of Flatland 4. Concerning the Women 5. Of our Methods of Recognizing one another 6. Of Recognition by Sight 7. Concerning Irregular Figures 8. Of the Ancient Practice of Painting 9. Of the Universal Colour Bill 10. Of the Suppression of the Chromatic Sedition 11. Concerning our Priests 12. Of the Doctrine of our Priests PART II: OTHER WORLDS 13. How I had a Vision of Lineland 14. How I vainly tried to explain the nature of Flatland 15. Concerning a Stranger from Spaceland 16. How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland 17. How the Sphere, having in vain tried words, resorted to deeds 18. How I came to Spaceland, and what I saw there 19. How, though the Sphere shewed me other mysteries of Spaceland, I still desired more; and what came of it 20. How the Sphere encouraged me in a Vision 21. How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success 22. How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result PART I: THIS WORLD "Be patient, for the world is broad and wide." Section 1. Of the Nature of Flatland I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space. Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows--only hard and with luminous edges--and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said "my universe": but now my mind has been opened to higher views of things. In such a country, you will perceive at once that it is impossible that there should be anything of what you call a "solid" kind; but I dare say you will suppose that we could at least distinguish by sight the Triangles, Squares, and other figures, moving about as I have described them. On the contrary, we could see nothing of the kind, not at least so as to distinguish one figure from another. Nothing was visible, nor could be visible, to us, except Straight Lines; and the necessity of this I will speedily demonstrate. Place a penny on the middle of one of your tables in Space; and leaning over it, look down upon it. It will appear a circle. But now, drawing back to the edge of the table, gradually lower your eye (thus bringing yourself more and more into the condition of the inhabitants of Flatland), and you will find the penny becoming more and more oval to your view, and at last when you have placed your eye exactly on the edge of the table (so that you are, as it were, actually a Flatlander) the penny will then have ceased to appear oval at all, and will have become, so far as you can see, a straight line. The same thing would happen if you were to treat in the same way a Triangle, or Square, or any other figure cut out of pasteboard. As soon as you look at it with your eye on the edge on the table, you will find that it ceases to appear to you a figure, and that it becomes in appearance a straight line. Take for example an equilateral Triangle--who represents with us a Tradesman of the respectable class. Fig. 1 represents the Tradesman as you would see him while you were bending over him from above; figs. 2 and 3 represent the Tradesman, as you would see him if your eye were close to the level, or all but on the level of the table; and if your eye were quite on the level of the table (and that is how we see him in Flatland) you would see nothing but a straight line. [Illustration 1] [ASCII approximation follows] (1) __________ (2) ___________ (3) _________ \ / --__ __-- --- \ / - \/ When I was in Spaceland I heard that your sailors have very similar experiences while they traverse your seas and discern some distant island or coast lying on the horizon. The far-off land may have bays, forelands, angles in and out to any number and extent; yet at a distance you see none of these (unless indeed your sun shines bright upon them revealing the projections and retirements by means of light and shade), nothing but a grey unbroken line upon the water. Well, that is just what we see when one of our triangular or other acquaintances comes toward us in Flatland. As there is neither sun with us, nor any light of such a kind as to make shadows, we have none of the helps to the sight that you have in Spaceland. If our friend comes closer to us we see his line becomes larger; if he leaves us it becomes smaller: but still he looks like a straight line; be he a Triangle, Square, Pentagon, Hexagon, Circle, what you will--a straight Line he looks and nothing else. You may perhaps ask how under these disadvantageous circumstances we are able to distinguish our friends from one another: but the answer to this very natural question will be more fitly and easily given when I come to describe the inhabitants of Flatland. For the present let me defer this subject, and say a word or two about the climate and houses in our country. Section 2. Of the Climate and Houses in Flatland As with you, so also with us, there are four points of the compass North, South, East, and West. There being no sun nor other heavenly bodies, it is impossible for us to determine the North in the usual way; but we have a method of our own. By a Law of Nature with us, there is a constant attraction to the South; and, although in temperate climates this is very slight--so that even a Woman in reasonable health can journey several furlongs northward without much difficulty--yet the hampering effect of the southward attraction is quite sufficient to serve as a compass in most parts of our earth. Moreover, the rain (which falls at stated intervals) coming always from the North, is an additional assistance; and in the towns we have the guidance of the houses, which of course have their side-walls running for the most part North and South, so that the roofs may keep off the rain from the North. In the country, where there are no houses, the trunks of the trees serve as some sort of guide. Altogether, we have not so much difficulty as might be expected in determining our bearings. Yet in our more temperate regions, in which the southward attraction is hardly felt, walking sometimes in a perfectly desolate plain where there have been no houses nor trees to guide me, I have been occasionally compelled to remain stationary for hours together, waiting till the rain came before continuing my journey. On the weak and aged, and especially on delicate Females, the force of attraction tells much more heavily than on the robust of the Male Sex, so that it is a point of breeding, if you meet a Lady in the street, always to give her the North side of the way--by no means an easy thing to do always at short notice when you are in rude health and in a climate where it is difficult to tell your North from your South. Windows there are none in our houses: for the light comes to us alike in our homes and out of them, by day and by night, equally at all times and in all places, whence we know not. It was in old days, with our learned men, an interesting and oft-investigated question, "What is the origin of light?" and the solution of it has been repeatedly attempted, with no other result than to crowd our lunatic asylums with the would-be solvers. Hence, after fruitless attempts to suppress such investigations indirectly by making them liable to a heavy tax, the Legislature, in comparatively recent times, absolutely prohibited them. I--alas, I alone in Flatland--know now only too well the true solution of this mysterious problem; but my knowledge cannot be made intelligible to a single one of my countrymen; and I am mocked at--I, the sole possessor of the truths of Space and of the theory of the introduction of Light from the world of three Dimensions--as if I were the maddest of the mad! But a truce to these painful digressions: let me return to our houses. The most common form for the construction of a house is five-sided or pentagonal, as in the annexed figure. The two Northern sides RO, OF, constitute the roof, and for the most part have no doors; on the East is a small door for the Women; on the West a much larger one for the Men; the South side or floor is usually doorless. Square and triangular houses are not allowed, and for this reason. The angles of a Square (and still more those of an equilateral Triangle), being much more pointed than those of a Pentagon, and the lines of inanimate objects (such as houses) being dimmer than the lines of Men and Women, it follows that there is no little danger lest the points of a square or triangular house residence might do serious injury to an inconsiderate or perhaps absent-minded traveller suddenly therefore, running against them: and as early as the eleventh century of our era, triangular houses were universally forbidden by Law, the only exceptions being fortifications, powder-magazines, barracks, and other state buildings, which it is not desirable that the general public should approach without circumspection. [Illustration 2] [ASCII approximation follows] O /\ / \ / \ / \ / \ R/ \F \_ / _/ Men's door _ Women's door _ / \____________/ A B At this period, square houses were still everywhere permitted, though discouraged by a special tax. But, about three centuries afterwards, the Law decided that in all towns containing a population above ten thousand, the angle of a Pentagon was the smallest house-angle that could be allowed consistently with the public safety. The good sense of the community has seconded the efforts of the Legislature; and now, even in the country, the pentagonal construction has superseded every other. It is only now and then in some very remote and backward agricultural district that an antiquarian may still discover a square house. Section 3. Concerning the Inhabitants of Flatland The greatest length or breadth of a full grown inhabitant of Flatland may be estimated at about eleven of your inches. Twelve inches may be regarded as a maximum. Our Women are Straight Lines. Our Soldiers and Lowest Classes of Workmen are Triangles with two equal sides, each about eleven inches long, and a base or third side so short (often not exceeding half an inch) that they form at their vertices a very sharp and formidable angle. Indeed when their bases are of the most degraded type (not more than the eighth part of an inch in size), they can hardly be distinguished from Straight Lines or Women; so extremely pointed are their vertices. With us, as with you, these Triangles are distinguished from others by being called Isosceles; and by this name I shall refer to them in the following pages. Our Middle Class consists of Equilateral or Equal-Sided Triangles. Our Professional Men and Gentlemen are Squares (to which class I myself belong) and Five-Sided Figures or Pentagons. Next above these come the Nobility, of whom there are several degrees, beginning at Six-Sided Figures, or Hexagons, and from thence rising in the number of their sides till they receive the honourable title of Polygonal, or many-sided. Finally when the number of the sides becomes so numerous, and the sides themselves so small, that the figure cannot be distinguished from a circle, he is included in the Circular or Priestly order; and this is the highest class of all. It is a Law of Nature with us that a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility. Thus the son of a Square is a Pentagon; the son of a Pentagon, a Hexagon; and so on. But this rule applies not always to the Tradesmen, and still less often to the Soldiers, and to the Workmen; who indeed can hardly be said to deserve the name of human Figures, since they have not all their sides equal. With them therefore the Law of Nature does not hold; and the son of an Isosceles (i.e. a Triangle with two sides equal) remains Isosceles still. Nevertheless, all hope is not shut out, even from the Isosceles, that his posterity may ultimately rise above his degraded condition. For, after a long series of military successes, or diligent and skilful labours, it is generally found that the more intelligent among the Artisan and Soldier classes manifest a slight increase of their third side or base, and a shrinkage of the two other sides. Intermarriages (arranged by the Priests) between the sons and daughters of these more intellectual members of the lower classes generally result in an offspring approximating still more to the type of the Equal-Sided Triangle. Rarely--in proportion to the vast numbers of Isosceles births--is a genuine and certifiable Equal-Sided Triangle produced from Isosceles parents. [Note: "What need of a certificate?" a Spaceland critic may ask: "Is not the procreation of a Square Son a certificate from Nature herself, proving the Equal-sidedness of the Father?" I reply that no Lady of any position will marry an uncertified Triangle. Square offspring has sometimes resulted from a slightly Irregular Triangle; but in almost every such case the Irregularity of the first generation is visited on the third; which either fails to attain the Pentagonal rank, or relapses to the Triangular.] Such a birth requires, as its antecedents, not only a series of carefully arranged intermarriages, but also a long, continued exercise of frugality and self-control on the part of the would-be ancestors of the coming Equilateral, and a patient, systematic, and continuous development of the Isosceles intellect through many generations. The birth of a True Equilateral Triangle from Isosceles parents is the subject of rejoicing in our country for many furlongs around. After a strict examination conducted by the Sanitary and Social Board, the infant, if certified as Regular, is with solemn ceremonial admitted into the class of Equilaterals. He is then immediately taken from his proud yet sorrowing parents and adopted by some childless Equilateral, who is bound by oath never to permit the child henceforth to enter his former home or so much as to look upon his relations again, for fear lest the freshly developed organism may, by force of unconscious imitation, fall back again into his hereditary level. The occasional emergence of an Equilateral from the ranks of his serf-born ancestors is welcomed, not only by the poor serfs themselves, as a gleam of light and hope shed upon the monotonous squalor of their existence, but also by the Aristocracy at large; for all the higher classes are well aware that these rare phenomena, while they do little or nothing to vulgarize their own privileges, serve as a most useful barrier against revolution from below. Had the acute-angled rabble been all, without exception, absolutely destitute of hope and of ambition, they might have found leaders in some of their many seditious outbreaks, so able as to render their superior numbers and strength too much even for the wisdom of the Circles. But a wise ordinance of Nature has decreed that, in proportion as the working-classes increase in intelligence, knowledge, and all virtue, in that same proportion their acute angle (which makes them physically terrible) shall increase also and approximate to the comparatively harmless angle of the Equilateral Triangle. Thus, in the most brutal and formidable of the soldier class--creatures almost on a level with women in their lack of intelligence--it is found that, as they wax in the mental ability necessary to employ their tremendous penetrating power to advantage, so do they wane in the power of penetration itself. How admirable is this Law of Compensation! And how perfect a proof of the natural fitness and, I may almost say, the divine origin of the aristocratic constitution of the States in Flatland! By a judicious use of this Law of Nature, the Polygons and Circles are almost always able to stifle sedition in its very cradle, taking advantage of the irrepressible and boundless hopefulness of the human mind. Art also comes to the aid of Law and Order. It is generally found possible--by a little artificial compression or expansion on the part of the State physicians--to make some of the more intelligent leaders of a rebellion perfectly Regular, and to admit them at once into the privileged classes; a much larger number, who are still below the standard, allured by the prospect of being ultimately ennobled, are induced to enter the State Hospitals, where they are kept in honourable confinement for life; one or two alone of the more obstinate, foolish, and hopelessly irregular are led to execution. Then the wretched rabble of the Isosceles, planless and leaderless, are either transfixed without resistance by the small body of their brethren whom the Chief Circle keeps in pay for emergencies of this kind; or else more often, by means of jealousies and suspicions skilfully fomented among them by the Circular party, they are stirred to mutual warfare, and perish by one another's angles. No less than one hundred and twenty rebellions are recorded in our annals, besides minor outbreaks numbered at two hundred and thirty-five; and they have all ended thus. Section 4. Concerning the Women If our highly pointed Triangles of the Soldier class are formidable, it may be readily inferred that far more formidable are our Women. For if a Soldier is a wedge, a Woman is a needle; being, so to speak, ALL point, at least at the two extremities. Add to this the power of making herself practically invisible at will, and you will perceive that a Female, in Flatland, is a creature by no means to be trifled with. But here, perhaps, some of my younger Readers may ask HOW a woman in Flatland can make herself invisible. This ought, I think, to be apparent without any explanation. However, a few words will make it clear to the most unreflecting. Place a needle on a table. Then, with your eye on the level of the table, look at it side-ways, and you see the whole length of it; but look at it end-ways, and you see nothing but a point, it has become practically invisible. Just so is it with one of our Women. When her side is turned towards us, we see her as a straight line; when the end containing her eye or mouth--for with us these two organs are identical--is the part that meets our eye, then we see nothing but a highly lustrous point; but when the back is presented to our view, then--being only sub-lustrous, and, indeed, almost as dim as an inanimate object--her hinder extremity serves her as a kind of Invisible Cap. The dangers to which we are exposed from our Women must now be manifest to the meanest capacity in Spaceland. If even the angle of a respectable Triangle in the middle class is not without its dangers; if to run against a Working Man involves a gash; if collision with an officer of the military class necessitates a serious wound; if a mere touch from the vertex of a Private Soldier brings with it danger of death;--what can it be to run against a Woman, except absolute and immediate destruction? And when a Woman is invisible, or visible only as a dim sub-lustrous point, how difficult must it be, even for the most cautious, always to avoid collision! Many are the enactments made at different times in the different States of Flatland, in order to minimize this peril; and in the Southern and less temperate climates where the force of gravitation is greater, and human beings more liable to casual and involuntary motions, the Laws concerning Women are naturally much more stringent. But a general view of the Code may be obtained from the following summary:-- 1. Every house shall have one entrance in the Eastern side, for the use of Females only; by which all females shall enter "in a becoming and respectful manner" and not by the Men's or Western door. [Note: When I was in Spaceland I understood that some of your Priestly circles have in the same way a separate entrance for Villagers, Farmers and Teachers of Board Schools (`Spectator', Sept. 1884, p. 1255) that they may "approach in a becoming and respectful manner."] 2. No Female shall walk in any public place without continually keeping up her Peace-cry, under penalty of death. 3. Any Female, duly certified to be suffering from St. Vitus's Dance, fits, chronic cold accompanied by violent sneezing, or any disease necessitating involuntary motions, shall be instantly destroyed. In some of the States there is an additional Law forbidding Females, under penalty of death, from walking or standing in any public place without moving their backs constantly from right to left so as to indicate their presence to those behind them; others oblige a Woman, when travelling, to be followed by one of her sons, or servants, or by her husband; others confine Women altogether to their houses except during the religious festivals. But it has been found by the wisest of our Circles or Statesmen that the multiplication of restrictions on Females tends not only to the debilitation and diminution of the race, but also to the increase of domestic murders to such an extent that a State loses more than it gains by a too prohibitive Code. For whenever the temper of the Women is thus exasperated by confinement at home or hampering regulations abroad, they are apt to vent their spleen upon their husbands and children; and in the less temperate climates the whole male population of a village has been sometimes destroyed in one or two hours of simultaneous female outbreak. Hence the Three Laws, mentioned above, suffice for the better regulated States, and may be accepted as a rough exemplification of our Female Code. After all, our principal safeguard is found, not in Legislature, but in the interests of the Women themselves. For, although they can inflict instantaneous death by a retrograde movement, yet unless they can at once disengage their stinging extremity from the struggling body of their victim, their own frail bodies are liable to be shattered. The power of Fashion is also on our side. I pointed out that in some less civilized States no female is suffered to stand in any public place without swaying her back from right to left. This practice has been universal among ladies of any pretensions to breeding in all well-governed States, as far back as the memory of Figures can reach. It is considered a disgrace to any State that legislation should have to enforce what ought to be, and is in every respectable female, a natural instinct. The rhythmical and, if I may so say, well-modulated undulation of the back in our ladies of Circular rank is envied and imitated by the wife of a common Equilateral, who can achieve nothing beyond a mere monotonous swing, like the ticking of a pendulum; and the regular tick of the Equilateral is no less admired and copied by the wife of the progressive and aspiring Isosceles, in the females of whose family no "back-motion" of any kind has become as yet a necessity of life. Hence, in every family of position and consideration, "back motion" is as prevalent as time itself; and the husbands and sons in these households enjoy immunity at least from invisible attacks. Not that it must be for a moment supposed that our Women are destitute of affection. But unfortunately the passion of the moment predominates, in the Frail Sex, over every other consideration. This is, of course, a necessity arising from their unfortunate conformation. For as they have no pretensions to an angle, being inferior in this respect to the very lowest of the Isosceles, they are consequently wholly devoid of brain-power, and have neither reflection, judgment nor forethought, and hardly any memory. Hence, in their fits of fury, they remember no claims and recognize no distinctions. I have actually known a case where a Woman has exterminated her whole household, and half an hour afterwards, when her rage was over and the fragments swept away, has asked what has become of her husband and her children. Obviously then a Woman is not to be irritated as long as she is in a position where she can turn round. When you have them in their apartments--which are constructed with a view to denying them that power--you can say and do what you like; for they are then wholly impotent for mischief, and will not remember a few minutes hence the incident for which they may be at this moment threatening you with death, nor the promises which you may have found it necessary to make in order to pacify their fury. On the whole we get on pretty smoothly in our domestic relations, except in the lower strata of the Military Classes. There the want of tact and discretion on the part of the husbands produces at times indescribable disasters. Relying too much on the offensive weapons of their acute angles instead of the defensive organs of good sense and seasonable simulation, these reckless creatures too often neglect the prescribed construction of the women's apartments, or irritate their wives by ill-advised expressions out of doors, which they refuse immediately to retract. Moreover a blunt and stolid regard for literal truth indisposes them to make those lavish promises by which the more judicious Circle can in a moment pacify his consort. The result is massacre; not, however, without its advantages, as it eliminates the more brutal and troublesome of the Isosceles; and by many of our Circles the destructiveness of the Thinner Sex is regarded as one among many providential arrangements for suppressing redundant population, and nipping Revolution in the bud. Yet even in our best regulated and most approximately Circular families I cannot say that the ideal of family life is so high as with you in Spaceland. There is peace, in so far as the absence of slaughter may be called by that name, but there is necessarily little harmony of tastes or pursuits; and the cautious wisdom of the Circles has ensured safety at the cost of domestic comfort. In every Circular or Polygonal household it has been a habit from time immemorial--and now has become a kind of instinct among the women of our higher classes--that the mothers and daughters should constantly keep their eyes and mouths towards their husband and his male friends; and for a lady in a family of distinction to turn her back upon her husband would be regarded as a kind of portent, involving loss of STATUS. But, as I shall soon shew, this custom, though it has the advantage of safety, is not without its disadvantages. In the house of the Working Man or respectable Tradesman--where the wife is allowed to turn her back upon her husband, while pursuing her household avocations--there are at least intervals of quiet, when the wife is neither seen nor heard, except for the humming sound of the continuous Peace-cry; but in the homes of the upper classes there is too often no peace. There the voluble mouth and bright penetrating eye are ever directed towards the Master of the household; and light itself is not more persistent than the stream of feminine discourse. The tact and skill which suffice to avert a Woman's sting are unequal to the task of stopping a Woman's mouth; and as the wife has absolutely nothing to say, and absolutely no constraint of wit, sense, or conscience to prevent her from saying it, not a few cynics have been found to aver that they prefer the danger of the death-dealing but inaudible sting to the safe sonorousness of a Woman's other end. To my readers in Spaceland the condition of our Women may seem truly deplorable, and so indeed it is. A Male of the lowest type of the Isosceles may look forward to some improvement of his angle, and to the ultimate elevation of the whole of his degraded caste; but no Woman can entertain such hopes for her sex. "Once a Woman, always a Woman" is a Decree of Nature; and the very Laws of Evolution seem suspended in her disfavour. Yet at least we can admire the wise Prearrangement which has ordained that, as they have no hopes, so they shall have no memory to recall, and no forethought to anticipate, the miseries and humiliations which are at once a necessity of their existence and the basis of the constitution of Flatland. Section 5. Of our Methods of Recognizing one another You, who are blessed with shade as well as light, you, who are gifted with two eyes, endowed with a knowledge of perspective, and charmed with the enjoyment of various colours, you, who can actually SEE an angle, and contemplate the complete circumference of a circle in the happy region of the Three Dimensions--how shall I make clear to you the extreme difficulty which we in Flatland experience in recognizing one another's configuration? Recall what I told you above. All beings in Flatland, animate or inanimate, no matter what their form, present TO OUR VIEW the same, or nearly the same, appearance, viz. that of a straight Line. How then can one be distinguished from another, where all appear the same? The answer is threefold. The first means of recognition is the sense of hearing; which with us is far more highly developed than with you, and which enables us not only to distinguish by the voice our personal friends, but even to discriminate between different classes, at least so far as concerns the three lowest orders, the Equilateral, the Square, and the Pentagon--for of the Isosceles I take no account. But as we ascend in the social scale, the process of discriminating and being discriminated by hearing increases in difficulty, partly because voices are assimilated, partly because the faculty of voice-discrimination is a plebeian virtue not much developed among the Aristocracy. And wherever there is any danger of imposture we cannot trust to this method. Amongst our lowest orders, the vocal organs are developed to a degree more than correspondent with those of hearing, so that an Isosceles can easily feign the voice of a Polygon, and, with some training, that of a Circle himself. A second method is therefore more commonly resorted to. FEELING is, among our Women and lower classes--about our upper classes I shall speak presently--the principal test of recognition, at all events between strangers, and when the question is, not as to the individual, but as to the class. What therefore "introduction" is among the higher classes in Spaceland, that the process of "feeling" is with us. "Permit me to ask you to feel and be felt by my friend Mr. So-and-so"--is still, among the more old-fashioned of our country gentlemen in districts remote from towns, the customary formula for a Flatland introduction. But in the towns, and among men of business, the words "be felt by" are omitted and the sentence is abbreviated to, "Let me ask you to feel Mr. So-and-so"; although it is assumed, of course, that the "feeling" is to be reciprocal. Among our still more modern and dashing young gentlemen--who are extremely averse to superfluous effort and supremely indifferent to the purity of their native language--the formula is still further curtailed by the use of "to feel" in a technical sense, meaning, "to recommend-for-the-purposes-of-feeling-and-being-felt"; and at this moment the "slang" of polite or fast society in the upper classes sanctions such a barbarism as "Mr. Smith, permit me to feel Mr. Jones." Let not my Reader however suppose that "feeling" is with us the tedious process that it would be with you, or that we find it necessary to feel right round all the sides of every individual before we determine the class to which he belongs. Long practice and training, begun in the schools and continued in the experience of daily life, enable us to discriminate at once by the sense of touch, between the angles of an equal-sided Triangle, Square, and Pentagon; and I need not say that the brainless vertex of an acute-angled Isosceles is obvious to the dullest touch. It is therefore not necessary, as a rule, to do more than feel a single angle of an individual; and this, once ascertained, tells us the class of the person whom we are addressing, unless indeed he belongs to the higher sections of the nobility. There the difficulty is much greater. Even a Master of Arts in our University of Wentbridge has been known to confuse a ten-sided with a twelve-sided Polygon; and there is hardly a Doctor of Science in or out of that famous University who could pretend to decide promptly and unhesitatingly between a twenty-sided and a twenty-four sided member of the Aristocracy. Those of my readers who recall the extracts I gave above from the Legislative code concerning Women, will readily perceive that the process of introduction by contact requires some care and discretion. Otherwise the angles might inflict on the unwary Feeler irreparable injury. It is essential for the safety of the Feeler that the Felt should stand perfectly still. A start, a fidgety shifting of the position, yes, even a violent sneeze, has been known before now to prove fatal to the incautious, and to nip in the bud many a promising friendship. Especially is this true among the lower classes of the Triangles. With them, the eye is situated so far from their vertex that they can scarcely take cognizance of what goes on at that extremity of their frame. They are, moreover, of a rough coarse nature, not sensitive to the delicate touch of the highly organized Polygon. What wonder then if an involuntary toss of the head has ere now deprived the State of a valuable life! I have heard that my excellent Grandfather--one of the least irregular of his unhappy Isosceles class, who indeed obtained, shortly before his decease, four out of seven votes from the Sanitary and Social Board for passing him into the class of the Equal-sided--often deplored, with a tear in his venerable eye, a miscarriage of this kind, which had occured to his great-great-great-Grandfather, a respectable Working Man with an angle or brain of 59 degrees 30 minutes. According to his account, my unfortunate Ancestor, being afflicted with rheumatism, and in the act of being felt by a Polygon, by one sudden start accidentally transfixed the Great Man through the diagonal; and thereby, partly in consequence of his long imprisonment and degradation, and partly because of the moral shock which pervaded the whole of my Ancestor's relations, threw back our family a degree and a half in their ascent towards better things. The result was that in the next generation the family brain was registered at only 58 degrees, and not till the lapse of five generations was the lost ground recovered, the full 60 degrees attained, and the Ascent from the Isosceles finally achieved. And all this series of calamities from one little accident in the process of Feeling. At this point I think I hear some of my better educated readers exclaim, "How could you in Flatland know anything about angles and degrees, or minutes? We can SEE an angle, because we, in the region of Space, can see two straight lines inclined to one another; but you, who can see nothing but one straight line at a time, or at all events only a number of bits of straight lines all in one straight line--how can you ever discern any angle, and much less register angles of different sizes?" I answer that though we cannot SEE angles, we can INFER them, and this with great precision. Our sense of touch, stimulated by necessity, and developed by long training, enables us to distinguish angles far more accurately than your sense of sight, when unaided by a rule or measure of angles. Nor must I omit to explain that we have great natural helps. It is with us a Law of Nature that the brain of the Isosceles class shall begin at half a degree, or thirty minutes, and shall increase (if it increases at all) by half a degree in every generation; until the goal of 60 degrees is reached, when the condition of serfdom is quitted, and the freeman enters the class of Regulars. Consequently, Nature herself supplies us with an ascending scale or Alphabet of angles for half a degree up to 60 degrees, Specimens of which are placed in every Elementary School throughout the land. Owing to occasional retrogressions, to still more frequent moral and intellectual stagnation, and to the extraordinary fecundity of the Criminal and Vagabond Classes, there is always a vast superfluity of individuals of the half degree and single degree class, and a fair abundance of Specimens up to 10 degrees. These are absolutely destitute of civic rights; and a great number of them, not having even intelligence enough for the purposes of warfare, are devoted by the States to the service of education. Fettered immovably so as to remove all possibility of danger, they are placed in the class rooms of our Infant Schools, and there they are utilized by the Board of Education for the purpose of imparting to the offspring of the Middle Classes that tact and intelligence of which these wretched creatures themselves are utterly devoid. In some States the Specimens are occasionally fed and suffered to exist for several years; but in the more temperate and better regulated regions, it is found in the long run more advantageous for the educational interests of the young, to dispense with food, and to renew the Specimens every month--which is about the average duration of the foodless existence of the Criminal class. In the cheaper schools, what is gained by the longer existence of the Specimen is lost, partly in the expenditure for food, and partly in the diminished accuracy of the angles, which are impaired after a few weeks of constant "feeling". Nor must we forget to add, in enumerating the advantages of the more expensive system, that it tends, though slightly yet perceptibly, to the diminution of the redundant Isosceles population--an object which every statesman in Flatland constantly keeps in view. On the whole therefore--although I am not ignorant that, in many popularly elected School Boards, there is a reaction in favour of "the cheap system" as it is called--I am myself disposed to think that this is one of the many cases in which expense is the truest economy. But I must not allow questions of School Board politics to divert me from my subject. Enough has been said, I trust, to shew that Recognition by Feeling is not so tedious or indecisive a process as might have been supposed; and it is obviously more trustworthy than Recognition by hearing. Still there remains, as has been pointed out above, the objection that this method is not without danger. For this reason many in the Middle and Lower classes, and all without exception in the Polygonal and Circular orders, prefer a third method, the description of which shall be reserved for the next section. Section 6. Of Recognition by Sight I am about to appear very inconsistent. In previous sections I have said that all figures in Flatland present the appearance of a straight line; and it was added or implied, that it is consequently impossible to distinguish by the visual organ between individuals of different classes: yet now I am about to explain to my Spaceland critics how we are able to recognize one another by the sense of sight. If however the Reader will take the trouble to refer to the passage in which Recognition by Feeling is stated to be universal, he will find this qualification--"among the lower classes". It is only among the higher classes and in our temperate climates that Sight Recognition is practised. That this power exists in any regions and for any classes is the result of Fog; which prevails during the greater part of the year in all parts save the torrid zones. That which is with you in Spaceland an unmixed evil, blotting out the landscape, depressing the spirits, and enfeebling the health, is by us recognized as a blessing scarcely inferior to air itself, and as the Nurse of arts and Parent of sciences. But let me explain my meaning, without further eulogies on this beneficent Element. If Fog were non-existent, all lines would appear equally and indistinguishably clear; and this is actually the case in those unhappy countries in which the atmosphere is perfectly dry and transparent. But wherever there is a rich supply of Fog objects that are at a distance, say of three feet, are appreciably dimmer than those at a distance of two feet eleven inches; and the result is that by careful and constant experimental observation of comparative dimness and clearness, we are enabled to infer with great exactness the configuration of the object observed. An instance will do more than a volume of generalities to make my meaning clear. Suppose I see two individuals approaching whose rank I wish to ascertain. They are, we will suppose, a Merchant and a Physician, or in other words, an Equilateral Triangle and a Pentagon: how am I to distinguish them? [Illustration 3] [ASCII approximation follows] C (1) |\ - _ D | \ ||- _ | \ || - _ | <--- >|| -----------+(> Eye-glance ___C' (2) | / A|| _ - ___--- \ - _D' | / ||_ - __--- \ || - _ |/ _ - E | \ || - _ B | \ || - _ | Eye-glance \ || - _ | <----------- A'>|| ------------------------+(> | / || _ - | / || _ - |__ / || _ - ---___ / || _ - ---___/ _ -E' B' It will be obvious, to every child in Spaceland who has touched the threshold of Geometrical Studies, that, if I can bring my eye so that its glance may bisect an angle (A) of the approaching stranger, my view will lie as it were evenly between his two sides that are next to me (viz. CA and AB), so that I shall contemplate the two impartially, and both will appear of the same size. Now in the case of (1) the Merchant, what shall I see? I shall see a straight line DAE, in which the middle point (A) will be very bright because it is nearest to me; but on either side the line will shade away RAPIDLY INTO DIMNESS, because the sides AC and AB RECEDE RAPIDLY INTO THE FOG and what appear to me as the Merchant's extremities, viz. D and E, will be VERY DIM INDEED. On the other hand in the case of (2) the Physician, though I shall here also see a line (D'A'E') with a bright centre (A'), yet it will shade away LESS RAPIDLY into dimness, because the sides (A'C', A'B') RECEDE LESS RAPIDLY INTO THE FOG: and what appear to me the Physician's extremities, viz. D' and E', will not be NOT SO DIM as the extremities of the Merchant. The Reader will probably understand from these two instances how--after a very long training supplemented by constant experience--it is possible for the well-educated classes among us to discriminate with fair accuracy between the middle and lowest orders, by the sense of sight. If my Spaceland Patrons have grasped this general conception, so far as to conceive the possibility of it and not to reject my account as altogether incredible--I shall have attained all I can reasonably expect. Were I to attempt further details I should only perplex. Yet for the sake of the young and inexperienced, who may perchance infer--from the two simple instances I have given above, of the manner in which I should recognize my Father and my Sons--that Recognition by sight is an easy affair, it may be needful to point out that in actual life most of the problems of Sight Recognition are far more subtle and complex. If for example, when my Father, the Triangle, approaches me, he happens to present his side to me instead of his angle, then, until I have asked him to rotate, or until I have edged my eye round him, I am for the moment doubtful whether he may not be a Straight Line, or, in other words, a Woman. Again, when I am in the company of one of my two hexagonal Grandsons, contemplating one of his sides (AB) full front, it will be evident from the accompanying diagram that I shall see one whole line (AB) in comparative brightness (shading off hardly at all at the ends) and two smaller lines (CA and BD) dim throughout and shading away into greater dimness towards the extremities C and D. [Illustration 4] [ASCII approximation follows] /\ - _ C / \ || _ / \ || - _ / \|| - _ | A || - _ | || -+(> (Eye) | B || _ - \ /|| _ - \ / || _ - \ / || - \/ _ - D But I must not give way to the temptation of enlarging on these topics. The meanest mathematician in Spaceland will readily believe me when I assert that the problems of life, which present themselves to the well-educated--when they are themselves in motion, rotating, advancing or retreating, and at the same time attempting to discriminate by the sense of sight between a number of Polygons of high rank moving in different directions, as for example in a ball-room or conversazione--must be of a nature to task the angularity of the most intellectual, and amply justify the rich endowments of the Learned Professors of Geometry, both Static and Kinetic, in the illustrious University of Wentbridge, where the Science and Art of Sight Recognition are regularly taught to large classes of the ELITE of the States. It is only a few of the scions of our noblest and wealthiest houses, who are able to give the time and money necessary for the thorough prosecution of this noble and valuable Art. Even to me, a Mathematician of no mean standing, and the Grandfather of two most hopeful and perfectly regular Hexagons, to find myself in the midst of a crowd of rotating Polygons of the higher classes, is occasionally very perplexing. And of course to a common Tradesman, or Serf, such a sight is almost as unintelligible as it would be to you, my Reader, were you suddenly transported into our country. In such a crowd you could see on all sides of you nothing but a Line, apparently straight, but of which the parts would vary irregularly and perpetually in brightness or dimness. Even if you had completed your third year in the Pentagonal and Hexagonal classes in the University, and were perfect in the theory of the subject, you would still find that there was need of many years of experience, before you could move in a fashionable crowd without jostling against your betters, whom it is against etiquette to ask to "feel", and who, by their superior culture and breeding, know all about your movements, while you know very little or nothing about theirs. In a word, to comport oneself with perfect propriety in Polygonal society, one ought to be a Polygon oneself. Such at least is the painful teaching of my experience. It is astonishing how much the Art--or I may almost call it instinct--of Sight Recognition is developed by the habitual practice of it and by the avoidance of the custom of "Feeling". Just as, with you, the deaf and dumb, if once allowed to gesticulate and to use the hand-alphabet, will never acquire the more difficult but far more valuable art of lipspeech and lip-reading, so it is with us as regards "Seeing" and "Feeling". None who in early life resort to "Feeling" will ever learn "Seeing" in perfection. For this reason, among our Higher Classes, "Feeling" is discouraged or absolutely forbidden. From the cradle their children, instead of going to the Public Elementary schools (where the art of Feeling is taught), are sent to higher Seminaries of an exclusive character; and at our illustrious University, to "feel" is regarded as a most serious fault, involving Rustication for the first offence, and Expulsion for the second. But among the lower classes the art of Sight Recognition is regarded as an unattainable luxury. A common Tradesman cannot afford to let his son spend a third of his life in abstract studies. The children of the poor are therefore allowed to "feel" from their earliest years, and they gain thereby a precocity and an early vivacity which contrast at first most favourably with the inert, undeveloped, and listless behaviour of the half-instructed youths of the Polygonal class; but when the latter have at last completed their University course, and are prepared to put their theory into practice, the change that comes over them may almost be described as a new birth, and in every art, science, and social pursuit they rapidly overtake and distance their Triangular competitors. Only a few of the Polygonal Class fail to pass the Final Test or Leaving Examination at the University. The condition of the unsuccessful minority is truly pitiable. Rejected from the higher class, they are also despised by the lower. They have neither the matured and systematically trained powers of the Polygonal Bachelors and Masters of Arts, nor yet the native precocity and mercurial versatility of the youthful Tradesman. The professions, the public services, are closed against them; and though in most States they are not actually debarred from marriage, yet they have the greatest difficulty in forming suitable alliances, as experience shews that the offspring of such unfortunate and ill-endowed parents is generally itself unfortunate, if not positively Irregular. It is from these specimens of the refuse of our Nobility that the great Tumults and Seditions of past ages have generally derived their leaders; and so great is the mischief thence arising that an increasing minority of our more progressive Statesmen are of opinion that true mercy would dictate their entire suppression, by enacting that all who fail to pass the Final Examination of the University should be either imprisoned for life, or extinguished by a painless death. But I find myself digressing into the subject of Irregularities, a matter of such vital interest that it demands a separate section. Section 7. Concerning Irregular Figures Throughout the previous pages I have been assuming--what perhaps should have been laid down at the beginning as a distinct and fundamental proposition--that every human being in Flatland is a Regular Figure, that is to say of regular construction. By this I mean that a Woman must not only be a line, but a straight line; that an Artisan or Soldier must have two of his sides equal; that Tradesmen must have three sides equal; Lawyers (of which class I am a humble member), four sides equal, and generally, that in every Polygon, all the sides must be equal. The size of the sides would of course depend upon the age of the individual. A Female at birth would be about an inch long, while a tall adult Woman might extend to a foot. As to the Males of every class, it may be roughly said that the length of an adult's sides, when added together, is two feet or a little more. But the size of our sides is not under consideration. I am speaking of the EQUALITY of sides, and it does not need much reflection to see that the whole of the social life in Flatland rests upon the fundamental fact that Nature wills all Figures to have their sides equal. If our sides were unequal our angles might be unequal. Instead of its being sufficient to feel, or estimate by sight, a single angle in order to determine the form of an individual, it would be necessary to ascertain each angle by the experiment of Feeling. But life would be too short for such a tedious grouping. The whole science and art of Sight Recognition would at once perish; Feeling, so far as it is an art, would not long survive; intercourse would become perilous or impossible; there would be an end to all confidence, all forethought; no one would be safe in making the most simple social arrangements; in a word, civilization would relapse into barbarism. Am I going too fast to carry my Readers with me to these obvious conclusions? Surely a moment's reflection, and a single instance from common life, must convince every one that our whole social system is based upon Regularity, or Equality of Angles. You meet, for example, two or three Tradesmen in the street, whom you recognize at once to be Tradesmen by a glance at their angles and rapidly bedimmed sides, and you ask them to step into your house to lunch. This you do at present with perfect confidence, because everyone knows to an inch or two the area occupied by an adult Triangle: but imagine that your Tradesman drags behind his regular and respectable vertex, a parallelogram of twelve or thirteen inches in diagonal:--what are you to do with such a monster sticking fast in your house door? But I am insulting the intelligence of my Readers by accumulating details which must be patent to everyone who enjoys the advantages of a Residence in Spaceland. Obviously the measurements of a single angle would no longer be sufficient under such portentous circumstances; one's whole life would be taken up in feeling or surveying the perimeter of one's acquaintances. Already the difficulties of avoiding a collision in a crowd are enough to tax the sagacity of even a well-educated Square; but if no one could calculate the Regularity of a single figure in the company, all would be chaos and confusion, and the slightest panic would cause serious injuries, or--if there happened to be any Women or Soldiers present--perhaps considerable loss of life. Expediency therefore concurs with Nature in stamping the seal of its approval upon Regularity of conformation: nor has the Law been backward in seconding their efforts. "Irregularity of Figure" means with us the same as, or more than, a combination of moral obliquity and criminality with you, and is treated accordingly. There are not wanting, it is true, some promulgators of paradoxes who maintain that there is no necessary connection between geometrical and moral Irregularity. "The Irregular", they say, "is from his birth scouted by his own parents, derided by his brothers and sisters, neglected by the domestics, scorned and suspected by society, and excluded from all posts of responsibility, trust, and useful activity. His every movement is jealously watched by the police till he comes of age and presents himself for inspection; then he is either destroyed, if he is found to exceed the fixed margin of deviation, or else immured in a Government Office as a clerk of the seventh class; prevented from marriage; forced to drudge at an uninteresting occupation for a miserable stipend; obliged to live and board at the office, and to take even his vacation under close supervision; what wonder that human nature, even in the best and purest, is embittered and perverted by such surroundings!" All this very plausible reasoning does not convince me, as it has not convinced the wisest of our Statesmen, that our ancestors erred in laying it down as an axiom of policy that the toleration of Irregularity is incompatible with the safety of the State. Doubtless, the life of an Irregular is hard; but the interests of the Greater Number require that it shall be hard. If a man with a triangular front and a polygonal back were allowed to exist and to propagate a still more Irregular posterity, what would become of the arts of life? Are the houses and doors and churches in Flatland to be altered in order to accommodate such monsters? Are our ticket-collectors to be required to measure every man's perimeter before they allow him to enter a theatre or to take his place in a lecture room? Is an Irregular to be exempted from the militia? And if not, how is he to be prevented from carrying desolation into the ranks of his comrades? Again, what irresistible temptations to fraudulent impostures must needs beset such a creature! How easy for him to enter a shop with his polygonal front foremost, and to order goods to any extent from a confiding tradesman! Let the advocates of a falsely called Philanthropy plead as they may for the abrogation of the Irregular Penal Laws, I for my part have never known an Irregular who was not also what Nature evidently intended him to be--a hypocrite, a misanthropist, and, up to the limits of his power, a perpetrator of all manner of mischief. Not that I should be disposed to recommend (at present) the extreme measures adopted by some States, where an infant whose angle deviates by half a degree from the correct angularity is summarily destroyed at birth. Some of our highest and ablest men, men of real genius, have during their earliest days laboured under deviations as great as, or even greater than, forty-five minutes: and the loss of their precious lives would have been an irreparable injury to the State. The art of healing also has achieved some of its most glorious triumphs in the compressions, extensions, trepannings, colligations, and other surgical or diaetetic operations by which Irregularity has been partly or wholly cured. Advocating therefore a VIA MEDIA, I would lay down no fixed or absolute line of demarcation; but at the period when the frame is just beginning to set, and when the Medical Board has reported that recovery is improbable, I would suggest that the Irregular offspring be painlessly and mercifully consumed. Section 8. Of the Ancient Practice of Painting If my Readers have followed me with any attention up to this point, they will not be surprised to hear that life is somewhat dull in Flatland. I do not, of course, mean that there are not battles, conspiracies, tumults, factions, and all those other phenomena which are supposed to make History interesting; nor would I deny that the strange mixture of the problems of life and the problems of Mathematics, continually inducing conjecture and giving the opportunity of immediate verification, imparts to our existence a zest which you in Spaceland can hardly comprehend. I speak now from the aesthetic and artistic point of view when I say that life with us is dull; aesthetically and artistically, very dull indeed. How can it be otherwise, when all one's prospect, all one's landscapes, historical pieces, portraits, flowers, still life, are nothing but a single line, with no varieties except degrees of brightness and obscurity? It was not always thus. Colour, if Tradition speaks the truth, once for the space of half a dozen centuries or more, threw a transient splendour over the lives of our ancestors in the remotest ages. Some private individual--a Pentagon whose name is variously reported--having casually discovered the constituents of the simpler colours and a rudimentary method of painting, is said to have begun decorating first his house, then his slaves, then his Father, his Sons, and Grandsons, lastly himself. The convenience as well as the beauty of the results commended themselves to all. Wherever Chromatistes,--for by that name the most trustworthy authorities concur in calling him,--turned his variegated frame, there he at once excited attention, and attracted respect. No one now needed to "feel" him; no one mistook his front for his back; all his movements were readily ascertained by his neighbours without the slightest strain on their powers of calculation; no one jostled him, or failed to make way for him; his voice was saved the labour of that exhausting utterance by which we colourless Squares and Pentagons are often forced to proclaim our individuality when we move amid a crowd of ignorant Isosceles. The fashion spread like wildfire. Before a week was over, every Square and Triangle in the district had copied the example of Chromatistes, and only a few of the more conservative Pentagons still held out. A month or two found even the Dodecagons infected with the innovation. A year had not elapsed before the habit had spread to all but the very highest of the Nobility. Needless to say, the custom soon made its way from the district of Chromatistes to surrounding regions; and within two generations no one in all Flatland was colourless except the Women and the Priests. Here Nature herself appeared to erect a barrier, and to plead against extending the innovation to these two classes. Many-sidedness was almost essential as a pretext for the Innovators. "Distinction of sides is intended by Nature to imply distinction of colours"--such was the sophism which in those days flew from mouth to mouth, converting whole towns at a time to the new culture. But manifestly to our Priests and Women this adage did not apply. The latter had only one side, and therefore--plurally and pedantically speaking--NO SIDES. The former--if at least they would assert their claim to be really and truly Circles, and not mere high-class Polygons with an infinitely large number of infinitesimally small sides--were in the habit of boasting (what Women confessed and deplored) that they also had no sides, being blessed with a perimeter of one line, or, in other words, a Circumference. Hence it came to pass that these two Classes could see no force in the so-called axiom about "Distinction of Sides implying Distinction of Colour"; and when all others had succumbed to the fascinations of corporal decoration, the Priests and the Women alone still remained pure from the pollution of paint. Immoral, licentious, anarchical, unscientific--call them by what names you will--yet, from an aesthetic point of view, those ancient days of the Colour Revolt were the glorious childhood of Art in Flatland--a childhood, alas, that never ripened into manhood, nor even reached the blossom of youth. To live was then in itself a delight, because living implied seeing. Even at a small party, the company was a pleasure to behold; the richly varied hues of the assembly in a church or theatre are said to have more than once proved too distracting for our greatest teachers and actors; but most ravishing of all is said to have been the unspeakable magnificence of a military review. The sight of a line of battle of twenty thousand Isosceles suddenly facing about, and exchanging the sombre black of their bases for the orange and purple of the two sides including their acute angle; the militia of the Equilateral Triangles tricoloured in red, white, and blue; the mauve, ultra-marine, gamboge, and burnt umber of the Square artillerymen rapidly rotating near their vermilion guns; the dashing and flashing of the five-coloured and six-coloured Pentagons and Hexagons careering across the field in their offices of surgeons, geometricians and aides-de-camp--all these may well have been sufficient to render credible the famous story how an illustrious Circle, overcome by the artistic beauty of the forces under his command, threw aside his marshal's baton and his royal crown, exclaiming that he henceforth exchanged them for the artist's pencil. How great and glorious the sensuous development of these days must have been is in part indicated by the very language and vocabulary of the period. The commonest utterances of the commonest citizens in the time of the Colour Revolt seem to have been suffused with a richer tinge of word or thought; and to that era we are even now indebted for our finest poetry and for whatever rhythm still remains in the more scientific utterance of these modern days. Section 9. Of the Universal Colour Bill But meanwhile the intellectual Arts were fast decaying. The Art of Sight Recognition, being no longer needed, was no longer practised; and the studies of Geometry, Statics, Kinetics, and other kindred subjects, came soon to be considered superfluous, and fell into disrespect and neglect even at our University. The inferior Art of Feeling speedily experienced the same fate at our Elementary Schools. Then the Isosceles classes, asserting that the Specimens were no longer used nor needed, and refusing to pay the customary tribute from the Criminal classes to the service of Education, waxed daily more numerous and more insolent on the strength of their immunity from the old burden which had formerly exercised the twofold wholesome effect of at once taming their brutal nature and thinning their excessive numbers. Year by year the Soldiers and Artisans began more vehemently to assert--and with increasing truth--that there was no great difference between them and the very highest class of Polygons, now that they were raised to an equality with the latter, and enabled to grapple with all the difficulties and solve all the problems of life, whether Statical or Kinetical, by the simple process of Colour Recognition. Not content with the natural neglect into which Sight Recognition was falling, they began boldly to demand the legal prohibition of all "monopolizing and aristocratic Arts" and the consequent abolition of all endowments for the studies of Sight Recognition, Mathematics, and Feeling. Soon, they began to insist that inasmuch as Colour, which was a second Nature, had destroyed the need of aristocratic distinctions, the Law should follow in the same path, and that henceforth all individuals and all classes should be recognized as absolutely equal and entitled to equal rights. Finding the higher Orders wavering and undecided, the leaders of the Revolution advanced still further in their requirements, and at last demanded that all classes alike, the Priests and the Women not excepted, should do homage to Colour by submitting to be painted. When it was objected that Priests and Women had no sides, they retorted that Nature and Expediency concurred in dictating that the front half of every human being (that is to say, the half containing his eye and mouth) should be distinguishable from his hinder half. They therefore brought before a general and extraordinary Assembly of all the States of Flatland a Bill proposing that in every Woman the half containing the eye and mouth should be coloured red, and the other half green. The Priests were to be painted in the same way, red being applied to that semicircle in which the eye and mouth formed the middle point; while the other or hinder semicircle was to be coloured green. There was no little cunning in this proposal, which indeed emanated not from any Isosceles--for no being so degraded would have had angularity enough to appreciate, much less to devise, such a model of state-craft--but from an Irregular Circle who, instead of being destroyed in his childhood, was reserved by a foolish indulgence to bring desolation on his country and destruction on myriads of his followers. On the one hand the proposition was calculated to bring the Women in all classes over to the side of the Chromatic Innovation. For by assigning to the Women the same two colours as were assigned to the Priests, the Revolutionists thereby ensured that, in certain positions, every Woman would appear like a Priest, and be treated with corresponding respect and deference--a prospect that could not fail to attract the Female Sex in a mass. But by some of my Readers the possibility of the identical appearance of Priests and Women, under the new Legislation, may not be recognized; if so, a word or two will make it obvious. Imagine a woman duly decorated, according to the new Code; with the front half (i.e. the half containing eye and mouth) red, and with the hinder half green. Look at her from one side. Obviously you will see a straight line, HALF RED, HALF GREEN. [Illustration 5] [ASCII approximation follows] [for simplicity's sake, the circle is approximated as an octogon] M _____ / \ - C_ / \|| - _ | || - _ A|- - - - - - -||B- - - - - -_-+(> (Eye) | || _ - \ /||_ - \ _____ / - D Now imagine a Priest, whose mouth is at M, and whose front semicircle (AMB) is consequently coloured red, while his hinder semicircle is green; so that the diameter AB divides the green from the red. If you contemplate the Great Man so as to have your eye in the same straight line as his dividing diameter (AB), what you will see will be a straight line (CBD), of which ONE HALF (CB) WILL BE RED, AND THE OTHER (BD) GREEN. The whole line (CD) will be rather shorter perhaps than that of a full-sized Woman, and will shade off more rapidly towards its extremities; but the identity of the colours would give you an immediate impression of identity of Class, making you neglectful of other details. Bear in mind the decay of Sight Recognition which threatened society at the time of the Colour Revolt; add too the certainty that Women would speedily learn to shade off their extremities so as to imitate the Circles; it must then be surely obvious to you, my dear Reader, that the Colour Bill placed us under a great danger of confounding a Priest with a young Woman. How attractive this prospect must have been to the Frail Sex may readily be imagined. They anticipated with delight the confusion that would ensue. At home they might hear political and ecclesiastical secrets intended not for them but for their husbands and brothers, and might even issue commands in the name of a priestly Circle; out of doors the striking combination of red and green, without addition of any other colours, would be sure to lead the common people into endless mistakes, and the Women would gain whatever the Circles lost, in the deference of the passers by. As for the scandal that would befall the Circular Class if the frivolous and unseemly conduct of the Women were imputed to them, and as to the consequent subversion of the Constitution, the Female Sex could not be expected to give a thought to these considerations. Even in the households of the Circles, the Women were all in favour of the Universal Colour Bill. The second object aimed at by the Bill was the gradual demoralization of the Circles themselves. In the general intellectual decay they still preserved their pristine clearness and strength of understanding. From their earliest childhood, familiarized in their Circular households with the total absence of Colour, the Nobles alone preserved the Sacred Art of Sight Recognition, with all the advantages that result from that admirable training of the intellect. Hence, up to the date of the introduction of the Universal Colour Bill, the Circles had not only held their own, but even increased their lead of the other classes by abstinence from the popular fashion. Now therefore the artful Irregular whom I described above as the real author of this diabolical Bill, determined at one blow to lower the status of the Hierarchy by forcing them to submit to the pollution of Colour, and at the same time to destroy their domestic opportunities of training in the Art of Sight Recognition, so as to enfeeble their intellects by depriving them of their pure and colourless homes. Once subjected to the chromatic taint, every parental and every childish Circle would demoralize each other. Only in discerning between the Father and the Mother would the Circular infant find problems for the exercise of its understanding--problems too often likely to be corrupted by maternal impostures with the result of shaking the child's faith in all logical conclusions. Thus by degrees the intellectual lustre of the Priestly Order would wane, and the road would then lie open for a total destruction of all Aristocratic Legislature and for the subversion of our Privileged Classes. Section 10. Of the Suppression of the Chromatic Sedition The agitation for the Universal Colour Bill continued for three years; and up to the last moment of that period it seemed as though Anarchy were destined to triumph. A whole army of Polygons, who turned out to fight as private soldiers, was utterly annihilated by a superior force of Isosceles Triangles--the Squares and Pentagons meanwhile remaining neutral. Worse than all, some of the ablest Circles fell a prey to conjugal fury. Infuriated by political animosity, the wives in many a noble household wearied their lords with prayers to give up their opposition to the Colour Bill; and some, finding their entreaties fruitless, fell on and slaughtered their innocent children and husband, perishing themselves in the act of carnage. It is recorded that during that triennial agitation no less than twenty-three Circles perished in domestic discord. Great indeed was the peril. It seemed as though the Priests had no choice between submission and extermination; when suddenly the course of events was completely changed by one of those picturesque incidents which Statesmen ought never to neglect, often to anticipate, and sometimes perhaps to originate, because of the absurdly disproportionate power with which they appeal to the sympathies of the populace. It happened that an Isosceles of a low type, with a brain little if at all above four degrees--accidentally dabbling in the colours of some Tradesman whose shop he had plundered--painted himself, or caused himself to be painted (for the story varies) with the twelve colours of a Dodecagon. Going into the Market Place he accosted in a feigned voice a maiden, the orphan daughter of a noble Polygon, whose affection in former days he had sought in vain; and by a series of deceptions--aided, on the one side, by a string of lucky accidents too long to relate, and on the other, by an almost inconceivable fatuity and neglect of ordinary precautions on the part of the relations of the bride--he succeeded in consummating the marriage. The unhappy girl committed suicide on discovering the fraud to which she had been subjected. When the news of this catastrophe spread from State to State the minds of the Women were violently agitated. Sympathy with the miserable victim and anticipations of similar deceptions for themselves, their sisters, and their daughters, made them now regard the Colour Bill in an entirely new aspect. Not a few openly avowed themselves converted to antagonism; the rest needed only a slight stimulus to make a similar avowal. Seizing this favourable opportunity, the Circles hastily convened an extraordinary Assembly of the States; and besides the usual guard of Convicts, they secured the attendance of a large number of reactionary Women. Amidst an unprecedented concourse, the Chief Circle of those days--by name Pantocyclus--arose to find himself hissed and hooted by a hundred and twenty thousand Isosceles. But he secured silence by declaring that henceforth the Circles would enter on a policy of Concession; yielding to the wishes of the majority, they would accept the Colour Bill. The uproar being at once converted to applause, he invited Chromatistes, the leader of the Sedition, into the centre of the hall, to receive in the name of his followers the submission of the Hierarchy. Then followed a speech, a masterpiece of rhetoric, which occupied nearly a day in the delivery, and to which no summary can do justice. With a grave appearance of impartiality he declared that as they were now finally committing themselves to Reform or Innovation, it was desirable that they should take one last view of the perimeter of the whole subject, its defects as well as its advantages. Gradually introducing the mention of the dangers to the Tradesmen, the Professional Classes and the Gentlemen, he silenced the rising murmurs of the Isosceles by reminding them that, in spite of all these defects, he was willing to accept the Bill if it was approved by the majority. But it was manifest that all, except the Isosceles, were moved by his words and were either neutral or averse to the Bill. Turning now to the Workmen he asserted that their interests must not be neglected, and that, if they intended to accept the Colour Bill, they ought at least to do so with full view of the consequences. Many of them, he said, were on the point of being admitted to the class of the Regular Triangles; others anticipated for their children a distinction they could not hope for themselves. That honourable ambition would now have to be sacrificed. With the universal adoption of Colour, all distinctions would cease; Regularity would be confused with Irregularity; development would give place to retrogression; the Workman would in a few generations be degraded to the level of the Military, or even the Convict Class; political power would be in the hands of the greatest number, that is to say the Criminal Classes, who were already more numerous than the Workmen, and would soon out-number all the other Classes put together when the usual Compensative Laws of Nature were violated. A subdued murmur of assent ran through the ranks of the Artisans, and Chromatistes, in alarm, attempted to step forward and address them. But he found himself encompassed with guards and forced to remain silent while the Chief Circle in a few impassioned words made a final appeal to the Women, exclaiming that, if the Colour Bill passed, no marriage would henceforth be safe, no woman's honour secure; fraud, deception, hypocrisy would pervade every household; domestic bliss would share the fate of the Constitution and pass to speedy perdition. "Sooner than this," he cried, "Come death." At these words, which were the preconcerted signal for action, the Isosceles Convicts fell on and transfixed the wretched Chromatistes; the Regular Classes, opening their ranks, made way for a band of Women who, under direction of the Circles, moved, back foremost, invisibly and unerringly upon the unconscious soldiers; the Artisans, imitating the example of their betters, also opened their ranks. Meantime bands of Convicts occupied every entrance with an impenetrable phalanx. The battle, or rather carnage, was of short duration. Under the skillful generalship of the Circles almost every Woman's charge was fatal and very many extracted their sting uninjured, ready for a second slaughter. But no second blow was needed; the rabble of the Isosceles did the rest of the business for themselves. Surprised, leader-less, attacked in front by invisible foes, and finding egress cut off by the Convicts behind them, they at once--after their manner--lost all presence of mind, and raised the cry of "treachery". This sealed their fate. Every Isosceles now saw and felt a foe in every other. In half an hour not one of that vast multitude was living; and the fragments of seven score thousand of the Criminal Class slain by one another's angles attested the triumph of Order. The Circles delayed not to push their victory to the uttermost. The Working Men they spared but decimated. The Militia of the Equilaterals was at once called out; and every Triangle suspected of Irregularity on reasonable grounds, was destroyed by Court Martial, without the formality of exact measurement by the Social Board. The homes of the Military and Artisan classes were inspected in a course of visitations extending through upwards of a year; and during that period every town, village, and hamlet was systematically purged of that excess of the lower orders which had been brought about by the neglect to pay the tribute of Criminals to the Schools and University, and by the violation of the other natural Laws of the Constitution of Flatland. Thus the balance of classes was again restored. Needless to say that henceforth the use of Colour was abolished, and its possession prohibited. Even the utterance of any word denoting Colour, except by the Circles or by qualified scientific teachers, was punished by a severe penalty. Only at our University in some of the very highest and most esoteric classes--which I myself have never been privileged to attend--it is understood that the sparing use of Colour is still sanctioned for the purpose of illustrating some of the deeper problems of mathematics. But of this I can only speak from hearsay. Elsewhere in Flatland, Colour is now non-existent. The art of making it is known to only one living person, the Chief Circle for the time being; and by him it is handed down on his death-bed to none but his Successor. One manufactory alone produces it; and, lest the secret should be betrayed, the Workmen are annually consumed, and fresh ones introduced. So great is the terror with which even now our Aristocracy looks back to the far-distant days of the agitation for the Universal Colour Bill. Section 11. Concerning our Priests It is high time that I should pass from these brief and discursive notes about things in Flatland to the central event of this book, my initiation into the mysteries of Space. THAT is my subject; all that has gone before is merely preface. For this reason I must omit many matters of which the explanation would not, I flatter myself, be without interest for my Readers: as for example, our method of propelling and stopping ourselves, although destitute of feet; the means by which we give fixity to structures of wood, stone, or brick, although of course we have no hands, nor can we lay foundations as you can, nor avail ourselves of the lateral pressure of the earth; the manner in which the rain originates in the intervals between our various zones, so that the northern regions do not intercept the moisture from falling on the southern; the nature of our hills and mines, our trees and vegetables, our seasons and harvests; our Alphabet and method of writing, adapted to our linear tablets; these and a hundred other details of our physical existence I must pass over, nor do I mention them now except to indicate to my readers that their omission proceeds not from forgetfulness on the part of the author, but from his regard for the time of the Reader. Yet before I proceed to my legitimate subject some few final remarks will no doubt be expected by my Readers upon those pillars and mainstays of the Constitution of Flatland, the controllers of our conduct and shapers of our destiny, the objects of universal homage and almost of adoration: need I say that I mean our Circles or Priests? When I call them Priests, let me not be understood as meaning no more than the term denotes with you. With us, our Priests are Administrators of all Business, Art, and Science; Directors of Trade, Commerce, Generalship, Architecture, Engineering, Education, Statesmanship, Legislature, Morality, Theology; doing nothing themselves, they are the Causes of everything worth doing, that is done by others. Although popularly everyone called a Circle is deemed a Circle, yet among the better educated Classes it is known that no Circle is really a Circle, but only a Polygon with a very large number of very small sides. As the number of the sides increases, a Polygon approximates to a Circle; and, when the number is very great indeed, say for example three or four hundred, it is extremely difficult for the most delicate touch to feel any polygonal angles. Let me say rather, it WOULD be difficult: for, as I have shown above, Recognition by Feeling is unknown among the highest society, and to FEEL a Circle would be considered a most audacious insult. This habit of abstention from Feeling in the best society enables a Circle the more easily to sustain the veil of mystery in which, from his earliest years, he is wont to enwrap the exact nature of his Perimeter or Circumference. Three feet being the average Perimeter it follows that, in a Polygon of three hundred sides each side will be no more than the hundredth part of a foot in length, or little more than the tenth part of an inch; and in a Polygon of six or seven hundred sides the sides are little larger than the diameter of a Spaceland pin-head. It is always assumed, by courtesy, that the Chief Circle for the time being has ten thousand sides. The ascent of the posterity of the Circles in the social scale is not restricted, as it is among the lower Regular classes, by the Law of Nature which limits the increase of sides to one in each generation. If it were so, the number of sides in a Circle would be a mere question of pedigree and arithmetic, and the four hundred and ninety-seventh descendant of an Equilateral Triangle would necessarily be a Polygon with five hundred sides. But this is not the case. Nature's Law prescribes two antagonistic decrees affecting Circular propagation; first, that as the race climbs higher in the scale of development, so development shall proceed at an accelerated pace; second, that in the same proportion, the race shall become less fertile. Consequently in the home of a Polygon of four or five hundred sides it is rare to find a son; more than one is never seen. On the other hand the son of a five-hundred-sided Polygon has been known to possess five hundred and fifty, or even six hundred sides. Art also steps in to help the process of the higher Evolution. Our physicians have discovered that the small and tender sides of an infant Polygon of the higher class can be fractured, and his whole frame re-set, with such exactness that a Polygon of two or three hundred sides sometimes--by no means always, for the process is attended with serious risk--but sometimes overleaps two or three hundred generations, and as it were doubles at a stroke, the number of his progenitors and the nobility of his descent. Many a promising child is sacrificed in this way. Scarcely one out of ten survives. Yet so strong is the parental ambition among those Polygons who are, as it were, on the fringe of the Circular class, that it is very rare to find a Nobleman of that position in society, who has neglected to place his first-born in the Circular Neo-Therapeutic Gymnasium before he has attained the age of a month. One year determines success or failure. At the end of that time the child has, in all probability, added one more to the tombstones that crowd the Neo-Therapeutic Cemetery; but on rare occasions a glad procession bears back the little one to his exultant parents, no longer a Polygon, but a Circle, at least by courtesy: and a single instance of so blessed a result induces multitudes of Polygonal parents to submit to similar domestic sacrifices, which have a dissimilar issue. Section 12. Of the Doctrine of our Priests As to the doctrine of the Circles it may briefly be summed up in a single maxim, "Attend to your Configuration." Whether political, ecclesiastical, or moral, all their teaching has for its object the improvement of individual and collective Configuration--with special reference of course to the Configuration of the Circles, to which all other objects are subordinated. It is the merit of the Circles that they have effectually suppressed those ancient heresies which led men to waste energy and sympathy in the vain belief that conduct depends upon will, effort, training, encouragement, praise, or anything else but Configuration. It was Pantocyclus--the illustrious Circle mentioned above, as the queller of the Colour Revolt--who first convinced mankind that Configuration makes the man; that if, for example, you are born an Isosceles with two uneven sides, you will assuredly go wrong unless you have them made even--for which purpose you must go to the Isosceles Hospital; similarly, if you are a Triangle, or Square, or even a Polygon, born with any Irregularity, you must be taken to one of the Regular Hospitals to have your disease cured; otherwise you will end your days in the State Prison or by the angle of the State Executioner. All faults or defects, from the slightest misconduct to the most flagitious crime, Pantocyclus attributed to some deviation from perfect Regularity in the bodily figure, caused perhaps (if not congenital) by some collision in a crowd; by neglect to take exercise, or by taking too much of it; or even by a sudden change of temperature, resulting in a shrinkage or expansion in some too susceptible part of the frame. Therefore, concluded that illustrious Philosopher, neither good conduct nor bad conduct is a fit subject, in any sober estimation, for either praise or blame. For why should you praise, for example, the integrity of a Square who faithfully defends the interests of his client, when you ought in reality rather to admire the exact precision of his right angles? Or again, why blame a lying, thievish Isosceles when you ought rather to deplore the incurable inequality of his sides? Theoretically, this doctrine is unquestionable; but it has practical drawbacks. In dealing with an Isosceles, if a rascal pleads that he cannot help stealing because of his unevenness, you reply that for that very reason, because he cannot help being a nuisance to his neighbours, you, the Magistrate, cannot help sentencing him to be consumed--and there's an end of the matter. But in little domestic difficulties, where the penalty of consumption, or death, is out of the question, this theory of Configuration sometimes comes in awkwardly; and I must confess that occasionally when one of my own Hexagonal Grandsons pleads as an excuse for his disobedience that a sudden change of the temperature has been too much for his Perimeter, and that I ought to lay the blame not on him but on his Configuration, which can only be strengthened by abundance of the choicest sweetmeats, I neither see my way logically to reject, nor practically to accept, his conclusions. For my own part, I find it best to assume that a good sound scolding or castigation has some latent and strengthening influence on my Grandson's Configuration; though I own that I have no grounds for thinking so. At all events I am not alone in my way of extricating myself from this dilemma; for I find that many of the highest Circles, sitting as Judges in law courts, use praise and blame towards Regular and Irregular Figures; and in their homes I know by experience that, when scolding their children, they speak about "right" or "wrong" as vehemently and passionately as if they believed that these names represented real existences, and that a human Figure is really capable of choosing between them. Constantly carrying out their policy of making Configuration the leading idea in every mind, the Circles reverse the nature of that Commandment which in Spaceland regulates the relations between parents and children. With you, children are taught to honour their parents; with us--next to the Circles, who are the chief object of universal homage--a man is taught to honour his Grandson, if he has one; or, if not, his Son. By "honour", however, is by no means meant "indulgence", but a reverent regard for their highest interests: and the Circles teach that the duty of fathers is to subordinate their own interests to those of posterity, thereby advancing the welfare of the whole State as well as that of their own immediate descendants. The weak point in the system of the Circles--if a humble Square may venture to speak of anything Circular as containing any element of weakness--appears to me to be found in their relations with Women. As it is of the utmost importance for Society that Irregular births should be discouraged, it follows that no Woman who has any Irregularities in her ancestry is a fit partner for one who desires that his posterity should rise by regular degrees in the social scale. Now the Irregularity of a Male is a matter of measurement; but as all Women are straight, and therefore visibly Regular so to speak, one has to devise some other means of ascertaining what I may call their invisible Irregularity, that is to say their potential Irregularities as regards possible offspring. This is effected by carefully-kept pedigrees, which are preserved and supervised by the State; and without a certified pedigree no Woman is allowed to marry. Now it might have been supposed that a Circle--proud of his ancestry and regardful for a posterity which might possibly issue hereafter in a Chief Circle--would be more careful than any other to choose a wife who had no blot on her escutcheon. But it is not so. The care in choosing a Regular wife appears to diminish as one rises in the social scale. Nothing would induce an aspiring Isosceles, who had hopes of generating an Equilateral Son, to take a wife who reckoned a single Irregularity among her Ancestors; a Square or Pentagon, who is confident that his family is steadily on the rise, does not inquire above the five-hundredth generation; a Hexagon or Dodecagon is even more careless of the wife's pedigree; but a Circle has been known deliberately to take a wife who has had an Irregular Great-Grandfather, and all because of some slight superiority of lustre, or because of the charms of a low voice--which, with us, even more than you, is thought "an excellent thing in Woman". Such ill-judged marriages are, as might be expected, barren, if they do not result in positive Irregularity or in diminution of sides; but none of these evils have hitherto proved sufficiently deterrent. The loss of a few sides in a highly-developed Polygon is not easily noticed, and is sometimes compensated by a successful operation in the Neo-Therapeutic Gymnasium, as I have described above; and the Circles are too much disposed to acquiesce in infecundity as a Law of the superior development. Yet, if this evil be not arrested, the gradual diminution of the Circular class may soon become more rapid, and the time may be not far distant when, the race being no longer able to produce a Chief Circle, the Constitution of Flatland must fall. One other word of warning suggests itself to me, though I cannot so easily mention a remedy; and this also refers to our relations with Women. About three hundred years ago, it was decreed by the Chief Circle that, since women are deficient in Reason but abundant in Emotion, they ought no longer to be treated as rational, nor receive any mental education. The consequence was that they were no longer taught to read, nor even to master Arithmetic enough to enable them to count the angles of their husband or children; and hence they sensibly declined during each generation in intellectual power. And this system of female non-education or quietism still prevails. My fear is that, with the best intentions, this policy has been carried so far as to react injuriously on the Male Sex. For the consequence is that, as things now are, we Males have to lead a kind of bi-lingual, and I may almost say bi-mental, existence. With Women, we speak of "love", "duty", "right", "wrong", "pity", "hope", and other irrational and emotional conceptions, which have no existence, and the fiction of which has no object except to control feminine exuberances; but among ourselves, and in our books, we have an entirely different vocabulary and I may almost say, idiom. "Love" then becomes "the anticipation of benefits"; "duty" becomes "necessity" or "fitness"; and other words are correspondingly transmuted. Moreover, among Women, we use language implying the utmost deference for their Sex; and they fully believe that the Chief Circle Himself is not more devoutly adored by us than they are: but behind their backs they are both regarded and spoken of--by all except the very young--as being little better than "mindless organisms". Our Theology also in the Women's chambers is entirely different from our Theology elsewhere. Now my humble fear is that this double training, in language as well as in thought, imposes somewhat too heavy a burden upon the young, especially when, at the age of three years old, they are taken from the maternal care and taught to unlearn the old language--except for the purpose of repeating it in the presence of their Mothers and Nurses--and to learn the vocabulary and idiom of science. Already methinks I discern a weakness in the grasp of mathematical truth at the present time as compared with the more robust intellect of our ancestors three hundred years ago. I say nothing of the possible danger if a Woman should ever surreptitiously learn to read and convey to her Sex the result of her perusal of a single popular volume; nor of the possibility that the indiscretion or disobedience of some infant Male might reveal to a Mother the secrets of the logical dialect. On the simple ground of the enfeebling of the Male intellect, I rest this humble appeal to the highest Authorities to reconsider the regulations of Female education. PART II: OTHER WORLDS "O brave new worlds, that have such people in them!" Section 13. How I had a Vision of Lineland It was the last day but one of the 1999th year of our era, and the first day of the Long Vacation. Having amused myself till a late hour with my favourite recreation of Geometry, I had retired to rest with an unsolved problem in my mind. In the night I had a dream. I saw before me a vast multitude of small Straight Lines (which I naturally assumed to be Women) interspersed with other Beings still smaller and of the nature of lustrous points--all moving to and fro in one and the same Straight Line, and, as nearly as I could judge, with the same velocity. A noise of confused, multitudinous chirping or twittering issued from them at intervals as long as they were moving; but sometimes they ceased from motion, and then all was silence. Approaching one of the largest of what I thought to be Women, I accosted her, but received no answer. A second and a third appeal on my part were equally ineffectual. Losing patience at what appeared to me intolerable rudeness, I brought my mouth into a position full in front of her mouth so as to intercept her motion, and loudly repeated my question, "Woman, what signifies this concourse, and this strange and confused chirping, and this monotonous motion to and fro in one and the same Straight Line?" [Illustration 6] [ASCII approximation follows] My view of Lineland --------- | | | Myself| | | My eye o-------- Women A boy Men The KING Men A boy Women + + + + - --- -- -- -- -- (>----<) -- -- -- -- --- - + + + + ^ ^ The KING'S eyes much larger than the reality shewing that HIS MAJESTY could see nothing but a point. "I am no Woman," replied the small Line. "I am the Monarch of the world. But thou, whence intrudest thou into my realm of Lineland?" Receiving this abrupt reply, I begged pardon if I had in any way startled or molested his Royal Highness; and describing myself as a stranger I besought the King to give me some account of his dominions. But I had the greatest possible difficulty in obtaining any information on points that really interested me; for the Monarch could not refrain from constantly assuming that whatever was familiar to him must also be known to me and that I was simulating ignorance in jest. However, by persevering questions I elicited the following facts: It seemed that this poor ignorant Monarch--as he called himself--was persuaded that the Straight Line which he called his Kingdom, and in which he passed his existence, constituted the whole of the world, and indeed the whole of Space. Not being able either to move or to see, save in his Straight Line, he had no conception of anything out of it. Though he had heard my voice when I first addressed him, the sounds had come to him in a manner so contrary to his experience that he had made no answer, "seeing no man", as he expressed it, "and hearing a voice as it were from my own intestines." Until the moment when I placed my mouth in his World, he had neither seen me, nor heard anything except confused sounds beating against--what I called his side, but what he called his INSIDE or STOMACH; nor had he even now the least conception of the region from which I had come. Outside his World, or Line, all was a blank to him; nay, not even a blank, for a blank implies Space; say, rather, all was non-existent. His subjects--of whom the small Lines were men and the Points Women--were all alike confined in motion and eye-sight to that single Straight Line, which was their World. It need scarcely be added that the whole of their horizon was limited to a Point; nor could any one ever see anything but a Point. Man, woman, child, thing--each was a Point to the eye of a Linelander. Only by the sound of the voice could sex or age be distinguished. Moreover, as each individual occupied the whole of the narrow path, so to speak, which constituted his Universe, and no one could move to the right or left to make way for passers by, it followed that no Linelander could ever pass another. Once neighbours, always neighbours. Neighbourhood with them was like marriage with us. Neighbours remained neighbours till death did them part. Such a life, with all vision limited to a Point, and all motion to a Straight Line, seemed to me inexpressibly dreary; and I was surprised to note the vivacity and cheerfulness of the King. Wondering whether it was possible, amid circumstances so unfavourable to domestic relations, to enjoy the pleasures of conjugal union, I hesitated for some time to question his Royal Highness on so delicate a subject; but at last I plunged into it by abruptly inquiring as to the health of his family. "My wives and children," he replied, "are well and happy." Staggered at this answer--for in the immediate proximity of the Monarch (as I had noted in my dream before I entered Lineland) there were none but Men--I ventured to reply, "Pardon me, but I cannot imagine how your Royal Highness can at any time either see or approach their Majesties, when there are at least half a dozen intervening individuals, whom you can neither see through, nor pass by? Is it possible that in Lineland proximity is not necessary for marriage and for the generation of children?" "How can you ask so absurd a question?" replied the Monarch. "If it were indeed as you suggest, the Universe would soon be depopulated. No, no; neighbourhood is needless for the union of hearts; and the birth of children is too important a matter to have been allowed to depend upon such an accident as proximity. You cannot be ignorant of this. Yet since you are pleased to affect ignorance, I will instruct you as if you were the veriest baby in Lineland. Know, then, that marriages are consummated by means of the faculty of sound and the sense of hearing. "You are of course aware that every Man has two mouths or voices--as well as two eyes--a bass at one and a tenor at the other of his extremities. I should not mention this, but that I have been unable to distinguish your tenor in the course of our conversation." I replied that I had but one voice, and that I had not been aware that his Royal Highness had two. "That confirms my impression," said the King, "that you are not a Man, but a feminine Monstrosity with a bass voice, and an utterly uneducated ear. But to continue. "Nature having herself ordained that every Man should wed two wives--" "Why two?" asked I. "You carry your affected simplicity too far", he cried. "How can there be a completely harmonious union without the combination of the Four in One, viz. the Bass and Tenor of the Man and the Soprano and Contralto of the two Women?" "But supposing," said I, "that a man should prefer one wife or three?" "It is impossible," he said; "it is as inconceivable as that two and one should make five, or that the human eye should see a Straight Line." I would have interrupted him; but he proceeded as follows: "Once in the middle of each week a Law of Nature compels us to move to and fro with a rhythmic motion of more than usual violence, which continues for the time you would take to count a hundred and one. In the midst of this choral dance, at the fifty-first pulsation, the inhabitants of the Universe pause in full career, and each individual sends forth his richest, fullest, sweetest strain. It is in this decisive moment that all our marriages are made. So exquisite is the adaptation of Bass to Treble, of Tenor to Contralto, that oftentimes the Loved Ones, though twenty thousand leagues away, recognize at once the responsive note of their destined Lover; and, penetrating the paltry obstacles of distance, Love unites the three. The marriage in that instant consummated results in a threefold Male and Female offspring which takes its place in Lineland." "What! Always threefold?" said I. "Must one wife then always have twins?" "Bass-voiced Monstrosity! yes," replied the King. "How else could the balance of the Sexes be maintained, if two girls were not born for every boy? Would you ignore the very Alphabet of Nature?" He ceased, speechless for fury; and some time elapsed before I could induce him to resume his narrative. "You will not, of course, suppose that every bachelor among us finds his mates at the first wooing in this universal Marriage Chorus. On the contrary, the process is by most of us many times repeated. Few are the hearts whose happy lot it is at once to recognize in each other's voices the partner intended for them by Providence, and to fly into a reciprocal and perfectly harmonious embrace. With most of us the courtship is of long duration. The Wooer's voices may perhaps accord with one of the future wives, but not with both; or not, at first, with either; or the Soprano and Contralto may not quite harmonize. In such cases Nature has provided that every weekly Chorus shall bring the three Lovers into closer harmony. Each trial of voice, each fresh discovery of discord, almost imperceptibly induces the less perfect to modify his or her vocal utterance so as to approximate to the more perfect. And after many trials and many approximations, the result is at last achieved. There comes a day at last, when, while the wonted Marriage Chorus goes forth from universal Lineland, the three far-off Lovers suddenly find themselves in exact harmony, and, before they are awake, the wedded Triplet is rapt vocally into a duplicate embrace; and Nature rejoices over one more marriage and over three more births." Section 14. How I vainly tried to explain the nature of Flatland Thinking that it was time to bring down the Monarch from his raptures to the level of common sense, I determined to endeavour to open up to him some glimpses of the truth, that is to say of the nature of things in Flatland. So I began thus: "How does your Royal Highness distinguish the shapes and positions of his subjects? I for my part noticed by the sense of sight, before I entered your Kingdom, that some of your people are Lines and others Points, and that some of the Lines are larger--" "You speak of an impossibility," interrupted the King; "you must have seen a vision; for to detect the difference between a Line and a Point by the sense of sight is, as every one knows, in the nature of things, impossible; but it can be detected by the sense of hearing, and by the same means my shape can be exactly ascertained. Behold me--I am a Line, the longest in Lineland, over six inches of Space--" "Of Length", I ventured to suggest. "Fool," said he, "Space is Length. Interrupt me again, and I have done." I apologized; but he continued scornfully, "Since you are impervious to argument, you shall hear with your ears how by means of my two voices I reveal my shape to my Wives, who are at this moment six thousand miles seventy yards two feet eight inches away, the one to the North, the other to the South. Listen, I call to them." He chirruped, and then complacently continued: "My wives at this moment receiving the sound of one of my voices, closely followed by the other, and perceiving that the latter reaches them after an interval in which sound can traverse 6.457 inches, infer that one of my mouths is 6.457 inches further from them than the other, and accordingly know my shape to be 6.457 inches. But you will of course understand that my wives do not make this calculation every time they hear my two voices. They made it, once for all, before we were married. But they COULD make it at any time. And in the same way I can estimate the shape of any of my Male subjects by the sense of sound." "But how," said I, "if a Man feigns a Woman's voice with one of his two voices, or so disguises his Southern voice that it cannot be recognized as the echo of the Northern? May not such deceptions cause great inconvenience? And have you no means of checking frauds of this kind by commanding your neighbouring subjects to feel one another?" This of course was a very stupid question, for feeling could not have answered the purpose; but I asked with the view of irritating the Monarch, and I succeeded perfectly. "What!" cried he in horror, "explain your meaning." "Feel, touch, come into contact," I replied. "If you mean by FEELING," said the King, "approaching so close as to leave no space between two individuals, know, Stranger, that this offence is punishable in my dominions by death. And the reason is obvious. The frail form of a Woman, being liable to be shattered by such an approximation, must be preserved by the State; but since Women cannot be distinguished by the sense of sight from Men, the Law ordains universally that neither Man nor Woman shall be approached so closely as to destroy the interval between the approximator and the approximated. "And indeed what possible purpose would be served by this illegal and unnatural excess of approximation which you call TOUCHING, when all the ends of so brutal and coarse a process are attained at once more easily and more exactly by the sense of hearing? As to your suggested danger of deception, it is non-existent: for the Voice, being the essence of one's Being, cannot be thus changed at will. But come, suppose that I had the power of passing through solid things, so that I could penetrate my subjects, one after another, even to the number of a billion, verifying the size and distance of each by the sense of FEELING: how much time and energy would be wasted in this clumsy and inaccurate method! Whereas now, in one moment of audition, I take as it were the census and statistics, local, corporeal, mental and spiritual, of every living being in Lineland. Hark, only hark!" So saying he paused and listened, as if in an ecstasy, to a sound which seemed to me no better than a tiny chirping from an innumerable multitude of lilliputian grasshoppers. "Truly," replied I, "your sense of hearing serves you in good stead, and fills up many of your deficiencies. But permit me to point out that your life in Lineland must be deplorably dull. To see nothing but a Point! Not even to be able to contemplate a Straight Line! Nay, not even to know what a Straight Line is! To see, yet be cut off from those Linear prospects which are vouchsafed to us in Flatland! Better surely to have no sense of sight at all than to see so little! I grant you I have not your discriminative faculty of hearing; for the concert of all Lineland which gives you such intense pleasure, is to me no better than a multitudinous twittering or chirping. But at least I can discern, by sight, a Line from a Point. And let me prove it. Just before I came into your kingdom, I saw you dancing from left to right, and then from right to left, with Seven Men and a Woman in your immediate proximity on the left, and eight Men and two Women on your right. Is not this correct?" "It is correct," said the King, "so far as the numbers and sexes are concerned, though I know not what you mean by 'right' and 'left'. But I deny that you saw these things. For how could you see the Line, that is to say the inside, of any Man? But you must have heard these things, and then dreamed that you saw them. And let me ask what you mean by those words 'left' and 'right'. I suppose it is your way of saying Northward and Southward." "Not so," replied I; "besides your motion of Northward and Southward, there is another motion which I call from right to left." KING. Exhibit to me, if you please, this motion from left to right. I. Nay, that I cannot do, unless you could step out of your Line altogether. KING. Out of my Line? Do you mean out of the world? Out of Space? I. Well, yes. Out of YOUR World. Out of YOUR Space. For your Space is not the true Space. True Space is a Plane; but your Space is only a Line. KING. If you cannot indicate this motion from left to right by yourself moving in it, then I beg you to describe it to me in words. I. If you cannot tell your right side from your left, I fear that no words of mine can make my meaning clear to you. But surely you cannot be ignorant of so simple a distinction. KING. I do not in the least understand you. I. Alas! How shall I make it clear? When you move straight on, does it not sometimes occur to you that you COULD move in some other way, turning your eye round so as to look in the direction towards which your side is now fronting? In other words, instead of always moving in the direction of one of your extremities, do you never feel a desire to move in the direction, so to speak, of your side? KING. Never. And what do you mean? How can a man's inside "front" in any direction? Or how can a man move in the direction of his inside? I. Well then, since words cannot explain the matter, I will try deeds, and will move gradually out of Lineland in the direction which I desire to indicate to you. At the word I began to move my body out of Lineland. As long as any part of me remained in his dominion and in his view, the King kept exclaiming, "I see you, I see you still; you are not moving." But when I had at last moved myself out of his Line, he cried in his shrillest voice, "She is vanished; she is dead." "I am not dead," replied I; "I am simply out of Lineland, that is to say, out of the Straight Line which you call Space, and in the true Space, where I can see things as they are. And at this moment I can see your Line, or side--or inside as you are pleased to call it; and I can see also the Men and Women on the North and South of you, whom I will now enumerate, describing their order, their size, and the interval between each." [Illustration 7] [ASCII approximation follows] My body just before I disappeared +---------+ |\ \ \ \ \| |\ \ \ \ \| |\ \ \ \ \| Lineland ----> |\ \ \ \ \| The King --------------------+---------+--------------======== When I had done this at great length, I cried triumphantly, "Does that at last convince you?" And, with that, I once more entered Lineland, taking up the same position as before. But the Monarch replied, "If you were a Man of sense--though, as you appear to have only one voice I have little doubt you are not a Man but a Woman--but, if you had a particle of sense, you would listen to reason. You ask me to believe that there is another Line besides that which my senses indicate, and another motion besides that of which I am daily conscious. I, in return, ask you to describe in words or indicate by motion that other Line of which you speak. Instead of moving, you merely exercise some magic art of vanishing and returning to sight; and instead of any lucid description of your new World, you simply tell me the numbers and sizes of some forty of my retinue, facts known to any child in my capital. Can anything be more irrational or audacious? Acknowledge your folly or depart from my dominions." Furious at his perversity, and especially indignant that he professed to be ignorant of my sex, I retorted in no measured terms, "Besotted Being! You think yourself the perfection of existence, while you are in reality the most imperfect and imbecile. You profess to see, whereas you can see nothing but a Point! You plume yourself on inferring the existence of a Straight Line; but I CAN SEE Straight Lines, and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. Why waste more words? Suffice it that I am the completion of your incomplete self. You are a Line, but I am a Line of Lines, called in my country a Square: and even I, infinitely superior though I am to you, am of little account among the great nobles of Flatland, whence I have come to visit you, in the hope of enlightening your ignorance." Hearing these words the King advanced towards me with a menacing cry as if to pierce me through the diagonal; and in that same moment there arose from myriads of his subjects a multitudinous war-cry, increasing in vehemence till at last methought it rivalled the roar of an army of a hundred thousand Isosceles, and the artillery of a thousand Pentagons. Spell-bound and motionless, I could neither speak nor move to avert the impending destruction; and still the noise grew louder, and the King came closer, when I awoke to find the breakfast-bell recalling me to the realities of Flatland. Section 15. Concerning a Stranger from Spaceland From dreams I proceed to facts. It was the last day of the 1999th year of our era. The pattering of the rain had long ago announced nightfall; and I was sitting in the company of my wife, musing on the events of the past and the prospects of the coming year, the coming century, the coming Millennium. [Note: When I say "sitting", of course I do not mean any change of attitude such as you in Spaceland signify by that word; for as we have no feet, we can no more "sit" nor "stand" (in your sense of the word) than one of your soles or flounders. Nevertheless, we perfectly well recognize the different mental states of volition implied in "lying", "sitting", and "standing", which are to some extent indicated to a beholder by a slight increase of lustre corresponding to the increase of volition. But on this, and a thousand other kindred subjects, time forbids me to dwell.] My four Sons and two orphan Grandchildren had retired to their several apartments; and my wife alone remained with me to see the old Millennium out and the new one in. I was rapt in thought, pondering in my mind some words that had casually issued from the mouth of my youngest Grandson, a most promising young Hexagon of unusual brilliancy and perfect angularity. His uncles and I had been giving him his usual practical lesson in Sight Recognition, turning ourselves upon our centres, now rapidly, now more slowly, and questioning him as to our positions; and his answers had been so satisfactory that I had been induced to reward him by giving him a few hints on Arithmetic, as applied to Geometry. Taking nine Squares, each an inch every way, I had put them together so as to make one large Square, with a side of three inches, and I had hence proved to my little Grandson that--though it was impossible for us to SEE the inside of the Square--yet we might ascertain the number of square inches in a Square by simply squaring the number of inches in the side: "and thus," said I, "we know that 3^2, or 9, represents the number of square inches in a Square whose side is 3 inches long." The little Hexagon meditated on this a while and then said to me; "But you have been teaching me to raise numbers to the third power: I suppose 3^3 must mean something in Geometry; what does it mean?" "Nothing at all," replied I, "not at least in Geometry; for Geometry has only Two Dimensions." And then I began to shew the boy how a Point by moving through a length of three inches makes a Line of three inches, which may be represented by 3; and how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches every way, which may be represented by 3^2. Upon this, my Grandson, again returning to his former suggestion, took me up rather suddenly and exclaimed, "Well, then, if a Point by moving three inches, makes a Line of three inches represented by 3; and if a straight Line of three inches, moving parallel to itself, makes a Square of three inches every way, represented by 3^2; it must be that a Square of three inches every way, moving somehow parallel to itself (but I don't see how) must make Something else (but I don't see what) of three inches every way--and this must be represented by 3^3." "Go to bed," said I, a little ruffled by this interruption: "if you would talk less nonsense, you would remember more sense." So my Grandson had disappeared in disgrace; and there I sat by my Wife's side, endeavouring to form a retrospect of the year 1999 and of the possibilities of the year 2000, but not quite able to shake off the thoughts suggested by the prattle of my bright little Hexagon. Only a few sands now remained in the half-hour glass. Rousing myself from my reverie I turned the glass Northward for the last time in the old Millennium; and in the act, I exclaimed aloud, "The boy is a fool." Straightway I became conscious of a Presence in the room, and a chilling breath thrilled through my very being. "He is no such thing," cried my Wife, "and you are breaking the Commandments in thus dishonouring your own Grandson." But I took no notice of her. Looking round in every direction I could see nothing; yet still I FELT a Presence, and shivered as the cold whisper came again. I started up. "What is the matter?" said my Wife, "there is no draught; what are you looking for? There is nothing." There was nothing; and I resumed my seat, again exclaiming, "The boy is a fool, I say; 3^3 can have no meaning in Geometry." At once there came a distinctly audible reply, "The boy is not a fool; and 3^3 has an obvious Geometrical meaning." My Wife as well as myself heard the words, although she did not understand their meaning, and both of us sprang forward in the direction of the sound. What was our horror when we saw before us a Figure! At the first glance it appeared to be a Woman, seen sideways; but a moment's observation shewed me that the extremities passed into dimness too rapidly to represent one of the Female Sex; and I should have thought it a Circle, only that it seemed to change its size in a manner impossible for a Circle or for any regular Figure of which I had had experience. But my Wife had not my experience, nor the coolness necessary to note these characteristics. With the usual hastiness and unreasoning jealousy of her Sex, she flew at once to the conclusion that a Woman had entered the house through some small aperture. "How comes this person here?" she exclaimed, "you promised me, my dear, that there should be no ventilators in our new house." "Nor are there any," said I; "but what makes you think that the stranger is a Woman? I see by my power of Sight Recognition----" "Oh, I have no patience with your Sight Recognition," replied she, "'Feeling is believing' and 'A Straight Line to the touch is worth a Circle to the sight'"--two Proverbs, very common with the Frailer Sex in Flatland. "Well," said I, for I was afraid of irritating her, "if it must be so, demand an introduction." Assuming her most gracious manner, my Wife advanced towards the Stranger, "Permit me, Madam, to feel and be felt by----" then, suddenly recoiling, "Oh! it is not a Woman, and there are no angles either, not a trace of one. Can it be that I have so misbehaved to a perfect Circle?" "I am indeed, in a certain sense a Circle," replied the Voice, "and a more perfect Circle than any in Flatland; but to speak more accurately, I am many Circles in one." Then he added more mildly, "I have a message, dear Madam, to your husband, which I must not deliver in your presence; and, if you would suffer us to retire for a few minutes----" But my Wife would not listen to the proposal that our august Visitor should so incommode himself, and assuring the Circle that the hour of her own retirement had long passed, with many reiterated apologies for her recent indiscretion, she at last retreated to her apartment. I glanced at the half-hour glass. The last sands had fallen. The third Millennium had begun. Section 16. How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland As soon as the sound of the Peace-cry of my departing Wife had died away, I began to approach the Stranger with the intention of taking a nearer view and of bidding him be seated: but his appearance struck me dumb and motionless with astonishment. Without the slightest symptoms of angularity he nevertheless varied every instant with gradations of size and brightness scarcely possible for any Figure within the scope of my experience. The thought flashed across me that I might have before me a burglar or cut-throat, some monstrous Irregular Isosceles, who, by feigning the voice of a Circle, had obtained admission somehow into the house, and was now preparing to stab me with his acute angle. In a sitting-room, the absence of Fog (and the season happened to be remarkably dry), made it difficult for me to trust to Sight Recognition, especially at the short distance at which I was standing. Desperate with fear, I rushed forward with an unceremonious, "You must permit me, Sir--" and felt him. My Wife was right. There was not the trace of an angle, not the slightest roughness or inequality: never in my life had I met with a more perfect Circle. He remained motionless while I walked round him, beginning from his eye and returning to it again. Circular he was throughout, a perfectly satisfactory Circle; there could not be a doubt of it. Then followed a dialogue, which I will endeavour to set down as near as I can recollect it, omitting only some of my profuse apologies--for I was covered with shame and humiliation that I, a Square, should have been guilty of the impertinence of feeling a Circle. It was commenced by the Stranger with some impatience at the lengthiness of my introductory process. STRANGER. Have you felt me enough by this time? Are you not introduced to me yet? I. Most illustrious Sir, excuse my awkwardness, which arises not from ignorance of the usages of polite society, but from a little surprise and nervousness, consequent on this somewhat unexpected visit. And I beseech you to reveal my indiscretion to no one, and especially not to my Wife. But before your Lordship enters into further communications, would he deign to satisfy the curiosity of one who would gladly know whence his Visitor came? STRANGER. From Space, from Space, Sir: whence else? I. Pardon me, my Lord, but is not your Lordship already in Space, your Lordship and his humble servant, even at this moment? STRANGER. Pooh! what do you know of Space? Define Space. I. Space, my Lord, is height and breadth indefinitely prolonged. STRANGER. Exactly: you see you do not even know what Space is. You think it is of Two Dimensions only; but I have come to announce to you a Third--height, breadth, and length. I. Your Lordship is pleased to be merry. We also speak of length and height, or breadth and thickness, thus denoting Two Dimensions by four names. STRANGER. But I mean not only three names, but Three Dimensions. I. Would your Lordship indicate or explain to me in what direction is the Third Dimension, unknown to me? STRANGER. I came from it. It is up above and down below. I. My Lord means seemingly that it is Northward and Southward. STRANGER. I mean nothing of the kind. I mean a direction in which you cannot look, because you have no eye in your side. I. Pardon me, my Lord, a moment's inspection will convince your Lordship that I have a perfect luminary at the juncture of two of my sides. STRANGER. Yes: but in order to see into Space you ought to have an eye, not on your Perimeter, but on your side, that is, on what you would probably call your inside; but we in Spaceland should call it your side. I. An eye in my inside! An eye in my stomach! Your Lordship jests. STRANGER. I am in no jesting humour. I tell you that I come from Space, or, since you will not understand what Space means, from the Land of Three Dimensions whence I but lately looked down upon your Plane which you call Space forsooth. From that position of advantage I discerned all that you speak of as SOLID (by which you mean "enclosed on four sides"), your houses, your churches, your very chests and safes, yes even your insides and stomachs, all lying open and exposed to my view. I. Such assertions are easily made, my Lord. STRANGER. But not easily proved, you mean. But I mean to prove mine. When I descended here, I saw your four Sons, the Pentagons, each in his apartment, and your two Grandsons the Hexagons; I saw your youngest Hexagon remain a while with you and then retire to his room, leaving you and your Wife alone. I saw your Isosceles servants, three in number, in the kitchen at supper, and the little Page in the scullery. Then I came here, and how do you think I came? I. Through the roof, I suppose. STRANGER. Not so. Your roof, as you know very well, has been recently repaired, and has no aperture by which even a Woman could penetrate. I tell you I come from Space. Are you not convinced by what I have told you of your children and household? I. Your Lordship must be aware that such facts touching the belongings of his humble servant might be easily ascertained by any one in the neighbourhood possessing your Lordship's ample means of obtaining information. STRANGER. (TO HIMSELF.) What must I do? Stay; one more argument suggests itself to me. When you see a Straight Line--your wife, for example--how many Dimensions do you attribute to her? I. Your Lordship would treat me as if I were one of the vulgar who, being ignorant of Mathematics, suppose that a Woman is really a Straight Line, and only of One Dimension. No, no, my Lord; we Squares are better advised, and are as well aware as your Lordship that a Woman, though popularly called a Straight Line, is, really and scientifically, a very thin Parallelogram, possessing Two Dimensions, like the rest of us, viz., length and breadth (or thickness). STRANGER. But the very fact that a Line is visible implies that it possesses yet another Dimension. I. My Lord, I have just acknowledged that a Woman is broad as well as long. We see her length, we infer her breadth; which, though very slight, is capable of measurement. STRANGER. You do not understand me. I mean that when you see a Woman, you ought--besides inferring her breadth--to see her length, and to SEE what we call her HEIGHT; although that last Dimension is infinitesimal in your country. If a Line were mere length without "height", it would cease to occupy Space and would become invisible. Surely you must recognize this? I. I must indeed confess that I do not in the least understand your Lordship. When we in Flatland see a Line, we see length and BRIGHTNESS. If the brightness disappears, the Line is extinguished, and, as you say, ceases to occupy Space. But am I to suppose that your Lordship gives to brightness the title of a Dimension, and that what we call "bright" you call "high"? STRANGER. No, indeed. By "height" I mean a Dimension like your length: only, with you, "height" is not so easily perceptible, being extremely small. I. My Lord, your assertion is easily put to the test. You say I have a Third Dimension, which you call "height". Now, Dimension implies direction and measurement. Do but measure my "height", or merely indicate to me the direction in which my "height" extends, and I will become your convert. Otherwise, your Lordship's own understanding must hold me excused. STRANGER. (TO HIMSELF.) I can do neither. How shall I convince him? Surely a plain statement of facts followed by ocular demonstration ought to suffice. --Now, Sir; listen to me. You are living on a Plane. What you style Flatland is the vast level surface of what I may call a fluid, on, or in, the top of which you and your countrymen move about, without rising above it or falling below it. I am not a plane Figure, but a Solid. You call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other. When I cut through your plane as I am now doing, I make in your plane a section which you, very rightly, call a Circle. For even a Sphere--which is my proper name in my own country--if he manifest himself at all to an inhabitant of Flatland--must needs manifest himself as a Circle. Do you not remember--for I, who see all things, discerned last night the phantasmal vision of Lineland written upon your brain--do you not remember, I say, how, when you entered the realm of Lineland, you were compelled to manifest yourself to the King, not as a Square, but as a Line, because that Linear Realm had not Dimensions enough to represent the whole of you, but only a slice or section of you? In precisely the same way, your country of Two Dimensions is not spacious enough to represent me, a being of Three, but can only exhibit a slice or section of me, which is what you call a Circle. The diminished brightness of your eye indicates incredulity. But now prepare to receive proof positive of the truth of my assertions. You cannot indeed see more than one of my sections, or Circles, at a time; for you have no power to raise your eye out of the plane of Flatland; but you can at least see that, as I rise in Space, so my sections become smaller. See now, I will rise; and the effect upon your eye will be that my Circle will become smaller and smaller till it dwindles to a point and finally vanishes. [Illustration 8] [ASCII approximation follows] The Sphere on the point of vanishing (2) __-----__ The Sphere with The Sphere rising / \ (3) his section __-----__ / \ at full size / \ | | __-----__ / \ | | / \ | | | | / __ - __ \ | | \ / My | -- -- | | __ --- __ | \ __ __ / Eye --|-----------------|----\--__-------__--/------------===----------+(> | -- __ __ -- | \ __ --- __ / \ - / ----- \ __ __ / ----- There was no "rising" that I could see; but he diminished and finally vanished. I winked once or twice to make sure that I was not dreaming. But it was no dream. For from the depths of nowhere came forth a hollow voice--close to my heart it seemed--"Am I quite gone? Are you convinced now? Well, now I will gradually return to Flatland and you shall see my section become larger and larger." Every reader in Spaceland will easily understand that my mysterious Guest was speaking the language of truth and even of simplicity. But to me, proficient though I was in Flatland Mathematics, it was by no means a simple matter. The rough diagram given above will make it clear to any Spaceland child that the Sphere, ascending in the three positions indicated there, must needs have manifested himself to me, or to any Flatlander, as a Circle, at first of full size, then small, and at last very small indeed, approaching to a Point. But to me, although I saw the facts before me, the causes were as dark as ever. All that I could comprehend was, that the Circle had made himself smaller and vanished, and that he had now reappeared and was rapidly making himself larger. When he regained his original size, he heaved a deep sigh; for he perceived by my silence that I had altogether failed to comprehend him. And indeed I was now inclining to the belief that he must be no Circle at all, but some extremely clever juggler; or else that the old wives' tales were true, and that after all there were such people as Enchanters and Magicians. After a long pause he muttered to himself, "One resource alone remains, if I am not to resort to action. I must try the method of Analogy." Then followed a still longer silence, after which he continued our dialogue. SPHERE. Tell me, Mr. Mathematician; if a Point moves Northward, and leaves a luminous wake, what name would you give to the wake? I. A straight Line. SPHERE. And a straight Line has how many extremities? I. Two. SPHERE. Now conceive the Northward straight Line moving parallel to itself, East and West, so that every point in it leaves behind it the wake of a straight Line. What name will you give to the Figure thereby formed? We will suppose that it moves through a distance equal to the original straight Line. --What name, I say? I. A Square. SPHERE. And how many sides has a Square? How many angles? I. Four sides and four angles. SPHERE. Now stretch your imagination a little, and conceive a Square in Flatland, moving parallel to itself upward. I. What? Northward? SPHERE. No, not Northward; upward; out of Flatland altogether. If it moved Northward, the Southern points in the Square would have to move through the positions previously occupied by the Northern points. But that is not my meaning. I mean that every Point in you--for you are a Square and will serve the purpose of my illustration--every Point in you, that is to say in what you call your inside, is to pass upwards through Space in such a way that no Point shall pass through the position previously occupied by any other Point; but each Point shall describe a straight Line of its own. This is all in accordance with Analogy; surely it must be clear to you. Restraining my impatience--for I was now under a strong temptation to rush blindly at my Visitor and to precipitate him into Space, or out of Flatland, anywhere, so that I could get rid of him--I replied:-- "And what may be the nature of the Figure which I am to shape out by this motion which you are pleased to denote by the word 'upward'? I presume it is describable in the language of Flatland." SPHERE. Oh, certainly. It is all plain and simple, and in strict accordance with Analogy--only, by the way, you must not speak of the result as being a Figure, but as a Solid. But I will describe it to you. Or rather not I, but Analogy. We began with a single Point, which of course--being itself a Point--has only ONE terminal Point. One Point produces a Line with TWO terminal Points. One Line produces a Square with FOUR terminal Points. Now you can give yourself the answer to your own question: 1, 2, 4, are evidently in Geometrical Progression. What is the next number? I. Eight. SPHERE. Exactly. The one Square produces a SOMETHING-WHICH- YOU-DO-NOT-AS-YET-KNOW-A-NAME-FOR-BUT-WHICH-WE-CALL-A-CUBE with EIGHT terminal Points. Now are you convinced? I. And has this Creature sides, as well as angles or what you call "terminal Points"? SPHERE. Of course; and all according to Analogy. But, by the way, not what YOU call sides, but what WE call sides. You would call them SOLIDS. I. And how many solids or sides will appertain to this Being whom I am to generate by the motion of my inside in an "upward" direction, and whom you call a Cube? SPHERE. How can you ask? And you a mathematician! The side of anything is always, if I may so say, one Dimension behind the thing. Consequently, as there is no Dimension behind a Point, a Point has 0 sides; a Line, if I may say, has 2 sides (for the Points of a Line may be called by courtesy, its sides); a Square has 4 sides; 0, 2, 4; what Progression do you call that? I. Arithmetical. SPHERE. And what is the next number? I. Six. SPHERE. Exactly. Then you see you have answered your own question. The Cube which you will generate will be bounded by six sides, that is to say, six of your insides. You see it all now, eh? "Monster," I shrieked, "be thou juggler, enchanter, dream, or devil, no more will I endure thy mockeries. Either thou or I must perish." And saying these words I precipitated myself upon him. Section 17. How the Sphere, having in vain tried words, resorted to deeds It was in vain. I brought my hardest right angle into violent collision with the Stranger, pressing on him with a force sufficient to have destroyed any ordinary Circle: but I could feel him slowly and unarrestably slipping from my contact; no edging to the right nor to the left, but moving somehow out of the world, and vanishing to nothing. Soon there was a blank. But still I heard the Intruder's voice. SPHERE. Why will you refuse to listen to reason? I had hoped to find in you--as being a man of sense and an accomplished mathematician--a fit apostle for the Gospel of the Three Dimensions, which I am allowed to preach once only in a thousand years: but now I know not how to convince you. Stay, I have it. Deeds, and not words, shall proclaim the truth. Listen, my friend. I have told you I can see from my position in Space the inside of all things that you consider closed. For example, I see in yonder cupboard near which you are standing, several of what you call boxes (but like everything else in Flatland, they have no tops nor bottoms) full of money; I see also two tablets of accounts. I am about to descend into that cupboard and to bring you one of those tablets. I saw you lock the cupboard half an hour ago, and I know you have the key in your possession. But I descend from Space; the doors, you see, remain unmoved. Now I am in the cupboard and am taking the tablet. Now I have it. Now I ascend with it. I rushed to the closet and dashed the door open. One of the tablets was gone. With a mocking laugh, the Stranger appeared in the other corner of the room, and at the same time the tablet appeared upon the floor. I took it up. There could be no doubt--it was the missing tablet. I groaned with horror, doubting whether I was not out of my senses; but the Stranger continued: "Surely you must now see that my explanation, and no other, suits the phenomena. What you call Solid things are really superficial; what you call Space is really nothing but a great Plane. I am in Space, and look down upon the insides of the things of which you only see the outsides. You could leave this Plane yourself, if you could but summon up the necessary volition. A slight upward or downward motion would enable you to see all that I can see. "The higher I mount, and the further I go from your Plane, the more I can see, though of course I see it on a smaller scale. For example, I am ascending; now I can see your neighbour the Hexagon and his family in their several apartments; now I see the inside of the Theatre, ten doors off, from which the audience is only just departing; and on the other side a Circle in his study, sitting at his books. Now I shall come back to you. And, as a crowning proof, what do you say to my giving you a touch, just the least touch, in your stomach? It will not seriously injure you, and the slight pain you may suffer cannot be compared with the mental benefit you will receive." Before I could utter a word of remonstrance, I felt a shooting pain in my inside, and a demoniacal laugh seemed to issue from within me. A moment afterwards the sharp agony had ceased, leaving nothing but a dull ache behind, and the Stranger began to reappear, saying, as he gradually increased in size, "There, I have not hurt you much, have I? If you are not convinced now, I don't know what will convince you. What say you?" My resolution was taken. It seemed intolerable that I should endure existence subject to the arbitrary visitations of a Magician who could thus play tricks with one's very stomach. If only I could in any way manage to pin him against the wall till help came! Once more I dashed my hardest angle against him, at the same time alarming the whole household by my cries for aid. I believe, at the moment of my onset, the Stranger had sunk below our Plane, and really found difficulty in rising. In any case he remained motionless, while I, hearing, as I thought, the sound of some help approaching, pressed against him with redoubled vigour, and continued to shout for assistance. A convulsive shudder ran through the Sphere. "This must not be," I thought I heard him say: "either he must listen to reason, or I must have recourse to the last resource of civilization." Then, addressing me in a louder tone, he hurriedly exclaimed, "Listen: no stranger must witness what you have witnessed. Send your Wife back at once, before she enters the apartment. The Gospel of Three Dimensions must not be thus frustrated. Not thus must the fruits of one thousand years of waiting be thrown away. I hear her coming. Back! back! Away from me, or you must go with me--whither you know not--into the Land of Three Dimensions!" "Fool! Madman! Irregular!" I exclaimed; "never will I release thee; thou shalt pay the penalty of thine impostures." "Ha! Is it come to this?" thundered the Stranger: "then meet your fate: out of your Plane you go. Once, twice, thrice! 'Tis done!" Section 18. How I came to Spaceland, and what I saw there An unspeakable horror seized me. There was a darkness; then a dizzy, sickening sensation of sight that was not like seeing; I saw a Line that was no Line; Space that was not Space: I was myself, and not myself. When I could find voice, I shrieked aloud in agony, "Either this is madness or it is Hell." "It is neither," calmly replied the voice of the Sphere, "it is Knowledge; it is Three Dimensions: open your eye once again and try to look steadily." I looked, and, behold, a new world! There stood before me, visibly incorporate, all that I had before inferred, conjectured, dreamed, of perfect Circular beauty. What seemed the centre of the Stranger's form lay open to my view: yet I could see no heart, nor lungs, nor arteries, only a beautiful harmonious Something--for which I had no words; but you, my Readers in Spaceland, would call it the surface of the Sphere. Prostrating myself mentally before my Guide, I cried, "How is it, O divine ideal of consummate loveliness and wisdom that I see thy inside, and yet cannot discern thy heart, thy lungs, thy arteries, thy liver?" "What you think you see, you see not," he replied; "it is not given to you, nor to any other Being to behold my internal parts. I am of a different order of Beings from those in Flatland. Were I a Circle, you could discern my intestines, but I am a Being, composed as I told you before, of many Circles, the Many in the One, called in this country a Sphere. And, just as the outside of a Cube is a Square, so the outside of a Sphere presents the appearance of a Circle." Bewildered though I was by my Teacher's enigmatic utterance, I no longer chafed against it, but worshipped him in silent adoration. He continued, with more mildness in his voice. "Distress not yourself if you cannot at first understand the deeper mysteries of Spaceland. By degrees they will dawn upon you. Let us begin by casting back a glance at the region whence you came. Return with me a while to the plains of Flatland, and I will shew you that which you have often reasoned and thought about, but never seen with the sense of sight--a visible angle." "Impossible!" I cried; but, the Sphere leading the way, I followed as if in a dream, till once more his voice arrested me: "Look yonder, and behold your own Pentagonal house, and all its inmates." I looked below, and saw with my physical eye all that domestic individuality which I had hitherto merely inferred with the understanding. And how poor and shadowy was the inferred conjecture in comparison with the reality which I now beheld! My four Sons calmly asleep in the North-Western rooms, my two orphan Grandsons to the South; the Servants, the Butler, my Daughter, all in their several apartments. Only my affectionate Wife, alarmed by my continued absence, had quitted her room and was roving up and down in the Hall, anxiously awaiting my return. Also the Page, aroused by my cries, had left his room, and under pretext of ascertaining whether I had fallen somewhere in a faint, was prying into the cabinet in my study. All this I could now SEE, not merely infer; and as we came nearer and nearer, I could discern even the contents of my cabinet, and the two chests of gold, and the tablets of which the Sphere had made mention. [Illustration 9] [ASCII approximation follows] /\ / |My \ / <> |Study \ /______ | ___ \ / <> My Sons\ \|The \ /______/ \ Page / \ N / <> \ / My \ ^ /______/ THE HALL \ Bedroom \ | \ <> My\ / | \____| /\Wife's / W--+--E \ My Wife / Apartment/ | ------- /\ --- \ WOMEN'S DOOR | MEN'S DOOR \My Daughter | /\ --== \ / The Scullion S \ My Grandsons \ -==# \/ The Footman \___ ___ _ _/ \-=#|/ The Butler \ <> | <> | |THE CELLAR \ / \____|____|_|____________/ ###===--- ---===### Policeman Policeman Touched by my Wife's distress, I would have sprung downward to reassure her, but I found myself incapable of motion. "Trouble not yourself about your Wife," said my Guide: "she will not be long left in anxiety; meantime, let us take a survey of Flatland." Once more I felt myself rising through space. It was even as the Sphere had said. The further we receded from the object we beheld, the larger became the field of vision. My native city, with the interior of every house and every creature therein, lay open to my view in miniature. We mounted higher, and lo, the secrets of the earth, the depths of mines and inmost caverns of the hills, were bared before me. Awestruck at the sight of the mysteries of the earth, thus unveiled before my unworthy eye, I said to my Companion, "Behold, I am become as a God. For the wise men in our country say that to see all things, or as they express it, OMNIVIDENCE, is the attribute of God alone." There was something of scorn in the voice of my Teacher as he made answer: "Is it so indeed? Then the very pick-pockets and cut-throats of my country are to be worshipped by your wise men as being Gods: for there is not one of them that does not see as much as you see now. But trust me, your wise men are wrong." I. Then is omnividence the attribute of others besides Gods? SPHERE. I do not know. But, if a pick-pocket or a cut-throat of our country can see everything that is in your country, surely that is no reason why the pick-pocket or cut-throat should be accepted by you as a God. This omnividence, as you call it--it is not a common word in Spaceland--does it make you more just, more merciful, less selfish, more loving? Not in the least. Then how does it make you more divine? I. "More merciful, more loving!" But these are the qualities of women! And we know that a Circle is a higher Being than a Straight Line, in so far as knowledge and wisdom are more to be esteemed than mere affection. SPHERE. It is not for me to classify human faculties according to merit. Yet many of the best and wisest in Spaceland think more of the affections than of the understanding, more of your despised Straight Lines than of your belauded Circles. But enough of this. Look yonder. Do you know that building? I looked, and afar off I saw an immense Polygonal structure, in which I recognized the General Assembly Hall of the States of Flatland, surrounded by dense lines of Pentagonal buildings at right angles to each other, which I knew to be streets; and I perceived that I was approaching the great Metropolis. "Here we descend," said my Guide. It was now morning, the first hour of the first day of the two thousandth year of our era. Acting, as was their wont, in strict accordance with precedent, the highest Circles of the realm were meeting in solemn conclave, as they had met on the first hour of the first day of the year 1000, and also on the first hour of the first day of the year 0. The minutes of the previous meetings were now read by one whom I at once recognized as my brother, a perfectly Symmetrical Square, and the Chief Clerk of the High Council. It was found recorded on each occasion that: "Whereas the States had been troubled by divers ill-intentioned persons pretending to have received revelations from another World, and professing to produce demonstrations whereby they had instigated to frenzy both themselves and others, it had been for this cause unanimously resolved by the Grand Council that on the first day of each millenary, special injunctions be sent to the Prefects in the several districts of Flatland, to make strict search for such misguided persons, and without formality of mathematical examination, to destroy all such as were Isosceles of any degree, to scourge and imprison any regular Triangle, to cause any Square or Pentagon to be sent to the district Asylum, and to arrest any one of higher rank, sending him straightway to the Capital to be examined and judged by the Council." "You hear your fate," said the Sphere to me, while the Council was passing for the third time the formal resolution. "Death or imprisonment awaits the Apostle of the Gospel of Three Dimensions." "Not so," replied I, "the matter is now so clear to me, the nature of real space so palpable, that methinks I could make a child understand it. Permit me but to descend at this moment and enlighten them." "Not yet," said my Guide, "the time will come for that. Meantime I must perform my mission. Stay thou there in thy place." Saying these words, he leaped with great dexterity into the sea (if I may so call it) of Flatland, right in the midst of the ring of Counsellors. "I come," cried he, "to proclaim that there is a land of Three Dimensions." I could see many of the younger Counsellors start back in manifest horror, as the Sphere's circular section widened before them. But on a sign from the presiding Circle--who shewed not the slightest alarm or surprise--six Isosceles of a low type from six different quarters rushed upon the Sphere. "We have him," they cried; "No; yes; we have him still! he's going! he's gone!" "My Lords," said the President to the Junior Circles of the Council, "there is not the slightest need for surprise; the secret archives, to which I alone have access, tell me that a similar occurrence happened on the last two millennial commencements. You will, of course, say nothing of these trifles outside the Cabinet." Raising his voice, he now summoned the guards. "Arrest the policemen; gag them. You know your duty." After he had consigned to their fate the wretched policemen--ill-fated and unwilling witnesses of a State-secret which they were not to be permitted to reveal--he again addressed the Counsellors. "My Lords, the business of the Council being concluded, I have only to wish you a happy New Year." Before departing, he expressed, at some length, to the Clerk, my excellent but most unfortunate brother, his sincere regret that, in accordance with precedent and for the sake of secrecy, he must condemn him to perpetual imprisonment, but added his satisfaction that, unless some mention were made by him of that day's incident, his life would be spared. Section 19. How, though the Sphere shewed me other mysteries of Spaceland, I still desired more; and what came of it When I saw my poor brother led away to imprisonment, I attempted to leap down into the Council Chamber, desiring to intercede on his behalf, or at least bid him farewell. But I found that I had no motion of my own. I absolutely depended on the volition of my Guide, who said in gloomy tones, "Heed not thy brother; haply thou shalt have ample time hereafter to condole with him. Follow me." [Illustration 10] [ASCII approximation follows] (1) (2) __________ __________ |\ |\ | \ | \ | \ | \ | \ ____|____\ | \ | | | | | | |_____|____| | | | \ | \ | \ | \ | \ | \ | \|_________\| \ __________| Once more we ascended into space. "Hitherto," said the Sphere, "I have shewn you naught save Plane Figures and their interiors. Now I must introduce you to Solids, and reveal to you the plan upon which they are constructed. Behold this multitude of moveable square cards. See, I put one on another, not, as you supposed, Northward of the other, but ON the other. Now a second, now a third. See, I am building up a Solid by a multitude of Squares parallel to one another. Now the Solid is complete, being as high as it is long and broad, and we call it a Cube." "Pardon me, my Lord," replied I; "but to my eye the appearance is as of an Irregular Figure whose inside is laid open to the view; in other words, methinks I see no Solid, but a Plane such as we infer in Flatland; only of an Irregularity which betokens some monstrous criminal, so that the very sight of it is painful to my eyes." "True," said the Sphere, "it appears to you a Plane, because you are not accustomed to light and shade and perspective; just as in Flatland a Hexagon would appear a Straight Line to one who has not the Art of Sight Recognition. But in reality it is a Solid, as you shall learn by the sense of Feeling." He then introduced me to the Cube, and I found that this marvellous Being was indeed no Plane, but a Solid; and that he was endowed with six plane sides and eight terminal points called solid angles; and I remembered the saying of the Sphere that just such a Creature as this would be formed by a Square moving, in Space, parallel to himself: and I rejoiced to think that so insignificant a Creature as I could in some sense be called the Progenitor of so illustrious an offspring. But still I could not fully understand the meaning of what my Teacher had told me concerning "light" and "shade" and "perspective"; and I did not hesitate to put my difficulties before him. Were I to give the Sphere's explanation of these matters, succinct and clear though it was, it would be tedious to an inhabitant of Space, who knows these things already. Suffice it, that by his lucid statements, and by changing the position of objects and lights, and by allowing me to feel the several objects and even his own sacred Person, he at last made all things clear to me, so that I could now readily distinguish between a Circle and a Sphere, a Plane Figure and a Solid. This was the Climax, the Paradise, of my strange eventful History. Henceforth I have to relate the story of my miserable Fall:--most miserable, yet surely most undeserved! For why should the thirst for knowledge be aroused, only to be disappointed and punished? My volition shrinks from the painful task of recalling my humiliation; yet, like a second Prometheus, I will endure this and worse, if by any means I may arouse in the interiors of Plane and Solid Humanity a spirit of rebellion against the Conceit which would limit our Dimensions to Two or Three or any number short of Infinity. Away then with all personal considerations! Let me continue to the end, as I began, without further digressions or anticipations, pursuing the plain path of dispassionate History. The exact facts, the exact words,--and they are burnt in upon my brain,--shall be set down without alteration of an iota; and let my Readers judge between me and Destiny. The Sphere would willingly have continued his lessons by indoctrinating me in the conformation of all regular Solids, Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured to interrupt him. Not that I was wearied of knowledge. On the contrary, I thirsted for yet deeper and fuller draughts than he was offering to me. "Pardon me," said I, "O Thou Whom I must no longer address as the Perfection of all Beauty; but let me beg thee to vouchsafe thy servant a sight of thine interior." SPHERE. My what? I. Thine interior: thy stomach, thy intestines. SPHERE. Whence this ill-timed impertinent request? And what mean you by saying that I am no longer the Perfection of all Beauty? I. My Lord, your own wisdom has taught me to aspire to One even more great, more beautiful, and more closely approximate to Perfection than yourself. As you yourself, superior to all Flatland forms, combine many Circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And even as we, who are now in Space, look down on Flatland and see the insides of all things, so of a certainty there is yet above us some higher, purer region, whither thou dost surely purpose to lead me--O Thou Whom I shall always call, everywhere and in all Dimensions, my Priest, Philosopher, and Friend--some yet more spacious Space, some more dimensionable Dimensionality, from the vantage-ground of which we shall look down together upon the revealed insides of Solid things, and where thine own intestines, and those of thy kindred Spheres, will lie exposed to the view of the poor wandering exile from Flatland, to whom so much has already been vouchsafed. SPHERE. Pooh! Stuff! Enough of this trifling! The time is short, and much remains to be done before you are fit to proclaim the Gospel of Three Dimensions to your blind benighted countrymen in Flatland. I. Nay, gracious Teacher, deny me not what I know it is in thy power to perform. Grant me but one glimpse of thine interior, and I am satisfied for ever, remaining henceforth thy docile pupil, thy unemancipable slave, ready to receive all thy teachings and to feed upon the words that fall from thy lips. SPHERE. Well, then, to content and silence you, let me say at once, I would shew you what you wish if I could; but I cannot. Would you have me turn my stomach inside out to oblige you? I. But my Lord has shewn me the intestines of all my countrymen in the Land of Two Dimensions by taking me with him into the Land of Three. What therefore more easy than now to take his servant on a second journey into the blessed region of the Fourth Dimension, where I shall look down with him once more upon this land of Three Dimensions, and see the inside of every three-dimensioned house, the secrets of the solid earth, the treasures of the mines in Spaceland, and the intestines of every solid living creature, even of the noble and adorable Spheres. SPHERE. But where is this land of Four Dimensions? I. I know not: but doubtless my Teacher knows. SPHERE. Not I. There is no such land. The very idea of it is utterly inconceivable. I. Not inconceivable, my Lord, to me, and therefore still less inconceivable to my Master. Nay, I despair not that, even here, in this region of Three Dimensions, your Lordship's art may make the Fourth Dimension visible to me; just as in the Land of Two Dimensions my Teacher's skill would fain have opened the eyes of his blind servant to the invisible presence of a Third Dimension, though I saw it not. Let me recall the past. Was I not taught below that when I saw a Line and inferred a Plane, I in reality saw a Third unrecognized Dimension, not the same as brightness, called "height"? And does it not now follow that, in this region, when I see a Plane and infer a Solid, I really see a Fourth unrecognized Dimension, not the same as colour, but existent, though infinitesimal and incapable of measurement? And besides this, there is the Argument from Analogy of Figures. SPHERE. Analogy! Nonsense: what analogy? I. Your Lordship tempts his servant to see whether he remembers the revelations imparted to him. Trifle not with me, my Lord; I crave, I thirst, for more knowledge. Doubtless we cannot SEE that other higher Spaceland now, because we we have no eye in our stomachs. But, just as there WAS the realm of Flatland, though that poor puny Lineland Monarch could neither turn to left nor right to discern it, and just as there WAS close at hand, and touching my frame, the land of Three Dimensions, though I, blind senseless wretch, had no power to touch it, no eye in my interior to discern it, so of a surety there is a Fourth Dimension, which my Lord perceives with the inner eye of thought. And that it must exist my Lord himself has taught me. Or can he have forgotten what he himself imparted to his servant? In One Dimension, did not a moving Point produce a Line with TWO terminal points? In Two Dimensions, did not a moving Line produce a Square with FOUR terminal points? In Three Dimensions, did not a moving Square produce--did not this eye of mine behold it--that blessed Being, a Cube, with EIGHT terminal points? And in Four Dimensions shall not a moving Cube--alas, for Analogy, and alas for the Progress of Truth, if it be not so--shall not, I say, the motion of a divine Cube result in a still more divine Organization with SIXTEEN terminal points? Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this--if I might quote my Lord's own words--"strictly according to Analogy"? Again, was I not taught by my Lord that as in a Line there are TWO bounding Points, and in a Square there are FOUR bounding Lines, so in a Cube there must be SIX bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, "strictly according to Analogy"? O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not knowing the facts; and I appeal to your Lordship to confirm or deny my logical anticipations. If I am wrong, I yield, and will no longer demand a fourth Dimension; but, if I am right, my Lord will listen to reason. I ask therefore, is it, or is it not, the fact, that ere now your countrymen also have witnessed the descent of Beings of a higher order than their own, entering closed rooms, even as your Lordship entered mine, without the opening of doors or windows, and appearing and vanishing at will? On the reply to this question I am ready to stake everything. Deny it, and I am henceforth silent. Only vouchsafe an answer. SPHERE. (AFTER A PAUSE). It is reported so. But men are divided in opinion as to the facts. And even granting the facts, they explain them in different ways. And in any case, however great may be the number of different explanations, no one has adopted or suggested the theory of a Fourth Dimension. Therefore, pray have done with this trifling, and let us return to business. I. I was certain of it. I was certain that my anticipations would be fulfilled. And now have patience with me and answer me yet one more question, best of Teachers! Those who have thus appeared--no one knows whence--and have returned--no one knows whither--have they also contracted their sections and vanished somehow into that more Spacious Space, whither I now entreat you to conduct me? SPHERE (MOODILY). They have vanished, certainly--if they ever appeared. But most people say that these visions arose from the thought--you will not understand me--from the brain; from the perturbed angularity of the Seer. I. Say they so? Oh, believe them not. Or if it indeed be so, that this other Space is really Thoughtland, then take me to that blessed Region where I in Thought shall see the insides of all solid things. There, before my ravished eye, a Cube, moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space, with a wake of its own--shall create a still more perfect perfection than himself, with sixteen terminal Extra-solid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth-- How long I should have continued I know not. In vain did the Sphere, in his voice of thunder, reiterate his command of silence, and threaten me with the direst penalties if I persisted. Nothing could stem the flood of my ecstatic aspirations. Perhaps I was to blame; but indeed I was intoxicated with the recent draughts of Truth to which he himself had introduced me. However, the end was not long in coming. My words were cut short by a crash outside, and a simultaneous crash inside me, which impelled me through space with a velocity that precluded speech. Down! down! down! I was rapidly descending; and I knew that return to Flatland was my doom. One glimpse, one last and never-to-be-forgotten glimpse I had of that dull level wilderness--which was now to become my Universe again--spread out before my eye. Then a darkness. Then a final, all-consummating thunder-peal; and, when I came to myself, I was once more a common creeping Square, in my Study at home, listening to the Peace-Cry of my approaching Wife. Section 20. How the Sphere encouraged me in a Vision Although I had less than a minute for reflection, I felt, by a kind of instinct, that I must conceal my experiences from my Wife. Not that I apprehended, at the moment, any danger from her divulging my secret, but I knew that to any Woman in Flatland the narrative of my adventures must needs be unintelligible. So I endeavoured to reassure her by some story, invented for the occasion, that I had accidentally fallen through the trap-door of the cellar, and had there lain stunned. The Southward attraction in our country is so slight that even to a Woman my tale necessarily appeared extraordinary and well-nigh incredible; but my Wife, whose good sense far exceeds that of the average of her Sex, and who perceived that I was unusually excited, did not argue with me on the subject, but insisted that I was ill and required repose. I was glad of an excuse for retiring to my chamber to think quietly over what had happened. When I was at last by myself, a drowsy sensation fell on me; but before my eyes closed I endeavoured to reproduce the Third Dimension, and especially the process by which a Cube is constructed through the motion of a Square. It was not so clear as I could have wished; but I remembered that it must be "Upward, and yet not Northward", and I determined steadfastly to retain these words as the clue which, if firmly grasped, could not fail to guide me to the solution. So mechanically repeating, like a charm, the words, "Upward, yet not Northward", I fell into a sound refreshing sleep. During my slumber I had a dream. I thought I was once more by the side of the Sphere, whose lustrous hue betokened that he had exchanged his wrath against me for perfect placability. We were moving together towards a bright but infinitesimally small Point, to which my Master directed my attention. As we approached, methought there issued from it a slight humming noise as from one of your Spaceland bluebottles, only less resonant by far, so slight indeed that even in the perfect stillness of the Vacuum through which we soared, the sound reached not our ears till we checked our flight at a distance from it of something under twenty human diagonals. "Look yonder," said my Guide, "in Flatland thou hast lived; of Lineland thou hast received a vision; thou hast soared with me to the heights of Spaceland; now, in order to complete the range of thy experience, I conduct thee downward to the lowest depth of existence, even to the realm of Pointland, the Abyss of No dimensions. "Behold yon miserable creature. That Point is a Being like ourselves, but confined to the non-dimensional Gulf. He is himself his own World, his own Universe; of any other than himself he can form no conception; he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality; for he is himself his One and All, being really Nothing. Yet mark his perfect self-contentment, and hence learn this lesson, that to be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy. Now listen." He ceased; and there arose from the little buzzing creature a tiny, low, monotonous, but distinct tinkling, as from one of your Spaceland phonographs, from which I caught these words, "Infinite beatitude of existence! It is; and there is none else beside It." "What," said I, "does the puny creature mean by 'it'?" "He means himself," said the Sphere: "have you not noticed before now, that babies and babyish people who cannot distinguish themselves from the world, speak of themselves in the Third Person? But hush!" "It fills all Space," continued the little soliloquizing Creature, "and what It fills, It is. What It thinks, that It utters; and what It utters, that It hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All. Ah, the happiness ah, the happiness of Being!" "Can you not startle the little thing out of its complacency?" said I. "Tell it what it really is, as you told me; reveal to it the narrow limitations of Pointland, and lead it up to something higher." "That is no easy task," said my Master; "try you." Hereon, raising my voice to the uttermost, I addressed the Point as follows: "Silence, silence, contemptible Creature. You call yourself the All in All, but you are the Nothing: your so-called Universe is a mere speck in a Line, and a Line is a mere shadow as compared with--" "Hush, hush, you have said enough," interrupted the Sphere, "now listen, and mark the effect of your harangue on the King of Pointland." The lustre of the Monarch, who beamed more brightly than ever upon hearing my words, shewed clearly that he retained his complacency; and I had hardly ceased when he took up his strain again. "Ah, the joy, ah, the joy of Thought! What can It not achieve by thinking! Its own Thought coming to Itself, suggestive of Its disparagement, thereby to enhance Its happiness! Sweet rebellion stirred up to result in triumph! Ah, the divine creative power of the All in One! Ah, the joy, the joy of Being!" "You see," said my Teacher, "how little your words have done. So far as the Monarch understands them at all, he accepts them as his own--for he cannot conceive of any other except himself--and plumes himself upon the variety of 'Its Thought' as an instance of creative Power. Let us leave this God of Pointland to the ignorant fruition of his omnipresence and omniscience: nothing that you or I can do can rescue him from his self-satisfaction." After this, as we floated gently back to Flatland, I could hear the mild voice of my Companion pointing the moral of my vision, and stimulating me to aspire, and to teach others to aspire. He had been angered at first--he confessed--by my ambition to soar to Dimensions above the Third; but, since then, he had received fresh insight, and he was not too proud to acknowledge his error to a Pupil. Then he proceeded to initiate me into mysteries yet higher than those I had witnessed, shewing me how to construct Extra-Solids by the motion of Solids, and Double Extra-Solids by the motion of Extra-Solids, and all "strictly according to Analogy", all by methods so simple, so easy, as to be patent even to the Female Sex. Section 21. How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success I awoke rejoicing, and began to reflect on the glorious career before me. I would go forth, methought, at once, and evangelize the whole of Flatland. Even to Women and Soldiers should the Gospel of Three Dimensions be proclaimed. I would begin with my Wife. Just as I had decided on the plan of my operations, I heard the sound of many voices in the street commanding silence. Then followed a louder voice. It was a herald's proclamation. Listening attentively, I recognized the words of the Resolution of the Council, enjoining the arrest, imprisonment, or execution of any one who should pervert the minds of the people by delusions, and by professing to have received revelations from another World. I reflected. This danger was not to be trifled with. It would be better to avoid it by omitting all mention of my Revelation, and by proceeding on the path of Demonstration--which after all, seemed so simple and so conclusive that nothing would be lost by discarding the former means. "Upward, not Northward"--was the clue to the whole proof. It had seemed to me fairly clear before I fell asleep; and when I first awoke, fresh from my dream, it had appeared as patent as Arithmetic; but somehow it did not seem to me quite so obvious now. Though my Wife entered the room opportunely just at that moment, I decided, after we had exchanged a few words of commonplace conversation, not to begin with her. My Pentagonal Sons were men of character and standing, and physicians of no mean reputation, but not great in mathematics, and, in that respect, unfit for my purpose. But it occurred to me that a young and docile Hexagon, with a mathematical turn, would be a most suitable pupil. Why therefore not make my first experiment with my little precocious Grandson, whose casual remarks on the meaning of 3^3 had met with the approval of the Sphere? Discussing the matter with him, a mere boy, I should be in perfect safety; for he would know nothing of the Proclamation of the Council; whereas I could not feel sure that my Sons--so greatly did their patriotism and reverence for the Circles predominate over mere blind affection--might not feel compelled to hand me over to the Prefect, if they found me seriously maintaining the seditious heresy of the Third Dimension. But the first thing to be done was to satisfy in some way the curiosity of my Wife, who naturally wished to know something of the reasons for which the Circle had desired that mysterious interview, and of the means by which he had entered the house. Without entering into the details of the elaborate account I gave her,--an account, I fear, not quite so consistent with truth as my Readers in Spaceland might desire,--I must be content with saying that I succeeded at last in persuading her to return quietly to her household duties without eliciting from me any reference to the World of Three Dimensions. This done, I immediately sent for my Grandson; for, to confess the truth, I felt that all that I had seen and heard was in some strange way slipping away from me, like the image of a half-grasped, tantalizing dream, and I longed to essay my skill in making a first disciple. When my Grandson entered the room I carefully secured the door. Then, sitting down by his side and taking our mathematical tablets,--or, as you would call them, Lines--I told him we would resume the lesson of yesterday. I taught him once more how a Point by motion in One Dimension produces a Line, and how a straight Line in Two Dimensions produces a Square. After this, forcing a laugh, I said, "And now, you scamp, you wanted to make me believe that a Square may in the same way by motion 'Upward, not Northward' produce another figure, a sort of extra Square in Three Dimensions. Say that again, you young rascal." At this moment we heard once more the herald's "O yes! O yes!" outside in the street proclaiming the Resolution of the Council. Young though he was, my Grandson--who was unusually intelligent for his age, and bred up in perfect reverence for the authority of the Circles--took in the situation with an acuteness for which I was quite unprepared. He remained silent till the last words of the Proclamation had died away, and then, bursting into tears, "Dear Grandpapa," he said, "that was only my fun, and of course I meant nothing at all by it; and we did not know anything then about the new Law; and I don't think I said anything about the Third Dimension; and I am sure I did not say one word about 'Upward, not Northward', for that would be such nonsense, you know. How could a thing move Upward, and not Northward? Upward and not Northward! Even if I were a baby, I could not be so absurd as that. How silly it is! Ha! ha! ha!" "Not at all silly," said I, losing my temper; "here for example, I take this Square," and, at the word, I grasped a moveable Square, which was lying at hand--"and I move it, you see, not Northward but--yes, I move it Upward--that is to say, not Northward, but I move it somewhere--not exactly like this, but somehow--" Here I brought my sentence to an inane conclusion, shaking the Square about in a purposeless manner, much to the amusement of my Grandson, who burst out laughing louder than ever, and declared that I was not teaching him, but joking with him; and so saying he unlocked the door and ran out of the room. Thus ended my first attempt to convert a pupil to the Gospel of Three Dimensions. Section 22. How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result My failure with my Grandson did not encourage me to communicate my secret to others of my household; yet neither was I led by it to despair of success. Only I saw that I must not wholly rely on the catch-phrase, "Upward, not Northward", but must rather endeavour to seek a demonstration by setting before the public a clear view of the whole subject; and for this purpose it seemed necessary to resort to writing. So I devoted several months in privacy to the composition of a treatise on the mysteries of Three Dimensions. Only, with the view of evading the Law, if possible, I spoke not of a physical Dimension, but of a Thoughtland whence, in theory, a Figure could look down upon Flatland and see simultaneously the insides of all things, and where it was possible that there might be supposed to exist a Figure environed, as it were, with six Squares, and containing eight terminal Points. But in writing this book I found myself sadly hampered by the impossibility of drawing such diagrams as were necessary for my purpose; for of course, in our country of Flatland, there are no tablets but Lines, and no diagrams but Lines, all in one straight Line and only distinguishable by difference of size and brightness; so that, when I had finished my treatise (which I entitled, "Through Flatland to Thoughtland") I could not feel certain that many would understand my meaning. Meanwhile my life was under a cloud. All pleasures palled upon me; all sights tantalized and tempted me to outspoken treason, because I could not but compare what I saw in Two Dimensions with what it really was if seen in Three, and could hardly refrain from making my comparisons aloud. I neglected my clients and my own business to give myself to the contemplation of the mysteries which I had once beheld, yet which I could impart to no one, and found daily more difficult to reproduce even before my own mental vision. One day, about eleven months after my return from Spaceland, I tried to see a Cube with my eye closed, but failed; and though I succeeded afterwards, I was not then quite certain (nor have I been ever afterwards) that I had exactly realized the original. This made me more melancholy than before, and determined me to take some step; yet what, I knew not. I felt that I would have been willing to sacrifice my life for the Cause, if thereby I could have produced conviction. But if I could not convince my Grandson, how could I convince the highest and most developed Circles in the land? And yet at times my spirit was too strong for me, and I gave vent to dangerous utterances. Already I was considered heterodox if not treasonable, and I was keenly alive to the danger of my position; nevertheless I could not at times refrain from bursting out into suspicious or half-seditious utterances, even among the highest Polygonal and Circular society. When, for example, the question arose about the treatment of those lunatics who said that they had received the power of seeing the insides of things, I would quote the saying of an ancient Circle, who declared that prophets and inspired people are always considered by the majority to be mad; and I could not help occasionally dropping such expressions as "the eye that discerns the interiors of things", and "the all-seeing land"; once or twice I even let fall the forbidden terms "the Third and Fourth Dimensions". At last, to complete a series of minor indiscretions, at a meeting of our Local Speculative Society held at the palace of the Prefect himself,--some extremely silly person having read an elaborate paper exhibiting the precise reasons why Providence has limited the number of Dimensions to Two, and why the attribute of omnividence is assigned to the Supreme alone--I so far forgot myself as to give an exact account of the whole of my voyage with the Sphere into Space, and to the Assembly Hall in our Metropolis, and then to Space again, and of my return home, and of everything that I had seen and heard in fact or vision. At first, indeed, I pretended that I was describing the imaginary experiences of a fictitious person; but my enthusiasm soon forced me to throw off all disguise, and finally, in a fervent peroration, I exhorted all my hearers to divest themselves of prejudice and to become believers in the Third Dimension. Need I say that I was at once arrested and taken before the Council? Next morning, standing in the very place where but a very few months ago the Sphere had stood in my company, I was allowed to begin and to continue my narration unquestioned and uninterrupted. But from the first I foresaw my fate; for the President, noting that a guard of the better sort of Policemen was in attendance, of angularity little, if at all, under 55 degrees, ordered them to be relieved before I began my defence, by an inferior class of 2 or 3 degrees. I knew only too well what that meant. I was to be executed or imprisoned, and my story was to be kept secret from the world by the simultaneous destruction of the officials who had heard it; and, this being the case, the President desired to substitute the cheaper for the more expensive victims. After I had concluded my defence, the President, perhaps perceiving that some of the junior Circles had been moved by my evident earnestness, asked me two questions:-- 1. Whether I could indicate the direction which I meant when I used the words "Upward, not Northward"? 2. Whether I could by any diagrams or descriptions (other than the enumeration of imaginary sides and angles) indicate the Figure I was pleased to call a Cube? I declared that I could say nothing more, and that I must commit myself to the Truth, whose cause would surely prevail in the end. The President replied that he quite concurred in my sentiment, and that I could not do better. I must be sentenced to perpetual imprisonment; but if the Truth intended that I should emerge from prison and evangelize the world, the Truth might be trusted to bring that result to pass. Meanwhile I should be subjected to no discomfort that was not necessary to preclude escape, and, unless I forfeited the privilege by misconduct, I should be occasionally permitted to see my brother who had preceded me to my prison. Seven years have elapsed and I am still a prisoner, and--if I except the occasional visits of my brother--debarred from all companionship save that of my jailers. My brother is one of the best of Squares, just, sensible, cheerful, and not without fraternal affection; yet I confess that my weekly interviews, at least in one respect, cause me the bitterest pain. He was present when the Sphere manifested himself in the Council Chamber; he saw the Sphere's changing sections; he heard the explanation of the phenomena then given to the Circles. Since that time, scarcely a week has passed during seven whole years, without his hearing from me a repetition of the part I played in that manifestation, together with ample descriptions of all the phenomena in Spaceland, and the arguments for the existence of Solid things derivable from Analogy. Yet--I take shame to be forced to confess it--my brother has not yet grasped the nature of the Third Dimension, and frankly avows his disbelief in the existence of a Sphere. Hence I am absolutely destitute of converts, and, for aught that I can see, the millennial Revelation has been made to me for nothing. Prometheus up in Spaceland was bound for bringing down fire for mortals, but I--poor Flatland Prometheus--lie here in prison for bringing down nothing to my countrymen. Yet I exist in the hope that these memoirs, in some manner, I know not how, may find their way to the minds of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality. That is the hope of my brighter moments. Alas, it is not always so. Heavily weighs on me at times the burdensome reflection that I cannot honestly say I am confident as to the exact shape of the once-seen, oft-regretted Cube; and in my nightly visions the mysterious precept, "Upward, not Northward", haunts me like a soul-devouring Sphinx. It is part of the martyrdom which I endure for the cause of the Truth that there are seasons of mental weakness, when Cubes and Spheres flit away into the background of scarce-possible existences; when the Land of Three Dimensions seems almost as visionary as the Land of One or None; nay, when even this hard wall that bars me from my freedom, these very tablets on which I am writing, and all the substantial realities of Flatland itself, appear no better than the offspring of a diseased imagination, or the baseless fabric of a dream. THE END of FLATLAND ----------------------------------------------------------------- | THE END of | | ______ | | / / /| ------ / /| /| / /-. | | /---- / /__| / / /__| / | / / / | | / /___ / | / /___ / | / |/ /__.-' | | | | The baseless fabric of my vision | | Melted into air into thin air | | Such stuff as dreams are made of | ----------------------------------------------------------------- 
16449	----	[*Transcriber's Note: The following errors found in the original have been left as is. Chapter I, 14th paragraph: drop double quote before 'It is said'; Chapter IV, 1st paragraph: 'so similar than' read 'so similar that'; Chapter IV, table of Hebrew numerals (near footnote 144): insert comma after 'shemoneh'; Chapter V, table of Tahuatan numerals (near footnote 201): 'tahi,' read 'tahi.'; Same table: ' 20,000. tufa' read '200,000. tufa'; Chapter VI, table of Bagrimma numerals (near footnote 259): 'marta = 5 + 2' read 'marta = 5 + 3'; Same table: 'do-so = [5] + 3' read 'do-so = [5] + 4'; Chapter VII, table of Nahuatl numerals (near footnote 365): '90-10' read '80-10'; In paragraph following that table: '+ (15 + 4) × 400 × 800' read '(15 + 4) × 20 × 400 × 8000 + (15 + 4) × 400 × 8000'; In text of footnote 297: 'II. I. p. 179' read 'II. i. p. 179'; *] THE MACMILLAN COMPANY NEW YORK · BOSTON · CHICAGO · DALLAS ATLANTA · SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON · BOMBAY · CALCUTTA MELBOURNE THE MACMILLAN COMPANY OF CANADA, LIMITED TORONTO THE NUMBER CONCEPT ITS ORIGIN AND DEVELOPMENT BY LEVI LEONARD CONANT, PH.D. ASSOCIATE PROFESSOR OF MATHEMATICS IN THE WORCESTER POLYTECHNIC INSTITUTE New York MACMILLAN AND CO. AND LONDON 1931 COPYRIGHT, 1896, BY THE MACMILLAN COMPANY. COPYRIGHT, 1924, BY EMMA B. CONANT. All rights reserved--no part of this book may be reproduced in any form without permission in writing from the publisher. Set up and electrotyped. Published July, 1896. Norwood Press J.S. Cushing Co.--Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE. In the selection of authorities which have been consulted in the preparation of this work, and to which reference is made in the following pages, great care has been taken. Original sources have been drawn upon in the majority of cases, and nearly all of these are the most recent attainable. Whenever it has not been possible to cite original and recent works, the author has quoted only such as are most standard and trustworthy. In the choice of orthography of proper names and numeral words, the forms have, in almost all cases, been written as they were found, with no attempt to reduce them to a systematic English basis. In many instances this would have been quite impossible; and, even if possible, it would have been altogether unimportant. Hence the forms, whether German, French, Italian, Spanish, or Danish in their transcription, are left unchanged. Diacritical marks are omitted, however, since the proper key could hardly be furnished in a work of this kind. With the above exceptions, this study will, it is hoped, be found to be quite complete; and as the subject here investigated has never before been treated in any thorough and comprehensive manner, it is hoped that this book may be found helpful. The collections of numeral systems illustrating the use of the binary, the quinary, and other number systems, are, taken together, believed to be the most extensive now existing in any language. Only the cardinal numerals have been considered. The ordinals present no marked peculiarities which would, in a work of this kind, render a separate discussion necessary. Accordingly they have, though with some reluctance, been omitted entirely. Sincere thanks are due to those who have assisted the author in the preparation of his materials. Especial acknowledgment should be made to Horatio Hale, Dr. D.G. Brinton, Frank Hamilton Cushing, and Dr. A.F. Chamberlain. WORCESTER, MASS., Nov. 12, 1895. CONTENTS. Chapter I. Counting 1 Chapter II. Number System Limits 21 Chapter III. Origin of Number Words 37 Chapter IV. Origin of Number Words (_continued_) 74 Chapter V. Miscellaneous Number Bases 100 Chapter VI. The Quinary System 134 Chapter VII. The Vigesimal System 176 * * * * * Index 211 THE NUMBER CONCEPT: ITS ORIGIN AND DEVELOPMENT. CHAPTER I. COUNTING. Among the speculative questions which arise in connection with the study of arithmetic from a historical standpoint, the origin of number is one that has provoked much lively discussion, and has led to a great amount of learned research among the primitive and savage languages of the human race. A few simple considerations will, however, show that such research must necessarily leave this question entirely unsettled, and will indicate clearly that it is, from the very nature of things, a question to which no definite and final answer can be given. Among the barbarous tribes whose languages have been studied, even in a most cursory manner, none have ever been discovered which did not show some familiarity with the number concept. The knowledge thus indicated has often proved to be most limited; not extending beyond the numbers 1 and 2, or 1, 2, and 3. Examples of this poverty of number knowledge are found among the forest tribes of Brazil, the native races of Australia and elsewhere, and they are considered in some detail in the next chapter. At first thought it seems quite inconceivable that any human being should be destitute of the power of counting beyond 2. But such is the case; and in a few instances languages have been found to be absolutely destitute of pure numeral words. The Chiquitos of Bolivia had no real numerals whatever,[1] but expressed their idea for "one" by the word _etama_, meaning alone. The Tacanas of the same country have no numerals except those borrowed from Spanish, or from Aymara or Peno, languages with which they have long been in contact.[2] A few other South American languages are almost equally destitute of numeral words. But even here, rudimentary as the number sense undoubtedly is, it is not wholly lacking; and some indirect expression, or some form of circumlocution, shows a conception of the difference between _one_ and _two_, or at least, between _one_ and _many_. These facts must of necessity deter the mathematician from seeking to push his investigation too far back toward the very origin of number. Philosophers have endeavoured to establish certain propositions concerning this subject, but, as might have been expected, have failed to reach any common ground of agreement. Whewell has maintained that "such propositions as that two and three make five are necessary truths, containing in them an element of certainty beyond that which mere experience can give." Mill, on the other hand, argues that any such statement merely expresses a truth derived from early and constant experience; and in this view he is heartily supported by Tylor.[3] But why this question should provoke controversy, it is difficult for the mathematician to understand. Either view would seem to be correct, according to the standpoint from which the question is approached. We know of no language in which the suggestion of number does not appear, and we must admit that the words which give expression to the number sense would be among the early words to be formed in any language. They express ideas which are, at first, wholly concrete, which are of the greatest possible simplicity, and which seem in many ways to be clearly understood, even by the higher orders of the brute creation. The origin of number would in itself, then, appear to lie beyond the proper limits of inquiry; and the primitive conception of number to be fundamental with human thought. In connection with the assertion that the idea of number seems to be understood by the higher orders of animals, the following brief quotation from a paper by Sir John Lubbock may not be out of place: "Leroy ... mentions a case in which a man was anxious to shoot a crow. 'To deceive this suspicious bird, the plan was hit upon of sending two men to the watch house, one of whom passed on, while the other remained; but the crow counted and kept her distance. The next day three went, and again she perceived that only two retired. In fine, it was found necessary to send five or six men to the watch house to put her out in her calculation. The crow, thinking that this number of men had passed by, lost no time in returning.' From this he inferred that crows could count up to four. Lichtenberg mentions a nightingale which was said to count up to three. Every day he gave it three mealworms, one at a time. When it had finished one it returned for another, but after the third it knew that the feast was over.... There is an amusing and suggestive remark in Mr. Galton's interesting _Narrative of an Explorer in Tropical South Africa_. After describing the Demara's weakness in calculations, he says: 'Once while I watched a Demara floundering hopelessly in a calculation on one side of me, I observed, "Dinah," my spaniel, equally embarrassed on the other; she was overlooking half a dozen of her new-born puppies, which had been removed two or three times from her, and her anxiety was excessive, as she tried to find out if they were all present, or if any were still missing. She kept puzzling and running her eyes over them backwards and forwards, but could not satisfy herself. She evidently had a vague notion of counting, but the figure was too large for her brain. Taking the two as they stood, dog and Demara, the comparison reflected no great honour on the man....' According to my bird-nesting recollections, which I have refreshed by more recent experience, if a nest contains four eggs, one may safely be taken; but if two are removed, the bird generally deserts. Here, then, it would seem as if we had some reason for supposing that there is sufficient intelligence to distinguish three from four. An interesting consideration arises with reference to the number of the victims allotted to each cell by the solitary wasps. One species of Ammophila considers one large caterpillar of _Noctua segetum_ enough; one species of Eumenes supplies its young with five victims; another 10, 15, and even up to 24. The number appears to be constant in each species. How does the insect know when her task is fulfilled? Not by the cell being filled, for if some be removed, she does not replace them. When she has brought her complement she considers her task accomplished, whether the victims are still there or not. How, then, does she know when she has made up the number 24? Perhaps it will be said that each species feels some mysterious and innate tendency to provide a certain number of victims. This would, under no circumstances, be any explanation; but it is not in accordance with the facts. In the genus Eumenes the males are much smaller than the females.... If the egg is male, she supplies five; if female, 10 victims. Does she count? Certainly this seems very like a commencement of arithmetic."[4] Many writers do not agree with the conclusions which Lubbock reaches; maintaining that there is, in all such instances, a perception of greater or less quantity rather than any idea of number. But a careful consideration of the objections offered fails entirely to weaken the argument. Example after example of a nature similar to those just quoted might be given, indicating on the part of animals a perception of the difference between 1 and 2, or between 2 and 3 and 4; and any reasoning which tends to show that it is quantity rather than number which the animal perceives, will apply with equal force to the Demara, the Chiquito, and the Australian. Hence the actual origin of number may safely be excluded from the limits of investigation, and, for the present, be left in the field of pure speculation. A most inviting field for research is, however, furnished by the primitive methods of counting and of giving visible expression to the idea of number. Our starting-point must, of course, be the sign language, which always precedes intelligible speech; and which is so convenient and so expressive a method of communication that the human family, even in its most highly developed branches, never wholly lays it aside. It may, indeed, be stated as a universal law, that some practical method of numeration has, in the childhood of every nation or tribe, preceded the formation of numeral words. Practical methods of numeration are many in number and diverse in kind. But the one primitive method of counting which seems to have been almost universal throughout all time is the finger method. It is a matter of common experience and observation that every child, when he begins to count, turns instinctively to his fingers; and, with these convenient aids as counters, tallies off the little number he has in mind. This method is at once so natural and obvious that there can be no doubt that it has always been employed by savage tribes, since the first appearance of the human race in remote antiquity. All research among uncivilized peoples has tended to confirm this view, were confirmation needed of anything so patent. Occasionally some exception to this rule is found; or some variation, such as is presented by the forest tribes of Brazil, who, instead of counting on the fingers themselves, count on the joints of their fingers.[5] As the entire number system of these tribes appears to be limited to _three_, this variation is no cause for surprise. The variety in practical methods of numeration observed among savage races, and among civilized peoples as well, is so great that any detailed account of them would be almost impossible. In one region we find sticks or splints used; in another, pebbles or shells; in another, simple scratches, or notches cut in a stick, Robinson Crusoe fashion; in another, kernels or little heaps of grain; in another, knots on a string; and so on, in diversity of method almost endless. Such are the devices which have been, and still are, to be found in the daily habit of great numbers of Indian, negro, Mongolian, and Malay tribes; while, to pass at a single step to the other extremity of intellectual development, the German student keeps his beer score by chalk marks on the table or on the wall. But back of all these devices, and forming a common origin to which all may be referred, is the universal finger method; the method with which all begin, and which all find too convenient ever to relinquish entirely, even though their civilization be of the highest type. Any such mode of counting, whether involving the use of the fingers or not, is to be regarded simply as an extraneous aid in the expression or comprehension of an idea which the mind cannot grasp, or cannot retain, without assistance. The German student scores his reckoning with chalk marks because he might otherwise forget; while the Andaman Islander counts on his fingers because he has no other method of counting,--or, in other words, of grasping the idea of number. A single illustration may be given which typifies all practical methods of numeration. More than a century ago travellers in Madagascar observed a curious but simple mode of ascertaining the number of soldiers in an army.[6] Each soldier was made to go through a passage in the presence of the principal chiefs; and as he went through, a pebble was dropped on the ground. This continued until a heap of 10 was obtained, when one was set aside and a new heap begun. Upon the completion of 10 heaps, a pebble was set aside to indicate 100; and so on until the entire army had been numbered. Another illustration, taken from the very antipodes of Madagascar, recently found its way into print in an incidental manner,[7] and is so good that it deserves a place beside de Flacourt's time-honoured example. Mom Cely, a Southern negro of unknown age, finds herself in debt to the storekeeper; and, unwilling to believe that the amount is as great as he represents, she proceeds to investigate the matter in her own peculiar way. She had "kept a tally of these purchases by means of a string, in which she tied commemorative knots." When her creditor "undertook to make the matter clear to Cely's comprehension, he had to proceed upon a system of her own devising. A small notch was cut in a smooth white stick for every dime she owed, and a large notch when the dimes amounted to a dollar; for every five dollars a string was tied in the fifth big notch, Cely keeping tally by the knots in her bit of twine; thus, when two strings were tied about the stick, the ten dollars were seen to be an indisputable fact." This interesting method of computing the amount of her debt, whether an invention of her own or a survival of the African life of her parents, served the old negro woman's purpose perfectly; and it illustrates, as well as a score of examples could, the methods of numeration to which the children of barbarism resort when any number is to be expressed which exceeds the number of counters with which nature has provided them. The fingers are, however, often employed in counting numbers far above the first decade. After giving the Il-Oigob numerals up to 60, Müller adds:[8] "Above 60 all numbers, indicated by the proper figure pantomime, are expressed by means of the word _ipi_." We know, moreover, that many of the American Indian tribes count one ten after another on their fingers; so that, whatever number they are endeavouring to indicate, we need feel no surprise if the savage continues to use his fingers throughout the entire extent of his counts. In rare instances we find tribes which, like the Mairassis of the interior of New Guinea, appear to use nothing but finger pantomime.[9] This tribe, though by no means destitute of the number sense, is said to have no numerals whatever, but to use the single word _awari_ with each show of fingers, no matter how few or how many are displayed. In the methods of finger counting employed by savages a considerable degree of uniformity has been observed. Not only does he use his fingers to assist him in his tally, but he almost always begins with the little finger of his left hand, thence proceeding towards the thumb, which is 5. From this point onward the method varies. Sometimes the second 5 also is told off on the left hand, the same order being observed as in the first 5; but oftener the fingers of the right hand are used, with a reversal of the order previously employed; _i.e._ the thumb denotes 6, the index finger 7, and so on to the little finger, which completes the count to 10. At first thought there would seem to be no good reason for any marked uniformity of method in finger counting. Observation among children fails to detect any such thing; the child beginning, with almost entire indifference, on the thumb or on the little finger of the left hand. My own observation leads to the conclusion that very young children have a slight, though not decided preference for beginning with the thumb. Experiments in five different primary rooms in the public schools of Worcester, Mass., showed that out of a total of 206 children, 57 began with the little finger and 149 with the thumb. But the fact that nearly three-fourths of the children began with the thumb, and but one-fourth with the little finger, is really far less significant than would appear at first thought. Children of this age, four to eight years, will count in either way, and sometimes seem at a loss themselves to know where to begin. In one school room where this experiment was tried the teacher incautiously asked one child to count on his fingers, while all the other children in the room watched eagerly to see what he would do. He began with the little finger--and so did every child in the room after him. In another case the same error was made by the teacher, and the child first asked began with the thumb. Every other child in the room did the same, each following, consciously or unconsciously, the example of the leader. The results from these two schools were of course rejected from the totals which are given above; but they serve an excellent purpose in showing how slight is the preference which very young children have in this particular. So slight is it that no definite law can be postulated of this age; but the tendency seems to be to hold the palm of the hand downward, and then begin with the thumb. The writer once saw a boy about seven years old trying to multiply 3 by 6; and his method of procedure was as follows: holding his left hand with its palm down, he touched with the forefinger of his right hand the thumb, forefinger, and middle finger successively of his left hand. Then returning to his starting-point, he told off a second three in the same manner. This process he continued until he had obtained 6 threes, and then he announced his result correctly. If he had been a few years older, he might not have turned so readily to his thumb as a starting-point for any digital count. The indifference manifested by very young children gradually disappears, and at the age of twelve or thirteen the tendency is decidedly in the direction of beginning with the little finger. Fully three-fourths of all persons above that age will be found to count from the little finger toward the thumb, thus reversing the proportion that was found to obtain in the primary school rooms examined. With respect to finger counting among civilized peoples, we fail, then, to find any universal law; the most that can be said is that more begin with the little finger than with the thumb. But when we proceed to the study of this slight but important particular among savages, we find them employing a certain order of succession with such substantial uniformity that the conclusion is inevitable that there must lie back of this some well-defined reason, or perhaps instinct, which guides them in their choice. This instinct is undoubtedly the outgrowth of the almost universal right-handedness of the human race. In finger counting, whether among children or adults, the beginning is made on the left hand, except in the case of left-handed individuals; and even then the start is almost as likely to be on the left hand as on the right. Savage tribes, as might be expected, begin with the left hand. Not only is this custom almost invariable, when tribes as a whole are considered, but the little finger is nearly always called into requisition first. To account for this uniformity, Lieutenant Gushing gives the following theory,[10] which is well considered, and is based on the results of careful study and observation among the Zuñi Indians of the Southwest: "Primitive man when abroad never lightly quit hold of his weapons. If he wanted to count, he did as the Zuñi afield does to-day; he tucked his instrument under his left arm, thus constraining the latter, but leaving the right hand free, that he might check off with it the fingers of the rigidly elevated left hand. From the nature of this position, however, the palm of the left hand was presented to the face of the counter, so that he had to begin his score on the little finger of it, and continue his counting from the right leftward. An inheritance of this may be detected to-day in the confirmed habit the Zuñi has of gesticulating from the right leftward, with the fingers of the right hand over those of the left, whether he be counting and summing up, or relating in any orderly manner." Here, then, is the reason for this otherwise unaccountable phenomenon. If savage man is universally right-handed, he will almost inevitably use the index finger of his right hand to mark the fingers counted, and he will begin his count just where it is most convenient. In his case it is with the little finger of the left hand. In the case of the child trying to multiply 3 by 6, it was with the thumb of the same hand. He had nothing to tuck under his arm; so, in raising his left hand to a position where both eye and counting finger could readily run over its fingers, he held the palm turned away from his face. The same choice of starting-point then followed as with the savage--the finger nearest his right hand; only in this case the finger was a thumb. The deaf mute is sometimes taught in this manner, which is for him an entirely natural manner. A left-handed child might be expected to count in a left-to-right manner, beginning, probably, with the thumb of his right hand. To the law just given, that savages begin to count on the little finger of the left hand, there have been a few exceptions noted; and it has been observed that the method of progression on the second hand is by no means as invariable as on the first. The Otomacs[11] of South America began their count with the thumb, and to express the number 3 would use the thumb, forefinger, and middle finger. The Maipures,[12] oddly enough, seem to have begun, in some cases at least, with the forefinger; for they are reported as expressing 3 by means of the fore, middle, and ring fingers. The Andamans[13] begin with the little finger of either hand, tapping the nose with each finger in succession. If they have but one to express, they use the forefinger of either hand, pronouncing at the same time the proper word. The Bahnars,[14] one of the native tribes of the interior of Cochin China, exhibit no particular order in the sequence of fingers used, though they employ their digits freely to assist them in counting. Among certain of the negro tribes of South Africa[15] the little finger of the right hand is used for 1, and their count proceeds from right to left. With them, 6 is the thumb of the left hand, 7 the forefinger, and so on. They hold the palm downward instead of upward, and thus form a complete and striking exception to the law which has been found to obtain with such substantial uniformity in other parts of the uncivilized world. In Melanesia a few examples of preference for beginning with the thumb may also be noticed. In the Banks Islands the natives begin by turning down the thumb of the right hand, and then the fingers in succession to the little finger, which is 5. This is followed by the fingers of the left hand, both hands with closed fists being held up to show the completed 10. In Lepers' Island, they begin with the thumb, but, having reached 5 with the little finger, they do not pass to the other hand, but throw up the fingers they have turned down, beginning with the forefinger and keeping the thumb for 10.[16] In the use of the single hand this people is quite peculiar. The second 5 is almost invariably told off by savage tribes on the second hand, though in passing from the one to the other primitive man does not follow any invariable law. He marks 6 with either the thumb or the little finger. Probably the former is the more common practice, but the statement cannot be made with any degree of certainty. Among the Zulus the sequence is from thumb to thumb, as is the case among the other South African tribes just mentioned; while the Veis and numerous other African tribes pass from thumb to little finger. The Eskimo, and nearly all the American Indian tribes, use the correspondence between 6 and the thumb; but this habit is by no means universal. Respecting progression from right to left or left to right on the toes, there is no general law with which the author is familiar. Many tribes never use the toes in counting, but signify the close of the first 10 by clapping the hands together, by a wave of the right hand, or by designating some object; after which the fingers are again used as before. One other detail in finger counting is worthy of a moment's notice. It seems to have been the opinion of earlier investigators that in his passage from one finger to the next, the savage would invariably bend down, or close, the last finger used; that is, that the count began with the fingers open and outspread. This opinion is, however, erroneous. Several of the Indian tribes of the West[17] begin with the hand clenched, and open the fingers one by one as they proceed. This method is much less common than the other, but that it exists is beyond question. In the Muralug Island, in the western part of Torres Strait, a somewhat remarkable method of counting formerly existed, which grew out of, and is to be regarded as an extension of, the digital method. Beginning with the little finger of the left hand, the natives counted up to 5 in the usual manner, and then, instead of passing to the other hand, or repeating the count on the same fingers, they expressed the numbers from 6 to 10 by touching and naming successively the left wrist, left elbow, left shoulder, left breast, and sternum. Then the numbers from 11 to 19 were indicated by the use, in inverse order, of the corresponding portions of the right side, arm, and hand, the little finger of the right hand signifying 19. The words used were in each case the actual names of the parts touched; the same word, for example, standing for 6 and 14; but they were never used in the numerical sense unless accompanied by the proper gesture, and bear no resemblance to the common numerals, which are but few in number. This method of counting is rapidly dying out among the natives of the island, and is at the present time used only by old people.[18] Variations on this most unusual custom have been found to exist in others of the neighbouring islands, but none were exactly similar to it. One is also reminded by it of a custom[19] which has for centuries prevailed among bargainers in the East, of signifying numbers by touching the joints of each other's fingers under a cloth. Every joint has a special signification; and the entire system is undoubtedly a development from finger counting. The buyer or seller will by this method express 6 or 60 by stretching out the thumb and little finger and closing the rest of the fingers. The addition of the fourth finger to the two thus used signifies 7 or 70; and so on. "It is said that between two brokers settling a price by thus snipping with the fingers, cleverness in bargaining, offering a little more, hesitating, expressing an obstinate refusal to go further, etc., are as clearly indicated as though the bargaining were being carried on in words. The place occupied, in the intellectual development of man, by finger counting and by the many other artificial methods of reckoning,--pebbles, shells, knots, the abacus, etc.,--seems to be this: The abstract processes of addition, subtraction, multiplication, division, and even counting itself, present to the mind a certain degree of difficulty. To assist in overcoming that difficulty, these artificial aids are called in; and, among savages of a low degree of development, like the Australians, they make counting possible. A little higher in the intellectual scale, among the American Indians, for example, they are employed merely as an artificial aid to what could be done by mental effort alone. Finally, among semi-civilized and civilized peoples, the same processes are retained, and form a part of the daily life of almost every person who has to do with counting, reckoning, or keeping tally in any manner whatever. They are no longer necessary, but they are so convenient and so useful that civilization can never dispense with them. The use of the abacus, in the form of the ordinary numeral frame, has increased greatly within the past few years; and the time may come when the abacus in its proper form will again find in civilized countries a use as common as that of five centuries ago. In the elaborate calculating machines of the present, such as are used by life insurance actuaries and others having difficult computations to make, we have the extreme of development in the direction of artificial aid to reckoning. But instead of appearing merely as an extraneous aid to a defective intelligence, it now presents itself as a machine so complex that a high degree of intellectual power is required for the mere grasp of its construction and method of working. CHAPTER II. NUMBER SYSTEM LIMITS. With respect to the limits to which the number systems of the various uncivilized races of the earth extend, recent anthropological research has developed many interesting facts. In the case of the Chiquitos and a few other native races of Bolivia we found no distinct number sense at all, as far as could be judged from the absence, in their language, of numerals in the proper sense of the word. How they indicated any number greater than _one_ is a point still requiring investigation. In all other known instances we find actual number systems, or what may for the sake of uniformity be dignified by that name. In many cases, however, the numerals existing are so few, and the ability to count is so limited, that the term _number system_ is really an entire misnomer. Among the rudest tribes, those whose mode of living approaches most nearly to utter savagery, we find a certain uniformity of method. The entire number system may consist of but two words, _one_ and _many_; or of three words, _one_, _two_, _many_. Or, the count may proceed to 3, 4, 5, 10, 20, or 100; passing always, or almost always, from the distinct numeral limit to the indefinite _many_ or several, which serves for the expression of any number not readily grasped by the mind. As a matter of fact, most races count as high as 10; but to this statement the exceptions are so numerous that they deserve examination in some detail. In certain parts of the world, notably among the native races of South America, Australia, and many of the islands of Polynesia and Melanesia, a surprising paucity of numeral words has been observed. The Encabellada of the Rio Napo have but two distinct numerals; _tey_, 1, and _cayapa_, 2.[20] The Chaco languages[21] of the Guaycuru stock are also notably poor in this respect. In the Mbocobi dialect of this language the only native numerals are _yña tvak_, 1, and _yfioaca_, 2. The Puris[22] count _omi_, 1, _curiri_, 2, _prica_, many; and the Botocudos[23] _mokenam_, 1, _uruhu_, many. The Fuegans,[24] supposed to have been able at one time to count to 10, have but three numerals,--_kaoueli_, 1, _compaipi_, 2, _maten_, 3. The Campas of Peru[25] possess only three separate words for the expression of number,--_patrio_, 1, _pitteni_, 2, _mahuani_, 3. Above 3 they proceed by combinations, as 1 and 3 for 4, 1 and 1 and 3 for 5. Counting above 10 is, however, entirely inconceivable to them, and any number beyond that limit they indicate by _tohaine_, many. The Conibos,[26] of the same region, had, before their contact with the Spanish, only _atchoupre_, 1, and _rrabui_, 2; though they made some slight progress above 2 by means of reduplication. The Orejones, one of the low, degraded tribes of the Upper Amazon,[27] have no names for number except _nayhay_, 1, _nenacome_, 2, _feninichacome_, 3, _ononoeomere_, 4. In the extensive vocabularies given by Von Martins,[28] many similar examples are found. For the Bororos he gives only _couai_, 1, _maeouai_, 2, _ouai_, 3. The last word, with the proper finger pantomime, serves also for any higher number which falls within the grasp of their comprehension. The Guachi manage to reach 5, but their numeration is of the rudest kind, as the following scale shows: _tamak_, 1, _eu-echo,_ 2, _eu-echo-kailau,_ 3, _eu-echo-way,_ 4, _localau_, 5. The Carajas counted by a scale equally rude, and their conception of number seemed equally vague, until contact with the neighbouring tribes furnished them with the means of going beyond their original limit. Their scale shows clearly the uncertain, feeble number sense which is so marked in the interior of South America. It contains _wadewo_, 1, _wadebothoa_, 2, _wadeboaheodo_, 3, _wadebojeodo_, 4, _wadewajouclay_, 5, _wadewasori_, 6, or many. Turning to the languages of the extinct, or fast vanishing, tribes of Australia, we find a still more noteworthy absence of numeral expressions. In the Gudang dialect[29] but two numerals are found--_pirman_, 1, and _ilabiu_, 2; in the Weedookarry, _ekkamurda_, 1, and _kootera_, 2; and in the Queanbeyan, _midjemban_, 1, and _bollan_, 2. In a score or more of instances the numerals stop at 3. The natives of Keppel Bay count _webben_, 1, _booli_, 2, _koorel_, 3; of the Boyne River, _karroon_, 1, _boodla_, 2, _numma_, 3; of the Flinders River, _kooroin_, 1, _kurto_, 2, _kurto kooroin_, 3; at the mouth of the Norman River, _lum_, 1, _buggar_, 2, _orinch_, 3; the Eaw tribe, _koothea_, 1, _woother_, 2, _marronoo_, 3; the Moree, _mal_, 1, _boolar_, 2, _kooliba_, 3; the Port Essington,[30] _erad_, 1, _nargarick_, 2, _nargarickelerad_, 3; the Darnly Islanders,[31] _netat_, 1, _naes_, 2, _naesa netat_, 3; and so on through a long list of tribes whose numeral scales are equally scanty. A still larger number of tribes show an ability to count one step further, to 4; but beyond this limit the majority of Australian and Tasmanian tribes do not go. It seems most remarkable that any human being should possess the ability to count to 4, and not to 5. The number of fingers on one hand furnishes so obvious a limit to any of these rudimentary systems, that positive evidence is needed before one can accept the statement. A careful examination of the numerals in upwards of a hundred Australian dialects leaves no doubt, however, that such is the fact. The Australians in almost all cases count by pairs; and so pronounced is this tendency that they pay but little attention to the fingers. Some tribes do not appear ever to count beyond 2--a single pair. Many more go one step further; but if they do, they are as likely as not to designate their next numeral as two-one, or possibly, one-two. If this step is taken, we may or may not find one more added to it, thus completing the second pair. Still, the Australian's capacity for understanding anything which pertains to number is so painfully limited that even here there is sometimes an indefinite expression formed, as many, heap, or plenty, instead of any distinct numeral; and it is probably true that no Australian language contains a pure, simple numeral for 4. Curr, the best authority on this subject, believes that, where a distinct word for 4 is given, investigators have been deceived in every case.[32] If counting is carried beyond 4, it is always by means of reduplication. A few tribes gave expressions for 5, fewer still for 6, and a very small number appeared able to reach 7. Possibly the ability to count extended still further; but if so, it consisted undoubtedly in reckoning one pair after another, without any consciousness whatever of the sum total save as a larger number. The numerals of a few additional tribes will show clearly that all distinct perception of number is lost as soon as these races attempt to count above 3, or at most, 4. The Yuckaburra[33] natives can go no further than _wigsin_, 1, _bullaroo_, 2, _goolbora_, 3. Above here all is referred to as _moorgha_, many. The Marachowies[34] have but three distinct numerals,--_cooma_, 1, _cootera_, 2, _murra_, 3. For 4 they say _minna_, many. At Streaky Bay we find a similar list, with the same words, _kooma_ and _kootera_, for 1 and 2, but entirely different terms, _karboo_ and _yalkata_ for 3 and many. The same method obtains in the Minnal Yungar tribe, where the only numerals are _kain_, 1, _kujal_, 2, _moa_, 3, and _bulla_, plenty. In the Pinjarra dialect we find _doombart_, 1, _gugal_, 2, _murdine_, 3, _boola_, plenty; and in the dialect described as belonging to "Eyre's Sand Patch," three definite terms are given--_kean_, 1, _koojal_, 2, _yalgatta_, 3, while a fourth, _murna_, served to describe anything greater. In all these examples the fourth numeral is indefinite; and the same statement is true of many other Australian languages. But more commonly still we find 4, and perhaps 3 also, expressed by reduplication. In the Port Mackay dialect[35] the latter numeral is compound, the count being _warpur_, 1, _boolera_, 2, _boolera warpur_, 3. For 4 the term is not given. In the dialect which prevailed between the Albert and Tweed rivers[36] the scale appears as _yaburu_, 1, _boolaroo_, 2, _boolaroo yaburu_, 3, and _gurul_ for 4 or anything beyond. The Wiraduroi[37] have _numbai_, 1, _bula_, 2, _bula numbai_, 3, _bungu_, 4, or many, and _bungu galan_ or _bian galan_, 5, or very many. The Kamilaroi[38] scale is still more irregular, compounding above 4 with little apparent method. The numerals are _mal_, 1, _bular_, 2, _guliba_, 3, _bular bular_, 4, _bular guliba_, 5, _guliba guliba_, 6. The last two numerals show that 5 is to these natives simply 2-3, and 6 is 3-3. For additional examples of a similar nature the extended list of Australian scales given in Chapter V. may be consulted. Taken as a whole, the Australian and Tasmanian tribes seem to have been distinctly inferior to those of South America in their ability to use and to comprehend numerals. In all but two or three cases the Tasmanians[39] were found to be unable to proceed beyond 2; and as the foregoing examples have indicated, their Australian neighbours were but little better off. In one or two instances we do find Australian numeral scales which reach 10, and perhaps we may safely say 20. One of these is given in full in a subsequent chapter, and its structure gives rise to the suspicion that it was originally as limited as those of kindred tribes, and that it underwent a considerable development after the natives had come in contact with the Europeans. There is good reason to believe that no Australian in his wild state could ever count intelligently to 7.[40] In certain portions of Asia, Africa, Melanesia, Polynesia, and North America, are to be found races whose number systems are almost and sometimes quite as limited as are those of the South. American and Australian tribes already cited, but nowhere else do we find these so abundant as in the two continents just mentioned, where example after example might be cited of tribes whose ability to count is circumscribed within the narrowest limits. The Veddas[41] of Ceylon have but two numerals, _ekkame[=i]_, 1, _dekkameï_, 2. Beyond this they count _otameekaï, otameekaï, otameekaï_, etc.; _i.e._ "and one more, and one more, and one more," and so on indefinitely. The Andamans,[42] inhabitants of a group of islands in the Bay of Bengal, are equally limited in their power of counting. They have _ubatulda_, 1, and _ikporda_, 2; but they can go no further, except in a manner similar to that of the Veddas. Above two they proceed wholly by means of the fingers, saying as they tap the nose with each successive finger, _anka_, "and this." Only the more intelligent of the Andamans can count at all, many of them seeming to be as nearly destitute of the number sense as it is possible for a human being to be. The Bushmen[43] of South Africa have but two numerals, the pronunciation of which can hardly be indicated without other resources than those of the English alphabet. Their word for 3 means, simply, many, as in the case of some of the Australian tribes. The Watchandies[44] have but two simple numerals, and their entire number system is _cooteon_, 1, _utaura_, 2, _utarra cooteoo_, 3, _atarra utarra_, 4. Beyond this they can only say, _booltha_, many, and _booltha bat_, very many. Although they have the expressions here given for 3 and 4, they are reluctant to use them, and only do so when absolutely required. The natives of Lower California[45] cannot count above 5. A few of the more intelligent among them understand the meaning of 2 fives, but this number seems entirely beyond the comprehension of the ordinary native. The Comanches, curiously enough, are so reluctant to employ their number words that they appear to prefer finger pantomime instead, thus giving rise to the impression which at one time became current, that they had no numerals at all for ordinary counting. Aside from the specific examples already given, a considerable number of sweeping generalizations may be made, tending to show how rudimentary the number sense may be in aboriginal life. Scores of the native dialects of Australia and South America have been found containing number systems but little more extensive than those alluded to above. The negro tribes of Africa give the same testimony, as do many of the native races of Central America, Mexico, and the Pacific coast of the United States and Canada, the northern part of Siberia, Greenland, Labrador, and the arctic archipelago. In speaking of the Eskimos of Point Barrow, Murdoch[46] says: "It was not easy to obtain any accurate information about the numeral system of these people, since in ordinary conversation they are not in the habit of specifying any numbers above five." Counting is often carried higher than this among certain of these northern tribes, but, save for occasional examples, it is limited at best. Dr. Franz Boas, who has travelled extensively among the Eskimos, and whose observations are always of the most accurate nature, once told the author that he never met an Eskimo who could count above 15. Their numerals actually do extend much higher; and a stray numeral of Danish origin is now and then met with, showing that the more intelligent among them are able to comprehend numbers of much greater magnitude than this. But as Dr. Boas was engaged in active work among them for three years, we may conclude that the Eskimo has an arithmetic but little more extended than that which sufficed for the Australians and the forest tribes of Brazil. Early Russian explorers among the northern tribes of Siberia noticed the same difficulty in ordinary, every-day reckoning among the natives. At first thought we might, then, state it as a general law that those races which are lowest in the scale of civilization, have the feeblest number sense also; or in other words, the least possible power of grasping the abstract idea of number. But to this law there are many and important exceptions. The concurrent testimony of explorers seems to be that savage races possess, in the great majority of cases, the ability to count at least as high as 10. This limit is often extended to 20, and not infrequently to 100. Again, we find 1000 as the limit; or perhaps 10,000; and sometimes the savage carries his number system on into the hundreds of thousands or millions. Indeed, the high limit to which some savage races carry their numeration is far more worthy of remark than the entire absence of the number sense exhibited by others of apparently equal intelligence. If the life of any tribe is such as to induce trade and barter with their neighbours, a considerable quickness in reckoning will be developed among them. Otherwise this power will remain dormant because there is but little in the ordinary life of primitive man to call for its exercise. In giving 1, 2, 3, 5, 10, or any other small number as a system limit, it must not be overlooked that this limit mentioned is in all cases the limit of the spoken numerals at the savage's command. The actual ability to count is almost always, and one is tempted to say always, somewhat greater than their vocabularies would indicate. The Bushman has no number word that will express for him anything higher than 2; but with the assistance of his fingers he gropes his way on as far as 10. The Veddas, the Andamans, the Guachi, the Botocudos, the Eskimos, and the thousand and one other tribes which furnish such scanty numeral systems, almost all proceed with more or less readiness as far as their fingers will carry them. As a matter of fact, this limit is frequently extended to 20; the toes, the fingers of a second man, or a recount of the savage's own fingers, serving as a tale for the second 10. Allusion is again made to this in a later chapter, where the subject of counting on the fingers and toes is examined more in detail. In saying that a savage can count to 10, to 20, or to 100, but little idea is given of his real mental conception of any except the smallest numbers. Want of familiarity with the use of numbers, and lack of convenient means of comparison, must result in extreme indefiniteness of mental conception and almost entire absence of exactness. The experience of Captain Parry,[47] who found that the Eskimos made mistakes before they reached 7, and of Humboldt,[48] who says that a Chayma might be made to say that his age was either 18 or 60, has been duplicated by all investigators who have had actual experience among savage races. Nor, on the other hand, is the development of a numeral system an infallible index of mental power, or of any real approach toward civilization. A continued use of the trading and bargaining faculties must and does result in a familiarity with numbers sufficient to enable savages to perform unexpected feats in reckoning. Among some of the West African tribes this has actually been found to be the case; and among the Yorubas of Abeokuta[49] the extraordinary saying, "You may seem very clever, but you can't tell nine times nine," shows how surprisingly this faculty has been developed, considering the general condition of savagery in which the tribe lived. There can be no doubt that, in general, the growth of the number sense keeps pace with the growth of the intelligence in other respects. But when it is remembered that the Tonga Islanders have numerals up to 100,000, and the Tembus, the Fingoes, the Pondos, and a dozen other South African tribes go as high as 1,000,000; and that Leigh Hunt never could learn the multiplication table, one must confess that this law occasionally presents to our consideration remarkable exceptions. While considering the extent of the savage's arithmetical knowledge, of his ability to count and to grasp the meaning of number, it may not be amiss to ask ourselves the question, what is the extent of the development of our own number sense? To what limit can we absorb the idea of number, with a complete appreciation of the idea of the number of units involved in any written or spoken quantity? Our perfect system of numeration enables us to express without difficulty any desired number, no matter how great or how small it be. But how much of actually clear comprehension does the number thus expressed convey to the mind? We say that one place is 100 miles from another; that A paid B 1000 dollars for a certain piece of property; that a given city contains 10,000 inhabitants; that 100,000 bushels of wheat were shipped from Duluth or Odessa on such a day; that 1,000,000 feet of lumber were destroyed by the fire of yesterday,--and as we pass from the smallest to the largest of the numbers thus instanced, and from the largest on to those still larger, we repeat the question just asked; and we repeat it with a new sense of our own mental limitation. The number 100 unquestionably stands for a distinct conception. Perhaps the same may be said for 1000, though this could not be postulated with equal certainty. But what of 10,000? If that number of persons were gathered together into a single hall or amphitheatre, could an estimate be made by the average onlooker which would approximate with any degree of accuracy the size of the assembly? Or if an observer were stationed at a certain point, and 10,000 persons were to pass him in single file without his counting them as they passed, what sort of an estimate would he make of their number? The truth seems to be that our mental conception of number is much more limited than is commonly thought, and that we unconsciously adopt some new unit as a standard of comparison when we wish to render intelligible to our minds any number of considerable magnitude. For example, we say that A has a fortune of $1,000,000. The impression is at once conveyed of a considerable degree of wealth, but it is rather from the fact that that fortune represents an annual income of $40,000 than, from the actual magnitude of the fortune itself. The number 1,000,000 is, in itself, so greatly in excess of anything that enters into our daily experience that we have but a vague conception of it, except as something very great. We are not, after all, so very much better off than the child who, with his arms about his mother's neck, informs her with perfect gravity and sincerity that he "loves her a million bushels." His idea is merely of some very great amount, and our own is often but little clearer when we use the expressions which are so easily represented by a few digits. Among the uneducated portions of civilized communities the limit of clear comprehension of number is not only relatively, but absolutely, very low. Travellers in Russia have informed the writer that the peasants of that country have no distinct idea of a number consisting of but a few hundred even. There is no reason to doubt this testimony. The entire life of a peasant might be passed without his ever having occasion to use a number as great as 500, and as a result he might have respecting that number an idea less distinct than a trained mathematician would have of the distance from the earth to the sun. De Quincey[50] incidentally mentions this characteristic in narrating a conversation which occurred while he was at Carnarvon, a little town in Wales. "It was on this occasion," he says, "that I learned how vague are the ideas of number in unpractised minds. 'What number of people do you think,' I said to an elderly person, 'will be assembled this day at Carnarvon?' 'What number?' rejoined the person addressed; 'what number? Well, really, now, I should reckon--perhaps a matter of four million.' Four millions of _extra_ people in little Carnarvon, that could barely find accommodation (I should calculate) for an extra four hundred!" So the Eskimo and the South American Indian are, after all, not so very far behind the "elderly person" of Carnarvon, in the distinct perception of a number which familiarity renders to us absurdly small. CHAPTER III. THE ORIGIN OF NUMBER WORDS. In the comparison of languages and the search for primitive root forms, no class of expressions has been subjected to closer scrutiny than the little cluster of words, found in each language, which constitutes a part of the daily vocabulary of almost every human being--the words with which we begin our counting. It is assumed, and with good reason, that these are among the earlier words to appear in any language; and in the mutations of human speech, they are found to suffer less than almost any other portion of a language. Kinship between tongues remote from each other has in many instances been detected by the similarity found to exist among the every-day words of each; and among these words one may look with a good degree of certainty for the 1, 2, 3, etc., of the number scale. So fruitful has been this line of research, that the attempt has been made, even, to establish a common origin for all the races of mankind by means of a comparison of numeral words.[51] But in this instance, as in so many others that will readily occur to the mind, the result has been that the theory has finally taken possession of the author and reduced him to complete subjugation, instead of remaining his servant and submitting to the legitimate results of patient and careful investigation. Linguistic research is so full of snares and pitfalls that the student must needs employ the greatest degree of discrimination before asserting kinship of race because of resemblances in vocabulary; or even relationship between words in the same language because of some chance likeness of form that may exist between them. Probably no one would argue that the English and the Babusessé of Central Africa were of the same primitive stock simply because in the language of the latter _five atano_ means 5, and _ten kumi_ means 10.[52] But, on the other hand, many will argue that, because the German _zehn_ means 10, and _zehen_ means toes, the ancestors of the Germans counted on their toes; and that with them, 10 was the complete count of the toes. It may be so. We certainly have no evidence with which to disprove this; but, before accepting it as a fact, or even as a reasonable hypothesis, we may be pardoned for demanding some evidence aside from the mere resemblance in the form of the words. If, in the study of numeral words, form is to constitute our chief guide, we must expect now and then to be confronted with facts which are not easily reconciled with any pet theory. The scope of the present work will admit of no more than a hasty examination of numeral forms, in which only actual and well ascertained meanings will be considered. But here we are at the outset confronted with a class of words whose original meanings appear to be entirely lost. They are what may be termed the numerals proper--the native, uncompounded words used to signify number. Such words are the one, two, three, etc., of English; the eins, zwei, drei, etc., of German; words which must at some time, in some prehistoric language, have had definite meanings entirely apart from those which they now convey to our minds. In savage languages it is sometimes possible to detect these meanings, and thus to obtain possession of the clue that leads to the development, in the barbarian's rude mind, of a count scale--a number system. But in languages like those of modern Europe, the pedigree claimed by numerals is so long that, in the successive changes through which they have passed, all trace of their origin seems to have been lost. The actual number of such words is, however, surprisingly small in any language. In English we count by simple words only to 10. From this point onward all our numerals except "hundred" and "thousand" are compounds and combinations of the names of smaller numbers. The words we employ to designate the higher orders of units, as million, billion, trillion, etc., are appropriated bodily from the Italian; and the native words _pair_, _tale_, _brace_, _dozen_, _gross_, and _score_, can hardly be classed as numerals in the strict sense of the word. German possesses exactly the same number of native words in its numeral scale as English; and the same may be said of the Teutonic languages generally, as well as of the Celtic, the Latin, the Slavonic, and the Basque. This is, in fact, the universal method observed in the formation of any numeral scale, though the actual number of simple words may vary. The Chiquito language has but one numeral of any kind whatever; English contains twelve simple terms; Sanskrit has twenty-seven, while Japanese possesses twenty-four, and the Chinese a number almost equally great. Very many languages, as might be expected, contain special numeral expressions, such as the German _dutzend_ and the French _dizaine_; but these, like the English _dozen_ and _score_, are not to be regarded as numerals proper. The formation of numeral words shows at a glance the general method in which any number scale has been built up. The primitive savage counts on his fingers until he has reached the end of one, or more probably of both, hands. Then, if he wishes to proceed farther, some mark is made, a pebble is laid aside, a knot tied, or some similar device employed to signify that all the counters at his disposal have been used. Then the count begins anew, and to avoid multiplication of words, as well as to assist the memory, the terms already used are again resorted to; and the name by which the first halting-place was designated is repeated with each new numeral. Hence the thirteen, fourteen, fifteen, etc., which are contractions of the fuller expressions three-and-ten, four-and-ten, five-and-ten, etc. The specific method of combination may not always be the same, as witness the _eighteen_, or eight-ten, in English, and _dix-huit,_ or ten-eight, in French; _forty-five_, or four-tens-five, in English, and _fünf und vierzig_, or five and four tens in German. But the general method is the same the world over, presenting us with nothing but local variations, which are, relatively speaking, entirely unimportant. With this fact in mind, we can cease to wonder at the small number of simple numerals in any language. It might, indeed, be queried, why do any languages, English and German, for example, have unusual compounds for 11 and 12? It would seem as though the regular method of compounding should begin with 10 and 1, instead of 10 and 3, in any language using a system with 10 as a base. An examination of several hundred numeral scales shows that the Teutonic languages are somewhat exceptional in this respect. The words _eleven_ and _twelve_ are undoubtedly combinations, but not in the same direct sense as _thirteen_, _twenty-five_, etc. The same may be said of the French _onze_, _douze_, _treize_, _quatorze_, _quinze_, and _seize_, which are obvious compounds, but not formed in the same manner as the numerals above that point. Almost all civilized languages, however, except the Teutonic, and practically all uncivilized languages, begin their direct numeral combinations as soon as they have passed their number base, whatever that may be. To give an illustration, selected quite at random from among the barbarous tribes of Africa, the Ki-Swahili numeral scale runs as follows:[53] 1. moyyi, 2. mbiri, 3. tato, 4. ena, 5. tano, 6. seta, 7. saba, 8. nani, 9. kenda, 10. kumi, 11. kumi na moyyi, 12. kumi na mbiri, 13. kumi na tato, etc. The words for 11, 12, and 13, are seen at a glance to signify ten-and-one, ten-and-two, ten-and-three, and the count proceeds, as might be inferred, in a similar manner as far as the number system extends. Our English combinations are a little closer than these, and the combinations found in certain other languages are, in turn, closer than those of the English; as witness the _once_, 11, _doce_, 12, _trece_, 13, etc., of Spanish. But the process is essentially the same, and the law may be accepted as practically invariable, that all numerals greater than the base of a system are expressed by compound words, except such as are necessary to establish some new order of unit, as hundred or thousand. In the scale just given, it will be noticed that the larger number precedes the smaller, giving 10 + 1, 10 + 2, etc., instead of 1 + 10, 2 + 10, etc. This seems entirely natural, and hardly calls for any comment whatever. But we have only to consider the formation of our English "teens" to see that our own method is, at its inception, just the reverse of this. Thirteen, 14, and the remaining numerals up to 19 are formed by prefixing the smaller number to the base; and it is only when we pass 20 that we return to the more direct and obvious method of giving precedence to the larger. In German and other Teutonic languages the inverse method is continued still further. Here 25 is _fünf und zwanzig_, 5 and 20; 92 is _zwei und neunzig_, 2 and 90, and so on to 99. Above 100 the order is made direct, as in English. Of course, this mode of formation between 20 and 100 is permissible in English, where "five and twenty" is just as correct a form as twenty-five. But it is archaic, and would soon pass out of the language altogether, were it not for the influence of some of the older writings which have had a strong influence in preserving for us many of older and more essentially Saxon forms of expression. Both the methods described above are found in all parts of the world, but what I have called the direct is far more common than the other. In general, where the smaller number precedes the larger it signifies multiplication instead of addition. Thus, when we say "thirty," _i.e._ three-ten, we mean 3 × 10; just as "three hundred" means 3 × 100. When the larger precedes the smaller, we must usually understand addition. But to both these rules there are very many exceptions. Among higher numbers the inverse order is very rarely used; though even here an occasional exception is found. The Taensa Indians, for example, place the smaller numbers before the larger, no matter how far their scale may extend. To say 1881 they make a complete inversion of our own order, beginning with 1 and ending with 1000. Their full numeral for this is _yeha av wabki mar-u-wab mar-u-haki_, which means, literally, 1 + 80 + 100 × 8 + 100 × 10.[54] Such exceptions are, however, quite rare. One other method of combination, that of subtraction, remains to be considered. Every student of Latin will recall at once the _duodeviginti_, 2 from 20, and _undeviginti_, 1 from 20, which in that language are the regular forms of expression for 18 and 19. At first they seem decidedly odd; but familiarity soon accustoms one to them, and they cease entirely to attract any special attention. This principle of subtraction, which, in the formation of numeral words, is quite foreign to the genius of English, is still of such common occurrence in other languages that the Latin examples just given cease to be solitary instances. The origin of numerals of this class is to be found in the idea of reference, not necessarily to the last, but to the nearest, halting-point in the scale. Many tribes seem to regard 9 as "almost 10," and to give it a name which conveys this thought. In the Mississaga, one of the numerous Algonquin languages, we have, for example, the word _cangaswi_, "incomplete 10," for 9.[55] In the Kwakiutl of British Columbia, 8 as well as 9 is formed in this way; these two numbers being _matlguanatl_, 10 - 2, and _nanema_, 10 - 1, respectively.[56] In many of the languages of British Columbia we find a similar formation for 8 and 9, or for 9 alone. The same formation occurs in Malay, resulting in the numerals _delapan_, 10 - 2, and _sambilan_ 10 - 1.[57] In Green Island, one of the New Ireland group, these become simply _andra-lua_, "less 2," and _andra-si_, "less 1."[58] In the Admiralty Islands this formation is carried back one step further, and not only gives us _shua-luea_, "less 2," and _shu-ri_, "less 1," but also makes 7 appear as _sua-tolu_, "less 3."[59] Surprising as this numeral is, it is more than matched by the Ainu scale, which carries subtraction back still another step, and calls 6, 10 - 4. The four numerals from 6 to 9 in this scale are respectively, _iwa_, 10 - 4, _arawa_, 10 - 3, _tupe-san_, 10 - 2, and _sinepe-san_, 10 - 1.[60] Numerous examples of this kind of formation will be found in later chapters of this work; but they will usually be found to occur in one or both of the numerals, 8 and 9. Occasionally they appear among the higher numbers; as in the Maya languages, where, for example, 99 years is "one single year lacking from five score years,"[61] and in the Arikara dialects, where 98 and 99 are "5 men minus" and "5 men 1 not."[62] The Welsh, Danish, and other languages less easily accessible than these to the general student, also furnish interesting examples of a similar character. More rarely yet are instances met with of languages which make use of subtraction almost as freely as addition, in the composition of numerals. Within the past few years such an instance has been noticed in the case of the Bellacoola language of British Columbia. In their numeral scale 15, "one foot," is followed by 16, "one man less 4"; 17, "one man less 3"; 18, "one man less 2"; 19, "one man less 1"; and 20, one man. Twenty-five is "one man and one hand"; 26, "one man and two hands less 4"; 36, "two men less 4"; and so on. This method of formation prevails throughout the entire numeral scale.[63] One of the best known and most interesting examples of subtraction as a well-defined principle of formation is found in the Maya scale. Up to 40 no special peculiarity appears; but as the count progresses beyond that point we find a succession of numerals which one is almost tempted to call 60 - 19, 60 - 18, 60 - 17, etc. Literally translated the meanings seem to be 1 to 60, 2 to 60, 3 to 60, etc. The point of reference is 60, and the thought underlying the words may probably be expressed by the paraphrases, "1 on the third score, 2 on the third score, 3 on the third score," etc. Similarly, 61 is 1 on the fourth score, 81 is one on the fifth score, 381 is 1 on the nineteenth score, and so on to 400. At 441 the same formation reappears; and it continues to characterize the system in a regular and consistent manner, no matter how far it is extended.[64] The Yoruba language of Africa is another example of most lavish use of subtraction; but it here results in a system much less consistent and natural than that just considered. Here we find not only 5, 10, and 20 subtracted from the next higher unit, but also 40, and even 100. For example, 360 is 400 - 40; 460 is 500 - 40; 500 is 600 - 100; 1300 is 1400 - 100, etc. One of the Yoruba units is 200; and all the odd hundreds up to 2000, the next higher unit, are formed by subtracting 100 from the next higher multiple of 200. The system is quite complex, and very artificial; and seems to have been developed by intercourse with traders.[65] It has already been stated that the primitive meanings of our own simple numerals have been lost. This is also true of the languages of nearly all other civilized peoples, and of numerous savage races as well. We are at liberty to suppose, and we do suppose, that in very many cases these words once expressed meanings closely connected with the names of the fingers, or with the fingers themselves, or both. Now and then a case is met with in which the numeral word frankly avows its meaning--as in the Botocudo language, where 1 is expressed by _podzik_, finger, and 2 by _kripo_, double finger;[66] and in the Eskimo dialect of Hudson's Bay, where _eerkitkoka_ means both 10 and little finger.[67] Such cases are, however, somewhat exceptional. In a few noteworthy instances, the words composing the numeral scale of a language have been carefully investigated and their original meanings accurately determined. The simple structure of many of the rude languages of the world should render this possible in a multitude of cases; but investigators are too often content with the mere numerals themselves, and make no inquiry respecting their meanings. But the following exposition of the Zuñi scale, given by Lieutenant Gushing[68] leaves nothing to be desired: 1. töpinte = taken to start with. 2. kwilli = put down together with. 3. ha'[=i] = the equally dividing finger. 4. awite = all the fingers all but done with. 5. öpte = the notched off. This finishes the list of original simple numerals, the Zuñi stopping, or "notching off," when he finishes the fingers of one hand. Compounding now begins. 6. topalïk'ya = another brought to add to the done with. 7. kwillilïk'ya = two brought to and held up with the rest. 8. hailïk'ye = three brought to and held up with the rest. 9. tenalïk'ya = all but all are held up with the rest. 10. ästem'thila = all the fingers. 11. ästem'thla topayä'thl'tona = all the fingers and another over above held. The process of formation indicated in 11 is used in the succeeding numerals up to 19. 20. kwillik'yënästem'thlan = two times all the fingers. 100. ässiästem'thlak'ya = the fingers all the fingers. 1000. ässiästem'thlanak'yënästem'thla = the fingers all the fingers times all the fingers. The only numerals calling for any special note are those for 11 and 9. For 9 we should naturally expect a word corresponding in structure and meaning to the words for 7 and 8. But instead of the "four brought to and held up with the rest," for which we naturally look, the Zuñi, to show that he has used all of his fingers but one, says "all but all are held up with the rest." To express 11 he cannot use a similar form of composition, since he has already used it in constructing his word for 6, so he says "all the fingers and another over above held." The one remarkable point to be noted about the Zuñi scale is, after all, the formation of the words for 1 and 2. While the savage almost always counts on his fingers, it does not seem at all certain that these words would necessarily be of finger formation. The savage can always distinguish between one object and two objects, and it is hardly reasonable to believe that any external aid is needed to arrive at a distinct perception of this difference. The numerals for 1 and 2 would be the earliest to be formed in any language, and in most, if not all, cases they would be formed long before the need would be felt for terms to describe any higher number. If this theory be correct, we should expect to find finger names for numerals beginning not lower than 3, and oftener with 5 than with any other number. The highest authority has ventured the assertion that all numeral words have their origin in the names of the fingers;[69] substantially the same conclusion was reached by Professor Pott, of Halle, whose work on numeral nomenclature led him deeply into the study of the origin of these words. But we have abundant evidence at hand to show that, universal as finger counting has been, finger origin for numeral words has by no means been universal. That it is more frequently met with than any other origin is unquestionably true; but in many instances, which will be more fully considered in the following chapter, we find strictly non-digital derivations, especially in the case of the lowest members of the scale. But in nearly all languages the origin of the words for 1, 2, 3, and 4 are so entirely unknown that speculation respecting them is almost useless. An excellent illustration of the ordinary method of formation which obtains among number scales is furnished by the Eskimos of Point Barrow,[70] who have pure numeral words up to 5, and then begin a systematic course of word formation from the names of their fingers. If the names of the first five numerals are of finger origin, they have so completely lost their original form, or else the names of the fingers themselves have so changed, that no resemblance is now to be detected between them. This scale is so interesting that it is given with considerable fulness, as follows: 1. atauzik. 2. madro. 3. pinasun. 4. sisaman. 5. tudlemut. 6. atautyimin akbinigin [tudlimu(t)] = 5 and 1 on the next. 7. madronin akbinigin = twice on the next. 8. pinasunin akbinigin = three times on the next. 9. kodlinotaila = that which has not its 10. 10. kodlin = the upper part--_i.e._ the fingers. 14. akimiaxotaityuna = I have not 15. 15. akimia. [This seems to be a real numeral word.] 20. inyuina = a man come to an end. 25. inyuina tudlimunin akbinidigin = a man come to an end and 5 on the next. 30. inyuina kodlinin akbinidigin = a man come to an end and 10 on the next. 35. inyuina akimiamin aipalin = a man come to an end accompanied by 1 fifteen times. 40. madro inyuina = 2 men come to an end. In this scale we find the finger origin appearing so clearly and so repeatedly that one feels some degree of surprise at finding 5 expressed by a pure numeral instead of by some word meaning _hand_ or _fingers of one hand_. In this respect the Eskimo dialects are somewhat exceptional among scales built up of digital words. The system of the Greenland Eskimos, though differing slightly from that of their Point Barrow cousins, shows the same peculiarity. The first ten numerals of this scale are:[71] 1. atausek. 2. mardluk. 3. pingasut. 4. sisamat. 5. tatdlimat. 6. arfinek-atausek = to the other hand 1. 7. arfinek-mardluk = to the other hand 2. 8. arfinek-pingasut = to the other hand 3. 9. arfinek-sisamat = to the other hand 4. 10. kulit. The same process is now repeated, only the feet instead of the hands are used; and the completion of the second 10 is marked by the word _innuk_, man. It may be that the Eskimo word for 5 is, originally, a digital word, but if so, the fact has not yet been detected. From the analogy furnished by other languages we are justified in suspecting that this may be the case; for whenever a number system contains digital words, we expect them to begin with _five_, as, for example, in the Arawak scale,[72] which runs: 1. abba. 2. biama. 3. kabbuhin. 4. bibiti. 5. abbatekkábe = 1 hand. 6. abbatiman = 1 of the other. 7. biamattiman = 2 of the other. 8. kabbuhintiman = 3 of the other. 9. bibitiman = 4 of the other. 10. biamantekábbe = 2 hands. 11. abba kutihibena = 1 from the feet. 20. abba lukku = hands feet. The four sets of numerals just given may be regarded as typifying one of the most common forms of primitive counting; and the words they contain serve as illustrations of the means which go to make up the number scales of savage races. Frequently the finger and toe origin of numerals is perfectly apparent, as in the Arawak system just given, which exhibits the simplest and clearest possible method of formation. Another even more interesting system is that of the Montagnais of northern Canada.[73] Here, as in the Zuñi scale, the words are digital from the outset. 1. inl'are = the end is bent. 2. nak'e = another is bent. 3. t'are = the middle is bent. 4. dinri = there are no more except this. 5. se-sunla-re = the row on the hand. 6. elkke-t'are = 3 from each side. 7.{ t'a-ye-oyertan = there are still 3 of them. { inl'as dinri = on one side there are 4 of them. 8. elkke-dinri = 4 on each side. 9. inl'a-ye-oyert'an = there is still 1 more. 10. onernan = finished on each side. 11. onernan inl'are ttcharidhel = 1 complete and 1. 12. onernan nak'e ttcharidhel = 1 complete and 2, etc. The formation of 6, 7, and 8 of this scale is somewhat different from that ordinarily found. To express 6, the Montagnais separates the thumb and forefinger from the three remaining fingers of the left hand, and bringing the thumb of the right hand close to them, says: "3 from each side." For 7 he either subtracts from 10, saying: "there are still 3 of them," or he brings the thumb and forefinger of the right hand up to the thumb of the left, and says: "on one side there are 4 of them." He calls 8 by the same name as many of the other Canadian tribes, that is, two 4's; and to show the proper number of fingers, he closes the thumb and little finger of the right hand, and then puts the three remaining fingers beside the thumb of the left hand. This method is, in some of these particulars, different from any other I have ever examined. It often happens that the composition of numeral words is less easily understood, and the original meanings more difficult to recover, than in the examples already given. But in searching for number systems which show in the formation of their words the influence of finger counting, it is not unusual to find those in which the derivation from native words signifying _finger, hand, toe, foot_, and _man_, is just as frankly obvious as in the case of the Zuñi, the Arawak, the Eskimo, or the Montagnais scale. Among the Tamanacs,[74] one of the numerous Indian tribes of the Orinoco, the numerals are as strictly digital as in any of the systems already examined. The general structure of the Tamanac scale is shown by the following numerals: 5. amgnaitone = 1 hand complete. 6. itacono amgna pona tevinitpe = 1 on the other hand. 10. amgna aceponare = all of the 2 hands. 11. puitta pona tevinitpe = 1 on the foot. 16. itacono puitta pona tevinitpe = 1 on the other foot. 20. tevin itoto = 1 man. 21. itacono itoto jamgnar bona tevinitpe = 1 on the hands of another man. In the Guarani[75] language of Paraguay the same method is found, with a different form of expression for 20. Here the numerals in question are 5. asepopetei = one hand. 10. asepomokoi = two hands. 20. asepo asepi abe = hands and feet. Another slight variation is furnished by the Kiriri language,[76] which is also one of the numerous South American Indian forms of speech, where we find the words to be 5. mi biche misa = one hand. 10. mikriba misa sai = both hands. 20. mikriba misa idecho ibi sai = both hands together with the feet. Illustrations of this kind might be multiplied almost indefinitely; and it is well to note that they may be drawn from all parts of the world. South America is peculiarly rich in native numeral words of this kind; and, as the examples above cited show, it is the field to which one instinctively turns when this subject is under discussion. The Zamuco numerals are, among others, exceedingly interesting, giving us still a new variation in method. They are[77] 1. tsomara. 2. gar. 3. gadiok. 4. gahagani. 5. tsuena yimana-ite = ended 1 hand. 6. tsomara-hi = 1 on the other. 7. gari-hi = 2 on the other. 8. gadiog-ihi = 3 on the other. 9. gahagani-hi = 4 on the other. 10. tsuena yimana-die = ended both hands. 11. tsomara yiri-tie = 1 on the foot. 12. gar yiritie = 2 on the foot. 20. tsuena yiri-die = ended both feet. As is here indicated, the form of progression from 5 to 10, which we should expect to be "hand-1," or "hand-and-1," or some kindred expression, signifying that one hand had been completed, is simply "1 on the other." Again, the expressions for 11, 12, etc., are merely "1 on the foot," "2 on the foot," etc., while 20 is "both feet ended." An equally interesting scale is furnished by the language of the Maipures[78] of the Orinoco, who count 1. papita. 2. avanume. 3. apekiva. 4. apekipaki. 5. papitaerri capiti = 1 only hand. 6. papita yana pauria capiti purena = 1 of the other hand we take. 10. apanumerri capiti = 2 hands. 11. papita yana kiti purena = 1 of the toes we take. 20. papita camonee = 1 man. 40. avanume camonee = 2 men. 60. apekiva camonee = 3 men, etc. In all the examples thus far given, 20 is expressed either by the equivalent of "man" or by some formula introducing the word "feet." Both these modes of expressing what our own ancestors termed a "score," are so common that one hesitates to say which is of the more frequent use. The following scale, from one of the Betoya dialects[79] of South America, is quite remarkable among digital scales, making no use of either "man" or "foot," but reckoning solely by fives, or hands, as the numerals indicate. 1. tey. 2. cayapa. 3. toazumba. 4. cajezea = 2 with plural termination. 5. teente = hand. 6. teyentetey = hand + 1. 7. teyente cayapa = hand + 2. 8. teyente toazumba = hand + 3. 9. teyente caesea = hand + 4. 10. caya ente, or caya huena = 2 hands. 11. caya ente-tey = 2 hands + 1. 15. toazumba-ente = 3 hands. 16. toazumba-ente-tey = 3 hands + 1. 20. caesea ente = 4 hands. In the last chapter mention was made of the scanty numeral systems of the Australian tribes, but a single scale was alluded to as reaching the comparatively high limit of 20. This system is that belonging to the Pikumbuls,[80] and the count runs thus: 1. mal. 2. bular. 3. guliba. 4. bularbular = 2-2. 5. mulanbu. 6. malmulanbu mummi = 1 and 5 added on. 7. bularmulanbu mummi = 2 and 5 added on. 8. gulibamulanbu mummi = 3 and 5 added on. 9. bularbularmulanbu mummi = 4 and 5 added on. 10. bularin murra = belonging to the 2 hands. 11. maldinna mummi = 1 of the toes added on (to the 10 fingers). 12. bular dinna mummi = 2 of the toes added on. 13. guliba dinna mummi = 3 of the toes added on. 14. bular bular dinna mummi = 4 of the toes added on. 15. mulanba dinna = 5 of the toes added on. 16. mal dinna mulanbu = 1 and 5 toes. 17. bular dinna mulanbu = 2 and 5 toes. 18. guliba dinna mulanbu = 3 and 5 toes. 19. bular bular dinna mulanbu = 4 and 5 toes. 20. bularin dinna = belonging to the 2 feet. As has already been stated, there is good ground for believing that this system was originally as limited as those obtained from other Australian tribes, and that its extension from 4, or perhaps from 5 onward, is of comparatively recent date. A somewhat peculiar numeral nomenclature is found in the language of the Klamath Indians of Oregon. The first ten words in the Klamath scale are:[81] 1. nash, or nas. 2. lap = hand. 3. ndan. 4. vunep = hand up. 5. tunep = hand away. 6. nadshkshapta = 1 I have bent over. 7. lapkshapta = 2 I have bent over. 8. ndankshapta = 3 I have bent over. 9. nadshskeksh = 1 left over. 10. taunep = hand hand? In describing this system Mr. Gatschet says: "If the origin of the Klamath numerals is thus correctly traced, their inventors must have counted only the four long fingers without the thumb, and 5 was counted while saying _hand away! hand off!_ The 'four,' or _hand high! hand up!_ intimates that the hand was held up high after counting its four digits; and some term expressing this gesture was, in the case of _nine_, substituted by 'one left over' ... which means to say, 'only one is left until all the fingers are counted.'" It will be observed that the Klamath introduces not only the ordinary finger manipulation, but a gesture of the entire hand as well. It is a common thing to find something of the kind to indicate the completion of 5 or 10, and in one or two instances it has already been alluded to. Sometimes one or both of the closed fists are held up; sometimes the open hand, with all the fingers extended, is used; and sometimes an entirely independent gesture is introduced. These are, in general, of no special importance; but one custom in vogue among some of the prairie tribes of Indians, to which my attention was called by Dr. J. Owen Dorsey,[82] should be mentioned. It is a gesture which signifies multiplication, and is performed by throwing the hand to the left. Thus, after counting 5, a wave of the hand to the left means 50. As multiplication is rather unusual among savage tribes, this is noteworthy, and would seem to indicate on the part of the Indian a higher degree of intelligence than is ordinarily possessed by uncivilized races. In the numeral scale as we possess it in English, we find it necessary to retain the name of the last unit of each kind used, in order to describe definitely any numeral employed. Thus, fifteen, one hundred forty-two, six thousand seven hundred twenty-seven, give in full detail the numbers they are intended to describe. In primitive scales this is not always considered necessary; thus, the Zamucos express their teens without using their word for 10 at all. They say simply, 1 on the foot, 2 on the foot, etc. Corresponding abbreviations are often met; so often, indeed, that no further mention of them is needed. They mark one extreme, the extreme of brevity, found in the savage method of building up hand, foot, and finger names for numerals; while the Zuñi scale marks the extreme of prolixity in the formation of such words. A somewhat ruder composition than any yet noticed is shown in the numerals of the Vilelo scale,[83] which are: 1. agit, or yaagit. 2. uke. 3. nipetuei. 4. yepkatalet. 5. isig-nisle-yaagit = hand fingers 1. 6. isig-teet-yaagit = hand with 1. 7. isig-teet-uke = hand with 2. 8. isig-teet-nipetuei = hand with 3. 9. isig-teet-yepkatalet = hand with 4. 10. isig-uke-nisle = second hand fingers (lit. hand-two-fingers). 11. isig-uke-nisle-teet-yaagit = second hand fingers with 1. 20. isig-ape-nisle-lauel = hand foot fingers all. In the examples thus far given, it will be noticed that the actual names of individual fingers do not appear. In general, such words as thumb, forefinger, little finger, are not found, but rather the hand-1, 1 on the next, or 1 over and above, which we have already seen, are the type forms for which we are to look. Individual finger names do occur, however, as in the scale of the Hudson's Bay Eskimos,[84] where the three following words are used both as numerals and as finger names: 8. kittukleemoot = middle finger. 9. mikkeelukkamoot = fourth finger. 10. eerkitkoka = little finger. Words of similar origin are found in the original Jiviro scale,[85] where the native numerals are: 1. ala. 2. catu. 3. cala. 4. encatu. 5. alacötegladu = 1 hand. 6. intimutu = thumb (of second hand). 7. tannituna = index finger. 8. tannituna cabiasu = the finger next the index finger. 9. bitin ötegla cabiasu = hand next to complete. 10. catögladu = 2 hands. As if to emphasize the rarity of this method of forming numerals, the Jiviros afterward discarded the last five of the above scale, replacing them by words borrowed from the Quichuas, or ancient Peruvians. The same process may have been followed by other tribes, and in this way numerals which were originally digital may have disappeared. But we have no evidence that this has ever happened in any extensive manner. We are, rather, impelled to accept the occasional numerals of this class as exceptions to the general rule, until we have at our disposal further evidence of an exact and critical nature, which would cause us to modify this opinion. An elaborate philological study by Dr. J.H. Trumbull[86] of the numerals used by many of the North American Indian tribes reveals the presence in the languages of these tribes of a few, but only a few, finger names which are used without change as numeral expressions also. Sometimes the finger gives a name not its own to the numeral with which it is associated in counting--as in the Chippeway dialect, which has _nawi-nindj_, middle of the hand, and _nisswi_, 3; and the Cheyenne, where _notoyos_, middle finger, and _na-nohhtu_, 8, are closely related. In other parts of the world isolated examples of the transference of finger names to numerals are also found. Of these a well-known example is furnished by the Zulu numerals, where "_tatisitupa_, taking the thumb, becomes a numeral for six. Then the verb _komba_, to point, indicating the forefinger, or 'pointer,' makes the next numeral, seven. Thus, answering the question, 'How much did your master give you?' a Zulu would say, '_U kombile_,' 'He pointed with his forefinger,' _i.e._ 'He gave me seven'; and this curious way of using the numeral verb is also shown in such an example as '_amahasi akombile_,' 'the horses have pointed,' _i.e._ 'there were seven of them.' In like manner, _Kijangalobili_, 'keep back two fingers,' _i.e._ eight, and _Kijangalolunje_, 'keep back one finger,' _i.e._ nine, lead on to _kumi_, ten."[87] Returning for a moment to the consideration of number systems in the formation of which the influence of the hand has been paramount, we find still further variations of the method already noticed of constructing names for the fives, tens, and twenties, as well as for the intermediate numbers. Instead of the simple words "hand," "foot," etc., we not infrequently meet with some paraphrase for one or for all these terms, the derivation of which is unmistakable. The Nengones,[88] an island tribe of the Indian Ocean, though using the word "man" for 20, do not employ explicit hand or foot words, but count 1. sa. 2. rewe. 3. tini. 4. etse. 5. se dono = the end (of the first hand). 6. dono ne sa = end and 1. 7. dono ne rewe = end and 2. 8. dono ne tini = end and 3. 9. dono ne etse = end and 4. 10. rewe tubenine = 2 series (of fingers). 11. rewe tubenine ne sa re tsemene = 2 series and 1 on the next? 20. sa re nome = 1 man. 30. sa re nome ne rewe tubenine = 1 man and 2 series. 40. rewe ne nome = 2 men. Examples like the above are not infrequent. The Aztecs used for 10 the word _matlactli_, hand-half, _i.e._ the hand half of a man, and for 20 _cempoalli_, one counting.[89] The Point Barrow Eskimos call 10 _kodlin_, the upper part, _i.e._ of a man. One of the Ewe dialects of Western Africa[90] has _ewo_, done, for 10; while, curiously enough, 9, _asieke_, is a digital word, meaning "to part (from) the hand." In numerous instances also some characteristic word not of hand derivation is found, like the Yoruba _ogodzi_, string, which becomes a numeral for 40, because 40 cowries made a "string"; and the Maori _tekau_, bunch, which signifies 10. The origin of this seems to have been the custom of counting yams and fish by "bunches" of ten each.[91] Another method of forming numeral words above 5 or 10 is found in the presence of such expressions as second 1, second 2, etc. In languages of rude construction and incomplete development the simple numeral scale is often found to end with 5, and all succeeding numerals to be formed from the first 5. The progression from that point may be 5-1, 5-2, etc., as in the numerous quinary scales to be noticed later, or it may be second 1, second 2, etc., as in the Niam Niam dialect of Central Africa, where the scale is[92] 1. sa. 2. uwi. 3. biata. 4. biama. 5. biswi. 6. batissa = 2d 1. 7. batiwwi = 2d 2. 8. batti-biata = 2d 3. 9. batti-biama = 2d 4. 10. bauwé = 2d 5. That this method of progression is not confined to the least developed languages, however, is shown by a most cursory examination of the numerals of our American Indian tribes, where numeral formation like that exhibited above is exceedingly common. In the Kootenay dialect,[93] of British Columbia, _qaetsa_, 4, and _wo-qaetsa,_ 8, are obviously related, the latter word probably meaning a second 4. Most of the native languages of British Columbia form their words for 7 and 8 from those which signify 2 and 3; as, for example, the Heiltsuk,[94] which shows in the following words a most obvious correspondence: 2. matl. 7. matlaaus. 3. yutq. 8. yutquaus. In the Choctaw language[95] the relation between 2 and 7, and 3 and 8, is no less clear. Here the words are: 2. tuklo. 7. untuklo. 3. tuchina. 8. untuchina. The Nez Percés[96] repeat the first three words of their scale in their 6, 7, and 8 respectively, as a comparison of these numerals will show. 1. naks. 6. oilaks. 2. lapit. 7. oinapt. 3. mitat. 8. oimatat. In all these cases the essential point of the method is contained in the repetition, in one way or another, of the numerals of the second quinate, without the use with each one of the word for 5. This may make 6, 7, 8, and 9 appear as second 1, second 2, etc., or another 1, another 2, etc.; or, more simply still, as 1 more, 2 more, etc. It is the method which was briefly discussed in the early part of the present chapter, and is by no means uncommon. In a decimal scale this repetition would begin with 11 instead of 6; as in the system found in use in Tagala and Pampanaga, two of the Philippine Islands, where, for example, 11, 12, and 13 are:[97] 11. labi-n-isa = over 1. 12. labi-n-dalaua = over 2. 13. labi-n-tatlo = over 3. A precisely similar method of numeral building is used by some of our Western Indian tribes. Selecting a few of the Assiniboine numerals[98] as an illustration, we have 11. ak kai washe = more 1. 12. ak kai noom pah = more 2. 13. ak kai yam me nee = more 3. 14. ak kai to pah = more 4. 15. ak kai zap tah = more 5. 16. ak kai shak pah = more 6, etc. A still more primitive structure is shown in the numerals of the Mboushas[99] of Equatorial Africa. Instead of using 5-1, 5-2, 5-3, 5-4, or 2d 1, 2d 2, 2d 3, 2d 4, in forming their numerals from 6 to 9, they proceed in the following remarkable and, at first thought, inexplicable manner to form their compound numerals: 1. ivoco. 2. beba. 3. belalo. 4. benai. 5. betano. 6. ivoco beba = 1-2. 7. ivoco belalo = 1-3. 8. ivoco benai = 1-4. 9. ivoco betano = 1-5. 10. dioum. No explanation is given by Mr. du Chaillu for such an apparently incomprehensible form of expression as, for example, 1-3, for 7. Some peculiar finger pantomime may accompany the counting, which, were it known, would enlighten us on the Mbousha's method of arriving at so anomalous a scale. Mere repetition in the second quinate of the words used in the first might readily be explained by supposing the use of fingers absolutely indispensable as an aid to counting, and that a certain word would have one meaning when associated with a certain finger of the left hand, and another meaning when associated with one of the fingers of the right. Such scales are, if the following are correct, actually in existence among the islands of the Pacific. BALAD.[100] UEA.[100] 1. parai. 1. tahi. 2. paroo. 2. lua. 3. pargen. 3. tolu. 4. parbai. 4. fa. 5. panim. 5. lima. 6. parai. 6. tahi. 7. paroo. 7. lua. 8. pargen. 8. tolu. 9. parbai. 9. fa. 10. panim. 10. lima. Such examples are, I believe, entirely unique among primitive number systems. In numeral scales where the formative process has been of the general nature just exhibited, irregularities of various kinds are of frequent occurrence. Hand numerals may appear, and then suddenly disappear, just where we should look for them with the greatest degree of certainty. In the Ende,[101] a dialect of the Flores Islands, 5, 6, and 7 are of hand formation, while 8 and 9 are of entirely different origin, as the scale shows. 1. sa. 2. zua. 3. telu. 4. wutu. 5. lima 6. lima sa = hand 1. 7. lima zua = hand 2. 8. rua butu = 2 × 4. 9. trasa = 10 - 1? 10. sabulu. One special point to be noticed in this scale is the irregularity that prevails between 7, 8, 9. The formation of 7 is of the most ordinary kind; 8 is 2 fours--common enough duplication; while 9 appears to be 10 - 1. All of these modes of compounding are, in their own way, regular; but the irregularity consists in using all three of them in connective numerals in the same system. But, odd as this jumble seems, it is more than matched by that found in the scale of the Karankawa Indians,[102] an extinct tribe formerly inhabiting the coast region of Texas. The first ten numerals of this singular array are: 1. natsa. 2. haikia. 3. kachayi. 4. hayo hakn = 2 × 2. 5. natsa behema = 1 father, _i.e._ of the fingers. 6. hayo haikia = 3 × 2? 7. haikia natsa = 2 + 5? 8. haikia behema = 2 fathers? 9. haikia doatn = 2d from 10? 10. doatn habe. Systems like the above, where chaos instead of order seems to be the ruling principle, are of occasional occurrence, but they are decidedly the exception. In some of the cases that have been adduced for illustration it is to be noticed that the process of combination begins with 7 instead of with 6. Among others, the scale of the Pigmies of Central Africa[103] and that of the Mosquitos[104] of Central America show this tendency. In the Pigmy scale the words for 1 and 6 are so closely akin that one cannot resist the impression that 6 was to them a new 1, and was thus named. MOSQUITO. PIGMY. 1. kumi. ujju. 2. wal. ibari. 3. niupa. ikaro. 4. wal-wal = 2-2. ikwanganya. 5. mata-sip = fingers of 1 hand. bumuti. 6. matlalkabe. ijju. 7. matlalkabe pura kumi = 6 and 1. bumutti-na-ibali = 5 and 2. 8. matlalkabe pura wal = 6 and 2. bumutti-na-ikaro = 5 and 3. 9. matlalkabe pura niupa = 6 and 3. bumutti-na-ikwanganya = 5 and 4. 10. mata wal sip = fingers of 2 hands. mabo = half man. The Mosquito scale is quite exceptional in forming 7, 8, and 9 from 6, instead of from 5. The usual method, where combinations appear between 6 and 10, is exhibited by the Pigmy scale. Still another species of numeral form, quite different from any that have already been noticed, is found in the Yoruba[105] scale, which is in many respects one of the most peculiar in existence. Here the words for 11, 12, etc., are formed by adding the suffix _-la_, great, to the words for 1, 2, etc., thus: 1. eni, or okan. 2. edzi. 3. eta. 4. erin. 5. arun. 6. efa. 7. edze. 8. edzo. 9. esan. 10. ewa. 11. okanla = great 1. 12. edzila = great 2. 13. etala = great 3. 14. erinla = great 4, etc. 40. ogodzi = string. 200. igba = heap. The word for 40 was adopted because cowrie shells, which are used for counting, were strung by forties; and _igba_, 200, because a heap of 200 shells was five strings, and thus formed a convenient higher unit for reckoning. Proceeding in this curious manner,[106] they called 50 strings 1 _afo_ or head; and to illustrate their singular mode of reckoning--the king of the Dahomans, having made war on the Yorubans, and attacked their army, was repulsed and defeated with a loss of "two heads, twenty strings, and twenty cowries" of men, or 4820. The number scale of the Abipones,[107] one of the low tribes of the Paraguay region, contains two genuine curiosities, and by reason of those it deserves a place among any collection of numeral scales designed to exhibit the formation of this class of words. It is: 1. initara = 1 alone. 2. inoaka. 3. inoaka yekaini = 2 and 1. 4. geyenknate = toes of an ostrich. 5. neenhalek = a five coloured, spotted hide, or hanambegen = fingers of 1 hand. 10. lanamrihegem = fingers of both hands. 20. lanamrihegem cat gracherhaka anamichirihegem = fingers of both hands together with toes of both feet. That the number sense of the Abipones is but little, if at all, above that of the native Australian tribes, is shown by their expressing 3 by the combination 2 and 1. This limitation, as we have already seen, is shared by the Botocudos, the Chiquitos, and many of the other native races of South America. But the Abipones, in seeking for words with which to enable themselves to pass beyond the limit 3, invented the singular terms just given for 4 and 5. The ostrich, having three toes in front and one behind on each foot presented them with a living example of 3 + 1; hence "toes of an ostrich" became their numeral for 4. Similarly, the number of colours in a certain hide being five, the name for that hide was adopted as their next numeral. At this point they began to resort to digital numeration also; and any higher number is expressed by that method. In the sense in which the word is defined by mathematicians, _number_ is a pure, abstract concept. But a moment's reflection will show that, as it originates among savage races, number is, and from the limitations of their intellect must be, entirely concrete. An abstract conception is something quite foreign to the essentially primitive mind, as missionaries and explorers have found to their chagrin. The savage can form no mental concept of what civilized man means by such a word as "soul"; nor would his idea of the abstract number 5 be much clearer. When he says _five_, he uses, in many cases at least, the same word that serves him when he wishes to say _hand_; and his mental concept when he says _five_ is of a hand. The concrete idea of a closed fist or an open hand with outstretched fingers, is what is upper-most in his mind. He knows no more and cares no more about the pure number 5 than he does about the law of the conservation of energy. He sees in his mental picture only the real, material image, and his only comprehension of the number is, "these objects are as many as the fingers on my hand." Then, in the lapse of the long interval of centuries which intervene between lowest barbarism and highest civilization, the abstract and the concrete become slowly dissociated, the one from the other. First the actual hand picture fades away, and the number is recognized without the original assistance furnished by the derivation of the word. But the number is still for a long time a certain number _of objects_, and not an independent concept. It is only when the savage ceases to be wholly an animal, and becomes a thinking human being, that number in the abstract can come within the grasp of his mind. It is at this point that mere reckoning ceases, and arithmetic begins. CHAPTER IV. THE ORIGIN OF NUMBER WORDS. (_CONTINUED_.) By the slow, and often painful, process incident to the extension and development of any mental conception in a mind wholly unused to abstractions, the savage gropes his way onward in his counting from 1, or more probably from 2, to the various higher numbers required to form his scale. The perception of unity offers no difficulty to his mind, though he is conscious at first of the object itself rather than of any idea of number associated with it. The concept of duality, also, is grasped with perfect readiness. This concept is, in its simplest form, presented to the mind as soon as the individual distinguishes himself from another person, though the idea is still essentially concrete. Perhaps the first glimmering of any real number thought in connection with 2 comes when the savage contrasts one single object with another--or, in other words, when he first recognizes the _pair_. At first the individuals composing the pair are simply "this one," and "that one," or "this and that"; and his number system now halts for a time at the stage when he can, rudely enough it may be, count 1, 2, many. There are certain cases where the forms of 1 and 2 are so similar than one may readily imagine that these numbers really were "this" and "that" in the savage's original conception of them; and the same likeness also occurs in the words for 3 and 4, which may readily enough have been a second "this" and a second "that." In the Lushu tongue the words for 1 and 2 are _tizi_ and _tazi_ respectively. In Koriak we find _ngroka_, 3, and _ngraka_, 4; in Kolyma, _niyokh_, 3, and _niyakh_, 4; and in Kamtschatkan, _tsuk_, 3, and _tsaak_, 4.[108] Sometimes, as in the case of the Australian races, the entire extent of the count is carried through by means of pairs. But the natural theory one would form is, that 2 is the halting place for a very long time; that up to this point the fingers may or may not have been used--probably not; and that when the next start is made, and 3, 4, 5, and so on are counted, the fingers first come into requisition. If the grammatical structure of the earlier languages of the world's history is examined, the student is struck with the prevalence of the dual number in them--something which tends to disappear as language undergoes extended development. The dual number points unequivocally to the time when 1 and 2 were _the_ numbers at mankind's disposal; to the time when his three numeral concepts, 1, 2, many, each demanded distinct expression. With increasing knowledge the necessity for this differentiatuin would pass away, and but two numbers, singular and plural, would remain. Incidentally it is to be noticed that the Indo-European words for 3--_three_, _trois_, _drei_, _tres_, _tri,_ etc., have the same root as the Latin _trans_, beyond, and give us a hint of the time when our Aryan ancestors counted in the manner I have just described. The first real difficulty which the savage experiences in counting, the difficulty which comes when he attempts to pass beyond 2, and to count 3, 4, and 5, is of course but slight; and these numbers are commonly used and readily understood by almost all tribes, no matter how deeply sunk in barbarism we find them. But the instances that have already been cited must not be forgotten. The Chiquitos do not, in their primitive state, properly count at all; the Andamans, the Veddas, and many of the Australian tribes have no numerals higher than 2; others of the Australians and many of the South Americans stop with 3 or 4; and tribes which make 5 their limit are still more numerous. Hence it is safe to assert that even this insignificant number is not always reached with perfect ease. Beyond 5 primitive man often proceeds with the greatest difficulty. Most savages, even those of the tribes just mentioned, can really count above here, even though they have no words with which to express their thought. But they do it with reluctance, and as they go on they quickly lose all sense of accuracy. This has already been commented on, but to emphasize it afresh the well-known example given by Mr. Oldfield from his own experience among the Watchandies may be quoted.[109] "I once wished to ascertain the exact number of natives who had been slain on a certain occasion. The individual of whom I made the inquiry began to think over the names ... assigning one of his fingers to each, and it was not until after many failures, and consequent fresh starts, that he was able to express so high a number, which he at length did by holding up his hand three times, thus giving me to understand that fifteen was the answer to this most difficult arithmetical question." This meagreness of knowledge in all things pertaining to numbers is often found to be sharply emphasized in the names adopted by savages for their numeral words. While discussing in a previous chapter the limits of number systems, we found many instances where anything above 2 or 3 was designated by some one of the comprehensive terms _much_, _many_, _very many_; these words, or such equivalents as _lot_, _heap_, or _plenty_, serving as an aid to the finger pantomime necessary to indicate numbers for which they have no real names. The low degree of intelligence and civilization revealed by such words is brought quite as sharply into prominence by the word occasionally found for 5. Whenever the fingers and hands are used at all, it would seem natural to expect for 5 some general expression signifying _hand_, for 10 _both hands_, and for 20 _man_. Such is, as we have already seen, the ordinary method of progression, but it is not universal. A drop in the scale of civilization takes us to a point where 10, instead of 20, becomes the whole man. The Kusaies,[110] of Strong's Island, call 10 _sie-nul_, 1 man, 30 _tol-nul_, 3 men, 40 _a naul_, 4 men, etc.; and the Ku-Mbutti[111] of central Africa have _mukko_, 10, and _moku_, man. If 10 is to be expressed by reference to the man, instead of his hands, it might appear more natural to employ some such expression as that adopted by the African Pigmies,[112] who call 10 _mabo_, and man _mabo-mabo_. With them, then, 10 is perhaps "half a man," as it actually is among the Towkas of South America; and we have already seen that with the Aztecs it was _matlactli_, the "hand half" of a man.[113] The same idea crops out in the expression used by the Nicobar Islanders for 30--_heam-umdjome ruktei_, 1 man (and a) half.[114] Such nomenclature is entirely natural, and it accords with the analogy offered by other words of frequent occurrence in the numeral scales of savage races. Still, to find 10 expressed by the term _man_ always conveys an impression of mental poverty; though it may, of course, be urged that this might arise from the fact that some races never use the toes in counting, but go over the fingers again, or perhaps bring into requisition the fingers of a second man to express the second 10. It is not safe to postulate an extremely low degree of civilization from the presence of certain peculiarities of numeral formation. Only the most general statements can be ventured on, and these are always subject to modification through some circumstance connected with environment, mode of living, or intercourse with other tribes. Two South American races may be cited, which seem in this respect to give unmistakable evidence of being sunk in deepest barbarism. These are the Juri and the Cayriri, who use the same word for man and for 5. The former express 5 by _ghomen apa_, 1 man,[115] and the latter by _ibicho_, person.[116] The Tasmanians of Oyster Bay use the native word of similar meaning, _puggana_, man,[117] for 5. Wherever the numeral 20 is expressed by the term _man_, it may be expected that 40 will be 2 men, 60, 3 men, etc. This form of numeration is usually, though not always, carried as far as the system extends; and it sometimes leads to curious terms, of which a single illustration will suffice. The San Blas Indians, like almost all the other Central and South American tribes, count by digit numerals, and form their twenties as follows:[118] 20. tula guena = man 1. 40. tula pogua = man 2. 100. tula atala = man 5. 120. tula nergua = man 6. 1000. tula wala guena = great 1 man. The last expression may, perhaps, be translated "great hundred," though the literal meaning is the one given. If 10, instead of 20, is expressed by the word "man," the multiples of 10 follow the law just given for multiples of 20. This is sufficiently indicated by the Kusaie scale; or equally well by the Api words for 100 and 200, which are[119] _duulimo toromomo_ = 10 times the whole man. _duulimo toromomo va juo_ = 10 times the whole man taken 2 times. As an illustration of the legitimate result which is produced by the attempt to express high numbers in this manner the term applied by educated native Greenlanders[120] for a thousand may be cited. This numeral, which is, of course, not in common use, is _inuit kulit tatdlima nik kuleriartut navdlugit_ = 10 men 5 times 10 times come to an end. It is worth noting that the word "great," which appears in the scale of the San Blas Indians, is not infrequently made use of in the formation of higher numeral words. The African Mabas[121] call 10 _atuk_, great 1; the Hottentots[122] and the Hidatsa Indians call 100 great 10, their words being _gei disi_ and _pitikitstia_ respectively. The Nicaraguans[123] express 100 by _guhamba_, great 10, and 400 by _dinoamba_, great 20; and our own familiar word "million," which so many modern languages have borrowed from the Italian, is nothing more nor less than a derivative of the Latin _mille_, and really means "great thousand." The Dakota[124] language shows the same origin for its expression of 1,000,000, which is _kick ta opong wa tunkah_, great 1000. The origin of such terms can hardly be ascribed to poverty of language. It is found, rather, in the mental association of the larger with the smaller unit, and the consequent repetition of the name of the smaller. Any unit, whether it be a single thing, a dozen, a score, a hundred, a thousand, or any other unit, is, whenever used, a single and complete group; and where the relation between them is sufficiently close, as in our "gross" and "great gross," this form of nomenclature is natural enough to render it a matter of some surprise that it has not been employed more frequently. An old English nursery rhyme makes use of this association, only in a manner precisely the reverse of that which appears now and then in numeral terms. In the latter case the process is always one of enlargement, and the associative word is "great." In the following rhyme, constructed by the mature for the amusement of the childish mind, the process is one of diminution, and the associative word is "little": One's none, Two's some, Three's a many, Four's a penny, Five's a little hundred.[125] Any real numeral formation by the use of "little," with the name of some higher unit, would, of course, be impossible. The numeral scale must be complete before the nursery rhyme can be manufactured. It is not to be supposed from the observations that have been made on the formation of savage numeral scales that all, or even the majority of tribes, proceed in the awkward and faltering manner indicated by many of the examples quoted. Some of the North American Indian tribes have numeral scales which are, as far as they go, as regular and almost as simple as our own. But where digital numeration is extensively resorted to, the expressions for higher numbers are likely to become complex, and to act as a real bar to the extension of the system. The same thing is true, to an even greater degree, of tribes whose number sense is so defective that they begin almost from the outset to use combinations. If a savage expresses the number 3 by the combination 2-1, it will at once be suspected that his numerals will, by the time he reaches 10 or 20, become so complex and confused that numbers as high as these will be expressed by finger pantomime rather than by words. Such is often the case; and the comment is frequently made by explorers that the tribes they have visited have no words for numbers higher than 3, 4, 5, 10, or 20, but that counting is carried beyond that point by the aid of fingers or other objects. So reluctant, in many cases, are savages to count by words, that limits have been assigned for spoken numerals, which subsequent investigation proved to fall far short of the real extent of the number systems to which they belonged. One of the south-western Indian tribes of the United States, the Comanches, was for a time supposed to have no numeral words below 10, but to count solely by the use of fingers. But the entire scale of this taciturn tribe was afterward discovered and published. To illustrate the awkward and inconvenient forms of expression which abound in primitive numeral nomenclature, one has only to draw from such scales as those of the Zuñi, or the Point Barrow Eskimos, given in the last chapter. Terms such as are found there may readily be duplicated from almost any quarter of the globe. The Soussous of Sierra Leone[126] call 99 _tongo solo manani nun solo manani_, _i.e._ to take (10 understood) 5 + 4 times and 5 + 4. The Malagasy expression for 1832 is[127] _roambistelo polo amby valonjato amby arivo_, 2 + 30 + 800 + 1000. The Aztec equivalent for 399 is[128] _caxtolli onnauh poalli ipan caxtolli onnaui_, (15 + 4) × 20 + 15 + 4; and the Sioux require for 29 the ponderous combination[129] _wick a chimen ne nompah sam pah nep e chu wink a._ These terms, long and awkward as they seem, are only the legitimate results which arise from combining the names of the higher and lower numbers, according to the peculiar genius of each language. From some of the Australian tribes are derived expressions still more complex, as for 6, _marh-jin-bang-ga-gudjir-gyn_, half the hands and 1; and for 15, _marh-jin-belli-belli-gudjir-jina-bang-ga_, the hand on either side and half the feet.[130] The Maré tribe, one of the numerous island tribes of Melanesia,[131] required for a translation of the numeral 38, which occurs in John v. 5, "had an infirmity thirty and eight years," the circumlocution, "one man and both sides five and three." Such expressions, curious as they seem at first thought, are no more than the natural outgrowth of systems built up by the slow and tedious process which so often obtains among primitive races, where digit numerals are combined in an almost endless variety of ways, and where mere reduplication often serves in place of any independent names for higher units. To what extent this may be carried is shown by the language of the Cayubabi,[132] who have for 10 the word _tunca_, and for 100 and 1000 the compounds _tunca tunca_, and _tunca tunca tunca_ respectively; or of the Sapibocones, who call 10 _bururuche_, hand hand, and 100 _buruche buruche_, hand hand hand hand.[133] More remarkable still is the Ojibwa language, which continues its numeral scale without limit, furnishing combinations which are really remarkable; as, _e.g._, that for 1,000,000,000, which is _me das wac me das wac as he me das wac_,[134] 1000 × 1000 × 1000. The Winnebago expression for the same number,[135] _ho ke he hhuta hhu chen a ho ke he ka ra pa ne za_ is no less formidable, but it has every appearance of being an honest, native combination. All such primitive terms for larger numbers must, however, be received with caution. Savages are sometimes eager to display a knowledge they do not possess, and have been known to invent numeral words on the spot for the sake of carrying their scales to as high a limit as possible. The Choctaw words for million and billion are obvious attempts to incorporate the corresponding English terms into their own language.[136] For million they gave the vocabulary-hunter the phrase _mil yan chuffa_, and for billion, _bil yan chuffa_. The word _chuffa_ signifies 1, hence these expressions are seen at a glance to be coined solely for the purpose of gratifying a little harmless Choctaw vanity. But this is innocence itself compared with the fraud perpetrated on Labillardière by the Tonga Islanders, who supplied the astonished and delighted investigator with a numeral vocabulary up to quadrillions. Their real limit was afterward found to be 100,000, and above that point they had palmed off as numerals a tolerably complete list of the obscene words of their language, together with a few nonsense terms. These were all accepted and printed in good faith, and the humiliating truth was not discovered until years afterward.[137] One noteworthy and interesting fact relating to numeral nomenclature is the variation in form which words of this class undergo when applied to different classes of objects. To one accustomed as we are to absolute and unvarying forms for numerals, this seems at first a novel and almost unaccountable linguistic freak. But it is not uncommon among uncivilized races, and is extensively employed by so highly enlightened a people, even, as the Japanese. This variation in form is in no way analogous to that produced by inflectional changes, such as occur in Hebrew, Greek, Latin, etc. It is sufficient in many cases to produce almost an entire change in the form of the word; or to result in compounds which require close scrutiny for the detection of the original root. For example, in the Carrier, one of the Déné dialects of western Canada, the word _tha_ means 3 things; _thane_, 3 persons; _that_, 3 times; _thatoen_, in 3 places; _thauh_, in 3 ways; _thailtoh_, all of the 3 things; _thahoeltoh_, all of the 3 persons; and _thahultoh_, all of the 3 times.[138] In the Tsimshian language of British Columbia we find seven distinct sets of numerals "which are used for various classes of objects that are counted. The first set is used in counting where there is no definite object referred to; the second class is used for counting flat objects and animals; the third for counting round objects and divisions of time; the fourth for counting men; the fifth for counting long objects, the numerals being composed with _kan_, tree; the sixth for counting canoes; and the seventh for measures. The last seem to be composed with _anon_, hand."[139] The first ten numerals of each of these classes is given in the following table: +----+---------+---------+---------+----------+------------+-------------+-------------+ |No. |Counting | Flat | Round | Men | Long | Canoes | Measures | | | | Objects | Objects | | Objects | | | +----+---------+---------+---------+----------+------------+-------------+-------------+ | 1 |gyak gak |g'erel |k'al |k'awutskan|k'amaet |k'al | | | 2 |t'epqat |t'epqat |goupel |t'epqadal |gaopskan |g'alp[=e]eltk|gulbel | | 3 |guant |guant |gutle |gulal |galtskan |galtskantk |guleont | | 4 |tqalpq |tqalpq |tqalpq |tqalpqdal |tqaapskan |tqalpqsk |tqalpqalont | | 5 |kct[=o]nc|kct[=o]nc|kct[=o]nc|kcenecal |k'etoentskan|kct[=o]onsk |kctonsilont | | 6 |k'alt |k'alt |k'alt |k'aldal |k'aoltskan |k'altk |k'aldelont | | 7 |t'epqalt |t'epqalt |t'epqalt |t'epqaldal|t'epqaltskan|t'epqaltk |t'epqaldelont| | 8 |guandalt |yuktalt |yuktalt |yuktleadal|ek'tlaedskan|yuktaltk |yuktaldelont | | 9 |kctemac |kctemac |kctemac |kctemacal |kctemaestkan|kctemack |kctemasilont | |10 |gy'ap |gy'ap |kp[=e]el |kpal |kp[=e]etskan|gy'apsk |kpeont | +----+---------+---------+---------+----------+------------+-------------+-------------+ Remarkable as this list may appear, it is by no means as extensive as that derived from many of the other British Columbian tribes. The numerals of the Shushwap, Stlatlumh, Okanaken, and other languages of this region exist in several different forms, and can also be modified by any of the innumerable suffixes of these tongues.[140] To illustrate the almost illimitable number of sets that may be formed, a table is given of "a few classes, taken from the Heiltsuk dialect.[141] It appears from these examples that the number of classes is unlimited." +-----------------------+-------------+--------------+--------------+ | | One. | Two. | Three. | +-----------------------+-------------+--------------+--------------+ |Animate. |menok |maalok |yutuk | |Round. |menskam |masem |yutqsem | |Long. |ments'ak |mats'ak |yututs'ak | |Flat. |menaqsa |matlqsa |yutqsa | |Day. |op'enequls |matlp'enequls |yutqp'enequls | |Fathom. |op'enkh |matlp'enkh |yutqp'enkh | |Grouped together. |---- |matloutl |yutoutl | |Groups of objects. |nemtsmots'utl|matltsmots'utl|yutqtsmots'utl| |Filled cup. |menqtlala |matl'aqtlala |yutqtlala | |Empty cup. |menqtla |matl'aqtla |yutqtla | |Full box. |menskamala |masemala |yutqsemala | |Empty box. |menskam |masem |yutqsem | |Loaded canoe. |mentsake |mats'ake |yututs'ake | |Canoe with crew. |ments'akis |mats'akla |yututs'akla | |Together on beach. |---- |maalis |---- | |Together in house, etc.|---- |maalitl |---- | +-----------------------+-------------+--------------+--------------+ Variation in numeral forms such as is exhibited in the above tables is not confined to any one quarter of the globe; but it is more universal among the British Columbian Indians than among any other race, and it is a more characteristic linguistic peculiarity of this than of any other region, either in the Old World or in the New. It was to some extent employed by the Aztecs,[142] and its use is current among the Japanese; in whose language Crawfurd finds fourteen different classes of numerals "without exhausting the list."[143] In examining the numerals of different languages it will be found that the tens of any ordinary decimal scale are formed in the same manner as in English. Twenty is simply 2 times 10; 30 is 3 times 10, and so on. The word "times" is, of course, not expressed, any more than in English; but the expressions briefly are, 2 tens, 3 tens, etc. But a singular exception to this method is presented by the Hebrew, and other of the Semitic languages. In Hebrew the word for 20 is the plural of the word for 10; and 30, 40, 50, etc. to 90 are plurals of 3, 4, 5, 6, 7, 8, 9. These numerals are as follows:[144] 10, eser, 20, eserim, 3, shalosh, 30, shaloshim, 4, arba, 40, arbaim, 5, chamesh, 50, chamishshim, 6, shesh, 60, sheshshim, 7, sheba, 70, shibim, 8, shemoneh 80, shemonim, 9, tesha, 90, tishim. The same formation appears in the numerals of the ancient Phoenicians,[145] and seems, indeed, to be a well-marked characteristic of the various branches of this division of the Caucasian race. An analogous method appears in the formation of the tens in the Bisayan,[146] one of the Malay numeral scales, where 30, 40, ... 90, are constructed from 3, 4, ... 9, by adding the termination _-an_. No more interesting contribution has ever been made to the literature of numeral nomenclature than that in which Dr. Trumbull embodies the results of his scholarly research among the languages of the native Indian tribes of this country.[147] As might be expected, we are everywhere confronted with a digital origin, direct or indirect, in the great body of the words examined. But it is clearly shown that such a derivation cannot be established for all numerals; and evidence collected by the most recent research fully substantiates the position taken by Dr. Trumbull. Nearly all the derivations established are such as to remind us of the meanings we have already seen recurring in one form or another in language after language. Five is the end of the finger count on one hand--as, the Micmac _nan_, and Mohegan _nunon_, gone, or spent; the Pawnee _sihuks_, hands half; the Dakota _zaptan_, hand turned down; and the Massachusetts _napanna_, on one side. Ten is the end of the finger count, but is not always expressed by the "both hands" formula so commonly met with. The Cree term for this number is _mitatat_, no further; and the corresponding word in Delaware is _m'tellen_, no more. The Dakota 10 is, like its 5, a straightening out of the fingers which have been turned over in counting, or _wickchemna_, spread out unbent. The same is true of the Hidatsa _pitika_, which signifies a smoothing out, or straightening. The Pawnee 4, _skitiks_, is unusual, signifying as it does "all the fingers," or more properly, "the fingers of the hand." The same meaning attaches to this numeral in a few other languages also, and reminds one of the habit some people have of beginning to count on the forefinger and proceeding from there to the little finger. Can this have been the habit of the tribes in question? A suggestion of the same nature is made by the Illinois and Miami words for 8, _parare_ and _polane_, which signify "nearly ended." Six is almost always digital in origin, though the derivation may be indirect, as in the Illinois _kakatchui_, passing beyond the middle; and the Dakota _shakpe_, 1 in addition. Some of these significations are well matched by numerals from the Ewe scales of western Africa, where we find the following:[148] 1. de = a going, _i.e._ a beginning. (Cf. the Zuñi _töpinte_, taken to start with.) 3. eto = the father (from the middle, or longest finger). 6. ade = the other going. 9. asieke = parting with the hands. 10. ewo = done. In studying the names for 2 we are at once led away from a strictly digital origin for the terms by which this number is expressed. These names seem to come from four different sources: (1) roots denoting separation or distinction; (2) likeness, equality, or opposition; (3) addition, _i.e._ putting to, or putting with; (4) coupling, pairing, or matching. They are often related to, and perhaps derived from, names of natural pairs, as feet, hands, eyes, arms, or wings. In the Dakota and Algonkin dialects 2 is almost always related to "arms" or "hands," and in the Athapaskan to "feet." But the relationship is that of common origin, rather than of derivation from these pair-names. In the Puri and Hottentot languages, 2 and "hand" are closely allied; while in Sanskrit, 2 may be expressed by any one of the words _kara_, hand, _bahu_, arm, _paksha_, wing, or _netra,_ eye.[149] Still more remote from anything digital in their derivation are the following, taken at random from a very great number of examples that might be cited to illustrate this point. The Assiniboines call 7, _shak ko we_, or _u she nah_, the odd number.[150] The Crow 1, _hamat,_ signifies "the least";[151] the Mississaga 1, _pecik_, a very small thing.[152] In Javanese, Malay, and Manadu, the words for 1, which are respectively _siji_, _satu_, and _sabuah_, signify 1 seed, 1 pebble, and 1 fruit respectively[153]--words as natural and as much to be expected at the beginning of a number scale as any finger name could possibly be. Among almost all savage races one form or another of palpable arithmetic is found, such as counting by seeds, pebbles, shells, notches, or knots; and the derivation of number words from these sources can constitute no ground for surprise. The Marquesan word for 4 is _pona_, knot, from the practice of tying breadfruit in knots of 4. The Maori 10 is _tekau_, bunch, or parcel, from the counting of yams and fish by parcels of 10.[154] The Javanese call 25, _lawe_, a thread, or string; 50, _ekat_, a skein of thread; 400, _samas_, a bit of gold; 800, _domas_, 2 bits of gold.[155] The Macassar and Butong term for 100 is _bilangan_, 1 tale or reckoning.[156] The Aztec 20 is _cem pohualli_, 1 count; 400 is _centzontli_, 1 hair of the head; and 8000 is _xiquipilli_, sack.[157] This sack was of such a size as to contain 8000 cacao nibs, or grains, hence the derivation of the word in its numeral sense is perfectly natural. In Japanese we find a large number of terms which, as applied to the different units of the number scale, seem almost purely fanciful. These words, with their meanings as given by a Japanese lexicon, are as follows: 10,000, or 10^4, män = enormous number. 10^8, oku = a compound of the words "man" and "mind." 10^12, chio = indication, or symptom. 10^16, kei = capital city. 10^20, si = a term referring to grains. 10^24, owi = ---- 10^28, jio = extent of land. 10^32, ko = canal. 10^36, kan = some kind of a body of water. 10^40, sai = justice. 10^44, s[=a] = support. 10^48, kioku = limit, or more strictly, ultimate. .01^2, rin = ---- .01^3, mo = hair (of some animal). .01^4, shi = thread. In addition to these, some of the lower fractional values are described by words meaning "very small," "very fine thread," "sand grain," "dust," and "very vague." Taken altogether, the Japanese number system is the most remarkable I have ever examined, in the extent and variety of the higher numerals with well-defined descriptive names. Most of the terms employed are such as to defy any attempt to trace the process of reasoning which led to their adoption. It is not improbable that the choice was, in some of these cases at least, either accidental or arbitrary; but still, the changes in word meanings which occur with the lapse of time may have differentiated significations originally alike, until no trace of kinship would appear to the casual observer. Our numerals "score" and "gross" are never thought of as having any original relation to what is conveyed by the other meanings which attach to these words. But the origin of each, which is easily traced, shows that, in the beginning, there existed a well-defined reason for the selection of these, rather than other terms, for the numbers they now describe. Possibly these remarkable Japanese terms may be accounted for in the same way, though the supposition is, for some reasons, quite improbable. The same may be said for the Malagasy 1000, _alina_, which also means "night," and the Hebrew 6, _shesh_, which has the additional signification "white marble," and the stray exceptions which now and then come to the light in this or that language. Such terms as these may admit of some logical explanation, but for the great mass of numerals whose primitive meanings can be traced at all, no explanation whatever is needed; the words are self-explanatory, as the examples already cited show. A few additional examples of natural derivation may still further emphasize the point just discussed. In Bambarese the word for 10, _tank_, is derived directly from _adang_, to count.[158] In the language of Mota, one of the islands of Melanesia, 100 is _mel nol_, used and done with, referring to the leaves of the cycas tree, with which the count had been carried on.[159] In many other Melanesian dialects[160] 100 is _rau_, a branch or leaf. In the Torres Straits we find the same number expressed by _na won_, the close; and in Eromanga it is _narolim narolim_ (2 × 5)(2 × 5).[161] This combination deserves remark only because of the involved form which seems to have been required for the expression of so small a number as 100. A compound instead of a simple term for any higher unit is never to be wondered at, so rude are some of the savage methods of expressing number; but "two fives (times) two fives" is certainly remarkable. Some form like that employed by the Nusqually[162] of Puget Sound for 1000, i.e. _paduts-subquätche_, ten hundred, is more in accordance with primitive method. But we are equally likely to find such descriptive phrases for this numeral as the _dor paka_, banyan roots, of the Torres Islands; _rau na hai_, leaves of a tree, of Vaturana; or _udolu_, all, of the Fiji Islands. And two curious phrases for 1000 are those of the Banks' Islands, _tar mataqelaqela_, eye blind thousand, _i.e._ many beyond count; and of Malanta, _warehune huto_, opossum's hairs, or _idumie one_, count the sand.[163] The native languages of India, Thibet, and portions of the Indian archipelago furnish us with abundant instances of the formation of secondary numeral scales, which were used only for special purposes, and without in any way interfering with the use of the number words already in use. "Thus the scholars of India, ages ago, selected a set of words for a memoria technica, in order to record dates and numbers. These words they chose for reasons which are still in great measure evident; thus 'moon' or 'earth' expressed 1, there being but one of each; 2 might be called 'eye,' 'wing,' 'arm,' 'jaw,' as going in pairs; for 3 they said 'Rama,' 'fire,' or 'quality,' there being considered to be three Ramas, three kinds of fire, three qualities (guna); for 4 were used 'veda,' 'age,' or 'ocean,' there being four of each recognized; 'season' for 6, because they reckoned six seasons; 'sage' or 'vowel,' for 7, from the seven sages and the seven vowels; and so on with higher numbers, 'sun' for 12, because of his twelve annual denominations, or 'zodiac' from his twelve signs, and 'nail' for 20, a word incidentally bringing in finger notation. As Sanskrit is very rich in synonyms, and as even the numerals themselves might be used, it became very easy to draw up phrases or nonsense verses to record series of numbers by this system of artificial memory."[164] More than enough has been said to show how baseless is the claim that all numeral words are derived, either directly or indirectly, from the names of fingers, hands, or feet. Connected with the origin of each number word there may be some metaphor, which cannot always be distinctly traced; and where the metaphor was born of the hand or of the foot, we inevitably associate it with the practice of finger counting. But races as fond of metaphor and of linguistic embellishment as are those of the East, or as are our American Indians even, might readily resort to some other source than that furnished by the members of the human body, when in want of a term with which to describe the 5, 10, or any other number of the numeral scale they were unconsciously forming. That the first numbers of a numeral scale are usually derived from other sources, we have some reason to believe; but that all above 2, 3, or at most 4, are almost universally of digital origin we must admit. Exception should properly be made of higher units, say 1000 or anything greater, which could not be expected to conform to any law of derivation governing the first few units of a system. Collecting together and comparing with one another the great mass of terms by which we find any number expressed in different languages, and, while admitting the great diversity of method practised by different tribes, we observe certain resemblances which were not at first supposed to exist. The various meanings of 1, where they can be traced at all, cluster into a little group of significations with which at last we come to associate the idea of unity. Similarly of 2, or 5, or 10, or any one of the little band which does picket duty for the advance guard of the great host of number words which are to follow. A careful examination of the first decade warrants the assertion that the probable meaning of any one of the units will be found in the list given below. The words selected are intended merely to serve as indications of the thought underlying the savage's choice, and not necessarily as the exact term by means of which he describes his number. Only the commonest meanings are included in the tabulation here given. 1 = existence, piece, group, beginning. 2 = repetition, division, natural pair. 3 = collection, many, two-one. 4 = two twos. 5 = hand, group, division, 6 = five-one, two threes, second one. 7 = five-two, second two, three from ten. 8 = five-three, second three, two fours, two from ten. 9 = five-four, three threes, one from ten. 10 = one (group), two fives (hands), half a man, one man. 15 = ten-five, one foot, three fives. 20 = two tens, one man, two feet.[165] CHAPTER V. MISCELLANEOUS NUMBER BASES. In the development and extension of any series of numbers into a systematic arrangement to which the term _system_ may be applied, the first and most indispensable step is the selection of some number which is to serve as a base. When the savage begins the process of counting he invents, one after another, names with which to designate the successive steps of his numerical journey. At first there is no attempt at definiteness in the description he gives of any considerable number. If he cannot show what he means by the use of his fingers, or perhaps by the fingers of a single hand, he unhesitatingly passes it by, calling it many, heap, innumerable, as many as the leaves on the trees, or something else equally expressive and equally indefinite. But the time comes at last when a greater degree of exactness is required. Perhaps the number 11 is to be indicated, and indicated precisely. A fresh mental effort is required of the ignorant child of nature; and the result is "all the fingers and one more," "both hands and one more," "one on another count," or some equivalent circumlocution. If he has an independent word for 10, the result will be simply ten-one. When this step has been taken, the base is established. The savage has, with entire unconsciousness, made all his subsequent progress dependent on the number 10, or, in other words, he has established 10 as the base of his number system. The process just indicated may be gone through with at 5, or at 20, thus giving us a quinary or a vigesimal, or, more probably, a mixed system; and, in rare instances, some other number may serve as the point of departure from simple into compound numeral terms. But the general idea is always the same, and only the details of formation are found to differ. Without the establishment of some base any _system_ of numbers is impossible. The savage has no means of keeping track of his count unless he can at each step refer himself to some well-defined milestone in his course. If, as has been pointed out in the foregoing chapters, confusion results whenever an attempt is made to count any number which carries him above 10, it must at once appear that progress beyond that point would be rendered many times more difficult if it were not for the fact that, at each new step, he has only to indicate the distance he has progressed beyond his base, and not the distance from his original starting-point. Some idea may, perhaps, be gained of the nature of this difficulty by imagining the numbers of our ordinary scale to be represented, each one by a single symbol different from that used to denote any other number. How long would it take the average intellect to master the first 50 even, so that each number could without hesitation be indicated by its appropriate symbol? After the first 50 were once mastered, what of the next 50? and the next? and the next? and so on. The acquisition of a scale for which we had no other means of expression than that just described would be a matter of the extremest difficulty, and could never, save in the most exceptional circumstances, progress beyond the attainment of a limit of a few hundred. If the various numbers in question were designated by words instead of by symbols, the difficulty of the task would be still further increased. Hence, the establishment of some number as a base is not only a matter of the very highest convenience, but of absolute necessity, if any save the first few numbers are ever to be used. In the selection of a base,--of a number from which he makes a fresh start, and to which he refers the next steps in his count,--the savage simply follows nature when he chooses 10, or perhaps 5 or 20. But it is a matter of the greatest interest to find that other numbers have, in exceptional cases, been used for this purpose. Two centuries ago the distinguished philosopher and mathematician, Leibnitz, proposed a binary system of numeration. The only symbols needed in such a system would be 0 and 1. The number which is now symbolized by the figure 2 would be represented by 10; while 3, 4, 5, 6, 7, 8, etc., would appear in the binary notation as 11, 100, 101, 110, 111, 1000, etc. The difficulty with such a system is that it rapidly grows cumbersome, requiring the use of so many figures for indicating any number. But Leibnitz found in the representation of all numbers by means of the two digits 0 and 1 a fitting symbolization of the creation out of chaos, or nothing, of the entire universe by the power of the Deity. In commemoration of this invention a medal was struck bearing on the obverse the words Numero Deus impari gaudet, and on the reverse, Omnibus ex nihilo ducendis sufficit Unum.[166] This curious system seems to have been regarded with the greatest affection by its inventor, who used every endeavour in his power to bring it to the notice of scholars and to urge its claims. But it appears to have been received with entire indifference, and to have been regarded merely as a mathematical curiosity. Unknown to Leibnitz, however, a binary method of counting actually existed during that age; and it is only at the present time that it is becoming extinct. In Australia, the continent that is unique in its flora, its fauna, and its general topography, we find also this anomaly among methods of counting. The natives, who are to be classed among the lowest and the least intelligent of the aboriginal races of the world, have number systems of the most rudimentary nature, and evince a decided tendency to count by twos. This peculiarity, which was to some extent shared by the Tasmanians, the island tribes of the Torres Straits, and other aboriginal races of that region, has by some writers been regarded as peculiar to their part of the world; as though a binary number system were not to be found elsewhere. This attempt to make out of the rude and unusual method of counting which obtained among the Australians a racial characteristic is hardly justified by fuller investigation. Binary number systems, which are given in full on another page, are found in South America. Some of the Dravidian scales are binary;[167] and the marked preference, not infrequently observed among savage races, for counting by pairs, is in itself a sufficient refutation of this theory. Still it is an unquestionable fact that this binary tendency is more pronounced among the Australians than among any other extensive number of kindred races. They seldom count in words above 4, and almost never as high as 7. One of the most careful observers among them expresses his doubt as to a native's ability to discover the loss of two pins, if he were first shown seven pins in a row, and then two were removed without his knowledge.[168] But he believes that if a single pin were removed from the seven, the Blackfellow would become conscious of its loss. This is due to his habit of counting by pairs, which enables him to discover whether any number within reasonable limit is odd or even. Some of the negro tribes of Africa, and of the Indian tribes of America, have the same habit. Progression by pairs may seem to some tribes as natural as progression by single units. It certainly is not at all rare; and in Australia its influence on spoken number systems is most apparent. Any number system which passes the limit 10 is reasonably sure to have either a quinary, a decimal, or a vigesimal structure. A binary scale could, as it is developed in primitive languages, hardly extend to 20, or even to 10, without becoming exceedingly cumbersome. A binary scale inevitably suggests a wretchedly low degree of mental development, which stands in the way of the formation of any number scale worthy to be dignified by the name of system. Take, for example, one of the dialects found among the western tribes of the Torres Straits, where, in general, but two numerals are found to exist. In this dialect the method of counting is:[169] 1. urapun. 2. okosa. 3. okosa urapun = 2-1. 4. okosa okosa = 2-2. 5. okosa okosa urapun = 2-2-1. 6. okosa okosa okosa = 2-2-2. Anything above 6 they call _ras_, a lot. For the sake of uniformity we may speak of this as a "system." But in so doing, we give to the legitimate meaning of the word a severe strain. The customs and modes of life of these people are not such as to require the use of any save the scanty list of numbers given above; and their mental poverty prompts them to call 3, the first number above a single pair, 2-1. In the same way, 4 and 6 are respectively 2 pairs and 3 pairs, while 5 is 1 more than 2 pairs. Five objects, however, they sometimes denote by _urapuni-getal_, 1 hand. A precisely similar condition is found to prevail respecting the arithmetic of all the Australian tribes. In some cases only two numerals are found, and in others three. But in a very great number of the native languages of that continent the count proceeds by pairs, if indeed it proceeds at all. Hence we at once reject the theory that Australian arithmetic, or Australian counting, is essentially peculiar. It is simply a legitimate result, such as might be looked for in any part of the world, of the barbarism in which the races of that quarter of the world were sunk, and in which they were content to live. The following examples of Australian and Tasmanian number systems show how scanty was the numerical ability possessed by these tribes, and illustrate fully their tendency to count by twos or pairs. MURRAY RIVER.[170] 1. enea. 2. petcheval. 3. petchevalenea = 2-1. 4. petcheval peteheval = 2-2. MAROURA. 1. nukee. 2. barkolo. 3. barkolo nuke = 2-1. 4. barkolo barkolo = 2-2. LAKE KOPPERAMANA. 1. ngerna. 2. mondroo. 3. barkooloo. 4. mondroo mondroo = 2-2. MORT NOULAR. 1. gamboden. 2. bengeroo. 3. bengeroganmel = 2-1. 4. bengeroovor bengeroo = 2 + 2. WIMMERA. 1. keyap. 2. pollit. 3. pollit keyap = 2-1. 4. pollit pollit = 2-2. POPHAM BAY. 1. motu. 2. lawitbari. 3. lawitbari-motu = 2-1. KAMILAROI.[171] 1. mal. 2. bularr. 3. guliba. 4. bularrbularr = 2-2. 5. bulaguliba = 2-3. 6. gulibaguliba = 3-3. PORT ESSINGTON.[172] 1. erad. 2. nargarik. 3. nargarikelerad = 2-1. 4. nargariknargarik = 2-2. WARREGO. 1. tarlina. 2. barkalo. 3. tarlina barkalo = 1-2. CROCKER ISLAND. 1. roka. 2. orialk. 3. orialkeraroka = 2-1. WARRIOR ISLAND.[173] 1. woorapoo. 2. ocasara. 3. ocasara woorapoo = 2-1. 4. ocasara ocasara = 2-2. DIPPIL.[174] 1. kalim. 2. buller. 3. boppa. 4. buller gira buller = 2 + 2. 5. buller gira buller kalim = 2 + 2 + 1. FRAZER'S ISLAND.[175] 1. kalim. 2. bulla. 3. goorbunda. 4. bulla-bulla = 2-2. MORETON'S BAY.[176] 1. kunner. 2. budela. 3. muddan. 4. budela berdelu = 2-2. ENCOUNTER BAY.[177] 1. yamalaitye. 2. ningenk. 3. nepaldar. 4. kuko kuko = 2-2, or pair pair. 5. kuko kuko ki = 2-2-1. 6. kuko kuko kuko = 2-2-2. 7. kuko kuko kuko ki = 2-2-2-1. ADELAIDE.[178] 1. kuma. 2. purlaitye, or bula. 3. marnkutye. 4. yera-bula = pair 2. 5. yera-bula kuma = pair 2-1. 6. yera-bula purlaitye = pair 2.2. WIRADUROI.[179] 1. numbai. 2. bula. 3. bula-numbai = 2-1. 4. bungu = many. 5. bungu-galan = very many. WIRRI-WIRRI.[180] 1. mooray. 2. boollar. 3. belar mooray = 2-1. 4. boollar boollar = 2-2. 5. mongoonballa. 6. mongun mongun. COOPER'S CREEK.[181] 1. goona. 2. barkoola. 3. barkoola goona = 2-1. 4. barkoola barkoola = 2-2. BOURKE, DARLING RIVER.[182] 1. neecha. 2. boolla. 4. boolla neecha = 2-1. 3. boolla boolla = 2-2. MURRAY RIVER, N.W. BEND.[183] 1. mata. 2. rankool. 3. rankool mata = 2-1. 4. rankool rankool = 2-2. YIT-THA.[184] 1. mo. 2. thral. 3. thral mo = 2-1. 4. thral thral = 2-2. PORT DARWIN.[185] 1. kulagook. 2. kalletillick. 3. kalletillick kulagook = 2-1. 4. kalletillick kalletillick = 2-2. CHAMPION BAY.[186] 1. kootea. 2. woothera. 3. woothera kootea = 2-1. 4. woothera woothera = 2-2. BELYANDO RIVER.[187] 1. wogin. 2. booleroo. 3. booleroo wogin = 2-1. 4. booleroo booleroo = 2-2. WARREGO RIVER. 1. onkera. 2. paulludy. 3. paulludy onkera = 2-1. 4. paulludy paulludy = 2-2. RICHMOND RIVER. 1. yabra. 2. booroora. 3. booroora yabra = 2-1. 4. booroora booroora = 2-2. PORT MACQUARIE. 1. warcol. 2. blarvo. 3. blarvo warcol = 2-1. 4. blarvo blarvo = 2-2. HILL END. 1. miko. 2. bullagut. 3. bullagut miko = 2-1. 4. bullagut bullagut = 2-2. MONEROO 1. boor. 2. wajala, blala. 3. blala boor = 2-1. 4. wajala wajala. GONN STATION. 1. karp. 2. pellige. 3. pellige karp = 2-1. 4. pellige pellige = 2-2. UPPER YARRA. 1. kaambo. 2. benjero. 3. benjero kaambo = 2-2. 4. benjero on benjero = 2-2. OMEO. 1. bore. 2. warkolala. 3. warkolala bore = 2-1. 4. warkolala warkolala = 2-2. SNOWY RIVER. 1. kootook. 2. boolong. 3. booloom catha kootook = 2 + 1. 4. booloom catha booloom = 2 + 2. NGARRIMOWRO. 1. warrangen. 2. platir. 3. platir warrangen = 2-1. 4. platir platir = 2-2. This Australian list might be greatly extended, but the scales selected may be taken as representative examples of Australian binary scales. Nearly all of them show a structure too clearly marked to require comment. In a few cases, however, the systems are to be regarded rather as showing a trace of binary structure, than as perfect examples of counting by twos. Examples of this nature are especially numerous in Curr's extensive list--the most complete collection of Australian vocabularies ever made. A few binary scales have been found in South America, but they show no important variation on the Australian systems cited above. The only ones I have been able to collect are the following: BAKAIRI.[188] 1. tokalole. 2. asage. 3. asage tokalo = 2-1. 4. asage asage = 2-2. ZAPARA.[189] 1. nuquaqui. 2. namisciniqui. 3. haimuckumarachi. 4. namisciniqui ckara maitacka = 2 + 2. 5. namisciniqui ckara maitacka nuquaqui = 2 pairs + 1. 6. haimuckumaracki ckaramsitacka = 3 pairs. APINAGES.[190] 1. pouchi. 2. at croudou. 3. at croudi-pshi = 2-1. 4. agontad-acroudo = 2-2. COTOXO.[191] 1. ihueto. 2. ize. 3. ize-te-hueto = 2-1. 4. ize-te-seze = 2-2. 5. ize-te-seze-hue = 2-2-1. MBAYI.[192] 1. uninitegui. 2. iniguata. 3. iniguata dugani = 2 over. 4. iniguata driniguata = 2-2. 5. oguidi = many. TAMA.[193] 1. teyo. 2. cayapa. 3. cho-teyo = 2 + 1. 4. cayapa-ria = 2 again. 5. cia-jente = hand. CURETU.[194] 1. tchudyu. 2. ap-adyu. 3. arayu. 4. apaedyái = 2 + 2. 5. tchumupa. If the existence of number systems like the above are to be accounted for simply on the ground of low civilization, one might reasonably expect to find ternary and and quaternary scales, as well as binary. Such scales actually exist, though not in such numbers as the binary. An example of the former is the Betoya scale,[195] which runs thus: 1. edoyoyoi. 2. edoi = another. 3. ibutu = beyond. 4. ibutu-edoyoyoi = beyond 1, or 3-1. 5. ru-mocoso = hand. The Kamilaroi scale, given as an example of binary formation, is partly ternary; and its word for 6, _guliba guliba_, 3-3, is purely ternary. An occasional ternary trace is also found in number systems otherwise decimal or quinary vigesimal; as the _dlkunoutl_, second 3, of the Haida Indians of British Columbia. The Karens of India[196] in a system otherwise strictly decimal, exhibit the following binary-ternary-quaternary vagary: 6. then tho = 3 × 2. 7. then tho ta = 3 × 2-1. 8. lwie tho = 4 × 2. 9. lwie tho ta = 4 × 2-1. In the Wokka dialect,[197] found on the Burnett River, Australia, a single ternary numeral is found, thus: 1. karboon. 2. wombura. 3. chrommunda. 4. chrommuda karboon = 3-1. Instances of quaternary numeration are less rare than are those of ternary, and there is reason to believe that this method of counting has been practised more extensively than any other, except the binary and the three natural methods, the quinary, the decimal, and the vigesimal. The number of fingers on one hand is, excluding the thumb, four. Possibly there have been tribes among which counting by fours arose as a legitimate, though unusual, result of finger counting; just as there are, now and then, individuals who count on their fingers with the forefinger as a starting-point. But no such practice has ever been observed among savages, and such theorizing is the merest guess-work. Still a definite tendency to count by fours is sometimes met with, whatever be its origin. Quaternary traces are repeatedly to be found among the Indian languages of British Columbia. In describing the Columbians, Bancroft says: "Systems of numeration are simple, proceeding by fours, fives, or tens, according to the different languages...."[198] The same preference for four is said to have existed in primitive times in the languages of Central Asia, and that this form of numeration, resulting in scores of 16 and 64, was a development of finger counting.[199] In the Hawaiian and a few other languages of the islands of the central Pacific, where in general the number systems employed are decimal, we find a most interesting case of the development, within number scales already well established, of both binary and quaternary systems. Their origin seems to have been perfectly natural, but the systems themselves must have been perfected very slowly. In Tahitian, Rarotongan, Mangarevan, and other dialects found in the neighbouring islands of those southern latitudes, certain of the higher units, _tekau_, _rau_, _mano_, which originally signified 10, 100, 1000, have become doubled in value, and now stand for 20, 200, 2000. In Hawaiian and other dialects they have again been doubled, and there they stand for 40, 400, 4000.[200] In the Marquesas group both forms are found, the former in the southern, the latter in the northern, part of the archipelago; and it seems probable that one or both of these methods of numeration are scattered somewhat widely throughout that region. The origin of these methods is probably to be found in the fact that, after the migration from the west toward the east, nearly all the objects the natives would ever count in any great numbers were small,--as yams, cocoanuts, fish, etc.,--and would be most conveniently counted by pairs. Hence the native, as he counted one pair, two pairs, etc., might readily say _one_, _two_, and so on, omitting the word "pair" altogether. Having much more frequent occasion to employ this secondary than the primary meaning of his numerals, the native would easily allow the original significations to fall into disuse, and in the lapse of time to be entirely forgotten. With a subsequent migration to the northward a second duplication might take place, and so produce the singular effect of giving to the same numeral word three different meanings in different parts of Oceania. To illustrate the former or binary method of numeration, the Tahuatan, one of the southern dialects of the Marquesas group, may be employed.[201] Here the ordinary numerals are: 1. tahi, 10. onohuu. 20. takau. 200. au. 2,000. mano. 20,000. tini. 20,000. tufa. 2,000,000. pohi. In counting fish, and all kinds of fruit, except breadfruit, the scale begins with _tauna_, pair, and then, omitting _onohuu_, they employ the same words again, but in a modified sense. _Takau_ becomes 10, _au_ 100, etc.; but as the word "pair" is understood in each case, the value is the same as before. The table formed on this basis would be: 2 (units) = 1 tauna = 2. 10 tauna = 1 takau = 20. 10 takau = 1 au = 200. 10 au = 1 mano = 2000. 10 mano = 1 tini = 20,000. 10 tini = 1 tufa = 200,000. 10 tufa = 1 pohi = 2,000,000. For counting breadfruit they use _pona_, knot, as their unit, breadfruit usually being tied up in knots of four. _Takau_ now takes its third signification, 40, and becomes the base of their breadfruit system, so to speak. For some unknown reason the next unit, 400, is expressed by _tauau_, while _au_, which is the term that would regularly stand for that number, has, by a second duplication, come to signify 800. The next unit, _mano_, has in a similar manner been twisted out of its original sense, and in counting breadfruit is made to serve for 8000. In the northern, or Nukuhivan Islands, the decimal-quaternary system is more regular. It is in the counting of breadfruit only,[202] 4 breadfruits = 1 pona = 4. 10 pona = 1 toha = 40. 10 toha = 1 au = 400. 10 au = 1 mano = 4000. 10 mano = 1 tini = 40,000. 10 tini = 1 tufa = 400,000. 10 tufa = 1 pohi = 4,000,000. In the Hawaiian dialect this scale is, with slight modification, the universal scale, used not only in counting breadfruit, but any other objects as well. The result is a complete decimal-quaternary system, such as is found nowhere else in the world except in this and a few of the neighbouring dialects of the Pacific. This scale, which is almost identical with the Nukuhivan, is[203] 4 units = 1 ha or tauna = 4. 10 tauna = 1 tanaha = 40. 10 tanaha = 1 lau = 400. 10 lau = 1 mano = 4000. 10 mano = 1 tini = 40,000. 10 tini = 1 lehu = 400,000. The quaternary element thus introduced has modified the entire structure of the Hawaiian number system. Fifty is _tanaha me ta umi_, 40 + 10; 76 is 40 + 20 + 10 + 6; 100 is _ua tanaha ma tekau_, 2 × 40 + 10; 200 is _lima tanaha_, 5 × 40; and 864,895 is 2 × 400,000 + 40,000 + 6 × 4000 + 2 × 400 + 2 × 40 + 10 + 5.[204] Such examples show that this secondary influence, entering and incorporating itself as a part of a well-developed decimal system, has radically changed it by the establishment of 4 as the primary number base. The role which 10 now plays is peculiar. In the natural formation of a quaternary scale new units would be introduced at 16, 64, 256, etc.; that is, at the square, the cube, and each successive power of the base. But, instead of this, the new units are introduced at 10 × 4, 100 × 4, 1000 × 4, etc.; that is, at the products of 4 by each successive power of the old base. This leaves the scale a decimal scale still, even while it may justly be called quaternary; and produces one of the most singular and interesting instances of number-system formation that has ever been observed. In this connection it is worth noting that these Pacific island number scales have been developed to very high limits--in some cases into the millions. The numerals for these large numbers do not seem in any way indefinite, but rather to convey to the mind of the native an idea as clear as can well be conveyed by numbers of such magnitude. Beyond the limits given, the islanders have indefinite expressions, but as far as can be ascertained these are only used when the limits given above have actually been passed. To quote one more example, the Hervey Islanders, who have a binary-decimal scale, count as follows: 5 kaviri (bunches of cocoanuts) = 1 takau = 20. 10 takau = 1 rau = 200. 10 rau = 1 mano = 2000. 10 mano = 1 kiu = 20,000. 10 kiu = 1 tini = 200,000. Anything above this they speak of in an uncertain way, as _mano mano_ or _tini tini_, which may, perhaps, be paralleled by our English phrases "myriads upon myriads," and "millions of millions."[205] It is most remarkable that the same quarter of the globe should present us with the stunted number sense of the Australians, and, side by side with it, so extended and intelligent an appreciation of numerical values as that possessed by many of the lesser tribes of Polynesia. The Luli of Paraguay[206] show a decided preference for the base 4. This preference gives way only when they reach the number 10, which is an ordinary digit numeral. All numbers above that point belong rather to decimal than to quaternary numeration. Their numerals are: 1. alapea. 2. tamop. 3. tamlip. 4. lokep. 5. lokep moile alapea = 4 with 1, or is-alapea = hand 1. 6. lokep moile tamop = 4 with 2. 7. lokep moile tamlip = 4 with 3. 8. lokep moile lokep = 4 with 4. 9. lokep moile lokep alapea = 4 with 4-1. 10. is yaoum = all the fingers of hand. 11. is yaoum moile alapea = all the fingers of hand with 1. 20. is elu yaoum = all the fingers of hand and foot. 30. is elu yaoum moile is-yaoum = all the fingers of hand and foot with all the fingers of hand. Still another instance of quaternary counting, this time carrying with it a suggestion of binary influence, is furnished by the Mocobi[207] of the Parana region. Their scale is exceedingly rude, and they use the fingers and toes almost exclusively in counting; only using their spoken numerals when, for any reason, they wish to dispense with the aid of their hands and feet. Their first eight numerals are: 1. iniateda. 2. inabaca. 3. inabacao caini = 2 above. 4. inabacao cainiba = 2 above 2; or natolatata. 5. inibacao cainiba iniateda = 2 above 2-1; or natolatata iniateda = 4-1. 6. natolatatata inibaca = 4-2. 7. natolata inibacao-caini = 4-2 above. 8. natolata-natolata = 4-4. There is probably no recorded instance of a number system formed on 6, 7, 8, or 9 as a base. No natural reason exists for the choice of any of these numbers for such a purpose; and it is hardly conceivable that any race should proceed beyond the unintelligent binary or quaternary stage, and then begin the formation of a scale for counting with any other base than one of the three natural bases to which allusion has already been made. Now and then some anomalous fragment is found imbedded in an otherwise regular system, which carries us back to the time when the savage was groping his way onward in his attempt to give expression to some number greater than any he had ever used before; and now and then one of these fragments is such as to lead us to the border land of the might-have-been, and to cause us to speculate on the possibility of so great a numerical curiosity as a senary or a septenary scale. The Bretons call 18 _triouec'h_, 3-6, but otherwise their language contains no hint of counting by sixes; and we are left at perfect liberty to theorize at will on the existence of so unusual a number word. Pott remarks[208] that the Bolans, of western Africa, appear to make some use of 6 as their number base, but their system, taken as a whole, is really a quinary-decimal. The language of the Sundas,[209] or mountaineers of Java, contains traces of senary counting. The Akra words for 7 and 8, _paggu_ and _paniu_, appear to mean 6-1 and 7-1, respectively; and the same is true of the corresponding Tambi words _pagu_ and _panjo_.[210] The Watji tribe[211] call 6 _andee_, and 7 _anderee_, which probably means 6-1. These words are to be regarded as accidental variations on the ordinary laws of formation, and are no more significant of a desire to count by sixes than is the Wallachian term _deu-maw_, which expresses 18 as 2-9, indicates the existence of a scale of which 9 is the base. One remarkably interesting number system is that exhibited by the Mosquito tribe[212] of Central America, who possess an extensive quinary-vigesimal scale containing one binary and three senary compounds. The first ten words of this singular scale, which has already been quoted, are: 1. kumi. 2. wal. 3. niupa. 4. wal-wal = 2-2. 5. mata-sip = fingers of one hand. 6. matlalkabe. 7. matlalkabe pura kumi = 6 + 1. 8. matlalkabe pura wal = 6 + 2. 9. matlalkabe pura niupa = 6 + 3. 10. mata-wal-sip = fingers of the second hand. In passing from 6 to 7, this tribe, also, has varied the almost universal law of progression, and has called 7 6-1. Their 8 and 9 are formed in a similar manner; but at 10 the ordinary method is resumed, and is continued from that point onward. Few number systems contain as many as three numerals which are associated with 6 as their base. In nearly all instances we find such numerals singly, or at most in pairs; and in the structure of any system as a whole, they are of no importance whatever. For example, in the Pawnee, a pure decimal scale, we find the following odd sequence:[213] 6. shekshabish. 7. petkoshekshabish = 2-6, _i.e._ 2d 6. 8. touwetshabish = 3-6, _i.e._ 3d 6. 9. loksherewa = 10 - 1. In the Uainuma scale the expressions for 7 and 8 are obviously referred to 6, though the meaning of 7 is not given, and it is impossible to guess what it really does signify. The numerals in question are:[214] 6. aira-ettagapi. 7. aira-ettagapi-hairiwigani-apecapecapsi. 8. aira-ettagapi-matschahma = 6 + 2. In the dialect of the Mille tribe a single trace of senary counting appears, as the numerals given below show:[215] 6. dildjidji. 7. dildjidji me djuun = 6 + 1. Finally, in the numerals used by the natives of the Marshall Islands, the following curiously irregular sequence also contains a single senary numeral:[216] 6. thil thino = 3 + 3. 7. thilthilim-thuon = 6 + 1. 8. rua-li-dok = 10 - 2. 9. ruathim-thuon = 10 - 2 + 1. Many years ago a statement appeared which at once attracted attention and awakened curiosity. It was to the effect that the Maoris, the aboriginal inhabitants of New Zealand, used as the basis of their numeral system the number 11; and that the system was quite extensively developed, having simple words for 121 and 1331, _i.e._ for the square and cube of 11. No apparent reason existed for this anomaly, and the Maori scale was for a long time looked upon as something quite exceptional and outside all ordinary rules of number-system formation. But a closer and more accurate knowledge of the Maori language and customs served to correct the mistake, and to show that this system was a simple decimal system, and that the error arose from the following habit. Sometimes when counting a number of objects the Maoris would put aside 1 to represent each 10, and then those so set aside would afterward be counted to ascertain the number of tens in the heap. Early observers among this people, seeing them count 10 and then set aside 1, at the same time pronouncing the word _tekau_, imagined that this word meant 11, and that the ignorant savage was making use of this number as his base. This misconception found its way into the early New Zealand dictionary, but was corrected in later editions. It is here mentioned only because of the wide diffusion of the error, and the interest it has always excited.[217] Aside from our common decimal scale, there exist in the English language other methods of counting, some of them formal enough to be dignified by the term _system_--as the sexagesimal method of measuring time and angular magnitude; and the duodecimal system of reckoning, so extensively used in buying and selling. Of these systems, other than decimal, two are noticed by Tylor,[218] and commented on at some length, as follows: "One is the well-known dicing set, _ace_, _deuce_, _tray_, _cater_, _cinque_, _size_; thus _size-ace_ is 6-1, _cinques_ or _sinks_, double 5. These came to us from France, and correspond with the common French numerals, except _ace_, which is Latin _as_, a word of great philological interest, meaning 'one.' The other borrowed set is to be found in the _Slang Dictionary_. It appears that the English street-folk have adopted as a means of secret communication a set of Italian numerals from the organ-grinders and image-sellers, or by other ways through which Italian or Lingua Franca is brought into the low neighbourhoods of London. In so doing they have performed a philological operation not only curious but instructive. By copying such expressions as _due soldi_, _tre soldi_, as equivalent to 'twopence,' 'threepence,' the word _saltee_ became a recognized slang term for 'penny'; and pence are reckoned as follows: oney saltee 1d. uno soldo. dooe saltee 2d. due soldi. tray saltee 3d. tre soldi. quarterer saltee 4d. quattro soldi. chinker saltee 5d. cinque soldi. say saltee 6d. sei soldi. say oney saltee, or setter saltee 7d. sette soldi. say dooe saltee, or otter saltee 8d. otto soldi. say tray saltee, or nobba saltee 9d. nove soldi. say quarterer saltee, or dacha saltee 10d. dieci soldi. say chinker saltee or dacha oney saltee 11d. undici soldi. oney beong 1s. a beong say saltee 1s. 6d. dooe beong say saltee, or madza caroon 2s. 6d. (half-crown, mezza corona). One of these series simply adopts Italian numerals decimally. But the other, when it has reached 6, having had enough of novelty, makes 7 by 6-1, and so forth. It is for no abstract reason that 6 is thus made the turning-point, but simply because the costermonger is adding pence up to the silver sixpence, and then adding pence again up to the shilling. Thus our duodecimal coinage has led to the practice of counting by sixes, and produced a philological curiosity, a real senary notation." In addition to the two methods of counting here alluded to, another may be mentioned, which is equally instructive as showing how readily any special method of reckoning may be developed out of the needs arising in connection with any special line of work. As is well known, it is the custom in ocean, lake, and river navigation to measure soundings by the fathom. On the Mississippi River, where constant vigilance is needed because of the rapid shifting of sand-bars, a special sounding nomenclature has come into vogue,[219] which the following terms will illustrate: 5 ft. = five feet. 6 ft. = six feet. 9 ft. = nine feet. 10-1/2 ft. = a quarter less twain; _i.e._ a quarter of a fathom less than 2. 12 ft. = mark twain. 13-1/2 ft. = a quarter twain. 16-1/2 ft. = a quarter less three. 18 ft. = mark three. 19-1/2 ft. = a quarter three. 24 ft. = deep four. As the soundings are taken, the readings are called off in the manner indicated in the table; 10-1/2 feet being "a quarter less twain," 12 feet "mark twain," etc. Any sounding above "deep four" is reported as "no bottom." In the Atlantic and Gulf waters on the coast of this country the same system prevails, only it is extended to meet the requirements of the deeper soundings there found, and instead of "six feet," "mark twain," etc., we find the fuller expressions, "by the mark one," "by the mark two," and so on, as far as the depth requires. This example also suggests the older and far more widely diffused method of reckoning time at sea by bells; a system in which "one bell," "two bells," "three bells," etc., mark the passage of time for the sailor as distinctly as the hands of the clock could do it. Other examples of a similar nature will readily suggest themselves to the mind. Two possible number systems that have, for purely theoretical reasons, attracted much attention, are the octonary and the duodecimal systems. In favour of the octonary system it is urged that 8 is an exact power of 2; or in other words, a large number of repeated halves can be taken with 8 as a starting-point, without producing a fractional result. With 8 as a base we should obtain by successive halvings, 4, 2, 1. A similar process in our decimal scale gives 5, 2-1/2, 1-1/4. All this is undeniably true, but, granting the argument up to this point, one is then tempted to ask "What of it?" A certain degree of simplicity would thereby be introduced into the Theory of Numbers; but the only persons sufficiently interested in this branch of mathematics to appreciate the benefit thus obtained are already trained mathematicians, who are concerned rather with the pure science involved, than with reckoning on any special base. A slightly increased simplicity would appear in the work of stockbrokers, and others who reckon extensively by quarters, eighths, and sixteenths. But such men experience no difficulty whatever in performing their mental computations in the decimal system; and they acquire through constant practice such quickness and accuracy of calculation, that it is difficult to see how octonary reckoning would materially assist them. Altogether, the reasons that have in the past been adduced in favour of this form of arithmetic seem trivial. There is no record of any tribe that ever counted by eights, nor is there the slightest likelihood that such a system could ever meet with any general favour. It is said that the ancient Saxons used the octonary system,[220] but how, or for what purposes, is not stated. It is not to be supposed that this was the common system of counting, for it is well known that the decimal scale was in use as far back as the evidence of language will take us. But the field of speculation into which one is led by the octonary scale has proved most attractive to some, and the conclusion has been soberly reached, that in the history of the Aryan race the octonary was to be regarded as the predecessor of the decimal scale. In support of this theory no direct evidence is brought forward, but certain verbal resemblances. Those ignes fatuii of the philologist are made to perform the duty of supporting an hypothesis which would never have existed but for their own treacherous suggestions. Here is one of the most attractive of them: Between the Latin words _novus_, new, and _novem_, nine, there exists a resemblance so close that it may well be more than accidental. Nine is, then, the _new_ number; that is, the first number on a new count, of which 8 must originally have been the base. Pursuing this thought by investigation into different languages, the same resemblance is found there. Hence the theory is strengthened by corroborative evidence. In language after language the same resemblance is found, until it seems impossible to doubt, that in prehistoric times, 9 _was_ the new number--the beginning of a second tale. The following table will show how widely spread is this coincidence: Sanskrit, navan = 9. nava = new. Persian, nuh = 9. nau = new. Greek, [Greek: ennea] = 9. [Greek: neos] = new. Latin, novem = 9. novus = new. German, neun = 9. neu = new. Swedish, nio = 9. ny = new. Dutch, negen = 9. nieuw = new. Danish, ni = 9. ny = new. Icelandic, nyr = 9. niu = new. English, nine = 9. new = new. French, neuf = 9. nouveau = new. Spanish, nueve = 9. neuvo = new. Italian, nove = 9. nuovo = new. Portuguese, nove = 9. novo = new. Irish, naoi = 9. nus = new. Welsh, naw = 9. newydd = new. Breton, nevez = 9. nuhue = new.[221] This table might be extended still further, but the above examples show how widely diffused throughout the Aryan languages is this resemblance. The list certainly is an impressive one, and the student is at first thought tempted to ask whether all these resemblances can possibly have been accidental. But a single consideration sweeps away the entire argument as though it were a cobweb. All the languages through which this verbal likeness runs are derived directly or indirectly from one common stock; and the common every-day words, "nine" and "new," have been transmitted from that primitive tongue into all these linguistic offspring with but little change. Not only are the two words in question akin in each individual language, but _they are akin in all the languages_. Hence all these resemblances reduce to a single resemblance, or perhaps identity, that between the Aryan words for "nine" and "new." This was probably an accidental resemblance, no more significant than any one of the scores of other similar cases occurring in every language. If there were any further evidence of the former existence of an Aryan octonary scale, the coincidence would possess a certain degree of significance; but not a shred has ever been produced which is worthy of consideration. If our remote ancestors ever counted by eights, we are entirely ignorant of the fact, and must remain so until much more is known of their language than scholars now have at their command. The word resemblances noted above are hardly more significant than those occurring in two Polynesian languages, the Fatuhivan and the Nakuhivan,[222] where "new" is associated with the number 7. In the former case 7 is _fitu_, and "new" is _fou_; in the latter 7 is _hitu_, and "new" is _hou_. But no one has, because of this likeness, ever suggested that these tribes ever counted by the senary method. Another equally trivial resemblance occurs in the Tawgy and the Kamassin languages,[223] thus: TAWGY. KAMASSIN. 8. siti-data = 2 × 4. 8. sin-the'de = 2 × 4. 9. nameaitjuma = another. 9. amithun = another. But it would be childish to argue, from this fact alone, that either 4 or 8 was the number base used. In a recent antiquarian work of considerable interest, the author examines into the question of a former octonary system of counting among the various races of the world, particularly those of Asia, and brings to light much curious and entertaining material respecting the use of this number. Its use and importance in China, India, and central Asia, as well as among some of the islands of the Pacific, and in Central America, leads him to the conclusion that there was a time, long before the beginning of recorded history, when 8 was the common number base of the world. But his conclusion has no basis in his own material even. The argument cannot be examined here, but any one who cares to investigate it can find there an excellent illustration of the fact that a pet theory may take complete possession of its originator, and reduce him finally to a state of infantile subjugation.[224] Of all numbers upon which a system could be based, 12 seems to combine in itself the greatest number of advantages. It is capable of division by 2, 3, 4, and 6, and hence admits of the taking of halves, thirds, quarters, and sixths of itself without the introduction of fractions in the result. From a commercial stand-point this advantage is very great; so great that many have seriously advocated the entire abolition of the decimal scale, and the substitution of the duodecimal in its stead. It is said that Charles XII. of Sweden was actually contemplating such a change in his dominions at the time of his death. In pursuance of this idea, some writers have gone so far as to suggest symbols for 10 and 11, and to recast our entire numeral nomenclature to conform to the duodecimal base.[225] Were such a change made, we should express the first nine numbers as at present, 10 and 11 by new, single symbols, and 12 by 10. From this point the progression would be regular, as in the decimal scale--only the same combination of figures in the different scales would mean very different things. Thus, 17 in the decimal scale would become 15 in the duodecimal; 144 in the decimal would become 100 in the duodecimal; and 1728, the cube of the new base, would of course be represented by the figures 1000. It is impossible that any such change can ever meet with general or even partial favour, so firmly has the decimal scale become intrenched in its position. But it is more than probable that a large part of the world of trade and commerce will continue to buy and sell by the dozen, the gross, or some multiple or fraction of the one or the other, as long as buying and selling shall continue. Such has been its custom for centuries, and such will doubtless be its custom for centuries to come. The duodecimal is not a natural scale in the same sense as are the quinary, the decimal, and the vigesimal; but it is a system which is called into being long after the complete development of one of the natural systems, solely because of the simple and familiar fractions into which its base is divided. It is the scale of civilization, just as the three common scales are the scales of nature. But an example of its use was long sought for in vain among the primitive races of the world. Humboldt, in commenting on the number systems of the various peoples he had visited during his travels, remarked that no race had ever used exclusively that best of bases, 12. But it has recently been announced[226] that the discovery of such a tribe had actually been made, and that the Aphos of Benuë, an African tribe, count to 12 by simple words, and then for 13 say 12-1, for 14, 12-2, etc. This report has yet to be verified, but if true it will constitute a most interesting addition to anthropological knowledge. CHAPTER VI. THE QUINARY SYSTEM. The origin of the quinary mode of counting has been discussed with some fulness in a preceding chapter, and upon that question but little more need be said. It is the first of the natural systems. When the savage has finished his count of the fingers of a single hand, he has reached this natural number base. At this point he ceases to use simple numbers, and begins the process of compounding. By some one of the numerous methods illustrated in earlier chapters, he passes from 5 to 10, using here the fingers of his second hand. He now has two fives; and, just as we say "twenty," _i.e._ two tens, he says "two hands," "the second hand finished," "all the fingers," "the fingers of both hands," "all the fingers come to an end," or, much more rarely, "one man." That is, he is, in one of the many ways at his command, saying "two fives." At 15 he has "three hands" or "one foot"; and at 20 he pauses with "four hands," "hands and feet," "both feet," "all the fingers of hands and feet," "hands and feet finished," or, more probably, "one man." All these modes of expression are strictly natural, and all have been found in the number scales which were, and in many cases still are, in daily use among the uncivilized races of mankind. In its structure the quinary is the simplest, the most primitive, of the natural systems. Its base is almost always expressed by a word meaning "hand," or by some equivalent circumlocution, and its digital origin is usually traced without difficulty. A consistent formation would require the expression of 10 by some phrase meaning "two fives," 15 by "three fives," etc. Such a scale is the one obtained from the Betoya language, already mentioned in Chapter III., where the formation of the numerals is purely quinary, as the following indicate:[227] 5. teente = 1 hand. 10. cayaente, or caya huena = 2 hands. 15. toazumba-ente = 3 hands. 20. caesa-ente = 4 hands. The same formation appears, with greater or less distinctness, in many of the quinary scales already quoted, and in many more of which mention might be made. Collecting the significant numerals from a few such scales, and tabulating them for the sake of convenience of comparison, we see this point clearly illustrated by the following: TAMANAC. 5. amnaitone = 1 hand. 10. amna atse ponare = 2 hands. ARAWAK, GUIANA. 5. abba tekkabe = 1 hand. 10. biamantekkabe = 2 hands. JIVIRO. 5. alacötegladu = 1 hand. 10. catögladu = 2 hands. NIAM NIAM 5. biswe 10. bauwe = 2d 5. NENGONES 5. se dono = the end (of the fingers of 1 hand). 10. rewe tubenine = 2 series (of fingers). SESAKE.[228] 5. lima = hand. 10. dua lima = 2 hands. AMBRYM.[229] 5. lim = hand. 10. ra-lim = 2 hands. PAMA.[229] 5. e-lime = hand. 10. ha-lua-lim = the 2 hands. DINKA.[230] 5. wdyets. 10. wtyer, or wtyar = 5 × 2. BARI 5. kanat 10. puök = 5 + 5? KANURI 5. ugu. 10. megu = 2 × 5. RIO NORTE AND SAN ANTONIO.[231] 5. juyopamauj. 10. juyopamauj ajte = 5 × 2. API.[232] 5. lima. 10. lua-lima = 2 × 5. ERROMANGO 5. suku-rim. 10. nduru-lim = 2 × 5. TLINGIT, BRITISH COLUMBIA.[233] 5. kedjin (from djin = hand). 10. djinkat = both hands? Thus far the quinary formation is simple and regular; and in view of the evidence with which these and similar illustrations furnish us, it is most surprising to find an eminent authority making the unequivocal statement that the number 10 is nowhere expressed by 2 fives[234]--that all tribes which begin their count on a quinary base express 10 by a simple word. It is a fact, as will be fully illustrated in the following pages, that quinary number systems, when extended, usually merge into either the decimal or the vigesimal. The result is, of course, a compound of two, and sometimes of three, systems in one scale. A pure quinary or vigesimal number system is exceedingly rare; but quinary scales certainly do exist in which, as far as we possess the numerals, no trace of any other influence appears. It is also to be noticed that some tribes, like the Eskimos of Point Barrow, though their systems may properly be classed as mixed systems, exhibit a decided preference for 5 as a base, and in counting objects, divided into groups of 5, obtaining the sum in this way.[235] But the savage, after counting up to 10, often finds himself unconsciously impelled to depart from his strict reckoning by fives, and to assume a new basis of reference. Take, for example, the Zuñi system, in which the first 2 fives are: 5. öpte = the notched off. 10. astem'thla = all the fingers. It will be noticed that the Zuñi does not say "two hands," or "the fingers of both hands," but simply "all the fingers." The 5 is no longer prominent, but instead the mere notion of one entire count of the fingers has taken its place. The division of the fingers into two sets of five each is still in his mind, but it is no longer the leading idea. As the count proceeds further, the quinary base may be retained, or it may be supplanted by a decimal or a vigesimal base. How readily the one or the other may predominate is seen by a glance at the following numerals: GALIBI.[236] 5. atoneigne oietonaï = 1 hand. 10. oia batoue = the other hand. 20. poupoupatoret oupoume = feet and hands. 40. opoupoume = twice the feet and hands. GUARANI.[237] 5. ace popetei = 1 hand. 10. ace pomocoi = 2 hands. 20. acepo acepiabe = hands and feet. FATE.[238] 5. lima = hand. 10. relima = 2 hands. 20. relima rua = (2 × 5) × 2. KIRIRI 5. mibika misa = 1 hand. 10. mikriba misa sai = both hands. 20. mikriba nusa ideko ibi sai = both hands together with the feet. ZAMUCO 5. tsuena yimana-ite = ended 1 hand. 10. tsuena yimana-die = ended both hands. 20. tsuena yiri-die = ended both feet. PIKUMBUL 5. mulanbu. 10. bularin murra = belonging to the two hands. 15. mulanba dinna = 5 toes added on (to the 10 fingers). 20. bularin dinna = belonging to the 2 feet. YARUROS.[239] 5. kani-iktsi-mo = 1 hand alone. 10. yowa-iktsi-bo = all the hands. 15. kani-tao-mo = 1 foot alone. 20. kani-pume = 1 man. By the time 20 is reached the savage has probably allowed his conception of any aggregate to be so far modified that this number does not present itself to his mind as 4 fives. It may find expression in some phraseology such as the Kiriris employ--"both hands together with the feet"--or in the shorter "ended both feet" of the Zamucos, in which case we may presume that he is conscious that his count has been completed by means of the four sets of fives which are furnished by his hands and feet. But it is at least equally probable that he instinctively divides his total into 2 tens, and thus passes unconsciously from the quinary into the decimal scale. Again, the summing up of the 10 fingers and 10 toes often results in the concept of a single whole, a lump sum, so to speak, and the savage then says "one man," or something that gives utterance to this thought of a new unit. This leads the quinary into the vigesimal scale, and produces the combination so often found in certain parts of the world. Thus the inevitable tendency of any number system of quinary origin is toward the establishment of another and larger base, and the formation of a number system in which both are used. Wherever this is done, the greater of the two bases is always to be regarded as the principal number base of the language, and the 5 as entirely subordinate to it. It is hardly correct to say that, as a number system is extended, the quinary element disappears and gives place to the decimal or vigesimal, but rather that it becomes a factor of quite secondary importance in the development of the scale. If, for example, 8 is expressed by 5-3 in a quinary decimal system, 98 will be 9 × 10 + 5-3. The quinary element does not disappear, but merely sinks into a relatively unimportant position. One of the purest examples of quinary numeration is that furnished by the Betoya scale, already given in full in Chapter III., and briefly mentioned at the beginning of this chapter. In the simplicity and regularity of its construction it is so noteworthy that it is worth repeating, as the first of the long list of quinary systems given in the following pages. No further comment is needed on it than that already made in connection with its digital significance. As far as given by Dr. Brinton the scale is: 1. tey. 2. cayapa. 3. toazumba. 4. cajezea = 2 with plural termination. 5. teente = hand. 6. teyente tey = hand 1. 7. teyente cayapa = hand 2. 8. teyente toazumba = hand 3. 9. teyente caesea = hand 4. 10. caya ente, or caya huena = 2 hands. 11. caya ente-tey = 2 hands 1. 15. toazumba-ente = 3 hands. 16. toazumba-ente-tey = 3 hands 1. 20. caesea ente = 4 hands. A far more common method of progression is furnished by languages which interrupt the quinary formation at 10, and express that number by a single word. Any scale in which this takes place can, from this point onward, be quinary only in the subordinate sense to which allusion has just been made. Examples of this are furnished in a more or less perfect manner by nearly all so-called quinary-vigesimal and quinary-decimal scales. As fairly representing this phase of number-system structure, I have selected the first 20 numerals from the following languages: WELSH.[240] 1. un. 2. dau. 3. tri. 4. pedwar. 5. pump. 6. chwech. 7. saith. 8. wyth. 9. naw. 10. deg. 11. un ar ddeg = 1 + 10. 12. deuddeg = 2 + 10. 13. tri ar ddeg = 3 + 10. 14. pedwar ar ddeg = 4 + 10. 15. pymtheg = 5 + 10. 16. un ar bymtheg = 1 + 5 + 10. 17. dau ar bymtheg = 2 + 5 + 10. 18. tri ar bymtheg = 3 + 5 + 10. 19. pedwar ar bymtheg = 4 + 5 + 10. 20. ugain. NAHUATL.[241] 1. ce. 2. ome. 3. yei. 4. naui. 5. macuilli. 6. chiquacen = [5] + 1. 7. chicome = [5] + 2. 8. chicuey = [5] + 3. 9. chiucnaui = [5] + 4. 10. matlactli. 11. matlactli oce = 10 + 1. 12. matlactli omome = 10 + 2. 13. matlactli omey = 10 + 3. 14. matlactli onnaui = 10 + 4. 15. caxtolli. 16. caxtolli oce = 15 + 1. 17. caxtolli omome = 15 + 2. 18. caxtolli omey = 15 + 3. 19. caxtolli onnaui = 15 + 4. 20. cempualli = 1 account. CANAQUE[242] NEW CALEDONIA. 1. chaguin. 2. carou. 3. careri. 4. caboue 5. cani. 6. cani-mon-chaguin = 5 + 1. 7. cani-mon-carou = 5 + 2. 8. cani-mon-careri = 5 + 3. 9. cani-mon-caboue = 5 + 4. 10. panrere. 11. panrere-mon-chaguin = 10 + 1. 12. panrere-mon-carou = 10 + 2. 13. panrere-mon-careri = 10 + 3. 14. panrere-mon-caboue = 10 + 4. 15. panrere-mon-cani = 10 + 5. 16. panrere-mon-cani-mon-chaguin = 10 + 5 + 1. 17. panrere-mon-cani-mon-carou = 10 + 5 + 2. 18. panrere-mon-cani-mon-careri = 10 + 5 + 3. 19. panrere-mon-cani-mon-caboue = 10 + 5 + 4. 20. jaquemo = 1 person. GUATO.[243] 1. cenai. 2. dououni. 3. coum. 4. dekai. 5. quinoui. 6. cenai-caicaira = 1 on the other? 7. dououni-caicaira = 2 on the other? 8. coum-caicaira = 3 on the other? 9. dekai-caicaira = 4 on the other? 10. quinoi-da = 5 × 2. 11. cenai-ai-caibo = 1 + (the) hands. 12. dououni-ai-caibo = 2 + 10. 13. coum-ai-caibo = 3 + 10. 14. dekai-ai-caibo = 4 + 10. 15. quin-oibo = 5 × 3. 16. cenai-ai-quacoibo = 1 + 15. 17. dououni-ai-quacoibo = 2 + 15. 18. coum-ai-quacoibo = 3 + 15. 19. dekai-ai-quacoibo = 4 + 15. 20. quinoui-ai-quacoibo = 5 + 15. The meanings assigned to the numerals 6 to 9 are entirely conjectural. They obviously mean 1, 2, 3, 4, taken a second time, and as the meanings I have given are often found in primitive systems, they have, at a venture, been given here. LIFU, LOYALTY ISLANDS.[244] 1. ca. 2. lue. 3. koeni. 4. eke. 5. tji pi. 6. ca ngemen = 1 above. 7. lue ngemen = 2 above. 8. koeni ngemen = 3 above. 9. eke ngemen = 4 above. 10. lue pi = 2 × 5. 11. ca ko. 12. lue ko. 13. koeni ko. 14. eke ko. 15. koeni pi = 3 × 5. 16. ca huai ano. 17. lua huai ano. 18. koeni huai ano. 19. eke huai ano. 20. ca atj = 1 man. BONGO.[245] 1. kotu. 2. ngorr. 3. motta. 4. neheo. 5. mui. 6. dokotu = [5] + 1. 7. dongorr = [5] + 2. 8. domotta = [5] + 3. 9. doheo = [5] + 4. 10. kih. 11. ki dokpo kotu = 10 + 1. 12. ki dokpo ngorr = 10 + 2. 13. ki dokpo motta = 10 + 3. 14. ki dokpo neheo = 10 + 4. 15. ki dokpo mui = 10 + 5. 16. ki dokpo mui do mui okpo kotu = 10 + 5 more, to 5, 1 more. 17. ki dokpo mui do mui okpo ngorr = 10 + 5 more, to 5, 2 more. 18. ki dokpo mui do mui okpo motta = 10 + 5 more, to 5, 3 more. 19. ki dokpo mui do mui okpo nehea = 10 + 5 more, to 5, 4 more. 20. mbaba kotu. Above 20, the Lufu and the Bongo systems are vigesimal, so that they are, as a whole, mixed systems. The Welsh scale begins as though it were to present a pure decimal structure, and no hint of the quinary element appears until it has passed 15. The Nahuatl, on the other hand, counts from 5 to 10 by the ordinary quinary method, and then appears to pass into the decimal form. But when 16 is reached, we find the quinary influence still persistent; and from this point to 20, the numeral words in both scales are such as to show that the notion of counting by fives is quite as prominent as the notion of referring to 10 as a base. Above 20 the systems become vigesimal, with a quinary or decimal structure appearing in all numerals except multiples of 20. Thus, in Welsh, 36 is _unarbymtheg ar ugain_, 1 + 5 + 10 + 20; and in Nahuatl the same number is _cempualli caxtolli oce_, 20 + 15 + 1. Hence these and similar number systems, though commonly alluded to as vigesimal, are really mixed scales, with 20 as their primary base. The Canaque scale differs from the Nahuatl only in forming a compound word for 15, instead of introducing a new and simple term. In the examples which follow, it is not thought best to extend the lists of numerals beyond 10, except in special instances where the illustration of some particular point may demand it. The usual quinary scale will be found, with a few exceptions like those just instanced, to have the following structure or one similar to it in all essential details: 1, 2, 3, 4, 5, 5-1, 5-2, 5-3, 5-4, 10, 10-1, 10-2, 10-3, 10-4, 10-5, 10-5-1, 10-5-2, 10-5-3, 10-5-4, 20. From these forms the entire system can readily be constructed as soon as it is known whether its principal base is to be 10 or 20. Turning first to the native African languages, I have selected the following quinary scales from the abundant material that has been collected by the various explorers of the "Dark Continent." In some cases the numerals of certain tribes, as given by one writer, are found to differ widely from the same numerals as reported by another. No attempt has been made at comparison of these varying forms of orthography, which are usually to be ascribed to difference of nationality on the part of the collectors. FELOOPS.[246] 1. enory. 2. sickaba, or cookaba. 3. sisajee. 4. sibakeer. 5. footuck. 6. footuck-enory = 5-1. 7. footuck-cookaba = 5-2. 8. footuck-sisajee = 5-3. 9. footuck-sibakeer = 5-4. 10. sibankonyen. KISSI.[247] 1. pili. 2. miu. 3. nga. 4. iol. 5. nguenu. 6. ngom-pum = 5-1. 7. ngom-miu = 5-2. 8. ngommag = 5-3. 9. nguenu-iol = 5-4. 10. to. ASHANTEE.[248] 1. tah. 2. noo. 3. sah. 4. nah. 5. taw. 6. torata = 5 + 1. 7. toorifeenoo = 5 + 2. 8. toorifeessa = 5 + 3. 9. toorifeena = 5 + 4. 10. nopnoo. BASA.[249] 1. do. 2. so. 3. ta. 4. hinye. 5. hum. 6. hum-le-do = 5 + 1. 7. hum-le-so = 5 + 2. 8. hum-le-ta = 5 + 3. 9. hum-le-hinyo = 5 + 4. 10. bla-bue. JALLONKAS.[250] 1. kidding. 2. fidding. 3. sarra. 4. nani. 5. soolo. 6. seni. 7. soolo ma fidding = 5 + 2. 8. soolo ma sarra = 5 + 3. 9. soolo ma nani = 5 + 4. 10. nuff. KRU. 1. da-do. 2. de-son. 3. de-tan. 4. de-nie. 5. de-mu. 6. dme-du = 5-1. 7. ne-son = [5] + 2. 8. ne-tan = [5] + 3. 9. sepadu = 10 - 1? 10. pua. JALOFFS.[251] 1. wean. 2. yar. 3. yat. 4. yanet. 5. judom. 6. judom-wean = 5-1. 7. judom-yar = 5-2. 8. judom-yat = 5-3. 9. judom yanet = 5-4. 10. fook. GOLO.[252] 1. mbali. 2. bisi. 3. bitta. 4. banda. 5. zonno. 6. tsimmi tongbali = 5 + 1. 7. tsimmi tobisi = 5 + 2. 8. tsimmi tobitta = 5 + 3. 9. tsimmi to banda = 5 + 4. 10. nifo. FOULAH.[253] 1. go. 2. deeddee. 3. tettee. 4. nee. 5. jouee. 6. jego = 5-1. 7. jedeeddee = 5-2. 8. je-tettee = 5-3. 9. je-nee = 5-4. 10. sappo. SOUSSOU.[254] 1. keren. 2. firing. 3. sarkan. 4. nani. 5. souli. 6. seni. 7. solo-fere = 5-2. 8. solo-mazarkan = 5 + 3. 9. solo-manani = 5 + 4. 10. fu. BULLOM.[255] 1. bul. 2. tin. 3. ra. 4. hyul. 5. men. 6. men-bul = 5-1. 7. men-tin = 5-2. 8. men-ra = 5-3. 9. men-hyul = 5-4. 10. won. VEI.[256] 1. dondo. 2. fera. 3. sagba. 4. nani. 5. soru. 6. sun-dondo = 5-1. 7. sum-fera = 5-2. 8. sun-sagba = 5-3. 9. sun-nani = 5-4. 10. tan. DINKA.[257] 1. tok. 2. rou. 3. dyak. 4. nuan. 5. wdyets. 6. wdetem = 5-1. 7. wderou = 5-2. 8. bet, bed = 5-3. 9. wdenuan = 5-4. 10. wtyer = 5 × 2. TEMNE. 1. in. 2. ran. 3. sas. 4. anle. 5. tr-amat. 6. tr-amat rok-in = 5 + 1. 7. tr-amat de ran = 5 + 2. 8. tr-amat re sas = 5 + 3. 9. tr-amat ro n-anle = 5 + 4. 10. tr-ofatr. ABAKER.[258] 1. kili. 2. bore. 3. dotla. 4. ashe. 5. ini. 6. im kili = 5-1. 7. im-bone = 5-2. 8. ini-dotta = 5-3. 9. tin ashe = 5-4. 10. chica. BAGRIMMA.[259] 1. kede. 2. sab. 3. muta. 4. so. 5. mi. 6. mi-ga = 5 + 1. 7. tsidi. 8. marta = 5 + 2. 9. do-so = [5] + 3 10. duk-keme. PAPAA.[260] 1. depoo. 2. auwi. 3. ottong. 4. enne. 5. attong. 6. attugo. 7. atjuwe = [5] + 2. 8. attiatong = [5] + 3. 9. atjeenne = [5] + 4. 10. awo. EFIK.[261] 1. kiet. 2. iba. 3. ita. 4. inan. 5. itiun. 6. itio-kiet = 5-1. 7. itia-ba = 5-2. 8. itia-eta = 5-3. 9. osu-kiet = 10 - 1? 10. duup. NUPE.[262] 1. nini. 2. gu-ba. 3. gu-ta. 4. gu-ni. 5. gu-tsun. 6. gu-sua-yin = 5 + 1. 7. gu-tua-ba = 5 + 2. 8. gu-tu-ta = 5 + 3. 9. gu-tua-ni = 5 + 4. 10. gu-wo. MOKKO.[263] 1. kiä. 2. iba. 3. itta. 4. inan. 5. üttin. 6. itjüekee = 5 + 1. 7. ittiaba = 5 + 2. 8. itteiata = 5 + 3. 9. huschukiet. 10. büb. KANURI.[264] 1. tilo. 2. ndi. 3. yasge. 4. dege. 5. ugu. 6. arasge = 5 + 1. 7. tulur. 8. wusge = 5 + 3. 9. legar. 10. megu = 2 × 5. BININ.[265] 1. bo. 2. be. 3. la. 4. nin. 5. tang. 6. tahu = 5 + 1? 7. tabi = 5 + 2. 8. tara = 5 + 3. 9. ianin (tanin?) = 5 + 4? 10. te. KREDY.[266] 1. baia. 2. rommu. 3. totto. 4. sosso. 5. saya. 6. yembobaia = [5] + 1. 7. yemborommu = [5] + 2. 8. yembototto = [5] + 3. 9. yembososso = [5] + 4. 10. puh. HERERO.[267] 1. mue. 2. vari. 3. tatu. 4. ne. 5. tano. 6. hambou-mue = [5] + 1. 7. hambou-vari = [5] + 2. 8. hambou-tatu = [5] + 3. 9. hambou-ne = [5] + 4. 10. KI-YAU.[268] 1. jumo. 2. wawiri. 3. watatu. 4. mcheche. 5. msano. 6. musano na jumo = 5 + 1. 7. musano na wiri = 5 + 2. 8. musano na watatu = 5 + 3. 9. musano na mcheche = 5 + 4. 10. ikumi. FERNANDO PO.[269] 1. muli. 2. mempa. 3. meta. 4. miene. 5. mimito. 6. mimito na muli = 5 + 1. 7. mimito na mempa = 5 + 2. 8. mimito na meta = 5 + 3. 9. mimito na miene = 5 + 4. 10. miemieu = 5-5? KI-NYASSA 1. kimodzi. 2. vi-wiri. 3. vi-tatu. 4. vinye. 5. visano. 6. visano na kimodzi = 5 + 1. 7. visano na vi-wiri = 5 + 2. 8. visano na vitatu = 5 + 3. 9. visano na vinye = 5 + 4. 10. chikumi. BALENGUE.[270] 1. guevoho. 2. ibare. 3. raro. 4. inaï. 5. itano. 6. itano na guevoho = 5 + 1. 7. itano na ibare = 5 + 2. 8. itano na raro = 5 + 3. 9. itano na inaï = 5 + 4. 10. ndioum, or nai-hinaï. KUNAMA.[271] 1. ella. 2. bare. 3. sadde. 4. salle. 5. kussume. 6. kon-t'-ella = hand 1. 7. kon-te-bare = hand 2. 8. kon-te-sadde = hand 3. 9. kon-te-salle = hand 4. 10. kol-lakada. GOLA.[272] 1. ngoumou. 2. ntie. 3. ntaï. 4. tina. 5. nonon. 6. diegoum = [5] + 1. 7. dientie = [5] + 2. 8. dietai = [5] + 3. 9. dectina = [5] + 4. 10. esia. BAREA.[273] 1. doko 2. arega. 3. sane. 4. sone. 5. oita. 6. data. 7. dz-ariga = 5 + 2. 8. dis-sena = 5 + 3. 9. lefete-mada = without 10. 10. lefek. MATIBANI.[274] 1. mosa. 2. pili. 3. taru. 4. teje. 5. taru. 6. tana mosa = 5-1. 7. tana pili = 5-2. 8. tana taru = 5-3. 9. loco. 10. loco nakege. BONZÉ.[275] 1. tan. 2. vele. 3. daba. 4. nani. 5. lolou. 6. maïda = [5] + 1. 7. maïfile = [5] + 2. 8. maïshaba = [5] + 3. 9. maïnan = [5] + 4. 10. bou. MPOVI 1. moueta. 2. bevali. 3. betata. 4. benaï. 5. betani. 6. betani moueta = 5-1. 7. betani bevali = 5-2. 8. betani betata = 5-3. 9. betani benai = 5-4. 10. nchinia. TRITON'S BAY, NEW QUINEA.[276] 1. samosi. 2. roueti. 3. tourou. 4. faat. 5. rimi. 6. rim-samosi = 5-1. 7. rim-roueti = 5-2. 8. rim-tourou = 5-3. 9. rim-faat = 5-4. 10. outsia. ENDE, OR FLORES.[277] 1. sa. 2. zua. 3. telu. 4. wutu. 5. lima = hand. 6. lima-sa = 5-1, or hand 1. 7. lima-zua = 5-2. 8. rua-butu = 2 × 4? 9. trasa = [10] - 1? 10. sabulu. MALLICOLO.[278] 1. tseekaee. 2. ery. 3. erei. 4. ebats. 5. ereem. 6. tsookaee = [5] + 1. 7. gooy = [5] + 2. 8. hoorey = [5] + 3. 9. goodbats = [5] + 4. 10. senearn. EBON, MARSHALL ISLANDS.[279] 1. iuwun. 2. drud. 3. chilu. 4. emer. 5. lailem. 6. chilchinu = 5 + 1. 7. chilchime = 5 + 2. 8. twalithuk = [10] - 2. 9. twahmejuwou = [10] - 1. 10. iungou. UEA, LOYALTY ISLAND.[280] 1. tahi. 2. lua. 3. tolu. 4. fa. 5. lima. 6. tahi. 7. lua. 8. tolu. 9. fa. 10. lima. UEA.[280]--[another dialect.] 1. hacha. 2. lo. 3. kuun. 4. thack. 5. thabumb. 6. lo-acha = 2d 1. 7. lo-alo = 2d 2. 8. lo-kuun = 2d 3. 9. lo-thack = 2d 4. 10. lebenetee. ISLE OF PINES.[281] 1. ta. 2. bo. 3. beti. 4. beu. 5. ta-hue. 6. no-ta = 2d 1. 7. no-bo = 2d 2. 8. no-beti = 2d 3. 9. no-beu = 2d 4. 10. de-kau. UREPARAPARA, BANKS ISLANDS.[282] 1. vo towa. 2. vo ro. 3. vo tol. 4. vo vet. 5. teveliem = 1 hand. 6. leve jea = other 1. 7. leve ro = other 2. 8. leve tol = other 3. 9. leve vet = other 4. 10. sanowul = 2 sets. MOTA, BANKS ISLANDS.[282] 1. tuwale. 2. nirua. 3. nitol. 4. nivat. 5. tavelima = 1 hand. 6. laveatea = other 1. 7. lavearua = other 2. 8. laveatol = other 3. 9. laveavat = other 4. 10. sanavul = 2 sets. NEW CALEDONIA.[283] 1. parai. 2. paroo. 3. parghen. 4. parbai. 5. panim. 6. panim-gha = 5-1. 7. panim-roo = 5-2. 8. panim-ghen = 5-3. 9. panim-bai = 5-4. 10. parooneek. YENGEN, NEW CAL.[284] 1. hets. 2. heluk. 3. heyen. 4. pobits. 5. nim = hand. 6. nim-wet = 5-1. 7. nim-weluk = 5-2. 8. nim-weyen = 5-3. 9. nim-pobit = 5-4. 10. pain-duk. ANEITEUM.[285] 1. ethi. 2. ero. 3. eseik. 4. manohwan. 5. nikman. 6. nikman cled et ethi = 5 + 1. 7. nikman cled et oro = 5 + 2. 8. nikman cled et eseik = 5 + 3. 9. nikman cled et manohwan = 5 + 4. 10. nikman lep ikman = 5 + 5. TANNA 1. riti. 2. karu. 3. kahar. 4. kefa. 5. krirum. 6. krirum riti = 5-1. 7. krirum karu = 5-2. 8. krirum kahar? = 5-3. 9. krirum kefa? = 5-4. 10. ---- EROMANGA 1. sai. 2. duru. 3. disil. 4. divat. 5. siklim = 1 hand. 6. misikai = other 1? 7. siklim naru = 5-2. 8. siklim disil = 5-3. 9. siklim mindivat = 5 + 4. 10. narolim = 2 hands. FATE, NEW HEB.[286] 1. iskei. 2. rua. 3. tolu. 4. bate. 5. lima = hand. 6. la tesa = other 1. 7. la rua = other 2. 8. la tolu = other 3. 9. la fiti = other 4. 10. relima = 2 hands. API, NEW HEB. 1. tai. 2. lua. 3. tolu. 4. vari. 5. lima = hand. 6. o rai = other 1. 7. o lua = other 2. 8. o tolo = other 3. 9. o vari = other 4. 10. lua lima = 2 hands. SESAKE, NEW HEB. 1. sikai. 2. dua. 3. dolu. 4. pati. 5. lima = hand. 6. la tesa = other 1. 7. la dua = other 2. 8. la dolu = other 3. 9. lo veti = other 4. 10. dua lima = 2 hands. PAMA, NEW HEB. 1. tai. 2. e lua. 3. e tolu. 4. e hati. 5. e lime = hand. 6. a hitai = other 1. 7. o lu = other 2. 8. o tolu = other 3. 9. o hati = other 4. 10. ha lua lim = 2 hands AURORA, NEW HEB. 1. tewa. 2. i rua. 3. i tol. 4. i vat. 5. tavalima = 1 hand. 6. lava tea = other 1. 7. lava rua = other 2. 8. lava tol = other 3. 9. la vat = other 4. 10. sanwulu = two sets. TOBI.[287] 1. yat. 2. glu. 3. ya. 4. uan. 5. yanim = 1 hand. 6. yawor = other 1. 7. yavic = other 2. 8. yawa = other 3. 9. yatu = other 4. 10. yasec. PALM ISLAND.[288] 1. yonkol. 2. yakka. 3. tetjora. 4. tarko. 5. yonkol mala = 1 hand. JAJOWERONG, VICTORIA.[288] 1. kiarp. 2. bulaits. 3. bulaits kiarp = 2-1. 4. bulaits bulaits = 2-2. 5. kiarp munnar = 1 hand. 6. bulaits bulaits bulaits = 2-2-2. 10. bulaits munnar = 2 hands. The last two scales deserve special notice. They are Australian scales, and the former is strongly binary, as are so many others of that continent. But both show an incipient quinary tendency in their names for 5 and 10. CAMBODIA.[289] 1. muy. 2. pir. 3. bey. 4. buon. 5. pram. 6. pram muy = 5-1. 7. pram pil = 5-2. 8. pram bey = 5-3. 9. pram buon = 5-4. 10. dap. TSCHUKSCHI.[290] 1. inen. 2. nirach. 3. n'roch. 4. n'rach. 5. miligen = hand. 6. inen miligen = 1-5. 7. nirach miligen = 2-5. 8. anwrotkin. 9. chona tsinki. 10. migitken = both hands. KOTTISCH[291] 1. hutsa. 2. ina. 3. tona. 4. sega. 5. chega. 6. chelutsa = 5 + 1. 7. chelina = 5 + 2. 8. chaltona = 5 + 3. 9. tsumnaga = 10 - 1. 10. haga. ESKIMO OF N.-W. ALASKA.[292] 1. a towshek. 2. hipah, or malho. 3. pingishute. 4. sesaimat. 5. talema. 6. okvinile, or ahchegaret = another 1? 7. talema-malronik = 5-two of them. 8. pingishu-okvingile = 2d 3? 9. kolingotalia = 10 - 1? 10. koleet. KAMTSCHATKA, SOUTH.[293] 1. dischak. 2. kascha. 3. tschook. 4. tschaaka. 5. kumnaka. 6. ky'lkoka. 7. itatyk = 2 + 5. 8. tschookotuk = 3 + 5. 9. tschuaktuk = 4 + 5. 10. kumechtuk = 5 + 5. ALEUTS[294] 1. ataqan. 2. aljak. 3. qankun. 4. sitsin. 5. tsan = my hand. 6. atun = 1 + 5. 7. ulun = 2 + 5. 8. qamtsin = 3 + 5. 9. sitsin = 4 + 5. 10. hatsiq. TCHIGLIT, MACKENZIE R.[295] 1. ataotçirkr. 2. aypak, or malloerok. 3. illaak, or piñatcut. 4. tçitamat. 5. tallemat. 6. arveneloerit. 7. arveneloerit-aypak = 5 + 2. 8. arveneloerit-illaak = 5 + 3. 9. arveneloerit-tçitamat = 5 + 4. 10. krolit. SAHAPTIN (NEZ PERCES).[296] 1. naks. 2. lapit. 3. mitat. 4. pi-lapt = 2 × 2. 5. pachat. 6. oi-laks = [5] + 1. 7. oi-napt = [5] + 2. 8. oi-matat = [5] + 3. 9. koits. 10. putimpt. GREENLAND.[297] 1. atauseq. 2. machdluq. 3. pinasut. 4. sisamat 5. tadlimat. 6. achfineq-atauseq = other hand 1. 7. achfineq-machdluq = other hand 2. 8. achfineq-pinasut = other hand 3. 9. achfineq-sisamat = other hand 4. 10. qulit. 11. achqaneq-atauseq = first foot 1. 12. achqaneq-machdluq = first foot 2. 13. achqaneq-pinasut = first foot 3. 14. achqaneq-sisamat = first foot 4. 15. achfechsaneq? 16. achfechsaneq-atauseq = other foot 1. 17. achfechsaneq-machdlup = other foot 2. 18. achfechsaneq-pinasut = other foot 3. 19. achfechsaneq-sisamat = other foot 4. 20. inuk navdlucho = a man ended. Up to this point the Greenlander's scale is almost purely quinary. Like those of which mention was made at the beginning of this chapter, it persists in progressing by fives until it reaches 20, when it announces a new base, which shows that the system will from now on be vigesimal. This scale is one of the most interesting of which we have any record, and will be noticed again in the next chapter. In many respects it is like the scale of the Point Barrow Eskimo, which was given early in Chapter III. The Eskimo languages are characteristically quinary-vigesimal in their number systems, but few of them present such perfect examples of that method of counting as do the two just mentioned. CHIPPEWAY.[298] 1. bejig. 2. nij. 3. nisswi. 4. niwin. 5. nanun. 6. ningotwasswi = 1 again? 7. nijwasswi = 2 again? 8. nishwasswi = 3 again? 9. jangasswi = 4 again? 10. midasswi = 5 again. MASSACHUSETTS.[299] 1. nequt. 2. neese. 3. nish. 4. yaw. 5. napanna = on one side, _i.e._ 1 hand. 6. nequttatash = 1 added. 7. nesausuk = 2 again? 8. shawosuk = 3 again? 9. pashoogun = it comes near, _i.e._ to 10. 10. puik. OJIBWA OF CHEGOIMEGON.[300] 1. bashik. 2. neensh. 3. niswe. 4. newin. 5. nanun. 6. ningodwaswe = 1 again? 7. nishwaswe = 2 again? 8. shouswe = 3 again? 9. shangaswe = 4 again? 10. medaswe = 5 again? OTTAWA. 1. ningotchau. 2. ninjwa. 3. niswa. 4. niwin. 5. nanau. 6. ningotwaswi = 1 again? 7. ninjwaswi = 2 again? 8. nichwaswi = 3 again? 9. shang. 10. kwetch. DELAWARE. 1. n'gutti. 2. niskha. 3. nakha. 4. newa. 5. nalan [akin to palenach, hand]. 6. guttash = 1 on the other side. 7. nishash = 2 on the other side. 8. khaash = 3 on the other side. 9. peshgonk = coming near. 10. tellen = no more. SHAWNOE. 1. negote. 2. neshwa. 3. nithuie. 4. newe. 5. nialinwe = gone. 6. negotewathwe = 1 further. 7. neshwathwe = 2 further. 8. sashekswa = 3 further? 9. chakatswe [akin to chagisse, "used up"]. 10. metathwe = no further. MICMAC.[301] 1. naiookt. 2. tahboo. 3. seest. 4. naioo. 5. nahn. 6. usoo-cum. 7. eloo-igunuk. 8. oo-gumoolchin. 9. pescoonaduk. 10. mtlin. One peculiarity of the Micmac numerals is most noteworthy. The numerals are real verbs, instead of adjectives, or, as is sometimes the case, nouns. They are conjugated through all the variations of mood, tense, person, and number. The forms given above are not those that would be used in counting, but are for specific use, being varied according to the thought it was intended to express. For example, _naiooktaich_ = there is 1, is present tense; _naiooktaichcus_, there was 1, is imperfect; and _encoodaichdedou_, there will be 1, is future. The variation in person is shown by the following inflection: PRESENT TENSE. 1st pers. tahboosee-ek = there are 2 of us. 2d pers. tahboosee-yok = there are 2 of you. 3d pers. tahboo-sijik = there are 2 of them. IMPERFECT TENSE. 1st pers. tahboosee-egup = there were 2 of us. 2d pers. tahboosee-yogup = there were 2 of you. 3d pers. tahboosee-sibunik = there were 2 of them. FUTURE TENSE. 3d pers. tahboosee-dak = there will be 2 of them, etc. The negative form is also comprehended in the list of possible variations. Thus, _tahboo-seekw_, there are not 2 of them; _mah tahboo-seekw_, there will not be 2 of them; and so on, through all the changes which the conjugation of the verb permits. OLD ALGONQUIN. 1. peygik. 2. ninsh. 3. nisswey. 4. neyoo. 5. nahran = gone. 6. ningootwassoo = 1 on the other side. 7. ninshwassoo = 2 on the other side. 8. nisswasso = 3 on the other side. 9. shangassoo [akin to chagisse, "used up"]. 10. mitassoo = no further. OMAHA. 1. meeachchee. 2. nomba. 3. rabeenee. 4. tooba. 5. satta = hand, _i.e._ all the fingers turned down. 6. shappai = 1 more. 7. painumba = fingers 2. 8. pairabeenee = fingers 3. 9. shonka = only 1 finger (remains). 10. kraibaira = unbent.[302] CHOCTAW. 1. achofee. 2. tuklo. 3. tuchina. 4. ushta. 5. tahlape = the first hand ends. 6. hanali. 7. untuklo = again 2. 8. untuchina = again 3. 9. chokali = soon the end; _i.e._ next the last. 10. pokoli. CADDOE. 1. kouanigh. 2. behit. 3. daho. 4. hehweh. 5. dihsehkon. 6. dunkeh. 7. bisekah = 5 + 2. 8. dousehka = 5 + 3. 9. hehwehsehka = 4 + hand. 10. behnehaugh. CHIPPEWAY. 1. payshik. 2. neesh. 3. neeswoy. 4. neon. 5. naman = gone. 6. nequtwosswoy = 1 on the other side. 7. neeshswosswoy = 2 on the other side. 8. swoswoy = 3 on the other side? 9. shangosswoy [akin to chagissi, "used up"]. 10. metosswoy = no further. ADAIZE. 1. nancas. 2. nass. 3. colle. 4. tacache. 5. seppacan. 6. pacanancus = 5 + 1. 7. pacaness = 5 + 2. 8. pacalcon = 5 + 3. 9. sickinish = hands minus? 10. neusne. PAWNEE. 1. askoo. 2. peetkoo. 3. touweet. 4. shkeetiksh. 5. sheeooksh = hands half. 6. sheekshabish = 5 + 1. 7. peetkoosheeshabish = 2 + 5. 8. touweetshabish = 3 + 5. 9. looksheereewa = 10 - 1. 10. looksheeree = 2d 5? MINSI. 1. gutti. 2. niskha. 3. nakba. 4. newa. 5. nulan = gone? 6. guttash = 1 added. 7. nishoash = 2 added. 8. khaash = 3 added. 9. noweli. 10. wimbat. KONLISCHEN. 1. tlek. 2. tech. 3. nezk. 4. taakun. 5. kejetschin. 6. klet uschu = 5 + 1. 7. tachate uschu = 5 + 2. 8. nesket uschu = 5 + 3. 9. kuschok = 10 - 1? 10. tschinkat. TLINGIT.[303] 1. tlek. 2. deq. 3. natsk. 4. dak'on = 2d 2. 5. kedjin = hand. 6. tle durcu = other 1. 7. daqa durcu = other 2. 8. natska durcu = other 3. 9. gocuk. 10. djinkat = both hands. RAPID, OR FALL, INDIANS. 1. karci. 2. neece. 3. narce. 4. nean. 5. yautune. 6. neteartuce = 1 over? 7. nesartuce = 2 over? 8. narswartuce = 3 over? 9. anharbetwartuce = 4 over? 10. mettartuce = no further? HEILTSUK.[304] 1. men. 2. matl. 3. yutq. 4. mu. 5. sky'a. 6. katla. 7. matlaaus = other 2? 8. yutquaus = other 3? 9. mamene = 10 - 1. 10. aiky'as. NOOTKA.[305] 1. nup. 2. atla. 3. katstsa. 4. mo. 5. sutca. 6. nopo = other 1? 7. atlpo = other 2? 8. atlakutl = 10 - 2. 9. ts'owakutl = 10 - 1. 10. haiu. TSIMSHIAN.[306] 1. gyak. 2. tepqat. 3. guant. 4. tqalpq. 5. kctonc (from _anon_, hand). 6. kalt = 2d 1. 7. t'epqalt = 2d 2. 8. guandalt = 2d 3? 9. kctemac. 10. gy'ap. BILQULA.[306] 1. (s)maotl. 2. tlnos. 3. asmost. 4. mos. 5. tsech. 6. tqotl = 2d 1? 7. nustlnos = 2d 2? 8. k'etlnos = 2 × 4. 9. k'esman. 10. tskchlakcht. MOLELE.[307] 1. mangu. 2. lapku. 3. mutka. 4. pipa. 5. pika. 6. napitka = 1 + 5. 7. lapitka = 2 + 5. 8. mutpitka = 3 + 5. 9. laginstshiatkus. 10. nawitspu. WAIILATPU.[308] 1. na. 2. leplin. 3. matnin. 4. piping. 5. tawit. 6. noina = [5] + 1. 7. noilip = [5] + 2. 8. noimat = [5] + 3. 9. tanauiaishimshim. 10. ningitelp. LUTUAMI.[307] 1. natshik. 2. lapit. 3. ntani. 4. wonip. 5. tonapni. 6. nakskishuptane = 1 + 5. 7. tapkishuptane = 2 + 5. 8. ndanekishuptane = 3 + 5. 9. natskaiakish = 10 - 1. 10. taunip. SASTE (SHASTA).[309] 1. tshiamu. 2. hoka. 3. hatski. 4. irahaia. 5. etsha. 6. tahaia. 7. hokaikinis = 2 + 5. 8. hatsikikiri = 3 + 5. 9. kirihariki-ikiriu. 10. etsehewi. CAHUILLO.[310] 1. supli. 2. mewi. 3. mepai. 4. mewittsu. 5. nomekadnun. 6. kadnun-supli = 5-1. 7. kan-munwi = 5-2. 8. kan-munpa = 5-3. 9. kan-munwitsu = 5-4. 10. nomatsumi. TIMUKUA.[311] 1. yaha. 2. yutsa. 3. hapu. 4. tseketa. 5. marua. 6. mareka = 5 + 1 7. pikitsa = 5 + 2 8. pikinahu = 5 + 3 9. peke-tsaketa = 5 + 4 10. tuma. OTOMI[312] 1. nara. 2. yocho. 3. chiu. 4. gocho. 5. kuto. 6. rato = 1 + 5. 7. yoto = 2 + 5. 8. chiato = 3 + 5. 9. guto = 4 + 5. 10. reta. TARASCO.[313] 1. ma. 2. dziman. 3. tanimo. 4. tamu. 5. yumu. 6. kuimu. 7. yun-dziman = [5] + 2. 8. yun-tanimo = [5] + 3. 9. yun-tamu = [5] + 4. 10. temben. MATLALTZINCAN.[314] 1. indawi. 2. inawi. 3. inyuhu. 4. inkunowi. 5. inkutaa. 6. inda-towi = 1 + 5. 7. ine-towi = 2 + 5. 8. ine-ukunowi = 2-4. 9. imuratadahata = 10 - 1? 10. inda-hata. CORA.[315] 1. ceaut. 2. huapoa. 3. huaeica. 4. moacua. 5. anxuvi. 6. a-cevi = [5] + 1. 7. a-huapoa = [5] + 2. 8. a-huaeica = [5] + 3. 9. a-moacua = [5] + 4. 10. tamoamata (akin to moamati, "hand"). AYMARA.[316] 1. maya. 2. paya. 3. kimsa. 4. pusi. 5. piska. 6. tsokta. 7. pa-kalko = 2 + 5. 8. kimsa-kalko = 3 + 5. 9. pusi-kalko = 4 + 5. 10. tunka. CARIBS OF ESSEQUIBO, GUIANA.[317] 1. oween. 2. oko. 3. oroowa. 4. oko-baimema. 5. wineetanee = 1 hand. 6. owee-puimapo = 1 again? 7. oko-puimapo = 2 again? 8. oroowa-puimapo = 3 again? 9. oko-baimema-puimapo = 4 again? 10. oween-abatoro. CARIB.[318] (ROUCOUYENNE?) 1. aban, amoin. 2. biama. 3. eleoua. 4. biam-bouri = 2 again? 5. ouacabo-apourcou-aban-tibateli. 6. aban laoyagone-ouacabo-apourcou. 7. biama laoyagone-ouacabo-apourcou. 8. eleoua laoyagone-ouacabo-apourcou. 9. ---- 10. chon noucabo. It is unfortunate that the meanings of these remarkable numerals cannot be given. The counting is evidently quinary, but the terms used must have been purely descriptive expressions, having their origin undoubtedly in certain gestures or finger motions. The numerals obtained from this region, and from the tribes to the south and east of the Carib country, are especially rich in digital terms, and an analysis of the above numerals would probably show clearly the mental steps through which this people passed in constructing the rude scale which served for the expression of their ideas of number. KIRIRI.[319] 1. biche. 2. watsani. 3. watsani dikie. 4. sumara oroba. 5. mi biche misa = 1 hand. 6. mirepri bu-biche misa sai. 7. mirepri watsani misa sai. 8. mirepri watsandikie misa sai. 9. mirepri sumara oraba sai. 10. mikriba misa sai = both hands. CAYUBABA[320] 1. pebi. 2. mbeta. 3. kimisa. 4. pusi. 5. pisika. 6. sukuta. 7. pa-kaluku = 2 again? 8. kimisa-kaluku = 3 again? 9. pusu-kaluku = 4 again? 10. tunka. SAPIBOCONA[320] 1. karata. 2. mitia. 3. kurapa. 4. tsada. 5. maidara (from _arue_, hand). 6. karata-rirobo = 1 hand with. 7. mitia-rirobo = 2 hand with. 8. kurapa-rirobo = 3 hand with. 9. tsada-rirobo = 4 hand with. 10. bururutse = hand hand. TICUNA.[321] 1. hueih. 2. tarepueh. 3. tomepueh. 4. aguemoujih 5. hueamepueh. 6. naïmehueapueh = 5 + 1. 7. naïmehueatareh = 5 + 2. 8. naïmehueatameapueh = 5 + 3. 9. gomeapueh = 10 - 1. 10. gomeh. YANUA.[322] 1. tckini. 2. nanojui. 3. munua. 4. naïrojuino = 2d 2. 5. tenaja. 6. teki-natea = 1 again? 7. nanojui-natea = 2 again? 8. munua-natea = 3 again? 9. naïrojuino-natea = 4 again? 10. huijejuino = 2 × 5? The foregoing examples will show with considerable fulness the wide dispersion of the quinary scale. Every part of the world contributes its share except Europe, where the only exceptions to the universal use of the decimal system are the half-dozen languages, which still linger on its confines, whose number base is the vigesimal. Not only is there no living European tongue possessing a quinary number system, but no trace of this method of counting is found in any of the numerals of the earlier forms of speech, which have now become obsolete. The only possible exceptions of which I can think are the Greek [Greek: pempazein], to count by fives, and a few kindred words which certainly do hint at a remote antiquity in which the ancestors of the Greeks counted on their fingers, and so grouped their units into fives. The Roman notation, the familiar I., II., III., IV. (originally IIII.), V., VI., etc., with equal certainty suggests quinary counting, but the Latin language contains no vestige of anything of the kind, and the whole range of Latin literature is silent on this point, though it contains numerous references to finger counting. It is quite within the bounds of possibility that the prehistoric nations of Europe possessed and used a quinary numeration. But of these races the modern world knows nothing save the few scanty facts that can be gathered from the stone implements which have now and then been brought to light. Their languages have perished as utterly as have the races themselves, and speculation concerning them is useless. Whatever their form of numeration may have been, it has left no perceptible trace on the languages by which they were succeeded. Even the languages of northern and central Europe which were contemporary with the Greek and Latin of classical times have, with the exception of the Celtic tongues of the extreme North-west, left behind them but meagre traces for the modern student to work on. We presume that the ancient Gauls and Goths, Huns and Scythians, and other barbarian tribes had the same method of numeration that their descendants now have; and it is a matter of certainty that the decimal scale was, at that time, not used with the universality which now obtains; but wherever the decimal was not used, the universal method was vigesimal; and that the quinary ever had anything of a foothold in Europe is only to be guessed from its presence to-day in almost all of the other corners of the world. From the fact that the quinary is that one of the three natural scales with the smallest base, it has been conjectured that all tribes possess, at some time in their history, a quinary numeration, which at a later period merges into either the decimal or the vigesimal, and thus disappears or forms with one of the latter a mixed system.[323] In support of this theory it is urged that extensive regions which now show nothing but decimal counting were, beyond all reasonable doubt, quinary. It is well known, for example, that the decimal system of the Malays has spread over almost the entire Polynesian region, displacing whatever native scales it encountered. The same phenomenon has been observed in Africa, where the Arab traders have disseminated their own numeral system very widely, the native tribes adopting it or modifying their own scales in such a manner that the Arab influence is detected without difficulty. In view of these facts, and of the extreme readiness with which a tribe would through its finger counting fall into the use of the quinary method, it does not at first seem improbable that the quinary was _the_ original system. But an extended study of the methods of counting in vogue among the uncivilized races of all parts of the world has shown that this theory is entirely untenable. The decimal scale is no less simple in its structure than the quinary; and the savage, as he extends the limit of his scale from 5 to 6, may call his new number 5-1, or, with equal probability, give it an entirely new name, independent in all respects of any that have preceded it. With the use of this new name there may be associated the conception of "5 and 1 more"; but in such multitudes of instances the words employed show no trace of any such meaning, that it is impossible for any one to draw, with any degree of safety, the inference that the signification was originally there, but that the changes of time had wrought changes in verbal form so great as to bury it past the power of recovery. A full discussion of this question need not be entered upon here. But it will be of interest to notice two or three numeral scales in which the quinary influence is so faint as to be hardly discernible. They are found in considerable numbers among the North American Indian languages, as may be seen by consulting the vocabularies that have been prepared and published during the last half century.[324] From these I have selected the following, which are sufficient to illustrate the point in question: QUAPPA. 1. milchtih. 2. nonnepah. 3. dahghenih. 4. tuah. 5. sattou. 6. schappeh. 7. pennapah. 8. pehdaghenih. 9. schunkkah. 10. gedeh bonah. TERRABA.[325] 1. krara. 2. krowü. 3. krom miah. 4. krob king. 5. krasch kingde. 6. terdeh. 7. kogodeh. 8. kwongdeh. 9. schkawdeh. 10. dwowdeh. MOHICAN 1. ngwitloh. 2. neesoh. 3. noghhoh. 4. nauwoh. 5. nunon. 6. ngwittus. 7. tupouwus. 8. ghusooh. 9. nauneeweh. 10. mtannit. In the Quappa scale 7 and 8 appear to be derived from 2 and 3, while 6 and 9 show no visible trace of kinship with 1 and 4. In Mohican, on the other hand, 6 and 9 seem to be derived from 1 and 4, while 7 and 8 have little or no claim to relationship with 2 and 3. In some scales a single word only is found in the second quinate to indicate that 5 was originally the base on which the system rested. It is hardly to be doubted, even, that change might affect each and every one of the numerals from 5 to 10 or 6 to 9, so that a dependence which might once have been easily detected is now unrecognizable. But if this is so, the natural and inevitable question follows--might not this have been the history of all numeral scales now purely decimal? May not the changes of time have altered the compounds which were once a clear indication of quinary counting, until no trace remains by which they can be followed back to their true origin? Perhaps so. It is not in the least degree probable, but its possibility may, of course, be admitted. But even then the universality of quinary counting for primitive peoples is by no means established. In Chapter II, examples were given of races which had no number base. Later on it was observed that in Australia and South America many tribes used 2 as their number base; in some cases counting on past 5 without showing any tendency to use that as a new unit. Again, through the habit of counting upon the finger joints, instead of the fingers themselves, the use of 3 as a base is brought into prominence, and 6 and 9 become 2 threes and 3 threes, respectively, instead of 5 + 1 and 5 + 4. The same may be noticed of 4. Counting by means of his fingers, without including the thumbs, the savage begins by dividing into fours instead of fives. Traces of this form of counting are somewhat numerous, especially among the North American aboriginal tribes. Hence the quinary form of counting, however widespread its use may be shown to be, can in no way be claimed as the universal method of any stage of development in the history of mankind. In the vast majority of cases, the passage from the base to the next succeeding number in any scale, is clearly defined. But among races whose intelligence is of a low order, or--if it be permissible to express it in this way--among races whose number sense is feeble, progression from one number to the next is not always in accordance with any well-defined law. After one or two distinct numerals the count may, as in the case of the Veddas and the Andamans, proceed by finger pantomime and by the repetition of the same word. Occasionally the same word is used for two successive numbers, some gesture undoubtedly serving to distinguish the one from the other in the savage's mind. Examples of this are not infrequent among the forest tribes of South America. In the Tariana dialect 9 and 10 are expressed by the same word, _paihipawalianuda;_ in Cobeu, 8 and 9 by _pepelicoloblicouilini;_ in Barre, 4, 5, and 9 by _ualibucubi._[326] In other languages the change from one numeral to the next is so slight that one instinctively concludes that the savage is forming in his own mind another, to him new, numeral immediately from the last. In such cases the entire number system is scanty, and the creeping hesitancy with which progress is made is visible in the forms which the numerals are made to take. A single illustration or two of this must suffice; but the ones chosen are not isolated cases. The scale of the Macunis,[327] one of the numerous tribes of Brazil, is 1. pocchaenang. 2. haihg. 3. haigunhgnill. 4. haihgtschating. 5. haihgtschihating = another 4? 6. hathig-stchihathing = 2-4? 7. hathink-tschihathing = 2-5? 8. hathink-tschihating = 2 × 4? The complete absence of--one is tempted to say--any rhyme or reason from this scale is more than enough to refute any argument which might tend to show that the quinary, or any other scale, was ever the sole number scale of primitive man. Irregular as this is, the system of the Montagnais fully matches it, as the subjoined numerals show:[328] 1. inl'are. 2. nak'e. 3. t'are. 4. dinri. 5. se-sunlare. 6. elkke-t'are = 2 × 3. 7. t'a-ye-oyertan = 10 - 3, or inl'as dinri = 4 + 3? 8. elkke-dinri = 2 × 4. 9. inl'a-ye-oyertan = 10 - 1. 10. onernan. CHAPTER VII. THE VIGESIMAL SYSTEM. In its ordinary development the quinary system is almost sure to merge into either the decimal or the vigesimal system, and to form, with one or the other or both of these, a mixed system of counting. In Africa, Oceanica, and parts of North America, the union is almost always with the decimal scale; while in other parts of the world the quinary and the vigesimal systems have shown a decided affinity for each other. It is not to be understood that any geographical law of distribution has ever been observed which governs this, but merely that certain families of races have shown a preference for the one or the other method of counting. These families, disseminating their characteristics through their various branches, have produced certain groups of races which exhibit a well-marked tendency, here toward the decimal, and there toward the vigesimal form of numeration. As far as can be ascertained, the choice of the one or the other scale is determined by no external circumstances, but depends solely on the mental characteristics of the tribes themselves. Environment does not exert any appreciable influence either. Both decimal and vigesimal numeration are found indifferently in warm and in cold countries; in fruitful and in barren lands; in maritime and in inland regions; and among highly civilized or deeply degraded peoples. Whether or not the principal number base of any tribe is to be 20 seems to depend entirely upon a single consideration; are the fingers alone used as an aid to counting, or are both fingers and toes used? If only the fingers are employed, the resulting scale must become decimal if sufficiently extended. If use is made of the toes in addition to the fingers, the outcome must inevitably be a vigesimal system. Subordinate to either one of these the quinary may and often does appear. It is never the principal base in any extended system. To the statement just made respecting the origin of vigesimal counting, exception may, of course, be taken. In the case of numeral scales like the Welsh, the Nahuatl, and many others where the exact meanings of the numerals cannot be ascertained, no proof exists that the ancestors of these peoples ever used either finger or toe counting; and the sweeping statement that any vigesimal scale is the outgrowth of the use of these natural counters is not susceptible of proof. But so many examples are met with in which the origin is clearly of this nature, that no hesitation is felt in putting the above forward as a general explanation for the existence of this kind of counting. Any other origin is difficult to reconcile with observed facts, and still more difficult to reconcile with any rational theory of number system development. Dismissing from consideration the quinary scale, let us briefly examine once more the natural process of evolution through which the decimal and the vigesimal scales come into being. After the completion of one count of the fingers the savage announces his result in some form which definitely states to his mind the fact that the end of a well-marked series has been reached. Beginning again, he now repeats his count of 10, either on his own fingers or on the fingers of another. With the completion of the second 10 the result is announced, not in a new unit, but by means of a duplication of the term already used. It is scarcely credible that the unit unconsciously adopted at the termination of the first count should now be dropped, and a new one substituted in its place. When the method here described is employed, 20 is not a natural unit to which higher numbers may be referred. It is wholly artificial; and it would be most surprising if it were adopted. But if the count of the second 10 is made on the toes in place of the fingers, the element of repetition which entered into the previous method is now wanting. Instead of referring each new number to the 10 already completed, the savage is still feeling his way along, designating his new terms by such phrases as "1 on the foot," "2 on the other foot," etc. And now, when 20 is reached, a single series is finished instead of a double series as before; and the result is expressed in one of the many methods already noticed--"one man," "hands and feet," "the feet finished," "all the fingers of hands and feet," or some equivalent formula. Ten is no longer the natural base. The number from which the new start is made is 20, and the resulting scale is inevitably vigesimal. If pebbles or sticks are used instead of fingers, the system will probably be decimal. But back of the stick and pebble counting the 10 natural counters always exist, and to them we must always look for the origin of this scale. In any collection of the principal vigesimal number systems of the world, one would naturally begin with those possessed by the Celtic races of Europe. These races, the earliest European peoples of whom we have any exact knowledge, show a preference for counting by twenties, which is almost as decided as that manifested by Teutonic races for counting by tens. It has been conjectured by some writers that the explanation for this was to be found in the ancient commercial intercourse which existed between the Britons and the Carthaginians and Phoenicians, whose number systems showed traces of a vigesimal tendency. Considering the fact that the use of vigesimal counting was universal among Celtic races, this explanation is quite gratuitous. The reason why the Celts used this method is entirely unknown, and need not concern investigators in the least. But the fact that they did use it is important, and commands attention. The five Celtic languages, Breton, Irish, Welsh, Manx, and Gaelic, contain the following well-defined vigesimal scales. Only the principal or characteristic numerals are given, those being sufficient to enable the reader to follow intelligently the growth of the systems. Each contains the decimal element also, and is, therefore, to be regarded as a mixed decimal-vigesimal system. IRISH.[329] 10. deic. 20. fice. 30. triocad = 3-10 40. da ficid = 2-20. 50. caogad = 5-10. 60. tri ficid = 3-20. 70. reactmoga = 7-10. 80. ceitqe ficid = 4-20. 90. nocad = 9-10. 100. cead. 1000. mile. GAELIC.[330] 10. deich. 20. fichead. 30. deich ar fichead = 10 + 20. 40. da fhichead = 2-20. 50. da fhichead is deich = 40 + 10. 60. tri fichead = 3-20. 70. tri fichead is deich = 60 + 10. 80. ceithir fichead = 4-20. 90. ceithir fichead is deich = 80 + 10. 100. ceud. 1000. mile. WELSH.[331] 10. deg. 20. ugain. 30. deg ar hugain = 10 + 20. 40. deugain = 2-20. 50. deg a deugain = 10 + 40. 60. trigain = 3-20. 70. deg a thrigain = 10 + 60. 80. pedwar ugain = 4-20. 90. deg a pedwar ugain = 80 + 10. 100. cant. MANX.[332] 10. jeih. 20. feed. 30. yn jeih as feed = 10 + 20. 40. daeed = 2-20. 50. jeih as daeed = 10 + 40. 60. three-feed = 3-20. 70. three-feed as jeih = 60 + 10. 80. kiare-feed = 4-20. 100. keead. 1000. thousane, or jeih cheead. BRETON.[333] 10. dec. 20. ueguend. 30. tregond = 3-10. 40. deu ueguend = 2-20. 50. hanter hand = half hundred. 60. tri ueguend = 3-20. 70. dec ha tri ueguend = 10 + 60. 80. piar ueguend = 4-20. 90. dec ha piar ueguend = 10 + 80. 100. cand. 120. hueh ueguend = 6-20. 140. seih ueguend = 7-20. 160. eih ueguend = 8-20. 180. nau ueguend = 9-20. 200. deu gand = 2-100. 240. deuzec ueguend = 12-20. 280. piarzec ueguend = 14-20. 300. tri hand, or pembzec ueguend. 400. piar hand = 4-100. 1000. mil. These lists show that the native development of the Celtic number systems, originally showing a strong preference for the vigesimal method of progression, has been greatly modified by intercourse with Teutonic and Latin races. The higher numerals in all these languages, and in Irish many of the lower also, are seen at a glance to be decimal. Among the scales here given the Breton, the legitimate descendant of the ancient Gallic, is especially interesting; but here, just as in the other Celtic tongues, when we reach 1000, the familiar Latin term for that number appears in the various corruptions of _mille_, 1000, which was carried into the Celtic countries by missionary and military influences. In connection with the Celtic language, mention must be made of the persistent vigesimal element which has held its place in French. The ancient Gauls, while adopting the language of their conquerors, so far modified the decimal system of Latin as to replace the natural _septante_, 70, _octante_, 80, _nonante_, 90, by _soixante-dix_, 60-10, _quatre-vingt_, 4-20, and _quatrevingt-dix_, 4-20-10. From 61 to 99 the French method of counting is wholly vigesimal, except for the presence of the one word _soixante_. In old French this element was still more pronounced. _Soixante_ had not yet appeared; and 60 and 70 were _treis vinz_, 3-20, and _treis vinz et dis_, 3-20 and 10 respectively. Also, 120 was _six vinz_, 6-20, 140 was _sept-vinz_, etc.[334] How far this method ever extended in the French language proper, it is, perhaps, impossible to say; but from the name of an almshouse, _les quinze-vingts_,[335] which formerly existed in Paris, and was designed as a home for 300 blind persons, and from the _pembzek-ueguent_, 15-20, of the Breton, which still survives, we may infer that it was far enough to make it the current system of common life. Europe yields one other example of vigesimal counting, in the number system of the Basques. Like most of the Celtic scales, the Basque seems to become decimal above 100. It does not appear to be related to any other European system, but to be quite isolated philologically. The higher units, as _mila_, 1000, are probably borrowed, and not native. The tens in the Basque scale are:[336] 10. hamar. 20. hogei. 30. hogei eta hamar = 20 + 10. 40. berrogei = 2-20. 50. berrogei eta hamar = 2-20 + 10. 60. hirurogei = 3-20. 70. hirurogei eta hamar = 3-20 + 10. 80. laurogei = 4-20. 90. laurogei eta hamar = 4-20 + 10. 100. ehun. 1000. _milla_. Besides these we find two or three numeral scales in Europe which contain distinct traces of vigesimal counting, though the scales are, as a whole, decidedly decimal. The Danish, one of the essentially Germanic languages, contains the following numerals: 30. tredive = 3-10. 40. fyrretyve = 4-10. 50. halvtredsindstyve = half (of 20) from 3-20. 60. tresindstyve = 3-20. 70. halvfierdsindstyve = half from 4-20. 80. fiirsindstyve = 4-20. 90. halvfemsindstyve = half from 5-20. 100. hundrede. Germanic number systems are, as a rule, pure decimal systems; and the Danish exception is quite remarkable. We have, to be sure, such expressions in English as _three score_, _four score_, etc., and the Swedish, Icelandic, and other languages of this group have similar terms. Still, these are not pure numerals, but auxiliary words rather, which belong to the same category as _pair_, _dozen_, _dizaine_, etc., while the Danish words just given are the ordinary numerals which form a part of the every-day vocabulary of that language. The method by which this scale expresses 50, 70, and 90 is especially noticeable. It will be met with again, and further examples of its occurrence given. In Albania there exists one single fragment of vigesimal numeration, which is probably an accidental compound rather than the remnant of a former vigesimal number system. With this single exception the Albanian scale is of regular decimal formation. A few of the numerals are given for the sake of comparison:[337] 30. tridgiete = 3-10. 40. dizet = 2-20. 50. pesedgiete = 5-10. 60. giastedgiete = 6-10, etc. Among the almost countless dialects of Africa we find a comparatively small number of vigesimal number systems. The powers of the negro tribes are not strongly developed in counting, and wherever their numeral scales have been taken down by explorers they have almost always been found to be decimal or quinary-decimal. The small number I have been able to collect are here given. They are somewhat fragmentary, but are as complete as it was possible to make them. AFFADEH.[338] 10. dekang. 20. degumm. 30. piaske. 40. tikkumgassih = 20 × 2. 50. tikkumgassigokang = 20 × 2 + 10. 60. tikkumgakro = 20 × 3. 70. dungokrogokang = 20 × 3 + 10. 80. dukumgade = 20 × 4. 90. dukumgadegokang = 20 × 4 + 10. 100. miah (borrowed from the Arabs). IBO.[339] 10. iri. 20. ogu. 30. ogu n-iri = 20 + 10, or iri ato = 10 × 3. 40. ogu abuo = 20 × 2, or iri anno = 10 × 4. 100. ogu ise = 20 × 5. VEI.[340] 10. tan. 20. mo bande = a person finished. 30. mo bande ako tan = 20 + 10. 40. mo fera bande = 2 × 20. 100. mo soru bande = 5 persons finished. YORUBA.[341] 10. duup. 20. ogu. 30. ogbo. 40. ogo-dzi = 20 × 2. 60. ogo-ta = 20 × 3. 80. ogo-ri = 20 × 4. 100. ogo-ru = 20 × 5. 120. ogo-fa = 20 × 6. 140. ogo-dze = 20 × 7. 160. ogo-dzo = 20 × 8, etc. EFIK.[342] 10. duup. 20. edip. 30. edip-ye-duup = 20 + 10. 40. aba = 20 × 2. 60. ata = 20 × 3. 80. anan = 20 × 4. 100. ikie. The Yoruba scale, to which reference has already been made, p. 70, again shows its peculiar structure, by continuing its vigesimal formation past 100 with no interruption in its method of numeral building. It will be remembered that none of the European scales showed this persistency, but passed at that point into decimal numeration. This will often be found to be the case; but now and then a scale will come to our notice whose vigesimal structure is continued, without any break, on into the hundreds and sometimes into the thousands. BONGO.[343] 10. kih. 20. mbaba kotu = 20 × 1. 40. mbaba gnorr = 20 × 2. 100. mbaba mui = 20 × 5. MENDE.[344] 10. pu. 20. nu yela gboyongo mai = a man finished. 30. nu yela gboyongo mahu pu = 20 + 10. 40. nu fele gboyongo = 2 men finished. 100. nu lolu gboyongo = 5 men finished. NUPE.[345] 10. gu-wo. 20. esin. 30. gbonwo. 40. si-ba = 2 × 20. 50. arota. 60. sita = 3 × 20. 70. adoni. 80. sini = 4 × 20. 90. sini be-guwo = 80 + 10. 100. sisun = 5 × 20. LOGONE.[346] 10. chkan. 20. tkam. 30. tkam ka chkan = 20 + 10. 40. tkam ksde = 20 × 2. 50. tkam ksde ka chkan = 40 + 10. 60. tkam gachkir = 20 × 3. 100. mia (from Arabic). 1000. debu. MUNDO.[347] 10. nujorquoi. 20. tiki bere. 30. tiki bire nujorquoi = 20 + 10. 40. tiki borsa = 20 × 2. 50. tike borsa nujorquoi = 40 + 10. MANDINGO.[348] 10. tang. 20. mulu. 30. mulu nintang = 20 + 10. 40. mulu foola = 20 × 2. 50. mulu foola nintang = 40 + 10. 60. mulu sabba = 20 × 3. 70. mulu sabba nintang = 60 + 10. 80. mulu nani = 20 × 4. 90. mulu nani nintang = 80 + 10. 100. kemi. This completes the scanty list of African vigesimal number systems that a patient and somewhat extended search has yielded. It is remarkable that the number is no greater. Quinary counting is not uncommon in the "Dark Continent," and there is no apparent reason why vigesimal reckoning should be any less common than quinary. Any one investigating African modes of counting with the material at present accessible, will find himself hampered by the fact that few explorers have collected any except the first ten numerals. This leaves the formation of higher terms entirely unknown, and shows nothing beyond the quinary or non-quinary character of the system. Still, among those which Stanley, Schweinfurth, Salt, and others have collected, by far the greatest number are decimal. As our knowledge of African languages is extended, new examples of the vigesimal method may be brought to light. But our present information leads us to believe that they will be few in number. In Asia the vigesimal system is to be found with greater frequency than in Europe or Africa, but it is still the exception. As Asiatic languages are much better known than African, it is probable that the future will add but little to our stock of knowledge on this point. New instances of counting by twenties may still be found in northern Siberia, where much ethnological work yet remains to be done, and where a tendency toward this form of numeration has been observed to exist. But the total number of Asiatic vigesimal scales must always remain small--quite insignificant in comparison with those of decimal formation. In the Caucasus region a group of languages is found, in which all but three or four contain vigesimal systems. These systems are as follows: ABKHASIA.[349] 10. zpha-ba. 20. gphozpha = 2 × 10. 30. gphozphei zphaba = 20 + 10. 40. gphin-gphozpha = 2 × 20. 60. chin-gphozpha = 3 × 20. 80. phsin-gphozpha = 4 × 20. 100. sphki. AVARI 10. antsh-go. 20. qo-go. 30. lebergo. 40. khi-qogo = 2 × 20. 50. khiqojalda antshgo = 40 + 10. 60. lab-qogo = 3 × 20. 70. labqojalda antshgo = 60 + 10. 80. un-qogo = 4 × 20. 100. nusgo. KURI 10. tshud. 20. chad. 30. channi tshud = 20 + 10. 40. jachtshur. 50. jachtshurni tshud = 40 + 10. 60. put chad = 3 × 20. 70. putchanni tshud = 60 + 10. 80. kud-chad = 4 × 20. 90. kudchanni tshud = 80 + 10. 100. wis. UDI 10. witsh. 20. qa. 30. sa-qo-witsh = 20 + 10. 40. pha-qo = 2 × 20. 50. pha-qo-witsh = 40 + 10. 60. chib-qo = 3 × 20. 70. chib-qo-witsh = 60 + 10. 80. bip-qo = 4 × 20. 90. bip-qo-witsh = 80 + 10. 100. bats. 1000. hazar (Persian). TCHETCHNIA 10. ith. 20. tqa. 30. tqe ith = 20 + 10. 40. sauz-tqa = 2 × 20. 50. sauz-tqe ith = 40 + 10. 60. chuz-tqa = 3 × 20. 70. chuz-tqe ith = 60 + 10. 80. w-iez-tqa = 4 × 20. 90. w-iez-tqe ith = 80 + 10. 100. b'e. 1000. ezir (akin to Persian). THUSCH 10. itt. 20. tqa. 30. tqa-itt = 20 + 10. 40. sauz-tq = 2 × 20. 50. sauz-tqa-itt = 40 + 10. 60. chouz-tq = 3 × 20. 70. chouz-tqa-itt = 60 + 10. 80. dhewuz-tq = 4 × 20. 90. dhewuz-tqa-itt = 80 + 10. 100. phchauz-tq = 5 × 20. 200. itsha-tq = 10 × 20. 300. phehiitsha-tq = 15 × 20. 1000. satsh tqauz-tqa itshatqa = 2 × 20 × 20 + 200. GEORGIA 10. athi. 20. otsi. 30. ots da athi = 20 + 10. 40. or-m-otsi = 2 × 20. 50. ormots da athi = 40 + 10. 60. sam-otsi = 3 × 20. 70. samots da athi = 60 + 10. 80. othch-m-otsi = 4 × 20. 90. othmots da athi = 80 + 10. 100. asi. 1000. ath-asi = 10 × 100. LAZI 10. wit. 20. öts. 30. öts do wit = 20 × 10. 40. dzur en öts = 2 × 20. 50. dzur en öts do wit = 40 + 10. 60. dzum en öts = 3 × 20. 70. dzum en öts do wit = 60 + 10. 80. otch-an-öts = 4 × 20. 100. os. 1000. silia (akin to Greek). CHUNSAG.[350] 10. ants-go. 20. chogo. 30. chogela antsgo = 20 + 10. 40. kichogo = 2 × 20. 50. kichelda antsgo = 40 + 10. 60. taw chago = 3 × 20. 70. taw chogelda antsgo = 60 + 10. 80. uch' chogo = 4 × 20. 90. uch' chogelda antsgo. 100. nusgo. 1000. asargo (akin to Persian). DIDO.[351] 10. zino. 20. ku. 30. kunozino. 40. kaeno ku = 2 × 20. 50. kaeno kuno zino = 40 + 10. 60. sonno ku = 3 × 20. 70. sonno kuno zino = 60 + 10. 80. uino ku = 4 × 20. 90. uino huno zino = 80 + 10. 100. bischon. 400. kaeno kuno zino = 40 × 10. AKARI 10. entzelgu. 20. kobbeggu. 30. lowergu. 40. kokawu = 2 × 20. 50. kikaldanske = 40 + 10. 60. secikagu. 70. kawalkaldansku = 3 × 20 + 10. 80. onkuku = 4 × 20. 90. onkordansku = 4 × 20 + 10. 100. nosku. 1000. askergu (from Persian). CIRCASSIA 10. psche. 20. to-tsch. 30. totsch-era-pschirre = 20 + 10. 40. ptl'i-sch = 4 × 10. 50. ptl'isch-era-pschirre = 40 + 10. 60. chi-tsch = 6 × 10. 70. chitsch-era-pschirre = 60 + 10. 80. toshitl = 20 × 4? 90. toshitl-era-pschirre = 80 + 10. 100. scheh. 1000. min (Tartar) or schi-psche = 100 × 10. The last of these scales is an unusual combination of decimal and vigesimal. In the even tens it is quite regularly decimal, unless 80 is of the structure suggested above. On the other hand, the odd tens are formed in the ordinary vigesimal manner. The reason for this anomaly is not obvious. I know of no other number system that presents the same peculiarity, and cannot give any hypothesis which will satisfactorily account for its presence here. In nearly all the examples given the decimal becomes the leading element in the formation of all units above 100, just as was the case in the Celtic scales already noticed. Among the northern tribes of Siberia the numeral scales appear to be ruder and less simple than those just examined, and the counting to be more consistently vigesimal than in any scale we have thus far met with. The two following examples are exceedingly interesting, as being among the best illustrations of counting by twenties that are to be found anywhere in the Old World. TSCHUKSCHI.[352] 10. migitken = both hands. 20. chlik-kin = a whole man. 30. chlikkin mingitkin parol = 20 + 10. 40. nirach chlikkin = 2 × 20. 100. milin chlikkin = 5 × 20. 200. mingit chlikkin = 10 × 20, _i.e._ 10 men. 1000. miligen chlin-chlikkin = 5 × 200, _i.e._ five (times) 10 men. AINO.[353] 10. wambi. 20. choz. 30. wambi i-doehoz = 10 from 40. 40. tochoz = 2 × 20. 50. wambi i-richoz = 10 from 60. 60. rechoz = 3 × 20. 70. wambi [i?] inichoz = 10 from 80. 80. inichoz = 4 × 20. 90. wambi aschikinichoz = 10 from 100. 100. aschikinichoz = 5 × 20. 110. wambi juwanochoz = 10 from 120. 120. juwano choz = 6 × 20. 130. wambi aruwanochoz = 10 from 140. 140. aruwano choz = 7 × 20. 150. wambi tubischano choz = 10 from 160. 160. tubischano choz = 8 × 20. 170. wambi schnebischano choz = 10 from 180. 180. schnebischano choz = 9 × 20. 190. wambi schnewano choz = 10 from 200. 200. schnewano choz = 10 × 20. 300. aschikinichoz i gaschima chnewano choz = 5 × 20 + 10 × 20. 400. toschnewano choz = 2 × (10 × 20). 500. aschikinichoz i gaschima toschnewano choz = 100 + 400. 600. reschiniwano choz = 3 × 200. 700. aschikinichoz i gaschima reschiniwano choz = 100 + 600. 800. inischiniwano choz = 4 × 200. 900. aschikinichoz i gaschima inischiniwano choz = 100 + 800. 1000. aschikini schinewano choz = 5 × 200. 2000. wanu schinewano choz = 10 × (10 × 20). This scale is in one sense wholly vigesimal, and in another way it is not to be regarded as pure, but as mixed. Below 20 it is quinary, and, however far it might be extended, this quinary element would remain, making the scale quinary-vigesimal. But in another sense, also, the Aino system is not pure. In any unmixed vigesimal scale the word for 400 must be a simple word, and that number must be taken as the vigesimal unit corresponding to 100 in the decimal scale. But the Ainos have no simple numeral word for any number above 20, forming all higher numbers by combinations through one or more of the processes of addition, subtraction, and multiplication. The only number above 20 which is used as a unit is 200, which is expressed merely as 10 twenties. Any even number of hundreds, or any number of thousands, is then indicated as being so many times 10 twenties; and the odd hundreds are so many times 10 twenties, plus 5 twenties more. This scale is an excellent example of the cumbersome methods used by uncivilized races in extending their number systems beyond the ordinary needs of daily life. In Central Asia a single vigesimal scale comes to light in the following fragment of the Leptscha scale, of the Himalaya region:[354] 10. kati. 40. kafali = 4 × 10, or kha nat = 2 × 20. 50. kafano = 5 × 10, or kha nat sa kati = 2 × 20 + 10. 100. gjo, or kat. Further to the south, among the Dravidian races, the vigesimal element is also found. The following will suffice to illustrate the number systems of these dialects, which, as far as the material at hand shows, are different from each other only in minor particulars: MUNDARI.[355] 10. gelea. 20. mi hisi. 30. mi hisi gelea = 20 + 10. 40. bar hisi = 2 × 20. 60. api hisi = 3 × 20. 80. upun hisi = 4 × 20. 100. mone hisi = 5 × 20. In the Nicobar Islands of the Indian Ocean a well-developed example of vigesimal numeration is found. The inhabitants of these islands are so low in the scale of civilization that a definite numeral system of any kind is a source of some surprise. Their neighbours, the Andaman Islanders, it will be remembered, have but two numerals at their command; their intelligence does not seem in any way inferior to that of the Nicobar tribes, and one is at a loss to account for the superior development of the number sense in the case of the latter. The intercourse of the coast tribes with traders might furnish an explanation of the difficulty were it not for the fact that the numeration of the inland tribes is quite as well developed as that of the coast tribes; and as the former never come in contact with traders and never engage in barter of any kind except in the most limited way, the conclusion seems inevitable that this is merely one of the phenomena of mental development among savage races for which we have at present no adequate explanation. The principal numerals of the inland and of the coast tribes are:[356] INLAND TRIBES COAST TRIBES 10. teya. 10. sham. 20. heng-inai. 20. heang-inai. 30. heng-inai-tain 30. heang-inai-tanai = 20 + 5 (couples). = 20 + 5 (couples). 40. au-inai = 2 × 20. 40. an-inai = 2 × 20. 100. tain-inai = 5 × 20. 100. tanai-inai = 5 × 20. 200. teya-inai = 10 × 20. 200. sham-inai = 10 × 20. 300. teya-tain-inai 300. heang-tanai-inai = (10 + 5) × 20. = (10 + 5) 20. 400. heng-teo. 400. heang-momchiama. In no other part of the world is vigesimal counting found so perfectly developed, and, among native races, so generally preferred, as in North and South America. In the eastern portions of North America and in the extreme western portions of South America the decimal or the quinary decimal scale is in general use. But in the northern regions of North America, in western Canada and northwestern United States, in Mexico and Central America, and in the northern and western parts of South America, the unit of counting among the great majority of the native races was 20. The ethnological affinities of these races are not yet definitely ascertained; and it is no part of the scope of this work to enter into any discussion of that involved question. But either through contact or affinity, this form of numeration spread in prehistoric times over half or more than half of the western hemisphere. It was the method employed by the rude Eskimos of the north and their equally rude kinsmen of Paraguay and eastern Brazil; by the forest Indians of Oregon and British Columbia, and by their more southern kinsmen, the wild tribes of the Rio Grande and of the Orinoco. And, most striking and interesting of all, it was the method upon which were based the numeral systems of the highly civilized races of Mexico, Yucatan, and New Granada. Some of the systems obtained from the languages of these peoples are perfect, extended examples of vigesimal counting, not to be duplicated in any other quarter of the globe. The ordinary unit was, as would be expected, "one man," and in numerous languages the words for 20 and man are identical. But in other cases the original meaning of that numeral word has been lost; and in others still it has a signification quite remote from that given above. These meanings will be noticed in connection with the scales themselves, which are given, roughly speaking, in their geographical order, beginning with the Eskimo of the far north. The systems of some of the tribes are as follows: ALASKAN ESKIMOS.[357] 10. koleet. 20. enuenok. 30. enuenok kolinik = 20 + 10. 40. malho kepe ak = 2 × 20. 50. malho-kepe ak-kolmik che pah ak to = 2 × 20 + 10. 60. pingi shu-kepe ak = 3 × 20. 100. tale ma-kepe ak = 5 × 20. 400. enue nok ke pe ak = 20 × 20. TCHIGLIT.[358] 10. krolit. 20. kroleti, or innun = man. 30. innok krolinik-tchikpalik = man + 2 hands. 40. innum mallerok = 2 men. 50. adjigaynarmitoat = as many times 10 as the fingers of the hand. 60. innumipit = 3 men. 70. innunmalloeronik arveneloerit = 7 men? 80. innun pinatçunik arveneloerit = 8 men? 90. innun tcitamanik arveneloerit = 9 men? 100. itchangnerkr. 1000. itchangner-park = great 100. The meanings for 70, 80, 90, are not given by Father Petitot, but are of such a form that the significations seem to be what are given above. Only a full acquaintance with the Tchiglit language would justify one in giving definite meanings to these words, or in asserting that an error had been made in the numerals. But it is so remarkable and anomalous to find the decimal and vigesimal scales mingled in this manner that one involuntarily suspects either incompleteness of form, or an actual mistake. TLINGIT.[359] 10. djinkat = both hands? 20. tle ka = 1 man. 30. natsk djinkat = 3 × 10. 40. dak'on djinkat = 4 × 10. 50. kedjin djinkat = 5 × 10. 60. tle durcu djinkat = 6 × 10. 70. daqa durcu djinkat = 7 × 10. 80. natska durcu djinkat = 8 × 10. 90. gocuk durcu djinkat = 9 × 10. 100. kedjin ka = 5 men, or 5 × 20. 200. djinkat ka = 10 × 20. 300. natsk djinkat ka = 30 men. 400. dak'on djinkat ka = 40 men. This scale contains a strange commingling of decimal and vigesimal counting. The words for 20, 100, and 200 are clear evidence of vigesimal, while 30 to 90, and the remaining hundreds, are equally unmistakable proof of decimal, numeration. The word _ka_, man, seems to mean either 10 or 20; a most unusual occurrence. The fact that a number system is partly decimal and partly vigesimal is found to be of such frequent occurrence that this point in the Tlingit scale need excite no special wonder. But it is remarkable that the same word should enter into numeral composition under such different meanings. NOOTKA.[360] 10. haiu. 20. tsakeits. 30. tsakeits ic haiu = 20 + 10. 40. atlek = 2 × 20. 60. katstsek = 3 × 20. 80. moyek = 4 × 20. 100. sutc'ek = 5 × 20. 120. nop'ok = 6 × 20. 140. atlpok = 7 × 20. 160. atlakutlek = 8 × 20. 180. ts'owakutlek = 9 × 20. 200. haiuk = 10 × 20. This scale is quinary-vigesimal, with no apparent decimal element in its composition. But the derivation of some of the terms used is detected with difficulty. In the following scale the vigesimal structure is still more obscure. TSIMSHIAN.[361] 10. gy'ap. 20. kyedeel = 1 man. 30. gulewulgy'ap. 40. t'epqadalgyitk, or tqalpqwulgyap. 50. kctoncwulgyap. 100. kcenecal. 200. k'pal. 300. k'pal te kcenecal = 200 + 100. 400. kyedal. 500. kyedal te kcenecal = 400 + 100. 600. gulalegyitk. 700. gulalegyitk te kcenecal = 600 + 100. 800. tqalpqtalegyitk. 900. tqalpqtalegyitk te kcenecal = 800 + 100. 1000. k'pal. To the unobservant eye this scale would certainly appear to contain no more than a trace of the vigesimal in its structure. But Dr. Boas, who is one of the most careful and accurate of investigators, says in his comment on this system: "It will be seen at once that this system is quinary-vigesimal.... In 20 we find the word _gyat_, man. The hundreds are identical with the numerals used in counting men (see p. 87), and then the quinary-vigesimal system is most evident." RIO NORTE INDIANS.[362] 20. taiguaco. 30. taiguaco co juyopamauj ajte = 20 + 2 × 5. 40. taiguaco ajte = 20 × 2. 50. taiguaco ajte co juyopamauj ajte = 20 × 2 + 5 × 2. CARIBS OF ESSIQUIBO, GUIANA 10. oween-abatoro. 20. owee-carena = 1 person. 40. oko-carena = 2 persons. 60. oroowa-carena = 3 persons. OTOMI 10. ra-tta. 20. na-te. 30. na-te-m'a-ratta = 20 + 10. 40. yo-te = 2 × 30. 50. yote-m'a-ratta = 2 × 20 + 10. 60. hiu-te = 3 × 20. 70. hiute-m'a-ratta = 3 × 20 + 10. 80. gooho-rate = 4 × 20. 90. gooho-rate-m'a ratta = 4 × 20 + 10. 100. cytta-te = 5 × 20, or nanthebe = 1 × 100. MAYA, YUCATAN.[363] 1. hun. 10. lahun = it is finished. 20. hunkal = a measure, or more correctly, a fastening together. 30. lahucakal = 40 - 10? 40. cakal = 2 × 20. 50. lahuyoxkal = 60 - 10. 60. oxkal = 3 × 20. 70. lahucankal = 80 - 10. 80. cankal = 4 × 20. 90. lahuyokal = 100 - 10. 100. hokal = 5 × 20. 110. lahu uackal = 120 - 10. 120. uackal = 6 × 20. 130. lahu uuckal = 140 - 10. 140. uuckal = 7 × 20. 200. lahuncal = 10 × 20. 300. holhukal = 15 × 20. 400. hunbak = 1 tying around. 500. hotubak. 600. lahutubak 800. calbak = 2 × 400. 900. hotu yoxbak. 1000. lahuyoxbak. 1200. oxbak = 3 × 400. 2000. capic (modern). 8000. hunpic = 1 sack. 16,000. ca pic (ancient). 160,000. calab = a filling full 3,200,000. kinchil. 64,000,000. hunalau. In the Maya scale we have one of the best and most extended examples of vigesimal numeration ever developed by any race. To show in a more striking and forcible manner the perfect regularity of the system, the following tabulation is made of the various Maya units, which will correspond to the "10 units make one ten, 10 tens make one hundred, 10 hundreds make one thousand," etc., which old-fashioned arithmetic compelled us to learn in childhood. The scale is just as regular by twenties in Maya as by tens in English. It is[364] 20 hun = 1 kal = 20. 20 kal = 1 bak = 400. 20 bak = 1 pic = 8000. 20 pic = 1 calab = 160,000. 20 calab = 1 { kinchil } = 3,200,000. { tzotzceh } 20 kinchil = 1 alau = 64,000,000. The original meaning of _pic_, given in the scale as "a sack," was rather "a short petticoat, somtimes used as a sack." The word _tzotzceh_ signified "deerskin." No reason can be given for the choice of this word as a numeral, though the appropriateness of the others is sufficiently manifest. No evidence of digital numeration appears in the first 10 units, but, judging from the almost universal practice of the Indian tribes of both North and South America, such may readily have been the origin of Maya counting. Whatever its origin, it certainly expanded and grew into a system whose perfection challenges our admiration. It was worthy of the splendid civilization of this unfortunate race, and, through its simplicity and regularity, bears ample testimony to the intellectual capacity which originated it. The only example of vigesimal reckoning which is comparable with that of the Mayas is the system employed by their northern neighbours, the Nahuatl, or, as they are more commonly designated, the Aztecs of Mexico. This system is quite as pure and quite as simple as the Maya, but differs from it in some important particulars. In its first 20 numerals it is quinary (see p. 141), and as a system must be regarded as quinary-vigesimal. The Maya scale is decimal through its first 20 numerals, and, if it is to be regarded as a mixed scale, must be characterized as decimal-vigesimal. But in both these instances the vigesimal element preponderates so strongly that these, in common with their kindred number systems of Mexico, Yucatan, and Central America, are always thought of and alluded to as vigesimal scales. On account of its importance, the Nahuatl system[365] is given in fuller detail than most of the other systems I have made use of. 10. matlactli = 2 hands. 20. cempoalli = 1 counting. 21. cempoalli once = 20-1. 22. cempoalli omome = 20-2. 30. cempoalli ommatlactli = 20-10. 31. cempoalli ommatlactli once = 20-10-1. 40. ompoalli = 2 × 20. 50. ompoalli ommatlactli = 40-10. 60. eipoalli, or epoalli, = 3 × 20. 70. epoalli ommatlactli = 60-10. 80. nauhpoalli = 4 × 20. 90. nauhpoalli ommatlactli = 90-10. 100. macuilpoalli = 5 × 20. 120. chiquacempoalli = 6 × 20. 140. chicompoalli = 7 × 20. 160. chicuepoalli = 8 × 20. 180. chiconauhpoalli = 9 × 20. 200. matlacpoalli = 10 × 20. 220. matlactli oncempoalli = 11 × 20. 240. matlactli omompoalli = 12 × 20. 260. matlactli omeipoalli = 13 × 20. 280. matlactli onnauhpoalli = 14 × 20. 300. caxtolpoalli = 15 × 20. 320. caxtolli oncempoalli. 399. caxtolli onnauhpoalli ipan caxtolli onnaui = 19 × 20 + 19. 400. centzontli = 1 bunch of grass, or 1 tuft of hair. 800. ometzontli = 2 × 400. 1200. eitzontli = 3 × 400. 7600. caxtolli onnauhtzontli = 19 × 400. 8000. cenxiquipilli, or cexiquipilli. 160,000. cempoalxiquipilli = 20 × 8000. 3,200,000. centzonxiquipilli = 400 × 8000. 64,000,000. cempoaltzonxiquipilli = 20 × 400 × 8000. Up to 160,000 the Nahuatl system is as simple and regular in its construction as the English. But at this point it fails in the formation of a new unit, or rather in the expression of its new unit by a simple word; and in the expression of all higher numbers it is forced to resort in some measure to compound terms, just as the English might have done had it not been able to borrow from the Italian. The higher numeral terms, under such conditions, rapidly become complex and cumbersome, as the following analysis of the number 1,279,999,999 shows.[366] The analysis will be readily understood when it is remembered that _ipan_ signifies plus. _Caxtolli onnauhpoaltzonxiquipilli ipan caxtolli onnauhtzonxiquipilli ipan caxtolli onnauhpoalxiquipilli ipan caxtolli onnauhxiquipilli ipan caxtolli onnauhtzontli ipan caxtolli onnauhpoalli ipan caxtolli onnaui;_ _i.e._ 1,216,000,000 + 60,800,000 + 3,040,000 + 152,000 + 7600 + 380 + 19. To show the compounding which takes place in the higher numerals, the analysis may be made more literally, thus: + (15 + 4) × 400 × 800 + (15 + 4) × 20 × 8000 + (15 + 4) × 8000 + (15 + 4) × 400 + (15 + 4) × 20 + 15 + 4. Of course this resolution suffers from the fact that it is given in digits arranged in accordance with decimal notation, while the Nahuatl numerals express values by a base twice as great. This gives the effect of a complexity and awkwardness greater than really existed in the actual use of the scale. Except for the presence of the quinary element the number just given is really expressed with just as great simplicity as it could be in English words if our words "million" and "billion" were replaced by "thousand thousand" and "thousand thousand thousand." If Mexico had remained undisturbed by Europeans, and science and commerce had been left to their natural growth and development, uncompounded words would undoubtedly have been found for the higher units, 160,000, 3,200,000, etc., and the system thus rendered as simple as it is possible for a quinary-vigesimal system to be. Other number scales of this region are given as follows: HUASTECA.[367] 10. laluh. 20. hum-inic = 1 man. 30. hum-inic-lahu = 1 man 10. 40. tzab-inic = 2 men. 50. tzab-inic-lahu = 2 men 10. 60. ox-inic = 3 men. 70. ox-inic-lahu = 3 men 10. 80. tze-tnic = 4 men. 90. tze-ynic-kal-laluh = 4 men and 10. 100. bo-inic = 5 men. 200. tzab-bo-inic = 2 × 5 men. 300. ox-bo-inic = 3 × 5 men. 400. tsa-bo-inic = 4 × 5 men. 600. acac-bo-inic = 6 × 5 men. 800. huaxic-bo-inic = 8 × 5 men. 1000. xi. 8000. huaxic-xi = 8-1000. The essentially vigesimal character of this system changes in the formation of some of the higher numerals, and a suspicion of the decimal enters. One hundred is _boinic_, 5 men; but 200, instead of being simply _lahuh-inic_, 10 men, is _tsa-bo-inic_, 2 × 100, or more strictly, 2 times 5 men. Similarly, 300 is 3 × 100, 400 is 4 × 100, etc. The word for 1000 is simple instead of compound, and the thousands appear to be formed wholly on the decimal base. A comparison of this scale with that of the Nahuatl shows how much inferior it is to the latter, both in simplicity and consistency. TOTONACO.[368] 10. cauh. 20. puxam. 30. puxamacauh = 20 + 10. 40. tipuxam = 2 × 20. 50. tipuxamacauh = 40 + 10. 60. totonpuxam = 3 × 20. 100. quitziz puxum = 5 × 20. 200. copuxam = 10 × 20. 400. tontaman. 1000. titamanacopuxam = 2 × 400 + 200. The essential character of the vigesimal element is shown by the last two numerals. _Tontamen_, the square of 20, is a simple word, and 1000 is, as it should be, 2 times 400, plus 200. It is most unfortunate that the numeral for 8000, the cube of 20, is not given. CORA.[369] 10. tamoamata. 20. cei-tevi. 30. ceitevi apoan tamoamata = 20 + 10. 40. huapoa-tevi = 2 × 20. 60. huaeica-tevi = 3 × 20. 100. anxu-tevi = 5 × 20. 400. ceitevi-tevi = 20 × 20. Closely allied with the Maya numerals and method of counting are those of the Quiches of Guatemala. The resemblance is so obvious that no detail in the Quiche scale calls for special mention. QUICHE.[370] 10. lahuh. 20. hu-uinac = 1 man. 30. hu-uinac-lahuh = 20 + 10. 40. ca-uinac = 2 men. 50. lahu-r-ox-kal = -10 + 3 × 20. 60. ox-kal = 3 × 20. 70. lahu-u-humuch = -10 + 80. 80. humuch. 90. lahu-r-ho-kal = -10 + 100. 100. hokal. 1000. o-tuc-rox-o-kal. Among South American vigesimal systems, the best known is that of the Chibchas or Muyscas of the Bogota region, which was obtained at an early date by the missionaries who laboured among them. This system is much less extensive than that of some of the more northern races; but it is as extensive as almost any other South American system with the exception of the Peruvian, which was, however, a pure decimal system. As has already been stated, the native races of South America were, as a rule, exceedingly deficient in regard to the number sense. Their scales are rude, and show great poverty, both in formation of numeral words and in the actual extent to which counting was carried. If extended as far as 20, these scales are likely to become vigesimal, but many stop far short of that limit, and no inconsiderable number of them fail to reach even 5. In this respect we are reminded of the Australian scales, which were so rudimentary as really to preclude any proper use of the word "system" in connection with them. Counting among the South American tribes was often equally limited, and even less regular. Following are the significant numerals of the scale in question: CHIBCHA, OR MUYSCA.[371] 10. hubchibica. 20. quihica ubchihica = thus says the foot, 10 = 10-10, or gueta = house. 30. guetas asaqui ubchihica = 20 + 10. 40. gue-bosa = 20 × 2. 60. gue-mica = 20 × 3. 80. gue-muyhica = 20 × 4. 100. gue-hisca = 20 × 5. NAGRANDA.[372] 10. guha. 20. dino. 30. 'badiñoguhanu = 20 + 10. 40. apudiño = 2 × 20. 50. apudiñoguhanu = 2 × 20 + 10. 60. asudiño = 3 × 20. 70. asudiñoguhanu = 3 × 20 + 10. 80. acudiño = 4 × 20. 90. acudiñoguhanu = 4 × 20 + 10. 100. huisudiño = 5 × 20, or guhamba = great 10. 200. guahadiño = 10 × 20. 400. diñoamba = great 20. 1000. guhaisudiño = 10 × 5 × 20. 2000. hisudiñoamba = 5 great 20's. 4000. guhadiñoamba = 10 great 20's. In considering the influence on the manners and customs of any people which could properly be ascribed to the use among them of any other base than 10, it must not be forgotten that no races, save those using that base, have ever attained any great degree of civilization, with the exception of the ancient Aztecs and their immediate neighbours, north and south. For reasons already pointed out, no highly civilized race has ever used an exclusively quinary system; and all that can be said of the influence of this mode of counting is that it gives rise to the habit of collecting objects in groups of five, rather than of ten, when any attempt is being made to ascertain their sum. In the case of the subsidiary base 12, for which the Teutonic races have always shown such a fondness, the dozen and gross of commerce, the divisions of English money, and of our common weights and measures are probably an outgrowth of this preference; and the Babylonian base, 60, has fastened upon the world forever a sexagesimal method of dividing time, and of measuring the circumference of the circle. The advanced civilization attained by the races of Mexico and Central America render it possible to see some of the effects of vigesimal counting, just as a single thought will show how our entire lives are influenced by our habit of counting by tens. Among the Aztecs the universal unit was 20. A load of cloaks, of dresses, or other articles of convenient size, was 20. Time was divided into periods of 20 days each. The armies were numbered by divisions of 8000;[373] and in countless other ways the vigesimal element of numbers entered into their lives, just as the decimal enters into ours; and it is to be supposed that they found it as useful and as convenient for all measuring purposes as we find our own system; as the tradesman of to-day finds the duodecimal system of commerce; or as the Babylonians of old found that singularly curious system, the sexagesimal. Habituation, the laws which the habits and customs of every-day life impose upon us, are so powerful, that our instinctive readiness to make use of any concept depends, not on the intrinsic perfection or imperfection which pertains to it, but on the familiarity with which previous use has invested it. Hence, while one race may use a decimal, another a quinary-vigesimal, and another a sexagesimal scale, and while one system may actually be inherently superior to another, no user of one method of reckoning need ever think of any other method as possessing practical inconveniences, of which those employing it are ever conscious. And, to cite a single instance which illustrates the unconscious daily use of two modes of reckoning in one scale, we have only to think of the singular vigesimal fragment which remains to this day imbedded in the numeral scale of the French. In counting from 70 to 100, or in using any number which lies between those limits, no Frenchman is conscious of employing a method of numeration less simple or less convenient in any particular, than when he is at work with the strictly decimal portions of his scale. He passes from the one style of counting to the other, and from the second back to the first again, entirely unconscious of any break or change; entirely unconscious, in fact, that he is using any particular system, except that which the daily habit of years has made a part himself. Deep regret must be felt by every student of philology, that the primitive meanings of simple numerals have been so generally lost. But, just as the pebble on the beach has been worn and rounded by the beating of the waves and by other pebbles, until no trace of its original form is left, and until we can say of it now only that it is quartz, or that it is diorite, so too the numerals of many languages have suffered from the attrition of the ages, until all semblance of their origin has been lost, and we can say of them only that they are numerals. Beyond a certain point we can carry the study neither of number nor of number words. At that point both the mathematician and the philologist must pause, and leave everything beyond to the speculations of those who delight in nothing else so much as in pure theory. THE END. INDEX OF AUTHORS. Adam, L., 44, 159, 166, 175. Armstrong, R.A., 180. Aymonier, A., 156. Bachofen, J.J., 131. Balbi, A., 151. Bancroft, H.H., 29, 47, 89, 93, 113, 199. Barlow, H., 108. Beauregard, O., 45, 83, 152. Bellamy, E.W., 9. Boas, F., 30, 45, 46, 65, 87, 88, 136, 163, 164, 171, 197, 198. Bonwick, J., 24, 27, 107, 108. Brinton, D.G., 2, 22, 46, 52, 57, 61, 111, 112, 140, 199, 200. Burton, R.F., 37, 71. Chamberlain, A.F., 45, 65, 93. Chase, P.E., 99. Clarke, H., 113. Codrington, R.H., 16, 95, 96, 136, 138, 145, 153, 154. Crawfurd, J., 89, 93, 130. Curr, E.M., 24-27, 104, 107-110, 112. Cushing, F.H., 13, 48. De Flacourt, 8, 9. De Quincey, T., 35. Deschamps, M., 28. Dobrizhoffer, M., 71. Dorsey, J.O., 59. Du Chaillu, P.B., 66, 67, 150, 151. Du Graty, A.M., 138. Ellis, A.A., 64, 91. Ellis, R., 37, 142. Ellis, W., 83, 119. Erskine, J.E., 153, 154. Flegel, R., 133. Gallatin, A., 136, 159, 166, 171, 199, 204, 206, 208. Galton, F., 4. Gatschet, A.S., 58, 59, 68. Gilij, F.S., 54. Gill, W.W., 18, 118. Goedel, M., 83, 147. Grimm, J.L.C., 48. Gröber, G., 182. Guillome, J., 181. Haddon, A.C., 18, 105. Hale, H., 61, 65, 93, 114-116, 122, 130, 156, 163, 164, 171. Hankel, H., 137. Haumonté, J.D., 44. Hervas, L., 170. Humboldt, A. von, 32, 207. Hyades, M., 22. Kelly, J.W., 157, 196. Kelly, J., 180. Kleinschmidt, S., 52, 80. Lang, J.D., 108. Lappenberg, J.M., 127. Latham, R.G., 24, 67, 107. Leibnitz, G.W. von, 102, 103. Lloyd, H.E., 7. Long, C.C., 148, 186. Long, S.H., 121. Lubbock, Sir J., 3, 5. Lull, E.P., 79. Macdonald, J., 15. Mackenzie, A., 26. Man, E.H., 28, 194. Mann, A., 47. Marcoy, P. (Saint Cricq), 23, 168. Mariner, A., 85. Martius, C.F. von, 23, 79, 111, 122, 138, 142, 174. Mason, 112. Mill, J.S., 3. Moncelon, M., 142. Morice, A., 15, 86. Müller, Fr., 10, 27, 28, 45, 48, 55, 56, 60, 63, 66, 69, 78, 80, 90, 108, 111, 121, 122, 130, 136, 139, 146-151, 156-158, 165-167, 185-187, 191, 193. Murdoch, J., 30, 49,137. Nystron, J.W., 132. O'Donovan, J., 180. Oldfield, A., 29, 77. Olmos, A. de, 141. Parisot, J., 44. Park, M., 145-147. Parry, W.E., 32. Peacock, G., 8, 56, 84, 111, 118, 119, 154, 186. Petitot, E., 53, 157, 196. Pott, A.F., 50, 68, 92, 120, 145, 148, 149, 152, 157, 166, 182, 184, 189, 191, 205. Pruner-Bey, 10, 104. Pughe, W.O., 141. Ralph, J., 125. Ray, S.H., 45, 78, 80. Ridley, W., 57. Roth, H.L., 79. Salt, H., 187. Sayce, A.H., 75. Schoolcraft, H.R., 66, 81, 83, 84, 159, 160. Schröder, P., 90. Schweinfurth, G., 143, 146, 149, 186, 187. Simeon, R., 201. Spix, J.B. von, 7. Spurrell, W., 180. Squier, G.E., 80, 207. Stanley, H.M., 38, 42, 64, 69, 78, 150, 187. Taplin, G., 106. Thiel, B.A., 172. Toy, C.H., 70. Turner, G., 152, 154. Tylor, E.B., 2, 3, 15, 18, 22, 63, 65, 78, 79, 81, 84, 97, 124. Van Eys, J.W., 182. Vignoli, T., 95. Wallace, A.R., 174. Wells, E.R., jr., 157, 196. Whewell, W., 3. Wickersham, J., 96. Wiener, C., 22. Williams, W.L., 123. INDEX OF SUBJECTS. Abacus, 19. Abeokuta, 33. Abipone, 71, 72. Abkhasia, 188. Aboker, 148. Actuary, Life ins., 19. Adaize, 162. Addition, 19, 43, 46, 92. Adelaide, 108. Admiralty Islands, 45. Affadeh, 184. Africa (African), 9, 16, 28, 29, 32, 33, 38, 42, 47, 64, 66, 69, 78, 80, 91, 105, 120, 145, 170, 176, 184, 187. Aino (Ainu), 45, 191. Akra, 120. Akari, 190. Alaska, 157, 196. Albania, 184. Albert River, 26. Aleut, 157. Algonkin (Algonquin), 45, 92, 161. Amazon, 23. Ambrym, 136. American, 10, 16, 19, 98, 105. Andaman, 8, 15, 28, 31, 76, 174, 193. Aneitum, 154. Animal, 3, 6. Anthropological, 21. Apho, 133. Api, 80, 136, 155. Apinage, 111. Arab, 170. Arawak, 52-54, 135. Arctic, 29. Arikara, 46. Arithmetic, 1, 5, 30, 33, 73, 93. Aryan, 76, 128-130. Ashantee, 145. Asia (Asiatic), 28, 113, 131, 187. Assiniboine, 66, 92. Athapaskan,92. Atlantic, 126. Aurora, 155. Australia (Australian), 2, 6, 19, 22, 24-30, 57, 58, 71, 75, 76, 84, 103, 105, 106, 110, 112, 118, 173, 206. Avari, 188. Aymara, 166. Aztec, 63, 78, 83, 89, 93, 201, 207, 208. Babusessé, 38. Babylonian, 208. Bagrimma, 148. Bahnars, 15. Bakairi, 111. Balad, 67. Balenque, 150. Bambarese, 95. Banks Islands, 16, 96, 153. Barea, 151. Bargaining, 18, 19, 32. Bari, 136. Barre, 174. Basa, 146. Basque, 40, 182. Bellacoola, see Bilqula. Belyando River, 109. Bengal, Bay of, 28. Benuë, 133. Betoya, 57, 112, 135, 140. Bilqula, 46, 164. Binary, chap. v. Binin, 149. Bird-nesting, 5. Bisaye, 90. Bogota, 206. Bolan, 120. Bolivia, 2, 21. Bongo, 143, 186. Bonzé, 151. Bororo, 23. Botocudo, 22, 31, 48, 71. Bourke, 108. Boyne River, 24. Brazil, 2, 7, 30, 174, 195. Bretagne (Breton), 120, 129, 181, 182. British Columbia, 45, 46, 65, 86, 88, 89, 112, 113, 195. Bullom, 147. Bunch, 64. Burnett River, 112. Bushman, 28, 31. Butong, 93. Caddoe, 162. Cahuillo, 165. Calculating machine, 19. Campa, 22. Canada, 29, 53, 54, 86, 195. Canaque, 142, 144. Caraja, 23. Carib, 166, 167, 199. Carnarvon, 35, 36. Carrier, 86. Carthaginian, 179. Caucasus, 188. Cayriri (see Kiriri), 79. Cayubaba (Cayubabi), 84, 167. Celtic, 40, 169, 179, 181, 190. Cely, Mom, 9. Central America, 29, 69, 79, 121, 131, 195, 201, 208. Ceylon, 28. Chaco, 22. Champion Bay, 109. Charles XII., 132. Cheyenne, 62. Chibcha, 206. China (Chinese), 40, 131. Chippeway, 62, 159, 162. Chiquito, 2, 6, 21, 40, 71, 76. Choctaw, 65, 85, 162. Chunsag, 189. Circassia, 190. Cobeu, 174. Cochin China, 15. Columbian, 113. Comanche, 29, 83. Conibo, 23. Cooper's Creek, 108. Cora, 166. Cotoxo, 111. Cowrie, 64, 70, 71. Cree, 91. Crocker Island, 107. Crow, 3, 4, 92. Crusoe, Robinson, 7. Curetu, 111. Dahomey, 71. Dakota, 81, 91, 92. Danish, 30, 46, 129, 183. Darnley Islands, 24. Delaware, 91, 160. Demara, 4, 6. Déné, 86. Dido, 189. Dinka, 136, 147. Dippil, 107. Division, 19. Dravidian, 104, 193. Dual number, 75. Duluth, 34. Duodecimal, chap. v. Dutch, 129. Eaw, 24. Ebon, 152. Efik, 148, 185. Encabellada, 22. Encounter Bay, 108. Ende, 68, 152. English, 28, 38-44, 60, 81, 85, 89, 118, 123, 124, 129, 183, 200, 203, 208. Eromanga, 96, 136, 154. Eskimo, 16, 30, 31, 32, 36, 48, 51, 52, 54, 61, 64, 83, 137, 157, 159, 195, 196. Essequibo, 166. Europe (European), 27, 39, 168, 169, 179, 182, 183, 185, 204. Eye, 14, 97. Eyer's Sand Patch, 26. Ewe, 64, 91. Fall, 163. Fate, 138, 155. Fatuhiva, 130. Feloop, 145. Fernando Po, 150. Fiji, 96. Finger pantomime, 10, 23, 29, 67, 82. Fingoe, 33. Fist, 16, 59, 72. Flinder's River, 24. Flores, 68, 152. Forefinger, 12, 15, 16, 54, 61, 91, 113. Foulah, 147. Fourth finger, 18. Frazer's Island, 108. French, 40, 41, 124, 129, 181, 182, 209. Fuegan, 22. Gaelic, 180. Galibi, 138. Gaul, 169, 182. Georgia, 189. German, 38-43, 129, 183. Gesture, 18, 59. Gola, 151. Golo, 146. Gonn Station, 110. Goth, 169. Greek, 86, 129, 168, 169. Green Island, 45. Greenland, 29, 52, 80, 158. Guachi, 23, 31. Guarani, 55, 138. Guatemala, 205. Guato, 142. Guaycuru, 22. Gudang, 24. Haida, 112. Hawaii, 113, 114, 116, 117. Head, 71. Heap, 8, 9, 25, 70, 77, 100. Hebrew, 86, 89, 95. Heiltsuk, 65, 88, 163. Herero, 150. Hervey Islands, 118. Hidatsa, 80, 91. Hill End, 109. Himalaya, 193. Hottentot, 80, 92. Huasteca, 204. Hudson's Bay, 48, 61. Hun, 169. Hunt, Leigh, 33. Ibo, 185. Icelandic, 129, 183. Illinois, 91. Index finger, 11, 14. India, 96, 112, 131. Indian, 8, 10, 13, 16, 17, 19, 32, 36, 54, 55, 59, 62, 65, 66, 79, 80, 82, 83, 89, 90, 98, 105, 112, 171, 201. Indian Ocean, 63, 193. Indo-European, 76. Irish, 129, 180. Italian, 39, 80, 124, 129, 203. Jajowerong, 156. Jallonkas, 146. Jaloff, 146. Japanese, 40, 86, 89, 93-95. Java, 93, 120. Jiviro, 61, 136. Joints of fingers, 7, 18, 173. Juri, 79. Kamassin, 130. Kamilaroi, 27, 107, 112. Kamtschatka, 75, 157. Kanuri, 136, 149. Karankawa, 68. Karen, 112. Keppel Bay, 24. Ki-Nyassa, 150. Kiriri, 55, 138, 139, 167. Kissi, 145. Ki-Swahili, 42. Ki-Yau, 150. Klamath, 58, 59. Knot, 7, 9, 19, 40, 93, 115. Kolyma, 75. Kootenay, 65. Koriak, 75. Kredy, 149. Kru, 146. Ku-Mbutti, 78. Kunama, 151. Kuri, 188. Kusaie, 78, 80. Kwakiutl, 45. Labillardière, 85. Labrador, 29. Lake Kopperamana, 107. Latin, 40, 44, 76, 81, 86, 124, 128, 168, 169, 181, 182. Lazi, 189. Left hand, 10-17, 54. Leper's Island, 16. Leptscha, 193. Lifu, 143. Little finger, 10-18, 48, 54, 61, 91. Logone, 186. London, 124. Lower California, 29. Luli, 118. Lutuami, 164. Maba, 80. Macassar, 93. Machine, Calculating, 19, 20. Mackenzie River, 157. Macuni, 174. Madagascar, 8, 9. Maipures, 15, 56. Mairassis, 10. Malagasy, 83, 95. Malanta, 96. Malay, 8, 45, 90, 93, 170. Mallicolo, 152. Manadu, 93. Mandingo, 186. Mangareva, 114. Manx, 180. Many, 2, 21-23, 25, 28, 100. Maori, 64, 93, 122. Marachowie, 26. Maré, 84. Maroura, 106. Marquesas, 93, 114, 115. Marshall Islands, 122, 152. Massachusetts, 91, 159. Mathematician, 2, 3, 35, 102, 127, 210. Matibani, 151. Matlaltzinca, 166. Maya, 45, 46, 199, 205. Mbayi, 111. Mbocobi, 22. Mbousha, 66. Melanesia, 16, 22, 28, 84, 95. Mende, 186. Mexico, 29, 195, 201, 204, 208. Miami, 91. Micmac, 90, 160. Middle finger, 12, 15, 62. Mille, 122. Minnal Yungar, 26. Minsi, 162. Mississaga, 44, 92. Mississippi, 125. Mocobi, 119. Mohegan, 91. Mohican, 172. Mokko, 149. Molele, 164. Moneroo, 109. Mongolian, 8. Montagnais, 53, 54, 175. Moree, 24. Moreton Bay, 108. Mort Noular, 107. Mosquito, 69, 70, 121. Mota, 95, 153. Mpovi, 152. Multiplication, 19, 33, 40, 43, 59. Mundari, 193. Mundo, 186. Muralug, 17. Murray River, 106, 109. Muysca, 206. Nagranda, 207. Nahuatl, 141, 144, 177, 201, 205. Nakuhiva, 116, 130. Negro, 8, 9, 15, 29, 184. Nengone, 63, 136. New, 128-130. New Caledonia, 154. New Granada, 195. New Guinea, 10, 152. New Hebrides, 155. New Ireland, 45. New Zealand, 123. Nez Perces, 65, 158. Ngarrimowro, 110. Niam Niam, 64, 136. Nicaragua, 80. Nicobar, 78, 193. Nightingale, 4. Nootka, 163, 198. Norman River, 24. North America, 28, 82, 171, 173, 176, 194, 201. Notch, 7, 9, 93. Numeral frame, 19. Nupe, 149, 186. Nusqually, 96. Oceania, 115, 176. Octonary, chap. v. Odessa, 34. Ojibwa, 84, 159. Okanaken, 88. Omaha, 161. Omeo, 110. Oregon, 58, 195. Orejone, 23. Orinoco, 54, 56, 195. Ostrich, 71, 72. Otomac, 15. Otomi, 165, 199. Ottawa, 159. Oyster Bay, 79. Pacific, 29, 113, 116, 117, 131. Palm (of the hand), 12, 14, 15. Palm Island, 156. Pama, 136, 155. Pampanaga, 66. Papaa, 148. Paraguay, 55, 71, 118, 195. Parana, 119. Paris, 182. Pawnee, 91, 121, 162. Pebble, 7-9, 19, 40, 93, 179. Peno, 2. Peru (Peruvian), 2, 22, 61, 206. Philippine, 66. Philology (Philologist), 128, 209, 210. Phoenician, 90, 179. Pigmy, 69, 70, 78. Pikumbul, 57, 138. Pines, Isle of, 153. Pinjarra, 26. Plenty, 25, 77. Point Barrow, 30, 51, 64, 83, 137, 159. Polynesia, 22, 28, 118, 130, 170. Pondo, 33. Popham Bay, 107. Port Darwin, 109. Port Essington, 24, 107. Port Mackay, 26. Port Macquarie, 109. Puget Sound, 96. Puri, 22, 92. Quappa, 171, 172. Quaternary, chap. v. Queanbeyan, 24. Quiche, 205. Quichua, 61. Rapid, 163. Rarotonga, 114. Richmond River, 109. Right hand, 10-18, 54. Right-handedness, 13, 14. Ring finger, 15. Rio Grande, 195. Rio Napo, 22. Rio Norte, 136, 199. Russia (Russian), 30, 35. Sahaptin, 158. San Antonio, 136. San Blas, 79, 80. Sanskrit, 40, 92, 97, 128. Sapibocone, 84, 167. Saste (Shasta), 165. Scratch, 7. Scythian, 169. Seed, 93. Semitic, 89. Senary, chap. v. Sesake, 136, 155. Several, 22. Sexagesimal, 124, 208. Shawnoe, 160. Shell, 7, 19, 70, 93. Shushwap, 88. Siberia, 29, 30, 187, 190. Sierra Leone, 83. Sign language, 6. Sioux, 83. Slang, 124. Slavonic, 40. Snowy River, 110. Soussou, 83, 147. South Africa, 4, 15, 28. South America, 2, 15, 22, 23, 27-29, 54, 57, 72, 76, 78, 79, 104, 110, 173, 174, 194, 201, 206. Spanish, 2, 23, 42. Splint, 7. Stick, 7, 179. Stlatlumh, 88. Streaky Bay, 26. String, 7, 9, 64, 71. Strong's Island, 78. Subtraction, 19, 44-47. Sunda, 120. Sweden (Swedish), 129, 132, 183. Tacona, 2. Taensa, 44. Tagala, 66. Tahiti, 114. Tahuata, 115. Tama, 111. Tamanac, 54, 135. Tambi, 120. Tanna, 154. Tarascan, 165. Tariana, 174. Tasmania, 24, 27, 79, 104, 106. Tawgy, 130. Tchetchnia, 188. Tchiglit, 157, 196. Tembu, 33. Temne, 148. Ternary, chap. v. Terraba, 172. Teutonic, 40, 41, 43, 179, 181, 208. Texas, 69. Thibet, 96. Thumb, 10-18, 54, 59, 61, 62, 113, 173. Thusch, 189. Ticuna, 168. Timukua, 165. Tlingit, 136, 163, 197. Tobi, 156. Tonga, 33, 85. Torres, 17, 96, 104, 105. Totonaco, 205. Towka, 78. Triton's Bay, 152. Tschukshi, 156, 191. Tsimshian, 86, 164, 198. Tweed River, 26. Uainuma, 122. Udi, 188. Uea, 67, 153. United States, 29, 83, 195. Upper Yarra, 110. Ureparapara, 153. Vaturana, 96. Vedda, 28, 31, 76, 174. Vei, 16, 147, 185. Victoria, 156. Vilelo, 60. Waiclatpu, 164. Wales (Welsh), 35, 46, 141, 144, 177, 180. Wallachia, 121. Warrego, 107, 109. Warrior Island, 107. Wasp, 5. Watchandie, 29, 77. Watji, 120. Weedookarry, 24. Wimmera, 107. Winnebago, 85. Wiraduroi, 27, 108. Wirri-Wirri, 108. Wokke, 112. Worcester, Mass., Schools of, 11. Yahua, 168. Yaruro, 139. Yengen, 154. Yit-tha, 109. Yoruba, 33, 47, 64, 70, 185. Yucatan, 195, 201. Yuckaburra, 26. Zamuco, 55, 60, 138, 139. Zapara, 111. Zulu, 16, 62. Zuñi, 13, 14, 48, 49, 53, 54, 60, 83, 137. FOOTNOTES: [1] Brinton, D.G., _Essays of an Americanist_, p. 406; and _American Race_, p. 359. [2] This information I received from Dr. Brinton by letter. [3] Tylor, _Primitive Culture_, Vol. I. p. 240. [4] _Nature_, Vol. XXXIII. p. 45. [5] Spix and Martius, _Travels in Brazil_, Tr. from German by H.E. Lloyd, Vol. II. p. 255. [6] De Flacourt, _Histoire de le grande Isle de Madagascar_, ch. xxviii. Quoted by Peacock, _Encyc. Met._, Vol. I. p. 393. [7] Bellamy, Elizabeth W., _Atlantic Monthly_, March, 1893, p. 317. [8] _Grundriss der Sprachwissenschaft_, Bd. III. Abt. i., p. 94. [9] Pruner-Bey, _Bulletin de la Société d'Anthr. de Paris_, 1861, p. 462. [10] "Manual Concepts," _Am. Anthropologist_, 1892, p. 292. [11] Tylor, _Primitive Culture_, Vol. I. p. 245. [12] _Op. cit._, _loc. cit._ [13] "Aboriginal Inhabitants of Andaman Islands," _Journ. Anth. Inst._, 1882, p. 100. [14] Morice, A., _Revue d'Anthropologie_, 1878, p. 634. [15] Macdonald, J., "Manners, Customs, etc., of South African Tribes," _Journ. Anthr. Inst._, 1889, p. 290. About a dozen tribes are enumerated by Mr. Macdonald: Pondos, Tembucs, Bacas, Tolas, etc. [16] Codrington, R.H., _Melanesians, their Anthropology and Folk-Lore_, p. 353. [17] _E.g._ the Zuñis. See Cushing's paper quoted above. [18] Haddon, A.C., "Ethnography Western Tribes Torres Strait," _Journ. Anth. Inst._, 1889, p. 305. For a similar method, see _Life in the Southern Isles_, by W.W. Gill. [19] Tylor, _Primitive Culture_, Vol. I. p. 246. [20] Brinton, D.G., Letter of Sept. 23, 1893. [21] _Ibid_. The reference for the Mbocobi, _infra_, is the same. See also Brinton's _American Race_, p. 361. [22] Tylor, _Primitive Culture_, Vol. I. p. 243. [23] _Op. cit._, _loc. cit._ [24] Hyades, _Bulletin de la Société d'Anthr. de Paris_, 1887, p. 340. [25] Wiener, C., _Pérou et Bolivie_, p. 360. [26] Marcoy, P., _Travels in South America_, Vol. II p. 47. According to the same authority, most of the tribes of the Upper Amazon cannot count above 2 or 3 except by reduplication. [27] _Op. cit._, Vol. II. p. 281. [28] _Glossaria Linguarum Brasiliensium_. Bororos, p. 15; Guachi, p. 133; Carajas, p. 265. [29] Curr, E.M., _The Australian Race_, Vol. I. p. 282. The next eight lists are, in order, from I. p. 294, III. p. 424, III. p. 114, III. p. 124, II. p. 344, II. p. 308, I. p. 314, III. p. 314, respectively. [30] Bonwick, J., _The Daily Life and Origin of the Tasmanians_, p. 144. [31] Latham, _Comparative Philology_, p. 336. [32] _The Australian Race_, Vol. I. p. 205. [33] Mackenzie, A., "Native Australian Langs.," _Journ. Anthr. Inst._, 1874, p. 263. [34] Curr, _The Australian Race_, Vol. II. p. 134. The next four lists are from II. p. 4, I. p. 322, I. p. 346, and I. p. 398, respectively. [35] Curr, _op. cit._, Vol. III. p. 50. [36] _Op. cit._, Vol. III. p. 236. [37] Müller, _Sprachwissenschaft_. II. i. p. 23. [38] _Op. cit._, II. i. p. 31. [39] Bonwick, _op. cit._, p. 143. [40] Curr, _op. cit._, Vol. I. p. 31. [41] Deschamps, _L'Anthropologie_, 1891, p. 318. [42] Man, E.H. _Aboriginal Inhabitants of the Andaman Islands_, p. 32. [43] Müller, _Sprachwissenschaft_, I. ii. p. 29. [44] Oldfield, A., Tr. Eth. Soc. Vol. III. p. 291. [45] Bancroft, H.H., _Native Races_, Vol. I. p. 564. [46] "Notes on Counting, etc., among the Eskimos of Point Barrow." _Am. Anthrop._, 1890, p. 38. [47] _Second Voyage_, p. 556. [48] _Personal Narrative_, Vol. I. p. 311. [49] Burton, B.F., _Mem. Anthr. Soc. of London_, Vol. I. p. 314. [50] _Confessions_. In collected works, Edinburgh, 1890, Vol. III. p. 337. [51] Ellis, Robert, _On Numerals as Signs of Primeval Unity_. See also _Peruvia Scythia_, by the same author. [52] Stanley, H.M., _In Darkest Africa_, Vol. II. p. 493. [53] Stanley, H.M., _Through the Dark Continent_, Vol. II. p. 486. [54] Haumontè, Parisot, Adam, _Grammaire et Vocabulaire de la Langue Taensa_, p. 20. [55] Chamberlain, A.F., _Lang. of the Mississaga Indians of Skugog. Vocab._ [56] Boas, Fr., _Sixth Report on the Indians of the Northwest_, p. 105. [57] Beauregard, O., _Bulletin de la Soc. d'Anthr. de Paris_, 1886, p. 526. [58] Ray, S.H., _Journ. Anthr. Inst._, 1891, p. 8. [59] _Op. cit._, p. 12. [60] Müller, _Sprachwissenschaft_, IV. i. p. 136. [61] Brinton, _The Maya Chronicles_, p. 50. [62] Trumbull, _On Numerals in Am. Ind. Lang._, p. 35. [63] Boas, Fr. This information was received directly from Dr. Boas. It has never before been published. [64] Bancroft, H.H., _Native Races_, Vol. II. p. 753. See also p. 199, _infra_. [65] Mann, A., "Notes on the Numeral Syst. of the Yoruba Nation," _Journ. Anth. Inst._, 1886, p. 59, _et seq._ [66] Müller, _Sprachwissenschaft_, IV. i. p. 202. [67] Trumbull, J.H., _On Numerals in Am. Ind. Langs._, p. 11. [68] Cushing, F.H., "Manual Concepts," _Am. Anthr._, 1892, p. 289. [69] Grimm, _Geschichte der deutschen Sprache_, Vol. I. p. 239. [70] Murdoch, J., _American Anthropologist_, 1890, p. 39. [71] Kleinschmidt, S., _Grammatik der Grönlandischen Sprache_, p. 37. [72] Brinton, _The Arawak Lang. of Guiana_, p. 4. [73] Petitot, E., _Dictionnaire de la langue Dènè-Dindjie_, p. lv. [74] Gilij, F.S., _Saggio di Storia Am._, Vol. II. p. 333. [75] Müller, _Sprachwissenschaft_, II. i. p. 389. [76] _Op. cit._, p. 395. [77] Müller, _Sprachwissenschaft_, II. i. p. 438. [78] Peacock, "Arithmetic," in _Encyc. Metropolitana_, 1, p. 480. [79] Brinton, D.G., "The Betoya Dialects," _Proc. Am. Philos. Soc._, 1892, p. 273. [80] Ridley, W., "Report on Australian Languages and Traditions." _Journ. Anth. Inst._, 1873, p. 262. [81] Gatschet, "Gram. Klamath Lang." _U.S. Geog. and Geol. Survey_, Vol. II. part 1, pp. 524 and 536. [82] Letter of Nov. 17, 1893. [83] Müller, _Sprachwissenschaft_, II. i. p. 439. [84] Hale, "Indians of No. West. Am.," _Tr. Am. Eth. Soc._, Vol. II. p. 82. [85] Brinton, D.G., _Studies in So. Am. Native Languages_, p. 25. [86] _Tr. Am. Philological Association_, 1874, p. 41. [87] Tylor, _Primitive Culture_, Vol. I. p. 251. [88] Müller, _Sprachwissenschaft_, IV. i. p. 27. [89] See _infra_, Chapter VII. [90] Ellis, A.B., _Ewe Speaking Peoples_, etc., p. 253. [91] Tylor, _Primitive Culture_, Vol. I. p. 256. [92] Stanley, _In Darkest Africa_, Vol. II. p. 493. [93] Chamberlain, A.F., _Proc. Brit. Ass. Adv. of Sci._, 1892, p. 599. [94] Boas, Fr., "Sixth Report on Northwestern Tribes of Canada," _Proc. Brit. Ass. Adv. Sci._, 1890, p. 657. [95] Hale, H., "Indians of Northwestern Am.," _Tr. Am. Eth. Soc._, Vol. II. p. 88. [96] _Op. cit._, p. 95. [97] Müller, _Sprachwissenschaft_, II. ii. p. 147. [98] Schoolcraft, _Archives of Aboriginal Knowledge_, Vol. IV. p. 429. [99] Du Chaillu, P.B., _Tr. Eth. Soc._, London, Vol. I. p. 315. [100] Latham, R.G., _Essays, chiefly Philological and Ethnographical_, p. 247. The above are so unlike anything else in the world, that they are not to be accepted without careful verification. [101] Pott, _Zählmethode_, p. 45. [102] Gatschet, A.S., _The Karankawa Indians, the Coast People of Texas_. The meanings of 6, 7, 8, and 9 are conjectural with me. [103] Stanley, H.M., _In Darkest Africa_, Vol. II. p. 492. [104] Müller, _Sprachwissenschaft_, II. i. p. 317. [105] Toy, C.H., _Trans. Am. Phil. Assn._, 1878, p. 29. [106] Burton, R.F., _Mem. Anthrop. Soc. of London_. 1, p. 314. In the illustration which follows, Burton gives 6820, instead of 4820; which is obviously a misprint. [107] Dobrizhoffer, _History of the Abipones_, Vol. II. p. 169. [108] Sayce, A.H., _Comparative Philology_, p. 254. [109] _Tr. Eth. Society of London _, Vol. III. p. 291. [110] Ray, S.H., _Journ. Anthr. Inst._, 1889, p. 501. [111] Stanley, _In Darkest Africa_, Vol. II. p. 492. [112] _Op. cit._, _loc. cit._ [113] Tylor, _Primitive Culture_, Vol. I. p. 249. [114] Müller, _Sprachwissenschaft_, IV. i. p. 36. [115] Martius, _Glos. Ling. Brasil._, p. 271. [116] Tylor, _Primitive Culture_, Vol. I. p. 248. [117] Roth, H. Ling, _Aborigines of Tasmania_, p. 146. [118] Lull, E.P., _Tr. Am. Phil, Soc._, 1873, p. 108. [119] Ray, S.H. "Sketch of Api Gram.," _Journ. Anthr. Inst._, 1888, p. 300. [120] Kleinschmidt, S., _Grammatik der Grönlandischen Spr._, p. 39. [121] Müller, _Sprachwissenschaft_, I. ii. p. 184. [122] _Op. cit._, I. ii. p. 18, and II. i. p. 222. [123] Squier, G.E., _Nicaragua_, Vol. II. p. 326. [124] Schoolcraft, H.R., _Archives of Aboriginal Knowledge_, Vol. II. p. 208. [125] Tylor, _Primitive Culture_, Vol. I. p. 264. [126] Goedel, "Ethnol. des Soussous," _Bull. de la Soc. d'Anthr. de Paris_, 1892, p. 185. [127] Ellis, W., _History of Madagascar_, Vol. I. p. 507. [128] Beauregard, O., _Bull. de la Soc. d'Anthr. de Paris_, 1886, p. 236. [129] Schoolcraft, H.R., _Archives of Aboriginal Knowledge_, Vol. II. p. 207. [130] Tylor, _Primitive Culture_, Vol. I. p. 249. [131] _Op. cit._ Vol. I. p. 250. [132] Peacock, _Encyc. Metropolitana_, 1, p. 478. [133] _Op. cit._, _loc. cit._ [134] Schoolcraft, H.R., _Archives of Aboriginal Knowledge_, Vol. II. p. 213. [135] _Op. cit._, p. 216. [136] _Op. cit._, p. 206. [137] Mariner, _Gram. Tonga Lang._, last part of book. [Not paged.] [138] Morice, A.G., "The Déné Langs," _Trans. Can. Inst._, March 1890, p. 186. [139] Boas, Fr., "Fifth Report on the Northwestern Tribes of Canada," _Proc. Brit. Ass. Adv. of Science_, 1889, p. 881. [140] _Do. Sixth Rep._, 1890, pp. 684, 686, 687. [141] _Op. cit._, p. 658. [142] Bancroft, H.H., _Native Races_, Vol. II. p. 499. [143] _Tr. Ethnological Soc. of London_, Vol. IV. p. 92. [144] Any Hebrew lexicon. [145] Schröder, P., _Die Phönizische Sprache, _p. 184 _et seq._ [146] Müller, _Sprachwissenschaft_, II. ii. p. 147. [147] _On Numerals in Am. Indian Languages._ [148] Ellis, A.B., _Ewe Speaking Peoples_, etc., p. 253. The meanings here given are partly conjectural. [149] Pott, _Zählmethode_, p. 29. [150] Schoolcraft, _op. cit._, Vol. IV. p. 429. [151] Trumbull, _op. cit._ [152] Chamberlain, A.F., _Lang, of the Mississaga Indians_, Vocab. [153] Crawfurd, _Hist. Ind. Archipelago_, 1, p. 258. [154] Hale, H., _Eth. and Philol._, Vol. VII.; Wilkes, _Expl. Expedition_, Phil. 1846, p. 172. [155] Crawfurd, _op. cit._, 1, p. 258. [156] _Op. cit._, _loc. cit._ [157] Bancroft, H.H., _Native Races_, Vol. II. p. 498. [158] Vignoli, T., _Myth and Science_, p. 203. [159] Codrington, R.H., _The Melanesian Languages_, p. 249. [160] _Op. cit._, _loc. cit._ [161] Codrington, R.H., _The Melanesian Languages_, p. 249. [162] Wickersham, J., "Japanese Art on Puget Sound," _Am. Antiq._, 1894, p. 79. [163] Codrington, R.H., _op. cit._, p. 250. [164] Tylor, _Primitive Culture_, Vol. I. p. 252. [165] Compare a similar table by Chase, _Proc. Amer. Philos. Soc._, 1865, p. 23. [166] _Leibnitzii Opera_, III. p. 346. [167] Pruner-Bey, _Bulletin de la Soc. d'Anthr. de Paris_, 1860, p. 486. [168] Curr, E.M., _The Australian Race_, Vol. I. p. 32. [169] Haddon, A.C., "Western Tribes of the Torres Straits," _Journ. Anthr. Inst._, 1889, p. 303. [170] Taplin, Rev. G., "Notes on a Table of Australian Languages," _Journ. Anthr. Inst.,_ 1872, p. 88. The first nine scales are taken from this source. [171] Latham, R.G., _Comparative Philology_, p. 352. [172] It will be observed that this list differs slightly from that given in Chapter II. [173] Curr, E.M., _The Australian Race_, Vol. III. p. 684. [174] Bonwick, _Tasmania_, p. 143. [175] Lang, J.D., _Queensland_, p. 435. [176] Bonwick, _Tasmania_, p. 143. [177] Müller, _Sprachwissenschaft_, II. i. p. 58. [178] _Op. cit._, II. i. p. 70. [179] _Op. cit._, II. i. p. 23. [180] Barlow, H., "Aboriginal Dialects of Queensland," _Journ. Anth. Inst._, 1873, p. 171. [181] Curr, E.M., _The Australian Race_, Vol. II. p. 26. [182] _Op. cit._, Vol. II. p. 208. [183] _Op. cit._, Vol. II. p. 278. [184] _Op. cit._, Vol. II. p. 288. [185] _Op. cit._, Vol. I. p. 258. [186] _Op. cit._, Vol. I. p. 316. [187] _Op. cit._, Vol. III. p. 32. The next ten lists are taken from the same volume, pp. 282, 288, 340, 376, 432, 506, 530, 558, 560, 588, respectively. [188] Brinton, _The American Race_, p. 351. [189] Martius, _Glossaria Ling. Brazil._, p. 307. [190] _Op. cit._, p. 148. [191] Müller, _Sprachwissenschaft_, II. i. p. 438. [192] Peacock, "Arithmetic," _Encyc. Metropolitana_, 1, p. 480. [193] Brinton, _Studies in So. Am. Native Langs._, p. 67. [194] _Op. cit._, _loc. cit._ [195] Brinton, _Studies in So. Am. Native Langs._, p. 67. The meanings of the numerals are from Peacock, _Encyc. Metropolitana_, 1, p. 480. [196] Mason, _Journ. As. Soc. of Bengal_, Vol. XXVI. p. 146. [197] Curr, E.M., _The Australian Race_, Vol. III. p. 108. [198] Bancroft, H.H., _Native Races_, Vol. I. p. 274. [199] Clarke, Hyde, _Journ. Anthr. Inst._, 1872, p. clvii. In the article from which this is quoted, no evidence is given to substantiate the assertion made. It is to be received with great caution. [200] Hale, H., _Wilkes Exploring Expedition_, Vol. VII. p. 172. [201] _Op. cit._, p. 248. [202] Hale, _Ethnography and Philology, _p. 247. [203] _Loc. cit._ [204] Ellis, _Polynesian Researches_, Vol. IV. p. 341. [205] Gill, W.W., _Myths and Songs of the South Pacific_, p. 325. [206] Peacock, "Arithmetic," _Encyc. Metropolitana_, 1, p. 479. [207] Peacock, _Encyc. Metropolitana_, 1, p. 480. [208] _Sprachverschiedenheit_, p. 30. [209] Crawfurd, _History of the Indian Archipelago_, Vol. I. p. 256. [210] Pott, _Zählmethode_, p. 39. [211] _Op. cit._, p. 41. [212] Müller, _Sprachwissenschaft_, II. i. p. 317. See also Chap. III., _supra_. [213] Long, S.H., _Expedition_, Vol. II. p. lxxviii. [214] Martius, _Glossaria Ling. Brasil._, p. 246. [215] Hale, _Ethnography and Philology_, p. 434. [216] Müller, _Sprachwissenschaft_, II. ii. p. 82. [217] The information upon which the above statements are based was obtained from Mr. W.L. Williams, of Gisborne, N.Z. [218] _Primitive Culture_, Vol. I. p. 268. [219] Ralph, Julian, _Harper's Monthly_, Vol. 86, p. 184. [220] Lappenberg, J.M., _History of Eng. under the Anglo-Saxon Kings_, Vol. I. p. 82. [221] The compilation of this table was suggested by a comparison found in the _Bulletin Soc. Anth. de Paris_, 1886, p. 90. [222] Hale, _Ethnography and Philology_, p. 126. [223] Müller, _Sprachwissenschaft_, II. ii. p. 183. [224] Bachofen, J.J., _Antiquarische Briefe_, Vol. I. pp. 101-115, and Vol. II. pp. 1-90. [225] An extended table of this kind may be found in the last part of Nystrom's _Mechanics_. [226] Schubert, H., quoting Robert Flegel, in Neumayer's _Anleitung zu Wissenschaftlichen Beobachtung auf Reisen_, Vol. II. p. 290. [227] These numerals, and those in all the sets immediately following, except those for which the authority is given, are to be found in Chapter III. [228] Codrington, _The Melanesian Languages_, p. 222. [229] Müller, _Sprachwissenschaft_, II. ii. p. 83. [230] _Op. cit._, I. ii. p. 55. The next two are the same, p. 83 and p. 210. The meaning given for the Bari _puök_ is wholly conjectural. [231] Gallatin, "Semi-civilized Nations," _Tr. Am. Eth. Soc._, Vol. I. p. 114. [232] Müller, _Sprachwissenschaft_, II. ii. p. 80. Erromango, the same. [233] Boas, Fr., _Proc. Brit. Ass'n. Adv. Science_, 1889, p. 857. [234] Hankel, H., _Geschichte der Mathematik_, p. 20. [235] Murdoch, J., "Eskimos of Point Barrow," _Am. Anthr._, 1890, p. 40. [236] Martius, _Glos. Ling. Brasil._, p. 360. [237] Du Graty, A.M., _La République du Paraguay_, p. 217. [238] Codrington, _The Melanesian Languages_, p. 221. [239] Müller, _Sprachwissenschaft_, II. i. p. 363. [240] Spurrell, W., _Welsh Grammar_, p. 59. [241] Olmos, André de, _Grammaire Nahuatl ou Mexicaine_, p. 191. [242] Moncelon, _Bull. Soc. d'Anthr. de Paris_, 1885, p. 354. This is a purely digital scale, but unfortunately M. Moncelon does not give the meanings of any of the numerals except the last. [243] Ellis, _Peruvia Scythia_, p. 37. Part of these numerals are from Martius, _Glos. Brasil._, p. 210. [244] Codrington, _The Melanesian Languages_, p. 236. [245] Schweinfurth, G., _Linguistische Ergebnisse einer Reise nach Centralafrika_, p. 25. [246] Park, M., _Travels in the Interior Districts of Africa_, p. 8. [247] Pott, _Zählmethode_, p. 37. [248] _Op. cit._, p. 39. [249] Müller, _Sprachwissenschaft_, IV. i. p. 101. The Kru scale, kindred with the Basa, is from the same page. [250] Park, in Pinkerton's _Voyages and Travels_, Vol. XVI. p. 902. [251] Park, _Travels_, Vol. I. p. 16. [252] Schweinfurth, G., _Linguistische Ergebnisse einer Reise nach Centralafrika_, p. 78. [253] Park, _Travels_, Vol. I. p. 58. [254] Goedel, "Ethnol. des Soussous," _Bull. Soc. Anth. Paris_, 1892, p. 185. [255] Müller, _Sprachwissenschaft_, I. ii. p. 114. The Temne scale is from the same page. These two languages are closely related. [256] _Op. cit._, I. ii. p. 155. [257] _Op. cit._, I. ii. p. 55. [258] Long, C.C., _Central Africa_, p. 330. [259] Müller, _Sprachwissenschaft_, IV. i. p. 105. [260] Pott, _Zählmethode_, p. 41. [261] Müller, _op. cit._, I. ii. p. 140. [262] Müller, _Sprachwissenschaft_, IV. i. p. 81. [263] Pott, _Zählmethode_, p. 41. [264] Müller, _op. cit._, I. ii., p. 210. [265] Pott, _Zählmethode_, p. 42. [266] Schweinfurth, _Linguistische Ergebnisse_, p. 59. [267] Müller, _Sprachwissenschaft_, I. ii. p. 261. The "ten" is not given. [268] Stanley, _Through the Dark Continent_, Vol. II. p. 490. Ki-Nyassa, the same page. [269] Müller, _op. cit._, I. ii. p. 261. [270] Du Chaillu, _Adventures in Equatorial Africa_, p. 534. [271] Müller, _Sprachwissenschaft_, III. i. p. 65. [272] Du Chaillu, _Adventures in Equatorial Africa_, p. 533. [273] Müller, _op. cit._, III. ii. p. 77. [274] Balbi, A., _L'Atlas Eth._, Vol. I. p. 226. In Balbi's text 7 and 8 are ansposed. _Taru_ for 5 is probably a misprint for _tana_. [275] Du Chaillu, _op. cit._, p. 533. The next scale is _op. cit._, p. 534. [276] Beauregard, O., _Bull. Soc. Anth. de Paris_, 1886, p. 526. [277] Pott, _Zählmethode_, p. 46. [278] _Op. cit._, p. 48. [279] Turner, _Nineteen Years in Polynesia_, p. 536. [280] Erskine, J.E., _Islands of the Western Pacific_, p. 341. [281] _Op. cit._, p. 400. [282] Codrington, _Melanesian Languages_, pp. 235, 236. [283] Peacock, _Encyc. Met._, Vol. 1. p. 385. Peacock does not specify the dialect. [284] Erskine, _Islands of the Western Pacific_, p. 360. [285] Turner, G., _Samoa a Hundred Years Ago_, p. 373. The next three scales are from the same page of this work. [286] Codrington, _Melanesian Languages_, p. 235. The next four scales are from the same page. Perhaps the meanings of the words for 6 to 9 are more properly "more 1," "more 2," etc. Codrington merely indicates their significations in a general way. [287] Hale, _Ethnography and Philology_, p. 429. The meanings of 6 to 9 in this and the preceding are my conjectures. [288] Müller, _Sprachwissenschaft_, IV. i. p. 124. [289] Aymonier, E., _Dictionnaire Francaise-Cambodgien_. [290] Müller, _Op. cit._, II. i. p. 139. [291] Müller, _Sprachwissenschaft_, II. i. p. 123. [292] Wells, E.R., Jr., and John W. Kelly, Bureau of Ed., Circ. of Inf., No. 2, 1890. [293] Pott, _Zählmethode_, p. 57. [294] Müller, _Op. cit._, II. i. p. 161. [295] Petitot, _Vocabulaire Française Esquimau_, p. lv. [296] Müller, _Sprachwissenschaft_, II. i. p. 253. [297] Müller, _Op. cit._, II. I. p. 179, and Kleinschmidt, _Grönlandisches Grammatik_. [298] Adam, L., _Congres Int. des Am._, 1877, p. 244 (see p. 162 _infra_). [299] Gallatin, "Synopsis of Indian Tribes," _Trans. Am. Antq. Soc._, 1836, p. 358. The next fourteen lists are, with the exception of the Micmac, from the same collection. The meanings are largely from Trumbull, _op. cit._ [300] Schoolcraft, _Archives of Aboriginal Knowledge_, Vol. II. p. 211. [301] Schoolcraft, _Archives of Aboriginal Knowledge_, Vol. V. p. 587. [302] In the Dakota dialects 10 is expressed, as here, by a word signifying that the fingers, which have been bent down in counting, are now straightened out. [303] Boas, _Fifth Report B.A.A.S._, 1889. Reprint, p. 61. [304] Boas, _Sixth Report B.A.A.S._, 1890. Reprint, p. 117. Dr. Boas does not give the meanings assigned to 7 and 8, but merely states that they are derived from 2 and 3. [305] _Op. cit._, p. 117. The derivations for 6 and 7 are obvious, but the meanings are conjectural. [306] Boas, _Sixth Report B.A.A.S._, 1889. Reprint, pp. 158, 160. The meanings assigned to the Tsimshian 8 and to Bilqula 6 to 8 are conjectural. [307] Hale, _Ethnography and Philology_, p. 619. [308] _Op. cit._, _loc. cit._ [309] Hale, _Ethnography and Philology_, p. 619. [310] Müller, _Sprachwissenschaft_, II. i. p. 436. [311] _Op. cit._, IV. i. p. 167. [312] _Op. cit._, II. i. p. 282. [313] _Op. cit._, II. i. p. 287. The meanings given for the words for 7, 8, 9 are conjectures of my own. [314] Müller, _Sprachwissenschaft_, II. i. p. 297. [315] Pott, _Zählmethode_, p. 90. [316] Müller, _op. cit._, II. i. p. 379. [317] Gallatin, "Semi-Civilized Nations of Mexico and Central America," _Tr. Am. Ethn. Soc._, Vol. I. p. 114. [318] Adam, Lucien, _Congres Internationale des Americanistes_, 1877, Vol. II. p. 244. [319] Müller, _Sprachwissenschaft_, II. i. p. 395. I can only guess at the meanings of 6 to 9. They are obviously circumlocutions for 5-1, 5-2, etc. [320] _Op. cit._, p. 438. Müller has transposed these two scales. See Brinton's _Am. Race_, p. 358. [321] Marcoy, P., _Tour du Monde_, 1866, 2ème sem. p. 148. [322] _Op. cit._, p. 132. The meanings are my own conjectures. [323] An elaborate argument in support of this theory is to be found in Hervas' celebrated work, _Arithmetica di quasi tutte le nazioni conosciute_. [324] See especially the lists of Hale, Gallatin, Trumbull, and Boas, to which references have been given above. [325] Thiel, B.A., "Vocab. der Indianier in Costa Rica," _Archiv für Anth._, xvi. p. 620. [326] These three examples are from A.R. Wallace's _Narrative of Travels on the Amazon and Rio Negro_, vocab. Similar illustrations may be found in Martius' _Glos. Brasil_. [327] Martius, _Glos. Brasil._, p. 176. [328] Adam, L., _Congres International des Americanistes_, 1877, Vol. II. p. 244. Given also _supra_, p. 53. [329] O'Donovan, _Irish Grammar_, p. 123. [330] Armstrong, R.A., _Gaelic Dict._, p. xxi. [331] Spurrell, _Welsh Dictionary_. [332] Kelly, _Triglot Dict._, pub. by the Manx Society. [333] Guillome, J., _Grammaire Française-Bretonne_, p. 27. [334] Gröber, G., _Grundriss der Romanischen Philologie_, Bd. I. p. 309. [335] Pott, _Zählmethode_, p. 88. [336] Van Eys, _Basque Grammar_, p. 27. [337] Pott, _Zählmethode_, p. 101. [338] _Op. cit._, p. 78. [339] Müller, _Sprachwissenschaft_, I. ii. p. 124. [340] _Op. cit._, p. 155. [341] _Op. cit._, p. 140. [342] _Op. cit._, _loc. cit._ [343] Schweinfurth, _Reise nach Centralafrika_, p. 25. [344] Müller, _Sprachwissenschaft_, IV. i. p. 83. [345] _Op. cit._, IV. i. p. 81. [346] _Op. cit._, I. ii. p. 166. [347] Long, C.C., _Central Africa_, p. 330. [348] Peacock, _Encyc. Met._, Vol. I. p. 388. [349] Müller, _Sprachwissenschaft_, III. ii. p. 64. The next seven scales are from _op. cit._, pp. 80, 137, 155, 182, 213. [350] Pott, _Zählmethode_, p. 83. [351] _Op. cit._, p. 83,--Akari, p. 84; Circassia, p. 85. [352] Müller, _Sprachwissenschaft_, II. i. p. 140. [353] Pott, _Zählmethode_, p. 87. [354] Müller, _Sprachwissenschaft_, II. ii. p. 346. [355] _Op. cit._, III. i. p. 130. [356] Man, E.H., "Brief Account of the Nicobar Islands," _Journ. Anthr. Inst._, 1885, p. 435. [357] Wells, E.R., Jr., and Kelly, J.W., "Eng. Esk. and Esk. Eng. Vocab.," Bureau of Education Circular of Information, No. 2, 1890, p. 65. [358] Petitot, E., _Vocabulaire Française Esquimau_, p. lv. [359] Boas, Fr., _Proc. Brit. Ass. Adv. Sci._, 1889, p. 857. [360] Boas, _Sixth Report on the Northwestern Tribes of Canada_, p. 117. [361] Boas, Fr., _Fifth Report on the Northwestern Tribes of Canada_, p. 85. [362] Gallatin, _Semi-Civilized Nations_, p. 114. References for the next two are the same. [363] Bancroft, H.H., _Native Races of the Pacific States_, Vol. II. p. 763. The meanings are from Brinton's _Maya Chronicles_, p. 38 _et seq._ [364] Brinton, _Maya Chronicles_, p. 44. [365] Siméon Rémi, _Dictionnaire de la langue nahuatl_, p. xxxii. [366] An error occurs on p. xxxiv of the work from which these numerals are taken, which makes the number in question appear as 279,999,999 instead of 1,279,999,999. [367] Gallatin, "Semi-Civilized Nations of Mexico and Central America," _Tr. Am. Ethn. Soc._ Vol. I. p. 114. [368] Pott, _Zählmethode_, p. 89. The Totonacos were the first race Cortez encountered after landing in Mexico. [369] _Op. cit._, p. 90. The Coras are of the Mexican state of Sonora. [370] Gallatin, _Semi-Civilized Nations_, p. 114. [371] Humboldt, _Recherches_, Vol. II. p. 112. [372] Squier, _Nicaragua_, Vol. II. p. 326. [373] Gallatin, _Semi-Civilized Nations_, p. 57. 
212	----	None 
254	----	MENO II A CONTINUATION OF SOCRATES' DIALOGUE WITH MENO IN WHICH THE BOY PROVES ROOT 2 IS IRRATIONAL By Socrates A Millennium Fulcrum Edition [Copyright 1995] Socrates: Well, here we are at the appointed time, Meno. Meno: Yes, and it looks like a fine day for it, too. Socrates: And I see our serving boy is also here. Boy: Yes, I am, and ready to do your bidding. Socrates: Wonderful. Now, Meno, I want you to be on your guard, as you were the other day, to insure that I teach nothing to the boy, but rather pull out of his mind the premises which are already there. Meno: I shall do my best, Socrates. Socrates: I can ask more of no man, Meno, and I am certain that you will do well, and I hope I will give you no call to halt me in my saying if I should say too much, in which you would feel I was actually teaching the boy the answer to this riddle. Meno: No, Socrates, I don't think I will have to call you on anything you might say today, for the most wondrously learned men of the group of Pythagoras have spent many hours, weeks, and even months and years toiling in their manner to arrive at the mystic solutions to the puzzles formed by the simple squares with which we worked the other day. Therefore, I am certain to regain my virtue, which I lost the other day, when I was so steadfastly proven by you to be in error in my statement that the root of a square with an area of two square feet was beyond this boy, who is a fine boy, whom we must make to understand that he should do his best here, and not feel that he has done any wrongness by causing me to lose my virtue to you the other day. Socrates: Meno, my friend, it is my opinion, and I hope it will soon be yours, that your virtue was increased the other day, rather than decreased. Meno: I fail to see how, when I was humiliated by seeing this young boy, of modest education, arrive in minutes at the highest mystic levels of the magic of the Pythagoreans. Most of all when I wagered as many dinners as you could eat at my house that this could not be the case. Socrates: First, friend Meno, let me assure you that I will promise never to eat you out of house and home, not that I could if I tried, for my tastes are simple and your wallet is large. Nevertheless, Meno, my friend, I would hasten to add that I will promise, if you like, not to ever come to your table uninvited. As a second reason you and your virtue should feel better after the events of the other day, because you were in error before, but are less in error now. And the path to virtue, at least one aspect of the path to virtue, is in finding and correcting error. Meno: Socrates, you know you are always welcome at my table, except when I am suffering from my ulcer, which you aggravate greatly, or at times when I am entertaining the highest nobles of the land, and you would appear out of place in your clothing. (Socrates was known for his simple attire, and for wearing his garments over and over till they wore out. However, the only surviving example of his writing is a laundry list, so we know he kept his clothes clean and somewhat presentable, though simple) Socrates: I would hope you would have me over because I was a good influence on your development, than for any other reason. I notice you did not respond to my claim to have increased your virtue, through the exorcism of your error. Meno: Well Socrates, you know that it is not always the easiest thing to give up one's ways, even though one has found them to be in error. Therefore, please forgive me if I am not sounding as grateful as you would like for your lessons. Socrates: The easier one finds it to give up the ways of error, the easier it is to replace the error with that which we hope is not in error. Is this not the way to virtue? Meno: Yes, Socrates, and you know the path is hard, and that we often stumble and fall. Socrates: Yes, but is it not true that we stumble and fall over the obstacles which we make for ourselves to trip over? Meno: Certainly that is most true, Socrates, in some cases. Socrates: Well, then, let us proceed, for I see the hour is upon us when I do my best thinking, and that hour shall be passed soon, and hopefully with it shall pass a bit of your ignorance. Meno: Well said, Socrates. I am with you. Socrates: And shall have we a wager on the events of today? Meno: Certainly, Socrates. Socrates: And what shall you wager against this boy proving that the length of the root of a square with an area of two square feet, cannot be made by the ratio of two whole numbers? Meno: You may have anything it is in my power to give, unless it cause harm to myself or to another to give it. Socrates: Well said, my friend Meno, and I shall leave it at that. And what shall I offer you as a return wager? Meno: Well, the easiest thing which comes to mind is to wager all those dinners you won from me the other day. Socrates: Very well, so be it. Meno: Now Socrates, since you are my friend, I must give you this friendly warning: you know that the Pythagoreans jealously guard their secrets with secret meetings, protected by secret handshakes, secret signs, passwords, and all that, do you not? Socrates: I have heard as much, friend Meno. Meno: Then be sure that they will seek revenge upon you for demystifying the ideas and concepts which they worked so long and hard and secretly to create and protect; for they are a jealous lot in the extreme, hiding in mountain caves, which are hardly fit to be called monasteries by even the most hardened monk. Socrates: I take your meaning, friend Meno, and thank you for your consideration, but I think that if I lose, that they will not bother me, and if I win, it will appear so simple to everyone, that if would be sheerest folly for anyone to make even the smallest gesture to protect its fallen mystic secrecy. Besides, I have a citizen's responsibility to Athens and to all Athenians to do my best to protect them and enlighten them. Meno: Very well, Socrates. Please do not ever say that I did not try to warn you, especially after they have nailed you to a cross in a public place, where anyone and everyone could hear you say that the fault of this lay in my name. Socrates: Do not worry, friend Meno, for if I were not to show this simple feat of logic to you, I should just walk down the street and find someone else, though not someone whose company and conversation I should enjoy as much as yours. Meno: Thank you, friend Socrates. Socrates: Now, boy, do you remember me, and the squares with which we worked and played the other day? Boy: Yes, sir, Socrates. Socrates: Please, Meno, instruct the boy to merely call me by my name, as does everyone else. Calling me "sir" merely puts me off my mental stride, and, besides, it will create a greater distance between me and the boy. Meno: You heard what Socrates, said, boy. Can you do it? Boy: Yes, sir. (Turning to Socrates) You know I like you very much, and that I call you "sir" not only out of relation of our positions in society, but also because of my true respect and admiration, especially after the events of the other day. Socrates: Yes, boy. And I will try to live up to your expectations. (Turning to Meno) Would you allow some reward for the boy, as well as that which is for myself, if he should prove to your satisfaction that the square root of two is irrational? Meno: Certainly, Socrates. Socrates: (taking the boy aside) What would you like the most in the whole world, boy? Boy: You mean anything? Socrates: Well, I can't guarantee to get it for you, but at least I can ask it, and it shouldn't hurt to ask; and besides, as you should know, it is very hard to expect someone to give you what you want, if you never let them know you want it. Boy: Well, Socrates... you know what I would want. Socrates: Do I? Boy: Better than I knew the square root of two the other day. Socrates: You want to be a free man, then, and a citizen. Boy: (looking down) Yes. Socrates: Don't look down, then, for that is an admirable desire for one to have, and speaks highly of him who has it. I will speak to Meno, while you hold your tongue. Boy: Yes, Socrates. (bows to kiss his hand, Socrates turns) Socrates: Friend Meno, how hard do you think it will be for this boy to prove the irrationality of the square root of two? Meno: You know that I think it is impossible, Socrates. Socrates: Well, how long did it take the Pythagoreans? Meno: I should think it took them years. Socrates: And how many of them were there? Meno: Quite a few, though not all worked equally, and some hardly at all, for they were most interested in triangles of the right and virtuous variety, and not in squares and their roots. Socrates: Can you give me an estimate? Meno: No, I can't say that I can. I am sorry, Socrates. Socrates: No problem, would you accept five thinkers as an estimate. Meno: I think that should be fair. Socrates: And shall we assume they worked for two years, that is the smallest number which retains the plural, and our assumption was that they worked for years. Meno: Two years is indeed acceptable to me, Socrates. Socrates: Very well then, Meno, it would appear that the Pythagoreans spent 10 total years of thinking time to solve the riddles of the square root of two. Meno: I agree. Socrates: And would you like to hire the Pythagoreans to run your household, Meno? Meno: Surely I would, Socrates, if they were only for hire, but, as you well know, they are a secret lot, and hire to no one. Socrates: Well, if I could get you one, perhaps one of the best of them, in fact the leader of the group that solved the square root of two, would you not hire him, and at high wages? Meno: Certainly, Socrates. I'd be a fool not to. Socrates: And you would put him in charge of your house. Meno: And all my lands, too, Socrates. Socrates: Possibly. Then I would like to propose, that if this boy should solve the proof of the square root of two being irrational, in the next few hours of our discussion, that he be given wages equal to those due to your most highly placed servant for ten years of service, as he shall perform ten years service for you in the next few hours, should he succeed. Meno: That sounds quite fair, Socrates, I like your logic. Socrates: (the boy tugs his tunic, to complain that he wants his freedom, not a mere bucket of gold) Hush, boy, did you not promise to hold your tongue? Boy: Yes, Socrates, but.... Socrates: (turning to Meno) And, of course, with the monetary rewards for such a position, go all the rest of it. Meno: Of course, Socrates. I never thought to cheat you. Socrates: I know that, friend Meno, but I merely ask for the boy's sake, who is not used to hearing about high finance and the powers and rank which accompany such things. Meno: Of course, Socrates. Shall I tell the boy what he shall receive? Socrates: You are very kind to do so, my friend Meno. Meno: (turns to the boy) You are aware that a servant may not own the amount of gold I would have to give you, should you win the day? Boy: Yes, sir. Meno: Therefore, I would have to give to you the freedom to own the money, before I could give you the money, would I not? Boy: Yes, sir. Meno: And in giving you freedom, I would be remiss if I did not give you a job and a coming out party of equal position with your wealth, would I not? Boy: I can't really say, sir, though I suppose so. Meno: You suppose correctly. I will feed you for a week of partying, and dress you in the finest garments, while you are introduced to the finest ladies and gentlemen of Athens, from whom you are free to select for your interests as friends, business partners, social acquaintances, connections, and perhaps even a wife, should you find someone you like for that. Do you now understand that there is nothing I would leave out that you would have to ask for, or that if you did have to ask, I would give it immediately, and ask your forgiveness for my error? Boy: It is hard to understand, but I take your word. Socrates: Now don't let this all go to your head, boy. This is something you could have figured out for yourself, if you had applied your mind to it as you did to squares the other day. Can you do as well, today? Boy: I should think and hope so, friend Socrates, for I see you are indeed my friend, and I should hope I am more capable today, for having learned some the other day. Socrates: We shall see, boy. Let us on to the test. Now you remember the squares we dealt with the other day. Boy: Yes, Socrates. Socrates: And the one particular square on the diagonal we made, whose area was two, do you remember that one? Boy: Yes, Socrates. Socrates: And you remember that the length of the side of a square, when multiplied by itself, yields the area of the square. Boy: Everyone at school knows that, Socrates. Socrates: Well, maybe. However, it is about that side, which when multiplied time itself yields an area of two, that I would like to speak further today. How is that with you? Boy: That is fine, Socrates. I remember that line, and I sort of liked it the best, if you know what I mean. Socrates: Good, then we should have a great time. Do you know how long that line is, boy? Boy: Well, I know that you both thought it wise when I said it was of a length which when made a square of, yielded a square with an area of two, so I suppose I should answer that way. Socrates: And a good answer it is, too. We are going to make it an even better answer as we proceed. Boy: Good. Socrates: Do you remember when you tripped up and fell on your face the other day, when you thought that the square of area nine was actually a square of area eight? Boy: Oh yes, Socrates! And I am sorely ashamed, because I still do not know enough to make sure I never make such an error again, and therefore I know my virtue and rightness are lacking. Socrates: They are not lacking so much that they cannot be improved, are they boy? Boy: I should hope and pray not. Socrates: Well today, you are going to tell us some things about that number, which when multiplied by itself gives us two. Boy: I will tell you everything I know, or think I know, Socrates, and hope that I am correct or can be corrected. Socrates: To Meno, surely he is a fine boy, eh Meno? Meno: Yes, I am proud to own him, but I don't see how he can be smart enough to do the work today that would take a Pythagorean monk ten years of cloistered life to accomplish. Socrates: We shall see. Boy, you are doing fine. I think I could even make a scholar of you, though I fear you might turn to wine and women with your new found wealth, if you succeed, rather than continue to polish the wit which should get you that reward. Boy: I don't think I would want to spend that much time with women or with wine, Socrates. Socrates: You will find something, no doubt. So, back to the number which when square gives us two. What can we say about such a number? Is it odd or even? Well it would have to be a whole number to be one of those, would it not, and we saw the other day what happens to whole numbers when they are squared? They give us 1,4,9 and 16 as square areas, did they not? Boy: Yes, Socrates, though I remember thinking that there should have been a number which would give eight, Socrates? Socrates: I think we shall find one, if we keep searching. Now, this number, do you remember if it had to be larger or smaller than one? Boy: Larger, Socrates. For one squared gives only an area of one, and we need and area of two, which is larger. Socrates: Good. And what of two? Boy: Two gives a square of four, which is too large. Socrates: Fine. So the square root of two is smaller than the side two which is the root of four, and larger than the side one which yields one? Boy: Yes, Socrates. Socrates: (Turning to Meno) So now he is as far as most of us get in determining the magnitude of the square root of two? And getting farther is largely a matter of guesswork, is it not? Meno: Yes, Socrates, but I don't see how he will do it. Socrates: Neither does he. But I do. Watch! (turning to the boy) Now I am going to tell you something you don't know, so Meno will listen very closely to make sure he agrees that I can tell you. You know multiplication, boy? Boy: I thought I had demonstrated that, Socrates? Socrates: So you have, my boy, has he not Meno? Meno: Yes, Socrates, I recall he did the other day. Socrates: And you know the way to undo multiplication? Boy: It is called division, but I do not know it as well as multiplication, since we have not studied it as long. Socrates: Well, I will not ask you to do much division, but rather I will ask you only whether certain answers may be called odd or even, and the like. Does that suit you? Boy: It suits me well, Socrates. Socrates: Then you know what odd and even are, boy? Boy: Yes, shall I tell you? Socrates: Please do. I would love to hear what they teach. Boy: (the boy recites) A number can only be odd or even if it is a whole number, that is has no parts but only wholes of what it measures. Even numbers are special in that they have only whole twos in them, with no ones left over, while odd numbers always have a one left over when all the twos are taken out. Socrates: An interesting, and somewhat effective definition. Do you agree, Meno. Meno: Yes, Socrates. Please continue. Socrates: Now boy, what do you get when you divide these odd and even numbers by other odd and even numbers. Boy: Sometimes you get whole numbers, especially when you divide an even number by an even number, but odd numbers sometimes give whole numbers, both odd and even, and sometimes they give numbers which are not whole numbers, but have parts. Socrates: Very good, and have your teachers ever called these numbers ratios? Boy: Sometimes, Socrates, but usually only with simple numbers which make one-half, one-third, two-thirds and the like. Socrates: Yes, that is usually what people mean by ratios. The learned people call numbers made from the ratios, rational. Does the name rational number suit you to call a number which can be expressed as the ratio of two whole numbers, whether they be odd or even whole numbers? Boy: You want me to call the numbers made from ratios of whole numbers something called rational? A ratio makes a rational number? Socrates: Yes boy, can you do that? Boy: Certainly, Socrates. Socrates: Do you agree with the way I told him this, Meno? Does it violate our agreement? Meno: You added -nal to the word ratio, just as we add -nal to the French word "jour" to create the word journal which means something that contains words of the "jour" or of today. So we now have a word which means a number made from a ratio. This is more than acceptable to me, Socrates. A sort of lesson in linguistics, perhaps, but certainly not in mathematics. No, I do not see that you have told him how to solve anything about the square root of two, but thank you for asking. I give you your journalistic license to do so. Socrates: Good. Now boy, I need your attention. Please get up and stretch, if it will help you stay and think for awhile. Boy: (stretches only a little) I am fine, Socrates. Socrates: Now think carefully, boy, what kind of ratios can we make from even numbers and odd numbers? Boy: We could make even numbers divided by odd numbers, and odd numbers divided by even numbers. Socrates: Yes, we could. Could we make any other kind? Boy: Well... we could make even numbers divided by even numbers, or odd numbers divided by odd. Socrates: Very good. Any other kind? Boy: I'm not sure, I can't think of any, but I might have to think a while to be sure. Socrates: (to Meno) Are you still satisfied. Meno: Yes, Socrates. He knows even and odd numbers, and ratios; as do all the school children his age. Socrates: Very well, boy. You have named four kinds of ratios: Even over odd, odd over even, even over even, odd over odd, and all the ratios make numbers we call rational numbers. Boy: That's what it looks like, Socrates. Socrates: Meno, have you anything to contribute here? Meno: No, Socrates, I am fine. Socrates: Very well. Now, boy, we are off in search of more about the square root of two. We have divided the rational numbers into four groups, odd/even, even/odd, even/even, odd/odd? Boy: Yes. Socrates: And if we find another group we can include them. Now, we want to find which one of these groups, if any, contains the number you found the other day, the one which squared is two. Would that be fun to try? Boy: Yes, Socrates, and also educational. Socrates: I think we can narrow these four groups down to three, and thus make the search easier. Would you like that? Boy: Certainly, Socrates. Socrates: Let's take even over even ratios. What are they? Boy: We know that both parts of the ratio have two in them. Socrates: Excellent. See, Meno, how well he has learned his lessons in school. His teacher must be proud, for I have taught him nothing of this, have I? Meno: No, I have not seen you teach it to him, therefore he must have been exposed to it elsewhere. Socrates: (back to the boy) And what have you learned about ratios of even numbers, boy? Boy: That both parts can be divided by two, to get the twos out, over and over, until one part becomes odd. Socrates: Very good. Do all school children know that, Meno? Meno: All the ones who stay awake in class. (he stretches) Socrates: So, boy, we can change the parts of the ratios, without changing the real meaning of the ratio itself? Boy: Yes, Socrates. I will demonstrate, as we do in class. Suppose I use 16 and 8, as we did the other day. If I make a ratio of 16 divided by 8, I can divide both the 16 and the 8 by two and get 8 divided by 4. We can see that 8 divided by 4 is the same as 16 divided by 8, each one is twice the other, as it should be. We can then divide by two again and get 4 over 2, and again to get 2 over 1. We can't do it again, so we say that this fraction has been reduced as far as it will go, and everything that is true of the other ways of expressing it is true of this. Socrates: Your demonstration is effective. Can you divide by other numbers than two? Boy: Yes, Socrates. We can divide by any number which goes as wholes into the parts which make up the ratio. We could have started by dividing by 8 before, but I divided by three times, each time by two, to show you the process, though now I feel ashamed because I realize you are both masters of this, and that I spoke to you in too simple a manner. Socrates: Better to speak too simply, than in a manner in which part or all of your audience gets lost, like the Sophists. Boy: I agree, but please stop me if I get too simple. Socrates: I am sure we can survive a simple explanation. (nudges Meno, who has been gazing elsewhere) But back to your simple proof: we know that a ratio of two even numbers can be divided until reduced until one or both its parts are odd? Boy: Yes, Socrates. Then it is a proper ratio. Socrates: So we can eliminate one of our four groups, the one where even was divided by even, and now we have odd/odd, odd/even and even/odd? Boy: Yes, Socrates. Socrates: Let's try odd over even next, shall we? Boy: Fine. Socrates: What happens when you multiply an even number by an even number, what kind of number do you get, even or odd? Boy: Even, of course. An even multiple of any whole number gives another even number. Socrates: Wonderful, you have answered two questions, but we need only one at the moment. We shall save the other. So, with odd over even, if we multiply any of these times themselves, we well get odd times odd over even times even, and therefore odd over even, since odd times odd is odd and even of even is even. Boy: Yes. A ratio of odd over even, when multiplied times itself, yields odd over even. Socrates: And can our square root of two be in that group? Boy: I don't know, Socrates. Have I failed? Socrates: Oh, you know, you just don't know that you know. Try this: after we multiply our number times itself, which the learned call "squaring" the number which is the root, we need to get a ratio in which the first or top number is twice as large as the second or bottom number. Is this much correct? Boy: A ratio which when "squared" as you called it, yields an area of two, must then yield one part which is two times the other part. That is the definition of a ratio of two to one. Socrates: So you agree that this is correct? Boy: Certainly. Socrates: Now if a number is to be twice as great as another, it must be two times that number? Boy: Certainly. Socrates: And if a number is two times any whole number, it must then be an even number, must it not? Boy: Yes, Socrates. Socrates: So, in our ratio we want to square to get two, the top number cannot be odd, can it? Boy: No, Socrates. Therefore, the group of odd over even rational numbers cannot have the square root of two in it! Nor can the group ratios of odd numbers over odd numbers. Socrates: Wonderful. We have just eliminated three of the four groups of rational numbers, first we eliminated the group of even over even numbers, then the ones with odd numbers divided by other numbers. However, these were the easier part, and we are now most of the way up the mountain, so we must rest and prepare to try even harder to conquer the rest, where the altitude is highest, and the terrain is rockiest. So let us sit and rest a minute, and look over what we have done, if you will. Boy: Certainly, Socrates, though I am much invigorated by the solution of two parts of the puzzle with one thought. It was truly wonderful to see such simple effectiveness. Are all great thoughts as simple as these, once you see them clearly? Socrates: What do you say, Meno? Do thoughts get simpler as they get greater? Meno: Well, it would appear that they do, for as the master of a great house, I can just order something be done, and it is; but if I were a master in a lesser house, I would have to watch over it much more closely to insure it got done. The bigger the decisions I have to make, the more help and advice I get in the making of them, so I would have to agree. Socrates: Glad to see that you are still agreeable, Meno, though I think there are some slight differences in the way each of us view the simplicity of great thought. Shall we go on? Meno: Yes, quite. Boy: Yes, Socrates. I am ready for the last group, the ratios of even numbers divided by the odd, though, I cannot yet see how we will figure these out, yet, somehow I have confidence that the walls of these numbers shall tumble before us, as did the three groups before them. Socrates: Let us review the three earlier groups, to prepare us for the fourth, and to make sure that we have not already broken the rules and therefore forfeited our wager. The four groups were even over even ratios, which we decided could be reduced in various manners to the other groups by dividing until one number of the ratio was no longer even; then we eliminated the two other groups which had odd numbers divided by either odd or even numbers, because the first or top number had to be twice the second or bottom number, and therefore could not be odd; this left the last group we are now to greet, even divided by odd. Boy: Wonderfully put, Socrates. It is amazing how neatly you put an hour of thinking into a minute. Perhaps we can, indeed, put ten years of thinking into this one day. Please continue in this manner, if you know how it can be done. Socrates: Would you have me continue, Meno? You know what shall have to happen if we solve this next group and do not find the square root of two in it. Meno: Socrates, you are my friend, and my teacher, and a good companion. I will not shirk my duty to you or to this fine boy, who appears to be growing beyond my head, even as we speak. However, I still do not see that his head has reached the clouds wherein lie the minds of the Pythagoreans. Socrates: Very well, on then, to even over odd. If we multiply these numbers times themselves, what do we get, boy? Boy: We will get a ratio of even over odd, Socrates. Socrates: And could an even number be double an odd number? Boy: Yes, Socrates. Socrates: So, indeed, this could be where we find a number such that when multiplied times itself yields an area of two? Boy: Yes, Socrates. It could very well be in this group. Socrates: So, the first, or top number, is the result of an even number times itself? Boy: Yes. Socrates: And the second, or bottom number, is the result of an odd number times itself? Boy: Yes. Socrates: And an even number is two times one whole number? Boy: Of course. Socrates: So if we use this even number twice in multiplication, as we have on top, we have two twos times two whole numbers? Boy: Yes, Socrates. Socrates: (nudges Meno) and therefore the top number is four times some whole number times that whole number again? Boy: Yes, Socrates. Socrates: And this number on top has to be twice the number on the bottom, if the even over odd number we began with is to give us two when multiplied by itself, or squared, as we call it? Boy: Yes, Socrates. Socrates: And if the top number is four times some whole number, then a number half as large would have to be two times that same whole number? Boy: Of course, Socrates. Socrates: So the number on the bottom is two times that whole number, whatever it is? Boy: Yes, Socrates. Socrates: (standing) And if it is two times a whole number, then it must be an even number, must it not? Boy: Yes. Socrates: Then is cannot be a member of the group which has an odd number on the bottom, can it? Boy: No, Socrates. Socrates: So can it be a member of the ratios created by an even number divided by an odd number and then used as a root to create a square? Boy: No, Socrates. And that must mean it can't be a member of the last group, doesn't it? Socrates: Yes, my boy, although I don't see how we can continue calling you boy, since you have now won your freedom, and are far richer than I will ever be. Boy: Are you sure we have proved this properly? Let me go over it again, so I can see it in my head. Socrates: Yes, my boy, er, ah, sir. Boy: We want to see if this square root of two we discovered the other day is a member of the rational numbers? Socrates: Yes. Boy: So we define the rational numbers as numbers made from the division into ratios of whole numbers, whether those whole numbers are even or odd. Socrates: Yes. Boy: We get four groups, even over even, which we don't use, odd over even, odd over odd, and even over odd. Socrates: Continue. Boy: We know the first number in the squared ratio cannot be odd because it must be twice the value of the second number, and therefore is must be an even number, two times a whole number. Therefore it cannot be a member of either of the next groups, because they both have whole numbers over odd numbers. Socrates: Wonderful! Boy: So we are left with one group, the evens over odds. Socrates: Yes. Boy: When we square an even over odd ratio, the first number becomes even times even, which is two times two times some other whole number, which means it is four times the whole number, and this number must be double the second number, which is odd, as it was made of odd times odd. But the top number cannot be double some bottom odd number because the top number is four times some whole number, and the bottom number is odd--but a number which is four times another whole number, cannot be odd when cut in half, so an even number times an even number can never be double what you would get from any odd number times another odd number... therefore none of these rational numbers, when multiplied times themselves, could possibly yield a ratio in which the top number was twice the bottom number. Amazing. We have proved that the square root of two is not a rational number. Fantastic! (He continues to wander up and down the stage, reciting various portions of the proof to himself, looking up, then down, then all around. He comes to Meno) Boy: Do you see? It's so simple, so clear. This is really wonderful! This is fantastic! Socrates: (lays an arm on Meno's arm) Tell him how happy you are for his new found thoughts, Meno, for you can easily tell he is not thinking at all of his newly won freedom and wealth. Meno: I quite agree with you, son, the clarity of your reasoning is truly astounding. I will leave you here with Socrates, as I go to prepare my household. I trust you will both be happy for the rest of the day without my assistance. [The party, the presentation of 10 years salary to the newly freed young man, is another story, as is the original story of the drawing in the sand the square with an area of two.] 
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22599	----	from the Cornell University Library: Historical Mathematics Monographs collection.) Transcriber's Note: The following codes are used for characters that are not present in the character set used for this version of the book. [=a] a with macron (etc.) [.g] g with dot above (etc.) ['s] s with acute accent [d.] d with dot below (etc.) [d=] d with line below [H)] H with breve below THE HINDU-ARABIC NUMERALS BY DAVID EUGENE SMITH AND LOUIS CHARLES KARPINSKI BOSTON AND LONDON GINN AND COMPANY, PUBLISHERS 1911 COPYRIGHT, 1911, BY DAVID EUGENE SMITH AND LOUIS CHARLES KARPINSKI ALL RIGHTS RESERVED 811.7 THE ATHENÆUM PRESS GINN AND COMPANY · PROPRIETORS BOSTON · U.S.A. * * * * * {iii} PREFACE So familiar are we with the numerals that bear the misleading name of Arabic, and so extensive is their use in Europe and the Americas, that it is difficult for us to realize that their general acceptance in the transactions of commerce is a matter of only the last four centuries, and that they are unknown to a very large part of the human race to-day. It seems strange that such a labor-saving device should have struggled for nearly a thousand years after its system of place value was perfected before it replaced such crude notations as the one that the Roman conqueror made substantially universal in Europe. Such, however, is the case, and there is probably no one who has not at least some slight passing interest in the story of this struggle. To the mathematician and the student of civilization the interest is generally a deep one; to the teacher of the elements of knowledge the interest may be less marked, but nevertheless it is real; and even the business man who makes daily use of the curious symbols by which we express the numbers of commerce, cannot fail to have some appreciation for the story of the rise and progress of these tools of his trade. This story has often been told in part, but it is a long time since any effort has been made to bring together the fragmentary narrations and to set forth the general problem of the origin and development of these {iv} numerals. In this little work we have attempted to state the history of these forms in small compass, to place before the student materials for the investigation of the problems involved, and to express as clearly as possible the results of the labors of scholars who have studied the subject in different parts of the world. We have had no theory to exploit, for the history of mathematics has seen too much of this tendency already, but as far as possible we have weighed the testimony and have set forth what seem to be the reasonable conclusions from the evidence at hand. To facilitate the work of students an index has been prepared which we hope may be serviceable. In this the names of authors appear only when some use has been made of their opinions or when their works are first mentioned in full in a footnote. If this work shall show more clearly the value of our number system, and shall make the study of mathematics seem more real to the teacher and student, and shall offer material for interesting some pupil more fully in his work with numbers, the authors will feel that the considerable labor involved in its preparation has not been in vain. We desire to acknowledge our especial indebtedness to Professor Alexander Ziwet for reading all the proof, as well as for the digest of a Russian work, to Professor Clarence L. Meader for Sanskrit transliterations, and to Mr. Steven T. Byington for Arabic transliterations and the scheme of pronunciation of Oriental names, and also our indebtedness to other scholars in Oriental learning for information. DAVID EUGENE SMITH LOUIS CHARLES KARPINSKI * * * * * {v} CONTENTS CHAPTER PRONUNCIATION OF ORIENTAL NAMES vi I. EARLY IDEAS OF THEIR ORIGIN 1 II. EARLY HINDU FORMS WITH NO PLACE VALUE 12 III. LATER HINDU FORMS, WITH A PLACE VALUE 38 IV. THE SYMBOL ZERO 51 V. THE QUESTION OF THE INTRODUCTION OF THE NUMERALS INTO EUROPE BY BOETHIUS 63 VI. THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS 91 VII. THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE 99 VIII. THE SPREAD OF THE NUMERALS IN EUROPE 128 INDEX 153 * * * * * {vi} PRONUNCIATION OF ORIENTAL NAMES (S) = in Sanskrit names and words; (A) = in Arabic names and words. B, D, F, G, H, J, L, M, N, P, SH (A), T, TH (A), V, W, X, Z, as in English. A, (S) like _u_ in _but_: thus _pandit_, pronounced _pundit_. (A) like _a_ in _ask_ or in _man_. [=A], as in _father_. C, (S) like _ch_ in _church_ (Italian _c_ in _cento_). [D.], [N.], [S.], [T.], (S) _d_, _n_, _sh_, _t_, made with the tip of the tongue turned up and back into the dome of the palate. [D.], [S.], [T.], [Z.], (A) _d_, _s_, _t_, _z_, made with the tongue spread so that the sounds are produced largely against the side teeth. Europeans commonly pronounce [D.], [N.], [S.], [T.], [Z.], both (S) and (A), as simple _d_, _n_, _sh_ (S) or _s_ (A), _t_, _z_. [D=] (A), like _th_ in _this_. E, (S) as in _they_. (A) as in _bed_. [.G], (A) a voiced consonant formed below the vocal cords; its sound is compared by some to a _g_, by others to a guttural _r_; in Arabic words adopted into English it is represented by _gh_ (e.g. _ghoul_), less often _r_ (e.g. _razzia_). H preceded by _b_, _c_, _t_, _[t.]_, etc. does not form a single sound with these letters, but is a more or less distinct _h_ sound following them; cf. the sounds in _abhor, boathook_, etc., or, more accurately for (S), the "bhoys" etc. of Irish brogue. H (A) retains its consonant sound at the end of a word. [H.], (A) an unvoiced consonant formed below the vocal cords; its sound is sometimes compared to German hard _ch_, and may be represented by an _h_ as strong as possible. In Arabic words adopted into English it is represented by _h_, e.g. in _sahib_, _hakeem_. [H.] (S) is final consonant _h_, like final _h_ (A). I, as in _pin_. [=I], as in _pique_. K, as in _kick_. KH, (A) the hard _ch_ of Scotch _loch_, German _ach_, especially of German as pronounced by the Swiss. [.M], [.N], (S) like French final _m_ or _n_, nasalizing the preceding vowel. [N.], see [D.]. Ñ, like _ng_ in _singing_. O, (S) as in _so_. (A) as in _obey_. Q, (A) like _k_ (or _c_) in _cook_; further back in the mouth than in _kick_. R, (S) English _r_, smooth and untrilled. (A) stronger. [R.], (S) r used as vowel, as in _apron_ when pronounced _aprn_ and not _apern_; modern Hindus say _ri_, hence our _amrita_, _Krishna_, for _a-m[r.]ta, K[r.][s.][n.]a_. S, as in _same_. [S.], see [D.]. ['S], (S) English _sh_ (German _sch_). [T.], see [D.]. U, as in _put_. [=U], as in _rule_. Y, as in _you_. [Z.], see [D.]. `, (A) a sound kindred to the spiritus lenis (that is, to our ears, the mere distinct separation of a vowel from the preceding sound, as at the beginning of a word in German) and to _[h.]_. The ` is a very distinct sound in Arabic, but is more nearly represented by the spiritus lenis than by any sound that we can produce without much special training. That is, it should be treated as silent, but the sounds that precede and follow it should not run together. In Arabic words adopted into English it is treated as silent, e.g. in _Arab_, _amber_, _Caaba_ (_`Arab_, _`anbar_, _ka`abah_). (A) A final long vowel is shortened before _al_ (_'l_) or _ibn_ (whose _i_ is then silent). Accent: (S) as if Latin; in determining the place of the accent _[.m]_ and _[.n]_ count as consonants, but _h_ after another consonant does not. (A), on the last syllable that contains a long vowel or a vowel followed by two consonants, except that a final long vowel is not ordinarily accented; if there is no long vowel nor two consecutive consonants, the accent falls on the first syllable. The words _al_ and _ibn_ are never accented. * * * * * {1} THE HINDU-ARABIC NUMERALS CHAPTER I EARLY IDEAS OF THEIR ORIGIN It has long been recognized that the common numerals used in daily life are of comparatively recent origin. The number of systems of notation employed before the Christian era was about the same as the number of written languages, and in some cases a single language had several systems. The Egyptians, for example, had three systems of writing, with a numerical notation for each; the Greeks had two well-defined sets of numerals, and the Roman symbols for number changed more or less from century to century. Even to-day the number of methods of expressing numerical concepts is much greater than one would believe before making a study of the subject, for the idea that our common numerals are universal is far from being correct. It will be well, then, to think of the numerals that we still commonly call Arabic, as only one of many systems in use just before the Christian era. As it then existed the system was no better than many others, it was of late origin, it contained no zero, it was cumbersome and little used, {2} and it had no particular promise. Not until centuries later did the system have any standing in the world of business and science; and had the place value which now characterizes it, and which requires a zero, been worked out in Greece, we might have been using Greek numerals to-day instead of the ones with which we are familiar. Of the first number forms that the world used this is not the place to speak. Many of them are interesting, but none had much scientific value. In Europe the invention of notation was generally assigned to the eastern shores of the Mediterranean until the critical period of about a century ago,--sometimes to the Hebrews, sometimes to the Egyptians, but more often to the early trading Phoenicians.[1] The idea that our common numerals are Arabic in origin is not an old one. The mediæval and Renaissance writers generally recognized them as Indian, and many of them expressly stated that they were of Hindu origin.[2] {3} Others argued that they were probably invented by the Chaldeans or the Jews because they increased in value from right to left, an argument that would apply quite as well to the Roman and Greek systems, or to any other. It was, indeed, to the general idea of notation that many of these writers referred, as is evident from the words of England's earliest arithmetical textbook-maker, Robert Recorde (c. 1542): "In that thinge all men do agree, that the Chaldays, whiche fyrste inuented thys arte, did set these figures as thei set all their letters. for they wryte backwarde as you tearme it, and so doo they reade. And that may appeare in all Hebrewe, Chaldaye and Arabike bookes ... where as the Greekes, Latines, and all nations of Europe, do wryte and reade from the lefte hand towarde the ryghte."[3] Others, and {4} among them such influential writers as Tartaglia[4] in Italy and Köbel[5] in Germany, asserted the Arabic origin of the numerals, while still others left the matter undecided[6] or simply dismissed them as "barbaric."[7] Of course the Arabs themselves never laid claim to the invention, always recognizing their indebtedness to the Hindus both for the numeral forms and for the distinguishing feature of place value. Foremost among these writers was the great master of the golden age of Bagdad, one of the first of the Arab writers to collect the mathematical classics of both the East and the West, preserving them and finally passing them on to awakening Europe. This man was Mo[h.]ammed the Son of Moses, from Khow[=a]rezm, or, more after the manner of the Arab, Mo[h.]ammed ibn M[=u]s[=a] al-Khow[=a]razm[=i],[8] a man of great {5} learning and one to whom the world is much indebted for its present knowledge of algebra[9] and of arithmetic. Of him there will often be occasion to speak; and in the arithmetic which he wrote, and of which Adelhard of Bath[10] (c. 1130) may have made the translation or paraphrase,[11] he stated distinctly that the numerals were due to the Hindus.[12] This is as plainly asserted by later Arab {6} writers, even to the present day.[13] Indeed the phrase _`ilm hind[=i]_, "Indian science," is used by them for arithmetic, as also the adjective _hind[=i]_ alone.[14] Probably the most striking testimony from Arabic sources is that given by the Arabic traveler and scholar Mohammed ibn A[h.]med, Ab[=u] 'l-R[=i][h.][=a]n al-B[=i]r[=u]n[=i] (973-1048), who spent many years in Hindustan. He wrote a large work on India,[15] one on ancient chronology,[16] the "Book of the Ciphers," unfortunately lost, which treated doubtless of the Hindu art of calculating, and was the author of numerous other works. Al-B[=i]r[=u]n[=i] was a man of unusual attainments, being versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as in astronomy, chronology, and mathematics. In his work on India he gives detailed information concerning the language and {7} customs of the people of that country, and states explicitly[17] that the Hindus of his time did not use the letters of their alphabet for numerical notation, as the Arabs did. He also states that the numeral signs called _a[.n]ka_[18] had different shapes in various parts of India, as was the case with the letters. In his _Chronology of Ancient Nations_ he gives the sum of a geometric progression and shows how, in order to avoid any possibility of error, the number may be expressed in three different systems: with Indian symbols, in sexagesimal notation, and by an alphabet system which will be touched upon later. He also speaks[19] of "179, 876, 755, expressed in Indian ciphers," thus again attributing these forms to Hindu sources. Preceding Al-B[=i]r[=u]n[=i] there was another Arabic writer of the tenth century, Mo[t.]ahhar ibn [T.][=a]hir,[20] author of the _Book of the Creation and of History_, who gave as a curiosity, in Indian (N[=a]gar[=i]) symbols, a large number asserted by the people of India to represent the duration of the world. Huart feels positive that in Mo[t.]ahhar's time the present Arabic symbols had not yet come into use, and that the Indian symbols, although known to scholars, were not current. Unless this were the case, neither the author nor his readers would have found anything extraordinary in the appearance of the number which he cites. Mention should also be made of a widely-traveled student, Al-Mas`[=u]d[=i] (885?-956), whose journeys carried him from Bagdad to Persia, India, Ceylon, and even {8} across the China sea, and at other times to Madagascar, Syria, and Palestine.[21] He seems to have neglected no accessible sources of information, examining also the history of the Persians, the Hindus, and the Romans. Touching the period of the Caliphs his work entitled _Meadows of Gold_ furnishes a most entertaining fund of information. He states[22] that the wise men of India, assembled by the king, composed the _Sindhind_. Further on[23] he states, upon the authority of the historian Mo[h.]ammed ibn `Al[=i] `Abd[=i], that by order of Al-Man[s.][=u]r many works of science and astrology were translated into Arabic, notably the _Sindhind_ (_Siddh[=a]nta_). Concerning the meaning and spelling of this name there is considerable diversity of opinion. Colebrooke[24] first pointed out the connection between _Siddh[=a]nta_ and _Sindhind_. He ascribes to the word the meaning "the revolving ages."[25] Similar designations are collected by Sédillot,[26] who inclined to the Greek origin of the sciences commonly attributed to the Hindus.[27] Casiri,[28] citing the _T[=a]r[=i]kh al-[h.]okam[=a]_ or _Chronicles of the Learned_,[29] refers to the work {9} as the _Sindum-Indum_ with the meaning "perpetuum æternumque." The reference[30] in this ancient Arabic work to Al-Khow[=a]razm[=i] is worthy of note. This _Sindhind_ is the book, says Mas`[=u]d[=i],[31] which gives all that the Hindus know of the spheres, the stars, arithmetic,[32] and the other branches of science. He mentions also Al-Khow[=a]razm[=i] and [H.]abash[33] as translators of the tables of the _Sindhind_. Al-B[=i]r[=u]n[=i][34] refers to two other translations from a work furnished by a Hindu who came to Bagdad as a member of the political mission which Sindh sent to the caliph Al-Man[s.][=u]r, in the year of the Hejira 154 (A.D. 771). The oldest work, in any sense complete, on the history of Arabic literature and history is the _Kit[=a]b al-Fihrist_, written in the year 987 A.D., by Ibn Ab[=i] Ya`q[=u]b al-Nad[=i]m. It is of fundamental importance for the history of Arabic culture. Of the ten chief divisions of the work, the seventh demands attention in this discussion for the reason that its second subdivision treats of mathematicians and astronomers.[35] {10} The first of the Arabic writers mentioned is Al-Kind[=i] (800-870 A.D.), who wrote five books on arithmetic and four books on the use of the Indian method of reckoning. Sened ibn `Al[=i], the Jew, who was converted to Islam under the caliph Al-M[=a]m[=u]n, is also given as the author of a work on the Hindu method of reckoning. Nevertheless, there is a possibility[36] that some of the works ascribed to Sened ibn `Al[=i] are really works of Al-Khow[=a]razm[=i], whose name immediately precedes his. However, it is to be noted in this connection that Casiri[37] also mentions the same writer as the author of a most celebrated work on arithmetic. To Al-[S.][=u]f[=i], who died in 986 A.D., is also credited a large work on the same subject, and similar treatises by other writers are mentioned. We are therefore forced to the conclusion that the Arabs from the early ninth century on fully recognized the Hindu origin of the new numerals. Leonard of Pisa, of whom we shall speak at length in the chapter on the Introduction of the Numerals into Europe, wrote his _Liber Abbaci_[38] in 1202. In this work he refers frequently to the nine Indian figures,[39] thus showing again the general consensus of opinion in the Middle Ages that the numerals were of Hindu origin. Some interest also attaches to the oldest documents on arithmetic in our own language. One of the earliest {11} treatises on algorism is a commentary[40] on a set of verses called the _Carmen de Algorismo_, written by Alexander de Villa Dei (Alexandra de Ville-Dieu), a Minorite monk of about 1240 A.D. The text of the first few lines is as follows: "Hec algorism' ars p'sens dicit' in qua Talib; indor[um] fruim bis quinq; figuris.[41] "This boke is called the boke of algorim or augrym after lewder use. And this boke tretys of the Craft of Nombryng, the quych crafte is called also Algorym. Ther was a kyng of Inde the quich heyth Algor & he made this craft.... Algorisms, in the quych we use teen figurys of Inde." * * * * * {12} CHAPTER II EARLY HINDU FORMS WITH NO PLACE VALUE While it is generally conceded that the scientific development of astronomy among the Hindus towards the beginning of the Christian era rested upon Greek[42] or Chinese[43] sources, yet their ancient literature testifies to a high state of civilization, and to a considerable advance in sciences, in philosophy, and along literary lines, long before the golden age of Greece. From the earliest times even up to the present day the Hindu has been wont to put his thought into rhythmic form. The first of this poetry--it well deserves this name, being also worthy from a metaphysical point of view[44]--consists of the Vedas, hymns of praise and poems of worship, collected during the Vedic period which dates from approximately 2000 B.C. to 1400 B.C.[45] Following this work, or possibly contemporary with it, is the Brahmanic literature, which is partly ritualistic (the Br[=a]hma[n.]as), and partly philosophical (the Upanishads). Our especial interest is {13} in the S[=u]tras, versified abridgments of the ritual and of ceremonial rules, which contain considerable geometric material used in connection with altar construction, and also numerous examples of rational numbers the sum of whose squares is also a square, i.e. "Pythagorean numbers," although this was long before Pythagoras lived. Whitney[46] places the whole of the Veda literature, including the Vedas, the Br[=a]hma[n.]as, and the S[=u]tras, between 1500 B.C. and 800 B.C., thus agreeing with Bürk[47] who holds that the knowledge of the Pythagorean theorem revealed in the S[=u]tras goes back to the eighth century B.C. The importance of the S[=u]tras as showing an independent origin of Hindu geometry, contrary to the opinion long held by Cantor[48] of a Greek origin, has been repeatedly emphasized in recent literature,[49] especially since the appearance of the important work of Von Schroeder.[50] Further fundamental mathematical notions such as the conception of irrationals and the use of gnomons, as well as the philosophical doctrine of the transmigration of souls,--all of these having long been attributed to the Greeks,--are shown in these works to be native to India. Although this discussion does not bear directly upon the {14} origin of our numerals, yet it is highly pertinent as showing the aptitude of the Hindu for mathematical and mental work, a fact further attested by the independent development of the drama and of epic and lyric poetry. It should be stated definitely at the outset, however, that we are not at all sure that the most ancient forms of the numerals commonly known as Arabic had their origin in India. As will presently be seen, their forms may have been suggested by those used in Egypt, or in Eastern Persia, or in China, or on the plains of Mesopotamia. We are quite in the dark as to these early steps; but as to their development in India, the approximate period of the rise of their essential feature of place value, their introduction into the Arab civilization, and their spread to the West, we have more or less definite information. When, therefore, we consider the rise of the numerals in the land of the Sindhu,[51] it must be understood that it is only the large movement that is meant, and that there must further be considered the numerous possible sources outside of India itself and long anterior to the first prominent appearance of the number symbols. No one attempts to examine any detail in the history of ancient India without being struck with the great dearth of reliable material.[52] So little sympathy have the people with any save those of their own caste that a general literature is wholly lacking, and it is only in the observations of strangers that any all-round view of scientific progress is to be found. There is evidence that primary schools {15} existed in earliest times, and of the seventy-two recognized sciences writing and arithmetic were the most prized.[53] In the Vedic period, say from 2000 to 1400 B.C., there was the same attention to astronomy that was found in the earlier civilizations of Babylon, China, and Egypt, a fact attested by the Vedas themselves.[54] Such advance in science presupposes a fair knowledge of calculation, but of the manner of calculating we are quite ignorant and probably always shall be. One of the Buddhist sacred books, the _Lalitavistara_, relates that when the B[=o]dhisattva[55] was of age to marry, the father of Gopa, his intended bride, demanded an examination of the five hundred suitors, the subjects including arithmetic, writing, the lute, and archery. Having vanquished his rivals in all else, he is matched against Arjuna the great arithmetician and is asked to express numbers greater than 100 kotis.[56] In reply he gave a scheme of number names as high as 10^{53}, adding that he could proceed as far as 10^{421},[57] all of which suggests the system of Archimedes and the unsettled question of the indebtedness of the West to the East in the realm of ancient mathematics.[58] Sir Edwin Arnold, {16} in _The Light of Asia_, does not mention this part of the contest, but he speaks of Buddha's training at the hands of the learned Vi[s.]vamitra: "And Viswamitra said, 'It is enough, Let us to numbers. After me repeat Your numeration till we reach the lakh,[59] One, two, three, four, to ten, and then by tens To hundreds, thousands.' After him the child Named digits, decads, centuries, nor paused, The round lakh reached, but softly murmured on, Then comes the k[=o]ti, nahut, ninnahut, Khamba, viskhamba, abab, attata, To kumuds, gundhikas, and utpalas, By pundar[=i]kas into padumas, Which last is how you count the utmost grains Of Hastagiri ground to finest dust;[60] But beyond that a numeration is, The K[=a]tha, used to count the stars of night, The K[=o]ti-K[=a]tha, for the ocean drops; Ingga, the calculus of circulars; Sarvanikchepa, by the which you deal With all the sands of Gunga, till we come To Antah-Kalpas, where the unit is The sands of the ten crore Gungas. If one seeks More comprehensive scale, th' arithmic mounts By the Asankya, which is the tale Of all the drops that in ten thousand years Would fall on all the worlds by daily rain; Thence unto Maha Kalpas, by the which The gods compute their future and their past.'" {17} Thereupon Vi[s.]vamitra [=A]c[=a]rya[61] expresses his approval of the task, and asks to hear the "measure of the line" as far as y[=o]jana, the longest measure bearing name. This given, Buddha adds: ... "'And master! if it please, I shall recite how many sun-motes lie From end to end within a y[=o]jana.' Thereat, with instant skill, the little prince Pronounced the total of the atoms true. But Viswamitra heard it on his face Prostrate before the boy; 'For thou,' he cried, 'Art Teacher of thy teachers--thou, not I, Art G[=u]r[=u].'" It is needless to say that this is far from being history. And yet it puts in charming rhythm only what the ancient _Lalitavistara_ relates of the number-series of the Buddha's time. While it extends beyond all reason, nevertheless it reveals a condition that would have been impossible unless arithmetic had attained a considerable degree of advancement. To this pre-Christian period belong also the _Ved[=a][.n]gas_, or "limbs for supporting the Veda," part of that great branch of Hindu literature known as _Sm[r.]iti_ (recollection), that which was to be handed down by tradition. Of these the sixth is known as _Jyoti[s.]a_ (astronomy), a short treatise of only thirty-six verses, written not earlier than 300 B.C., and affording us some knowledge of the extent of number work in that period.[62] The Hindus {18} also speak of eighteen ancient Siddh[=a]ntas or astronomical works, which, though mostly lost, confirm this evidence.[63] As to authentic histories, however, there exist in India none relating to the period before the Mohammedan era (622 A.D.). About all that we know of the earlier civilization is what we glean from the two great epics, the Mah[=a]bh[=a]rata[64] and the R[=a]m[=a]yana, from coins, and from a few inscriptions.[65] It is with this unsatisfactory material, then, that we have to deal in searching for the early history of the Hindu-Arabic numerals, and the fact that many unsolved problems exist and will continue to exist is no longer strange when we consider the conditions. It is rather surprising that so much has been discovered within a century, than that we are so uncertain as to origins and dates and the early spread of the system. The probability being that writing was not introduced into India before the close of the fourth century B.C., and literature existing only in spoken form prior to that period,[66] the number work was doubtless that of all primitive peoples, palpable, merely a matter of placing sticks or cowries or pebbles on the ground, of marking a sand-covered board, or of cutting notches or tying cords as is still done in parts of Southern India to-day.[67] {19} The early Hindu numerals[68] may be classified into three great groups, (1) the Kharo[s.][t.]h[=i], (2) the Br[=a]hm[=i], and (3) the word and letter forms; and these will be considered in order. The Kharo[s.][t.]h[=i] numerals are found in inscriptions formerly known as Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandh[=a]ra, now eastern Afghanistan and northern Punjab. The alphabet of the language is found in inscriptions dating from the fourth century B.C. to the third century A.D., and from the fact that the words are written from right to left it is assumed to be of Semitic origin. No numerals, however, have been found in the earliest of these inscriptions, number-names probably having been written out in words as was the custom with many ancient peoples. Not until the time of the powerful King A['s]oka, in the third century B.C., do numerals appear in any inscriptions thus far discovered; and then only in the primitive form of marks, quite as they would be found in Egypt, Greece, Rome, or in {20} various other parts of the world. These A['s]oka[69] inscriptions, some thirty in all, are found in widely separated parts of India, often on columns, and are in the various vernaculars that were familiar to the people. Two are in the Kharo[s.][t.]h[=i] characters, and the rest in some form of Br[=a]hm[=i]. In the Kharo[s.][t.]h[=i] inscriptions only four numerals have been found, and these are merely vertical marks for one, two, four, and five, thus: | || ||| |||| In the so-called ['S]aka inscriptions, possibly of the first century B.C., more numerals are found, and in more highly developed form, the right-to-left system appearing, together with evidences of three different scales of counting,--four, ten, and twenty. The numerals of this period are as follows: [Illustration] There are several noteworthy points to be observed in studying this system. In the first place, it is probably not as early as that shown in the N[=a]n[=a] Gh[=a]t forms hereafter given, although the inscriptions themselves at N[=a]n[=a] Gh[=a]t are later than those of the A['s]oka period. The {21} four is to this system what the X was to the Roman, probably a canceling of three marks as a workman does to-day for five, or a laying of one stick across three others. The ten has never been satisfactorily explained. It is similar to the A of the Kharo[s.][t.]h[=i] alphabet, but we have no knowledge as to why it was chosen. The twenty is evidently a ligature of two tens, and this in turn suggested a kind of radix, so that ninety was probably written in a way reminding one of the quatre-vingt-dix of the French. The hundred is unexplained, although it resembles the letter _ta_ or _tra_ of the Br[=a]hm[=i] alphabet with 1 before (to the right of) it. The two hundred is only a variant of the symbol for hundred, with two vertical marks.[70] This system has many points of similarity with the Nabatean numerals[71] in use in the first centuries of the Christian era. The cross is here used for four, and the Kharo[s.][t.]h[=i] form is employed for twenty. In addition to this there is a trace of an analogous use of a scale of twenty. While the symbol for 100 is quite different, the method of forming the other hundreds is the same. The correspondence seems to be too marked to be wholly accidental. It is not in the Kharo[s.][t.]h[=i] numerals, therefore, that we can hope to find the origin of those used by us, and we turn to the second of the Indian types, the Br[=a]hm[=i] characters. The alphabet attributed to Brahm[=a] is the oldest of the several known in India, and was used from the earliest historic times. There are various theories of its origin, {22} none of which has as yet any wide acceptance,[72] although the problem offers hope of solution in due time. The numerals are not as old as the alphabet, or at least they have not as yet been found in inscriptions earlier than those in which the edicts of A['s]oka appear, some of these having been incised in Br[=a]hm[=i] as well as Kharo[s.][t.]h[=i]. As already stated, the older writers probably wrote the numbers in words, as seems to have been the case in the earliest Pali writings of Ceylon.[73] The following numerals are, as far as known, the only ones to appear in the A['s]oka edicts:[74] [Illustration] These fragments from the third century B.C., crude and unsatisfactory as they are, are the undoubted early forms from which our present system developed. They next appear in the second century B.C. in some inscriptions in the cave on the top of the N[=a]n[=a] Gh[=a]t hill, about seventy-five miles from Poona in central India. These inscriptions may be memorials of the early Andhra dynasty of southern India, but their chief interest lies in the numerals which they contain. The cave was made as a resting-place for travelers ascending the hill, which lies on the road from Kaly[=a]na to Junar. It seems to have been cut out by a descendant {23} of King ['S][=a]tav[=a]hana,[75] for inside the wall opposite the entrance are representations of the members of his family, much defaced, but with the names still legible. It would seem that the excavation was made by order of a king named Vedisiri, and "the inscription contains a list of gifts made on the occasion of the performance of several _yagnas_ or religious sacrifices," and numerals are to be seen in no less than thirty places.[76] There is considerable dispute as to what numerals are really found in these inscriptions, owing to the difficulty of deciphering them; but the following, which have been copied from a rubbing, are probably number forms:[77] [Illustration] The inscription itself, so important as containing the earliest considerable Hindu numeral system connected with our own, is of sufficient interest to warrant reproducing part of it in facsimile, as is done on page 24. {24} [Illustration] The next very noteworthy evidence of the numerals, and this quite complete as will be seen, is found in certain other cave inscriptions dating back to the first or second century A.D. In these, the Nasik[78] cave inscriptions, the forms are as follows: [Illustration] From this time on, until the decimal system finally adopted the first nine characters and replaced the rest of the Br[=a]hm[=i] notation by adding the zero, the progress of these forms is well marked. It is therefore well to present synoptically the best-known specimens that have come down to us, and this is done in the table on page 25.[79] {25} TABLE SHOWING THE PROGRESS OF NUMBER FORMS IN INDIA NUMERALS 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 1000 A['s]oka[80] [Illustration] ['S]aka[81] [Illustration] A['s]oka[82] [Illustration] N[=a]gar[=i][83] [Illustration] Nasik[84] [Illustration] K[s.]atrapa[85] [Illustration] Ku[s.]ana [86] [Illustration] Gupta[87] [Illustration] Valhab[=i][88] [Illustration] Nepal [89] [Illustration] Kali[.n]ga[90] [Illustration] V[=a]k[=a][t.]aka[91] [Illustration] [Most of these numerals are given by Bühler, loc. cit., Tafel IX.] {26} With respect to these numerals it should first be noted that no zero appears in the table, and as a matter of fact none existed in any of the cases cited. It was therefore impossible to have any place value, and the numbers like twenty, thirty, and other multiples of ten, one hundred, and so on, required separate symbols except where they were written out in words. The ancient Hindus had no less than twenty of these symbols,[92] a number that was afterward greatly increased. The following are examples of their method of indicating certain numbers between one hundred and one thousand: [93] [Numerals] for 174 [94] [Numerals] for 191 [95] [Numerals] for 269 [96] [Numerals] for 252 [97] [Numerals] for 400 [98] [Numerals] for 356 {27} To these may be added the following numerals below one hundred, similar to those in the table: [Numerals][99] for 90 [Numerals][100] for 70 We have thus far spoken of the Kharo[s.][t.]h[=i] and Br[=a]hm[=i] numerals, and it remains to mention the third type, the word and letter forms. These are, however, so closely connected with the perfecting of the system by the invention of the zero that they are more appropriately considered in the next chapter, particularly as they have little relation to the problem of the origin of the forms known as the Arabic. Having now examined types of the early forms it is appropriate to turn our attention to the question of their origin. As to the first three there is no question. The [1 vertical stroke] or [1 horizontal stroke] is simply one stroke, or one stick laid down by the computer. The [2 vertical strokes] or [2 horizontal strokes] represents two strokes or two sticks, and so for the [3 vertical strokes] and [3 horizontal strokes]. From some primitive [2 vertical strokes] came the two of Egypt, of Rome, of early Greece, and of various other civilizations. It appears in the three Egyptian numeral systems in the following forms: Hieroglyphic [2 vertical strokes] Hieratic [Hieratic 2] Demotic [Demotic 2] The last of these is merely a cursive form as in the Arabic [Arabic 2], which becomes our 2 if tipped through a right angle. From some primitive [2 horizontal strokes] came the Chinese {28} symbol, which is practically identical with the symbols found commonly in India from 150 B.C. to 700 A.D. In the cursive form it becomes [2 horizontal strokes joined], and this was frequently used for two in Germany until the 18th century. It finally went into the modern form 2, and the [3 horizontal strokes] in the same way became our 3. There is, however, considerable ground for interesting speculation with respect to these first three numerals. The earliest Hindu forms were perpendicular. In the N[=a]n[=a] Gh[=a]t inscriptions they are vertical. But long before either the A['s]oka or the N[=a]n[=a] Gh[=a]t inscriptions the Chinese were using the horizontal forms for the first three numerals, but a vertical arrangement for four.[101] Now where did China get these forms? Surely not from India, for she had them, as her monuments and literature[102] show, long before the Hindus knew them. The tradition is that China brought her civilization around the north of Tibet, from Mongolia, the primitive habitat being Mesopotamia, or possibly the oases of Turkestan. Now what numerals did Mesopotamia use? The Babylonian system, simple in its general principles but very complicated in many of its details, is now well known.[103] In particular, one, two, and three were represented by vertical arrow-heads. Why, then, did the Chinese write {29} theirs horizontally? The problem now takes a new interest when we find that these Babylonian forms were not the primitive ones of this region, but that the early Sumerian forms were horizontal.[104] What interpretation shall be given to these facts? Shall we say that it was mere accident that one people wrote "one" vertically and that another wrote it horizontally? This may be the case; but it may also be the case that the tribal migrations that ended in the Mongol invasion of China started from the Euphrates while yet the Sumerian civilization was prominent, or from some common source in Turkestan, and that they carried to the East the primitive numerals of their ancient home, the first three, these being all that the people as a whole knew or needed. It is equally possible that these three horizontal forms represent primitive stick-laying, the most natural position of a stick placed in front of a calculator being the horizontal one. When, however, the cuneiform writing developed more fully, the vertical form may have been proved the easier to make, so that by the time the migrations to the West began these were in use, and from them came the upright forms of Egypt, Greece, Rome, and other Mediterranean lands, and those of A['s]oka's time in India. After A['s]oka, and perhaps among the merchants of earlier centuries, the horizontal forms may have come down into India from China, thus giving those of the N[=a]n[=a] Gh[=a]t cave and of later inscriptions. This is in the realm of speculation, but it is not improbable that further epigraphical studies may confirm the hypothesis. {30} As to the numerals above three there have been very many conjectures. The figure one of the Demotic looks like the one of the Sanskrit, the two (reversed) like that of the Arabic, the four has some resemblance to that in the Nasik caves, the five (reversed) to that on the K[s.]atrapa coins, the nine to that of the Ku[s.]ana inscriptions, and other points of similarity have been imagined. Some have traced resemblance between the Hieratic five and seven and those of the Indian inscriptions. There have not, therefore, been wanting those who asserted an Egyptian origin for these numerals.[105] There has already been mentioned the fact that the Kharo[s.][t.]h[=i] numerals were formerly known as Bactrian, Indo-Bactrian, and Aryan. Cunningham[106] was the first to suggest that these numerals were derived from the alphabet of the Bactrian civilization of Eastern Persia, perhaps a thousand years before our era, and in this he was supported by the scholarly work of Sir E. Clive Bayley,[107] who in turn was followed by Canon Taylor.[108] The resemblance has not proved convincing, however, and Bayley's drawings {31} have been criticized as being affected by his theory. The following is part of the hypothesis:[109] _Numeral_ _Hindu_ _Bactrian_ _Sanskrit_ 4 [Symbol] [Symbol] = ch chatur, Lat. quattuor 5 [Symbol] [Symbol] = p pancha, Gk. [Greek:p/ente] 6 [Symbol] [Symbol] = s [s.]a[s.] 7 [Symbol] [Symbol] = [s.] sapta ( the s and [s.] are interchanged as occasionally in N. W. India) Bühler[110] rejects this hypothesis, stating that in four cases (four, six, seven, and ten) the facts are absolutely against it. While the relation to ancient Bactrian forms has been generally doubted, it is agreed that most of the numerals resemble Br[=a]hm[=i] letters, and we would naturally expect them to be initials.[111] But, knowing the ancient pronunciation of most of the number names,[112] we find this not to be the case. We next fall back upon the hypothesis {32} that they represent the order of letters[113] in the ancient alphabet. From what we know of this order, however, there seems also no basis for this assumption. We have, therefore, to confess that we are not certain that the numerals were alphabetic at all, and if they were alphabetic we have no evidence at present as to the basis of selection. The later forms may possibly have been alphabetical expressions of certain syllables called _ak[s.]aras_, which possessed in Sanskrit fixed numerical values,[114] but this is equally uncertain with the rest. Bayley also thought[115] that some of the forms were Phoenician, as notably the use of a circle for twenty, but the resemblance is in general too remote to be convincing. There is also some slight possibility that Chinese influence is to be seen in certain of the early forms of Hindu numerals.[116] {33} More absurd is the hypothesis of a Greek origin, supposedly supported by derivation of the current symbols from the first nine letters of the Greek alphabet.[117] This difficult feat is accomplished by twisting some of the letters, cutting off, adding on, and effecting other changes to make the letters fit the theory. This peculiar theory was first set up by Dasypodius[118] (Conrad Rauhfuss), and was later elaborated by Huet.[119] {34} A bizarre derivation based upon early Arabic (c. 1040 A.D.) sources is given by Kircher in his work[120] on number mysticism. He quotes from Abenragel,[121] giving the Arabic and a Latin translation[122] and stating that the ordinary Arabic forms are derived from sectors of a circle, [circle]. Out of all these conflicting theories, and from all the resemblances seen or imagined between the numerals of the West and those of the East, what conclusions are we prepared to draw as the evidence now stands? Probably none that is satisfactory. Indeed, upon the evidence at {35} hand we might properly feel that everything points to the numerals as being substantially indigenous to India. And why should this not be the case? If the king Srong-tsan-Gampo (639 A.D.), the founder of Lh[=a]sa,[123] could have set about to devise a new alphabet for Tibet, and if the Siamese, and the Singhalese, and the Burmese, and other peoples in the East, could have created alphabets of their own, why should not the numerals also have been fashioned by some temple school, or some king, or some merchant guild? By way of illustration, there are shown in the table on page 36 certain systems of the East, and while a few resemblances are evident, it is also evident that the creators of each system endeavored to find original forms that should not be found in other systems. This, then, would seem to be a fair interpretation of the evidence. A human mind cannot readily create simple forms that are absolutely new; what it fashions will naturally resemble what other minds have fashioned, or what it has known through hearsay or through sight. A circle is one of the world's common stock of figures, and that it should mean twenty in Phoenicia and in India is hardly more surprising than that it signified ten at one time in Babylon.[124] It is therefore quite probable that an extraneous origin cannot be found for the very sufficient reason that none exists. Of absolute nonsense about the origin of the symbols which we use much has been written. Conjectures, {36} however, without any historical evidence for support, have no place in a serious discussion of the gradual evolution of the present numeral forms.[125] TABLE OF CERTAIN EASTERN SYSTEMS Siam [Illustration: numerals] Burma[126] [Illustration: numerals] Malabar[127] [Illustration: numerals] Tibet[128] [Illustration: numerals] Ceylon[129] [Illustration: numerals] Malayalam[129] [Illustration: numerals] {37} We may summarize this chapter by saying that no one knows what suggested certain of the early numeral forms used in India. The origin of some is evident, but the origin of others will probably never be known. There is no reason why they should not have been invented by some priest or teacher or guild, by the order of some king, or as part of the mysticism of some temple. Whatever the origin, they were no better than scores of other ancient systems and no better than the present Chinese system when written without the zero, and there would never have been any chance of their triumphal progress westward had it not been for this relatively late symbol. There could hardly be demanded a stronger proof of the Hindu origin of the character for zero than this, and to it further reference will be made in Chapter IV. * * * * * {38} CHAPTER III LATER HINDU FORMS, WITH A PLACE VALUE Before speaking of the perfected Hindu numerals with the zero and the place value, it is necessary to consider the third system mentioned on page 19,--the word and letter forms. The use of words with place value began at least as early as the 6th century of the Christian era. In many of the manuals of astronomy and mathematics, and often in other works in mentioning dates, numbers are represented by the names of certain objects or ideas. For example, zero is represented by "the void" (_['s][=u]nya_), or "heaven-space" (_ambara [=a]k[=a]['s]a_); one by "stick" (_rupa_), "moon" (_indu ['s]a['s]in_), "earth" (_bh[=u]_), "beginning" (_[=a]di_), "Brahma," or, in general, by anything markedly unique; two by "the twins" (_yama_), "hands" (_kara_), "eyes" (_nayana_), etc.; four by "oceans," five by "senses" (_vi[s.]aya_) or "arrows" (the five arrows of K[=a]mad[=e]va); six by "seasons" or "flavors"; seven by "mountain" (_aga_), and so on.[130] These names, accommodating themselves to the verse in which scientific works were written, had the additional advantage of not admitting, as did the figures, easy alteration, since any change would tend to disturb the meter. {39} As an example of this system, the date "['S]aka Sa[m.]vat, 867" (A.D. 945 or 946), is given by "_giri-ra[s.]a-vasu_," meaning "the mountains" (seven), "the flavors" (six), and the gods "_Vasu_" of which there were eight. In reading the date these are read from right to left.[131] The period of invention of this system is uncertain. The first trace seems to be in the _['S]rautas[=u]tra_ of K[=a]ty[=a]yana and L[=a][t.]y[=a]yana.[132] It was certainly known to Var[=a]ha-Mihira (d. 587),[133] for he used it in the _B[r.]hat-Sa[m.]hit[=a]._[134] It has also been asserted[135] that [=A]ryabha[t.]a (c. 500 A.D.) was familiar with this system, but there is nothing to prove the statement.[136] The earliest epigraphical examples of the system are found in the Bayang (Cambodia) inscriptions of 604 and 624 A.D.[137] Mention should also be made, in this connection, of a curious system of alphabetic numerals that sprang up in southern India. In this we have the numerals represented by the letters as given in the following table: 1 2 3 4 5 6 7 8 9 0 k kh g gh [.n] c ch j jh ñ [t.] [t.]h [d.] [d.]h [n.] t th d th n p ph b bh m y r l v ['s] [s.] s h l {40} By this plan a numeral might be represented by any one of several letters, as shown in the preceding table, and thus it could the more easily be formed into a word for mnemonic purposes. For example, the word 2 3 1 5 6 5 1 _kha_ _gont_ _yan_ _me_ _[s.]a_ _m[=a]_ _pa_ has the value 1,565,132, reading from right to left.[138] This, the oldest specimen (1184 A.D.) known of this notation, is given in a commentary on the Rigveda, representing the number of days that had elapsed from the beginning of the Kaliyuga. Burnell[139] states that this system is even yet in use for remembering rules to calculate horoscopes, and for astronomical tables. A second system of this kind is still used in the pagination of manuscripts in Ceylon, Siam, and Burma, having also had its rise in southern India. In this the thirty-four consonants when followed by _a_ (as _ka_ ... _la_) designate the numbers 1-34; by _[=a]_ (as _k[=a]_ ... _l[=a]_), those from 35 to 68; by _i_ (_ki_ ... _li_), those from 69 to 102, inclusive; and so on.[140] As already stated, however, the Hindu system as thus far described was no improvement upon many others of the ancients, such as those used by the Greeks and the Hebrews. Having no zero, it was impracticable to designate the tens, hundreds, and other units of higher order by the same symbols used for the units from one to nine. In other words, there was no possibility of place value without some further improvement. So the N[=a]n[=a] Gh[=a]t {41} symbols required the writing of "thousand seven twenty-four" about like T 7, tw, 4 in modern symbols, instead of 7024, in which the seven of the thousands, the two of the tens (concealed in the word twenty, being originally "twain of tens," the _-ty_ signifying ten), and the four of the units are given as spoken and the order of the unit (tens, hundreds, etc.) is given by the place. To complete the system only the zero was needed; but it was probably eight centuries after the N[=a]n[=a] Gh[=a]t inscriptions were cut, before this important symbol appeared; and not until a considerably later period did it become well known. Who it was to whom the invention is due, or where he lived, or even in what century, will probably always remain a mystery.[141] It is possible that one of the forms of ancient abacus suggested to some Hindu astronomer or mathematician the use of a symbol to stand for the vacant line when the counters were removed. It is well established that in different parts of India the names of the higher powers took different forms, even the order being interchanged.[142] Nevertheless, as the significance of the name of the unit was given by the order in reading, these variations did not lead to error. Indeed the variation itself may have necessitated the introduction of a word to signify a vacant place or lacking unit, with the ultimate introduction of a zero symbol for this word. To enable us to appreciate the force of this argument a large number, 8,443,682,155, may be considered as the Hindus wrote and read it, and then, by way of contrast, as the Greeks and Arabs would have read it. {42} _Modern American reading_, 8 billion, 443 million, 682 thousand, 155. _Hindu_, 8 padmas, 4 vyarbudas, 4 k[=o][t.]is, 3 prayutas, 6 lak[s.]as, 8 ayutas, 2 sahasra, 1 ['s]ata, 5 da['s]an, 5. _Arabic and early German_, eight thousand thousand thousand and four hundred thousand thousand and forty-three thousand thousand, and six hundred thousand and eighty-two thousand and one hundred fifty-five (or five and fifty). _Greek_, eighty-four myriads of myriads and four thousand three hundred sixty-eight myriads and two thousand and one hundred fifty-five. As Woepcke[143] pointed out, the reading of numbers of this kind shows that the notation adopted by the Hindus tended to bring out the place idea. No other language than the Sanskrit has made such consistent application, in numeration, of the decimal system of numbers. The introduction of myriads as in the Greek, and thousands as in Arabic and in modern numeration, is really a step away from a decimal scheme. So in the numbers below one hundred, in English, eleven and twelve are out of harmony with the rest of the -teens, while the naming of all the numbers between ten and twenty is not analogous to the naming of the numbers above twenty. To conform to our written system we should have ten-one, ten-two, ten-three, and so on, as we have twenty-one, twenty-two, and the like. The Sanskrit is consistent, the units, however, preceding the tens and hundreds. Nor did any other ancient people carry the numeration as far as did the Hindus.[144] {43} When the _a[.n]kapalli_,[145] the decimal-place system of writing numbers, was perfected, the tenth symbol was called the _['s][=u]nyabindu_, generally shortened to _['s][=u]nya_ (the void). Brockhaus[146] has well said that if there was any invention for which the Hindus, by all their philosophy and religion, were well fitted, it was the invention of a symbol for zero. This making of nothingness the crux of a tremendous achievement was a step in complete harmony with the genius of the Hindu. It is generally thought that this _['s][=u]nya_ as a symbol was not used before about 500 A.D., although some writers have placed it earlier.[147] Since [=A]ryabha[t.]a gives our common method of extracting roots, it would seem that he may have known a decimal notation,[148] although he did not use the characters from which our numerals are derived.[149] Moreover, he frequently speaks of the {44} void.[150] If he refers to a symbol this would put the zero as far back as 500 A.D., but of course he may have referred merely to the concept of nothingness. A little later, but also in the sixth century, Var[=a]ha-Mihira[151] wrote a work entitled _B[r.]hat Sa[m.]hit[=a]_[152] in which he frequently uses _['s][=u]nya_ in speaking of numerals, so that it has been thought that he was referring to a definite symbol. This, of course, would add to the probability that [=A]ryabha[t.]a was doing the same. It should also be mentioned as a matter of interest, and somewhat related to the question at issue, that Var[=a]ha-Mihira used the word-system with place value[153] as explained above. The first kind of alphabetic numerals and also the word-system (in both of which the place value is used) are plays upon, or variations of, position arithmetic, which would be most likely to occur in the country of its origin.[154] At the opening of the next century (c. 620 A.D.) B[=a][n.]a[155] wrote of Subandhus's _V[=a]savadatt[=a]_ as a celebrated work, {45} and mentioned that the stars dotting the sky are here compared with zeros, these being points as in the modern Arabic system. On the other hand, a strong argument against any Hindu knowledge of the symbol zero at this time is the fact that about 700 A.D. the Arabs overran the province of Sind and thus had an opportunity of knowing the common methods used there for writing numbers. And yet, when they received the complete system in 776 they looked upon it as something new.[156] Such evidence is not conclusive, but it tends to show that the complete system was probably not in common use in India at the beginning of the eighth century. On the other hand, we must bear in mind the fact that a traveler in Germany in the year 1700 would probably have heard or seen nothing of decimal fractions, although these were perfected a century before that date. The élite of the mathematicians may have known the zero even in [=A]ryabha[t.]a's time, while the merchants and the common people may not have grasped the significance of the novelty until a long time after. On the whole, the evidence seems to point to the west coast of India as the region where the complete system was first seen.[157] As mentioned above, traces of the numeral words with place value, which do not, however, absolutely require a decimal place-system of symbols, are found very early in Cambodia, as well as in India. Concerning the earliest epigraphical instances of the use of the nine symbols, plus the zero, with place value, there {46} is some question. Colebrooke[158] in 1807 warned against the possibility of forgery in many of the ancient copper-plate land grants. On this account Fleet, in the _Indian Antiquary_,[159] discusses at length this phase of the work of the epigraphists in India, holding that many of these forgeries were made about the end of the eleventh century. Colebrooke[160] takes a more rational view of these forgeries than does Kaye, who seems to hold that they tend to invalidate the whole Indian hypothesis. "But even where that may be suspected, the historical uses of a monument fabricated so much nearer to the times to which it assumes to belong, will not be entirely superseded. The necessity of rendering the forged grant credible would compel a fabricator to adhere to history, and conform to established notions: and the tradition, which prevailed in his time, and by which he must be guided, would probably be so much nearer to the truth, as it was less remote from the period which it concerned."[161] Bühler[162] gives the copper-plate Gurjara inscription of Cedi-sa[m.]vat 346 (595 A.D.) as the oldest epigraphical use of the numerals[163] "in which the symbols correspond to the alphabet numerals of the period and the place." Vincent A. Smith[164] quotes a stone inscription of 815 A.D., dated Sa[m.]vat 872. So F. Kielhorn in the _Epigraphia Indica_[165] gives a Pathari pillar inscription of Parabala, dated Vikrama-sa[m.]vat 917, which corresponds to 861 A.D., {47} and refers also to another copper-plate inscription dated Vikrama-sa[m.]vat 813 (756 A.D.). The inscription quoted by V. A. Smith above is that given by D. R. Bhandarkar,[166] and another is given by the same writer as of date Saka-sa[m.]vat 715 (798 A.D.), being incised on a pilaster. Kielhorn[167] also gives two copper-plate inscriptions of the time of Mahendrapala of Kanauj, Valhab[=i]-sa[m.]vat 574 (893 A.D.) and Vikrama-sa[m.]vat 956 (899 A.D.). That there should be any inscriptions of date as early even as 750 A.D., would tend to show that the system was at least a century older. As will be shown in the further development, it was more than two centuries after the introduction of the numerals into Europe that they appeared there upon coins and inscriptions. While Thibaut[168] does not consider it necessary to quote any specific instances of the use of the numerals, he states that traces are found from 590 A.D. on. "That the system now in use by all civilized nations is of Hindu origin cannot be doubted; no other nation has any claim upon its discovery, especially since the references to the origin of the system which are found in the nations of western Asia point unanimously towards India."[169] The testimony and opinions of men like Bühler, Kielhorn, V. A. Smith, Bhandarkar, and Thibaut are entitled to the most serious consideration. As authorities on ancient Indian epigraphy no others rank higher. Their work is accepted by Indian scholars the world over, and their united judgment as to the rise of the system with a place value--that it took place in India as early as the {48} sixth century A.D.--must stand unless new evidence of great weight can be submitted to the contrary. Many early writers remarked upon the diversity of Indian numeral forms. Al-B[=i]r[=u]n[=i] was probably the first; noteworthy is also Johannes Hispalensis,[170] who gives the variant forms for seven and four. We insert on p. 49 a table of numerals used with place value. While the chief authority for this is Bühler,[171] several specimens are given which are not found in his work and which are of unusual interest. The ['S][=a]rad[=a] forms given in the table use the circle as a symbol for 1 and the dot for zero. They are taken from the paging and text of _The Kashmirian Atharva-Veda_[172], of which the manuscript used is certainly four hundred years old. Similar forms are found in a manuscript belonging to the University of Tübingen. Two other series presented are from Tibetan books in the library of one of the authors. For purposes of comparison the modern Sanskrit and Arabic numeral forms are added. Sanskrit, [Illustration] Arabic, [Illustration] {49} NUMERALS USED WITH PLACE VALUE 1 2 3 4 5 6 7 8 9 0 a[173] [Illustration] b[174] [Illustration] c[175] [Illustration] d[176] [Illustration] e[177] [Illustration] f[178] [Illustration] g[179] [Illustration] h[180] [Illustration] i[180] [Illustration] j[181] [Illustration] k[181] [Illustration] l[182] [Illustration] m[183] [Illustration] n[184] [Illustration] * * * * * {51} CHAPTER IV THE SYMBOL ZERO What has been said of the improved Hindu system with a place value does not touch directly the origin of a symbol for zero, although it assumes that such a symbol exists. The importance of such a sign, the fact that it is a prerequisite to a place-value system, and the further fact that without it the Hindu-Arabic numerals would never have dominated the computation system of the western world, make it proper to devote a chapter to its origin and history. It was some centuries after the primitive Br[=a]hm[=i] and Kharo[s.][t.]h[=i] numerals had made their appearance in India that the zero first appeared there, although such a character was used by the Babylonians[185] in the centuries immediately preceding the Christian era. The symbol is [Babylonian zero symbol] or [Babylonian zero symbol], and apparently it was not used in calculation. Nor does it always occur when units of any order are lacking; thus 180 is written [Babylonian numerals 180] with the meaning three sixties and no units, since 181 immediately following is [Babylonian numerals 181], three sixties and one unit.[186] The main {52} use of this Babylonian symbol seems to have been in the fractions, 60ths, 3600ths, etc., and somewhat similar to the Greek use of [Greek: o], for [Greek: ouden], with the meaning _vacant_. "The earliest undoubted occurrence of a zero in India is an inscription at Gwalior, dated Samvat 933 (876 A.D.). Where 50 garlands are mentioned (line 20), 50 is written [Gwalior numerals 50]. 270 (line 4) is written [Gwalior numerals 270]."[187] The Bakh[s.][=a]l[=i] Manuscript[188] probably antedates this, using the point or dot as a zero symbol. Bayley mentions a grant of Jaika Rashtrakúta of Bharuj, found at Okamandel, of date 738 A.D., which contains a zero, and also a coin with indistinct Gupta date 707 (897 A.D.), but the reliability of Bayley's work is questioned. As has been noted, the appearance of the numerals in inscriptions and on coins would be of much later occurrence than the origin and written exposition of the system. From the period mentioned the spread was rapid over all of India, save the southern part, where the Tamil and Malayalam people retain the old system even to the present day.[189] Aside from its appearance in early inscriptions, there is still another indication of the Hindu origin of the symbol in the special treatment of the concept zero in the early works on arithmetic. Brahmagupta, who lived in Ujjain, the center of Indian astronomy,[190] in the early part {53} of the seventh century, gives in his arithmetic[191] a distinct treatment of the properties of zero. He does not discuss a symbol, but he shows by his treatment that in some way zero had acquired a special significance not found in the Greek or other ancient arithmetics. A still more scientific treatment is given by Bh[=a]skara,[192] although in one place he permits himself an unallowed liberty in dividing by zero. The most recently discovered work of ancient Indian mathematical lore, the Ganita-S[=a]ra-Sa[.n]graha[193] of Mah[=a]v[=i]r[=a]c[=a]rya (c. 830 A.D.), while it does not use the numerals with place value, has a similar discussion of the calculation with zero. What suggested the form for the zero is, of course, purely a matter of conjecture. The dot, which the Hindus used to fill up lacunæ in their manuscripts, much as we indicate a break in a sentence,[194] would have been a more natural symbol; and this is the one which the Hindus first used[195] and which most Arabs use to-day. There was also used for this purpose a cross, like our X, and this is occasionally found as a zero symbol.[196] In the Bakh[s.][=a]l[=i] manuscript above mentioned, the word _['s][=u]nya_, with the dot as its symbol, is used to denote the unknown quantity, as well as to denote zero. An analogous use of the {54} zero, for the unknown quantity in a proportion, appears in a Latin manuscript of some lectures by Gottfried Wolack in the University of Erfurt in 1467 and 1468.[197] The usage was noted even as early as the eighteenth century.[198] The small circle was possibly suggested by the spurred circle which was used for ten.[199] It has also been thought that the omicron used by Ptolemy in his _Almagest_, to mark accidental blanks in the sexagesimal system which he employed, may have influenced the Indian writers.[200] This symbol was used quite generally in Europe and Asia, and the Arabic astronomer Al-Batt[=a]n[=i][201] (died 929 A.D.) used a similar symbol in connection with the alphabetic system of numerals. The occasional use by Al-Batt[=a]n[=i] of the Arabic negative, _l[=a]_, to indicate the absence of minutes {55} (or seconds), is noted by Nallino.[202] Noteworthy is also the use of the [Circle] for unity in the ['S][=a]rad[=a] characters of the Kashmirian Atharva-Veda, the writing being at least 400 years old. Bh[=a]skara (c. 1150) used a small circle above a number to indicate subtraction, and in the Tartar writing a redundant word is removed by drawing an oval around it. It would be interesting to know whether our score mark [score mark], read "four in the hole," could trace its pedigree to the same sources. O'Creat[203] (c. 1130), in a letter to his teacher, Adelhard of Bath, uses [Greek: t] for zero, being an abbreviation for the word _teca_ which we shall see was one of the names used for zero, although it could quite as well be from [Greek: tziphra]. More rarely O'Creat uses [circle with bar], applying the name _cyfra_ to both forms. Frater Sigsboto[204] (c. 1150) uses the same symbol. Other peculiar forms are noted by Heiberg[205] as being in use among the Byzantine Greeks in the fifteenth century. It is evident from the text that some of these writers did not understand the import of the new system.[206] Although the dot was used at first in India, as noted above, the small circle later replaced it and continues in use to this day. The Arabs, however, did not adopt the {56} circle, since it bore some resemblance to the letter which expressed the number five in the alphabet system.[207] The earliest Arabic zero known is the dot, used in a manuscript of 873 A.D.[208] Sometimes both the dot and the circle are used in the same work, having the same meaning, which is the case in an Arabic MS., an abridged arithmetic of Jamshid,[209] 982 A.H. (1575 A.D.). As given in this work the numerals are [symbols]. The form for 5 varies, in some works becoming [symbol] or [symbol]; [symbol] is found in Egypt and [symbol] appears in some fonts of type. To-day the Arabs use the 0 only when, under European influence, they adopt the ordinary system. Among the Chinese the first definite trace of zero is in the work of Tsin[210] of 1247 A.D. The form is the circular one of the Hindus, and undoubtedly was brought to China by some traveler. The name of this all-important symbol also demands some attention, especially as we are even yet quite undecided as to what to call it. We speak of it to-day as _zero, naught_, and even _cipher_; the telephone operator often calls it _O_, and the illiterate or careless person calls it _aught_. In view of all this uncertainty we may well inquire what it has been called in the past.[211] {57} As already stated, the Hindus called it _['s][=u]nya_, "void."[212] This passed over into the Arabic as _a[s.]-[s.]ifr_ or _[s.]ifr_.[213] When Leonard of Pisa (1202) wrote upon the Hindu numerals he spoke of this character as _zephirum_.[214] Maximus Planudes (1330), writing under both the Greek and the Arabic influence, called it _tziphra_.[215] In a treatise on arithmetic written in the Italian language by Jacob of Florence[216] {58} (1307) it is called _zeuero_,[217] while in an arithmetic of Giovanni di Danti of Arezzo (1370) the word appears as _çeuero_.[218] Another form is _zepiro_,[219] which was also a step from _zephirum_ to zero.[220] Of course the English _cipher_, French _chiffre_, is derived from the same Arabic word, _a[s.]-[s.]ifr_, but in several languages it has come to mean the numeral figures in general. A trace of this appears in our word _ciphering_, meaning figuring or computing.[221] Johann Huswirt[222] uses the word with both meanings; he gives for the tenth character the four names _theca, circulus, cifra_, and _figura nihili_. In this statement Huswirt probably follows, as did many writers of that period, the _Algorismus_ of Johannes de Sacrobosco (c. 1250 A.D.), who was also known as John of Halifax or John of Holywood. The commentary of {59} Petrus de Dacia[223] (c. 1291 A.D.) on the _Algorismus vulgaris_ of Sacrobosco was also widely used. The widespread use of this Englishman's work on arithmetic in the universities of that time is attested by the large number[224] of MSS. from the thirteenth to the seventeenth century still extant, twenty in Munich, twelve in Vienna, thirteen in Erfurt, several in England given by Halliwell,[225] ten listed in Coxe's _Catalogue of the Oxford College Library_, one in the Plimpton collection,[226] one in the Columbia University Library, and, of course, many others. From _a[s.]-[s.]ifr _has come _zephyr, cipher,_ and finally the abridged form _zero_. The earliest printed work in which is found this final form appears to be Calandri's arithmetic of 1491,[227] while in manuscript it appears at least as early as the middle of the fourteenth century.[228] It also appears in a work, _Le Kadran des marchans_, by Jehan {60} Certain,[229] written in 1485. This word soon became fairly well known in Spain[230] and France.[231] The medieval writers also spoke of it as the _sipos_,[232] and occasionally as the _wheel_,[233] _circulus_[234] (in German _das Ringlein_[235]), _circular {61} note_,[236] _theca_,[237] long supposed to be from its resemblance to the Greek theta, but explained by Petrus de Dacia as being derived from the name of the iron[238] used to brand thieves and robbers with a circular mark placed on the forehead or on the cheek. It was also called _omicron_[239] (the Greek _o_), being sometimes written õ or [Greek: ph] to distinguish it from the letter _o_. It also went by the name _null_[240] (in the Latin books {62} _nihil_[241] or _nulla_,[242] and in the French _rien_[243]), and very commonly by the name _cipher_.[244] Wallis[245] gives one of the earliest extended discussions of the various forms of the word, giving certain other variations worthy of note, as _ziphra_, _zifera_, _siphra_, _ciphra_, _tsiphra_, _tziphra,_ and the Greek [Greek: tziphra].[246] * * * * * {63} CHAPTER V THE QUESTION OF THE INTRODUCTION OF THE NUMERALS INTO EUROPE BY BOETHIUS Just as we were quite uncertain as to the origin of the numeral forms, so too are we uncertain as to the time and place of their introduction into Europe. There are two general theories as to this introduction. The first is that they were carried by the Moors to Spain in the eighth or ninth century, and thence were transmitted to Christian Europe, a theory which will be considered later. The second, advanced by Woepcke,[247] is that they were not brought to Spain by the Moors, but that they were already in Spain when the Arabs arrived there, having reached the West through the Neo-Pythagoreans. There are two facts to support this second theory: (1) the forms of these numerals are characteristic, differing materially from those which were brought by Leonardo of Pisa from Northern Africa early in the thirteenth century (before 1202 A.D.); (2) they are essentially those which {64} tradition has so persistently assigned to Boethius (c. 500 A.D.), and which he would naturally have received, if at all, from these same Neo-Pythagoreans or from the sources from which they derived them. Furthermore, Woepcke points out that the Arabs on entering Spain (711 A.D.) would naturally have followed their custom of adopting for the computation of taxes the numerical systems of the countries they conquered,[248] so that the numerals brought from Spain to Italy, not having undergone the same modifications as those of the Eastern Arab empire, would have differed, as they certainly did, from those that came through Bagdad. The theory is that the Hindu system, without the zero, early reached Alexandria (say 450 A.D.), and that the Neo-Pythagorean love for the mysterious and especially for the Oriental led to its use as something bizarre and cabalistic; that it was then passed along the Mediterranean, reaching Boethius in Athens or in Rome, and to the schools of Spain, being discovered in Africa and Spain by the Arabs even before they themselves knew the improved system with the place value. {65} A recent theory set forth by Bubnov[249] also deserves mention, chiefly because of the seriousness of purpose shown by this well-known writer. Bubnov holds that the forms first found in Europe are derived from ancient symbols used on the abacus, but that the zero is of Hindu origin. This theory does not seem tenable, however, in the light of the evidence already set forth. Two questions are presented by Woepcke's theory: (1) What was the nature of these Spanish numerals, and how were they made known to Italy? (2) Did Boethius know them? The Spanish forms of the numerals were called the _[h.]ur[=u]f al-[.g]ob[=a]r_, the [.g]ob[=a]r or dust numerals, as distinguished from the _[h.]ur[=u]f al-jumal_ or alphabetic numerals. Probably the latter, under the influence of the Syrians or Jews,[250] were also used by the Arabs. The significance of the term [.g]ob[=a]r is doubtless that these numerals were written on the dust abacus, this plan being distinct from the counter method of representing numbers. It is also worthy of note that Al-B[=i]r[=u]n[=i] states that the Hindus often performed numerical computations in the sand. The term is found as early as c. 950, in the verses of an anonymous writer of Kairw[=a]n, in Tunis, in which the author speaks of one of his works on [.g]ob[=a]r calculation;[251] and, much later, the Arab writer Ab[=u] Bekr Mo[h.]ammed ibn `Abdall[=a]h, surnamed al-[H.]a[s.][s.][=a]r {66} (the arithmetician), wrote a work of which the second chapter was "On the dust figures."[252] The [.g]ob[=a]r numerals themselves were first made known to modern scholars by Silvestre de Sacy, who discovered them in an Arabic manuscript from the library of the ancient abbey of St.-Germain-des-Prés.[253] The system has nine characters, but no zero. A dot above a character indicates tens, two dots hundreds, and so on, [5 with dot] meaning 50, and [5 with 3 dots] meaning 5000. It has been suggested that possibly these dots, sprinkled like dust above the numerals, gave rise to the word _[.g]ob[=a]r_,[254] but this is not at all probable. This system of dots is found in Persia at a much later date with numerals quite like the modern Arabic;[255] but that it was used at all is significant, for it is hardly likely that the western system would go back to Persia, when the perfected Hindu one was near at hand. At first sight there would seem to be some reason for believing that this feature of the [.g]ob[=a]r system was of {67} Arabic origin, and that the present zero of these people,[256] the dot, was derived from it. It was entirely natural that the Semitic people generally should have adopted such a scheme, since their diacritical marks would suggest it, not to speak of the possible influence of the Greek accents in the Hellenic number system. When we consider, however, that the dot is found for zero in the Bakh[s.][=a]l[=i] manuscript,[257] and that it was used in subscript form in the _Kit[=a]b al-Fihrist_[258] in the tenth century, and as late as the sixteenth century,[259] although in this case probably under Arabic influence, we are forced to believe that this form may also have been of Hindu origin. The fact seems to be that, as already stated,[260] the Arabs did not immediately adopt the Hindu zero, because it resembled their 5; they used the superscript dot as serving their purposes fairly well; they may, indeed, have carried this to the west and have added it to the [.g]ob[=a]r forms already there, just as they transmitted it to the Persians. Furthermore, the Arab and Hebrew scholars of Northern Africa in the tenth century knew these numerals as Indian forms, for a commentary on the _S[=e]fer Ye[s.][=i]r[=a]h_ by Ab[=u] Sahl ibn Tamim (probably composed at Kairw[=a]n, c. 950) speaks of "the Indian arithmetic known under the name of _[.g]ob[=a]r_ or dust calculation."[261] All this suggests that the Arabs may very {68} likely have known the [.g]ob[=a]r forms before the numerals reached them again in 773.[262] The term "[.g]ob[=a]r numerals" was also used without any reference to the peculiar use of dots.[263] In this connection it is worthy of mention that the Algerians employed two different forms of numerals in manuscripts even of the fourteenth century,[264] and that the Moroccans of to-day employ the European forms instead of the present Arabic. The Indian use of subscript dots to indicate the tens, hundreds, thousands, etc., is established by a passage in the _Kit[=a]b al-Fihrist_[265] (987 A.D.) in which the writer discusses the written language of the people of India. Notwithstanding the importance of this reference for the early history of the numerals, it has not been mentioned by previous writers on this subject. The numeral forms given are those which have usually been called Indian,[266] in opposition to [.g]ob[=a]r. In this document the dots are placed below the characters, instead of being superposed as described above. The significance was the same. In form these [.g]ob[=a]r numerals resemble our own much more closely than the Arab numerals do. They varied more or less, but were substantially as follows: {69} 1[267][Illustration] 2[268][Illustration] 3[269][Illustration] 4[270][Illustration] 5[271][Illustration] 6[271][Illustration] The question of the possible influence of the Egyptian demotic and hieratic ordinal forms has been so often suggested that it seems well to introduce them at this point, for comparison with the [.g]ob[=a]r forms. They would as appropriately be used in connection with the Hindu forms, and the evidence of a relation of the first three with all these systems is apparent. The only further resemblance is in the Demotic 4 and in the 9, so that the statement that the Hindu forms in general came from {70} this source has no foundation. The first four Egyptian cardinal numerals[272] resemble more the modern Arabic. [Illustration: DEMOTIC AND HIERATIC ORDINALS] This theory of the very early introduction of the numerals into Europe fails in several points. In the first place the early Western forms are not known; in the second place some early Eastern forms are like the [.g]ob[=a]r, as is seen in the third line on p. 69, where the forms are from a manuscript written at Shiraz about 970 A.D., and in which some western Arabic forms, e.g. [symbol] for 2, are also used. Probably most significant of all is the fact that the [.g]ob[=a]r numerals as given by Sacy are all, with the exception of the symbol for eight, either single Arabic letters or combinations of letters. So much for the Woepcke theory and the meaning of the [.g]ob[=a]r numerals. We now have to consider the question as to whether Boethius knew these [.g]ob[=a]r forms, or forms akin to them. This large question[273] suggests several minor ones: (1) Who was Boethius? (2) Could he have known these numerals? (3) Is there any positive or strong circumstantial evidence that he did know them? (4) What are the probabilities in the case? {71} First, who was Boethius,--Divus[274] Boethius as he was called in the Middle Ages? Anicius Manlius Severinus Boethius[275] was born at Rome c. 475. He was a member of the distinguished family of the Anicii,[276] which had for some time before his birth been Christian. Early left an orphan, the tradition is that he was taken to Athens at about the age of ten, and that he remained there eighteen years.[277] He married Rusticiana, daughter of the senator Symmachus, and this union of two such powerful families allowed him to move in the highest circles.[278] Standing strictly for the right, and against all iniquity at court, he became the object of hatred on the part of all the unscrupulous element near the throne, and his bold defense of the ex-consul Albinus, unjustly accused of treason, led to his imprisonment at Pavia[279] and his execution in 524.[280] Not many generations after his death, the period being one in which historical criticism was at its lowest ebb, the church found it profitable to look upon his execution as a martyrdom.[281] He was {72} accordingly looked upon as a saint,[282] his bones were enshrined,[283] and as a natural consequence his books were among the classics in the church schools for a thousand years.[284] It is pathetic, however, to think of the medieval student trying to extract mental nourishment from a work so abstract, so meaningless, so unnecessarily complicated, as the arithmetic of Boethius. He was looked upon by his contemporaries and immediate successors as a master, for Cassiodorus[285] (c. 490-c. 585 A.D.) says to him: "Through your translations the music of Pythagoras and the astronomy of Ptolemy are read by those of Italy, and the arithmetic of Nicomachus and the geometry of Euclid are known to those of the West."[286] Founder of the medieval scholasticism, {73} distinguishing the trivium and quadrivium,[287] writing the only classics of his time, Gibbon well called him "the last of the Romans whom Cato or Tully could have acknowledged for their countryman."[288] The second question relating to Boethius is this: Could he possibly have known the Hindu numerals? In view of the relations that will be shown to have existed between the East and the West, there can only be an affirmative answer to this question. The numerals had existed, without the zero, for several centuries; they had been well known in India; there had been a continued interchange of thought between the East and West; and warriors, ambassadors, scholars, and the restless trader, all had gone back and forth, by land or more frequently by sea, between the Mediterranean lands and the centers of Indian commerce and culture. Boethius could very well have learned one or more forms of Hindu numerals from some traveler or merchant. To justify this statement it is necessary to speak more fully of these relations between the Far East and Europe. It is true that we have no records of the interchange of learning, in any large way, between eastern Asia and central Europe in the century preceding the time of Boethius. But it is one of the mistakes of scholars to believe that they are the sole transmitters of knowledge. {74} As a matter of fact there is abundant reason for believing that Hindu numerals would naturally have been known to the Arabs, and even along every trade route to the remote west, long before the zero entered to make their place-value possible, and that the characters, the methods of calculating, the improvements that took place from time to time, the zero when it appeared, and the customs as to solving business problems, would all have been made known from generation to generation along these same trade routes from the Orient to the Occident. It must always be kept in mind that it was to the tradesman and the wandering scholar that the spread of such learning was due, rather than to the school man. Indeed, Avicenna[289] (980-1037 A.D.) in a short biography of himself relates that when his people were living at Bokh[=a]ra his father sent him to the house of a grocer to learn the Hindu art of reckoning, in which this grocer (oil dealer, possibly) was expert. Leonardo of Pisa, too, had a similar training. The whole question of this spread of mercantile knowledge along the trade routes is so connected with the [.g]ob[=a]r numerals, the Boethius question, Gerbert, Leonardo of Pisa, and other names and events, that a digression for its consideration now becomes necessary.[290] {75} Even in very remote times, before the Hindu numerals were sculptured in the cave of N[=a]n[=a] Gh[=a]t, there were trade relations between Arabia and India. Indeed, long before the Aryans went to India the great Turanian race had spread its civilization from the Mediterranean to the Indus.[291] At a much later period the Arabs were the intermediaries between Egypt and Syria on the west, and the farther Orient.[292] In the sixth century B.C., Hecatæus,[293] the father of geography, was acquainted not only with the Mediterranean lands but with the countries as far as the Indus,[294] and in Biblical times there were regular triennial voyages to India. Indeed, the story of Joseph bears witness to the caravan trade from India, across Arabia, and on to the banks of the Nile. About the same time as Hecatæus, Scylax, a Persian admiral under Darius, from Caryanda on the coast of Asia Minor, traveled to {76} northwest India and wrote upon his ventures.[295] He induced the nations along the Indus to acknowledge the Persian supremacy, and such number systems as there were in these lands would naturally have been known to a man of his attainments. A century after Scylax, Herodotus showed considerable knowledge of India, speaking of its cotton and its gold,[296] telling how Sesostris[297] fitted out ships to sail to that country, and mentioning the routes to the east. These routes were generally by the Red Sea, and had been followed by the Phoenicians and the Sabæans, and later were taken by the Greeks and Romans.[298] In the fourth century B.C. the West and East came into very close relations. As early as 330, Pytheas of Massilia (Marseilles) had explored as far north as the northern end of the British Isles and the coasts of the German Sea, while Macedon, in close touch with southern France, was also sending her armies under Alexander[299] through Afghanistan as far east as the Punjab.[300] Pliny tells us that Alexander the Great employed surveyors to measure {77} the roads of India; and one of the great highways is described by Megasthenes, who in 295 B.C., as the ambassador of Seleucus, resided at P[=a]tal[=i]pu[t.]ra, the present Patna.[301] The Hindus also learned the art of coining from the Greeks, or possibly from the Chinese, and the stores of Greco-Hindu coins still found in northern India are a constant source of historical information.[302] The R[=a]m[=a]yana speaks of merchants traveling in great caravans and embarking by sea for foreign lands.[303] Ceylon traded with Malacca and Siam, and Java was colonized by Hindu traders, so that mercantile knowledge was being spread about the Indies during all the formative period of the numerals. Moreover the results of the early Greek invasion were embodied by Dicæarchus of Messana (about 320 B.C.) in a map that long remained a standard. Furthermore, Alexander did not allow his influence on the East to cease. He divided India into three satrapies,[304] placing Greek governors over two of them and leaving a Hindu ruler in charge of the third, and in Bactriana, a part of Ariana or ancient Persia, he left governors; and in these the western civilization was long in evidence. Some of the Greek and Roman metrical and astronomical terms {78} found their way, doubtless at this time, into the Sanskrit language.[305] Even as late as from the second to the fifth centuries A.D., Indian coins showed the Hellenic influence. The Hindu astronomical terminology reveals the same relationship to western thought, for Var[=a]ha-Mihira (6th century A.D.), a contemporary of [=A]ryabha[t.]a, entitled a work of his the _B[r.]hat-Sa[m.]hit[=a]_, a literal translation of [Greek: megalê suntaxis] of Ptolemy;[306] and in various ways is this interchange of ideas apparent.[307] It could not have been at all unusual for the ancient Greeks to go to India, for Strabo lays down the route, saying that all who make the journey start from Ephesus and traverse Phrygia and Cappadocia before taking the direct road.[308] The products of the East were always finding their way to the West, the Greeks getting their ginger[309] from Malabar, as the Phoenicians had long before brought gold from Malacca. Greece must also have had early relations with China, for there is a notable similarity between the Greek and Chinese life, as is shown in their houses, their domestic customs, their marriage ceremonies, the public story-tellers, the puppet shows which Herodotus says were introduced from Egypt, the street jugglers, the games of dice,[310] the game of finger-guessing,[311] the water clock, the {79} music system, the use of the myriad,[312] the calendars, and in many other ways.[313] In passing through the suburbs of Peking to-day, on the way to the Great Bell temple, one is constantly reminded of the semi-Greek architecture of Pompeii, so closely does modern China touch the old classical civilization of the Mediterranean. The Chinese historians tell us that about 200 B.C. their arms were successful in the far west, and that in 180 B.C. an ambassador went to Bactria, then a Greek city, and reported that Chinese products were on sale in the markets there.[314] There is also a noteworthy resemblance between certain Greek and Chinese words,[315] showing that in remote times there must have been more or less interchange of thought. The Romans also exchanged products with the East. Horace says, "A busy trader, you hasten to the farthest Indies, flying from poverty over sea, over crags, over fires."[316] The products of the Orient, spices and jewels from India, frankincense from Persia, and silks from China, being more in demand than the exports from the Mediterranean lands, the balance of trade was against the West, and thus Roman coin found its way eastward. In 1898, for example, a number of Roman coins dating from 114 B.C. to Hadrian's time were found at Pakl[=i], a part of the Haz[=a]ra district, sixteen miles north of Abbott[=a]b[=a]d,[317] and numerous similar discoveries have been made from time to time. {80} Augustus speaks of envoys received by him from India, a thing never before known,[318] and it is not improbable that he also received an embassy from China.[319] Suetonius (first century A.D.) speaks in his history of these relations,[320] as do several of his contemporaries,[321] and Vergil[322] tells of Augustus doing battle in Persia. In Pliny's time the trade of the Roman Empire with Asia amounted to a million and a quarter dollars a year, a sum far greater relatively then than now,[323] while by the time of Constantine Europe was in direct communication with the Far East.[324] In view of these relations it is not beyond the range of possibility that proof may sometime come to light to show that the Greeks and Romans knew something of the {81} number system of India, as several writers have maintained.[325] Returning to the East, there are many evidences of the spread of knowledge in and about India itself. In the third century B.C. Buddhism began to be a connecting medium of thought. It had already permeated the Himalaya territory, had reached eastern Turkestan, and had probably gone thence to China. Some centuries later (in 62 A.D.) the Chinese emperor sent an ambassador to India, and in 67 A.D. a Buddhist monk was invited to China.[326] Then, too, in India itself A['s]oka, whose name has already been mentioned in this work, extended the boundaries of his domains even into Afghanistan, so that it was entirely possible for the numerals of the Punjab to have worked their way north even at that early date.[327] Furthermore, the influence of Persia must not be forgotten in considering this transmission of knowledge. In the fifth century the Persian medical school at Jondi-Sapur admitted both the Hindu and the Greek doctrines, and Firdus[=i] tells us that during the brilliant reign of {82} Khosr[=u] I,[328] the golden age of Pahlav[=i] literature, the Hindu game of chess was introduced into Persia, at a time when wars with the Greeks were bringing prestige to the Sassanid dynasty. Again, not far from the time of Boethius, in the sixth century, the Egyptian monk Cosmas, in his earlier years as a trader, made journeys to Abyssinia and even to India and Ceylon, receiving the name _Indicopleustes_ (the Indian traveler). His map (547 A.D.) shows some knowledge of the earth from the Atlantic to India. Such a man would, with hardly a doubt, have observed every numeral system used by the people with whom he sojourned,[329] and whether or not he recorded his studies in permanent form he would have transmitted such scraps of knowledge by word of mouth. As to the Arabs, it is a mistake to feel that their activities began with Mohammed. Commerce had always been held in honor by them, and the Qoreish[330] had annually for many generations sent caravans bearing the spices and textiles of Yemen to the shores of the Mediterranean. In the fifth century they traded by sea with India and even with China, and [H.]ira was an emporium for the wares of the East,[331] so that any numeral system of any part of the trading world could hardly have remained isolated. Long before the warlike activity of the Arabs, Alexandria had become the great market-place of the world. From this center caravans traversed Arabia to Hadramaut, where they met ships from India. Others went north to Damascus, while still others made their way {83} along the southern shores of the Mediterranean. Ships sailed from the isthmus of Suez to all the commercial ports of Southern Europe and up into the Black Sea. Hindus were found among the merchants[332] who frequented the bazaars of Alexandria, and Brahmins were reported even in Byzantium. Such is a very brief résumé of the evidence showing that the numerals of the Punjab and of other parts of India as well, and indeed those of China and farther Persia, of Ceylon and the Malay peninsula, might well have been known to the merchants of Alexandria, and even to those of any other seaport of the Mediterranean, in the time of Boethius. The Br[=a]hm[=i] numerals would not have attracted the attention of scholars, for they had no zero so far as we know, and therefore they were no better and no worse than those of dozens of other systems. If Boethius was attracted to them it was probably exactly as any one is naturally attracted to the bizarre or the mystic, and he would have mentioned them in his works only incidentally, as indeed they are mentioned in the manuscripts in which they occur. In answer therefore to the second question, Could Boethius have known the Hindu numerals? the reply must be, without the slightest doubt, that he could easily have known them, and that it would have been strange if a man of his inquiring mind did not pick up many curious bits of information of this kind even though he never thought of making use of them. Let us now consider the third question, Is there any positive or strong circumstantial evidence that Boethius did know these numerals? The question is not new, {84} nor is it much nearer being answered than it was over two centuries ago when Wallis (1693) expressed his doubts about it[333] soon after Vossius (1658) had called attention to the matter.[334] Stated briefly, there are three works on mathematics attributed to Boethius:[335] (1) the arithmetic, (2) a work on music, and (3) the geometry.[336] The genuineness of the arithmetic and the treatise on music is generally recognized, but the geometry, which contains the Hindu numerals with the zero, is under suspicion.[337] There are plenty of supporters of the idea that Boethius knew the numerals and included them in this book,[338] and on the other hand there are as many who {85} feel that the geometry, or at least the part mentioning the numerals, is spurious.[339] The argument of those who deny the authenticity of the particular passage in question may briefly be stated thus: 1. The falsification of texts has always been the subject of complaint. It was so with the Romans,[340] it was common in the Middle Ages,[341] and it is much more prevalent {86} to-day than we commonly think. We have but to see how every hymn-book compiler feels himself authorized to change at will the classics of our language, and how unknown editors have mutilated Shakespeare, to see how much more easy it was for medieval scribes to insert or eliminate paragraphs without any protest from critics.[342] 2. If Boethius had known these numerals he would have mentioned them in his arithmetic, but he does not do so.[343] 3. If he had known them, and had mentioned them in any of his works, his contemporaries, disciples, and successors would have known and mentioned them. But neither Capella (c. 475)[344] nor any of the numerous medieval writers who knew the works of Boethius makes any reference to the system.[345] {87} 4. The passage in question has all the appearance of an interpolation by some scribe. Boethius is speaking of angles, in his work on geometry, when the text suddenly changes to a discussion of classes of numbers.[346] This is followed by a chapter in explanation of the abacus,[347] in which are described those numeral forms which are called _apices_ or _caracteres_.[348] The forms[349] of these characters vary in different manuscripts, but in general are about as shown on page 88. They are commonly written with the 9 at the left, decreasing to the unit at the right, numerous writers stating that this was because they were derived from Semitic sources in which the direction of writing is the opposite of our own. This practice continued until the sixteenth century.[350] The writer then leaves the subject entirely, using the Roman numerals for the rest of his discussion, a proceeding so foreign to the method of Boethius as to be inexplicable on the hypothesis of authenticity. Why should such a scholarly writer have given them with no mention of their origin or use? Either he would have mentioned some historical interest attaching to them, or he would have used them in some discussion; he certainly would not have left the passage as it is. {88} FORMS OF THE NUMERALS, LARGELY FROM WORKS ON THE ABACUS[351] a[352] [Illustration] b[353] [Illustration] c[354] [Illustration] d[355] [Illustration] e[356] [Illustration] f[357] [Illustration] g[358] [Illustration] h[359] [Illustration] i[360] [Illustration] {89} Sir E. Clive Bayley has added[361] a further reason for believing them spurious, namely that the 4 is not of the N[=a]n[=a] Gh[=a]t type, but of the Kabul form which the Arabs did not receive until 776;[362] so that it is not likely, even if the characters were known in Europe in the time of Boethius, that this particular form was recognized. It is worthy of mention, also, that in the six abacus forms from the chief manuscripts as given by Friedlein,[363] each contains some form of zero, which symbol probably originated in India about this time or later. It could hardly have reached Europe so soon. As to the fourth question, Did Boethius probably know the numerals? It seems to be a fair conclusion, according to our present evidence, that (1) Boethius might very easily have known these numerals without the zero, but, (2) there is no reliable evidence that he did know them. And just as Boethius might have come in contact with them, so any other inquiring mind might have done so either in his time or at any time before they definitely appeared in the tenth century. These centuries, five in number, represented the darkest of the Dark Ages, and even if these numerals were occasionally met and studied, no trace of them would be likely to show itself in the {90} literature of the period, unless by chance it should get into the writings of some man like Alcuin. As a matter of fact, it was not until the ninth or tenth century that there is any tangible evidence of their presence in Christendom. They were probably known to merchants here and there, but in their incomplete state they were not of sufficient importance to attract any considerable attention. As a result of this brief survey of the evidence several conclusions seem reasonable: (1) commerce, and travel for travel's sake, never died out between the East and the West; (2) merchants had every opportunity of knowing, and would have been unreasonably stupid if they had not known, the elementary number systems of the peoples with whom they were trading, but they would not have put this knowledge in permanent written form; (3) wandering scholars would have known many and strange things about the peoples they met, but they too were not, as a class, writers; (4) there is every reason a priori for believing that the [.g]ob[=a]r numerals would have been known to merchants, and probably to some of the wandering scholars, long before the Arabs conquered northern Africa; (5) the wonder is not that the Hindu-Arabic numerals were known about 1000 A.D., and that they were the subject of an elaborate work in 1202 by Fibonacci, but rather that more extended manuscript evidence of their appearance before that time has not been found. That they were more or less known early in the Middle Ages, certainly to many merchants of Christian Europe, and probably to several scholars, but without the zero, is hardly to be doubted. The lack of documentary evidence is not at all strange, in view of all of the circumstances. * * * * * {91} CHAPTER VI THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS If the numerals had their origin in India, as seems most probable, when did the Arabs come to know of them? It is customary to say that it was due to the influence of Mohammedanism that learning spread through Persia and Arabia; and so it was, in part. But learning was already respected in these countries long before Mohammed appeared, and commerce flourished all through this region. In Persia, for example, the reign of Khosr[=u] Nu['s][=i]rw[=a]n,[364] the great contemporary of Justinian the law-maker, was characterized not only by an improvement in social and economic conditions, but by the cultivation of letters. Khosr[=u] fostered learning, inviting to his court scholars from Greece, and encouraging the introduction of culture from the West as well as from the East. At this time Aristotle and Plato were translated, and portions of the _Hito-pad[=e]['s]a_, or Fables of Pilpay, were rendered from the Sanskrit into Persian. All this means that some three centuries before the great intellectual ascendancy of Bagdad a similar fostering of learning was taking place in Persia, and under pre-Mohammedan influences. {92} The first definite trace that we have of the introduction of the Hindu system into Arabia dates from 773 A.D.,[365] when an Indian astronomer visited the court of the caliph, bringing with him astronomical tables which at the caliph's command were translated into Arabic by Al-Faz[=a]r[=i].[366] Al-Khow[=a]razm[=i] and [H.]abash (A[h.]med ibn `Abdall[=a]h, died c. 870) based their well-known tables upon the work of Al-F[=a]zar[=i]. It may be asserted as highly probable that the numerals came at the same time as the tables. They were certainly known a few decades later, and before 825 A.D., about which time the original of the _Algoritmi de numero Indorum_ was written, as that work makes no pretense of being the first work to treat of the Hindu numerals. The three writers mentioned cover the period from the end of the eighth to the end of the ninth century. While the historians Al-Ma['s]`[=u]d[=i] and Al-B[=i]r[=u]n[=i] follow quite closely upon the men mentioned, it is well to note again the Arab writers on Hindu arithmetic, contemporary with Al-Khow[=a]razm[=i], who were mentioned in chapter I, viz. Al-Kind[=i], Sened ibn `Al[=i], and Al-[S.][=u]f[=i]. For over five hundred years Arabic writers and others continued to apply to works on arithmetic the name "Indian." In the tenth century such writers are `Abdall[=a]h ibn al-[H.]asan, Ab[=u] 'l-Q[=a]sim[367] (died 987 A.D.) of Antioch, and Mo[h.]ammed ibn `Abdall[=a]h, Ab[=u] Na[s.]r[368] (c. 982), of Kalw[=a]d[=a] near Bagdad. Others of the same period or {93} earlier (since they are mentioned in the _Fihrist_,[369] 987 A.D.), who explicitly use the word "Hindu" or "Indian," are Sin[=a]n ibn al-Fat[h.][370] of [H.]arr[=a]n, and Ahmed ibn `Omar, al-Kar[=a]b[=i]s[=i].[371] In the eleventh century come Al-B[=i]r[=u]n[=i][372] (973-1048) and `Ali ibn A[h.]med, Ab[=u] 'l-[H.]asan, Al-Nasaw[=i][373] (c. 1030). The following century brings similar works by Ish[=a]q ibn Y[=u]suf al-[S.]ardaf[=i][374] and Sam[=u]'[=i]l ibn Ya[h.]y[=a] ibn `Abb[=a]s al-Ma[.g]reb[=i] al-Andalus[=i][375] (c. 1174), and in the thirteenth century are `Abdallat[=i]f ibn Y[=u]suf ibn Mo[h.]ammed, Muwaffaq al-D[=i]n Ab[=u] Mo[h.]ammed al-Ba[.g]d[=a]d[=i][376] (c. 1231), and Ibn al-Bann[=a].[377] The Greek monk Maximus Planudes, writing in the first half of the fourteenth century, followed the Arabic usage in calling his work _Indian Arithmetic_.[378] There were numerous other Arabic writers upon arithmetic, as that subject occupied one of the high places among the sciences, but most of them did not feel it necessary to refer to the origin of the symbols, the knowledge of which might well have been taken for granted. {94} One document, cited by Woepcke,[379] is of special interest since it shows at an early period, 970 A.D., the use of the ordinary Arabic forms alongside the [.g]ob[=a]r. The title of the work is _Interesting and Beautiful Problems on Numbers_ copied by A[h.]med ibn Mo[h.]ammed ibn `Abdaljal[=i]l, Ab[=u] Sa`[=i]d, al-Sijz[=i],[380] (951-1024) from a work by a priest and physician, Na[z.][=i]f ibn Yumn,[381] al-Qass (died c. 990). Suter does not mention this work of Na[z.][=i]f. The second reason for not ascribing too much credit to the purely Arab influence is that the Arab by himself never showed any intellectual strength. What took place after Mo[h.]ammed had lighted the fire in the hearts of his people was just what always takes place when different types of strong races blend,--a great renaissance in divers lines. It was seen in the blending of such types at Miletus in the time of Thales, at Rome in the days of the early invaders, at Alexandria when the Greek set firm foot on Egyptian soil, and we see it now when all the nations mingle their vitality in the New World. So when the Arab culture joined with the Persian, a new civilization rose and flourished.[382] The Arab influence came not from its purity, but from its intermingling with an influence more cultured if less virile. As a result of this interactivity among peoples of diverse interests and powers, Mohammedanism was to the world from the eighth to the thirteenth century what Rome and Athens and the Italo-Hellenic influence generally had {95} been to the ancient civilization. "If they did not possess the spirit of invention which distinguished the Greeks and the Hindus, if they did not show the perseverance in their observations that characterized the Chinese astronomers, they at least possessed the virility of a new and victorious people, with a desire to understand what others had accomplished, and a taste which led them with equal ardor to the study of algebra and of poetry, of philosophy and of language."[383] It was in 622 A.D. that Mo[h.]ammed fled from Mecca, and within a century from that time the crescent had replaced the cross in Christian Asia, in Northern Africa, and in a goodly portion of Spain. The Arab empire was an ellipse of learning with its foci at Bagdad and Cordova, and its rulers not infrequently took pride in demanding intellectual rather than commercial treasure as the result of conquest.[384] It was under these influences, either pre-Mohammedan or later, that the Hindu numerals found their way to the North. If they were known before Mo[h.]ammed's time, the proof of this fact is now lost. This much, however, is known, that in the eighth century they were taken to Bagdad. It was early in that century that the Mohammedans obtained their first foothold in northern India, thus foreshadowing an epoch of supremacy that endured with varied fortunes until after the golden age of Akbar the Great (1542-1605) and Shah Jehan. They also conquered Khorassan and Afghanistan, so that the learning and the commercial customs of India at once found easy {96} access to the newly-established schools and the bazaars of Mesopotamia and western Asia. The particular paths of conquest and of commerce were either by way of the Khyber Pass and through Kabul, Herat and Khorassan, or by sea through the strait of Ormuz to Basra (Busra) at the head of the Persian Gulf, and thence to Bagdad. As a matter of fact, one form of Arabic numerals, the one now in use by the Arabs, is attributed to the influence of Kabul, while the other, which eventually became our numerals, may very likely have reached Arabia by the other route. It is in Bagdad,[385] D[=a]r al-Sal[=a]m--"the Abode of Peace," that our special interest in the introduction of the numerals centers. Built upon the ruins of an ancient town by Al-Man[s.][=u]r[386] in the second half of the eighth century, it lies in one of those regions where the converging routes of trade give rise to large cities.[387] Quite as well of Bagdad as of Athens might Cardinal Newman have said:[388] "What it lost in conveniences of approach, it gained in its neighborhood to the traditions of the mysterious East, and in the loveliness of the region in which it lay. Hither, then, as to a sort of ideal land, where all archetypes of the great and the fair were found in substantial being, and all departments of truth explored, and all diversities of intellectual power exhibited, where taste and philosophy were majestically enthroned as in a royal court, where there was no sovereignty but that of mind, and no nobility but that of genius, where professors were {97} rulers, and princes did homage, thither flocked continually from the very corners of the _orbis terrarum_ the many-tongued generation, just rising, or just risen into manhood, in order to gain wisdom." For here it was that Al-Man[s.][=u]r and Al-M[=a]m[=u]n and H[=a]r[=u]n al-Rash[=i]d (Aaron the Just) made for a time the world's center of intellectual activity in general and in the domain of mathematics in particular.[389] It was just after the _Sindhind_ was brought to Bagdad that Mo[h.]ammed ibn M[=u]s[=a] al-Khow[=a]razm[=i], whose name has already been mentioned,[390] was called to that city. He was the most celebrated mathematician of his time, either in the East or West, writing treatises on arithmetic, the sundial, the astrolabe, chronology, geometry, and algebra, and giving through the Latin transliteration of his name, _algoritmi_, the name of algorism to the early arithmetics using the new Hindu numerals.[391] Appreciating at once the value of the position system so recently brought from India, he wrote an arithmetic based upon these numerals, and this was translated into Latin in the time of Adelhard of Bath (c. 1180), although possibly by his contemporary countryman Robert Cestrensis.[392] This translation was found in Cambridge and was published by Boncompagni in 1857.[393] Contemporary with Al-Khow[=a]razm[=i], and working also under Al-M[=a]m[=u]n, was a Jewish astronomer, Ab[=u] 'l-[T.]eiyib, {98} Sened ibn `Al[=i], who is said to have adopted the Mohammedan religion at the caliph's request. He also wrote a work on Hindu arithmetic,[394] so that the subject must have been attracting considerable attention at that time. Indeed, the struggle to have the Hindu numerals replace the Arabic did not cease for a long time thereafter. `Al[=i] ibn A[h.]med al-Nasaw[=i], in his arithmetic of c. 1025, tells us that the symbolism of number was still unsettled in his day, although most people preferred the strictly Arabic forms.[395] We thus have the numerals in Arabia, in two forms: one the form now used there, and the other the one used by Al-Khow[=a]razm[=i]. The question then remains, how did this second form find its way into Europe? and this question will be considered in the next chapter. * * * * * {99} CHAPTER VII THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE It being doubtful whether Boethius ever knew the Hindu numeral forms, certainly without the zero in any case, it becomes necessary now to consider the question of their definite introduction into Europe. From what has been said of the trade relations between the East and the West, and of the probability that it was the trader rather than the scholar who carried these numerals from their original habitat to various commercial centers, it is evident that we shall never know when they first made their inconspicuous entrance into Europe. Curious customs from the East and from the tropics,--concerning games, social peculiarities, oddities of dress, and the like,--are continually being related by sailors and traders in their resorts in New York, London, Hamburg, and Rotterdam to-day, customs that no scholar has yet described in print and that may not become known for many years, if ever. And if this be so now, how much more would it have been true a thousand years before the invention of printing, when learning was at its lowest ebb. It was at this period of low esteem of culture that the Hindu numerals undoubtedly made their first appearance in Europe. There were many opportunities for such knowledge to reach Spain and Italy. In the first place the Moors went into Spain as helpers of a claimant of the throne, and {100} remained as conquerors. The power of the Goths, who had held Spain for three centuries, was shattered at the battle of Jerez de la Frontera in 711, and almost immediately the Moors became masters of Spain and so remained for five hundred years, and masters of Granada for a much longer period. Until 850 the Christians were absolutely free as to religion and as to holding political office, so that priests and monks were not infrequently skilled both in Latin and Arabic, acting as official translators, and naturally reporting directly or indirectly to Rome. There was indeed at this time a complaint that Christian youths cultivated too assiduously a love for the literature of the Saracen, and married too frequently the daughters of the infidel.[396] It is true that this happy state of affairs was not permanent, but while it lasted the learning and the customs of the East must have become more or less the property of Christian Spain. At this time the [.g]ob[=a]r numerals were probably in that country, and these may well have made their way into Europe from the schools of Cordova, Granada, and Toledo. Furthermore, there was abundant opportunity for the numerals of the East to reach Europe through the journeys of travelers and ambassadors. It was from the records of Suleim[=a]n the Merchant, a well-known Arab trader of the ninth century, that part of the story of Sindb[=a]d the Sailor was taken.[397] Such a merchant would have been particularly likely to know the numerals of the people whom he met, and he is a type of man that may well have taken such symbols to European markets. A little later, {101} Ab[=u] 'l-[H.]asan `Al[=i] al-Mas`[=u]d[=i] (d. 956) of Bagdad traveled to the China Sea on the east, at least as far south as Zanzibar, and to the Atlantic on the west,[398] and he speaks of the nine figures with which the Hindus reckoned.[399] There was also a Bagdad merchant, one Ab[=u] 'l-Q[=a]sim `Obeidall[=a]h ibn A[h.]med, better known by his Persian name Ibn Khord[=a][d.]beh,[400] who wrote about 850 A.D. a work entitled _Book of Roads and Provinces_[401] in which the following graphic account appears:[402] "The Jewish merchants speak Persian, Roman (Greek and Latin), Arabic, French, Spanish, and Slavic. They travel from the West to the East, and from the East to the West, sometimes by land, sometimes by sea. They take ship from France on the Western Sea, and they voyage to Farama (near the ruins of the ancient Pelusium); there they transfer their goods to caravans and go by land to Colzom (on the Red Sea). They there reëmbark on the Oriental (Red) Sea and go to Hejaz and to Jiddah, and thence to the Sind, India, and China. Returning, they bring back the products of the oriental lands.... These journeys are also made by land. The merchants, leaving France and Spain, cross to Tangier and thence pass through the African provinces and Egypt. They then go to Ramleh, visit Damascus, Kufa, Bagdad, and Basra, penetrate into Ahwaz, Fars, Kerman, Sind, and thus reach India and China." Such travelers, about 900 A.D., must necessarily have spread abroad a knowledge of all number {102} systems used in recording prices or in the computations of the market. There is an interesting witness to this movement, a cruciform brooch now in the British Museum. It is English, certainly as early as the eleventh century, but it is inlaid with a piece of paste on which is the Mohammedan inscription, in Kufic characters, "There is no God but God." How did such an inscription find its way, perhaps in the time of Alcuin of York, to England? And if these Kufic characters reached there, then why not the numeral forms as well? Even in literature of the better class there appears now and then some stray proof of the important fact that the great trade routes to the far East were never closed for long, and that the customs and marks of trade endured from generation to generation. The _Gulist[=a]n_ of the Persian poet Sa`d[=i][403] contains such a passage: "I met a merchant who owned one hundred and forty camels, and fifty slaves and porters.... He answered to me: 'I want to carry sulphur of Persia to China, which in that country, as I hear, bears a high price; and thence to take Chinese ware to Roum; and from Roum to load up with brocades for Hind; and so to trade Indian steel (_pûlab_) to Halib. From Halib I will convey its glass to Yeman, and carry the painted cloths of Yeman back to Persia.'"[404] On the other hand, these men were not of the learned class, nor would they preserve in treatises any knowledge that they might have, although this knowledge would occasionally reach the ears of the learned as bits of curious information. {103} There were also ambassadors passing back and forth from time to time, between the East and the West, and in particular during the period when these numerals probably began to enter Europe. Thus Charlemagne (c. 800) sent emissaries to Bagdad just at the time of the opening of the mathematical activity there.[405] And with such ambassadors must have gone the adventurous scholar, inspired, as Alcuin says of Archbishop Albert of York (766-780),[406] to seek the learning of other lands. Furthermore, the Nestorian communities, established in Eastern Asia and in India at this time, were favored both by the Persians and by their Mohammedan conquerors. The Nestorian Patriarch of Syria, Timotheus (778-820), sent missionaries both to India and to China, and a bishop was appointed for the latter field. Ibn Wahab, who traveled to China in the ninth century, found images of Christ and the apostles in the Emperor's court.[407] Such a learned body of men, knowing intimately the countries in which they labored, could hardly have failed to make strange customs known as they returned to their home stations. Then, too, in Alfred's time (849-901) emissaries went {104} from England as far as India,[408] and generally in the Middle Ages groceries came to Europe from Asia as now they come from the colonies and from America. Syria, Asia Minor, and Cyprus furnished sugar and wool, and India yielded her perfumes and spices, while rich tapestries for the courts and the wealthy burghers came from Persia and from China.[409] Even in the time of Justinian (c. 550) there seems to have been a silk trade with China, which country in turn carried on commerce with Ceylon,[410] and reached out to Turkestan where other merchants transmitted the Eastern products westward. In the seventh century there was a well-defined commerce between Persia and India, as well as between Persia and Constantinople.[411] The Byzantine _commerciarii_ were stationed at the outposts not merely as customs officers but as government purchasing agents.[412] Occasionally there went along these routes of trade men of real learning, and such would surely have carried the knowledge of many customs back and forth. Thus at a period when the numerals are known to have been partly understood in Italy, at the opening of the eleventh century, one Constantine, an African, traveled from Italy through a great part of Africa and Asia, even on to India, for the purpose of learning the sciences of the Orient. He spent thirty-nine years in travel, having been hospitably received in Babylon, and upon his return he was welcomed with great honor at Salerno.[413] A very interesting illustration of this intercourse also appears in the tenth century, when the son of Otto I {105} (936-973) married a princess from Constantinople. This monarch was in touch with the Moors of Spain and invited to his court numerous scholars from abroad,[414] and his intercourse with the East as well as the West must have brought together much of the learning of each. Another powerful means for the circulation of mysticism and philosophy, and more or less of culture, took its start just before the conversion of Constantine (c. 312), in the form of Christian pilgrim travel. This was a feature peculiar to the zealots of early Christianity, found in only a slight degree among their Jewish predecessors in the annual pilgrimage to Jerusalem, and almost wholly wanting in other pre-Christian peoples. Chief among these early pilgrims were the two Placentians, John and Antonine the Elder (c. 303), who, in their wanderings to Jerusalem, seem to have started a movement which culminated centuries later in the crusades.[415] In 333 a Bordeaux pilgrim compiled the first Christian guide-book, the _Itinerary from Bordeaux to Jerusalem_,[416] and from this time on the holy pilgrimage never entirely ceased. Still another certain route for the entrance of the numerals into Christian Europe was through the pillaging and trading carried on by the Arabs on the northern shores of the Mediterranean. As early as 652 A.D., in the thirtieth year of the Hejira, the Mohammedans descended upon the shores of Sicily and took much spoil. Hardly had the wretched Constans given place to the {106} young Constantine IV when they again attacked the island and plundered ancient Syracuse. Again in 827, under Asad, they ravaged the coasts. Although at this time they failed to conquer Syracuse, they soon held a good part of the island, and a little later they successfully besieged the city. Before Syracuse fell, however, they had plundered the shores of Italy, even to the walls of Rome itself; and had not Leo IV, in 849, repaired the neglected fortifications, the effects of the Moslem raid of that year might have been very far-reaching. Ibn Khord[=a][d.]beh, who left Bagdad in the latter part of the ninth century, gives a picture of the great commercial activity at that time in the Saracen city of Palermo. In this same century they had established themselves in Piedmont, and in 906 they pillaged Turin.[417] On the Sorrento peninsula the traveler who climbs the hill to the beautiful Ravello sees still several traces of the Arab architecture, reminding him of the fact that about 900 A.D. Amalfi was a commercial center of the Moors.[418] Not only at this time, but even a century earlier, the artists of northern India sold their wares at such centers, and in the courts both of H[=a]r[=u]n al-Rash[=i]d and of Charlemagne.[419] Thus the Arabs dominated the Mediterranean Sea long before Venice "held the gorgeous East in fee And was the safeguard of the West," and long before Genoa had become her powerful rival.[420] {107} Only a little later than this the brothers Nicolo and Maffeo Polo entered upon their famous wanderings.[421] Leaving Constantinople in 1260, they went by the Sea of Azov to Bokhara, and thence to the court of Kublai Khan, penetrating China, and returning by way of Acre in 1269 with a commission which required them to go back to China two years later. This time they took with them Nicolo's son Marco, the historian of the journey, and went across the plateau of Pamir; they spent about twenty years in China, and came back by sea from China to Persia. The ventures of the Poli were not long unique, however: the thirteenth century had not closed before Roman missionaries and the merchant Petrus de Lucolongo had penetrated China. Before 1350 the company of missionaries was large, converts were numerous, churches and Franciscan convents had been organized in the East, travelers were appealing for the truth of their accounts to the "many" persons in Venice who had been in China, Tsuan-chau-fu had a European merchant community, and Italian trade and travel to China was a thing that occupied two chapters of a commercial handbook.[422] {108} It is therefore reasonable to conclude that in the Middle Ages, as in the time of Boethius, it was a simple matter for any inquiring scholar to become acquainted with such numerals of the Orient as merchants may have used for warehouse or price marks. And the fact that Gerbert seems to have known only the forms of the simplest of these, not comprehending their full significance, seems to prove that he picked them up in just this way. Even if Gerbert did not bring his knowledge of the Oriental numerals from Spain, he may easily have obtained them from the marks on merchant's goods, had he been so inclined. Such knowledge was probably obtainable in various parts of Italy, though as parts of mere mercantile knowledge the forms might soon have been lost, it needing the pen of the scholar to preserve them. Trade at this time was not stagnant. During the eleventh and twelfth centuries the Slavs, for example, had very great commercial interests, their trade reaching to Kiev and Novgorod, and thence to the East. Constantinople was a great clearing-house of commerce with the Orient,[423] and the Byzantine merchants must have been entirely familiar with the various numerals of the Eastern peoples. In the eleventh century the Italian town of Amalfi established a factory[424] in Constantinople, and had trade relations with Antioch and Egypt. Venice, as early as the ninth century, had a valuable trade with Syria and Cairo.[425] Fifty years after Gerbert died, in the time of Cnut, the Dane and the Norwegian pushed their commerce far beyond the northern seas, both by caravans through Russia to the Orient, and by their venturesome barks which {109} sailed through the Strait of Gibraltar into the Mediterranean.[426] Only a little later, probably before 1200 A.D., a clerk in the service of Thomas à Becket, present at the latter's death, wrote a life of the martyr, to which (fortunately for our purposes) he prefixed a brief eulogy of the city of London.[427] This clerk, William Fitz Stephen by name, thus speaks of the British capital: Aurum mittit Arabs: species et thura Sabæus: Arma Sythes: oleum palmarum divite sylva Pingue solum Babylon: Nilus lapides pretiosos: Norwegi, Russi, varium grisum, sabdinas: Seres, purpureas vestes: Galli, sua vina. Although, as a matter of fact, the Arabs had no gold to send, and the Scythians no arms, and Egypt no precious stones save only the turquoise, the Chinese (_Seres_) may have sent their purple vestments, and the north her sables and other furs, and France her wines. At any rate the verses show very clearly an extensive foreign trade. Then there were the Crusades, which in these times brought the East in touch with the West. The spirit of the Orient showed itself in the songs of the troubadours, and the _baudekin_,[428] the canopy of Bagdad,[429] became common in the churches of Italy. In Sicily and in Venice the textile industries of the East found place, and made their way even to the Scandinavian peninsula.[430] We therefore have this state of affairs: There was abundant intercourse between the East and West for {110} some centuries before the Hindu numerals appear in any manuscripts in Christian Europe. The numerals must of necessity have been known to many traders in a country like Italy at least as early as the ninth century, and probably even earlier, but there was no reason for preserving them in treatises. Therefore when a man like Gerbert made them known to the scholarly circles, he was merely describing what had been familiar in a small way to many people in a different walk of life. Since Gerbert[431] was for a long time thought to have been the one to introduce the numerals into Italy,[432] a brief sketch of this unique character is proper. Born of humble parents,[433] this remarkable man became the counselor and companion of kings, and finally wore the papal tiara as Sylvester II, from 999 until his death in 1003.[434] He was early brought under the influence of the monks at Aurillac, and particularly of Raimund, who had been a pupil of Odo of Cluny, and there in due time he himself took holy orders. He visited Spain in about 967 in company with Count Borel,[435] remaining there three years, {111} and studying under Bishop Hatto of Vich,[436] a city in the province of Barcelona,[437] then entirely under Christian rule. Indeed, all of Gerbert's testimony is as to the influence of the Christian civilization upon his education. Thus he speaks often of his study of Boethius,[438] so that if the latter knew the numerals Gerbert would have learned them from him.[439] If Gerbert had studied in any Moorish schools he would, under the decree of the emir Hish[=a]m (787-822), have been obliged to know Arabic, which would have taken most of his three years in Spain, and of which study we have not the slightest hint in any of his letters.[440] On the other hand, Barcelona was the only Christian province in immediate touch with the Moorish civilization at that time.[441] Furthermore we know that earlier in the same century King Alonzo of Asturias (d. 910) confided the education of his son Ordoño to the Arab scholars of the court of the {112} w[=a]l[=i] of Saragossa,[442] so that there was more or less of friendly relation between Christian and Moor. After his three years in Spain, Gerbert went to Italy, about 970, where he met Pope John XIII, being by him presented to the emperor Otto I. Two years later (972), at the emperor's request, he went to Rheims, where he studied philosophy, assisting to make of that place an educational center; and in 983 he became abbot at Bobbio. The next year he returned to Rheims, and became archbishop of that diocese in 991. For political reasons he returned to Italy in 996, became archbishop of Ravenna in 998, and the following year was elected to the papal chair. Far ahead of his age in wisdom, he suffered as many such scholars have even in times not so remote by being accused of heresy and witchcraft. As late as 1522, in a biography published at Venice, it is related that by black art he attained the papacy, after having given his soul to the devil.[443] Gerbert was, however, interested in astrology,[444] although this was merely the astronomy of that time and was such a science as any learned man would wish to know, even as to-day we wish to be reasonably familiar with physics and chemistry. That Gerbert and his pupils knew the [.g]ob[=a]r numerals is a fact no longer open to controversy.[445] Bernelinus and Richer[446] call them by the well-known name of {113} "caracteres," a word used by Radulph of Laon in the same sense a century later.[447] It is probable that Gerbert was the first to describe these [.g]ob[=a]r numerals in any scientific way in Christian Europe, but without the zero. If he knew the latter he certainly did not understand its use.[448] The question still to be settled is as to where he found these numerals. That he did not bring them from Spain is the opinion of a number of careful investigators.[449] This is thought to be the more probable because most of the men who made Spain famous for learning lived after Gerbert was there. Such were Ibn S[=i]n[=a] (Avicenna) who lived at the beginning, and Gerber of Seville who flourished in the middle, of the eleventh century, and Ab[=u] Roshd (Averroës) who lived at the end of the twelfth.[450] Others hold that his proximity to {114} the Arabs for three years makes it probable that he assimilated some of their learning, in spite of the fact that the lines between Christian and Moor at that time were sharply drawn.[451] Writers fail, however, to recognize that a commercial numeral system would have been more likely to be made known by merchants than by scholars. The itinerant peddler knew no forbidden pale in Spain, any more than he has known one in other lands. If the [.g]ob[=a]r numerals were used for marking wares or keeping simple accounts, it was he who would have known them, and who would have been the one rather than any Arab scholar to bring them to the inquiring mind of the young French monk. The facts that Gerbert knew them only imperfectly, that he used them solely for calculations, and that the forms are evidently like the Spanish [.g]ob[=a]r, make it all the more probable that it was through the small tradesman of the Moors that this versatile scholar derived his knowledge. Moreover the part of the geometry bearing his name, and that seems unquestionably his, shows the Arab influence, proving that he at least came into contact with the transplanted Oriental learning, even though imperfectly.[452] There was also the persistent Jewish merchant trading with both peoples then as now, always alive to the acquiring of useful knowledge, and it would be very natural for a man like Gerbert to welcome learning from such a source. On the other hand, the two leading sources of information as to the life of Gerbert reveal practically nothing to show that he came within the Moorish sphere of influence during his sojourn in Spain. These sources {115} are his letters and the history written by Richer. Gerbert was a master of the epistolary art, and his exalted position led to the preservation of his letters to a degree that would not have been vouchsafed even by their classic excellence.[453] Richer was a monk at St. Remi de Rheims, and was doubtless a pupil of Gerbert. The latter, when archbishop of Rheims, asked Richer to write a history of his times, and this was done. The work lay in manuscript, entirely forgotten until Pertz discovered it at Bamberg in 1833.[454] The work is dedicated to Gerbert as archbishop of Rheims,[455] and would assuredly have testified to such efforts as he may have made to secure the learning of the Moors. Now it is a fact that neither the letters nor this history makes any statement as to Gerbert's contact with the Saracens. The letters do not speak of the Moors, of the Arab numerals, nor of Cordova. Spain is not referred to by that name, and only one Spanish scholar is mentioned. In one of his letters he speaks of Joseph Ispanus,[456] or Joseph Sapiens, but who this Joseph the Wise of Spain may have been we do not know. Possibly {116} it was he who contributed the morsel of knowledge so imperfectly assimilated by the young French monk.[457] Within a few years after Gerbert's visit two young Spanish monks of lesser fame, and doubtless with not that keen interest in mathematical matters which Gerbert had, regarded the apparently slight knowledge which they had of the Hindu numeral forms as worthy of somewhat permanent record[458] in manuscripts which they were transcribing. The fact that such knowledge had penetrated to their modest cloisters in northern Spain--the one Albelda or Albaida--indicates that it was rather widely diffused. Gerbert's treatise _Libellus de numerorum divisione_[459] is characterized by Chasles as "one of the most obscure documents in the history of science."[460] The most complete information in regard to this and the other mathematical works of Gerbert is given by Bubnov,[461] who considers this work to be genuine.[462] {117} So little did Gerbert appreciate these numerals that in his works known as the _Regula de abaco computi_ and the _Libellus_ he makes no use of them at all, employing only the Roman forms.[463] Nevertheless Bernelinus[464] refers to the nine [.g]ob[=a]r characters.[465] These Gerbert had marked on a thousand _jetons_ or counters,[466] using the latter on an abacus which he had a sign-maker prepare for him.[467] Instead of putting eight counters in say the tens' column, Gerbert would put a single counter marked 8, and so for the other places, leaving the column empty where we would place a zero, but where he, lacking the zero, had no counter to place. These counters he possibly called _caracteres_, a name which adhered also to the figures themselves. It is an interesting speculation to consider whether these _apices_, as they are called in the Boethius interpolations, were in any way suggested by those Roman jetons generally known in numismatics as _tesserae_, and bearing the figures I-XVI, the sixteen referring to the number of _assi_ in a _sestertius_.[468] The {118} name _apices_ adhered to the Hindu-Arabic numerals until the sixteenth century.[469] To the figures on the _apices_ were given the names Igin, andras, ormis, arbas, quimas, calctis or caltis, zenis, temenias, celentis, sipos,[470] the origin and meaning of which still remain a mystery. The Semitic origin of several of the words seems probable. _Wahud_, _thaneine_, {119} _thalata_, _arba_, _kumsa_, _setta_, _sebba_, _timinia_, _taseud_ are given by the Rev. R. Patrick[471] as the names, in an Arabic dialect used in Morocco, for the numerals from one to nine. Of these the words for four, five, and eight are strikingly like those given above. The name _apices_ was not, however, a common one in later times. _Notae_ was more often used, and it finally gave the name to notation.[472] Still more common were the names _figures_, _ciphers_, _signs_, _elements_, and _characters_.[473] So little effect did the teachings of Gerbert have in making known the new numerals, that O'Creat, who lived a century later, a friend and pupil of Adelhard {120} of Bath, used the zero with the Roman characters, in contrast to Gerbert's use of the [.g]ob[=a]r forms without the zero.[474] O'Creat uses three forms for zero, o, [=o], and [Greek: t], as in Maximus Planudes. With this use of the zero goes, naturally, a place value, for he writes III III for 33, ICCOO and I. II. [tau]. [tau] for 1200, I. O. VIII. IX for 1089, and I. IIII. IIII. [tau][tau][tau][tau] for the square of 1200. The period from the time of Gerbert until after the appearance of Leonardo's monumental work may be called the period of the abacists. Even for many years after the appearance early in the twelfth century of the books explaining the Hindu art of reckoning, there was strife between the abacists, the advocates of the abacus, and the algorists, those who favored the new numerals. The words _cifra_ and _algorismus cifra_ were used with a somewhat derisive significance, indicative of absolute uselessness, as indeed the zero is useless on an abacus in which the value of any unit is given by the column which it occupies.[475] So Gautier de Coincy (1177-1236) in a work on the miracles of Mary says: A horned beast, a sheep, An algorismus-cipher, Is a priest, who on such a feast day Does not celebrate the holy Mother.[476] So the abacus held the field for a long time, even against the new algorism employing the new numerals. {121} Geoffrey Chaucer[477] describes in _The Miller's Tale_ the clerk with "His Almageste and bokes grete and smale, His astrelabie, longinge for his art, His augrim-stones layen faire apart On shelves couched at his beddes heed." So, too, in Chaucer's explanation of the astrolabe,[478] written for his son Lewis, the number of degrees is expressed on the instrument in Hindu-Arabic numerals: "Over the whiche degrees ther ben noumbres of augrim, that devyden thilke same degrees fro fyve to fyve," and "... the nombres ... ben writen in augrim," meaning in the way of the algorism. Thomas Usk about 1387 writes:[479] "a sypher in augrim have no might in signification of it-selve, yet he yeveth power in signification to other." So slow and so painful is the assimilation of new ideas. Bernelinus[480] states that the abacus is a well-polished board (or table), which is covered with blue sand and used by geometers in drawing geometrical figures. We have previously mentioned the fact that the Hindus also performed mathematical computations in the sand, although there is no evidence to show that they had any column abacus.[481] For the purposes of computation, Bernelinus continues, the board is divided into thirty vertical columns, three of which are reserved for fractions. Beginning with the units columns, each set of {122} three columns (_lineae_ is the word which Bernelinus uses) is grouped together by a semicircular arc placed above them, while a smaller arc is placed over the units column and another joins the tens and hundreds columns. Thus arose the designation _arcus pictagore_[482] or sometimes simply _arcus_.[483] The operations of addition, subtraction, and multiplication upon this form of the abacus required little explanation, although they were rather extensively treated, especially the multiplication of different orders of numbers. But the operation of division was effected with some difficulty. For the explanation of the method of division by the use of the complementary difference,[484] long the stumbling-block in the way of the medieval arithmetician, the reader is referred to works on the history of mathematics[485] and to works relating particularly to the abacus.[486] Among the writers on the subject may be mentioned Abbo[487] of Fleury (c. 970), Heriger[488] of Lobbes or Laubach {123} (c. 950-1007), and Hermannus Contractus[489] (1013-1054), all of whom employed only the Roman numerals. Similarly Adelhard of Bath (c. 1130), in his work _Regulae Abaci_,[490] gives no reference to the new numerals, although it is certain that he knew them. Other writers on the abacus who used some form of Hindu numerals were Gerland[491] (first half of twelfth century) and Turchill[492] (c. 1200). For the forms used at this period the reader is referred to the plate on page 88. After Gerbert's death, little by little the scholars of Europe came to know the new figures, chiefly through the introduction of Arab learning. The Dark Ages had passed, although arithmetic did not find another advocate as prominent as Gerbert for two centuries. Speaking of this great revival, Raoul Glaber[493] (985-c. 1046), a monk of the great Benedictine abbey of Cluny, of the eleventh century, says: "It was as though the world had arisen and tossed aside the worn-out garments of ancient time, and wished to apparel itself in a white robe of churches." And with this activity in religion came a corresponding interest in other lines. Algorisms began to appear, and knowledge from the outside world found {124} interested listeners. Another Raoul, or Radulph, to whom we have referred as Radulph of Laon,[494] a teacher in the cloister school of his city, and the brother of Anselm of Laon[495] the celebrated theologian, wrote a treatise on music, extant but unpublished, and an arithmetic which Nagl first published in 1890.[496] The latter work, preserved to us in a parchment manuscript of seventy-seven leaves, contains a curious mixture of Roman and [.g]ob[=a]r numerals, the former for expressing large results, the latter for practical calculation. These [.g]ob[=a]r "caracteres" include the sipos (zero), [Symbol], of which, however, Radulph did not know the full significance; showing that at the opening of the twelfth century the system was still uncertain in its status in the church schools of central France. At the same time the words _algorismus_ and _cifra_ were coming into general use even in non-mathematical literature. Jordan [497] cites numerous instances of such use from the works of Alanus ab Insulis[498] (Alain de Lille), Gautier de Coincy (1177-1236), and others. Another contributor to arithmetic during this interesting period was a prominent Spanish Jew called variously John of Luna, John of Seville, Johannes Hispalensis, Johannes Toletanus, and Johannes Hispanensis de Luna.[499] {125} His date is rather closely fixed by the fact that he dedicated a work to Raimund who was archbishop of Toledo between 1130 and 1150.[500] His interests were chiefly in the translation of Arabic works, especially such as bore upon the Aristotelian philosophy. From the standpoint of arithmetic, however, the chief interest centers about a manuscript entitled _Joannis Hispalensis liber Algorismi de Practica Arismetrice_ which Boncompagni found in what is now the _Bibliothèque nationale_ at Paris. Although this distinctly lays claim to being Al-Khow[=a]razm[=i]'s work,[501] the evidence is altogether against the statement,[502] but the book is quite as valuable, since it represents the knowledge of the time in which it was written. It relates to the operations with integers and sexagesimal fractions, including roots, and contains no applications.[503] Contemporary with John of Luna, and also living in Toledo, was Gherard of Cremona,[504] who has sometimes been identified, but erroneously, with Gernardus,[505] the {126} author of a work on algorism. He was a physician, an astronomer, and a mathematician, translating from the Arabic both in Italy and in Spain. In arithmetic he was influential in spreading the ideas of algorism. Four Englishmen--Adelhard of Bath (c. 1130), Robert of Chester (Robertus Cestrensis, c. 1143), William Shelley, and Daniel Morley (1180)--are known[506] to have journeyed to Spain in the twelfth century for the purpose of studying mathematics and Arabic. Adelhard of Bath made translations from Arabic into Latin of Al-Khow[=a]razm[=i]'s astronomical tables[507] and of Euclid's Elements,[508] while Robert of Chester is known as the translator of Al-Khow[=a]razm[=i]'s algebra.[509] There is no reason to doubt that all of these men, and others, were familiar with the numerals which the Arabs were using. The earliest trace we have of computation with Hindu numerals in Germany is in an Algorismus of 1143, now in the Hofbibliothek in Vienna.[510] It is bound in with a {127} _Computus_ by the same author and bearing the date given. It contains chapters "De additione," "De diminutione," "De mediatione," "De divisione," and part of a chapter on multiplication. The numerals are in the usual medieval forms except the 2 which, as will be seen from the illustration,[511] is somewhat different, and the 3, which takes the peculiar shape [Symbol], a form characteristic of the twelfth century. It was about the same time that the _Sefer ha-Mispar_,[512] the Book of Number, appeared in the Hebrew language. The author, Rabbi Abraham ibn Meïr ibn Ezra,[513] was born in Toledo (c. 1092). In 1139 he went to Egypt, Palestine, and the Orient, spending also some years in Italy. Later he lived in southern France and in England. He died in 1167. The probability is that he acquired his knowledge of the Hindu arithmetic[514] in his native town of Toledo, but it is also likely that the knowledge of other systems which he acquired on travels increased his appreciation of this one. We have mentioned the fact that he used the first letters of the Hebrew alphabet, [Hebrew: A B G D H W Z CH T`], for the numerals 9 8 7 6 5 4 3 2 1, and a circle for the zero. The quotation in the note given below shows that he knew of the Hindu origin; but in his manuscript, although he set down the Hindu forms, he used the above nine Hebrew letters with place value for all computations. * * * * * {128} CHAPTER VIII THE SPREAD OF THE NUMERALS IN EUROPE Of all the medieval writers, probably the one most influential in introducing the new numerals to the scholars of Europe was Leonardo Fibonacci, of Pisa.[515] This remarkable man, the most noteworthy mathematical genius of the Middle Ages, was born at Pisa about 1175.[516] The traveler of to-day may cross the Via Fibonacci on his way to the Campo Santo, and there he may see at the end of the long corridor, across the quadrangle, the statue of Leonardo in scholars garb. Few towns have honored a mathematician more, and few mathematicians have so distinctly honored their birthplace. Leonardo was born in the golden age of this city, the period of its commercial, religious, and intellectual prosperity.[517] {129} Situated practically at the mouth of the Arno, Pisa formed with Genoa and Venice the trio of the greatest commercial centers of Italy at the opening of the thirteenth century. Even before Venice had captured the Levantine trade, Pisa had close relations with the East. An old Latin chronicle relates that in 1005 "Pisa was captured by the Saracens," that in the following year "the Pisans overthrew the Saracens at Reggio," and that in 1012 "the Saracens came to Pisa and destroyed it." The city soon recovered, however, sending no fewer than a hundred and twenty ships to Syria in 1099,[518] founding a merchant colony in Constantinople a few years later,[519] and meanwhile carrying on an interurban warfare in Italy that seemed to stimulate it to great activity.[520] A writer of 1114 tells us that at that time there were many heathen people--Turks, Libyans, Parthians, and Chaldeans--to be found in Pisa. It was in the midst of such wars, in a cosmopolitan and commercial town, in a center where literary work was not appreciated,[521] that the genius of Leonardo appears as one of the surprises of history, warning us again that "we should draw no horoscope; that we should expect little, for what we expect will not come to pass."[522] Leonardo's father was one William,[523] and he had a brother named Bonaccingus,[524] but nothing further is {130} known of his family. As to Fibonacci, most writers[525] have assumed that his father's name was Bonaccio,[526] whence _filius Bonaccii_, or Fibonacci. Others[527] believe that the name, even in the Latin form of _filius Bonaccii_ as used in Leonardo's work, was simply a general one, like our Johnson or Bronson (Brown's son); and the only contemporary evidence that we have bears out this view. As to the name Bigollo, used by Leonardo, some have thought it a self-assumed one meaning blockhead, a term that had been applied to him by the commercial world or possibly by the university circle, and taken by him that he might prove what a blockhead could do. Milanesi,[528] however, has shown that the word Bigollo (or Pigollo) was used in Tuscany to mean a traveler, and was naturally assumed by one who had studied, as Leonardo had, in foreign lands. Leonardo's father was a commercial agent at Bugia, the modern Bougie,[529] the ancient Saldae on the coast of Barbary,[530] a royal capital under the Vandals and again, a century before Leonardo, under the Beni Hammad. It had one of the best harbors on the coast, sheltered as it is by Mt. Lalla Guraia,[531] and at the close of the twelfth century it was a center of African commerce. It was here that Leonardo was taken as a child, and here he went to school to a Moorish master. When he reached the years of young manhood he started on a tour of the Mediterranean Sea, and visited Egypt, Syria, Greece, Sicily, and Provence, meeting with scholars as well as with {131} merchants, and imbibing a knowledge of the various systems of numbers in use in the centers of trade. All these systems, however, he says he counted almost as errors compared with that of the Hindus.[532] Returning to Pisa, he wrote his _Liber Abaci_[533] in 1202, rewriting it in 1228.[534] In this work the numerals are explained and are used in the usual computations of business. Such a treatise was not destined to be popular, however, because it was too advanced for the mercantile class, and too novel for the conservative university circles. Indeed, at this time mathematics had only slight place in the newly established universities, as witness the oldest known statute of the Sorbonne at Paris, dated 1215, where the subject is referred to only in an incidental way.[535] The period was one of great commercial activity, and on this very {132} account such a book would attract even less attention than usual.[536] It would now be thought that the western world would at once adopt the new numerals which Leonardo had made known, and which were so much superior to anything that had been in use in Christian Europe. The antagonism of the universities would avail but little, it would seem, against such an improvement. It must be remembered, however, that there was great difficulty in spreading knowledge at this time, some two hundred and fifty years before printing was invented. "Popes and princes and even great religious institutions possessed far fewer books than many farmers of the present age. The library belonging to the Cathedral Church of San Martino at Lucca in the ninth century contained only nineteen volumes of abridgments from ecclesiastical commentaries."[537] Indeed, it was not until the early part of the fifteenth century that Palla degli Strozzi took steps to carry out the project that had been in the mind of Petrarch, the founding of a public library. It was largely by word of mouth, therefore, that this early knowledge had to be transmitted. Fortunately the presence of foreign students in Italy at this time made this transmission feasible. (If human nature was the same then as now, it is not impossible that the very opposition of the faculties to the works of Leonardo led the students to investigate {133} them the more zealously.) At Vicenza in 1209, for example, there were Bohemians, Poles, Frenchmen, Burgundians, Germans, and Spaniards, not to speak of representatives of divers towns of Italy; and what was true there was also true of other intellectual centers. The knowledge could not fail to spread, therefore, and as a matter of fact we find numerous bits of evidence that this was the case. Although the bankers of Florence were forbidden to use these numerals in 1299, and the statutes of the university of Padua required stationers to keep the price lists of books "non per cifras, sed per literas claros,"[538] the numerals really made much headway from about 1275 on. It was, however, rather exceptional for the common people of Germany to use the Arabic numerals before the sixteenth century, a good witness to this fact being the popular almanacs. Calendars of 1457-1496[539] have generally the Roman numerals, while Köbel's calendar of 1518 gives the Arabic forms as subordinate to the Roman. In the register of the Kreuzschule at Dresden the Roman forms were used even until 1539. While not minimizing the importance of the scientific work of Leonardo of Pisa, we may note that the more popular treatises by Alexander de Villa Dei (c. 1240 A.D.) and John of Halifax (Sacrobosco, c. 1250 A.D.) were much more widely used, and doubtless contributed more to the spread of the numerals among the common people. {134} The _Carmen de Algorismo_[540] of Alexander de Villa Dei was written in verse, as indeed were many other textbooks of that time. That it was widely used is evidenced by the large number of manuscripts[541] extant in European libraries. Sacrobosco's _Algorismus_,[542] in which some lines from the Carmen are quoted, enjoyed a wide popularity as a textbook for university instruction.[543] The work was evidently written with this end in view, as numerous commentaries by university lecturers are found. Probably the most widely used of these was that of Petrus de Dacia[544] written in 1291. These works throw an interesting light upon the method of instruction in mathematics in use in the universities from the thirteenth even to the sixteenth century. Evidently the text was first read and copied by students.[545] Following this came line by line an exposition of the text, such as is given in Petrus de Dacia's commentary. Sacrobosco's work is of interest also because it was probably due to the extended use of this work that the {135} term _Arabic numerals_ became common. In two places there is mention of the inventors of this system. In the introduction it is stated that this science of reckoning was due to a philosopher named Algus, whence the name _algorismus_,[546] and in the section on numeration reference is made to the Arabs as the inventors of this science.[547] While some of the commentators, Petrus de Dacia[548] among them, knew of the Hindu origin, most of them undoubtedly took the text as it stood; and so the Arabs were credited with the invention of the system. The first definite trace that we have of an algorism in the French language is found in a manuscript written about 1275.[549] This interesting leaf, for the part on algorism consists of a single folio, was noticed by the Abbé Leboeuf as early as 1741,[550] and by Daunou in 1824.[551] It then seems to have been lost in the multitude of Paris manuscripts; for although Chasles[552] relates his vain search for it, it was not rediscovered until 1882. In that year M. Ch. Henry found it, and to his care we owe our knowledge of the interesting manuscript. The work is anonymous and is devoted almost entirely to geometry, only {136} two pages (one folio) relating to arithmetic. In these the forms of the numerals are given, and a very brief statement as to the operations, it being evident that the writer himself had only the slightest understanding of the subject. Once the new system was known in France, even thus superficially, it would be passed across the Channel to England. Higden,[553] writing soon after the opening of the fourteenth century, speaks of the French influence at that time and for some generations preceding:[554] "For two hundred years children in scole, agenst the usage and manir of all other nations beeth compelled for to leave hire own language, and for to construe hir lessons and hire thynges in Frensche.... Gentilmen children beeth taught to speke Frensche from the tyme that they bith rokked in hir cradell; and uplondissche men will likne himself to gentylmen, and fondeth with greet besynesse for to speke Frensche." The question is often asked, why did not these new numerals attract more immediate attention? Why did they have to wait until the sixteenth century to be generally used in business and in the schools? In reply it may be said that in their elementary work the schools always wait upon the demands of trade. That work which pretends to touch the life of the people must come reasonably near doing so. Now the computations of business until about 1500 did not demand the new figures, for two reasons: First, cheap paper was not known. Paper-making of any kind was not introduced into Europe until {137} the twelfth century, and cheap paper is a product of the nineteenth. Pencils, too, of the modern type, date only from the sixteenth century. In the second place, modern methods of operating, particularly of multiplying and dividing (operations of relatively greater importance when all measures were in compound numbers requiring reductions at every step), were not yet invented. The old plan required the erasing of figures after they had served their purpose, an operation very simple with counters, since they could be removed. The new plan did not as easily permit this. Hence we find the new numerals very tardily admitted to the counting-house, and not welcomed with any enthusiasm by teachers.[555] Aside from their use in the early treatises on the new art of reckoning, the numerals appeared from time to time in the dating of manuscripts and upon monuments. The oldest definitely dated European document known {138} to contain the numerals is a Latin manuscript,[556] the Codex Vigilanus, written in the Albelda Cloister not far from Logroño in Spain, in 976 A.D. The nine characters (of [.g]ob[=a]r type), without the zero, are given as an addition to the first chapters of the third book of the _Origines_ by Isidorus of Seville, in which the Roman numerals are under discussion. Another Spanish copy of the same work, of 992 A.D., contains the numerals in the corresponding section. The writer ascribes an Indian origin to them in the following words: "Item de figuris arithmetic[e,]. Scire debemus in Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et geometria et ceteris liberalibus disciplinis concedere. Et hoc manifestum est in nobem figuris, quibus designant unumquemque gradum cuiuslibet gradus. Quarum hec sunt forma." The nine [.g]ob[=a]r characters follow. Some of the abacus forms[557] previously given are doubtless also of the tenth century. The earliest Arabic documents containing the numerals are two manuscripts of 874 and 888 A.D.[558] They appear about a century later in a work[559] written at Shiraz in 970 A.D. There is also an early trace of their use on a pillar recently discovered in a church apparently destroyed as early as the tenth century, not far from the Jeremias Monastery, in Egypt. {139} A graffito in Arabic on this pillar has the date 349 A.H., which corresponds to 961 A.D.[560] For the dating of Latin documents the Arabic forms were used as early as the thirteenth century.[561] On the early use of these numerals in Europe the only scientific study worthy the name is that made by Mr. G. F. Hill of the British Museum.[562] From his investigations it appears that the earliest occurrence of a date in these numerals on a coin is found in the reign of Roger of Sicily in 1138.[563] Until recently it was thought that the earliest such date was 1217 A.D. for an Arabic piece and 1388 for a Turkish one.[564] Most of the seals and medals containing dates that were at one time thought to be very early have been shown by Mr. Hill to be of relatively late workmanship. There are, however, in European manuscripts, numerous instances of the use of these numerals before the twelfth century. Besides the example in the Codex Vigilanus, another of the tenth century has been found in the St. Gall MS. now in the University Library at Zürich, the forms differing materially from those in the Spanish codex. The third specimen in point of time in Mr. Hill's list is from a Vatican MS. of 1077. The fourth and fifth specimens are from the Erlangen MS. of Boethius, of the same {140} (eleventh) century, and the sixth and seventh are also from an eleventh-century MS. of Boethius at Chartres. These and other early forms are given by Mr. Hill in this table, which is reproduced with his kind permission. EARLIEST MANUSCRIPT FORMS [Illustration] This is one of more than fifty tables given in Mr. Hill's valuable paper, and to this monograph students {141} are referred for details as to the development of number-forms in Europe from the tenth to the sixteenth century. It is of interest to add that he has found that among the earliest dates of European coins or medals in these numerals, after the Sicilian one already mentioned, are the following: Austria, 1484; Germany, 1489 (Cologne); Switzerland, 1424 (St. Gall); Netherlands, 1474; France, 1485; Italy, 1390.[565] The earliest English coin dated in these numerals was struck in 1551,[566] although there is a Scotch piece of 1539.[567] In numbering pages of a printed book these numerals were first used in a work of Petrarch's published at Cologne in 1471.[568] The date is given in the following form in the _Biblia Pauperum_,[569] a block-book of 1470, [Illustration] while in another block-book which possibly goes back to c. 1430[570] the numerals appear in several illustrations, with forms as follows: [Illustration] Many printed works anterior to 1471 have pages or chapters numbered by hand, but many of these numerals are {142} of date much later than the printing of the work. Other works were probably numbered directly after printing. Thus the chapters 2, 3, 4, 5, 6 in a book of 1470[571] are numbered as follows: Capitulem [Symbol 2]m.,... [Symbol 3]m.,... 4m.,... v,... vi, and followed by Roman numerals. This appears in the body of the text, in spaces left by the printer to be filled in by hand. Another book[572] of 1470 has pages numbered by hand with a mixture of Roman and Hindu numerals, thus, [Illustration] for 125 [Illustration] for 150 [Illustration] for 147 [Illustration] for 202 As to monumental inscriptions,[573] there was once thought to be a gravestone at Katharein, near Troppau, with the date 1007, and one at Biebrich of 1299. There is no doubt, however, of one at Pforzheim of 1371 and one at Ulm of 1388.[574] Certain numerals on Wells Cathedral have been assigned to the thirteenth century, but they are undoubtedly considerably later.[575] The table on page 143 will serve to supplement that from Mr. Hill's work.[576] {143} EARLY MANUSCRIPT FORMS [577] [Illustration] Twelfth century A.D. [578] [Illustration] 1197 A.D. [579] [Illustration] 1275 A.D. [580] [Illustration] c. 1294 A.D. [581] [Illustration] c. 1303 A.D. [582] [Illustration] c. 1360 A.D. [583] [Illustration] c. 1442 A.D. {144} [Illustration] For the sake of further comparison, three illustrations from works in Mr. Plimpton's library, reproduced from the _Rara Arithmetica_, may be considered. The first is from a Latin manuscript on arithmetic,[584] of which the original was written at Paris in 1424 by Rollandus, a Portuguese physician, who prepared the work at the command of John of Lancaster, Duke of Bedford, at one time Protector of England and Regent of France, to whom the work is dedicated. The figures show the successive powers of 2. The second illustration is from Luca da Firenze's _Inprencipio darte dabacho_,[585] c. 1475, and the third is from an anonymous manuscript[586] of about 1500. [Illustration] As to the forms of the numerals, fashion played a leading part until printing was invented. This tended to fix these forms, although in writing there is still a great variation, as witness the French 5 and the German 7 and 9. Even in printing there is not complete uniformity, {145} and it is often difficult for a foreigner to distinguish between the 3 and 5 of the French types. [Illustration] As to the particular numerals, the following are some of the forms to be found in the later manuscripts and in the early printed books. 1. In the early printed books "one" was often i, perhaps to save types, just as some modern typewriters use the same character for l and 1.[587] In the manuscripts the "one" appears in such forms as[588] [Illustration] 2. "Two" often appears as z in the early printed books, 12 appearing as iz.[589] In the medieval manuscripts the following forms are common:[590] [Illustration] {146} It is evident, from the early traces, that it is merely a cursive form for the primitive [2 horizontal strokes], just as 3 comes from [3 horizontal strokes], as in the N[=a]n[=a] Gh[=a]t inscriptions. 3. "Three" usually had a special type in the first printed books, although occasionally it appears as [Symbol].[591] In the medieval manuscripts it varied rather less than most of the others. The following are common forms:[592] [Illustration] 4. "Four" has changed greatly; and one of the first tests as to the age of a manuscript on arithmetic, and the place where it was written, is the examination of this numeral. Until the time of printing the most common form was [Symbol], although the Florentine manuscript of Leonard of Pisa's work has the form [Symbol];[593] but the manuscripts show that the Florentine arithmeticians and astronomers rather early began to straighten the first of these forms up to forms like [Symbol][594] and [Symbol][594] or [Symbol],[595] more closely resembling our own. The first printed books generally used our present form[596] with the closed top [Symbol], the open top used in writing ( [Symbol]) being {147} purely modern. The following are other forms of the four, from various manuscripts:[597] [Illustration] 5. "Five" also varied greatly before the time of printing. The following are some of the forms:[598] [Illustration] 6. "Six" has changed rather less than most of the others. The chief variation has been in the slope of the top, as will be seen in the following:[599] [Illustration] 7. "Seven," like "four," has assumed its present erect form only since the fifteenth century. In medieval times it appeared as follows:[600] [Illustration] {148} 8. "Eight," like "six," has changed but little. In medieval times there are a few variants of interest as follows:[601] [Illustration] In the sixteenth century, however, there was manifested a tendency to write it [Symbol].[602] 9. "Nine" has not varied as much as most of the others. Among the medieval forms are the following:[603] [Illustration] 0. The shape of the zero also had a varied history. The following are common medieval forms:[604] [Illustration] The explanation of the place value was a serious matter to most of the early writers. If they had been using an abacus constructed like the Russian chotü, and had placed this before all learners of the positional system, there would have been little trouble. But the medieval {149} line-reckoning, where the lines stood for powers of 10 and the spaces for half of such powers, did not lend itself to this comparison. Accordingly we find such labored explanations as the following, from _The Crafte of Nombrynge_: "Euery of these figuris bitokens hym selfe & no more, yf he stonde in the first place of the rewele.... "If it stonde in the secunde place of the rewle, he betokens ten tymes hym selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is twenty, for he hym selfe betokens tweyne, & ten tymes twene is twenty. And for he stondis on the lyft side & in the secunde place, he betokens ten tyme hym selfe. And so go forth.... "Nil cifra significat sed dat signare sequenti. Expone this verse. A cifre tokens no[gh]t, bot he makes the figure to betoken that comes after hym more than he shuld & he were away, as thus 10. here the figure of one tokens ten, & yf the cifre were away & no figure byfore hym he schuld token bot one, for than he schuld stonde in the first place...."[605] It would seem that a system that was thus used for dating documents, coins, and monuments, would have been generally adopted much earlier than it was, particularly in those countries north of Italy where it did not come into general use until the sixteenth century. This, however, has been the fate of many inventions, as witness our neglect of logarithms and of contracted processes to-day. As to Germany, the fifteenth century saw the rise of the new symbolism; the sixteenth century saw it slowly {150} gain the mastery; the seventeenth century saw it finally conquer the system that for two thousand years had dominated the arithmetic of business. Not a little of the success of the new plan was due to Luther's demand that all learning should go into the vernacular.[606] During the transition period from the Roman to the Arabic numerals, various anomalous forms found place. For example, we have in the fourteenth century c[alpha] for 104;[607] 1000. 300. 80 et 4 for 1384;[608] and in a manuscript of the fifteenth century 12901 for 1291.[609] In the same century m. cccc. 8II appears for 1482,[610] while M^oCCCC^o50 (1450) and MCCCCXL6 (1446) are used by Theodoricus Ruffi about the same time.[611] To the next century belongs the form 1vojj for 1502. Even in Sfortunati's _Nuovo lume_[612] the use of ordinals is quite confused, the propositions on a single page being numbered "tertia," "4," and "V." Although not connected with the Arabic numerals in any direct way, the medieval astrological numerals may here be mentioned. These are given by several early writers, but notably by Noviomagus (1539),[613] as follows[614]: [Illustration] {151} Thus we find the numerals gradually replacing the Roman forms all over Europe, from the time of Leonardo of Pisa until the seventeenth century. But in the Far East to-day they are quite unknown in many countries, and they still have their way to make. In many parts of India, among the common people of Japan and China, in Siam and generally about the Malay Peninsula, in Tibet, and among the East India islands, the natives still adhere to their own numeral forms. Only as Western civilization is making its way into the commercial life of the East do the numerals as used by us find place, save as the Sanskrit forms appear in parts of India. It is therefore with surprise that the student of mathematics comes to realize how modern are these forms so common in the West, how limited is their use even at the present time, and how slow the world has been and is in adopting such a simple device as the Hindu-Arabic numerals. * * * * * {153} INDEX _Transcriber's note: many of the entries refer to footnotes linked from the page numbers given._ Abbo of Fleury, 122 `Abdall[=a]h ibn al-[H.]asan, 92 `Abdallat[=i]f ibn Y[=u]suf, 93 `Abdalq[=a]dir ibn `Al[=i] al-Sakh[=a]w[=i], 6 Abenragel, 34 Abraham ibn Meïr ibn Ezra, _see_ Rabbi ben Ezra Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a], 74 Ab[=u] 'l-[H.]asan, 93, 100 Ab[=u] 'l-Q[=a]sim, 92 Ab[=u] 'l-[T.]eiyib, 97 Ab[=u] Na[s.]r, 92 Ab[=u] Roshd, 113 Abu Sahl Dunash ibn Tamim, 65, 67 Adelhard of Bath, 5, 55, 97, 119, 123, 126 Adhemar of Chabanois, 111 A[h.]med al-Nasaw[=i], 98 A[h.]med ibn `Abdall[=a]h, 9, 92 A[h.]med ibn Mo[h.]ammed, 94 A[h.]med ibn `Omar, 93 Ak[s.]aras, 32 Alanus ab Insulis, 124 Al-Ba[.g]d[=a]d[=i], 93 Al-Batt[=a]n[=i], 54 Albelda (Albaida) MS., 116 Albert, J., 62 Albert of York, 103 Al-B[=i]r[=u]n[=i], 6, 41, 49, 65, 92, 93 Alcuin, 103 Alexander the Great, 76 Alexander de Villa Dei, 11, 133 Alexandria, 64, 82 Al-Faz[=a]r[=i], 92 Alfred, 103 Algebra, etymology, 5 Algerian numerals, 68 Algorism, 97 Algorismus, 124, 126, 135 Algorismus cifra, 120 Al-[H.]a[s.][s.][=a]r, 65 `Al[=i] ibn Ab[=i] Bekr, 6 `Al[=i] ibn A[h.]med, 93, 98 Al-Kar[=a]b[=i]s[=i], 93 Al-Khow[=a]razm[=i], 4, 9, 10, 92, 97, 98, 125, 126 Al-Kind[=i], 10, 92 Almagest, 54 Al-Ma[.g]reb[=i], 93 Al-Ma[h.]all[=i], 6 Al-M[=a]m[=u]n, 10, 97 Al-Man[s.][=u]r, 96, 97 Al-Mas`[=u]d[=i], 7, 92 Al-Nad[=i]m, 9 Al-Nasaw[=i], 93, 98 Alphabetic numerals, 39, 40, 43 Al-Q[=a]sim, 92 Al-Qass, 94 Al-Sakh[=a]w[=i], 6 Al-[S.]ardaf[=i], 93 Al-Sijz[=i], 94 Al-S[=u]f[=i], 10, 92 Ambrosoli, 118 A[.n]kapalli, 43 Apices, 87, 117, 118 Arabs, 91-98 Arbuthnot, 141 {154} Archimedes, 15, 16 Arcus Pictagore, 122 Arjuna, 15 Arnold, E., 15, 102 Ars memorandi, 141 [=A]ryabha[t.]a, 39, 43, 44 Aryan numerals, 19 Aschbach, 134 Ashmole, 134 A['s]oka, 19, 20, 22, 81 A[s.]-[s.]ifr, 57, 58 Astrological numerals, 150 Atharva-Veda, 48, 49, 55 Augustus, 80 Averroës, 113 Avicenna, 58, 74, 113 Babylonian numerals, 28 Babylonian zero, 51 Bacon, R., 131 Bactrian numerals, 19, 30 Bæda, 2, 72 Bagdad, 4, 96 Bakh[s.][=a]l[=i] manuscript, 43, 49, 52, 53 Ball, C. J., 35 Ball, W. W. R., 36, 131 B[=a][n.]a, 44 Barth, A., 39 Bayang inscriptions, 39 Bayer, 33 Bayley, E. C., 19, 23, 30, 32, 52, 89 Beazley, 75 Bede, _see_ Bæda Beldomandi, 137 Beloch, J., 77 Bendall, 25, 52 Benfey, T., 26 Bernelinus, 88, 112, 117, 121 Besagne, 128 Besant, W., 109 Bettino, 36 Bhandarkar, 18, 47, 49 Bh[=a]skara, 53, 55 Biernatzki, 32 Biot, 32 Björnbo, A. A., 125, 126 Blassière, 119 Bloomfield, 48 Blume, 85 Boeckh, 62 Boehmer, 143 Boeschenstein, 119 Boethius, 63, 70-73, 83-90 Boissière, 63 Bombelli, 81 Bonaini, 128 Boncompagni, 5, 6, 10, 48, 49, 123, 125 Borghi, 59 Borgo, 119 Bougie, 130 Bowring, J., 56 Brahmagupta, 52 Br[=a]hma[n.]as, 12, 13 Br[=a]hm[=i], 19, 20, 31, 83 Brandis, J., 54 B[r.]hat-Sa[m.]hita, 39, 44, 78 Brockhaus, 43 Bubnov, 65, 84, 110, 116 Buddha, education of, 15, 16 Büdinger, 110 Bugia, 130 Bühler, G., 15, 19, 22, 31, 44, 49 Burgess, 25 Bürk, 13 Burmese numerals, 36 Burnell, A. C., 18, 40 Buteo, 61 Calandri, 59, 81 Caldwell, R., 19 Calendars, 133 Calmet, 34 Cantor, M., 5, 13, 30, 43, 84 {155} Capella, 86 Cappelli, 143 Caracteres, 87, 113, 117, 119 Cardan, 119 Carmen de Algorismo, 11, 134 Casagrandi, 132 Casiri, 8, 10 Cassiodorus, 72 Cataldi, 62 Cataneo, 3 Caxton, 143, 146 Ceretti, 32 Ceylon numerals, 36 Chalfont, F. H., 28 Champenois, 60 Characters, _see_ Caracteres Charlemagne, 103 Chasles, 54, 60, 85, 116, 122, 135 Chassant, L. A., 142 Chaucer, 121 Chiarini, 145, 146 Chiffre, 58 Chinese numerals, 28, 56 Chinese zero, 56 Cifra, 120, 124 Cipher, 58 Circulus, 58, 60 Clichtoveus, 61, 119, 145 Codex Vigilanus, 138 Codrington, O., 139 Coins dated, 141 Colebrooke, 8, 26, 46, 53 Constantine, 104, 105 Cosmas, 82 Cossali, 5 Counters, 117 Courteille, 8 Coxe, 59 Crafte of Nombrynge, 11, 87, 149 Crusades, 109 Cunningham, A., 30, 75 Curtze, 55, 59, 126, 134 Cyfra, 55 Dagomari, 146 D'Alviella, 15 Dante, 72 Dasypodius, 33, 67, 63 Daunou, 135 Delambre, 54 Devan[=a]gar[=i], 7 Devoulx, A., 68 Dhruva, 49 Dicæarchus of Messana, 77 Digits, 119 Diodorus Siculus, 76 Du Cange, 62 Dumesnil, 36 Dutt, R. C., 12, 15, 18, 75 Dvived[=i], 44 East and West, relations, 73-81, 100-109 Egyptian numerals, 27 Eisenlohr, 28 Elia Misrachi, 57 Enchiridion Algorismi, 58 Eneström, 5, 48, 59, 97, 125, 128 Europe, numerals in, 63, 99, 128, 136 Eusebius Caesariensis, 142 Euting, 21 Ewald, P., 116 Fazzari, 53, 54 Fibonacci, _see_ Leonardo of Pisa Figura nihili, 58 Figures, 119. _See_ numerals. Fihrist, 67, 68, 93 Finaeus, 57 Firdus[=i], 81 Fitz Stephen, W., 109 Fleet, J. C., 19, 20, 49 {156} Florus, 80 Flügel, G., 68 Francisco de Retza, 142 François, 58 Friedlein, G., 84, 113, 116, 122 Froude, J. A., 129 Gandh[=a]ra, 19 Garbe, 48 Gasbarri, 58 Gautier de Coincy, 120, 124 Gemma Frisius, 2, 3, 119 Gerber, 113 Gerbert, 108, 110-120, 122 Gerhardt, C. I., 43, 56, 93, 118 Gerland, 88, 123 Gherard of Cremona, 125 Gibbon, 72 Giles, H. A., 79 Ginanni, 81 Giovanni di Danti, 58 Glareanus, 4, 119 Gnecchi, 71, 117 [.G]ob[=a]r numerals, 65, 100, 112, 124, 138 Gow, J., 81 Grammateus, 61 Greek origin, 33 Green, J. R., 109 Greenwood, I., 62, 119 Guglielmini, 128 Gulist[=a]n, 102 Günther, S., 131 Guyard, S., 82 [H.]abash, 9, 92 Hager, J. (G.), 28, 32 Halliwell, 59, 85 Hankel, 93 H[=a]r[=u]n al-Rash[=i]d, 97, 106 Havet, 110 Heath, T. L., 125 Hebrew numerals, 127 Hecatæus, 75 Heiberg, J. L., 55, 85, 148 Heilbronner, 5 Henry, C., 5, 31, 55, 87, 120, 135 Heriger, 122 Hermannus Contractus, 123 Herodotus, 76, 78 Heyd, 75 Higden, 136 Hill, G. F., 52, 139, 142 Hillebrandt, A., 15, 74 Hilprecht, H. V., 28 Hindu forms, early, 12 Hindu number names, 42 Hodder, 62 Hoernle, 43, 49 Holywood, _see_ Sacrobosco Hopkins, E. W., 12 Horace, 79, 80 [H.]osein ibn Mo[h.]ammed al-Ma[h.]all[=i], 6 Hostus, M., 56 Howard, H. H., 29 Hrabanus Maurus, 72 Huart, 7 Huet, 33 Hugo, H., 57 Humboldt, A. von, 62 Huswirt, 58 Iamblichus, 81 Ibn Ab[=i] Ya`q[=u]b, 9 Ibn al-Adam[=i], 92 Ibn al-Bann[=a], 93 Ibn Khord[=a][d.]beh, 101, 106 Ibn Wahab, 103 India, history of, 14 writing in, 18 Indicopleustes, 83 Indo-Bactrian numerals, 19 {157} Indr[=a]j[=i], 23 Is[h.][=a]q ibn Y[=u]suf al-[S.]ardaf[=i], 93 Jacob of Florence, 57 Jacquet, E., 38 Jamshid, 56 Jehan Certain, 59 Jetons, 58, 117 Jevons, F. B., 76 Johannes Hispalensis, 48, 88, 124 John of Halifax, _see_ Sacrobosco John of Luna, _see_ Johannes Hispalensis Jordan, L., 58, 124 Joseph Ispanus (Joseph Sapiens), 115 Justinian, 104 Kále, M. R., 26 Karabacek, 56 Karpinski, L. C., 126, 134, 138 K[=a]ty[=a]yana, 39 Kaye, C. R., 6, 16, 43, 46, 121 Keane, J., 75, 82 Keene, H. G., 15 Kern, 44 Kharo[s.][t.]h[=i], 19, 20 Khosr[=u], 82, 91 Kielhorn, F., 46, 47 Kircher, A., 34 Kit[=a]b al-Fihrist, _see_ Fihrist Kleinwächter, 32 K[=l]os, 62 Köbel, 4, 58, 60, 119, 123 Krumbacher, K., 57 Kuckuck, 62, 133 Kugler, F. X., 51 Lachmann, 85 Lacouperie, 33, 35 Lalitavistara, 15, 17 Lami, G., 57 La Roche, 61 Lassen, 39 L[=a][t.]y[=a]yana, 39 Leboeuf, 135 Leonardo of Pisa, 5, 10, 57, 64, 74, 120, 128-133 Lethaby, W. R., 142 Levi, B., 13 Levias, 3 Libri, 73, 85, 95 Light of Asia, 16 Luca da Firenze, 144 Lucas, 128 Mah[=a]bh[=a]rata, 18 Mah[=a]v[=i]r[=a]c[=a]rya, 53 Malabar numerals, 36 Malayalam numerals, 36 Mannert, 81 Margarita Philosophica, 146 Marie, 78 Marquardt, J., 85 Marshman, J. C., 17 Martin, T. H., 30, 62, 85, 113 Martines, D. C., 58 M[=a]sh[=a]ll[=a]h, 3 Maspero, 28 Mauch, 142 Maximus Planudes, 2, 57, 66, 93, 120 Megasthenes, 77 Merchants, 114 Meynard, 8 Migne, 87 Mikami, Y., 56 Milanesi, 128 Mo[h.]ammed ibn `Abdall[=a]h, 92 Mo[h.]ammed ibn A[h.]med, 6 Mo[h.]ammed ibn `Al[=i] `Abd[=i], 8 Mo[h.]ammed ibn M[=u]s[=a], _see_ Al-Khow[=a]razm[=i] Molinier, 123 Monier-Williams, 17 {158} Morley, D., 126 Moroccan numerals, 68, 119 Mortet, V., 11 Moseley, C. B., 33 Mo[t.]ahhar ibn [T.][=a]hir, 7 Mueller, A., 68 Mumford, J. K., 109 Muwaffaq al-D[=i]n, 93 Nabatean forms, 21 Nallino, 4, 54, 55 Nagl, A., 55, 110, 113, 126 N[=a]n[=a] Gh[=a]t inscriptions, 20, 22, 23, 40 Narducci, 123 Nasik cave inscriptions, 24 Na[z.][=i]f ibn Yumn, 94 Neander, A., 75 Neophytos, 57, 62 Neo-Pythagoreans, 64 Nesselmann, 58 Newman, Cardinal, 96 Newman, F. W., 131 Nöldeke, Th., 91 Notation, 61 Note, 61, 119 Noviomagus, 45, 61, 119, 150 Null, 61 Numerals, Algerian, 68 astrological, 150 Br[=a]hm[=i], 19-22, 83 early ideas of origin, 1 Hindu, 26 Hindu, classified, 19, 38 Kharo[s.][t.]h[=i], 19-22 Moroccan, 68 Nabatean, 21 origin, 27, 30, 31, 37 supposed Arabic origin, 2 supposed Babylonian origin, 28 supposed Chaldean and Jewish origin, 3 supposed Chinese origin, 28, 32 supposed Egyptian origin, 27, 30, 69, 70 supposed Greek origin, 33 supposed Phoenician origin, 32 tables of, 22-27, 36, 48, 49, 69, 88, 140, 143, 145-148 O'Creat, 5, 55, 119, 120 Olleris, 110, 113 Oppert, G., 14, 75 Pali, 22 Pañcasiddh[=a]ntik[=a], 44 Paravey, 32, 57 P[=a]tal[=i]pu[t.]ra, 77 Patna, 77 Patrick, R., 119 Payne, E. J., 106 Pegolotti, 107 Peletier, 2, 62 Perrot, 80 Persia, 66, 91, 107 Pertz, 115 Petrus de Dacia, 59, 61, 62 Pez, P. B., 117 "Philalethes," 75 Phillips, G., 107 Picavet, 105 Pichler, F., 141 Pihan, A. P., 36 Pisa, 128 Place value, 26, 42, 46, 48 Planudes, _see_ Maximus Planudes Plimpton, G. A., 56, 59, 85, 143, 144, 145, 148 Pliny, 76 Polo, N. and M., 107 {159} Prändel, J. G., 54 Prinsep, J., 20, 31 Propertius, 80 Prosdocimo de' Beldomandi, 137 Prou, 143 Ptolemy, 54, 78 Putnam, 103 Pythagoras, 63 Pythagorean numbers, 13 Pytheas of Massilia, 76 Rabbi ben Ezra, 60, 127 Radulph of Laon, 60, 113, 118, 124 Raets, 62 Rainer, _see_ Gemma Frisius R[=a]m[=a]yana, 18 Ramus, 2, 41, 60, 61 Raoul Glaber, 123 Rapson, 77 Rauhfuss, _see_ Dasypodius Raumer, K. von, 111 Reclus, E., 14, 96, 130 Recorde, 3, 58 Reinaud, 67, 74, 80 Reveillaud, 36 Richer, 110, 112, 115 Riese, A., 119 Robertson, 81 Robertus Cestrensis, 97, 126 Rodet, 5, 44 Roediger, J., 68 Rollandus, 144 Romagnosi, 81 Rosen, F., 5 Rotula, 60 Rudolff, 85 Rudolph, 62, 67 Ruffi, 150 Sachau, 6 Sacrobosco, 3, 58, 133 Sacy, S. de, 66, 70 Sa`d[=i], 102 ['S]aka inscriptions, 20 Sam[=u]'[=i]l ibn Ya[h.]y[=a], 93 ['S][=a]rad[=a] characters, 55 Savonne, 60 Scaliger, J. C., 73 Scheubel, 62 Schlegel, 12 Schmidt, 133 Schonerus, 87, 119 Schroeder, L. von, 13 Scylax, 75 Sedillot, 8, 34 Senart, 20, 24, 25 Sened ibn `Al[=i], 10, 98 Sfortunati, 62, 150 Shelley, W., 126 Siamese numerals, 36 Siddh[=a]nta, 8, 18 [S.]ifr, 57 Sigsboto, 55 Sih[=a]b al-D[=i]n, 67 Silberberg, 60 Simon, 13 Sin[=a]n ibn al-Fat[h.], 93 Sindbad, 100 Sindhind, 97 Sipos, 60 Sirr, H. C., 75 Skeel, C. A., 74 Smith, D. E., 11, 17, 53, 86, 141, 143 Smith, V. A., 20, 35, 46, 47 Smith, Wm., 75 Sm[r.]ti, 17 Spain, 64, 65, 100 Spitta-Bey, 5 Sprenger, 94 ['S]rautas[=u]tra, 39 Steffens, F., 116 Steinschneider, 5, 57, 65, 66, 98, 126 Stifel, 62 {160} Subandhus, 44 Suetonius, 80 Suleim[=a]n, 100 ['S][=u]nya, 43, 53, 57 Suter, 5, 9, 68, 69, 93, 116, 131 S[=u]tras, 13 Sykes, P. M., 75 Sylvester II, _see_ Gerbert Symonds, J. A., 129 Tannery, P., 62, 84, 85 Tartaglia, 4, 61 Taylor, I., 19, 30 Teca, 55, 61 Tennent, J. E., 75 Texada, 60 Theca, 58, 61 Theophanes, 64 Thibaut, G., 12, 13, 16, 44, 47 Tibetan numerals, 36 Timotheus, 103 Tonstall, C., 3, 61 Trenchant, 60 Treutlein, 5, 63, 123 Trevisa, 136 Treviso arithmetic, 145 Trivium and quadrivium, 73 Tsin, 56 Tunis, 65 Turchill, 88, 118, 123 Turnour, G., 75 Tziphra, 57, 62 [Greek: tziphra], 55, 57, 62 Tzwivel, 61, 118, 145 Ujjain, 32 Unger, 133 Upanishads, 12 Usk, 121 Valla, G., 61 Van der Schuere, 62 Var[=a]ha-Mihira, 39, 44, 78 V[=a]savadatt[=a], 44 Vaux, Carra de, 9, 74 Vaux, W. S. W., 91 Ved[=a][.n]gas, 17 Vedas, 12, 15, 17 Vergil, 80 Vincent, A. J. H., 57 Vogt, 13 Voizot, P., 36 Vossius, 4, 76, 81, 84 Wallis, 3, 62, 84, 116 Wappler, E., 54, 126 Wäschke, H., 2, 93 Wattenbach, 143 Weber, A., 31 Weidler, I. F., 34, 66 Weidler, I. F. and G. I., 63, 66 Weissenborn, 85, 110 Wertheim, G., 57, 61 Whitney, W. D., 13 Wilford, F., 75 Wilkens, 62 Wilkinson, J. G., 70 Willichius, 3 Woepcke, 3, 6, 42, 63, 64, 65, 67, 69, 70, 94, 113, 138 Wolack, G., 54 Woodruff, C. E., 32 Word and letter numerals, 38, 44 Wüstenfeld, 74 Yule, H., 107 Zephirum, 57, 58 Zephyr, 59 Zepiro, 58 Zero, 26, 38, 40, 43, 45, 49, 51-62, 67 Zeuero, 58 * * * * * ANNOUNCEMENTS * * * * * WENTWORTH'S COLLEGE ALGEBRA REVISED EDITION 12mo. Half morocco. 530 pages. List price, $1.50; mailing price, $1.65 * * * * * This book is a thorough revision of the author's "College Algebra." Some chapters of the old edition have been wholly rewritten, and the other chapters have been rewritten in part and greatly improved. The order of topics has been changed to a certain extent; the plan is to have each chapter as complete in itself as possible, so that the teacher may vary the order of succession at his discretion. As the name implies, the work is intended for colleges and scientific schools. 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So in a commentary by [H.]osein ibn Mo[h.]ammed al-Ma[h.]all[=i] (died in 1756) on the _Mokhta[s.]ar f[=i]`ilm el-[h.]is[=a]b_ (Extract from Arithmetic) by `Abdalq[=a]dir ibn `Al[=i] al-Sakh[=a]w[=i] (died c. 1000) it is related that "the preface treats of the forms of the figures of Hindu signs, such as were established by the Hindu nation." [Woepcke, _Propagation_, p. 63.]] which, of course, are interpolations. An interesting example of a forgery in ecclesiastical matters is in the charter said to have been given by St. Patrick, granting indulgences to the benefactors of Glastonbury, dated "In nomine domini nostri Jhesu Christi Ego Patricius humilis servunculus Dei anno incarnationis ejusdem ccccxxx." Now if the Benedictines are right in saying that Dionysius Exiguus, a Scythian monk, first arranged the Christian chronology c. 532 A.D., this can hardly be other than spurious. See Arbuthnot, loc. cit., p. 38. [1] "_Discipulus._ Quis primus invenit numerum apud Hebræos et Ægyptios? _Magister._ Abraham primus invenit numerum apud Hebræos, deinde Moses; et Abraham tradidit istam scientiam numeri ad Ægyptios, et docuit eos: deinde Josephus." [Bede, _De computo dialogus_ (doubtfully assigned to him), _Opera omnia_, Paris, 1862, Vol. I, p. 650.] "Alii referunt ad Phoenices inventores arithmeticæ, propter eandem commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus sepulchrum est Lutetiæ in comitio Maturinensi, refert ad Arabes." [Ramus, _Arithmeticæ libri dvo_, Basel, 1569, p. 112.] Similar notes are given by Peletarius in his commentary on the arithmetic of Gemma Frisius (1563 ed., fol. 77), and in his own work (1570 Lyons ed., p. 14): "La valeur des Figures commence au coste dextre tirant vers le coste senestre: au rebours de notre maniere d'escrire par ce que la premiere prattique est venue des Chaldees: ou des Pheniciens, qui ont été les premiers traffiquers de marchandise." [2] Maximus Planudes (c. 1330) states that "the nine symbols come from the Indians." [Wäschke's German translation, Halle, 1878, p. 3.] Willichius speaks of the "Zyphræ Indicæ," in his _Arithmeticæ libri tres_ (Strasburg, 1540, p. 93), and Cataneo of "le noue figure de gli Indi," in his _Le pratiche delle dve prime mathematiche_ (Venice, 1546, fol. 1). Woepcke is not correct, therefore, in saying ("Mémoire sur la propagation des chiffres indiens," hereafter referred to as _Propagation_ [_Journal Asiatique_, Vol. I (6), 1863, p. 34]) that Wallis (_A Treatise on Algebra, both historical and practical_, London, 1685, p. 13, and _De algebra tractatus_, Latin edition in his _Opera omnia_, 1693, Vol. II, p. 10) was one of the first to give the Hindu origin. [3] From the 1558 edition of _The Grovnd of Artes_, fol. C, 5. Similarly Bishop Tonstall writes: "Qui a Chaldeis primum in finitimos, deinde in omnes pene gentes fluxit.... Numerandi artem a Chaldeis esse profectam: qui dum scribunt, a dextra incipiunt, et in leuam progrediuntur." [_De arte supputandi_, London, 1522, fol. B, 3.] Gemma Frisius, the great continental rival of Recorde, had the same idea: "Primùm autem appellamus dexterum locum, eo quòd haec ars vel à Chaldæis, vel ab Hebræis ortum habere credatur, qui etiam eo ordine scribunt"; but this refers more evidently to the Arabic numerals. [_Arithmeticæ practicæ methodvs facilis_, Antwerp, 1540, fol. 4 of the 1563 ed.] Sacrobosco (c. 1225) mentions the same thing. Even the modern Jewish writers claim that one of their scholars, M[=a]sh[=a]ll[=a]h (c. 800), introduced them to the Mohammedan world. [C. Levias, _The Jewish Encyclopedia_, New York, 1905, Vol. IX, p. 348.] [4] "... & que esto fu trouato di fare da gli Arabi con diece figure." [_La prima parte del general trattato di nvmeri, et misvre_, Venice, 1556, fol. 9 of the 1592 edition.] [5] "Vom welchen Arabischen auch disz Kunst entsprungen ist." [_Ain nerv geordnet Rechenbiechlin_, Augsburg, 1514, fol. 13 of the 1531 edition. The printer used the letters _rv_ for _w_ in "new" in the first edition, as he had no _w_ of the proper font.] [6] Among them Glareanus: "Characteres simplices sunt nouem significatiui, ab Indis usque, siue Chaldæis asciti .1.2.3.4.5.6.7.8.9. Est item unus .0 circulus, qui nihil significat." [_De VI. Arithmeticae practicae speciebvs_, Paris, 1539, fol. 9 of the 1543 edition.] [7] "Barbarische oder gemeine Ziffern." [Anonymous, _Das Einmahl Eins cum notis variorum_, Dresden, 1703, p. 3.] So Vossius (_De universae matheseos natura et constitutione liber_, Amsterdam, 1650, p. 34) calls them "Barbaras numeri notas." The word at that time was possibly synonymous with Arabic. [8] His full name was `Ab[=u] `Abdall[=a]h Mo[h.]ammed ibn M[=u]s[=a] al-Khow[=a]razm[=i]. He was born in Khow[=a]rezm, "the lowlands," the country about the present Khiva and bordering on the Oxus, and lived at Bagdad under the caliph al-M[=a]m[=u]n. He died probably between 220 and 230 of the Mohammedan era, that is, between 835 and 845 A.D., although some put the date as early as 812. The best account of this great scholar may be found in an article by C. Nallino, "Al-[H)]uw[=a]rizm[=i]" in the _Atti della R. Accad. dei Lincei_, Rome, 1896. See also _Verhandlungen des 5. Congresses der Orientalisten_, Berlin, 1882, Vol. II, p. 19; W. Spitta-Bey in the _Zeitschrift der deutschen Morgenländ. Gesellschaft_, Vol. XXXIII, p. 224; Steinschneider in the _Zeitschrift der deutschen Morgenländ. Gesellschaft_, Vol. L, p. 214; Treutlein in the _Abhandlungen zur Geschichte der Mathematik_, Vol. I, p. 5; Suter, "Die Mathematiker und Astronomen der Araber und ihre Werke," _Abhandlungen zur Geschichte der Mathematik_, Vol. X, Leipzig, 1900, p. 10, and "Nachträge," in Vol. XIV, p. 158; Cantor, _Geschichte der Mathematik_, Vol. I, 3d ed., pp. 712-733 etc.; F. Woepcke in _Propagation_, p. 489. So recently has he become known that Heilbronner, writing in 1742, merely mentions him as "Ben-Musa, inter Arabes celebris Geometra, scripsit de figuris planis & sphericis." [_Historia matheseos universæ_, Leipzig, 1742, p. 438.] In this work most of the Arabic names will be transliterated substantially as laid down by Suter in his work _Die Mathematiker_ etc., except where this violates English pronunciation. The scheme of pronunciation of oriental names is set forth in the preface. [9] Our word _algebra_ is from the title of one of his works, Al-jabr wa'l-muq[=a]balah, Completion and Comparison. The work was translated into English by F. Rosen, London, 1831, and treated in _L'Algèbre d'al-Kh[=a]rizmi et les méthodes indienne et grecque_, Léon Rodet, Paris, 1878, extract from the _Journal Asiatique_. For the derivation of the word _algebra_, see Cossali, _Scritti Inediti_, pp. 381-383, Rome, 1857; Leonardo's _Liber Abbaci_ (1202), p. 410, Rome, 1857; both published by B. Boncompagni. "Almuchabala" also was used as a name for algebra. [10] This learned scholar, teacher of O'Creat who wrote the _Helceph_ ("_Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum_"), studied in Toledo, learned Arabic, traveled as far east as Egypt, and brought from the Levant numerous manuscripts for study and translation. See Henry in the _Abhandlungen zur Geschichte der Mathematik_, Vol. III, p. 131; Woepcke in _Propagation_, p. 518. [11] The title is _Algoritmi de numero Indorum_. That he did not make this translation is asserted by Eneström in the _Bibliotheca Mathematica_, Vol. I (3), p. 520. [12] Thus he speaks "de numero indorum per .IX. literas," and proceeds: "Dixit algoritmi: Cum uidissem yndos constituisse .IX. literas in uniuerso numero suo, propter dispositionem suam quam posuerunt, uolui patefacere de opera quod fit per eas aliquid quod esset leuius discentibus, si deus uoluerit." [Boncompagni, _Trattati d'Aritmetica_, Rome, 1857.] Discussed by F. Woepcke, _Sur l'introduction de l'arithmétique indienne en Occident_, Rome, 1859. [13] Thus in a commentary by `Al[=i] ibn Ab[=i] Bekr ibn al-Jam[=a]l al-An[s.][=a]r[=i [14] See also Woepcke, _Propagation_, p. 505. The origin is discussed at much length by G. R. Kaye, "Notes on Indian Mathematics.--Arithmetical Notation," _Journ. and Proc. of the Asiatic Soc. of Bengal_, Vol. III, 1907, p. 489. [15] _Alberuni's India_, Arabic version, London, 1887; English translation, ibid., 1888. [16] _Chronology of Ancient Nations_, London, 1879. Arabic and English versions, by C. E. Sachau. [17] _India_, Vol. I, chap. xvi. [18] The Hindu name for the symbols of the decimal place system. [19] Sachau's English edition of the _Chronology_, p. 64. [20] _Littérature arabe_, Cl. Huart, Paris, 1902. [21] Huart, _History of Arabic Literature_, English ed., New York, 1903, p. 182 seq. [22] Al-Mas`[=u]d[=i]'s _Meadows of Gold_, translated in part by Aloys Sprenger, London, 1841; _Les prairies d'or_, trad. par C. Barbier de Meynard et Pavet de Courteille, Vols. I to IX, Paris, 1861-1877. [23] _Les prairies d'or_, Vol. VIII, p. 289 seq. [24] _Essays_, Vol. II, p. 428. [25] Loc. cit., p. 504. [26] _Matériaux pour servir à l'histoire comparée des sciences mathématiques chez les Grecs et les Orientaux_, 2 vols., Paris, 1845-1849, pp. 438-439. [27] He made an exception, however, in favor of the numerals, loc. cit., Vol. II, p. 503. [28] _Bibliotheca Arabico-Hispana Escurialensis_, Madrid, 1760-1770, pp. 426-427. [29] The author, Ibn al-Qif[t.][=i], flourished A.D. 1198 [Colebrooke, loc. cit., note Vol. II, p. 510]. [30] "Liber Artis Logisticae à Mohamado Ben Musa _Alkhuarezmita_ exornatus, qui ceteros omnes brevitate methodi ac facilitate praestat, Indorum que in praeclarissimis inventis ingenium & acumen ostendit." [Casiri, loc. cit., p. 427.] [31] Maçoudi, _Le livre de l'avertissement et de la révision_. Translation by B. Carra de Vaux, Paris, 1896. [32] Verifying the hypothesis of Woepcke, _Propagation_, that the Sindhind included a treatment of arithmetic. [33] A[h.]med ibn `Abdall[=a]h, Suter, _Die Mathematiker_, etc., p. 12. [34] _India_, Vol. II, p. 15. [35] See H. Suter, "Das Mathematiker-Verzeichniss im Fihrist," _Abhandlungen zur Geschichte der Mathematik_, Vol. VI, Leipzig, 1892. For further references to early Arabic writers the reader is referred to H. Suter, _Die Mathematiker und Astronomen der Araber und ihre Werke_. Also "Nachträge und Berichtigungen" to the same (_Abhandlungen_, Vol. XIV, 1902, pp. 155-186). [36] Suter, loc. cit., note 165, pp. 62-63. [37] "Send Ben Ali,... tùm arithmetica scripta maximè celebrata, quae publici juris fecit." [Loc. cit., p. 440.] [38] _Scritti di Leonardo Pisano_, Vol. I, _Liber Abbaci_ (1857); Vol. II, _Scritti_ (1862); published by Baldassarre Boncompagni, Rome. Also _Tre Scritti Inediti_, and _Intorno ad Opere di Leonardo Pisano_, Rome, 1854. [39] "Ubi ex mirabili magisterio in arte per novem figuras indorum introductus" etc. In another place, as a heading to a separate division, he writes, "De cognitione novem figurarum yndorum" etc. "Novem figure indorum he sunt 9 8 7 6 5 4 3 2 1." [40] See _An Ancient English Algorism_, by David Eugene Smith, in _Festschrift Moritz Cantor_, Leipzig, 1909. See also Victor Mortet, "Le plus ancien traité francais d'algorisme," _Bibliotheca Mathematica_, Vol. IX (3), pp. 55-64. [41] These are the two opening lines of the _Carmen de Algorismo_ that the anonymous author is explaining. They should read as follows: Haec algorismus ars praesens dicitur, in qua Talibus Indorum fruimur bis quinque figuris. What follows is the translation. [42] Thibaut, _Astronomie, Astrologie und Mathematik_, Strassburg, 1899. [43] Gustave Schlegel, _Uranographie chinoise ou preuves directes que l'astronomie primitive est originaire de la Chine, et qu'elle a été empruntée par les anciens peuples occidentaux à la sphère chinoise; ouvrage accompagné d'un atlas céleste chinois et grec_, The Hague and Leyden, 1875. [44] E. W. Hopkins, _The Religions of India_, Boston, 1898, p. 7. [45] R. C. Dutt, _History of India_, London, 1906. [46] W. D. Whitney, _Sanskrit Grammar_, 3d ed., Leipzig, 1896. [47] "Das [=A]pastamba-['S]ulba-S[=u]tra," _Zeitschrift der deutschen Morgenländischen Gesellschaft_, Vol. LV, p. 543, and Vol. LVI, p. 327. [48] _Geschichte der Math._, Vol. I, 2d ed., p. 595. [49] L. von Schroeder, _Pythagoras und die Inder_, Leipzig, 1884; H. Vogt, "Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt?" _Bibliotheca Mathematica_, Vol. VII (3), pp. 6-20; A. Bürk, loc. cit.; Max Simon, _Geschichte der Mathematik im Altertum_, Berlin, 1909, pp. 137-165; three S[=u]tras are translated in part by Thibaut, _Journal of the Asiatic Society of Bengal_, 1875, and one appeared in _The Pandit_, 1875; Beppo Levi, "Osservazioni e congetture sopra la geometria degli indiani," _Bibliotheca Mathematica_, Vol. IX (3), 1908, pp. 97-105. [50] Loc. cit.; also _Indiens Literatur und Cultur_, Leipzig, 1887. [51] It is generally agreed that the name of the river Sindhu, corrupted by western peoples to Hindhu, Indos, Indus, is the root of Hindustan and of India. Reclus, _Asia_, English ed., Vol. III, p. 14. [52] See the comments of Oppert, _On the Original Inhabitants of Bharatavar[s.]a or India_, London, 1893, p. 1. [53] A. Hillebrandt, _Alt-Indien_, Breslau, 1899, p. 111. Fragmentary records relate that Kh[=a]ravela, king of Kali[.n]ga, learned as a boy _lekh[=a]_ (writing), _ga[n.]an[=a]_ (reckoning), and _r[=u]pa_ (arithmetic applied to monetary affairs and mensuration), probably in the 5th century B.C. [Bühler, _Indische Palaeographie_, Strassburg, 1896, p. 5.] [54] R. C. Dutt, _A History of Civilization in Ancient India_, London, 1893, Vol. I, p. 174. [55] The Buddha. The date of his birth is uncertain. Sir Edwin Arnold put it c. 620 B.C. [56] I.e. 100·10^7. [57] There is some uncertainty about this limit. [58] This problem deserves more study than has yet been given it. A beginning may be made with Comte Goblet d'Alviella, _Ce que l'Inde doit à la Grèce_, Paris, 1897, and H. G. Keene's review, "The Greeks in India," in the _Calcutta Review_, Vol. CXIV, 1902, p. 1. See also F. Woepeke, _Propagation_, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The Source of Hindu Mathematics," _Journal of the Royal Asiatic Society_, July, 1910, pp. 749-760; G. Thibaut, _Astronomie, Astrologie und Mathematik_, pp. 43-50 and 76-79. It will be discussed more fully in Chapter VI. [59] I.e. to 100,000. The lakh is still the common large unit in India, like the myriad in ancient Greece and the million in the West. [60] This again suggests the _Psammites_, or _De harenae numero_ as it is called in the 1544 edition of the _Opera_ of Archimedes, a work in which the great Syracusan proposes to show to the king "by geometric proofs which you can follow, that the numbers which have been named by us ... are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe." For a list of early editions of this work see D. E. Smith, _Rara Arithmetica_, Boston, 1909, p. 227. [61] I.e. the Wise. [62] Sir Monier Monier-Williams, _Indian Wisdom_, 4th ed., London, 1893, pp. 144, 177. See also J. C. Marshman, _Abridgment of the History of India_, London, 1893, p. 2. [63] For a list and for some description of these works see R. C. Dutt, _A History of Civilization in Ancient India_, Vol. II, p. 121. [64] Professor Ramkrishna Gopal Bhandarkar fixes the date as the fifth century B.C. ["Consideration of the Date of the Mah[=a]bh[=a]rata," in the _Journal of the Bombay Branch of the R. A. Soc._, Bombay, 1873, Vol. X, p. 2.]. [65] Marshman, loc. cit., p. 2. [66] A. C. Burnell, _South Indian Palæography_, 2d ed., London, 1878, p. 1, seq. [67] This extensive subject of palpable arithmetic, essentially the history of the abacus, deserves to be treated in a work by itself. [68] The following are the leading sources of information upon this subject: G. Bühler, _Indische Palaeographie_, particularly chap. vi; A. C. Burnell, _South Indian Palæography_, 2d ed., London, 1878, where tables of the various Indian numerals are given in Plate XXIII; E. C. Bayley, "On the Genealogy of Modern Numerals," _Journal of the Royal Asiatic Society_, Vol. XIV, part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in _The Academy_, January 28, 1882, with a repetition of his argument in his work _The Alphabet_, London, 1883, Vol. II, p. 265, based on Bayley; G. R. Kaye, loc. cit., in some respects one of the most critical articles thus far published; J. C. Fleet, _Corpus inscriptionum Indicarum_, London, 1888, Vol. III, with facsimiles of many Indian inscriptions, and _Indian Epigraphy_, Oxford, 1907, reprinted from the _Imperial Gazetteer of India_, Vol. II, pp. 1-88, 1907; G. Thibaut, loc. cit., _Astronomie_ etc.; R. Caldwell, _Comparative Grammar of the Dravidian Languages_, London, 1856, p. 262 seq.; and _Epigraphia Indica_ (official publication of the government of India), Vols. I-IX. Another work of Bühler's, _On the Origin of the Indian Br[=a]hma Alphabet_, is also of value. [69] The earliest work on the subject was by James Prinsep, "On the Inscriptions of Piyadasi or A['s]oka," etc., _Journal of the Asiatic Society of Bengal_, 1838, following a preliminary suggestion in the same journal in 1837. See also "A['s]oka Notes," by V. A. Smith, _The Indian Antiquary_, Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159, June, 1909; _The Early History of India_, 2d ed., Oxford, 1908, p. 154; J. F. Fleet, "The Last Words of A['s]oka," _Journal of the Royal Asiatic Society_, October, 1909, pp. 981-1016; E. Senart, _Les inscriptions de Piyadasi_, 2 vols., Paris, 1887. [70] For a discussion of the minor details of this system, see Bühler, loc. cit., p. 73. [71] Julius Euting, _Nabatäische Inschriften aus Arabien_, Berlin, 1885, pp. 96-97, with a table of numerals. [72] For the five principal theories see Bühler, loc. cit., p. 10. [73] Bayley, loc. cit., reprint p. 3. [74] Bühler, loc. cit.; _Epigraphia Indica_, Vol. III, p. 134; _Indian Antiquary_, Vol. VI, p. 155 seq., and Vol. X, p. 107. [75] Pandit Bhagav[=a]nl[=a]l Indr[=a]j[=i], "On Ancient N[=a]g[=a]ri Numeration; from an Inscription at N[=a]negh[=a]t," _Journal of the Bombay Branch of the Royal Asiatic Society_, 1876, Vol. XII, p. 404. [76] Ib., p. 405. He gives also a plate and an interpretation of each numeral. [77] These may be compared with Bühler's drawings, loc. cit.; with Bayley, loc. cit., p. 337 and plates; and with Bayley's article in the _Encyclopædia Britannica_, 9th ed., art. "Numerals." [78] E. Senart, "The Inscriptions in the Caves at Nasik," _Epigraphia Indica_, Vol. VIII, pp. 59-96; "The Inscriptions in the Cave at Karle," _Epigraphia Indica_, Vol. VII, pp. 47-74; Bühler, _Palaeographie_, Tafel IX. [79] See Fleet, loc. cit. See also T. Benfey, _Sanskrit Grammar_, London, 1863, p. 217; M. R. Kále, _Higher Sanskrit Grammar_, 2d ed., Bombay, 1898, p. 110, and other authorities as cited. [80] Kharo[s.][t.]h[=i] numerals, A['s]oka inscriptions, c. 250 B.C. Senart, _Notes d'épigraphie indienne_. Given by Bühler, loc. cit., Tafel I. [81] Same, ['S]aka inscriptions, probably of the first century B.C. Senart, loc. cit.; Bühler, loc. cit. [82] Br[=a]hm[=i] numerals, A['s]oka inscriptions, c. 250 B.C. _Indian Antiquary_, Vol. VI, p. 155 seq. [83] Same, N[=a]n[=a] Gh[=a]t inscriptions, c. 150 B.C. Bhagav[=a]nl[=a]l Indr[=a]j[=i], _On Ancient N[=a]gar[=i] Numeration_, loc. cit. Copied from a squeeze of the original. [84] Same, Nasik inscription, c. 100 B.C. Burgess, _Archeological Survey Report, Western India_; Senart, _Epigraphia Indica_, Vol. VII, pp. 47-79, and Vol. VIII, pp. 59-96. [85] K[s.]atrapa coins, c. 200 A.D. _Journal of the Royal Asiatic Society_, 1890, p. 639. [86] Ku[s.]ana inscriptions, c. 150 A.D. _Epigraphia Indica_, Vol. I, p. 381, and Vol. II, p. 201. [87] Gupta Inscriptions, c. 300 A.D. to 450 A.D. Fleet, loc. cit., Vol. III. [88] Valhab[=i], c. 600 A.D. _Corpus_, Vol. III. [89] Bendall's Table of Numerals, in _Cat. Sansk. Budd. MSS._, British Museum. [90] _Indian Antiquary_, Vol. XIII, 120; _Epigraphia Indica_, Vol. III, 127 ff. [91] Fleet, loc. cit. [92] Bayley, loc. cit., p. 335. [93] From a copper plate of 493 A.D., found at K[=a]r[=i]tal[=a][=i], Central India. [Fleet, loc. cit., Plate XVI.] It should be stated, however, that many of these copper plates, being deeds of property, have forged dates so as to give the appearance of antiquity of title. On the other hand, as Colebrooke long ago pointed out, a successful forgery has to imitate the writing of the period in question, so that it becomes evidence well worth considering, as shown in Chapter III. [94] From a copper plate of 510 A.D., found at Majhgaw[=a]in, Central India. [Fleet, loc. cit., Plate XIV.] [95] From an inscription of 588 A.D., found at B[=o]dh-Gay[=a], Bengal Presidency. [Fleet, loc. cit., Plate XXIV.] [96] From a copper plate of 571 A.D., found at M[=a]liy[=a], Bombay Presidency. [Fleet, loc. cit., Plate XXIV.] [97] From a Bijayaga[d.]h pillar inscription of 372 A.D. [Fleet, loc. cit., Plate XXXVI, C.] [98] From a copper plate of 434 A.D. [_Indian Antiquary_, Vol. I, p. 60.] [99] Gadhwa inscription, c. 417 A.D. [Fleet, loc. cit., Plate IV, D.] [100] K[=a]r[=i]tal[=a][=i] plate of 493 A.D., referred to above. [101] It seems evident that the Chinese four, curiously enough called "eight in the mouth," is only a cursive [4 vertical strokes]. [102] Chalfont, F. H., _Memoirs of the Carnegie Museum_, Vol. IV, no. 1; J. Hager, _An Explanation of the Elementary Characters of the Chinese_, London, 1801. [103] H. V. Hilprecht, _Mathematical, Metrological and Chronological Tablets from the Temple Library at Nippur_, Vol. XX, part I, of Series A, Cuneiform Texts Published by the Babylonian Expedition of the University of Pennsylvania, 1906; A. Eisenlohr, _Ein altbabylonischer Felderplan_, Leipzig, 1906; Maspero, _Dawn of Civilization_, p. 773. [104] Sir H. H. Howard, "On the Earliest Inscriptions from Chaldea," _Proceedings of the Society of Biblical Archæology_, XXI, p. 301, London, 1899. [105] For a bibliography of the principal hypotheses of this nature see Bühler, loc. cit., p. 77. Bühler (p. 78) feels that of all these hypotheses that which connects the Br[=a]hm[=i] with the Egyptian numerals is the most plausible, although he does not adduce any convincing proof. Th. Henri Martin, "Les signes numéraux et l'arithmétique chez les peuples de l'antiquité et du moyen âge" (being an examination of Cantor's _Mathematische Beiträge zum Culturleben der Völker_), _Annali di matematica pura ed applicata_, Vol. V, Rome, 1864, pp. 8, 70. Also, same author, "Recherches nouvelles sur l'origine de notre système de numération écrite," _Revue Archéologique_, 1857, pp. 36, 55. See also the tables given later in this work. [106] _Journal of the Royal Asiatic Society, Bombay Branch_, Vol. XXIII. [107] Loc. cit., reprint, Part I, pp. 12, 17. Bayley's deductions are generally regarded as unwarranted. [108] _The Alphabet_; London, 1883, Vol. II, pp. 265, 266, and _The Academy_ of Jan. 28, 1882. [109] Taylor, _The Alphabet_, loc. cit., table on p. 266. [110] Bühler, _On the Origin of the Indian Br[=a]hma Alphabet_, Strassburg, 1898, footnote, pp. 52, 53. [111] Albrecht Weber, _History of Indian Literature_, English ed., Boston, 1878, p. 256: "The Indian figures from 1-9 are abbreviated forms of the initial letters of the numerals themselves...: the zero, too, has arisen out of the first letter of the word _[s.]unya_ (empty) (it occurs even in Piñgala). It is the decimal place value of these figures which gives them significance." C. Henry, "Sur l'origine de quelques notations mathématiques," _Revue Archéologique_, June and July, 1879, attempts to derive the Boethian forms from the initials of Latin words. See also J. Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and Dhauli in Cuttach," _Journal of the Asiatic Society of Bengal_, 1838, especially Plate XX, p. 348; this was the first work on the subject. [112] Bühler, _Palaeographie_, p. 75, gives the list, with the list of letters (p. 76) corresponding to the number symbols. [113] For a general discussion of the connection between the numerals and the different kinds of alphabets, see the articles by U. Ceretti, "Sulla origine delle cifre numerali moderne," _Rivista di fisica, matematica e scienze naturali_, Pisa and Pavia, 1909, anno X, numbers 114, 118, 119, and 120, and continuation in 1910. [114] This is one of Bühler's hypotheses. See Bayley, loc. cit., reprint p. 4; a good bibliography of original sources is given in this work, p. 38. [115] Loc. cit., reprint, part I, pp. 12, 17. See also Burnell, loc. cit., p. 64, and tables in plate XXIII. [116] This was asserted by G. Hager (_Memoria sulle cifre arabiche_, Milan, 1813, also published in _Fundgruben des Orients_, Vienna, 1811, and in _Bibliothèque Britannique_, Geneva, 1812). See also the recent article by Major Charles E. Woodruff, "The Evolution of Modern Numerals from Tally Marks," _American Mathematical Monthly_, August-September, 1909. Biernatzki, "Die Arithmetik der Chinesen," _Crelle's Journal für die reine und angewandte Mathematik_, Vol. LII, 1857, pp. 59-96, also asserts the priority of the Chinese claim for a place system and the zero, but upon the flimsiest authority. Ch. de Paravey, _Essai sur l'origine unique et hiéroglyphique des chiffres et des lettres de tous les peuples_, Paris, 1826; G. Kleinwächter, "The Origin of the Arabic Numerals," _China Review_, Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la connaissance que les Chinois ont eue de la valeur de position des chiffres," _Journal Asiatique_, 1839, pp. 497-502. A. Terrien de Lacouperie, "The Old Numerals, the Counting-Rods and the Swan-Pan in China," _Numismatic Chronicle_, Vol. III (3), pp. 297-340, and Crowder B. Moseley, "Numeral Characters: Theory of Origin and Development," _American Antiquarian_, Vol. XXII, pp. 279-284, both propose to derive our numerals from Chinese characters, in much the same way as is done by Major Woodruff, in the article above cited. [117] The Greeks, probably following the Semitic custom, used nine letters of the alphabet for the numerals from 1 to 9, then nine others for 10 to 90, and further letters to represent 100 to 900. As the ordinary Greek alphabet was insufficient, containing only twenty-four letters, an alphabet of twenty-seven letters was used. [118] _Institutiones mathematicae_, 2 vols., Strassburg, 1593-1596, a somewhat rare work from which the following quotation is taken: "_Quis est harum Cyphrarum autor?_ "A quibus hae usitatae syphrarum notae sint inventae: hactenus incertum fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis librarijs (quorum olim magna fuit copia) literae Graecorum quibus veteres Graeci tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his licet videre. "Graecorum Literae corruptae. [Illustration] _"Sed qua ratione graecorum literae ita fuerunt corruptae?_ "Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt in nota binarij numeri, vel additione vt in ternarij, vel inuersione vt in septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his sola differunt elegantia, vt apparet." See also Bayer, _Historia regni Graecorum Bactriani_, St. Petersburg, 1788, pp. 129-130, quoted by Martin, _Recherches nouvelles_, etc., loc. cit. [119] P. D. Huet, _Demonstratio evangelica_, Paris, 1769, note to p. 139 on p. 647: "Ab Arabibus vel ab Indis inventas esse, non vulgus eruditorum modo, sed doctissimi quique ad hanc diem arbitrati sunt. Ego vero falsum id esse, merosque esse Graecorum characteres aio; à librariis Graecae linguae ignaris interpolatos, et diuturna scribendi consuetudine corruptos. Nam primum 1 apex fuit, seu virgula, nota [Greek: monados]. 2, est ipsum [beta] extremis suis truncatum. [gamma], si in sinistram partem inclinaveris & cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. Res ipsa loquitur 4 ipsissimum esse [Delta], cujus crus sinistrum erigitur [Greek: kata katheton], & infra basim descendit; basis vero ipsa ultra crus producta eminet. Vides quam 5 simile sit [Greek: tôi] [epsilon]; infimo tantum semicirculo, qui sinistrorsum patebat, dextrorsum converso. [Greek: episêmon bau] quod ita notabatur [digamma], rotundato ventre, pede detracto, peperit [Greek: to] 6. Ex [Zeta] basi sua mutilato, ortum est [Greek: to] 7. Si [Eta] inflexis introrsum apicibus in rotundiorem & commodiorem formam mutaveris, exurget [Greek: to] 8. At 9 ipsissimum est [alt theta]." I. Weidler, _Spicilegium observationum ad historiam notarum numeralium_, Wittenberg, 1755, derives them from the Hebrew letters; Dom Augustin Calmet, "Recherches sur l'origine des chiffres d'arithmétique," _Mémoires pour l'histoire des sciences et des beaux arts_, Trévoux, 1707 (pp. 1620-1635, with two plates), derives the current symbols from the Romans, stating that they are relics of the ancient "Notae Tironianae." These "notes" were part of a system of shorthand invented, or at least perfected, by Tiro, a slave who was freed by Cicero. L. A. Sedillot, "Sur l'origine de nos chiffres," _Atti dell' Accademia pontificia dei nuovi Lincei_, Vol. XVIII, 1864-1865, pp. 316-322, derives the Arabic forms from the Roman numerals. [120] Athanasius Kircher, _Arithmologia sive De abditis Numerorum, mysterijs qua origo, antiquitas & fabrica Numerorum exponitur_, Rome, 1665. [121] See Suter, _Die Mathematiker und Astronomen der Araber_, p. 100. [122] "Et hi numeri sunt numeri Indiani, a Brachmanis Indiae Sapientibus ex figura circuli secti inuenti." [123] V. A. Smith, _The Early History of India_, Oxford, 2d ed., 1908, p. 333. [124] C. J. Ball, "An Inscribed Limestone Tablet from Sippara," _Proceedings of the Society of Biblical Archæology_, Vol. XX, p. 25 (London, 1898). Terrien de Lacouperie states that the Chinese used the circle for 10 before the beginning of the Christian era. [_Catalogue of Chinese Coins_, London, 1892, p. xl.] [125] For a purely fanciful derivation from the corresponding number of strokes, see W. W. R. Ball, _A Short Account of the History of Mathematics_, 1st ed., London, 1888, p. 147; similarly J. B. Reveillaud, _Essai sur les chiffres arabes_, Paris, 1883; P. Voizot, "Les chiffres arabes et leur origine," _La Nature_, 1899, p. 222; G. Dumesnil, "De la forme des chiffres usuels," _Annales de l'université de Grenoble_, 1907, Vol. XIX, pp. 657-674, also a note in _Revue Archéologique_, 1890, Vol. XVI (3), pp. 342-348; one of the earliest references to a possible derivation from points is in a work by Bettino entitled _Apiaria universae philosophiae mathematicae in quibus paradoxa et noua machinamenta ad usus eximios traducta, et facillimis demonstrationibus confirmata_, Bologna, 1545, Vol. II, Apiarium XI, p. 5. [126] _Alphabetum Barmanum_, Romae, MDCCLXXVI, p. 50. The 1 is evidently Sanskrit, and the 4, 7, and possibly 9 are from India. [127] _Alphabetum Grandonico-Malabaricum_, Romae, MDCCLXXII, p. 90. The zero is not used, but the symbols for 10, 100, and so on, are joined to the units to make the higher numbers. [128] _Alphabetum Tangutanum_, Romae, MDCCLXXIII, p. 107. In a Tibetan MS. in the library of Professor Smith, probably of the eighteenth century, substantially these forms are given. [129] Bayley, loc. cit., plate II. Similar forms to these here shown, and numerous other forms found in India, as well as those of other oriental countries, are given by A. P. Pihan, _Exposé des signes de numération usités chez les peuples orientaux anciens et modernes_, Paris, 1860. [130] Bühler, loc. cit., p. 80; J. F. Fleet, _Corpus inscriptionum Indicarum_, Vol. III, Calcutta, 1888. Lists of such words are given also by Al-B[=i]r[=u]n[=i] in his work _India_; by Burnell, loc. cit.; by E. Jacquet, "Mode d'expression symbolique des nombres employé par les Indiens, les Tibétains et les Javanais," _Journal Asiatique_, Vol. XVI, Paris, 1835. [131] This date is given by Fleet, loc. cit., Vol. III, p. 73, as the earliest epigraphical instance of this usage in India proper. [132] Weber, _Indische Studien_, Vol. VIII, p. 166 seq. [133] _Journal of the Royal Asiatic Society_, Vol. I (N.S.), p. 407. [134] VIII, 20, 21. [135] Th. H. Martin, _Les signes numéraux_ ..., Rome, 1864; Lassen, _Indische Alterthumskunde_, Vol. II, 2d ed., Leipzig and London, 1874, p. 1153. [136] But see Burnell, loc. cit., and Thibaut, _Astronomie, Astrologie und Mathematik_, p. 71. [137] A. Barth, "Inscriptions Sanscrites du Cambodge," in the _Notices et extraits des Mss. de la Bibliothèque nationale_, Vol. XXVII, Part I, pp. 1-180, 1885; see also numerous articles in _Journal Asiatique_, by Aymonier. [138] Bühler, loc. cit., p. 82. [139] Loc. cit., p. 79. [140] Bühler, loc. cit., p. 83. The Hindu astrologers still use an alphabetical system of numerals. [Burnell, loc. cit., p. 79.] [141] Well could Ramus say, "Quicunq; autem fuerit inventor decem notarum laudem magnam meruit." [142] Al-B[=i]r[=u]n[=i] gives lists. [143] _Propagation_, loc. cit., p. 443. [144] See the quotation from _The Light of Asia_ in Chapter II, p. 16. [145] The nine ciphers were called _a[.n]ka_. [146] "Zur Geschichte des indischen Ziffernsystems," _Zeitschrift für die Kunde des Morgenlandes_, Vol. IV, 1842, pp. 74-83. [147] It is found in the Bakh[s.][=a]l[=i] MS. of an elementary arithmetic which Hoernle placed, at first, about the beginning of our era, but the date is much in question. G. Thibaut, loc. cit., places it between 700 and 900 A.D.; Cantor places the body of the work about the third or fourth century A.D., _Geschichte der Mathematik_, Vol. I (3), p. 598. [148] For the opposite side of the case see G. R. Kaye, "Notes on Indian Mathematics, No. 2.--[=A]ryabha[t.]a," _Journ. and Proc. of the Asiatic Soc. of Bengal_, Vol. IV, 1908, pp. 111-141. [149] He used one of the alphabetic systems explained above. This ran up to 10^{18} and was not difficult, beginning as follows: [Illustration] the same letter (_ka_) appearing in the successive consonant forms, _ka_, _kha_, _ga_, _gha_, etc. See C. I. Gerhardt, _Über die Entstehung und Ausbreitung des dekadischen Zahlensystems_, Programm, p. 17, Salzwedel, 1853, and _Études historiques sur l'arithmétique de position_, Programm, p. 24, Berlin, 1856; E. Jacquet, _Mode d'expression symbolique des nombres_, loc. cit., p. 97; L. Rodet, "Sur la véritable signification de la notation numérique inventée par [=A]ryabhata," _Journal Asiatique_, Vol. XVI (7), pp. 440-485. On the two [=A]ryabha[t.]as see Kaye, _Bibl. Math._, Vol. X (3), p. 289. [150] Using _kha_, a synonym of _['s][=u]nya_. [Bayley, loc. cit., p. 22, and L. Rodet, _Journal Asiatique_, Vol. XVI (7), p. 443.] [151] Var[=a]ha-Mihira, _Pañcasiddh[=a]ntik[=a]_, translated by G. Thibaut and M. S. Dvived[=i], Benares, 1889; see Bühler, loc. cit., p. 78; Bayley, loc. cit., p. 23. [152] _B[r.]hat Sa[m.]hit[=a]_, translated by Kern, _Journal of the Royal Asiatic Society_, 1870-1875. [153] It is stated by Bühler in a personal letter to Bayley (loc. cit., p. 65) that there are hundreds of instances of this usage in the _B[r.]hat Sa[m.]hit[=a]_. The system was also used in the _Pañcasiddh[=a]ntik[=a]_ as early as 505 A.D. [Bühler, _Palaeographie_, p. 80, and Fleet, _Journal of the Royal Asiatic Society_, 1910, p. 819.] [154] Cantor, _Geschichte der Mathematik_, Vol. I (3), p. 608. [155] Bühler, loc. cit., p. 78. [156] Bayley, p. 38. [157] Noviomagus, in his _De numeris libri duo_, Paris, 1539, confesses his ignorance as to the origin of the zero, but says: "D. Henricus Grauius, vir Graecè & Hebraicè eximè doctus, Hebraicam originem ostendit," adding that Valla "Indis Orientalibus gentibus inventionem tribuit." [158] See _Essays_, Vol. II, pp. 287 and 288. [159] Vol. XXX, p. 205 seqq. [160] Loc. cit., p. 284 seqq. [161] Colebrooke, loc. cit., p. 288. [162] Loc. cit., p. 78. [163] Hereafter, unless expressly stated to the contrary, we shall use the word "numerals" to mean numerals with place value. [164] "The Gurjaras of R[=a]jput[=a]na and Kanauj," in _Journal of the Royal Asiatic Society_, January and April, 1909. [165] Vol. IX, 1908, p. 248. [166] _Epigraphia Indica_, Vol. IX, pp. 193 and 198. [167] _Epigraphia Indica_, Vol. IX, p. 1. [168] Loc. cit., p. 71. [169] Thibaut, p. 71. [170] "Est autem in aliquibus figurarum istaram apud multos diuersitas. Quidam enim septimam hanc figuram representant," etc. [Boncompagni, _Trattati_, p. 28.] Eneström has shown that very likely this work is incorrectly attributed to Johannes Hispalensis. [_Bibliotheca Mathematica_, Vol. IX (3), p. 2.] [171] _Indische Palaeographie_, Tafel IX. [172] Edited by Bloomfield and Garbe, Baltimore, 1901, containing photographic reproductions of the manuscript. [173] Bakh[s.][=a]l[=i] MS. See page 43; Hoernle, R., _The Indian Antiquary_, Vol. XVII, pp. 33-48, 1 plate; Hoernle, _Verhandlungen des VII. Internationalen Orientalisten-Congresses, Arische Section_, Vienna, 1888, "On the Baksh[=a]l[=i] Manuscript," pp. 127-147, 3 plates; Bühler, loc. cit. [174] 3, 4, 6, from H. H. Dhruva, "Three Land-Grants from Sankheda," _Epigraphia Indica_, Vol. II, pp. 19-24 with plates; date 595 A.D. 7, 1, 5, from Bhandarkar, "Daulatabad Plates," _Epigraphia Indica_, Vol. IX, part V; date c. 798 A.D. [175] 8, 7, 2, from "Buckhala Inscription of Nagabhatta," Bhandarkar, _Epigraphia Indica_, Vol. IX, part V; date 815 A.D. 5 from "The Morbi Copper-Plate," Bhandarkar, _The Indian Antiquary_, Vol. II, pp. 257-258, with plate; date 804 A.D. See Bühler, loc. cit. [176] 8 from the above Morbi Copper-Plate. 4, 5, 7, 9, and 0, from "Asni Inscription of Mahipala," _The Indian Antiquary_, Vol. XVI, pp. 174-175; inscription is on red sandstone, date 917 A.D. See Bühler. [177] 8, 9, 4, from "Rashtrakuta Grant of Amoghavarsha," J. F. Fleet, _The Indian Antiquary_, Vol. XII, pp. 263-272; copper-plate grant of date c. 972 A.D. See Bühler. 7, 3, 5, from "Torkhede Copper-Plate Grant of the Time of Govindaraja of Gujerat," Fleet, _Epigraphia Indica_, Vol. III, pp. 53-58. See Bühler. [178] From "A Copper-Plate Grant of King Tritochanapâla Chanlukya of L[=a][t.]ade['s]a," H.H. Dhruva, _Indian Antiquary_, Vol. XII, pp. 196-205; date 1050 A.D. See Bühler. [179] Burnell, A. C., _South Indian Palæography_, plate XXIII, Telugu-Canarese numerals of the eleventh century. See Bühler. [180] From a manuscript of the second half of the thirteenth century, reproduced in "Della vita e delle opere di Leonardo Pisano," Baldassare Boncompagni, Rome, 1852, in _Atti dell' Accademia Pontificia dei nuovi Lincei_, anno V. [181] From a fourteenth-century manuscript, as reproduced in _Della vita_ etc., Boncompagni, loc. cit. [182] From a Tibetan MS. in the library of D. E. Smith. [183] From a Tibetan block-book in the library of D. E. Smith. [184] ['S][=a]rad[=a] numerals from _The Kashmirian Atharva-Veda, reproduced by chromophotography from the manuscript in the University Library at Tübingen_, Bloomfield and Garbe, Baltimore, 1901. Somewhat similar forms are given under "Numération Cachemirienne," by Pihan, _Exposé_ etc., p. 84. [185] Franz X. Kugler, _Die Babylonische Mondrechnung_, Freiburg i. Br., 1900, in the numerous plates at the end of the book; practically all of these contain the symbol to which reference is made. Cantor, _Geschichte_, Vol. I, p. 31. [186] F. X. Kugler, _Sternkunde und Sterndienst in Babel_, I. Buch, from the beginnings to the time of Christ, Münster i. Westfalen, 1907. It also has numerous tables containing the above zero. [187] From a letter to D. E. Smith, from G. F. Hill of the British Museum. See also his monograph "On the Early Use of Arabic Numerals in Europe," in _Archæologia_, Vol. LXII (1910), p. 137. [188] R. Hoernle, "The Baksh[=a]l[=i] Manuscript," _Indian Antiquary_, Vol. XVII, pp. 33-48 and 275-279, 1888; Thibaut, _Astronomie, Astrologie und Mathematik_, p. 75; Hoernle, _Verhandlungen_, loc. cit., p. 132. [189] Bayley, loc. cit., Vol. XV, p. 29. Also Bendall, "On a System of Numerals used in South India," _Journal of the Royal Asiatic Society_, 1896, pp. 789-792. [190] V. A. Smith, _The Early History of India_, 2d ed., Oxford, 1908, p. 14. [191] Colebrooke, _Algebra, with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and Bháscara_, London, 1817, pp. 339-340. [192] Ibid., p. 138. [193] D. E. Smith, in the _Bibliotheca Mathematica_, Vol. IX (3), pp. 106-110. [194] As when we use three dots (...). [195] "The Hindus call the nought explicitly _['s][=u]nyabindu_ 'the dot marking a blank,' and about 500 A.D. they marked it by a simple dot, which latter is commonly used in inscriptions and MSS. in order to mark a blank, and which was later converted into a small circle." [Bühler, _On the Origin of the Indian Alphabet_, p. 53, note.] [196] Fazzari, _Dell' origine delle parole zero e cifra_, Naples, 1903. [197] E. Wappler, "Zur Geschichte der Mathematik im 15. Jahrhundert," in the _Zeitschrift für Mathematik und Physik_, Vol. XLV, _Hist.-lit. Abt._, p. 47. The manuscript is No. C. 80, in the Dresden library. [198] J. G. Prändel, _Algebra nebst ihrer literarischen Geschichte_, p. 572, Munich, 1795. [199] See the table, p. 23. Does the fact that the early European arithmetics, following the Arab custom, always put the 0 after the 9, suggest that the 0 was derived from the old Hindu symbol for 10? [200] Bayley, loc. cit., p. 48. From this fact Delambre (_Histoire de l'astronomie ancienne_) inferred that Ptolemy knew the zero, a theory accepted by Chasles, _Aperçu historique sur l'origine et le développement des méthodes en géométrie_, 1875 ed., p. 476; Nesselmann, however, showed (_Algebra der Griechen_, 1842, p. 138), that Ptolemy merely used [Greek: o] for [Greek: ouden], with no notion of zero. See also G. Fazzari, "Dell' origine delle parole zero e cifra," _Ateneo_, Anno I, No. 11, reprinted at Naples in 1903, where the use of the point and the small cross for zero is also mentioned. Th. H. Martin, _Les signes numéraux_ etc., reprint p. 30, and J. Brandis, _Das Münz-, Mass- und Gewichtswesen in Vorderasien bis auf Alexander den Grossen_, Berlin, 1866, p. 10, also discuss this usage of [Greek: o], without the notion of place value, by the Greeks. [201] _Al-Batt[=a]n[=i] sive Albatenii opus astronomicum_. Ad fidem codicis escurialensis arabice editum, latine versum, adnotationibus instructum a Carolo Alphonso Nallino, 1899-1907. Publicazioni del R. Osservatorio di Brera in Milano, No. XL. [202] Loc. cit., Vol. II, p. 271. [203] C. Henry, "Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum," _Abhandlungen zur Geschichte der Mathematik_, Vol. III, 1880. [204] Max. Curtze, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts," _Abhandlungen zur Geschichte der Mathematik_, Vol. VIII, 1898, pp. 1-27; Alfred Nagl, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," _Zeitschrift für Mathematik und Physik, Hist.-lit. Abth._, Vol. XXXIV, pp. 129-146 and 161-170, with one plate. [205] "Byzantinische Analekten," _Abhandlungen zur Geschichte der Mathematik_, Vol. IX, pp. 161-189. [206] [symbol] or [symbol] for 0. [symbol] also used for 5. [symbols] for 13. [Heiberg, loc. cit.] [207] Gerhardt, _Études historiques sur l'arithmétique de position_, Berlin, 1856, p. 12; J. Bowring, _The Decimal System in Numbers, Coins, & Accounts_, London, 1854, p. 33. [208] Karabacek, _Wiener Zeitschrift für die Kunde des Morgenlandes_, Vol. XI, p. 13; _Führer durch die Papyrus-Ausstellung Erzherzog Rainer_, Vienna, 1894, p. 216. [209] In the library of G. A. Plimpton, Esq. [210] Cantor, _Geschichte_, Vol. I (3), p. 674; Y. Mikami, "A Remark on the Chinese Mathematics in Cantor's Geschichte der Mathematik," _Archiv der Mathematik und Physik_, Vol. XV (3), pp. 68-70. [211] Of course the earlier historians made innumerable guesses as to the origin of the word _cipher_. E.g. Matthew Hostus, _De numeratione emendata_, Antwerp, 1582, p. 10, says: "Siphra vox Hebræam originem sapit refértque: & ut docti arbitrantur, à verbo saphar, quod Ordine numerauit significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi verò gens Iudaica his notis, quæ hodie Siphræ vocantur, usa non fuit: mansit tamen rei appellatio apud multas gentes." Dasypodius, _Institutiones mathematicae_, Vol. I, 1593, gives a large part of this quotation word for word, without any mention of the source. Hermannus Hugo, _De prima scribendi origine_, Trajecti ad Rhenum, 1738, pp. 304-305, and note, p. 305; Karl Krumbacher, "Woher stammt das Wort Ziffer (Chiffre)?", _Études de philologie néo-grecque_, Paris, 1892. [212] Bühler, loc. cit., p. 78 and p. 86. [213] Fazzari, loc. cit., p. 4. So Elia Misrachi (1455-1526) in his posthumous _Book of Number_, Constantinople, 1534, explains _sifra_ as being Arabic. See also Steinschneider, _Bibliotheca Mathematica_, 1893, p. 69, and G. Wertheim, _Die Arithmetik des Elia Misrachi_, Programm, Frankfurt, 1893. [214] "Cum his novem figuris, et cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet numerus." [215] [Greek: tziphra], a form also used by Neophytos (date unknown, probably c. 1330). It is curious that Finaeus (1555 ed., f. 2) used the form _tziphra_ throughout. A. J. H. Vincent ["Sur l'origine de nos chiffres," _Notices et Extraits des MSS._, Paris, 1847, pp. 143-150] says: "Ce cercle fut nommé par les uns, _sipos, rota, galgal_ ...; par les autres _tsiphra_ (de [Hebrew: TSPR], _couronne_ ou _diadème_) ou _ciphra_ (de [Hebrew: SPR], _numération_)." Ch. de Paravey, _Essai sur l'origine unique et hiéroglyphique des chiffres et des lettres de tous les peuples_, Paris, 1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et fermé par un couvercle, qui est le symbole de la 10^e Heure, [symbol]," among the Chinese; also "Tsiphron Zéron, ou tout à fait vide en arabe, [Greek: tziphra] en grec ... d'où chiffre (qui dérive plutôt, suivant nous, de l'Hébreu _Sepher_, compter.") [216] "Compilatus a Magistro Jacobo de Florentia apud montem pesalanum," and described by G. Lami in his _Catalogus codicum manuscriptorum qui in bibliotheca Riccardiana Florentiæ adservantur_. See Fazzari, loc. cit., p. 5. [217] "Et doveto sapere chel zeuero per se solo non significa nulla ma è potentia di fare significare, ... Et decina o centinaia o migliaia non si puote scrivere senza questo segno 0. la quale si chiama zeuero." [Fazzari, loc. cit., p. 5.] [218] Ibid., p. 6. [219] Avicenna (980-1036), translation by Gasbarri et François, "più il punto (gli Arabi adoperavano il punto in vece dello zero il cui segno 0 in arabo si chiama _zepiro_ donde il vocabolo zero), che per sè stesso non esprime nessun numero." This quotation is taken from D. C. Martines, _Origine e progressi dell' aritmetica_, Messina, 1865. [220] Leo Jordan, "Materialien zur Geschichte der arabischen Zahlzeichen in Frankreich," _Archiv für Kulturgeschichte_, Berlin, 1905, pp. 155-195, gives the following two schemes of derivation, (1) "zefiro, zeviro, zeiro, zero," (2) "zefiro, zefro, zevro, zero." [221] Köbel (1518 ed., f. A_4) speaks of the numerals in general as "die der gemain man Zyfer nendt." Recorde (_Grounde of Artes_, 1558 ed., f. B_6) says that the zero is "called priuatly a Cyphar, though all the other sometimes be likewise named." [222] "Decimo X 0 theca, circul[us] cifra sive figura nihili appelat'." [_Enchiridion Algorismi_, Cologne, 1501.] Later, "quoniam de integris tam in cifris quam in proiectilibus,"--the word _proiectilibus_ referring to markers "thrown" and used on an abacus, whence the French _jetons_ and the English expression "to _cast_ an account." [223] "Decima vero o dicitur teca, circulus, vel cyfra vel figura nichili." [Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco commentarius, una cum Algorismo ipso_, Copenhagen, 1897, p. 2.] Curtze cites five manuscripts (fourteenth and fifteenth centuries) of Dacia's commentary in the libraries at Erfurt, Leipzig, and Salzburg, in addition to those given by Eneström, _Öfversigt af Kongl. Vetenskaps-Akademiens Förhandlingar_, 1885, pp. 15-27, 65-70; 1886, pp. 57-60. [224] Curtze, loc. cit., p. VI. [225] _Rara Mathematica_, London, 1841, chap, i, "Joannis de Sacro-Bosco Tractatus de Arte Numerandi." [226] Smith, _Rara Arithmetica_, Boston, 1909. [227] In the 1484 edition, Borghi uses the form "çefiro: ouero nulla:" while in the 1488 edition he uses "zefiro: ouero nulla," and in the 1540 edition, f. 3, appears "Chiamata zero, ouero nulla." Woepcke asserted that it first appeared in Calandri (1491) in this sentence: "Sono dieci le figure con le quali ciascuno numero si può significare: delle quali n'è una che si chiama zero: et per se sola nulla significa." (f. 4). [See _Propagation_, p. 522.] [228] Boncompagni _Bulletino_, Vol. XVI, pp. 673-685. [229] Leo Jordan, loc. cit. In the _Catalogue of MSS., Bibl. de l'Arsenal_, Vol. III, pp. 154-156, this work is No. 2904 (184 S.A.F.), Bibl. Nat., and is also called _Petit traicté de algorisme_. [230] Texada (1546) says that there are "nueue letros yvn zero o cifra" (f. 3). [231] Savonne (1563, 1751 ed., f. 1): "Vne ansi formee (o) qui s'appelle nulle, & entre marchans zero," showing the influence of Italian names on French mercantile customs. Trenchant (Lyons, 1566, 1578 ed., p. 12) also says: "La derniere qui s'apele nulle, ou zero;" but Champenois, his contemporary, writing in Paris in 1577 (although the work was not published until 1578), uses "cipher," the Italian influence showing itself less in this center of university culture than in the commercial atmosphere of Lyons. [232] Thus Radulph of Laon (c. 1100): "Inscribitur in ultimo ordine et figura [symbol] sipos nomine, quae, licet numerum nullum signitet, tantum ad alia quaedam utilis, ut insequentibus declarabitur." ["Der Arithmetische Tractat des Radulph von Laon," _Abhandlungen zur Geschichte der Mathematik_, Vol. V, p. 97, from a manuscript of the thirteenth century.] Chasles (_Comptes rendus_, t. 16, 1843, pp. 1393, 1408) calls attention to the fact that Radulph did not know how to use the zero, and he doubts if the sipos was really identical with it. Radulph says: "... figuram, cui sipos nomen est [symbol] in motum rotulae formatam nullius numeri significatione inscribi solere praediximus," and thereafter uses _rotula_. He uses the sipos simply as a kind of marker on the abacus. [233] Rabbi ben Ezra (1092-1168) used both [Hebrew: GLGL], _galgal_ (the Hebrew for _wheel_), and [Hebrew: SPR'], _sifra_. See M. Steinschneider, "Die Mathematik bei den Juden," in _Bibliotheca Mathematica_, 1893, p. 69, and Silberberg, _Das Buch der Zahl des R. Abraham ibn Esra_, Frankfurt a. M., 1895, p. 96, note 23; in this work the Hebrew letters are used for numerals with place value, having the zero. [234] E.g., in the twelfth-century _Liber aligorismi_ (see Boncompagni's _Trattati_, II, p. 28). So Ramus (_Libri II_, 1569 ed., p. 1) says: "Circulus quæ nota est ultima: nil per se significat." (See also the Schonerus ed. of Ramus, 1586, p. 1.) [235] "Und wirt das ringlein o. die Ziffer genant die nichts bedeut." [Köbel's _Rechenbuch_, 1549 ed., f. 10, and other editions.] [236] I.e. "circular figure," our word _notation_ having come from the medieval _nota_. Thus Tzwivel (1507, f. 2) says: "Nota autem circularis .o. per se sumpta nihil vsus habet. alijs tamen adiuncta earum significantiam et auget et ordinem permutat quantum quo ponit ordinem. vt adiuncta note binarij hoc modo 20 facit eam significare bis decem etc." Also (ibid., f. 4), "figura circularis," "circularis nota." Clichtoveus (1503 ed., f. XXXVII) calls it "nota aut circularis o," "circularis nota," and "figura circularis." Tonstall (1522, f. B_3) says of it: "Decimo uero nota ad formam [symbol] litteræ circulari figura est: quam alij circulum, uulgus cyphram uocat," and later (f. C_4) speaks of the "circulos." Grammateus, in his _Algorismus de integris_ (Erfurt, 1523, f. A_2), speaking of the nine significant figures, remarks: "His autem superadditur decima figura circularis ut 0 existens que ratione sua nihil significat." Noviomagus (_De Numeris libri II_, Paris, 1539, chap. xvi, "De notis numerorum, quas zyphras vocant") calls it "circularis nota, quam ex his solam, alij sipheram, Georgius Valla zyphram." [237] Huswirt, as above. Ramus (_Scholae mathematicae_, 1569 ed., p. 112) discusses the name interestingly, saying: "Circulum appellamus cum multis, quam alii thecam, alii figuram nihili, alii figuram privationis, seu figuram nullam vocant, alii ciphram, cùm tamen hodie omnes hæ notæ vulgò ciphræ nominentur, & his notis numerare idem sit quod ciphrare." Tartaglia (1592 ed., f. 9) says: "si chiama da alcuni tecca, da alcuni circolo, da altri cifra, da altri zero, & da alcuni altri nulla." [238] "Quare autem aliis nominibus vocetur, non dicit auctor, quia omnia alia nomina habent rationem suae lineationis sive figurationis. Quia rotunda est, dicitur haec figura teca ad similitudinem tecae. Teca enim est ferrum figurae rotundae, quod ignitum solet in quibusdam regionibus imprimi fronti vel maxillae furis seu latronum." [Loc. cit., p. 26.] But in Greek _theca_ ([THEKE], [Greek: thêkê]) is a place to put something, a receptacle. If a vacant column, e.g. in the abacus, was so called, the initial might have given the early forms [symbol] and [symbol] for the zero. [239] Buteo, _Logistica_, Lyons, 1559. See also Wertheim in the _Bibliotheca Mathematica_, 1901, p. 214. [240] "0 est appellee chiffre ou nulle ou figure de nulle valeur." [La Roche, _L'arithmétique_, Lyons, 1520.] [241] "Decima autem figura nihil uocata," "figura nihili (quam etiam cifram uocant)." [Stifel, _Arithmetica integra_, 1544, f. 1.] [242] "Zifra, & Nulla uel figura Nihili." [Scheubel, 1545, p. 1 of ch. 1.] _Nulla_ is also used by Italian writers. Thus Sfortunati (1545 ed., f. 4) says: "et la decima nulla & e chiamata questa decima zero;" Cataldi (1602, p. 1): "La prima, che è o, si chiama nulla, ouero zero, ouero niente." It also found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed., f. A_3): "Nullo dat ist niet;" Van der Schuere (1600, 1624 ed., f. 7); Wilkens (1669 ed., p. 1). In Germany Johann Albert (Wittenberg, 1534) and Rudolff (1526) both adopted the Italian _nulla_ and popularized it. (See also Kuckuck, _Die Rechenkunst im sechzehnten Jahrhundert_, Berlin, 1874, p. 7; Günther, _Geschichte_, p. 316.) [243] "La dixième s'appelle chifre vulgairement: les vns l'appellant zero: nous la pourrons appeller vn Rien." [Peletier, 1607 ed., p. 14.] [244] It appears in the Polish arithmetic of K[=l]os (1538) as _cyfra_. "The Ciphra 0 augmenteth places, but of himselfe signifieth not," Digges, 1579, p. 1. Hodder (10th ed., 1672, p. 2) uses only this word (cypher or cipher), and the same is true of the first native American arithmetic, written by Isaac Greenwood (1729, p. 1). Petrus de Dacia derives _cyfra_ from circumference. "Vocatur etiam cyfra, quasi circumfacta vel circumferenda, quod idem est, quod circulus non habito respectu ad centrum." [Loc. cit., p. 26.] [245] _Opera mathematica_, 1695, Oxford, Vol. I, chap. ix, _Mathesis universalis_, "De figuris numeralibus," pp. 46-49; Vol. II, _Algebra_, p. 10. [246] Martin, _Origine de notre système de numération écrite_, note 149, p. 36 of reprint, spells [Greek: tsiphra] from Maximus Planudes, citing Wallis as an authority. This is an error, for Wallis gives the correct form as above. Alexander von Humboldt, "Über die bei verschiedenen Völkern üblichen Systeme von Zahlzeichen und über den Ursprung des Stellenwerthes in den indischen Zahlen," Crelle's _Journal für reine und angewandte Mathematik_, Vol. IV, 1829, called attention to the work [Greek: arithmoi Indikoi] of the monk Neophytos, supposed to be of the fourteenth century. In this work the forms [Greek: tzuphra] and [Greek: tzumphra] appear. See also Boeckh, _De abaco Graecorum_, Berlin, 1841, and Tannery, "Le Scholie du moine Néophytos," _Revue Archéologique_, 1885, pp. 99-102. Jordan, loc. cit., gives from twelfth and thirteenth century manuscripts the forms _cifra_, _ciffre_, _chifras_, and _cifrus_. Du Cange, _Glossarium mediae et infimae Latinitatis_, Paris, 1842, gives also _chilerae_. Dasypodius, _Institutiones Mathematicae_, Strassburg, 1593-1596, adds the forms _zyphra_ and _syphra_. Boissière, _L'art d'arythmetique contenant toute dimention, tres-singulier et commode, tant pour l'art militaire que autres calculations_, Paris, 1554: "Puis y en a vn autre dict zero lequel ne designe nulle quantité par soy, ains seulement les loges vuides." [247] _Propagation_, pp. 27, 234, 442. Treutlein, "Das Rechnen im 16. Jahrhundert," _Abhandlungen zur Geschichte der Mathematik_, Vol. I, p. 5, favors the same view. It is combated by many writers, e.g. A. C. Burnell, loc. cit., p. 59. Long before Woepcke, I. F. and G. I. Weidler, _De characteribus numerorum vulgaribus et eorum aetatibus_, Wittenberg, 1727, asserted the possibility of their introduction into Greece by Pythagoras or one of his followers: "Potuerunt autem ex oriente, uel ex phoenicia, ad graecos traduci, uel Pythagorae, uel eius discipulorum auxilio, cum aliquis eo, proficiendi in literis causa, iter faceret, et hoc quoque inuentum addisceret." [248] E.g., they adopted the Greek numerals in use in Damascus and Syria, and the Coptic in Egypt. Theophanes (758-818 A.D.), _Chronographia_, Scriptores Historiae Byzantinae, Vol. XXXIX, Bonnae, 1839, p. 575, relates that in 699 A.D. the caliph Wal[=i]d forbade the use of the Greek language in the bookkeeping of the treasury of the caliphate, but permitted the use of the Greek alphabetic numerals, since the Arabs had no convenient number notation: [Greek: kai ekôluse graphesthai Hellênisti tous dêmosious tôn logothesiôn kôdikas, all' Arabiois auta parasêmainesthai, chôris tôn psêphôn, epeidê adunaton têi ekeinôn glôssêi monada ê duada ê triada ê oktô hêmisu ê tria graphesthai; dio kai heôs sêmeron eisin sun autois notarioi Christianoi.] The importance of this contemporaneous document was pointed out by Martin, loc. cit. Karabacek, "Die Involutio im arabischen Schriftwesen," Vol. CXXXV of _Sitzungsberichte d. phil.-hist. Classe d. k. Akad. d. Wiss._, Vienna, 1896, p. 25, gives an Arabic date of 868 A.D. in Greek letters. [249] _The Origin and History of Our Numerals_ (in Russian), Kiev, 1908; _The Independence of European Arithmetic_ (in Russian), Kiev. [250] Woepcke, loc. cit., pp. 462, 262. [251] Woepcke, loc. cit., p. 240. _[H.]is[=a]b-al-[.G]ob[=a]r_, by an anonymous author, probably Ab[=u] Sahl Dunash ibn Tamim, is given by Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca Mathematica_, 1896, p. 26. [252] Steinschneider in the _Abhandlungen_, Vol. III, p. 110. [253] See his _Grammaire arabe_, Vol. I, Paris, 1810, plate VIII; Gerhardt, _Études_, pp. 9-11, and _Entstehung_ etc., p. 8; I. F. Weidler, _Spicilegium observationum ad historiam notarum numeralium pertinentium_, Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being different from that of the "characterum Boethianorum," which are similar to the "vulgar" or common numerals; see also Humboldt, loc. cit. [254] Gerhardt mentions it in his _Entstehung_ etc., p. 8; Woepcke, _Propagation_, states that these numerals were used not for calculation, but very much as we use Roman numerals. These superposed dots are found with both forms of numerals (_Propagation_, pp. 244-246). [255] Gerhardt (_Études_, p. 9) from a manuscript in the Bibliothèque Nationale. The numeral forms are [symbols], 20 being indicated by [symbol with dot] and 200 by [symbol with 2 dots]. This scheme of zero dots was also adopted by the Byzantine Greeks, for a manuscript of Planudes in the Bibliothèque Nationale has numbers like [pi alpha with 4 dots] for 8,100,000,000. See Gerhardt, _Études_, p. 19. Pihan, _Exposé_ etc., p. 208, gives two forms, Asiatic and Maghrebian, of "Ghob[=a]r" numerals. [256] See Chap. IV. [257] Possibly as early as the third century A.D., but probably of the eighth or ninth. See Cantor, I (3), p. 598. [258] Ascribed by the Arabic writer to India. [259] See Woepcke's description of a manuscript in the Chasles library, "Recherches sur l'histoire des sciences mathématiques chez les orientaux," _Journal Asiatique_, IV (5), 1859, p. 358, note. [260] P. 56. [261] Reinaud, _Mémoire sur l'Inde_, p. 399. In the fourteenth century one Sih[=a]b al-D[=i]n wrote a work on which, a scholiast to the Bodleian manuscript remarks: "The science is called Algobar because the inventor had the habit of writing the figures on a tablet covered with sand." [Gerhardt, _Études, _p. 11, note.] [262] Gerhardt, _Entstehung _etc., p. 20. [263] H. Suter, "Das Rechenbuch des Ab[=u] Zakar[=i]j[=a] el-[H.]a[s.][s.][=a]r," _Bibliotheca Mathematica_, Vol. II (3), p. 15. [264] A. Devoulx, "Les chiffres arabes," _Revue Africaine_, Vol. XVI, pp. 455-458. [265] _Kit[=a]b al-Fihrist_, G. Flügel, Leipzig, Vol. I, 1871, and Vol. II, 1872. This work was published after Professor Flügel's death by J. Roediger and A. Mueller. The first volume contains the Arabic text and the second volume contains critical notes upon it. [266] Like those of line 5 in the illustration on page 69. [267] Woepcke, _Recherches sur l'histoire des sciences mathématiques chez les orientaux_, loc. cit.; _Propagation, _p. 57. [268] Al-[H.]a[s.][s.][=a]r's forms, Suter, _Bibliotheca Mathematica_, Vol. II (3), p. 15. [269] Woepcke, _Sur une donnée historique_, etc., loc. cit. The name _[.g]ob[=a]r_ is not used in the text. The manuscript from which these are taken is the oldest (970 A.D.) Arabic document known to contain all of the numerals. [270] Silvestre de Sacy, loc. cit. He gives the ordinary modern Arabic forms, calling them _Indien_. [271] Woepcke, "Introduction au calcul Gob[=a]r[=i] et Haw[=a][=i]," _Atti dell' accademia pontificia dei nuovi Lincei_, Vol. XIX. The adjective applied to the forms in 5 is _gob[=a]r[=i]_ and to those in 6 _indienne_. This is the direct opposite of Woepcke's use of these adjectives in the _Recherches sur l'histoire_ cited above, in which the ordinary Arabic forms (like those in row 5) are called _indiens_. These forms are usually written from right to left. [272] J. G. Wilkinson, _The Manners and Customs of the Ancient Egyptians_, revised by S. Birch, London, 1878, Vol. II, p. 493, plate XVI. [273] There is an extensive literature on this "Boethius-Frage." The reader who cares to go fully into it should consult the various volumes of the _Jahrbuch über die Fortschritte der Mathematik_. [274] This title was first applied to Roman emperors in posthumous coins of Julius Cæsar. Subsequently the emperors assumed it during their own lifetimes, thus deifying themselves. See F. Gnecchi, _Monete romane_, 2d ed., Milan, 1900, p. 299. [275] This is the common spelling of the name, although the more correct Latin form is Boëtius. See Harper's _Dict. of Class. Lit. and Antiq._, New York, 1897, Vol. I, p. 213. There is much uncertainty as to his life. A good summary of the evidence is given in the last two editions of the _Encyclopædia Britannica_. [276] His father, Flavius Manlius Boethius, was consul in 487. [277] There is, however, no good historic evidence of this sojourn in Athens. [278] His arithmetic is dedicated to Symmachus: "Domino suo patricio Symmacho Boetius." [Friedlein ed., p. 3.] [279] It was while here that he wrote _De consolatione philosophiae_. [280] It is sometimes given as 525. [281] There was a medieval tradition that he was executed because of a work on the Trinity. [282] Hence the _Divus_ in his name. [283] Thus Dante, speaking of his burial place in the monastery of St. Pietro in Ciel d'Oro, at Pavia, says: "The saintly soul, that shows The world's deceitfulness, to all who hear him, Is, with the sight of all the good that is, Blest there. The limbs, whence it was driven, lie Down in Cieldauro; and from martyrdom And exile came it here."--_Paradiso_, Canto X. [284] Not, however, in the mercantile schools. The arithmetic of Boethius would have been about the last book to be thought of in such institutions. While referred to by Bæda (672-735) and Hrabanus Maurus (c. 776-856), it was only after Gerbert's time that the _Boëtii de institutione arithmetica libri duo_ was really a common work. [285] Also spelled Cassiodorius. [286] As a matter of fact, Boethius could not have translated any work by Pythagoras on music, because there was no such work, but he did make the theories of the Pythagoreans known. Neither did he translate Nicomachus, although he embodied many of the ideas of the Greek writer in his own arithmetic. Gibbon follows Cassiodorus in these statements in his _Decline and Fall of the Roman Empire_, chap. xxxix. Martin pointed out with positiveness the similarity of the first book of Boethius to the first five books of Nicomachus. [_Les signes numéraux_ etc., reprint, p. 4.] [287] The general idea goes back to Pythagoras, however. [288] J. C. Scaliger in his _Poëtice_ also said of him: "Boethii Severini ingenium, eruditio, ars, sapientia facile provocat omnes auctores, sive illi Graeci sint, sive Latini" [Heilbronner, _Hist. math. univ._, p. 387]. Libri, speaking of the time of Boethius, remarks: "Nous voyons du temps de Théodoric, les lettres reprendre une nouvelle vie en Italie, les écoles florissantes et les savans honorés. Et certes les ouvrages de Boëce, de Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du siècle précédent." [_Histoire des mathématiques_, Vol. I, p. 78.] [289] Carra de Vaux, _Avicenne_, Paris, 1900; Woepcke, _Sur l'introduction_, etc.; Gerhardt, _Entstehung_ etc., p. 20. Avicenna is a corruption from Ibn S[=i]n[=a], as pointed out by Wüstenfeld, _Geschichte der arabischen Aerzte und Naturforscher_, Göttingen, 1840. His full name is Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a]. For notes on Avicenna's arithmetic, see Woepcke, _Propagation_, p. 502. [290] On the early travel between the East and the West the following works may be consulted: A. Hillebrandt, _Alt-Indien_, containing "Chinesische Reisende in Indien," Breslau, 1899, p. 179; C. A. Skeel, _Travel in the First Century after Christ_, Cambridge, 1901, p. 142; M. Reinaud, "Relations politiques et commerciales de l'empire romain avec l'Asie orientale," in the _Journal Asiatique_, Mars-Avril, 1863, Vol. I (6), p. 93; Beazley, _Dawn of Modern Geography, a History of Exploration and Geographical Science from the Conversion of the Roman Empire to A.D. 1420_, London, 1897-1906, 3 vols.; Heyd, _Geschichte des Levanthandels im Mittelalter_, Stuttgart, 1897; J. Keane, _The Evolution of Geography_, London, 1899, p. 38; A. Cunningham, _Corpus inscriptionum Indicarum_, Calcutta, 1877, Vol. I; A. Neander, _General History of the Christian Religion and Church_, 5th American ed., Boston, 1855, Vol. III, p. 89; R. C. Dutt, _A History of Civilization in Ancient India_, Vol. II, Bk. V, chap, ii; E. C. Bayley, loc. cit., p. 28 et seq.; A. C. Burnell, loc. cit., p. 3; J. E. Tennent, _Ceylon_, London, 1859, Vol. I, p. 159; Geo. Turnour, _Epitome of the History of Ceylon_, London, n.d., preface; "Philalethes," _History of Ceylon_, London, 1816, chap, i; H. C. Sirr, _Ceylon and the Cingalese_, London, 1850, Vol. I, chap. ix. On the Hindu knowledge of the Nile see F. Wilford, _Asiatick Researches_, Vol. III, p. 295, Calcutta, 1792. [291] G. Oppert, _On the Ancient Commerce of India_, Madras, 1879, p. 8. [292] Gerhardt, _Études_ etc., pp. 8, 11. [293] See Smith's _Dictionary of Greek and Roman Biography and Mythology_. [294] P. M. Sykes, _Ten Thousand Miles in Persia, or Eight Years in Irán_, London, 1902, p. 167. Sykes was the first European to follow the course of Alexander's army across eastern Persia. [295] Bühler, _Indian Br[=a]hma Alphabet_, note, p. 27; _Palaeographie_, p. 2; _Herodoti Halicarnassei historia_, Amsterdam, 1763, Bk. IV, p. 300; Isaac Vossius, _Periplus Scylacis Caryandensis_, 1639. It is doubtful whether the work attributed to Scylax was written by him, but in any case the work dates back to the fourth century B.C. See Smith's _Dictionary of Greek and Roman Biography_. [296] Herodotus, Bk. III. [297] Rameses II(?), the _Sesoosis_ of Diodorus Siculus. [298] _Indian Antiquary_, Vol. I, p. 229; F. B. Jevons, _Manual of Greek Antiquities_, London, 1895, p. 386. On the relations, political and commercial, between India and Egypt c. 72 B.C., under Ptolemy Auletes, see the _Journal Asiatique_, 1863, p. 297. [299] Sikandar, as the name still remains in northern India. [300] _Harper's Classical Dict._, New York, 1897, Vol. I, p. 724; F. B. Jevons, loc. cit., p. 389; J. C. Marshman, _Abridgment of the History of India_, chaps. i and ii. [301] Oppert, loc. cit., p. 11. It was at or near this place that the first great Indian mathematician, [=A]ryabha[t.]a, was born in 476 A.D. [302] Bühler, _Palaeographie_, p. 2, speaks of Greek coins of a period anterior to Alexander, found in northern India. More complete information may be found in _Indian Coins_, by E. J. Rapson, Strassburg, 1898, pp. 3-7. [303] Oppert, loc. cit., p. 14; and to him is due other similar information. [304] J. Beloch, _Griechische Geschichte_, Vol. III, Strassburg, 1904, pp. 30-31. [305] E.g., the denarius, the words for hour and minute ([Greek: hôra, lepton]), and possibly the signs of the zodiac. [R. Caldwell, _Comparative Grammar of the Dravidian Languages_, London, 1856, p. 438.] On the probable Chinese origin of the zodiac see Schlegel, loc. cit. [306] Marie, Vol. II, p. 73; R. Caldwell, loc. cit. [307] A. Cunningham, loc. cit., p. 50. [308] C. A. J. Skeel, _Travel_, loc. cit., p. 14. [309] _Inchiver_, from _inchi_, "the green root." [_Indian Antiquary_, Vol. I, p. 352.] [310] In China dating only from the second century A.D., however. [311] The Italian _morra_. [312] J. Bowring, _The Decimal System_, London, 1854, p. 2. [313] H. A. Giles, lecture at Columbia University, March 12, 1902, on "China and Ancient Greece." [314] Giles, loc. cit. [315] E.g., the names for grape, radish (_la-po_, [Greek: rhaphê]), water-lily (_si-kua_, "west gourds"; [Greek: sikua], "gourds"), are much alike. [Giles, loc. cit.] [316] _Epistles_, I, 1, 45-46. On the Roman trade routes, see Beazley, loc. cit., Vol. I, p. 179. [317] _Am. Journ. of Archeol._, Vol. IV, p. 366. [318] M. Perrot gives this conjectural restoration of his words: "Ad me ex India regum legationes saepe missi sunt numquam antea visae apud quemquam principem Romanorum." [M. Reinaud, "Relations politiques et commerciales de l'empire romain avec l'Asie orientale," _Journ. Asiat._, Vol. I (6), p. 93.] [319] Reinaud, loc. cit., p. 189. Florus, II, 34 (IV, 12), refers to it: "Seres etiam habitantesque sub ipso sole Indi, cum gemmis et margaritis elephantes quoque inter munera trahentes nihil magis quam longinquitatem viae imputabant." Horace shows his geographical knowledge by saying: "Not those who drink of the deep Danube shall now break the Julian edicts; not the Getae, not the Seres, nor the perfidious Persians, nor those born on the river Tanaïs." [_Odes_, Bk. IV, Ode 15, 21-24.] [320] "Qua virtutis moderationisque fama Indos etiam ac Scythas auditu modo cognitos pellexit ad amicitiam suam populique Romani ultro per legatos petendam." [Reinaud, loc. cit., p. 180.] [321] Reinaud, loc. cit., p. 180. [322] _Georgics_, II, 170-172. So Propertius (_Elegies_, III, 4): Arma deus Caesar dites meditatur ad Indos Et freta gemmiferi findere classe maris. "The divine Cæsar meditated carrying arms against opulent India, and with his ships to cut the gem-bearing seas." [323] Heyd, loc. cit., Vol. I, p. 4. [324] Reinaud, loc. cit., p. 393. [325] The title page of Calandri (1491), for example, represents Pythagoras with these numerals before him. [Smith, _Rara Arithmetica_, p. 46.] Isaacus Vossius, _Observationes ad Pomponium Melam de situ orbis_, 1658, maintained that the Arabs derived these numerals from the west. A learned dissertation to this effect, but deriving them from the Romans instead of the Greeks, was written by Ginanni in 1753 (_Dissertatio mathematica critica de numeralium notarum minuscularum origine_, Venice, 1753). See also Mannert, _De numerorum quos arabicos vocant vera origine Pythagorica_, Nürnberg, 1801. Even as late as 1827 Romagnosi (in his supplement to _Ricerche storiche sull' India_ etc., by Robertson, Vol. II, p. 580, 1827) asserted that Pythagoras originated them. [R. Bombelli, _L'antica numerazione italica_, Rome, 1876, p. 59.] Gow (_Hist. of Greek Math._, p. 98) thinks that Iamblichus must have known a similar system in order to have worked out certain of his theorems, but this is an unwarranted deduction from the passage given. [326] A. Hillebrandt, _Alt-Indien_, p. 179. [327] J. C. Marshman, loc. cit., chaps. i and ii. [328] He reigned 631-579 A.D.; called Nu['s][=i]rw[=a]n, _the holy one_. [329] J. Keane, _The Evolution of Geography_, London, 1899, p. 38. [330] The Arabs who lived in and about Mecca. [331] S. Guyard, in _Encyc. Brit._, 9th ed., Vol. XVI, p. 597. [332] Oppert, loc. cit., p. 29. [333] "At non credendum est id in Autographis contigisse, aut vetustioribus Codd. MSS." [Wallis, _Opera omnia_, Vol. II, p. 11.] [334] In _Observationes ad Pomponium Melam de situ orbis_. The question was next taken up in a large way by Weidler, loc. cit., _De characteribus_ etc., 1727, and in _Spicilegium_ etc., 1755. [335] The best edition of these works is that of G. Friedlein, _Anicii Manlii Torquati Severini Boetii de institutione arithmetica libri duo, de institutione musica libri quinque. Accedit geometria quae fertur Boetii_.... Leipzig.... MDCCCLXVII. [336] See also P. Tannery, "Notes sur la pseudo-géometrie de Boèce," in _Bibliotheca Mathematica_, Vol. I (3), p. 39. This is not the geometry in two books in which are mentioned the numerals. There is a manuscript of this pseudo-geometry of the ninth century, but the earliest one of the other work is of the eleventh century (Tannery), unless the Vatican codex is of the tenth century as Friedlein (p. 372) asserts. [337] Friedlein feels that it is partly spurious, but he says: "Eorum librorum, quos Boetius de geometria scripsisse dicitur, investigare veram inscriptionem nihil aliud esset nisi operam et tempus perdere." [Preface, p. v.] N. Bubnov in the Russian _Journal of the Ministry of Public Instruction_, 1907, in an article of which a synopsis is given in the _Jahrbuch über die Fortschritte der Mathematik_ for 1907, asserts that the geometry was written in the eleventh century. [338] The most noteworthy of these was for a long time Cantor (_Geschichte_, Vol. I., 3d ed., pp. 587-588), who in his earlier days even believed that Pythagoras had known them. Cantor says (_Die römischen Agrimensoren_, Leipzig, 1875, p. 130): "Uns also, wir wiederholen es, ist die Geometrie des Boetius echt, dieselbe Schrift, welche er nach Euklid bearbeitete, von welcher ein Codex bereits in Jahre 821 im Kloster Reichenau vorhanden war, von welcher ein anderes Exemplar im Jahre 982 zu Mantua in die Hände Gerbert's gelangte, von welcher mannigfache Handschriften noch heute vorhanden sind." But against this opinion of the antiquity of MSS. containing these numerals is the important statement of P. Tannery, perhaps the most critical of modern historians of mathematics, that none exists earlier than the eleventh century. See also J. L. Heiberg in _Philologus, Zeitschrift f. d. klass. Altertum_, Vol. XLIII, p. 508. Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his argument for authenticity appearing in the _Revue Archéologique_ for 1856-1857, and in his treatise _Les signes numéraux_ etc. See also M. Chasles, "De la connaissance qu'ont eu les anciens d'une numération décimale écrite qui fait usage de neuf chiffres prenant les valeurs de position," _Comptes rendus_, Vol. VI, pp. 678-680; "Sur l'origine de notre système de numération," _Comptes rendus_, Vol. VIII, pp. 72-81; and note "Sur le passage du premier livre de la géométrie de Boèce, relatif à un nouveau système de numération," in his work _Aperçu historique sur l'origine et le devéloppement des méthodes en géométrie_, of which the first edition appeared in 1837. [339] J. L. Heiberg places the book in the eleventh century on philological grounds, _Philologus_, loc. cit.; Woepcke, in _Propagation_, p. 44; Blume, Lachmann, and Rudorff, _Die Schriften der römischen Feldmesser_, Berlin, 1848; Boeckh, _De abaco graecorum_, Berlin, 1841; Friedlein, in his Leipzig edition of 1867; Weissenborn, _Abhandlungen_, Vol. II, p. 185, his _Gerbert_, pp. 1, 247, and his _Geschichte der Einführung der jetzigen Ziffern in Europa durch Gerbert_, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59; Gerhardt, _Études_, p. 17, _Entstehung und Ausbreitung_, p. 14; Nagl, _Gerbert_, p. 57; Bubnov, loc. cit. See also the discussion by Chasles, Halliwell, and Libri, in the _Comptes rendus_, 1839, Vol. IX, p. 447, and in Vols. VIII, XVI, XVII of the same journal. [340] J. Marquardt, _La vie privée des Romains_, Vol. II (French trans.), p. 505, Paris, 1893. [341] In a Plimpton manuscript of the arithmetic of Boethius of the thirteenth century, for example, the Roman numerals are all replaced by the Arabic, and the same is true in the first printed edition of the book. (See Smith's _Rara Arithmetica_, pp. 434, 25-27.) D. E. Smith also copied from a manuscript of the arithmetic in the Laurentian library at Florence, of 1370, the following forms, [Forged numerals [342] Halliwell, in his _Rara Mathematica, _p. 107, states that the disputed passage is not in a manuscript belonging to Mr. Ames, nor in one at Trinity College. See also Woepcke, in _Propagation_, pp. 37 and 42. It was the evident corruption of the texts in such editions of Boethius as those of Venice, 1499, Basel, 1546 and 1570, that led Woepcke to publish his work _Sur l'introduction de l'arithmétique indienne en Occident_. [343] They are found in none of the very ancient manuscripts, as, for example, in the ninth-century (?) codex in the Laurentian library which one of the authors has examined. It should be said, however, that the disputed passage was written after the arithmetic, for it contains a reference to that work. See the Friedlein ed., p. 397. [344] Smith, _Rara Arithmetica_, p. 66. [345] J. L. Heiberg, _Philologus_, Vol. XLIII, p. 507. [346] "Nosse autem huius artis dispicientem, quid sint digiti, quid articuli, quid compositi, quid incompositi numeri." [Friedlein ed., p. 395.] [347] _De ratione abaci._ In this he describes "quandam formulam, quam ob honorem sui praeceptoris mensam Pythagoream nominabant ... a posterioribus appellabatur abacus." This, as pictured in the text, is the common Gerbert abacus. In the edition in Migne's _Patrologia Latina_, Vol. LXIII, an ordinary multiplication table (sometimes called Pythagorean abacus) is given in the illustration. [348] "Habebant enim diverse formatos apices vel caracteres." See the reference to Gerbert on p. 117. [349] C. Henry, "Sur l'origine de quelques notations mathématiques," _Revue Archéologique_, 1879, derives these from the initial letters used as abbreviations for the names of the numerals, a theory that finds few supporters. [350] E.g., it appears in Schonerus, _Algorithmus Demonstratus_, Nürnberg, 1534, f. A4. In England it appeared in the earliest English arithmetical manuscript known, _The Crafte of Nombrynge_: "¶ fforthermore ye most vndirstonde that in this craft ben vsid teen figurys, as here bene writen for ensampul, [Numerals] ... in the quych we vse teen figurys of Inde. Questio. ¶ why ten fyguris of Inde? Solucio. for as I have sayd afore thei were fonde fyrst in Inde of a kynge of that Cuntre, that was called Algor." See Smith, _An Early English Algorism_, loc. cit. [351] Friedlein ed., p. 397. [352] Carlsruhe codex of Gerlando. [353] Munich codex of Gerlando. [354] Carlsruhe codex of Bernelinus. [355] Munich codex of Bernelinus. [356] Turchill, c. 1200. [357] Anon. MS., thirteenth century, Alexandrian Library, Rome. [358] Twelfth-century Boethius, Friedlein, p. 396. [359] Vatican codex, tenth century, Boethius. [360] a, h, i, are from the Friedlein ed.; the original in the manuscript from which a is taken contains a zero symbol, as do all of the six plates given by Friedlein. b-e from the Boncompagni _Bulletino_, Vol. X, p. 596; f ibid., Vol. XV, p. 186; g _Memorie della classe di sci., Reale Acc. dei Lincei_, An. CCLXXIV (1876-1877), April, 1877. A twelfth-century arithmetician, possibly John of Luna (Hispalensis, of Seville, c. 1150), speaks of the great diversity of these forms even in his day, saying: "Est autem in aliquibus figuram istarum apud multos diuersitas. Quidam enim septimam hanc figuram representant [Symbol] alii autem sic [Symbol], uel sic [Symbol]. Quidam vero quartam sic [Symbol]." [Boncompagni, _Trattati_, Vol. II, p. 28.] [361] Loc. cit., p. 59. [362] Ibid., p. 101. [363] Loc. cit., p. 396. [364] Khosr[=u] I, who began to reign in 531 A.D. See W. S. W Vaux, _Persia, _London, 1875, p. 169; Th. Nöldeke, _Aufsätze zur persichen Geschichte_, Leipzig, 1887, p. 113, and his article in the ninth edition of the _Encyclopædia Britannica_. [365] Colebrooke, _Essays_, Vol. II, p. 504, on the authority of Ibn al-Adam[=i], astronomer, in a work published by his continuator Al-Q[=a]sim in 920 A.D.; Al-B[=i]r[=u]n[=i], _India, _Vol. II, p. 15. [366] H. Suter, _Die Mathematiker_ etc., pp. 4-5, states that Al-Faz[=a]r[=i] died between 796 and 806. [367] Suter, loc. cit., p. 63. [368] Suter, loc. cit., p. 74. [369] Suter, _Das Mathematiker-Verzeichniss im Fihrist_. The references to Suter, unless otherwise stated, are to his later work _Die Mathematiker und Astronomen der Araber_ etc. [370] Suter, _Fihrist_, p. 37, no date. [371] Suter, _Fihrist_, p. 38, no date. [372] Possibly late tenth, since he refers to one arithmetical work which is entitled _Book of the Cyphers_ in his _Chronology_, English ed., p. 132. Suter, _Die Mathematiker_ etc., pp. 98-100, does not mention this work; see the _Nachträge und Berichtigungen_, pp. 170-172. [373] Suter, pp. 96-97. [374] Suter, p. 111. [375] Suter, p. 124. As the name shows, he came from the West. [376] Suter, p. 138. [377] Hankel, _Zur Geschichte der Mathematik_, p. 256, refers to him as writing on the Hindu art of reckoning; Suter, p. 162. [378] [Greek: Psêphophoria kat' Indous], Greek ed., C. I. Gerhardt, Halle, 1865; and German translation, _Das Rechenbuch des Maximus Planudes_, H. Wäschke, Halle, 1878. [379] "Sur une donnée historique relative à l'emploi des chiffres indiens par les Arabes," Tortolini's _Annali di scienze mat. e fis._, 1855. [380] Suter, p. 80. [381] Suter, p. 68. [382] Sprenger also calls attention to this fact, in the _Zeitschrift d. deutschen morgenländ. Gesellschaft_, Vol. XLV, p. 367. [383] Libri, _Histoire des mathématiques_, Vol. I, p. 147. [384] "Dictant la paix à l'empereur de Constantinople, l'Arabe victorieux demandait des manuscrits et des savans." [Libri, loc. cit., p. 108.] [385] Persian _bagadata_, "God-given." [386] One of the Abbassides, the (at least pretended) descendants of `Al-Abb[=a]s, uncle and adviser of Mo[h.]ammed. [387] E. Reclus, _Asia_, American ed., N. Y., 1891, Vol. IV, p. 227. [388] _Historical Sketches_, Vol. III, chap. iii. [389] On its prominence at that period see Villicus, p. 70. [390] See pp. 4-5. [391] Smith, D. E., in the _Cantor Festschrift_, 1909, note pp. 10-11. See also F. Woepcke, _Propagation_. [392] Eneström, in _Bibliotheca Mathematica_, Vol. I (3), p. 499; Cantor, _Geschichte_, Vol. I (3), p. 671. [393] Cited in Chapter I. It begins: "Dixit algoritmi: laudes deo rectori nostro atque defensori dicamus dignas." It is devoted entirely to the fundamental operations and contains no applications. [394] M. Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca Mathematica_, Vol. VIII (2), p. 99. See also the reference to this writer in Chapter I. [395] Part of this work has been translated from a Leyden MS. by F. Woepcke, _Propagation_, and more recently by H. Suter, _Bibliotheca Mathematica_, Vol. VII (3), pp. 113-119. [396] A. Neander, _General History of the Christian Religion and Church_, 5th American ed., Boston, 1855, Vol. III, p. 335. [397] Beazley, loc. cit., Vol. I, p. 49. [398] Beazley, loc. cit., Vol. I, pp. 50, 460. [399] See pp. 7-8. [400] The name also appears as Mo[h.]ammed Ab[=u]'l-Q[=a]sim, and Ibn Hauqal. Beazley, loc. cit., Vol. I, p. 45. [401] _Kit[=a]b al-mas[=a]lik wa'l-mam[=a]lik._ [402] Reinaud, _Mém. sur l'Inde_; in Gerhardt, _Études_, p. 18. [403] Born at Shiraz in 1193. He himself had traveled from India to Europe. [404] _Gulistan_ (_Rose Garden_), Gateway the third, XXII. Sir Edwin Arnold's translation, N. Y., 1899, p. 177. [405] Cunningham, loc. cit., p. 81. [406] Putnam, _Books_, Vol. I, p. 227: "Non semel externas peregrino tramite terras Jam peragravit ovans, sophiae deductus amore, Si quid forte novi librorum seu studiorum Quod secum ferret, terris reperiret in illis. Hic quoque Romuleum venit devotus ad urbem." ("More than once he has traveled joyfully through remote regions and by strange roads, led on by his zeal for knowledge and seeking to discover in foreign lands novelties in books or in studies which he could take back with him. And this zealous student journeyed to the city of Romulus.") [407] A. Neander, _General History of the Christian Religion and Church_, 5th American ed., Boston, 1855, Vol. III, p. 89, note 4; Libri, _Histoire_, Vol. I, p. 143. [408] Cunningham, loc. cit., p. 81. [409] Heyd, loc. cit., Vol. I, p. 4. [410] Ibid., p. 5. [411] Ibid., p. 21. [412] Ibid., p. 23. [413] Libri, _Histoire_, Vol. I, p. 167. [414] Picavet, _Gerbert, un pape philosophe, d'après l'histoire et d'après la légende_, Paris, 1897, p. 19. [415] Beazley, loc. cit., Vol. I, chap, i, and p. 54 seq. [416] Ibid., p. 57. [417] Libri, _Histoire_, Vol. I, p. 110, n., citing authorities, and p. 152. [418] Possibly the old tradition, "Prima dedit nautis usum magnetis Amalphis," is true so far as it means the modern form of compass card. See Beazley, loc. cit., Vol. II, p. 398. [419] R. C. Dutt, loc. cit., Vol. II, p. 312. [420] E. J. Payne, in _The Cambridge Modern History_, London, 1902, Vol. I, chap. i. [421] Geo. Phillips, "The Identity of Marco Polo's Zaitun with Changchau, in T'oung pao," _Archives pour servir à l'étude de l'histoire de l'Asie orientale_, Leyden, 1890, Vol. I, p. 218. W. Heyd, _Geschichte des Levanthandels im Mittelalter_, Vol. II, p. 216. The Palazzo dei Poli, where Marco was born and died, still stands in the Corte del Milione, in Venice. The best description of the Polo travels, and of other travels of the later Middle Ages, is found in C. R. Beazley's _Dawn of Modern Geography_, Vol. III, chap, ii, and Part II. [422] Heyd, loc. cit., Vol. II, p. 220; H. Yule, in _Encyclopædia Britannica_, 9th (10th) or 11th ed., article "China." The handbook cited is Pegolotti's _Libro di divisamenti di paesi_, chapters i-ii, where it is implied that $60,000 would be a likely amount for a merchant going to China to invest in his trip. [423] Cunningham, loc. cit., p. 194. [424] I.e. a commission house. [425] Cunningham, loc. cit., p. 186. [426] J. R. Green, _Short History of the English People_, New York, 1890, p. 66. [427] W. Besant, _London_, New York, 1892, p. 43. [428] _Baldakin_, _baldekin_, _baldachino_. [429] Italian _Baldacco_. [430] J. K. Mumford, _Oriental Rugs_, New York, 1901, p. 18. [431] Or Girbert, the Latin forms _Gerbertus_ and _Girbertus_ appearing indifferently in the documents of his time. [432] See, for example, J. C. Heilbronner, _Historia matheseos universæ_, p. 740. [433] "Obscuro loco natum," as an old chronicle of Aurillac has it. [434] N. Bubnov, _Gerberti postea Silvestri II papae opera mathematica_, Berlin, 1899, is the most complete and reliable source of information; Picavet, loc. cit., _Gerbert_ etc.; Olleris, _Oeuvres de Gerbert_, Paris, 1867; Havet, _Lettres de Gerbert_, Paris, 1889 ; H. Weissenborn, _Gerbert; Beiträge zur Kenntnis der Mathematik des Mittelalters_, Berlin, 1888, and _Zur Geschichte der Einführung der jetzigen Ziffern in Europa durch Gerbert_, Berlin, 1892; Büdinger, _Ueber Gerberts wissenschaftliche und politische Stellung_, Cassel, 1851; Richer, "Historiarum liber III," in Bubnov, loc. cit., pp. 376-381; Nagl, _Gerbert und die Rechenkunst des 10. Jahrhunderts_, Vienna, 1888. [435] Richer tells of the visit to Aurillac by Borel, a Spanish nobleman, just as Gerbert was entering into young manhood. He relates how affectionately the abbot received him, asking if there were men in Spain well versed in the arts. Upon Borel's reply in the affirmative, the abbot asked that one of his young men might accompany him upon his return, that he might carry on his studies there. [436] Vicus Ausona. Hatto also appears as Atton and Hatton. [437] This is all that we know of his sojourn in Spain, and this comes from his pupil Richer. The stories told by Adhemar of Chabanois, an apparently ignorant and certainly untrustworthy contemporary, of his going to Cordova, are unsupported. (See e.g. Picavet, p. 34.) Nevertheless this testimony is still accepted: K. von Raumer, for example (_Geschichte der Pädagogik_, 6th ed., 1890, Vol. I, p. 6), says "Mathematik studierte man im Mittelalter bei den Arabern in Spanien. Zu ihnen gieng Gerbert, nachmaliger Pabst Sylvester II." [438] Thus in a letter to Aldaberon he says: "Quos post repperimus speretis, id est VIII volumina Boeti de astrologia, praeclarissima quoque figurarum geometriæ, aliaque non minus admiranda" (Epist. 8). Also in a letter to Rainard (Epist. 130), he says: "Ex tuis sumptibus fac ut michi scribantur M. Manlius (Manilius in one MS.) de astrologia." [439] Picavet, loc. cit., p. 31. [440] Picavet, loc. cit., p. 36. [441] Havet, loc. cit., p. vii. [442] Picavet, loc. cit., p. 37. [443] "Con sinistre arti conseguri la dignita del Pontificato.... Lasciato poi l' abito, e 'l monasterio, e datosi tutto in potere del diavolo." [Quoted in Bombelli, _L'antica numerazione Italica_, Rome, 1876, p. 41 n.] [444] He writes from Rheims in 984 to one Lupitus, in Barcelona, saying: "Itaque librum de astrologia translatum a te michi petenti dirige," presumably referring to some Arabic treatise. [Epist. no. 24 of the Havet collection, p. 19.] [445] See Bubnov, loc. cit., p. x. [446] Olleris, loc. cit., p. 361, l. 15, for Bernelinus; and Bubnov, loc. cit., p. 381, l. 4, for Richer. [447] Woepcke found this in a Paris MS. of Radulph of Laon, c. 1100. [_Propagation_, p. 246.] "Et prima quidem trium spaciorum superductio unitatis caractere inscribitur, qui chaldeo nomine dicitur igin." See also Alfred Nagl, "Der arithmetische Tractat des Radulph von Laon" (_Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133), p. 97. [448] Weissenborn, loc. cit., p. 239. When Olleris (_Oeuvres de Gerbert_, Paris, 1867, p. cci) says, "C'est à lui et non point aux Arabes, que l'Europe doit son système et ses signes de numération," he exaggerates, since the evidence is all against his knowing the place value. Friedlein emphasizes this in the _Zeitschrift für Mathematik und Physik_, Vol. XII (1867), _Literaturzeitung_, p. 70: "Für das _System_ unserer Numeration ist die _Null_ das wesentlichste Merkmal, und diese kannte Gerbert nicht. Er selbst schrieb alle Zahlen mit den römischen Zahlzeichen und man kann ihm also nicht verdanken, was er selbst nicht kannte." [449] E.g., Chasles, Büdinger, Gerhardt, and Richer. So Martin (_Recherches nouvelles_ etc.) believes that Gerbert received them from Boethius or his followers. See Woepcke, _Propagation_, p. 41. [450] Büdinger, loc. cit., p. 10. Nevertheless, in Gerbert's time one Al-Man[s.][=u]r, governing Spain under the name of Hish[=a]m (976-1002), called from the Orient Al-Be[.g][=a]n[=i] to teach his son, so that scholars were recognized. [Picavet, p. 36.] [451] Weissenborn, loc. cit., p. 235. [452] Ibid., p. 234. [453] These letters, of the period 983-997, were edited by Havet, loc. cit., and, less completely, by Olleris, loc. cit. Those touching mathematical topics were edited by Bubnov, loc. cit., pp. 98-106. [454] He published it in the _Monumenta Germaniae historica_, "Scriptores," Vol. III, and at least three other editions have since appeared, viz. those by Guadet in 1845, by Poinsignon in 1855, and by Waitz in 1877. [455] Domino ac beatissimo Patri Gerberto, Remorum archiepiscopo, Richerus Monchus, Gallorum congressibus in volumine regerendis, imperii tui, pater sanctissime Gerberte, auctoritas seminarium dedit. [456] In epistle 17 (Havet collection) he speaks of the "De multiplicatione et divisione numerorum libellum a Joseph Ispano editum abbas Warnerius" (a person otherwise unknown). In epistle 25 he says: "De multiplicatione et divisione numerorum, Joseph Sapiens sententias quasdam edidit." [457] H. Suter, "Zur Frage über den Josephus Sapiens," _Bibliotheca Mathematica_, Vol. VIII (2), p. 84; Weissenborn, _Einführung_, p. 14; also his _Gerbert_; M. Steinschneider, in _Bibliotheca Mathematica_, 1893, p. 68. Wallis (_Algebra_, 1685, chap. 14) went over the list of Spanish Josephs very carefully, but could find nothing save that "Josephus Hispanus seu Josephus sapiens videtur aut Maurus fuisse aut alius quis in Hispania." [458] P. Ewald, _Mittheilungen, Neues Archiv d. Gesellschaft für ältere deutsche Geschichtskunde_, Vol. VIII, 1883, pp. 354-364. One of the manuscripts is of 976 A.D. and the other of 992 A.D. See also Franz Steffens, _Lateinische Paläographie_, Freiburg (Schweiz), 1903, pp. xxxix-xl. The forms are reproduced in the plate on page 140. [459] It is entitled _Constantino suo Gerbertus scolasticus_, because it was addressed to Constantine, a monk of the Abbey of Fleury. The text of the letter to Constantine, preceding the treatise on the Abacus, is given in the _Comptes rendus_, Vol. XVI (1843), p. 295. This book seems to have been written c. 980 A.D. [Bubnov, loc. cit., p. 6.] [460] "Histoire de l'Arithmétique," _Comptes rendus_, Vol. XVI (1843), pp. 156, 281. [461] Loc. cit., _Gerberti Opera_ etc. [462] Friedlein thought it spurious. See _Zeitschrift für Mathematik und Physik_, Vol. XII (1867), Hist.-lit. suppl., p. 74. It was discovered in the library of the Benedictine monastry of St. Peter, at Salzburg, and was published by Peter Bernhard Pez in 1721. Doubt was first cast upon it in the Olleris edition (_Oeuvres de Gerbert_). See Weissenborn, _Gerbert_, pp. 2, 6, 168, and Picavet, p. 81. Hock, Cantor, and Th. Martin place the composition of the work at c. 996 when Gerbert was in Germany, while Olleris and Picavet refer it to the period when he was at Rheims. [463] Picavet, loc. cit., p. 182. [464] Who wrote after Gerbert became pope, for he uses, in his preface, the words, "a domino pape Gerberto." He was quite certainly not later than the eleventh century; we do not have exact information about the time in which he lived. [465] Picavet, loc. cit., p. 182. Weissenborn, _Gerbert_, p. 227. In Olleris, _Liber Abaci_ (of Bernelinus), p. 361. [466] Richer, in Bubnov, loc. cit., p. 381. [467] Weissenborn, _Gerbert_, p. 241. [468] Writers on numismatics are quite uncertain as to their use. See F. Gnecchi, _Monete Romane_, 2d ed., Milan, 1900, cap. XXXVII. For pictures of old Greek tesserae of Sarmatia, see S. Ambrosoli, _Monete Greche_, Milan, 1899, p. 202. [469] Thus Tzwivel's arithmetic of 1507, fol. 2, v., speaks of the ten figures as "characteres sive numerorum apices a diuo Seuerino Boetio." [470] Weissenborn uses _sipos_ for 0. It is not given by Bernelinus, and appears in Radulph of Laon, in the twelfth century. See Günther's _Geschichte_, p. 98, n.; Weissenborn, p. 11; Pihan, _Exposé_ etc., pp. xvi-xxii. In Friedlein's _Boetius_, p. 396, the plate shows that all of the six important manuscripts from which the illustrations are taken contain the symbol, while four out of five which give the words use the word _sipos_ for 0. The names appear in a twelfth-century anonymous manuscript in the Vatican, in a passage beginning Ordine primigeno sibi nomen possidet igin. Andras ecce locum mox uendicat ipse secundum Ormis post numeros incompositus sibi primus. [Boncompagni _Buttetino_, XV, p. 132.] Turchill (twelfth century) gives the names Igin, andras, hormis, arbas, quimas, caletis, zenis, temenias, celentis, saying: "Has autem figuras, ut donnus [dominus] Gvillelmus Rx testatur, a pytagoricis habemus, nomina uero ab arabibus." (Who the William R. was is not known. Boncompagni _Bulletino_ XV, p. 136.) Radulph of Laon (d. 1131) asserted that they were Chaldean (_Propagation_, p. 48 n.). A discussion of the whole question is also given in E. C. Bayley, loc. cit. Huet, writing in 1679, asserted that they were of Semitic origin, as did Nesselmann in spite of his despair over ormis, calctis, and celentis; see Woepcke, _Propagation_, p. 48. The names were used as late as the fifteenth century, without the zero, but with the superscript dot for 10's, two dots for 100's, etc., as among the early Arabs. Gerhardt mentions having seen a fourteenth or fifteenth century manuscript in the Bibliotheca Amploniana with the names "Ingnin, andras, armis, arbas, quinas, calctis, zencis, zemenias, zcelentis," and the statement "Si unum punctum super ingnin ponitur, X significat.... Si duo puncta super ... figuras superponunter, fiet decuplim illius quod cum uno puncto significabatur," in _Monatsberichte der K. P. Akad. d. Wiss._, Berlin, 1867, p. 40. [471] _A chart of ten numerals in 200 tongues_, by Rev. R. Patrick, London, 1812. [472] "Numeratio figuralis est cuiusuis numeri per notas, et figuras numerates descriptio." [Clichtoveus, edition of c. 1507, fol. C ii, v.] "Aristoteles enim uoces rerum [Greek: sumbola] uocat: id translatum, sonat notas." [Noviomagus, _De Numeris Libri II_, cap. vi.] "Alphabetum decem notarum." [Schonerus, notes to Ramus, 1586, p. 3 seq.] Richer says: "novem numero notas omnem numerum significantes." [Bubnov, loc. cit., p. 381.] [473] "Il y a dix Characteres, autrement Figures, Notes, ou Elements." [Peletier, edition of 1607, p. 13.] "Numerorum notas alij figuras, alij signa, alij characteres uocant." [Glareanus, 1545 edition, f. 9, r.] "Per figuras (quas zyphras uocant) assignationem, quales sunt hæ notulæ, 1. 2. 3. 4...." [Noviomagus, _De Numeris Libri II_, cap. vi.] Gemma Frisius also uses _elementa_ and Cardan uses _literae_. In the first arithmetic by an American (Greenwood, 1729) the author speaks of "a few Arabian _Charecters_ or Numeral Figures, called _Digits_" (p. 1), and as late as 1790, in the third edition of J. J. Blassière's arithmetic (1st ed. 1769), the name _characters_ is still in use, both for "de Latynsche en de Arabische" (p. 4), as is also the term "Cyfferletters" (p. 6, n.). _Ziffer_, the modern German form of cipher, was commonly used to designate any of the nine figures, as by Boeschenstein and Riese, although others, like Köbel, used it only for the zero. So _zifre_ appears in the arithmetic by Borgo, 1550 ed. In a Munich codex of the twelfth century, attributed to Gerland, they are called _characters_ only: "Usque ad VIIII. enim porrigitur omnis numerus et qui supercrescit eisdem designator Karacteribus." [Boncompagni _Bulletino_, Vol. X. p. 607.] [474] The title of his work is _Prologus N. Ocreati in Helceph_ (Arabic _al-qeif_, investigation or memoir) _ad Adelardum Batensem magistrum suum_. The work was made known by C. Henry, in the _Zeitschrift für Mathematik und Physik_, Vol. XXV, p. 129, and in the _Abhandlungen zur Geschichte der Mathematik_, Vol. III; Weissenborn, _Gerbert_, p. 188. [475] The zero is indicated by a vacant column. [476] Leo Jordan, loc. cit., p. 170. "Chifre en augorisme" is the expression used, while a century later "giffre en argorisme" and "cyffres d'augorisme" are similarly used. [477] _The Works of Geoffrey Chaucer_, edited by W. W. Skeat, Vol. IV, Oxford, 1894, p. 92. [478] Loc. cit., Vol. III, pp. 179 and 180. [479] In Book II, chap, vii, of _The Testament of Love_, printed with Chaucer's Works, loc. cit., Vol. VII, London, 1897. [480] _Liber Abacci_, published in Olleris, _Oeuvres de Gerbert_, pp. 357-400. [481] G. R. Kaye, "The Use of the Abacus in Ancient India," _Journal and Proceedings of the Asiatic Society of Bengal_, 1908, pp. 293-297. [482] _Liber Abbaci_, by Leonardo Pisano, loc. cit., p. 1. [483] Friedlein, "Die Entwickelung des Rechnens mit Columnen," _Zeitschrift für Mathematik und Physik_, Vol. X, p. 247. [484] The divisor 6 or 16 being increased by the difference 4, to 10 or 20 respectively. [485] E.g. Cantor, Vol. I, p. 882. [486] Friedlein, loc. cit.; Friedlein, "Gerbert's Regeln der Division" and "Das Rechnen mit Columnen vor dem 10. Jahrhundert," _Zeitschrift für Mathematik und Physik_, Vol. IX; Bubnov, loc. cit., pp. 197-245; M. Chasles, "Histoire de l'arithmétique. Recherches des traces du système de l'abacus, après que cette méthode a pris le nom d'Algorisme.--Preuves qu'à toutes les époques, jusq'au XVI^e siècle, on a su que l'arithmétique vulgaire avait pour origine cette méthode ancienne," _Comptes rendus_, Vol. XVII, pp. 143-154, also "Règles de l'abacus," _Comptes rendus_, Vol. XVI, pp. 218-246, and "Analyse et explication du traité de Gerbert," _Comptes rendus_, Vol. XVI, pp. 281-299. [487] Bubnov, loc. cit., pp. 203-204, "Abbonis abacus." [488] "Regulae de numerorum abaci rationibus," in Bubnov, loc. cit., pp. 205-225. [489] P. Treutlein, "Intorno ad alcuni scritti inediti relativi al calcolo dell' abaco," _Bulletino di bibliografia e di storia delle scienze matematiche e fisiche_, Vol. X, pp. 589-647. [490] "Intorno ad uno scritto inedito di Adelhardo di Bath intitolato 'Regulae Abaci,'" B. Boncompagni, in his _Bulletino_, Vol. XIV, pp. 1-134. [491] Treutlein, loc. cit.; Boncompagni, "Intorno al Tractatus de Abaco di Gerlando," _Bulletino_, Vol. X, pp. 648-656. [492] E. Narducci, "Intorno a due trattati inediti d'abaco contenuti in due codici Vaticani del secolo XII," Boncompagni _Bulletino_, Vol. XV, pp. 111-162. [493] See Molinier, _Les sources de l'histoire de France_, Vol. II, Paris, 1902, pp. 2, 3. [494] Cantor, _Geschichte_, Vol. I, p. 762. A. Nagl in the _Abhandlungen zur Geschichte der Mathematik_, Vol. V, p. 85. [495] 1030-1117. [496] _Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133. The work begins "Incipit Liber Radulfi laudunensis de abaco." [497] _Materialien zur Geschichte der arabischen Zahlzeichen in Frankreich_, loc. cit. [498] Who died in 1202. [499] Cantor, _Geschichte_, Vol. I (3), pp. 800-803; Boncompagni, _Trattati_, Part II. M. Steinschneider ("Die Mathematik bei den Juden," _Bibliotheca Mathematica_, Vol. X (2), p. 79) ingeniously derives another name by which he is called (Abendeuth) from Ibn Da[=u]d (Son of David). See also _Abhandlungen_, Vol. III, p. 110. [500] John is said to have died in 1157. [501] For it says, "Incipit prologus in libro alghoarismi de practica arismetrice. Qui editus est a magistro Johanne yspalensi." It is published in full in the second part of Boncompagni's _Trattati d'aritmetica_. [502] Possibly, indeed, the meaning of "libro alghoarismi" is not "to Al-Khow[=a]razm[=i]'s book," but "to a book of algorism." John of Luna says of it: "Hoc idem est illud etiam quod ... alcorismus dicere videtur." [_Trattati_, p. 68.] [503] For a résumé, see Cantor, Vol. I (3), pp. 800-803. As to the author, see Eneström in the _Bibliotheca Mathematica_, Vol. VI (3), p. 114, and Vol. IX (3), p. 2. [504] Born at Cremona (although some have asserted at Carmona, in Andalusia) in 1114; died at Toledo in 1187. Cantor, loc. cit.; Boncompagni, _Atti d. R. Accad. d. n. Lincei_, 1851. [505] See _Abhandlungen zur Geschichte der Mathematik_, Vol. XIV, p. 149; _Bibliotheca Mathematica_, Vol. IV (3), p. 206. Boncompagni had a fourteenth-century manuscript of his work, _Gerardi Cremonensis artis metrice practice_. See also T. L. Heath, _The Thirteen Books of Euclid's Elements_, 3 vols., Cambridge, 1908, Vol. I, pp. 92-94 ; A. A. Björnbo, "Gerhard von Cremonas Übersetzung von Alkwarizmis Algebra und von Euklids Elementen," _Bibliotheca Mathematica_, Vol. VI (3), pp. 239-248. [506] Wallis, _Algebra_, 1685, p. 12 seq. [507] Cantor, _Geschichte_, Vol. I (3), p. 906; A. A. Björnbo, "Al-Chw[=a]rizm[=i]'s trigonometriske Tavler," _Festskrift til H. G. Zeuthen_, Copenhagen, 1909, pp. 1-17. [508] Heath, loc. cit., pp. 93-96. [509] M. Steinschneider, _Zeitschrift der deutschen morgenländischen Gesellschaft_, Vol. XXV, 1871, p. 104, and _Zeitschrift für Mathematik und Physik_, Vol. XVI, 1871, pp. 392-393; M. Curtze, _Centralblatt für Bibliothekswesen_, 1899, p. 289; E. Wappler, _Zur Geschichte der deutschen Algebra im 15. Jahrhundert_, Programm, Zwickau, 1887; L. C. Karpinski, "Robert of Chester's Translation of the Algebra of Al-Khow[=a]razm[=i]," _Bibliotheca Mathematica_, Vol. XI (3), p. 125. He is also known as Robertus Retinensis, or Robert of Reading. [510] Nagl, A., "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," in the _Zeitschrift für Mathematik und Physik, Hist.-lit. Abth._, Vol. XXXIV, p. 129. Curtze, _Abhandlungen zur Geschichte der Mathematik_, Vol. VIII, pp. 1-27. [511] See line _a_ in the plate on p. 143. [512] _Sefer ha-Mispar, Das Buch der Zahl, ein hebräisch-arithmetisches Werk des R. Abraham ibn Esra_, Moritz Silberberg, Frankfurt a. M., 1895. [513] Browning's "Rabbi ben Ezra." [514] "Darum haben auch die Weisen Indiens all ihre Zahlen durch neun bezeichnet und Formen für die 9 Ziffern gebildet." [_Sefer ha-Mispar_, loc. cit., p. 2.] [515] F. Bonaini, "Memoria unica sincrona di Leonardo Fibonacci," Pisa, 1858, republished in 1867, and appearing in the _Giornale Arcadico_, Vol. CXCVII (N.S. LII); Gaetano Milanesi, _Documento inedito e sconosciuto a Lionardo Fibonacci_, Roma, 1867; Guglielmini, _Elogio di Lionardo Pisano_, Bologna, 1812, p. 35; Libri, _Histoire des sciences mathématiques_, Vol. II, p. 25; D. Martines, _Origine e progressi dell' aritmetica_, Messina, 1865, p. 47; Lucas, in Boncompagni _Bulletino_, Vol. X, pp. 129, 239; Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as cited in Chap. I; G. Eneström, "Ueber zwei angebliche mathematische Schulen im christlichen Mittelalter," _Bibliotheca Mathematica_, Vol. VIII (3), pp. 252-262; Boncompagni, "Della vita e delle opere di Leonardo Pisano," loc. cit. [516] The date is purely conjectural. See the _Bibliotheca Mathematica_, Vol. IV (3), p. 215. [517] An old chronicle relates that in 1063 Pisa fought a great battle with the Saracens at Palermo, capturing six ships, one being "full of wondrous treasure," and this was devoted to building the cathedral. [518] Heyd, loc. cit., Vol. I, p. 149. [519] Ibid., p. 211. [520] J. A. Symonds, _Renaissance in Italy. The Age of Despots._ New York, 1883, p. 62. [521] Symonds, loc. cit., p. 79. [522] J. A. Froude, _The Science of History_, London, 1864. "Un brevet d'apothicaire n'empêcha pas Dante d'être le plus grand poète de l'Italie, et ce fut un petit marchand de Pise qui donna l'algèbre aux Chrétiens." [Libri, _Histoire_, Vol. I, p. xvi.] [523] A document of 1226, found and published in 1858, reads: "Leonardo bigollo quondam Guilielmi." [524] "Bonaccingo germano suo." [525] E.g. Libri, Guglielmini, Tiraboschi. [526] Latin, _Bonaccius_. [527] Boncompagni and Milanesi. [528] Reprint, p. 5. [529] Whence the French name for candle. [530] Now part of Algiers. [531] E. Reclus, _Africa_, New York, 1893, Vol. II, p. 253. [532] "Sed hoc totum et algorismum atque arcus pictagore quasi errorem computavi respectu modi indorum." Woepcke, _Propagation_ etc., regards this as referring to two different systems, but the expression may very well mean algorism as performed upon the Pythagorean arcs (or table). [533] "Book of the Abacus," this term then being used, and long afterwards in Italy, to mean merely the arithmetic of computation. [534] "Incipit liber Abaci a Leonardo filio Bonacci compositus anno 1202 et correctus ab eodem anno 1228." Three MSS. of the thirteenth century are known, viz. at Milan, at Siena, and in the Vatican library. The work was first printed by Boncompagni in 1857. [535] I.e. in relation to the quadrivium. "Non legant in festivis diebus, nisi Philosophos et rhetoricas et quadrivalia et barbarismum et ethicam, si placet." Suter, _Die Mathematik auf den Universitäten des Mittelalters_, Zürich, 1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford in his time in his _Opus minus_, in the _Rerum Britannicarum medii aevi scriptores_, London, 1859, Vol. I, p. 327. For a picture of Cambridge at this time consult F. W. Newman, _The English Universities, translated from the German of V. A. Huber_, London, 1843, Vol. I, p. 61; W. W. R. Ball, _History of Mathematics at Cambridge_, 1889; S. Günther, _Geschichte des mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525_, Berlin, 1887, being Vol. III of _Monumenta Germaniae paedagogica_. [536] On the commercial activity of the period, it is known that bills of exchange passed between Messina and Constantinople in 1161, and that a bank was founded at Venice in 1170, the Bank of San Marco being established in the following year. The activity of Pisa was very manifest at this time. Heyd, loc. cit., Vol. II, p. 5; V. Casagrandi, _Storia e cronologia_, 3d ed., Milan, 1901, p. 56. [537] J. A. Symonds, loc. cit., Vol. II, p. 127. [538] I. Taylor, _The Alphabet_, London, 1883, Vol. II, p. 263. [539] Cited by Unger's History, p. 15. The Arabic numerals appear in a Regensburg chronicle of 1167 and in Silesia in 1340. See Schmidt's _Encyclopädie der Erziehung_, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst im sechzehnten Jahrhundert," _Festschrift zur dritten Säcularfeier des Berlinischen Gymnasiums zum grauen Kloster_, Berlin, 1874, p. 4. [540] The text is given in Halliwell, _Rara Mathematica_, London, 1839. [541] Seven are given in Ashmole's _Catalogue of Manuscripts in the Oxford Library_, 1845. [542] Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco commentarius, una cum Algorismo ipso_, Copenhagen, 1897; L. C. Karpinski, "Jordanus Nemorarius and John of Halifax," _American Mathematical Monthly_, Vol. XVII, pp. 108-113. [543] J. Aschbach, _Geschichte der Wiener Universität im ersten Jahrhunderte ihres Bestehens_, Wien, 1865, p. 93. [544] Curtze, loc. cit., gives the text. [545] Curtze, loc. cit., found some forty-five copies of the _Algorismus_ in three libraries of Munich, Venice, and Erfurt (Amploniana). Examination of two manuscripts from the Plimpton collection and the Columbia library shows such marked divergence from each other and from the text published by Curtze that the conclusion seems legitimate that these were students' lecture notes. The shorthand character of the writing further confirms this view, as it shows that they were written largely for the personal use of the writers. [546] "Quidam philosophus edidit nomine Algus, unde et Algorismus nuncupatur." [Curtze, loc. cit., p. 1.] [547] "Sinistrorsum autera scribimus in hac arte more arabico sive iudaico, huius scientiae inventorum." [Curtze, loc. cit., p. 7.] The Plimpton manuscript omits the words "sive iudaico." [548] "Non enim omnis numerus per quascumque figuras Indorum repraesentatur, sed tantum determinatus per determinatam, ut 4 non per 5,..." [Curtze, loc. cit., p. 25.] [549] C. Henry, "Sur les deux plus anciens traités français d'algorisme et de géométrie," Boncompagni _Bulletino_, Vol. XV, p. 49; Victor Mortet, "Le plus ancien traité français d'algorisme," loc. cit. [550] _L'État des sciences en France, depute la mort du Roy Robert, arrivée en 1031, jusqu'à celle de Philippe le Bel, arrivée en 1314_, Paris, 1741. [551] _Discours sur l'état des lettres en France au XIII^e siecle_, Paris, 1824. [552] _Aperçu historique_, Paris, 1876 ed., p. 464. [553] Ranulf Higden, a native of the west of England, entered St. Werburgh's monastery at Chester in 1299. He was a Benedictine monk and chronicler, and died in 1364. His _Polychronicon_, a history in seven books, was printed by Caxton in 1480. [554] Trevisa's translation, Higden having written in Latin. [555] An illustration of this feeling is seen in the writings of Prosdocimo de' Beldomandi (b. c. 1370-1380, d. 1428): "Inveni in quam pluribus libris algorismi nuncupatis mores circa numeros operandi satis varios atque diversos, qui licet boni existerent atque veri erant, tamen fastidiosi, tum propter ipsarum regularum multitudinem, tum propter earum deleationes, tum etiam propter ipsarum operationum probationes, utrum si bone fuerint vel ne. Erant et etiam isti modi interim fastidiosi, quod si in aliquo calculo astroloico error contigisset, calculatorem operationem suam a capite incipere oportebat, dato quod error suus adhuc satis propinquus existeret; et hoc propter figuras in sua operatione deletas. Indigebat etiam calculator semper aliquo lapide vel sibi conformi, super quo scribere atque faciliter delere posset figuras cum quibus operabatur in calculo suo. Et quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui libellum edere in quo omnia ista abicerentur: qui etiam algorismus sive liber de numeris denominari poterit. Scias tamen quod in hoc libello ponere non intendo nisi ea quae ad calculum necessaria sunt, alia quae in aliis libris practice arismetrice tanguntur, ad calculum non necessaria, propter brevitatem dimitendo." [Quoted by A. Nagl, _Zeitschrift für Mathematik und Physik, Hist.-lit. Abth._, Vol. XXXIV, p. 143; Smith, _Rara Arithmetica_, p. 14, in facsimile.] [556] P. Ewald, loc. cit.; Franz Steffens, _Lateinische Paläographie_, pp. xxxix-xl. We are indebted to Professor J. M. Burnam for a photograph of this rare manuscript. [557] See the plate of forms on p. 88. [558] Karabacek, loc. cit., p. 56; Karpinski, "Hindu Numerals in the Fihrist," _Bibliotheca Mathematica_, Vol. XI (3), p. 121. [559] Woepcke, "Sur une donnée historique," etc., loc. cit., and "Essai d'une restitution de travaux perdus d'Apollonius sur les quantités irrationnelles, d'après des indications tirées d'un manuscrit arabe," _Tome XIV des Mémoires présentés par divers savants à l'Académie des sciences_, Paris, 1856, note, pp. 6-14. [560] _Archeological Report of the Egypt Exploration Fund for 1908-1909_, London, 1910, p. 18. [561] There was a set of astronomical tables in Boncompagni's library bearing this date: "Nota quod anno d[=n]i [=n]ri ihû x[=p]i. 1264. perfecto." See Narducci's _Catalogo_, p. 130. [562] "On the Early use of Arabic Numerals in Europe," read before the Society of Antiquaries April 14, 1910, and published in _Archæologia_ in the same year. [563] Ibid., p. 8, n. The date is part of an Arabic inscription. [564] O. Codrington, _A Manual of Musalman Numismatics_, London, 1904. [565] See Arbuthnot, _The Mysteries of Chronology_, London, 1900, pp. 75, 78, 98; F. Pichler, _Repertorium der steierischen Münzkunde_, Grätz, 1875, where the claim is made of an Austrian coin of 1458; _Bibliotheca Mathematica_, Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a Brabant piece of 1478 in the collection of D. E. Smith. [566] A specimen is in the British Museum. [Arbuthnot, p. 79.] [567] Ibid., p. 79. [568] _Liber de Remediis utriusque fortunae Coloniae._ [569] Fr. Walthern et Hans Hurning, Nördlingen. [570] _Ars Memorandi_, one of the oldest European block-books. [571] Eusebius Caesariensis, _De praeparatione evangelica_, Venice, Jenson, 1470. The above statement holds for copies in the Astor Library and in the Harvard University Library. [572] Francisco de Retza, _Comestorium vitiorum_, Nürnberg, 1470. The copy referred to is in the Astor Library. [573] See Mauch, "Ueber den Gebrauch arabischer Ziffern und die Veränderungen derselben," _Anzeiger für Kunde der deutschen Vorzeit_, 1861, columns 46, 81, 116, 151, 189, 229, and 268; Calmet, _Recherches sur l'origine des chiffres d'arithmétique_, plate, loc. cit. [574] Günther, _Geschichte_, p. 175, n.; Mauch, loc. cit. [575] These are given by W. R. Lethaby, from drawings by J. T. Irvine, in the _Proceedings of the Society of Antiquaries_, 1906, p. 200. [576] There are some ill-tabulated forms to be found in J. Bowring, _The Decimal System_, London, 1854, pp. 23, 25, and in L. A. Chassant, _Dictionnaire des abréviations latines et françaises ... du moyen âge_, Paris, MDCCCLXVI, p. 113. The best sources we have at present, aside from the Hill monograph, are P. Treutlein, _Geschichte unserer Zahlzeichen_, Karlsruhe, 1875; Cantor's _Geschichte_, Vol. I, table; M. Prou, _Manuel de paléographie latine et française_, 2d ed., Paris, 1892, p. 164; A. Cappelli, _Dizionario di abbreviature latine ed italiane_, Milan, 1899. An interesting early source is found in the rare Caxton work of 1480, _The Myrrour of the World_. In Chap. X is a cut with the various numerals, the chapter beginning "The fourth scyence is called arsmetrique." Two of the fifteen extant copies of this work are at present in the library of Mr. J. P. Morgan, in New York. [577] From the twelfth-century manuscript on arithmetic, Curtze, loc. cit., _Abhandlungen_, and Nagl, loc. cit. The forms are copied from Plate VII in _Zeitschrift für Mathematik und Physik_, Vol. XXXIV. [578] From the Regensburg chronicle. Plate containing some of these numerals in _Monumenta Germaniae historica_, "Scriptores" Vol. XVII, plate to p. 184; Wattenbach, _Anleitung zur lateinischen Palaeographie_, Leipzig, 1886, p. 102; Boehmer, _Fontes rerum Germanicarum_, Vol. III, Stuttgart, 1852, p. lxv. [579] French Algorismus of 1275; from an unpublished photograph of the original, in the possession of D. E. Smith. See also p. 135. [580] From a manuscript of Boethius c. 1294, in Mr. Plimpton's library. Smith, _Rara Arithmetica_, Plate I. [581] Numerals in a 1303 manuscript in Sigmaringen, copied from Wattenbach, loc. cit., p. 102. [582] From a manuscript, Add. Manuscript 27,589, British Museum, 1360 A.D. The work is a computus in which the date 1360 appears, assigned in the British Museum catalogue to the thirteenth century. [583] From the copy of Sacrabosco's _Algorismus_ in Mr. Plimpton's library. Date c. 1442. See Smith, _Rara Arithmetica_, p. 450. [584] See _Rara Arithmetica_, pp. 446-447. [585] Ibid., pp. 469-470. [586] Ibid., pp. 477-478. [587] The i is used for "one" in the Treviso arithmetic (1478), Clichtoveus (c. 1507 ed., where both i and j are so used), Chiarini (1481), Sacrobosco (1488 ed.), and Tzwivel (1507 ed., where jj and jz are used for 11 and 12). This was not universal, however, for the _Algorithmus linealis_ of c. 1488 has a special type for 1. In a student's notebook of lectures taken at the University of Würzburg in 1660, in Mr. Plimpton's library, the ones are all in the form of i. [588] Thus the date [Numerals 1580], for 1580, appears in a MS. in the Laurentian library at Florence. The second and the following five characters are taken from Cappelli's _Dizionario_, p. 380, and are from manuscripts of the twelfth, thirteenth, fourteenth, sixteenth, seventeenth, and eighteenth centuries, respectively. [589] E.g. Chiarini's work of 1481; Clichtoveus (c. 1507). [590] The first is from an algorismus of the thirteenth century, in the Hannover Library. [See Gerhardt, "Ueber die Entstehung und Ausbreitung des dekadischen Zahlensystems," loc. cit., p. 28.] The second character is from a French algorismus, c. 1275. [Boncompagni _Bulletino_, Vol. XV, p. 51.] The third and the following sixteen characters are given by Cappelli, loc. cit., and are from manuscripts of the twelfth (1), thirteenth (2), fourteenth (7), fifteenth (3), sixteenth (1), seventeenth (2), and eighteenth (1) centuries, respectively. [591] Thus Chiarini (1481) has [Symbol] for 23. [592] The first of these is from a French algorismus, c. 1275. The second and the following eight characters are given by Cappelli, loc. cit., and are from manuscripts of the twelfth (2), thirteenth, fourteenth, fifteenth (3), seventeenth, and eighteenth centuries, respectively. [593] See Nagl, loc. cit. [594] Hannover algorismus, thirteenth century. [595] See the Dagomari manuscript, in _Rara Arithmetica_, pp. 435, 437-440. [596] But in the woodcuts of the _Margarita Philosophica_ (1503) the old forms are used, although the new ones appear in the text. In Caxton's _Myrrour of the World_ (1480) the old form is used. [597] Cappelli, loc. cit. They are partly from manuscripts of the tenth, twelfth, thirteenth (3), fourteenth (7), fifteenth (6), and eighteenth centuries, respectively. Those in the third line are from Chassant's _Dictionnaire_, p. 113, without mention of dates. [598] The first is from the Hannover algorismus, thirteenth century. The second is taken from the Rollandus manuscript, 1424. The others in the first two lines are from Cappelli, twelfth (3), fourteenth (6), fifteenth (13) centuries, respectively. The third line is from Chassant, loc. cit., p. 113, no mention of dates. [599] The first of these forms is from the Hannover algorismus, thirteenth century. The following are from Cappelli, fourteenth (3), fifteenth, sixteenth (2), and eighteenth centuries, respectively. [600] The first of these is taken from the Hannover algorismus, thirteenth century. The following forms are from Cappelli, twelfth, thirteenth, fourteenth (5), fifteenth (2), seventeenth, and eighteenth centuries, respectively. [601] All of these are given by Cappelli, thirteenth, fourteenth, fifteenth (2), and sixteenth centuries, respectively. [602] Smith, _Rara Arithmetica_, p. 489. This is also seen in several of the Plimpton manuscripts, as in one written at Ancona in 1684. See also Cappelli, loc. cit. [603] French algorismus, c. 1275, for the first of these forms. Cappelli, thirteenth, fourteenth, fifteenth (3), and seventeenth centuries, respectively. The last three are taken from _Byzantinische Analekten_, J. L. Heiberg, being forms of the fifteenth century, but not at all common. [Symbol: Qoppa] was the old Greek symbol for 90. [604] For the first of these the reader is referred to the forms ascribed to Boethius, in the illustration on p. 88; for the second, to Radulph of Laon, see p. 60. The third is used occasionally in the Rollandus (1424) manuscript, in Mr. Plimpton's library. The remaining three are from Cappelli, fourteenth (2) and seventeenth centuries. [605] Smith, _An Early English Algorism_. [606] Kuckuck, p. 5. [607] A. Cappelli, loc. cit., p. 372. [608] Smith, _Rara Arithmetica_, p. 443. [609] Curtze, _Petri Philomeni de Dacia_ etc., p. IX. [610] Cappelli, loc. cit., p. 376. [611] Curtze, loc. cit., pp. VIII-IX, note. [612] Edition of 1544-1545, f. 52. [613] _De numeris libri II_, 1544 ed., cap. XV. Heilbronner, loc. cit., p. 736, also gives them, and compares this with other systems. [614] Noviomagus says of them: "De quibusdam Astrologicis, sive Chaldaicis numerorum notis.... Sunt & aliæ quædam notæ, quibus Chaldaei & Astrologii quemlibet numerum artificiose & arguté describunt, scitu periucundae, quas nobis communicauit Rodolphus Paludanus Nouiomagus." 
26752	----	THE WAY TO GEOMETRY. Being necessary and usefull, For Astronomers. Engineres. Geographers. Architecks. Land-meaters. Carpenters. Sea-men. Paynters. Carvers, &c. Written by Peter Ramus Translated by William Bedwell Note from submitter: Because of the heavy dependence of this book on its diagrams and illustrations, a text version was not prepared. 
27468	----	Transcriber's Note On page 3 "Left" was changed to "Page 2" because of the e-text format. On pages 5 and 8 a registered symbol is represented by ^[registered]. IBM 1401 Programming Systems P PR PRO SYSTEMS PROG YSTEMS PROGR STEMS PROGRA TEMS PROGRAM EMS PROGRAMM MS PROGRAMMI S PROGRAMMIN PROGRAMMING [Illustration] _When companies order an IBM 1401 Data Processing System, methods-programming staffs are given the responsibility of translating the requirements of management into finished applications. 1401 Programming Systems are helping cut the costs of getting the computer into operation by simplifying and expediting the work of these methods staffs._ _Modern, high-speed computers, such as the 1401, are marvelous electronic instruments, but they represent only portions of data processing systems. Well-tested programming languages for communication with computers must accompany the systems. It is through these languages that the computer itself is used to perform many of the tedious functions that the programmer would otherwise have to perform. A few minutes of computer time in translating the program can be equal to many, many hours of staff time in writing instructions coded in the language of the computer._ _The combination of a modern computer plus modern programming languages is the key to profitable data processing. This brochure explains modern IBM Programming Languages and their significance to management._ Page 2: Here an operator points to machine language instructions for a new application being generated by the 1401 system on the 1403 high-speed printer. Statements about the application which were written by the programmer are being translated internally to machine-coded language. "=What Is A 1401 Program?=" A program is a series of instructions that direct the 1401 as it solves an application. "=What Is A Stored Program Machine?=" A stored program machine is one which stores its own instructions in magnetic form and is capable of acting on those instructions to complete the application assigned. The 1401 uses a stored program. "=What Are 1401 Programming Systems?=" There are two types: (1) Systems that provide the programmer with a simplified vocabulary of statements to use in writing programs, and (2) Pre-written programs, which take care of many of the everyday operations of the 1401. =What 1401 Programming Systems Mean To Management:= INCREASED PROGRAMMING EFFICIENCY Programmers can concentrate on the application and results rather than on a multitude of "bookkeeping" functions, such as keeping track of storage locations. FASTER TRANSLATION OF MANAGEMENT REQUIREMENTS INTO USABLE RESULTS Simplified programming routines allow programmers to write more instructions in less time. SHORTER TRAINING PERIODS Programmers use a language more familiar to them rather than having to learn detailed machine codes. REDUCED PROGRAMMING COSTS Many pre-written programs are supplied by IBM, eliminating necessity of customers' staffs writing their own. MORE AVAILABLE 1401 TIME Pre-written programs have already been tested by IBM, reducing tedious checking operations on the computer. EASIER TO UNDERSTAND PROGRAMS Programs are written in symbolic or application-oriented form instead of computer language. This enables management to communicate more easily with the programming staff. FASTER REPORTS ON OPERATIONS Routines such as those designed for report writing permit faster translation of management requirements into usable information. IBM Programming Systems: _Symbolic Programming Systems_ These systems permit programs to be written using meaningful names (symbols) rather than actual machine language. _Autocoder_ This is an advanced symbolic programming system. It allows generation of multiple machine instructions from one source statement, free-form coding, and an automatic assembly process through magnetic tape. _COBOL_ COBOL is a problem-oriented programming language for commercial applications.[A] COBOL permits a programmer to use language based on English words and phrases in describing an application. _Input/Output Control System_ This system provides the programmer with a packaged means of accomplishing input and output requirements. _Utility Programs_ These are pre-written instructions to perform many of the everyday operations of an installation. _Subroutines_ These are routines for multiplication, division, dozens conversion, and program error detection aids. _Tape Utilities_ These are generalized instructions, particularly useful to 1401 customers who also use larger data processing systems. They facilitate the transfer of data between IBM cards, magnetic tapes, and printers. They also provide for some 1401 processing while the transfer of data is taking place. _Tape Sort Programs_ Data can be sorted and classified at high speed for further processing by use of these generalized sorting routines. _Report Program Generator_ The programmer uses simplified, descriptive language with which he is already familiar to obtain reports swiftly and efficiently. FORTRAN (_Contraction of_ FOR_mula_ TRAN_slator_) Engineers and mathematicians state problems in familiar algebraic language for solution by the computer. RAMAC^[registered] _File Organization_ Routines are supplied for simplifying organization of records for storage in the 1401 Random Access File. =Here's how one of the 1401 programming systems--Report Program Generator--works to increase programming efficiency= 1401 computers produce important reports for management in record time because of their outstanding processing and printing abilities. In addition to this rapid machine processing of _input data_ used in reports, still more speed is achieved by the rapid preparation of _programs_ to produce the reports. This is possible because of the IBM Report Program Generator, a unique system which permits programs to be created with a minimum of time and effort. This example illustrates how the Report Program Generator simplifies the preparation of one part of an Expense Distribution Report (The Major Total Line): [Illustration] Without the Report Program Generator, the program to get the Major Total Line would be written out in detail, step by step: [Illustration] But with the Report Program Generator, all the programmer has to write are these two statements: [Illustration] It's just as easy to write the statements to generate the rest of the report! The 1401 itself does the work of converting the programmers' statements into the detailed instructions. The Report Program Generator is an example of what IBM Programming Systems can accomplish. With IBM you can be certain of total systems support for maximum profitability. IBM ^[registered] Stands For Service Service that begins long before the delivery of a computer ... and continues in depth long after. Service that has been _proven_ by years of data processing experience. New IBM Services include: _Programmed Applications Library_ Pre-tested computer programs designed to handle various major data processing functions common to firms within a specific industry. _Programming Systems Support_ To keep customers up-to-date on the availability and use of all new programming systems. To assist the IBM programming staff in reflecting customer requirements in the specification of new programming systems. Other services available to every IBM customer: _Program Library_ A library of 1401 programs will be established to aid all 1401 customers in solving specific applications, scientific as well as commercial. These will include programs written by customers and programs written by IBM. _Schools and Seminars_ Executive schools for management personnel. Programming schools for methods personnel. Industry seminars where customers meet to discuss subjects of common interest. _Branch Offices_ More than 200 branch offices serve customer needs promptly and efficiently. _Sales and Systems Representatives_ Experienced, highly trained individuals work with customers in applying IBM methods to their requirements. These are just a few of the many IBM services. Your IBM Sales Representative will be pleased to discuss all of them with you. International Business Machines Corporation Data Processing Division, 112 East Post Road White Plains, New York [Footnote A: COBOL specifications were developed by the Conference on Data Systems Languages, a voluntary cooperative effort of users, and manufacturers of data processing systems.] 
58225	----	Primes to One Trillion The Gutenberg text "The First 100,000 Prime Numbers", EBook #65, lists the primes up to 1,318,699. This somewhat more ambitious version lists the primes up to one trillion (1,000,000,000,000 or 1E12). Introduction I became interested in prime numbers after hearing about Goldbach's Conjecture, "Every even integer greater than 2 can be expressed as the sum of two primes". Verifying this requires a source of primes. Short lists (or programs to generate them) are widely available. Really long lists are scarce, except for primos.mat.br. To make these lists more accessible, I have reformatted them to a size easily manageable by ordinary text editors and viewers--about 55MB. The file names correspond to the range of primes the file contains: 00000000000_to_00100000000.txt Zero to 100 Million 00100000000_to_00200000000.txt 100 Million to 200 Million etc. The leading zero digits cause the file names to collate in order of their content. Longer lists can be composed with the DOS copy command. Move the required prime txt files to a temporary directory and use: copy *.txt longList.txt Prime Text Files This is a collection of 10,000 files, occupying about 486GB of disk space in their unzipped native txt format. Since adjacent primes have about 90% identical leading digits, the compressed (zip) versions total 61GB. Each zip file contains 100 txt files. Do not use Windows Explorer to copy or move large numbers of files at a time. Use DOS copy or xcopy for large copies. I find Beyond Compare (Scooter Software) handy for keeping track of large numbers of files. 0000.zip 000G to 010G (0 to 1E10) 0010.zip 010G to 020G 0020.zip 020G to 030G 0030.zip 030G to 040G 0040.zip 040G to 050G 0050.zip 050G to 060G 0060.zip 060G to 070G 0070.zip 070G to 080G 0080.zip 080G to 090G 0090.zip 090G to 100G 0100.zip 100G to 110G 0110.zip 110G to 120G 0120.zip 120G to 130G 0130.zip 130G to 140G 0140.zip 140G to 150G 0150.zip 150G to 160G 0160.zip 160G to 170G 0170.zip 170G to 180G 0180.zip 180G to 190G 0190.zip 190G to 100G 0200.zip 200G to 210G 0210.zip 210G to 220G 0220.zip 220G to 230G 0230.zip 230G to 240G 0240.zip 240G to 250G 0250.zip 250G to 260G 0260.zip 260G to 270G 0270.zip 270G to 280G 0280.zip 280G to 290G 0290.zip 290G to 200G 0300.zip 300G to 310G 0310.zip 310G to 320G 0320.zip 320G to 330G 0330.zip 330G to 340G 0340.zip 340G to 350G 0350.zip 350G to 360G 0360.zip 360G to 370G 0370.zip 370G to 380G 0380.zip 380G to 390G 0390.zip 390G to 300G 0400.zip 400G to 410G 0410.zip 410G to 420G 0420.zip 420G to 430G 0430.zip 430G to 440G 0440.zip 440G to 450G 0450.zip 450G to 460G 0460.zip 460G to 470G 0470.zip 470G to 480G 0480.zip 480G to 490G 0490.zip 490G to 400G 0500.zip 500G to 510G 0510.zip 510G to 520G 0520.zip 520G to 530G 0530.zip 530G to 540G 0540.zip 540G to 550G 0550.zip 550G to 560G 0560.zip 560G to 570G 0570.zip 570G to 580G 0580.zip 580G to 590G 0590.zip 590G to 500G 0600.zip 600G to 610G 0610.zip 610G to 620G 0620.zip 620G to 630G 0630.zip 630G to 640G 0640.zip 640G to 650G 0650.zip 650G to 660G 0660.zip 660G to 670G 0670.zip 670G to 680G 0680.zip 680G to 690G 0690.zip 690G to 600G 0700.zip 700G to 710G 0710.zip 710G to 720G 0720.zip 720G to 730G 0730.zip 730G to 740G 0740.zip 740G to 750G 0750.zip 750G to 760G 0760.zip 760G to 770G 0770.zip 770G to 780G 0780.zip 780G to 790G 0790.zip 790G to 700G 0800.zip 800G to 810G 0810.zip 810G to 820G 0820.zip 820G to 830G 0830.zip 830G to 840G 0840.zip 840G to 850G 0850.zip 850G to 860G 0860.zip 860G to 870G 0870.zip 870G to 880G 0880.zip 880G to 890G 0890.zip 890G to 800G 0900.zip 900G to 910G 0910.zip 910G to 920G 0920.zip 920G to 930G 0930.zip 930G to 940G 0940.zip 940G to 950G 0950.zip 950G to 960G 0960.zip 960G to 970G 0970.zip 970G to 980G 0980.zip 980G to 990G 0990.zip 990G to 1000G Additional prime files will be posted here. PrimeC File Format and Miscellaneous C++ Programs While working with primes, I developed the primec format, a file or array representation for primes that is roughly the same size of the compressed (zipped) txt representation, and supports fast access, both sequential and direct. The exact location of the primality specification of any number in the file (or memory array) is computed with a few instructions and no search. If you wish to examine and experiment with the C++ programs used to reformat these prime lists and test the Goldbach Conjecture, download the "programs.zip" package. It contains Generating and Analyzing Prime Numbers, a description of the content and use of these files, including the primec file format. Primec Format The primec format exploits the fact that all primes greater than 5 end in the decimal digits 1, 3, 7, or 9. Thus, the primality of 20 successive numbers can be specified in one 8 bit byte. The file begins with the complete binary representation of: The beginning of the sequence The end of the sequence A check sum of all data bytes (All three are 8 bytes for this implementation). The first and last values are a multiple of twenty, thus are never primes. There is no overlap of primes between successive files that use the same number for the upper boundary of the first file, and the lower boundary of the second file. The primality of any number in the range of the file is determined as follows: If the number ends in 0, 2, 4, 5, 6, or 8, it is not prime. Otherwise, the location of the specifying byte is at offset: ( value - start ) / 20 Within that byte, the primality of the value is specified by the bit as shown in the following table. The only tedious programming tasks were: Special case code for values less than 20, which include 2 and 5, and exclude 1 and 9. All larger values follow the same simple pattern. The increment and decrement operators for the corresponding iterators must search forward (or backward) for the next true bit, specifying the next prime number. This table shows the layout and content for a file containing 20 to 60. The first 24 bytes (start value, end value, check sum) are not shown. Byte 0------------------------------| 1---------------------------- Bit 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Value 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 Prime F T F T T F T F T T T F F T F T Hex 5A E5 As primes become larger, the density of primes becomes smaller as 1/ln(n). Thus the density of true bits also falls off. The number of digits (binary or decimal) to represent the primes grows as ln(n). Thus, a sequence of primes represented as primec is always competitive in size with the corresponding sequence in ASCI text or binary, besides providing fast direct access by value: bool isPrime(value). The results for the sequence of the largest 64 bit primes (18446744073707000000 to 18446744073709551558) is: Format Size (KB) 64 Bit Binary 445 Txt 1150 Zip Txt 114 PrimeC 125 Programs Among the programs in the "program.zip" package are: BuildTxtPrime Create file of primes, txt format TxtToPrimeC Convert txt to primec format. Goldbach Verify Goldbach's conjecture for zero to 1E12 Among the more than 15 classes and utilities are: PrimeGenerator Create prime numbers in a given range. Directory A vector of strings containing the names of files in a file directory. Progress A class to manage the periodic reporting of program activity. PrimeCVector Abstract class providing the algorithms to access primec data. PrimeCFileWriter Create a primec file. PrimeCFileReader Read a primec file. I hope you find them useful. If you have any questions, observations or bug reports concerning the C++ programming or the content of the prime files, send an email (after changing "at" to "@". primes1e12 at earthlink.net I embarked on this project as a programming challenge. I am not a mathematician. I have no deep insight into prime number theory. Please confine messages to programming issues. Here are some references: Prime Numbers: http://www.primos.mat.br/indexen.html Wikipedia: List of Prime Numbers (with numerous references): https://en.wikipedia.org/wiki/List_of_prime_numbers The Math Forum: http://mathforum.org/dr.math/faq/faq.prime.num.html The Prime Pages https://primes.utm.edu The program files are also posted on http://home.earthlink.net/~primes1E12. Corrections and additions will be posted there as they occur. Don Kostuch October, 2018. 
29042	----	Music transcribed by Linda Cantoni. Transcriber's note Minor punctuation errors have been changed without notice. Printer errors have been changed and are listed at the end. All other inconsistencies are as in the original. In this version the square root symbol is indicated by [** sqrt], the superscript by ^, and the therefore symbol by [** therefore]. A TANGLED TALE [Decoration] [Illustration: "AT A PACE OF SIX MILES IN THE HOUR." _Frontispiece._] A TANGLED TALE BY LEWIS CARROLL _WITH SIX ILLUSTRATIONS_ BY ARTHUR B. FROST Hoc meum tale quale est accipe. _SECOND THOUSAND._ London MACMILLAN AND CO. 1885 [_All Rights Reserved_] RICHARD CLAY & SONS, BREAD STREET HILL, LONDON, E.C. _And Bungay, Suffolk_. To My Pupil. Beloved pupil! Tamed by thee, Addish-, Subtrac-, Multiplica-tion, Division, Fractions, Rule of Three, Attest thy deft manipulation! Then onward! Let the voice of Fame From Age to Age repeat thy story, Till thou hast won thyself a name Exceeding even Euclid's glory! PREFACE. This Tale originally appeared as a serial in _The Monthly Packet_, beginning in April, 1880. The writer's intention was to embody in each Knot (like the medicine so dexterously, but ineffectually, concealed in the jam of our early childhood) one or more mathematical questions--in Arithmetic, Algebra, or Geometry, as the case might be--for the amusement, and possible edification, of the fair readers of that Magazine. L. C. _October, 1885._ CONTENTS. KNOT PAGE I. EXCELSIOR 1 II. ELIGIBLE APARTMENTS 4 III. MAD MATHESIS 13 IV. THE DEAD RECKONING 19 V. OUGHTS AND CROSSES 27 VI. HER RADIANCY 34 VII. PETTY CASH 43 VIII. DE OMNIBUS REBUS 52 IX. A SERPENT WITH CORNERS 58 X. CHELSEA BUNS 66 ANSWERS TO KNOT I. 77 " " II. 84 " " III. 90 " " IV. 96 " " V. 102 " " VI. 106 " " VII. 112 " " VIII. 132 " " IX. 135 " " X. 142 A TANGLED TALE. KNOT I. EXCELSIOR. "Goblin, lead them up and down." The ruddy glow of sunset was already fading into the sombre shadows of night, when two travellers might have been observed swiftly--at a pace of six miles in the hour--descending the rugged side of a mountain; the younger bounding from crag to crag with the agility of a fawn, while his companion, whose aged limbs seemed ill at ease in the heavy chain armour habitually worn by tourists in that district, toiled on painfully at his side. As is always the case under such circumstances, the younger knight was the first to break the silence. "A goodly pace, I trow!" he exclaimed. "We sped not thus in the ascent!" "Goodly, indeed!" the other echoed with a groan. "We clomb it but at three miles in the hour." "And on the dead level our pace is----?" the younger suggested; for he was weak in statistics, and left all such details to his aged companion. "Four miles in the hour," the other wearily replied. "Not an ounce more," he added, with that love of metaphor so common in old age, "and not a farthing less!" "'Twas three hours past high noon when we left our hostelry," the young man said, musingly. "We shall scarce be back by supper-time. Perchance mine host will roundly deny us all food!" "He will chide our tardy return," was the grave reply, "and such a rebuke will be meet." "A brave conceit!" cried the other, with a merry laugh. "And should we bid him bring us yet another course, I trow his answer will be tart!" "We shall but get our deserts," sighed the elder knight, who had never seen a joke in his life, and was somewhat displeased at his companion's untimely levity. "'Twill be nine of the clock," he added in an undertone, "by the time we regain our hostelry. Full many a mile shall we have plodded this day!" "How many? How many?" cried the eager youth, ever athirst for knowledge. The old man was silent. "Tell me," he answered, after a moment's thought, "what time it was when we stood together on yonder peak. Not exact to the minute!" he added hastily, reading a protest in the young man's face. "An' thy guess be within one poor half-hour of the mark, 'tis all I ask of thy mother's son! Then will I tell thee, true to the last inch, how far we shall have trudged betwixt three and nine of the clock." A groan was the young man's only reply; while his convulsed features and the deep wrinkles that chased each other across his manly brow, revealed the abyss of arithmetical agony into which one chance question had plunged him. KNOT II. ELIGIBLE APARTMENTS. "Straight down the crooked lane, And all round the square." "Let's ask Balbus about it," said Hugh. "All right," said Lambert. "_He_ can guess it," said Hugh. "Rather," said Lambert. No more words were needed: the two brothers understood each other perfectly. [Illustration: "BALBUS WAS ASSISTING HIS MOTHER-IN-LAW TO CONVINCE THE DRAGON."] Balbus was waiting for them at the hotel: the journey down had tired him, he said: so his two pupils had been the round of the place, in search of lodgings, without the old tutor who had been their inseparable companion from their childhood. They had named him after the hero of their Latin exercise-book, which overflowed with anecdotes of that versatile genius--anecdotes whose vagueness in detail was more than compensated by their sensational brilliance. "Balbus has overcome all his enemies" had been marked by their tutor, in the margin of the book, "Successful Bravery." In this way he had tried to extract a moral from every anecdote about Balbus--sometimes one of warning, as in "Balbus had borrowed a healthy dragon," against which he had written "Rashness in Speculation"--sometimes of encouragement, as in the words "Influence of Sympathy in United Action," which stood opposite to the anecdote "Balbus was assisting his mother-in-law to convince the dragon"--and sometimes it dwindled down to a single word, such as "Prudence," which was all he could extract from the touching record that "Balbus, having scorched the tail of the dragon, went away." His pupils liked the short morals best, as it left them more room for marginal illustrations, and in this instance they required all the space they could get to exhibit the rapidity of the hero's departure. Their report of the state of things was discouraging. That most fashionable of watering-places, Little Mendip, was "chockfull" (as the boys expressed it) from end to end. But in one Square they had seen no less than four cards, in different houses, all announcing in flaming capitals "ELIGIBLE APARTMENTS." "So there's plenty of choice, after all, you see," said spokesman Hugh in conclusion. "That doesn't follow from the data," said Balbus, as he rose from the easy chair, where he had been dozing over _The Little Mendip Gazette_. "They may be all single rooms. However, we may as well see them. I shall be glad to stretch my legs a bit." An unprejudiced bystander might have objected that the operation was needless, and that this long, lank creature would have been all the better with even shorter legs: but no such thought occurred to his loving pupils. One on each side, they did their best to keep up with his gigantic strides, while Hugh repeated the sentence in their father's letter, just received from abroad, over which he and Lambert had been puzzling. "He says a friend of his, the Governor of----_what_ was that name again, Lambert?" ("Kgovjni," said Lambert.) "Well, yes. The Governor of----what-you-may-call-it----wants to give a _very_ small dinner-party, and he means to ask his father's brother-in-law, his brother's father-in-law, his father-in-law's brother, and his brother-in-law's father: and we're to guess how many guests there will be." There was an anxious pause. "_How_ large did he say the pudding was to be?" Balbus said at last. "Take its cubical contents, divide by the cubical contents of what each man can eat, and the quotient----" "He didn't say anything about pudding," said Hugh, "--and here's the Square," as they turned a corner and came into sight of the "eligible apartments." "It _is_ a Square!" was Balbus' first cry of delight, as he gazed around him. "Beautiful! Beau-ti-ful! Equilateral! _And_ rectangular!" The boys looked round with less enthusiasm. "Number nine is the first with a card," said prosaic Lambert; but Balbus would not so soon awake from his dream of beauty. "See, boys!" he cried. "Twenty doors on a side! What symmetry! Each side divided into twenty-one equal parts! It's delicious!" "Shall I knock, or ring?" said Hugh, looking in some perplexity at a square brass plate which bore the simple inscription "RING ALSO." "Both," said Balbus. "That's an Ellipsis, my boy. Did you never see an Ellipsis before?" "I couldn't hardly read it," said Hugh, evasively. "It's no good having an Ellipsis, if they don't keep it clean." "Which there is _one_ room, gentlemen," said the smiling landlady. "And a sweet room too! As snug a little back-room----" "We will see it," said Balbus gloomily, as they followed her in. "I knew how it would be! One room in each house! No view, I suppose?" "Which indeed there _is_, gentlemen!" the landlady indignantly protested, as she drew up the blind, and indicated the back garden. "Cabbages, I perceive," said Balbus. "Well, they're green, at any rate." "Which the greens at the shops," their hostess explained, "are by no means dependable upon. Here you has them on the premises, _and_ of the best." "Does the window open?" was always Balbus' first question in testing a lodging: and "Does the chimney smoke?" his second. Satisfied on all points, he secured the refusal of the room, and they moved on to Number Twenty-five. This landlady was grave and stern. "I've nobbut one room left," she told them: "and it gives on the back-gyardin." "But there are cabbages?" Balbus suggested. The landlady visibly relented. "There is, sir," she said: "and good ones, though I say it as shouldn't. We can't rely on the shops for greens. So we grows them ourselves." "A singular advantage," said Balbus: and, after the usual questions, they went on to Fifty-two. "And I'd gladly accommodate you all, if I could," was the greeting that met them. "We are but mortal," ("Irrelevant!" muttered Balbus) "and I've let all my rooms but one." "Which one is a back-room, I perceive," said Balbus: "and looking out on--on cabbages, I presume?" "Yes, indeed, sir!" said their hostess. "Whatever _other_ folks may do, _we_ grows our own. For the shops----" "An excellent arrangement!" Balbus interrupted. "Then one can really depend on their being good. Does the window open?" The usual questions were answered satisfactorily: but this time Hugh added one of his own invention--"Does the cat scratch?" The landlady looked round suspiciously, as if to make sure the cat was not listening, "I will not deceive you, gentlemen," she said. "It _do_ scratch, but not without you pulls its whiskers! It'll never do it," she repeated slowly, with a visible effort to recall the exact words of some written agreement between herself and the cat, "without you pulls its whiskers!" "Much may be excused in a cat so treated," said Balbus, as they left the house and crossed to Number Seventy-three, leaving the landlady curtseying on the doorstep, and still murmuring to herself her parting words, as if they were a form of blessing, "---- not without you pulls its whiskers!" At Number Seventy-three they found only a small shy girl to show the house, who said "yes'm" in answer to all questions. "The usual room," said Balbus, as they marched in: "the usual back-garden, the usual cabbages. I suppose you can't get them good at the shops?" "Yes'm," said the girl. "Well, you may tell your mistress we will take the room, and that her plan of growing her own cabbages is simply _admirable_!" "Yes'm," said the girl, as she showed them out. "One day-room and three bed-rooms," said Balbus, as they returned to the hotel. "We will take as our day-room the one that gives us the least walking to do to get to it." "Must we walk from door to door, and count the steps?" said Lambert. "No, no! Figure it out, my boys, figure it out!" Balbus gaily exclaimed, as he put pens, ink, and paper before his hapless pupils, and left the room. "I say! It'll be a job!" said Hugh. "Rather!" said Lambert. KNOT III. MAD MATHESIS. "I waited for the train." "Well, they call me so because I _am_ a little mad, I suppose," she said, good-humouredly, in answer to Clara's cautiously-worded question as to how she came by so strange a nick-name. "You see, I never do what sane people are expected to do now-a-days. I never wear long trains, (talking of trains, that's the Charing Cross Metropolitan Station--I've something to tell you about _that_), and I never play lawn-tennis. I can't cook an omelette. I can't even set a broken limb! _There's_ an ignoramus for you!" Clara was her niece, and full twenty years her junior; in fact, she was still attending a High School--an institution of which Mad Mathesis spoke with undisguised aversion. "Let a woman be meek and lowly!" she would say. "None of your High Schools for me!" But it was vacation-time just now, and Clara was her guest, and Mad Mathesis was showing her the sights of that Eighth Wonder of the world--London. "The Charing Cross Metropolitan Station!" she resumed, waving her hand towards the entrance as if she were introducing her niece to a friend. "The Bayswater and Birmingham Extension is just completed, and the trains now run round and round continuously--skirting the border of Wales, just touching at York, and so round by the east coast back to London. The way the trains run is _most_ peculiar. The westerly ones go round in two hours; the easterly ones take three; but they always manage to start two trains from here, opposite ways, punctually every quarter-of-an-hour." "They part to meet again," said Clara, her eyes filling with tears at the romantic thought. "No need to cry about it!" her aunt grimly remarked. "They don't meet on the same line of rails, you know. Talking of meeting, an idea strikes me!" she added, changing the subject with her usual abruptness. "Let's go opposite ways round, and see which can meet most trains. No need for a chaperon--ladies' saloon, you know. You shall go whichever way you like, and we'll have a bet about it!" "I never make bets," Clara said very gravely. "Our excellent preceptress has often warned us----" "You'd be none the worse if you did!" Mad Mathesis interrupted. "In fact, you'd be the better, I'm certain!" "Neither does our excellent preceptress approve of puns," said Clara. "But we'll have a match, if you like. Let me choose my train," she added after a brief mental calculation, "and I'll engage to meet exactly half as many again as you do." "Not if you count fair," Mad Mathesis bluntly interrupted. "Remember, we only count the trains we meet _on the way_. You mustn't count the one that starts as you start, nor the one that arrives as you arrive." "That will only make the difference of _one_ train," said Clara, as they turned and entered the station. "But I never travelled alone before. There'll be no one to help me to alight. However, I don't mind. Let's have a match." A ragged little boy overheard her remark, and came running after her. "Buy a box of cigar-lights, Miss!" he pleaded, pulling her shawl to attract her attention. Clara stopped to explain. "I never smoke cigars," she said in a meekly apologetic tone. "Our excellent preceptress----," but Mad Mathesis impatiently hurried her on, and the little boy was left gazing after her with round eyes of amazement. The two ladies bought their tickets and moved slowly down the central platform, Mad Mathesis prattling on as usual--Clara silent, anxiously reconsidering the calculation on which she rested her hopes of winning the match. "Mind where you go, dear!" cried her aunt, checking her just in time. "One step more, and you'd have been in that pail of cold water!" "I know, I know," Clara said, dreamily. "The pale, the cold, and the moony----" "Take your places on the spring-boards!" shouted a porter. "What are _they_ for!" Clara asked in a terrified whisper. "Merely to help us into the trains." The elder lady spoke with the nonchalance of one quite used to the process. "Very few people can get into a carriage without help in less than three seconds, and the trains only stop for one second." At this moment the whistle was heard, and two trains rushed into the station. A moment's pause, and they were gone again; but in that brief interval several hundred passengers had been shot into them, each flying straight to his place with the accuracy of a Minie bullet--while an equal number were showered out upon the side-platforms. Three hours had passed away, and the two friends met again on the Charing Cross platform, and eagerly compared notes. Then Clara turned away with a sigh. To young impulsive hearts, like hers, disappointment is always a bitter pill. Mad Mathesis followed her, full of kindly sympathy. "Try again, my love!" she said, cheerily. "Let us vary the experiment. We will start as we did before, but not to begin counting till our trains meet. When we see each other, we will say 'One!' and so count on till we come here again." Clara brightened up. "I shall win _that_," she exclaimed eagerly, "if I may choose my train!" Another shriek of engine whistles, another upheaving of spring-boards, another living avalanche plunging into two trains as they flashed by: and the travellers were off again. Each gazed eagerly from her carriage window, holding up her handkerchief as a signal to her friend. A rush and a roar. Two trains shot past each other in a tunnel, and two travellers leaned back in their corners with a sigh--or rather with _two_ sighs--of relief. "One!" Clara murmured to herself. "Won! It's a word of good omen. _This_ time, at any rate, the victory will be mine!" But _was_ it? KNOT IV. THE DEAD RECKONING. "I did dream of money-bags to-night." Noonday on the open sea within a few degrees of the Equator is apt to be oppressively warm; and our two travellers were now airily clad in suits of dazzling white linen, having laid aside the chain-armour which they had found not only endurable in the cold mountain air they had lately been breathing, but a necessary precaution against the daggers of the banditti who infested the heights. Their holiday-trip was over, and they were now on their way home, in the monthly packet which plied between the two great ports of the island they had been exploring. Along with their armour, the tourists had laid aside the antiquated speech it had pleased them to affect while in knightly disguise, and had returned to the ordinary style of two country gentlemen of the Twentieth Century. Stretched on a pile of cushions, under the shade of a huge umbrella, they were lazily watching some native fishermen, who had come on board at the last landing-place, each carrying over his shoulder a small but heavy sack. A large weighing-machine, that had been used for cargo at the last port, stood on the deck; and round this the fishermen had gathered, and, with much unintelligible jabber, seemed to be weighing their sacks. "More like sparrows in a tree than human talk, isn't it?" the elder tourist remarked to his son, who smiled feebly, but would not exert himself so far as to speak. The old man tried another listener. "What have they got in those sacks, Captain?" he inquired, as that great being passed them in his never ending parade to and fro on the deck. The Captain paused in his march, and towered over the travellers--tall, grave, and serenely self-satisfied. "Fishermen," he explained, "are often passengers in My ship. These five are from Mhruxi--the place we last touched at--and that's the way they carry their money. The money of this island is heavy, gentlemen, but it costs little, as you may guess. We buy it from them by weight--about five shillings a pound. I fancy a ten pound-note would buy all those sacks." By this time the old man had closed his eyes--in order, no doubt, to concentrate his thoughts on these interesting facts; but the Captain failed to realise his motive, and with a grunt resumed his monotonous march. Meanwhile the fishermen were getting so noisy over the weighing-machine that one of the sailors took the precaution of carrying off all the weights, leaving them to amuse themselves with such substitutes in the form of winch-handles, belaying-pins, &c., as they could find. This brought their excitement to a speedy end: they carefully hid their sacks in the folds of the jib that lay on the deck near the tourists, and strolled away. When next the Captain's heavy footfall passed, the younger man roused himself to speak. "_What_ did you call the place those fellows came from, Captain?" he asked. "Mhruxi, sir." "And the one we are bound for?" The Captain took a long breath, plunged into the word, and came out of it nobly. "They call it Kgovjni, sir." "K--I give it up!" the young man faintly said. He stretched out his hand for a glass of iced water which the compassionate steward had brought him a minute ago, and had set down, unluckily, just outside the shadow of the umbrella. It was scalding hot, and he decided not to drink it. The effort of making this resolution, coming close on the fatiguing conversation he had just gone through, was too much for him: he sank back among the cushions in silence. His father courteously tried to make amends for his _nonchalance_. "Whereabouts are we now, Captain?" said he, "Have you any idea?" The Captain cast a pitying look on the ignorant landsman. "I could tell you _that_, sir," he said, in a tone of lofty condescension, "to an inch!" "You don't say so!" the old man remarked, in a tone of languid surprise. "And mean so," persisted the Captain. "Why, what do you suppose would become of My ship, if I were to lose My Longitude and My Latitude? Could _you_ make anything of My Dead Reckoning?" "Nobody could, I'm sure!" the other heartily rejoined. But he had overdone it. "It's _perfectly_ intelligible," the Captain said, in an offended tone, "to any one that understands such things." With these words he moved away, and began giving orders to the men, who were preparing to hoist the jib. Our tourists watched the operation with such interest that neither of them remembered the five money-bags, which in another moment, as the wind filled out the jib, were whirled overboard and fell heavily into the sea. But the poor fishermen had not so easily forgotten their property. In a moment they had rushed to the spot, and stood uttering cries of fury, and pointing, now to the sea, and now to the sailors who had caused the disaster. The old man explained it to the Captain. "Let us make it up among us," he added in conclusion. "Ten pounds will do it, I think you said?" [Illustration] But the Captain put aside the suggestion with a wave of the hand. "No, sir!" he said, in his grandest manner. "You will excuse Me, I am sure; but these are My passengers. The accident has happened on board My ship, and under My orders. It is for Me to make compensation." He turned to the angry fishermen. "Come here, my men!" he said, in the Mhruxian dialect. "Tell me the weight of each sack. I saw you weighing them just now." Then ensued a perfect Babel of noise, as the five natives explained, all screaming together, how the sailors had carried off the weights, and they had done what they could with whatever came handy. Two iron belaying-pins, three blocks, six holystones, four winch-handles, and a large hammer, were now carefully weighed, the Captain superintending and noting the results. But the matter did not seem to be settled, even then: an angry discussion followed, in which the sailors and the five natives all joined: and at last the Captain approached our tourists with a disconcerted look, which he tried to conceal under a laugh. "It's an absurd difficulty," he said. "Perhaps one of you gentlemen can suggest something. It seems they weighed the sacks two at a time!" "If they didn't have five separate weighings, of course you can't value them separately," the youth hastily decided. "Let's hear all about it," was the old man's more cautious remark. "They _did_ have five separate weighings," the Captain said, "but--Well, it beats _me_ entirely!" he added, in a sudden burst of candour. "Here's the result. First and second sack weighed twelve pounds; second and third, thirteen and a half; third and fourth, eleven and a half; fourth and fifth, eight: and then they say they had only the large hammer left, and it took _three_ sacks to weigh it down--that's the first, third and fifth--and _they_ weighed sixteen pounds. There, gentlemen! Did you ever hear anything like _that_?" The old man muttered under his breath "If only my sister were here!" and looked helplessly at his son. His son looked at the five natives. The five natives looked at the Captain. The Captain looked at nobody: his eyes were cast down, and he seemed to be saying softly to himself "Contemplate one another, gentlemen, if such be your good pleasure. _I_ contemplate _Myself_!" KNOT V. OUGHTS AND CROSSES. "Look here, upon this picture, and on this." "And what made you choose the first train, Goosey?" said Mad Mathesis, as they got into the cab. "Couldn't you count better than _that_?" "I took an extreme case," was the tearful reply. "Our excellent preceptress always says 'When in doubt, my dears, take an extreme case.' And I _was_ in doubt." "Does it always succeed?" her aunt enquired. Clara sighed. "Not _always_," she reluctantly admitted. "And I can't make out why. One day she was telling the little girls--they make such a noise at tea, you know--'The more noise you make, the less jam you will have, and _vice versâ_.' And I thought they wouldn't know what '_vice versâ_' meant: so I explained it to them. I said 'If you make an infinite noise, you'll get no jam: and if you make no noise, you'll get an infinite lot of jam.' But our excellent preceptress said that wasn't a good instance. _Why_ wasn't it?" she added plaintively. Her aunt evaded the question. "One sees certain objections to it," she said. "But how did you work it with the Metropolitan trains? None of them go infinitely fast, I believe." "I called them hares and tortoises," Clara said--a little timidly, for she dreaded being laughed at. "And I thought there couldn't be so many hares as tortoises on the Line: so I took an extreme case--one hare and an infinite number of tortoises." "An extreme case, indeed," her aunt remarked with admirable gravity: "and a most dangerous state of things!" "And I thought, if I went with a tortoise, there would be only _one_ hare to meet: but if I went with the hare--you know there were _crowds_ of tortoises!" "It wasn't a bad idea," said the elder lady, as they left the cab, at the entrance of Burlington House. "You shall have another chance to-day. We'll have a match in marking pictures." Clara brightened up. "I should like to try again, very much," she said. "I'll take more care this time. How are we to play?" To this question Mad Mathesis made no reply: she was busy drawing lines down the margins of the catalogue. "See," she said after a minute, "I've drawn three columns against the names of the pictures in the long room, and I want you to fill them with oughts and crosses--crosses for good marks and oughts for bad. The first column is for choice of subject, the second for arrangement, the third for colouring. And these are the conditions of the match. You must give three crosses to two or three pictures. You must give two crosses to four or five----" "Do you mean _only_ two crosses?" said Clara. "Or may I count the three-cross pictures among the two-cross pictures?" "Of course you may," said her aunt. "Any one, that has _three_ eyes, may be said to have _two_ eyes, I suppose?" Clara followed her aunt's dreamy gaze across the crowded gallery, half-dreading to find that there was a three-eyed person in sight. "And you must give one cross to nine or ten." "And which wins the match?" Clara asked, as she carefully entered these conditions on a blank leaf in her catalogue. "Whichever marks fewest pictures." "But suppose we marked the same number?" "Then whichever uses most marks." Clara considered. "I don't think it's much of a match," she said. "I shall mark nine pictures, and give three crosses to three of them, two crosses to two more, and one cross each to all the rest." "Will you, indeed?" said her aunt. "Wait till you've heard all the conditions, my impetuous child. You must give three oughts to one or two pictures, two oughts to three or four, and one ought to eight or nine. I don't want you to be _too_ hard on the R.A.'s." Clara quite gasped as she wrote down all these fresh conditions. "It's a great deal worse than Circulating Decimals!" she said. "But I'm determined to win, all the same!" Her aunt smiled grimly. "We can begin _here_," she said, as they paused before a gigantic picture, which the catalogue informed them was the "Portrait of Lieutenant Brown, mounted on his favorite elephant." "He looks awfully conceited!" said Clara. "I don't think he was the elephant's favorite Lieutenant. What a hideous picture it is! And it takes up room enough for twenty!" "Mind what you say, my dear!" her aunt interposed. "It's by an R.A.!" But Clara was quite reckless. "I don't care who it's by!" she cried. "And I shall give it three bad marks!" Aunt and niece soon drifted away from each other in the crowd, and for the next half-hour Clara was hard at work, putting in marks and rubbing them out again, and hunting up and down for suitable pictures. This she found the hardest part of all. "I _can't_ find the one I want!" she exclaimed at last, almost crying with vexation. "What is it you want to find, my dear?" The voice was strange to Clara, but so sweet and gentle that she felt attracted to the owner of it, even before she had seen her; and when she turned, and met the smiling looks of two little old ladies, whose round dimpled faces, exactly alike, seemed never to have known a care, it was as much as she could do--as she confessed to Aunt Mattie afterwards--to keep herself from hugging them both. "I was looking for a picture," she said, "that has a good subject--and that's well arranged--but badly coloured." The little old ladies glanced at each other in some alarm. "Calm yourself, my dear," said the one who had spoken first, "and try to remember which it was. What _was_ the subject?" "Was it an elephant, for instance?" the other sister suggested. They were still in sight of Lieutenant Brown. "I don't know, indeed!" Clara impetuously replied. "You know it doesn't matter a bit what the subject _is_, so long as it's a good one!" Once more the sisters exchanged looks of alarm, and one of them whispered something to the other, of which Clara caught only the one word "mad." "They mean Aunt Mattie, of course," she said to herself--fancying, in her innocence, that London was like her native town, where everybody knew everybody else. "If you mean my aunt," she added aloud, "she's _there_--just three pictures beyond Lieutenant Brown." "Ah, well! Then you'd better go to her, my dear!" her new friend said, soothingly. "_She'll_ find you the picture you want. Good-bye, dear!" "Good-bye, dear!" echoed the other sister, "Mind you don't lose sight of your aunt!" And the pair trotted off into another room, leaving Clara rather perplexed at their manner. "They're real darlings!" she soliloquised. "I wonder why they pity me so!" And she wandered on, murmuring to herself "It must have two good marks, and----" KNOT VI. HER RADIANCY. "One piecee thing that my have got, Maskee[A] that thing my no can do. You talkee you no sabey what? Bamboo." They landed, and were at once conducted to the Palace. About half way they were met by the Governor, who welcomed them in English--a great relief to our travellers, whose guide could speak nothing but Kgovjnian. "I don't half like the way they grin at us as we go by!" the old man whispered to his son. "And why do they say 'Bamboo!' so often?" "It alludes to a local custom," replied the Governor, who had overheard the question. "Such persons as happen in any way to displease Her Radiancy are usually beaten with rods." [Illustration: "WHY DO THEY SAY 'BAMBOO!' SO OFTEN?"] The old man shuddered. "A most objectional local custom!" he remarked with strong emphasis. "I wish we had never landed! Did you notice that black fellow, Norman, opening his great mouth at us? I verily believe he would like to eat us!" Norman appealed to the Governor, who was walking at his other side. "Do they often eat distinguished strangers here?" he said, in as indifferent a tone as he could assume. "Not often--not ever!" was the welcome reply. "They are not good for it. Pigs we eat, for they are fat. This old man is thin." "And thankful to be so!" muttered the elder traveller. "Beaten we shall be without a doubt. It's a comfort to know it won't be Beaten without the B! My dear boy, just look at the peacocks!" They were now walking between two unbroken lines of those gorgeous birds, each held in check, by means of a golden collar and chain, by a black slave, who stood well behind, so as not to interrupt the view of the glittering tail, with its network of rustling feathers and its hundred eyes. The Governor smiled proudly. "In your honour," he said, "Her Radiancy has ordered up ten thousand additional peacocks. She will, no doubt, decorate you, before you go, with the usual Star and Feathers." "It'll be Star without the S!" faltered one of his hearers. "Come, come! Don't lose heart!" said the other. "All this is full of charm for me." "You are young, Norman," sighed his father; "young and light-hearted. For me, it is Charm without the C." "The old one is sad," the Governor remarked with some anxiety. "He has, without doubt, effected some fearful crime?" "But I haven't!" the poor old gentleman hastily exclaimed. "Tell him I haven't, Norman!" "He has not, as yet," Norman gently explained. And the Governor repeated, in a satisfied tone, "Not as yet." "Yours is a wondrous country!" the Governor resumed, after a pause. "Now here is a letter from a friend of mine, a merchant, in London. He and his brother went there a year ago, with a thousand pounds apiece; and on New-Year's-day they had sixty thousand pounds between them!" "How did they do it?" Norman eagerly exclaimed. Even the elder traveller looked excited. The Governor handed him the open letter. "Anybody can do it, when once they know how," so ran this oracular document. "We borrowed nought: we stole nought. We began the year with only a thousand pounds apiece: and last New-Year's-day we had sixty thousand pounds between us--sixty thousand golden sovereigns!" Norman looked grave and thoughtful as he handed back the letter. His father hazarded one guess. "Was it by gambling?" "A Kgovjnian never gambles," said the Governor gravely, as he ushered them through the palace gates. They followed him in silence down a long passage, and soon found themselves in a lofty hall, lined entirely with peacocks' feathers. In the centre was a pile of crimson cushions, which almost concealed the figure of Her Radiancy--a plump little damsel, in a robe of green satin dotted with silver stars, whose pale round face lit up for a moment with a half-smile as the travellers bowed before her, and then relapsed into the exact expression of a wax doll, while she languidly murmured a word or two in the Kgovjnian dialect. The Governor interpreted. "Her Radiancy welcomes you. She notes the Impenetrable Placidity of the old one, and the Imperceptible Acuteness of the youth." Here the little potentate clapped her hands, and a troop of slaves instantly appeared, carrying trays of coffee and sweetmeats, which they offered to the guests, who had, at a signal from the Governor, seated themselves on the carpet. "Sugar-plums!" muttered the old man. "One might as well be at a confectioner's! Ask for a penny bun, Norman!" "Not so loud!" his son whispered. "Say something complimentary!" For the Governor was evidently expecting a speech. "We thank Her Exalted Potency," the old man timidly began. "We bask in the light of her smile, which----" "The words of old men are weak!" the Governor interrupted angrily. "Let the youth speak!" "Tell her," cried Norman, in a wild burst of eloquence, "that, like two grasshoppers in a volcano, we are shrivelled up in the presence of Her Spangled Vehemence!" "It is well," said the Governor, and translated this into Kgovjnian. "I am now to tell you," he proceeded, "what Her Radiancy requires of you before you go. The yearly competition for the post of Imperial Scarf-maker is just ended; you are the judges. You will take account of the rate of work, the lightness of the scarves, and their warmth. Usually the competitors differ in one point only. Thus, last year, Fifi and Gogo made the same number of scarves in the trial-week, and they were equally light; but Fifi's were twice as warm as Gogo's and she was pronounced twice as good. But this year, woe is me, who can judge it? Three competitors are here, and they differ in all points! While you settle their claims, you shall be lodged, Her Radiancy bids me say, free of expense--in the best dungeon, and abundantly fed on the best bread and water." The old man groaned. "All is lost!" he wildly exclaimed. But Norman heeded him not: he had taken out his note-book, and was calmly jotting down the particulars. "Three they be," the Governor proceeded, "Lolo, Mimi, and Zuzu. Lolo makes 5 scarves while Mimi makes 2; but Zuzu makes 4 while Lolo makes 3! Again, so fairylike is Zuzu's handiwork, 5 of her scarves weigh no more than one of Lolo's; yet Mimi's is lighter still--5 of hers will but balance 3 of Zuzu's! And for warmth one of Mimi's is equal to 4 of Zuzu's; yet one of Lolo's is as warm as 3 of Mimi's!" Here the little lady once more clapped her hands. "It is our signal of dismissal!" the Governor hastily said. "Pay Her Radiancy your farewell compliments--and walk out backwards." The walking part was all the elder tourist could manage. Norman simply said "Tell Her Radiancy we are transfixed by the spectacle of Her Serene Brilliance, and bid an agonized farewell to her Condensed Milkiness!" "Her Radiancy is pleased," the Governor reported, after duly translating this. "She casts on you a glance from Her Imperial Eyes, and is confident that you will catch it!" "That I warrant we shall!" the elder traveller moaned to himself distractedly. Once more they bowed low, and then followed the Governor down a winding staircase to the Imperial Dungeon, which they found to be lined with coloured marble, lighted from the roof, and splendidly though not luxuriously furnished with a bench of polished malachite. "I trust you will not delay the calculation," the Governor said, ushering them in with much ceremony. "I have known great inconvenience--great and serious inconvenience--result to those unhappy ones who have delayed to execute the commands of Her Radiancy! And on this occasion she is resolute: she says the thing must and shall be done: and she has ordered up ten thousand additional bamboos!" With these words he left them, and they heard him lock and bar the door on the outside. "I told you how it would end!" moaned the elder traveller, wringing his hands, and quite forgetting in his anguish that he had himself proposed the expedition, and had never predicted anything of the sort. "Oh that we were well out of this miserable business!" "Courage!" cried the younger cheerily. "_Hæc olim meminisse juvabit!_ The end of all this will be glory!" "Glory without the L!" was all the poor old man could say, as he rocked himself to and fro on the malachite bench. "Glory without the L!" FOOTNOTE: [Footnote A: "_Maskee_," in Pigeon-English, means "_without_."] KNOT VII. PETTY CASH. "Base is the slave that pays." "Aunt Mattie!" "My child?" "_Would_ you mind writing it down at once? I shall be quite _certain_ to forget it if you don't!" "My dear, we really must wait till the cab stops. How can I possibly write anything in the midst of all this jolting?" "But _really_ I shall be forgetting it!" Clara's voice took the plaintive tone that her aunt never knew how to resist, and with a sigh the old lady drew forth her ivory tablets and prepared to record the amount that Clara had just spent at the confectioner's shop. Her expenditure was always made out of her aunt's purse, but the poor girl knew, by bitter experience, that sooner or later "Mad Mathesis" would expect an exact account of every penny that had gone, and she waited, with ill-concealed impatience, while the old lady turned the tablets over and over, till she had found the one headed "PETTY CASH." "Here's the place," she said at last, "and here we have yesterday's luncheon duly entered. _One glass lemonade_ (Why can't you drink water, like me?) _three sandwiches_ (They never put in half mustard enough. I told the young woman so, to her face; and she tossed her head--like her impudence!) _and seven biscuits_. _Total one-and-two-pence._ Well, now for to-day's?" "One glass of lemonade----" Clara was beginning to say, when suddenly the cab drew up, and a courteous railway-porter was handing out the bewildered girl before she had had time to finish her sentence. Her aunt pocketed the tablets instantly. "Business first," she said: "petty cash--which is a form of pleasure, whatever _you_ may think--afterwards." And she proceeded to pay the driver, and to give voluminous orders about the luggage, quite deaf to the entreaties of her unhappy niece that she would enter the rest of the luncheon account. "My dear, you really must cultivate a more capacious mind!" was all the consolation she vouchsafed to the poor girl. "Are not the tablets of your memory wide enough to contain the record of one single luncheon?" "Not wide enough! Not half wide enough!" was the passionate reply. The words came in aptly enough, but the voice was not that of Clara, and both ladies turned in some surprise to see who it was that had so suddenly struck into their conversation. A fat little old lady was standing at the door of a cab, helping the driver to extricate what seemed an exact duplicate of herself: it would have been no easy task to decide which was the fatter, or which looked the more good-humoured of the two sisters. "I tell you the cab-door isn't half wide enough!" she repeated, as her sister finally emerged, somewhat after the fashion of a pellet from a pop-gun, and she turned to appeal to Clara. "Is it, dear?" she said, trying hard to bring a frown into a face that dimpled all over with smiles. "Some folks is too wide for 'em," growled the cab-driver. [Illustration: "I TELL YOU THE CAB-DOOR ISN'T HALF WIDE ENOUGH!"] "Don't provoke me, man!" cried the little old lady, in what she meant for a tempest of fury. "Say another word and I'll put you into the County Court, and sue you for a _Habeas Corpus_!" The cabman touched his hat, and marched off, grinning. "Nothing like a little Law to cow the ruffians, my dear!" she remarked confidentially to Clara. "You saw how he quailed when I mentioned the _Habeas Corpus_? Not that I've any idea what it means, but it sounds very grand, doesn't it?" "It's very provoking," Clara replied, a little vaguely. "Very!" the little old lady eagerly repeated. "And we're very much provoked indeed. Aren't we, sister?" "I never was so provoked in all my life!" the fatter sister assented, radiantly. By this time Clara had recognised her picture-gallery acquaintances, and, drawing her aunt aside, she hastily whispered her reminiscences. "I met them first in the Royal Academy--and they were very kind to me--and they were lunching at the next table to us, just now, you know--and they tried to help me to find the picture I wanted--and I'm sure they're dear old things!" "Friends of yours, are they?" said Mad Mathesis. "Well, I like their looks. You can be civil to them, while I get the tickets. But do try and arrange your ideas a little more chronologically!" And so it came to pass that the four ladies found themselves seated side by side on the same bench waiting for the train, and chatting as if they had known one another for years. "Now this I call quite a remarkable coincidence!" exclaimed the smaller and more talkative of the two sisters--the one whose legal knowledge had annihilated the cab-driver. "Not only that we should be waiting for the same train, and at the same station--_that_ would be curious enough--but actually on the same day, and the same hour of the day! That's what strikes _me_ so forcibly!" She glanced at the fatter and more silent sister, whose chief function in life seemed to be to support the family opinion, and who meekly responded-- "And me too, sister!" "Those are not _independent_ coincidences----" Mad Mathesis was just beginning, when Clara ventured to interpose. "There's no jolting here," she pleaded meekly. "_Would_ you mind writing it down now?" Out came the ivory tablets once more. "What was it, then?" said her aunt. "One glass of lemonade, one sandwich, one biscuit--Oh dear me!" cried poor Clara, the historical tone suddenly changing to a wail of agony. "Toothache?" said her aunt calmly, as she wrote down the items. The two sisters instantly opened their reticules and produced two different remedies for neuralgia, each marked "unequalled." "It isn't that!" said poor Clara. "Thank you very much. It's only that I _can't_ remember how much I paid!" "Well, try and make it out, then," said her aunt. "You've got yesterday's luncheon to help you, you know. And here's the luncheon we had the day before--the first day we went to that shop--_one glass lemonade_, _four sandwiches_, _ten biscuits_. _Total, one-and-fivepence._" She handed the tablets to Clara, who gazed at them with eyes so dim with tears that she did not at first notice that she was holding them upside down. The two sisters had been listening to all this with the deepest interest, and at this juncture the smaller one softly laid her hand on Clara's arm. "Do you know, my dear," she said coaxingly, "my sister and I are in the very same predicament! Quite identically the very same predicament! Aren't we, sister?" "Quite identically and absolutely the very----" began the fatter sister, but she was constructing her sentence on too large a scale, and the little one would not wait for her to finish it. "Yes, my dear," she resumed; "we were lunching at the very same shop as you were--and we had two glasses of lemonade and three sandwiches and five biscuits--and neither of us has the least idea what we paid. Have we, sister?" "Quite identically and absolutely----" murmured the other, who evidently considered that she was now a whole sentence in arrears, and that she ought to discharge one obligation before contracting any fresh liabilities; but the little lady broke in again, and she retired from the conversation a bankrupt. "_Would_ you make it out for us, my dear?" pleaded the little old lady. "You can do Arithmetic, I trust?" her aunt said, a little anxiously, as Clara turned from one tablet to another, vainly trying to collect her thoughts. Her mind was a blank, and all human expression was rapidly fading out of her face. A gloomy silence ensued. KNOT VIII. DE OMNIBUS REBUS. "This little pig went to market: This little pig staid at home." "By Her Radiancy's express command," said the Governor, as he conducted the travellers, for the last time, from the Imperial presence, "I shall now have the ecstasy of escorting you as far as the outer gate of the Military Quarter, where the agony of parting--if indeed Nature can survive the shock--must be endured! From that gate grurmstipths start every quarter of an hour, both ways----" "Would you mind repeating that word?" said Norman. "Grurm----?" "Grurmstipths," the Governor repeated. "You call them omnibuses in England. They run both ways, and you can travel by one of them all the way down to the harbour." The old man breathed a sigh of relief; four hours of courtly ceremony had wearied him, and he had been in constant terror lest something should call into use the ten thousand additional bamboos. In another minute they were crossing a large quadrangle, paved with marble, and tastefully decorated with a pigsty in each corner. Soldiers, carrying pigs, were marching in all directions: and in the middle stood a gigantic officer giving orders in a voice of thunder, which made itself heard above all the uproar of the pigs. "It is the Commander-in-Chief!" the Governor hurriedly whispered to his companions, who at once followed his example in prostrating themselves before the great man. The Commander gravely bowed in return. He was covered with gold lace from head to foot: his face wore an expression of deep misery: and he had a little black pig under each arm. Still the gallant fellow did his best, in the midst of the orders he was every moment issuing to his men, to bid a courteous farewell to the departing guests. "Farewell, oh old one--carry these three to the South corner--and farewell to thee, thou young one--put this fat one on the top of the others in the Western sty--may your shadows never be less--woe is me, it is wrongly done! Empty out all the sties, and begin again!" And the soldier leant upon his sword, and wiped away a tear. "He is in distress," the Governor explained as they left the court. "Her Radiancy has commanded him to place twenty-four pigs in those four sties, so that, as she goes round the court, she may always find the number in each sty nearer to ten than the number in the last." "Does she call ten nearer to ten than nine is?" said Norman. "Surely," said the Governor. "Her Radiancy would admit that ten is nearer to ten than nine is--and also nearer than eleven is." "Then I think it can be done," said Norman. The Governor shook his head. "The Commander has been transferring them in vain for four months," he said. "What hope remains? And Her Radiancy has ordered up ten thousand additional----" "The pigs don't seem to enjoy being transferred," the old man hastily interrupted. He did not like the subject of bamboos. "They are only _provisionally_ transferred, you know," said the Governor. "In most cases they are immediately carried back again: so they need not mind it. And all is done with the greatest care, under the personal superintendence of the Commander-in-Chief." "Of course she would only go _once_ round?" said Norman. "Alas, no!" sighed their conductor. "Round and round. Round and round. These are Her Radiancy's own words. But oh, agony! Here is the outer gate, and we must part!" He sobbed as he shook hands with them, and the next moment was briskly walking away. "He _might_ have waited to see us off!" said the old man, piteously. "And he needn't have begun whistling the very _moment_ he left us!" said the young one, severely. "But look sharp--here are two what's-his-names in the act of starting!" Unluckily, the sea-bound omnibus was full. "Never mind!" said Norman, cheerily. "We'll walk on till the next one overtakes us." They trudged on in silence, both thinking over the military problem, till they met an omnibus coming from the sea. The elder traveller took out his watch. "Just twelve minutes and a half since we started," he remarked in an absent manner. Suddenly the vacant face brightened; the old man had an idea. "My boy!" he shouted, bringing his hand down upon Norman's shoulder so suddenly as for a moment to transfer his centre of gravity beyond the base of support. Thus taken off his guard, the young man wildly staggered forwards, and seemed about to plunge into space: but in another moment he had gracefully recovered himself. "Problem in Precession and Nutation," he remarked--in tones where filial respect only just managed to conceal a shade of annoyance. "What is it?" he hastily added, fearing his father might have been taken ill. "Will you have some brandy?" "When will the next omnibus overtake us? When? When?" the old man cried, growing more excited every moment. Norman looked gloomy. "Give me time," he said. "I must think it over." And once more the travellers passed on in silence--a silence only broken by the distant squeals of the unfortunate little pigs, who were still being provisionally transferred from sty to sty, under the personal superintendence of the Commander-in-Chief. KNOT IX. A SERPENT WITH CORNERS. "Water, water, every where, Nor any drop to drink." "It'll just take one more pebble." "What ever _are_ you doing with those buckets?" The speakers were Hugh and Lambert. Place, the beach of Little Mendip. Time, 1.30, P.M. Hugh was floating a bucket in another a size larger, and trying how many pebbles it would carry without sinking. Lambert was lying on his back, doing nothing. For the next minute or two Hugh was silent, evidently deep in thought. Suddenly he started. "I say, look here, Lambert!" he cried. "If it's alive, and slimy, and with legs, I don't care to," said Lambert. "Didn't Balbus say this morning that, if a body is immersed in liquid, it displaces as much liquid as is equal to its own bulk?" said Hugh. "He said things of that sort," Lambert vaguely replied. "Well, just look here a minute. Here's the little bucket almost quite immersed: so the water displaced ought to be just about the same bulk. And now just look at it!" He took out the little bucket as he spoke, and handed the big one to Lambert. "Why, there's hardly a teacupful! Do you mean to say _that_ water is the same bulk as the little bucket?" "Course it is," said Lambert. "Well, look here again!" cried Hugh, triumphantly, as he poured the water from the big bucket into the little one. "Why, it doesn't half fill it!" "That's _its_ business," said Lambert. "If Balbus says it's the same bulk, why, it _is_ the same bulk, you know." "Well, I don't believe it," said Hugh. "You needn't," said Lambert. "Besides, it's dinner-time. Come along." They found Balbus waiting dinner for them, and to him Hugh at once propounded his difficulty. "Let's get you helped first," said Balbus, briskly cutting away at the joint. "You know the old proverb 'Mutton first, mechanics afterwards'?" The boys did _not_ know the proverb, but they accepted it in perfect good faith, as they did every piece of information, however startling, that came from so infallible an authority as their tutor. They ate on steadily in silence, and, when dinner was over, Hugh set out the usual array of pens, ink, and paper, while Balbus repeated to them the problem he had prepared for their afternoon's task. "A friend of mine has a flower-garden--a very pretty one, though no great size--" "How big is it?" said Hugh. "That's what _you_ have to find out!" Balbus gaily replied. "All _I_ tell you is that it is oblong in shape--just half a yard longer than its width--and that a gravel-walk, one yard wide, begins at one corner and runs all round it." "Joining into itself?" said Hugh. "_Not_ joining into itself, young man. Just before doing _that_, it turns a corner, and runs round the garden again, alongside of the first portion, and then inside that again, winding in and in, and each lap touching the last one, till it has used up the whole of the area." "Like a serpent with corners?" said Lambert. "Exactly so. And if you walk the whole length of it, to the last inch, keeping in the centre of the path, it's exactly two miles and half a furlong. Now, while you find out the length and breadth of the garden, I'll see if I can think out that sea-water puzzle." "You said it was a flower-garden?" Hugh inquired, as Balbus was leaving the room. "I did," said Balbus. "Where do the flowers grow?" said Hugh. But Balbus thought it best not to hear the question. He left the boys to their problem, and, in the silence of his own room, set himself to unravel Hugh's mechanical paradox. "To fix our thoughts," he murmured to himself, as, with hands deep-buried in his pockets, he paced up and down the room, "we will take a cylindrical glass jar, with a scale of inches marked up the side, and fill it with water up to the 10-inch mark: and we will assume that every inch depth of jar contains a pint of water. We will now take a solid cylinder, such that every inch of it is equal in bulk to _half_ a pint of water, and plunge 4 inches of it into the water, so that the end of the cylinder comes down to the 6-inch mark. Well, that displaces 2 pints of water. What becomes of them? Why, if there were no more cylinder, they would lie comfortably on the top, and fill the jar up to the 12-inch mark. But unfortunately there _is_ more cylinder, occupying half the space between the 10-inch and the 12-inch marks, so that only _one_ pint of water can be accommodated there. What becomes of the other pint? Why, if there were no more cylinder, it would lie on the top, and fill the jar up to the 13-inch mark. But unfortunately----Shade of Newton!" he exclaimed, in sudden accents of terror. "When _does_ the water stop rising?" A bright idea struck him. "I'll write a little essay on it," he said. * * * * * _Balbus's Essay._ "When a solid is immersed in a liquid, it is well known that it displaces a portion of the liquid equal to itself in bulk, and that the level of the liquid rises just so much as it would rise if a quantity of liquid had been added to it, equal in bulk to the solid. Lardner says, precisely the same process occurs when a solid is _partially_ immersed: the quantity of liquid displaced, in this case, equalling the portion of the solid which is immersed, and the rise of the level being in proportion. "Suppose a solid held above the surface of a liquid and partially immersed: a portion of the liquid is displaced, and the level of the liquid rises. But, by this rise of level, a little bit more of the solid is of course immersed, and so there is a new displacement of a second portion of the liquid, and a consequent rise of level. Again, this second rise of level causes a yet further immersion, and by consequence another displacement of liquid and another rise. It is self-evident that this process must continue till the entire solid is immersed, and that the liquid will then begin to immerse whatever holds the solid, which, being connected with it, must for the time be considered a part of it. If you hold a stick, six feet long, with its end in a tumbler of water, and wait long enough, you must eventually be immersed. The question as to the source from which the water is supplied--which belongs to a high branch of mathematics, and is therefore beyond our present scope--does not apply to the sea. Let us therefore take the familiar instance of a man standing at the edge of the sea, at ebb-tide, with a solid in his hand, which he partially immerses: he remains steadfast and unmoved, and we all know that he must be drowned. The multitudes who daily perish in this manner to attest a philosophical truth, and whose bodies the unreasoning wave casts sullenly upon our thankless shores, have a truer claim to be called the martyrs of science than a Galileo or a Kepler. To use Kossuth's eloquent phrase, they are the unnamed demigods of the nineteenth century."[B] * * * * * "There's a fallacy _somewhere_," he murmured drowsily, as he stretched his long legs upon the sofa. "I must think it over again." He closed his eyes, in order to concentrate his attention more perfectly, and for the next hour or so his slow and regular breathing bore witness to the careful deliberation with which he was investigating this new and perplexing view of the subject. [Illustration: "HE REMAINS STEADFAST AND UNMOVED."] FOOTNOTE: [Footnote B: _Note by the writer._--For the above Essay I am indebted to a dear friend, now deceased.] KNOT X. CHELSEA BUNS. "Yea, buns, and buns, and buns!" OLD SONG. "How very, very sad!" exclaimed Clara; and the eyes of the gentle girl filled with tears as she spoke. "Sad--but very curious when you come to look at it arithmetically," was her aunt's less romantic reply. "Some of them have lost an arm in their country's service, some a leg, some an ear, some an eye----" "And some, perhaps, _all_!" Clara murmured dreamily, as they passed the long rows of weather-beaten heroes basking in the sun. "Did you notice that very old one, with a red face, who was drawing a map in the dust with his wooden leg, and all the others watching? I _think_ it was a plan of a battle----" "The battle of Trafalgar, no doubt," her aunt interrupted, briskly. "Hardly that, I think," Clara ventured to say. "You see, in that case, he couldn't well be alive----" "Couldn't well be alive!" the old lady contemptuously repeated. "He's as lively as you and me put together! Why, if drawing a map in the dust--with one's wooden leg--doesn't prove one to be alive, perhaps you'll kindly mention what _does_ prove it!" Clara did not see her way out of it. Logic had never been her _forte_. "To return to the arithmetic," Mad Mathesis resumed--the eccentric old lady never let slip an opportunity of driving her niece into a calculation--"what percentage do you suppose must have lost all four--a leg, an arm, an eye, and an ear?" "How _can_ I tell?" gasped the terrified girl. She knew well what was coming. "You can't, of course, without _data_," her aunt replied: "but I'm just going to give you----" "Give her a Chelsea bun, Miss! That's what most young ladies likes best!" The voice was rich and musical, and the speaker dexterously whipped back the snowy cloth that covered his basket, and disclosed a tempting array of the familiar square buns, joined together in rows, richly egged and browned, and glistening in the sun. "No, sir! I shall give her nothing so indigestible! Be off!" The old lady waved her parasol threateningly: but nothing seemed to disturb the good-humour of the jolly old man, who marched on, chanting his melodious refrain:-- [Music: Chel-sea buns! Chel-sea buns hot! Chel-sea buns! Pi-ping hot! Chel-sea buns hot! Chel-sea buns!] "Far too indigestible, my love!" said the old lady. "Percentages will agree with you ever so much better!" Clara sighed, and there was a hungry look in her eyes as she watched the basket lessening in the distance: but she meekly listened to the relentless old lady, who at once proceeded to count off the _data_ on her fingers. "Say that 70 per cent. have lost an eye--75 per cent. an ear--80 per cent. an arm--85 per cent. a leg--that'll do it beautifully. Now, my dear, what percentage, _at least_, must have lost all four?" No more conversation occurred--unless a smothered exclamation of "Piping hot!" which escaped from Clara's lips as the basket vanished round a corner could be counted as such--until they reached the old Chelsea mansion, where Clara's father was then staying, with his three sons and their old tutor. Balbus, Lambert, and Hugh had entered the house only a few minutes before them. They had been out walking, and Hugh had been propounding a difficulty which had reduced Lambert to the depths of gloom, and had even puzzled Balbus. "It changes from Wednesday to Thursday at midnight, doesn't it?" Hugh had begun. "Sometimes," said Balbus, cautiously. "Always," said Lambert, decisively. "_Sometimes_," Balbus gently insisted. "Six midnights out of seven, it changes to some other name." "I meant, of course," Hugh corrected himself, "when it _does_ change from Wednesday to Thursday, it does it at midnight--and _only_ at midnight." "Surely," said Balbus. Lambert was silent. "Well, now, suppose it's midnight here in Chelsea. Then it's Wednesday _west_ of Chelsea (say in Ireland or America) where midnight hasn't arrived yet: and it's Thursday _east_ of Chelsea (say in Germany or Russia) where midnight has just passed by?" "Surely," Balbus said again. Even Lambert nodded this time. "But it isn't midnight, anywhere else; so it can't be changing from one day to another anywhere else. And yet, if Ireland and America and so on call it Wednesday, and Germany and Russia and so on call it Thursday, there _must_ be some place--not Chelsea--that has different days on the two sides of it. And the worst of it is, the people _there_ get their days in the wrong order: they've got Wednesday _east_ of them, and Thursday _west_--just as if their day had changed from Thursday to Wednesday!" "I've heard that puzzle before!" cried Lambert. "And I'll tell you the explanation. When a ship goes round the world from east to west, we know that it loses a day in its reckoning: so that when it gets home, and calls its day Wednesday, it finds people here calling it Thursday, because we've had one more midnight than the ship has had. And when you go the other way round you gain a day." "I know all that," said Hugh, in reply to this not very lucid explanation: "but it doesn't help me, because the ship hasn't proper days. One way round, you get more than twenty-four hours to the day, and the other way you get less: so of course the names get wrong: but people that live on in one place always get twenty-four hours to the day." "I suppose there _is_ such a place," Balbus said, meditatively, "though I never heard of it. And the people must find it very queer, as Hugh says, to have the old day _east_ of them, and the new one _west_: because, when midnight comes round to them, with the new day in front of it and the old one behind it, one doesn't see exactly what happens. I must think it over." So they had entered the house in the state I have described--Balbus puzzled, and Lambert buried in gloomy thought. "Yes, m'm, Master _is_ at home, m'm," said the stately old butler. (N.B.--It is only a butler of experience who can manage a series of three M's together, without any interjacent vowels.) "And the _ole_ party is a-waiting for you in the libery." "I don't like his calling your father an _old_ party," Mad Mathesis whispered to her niece, as they crossed the hall. And Clara had only just time to whisper in reply "he meant the _whole_ party," before they were ushered into the library, and the sight of the five solemn faces there assembled chilled her into silence. Her father sat at the head of the table, and mutely signed to the ladies to take the two vacant chairs, one on each side of him. His three sons and Balbus completed the party. Writing materials had been arranged round the table, after the fashion of a ghostly banquet: the butler had evidently bestowed much thought on the grim device. Sheets of quarto paper, each flanked by a pen on one side and a pencil on the other, represented the plates--penwipers did duty for rolls of bread--while ink-bottles stood in the places usually occupied by wine-glasses. The _pièce de resistance_ was a large green baize bag, which gave forth, as the old man restlessly lifted it from side to side, a charming jingle, as of innumerable golden guineas. "Sister, daughter, sons--and Balbus--," the old man began, so nervously, that Balbus put in a gentle "Hear, hear!" while Hugh drummed on the table with his fists. This disconcerted the unpractised orator. "Sister--" he began again, then paused a moment, moved the bag to the other side, and went on with a rush, "I mean--this being--a critical occasion--more or less--being the year when one of my sons comes of age--" he paused again in some confusion, having evidently got into the middle of his speech sooner than he intended: but it was too late to go back. "Hear, hear!" cried Balbus. "Quite so," said the old gentleman, recovering his self-possession a little: "when first I began this annual custom--my friend Balbus will correct me if I am wrong--" (Hugh whispered "with a strap!" but nobody heard him except Lambert, who only frowned and shook his head at him) "--this annual custom of giving each of my sons as many guineas as would represent his age--it was a critical time--so Balbus informed me--as the ages of two of you were together equal to that of the third--so on that occasion I made a speech----" He paused so long that Balbus thought it well to come to the rescue with the words "It was a most----" but the old man checked him with a warning look: "yes, made a speech," he repeated. "A few years after that, Balbus pointed out--I say pointed out--" ("Hear, hear"! cried Balbus. "Quite so," said the grateful old man.) "--that it was _another_ critical occasion. The ages of two of you were together _double_ that of the third. So I made another speech--another speech. And now again it's a critical occasion--so Balbus says--and I am making----" (Here Mad Mathesis pointedly referred to her watch) "all the haste I can!" the old man cried, with wonderful presence of mind. "Indeed, sister, I'm coming to the point now! The number of years that have passed since that first occasion is just two-thirds of the number of guineas I then gave you. Now, my boys, calculate your ages from the _data_, and you shall have the money!" "But we _know_ our ages!" cried Hugh. "Silence, sir!" thundered the old man, rising to his full height (he was exactly five-foot five) in his indignation. "I say you must use the _data_ only! You mustn't even assume _which_ it is that comes of age!" He clutched the bag as he spoke, and with tottering steps (it was about as much as he could do to carry it) he left the room. "And _you_ shall have a similar _cadeau_," the old lady whispered to her niece, "when you've calculated that percentage!" And she followed her brother. Nothing could exceed the solemnity with which the old couple had risen from the table, and yet was it--was it a _grin_ with which the father turned away from his unhappy sons? Could it be--could it be a _wink_ with which the aunt abandoned her despairing niece? And were those--were those sounds of suppressed _chuckling_ which floated into the room, just before Balbus (who had followed them out) closed the door? Surely not: and yet the butler told the cook--but no, that was merely idle gossip, and I will not repeat it. The shades of evening granted their unuttered petition, and "closed not o'er" them (for the butler brought in the lamp): the same obliging shades left them a "lonely bark" (the wail of a dog, in the back-yard, baying the moon) for "awhile": but neither "morn, alas," (nor any other epoch) seemed likely to "restore" them--to that peace of mind which had once been theirs ere ever these problems had swooped upon them, and crushed them with a load of unfathomable mystery! "It's hardly fair," muttered Hugh, "to give us such a jumble as this to work out!" "Fair?" Clara echoed, bitterly. "Well!" And to all my readers I can but repeat the last words of gentle Clara-- FARE-WELL! APPENDIX. "A knot!" said Alice. "Oh, do let me help to undo it!" ANSWERS TO KNOT I. _Problem._--"Two travellers spend from 3 o'clock till 9 in walking along a level road, up a hill, and home again: their pace on the level being 4 miles an hour, up hill 3, and down hill 6. Find distance walked: also (within half an hour) time of reaching top of hill." _Answer._--"24 miles: half-past 6." * * * * * _Solution._--A level mile takes 1/4 of an hour, up hill 1/3, down hill 1/6. Hence to go and return over the same mile, whether on the level or on the hill-side, takes 1/2 an hour. Hence in 6 hours they went 12 miles out and 12 back. If the 12 miles out had been nearly all level, they would have taken a little over 3 hours; if nearly all up hill, a little under 4. Hence 3-1/2 hours must be within 1/2 an hour of the time taken in reaching the peak; thus, as they started at 3, they got there within 1/2 an hour of 1/2 past 6. * * * * * Twenty-seven answers have come in. Of these, 9 are right, 16 partially right, and 2 wrong. The 16 give the _distance_ correctly, but they have failed to grasp the fact that the top of the hill might have been reached at _any_ moment between 6 o'clock and 7. The two wrong answers are from GERTY VERNON and A NIHILIST. The former makes the distance "23 miles," while her revolutionary companion puts it at "27." GERTY VERNON says "they had to go 4 miles along the plain, and got to the foot of the hill at 4 o'clock." They _might_ have done so, I grant; but you have no ground for saying they _did_ so. "It was 7-1/2 miles to the top of the hill, and they reached that at 1/4 before 7 o'clock." Here you go wrong in your arithmetic, and I must, however reluctantly, bid you farewell. 7-1/2 miles, at 3 miles an hour, would _not_ require 2-3/4 hours. A NIHILIST says "Let _x_ denote the whole number of miles; _y_ the number of hours to hill-top; [** therefore] 3_y_ = number of miles to hill-top, and _x_-3_y_ = number of miles on the other side." You bewilder me. The other side of _what_? "Of the hill," you say. But then, how did they get home again? However, to accommodate your views we will build a new hostelry at the foot of the hill on the opposite side, and also assume (what I grant you is _possible_, though it is not _necessarily_ true) that there was no level road at all. Even then you go wrong. You say "_y_ = 6 - (_x_ - 3_y_)/6, ..... (i); _x_/4-1/2 = 6 ..... (ii)." I grant you (i), but I deny (ii): it rests on the assumption that to go _part_ of the time at 3 miles an hour, and the rest at 6 miles an hour, comes to the same result as going the _whole_ time at 4-1/2 miles an hour. But this would only be true if the "_part_" were an exact _half_, i.e., if they went up hill for 3 hours, and down hill for the other 3: which they certainly did _not_ do. The sixteen, who are partially right, are AGNES BAILEY, F. K., FIFEE, G. E. B., H. P., KIT, M. E. T., MYSIE, A MOTHER'S SON, NAIRAM, A REDRUTHIAN, A SOCIALIST, SPEAR MAIDEN, T. B. C., VIS INERTI�, and YAK. Of these, F. K., FIFEE, T. B. C., and VIS INERTI� do not attempt the second part at all. F. K. and H. P. give no working. The rest make particular assumptions, such as that there was no level road--that there were 6 miles of level road--and so on, all leading to _particular_ times being fixed for reaching the hill-top. The most curious assumption is that of AGNES BAILEY, who says "Let _x_ = number of hours occupied in ascent; then _x_/2 = hours occupied in descent; and 4_x_/3 = hours occupied on the level." I suppose you were thinking of the relative _rates_, up hill and on the level; which we might express by saying that, if they went _x_ miles up hill in a certain time, they would go 4_x_/3 miles on the level _in the same time_. You have, in fact, assumed that they took _the same time_ on the level that they took in ascending the hill. FIFEE assumes that, when the aged knight said they had gone "four miles in the hour" on the level, he meant that four miles was the _distance_ gone, not merely the rate. This would have been--if FIFEE will excuse the slang expression--a "sell," ill-suited to the dignity of the hero. And now "descend, ye classic Nine!" who have solved the whole problem, and let me sing your praises. Your names are BLITHE, E. W., L. B., A MARLBOROUGH BOY, O. V. L., PUTNEY WALKER, ROSE, SEA BREEZE, SIMPLE SUSAN, and MONEY SPINNER. (These last two I count as one, as they send a joint answer.) ROSE and SIMPLE SUSAN and CO. do not actually state that the hill-top was reached some time between 6 and 7, but, as they have clearly grasped the fact that a mile, ascended and descended, took the same time as two level miles, I mark them as "right." A MARLBOROUGH BOY and PUTNEY WALKER deserve honourable mention for their algebraical solutions being the only two who have perceived that the question leads to _an indeterminate equation_. E. W. brings a charge of untruthfulness against the aged knight--a serious charge, for he was the very pink of chivalry! She says "According to the data given, the time at the summit affords no clue to the total distance. It does not enable us to state precisely to an inch how much level and how much hill there was on the road." "Fair damsel," the aged knight replies, "--if, as I surmise, thy initials denote Early Womanhood--bethink thee that the word 'enable' is thine, not mine. I did but ask the time of reaching the hill-top as my _condition_ for further parley. If _now_ thou wilt not grant that I am a truth-loving man, then will I affirm that those same initials denote Envenomed Wickedness!" CLASS LIST. I. A MARLBOROUGH BOY. PUTNEY WALKER. II. BLITHE. E. W. L. B. O. V. L. ROSE. SEA BREEZE. {SIMPLE SUSAN. {MONEY-SPINNER. BLITHE has made so ingenious an addition to the problem, and SIMPLE SUSAN and CO. have solved it in such tuneful verse, that I record both their answers in full. I have altered a word or two in BLITHE'S--which I trust she will excuse; it did not seem quite clear as it stood. * * * * * "Yet stay," said the youth, as a gleam of inspiration lighted up the relaxing muscles of his quiescent features. "Stay. Methinks it matters little _when_ we reached that summit, the crown of our toil. For in the space of time wherein we clambered up one mile and bounded down the same on our return, we could have trudged the _twain_ on the level. We have plodded, then, four-and-twenty miles in these six mortal hours; for never a moment did we stop for catching of fleeting breath or for gazing on the scene around!" "Very good," said the old man. "Twelve miles out and twelve miles in. And we reached the top some time between six and seven of the clock. Now mark me! For every five minutes that had fled since six of the clock when we stood on yonder peak, so many miles had we toiled upwards on the dreary mountainside!" The youth moaned and rushed into the hostel. BLITHE. The elder and the younger knight, They sallied forth at three; How far they went on level ground It matters not to me; What time they reached the foot of hill, When they began to mount, Are problems which I hold to be Of very small account. The moment that each waved his hat Upon the topmost peak-- To trivial query such as this No answer will I seek. Yet can I tell the distance well They must have travelled o'er: On hill and plain, 'twixt three and nine, The miles were twenty-four. Four miles an hour their steady pace Along the level track, Three when they climbed--but six when they Came swiftly striding back Adown the hill; and little skill It needs, methinks, to show, Up hill and down together told, Four miles an hour they go. For whether long or short the time Upon the hill they spent, Two thirds were passed in going up, One third in the descent. Two thirds at three, one third at six, If rightly reckoned o'er, Will make one whole at four--the tale Is tangled now no more. SIMPLE SUSAN. MONEY SPINNER. ANSWERS TO KNOT II. § 1. THE DINNER PARTY. _Problem._--"The Governor of Kgovjni wants to give a very small dinner party, and invites his father's brother-in-law, his brother's father-in-law, his father-in-law's brother, and his brother-in-law's father. Find the number of guests." _Answer._--"One." * * * * * In this genealogy, males are denoted by capitals, and females by small letters. The Governor is E and his guest is C. A = a | +------+-+----+ | | | b = B D = d C = c | | | | +---++--+ +-+-+ | | | | | | e = E | g = G | F ========= f Ten answers have been received. Of these, one is wrong, GALANTHUS NIVALIS MAJOR, who insists on inviting _two_ guests, one being the Governor's _wife's brother's father_. If she had taken his _sister's husband's father_ instead, she would have found it possible to reduce the guests to _one_. Of the nine who send right answers, SEA-BREEZE is the very faintest breath that ever bore the name! She simply states that the Governor's uncle might fulfill all the conditions "by intermarriages"! "Wind of the western sea," you have had a very narrow escape! Be thankful to appear in the Class-list at all! BOG-OAK and BRADSHAW OF THE FUTURE use genealogies which require 16 people instead of 14, by inviting the Governor's _father's sister's husband_ instead of his _father's wife's brother_. I cannot think this so good a solution as one that requires only 14. CAIUS and VALENTINE deserve special mention as the only two who have supplied genealogies. CLASS LIST. I. BEE. CAIUS. M. M. MATTHEW MATTICKS. OLD CAT. VALENTINE. II. BOG-OAK. BRADSHAW OF THE FUTURE. III. SEA-BREEZE. § 2. THE LODGINGS. _Problem._--"A Square has 20 doors on each side, which contains 21 equal parts. They are numbered all round, beginning at one corner. From which of the four, Nos. 9, 25, 52, 73, is the sum of the distances, to the other three, least?" _Answer._--"From No. 9." * * * * * [Illustration] Let A be No. 9, B No. 25, C No. 52, and D No. 73. Then AB = [** sqrt](12^{2} + 5^{2}) = [** sqrt]169 = 13; AC = 21; AD = [** sqrt](9^{2} + 8^{2}) = [** sqrt]145 = 12+ (N.B. _i.e._ "between 12 and 13.") BC = [** sqrt](16^{2} + 12^{2}) = [** sqrt]400 = 20; BD = [** sqrt](3^{2} + 21^{2}) = [** sqrt]450 = 21+; CD = [** sqrt](9^{2} + 13^{2}) = [** sqrt]250 = 15+; Hence sum of distances from A is between 46 and 47; from B, between 54 and 55; from C, between 56 and 57; from D, between 48 and 51. (Why not "between 48 and 49"? Make this out for yourselves.) Hence the sum is least for A. * * * * * Twenty-five solutions have been received. Of these, 15 must be marked "0," 5 are partly right, and 5 right. Of the 15, I may dismiss ALPHABETICAL PHANTOM, BOG-OAK, DINAH MITE, FIFEE, GALANTHUS NIVALIS MAJOR (I fear the cold spring has blighted our SNOWDROP), GUY, H.M.S. PINAFORE, JANET, and VALENTINE with the simple remark that they insist on the unfortunate lodgers _keeping to the pavement_. (I used the words "crossed to Number Seventy-three" for the special purpose of showing that _short cuts_ were possible.) SEA-BREEZE does the same, and adds that "the result would be the same" even if they crossed the Square, but gives no proof of this. M. M. draws a diagram, and says that No. 9 is the house, "as the diagram shows." I cannot see _how_ it does so. OLD CAT assumes that the house _must_ be No. 9 or No. 73. She does not explain how she estimates the distances. BEE's Arithmetic is faulty: she makes [** sqrt]169 + [** sqrt]442 + [** sqrt]130 = 741. (I suppose you mean [** sqrt]741, which would be a little nearer the truth. But roots cannot be added in this manner. Do you think [** sqrt]9 + [** sqrt]16 is 25, or even [** sqrt]25?) But AYR'S state is more perilous still: she draws illogical conclusions with a frightful calmness. After pointing out (rightly) that AC is less than BD she says, "therefore the nearest house to the other three must be A or C." And again, after pointing out (rightly) that B and D are both within the half-square containing A, she says "therefore" AB + AD must be less than BC + CD. (There is no logical force in either "therefore." For the first, try Nos. 1, 21, 60, 70: this will make your premiss true, and your conclusion false. Similarly, for the second, try Nos. 1, 30, 51, 71.) Of the five partly-right solutions, RAGS AND TATTERS and MAD HATTER (who send one answer between them) make No. 25 6 units from the corner instead of 5. CHEAM, E. R. D. L., and MEGGY POTTS leave openings at the corners of the Square, which are not in the _data_: moreover CHEAM gives values for the distances without any hint that they are only _approximations_. CROPHI AND MOPHI make the bold and unfounded assumption that there were really 21 houses on each side, instead of 20 as stated by Balbus. "We may assume," they add, "that the doors of Nos. 21, 42, 63, 84, are invisible from the centre of the Square"! What is there, I wonder, that CROPHI AND MOPHI would _not_ assume? Of the five who are wholly right, I think BRADSHAW OF THE FUTURE, CAIUS, CLIFTON C., and MARTREB deserve special praise for their full _analytical_ solutions. MATTHEW MATTICKS picks out No. 9, and proves it to be the right house in two ways, very neatly and ingeniously, but _why_ he picks it out does not appear. It is an excellent _synthetical_ proof, but lacks the analysis which the other four supply. CLASS LIST. I. BRADSHAW OF THE FUTURE CAIUS. CLIFTON C. MARTREB. II. MATTHEW MATTICKS. III. CHEAM. CROPHI AND MOPHI. E. R. D. L. MEGGY POTTS. {RAGS AND TATTERS. {MAD HATTER. A remonstrance has reached me from SCRUTATOR on the subject of KNOT I., which he declares was "no problem at all." "Two questions," he says, "are put. To solve one there is no data: the other answers itself." As to the first point, SCRUTATOR is mistaken; there _are_ (not "is") data sufficient to answer the question. As to the other, it is interesting to know that the question "answers itself," and I am sure it does the question great credit: still I fear I cannot enter it on the list of winners, as this competition is only open to human beings. ANSWERS TO KNOT III. _Problem._--(1) "Two travellers, starting at the same time, went opposite ways round a circular railway. Trains start each way every 15 minutes, the easterly ones going round in 3 hours, the westerly in 2. How many trains did each meet on the way, not counting trains met at the terminus itself?" (2) "They went round, as before, each traveller counting as 'one' the train containing the other traveller. How many did each meet?" _Answers._--(1) 19. (2) The easterly traveller met 12; the other 8. * * * * * The trains one way took 180 minutes, the other way 120. Let us take the L. C. M., 360, and divide the railway into 360 units. Then one set of trains went at the rate of 2 units a minute and at intervals of 30 units; the other at the rate of 3 units a minute and at intervals of 45 units. An easterly train starting has 45 units between it and the first train it will meet: it does 2-5ths of this while the other does 3-5ths, and thus meets it at the end of 18 units, and so all the way round. A westerly train starting has 30 units between it and the first train it will meet: it does 3-5ths of this while the other does 2-5ths, and thus meets it at the end of 18 units, and so all the way round. Hence if the railway be divided, by 19 posts, into 20 parts, each containing 18 units, trains meet at every post, and, in (1), each traveller passes 19 posts in going round, and so meets 19 trains. But, in (2), the easterly traveller only begins to count after traversing 2-5ths of the journey, _i.e._, on reaching the 8th post, and so counts 12 posts: similarly the other counts 8. They meet at the end of 2-5ths of 3 hours, or 3-5ths of 2 hours, _i.e._, 72 minutes. * * * * * Forty-five answers have been received. Of these 12 are beyond the reach of discussion, as they give no working. I can but enumerate their names. ARDMORE, E. A., F. A. D., L. D., MATTHEW MATTICKS, M. E. T., POO-POO, and THE RED QUEEN are all wrong. BETA and ROWENA have got (1) right and (2) wrong. CHEEKY BOB and NAIRAM give the right answers, but it may perhaps make the one less cheeky, and induce the other to take a less inverted view of things, to be informed that, if this had been a competition for a prize, they would have got no marks. [N.B.--I have not ventured to put E. A.'s name in full, as she only gave it provisionally, in case her answer should prove right.] Of the 33 answers for which the working is given, 10 are wrong; 11 half-wrong and half-right; 3 right, except that they cherish the delusion that it was _Clara_ who travelled in the easterly train--a point which the data do not enable us to settle; and 9 wholly right. The 10 wrong answers are from BO-PEEP, FINANCIER, I. W. T., KATE B., M. A. H., Q. Y. Z., SEA-GULL, THISTLEDOWN, TOM-QUAD, and an unsigned one. BO-PEEP rightly says that the easterly traveller met all trains which started during the 3 hours of her trip, as well as all which started during the previous 2 hours, _i.e._, all which started at the commencements of 20 periods of 15 minutes each; and she is right in striking out the one she met at the moment of starting; but wrong in striking out the _last_ train, for she did not meet this at the terminus, but 15 minutes before she got there. She makes the same mistake in (2). FINANCIER thinks that any train, met for the second time, is not to be counted. I. W. T. finds, by a process which is not stated, that the travellers met at the end of 71 minutes and 26-1/2 seconds. KATE B. thinks the trains which are met on starting and on arriving are _never_ to be counted, even when met elsewhere. Q. Y. Z. tries a rather complex algebraical solution, and succeeds in finding the time of meeting correctly: all else is wrong. SEA-GULL seems to think that, in (1), the easterly train _stood still_ for 3 hours; and says that, in (2), the travellers met at the end of 71 minutes 40 seconds. THISTLEDOWN nobly confesses to having tried no calculation, but merely having drawn a picture of the railway and counted the trains; in (1), she counts wrong; in (2) she makes them meet in 75 minutes. TOM-QUAD omits (1): in (2) he makes Clara count the train she met on her arrival. The unsigned one is also unintelligible; it states that the travellers go "1-24th more than the total distance to be traversed"! The "Clara" theory, already referred to, is adopted by 5 of these, viz., BO-PEEP, FINANCIER, KATE B., TOM-QUAD, and the nameless writer. The 11 half-right answers are from BOG-OAK, BRIDGET, CASTOR, CHESHIRE CAT, G. E. B., GUY, MARY, M. A. H., OLD MAID, R. W., and VENDREDI. All these adopt the "Clara" theory. CASTOR omits (1). VENDREDI gets (1) right, but in (2) makes the same mistake as BO-PEEP. I notice in your solution a marvellous proportion-sum:--"300 miles: 2 hours :: one mile: 24 seconds." May I venture to advise your acquiring, as soon as possible, an utter disbelief in the possibility of a ratio existing between _miles_ and _hours_? Do not be disheartened by your two friends' sarcastic remarks on your "roundabout ways." Their short method, of adding 12 and 8, has the slight disadvantage of bringing the answer wrong: even a "roundabout" method is better than _that_! M. A. H., in (2), makes the travellers count "one" _after_ they met, not _when_ they met. CHESHIRE CAT and OLD MAID get "20" as answer for (1), by forgetting to strike out the train met on arrival. The others all get "18" in various ways. BOG-OAK, GUY, and R. W. divide the trains which the westerly traveller has to meet into 2 sets, viz., those already on the line, which they (rightly) make "11," and those which started during her 2 hours' journey (exclusive of train met on arrival), which they (wrongly) make "7"; and they make a similar mistake with the easterly train. BRIDGET (rightly) says that the westerly traveller met a train every 6 minutes for 2 hours, but (wrongly) makes the number "20"; it should be "21." G. E. B. adopts BO-PEEP'S method, but (wrongly) strikes out (for the easterly traveller) the train which started at the _commencement_ of the previous 2 hours. MARY thinks a train, met on arrival, must not be counted, even when met on a _previous_ occasion. The 3, who are wholly right but for the unfortunate "Clara" theory, are F. LEE, G. S. C., and X. A. B. And now "descend, ye classic Ten!" who have solved the whole problem. Your names are AIX-LES-BAINS, ALGERNON BRAY (thanks for a friendly remark, which comes with a heart-warmth that not even the Atlantic could chill), ARVON, BRADSHAW OF THE FUTURE, FIFEE, H. L. R., J. L. O., OMEGA, S. S. G., and WAITING FOR THE TRAIN. Several of these have put Clara, provisionally, into the easterly train: but they seem to have understood that the data do not decide that point. CLASS LIST. I. AIX-LES-BAINS. ALGERNON BRAY. BRADSHAW OF THE FUTURE. FIFEE. H. L. R. OMEGA. S. S. G. WAITING FOR THE TRAIN. II. ARVON. J. L. O. III. F. LEE. G. S. C. X. A. B. ANSWERS TO KNOT IV. _Problem._--"There are 5 sacks, of which Nos. 1, 2, weigh 12 lbs.; Nos. 2, 3, 13-1/2 lbs.; Nos. 3, 4, 11-1/2 lbs.; Nos. 4, 5, 8 lbs.; Nos. 1, 3, 5, 16 lbs. Required the weight of each sack." _Answer._--"5-1/2, 6-1/2, 7, 4-1/2, 3-1/2." * * * * * The sum of all the weighings, 61 lbs., includes sack No. 3 _thrice_ and each other _twice_. Deducting twice the sum of the 1st and 4th weighings, we get 21 lbs. for _thrice_ No. 3, _i.e._, 7 lbs. for No. 3. Hence, the 2nd and 3rd weighings give 6-1/2 lbs., 4-1/2 lbs. for Nos. 2, 4; and hence again, the 1st and 4th weighings give 5-1/2 lbs., 3-1/2 lbs., for Nos. 1, 5. * * * * * Ninety-seven answers have been received. Of these, 15 are beyond the reach of discussion, as they give no working. I can but enumerate their names, and I take this opportunity of saying that this is the last time I shall put on record the names of competitors who give no sort of clue to the process by which their answers were obtained. In guessing a conundrum, or in catching a flea, we do not expect the breathless victor to give us afterwards, in cold blood, a history of the mental or muscular efforts by which he achieved success; but a mathematical calculation is another thing. The names of this "mute inglorious" band are COMMON SENSE, D. E. R., DOUGLAS, E. L., ELLEN, I. M. T., J. M. C., JOSEPH, KNOT I, LUCY, MEEK, M. F. C., PYRAMUS, SHAH, VERITAS. Of the eighty-two answers with which the working, or some approach to it, is supplied, one is wrong: seventeen have given solutions which are (from one cause or another) practically valueless: the remaining sixty-four I shall try to arrange in a Class-list, according to the varying degrees of shortness and neatness to which they seem to have attained. The solitary wrong answer is from NELL. To be thus "alone in the crowd" is a distinction--a painful one, no doubt, but still a distinction. I am sorry for you, my dear young lady, and I seem to hear your tearful exclamation, when you read these lines, "Ah! This is the knell of all my hopes!" Why, oh why, did you assume that the 4th and 5th bags weighed 4 lbs. each? And why did you not test your answers? However, please try again: and please don't change your _nom-de-plume_: let us have NELL in the First Class next time! The seventeen whose solutions are practically valueless are ARDMORE, A READY RECKONER, ARTHUR, BOG-LARK, BOG-OAK, BRIDGET, FIRST ATTEMPT, J. L. C., M. E. T., ROSE, ROWENA, SEA-BREEZE, SYLVIA, THISTLEDOWN, THREE-FIFTHS ASLEEP, VENDREDI, and WINIFRED. BOG-LARK tries it by a sort of "rule of false," assuming experimentally that Nos. 1, 2, weigh 6 lbs. each, and having thus produced 17-1/2, instead of 16, as the weight of 1, 3, and 5, she removes "the superfluous pound and a half," but does not explain how she knows from which to take it. THREE-FIFTHS ASLEEP says that (when in that peculiar state) "it seemed perfectly clear" to her that, "3 out of the 5 sacks being weighed twice over, 3/5 of 45 = 27, must be the total weight of the 5 sacks." As to which I can only say, with the Captain, "it beats me entirely!" WINIFRED, on the plea that "one must have a starting-point," assumes (what I fear is a mere guess) that No. 1 weighed 5-1/2 lbs. The rest all do it, wholly or partly, by guess-work. The problem is of course (as any Algebraist sees at once) a case of "simultaneous simple equations." It is, however, easily soluble by Arithmetic only; and, when this is the case, I hold that it is bad workmanship to use the more complex method. I have not, this time, given more credit to arithmetical solutions; but in future problems I shall (other things being equal) give the highest marks to those who use the simplest machinery. I have put into Class I. those whose answers seemed specially short and neat, and into Class III. those that seemed specially long or clumsy. Of this last set, A. C. M., FURZE-BUSH, JAMES, PARTRIDGE, R. W., and WAITING FOR THE TRAIN, have sent long wandering solutions, the substitutions having no definite method, but seeming to have been made to see what would come of it. CHILPOME and DUBLIN BOY omit some of the working. ARVON MARLBOROUGH BOY only finds the weight of _one_ sack. CLASS LIST I. B. E. D. C. H. CONSTANCE JOHNSON. GREYSTEAD. GUY. HOOPOE. J. F. A. M. A. H. NUMBER FIVE. PEDRO. R. E. X. SEVEN OLD MEN. VIS INERTI�. WILLY B. YAHOO. II. AMERICAN SUBSCRIBER. AN APPRECIATIVE SCHOOLMA'AM. AYR. BRADSHAW OF THE FUTURE. CHEAM. C. M. G. DINAH MITE. DUCKWING. E. C. M. E. N. Lowry. ERA. EUROCLYDON. F. H. W. FIFEE. G. E. B. HARLEQUIN. HAWTHORN. HOUGH GREEN. J. A. B. JACK TAR. J. B. B. KGOVJNI. LAND LUBBER. L. D. MAGPIE. MARY. MHRUXI. MINNIE. MONEY-SPINNER. NAIRAM. OLD CAT. POLICHINELLE. SIMPLE SUSAN. S. S. G. THISBE. VERENA. WAMBA. WOLFE. WYKEHAMICUS. Y. M. A. H. III. A. C. M. ARVON MARLBOROUGH BOY. CHILPOME. DUBLIN BOY. FURZE-BUSH. JAMES. PARTRIDGE. R. W. WAITING FOR THE TRAIN. ANSWERS TO KNOT V. _Problem._--To mark pictures, giving 3 x's to 2 or 3, 2 to 4 or 5, and 1 to 9 or 10; also giving 3 o's to 1 or 2, 2 to 3 or 4 and 1 to 8 or 9; so as to mark the smallest possible number of pictures, and to give them the largest possible number of marks. _Answer._--10 pictures; 29 marks; arranged thus:-- x x x x x x x x x o x x x x x o o o o x x o o o o o o o o _Solution._--By giving all the x's possible, putting into brackets the optional ones, we get 10 pictures marked thus:-- x x x x x x x x x (x) x x x x (x) x x (x) By then assigning o's in the same way, beginning at the other end, we get 9 pictures marked thus:-- (o) o (o) o o o (o) o o o o o o o o All we have now to do is to run these two wedges as close together as they will go, so as to get the minimum number of pictures----erasing optional marks where by so doing we can run them closer, but otherwise letting them stand. There are 10 necessary marks in the 1st row, and in the 3rd; but only 7 in the 2nd. Hence we erase all optional marks in the 1st and 3rd rows, but let them stand in the 2nd. * * * * * Twenty-two answers have been received. Of these 11 give no working; so, in accordance with what I announced in my last review of answers, I leave them unnamed, merely mentioning that 5 are right and 6 wrong. Of the eleven answers with which some working is supplied, 3 are wrong. C. H. begins with the rash assertion that under the given conditions "the sum is impossible. For," he or she adds (these initialed correspondents are dismally vague beings to deal with: perhaps "it" would be a better pronoun), "10 is the least possible number of pictures" (granted): "therefore we must either give 2 x's to 6, or 2 o's to 5." Why "must," oh alphabetical phantom? It is nowhere ordained that every picture "must" have 3 marks! FIFEE sends a folio page of solution, which deserved a better fate: she offers 3 answers, in each of which 10 pictures are marked, with 30 marks; in one she gives 2 x's to 6 pictures; in another to 7; in the 3rd she gives 2 o's to 5; thus in every case ignoring the conditions. (I pause to remark that the condition "2 x's to 4 or 5 pictures" can only mean "_either_ to 4 _or else_ to 5": if, as one competitor holds, it might mean _any_ number not less than 4, the words "_or_ 5" would be superfluous.) I. E. A. (I am happy to say that none of these bloodless phantoms appear this time in the class-list. Is it IDEA with the "D" left out?) gives 2 x's to 6 pictures. She then takes me to task for using the word "ought" instead of "nought." No doubt, to one who thus rebels against the rules laid down for her guidance, the word must be distasteful. But does not I. E. A. remember the parallel case of "adder"? That creature was originally "a nadder": then the two words took to bandying the poor "n" backwards and forwards like a shuttlecock, the final state of the game being "an adder." May not "a nought" have similarly become "an ought"? Anyhow, "oughts and crosses" is a very old game. I don't think I ever heard it called "noughts and crosses." In the following Class-list, I hope the solitary occupant of III. will sheathe her claws when she hears how narrow an escape she has had of not being named at all. Her account of the process by which she got the answer is so meagre that, like the nursery tale of "Jack-a-Minory" (I trust I. E. A. will be merciful to the spelling), it is scarcely to be distinguished from "zero." CLASS LIST. I. GUY. OLD CAT. SEA-BREEZE. II. AYR. BRADSHAW OF THE FUTURE. F. LEE. H. VERNON. III. CAT. ANSWERS TO KNOT VI. _Problem 1._--_A_ and _B_ began the year with only 1,000_l._ a-piece. They borrowed nought; they stole nought. On the next New-Year's Day they had 60,000_l._ between them. How did they do it? _Solution._--They went that day to the Bank of England. _A_ stood in front of it, while _B_ went round and stood behind it. * * * * * Two answers have been received, both worthy of much honour. ADDLEPATE makes them borrow "0" and steal "0," and uses both cyphers by putting them at the right-hand end of the 1,000_l._, thus producing 100,000_l._, which is well over the mark. But (or to express it in Latin) AT SPES INFRACTA has solved it even more ingeniously: with the first cypher she turns the "1" of the 1,000_l._ into a "9," and adds the result to the original sum, thus getting 10,000_l._: and in this, by means of the other "0," she turns the "1" into a "6," thus hitting the exact 60,000_l._ CLASS LIST I. AT SPES INFRACTA. II. ADDLEPATE. * * * * * _Problem 2._--_L_ makes 5 scarves, while _M_ makes 2: _Z_ makes 4 while _L_ makes 3. Five scarves of _Z_'s weigh one of _L_'s; 5 of _M_'s weigh 3 of _Z_'s. One of _M_'s is as warm as 4 of _Z_'s: and one of _L_'s as warm as 3 of _M_'s. Which is best, giving equal weight in the result to rapidity of work, lightness, and warmth? _Answer._--The order is _M_, _L_, _Z_. * * * * * _Solution._--As to rapidity (other things being constant) _L_'s merit is to _M_'s in the ratio of 5 to 2: _Z_'s to _L_'s in the ratio of 4 to 3. In order to get one set of 3 numbers fulfilling these conditions, it is perhaps simplest to take the one that occurs _twice_ as unity, and reduce the others to fractions: this gives, for _L_, _M_, and _Z_, the marks 1, 2/5, 4/3. In estimating for _lightness_, we observe that the greater the weight, the less the merit, so that _Z_'s merit is to _L_'s as 5 to 1. Thus the marks for _lightness_ are 1/5, 5/3, 1. And similarly, the marks for warmth are 3, 1, 1/4. To get the total result, we must _multiply_ _L_'s 3 marks together, and do the same for _M_ and for _Z_. The final numbers are 1 � 1/5 � 3, 2/5 � 5/3 � 1, 4/3 � 1 � 1/4; _i.e._ 3/5, 2/3, 1/3; _i.e._ multiplying throughout by 15 (which will not alter the proportion), 9, 10, 5; showing the order of merit to be _M_, _L_, _Z_. * * * * * Twenty-nine answers have been received, of which five are right, and twenty-four wrong. These hapless ones have all (with three exceptions) fallen into the error of _adding_ the proportional numbers together, for each candidate, instead of _multiplying_. _Why_ the latter is right, rather than the former, is fully proved in text-books, so I will not occupy space by stating it here: but it can be _illustrated_ very easily by the case of length, breadth, and depth. Suppose _A_ and _B_ are rival diggers of rectangular tanks: the amount of work done is evidently measured by the number of _cubical feet_ dug out. Let _A_ dig a tank 10 feet long, 10 wide, 2 deep: let _B_ dig one 6 feet long, 5 wide, 10 deep. The cubical contents are 200, 300; _i.e._ _B_ is best digger in the ratio of 3 to 2. Now try marking for length, width, and depth, separately; giving a maximum mark of 10 to the best in each contest, and then _adding_ the results! Of the twenty-four malefactors, one gives no working, and so has no real claim to be named; but I break the rule for once, in deference to its success in Problem 1: he, she, or it, is ADDLEPATE. The other twenty-three may be divided into five groups. First and worst are, I take it, those who put the rightful winner _last_; arranging them as "Lolo, Zuzu, Mimi." The names of these desperate wrong-doers are AYR, BRADSHAW OF THE FUTURE, FURZE-BUSH and POLLUX (who send a joint answer), GREYSTEAD, GUY, OLD HEN, and SIMPLE SUSAN. The latter was _once_ best of all; the Old Hen has taken advantage of her simplicity, and beguiled her with the chaff which was the bane of her own chickenhood. Secondly, I point the finger of scorn at those who have put the worst candidate at the top; arranging them as "Zuzu, Mimi, Lolo." They are GRAECIA, M. M., OLD CAT, and R. E. X. "'Tis Greece, but----." The third set have avoided both these enormities, and have even succeeded in putting the worst last, their answer being "Lolo, Mimi, Zuzu." Their names are AYR (who also appears among the "quite too too"), CLIFTON C., F. B., FIFEE, GRIG, JANET, and MRS. SAIREY GAMP. F. B. has not fallen into the common error; she _multiplies_ together the proportionate numbers she gets, but in getting them she goes wrong, by reckoning warmth as a _de_-merit. Possibly she is "Freshly Burnt," or comes "From Bombay." JANET and MRS. SAIREY GAMP have also avoided this error: the method they have adopted is shrouded in mystery--I scarcely feel competent to criticize it. MRS. GAMP says "if Zuzu makes 4 while Lolo makes 3, Zuzu makes 6 while Lolo makes 5 (bad reasoning), while Mimi makes 2." From this she concludes "therefore Zuzu excels in speed by 1" (_i.e._ when compared with Lolo; but what about Mimi?). She then compares the 3 kinds of excellence, measured on this mystic scale. JANET takes the statement, that "Lolo makes 5 while Mimi makes 2," to prove that "Lolo makes 3 while Mimi makes 1 and Zuzu 4" (worse reasoning than MRS. GAMP'S), and thence concludes that "Zuzu excels in speed by 1/8"! JANET should have been ADELINE, "mystery of mysteries!" The fourth set actually put Mimi at the top, arranging them as "Mimi, Zuzu, Lolo." They are MARQUIS AND CO., MARTREB, S. B. B. (first initial scarcely legible: _may_ be meant for "J"), and STANZA. The fifth set consist of AN ANCIENT FISH and CAMEL. These ill-assorted comrades, by dint of foot and fin, have scrambled into the right answer, but, as their method is wrong, of course it counts for nothing. Also AN ANCIENT FISH has very ancient and fishlike ideas as to _how_ numbers represent merit: she says "Lolo gains 2-1/2 on Mimi." Two and a half _what_? Fish, fish, art thou in thy duty? Of the five winners I put BALBUS and THE ELDER TRAVELLER slightly below the other three--BALBUS for defective reasoning, the other for scanty working. BALBUS gives two reasons for saying that _addition_ of marks is _not_ the right method, and then adds "it follows that the decision must be made by _multiplying_ the marks together." This is hardly more logical than to say "This is not Spring: _therefore_ it must be Autumn." CLASS LIST. I. DINAH MITE. E. B. D. L. JORAM. II. BALBUS. THE ELDER TRAVELLER. * * * * * With regard to Knot V., I beg to express to VIS INERTI� and to any others who, like her, understood the condition to be that _every_ marked picture must have _three_ marks, my sincere regret that the unfortunate phrase "_fill_ the columns with oughts and crosses" should have caused them to waste so much time and trouble. I can only repeat that a _literal_ interpretation of "fill" would seem to _me_ to require that _every_ picture in the gallery should be marked. VIS INERTI� would have been in the First Class if she had sent in the solution she now offers. ANSWERS TO KNOT VII. _Problem._--Given that one glass of lemonade, 3 sandwiches, and 7 biscuits, cost 1_s._ 2_d._; and that one glass of lemonade, 4 sandwiches, and 10 biscuits, cost 1_s._ 5_d._: find the cost of (1) a glass of lemonade, a sandwich, and a biscuit; and (2) 2 glasses of lemonade, 3 sandwiches, and 5 biscuits. _Answer._--(1) 8_d._; (2) 1_s._ 7_d._ _Solution._--This is best treated algebraically. Let _x_ = the cost (in pence) of a glass of lemonade, _y_ of a sandwich, and _z_ of a biscuit. Then we have _x_ + 3_y_ + 7_z_ = 14, and _x_ + 4_y_ + 10_z_ = 17. And we require the values of _x_ + _y_ + _z_, and of 2_x_ + 3_y_ + 5_z_. Now, from _two_ equations only, we cannot find, _separately_, the values of _three_ unknowns: certain _combinations_ of them may, however, be found. Also we know that we can, by the help of the given equations, eliminate 2 of the 3 unknowns from the quantity whose value is required, which will then contain one only. If, then, the required value is ascertainable at all, it can only be by the 3rd unknown vanishing of itself: otherwise the problem is impossible. Let us then eliminate lemonade and sandwiches, and reduce everything to biscuits--a state of things even more depressing than "if all the world were apple-pie"--by subtracting the 1st equation from the 2nd, which eliminates lemonade, and gives _y_ + 3_z_ = 3, or _y_ = 3-3_z_; and then substituting this value of _y_ in the 1st, which gives _x_-2_z_ = 5, _i.e._ _x_ = 5 + 2_z_. Now if we substitute these values of _x_, _y_, in the quantities whose values are required, the first becomes (5 + 2_z_) + (3-3_z_) + _z_, _i.e._ 8: and the second becomes 2(5 + 2_z_) + 3(3-3_z_) + 5_z_, _i.e._ 19. Hence the answers are (1) 8_d._, (2) 1_s._ 7_d._ * * * * * The above is a _universal_ method: that is, it is absolutely certain either to produce the answer, or to prove that no answer is possible. The question may also be solved by combining the quantities whose values are given, so as to form those whose values are required. This is merely a matter of ingenuity and good luck: and as it _may_ fail, even when the thing is possible, and is of no use in proving it _im_possible, I cannot rank this method as equal in value with the other. Even when it succeeds, it may prove a very tedious process. Suppose the 26 competitors, who have sent in what I may call _accidental_ solutions, had had a question to deal with where every number contained 8 or 10 digits! I suspect it would have been a case of "silvered is the raven hair" (see "Patience") before any solution would have been hit on by the most ingenious of them. Forty-five answers have come in, of which 44 give, I am happy to say, some sort of _working_, and therefore deserve to be mentioned by name, and to have their virtues, or vices as the case may be, discussed. Thirteen have made assumptions to which they have no right, and so cannot figure in the Class-list, even though, in 10 of the 13 cases, the answer is right. Of the remaining 28, no less than 26 have sent in _accidental_ solutions, and therefore fall short of the highest honours. I will now discuss individual cases, taking the worst first, as my custom is. FROGGY gives no working--at least this is all he gives: after stating the given equations, he says "therefore the difference, 1 sandwich + 3 biscuits, = 3_d._": then follow the amounts of the unknown bills, with no further hint as to how he got them. FROGGY has had a _very_ narrow escape of not being named at all! Of those who are wrong, VIS INERTI� has sent in a piece of incorrect working. Peruse the horrid details, and shudder! She takes _x_ (call it "_y_") as the cost of a sandwich, and concludes (rightly enough) that a biscuit will cost (3-_y_)/3. She then subtracts the second equation from the first, and deduces 3_y_ + 7 � (3-_y_)/3-4_y_ + 10 � (3-_y_)/3 = 3. By making two mistakes in this line, she brings out _y_ = 3/2. Try it again, oh VIS INERTI�! Away with INERTI�: infuse a little more VIS: and you will bring out the correct (though uninteresting) result, 0 = 0! This will show you that it is hopeless to try to coax any one of these 3 unknowns to reveal its _separate_ value. The other competitor, who is wrong throughout, is either J. M. C. or T. M. C.: but, whether he be a Juvenile Mis-Calculator or a True Mathematician Confused, he makes the answers 7_d._ and 1_s._ 5_d._ He assumes, with Too Much Confidence, that biscuits were 1/2_d._ each, and that Clara paid for 8, though she only ate 7! We will now consider the 13 whose working is wrong, though the answer is right: and, not to measure their demerits too exactly, I will take them in alphabetical order. ANITA finds (rightly) that "1 sandwich and 3 biscuits cost 3_d._," and proceeds "therefore 1 sandwich = 1-1/2_d._, 3 biscuits = 1-1/2_d._, 1 lemonade = 6_d._" DINAH MITE begins like ANITA: and thence proves (rightly) that a biscuit costs less than a 1_d._: whence she concludes (wrongly) that it _must_ cost 1/2_d._ F. C. W. is so beautifully resigned to the certainty of a verdict of "guilty," that I have hardly the heart to utter the word, without adding a "recommended to mercy owing to extenuating circumstances." But really, you know, where _are_ the extenuating circumstances? She begins by assuming that lemonade is 4_d._ a glass, and sandwiches 3_d._ each, (making with the 2 given equations, _four_ conditions to be fulfilled by _three_ miserable unknowns!). And, having (naturally) developed this into a contradiction, she then tries 5_d._ and 2_d._ with a similar result. (N.B. _This_ process might have been carried on through the whole of the Tertiary Period, without gratifying one single Megatherium.) She then, by a "happy thought," tries half-penny biscuits, and so obtains a consistent result. This may be a good solution, viewing the problem as a conundrum: but it is _not_ scientific. JANET identifies sandwiches with biscuits! "One sandwich + 3 biscuits" she makes equal to "4." Four _what_? MAYFAIR makes the astounding assertion that the equation, _s_ + 3_b_ = 3, "is evidently only satisfied by _s_ = 3/2, _b_ = 1/2"! OLD CAT believes that the assumption that a sandwich costs 1-1/2_d._ is "the only way to avoid unmanageable fractions." But _why_ avoid them? Is there not a certain glow of triumph in taming such a fraction? "Ladies and gentlemen, the fraction now before you is one that for years defied all efforts of a refining nature: it was, in a word, hopelessly vulgar. Treating it as a circulating decimal (the treadmill of fractions) only made matters worse. As a last resource, I reduced it to its lowest terms, and extracted its square root!" Joking apart, let me thank OLD CAT for some very kind words of sympathy, in reference to a correspondent (whose name I am happy to say I have now forgotten) who had found fault with me as a discourteous critic. O. V. L. is beyond my comprehension. He takes the given equations as (1) and (2): thence, by the process [(2)-(1)] deduces (rightly) equation (3) viz. _s_ + 3_b_ = 3: and thence again, by the process [�33] (a hopeless mystery), deduces 3_s_ + 4_b_ = 4. I have nothing to say about it: I give it up. SEA-BREEZE says "it is immaterial to the answer" (why?) "in what proportion 3_d._ is divided between the sandwich and the 3 biscuits": so she assumes _s_ = l-1/2_d._, _b_ = 1/2_d._ STANZA is one of a very irregular metre. At first she (like JANET) identifies sandwiches with biscuits. She then tries two assumptions (_s_ = 1, _b_ = 2/3, and _s_ = 1/2 _b_ = 5/6), and (naturally) ends in contradictions. Then she returns to the first assumption, and finds the 3 unknowns separately: _quod est absurdum_. STILETTO identifies sandwiches and biscuits, as "articles." Is the word ever used by confectioners? I fancied "What is the next article, Ma'am?" was limited to linendrapers. TWO SISTERS first assume that biscuits are 4 a penny, and then that they are 2 a penny, adding that "the answer will of course be the same in both cases." It is a dreamy remark, making one feel something like Macbeth grasping at the spectral dagger. "Is this a statement that I see before me?" If you were to say "we both walked the same way this morning," and _I_ were to say "_one_ of you walked the same way, but the other didn't," which of the three would be the most hopelessly confused? TURTLE PYATE (what _is_ a Turtle Pyate, please?) and OLD CROW, who send a joint answer, and Y. Y., adopt the same method. Y. Y. gets the equation _s_ + 3_b_ = 3: and then says "this sum must be apportioned in one of the three following ways." It _may_ be, I grant you: but Y. Y. do you say "must"? I fear it is _possible_ for Y. Y. to be _two_ Y's. The other two conspirators are less positive: they say it "can" be so divided: but they add "either of the three prices being right"! This is bad grammar and bad arithmetic at once, oh mysterious birds! Of those who win honours, THE SHETLAND SNARK must have the 3rd class all to himself. He has only answered half the question, viz. the amount of Clara's luncheon: the two little old ladies he pitilessly leaves in the midst of their "difficulty." I beg to assure him (with thanks for his friendly remarks) that entrance-fees and subscriptions are things unknown in that most economical of clubs, "The Knot-Untiers." The authors of the 26 "accidental" solutions differ only in the number of steps they have taken between the _data_ and the answers. In order to do them full justice I have arranged the 2nd class in sections, according to the number of steps. The two Kings are fearfully deliberate! I suppose walking quick, or taking short cuts, is inconsistent with kingly dignity: but really, in reading THESEUS' solution, one almost fancied he was "marking time," and making no advance at all! The other King will, I hope, pardon me for having altered "Coal" into "Cole." King Coilus, or Coil, seems to have reigned soon after Arthur's time. Henry of Huntingdon identifies him with the King Coël who first built walls round Colchester, which was named after him. In the Chronicle of Robert of Gloucester we read:-- "Aftur Kyng Aruirag, of wam we habbeth y told, Marius ys sone was kyng, quoynte mon & bold. And ys sone was aftur hym, _Coil_ was ys name, Bothe it were quoynte men, & of noble fame." BALBUS lays it down as a general principle that "in order to ascertain the cost of any one luncheon, it must come to the same amount upon two different assumptions." (_Query._ Should not "it" be "we"? Otherwise the _luncheon_ is represented as wishing to ascertain its own cost!) He then makes two assumptions--one, that sandwiches cost nothing; the other, that biscuits cost nothing, (either arrangement would lead to the shop being inconveniently crowded!)--and brings out the unknown luncheons as 8_d._ and 19_d._, on each assumption. He then concludes that this agreement of results "shows that the answers are correct." Now I propose to disprove his general law by simply giving _one_ instance of its failing. One instance is quite enough. In logical language, in order to disprove a "universal affirmative," it is enough to prove its contradictory, which is a "particular negative." (I must pause for a digression on Logic, and especially on Ladies' Logic. The universal affirmative "everybody says he's a duck" is crushed instantly by proving the particular negative "Peter says he's a goose," which is equivalent to "Peter does _not_ say he's a duck." And the universal negative "nobody calls on her" is well met by the particular affirmative "_I_ called yesterday." In short, either of two contradictories disproves the other: and the moral is that, since a particular proposition is much more easily proved than a universal one, it is the wisest course, in arguing with a Lady, to limit one's _own_ assertions to "particulars," and leave _her_ to prove the "universal" contradictory, if she can. You will thus generally secure a _logical_ victory: a _practical_ victory is not to be hoped for, since she can always fall back upon the crushing remark "_that_ has nothing to do with it!"--a move for which Man has not yet discovered any satisfactory answer. Now let us return to BALBUS.) Here is my "particular negative," on which to test his rule. Suppose the two recorded luncheons to have been "2 buns, one queen-cake, 2 sausage-rolls, and a bottle of Zoëdone: total, one-and-ninepence," and "one bun, 2 queen-cakes, a sausage-roll, and a bottle of Zoëdone: total, one-and-fourpence." And suppose Clara's unknown luncheon to have been "3 buns, one queen-cake, one sausage-roll, and 2 bottles of Zoëdone:" while the two little sisters had been indulging in "8 buns, 4 queen-cakes, 2 sausage-rolls, and 6 bottles of Zoëdone." (Poor souls, how thirsty they must have been!) If BALBUS will kindly try this by his principle of "two assumptions," first assuming that a bun is 1_d._ and a queen-cake 2_d._, and then that a bun is 3_d._ and a queen-cake 3_d._, he will bring out the other two luncheons, on each assumption, as "one-and-nine-pence" and "four-and-ten-pence" respectively, which harmony of results, he will say, "shows that the answers are correct." And yet, as a matter of fact, the buns were 2_d._ each, the queen-cakes 3_d._, the sausage-rolls 6_d._, and the Zoëdone 2_d._ a bottle: so that Clara's third luncheon had cost one-and-sevenpence, and her thirsty friends had spent four-and-fourpence! Another remark of BALBUS I will quote and discuss: for I think that it also may yield a moral for some of my readers. He says "it is the same thing in substance whether in solving this problem we use words and call it Arithmetic, or use letters and signs and call it Algebra." Now this does not appear to me a correct description of the two methods: the Arithmetical method is that of "synthesis" only; it goes from one known fact to another, till it reaches its goal: whereas the Algebraical method is that of "analysis": it begins with the goal, symbolically represented, and so goes backwards, dragging its veiled victim with it, till it has reached the full daylight of known facts, in which it can tear off the veil and say "I know you!" Take an illustration. Your house has been broken into and robbed, and you appeal to the policeman who was on duty that night. "Well, Mum, I did see a chap getting out over your garden-wall: but I was a good bit off, so I didn't chase him, like. I just cut down the short way to the Chequers, and who should I meet but Bill Sykes, coming full split round the corner. So I just ups and says 'My lad, you're wanted.' That's all I says. And he says 'I'll go along quiet, Bobby,' he says, 'without the darbies,' he says." There's your _Arithmetical_ policeman. Now try the other method. "I seed somebody a running, but he was well gone or ever _I_ got nigh the place. So I just took a look round in the garden. And I noticed the foot-marks, where the chap had come right across your flower-beds. They was good big foot-marks sure-ly. And I noticed as the left foot went down at the heel, ever so much deeper than the other. And I says to myself 'The chap's been a big hulking chap: and he goes lame on his left foot.' And I rubs my hand on the wall where he got over, and there was soot on it, and no mistake. So I says to myself 'Now where can I light on a big man, in the chimbley-sweep line, what's lame of one foot?' And I flashes up permiscuous: and I says 'It's Bill Sykes!' says I." There is your _Algebraical_ policeman--a higher intellectual type, to my thinking, than the other. LITTLE JACK'S solution calls for a word of praise, as he has written out what really is an algebraical proof _in words_, without representing any of his facts as equations. If it is all his own, he will make a good algebraist in the time to come. I beg to thank SIMPLE SUSAN for some kind words of sympathy, to the same effect as those received from OLD CAT. HECLA and MARTREB are the only two who have used a method _certain_ either to produce the answer, or else to prove it impossible: so they must share between them the highest honours. CLASS LIST. I. HECLA. MARTREB. II. § 1 (2 _steps_). ADELAIDE. CLIFTON C.... E. K. C. GUY. L'INCONNU. LITTLE JACK. NIL DESPERANDUM. SIMPLE SUSAN. YELLOW-HAMMER. WOOLLY ONE. § 2 (3 _steps_). A. A. A CHRISTMAS CAROL. AFTERNOON TEA. AN APPRECIATIVE SCHOOLMA'AM. BABY. BALBUS. BOG-OAK. THE RED QUEEN. WALL-FLOWER. § 3 (4 _steps_). HAWTHORN. JORAM. S. S. G. § 4 (5 _steps_). A STEPNEY COACH. § 5 (6 _steps_). BAY LAUREL. BRADSHAW OF THE FUTURE. § 6 (9 _steps_). OLD KING COLE. § 7 (14 _steps_). THESEUS. ANSWERS TO CORRESPONDENTS. I have received several letters on the subjects of Knots II. and VI., which lead me to think some further explanation desirable. In Knot II., I had intended the numbering of the houses to begin at one corner of the Square, and this was assumed by most, if not all, of the competitors. TROJANUS however says "assuming, in default of any information, that the street enters the square in the middle of each side, it may be supposed that the numbering begins at a street." But surely the other is the more natural assumption? In Knot VI., the first Problem was of course a mere _jeu de mots_, whose presence I thought excusable in a series of Problems whose aim is to entertain rather than to instruct: but it has not escaped the contemptuous criticisms of two of my correspondents, who seem to think that Apollo is in duty bound to keep his bow always on the stretch. Neither of them has guessed it: and this is true human nature. Only the other day--the 31st of September, to be quite exact--I met my old friend Brown, and gave him a riddle I had just heard. With one great effort of his colossal mind, Brown guessed it. "Right!" said I. "Ah," said he, "it's very neat--very neat. And it isn't an answer that would occur to everybody. Very neat indeed." A few yards further on, I fell in with Smith and to him I propounded the same riddle. He frowned over it for a minute, and then gave it up. Meekly I faltered out the answer. "A poor thing, sir!" Smith growled, as he turned away. "A very poor thing! I wonder you care to repeat such rubbish!" Yet Smith's mind is, if possible, even more colossal than Brown's. The second Problem of Knot VI. is an example in ordinary Double Rule of Three, whose essential feature is that the result depends on the variation of several elements, which are so related to it that, if all but one be constant, it varies as that one: hence, if none be constant, it varies as their product. Thus, for example, the cubical contents of a rectangular tank vary as its length, if breadth and depth be constant, and so on; hence, if none be constant, it varies as the product of the length, breadth, and depth. When the result is not thus connected with the varying elements, the Problem ceases to be Double Rule of Three and often becomes one of great complexity. To illustrate this, let us take two candidates for a prize, _A_ and _B_, who are to compete in French, German, and Italian: (_a_) Let it be laid down that the result is to depend on their _relative_ knowledge of each subject, so that, whether their marks, for French, be "1, 2" or "100, 200," the result will be the same: and let it also be laid down that, if they get equal marks on 2 papers, the final marks are to have the same ratio as those of the 3rd paper. This is a case of ordinary Double Rule of Three. We multiply _A_'s 3 marks together, and do the same for _B_. Note that, if _A_ gets a single "0," his final mark is "0," even if he gets full marks for 2 papers while _B_ gets only one mark for each paper. This of course would be very unfair on _A_, though a correct solution under the given conditions. (_b_) The result is to depend, as before, on _relative_ knowledge; but French is to have twice as much weight as German or Italian. This is an unusual form of question. I should be inclined to say "the resulting ratio is to be nearer to the French ratio than if we multiplied as in (_a_), and so much nearer that it would be necessary to use the other multipliers _twice_ to produce the same result as in (_a_):" _e.g._ if the French Ratio were 9/10, and the others 4/9, 1/9 so that the ultimate ratio, by method (_a_), would be 2/45, I should multiply instead by 2/3, 1/3, giving the result, 1/3 which is nearer to 9/10 than if he had used method (_a_). (_c_) The result is to depend on _actual_ amount of knowledge of the 3 subjects collectively. Here we have to ask two questions. (1) What is to be the "unit" (_i.e._ "standard to measure by") in each subject? (2) Are these units to be of equal, or unequal value? The usual "unit" is the knowledge shown by answering the whole paper correctly; calling this "100," all lower amounts are represented by numbers between "0" and "100." Then, if these units are to be of equal value, we simply add _A_'s 3 marks together, and do the same for _B_. (_d_) The conditions are the same as (_c_), but French is to have double weight. Here we simply double the French marks, and add as before. (_e_) French is to have such weight, that, if other marks be equal, the ultimate ratio is to be that of the French paper, so that a "0" in this would swamp the candidate: but the other two subjects are only to affect the result collectively, by the amount of knowledge shown, the two being reckoned of equal value. Here I should add _A_'s German and Italian marks together, and multiply by his French mark. But I need not go on: the problem may evidently be set with many varying conditions, each requiring its own method of solution. The Problem in Knot VI. was meant to belong to variety (_a_), and to make this clear, I inserted the following passage: "Usually the competitors differ in one point only. Thus, last year, Fifi and Gogo made the same number of scarves in the trial week, and they were equally light; but Fifi's were twice as warm as Gogo's, and she was pronounced twice as good." What I have said will suffice, I hope, as an answer to BALBUS, who holds that (_a_) and (_c_) are the only possible varieties of the problem, and that to say "We cannot use addition, therefore we must be intended to use multiplication," is "no more illogical than, from knowledge that one was not born in the night, to infer that he was born in the daytime"; and also to FIFEE, who says "I think a little more consideration will show you that our 'error of _adding_ the proportional numbers together for each candidate instead of _multiplying_' is no error at all." Why, even if addition _had_ been the right method to use, not one of the writers (I speak from memory) showed any consciousness of the necessity of fixing a "unit" for each subject. "No error at all!" They were positively steeped in error! One correspondent (I do not name him, as the communication is not quite friendly in tone) writes thus:--"I wish to add, very respectfully, that I think it would be in better taste if you were to abstain from the very trenchant expressions which you are accustomed to indulge in when criticising the answer. That such a tone must not be" ("be not"?) "agreeable to the persons concerned who have made mistakes may possibly have no great weight with you, but I hope you will feel that it would be as well not to employ it, _unless you are quite certain of being correct yourself_." The only instances the writer gives of the "trenchant expressions" are "hapless" and "malefactors." I beg to assure him (and any others who may need the assurance: I trust there are none) that all such words have been used in jest, and with no idea that they could possibly annoy any one, and that I sincerely regret any annoyance I may have thus inadvertently given. May I hope that in future they will recognise the distinction between severe language used in sober earnest, and the "words of unmeant bitterness," which Coleridge has alluded to in that lovely passage beginning "A little child, a limber elf"? If the writer will refer to that passage, or to the preface to "Fire, Famine, and Slaughter," he will find the distinction, for which I plead, far better drawn out than I could hope to do in any words of mine. The writer's insinuation that I care not how much annoyance I give to my readers I think it best to pass over in silence; but to his concluding remark I must entirely demur. I hold that to use language likely to annoy any of my correspondents would not be in the least justified by the plea that I was "quite certain of being correct." I trust that the knot-untiers and I are not on such terms as those! I beg to thank _G. B._ for the offer of a puzzle--which, however, is too like the old one "Make four 9's into 100." ANSWERS TO KNOT VIII. § 1. THE PIGS. _Problem._--Place twenty-four pigs in four sties so that, as you go round and round, you may always find the number in each sty nearer to ten than the number in the last. _Answer._--Place 8 pigs in the first sty, 10 in the second, nothing in the third, and 6 in the fourth: 10 is nearer ten than 8; nothing is nearer ten than 10; 6 is nearer ten than nothing; and 8 is nearer ten than 6. * * * * * This problem is noticed by only two correspondents. BALBUS says "it certainly cannot be solved mathematically, nor do I see how to solve it by any verbal quibble." NOLENS VOLENS makes Her Radiancy change the direction of going round; and even then is obliged to add "the pigs must be carried in front of her"! § 2. THE GRURMSTIPTHS. _Problem._--Omnibuses start from a certain point, both ways, every 15 minutes. A traveller, starting on foot along with one of them, meets one in 12-1/2 minutes: when will he be overtaken by one? _Answer._--In 6-1/4 minutes. * * * * * _Solution._--Let "_a_" be the distance an omnibus goes in 15 minutes, and "_x_" the distance from the starting-point to where the traveller is overtaken. Since the omnibus met is due at the starting-point in 2-1/2 minutes, it goes in that time as far as the traveller walks in 12-1/2; _i.e._ it goes 5 times as fast. Now the overtaking omnibus is "_a_" behind the traveller when he starts, and therefore goes "_a_ + _x_" while he goes "_x_." Hence _a_ + _x_ = 5_x_; _i.e._ 4_x_ = _a_, and _x_ = _a_/4. This distance would be traversed by an omnibus in 15/4 minutes, and therefore by the traveller in 5 � 15/4. Hence he is overtaken in 18-3/4 minutes after starting, _i.e._ in 6-1/4 minutes after meeting the omnibus. Four answers have been received, of which two are wrong. DINAH MITE rightly states that the overtaking omnibus reached the point where they met the other omnibus 5 minutes after they left, but wrongly concludes that, going 5 times as fast, it would overtake them in another minute. The travellers are 5-minutes-walk ahead of the omnibus, and must walk 1-4th of this distance farther before the omnibus overtakes them, which will be 1-5th of the distance traversed by the omnibus in the same time: this will require 1-1/4 minutes more. NOLENS VOLENS tries it by a process like "Achilles and the Tortoise." He rightly states that, when the overtaking omnibus leaves the gate, the travellers are 1-5th of "_a_" ahead, and that it will take the omnibus 3 minutes to traverse this distance; "during which time" the travellers, he tells us, go 1-15th of "_a_" (this should be 1-25th). The travellers being now 1-15th of "_a_" ahead, he concludes that the work remaining to be done is for the travellers to go 1-60th of "_a_," while the omnibus goes 1-12th. The _principle_ is correct, and might have been applied earlier. CLASS LIST. I. BALBUS. DELTA. ANSWERS TO KNOT IX. § 1. THE BUCKETS. _Problem._--Lardner states that a solid, immersed in a fluid, displaces an amount equal to itself in bulk. How can this be true of a small bucket floating in a larger one? _Solution._--Lardner means, by "displaces," "occupies a space which might be filled with water without any change in the surroundings." If the portion of the floating bucket, which is above the water, could be annihilated, and the rest of it transformed into water, the surrounding water would not change its position: which agrees with Lardner's statement. * * * * * Five answers have been received, none of which explains the difficulty arising from the well-known fact that a floating body is the same weight as the displaced fluid. HECLA says that "only that portion of the smaller bucket which descends below the original level of the water can be properly said to be immersed, and only an equal bulk of water is displaced." Hence, according to HECLA, a solid, whose weight was equal to that of an equal bulk of water, would not float till the whole of it was below "the original level" of the water: but, as a matter of fact, it would float as soon as it was all under water. MAGPIE says the fallacy is "the assumption that one body can displace another from a place where it isn't," and that Lardner's assertion is incorrect, except when the containing vessel "was originally full to the brim." But the question of floating depends on the present state of things, not on past history. OLD KING COLE takes the same view as HECLA. TYMPANUM and VINDEX assume that "displaced" means "raised above its original level," and merely explain how it comes to pass that the water, so raised, is less in bulk than the immersed portion of bucket, and thus land themselves--or rather set themselves floating--in the same boat as HECLA. I regret that there is no Class-list to publish for this Problem. * * * * * § 2. BALBUS' ESSAY. _Problem._--Balbus states that if a certain solid be immersed in a certain vessel of water, the water will rise through a series of distances, two inches, one inch, half an inch, &c., which series has no end. He concludes that the water will rise without limit. Is this true? _Solution._--No. This series can never reach 4 inches, since, however many terms we take, we are always short of 4 inches by an amount equal to the last term taken. * * * * * Three answers have been received--but only two seem to me worthy of honours. TYMPANUM says that the statement about the stick "is merely a blind, to which the old answer may well be applied, _solvitur ambulando_, or rather _mergendo_." I trust TYMPANUM will not test this in his own person, by taking the place of the man in Balbus' Essay! He would infallibly be drowned. OLD KING COLE rightly points out that the series, 2, 1, &c., is a decreasing Geometrical Progression: while VINDEX rightly identifies the fallacy as that of "Achilles and the Tortoise." CLASS LIST. I. OLD KING COLE. VINDEX. * * * * * § 3. THE GARDEN. _Problem._--An oblong garden, half a yard longer than wide, consists entirely of a gravel-walk, spirally arranged, a yard wide and 3,630 yards long. Find the dimensions of the garden. _Answer._--60, 60-1/2. _Solution._--The number of yards and fractions of a yard traversed in walking along a straight piece of walk, is evidently the same as the number of square-yards and fractions of a square-yard, contained in that piece of walk: and the distance, traversed in passing through a square-yard at a corner, is evidently a yard. Hence the area of the garden is 3,630 square-yards: _i.e._, if _x_ be the width, _x_ (_x_ + 1/2) = 3,630. Solving this Quadratic, we find _x_ = 60. Hence the dimensions are 60, 60-1/2. * * * * * Twelve answers have been received--seven right and five wrong. C. G. L., NABOB, OLD CROW, and TYMPANUM assume that the number of yards in the length of the path is equal to the number of square-yards in the garden. This is true, but should have been proved. But each is guilty of darker deeds. C. G. L.'s "working" consists of dividing 3,630 by 60. Whence came this divisor, oh Segiel? Divination? Or was it a dream? I fear this solution is worth nothing. OLD CROW'S is shorter, and so (if possible) worth rather less. He says the answer "is at once seen to be 60 � 60-1/2"! NABOB'S calculation is short, but "as rich as a Nabob" in error. He says that the square root of 3,630, multiplied by 2, equals the length plus the breadth. That is 60.25 � 2 = 120-1/2. His first assertion is only true of a _square_ garden. His second is irrelevant, since 60.25 is _not_ the square-root of 3,630! Nay, Bob, this will _not_ do! TYMPANUM says that, by extracting the square-root of 3,630, we get 60 yards with a remainder of 30/60, or half-a-yard, which we add so as to make the oblong 60 � 60-1/2. This is very terrible: but worse remains behind. TYMPANUM proceeds thus:--"But why should there be the half-yard at all? Because without it there would be no space at all for flowers. By means of it, we find reserved in the very centre a small plot of ground, two yards long by half-a-yard wide, the only space not occupied by walk." But Balbus expressly said that the walk "used up the whole of the area." Oh, TYMPANUM! My tympa is exhausted: my brain is num! I can say no more. HECLA indulges, again and again, in that most fatal of all habits in computation--the making _two_ mistakes which cancel each other. She takes _x_ as the width of the garden, in yards, and _x_ + 1/2 as its length, and makes her first "coil" the sum of _x_-1/2, _x_-1/2, _x_-1, _x_-1, _i.e._ 4_x_-3: but the fourth term should be _x_-1-1/2, so that her first coil is 1/2 a yard too long. Her second coil is the sum of _x_-2-1/2, _x_-2-1/2, _x_-3, _x_-3: here the first term should be _x_-2 and the last _x_-3-1/2: these two mistakes cancel, and this coil is therefore right. And the same thing is true of every other coil but the last, which needs an extra half-yard to reach the _end_ of the path: and this exactly balances the mistake in the first coil. Thus the sum total of the coils comes right though the working is all wrong. Of the seven who are right, DINAH MITE, JANET, MAGPIE, and TAFFY make the same assumption as C. G. L. and Co. They then solve by a Quadratic. MAGPIE also tries it by Arithmetical Progression, but fails to notice that the first and last "coils" have special values. ALUMNUS ETON� attempts to prove what C. G. L. assumes by a particular instance, taking a garden 6 by 5-1/2. He ought to have proved it generally: what is true of one number is not always true of others. OLD KING COLE solves it by an Arithmetical Progression. It is right, but too lengthy to be worth as much as a Quadratic. VINDEX proves it very neatly, by pointing out that a yard of walk measured along the middle represents a square yard of garden, "whether we consider the straight stretches of walk or the square yards at the angles, in which the middle line goes half a yard in one direction and then turns a right angle and goes half a yard in another direction." CLASS LIST. I. VINDEX. II. ALUMNUS ETON�. OLD KING COLE. III. DINAH MITE. JANET. MAGPIE. TAFFY. ANSWERS TO KNOT X. § 1. THE CHELSEA PENSIONERS. _Problem._--If 70 per cent. have lost an eye, 75 per cent. an ear, 80 per cent. an arm, 85 per cent. a leg: what percentage, _at least_, must have lost all four? _Answer._--Ten. * * * * * _Solution._--(I adopt that of POLAR STAR, as being better than my own). Adding the wounds together, we get 70 + 75 + 80 + 85 = 310, among 100 men; which gives 3 to each, and 4 to 10 men. Therefore the least percentage is 10. * * * * * Nineteen answers have been received. One is "5," but, as no working is given with it, it must, in accordance with the rule, remain "a deed without a name." JANET makes it "35 and 7/10ths." I am sorry she has misunderstood the question, and has supposed that those who had lost an ear were 75 per cent. _of those who had lost an eye_; and so on. Of course, on this supposition, the percentages must all be multiplied together. This she has done correctly, but I can give her no honours, as I do not think the question will fairly bear her interpretation, THREE SCORE AND TEN makes it "19 and 3/8ths." Her solution has given me--I will not say "many anxious days and sleepless nights," for I wish to be strictly truthful, but--some trouble in making any sense at all of it. She makes the number of "pensioners wounded once" to be 310 ("per cent.," I suppose!): dividing by 4, she gets 77 and a half as "average percentage:" again dividing by 4, she gets 19 and 3/8ths as "percentage wounded four times." Does she suppose wounds of different kinds to "absorb" each other, so to speak? Then, no doubt, the _data_ are equivalent to 77 pensioners with one wound each, and a half-pensioner with a half-wound. And does she then suppose these concentrated wounds to be _transferable_, so that 3/4ths of these unfortunates can obtain perfect health by handing over their wounds to the remaining 1/4th? Granting these suppositions, her answer is right; or rather, _if_ the question had been "A road is covered with one inch of gravel, along 77 and a half per cent. of it. How much of it could be covered 4 inches deep with the same material?" her answer _would_ have been right. But alas, that _wasn't_ the question! DELTA makes some most amazing assumptions: "let every one who has not lost an eye have lost an ear," "let every one who has not lost both eyes and ears have lost an arm." Her ideas of a battle-field are grim indeed. Fancy a warrior who would continue fighting after losing both eyes, both ears, and both arms! This is a case which she (or "it?") evidently considers _possible_. Next come eight writers who have made the unwarrantable assumption that, because 70 per cent. have lost an eye, _therefore_ 30 per cent. have _not_ lost one, so that they have _both_ eyes. This is illogical. If you give me a bag containing 100 sovereigns, and if in an hour I come to you (my face _not_ beaming with gratitude nearly so much as when I received the bag) to say "I am sorry to tell you that 70 of these sovereigns are bad," do I thereby guarantee the other 30 to be good? Perhaps I have not tested them yet. The sides of this illogical octagon are as follows, in alphabetical order:--ALGERNON BRAY, DINAH MITE, G. S. C., JANE E., J. D. W., MAGPIE (who makes the delightful remark "therefore 90 per cent. have two of something," recalling to one's memory that fortunate monarch, with whom Xerxes was so much pleased that "he gave him ten of everything!"), S. S. G., and TOKIO. BRADSHAW OF THE FUTURE and T. R. do the question in a piecemeal fashion--on the principle that the 70 per cent. and the 75 per cent., though commenced at opposite ends of the 100, must overlap by _at least_ 45 per cent.; and so on. This is quite correct working, but not, I think, quite the best way of doing it. The other five competitors will, I hope, feel themselves sufficiently glorified by being placed in the first class, without my composing a Triumphal Ode for each! CLASS LIST. I. OLD CAT. OLD HEN. POLAR STAR. SIMPLE SUSAN. WHITE SUGAR. II. BRADSHAW OF THE FUTURE. T. R. III. ALGERNON BRAY. DINAH MITE. G. S. C. JANE E. J. D. W. MAGPIE. S. S. G. TOKIO. § 2. CHANGE OF DAY. I must postpone, _sine die_, the geographical problem--partly because I have not yet received the statistics I am hoping for, and partly because I am myself so entirely puzzled by it; and when an examiner is himself dimly hovering between a second class and a third how is he to decide the position of others? § 3. THE SONS' AGES. _Problem._--"At first, two of the ages are together equal to the third. A few years afterwards, two of them are together double of the third. When the number of years since the first occasion is two-thirds of the sum of the ages on that occasion, one age is 21. What are the other two? _Answer._--"15 and 18." * * * * * _Solution._--Let the ages at first be _x_, _y_, (_x_ + _y_). Now, if _a_ + _b_ = 2_c_, then (_a_-_n_) + (_b_-_n_) = 2(_c_-_n_), whatever be the value of _n_. Hence the second relationship, if _ever_ true, was _always_ true. Hence it was true at first. But it cannot be true that _x_ and _y_ are together double of (_x_ + _y_). Hence it must be true of (_x_ + _y_), together with _x_ or _y_; and it does not matter which we take. We assume, then, (_x_ + _y_) + _x_ = 2_y_; _i.e._ _y_ = 2_x_. Hence the three ages were, at first, _x_, 2_x_, 3_x_; and the number of years, since that time is two-thirds of 6_x_, _i.e._ is 4_x_. Hence the present ages are 5_x_, 6_x_, 7_x_. The ages are clearly _integers_, since this is only "the year when one of my sons comes of age." Hence 7_x_ = 21, _x_ = 3, and the other ages are 15, 18. * * * * * Eighteen answers have been received. One of the writers merely asserts that the first occasion was 12 years ago, that the ages were then 9, 6, and 3; and that on the second occasion they were 14, 11, and 8! As a Roman father, I _ought_ to withhold the name of the rash writer; but respect for age makes me break the rule: it is THREE SCORE AND TEN. JANE E. also asserts that the ages at first were 9, 6, 3: then she calculates the present ages, leaving the _second_ occasion unnoticed. OLD HEN is nearly as bad; she "tried various numbers till I found one that fitted _all_ the conditions"; but merely scratching up the earth, and pecking about, is _not_ the way to solve a problem, oh venerable bird! And close after OLD HEN prowls, with hungry eyes, OLD CAT, who calmly assumes, to begin with, that the son who comes of age is the _eldest_. Eat your bird, Puss, for you will get nothing from me! There are yet two zeroes to dispose of. MINERVA assumes that, on _every_ occasion, a son comes of age; and that it is only such a son who is "tipped with gold." Is it wise thus to interpret "now, my boys, calculate your ages, and you shall have the money"? BRADSHAW OF THE FUTURE says "let" the ages at first be 9, 6, 3, then assumes that the second occasion was 6 years afterwards, and on these baseless assumptions brings out the right answers. Guide _future_ travellers, an thou wilt: thou art no Bradshaw for _this_ Age! Of those who win honours, the merely "honourable" are two. DINAH MITE ascertains (rightly) the relationship between the three ages at first, but then _assumes_ one of them to be "6," thus making the rest of her solution tentative. M. F. C. does the algebra all right up to the conclusion that the present ages are 5_z_, 6_z_, and 7_z_; it then assumes, without giving any reason, that 7_z_ = 21. Of the more honourable, DELTA attempts a novelty--to discover _which_ son comes of age by elimination: it assumes, successively, that it is the middle one, and that it is the youngest; and in each case it _apparently_ brings out an absurdity. Still, as the proof contains the following bit of algebra, "63 = 7_x_ + 4_y_; [** therefore] 21 = _x_ + 4 sevenths of _y_," I trust it will admit that its proof is not _quite_ conclusive. The rest of its work is good. MAGPIE betrays the deplorable tendency of her tribe--to appropriate any stray conclusion she comes across, without having any _strict_ logical right to it. Assuming _A_, _B_, _C_, as the ages at first, and _D_ as the number of the years that have elapsed since then, she finds (rightly) the 3 equations, 2_A_ = _B_, _C_ = _B_ + _A_, _D_ = 2_B_. She then says "supposing that _A_ = 1, then _B_ = 2, _C_ = 3, and _D_ = 4. Therefore for _A_, _B_, _C_, _D_, four numbers are wanted which shall be to each other as 1:2:3:4." It is in the "therefore" that I detect the unconscientiousness of this bird. The conclusion _is_ true, but this is only because the equations are "homogeneous" (_i.e._ having one "unknown" in each term), a fact which I strongly suspect had not been grasped--I beg pardon, clawed--by her. Were I to lay this little pitfall, "_A_ + 1 = _B_, _B_ + 1 = _C_; supposing _A_ = 1, then _B_ = 2 and _C_ = 3. _Therefore_ for _A_, _B_, _C_, three numbers are wanted which shall be to one another as 1:2:3," would you not flutter down into it, oh MAGPIE, as amiably as a Dove? SIMPLE SUSAN is anything but simple to _me_. After ascertaining that the 3 ages at first are as 3:2:1, she says "then, as two-thirds of their sum, added to one of them, = 21, the sum cannot exceed 30, and consequently the highest cannot exceed 15." I suppose her (mental) argument is something like this:--"two-thirds of sum, + one age, = 21; [** therefore] sum, + 3 halves of one age, = 31 and a half. But 3 halves of one age cannot be less than 1 and-a-half (here I perceive that SIMPLE SUSAN would on no account present a guinea to a new-born baby!) hence the sum cannot exceed 30." This is ingenious, but her proof, after that, is (as she candidly admits) "clumsy and roundabout." She finds that there are 5 possible sets of ages, and eliminates four of them. Suppose that, instead of 5, there had been 5 million possible sets? Would SIMPLE SUSAN have courageously ordered in the necessary gallon of ink and ream of paper? The solution sent in by C. R. is, like that of SIMPLE SUSAN, partly tentative, and so does not rise higher than being Clumsily Right. Among those who have earned the highest honours, ALGERNON BRAY solves the problem quite correctly, but adds that there is nothing to exclude the supposition that all the ages were _fractional_. This would make the number of answers infinite. Let me meekly protest that I _never_ intended my readers to devote the rest of their lives to writing out answers! E. M. RIX points out that, if fractional ages be admissible, any one of the three sons might be the one "come of age"; but she rightly rejects this supposition on the ground that it would make the problem indeterminate. WHITE SUGAR is the only one who has detected an oversight of mine: I had forgotten the possibility (which of course ought to be allowed for) that the son, who came of age that _year_, need not have done so by that _day_, so that he _might_ be only 20. This gives a second solution, viz., 20, 24, 28. Well said, pure Crystal! Verily, thy "fair discourse hath been as sugar"! CLASS LIST. I. ALGERNON BRAY. AN OLD FOGEY. E. M. RIX. G. S. C. S. S. G. TOKIO. T. R. WHITE SUGAR. II. C. R. DELTA. MAGPIE. SIMPLE SUSAN. III. DINAH MITE. M. F. C. * * * * * I have received more than one remonstrance on my assertion, in the Chelsea Pensioners' problem, that it was illogical to assume, from the _datum_ "70 p. c. have lost an eye," that 30 p. c. have _not_. ALGERNON BRAY states, as a parallel case, "suppose Tommy's father gives him 4 apples, and he eats one of them, how many has he left?" and says "I think we are justified in answering, 3." I think so too. There is no "must" here, and the _data_ are evidently meant to fix the answer _exactly_: but, if the question were set me "how many _must_ he have left?", I should understand the _data_ to be that his father gave him 4 _at least_, but _may_ have given him more. I take this opportunity of thanking those who have sent, along with their answers to the Tenth Knot, regrets that there are no more Knots to come, or petitions that I should recall my resolution to bring them to an end. I am most grateful for their kind words; but I think it wisest to end what, at best, was but a lame attempt. "The stretched metre of an antique song" is beyond my compass; and my puppets were neither distinctly _in_ my life (like those I now address), nor yet (like Alice and the Mock Turtle) distinctly _out_ of it. Yet let me at least fancy, as I lay down the pen, that I carry with me into my silent life, dear reader, a farewell smile from your unseen face, and a kindly farewell pressure from your unfelt hand! And so, good night! Parting is such sweet sorrow, that I shall say "good night!" till it be morrow. THE END LONDON: RICHARD CLAY AND SONS, PRINTERS. [TURN OVER. WORKS BY LEWIS CARROLL. ALICE'S ADVENTURES IN WONDERLAND. With Forty-two Illustrations by TENNIEL. Crown 8vo, cloth, gilt edges, price 6_s._ Seventy-fifth Thousand. TRANSLATIONS OF THE SAME--into French, by HENRI BU�--into German, by ANTONIE ZIMMERMANN--and into Italian, by T. PIETROC�LA ROSSETTI--with TENNIEL'S Illustrations. Crown 8vo, cloth, gilt edges, price 6_s._ each. THROUGH THE LOOKING-GLASS, AND WHAT ALICE FOUND THERE. With Fifty Illustrations by TENNIEL. Crown 8vo, cloth, gilt edges, price 6_s._ Fifty-sixth Thousand. RHYME? AND REASON? With Sixty-five Illustrations by ARTHUR B. FROST, and Nine by HENRY HOLIDAY. (This book is a reprint, with a few additions, of the comic portion of "Phantasmagoria and other Poems," and of "The Hunting of the Snark." Mr. Frost's pictures are new.) Crown 8vo, cloth, coloured edges, price 7_s._ Fifty Thousand. A TANGLED TALE. Reprinted from _The Monthly Packet_. With Six Illustrations by Arthur B. Frost. Crown 8vo, 4_s._ 6_d._ * * * * * N.B. In selling the above-mentioned books to the Trade, Messrs. Macmillan and Co. will abate 2_d._ in the shilling (no odd copies), and allow 5 per cent. discount for payment within six months, and 10 per cent. for cash. In selling them to the Public (for cash only) they will allow 10 per cent. discount. * * * * * MR. LEWIS CARROLL, having been requested to allow "AN EASTER GREETING" (a leaflet, addressed to children, and frequently given with his books) to be sold separately, has arranged with Messrs. Harrison, of 59, Pall Mall, who will supply a single copy for 1_d._, or 12 for 9_d._, or 100 for 5_s._ MACMILLAN AND CO., LONDON. LONDON: RICHARD CLAY AND SONS, PRINTERS. * * * * * Transcriber's note The following changes have been made to the text: Page 88: "he corners of the" changed to "the corners of the". Page 95: "Aix-le-Bains" changed to "Aix-les-Bains". Page 108: "3/5, 2, 1/3" changed to "3/5, 2/3, 1/3". Page 114: "10 of the 12 cases" changed to "10 of the 13 cases". Page 121: "four-and fourpence" changed to "four-and-fourpence". Last page: "Fifth Thousand" changed to "Fifty Thousand". 
302	----	Michael Husted H�jagerparken 95, 2. 2750 Ballerup Denmark husted@login.dknet.dk Phone: +45 44 68 30 08 The following "etext" has been created with a "homemade" program. ------------------------------------------------------------------------ Fibonacci's Numbers - the first 1000. 1 (1 digit) : 1 2 (1 digit) : 1 3 (1 digit) : 2 4 (1 digit) : 3 5 (1 digit) : 5 6 (1 digit) : 8 7 (2 digits) : 13 8 (2 digits) : 21 9 (2 digits) : 34 10 (2 digits) : 55 11 (2 digits) : 89 12 (3 digits) : 144 13 (3 digits) : 233 14 (3 digits) : 377 15 (3 digits) : 610 16 (3 digits) : 987 17 (4 digits) : 1597 18 (4 digits) : 2584 19 (4 digits) : 4181 20 (4 digits) : 6765 21 (5 digits) : 10946 22 (5 digits) : 17711 23 (5 digits) : 28657 24 (5 digits) : 46368 25 (5 digits) : 75025 26 (6 digits) : 12139 3 27 (6 digits) : 19641 8 28 (6 digits) : 31781 1 29 (6 digits) : 51422 9 30 (6 digits) : 83204 0 31 (7 digits) : 13462 69 32 (7 digits) : 21783 09 33 (7 digits) : 35245 78 34 (7 digits) : 57028 87 35 (7 digits) : 92274 65 36 (8 digits) : 14930 352 37 (8 digits) : 24157 817 38 (8 digits) : 39088 169 39 (8 digits) : 63245 986 40 (9 digits) : 10233 4155 41 (9 digits) : 16558 0141 42 (9 digits) : 26791 4296 43 (9 digits) : 43349 4437 44 (9 digits) : 70140 8733 45 (10 digits) : 11349 03170 46 (10 digits) : 18363 11903 47 (10 digits) : 29712 15073 48 (10 digits) : 48075 26976 49 (10 digits) : 77787 42049 50 (11 digits) : 12586 26902 5 51 (11 digits) : 20365 01107 4 52 (11 digits) : 32951 28009 9 53 (11 digits) : 53316 29117 3 54 (11 digits) : 86267 57127 2 55 (12 digits) : 13958 38624 45 56 (12 digits) : 22585 14337 17 57 (12 digits) : 36543 52961 62 58 (12 digits) : 59128 67298 79 59 (12 digits) : 95672 20260 41 60 (13 digits) : 15480 08755 920 61 (13 digits) : 25047 30781 961 62 (13 digits) : 40527 39537 881 63 (13 digits) : 65574 70319 842 64 (14 digits) : 10610 20985 7723 65 (14 digits) : 17167 68017 7565 66 (14 digits) : 27777 89003 5288 67 (14 digits) : 44945 57021 2853 68 (14 digits) : 72723 46024 8141 69 (15 digits) : 11766 90304 60994 70 (15 digits) : 19039 24907 09135 71 (15 digits) : 30806 15211 70129 72 (15 digits) : 49845 40118 79264 73 (15 digits) : 80651 55330 49393 74 (16 digits) : 13049 69544 92865 7 75 (16 digits) : 21114 85077 97805 0 76 (16 digits) : 34164 54622 90670 7 77 (16 digits) : 55279 39700 88475 7 78 (16 digits) : 89443 94323 79146 4 79 (17 digits) : 14472 33402 46762 21 80 (17 digits) : 23416 72834 84676 85 81 (17 digits) : 37889 06237 31439 06 82 (17 digits) : 61305 79072 16115 91 83 (17 digits) : 99194 85309 47554 97 84 (18 digits) : 16050 06438 16367 088 85 (18 digits) : 25969 54969 11122 585 86 (18 digits) : 42019 61407 27489 673 87 (18 digits) : 67989 16376 38612 258 88 (19 digits) : 11000 87778 36610 1931 89 (19 digits) : 17799 79416 00471 4189 90 (19 digits) : 28800 67194 37081 6120 91 (19 digits) : 46600 46610 37553 0309 92 (19 digits) : 75401 13804 74634 6429 93 (20 digits) : 12200 16041 51218 76738 94 (20 digits) : 19740 27421 98682 23167 95 (20 digits) : 31940 43463 49900 99905 96 (20 digits) : 51680 70885 48583 23072 97 (20 digits) : 83621 14348 98484 22977 98 (21 digits) : 13530 18523 44706 74604 9 99 (21 digits) : 21892 29958 34555 16902 6 100 (21 digits) : 35422 48481 79261 91507 5 101 (21 digits) : 57314 78440 13817 08410 1 102 (21 digits) : 92737 26921 93078 99917 6 103 (22 digits) : 15005 20536 20689 60832 77 104 (22 digits) : 24278 93228 39997 50824 53 105 (22 digits) : 39284 13764 60687 11657 30 106 (22 digits) : 63563 06993 00684 62481 83 107 (23 digits) : 10284 72075 76137 17413 913 108 (23 digits) : 16641 02775 06205 63662 096 109 (23 digits) : 26925 74850 82342 81076 009 110 (23 digits) : 43566 77625 88548 44738 105 111 (23 digits) : 70492 52476 70891 25814 114 112 (24 digits) : 11405 93010 25943 97055 2219 113 (24 digits) : 18455 18257 93033 09636 6333 114 (24 digits) : 29861 11268 18977 06691 8552 115 (24 digits) : 48316 29526 12010 16328 4885 116 (24 digits) : 78177 40794 30987 23020 3437 117 (25 digits) : 12649 37032 04299 73934 88322 118 (25 digits) : 20467 11111 47398 46236 91759 119 (25 digits) : 33116 48143 51698 20171 80081 120 (25 digits) : 53583 59254 99096 66408 71840 121 (25 digits) : 86700 07398 50794 86580 51921 122 (26 digits) : 14028 36665 34989 15298 92376 1 123 (26 digits) : 22698 37405 20068 63956 97568 2 124 (26 digits) : 36726 74070 55057 79255 89944 3 125 (26 digits) : 59425 11475 75126 43212 87512 5 126 (26 digits) : 96151 85546 30184 22468 77456 8 127 (27 digits) : 15557 69702 20531 06568 16496 93 128 (27 digits) : 25172 88256 83549 48815 04242 61 129 (27 digits) : 40730 57959 04080 55383 20739 54 130 (27 digits) : 65903 46215 87630 04198 24982 15 131 (28 digits) : 10663 40417 49171 05958 14572 169 132 (28 digits) : 17253 75039 07934 06377 97070 384 133 (28 digits) : 27917 15456 57105 12336 11642 553 134 (28 digits) : 45170 90495 65039 18714 08712 937 135 (28 digits) : 73088 05952 22144 31050 20355 490 136 (29 digits) : 11825 89644 78718 34976 42906 8427 137 (29 digits) : 19134 70240 00932 78081 44942 3917 138 (29 digits) : 30960 59884 79651 13057 87849 2344 139 (29 digits) : 50095 30124 80583 91139 32791 6261 140 (29 digits) : 81055 90009 60235 04197 20640 8605 141 (30 digits) : 13115 12013 44081 89533 65343 24866 142 (30 digits) : 21220 71014 40105 39953 37407 33471 143 (30 digits) : 34335 83027 84187 29487 02750 58337 144 (30 digits) : 55556 54042 24292 69440 40157 91808 145 (30 digits) : 89892 37070 08479 98927 42908 50145 146 (31 digits) : 14544 89111 23277 26836 78306 64195 3 147 (31 digits) : 23534 12818 24125 26729 52597 49209 8 148 (31 digits) : 38079 01929 47402 53566 30904 13405 1 149 (31 digits) : 61613 14747 71527 80295 83501 62614 9 150 (31 digits) : 99692 16677 18930 33862 14405 76020 0 151 (32 digits) : 16130 53142 49045 81415 79790 73863 49 152 (32 digits) : 26099 74810 20938 84802 01231 31465 49 153 (32 digits) : 42230 27952 69984 66217 81022 05328 98 154 (32 digits) : 68330 02762 90923 51019 82253 36794 47 155 (33 digits) : 11056 03071 56090 81723 76327 54212 345 156 (33 digits) : 17889 03347 85183 16825 74552 87891 792 157 (33 digits) : 28945 06419 41273 98549 50880 42104 137 158 (33 digits) : 46834 09767 26457 15375 25433 29995 929 159 (33 digits) : 75779 16186 67731 13924 76313 72100 066 160 (34 digits) : 12261 32595 39418 82930 00174 70209 5995 161 (34 digits) : 19839 24214 06191 94322 47806 07419 6061 162 (34 digits) : 32100 56809 45610 77252 47980 77629 2056 163 (34 digits) : 51939 81023 51802 71574 95786 85048 8117 164 (34 digits) : 84040 37832 97413 48827 43767 62678 0173 165 (35 digits) : 13598 01885 64921 62040 23955 44772 68290 166 (35 digits) : 22002 05668 94662 96922 98332 21040 48463 167 (35 digits) : 35600 07554 59584 58963 22287 65813 16753 168 (35 digits) : 57602 13223 54247 55886 20619 86853 65216 169 (35 digits) : 93202 20778 13832 14849 42907 52666 81969 170 (36 digits) : 15080 43400 16807 97073 56352 73952 04718 5 171 (36 digits) : 24400 65477 98191 18558 50643 49218 72915 4 172 (36 digits) : 39481 08878 14999 15632 06996 23170 77633 9 173 (36 digits) : 63881 74356 13190 34190 57639 72389 50549 3 174 (37 digits) : 10336 28323 42818 94982 26463 59556 02818 32 175 (37 digits) : 16724 45759 04137 98401 32227 56794 97873 25 176 (37 digits) : 27060 74082 46956 93383 58691 16351 00691 57 177 (37 digits) : 43785 19841 51094 91784 90918 73145 98564 82 178 (37 digits) : 70845 93923 98051 85168 49609 89496 99256 39 179 (38 digits) : 11463 11376 54914 67695 34052 86264 29782 121 180 (38 digits) : 18547 70768 94719 86212 19013 85213 99707 760 181 (38 digits) : 30010 82145 49634 53907 53066 71478 29489 881 182 (38 digits) : 48558 52914 44354 40119 72080 56692 29197 641 183 (38 digits) : 78569 35059 93988 94027 25147 28170 58687 522 184 (39 digits) : 12712 78797 43834 33414 69722 78486 28788 5163 185 (39 digits) : 20569 72303 43233 22817 42237 51303 34657 2685 186 (39 digits) : 33282 51100 87067 56232 11960 29789 63445 7848 187 (39 digits) : 53852 23404 30300 79049 54197 81092 98103 0533 188 (39 digits) : 87134 74505 17368 35281 66158 10882 61548 8381 189 (40 digits) : 14098 69790 94766 91433 12035 59197 55965 18914 190 (40 digits) : 22812 17241 46503 74961 28651 40285 82120 07295 191 (40 digits) : 36910 87032 41270 66394 40686 99483 38085 26209 192 (40 digits) : 59723 04273 87774 41355 69338 39769 20205 33504 193 (40 digits) : 96633 91306 29045 07750 10025 39252 58290 59713 194 (41 digits) : 15635 69558 01681 94910 57936 37902 17849 59321 7 195 (41 digits) : 25299 08688 64586 45685 58938 91827 43678 65293 0 196 (41 digits) : 40934 78246 66268 40596 16875 29729 61528 24614 7 197 (41 digits) : 66233 86935 30854 86281 75814 21557 05206 89907 7 198 (42 digits) : 10716 86518 19712 32687 79268 95128 66673 51452 24 199 (42 digits) : 17340 25211 72797 81315 96850 37284 37194 20443 01 200 (42 digits) : 28057 11729 92510 14003 76119 32413 03867 71895 25 201 (42 digits) : 45397 36941 65307 95319 72969 69697 41061 92338 26 202 (42 digits) : 73454 48671 57818 09323 49089 02110 44929 64233 51 203 (43 digits) : 11885 18561 32312 60464 32205 87180 78599 15657 177 204 (43 digits) : 19230 63428 48094 41396 67114 77391 83092 12080 528 205 (43 digits) : 31115 81989 80407 01860 99320 64572 61691 27737 705 206 (43 digits) : 50346 45418 28501 43257 66435 41964 44783 39818 233 207 (43 digits) : 81462 27408 08908 45118 65756 06537 06474 67555 938 208 (44 digits) : 13180 87282 63740 98837 63219 14850 15125 80737 4171 209 (44 digits) : 21327 10023 44631 83349 49794 75503 85773 27493 0109 210 (44 digits) : 34507 97306 08372 82187 13013 90354 00899 08230 4280 211 (44 digits) : 55835 07329 53004 65536 62808 65857 86672 35723 4389 212 (44 digits) : 90343 04635 61377 47723 75822 56211 87571 43953 8669 213 (45 digits) : 14617 81196 51438 21326 03863 12206 97424 37967 73058 214 (45 digits) : 23652 11660 07575 96098 41445 37828 16181 52363 11727 215 (45 digits) : 38269 92856 59014 17424 45308 50035 13605 90330 84785 216 (45 digits) : 61922 04516 66590 13522 86753 87863 29787 42693 96512 217 (46 digits) : 10019 19737 32560 43094 73206 23789 84339 33302 48129 7 218 (46 digits) : 16211 40188 99219 44447 01881 62576 17318 07571 87780 9 219 (46 digits) : 26230 59926 31779 87541 75087 86366 01657 40874 35910 6 220 (46 digits) : 42442 00115 30999 31988 76969 48942 18975 48446 23691 5 221 (46 digits) : 68672 60041 62779 19530 52057 35308 20632 89320 59602 1 222 (47 digits) : 11111 46015 69377 85151 92902 68425 03960 83776 68329 36 223 (47 digits) : 17978 72019 85655 77104 98108 41955 86024 12708 74289 57 224 (47 digits) : 29090 18035 55033 62256 91011 10380 89984 96485 42618 93 225 (47 digits) : 47068 90055 40689 39361 89119 52336 76009 09194 16908 50 226 (47 digits) : 76159 08090 95723 01618 80130 62717 65994 05679 59527 43 227 (48 digits) : 12322 79814 63641 24098 06925 01505 44200 31487 37643 593 228 (48 digits) : 19938 70623 73213 54259 94938 07777 20799 72055 33596 336 229 (48 digits) : 32261 50438 36854 78358 01863 09282 65000 03542 71239 929 230 (48 digits) : 52200 21062 10068 32617 96801 17059 85799 75598 04836 265 231 (48 digits) : 84461 71500 46923 10975 98664 26342 50799 79140 76076 194 232 (49 digits) : 13666 19256 25699 14359 39546 54340 23659 95473 88091 2459 233 (49 digits) : 22112 36406 30391 45456 99412 96974 48739 93387 95698 8653 234 (49 digits) : 35778 55662 56090 59816 38959 51314 72399 88861 83790 1112 235 (49 digits) : 57890 92068 86482 05273 38372 48289 21139 82249 79488 9765 236 (49 digits) : 93669 47731 42572 65089 77331 99603 93539 71111 63279 0877 237 (50 digits) : 15156 03980 02905 47036 31570 44789 31467 95336 14276 80642 238 (50 digits) : 24522 98753 17162 73545 29303 64749 70821 92447 30604 71519 239 (50 digits) : 39679 02733 20068 20581 60874 09539 02289 87783 44881 52161 240 (50 digits) : 64202 01486 37230 94126 90177 74288 73111 80230 75486 23680 241 (51 digits) : 10388 10421 95729 91470 85105 18382 77540 16801 42036 77584 1 242 (51 digits) : 16808 30570 59453 00883 54122 95811 64851 34824 49585 39952 1 243 (51 digits) : 27196 40992 55182 92354 39228 14194 42391 51625 91622 17536 2 244 (51 digits) : 44004 71563 14635 93237 93351 10006 07242 86450 41207 57488 3 245 (51 digits) : 71201 12555 69818 85592 32579 24200 49634 38076 32829 75024 5 246 (52 digits) : 11520 58411 88445 47883 02593 03420 65687 72452 67403 73251 28 247 (52 digits) : 18640 69667 45427 36442 25850 95840 70651 16260 30686 70753 73 248 (52 digits) : 30161 28079 33872 84325 28443 99261 36338 88712 98090 44005 01 249 (52 digits) : 48801 97746 79300 20767 54294 95102 06990 04973 28777 14758 74 250 (52 digits) : 78963 25826 13173 05092 82738 94363 43328 93686 26867 58763 75 251 (53 digits) : 12776 52357 29247 32586 03703 38946 55031 89865 95564 47352 249 252 (53 digits) : 20672 84939 90564 63095 31977 28382 89364 79234 58251 23228 624 253 (53 digits) : 33449 37297 19811 95681 35680 67329 44396 69100 53815 70580 873 254 (53 digits) : 54122 22237 10376 58776 67657 95712 33761 48335 12066 93809 497 255 (53 digits) : 87571 59534 30188 54458 03338 63041 78158 17435 65882 64390 370 256 (54 digits) : 14169 38177 14056 51323 47099 65875 41191 96577 07794 95819 9867 257 (54 digits) : 22926 54130 57075 36769 27433 52179 59007 78320 64383 22259 0237 258 (54 digits) : 37095 92307 71131 88092 74533 18055 00199 74897 72178 18079 0104 259 (54 digits) : 60022 46438 28207 24862 01966 70234 59207 53218 36561 40338 0341 260 (54 digits) : 97118 38745 99339 12954 76499 88289 59407 28116 08739 58417 0445 261 (55 digits) : 15714 08518 42754 63781 67846 65852 41861 48133 44530 09875 50786 262 (55 digits) : 25425 92393 02688 55077 15496 64681 37802 20945 05404 05717 21231 263 (55 digits) : 41140 00911 45443 18858 83343 30533 79663 69078 49934 15592 72017 264 (55 digits) : 66565 93304 48131 73935 98839 95215 17465 90023 55338 21309 93248 265 (56 digits) : 10770 59421 59357 49279 48218 32574 89712 95910 20527 23690 26526 5 266 (56 digits) : 17427 18752 04170 66673 08102 32096 41459 54912 56061 05821 25851 3 267 (56 digits) : 28197 78173 63528 15952 56320 64671 31172 50822 76588 29511 52377 8 268 (56 digits) : 45624 96925 67698 82625 64422 96767 72632 05735 32649 35332 78229 1 269 (56 digits) : 73822 75099 31226 98578 20743 61439 03804 56558 09237 64844 30606 9 270 (57 digits) : 11944 77202 49892 58120 38516 65820 67643 66229 34188 70017 70883 60 271 (57 digits) : 19327 04712 43015 27978 20591 01964 58024 11885 15112 46502 13944 29 272 (57 digits) : 31271 81914 92907 86098 59107 67785 25667 78114 49301 16519 84827 89 273 (57 digits) : 50598 86627 35923 14076 79698 69749 83691 89999 64413 63021 98772 18 274 (57 digits) : 81870 68542 28831 00175 38806 37535 09359 68114 13714 79541 83600 07 275 (58 digits) : 13246 95516 96475 41425 21850 50728 49305 15811 37812 84256 38237 225 276 (58 digits) : 21434 02371 19358 51442 75731 14482 00241 12622 79184 32210 56597 232 277 (58 digits) : 34680 97888 15833 92867 97581 65210 49546 28434 16997 16466 94834 457 278 (58 digits) : 56115 00259 35192 44310 73312 79692 49787 41056 96181 48677 51431 689 279 (58 digits) : 90795 98147 51026 37178 70894 44902 99333 69491 13178 65144 46266 146 280 (59 digits) : 14691 09840 68621 88148 94420 72459 54912 11054 80936 01382 19769 7835 281 (59 digits) : 23770 69655 43724 51866 81510 16949 84845 48003 92253 87896 64396 3981 282 (59 digits) : 38461 79496 12346 40015 75930 89409 39757 59058 73189 89278 84166 1816 283 (59 digits) : 62232 49151 56070 91882 57441 06359 24603 07062 65443 77175 48562 5797 284 (60 digits) : 10069 42864 76841 73189 83337 19576 86436 06612 13863 36645 43272 87613 285 (60 digits) : 16292 67779 92448 82378 09081 30212 78896 37318 40407 74362 98129 13410 286 (60 digits) : 26362 10644 69290 55567 92418 49789 65332 43930 54271 11008 41402 01023 287 (60 digits) : 42654 78424 61739 37946 01499 80002 44228 81248 94678 85371 39531 14433 288 (60 digits) : 69016 89069 31029 93513 93918 29792 09561 25179 48949 96379 80933 15456 289 (61 digits) : 11167 16749 39276 93145 99541 80979 45379 00642 84362 88175 12046 42988 9 290 (61 digits) : 18068 85656 32379 92497 38933 63958 66335 13160 79257 87813 10139 74534 5 291 (61 digits) : 29236 02405 71656 85643 38475 44938 11714 13803 63620 75988 22186 17523 4 292 (61 digits) : 47304 88062 04036 78140 77409 08896 78049 26964 42878 63801 32325 92057 9 293 (61 digits) : 76540 90467 75693 63784 15884 53834 89763 40768 06499 39789 54512 09581 3 294 (62 digits) : 12384 57852 97973 04192 49329 36273 16781 26773 24937 80359 08683 80163 92 295 (62 digits) : 20038 66899 75542 40570 90917 81656 65757 60850 05587 74338 04135 01122 05 296 (62 digits) : 32423 24752 73515 44763 40247 17929 82538 87623 30525 54697 12818 81285 97 297 (62 digits) : 52461 91652 49057 85334 31164 99586 48296 48473 36113 29035 16953 82408 02 298 (62 digits) : 84885 16405 22573 30097 71412 17516 30835 36096 66638 83732 29772 63693 99 299 (63 digits) : 13734 70805 77163 11543 20257 71710 27913 18457 00275 21276 74672 64610 201 300 (63 digits) : 22223 22446 29420 44552 97398 93461 90996 72066 66939 09649 97649 90979 600 301 (63 digits) : 35957 93252 06583 56096 17656 65172 18909 90523 67214 30926 72322 55589 801 302 (63 digits) : 58181 15698 36004 00649 15055 58634 09906 62590 34153 40576 69972 46569 401 303 (63 digits) : 94139 08950 42587 56745 32712 23806 28816 53114 01367 71503 42295 02159 202 304 (64 digits) : 15232 02464 87859 15739 44776 78244 03872 31570 43552 11208 01226 74872 8603 305 (64 digits) : 24645 93359 92117 91413 98048 00624 66753 96881 83688 88358 35456 25088 7805 306 (64 digits) : 39877 95824 79977 07153 42824 78868 70626 28452 27240 99566 36682 99961 6408 307 (64 digits) : 64523 89184 72094 98567 40872 79493 37380 25334 10929 87924 72139 25050 4213 308 (65 digits) : 10440 18500 95207 20572 08369 75836 20800 65378 63817 08749 10882 22501 20621 309 (65 digits) : 16892 57419 42416 70428 82457 03785 54538 67912 04910 07541 58096 15006 24834 310 (65 digits) : 27332 75920 37623 91000 90826 79621 75339 33290 68727 16290 68978 37507 45455 311 (65 digits) : 44225 33339 80040 61429 73283 83407 29878 01202 73637 23832 27074 52513 70289 312 (65 digits) : 71558 09260 17664 52430 64110 63029 05217 34493 42364 40122 96052 90021 15744 313 (66 digits) : 11578 34259 99770 51386 03739 44643 63509 53569 61600 16395 52312 74253 48603 3 314 (66 digits) : 18734 15186 01536 96629 10150 50946 54031 27018 95836 60407 81918 03255 60177 7 315 (66 digits) : 30312 49446 01307 48015 13889 95590 17540 80588 57436 76803 34230 77509 08781 0 316 (66 digits) : 49046 64632 02844 44644 24040 46536 71572 07607 53273 37211 16148 80764 68958 7 317 (66 digits) : 79359 14078 04151 92659 37930 42126 89112 88196 10710 14014 50379 58273 77739 7 318 (67 digits) : 12840 57871 00699 63730 36197 08866 36068 49580 36398 35122 56652 83903 84669 84 319 (67 digits) : 20776 49278 81114 82996 29990 13079 04979 78399 97469 36524 01690 79731 22443 81 320 (67 digits) : 33617 07149 81814 46726 66187 21945 41048 27980 33867 71646 58343 63635 07113 65 321 (67 digits) : 54393 56428 62929 29722 96177 35024 46028 06380 31337 08170 60034 43366 29557 46 322 (67 digits) : 88010 63578 44743 76449 62364 56969 87076 34360 65204 79817 18378 07001 36671 11 323 (68 digits) : 14240 42000 70767 30617 25854 19199 43310 44074 09654 18798 77841 25036 76622 857 324 (68 digits) : 23041 48358 55241 68262 22090 64896 42018 07510 16174 66780 49679 05736 90289 968 325 (68 digits) : 37281 90359 26008 98879 47944 84095 85328 51584 25828 85579 27520 30773 66912 825 326 (68 digits) : 60323 38717 81250 67141 70035 48992 27346 59094 42003 52359 77199 36510 57202 793 327 (68 digits) : 97605 29077 07259 66021 17980 33088 12675 10678 67832 37939 04719 67284 24115 618 328 (69 digits) : 15792 86779 48851 03316 28801 58208 04002 16977 30983 59029 88191 90379 48131 8411 329 (69 digits) : 25553 39687 19576 99918 40599 61516 85269 68045 17766 82823 78663 87107 90543 4029 330 (69 digits) : 41346 26466 68428 03234 69401 19724 89271 85022 48750 41853 66855 77487 38675 2440 331 (69 digits) : 66899 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19600	----	JEROME CARDAN [Illustration] JEROME CARDAN _A BIOGRAPHICAL STUDY_ BY W.G. WATERS "To be content that times to come should only know there was such a man, not caring whether they knew more of him, was a frigid ambition in Cardan."--SIR THOMAS BROWNE. [Illustration] LAWRENCE & BULLEN, Limited, 16 Henrietta Street, Covent Garden, London, MDCCCXCVIII. RICHARD CLAY & SONS, LIMITED, LONDON & BUNGAY. PREFACE No attempt is made in the following pages to submit to historical treatment the vast and varied mass of printed matter which Cardan left as his contribution to letters and science, except in the case of those works which are, in purpose or incidentally, autobiographical, or of those which furnish in themselves effective contributions towards the framing of an estimate of the genius and character of the writer. Neither has it seemed worth while to offer to the public another biography constructed on the lines of the one brought out by Professor Henry Morley in 1854, for the reason that the circumstances of Cardan's life, the character of his work, and of the times in which he lived, all appeared to be susceptible of more succinct and homogeneous treatment than is possible in a chronicle of the passing years, and of the work that each one saw accomplished. At certain junctures the narrative form is inevitable, but an attempt has been made to treat the more noteworthy episodes of Cardan's life and work, and the contemporary aspect of the republic of letters, in relation to existing tendencies and conditions, whenever such a course has seemed possible. Professor Morley's book, _The Life of Girolamo Cardano, of Milan, physician_, has been for some time out of print. This industrious writer gathered together a large quantity of material, dealing almost as fully with the more famous of the contemporary men of mark, with whom Cardan was brought into contact, as with Cardan himself. The translations and analyses of some of Cardan's more popular works which Professor Morley gives are admirable in their way, but the space they occupy in the biography is somewhat excessive. Had sufficient leisure for revision and condensation been allowed, Professor Morley's book would have taken a high place in biographical literature. As it stands it is a noteworthy performance; and, by reason of its wide and varied stores of information and its excellent index, it must always prove a valuable magazine of _mémoires pour servir_ for any future students who may be moved to write afresh, concerning the life and work of the great Milanese physician. An apology may be needed for the occurrence here and there of passages translated from the _De Vita Propria_ and the _De Utilitate ex Adversis capienda_, passages which some readers may find too frequent and too lengthy, but contemporary opinion is strongly in favour of letting the subject speak for himself as far as may be possible. The date and place of Cardan's quoted works are given in the first citation therefrom; those of his writings which have not been available in separate form have been consulted in the collected edition of his works in ten volumes, edited by Spon, and published at Lyons in 1663. The author desires to acknowledge with gratitude the valuable assistance in the way of suggestion and emendation which he received from Mr. R.C. Christie during the final revision of the proofs. _London, October 1898._ JEROME CARDAN CHAPTER I LIKE certain others of the illustrious personages who flourished in his time, Girolamo Cardano, or, as he has become to us by the unwritten law of nomenclature, Jerome Cardan, was fated to suffer the burden and obloquy of bastardy.[1] He was born at Pavia from the illicit union of Fazio Cardano, a Milanese jurisconsult and mathematician of considerable repute, and a young widow, whose maiden name had been Chiara Micheria, his father being fifty-six, and his mother thirty-seven years of age at his birth. The family of Fazio was settled at Gallarate, a town in Milanese territory, and was one which, according to Jerome's contention, could lay claim to considerable antiquity and distinction. He prefers a claim of descent from the house of Castillione, founding the same upon an inscription on the apse of the principal church at Gallarate.[2] He asserts that as far back as 1189 Milo Cardano was Governor of Milan for more than seven years, and according to tradition Franco Cardano, the commander of the forces of Matteo Visconti,[3] was a member of the family. If the claim of the Castillione ancestry be allowed the archives of the race would be still farther enriched by the name of Pope Celestine IV., Godfrey of Milan, who was elected Pope in 1241, and died the same year. Cardan's immediate ancestors were long-lived. The sons of Fazio Cardano, his great-grandfather, Joanni, Aldo, and Antonio, lived to be severally ninety-four, eighty-eight, and eighty-six years of age. Of these Joanni begat two sons: Antonio, who lived eighty-eight years, and Angelo, who reached the age of eighty-six. To Aldo were born Jacopo, who died at seventy-two; Gottardo, who died at eighty-four; and Fazio, the father of Jerome, who died at eighty.[4] Fazio, albeit he came of such a long-lived stock, and lived himself to be fourscore, suffered much physical trouble during his life. On account of a wound which he had received when he was a youth, some of the bones of his skull had to be removed, and from this time forth he never dared to remain long with his head uncovered. When he was fifty-nine he swallowed a certain corrosive poison, which did not kill him, but left him toothless. He was likewise round-shouldered, a stammerer, and subject to constant palpitation of the heart; but in compensation for these defects he had eyes which could see in the dark and which needed not spectacles even in advanced age. Of Jerome's mother little is known. Her family seems to have been as tenacious of life as that of Fazio, for her father Jacopo lived to be seventy-five years of age. Of his maternal grandfather Jerome remarks that he was a highly skilled mathematician, and that when he was about seventy years of age, he was cast into prison for some offence against the law. He speaks of his mother as choleric in temper, well dowered with memory and mental parts, small in stature and fat, and of a pious disposition,[5] and declares that she and his father were alike in one respect, to wit that they were easily moved to anger and were wont to manifest but lukewarm and intermittent affection for their child. Nevertheless they were in a way indulgent to him. His father permitted him to remain in bed till the second hour of the day had struck, or rather forbade him to rise before this time--an indulgence which worked well for the preservation of his health. He adds that in after times he always thought of his father as possessing the kindlier nature of the two.[6] It would seem from the passage above written, as well as from certain others subsequent, that Jerome had little affection for his mother; and albeit he neither chides nor reproaches her, he never refers to her in terms so appreciative and loving as those which he uses in lamenting the death of his harsh and tyrannical father. In the _Geniturarum Exempla_[7] he says that, seeing he is writing of a woman, he will confine his remarks to saying that she was ingenious, of good parts, generous, upright, and loving towards her children. Perhaps the fact that his father died early, while his mother lived on for many years, and was afterwards a member of his household--together with his wife--may account for the colder tone of his remarks while writing about her. She was the widow of a certain Antonio Alberio,[8] and during her marriage had borne him three children, Tommaso, Catilina, and Joanni Ambrogio; but when Jerome was a year old all three of these died of the plague within the space of a few weeks.[9] He himself narrowly escaped death from the same cause, and this attack he attributes to an inherited tendency from his mother, she having suffered from the same disease during her girlhood. There seems to have been born to Fazio and Chiara another son, who died at birth.[10] Jerome Cardan was born on September 24, 1501, between half-past six o'clock and a quarter to seven in the evening. In the second chapter of his autobiography he gives the year as 1500, and in _De Utilitate_, p. 347, he writes the date as September 23, but on all other occasions the date first written is used. Before he saw the light malefic influences were at work against him. His mother, urged on no doubt by the desire to conceal her shame, and persuaded by evil counsellors, drank a potion of abortive drugs in order to produce miscarriage,[11] but Nature on this occasion was not to be baulked. In recording the circumstances of his birth he writes at some length in the jargon of astrology to show how the celestial bodies were leagued together so as to mar him both in body and mind. "Wherefore I ought, according to every rule, to have been born a monster, and, under the circumstances, it was no marvel that it was found necessary to tear me from the womb in order to bring me into the world. Thus was I born, or rather dragged from my mother's body. I was to all outward seeming dead, with my head covered with black curly hair. I was brought round by being plunged in a bath of heated wine, a remedy which might well have proved hurtful to any other infant. My mother lay three whole days in labour, but at last gave birth to me, a living child."[12] The sinister influences of the stars soon began to manifest their power. Before Jerome had been many days in the world the woman into whose charge he had been given was seized with the plague and died the same day, whereupon his mother took him home with her. The first of his bodily ailments,--the catalogue of the same which he subsequently gives is indeed a portentous one,[13]--was an eruption of carbuncles on the face in the form of a cross, one of the sores being set on the tip of the nose; and when these disappeared, swellings came. Before the boy was two months old his godfather, Isidore di Resta of Ticino, gave him into the care of another nurse who lived at Moirago, a town about seven miles from Milan, but here again ill fortune attended him. His body began to waste and his stomach to swell because the nurse who gave him suck was herself pregnant.[14] A third foster-mother was found for him, and he remained with her till he was weaned in his third year. When he was four years of age he was taken to Milan to be under the care of his mother, who, with her sister, Margarita, was living in Fazio's house; but whether she was at this time legally married to him or not there is no evidence to show. In recording this change he remarks that he now came under a gentler discipline from the hands of his mother and his aunt, but immediately afterwards proclaims his belief that the last-named must have been born without a gall bladder, a remark somewhat difficult to apply, seeing he frequently complains afterwards of her harshness. It must be remembered, however, that these details are taken from a record of the writer's fifth year set down when he was past seventy.[15] He quotes certain lapses from kindly usage, as for instance when it happened that he was beaten by his father or his mother without a cause. After much chastisement he always fell sick, and lay some time in mortal danger. "When I was seven years old my father and my mother were then living apart--my kinsfolk determined, for some reason or other, to give over beating me, though perchance a touch of the whip might then have done me no harm. But ill-fortune was ever hovering around me; she let my tribulation take a different shape, but she did not remove it. My father, having hired a house, took me and my mother and my aunt to live with him, and made me always accompany him in his rounds about the city. On this account I, being taken at this tender age with my weak body from a life of absolute rest and put to hard and constant work, was seized at the beginning of my eighth year with dysentery and fever, an ailment which was at that time epidemic in our city. Moreover I had eaten by stealth a vast quantity of sour grapes. But after I had been visited by the physicians, Bernabo della Croce and Angelo Gyra, there seemed to be some hope of my recovery, albeit both my parents, and my aunt as well, had already bewept me as one dead. "At this season my father, who was at heart a man of piety, was minded to invoke the divine assistance of San Girolamo (commending me to the care of the Saint in his prayers) rather than trust to the working of that familiar spirit which, as he was wont to declare openly, was constantly in attendance upon him. The reason of this change in his treatment of me I never cared to inquire. It was during the time of my recovery from this sickness, that the French celebrated their triumph after defeating the Venetians on the banks of the Adda, which spectacle I was allowed to witness from my window.[16] After this my father freed me of the task of going with him on his rounds. But the anger of Juno was not yet exhausted; for, before I had fully recovered my health, I fell down-stairs (we were then living in the Via dei Maini), with a hammer in my hand, and by this accident I hurt the left side of my forehead, injuring the bone and causing a scar which remains to this day. Before I had recovered from this mishap I was sitting on the threshold of the house when a stone, about as long and as broad as a nut, fell down from the top of a high house next door and wounded my head just where my hair grew very thickly on the left side. "At the beginning of my tenth year my father changed this house, which had proved a very unlucky one for me, for another in the same street, and there I abode for three whole years. But my ill luck still followed me, for my father once more caused me to go about with him as his _famulus_, and would never allow me on any pretext to escape this task. I should hesitate to say that he did this through cruelty; for, taking into consideration what ensued, you may perchance be brought to see that this action of his came to pass rather through the will of Heaven than through any failing of his own. I must add too that my mother and my aunt were fully in agreement with him in his treatment of me. In after times, however, he dealt with me in much milder fashion, for he took to live with him two of his nephews, wherefore my own labour was lessened by the amount of service he exacted from these. Either I did not go out at all, or if we all went out together the task was less irksome. "When I had completed my sixteenth year--up to which time I served my father constantly--we once more changed our house, and dwelt with Alessandro Cardano next door to the bakery of the Bossi. My father had two other nephews, sons of a sister of his, one named Evangelista, a member of the Franciscan Order, and nearly seventy years of age, and the other Otto Cantone, a farmer of the taxes, and very rich. The last-named, before he died, wished to leave me his sole heir; but this my father forbad, saying that Otto's wealth had been ill gotten; wherefore the estate was distributed according to the directions of the surviving brother."[17] This, told as nearly as may be in his own words, is the story of Cardan's birth and childhood and early discipline, a discipline ill calculated to let him grow up to useful and worthy manhood. It must have been a wretched spring of life. Many times he refers to the hard slavery he underwent in the days when he was forced to carry his father's bag about the town, and tells how he had to listen to words of insult cast at his mother's name.[18] Like most boys who lead solitary lives, unrelieved by the companionship of other children, he was driven in upon himself, and grew up into a fanciful imaginative youth, a lover of books rather than of games, with an old head upon his young shoulders. After such a training it was only natural that he should be transformed from a nervous hysterical child into an embittered, cross-grained man, profligate and superstitious at the same time. Abundant light is thrown upon every stage of his career, for few men have left a clearer picture of themselves in their written words, and nowhere is Cardan, from the opening to the closing scene, so plainly exhibited as in the _De Vita Propria_, almost the last work which came from his pen. It has been asserted that this book, written in the twilight of senility by an old man with his heart cankered by misfortune and ill-usage, and his brain upset by the dread of real or fancied assaults of foes who lay in wait for him at every turn, is no trustworthy guide, even when bare facts are in question, and undoubtedly it would be undesirable to trust this record without seeking confirmation elsewhere. This confirmation is nearly always at hand, for there is hardly a noteworthy event in his career which he does not refer to constantly in the more autobiographic of his works. The _De Vita Propria_ is indeed ill arranged and full of inconsistencies, but in spite of its imperfections, it presents its subject as clearly and effectively as Benvenuto Cellini is displayed in his own work. The rough sketch of a great master often performs its task more thoroughly than the finished painting, and Cardan's autobiography is a fragment of this sort. It lets pass in order of procession the moody neglected boy in Fazio's ill-ordered house, the student at Pavia, the youthful Rector of the Paduan Gymnasium, plunging when just across the threshold of life into criminal excess of Sardanapalean luxury, the country doctor at Sacco and afterwards at Gallarate, starving amongst his penniless patients, the University professor, the famous physician for whose services the most illustrious monarchs in Europe came as suppliants in vain, the father broken by family disgrace and calamity, and the old man, disgraced and suspected and harassed by persecutors who shot their arrows in the dark, but at the same time tremblingly anxious to set down the record of his days before the night should descend. Until he had completed his nineteenth year Jerome continued to dwell under the roof which for the time being might give shelter to his parents. The emoluments which Fazio drew from his profession were sufficient for the family wants--he himself being a man of simple tastes; wherefore Jerome was not forced, in addition to his other youthful troubles, to submit to that _execrata paupertas_ and its concomitant miseries which vexed him in later years. To judge from his conduct in the matter of Otto Cantone's estate, Fazio seems to have been as great a despiser of wealth as his son proved to be afterwards. His virtue, such as it was, must have been the outcome of one of those hard cold natures, with wants few and trifling, and none of those tastes which cry out daily for some new toy, only to be procured by money. The fact that he made his son run after him through the streets of Milan in place of a servant is not a conclusive proof of avarice; it may just as likely mean that the old man was indifferent and callous to whatever suffering he might inflict upon his young son, and indisposed to trouble himself about searching for a hireling to carry his bag. The one indication we gather of his worldly wisdom is his dissatisfaction that his son was firmly set to follow medicine rather than jurisprudence, a step which would involve the loss of the stipend of one hundred crowns a year which he drew for his lectureship, an income which he had hoped might be continued to a son of his after his death.[19] Amidst the turmoil and discomfort of what must at the best have been a most ill-regulated household, the boy's education was undertaken by his father in such odds and ends of time as he might find to spare for the task.[20] What with the hardness and irritability of the teacher, and the peevishness inseparable from the pupil's physical feebleness and morbid overwrought mental habit, these hours of lessons must have been irksome to both, and of little benefit. "In the meantime my father taught me orally the Latin tongue as well as the rudiments of Arithmetic, Geometry, and Astrology. But he allowed me to sleep well into the day, and he himself would always remain abed till nine o'clock. But one habit of his appeared to me likely to lead to grave consequences, to wit the way he had of lending to others anything which belonged to him. Part of these loans, which were made to insolvents, he lost altogether; and the residue, lent to divers persons in high places, could only be recovered with much trouble and no little danger, and with loss of all interest on the same. I know not whether he acted in this wise by the advice of that familiar spirit[21] whose services he retained for eight-and-thirty years. What afterwards came to pass showed that my father treated me, his son, rightly in all things relating to education, seeing that I had a keen intelligence. For with boys of this sort it is well to make use of the bit as though you were dealing with mules. Beyond this he was witty and diverting in his conversation, and given to the telling of stories and strange occurrences well worth notice. He told me many things about familiar spirits, but what part of these were true I know not; but assuredly tales of this sort, wonderful in themselves and artfully put together, delighted me marvellously. "But what chiefly deserved condemnation in my father was that he brought up certain other youths with the intention of leaving to them his goods in case I should die; which thing, in sooth, meant nothing less than the exposure of myself to open danger through plots of the parents of the boys aforesaid, on account of the prize offered. Over this affair my father and my mother quarrelled grievously, and finally decided to live apart. Whereupon my mother, stricken by this mental vexation, and troubled at intervals with what I deem to have been an hysterical affection, fell one day full on the back of her neck, and struck her head upon the floor, which was composed of tiles. It was two or three hours before she came round, and indeed her recovery was little short of miraculous, especially as at the end of her seizure she foamed much at the mouth. "In the meantime I altered the whole drift of this tragedy by a pretended adoption of the religious life, for I became for a time a member of the mendicant Franciscan brotherhood. But at the beginning of my twenty-first year[22] I went to the Gymnasium at Pavia, whereupon my father, feeling my absence, was softened towards me, and a reconciliation between him and my mother took place. "Before this time I had learnt music, my mother and even my father having secretly given me money for the same; my father likewise paid for my instruction in dialectics. I became so proficient in this art that I taught it to certain other youths before I went to the University. Thus he sent me there endowed with the means of winning an honest living; but he never once spake a word to me concerning this matter, bearing himself always towards me in considerate, kindly, and pious wise. "For the residue of his days (and he lived on well-nigh four more years) his life was a sad one, as if he would fain let it be known to the world how much he loved me.[23] Moreover, when by the working of fate I returned home while he lay sick, he besought, he commanded, nay he even forced me, all unwilling, to depart thence, what though he knew his last hour was nigh, for the reason that the plague was in the city, and he was fain that I should put myself beyond danger from the same. Even now my tears rise when I think of his goodwill towards me. But, my father, I will do all the justice I can to thy merit and to thy paternal care; and, as long as these pages may be read, so long shall thy name and thy virtues be celebrated. He was a man not to be corrupted by any offering whatsoever, and indeed a saint. But I myself was left after his death involved in many lawsuits, having nothing clearly secured except one small house."[24] Fazio contracted a close intimacy with a certain Galeazzo Rosso, a man clever as a smith, and endowed with mechanical tastes which no doubt helped to secure him Fazio's friendship. Galeazzo discovered the principle of the water-screw of Archimedes before the description of the same, written in the books of the inventor, had been published. He also made swords which could be bent as if they were of lead, and sharp enough to cut iron like wood. He performed a more wonderful feat in fashioning iron breast-plates which would resist the impact of red-hot missiles. In the _De Sapientia_, Cardan records that when Galeazzo perfected his water-screw, he lost his wits for joy. Fazio took no trouble to teach his son Latin,[25] though the learned language would have been just as necessary for the study of jurisprudence as for any other liberal calling, and Jerome did not begin to study it systematically till he was past nineteen years of age. Through some whim or prejudice the old man refused for some time to allow the boy to go to the University, and when at last he gave his consent he still fought hard to compel Jerome to qualify himself in jurisprudence; but here he found himself at issue with a will more stubborn than his own. Cardan writes: "From my earliest youth I let every action of mine be regulated in view of the after course of my life, and I deemed that as a career medicine would serve my purpose far better than law, being more appropriate for the end I had in view, of greater interest to the world at large, and likely to last as long as time itself. At the same time I regarded it as a study which embodied the nobler principles, and rested upon the ground of reason (that is upon the eternal laws of Nature) rather than upon the sanction of human opinion. On this account I took up medicine rather than jurisprudence, nay I almost entirely cast aside, or even fled from the company of those friends of mine who followed the law, rejecting at the same time wealth and power and honour. My father, when he heard that I had abandoned the study of law to follow philosophy, wept in my presence, and grieved amain that I would not settle down to the study of his own subject. He deemed it the more salutary discipline--proofs of which opinion he would often bring forward out of Aristotle--that it was better adapted for the acquisition of power and riches; and that it would help me more efficiently in restoring the fortunes of our house. He perceived moreover that the office of teaching in the schools of the city, together with its accompanying salary of a hundred crowns which he had enjoyed for so many years, would not be handed on to me, as he had hoped, and he saw that a stranger would succeed to the same. Nor was that commentary of his destined ever to see the light or to be illustrated by my notes. Earlier in life he had nourished a hope that his name might become illustrious as the emendator of the 'Commentaries of John, Archbishop of Canterbury on Optics and Perspective.'[26] Indeed the following verses were printed thereanent: 'Hoc Cardana viro gaudet domus: omnia novit Unus: habent nullum saecula nostra parem.' "These words may be taken as a sort of augury referring rather to certain other men about to set forth to do their work in the world, than to my father, who, except in the department of jurisprudence (of which indeed rumour says that he was a master), never let his mind take in aught that was new. The rudiments of mathematics were all that he possessed, and he gathered no fresh knowledge from the store-houses of Greek learning. This disposition in him was probably produced by the vast multitude of subjects to be mastered, and by his infirmity of purpose, rather than by any lack of natural parts, or by idleness or by defect of judgment; vices to which he was in no way addicted. But I, being firmly set upon the object of my wishes, for the reasons given above, and because I perceived that my father had achieved only moderate success--though he had encountered but few hindrances--remained unconvinced by any of his exhortations."[27] FOOTNOTES: [1] Bayle is unwilling to admit Cardan's illegitimate birth. In _De Consolatione_, Opera, tom. i. p. 619 (Lyons, 1663), Cardan writes in reference to the action of the Milanese College of Physicians: "Medicorum collegium, suspitione obortâ, quòd (tam malè à patre tractatus) spurius essem, repellebat." Bayle apparently had not read the _De Consolatione_, as he quotes the sentence as the work of a modern writer, and affirms that the word "suspitio" would not have been used had the fact been notorious. But in the _Dialogus de Morte_, Opera, tom. i. p. 676, Cardan declares that his father openly spoke of him as a bastard. [2] _De Utilitate ex adversis Capienda_ (Franeker, 1648), p. 357. [3] Matteo Visconti was born in 1250, and died in 1322. He was lord of Novara Vercello Como and Monferrato, and was made Vicar Imperial by Adolphus of Nassau. Though he was worsted in his conflict with John XXII. he did much to lay the foundations of his family. [4] _De Vita Propria_ (Amsterdam, 1654), ch. i. p. 4. [5] Cardan makes a statement in _De Consolatione_, Opera, tom. i. p. 605, which indicates that her disposition was not a happy one. "Matrem meam Claram Micheriam, juvenem vidi, cum admodum puer essem, meminique hanc dicere solitam, Utinam si Deo placuisset, extincta forem in infantia." [6] _De Vita Propria_, ch. i. p. 4. [7] _Geniturarum Exempla_ (Basil, 1554), p. 436. [8] _De Rerum Varietate_ (Basil, 1557), p. 655. [9] _De Utilitate_, p. 347. There is a passage in _Geniturarum Exempla_, p. 435, dealing with Fazio's horoscope, which may be taken to mean that these children were his. "Alios habuisse filios qui obierint ipsa genitura dem[o=]strat, me solo diu post eti[a=] illius mort[e=] superstite." [10] With regard to the union of his parents he writes: "Uxorem vix duxit ob Lunam afflictam et eam in senectute."--_Geniturarum Exempla_, p. 435. [11] "Igitur ut ab initio exordiar, in pestilentia conceptus, matrem, nondum natus (ut puto) mearum calamitatum participem, profugam habui."--_Opera_, tom. i. p. 618. "Mater ut abortiret medicamentum abortivum dum in utero essem, alieno mandato bibit."--_De Utilitate_, p. 347. [12] _De Vita Propria_, ch. ii. p. 6. [13] In one passage, _De Utilitate_, p. 348, he sums up his physical misfortunes: "Hydrope, febribus, aliisque morbis conflictatus sum, donec sub fine octavi anni ex dysenteria ac febre usque ad mortis limina perveni, pulsavi ostium sed non aperuere qui intro erant." [14] "Inde lac praegnantis hausi per varias nutrices lactatus ac jactatus."--_De Utilitate_, p. 348. [15] The _De Vita Propria_, the chief authority for these remarks, was written by Cardan in Rome shortly before his death. [16] The illness would have occurred about October 1508, and the victory of the Adda was on May 14, 1509. This fact fixes his birth in 1501, and shows that his illness must have lasted six or seven months. [17] _De Vita Propria_, ch. iv. p. ii. [18] _Opera_, tom. i. p. 676. [19] "Quod munus profitendi institutiones in urbe ipsa cum honorario centum coronatorum, quo jam tot annis gaudebat, non in me (ut speraverat) transiturum intelligebat."--_De Vita Propria_, ch. x. p. 35. [20] "Pater jam antè concesserat ut Geometriæ et Dialecticæ operam darem, in quo (quanquam præter paucas admonitiones, librosque, ac licentiam, nullum aliud auxilium præbuerit) eas tamen ego (succicivis temporibus studens) interim feliciter sum assecutus."--_De Consolatione_, Opera, tom. i. p. 619. [21] "Facius Cardanus dæmonem ætherium, ut ipse dicebat, diu familiarem habuit; qui quamdiu conjuratione usus est, vera illi dabat responsa, cùm autem illam exussisset, veniebat quidem, sed responsa falsa dabat. Tenuit igitur annis, ni fallor, vinginti octo cum conjuratione, solutum autem circiter quinque."--_De Varietate_, p. 629. In the _Dialogus Tetim_ (_Opera_, tom. i. p. 672), Cardan writes: "Pater honeste obiit et ex senio, sed multo antea eum Genius ille reliquerat." [22] There is a discrepancy between this date and the one given in _De Vita Propria_, ch. iv. p. 11. "Anno exacto XIX contuli me in Ticinensem Academiam." [23] "Inde (desiderium augente absentiâ) mortuus est, sæviente peste, cùm primum me diligere coepisset."--_De Consolatione_, Opera, tom. i. p. 619. [24] _De Utilitate_, p. 348. [25] "Nimis satis fuit defuisse tot, memoriam, linguam Latinam per adolescentiam."--_De Vita Propria_, ch. li. p. 218. [26] John Peckham was a Franciscan friar, and was nominated to the see of Canterbury by Nicholas III. in 1279. He had spent much time in the convent of his Order at Oxford, and there is a legend connecting him with a Johannes Juvenis or John of London, a youth who had attracted the attention and benevolence of Roger Bacon. This Johannes became one of the first mathematicians and opticians of the age, and was sent to Rome by Bacon, who entrusted to him the works which he was sending to Pope Clement IV. There is no reason for this view beyond the fact that both were called John, and distinguished in the same branches of learning. The _Perspectiva Communis_ was his principal work; it does not deal with perspective as now understood, but with elementary propositions of optics. It was first printed in Milan in or about 1482. [27] _De Vita Propria_, ch. x. p. 34. A remark in _De Sapientia_, Opera, tom. i. p. 578, suggests that Fazio began life as a physician: "Pater meus Facius Cardanus Medicus primò, inde Jurisconsultus factus est." CHAPTER II THE University of Pavia to which Jerome now betook himself was by tradition one of the learned foundations of Charlemagne.[28] It had certainly enjoyed a high reputation all through the Middle Ages, and had recently had the honour of numbering Laurentius Valla amongst its professors. In 1362, Galeazzo Visconti had obtained a charter for it from the Emperor Charles IV., and that it had become a place of consequence in 1400 is proved by the fact that, besides maintaining several professors in the Canon Law, it supported thirteen in Civil Law, five in Medicine, three in Philosophy, and one each in Astrology, Greek, and Eloquence. Like all the other Universities of Northern Italy, it suffered occasional eclipse or even extinction on account of the constant war and desolation which vexed these parts almost without intermission during the years following the formation of the League of Cambrai. Indeed, as recently as 1500, the famous library collected by Petrarch, and presented by Gian Galeazzo Visconti to the University, was carried off by the French.[29] To judge from the pictures which the Pavian student, writing in after years, gives of his physical self, it may be inferred that he was ill-endowed by the Graces. "I am of middle height. My chest is somewhat narrow and my arms exceedingly thin: my right hand is the more grossly fashioned of the two, so that a chiromantist might have set me down as rude or doltish: indeed, should such an one examine my hand, he would be ashamed to say what he thought. In it the line of life is short, and that named after Saturn long and well marked. My left hand, however, is seemly, with fingers long, tapering, and well-set, and shining nails. My neck is longer and thinner than the rule, my chin is divided, my lower lip thick and pendulous, my eyes are very small, and it is my wont to keep them half-closed, peradventure lest I should discern things over clearly. My forehead is wide and bare of hair where it meets the temples. My hair and beard are both of them yellow in tint, and both as a rule kept close cut. My chin, which as I have said already is marked by a division, is covered in its lower part with a thick growth of long hair. My habit is to speak in a highly-pitched voice, so that my friends sometimes rebuke me thereanent; but, harsh and loud as is my voice, it cannot be heard at any great distance while I am lecturing. I am wont to talk too much, and in none too urbane a tone. The look of my eyes is fixed, like that of one in deep thought. My front teeth are large, and my complexion red and white: the form of my countenance being somewhat elongated, and my head is finished off in narrow wise at the back, like to a small sphere. Indeed, it was no rare thing for the painters, who came from distant countries to paint my portrait, to affirm that they could find no special characteristic which they could use for the rendering of my likeness, so that I might be known by the same."[30] After giving this account of his person, Cardan writes down a catalogue of the various diseases which vexed him from time to time, a chapter of autobiography which looks like a transcript from a dictionary of Nosology. More interesting is the sketch which he makes of his mental state during these early years. Boys brought up in company of their elders often show a tendency to introspection, and fall into a dreamy whimsical mood, and his case is a striking example. "By the command of my father I used to lie abed until nine o'clock,[31] and, if perchance I lay awake any time before the wonted hour of rising, it was my habit to spend the same by conjuring up to sight all sorts of pleasant visions, nor can I remember that I ever summoned these in vain. I used to behold figures of divers kinds like airy bodies. Meseemed they were made up of tiny rings, like those in coats of chain-armour, though at this time I had seen nought of the kind. They would rise at the bottom of the bed, from the right-hand corner; and, moving in a semi-circle, would pass slowly on and disappear in the left. Moreover I beheld the shapes of castles and houses, of horses and riders, of plants, trees, musical instruments, theatres, dresses of men of all sorts, and flute-players who seemed to be playing upon their instruments, but neither voice nor sound was heard therefrom. And besides these things I beheld soldiers, and crowds of men, and fields, and certain bodily forms, which seem hateful to me even now: groves and forests, and divers other things which I now forget. In all this I took no small delight, and with straining eyes I would gaze upon these marvels; wherefore my Aunt Margaret asked me more than once whether I saw anything. I, though I was then only a child, deliberated over this question of hers before I replied, saying to myself: 'If I tell her the facts she will be wroth at the thing--whatever it may be--which is the cause of these phantasms, and will deprive me of this delight.' And then I seemed to see flowers of all kinds, and four-footed beasts, and birds; but all these, though they were fashioned most beautifully, were lacking in colour, for they were things of air. Therefore I, who neither as a boy nor as an old man ever learned to lie, stood silent for some time. Then my aunt said--'Boy, what makes you stare thus and stand silent?' I know not what answer I made, but I think I said nothing at all. In my dreams I frequently saw what seemed to be a cock, which I feared might speak to me in a human voice. This in sooth came to pass later on, and the words it spake were threatening ones, but I cannot now recall what I may have heard on these occasions."[32] With a brain capable of such remarkable exercises as the above-written vision, living his life in an atmosphere of books, and with all games and relaxations dear to boys of his age denied to him, it was no marvel that Jerome should make an early literary essay on his own account. The death of a young kinsman, Niccolo Cardano,[33] suggested to him a theme which he elaborated in a tract called _De immortalitate paranda_, a work which perished unlamented by its author, and a little later he wrote a treatise on the calculation of the distances between the various heavenly bodies.[34] But he put his mathematical skill to other and more sinister uses than this; for, having gained practical experience at the gaming-tables, he combined this experience with his knowledge of the properties of numbers, and wrote a tract on games of chance. Afterwards he amplified this into his book, _Liber de Ludo Aleæ_. With this equipment and discipline Jerome went to Pavia in 1520. He found lodging in the house of Giovanni Ambrogio Targio, and until the end of his twenty-first year he spent all his time between Pavia and Milan. By this date he had made sufficiently good use of his time to let the world see of what metal he was formed, for in the year following he had advanced far enough in learning to dispute in public, to teach Euclid in the Gymnasium, and to take occasional classes in Dialectics and Elementary Philosophy. At the end of his twenty-second year the country was convulsed by the wars between the Spaniards and the French under Lautrec, which ended in the expulsion of the last-named and the establishment of the Imperial power in Milan. Another result of the war, more germane to this history, was the closing of the University of Pavia through lack of funds. In consequence of this calamity Jerome remained some time in Milan, and during these months he worked hard at mathematics; but he was not destined to return to Pavia as a student. The schools there remained some long time in confusion, so in 1524 he went with his father's consent to Padua. In the autumn of that same year he was summoned back to Milan to find Fazio in the grip of his dying illness. "Whereupon he, careful of my weal rather than his own, bade me return to Padua at once, being well pleased to hear that I had taken at the Venetian College the Baccalaureat of Arts.[35] After my return to Padua, letters were brought to me which told me that he had died on the ninth day after he had refused nourishment. He died on the twenty-eighth of August, having last eaten on Sunday the twentieth of the month. Towards the close of my twenty-fourth year I was chosen Rector of the Academy at Padua,[36] and at the end of the next was made Doctor of Medicine. For the first-named office I came out the victor by one vote, the suffrages having to be cast a second time; and for the Doctorate of Medicine my name had already twice come forth from the ballot with forty-seven votes cast against me (a circumstance which forbade another voting after the third), when, at the third trial, I came out the winner, with only nine votes against me (previously only this same number had been cast for me), and with forty-eight in my favour. "Though I know well enough that affairs like these must needs be of small account, I have set them down in the order in which they came to pass for no other reason than that I give pleasure to myself who write these words by so doing: and I do not write for the gratification of others. At the same time those people who read what I write--if indeed any one should ever be so minded--may learn hereby that the beginnings and the outcomes of great events may well be found difficult to trace, because in sooth it is the way of such things to come to the notice of anybody rather than of those who would rightly observe them."[37] Padua cannot claim for its University an antiquity as high as that which may be conceded to Pavia, but in spite of its more recent origin, there is no little obscurity surrounding its rise. The one fact which may be put down as certain is that it sprang originally from the University of Bologna. Early in the thirteenth century violent discords arose between the citizens of Bologna and the students, and there is a tradition that the general school of teaching was transferred to Padua in 1222. What happened was probably a large migration of students, part of whom remained behind when peace between town and gown in Bologna was restored. The orthodox origin of the University is a charter granted by Frederic II. in 1238. Frederic at this time was certainly trying to injure Bologna, actuated by a desire to help on his own University at Naples, and to crush Bologna as a member of the Lombard League.[38] Padua, however, was also a member of this league, so his benevolent action towards it is difficult to understand. In 1228 the students had quarrelled with the Paduan citizens, and there was a movement to migrate to Vercelli; but, whether this really took place or not, the Paduan school did not suffer: its ruin and extinction was deferred till the despotism of the Ezzelini. In 1260 it was again revived by a second migration from Bologna, and this movement was increased on account of the interdict laid by the Pope upon Bologna in 1306 after the expulsion of the Papal Legate by the citizens. In the early days Medicine and Arts were entirely subordinate to the schools of canon and civil law; but by the end of the fourteenth century these first-named Faculties had obtained a certain degree of independence, and were allowed an equal share in appointing the Rector.[39] The first College was founded in 1363, and after 1500 the number rapidly increased. The dominion of the Dukes of Carrara after 1322 was favourable to the growth of the University, which, however, did not attain its highest point till it came under Venetian rule in 1404. The Venetian government raised the stipends of the professors, and allowed four Paduan citizens to act as _Tutores Studii_; the election of the professors being vested in the students, which custom obtained until the end of the sixteenth century.[40] The Rector was allowed to wear a robe of purple and gold; and, when he retired, the degree of Doctor was granted to him, together with the right to wear the golden collar of the order of Saint Mark. Padua like Athens humanized its conquerors. It became the University town of Venice, as Pavia was of Milan, and it was for a long time protected from the assaults of the Catholic reaction by its rulers, who possibly were instigated rather by political jealousy of the Papacy as a temporal power, than by any enthusiasm for the humanist and scientific studies of which Padua was the most illustrious home south of the Alps; studies which the powers of the Church began already to recognize as their most dangerous foes. Such was the University of Padua at the height of its glory, and it will be apparent at once that Padua must have fallen considerably in its fortunes when it installed as its Rector an obscure student, only twenty-four years of age, and of illegitimate birth, and conferred upon him the right to go clad in purple and gold, and to claim, as his retiring gift, the degree of Doctor and the cross of Saint Mark. In 1508 the League of Cambrai had been formed, and Venice, not yet recovered from the effects of its disastrous wars with Bajazet II., was forced to meet the combined assault of the Pope, the Emperor, and the King of France. Padua was besieged by the Imperial forces, a motley horde of Germans, Swiss, and Spaniards, and the surrounding country was pillaged and devastated by these savages with a cruelty which recalled the days of Attila. It is not wonderful that the University closed its doors in such a time. When the confederates began to fight amongst themselves the class-rooms were reopened, intermittently at first, but after 1515 the teaching seems to have been continuous. Still the prevalent turmoil and poverty rendered it necessary to curtail all the mere honorary and ornamental adjuncts of the schools, and for several years no Rector was appointed, for the good and sufficient reason that no man of due position and wealth and character could be found to undertake the rectorial duties, with the Academy just emerging from complete disorganization. These duties were many and important, albeit the Rector could, if he willed, appoint a deputy, and the calls upon the purse of the holder must have been very heavy. It would be hard to imagine any one less fitted to fill such a post than Cardan, and assuredly no office could befit him less than this pseudo-rectorship.[41] It must ever remain a mystery why he was preferred, why he was elected, and why he consented to serve: though, as to the last-named matter, he hints in a passage lately cited from _De Utilitate_, that it was through the persuasions of his mother that he took upon himself this disastrous honour. Many pasages in his writings suggest that Chiara was an indulgent parent. She let Fazio have no peace till he consented to allow the boy to go to college; she paid secretly for music-lessons, so that Jerome was enabled to enjoy the relaxation he loved better than anything else in the world--except gambling; she paid all his charges during his student life at Padua; and now, quite naturally, she would have shed her heart's blood rather than let this son of hers--ugly duckling as he was--miss what she deemed to be the crowning honour of the rectorship; but after all the sacrifices Chiara made, after all the misfortunes which attended Jerome's ill-directed ambition, there is a doubt as to whether he ever was Rector in the full sense of the term. Many times and in divers works he affirms that once upon a time he was Rector, and over and beyond this he sets down in black and white the fact, more than once, that he never told a lie; so it is only polite to accept this legend for what it is worth. But it must likewise be noted that in the extant records of the University there is no mention of his name in the lists of Rectors.[42] Jerome has left very few details as to his life at Padua. Of those which he notices the following are the most interesting: "In 1525, the year in which I became Rector, I narrowly escaped drowning in the Lago di Garda. I went on board the boat, unwillingly enough, which carried likewise some hired horses; and, as we sailed on, the mast and the rudder, and one of the two oars we had with us, were broken by the wind. The sails, even those on the smaller mast, were split, and the night came on. We landed at last safe and sound at Sirmio, but not before all my companions had given up hope, and I myself was beginning to despair. Indeed, had we been a minute later we must have perished, for the tempest was so violent that the iron hinges of the inn windows were bent thereby. I, though I had been sore afraid ever since the wind began to blow, fell to supper with a good heart when the host set upon the board a mighty pike, but none of the others had any stomach for food, except the one passenger who had advised us to make trial of this perilous adventure, and who had proved to be an able and courageous helper in our hour of distress. "Again, once when I was in Venice on the birthday of the Virgin, I lost some money at dicing, and on the day following all that was left me went the same way. This happened in the house of the man with whom I was gambling, and in the course of play I noticed that the cards were marked, whereupon I struck him in the face with my dagger, wounding him slightly. Two of his servants were present at the time; some spears hung all ready from the beams of the roof, and besides this the house door was fastened. But when I had taken from him all the money he had about him--his own as well as that which he had won from me by cheating, and my cloak and the rings which I had lost to him the day before--I was satisfied that I had got back all my possessions. The chattels I sent home by my servant at once, but a portion of the money I tossed back to the fellow when I saw that I had drawn blood of him. Then I attacked the servants who were standing by; and, as they knew not how to use their weapons and besought my mercy, I granted this on the condition that they should unlock the door. Their master, taking account of the uproar and confusion, and mistrusting his safety in case the affair should not be settled forthwith (I suspect he was alarmed about the marked cards), commanded the servants to open the door, whereupon I went my way. "That very same evening, while I was doing my best to escape the notice of the officers of justice on account of the wound I had given to this Senator, I lost my footing and fell into a canal, having arms under my cloak the while. In my fall I did not lose my nerve, but flinging out my right arm, I grasped the thwart of a passing boat and was rescued by those on board. When I had been hauled into the boat I discovered--wonderful to relate--that the man with whom I had lately played cards was likewise on board, with his face bandaged by reason of the wounds I had given him. Now of his own accord he brought out a suit of clothes, fitted for seafaring, and, having clad myself in them, I journeyed with him as far as Padua."[43] Cardan's life from rise to set cannot be estimated otherwise than an unhappy one, and its least fortunate years were probably those lying between his twenty-first and his thirty-first year of age. During this period he was guilty of that crowning folly, the acceptance of the Rectorship of the Gymnasium at Padua, he felt the sharpest stings of poverty, and his life was overshadowed by dire physical misfortune. He gives a rapid sketch of the year following his father's death. "Then, my father having breathed his last and my term of office come to an end, I went, at the beginning of my twenty-sixth year, to reside at Sacco, a town distant ten miles from Padua and twenty-five from Venice. I fixed on this place by the advice of Francesco Buonafidei, a physician of Padua, who, albeit I brought no profit to him--not even being one of those who attended his public teaching--helped me and took a liking for me, being moved to this benevolence by his exceeding goodness of heart. In this place I lived while our State was being vexed by every sort of calamity. In 1524 by a raging pestilence and by a two-fold change of ruler. In 1526 and 1527 by a destructive scarcity of the fruits of the earth. It was hard to get corn in exchange for money of any kind, and over and beyond this was the intolerable weight of taxation. In 1528 the land was visited by divers diseases and by the plague as well, but these afflictions seemed the easier to bear because all other parts were likewise suffering from the same. In 1529 I ventured to return to Milan--these ill-starred troubles being in some degree abated--but I was refused membership by the College of Physicians there, I was unable to settle my lawsuit with the Barbiani, and I found my mother in a very ill humour, so I went back to my village home, having suffered greatly in health during my absence. For what with cruel vexations, and struggles, and cares which I saw impending, and a troublesome cough and pleurisy aggravated by a copious discharge of humour, I was brought into a condition such as few men exchange for aught else besides a coffin."[44] The closing words of his eulogy on his father tell how the son, on the father's death, found that one small house was all he could call his own. The explanation of this seems to be that the old man, being of a careless disposition and litigious to boot, had left his affairs in piteous disorder. In consequence of this neglect Jerome was involved in lawsuits for many years, and the one afore-mentioned with the Barbiani was one of them. This case was subsequently settled in Jerome's favour. FOOTNOTES: [28] Pavia, like certain modern universities, did not spend all its time over study. "Aggressus sum Mediolani vacationibus quadragenariæ, seu Bacchanalium potius, anni MDLXI. Ita enim non obscurum est, nostra ætate celebrari ante quadragenariam vacationes, in quibus ludunt, convivantur, personati ac larvati incedunt, denique nullum luxus ac lascivæ genus omittunt: Sybaritæ et Lydi Persæque vincuntur." _Opera_, tom. i. p. 118. [29] These books were taken to Blois. They were subsequently removed by Francis I. to Fontainebleau, and with the other collections formed the nucleus of the Bibliothèque Nationale. [30] _De Vita Propria_, ch. v. p. 18. [31] The time covered by this experience was from his fourth to his seventh year. [32] _De Vita Propria_, ch. xxxvii. p. 114; _De Rerum Subtilitate_ (Basil, 1554), p. 524. [33] _Opera_, tom. i. p. 61. [34] "Erat liber exiguus, rem tamen probe absolvebat: nam tunc forte in manus meas inciderat, Gebri Hispani liber, cujus auxilio non parum adjutus sum."--_Opera_, tom. i. p. 56. [35] "Initio multi quidem paupertate aliave causa quum se nolunt subjicere rigoroso examini Cl. Collegii in artibus Medicinae vel in Jure, Baccalaureatus, vel Doctoratus gradum a Comitibus Palatinis aut Lateranensibus sumebant. Postea vero, sublata hac consuetudine, Gymnasii Rector, sive substitutus, convocatis duobus professoribus, bina puncta dabantur, iisque recitatis et diligentis [_sic_] excussis, illis gradus Baccalaureatus conferebatur."--_Gymnasium Patavinum_ (1654), p. 200. [36] He constantly bewails this step as the chief folly of his life: "Stulte vero id egi, quod Rector Gymnasii Patavini effectus sum, tum, cum, inops essem, et in patria maxime bella vigerent, et tributa intolerabilia. Matris tamen solicitudine effectum est, ut pondus impensarum, quamvis aegre, sustinuerim."--_De Utilitate_, p. 350. [37] _De Vita Propria_, ch. iv. p. 11. [38] Muratori, _Chron. di Bologna_, xviii. 254. [39] The stipends paid to teachers of jurisprudence were much more liberal than those paid to humanists. In the Diary of Sanudo it is recorded that a jurist professor at Padua received a thousand ducats per annum. Lauro Quirino, a professor of rhetoric, meantime received only forty ducats, and Laurentius Valla at Pavia received fifty sequins.--Muratori, xxii. 990. [40] Tomasinus, _Gymnasium Patavinam_ (1654), p. 136. [41] Tomasinus writes that the Rector should be "Virum illustrem, providum, eloquentem ac divitem, quique eo pollet rerum usu ut Gymnasi decora ipsius gubernatione et splendore augeantur."--_Gymnasium Patavinum_, p. 54. He likewise gives a portrait of the Rector in his robes of office, and devotes several chapters to an account of his duties. [42] "Ab anno 1509 usque ad annum 1515 ob bellum Cameracense Gymn. interrmissum fuit."--_Elenchus nominum Patavii_ (1706), p. 28. The first names given after this interregnum are Dom. Jo. Maria de Zaffaris, Rector in Arts, and Dom. Marinus de Ongaris, Rector in Jurisprudence in 1527. Papadapoli (_Historia Gymn. Patav._) gives the name of Ascanius Serra as pro-Rector in 1526: no Rector being mentioned at all. [43] _De Vita Propria_, ch. xxx. p. 79. [44] _De Vita Propria_, ch. iv. p. 13. CHAPTER III DURING his life at Padua it would appear that Cardan, over and above the allowance made to him by his mother, had no other source of income than the gaming-table.[45] However futile and disastrous his sojourn at this University may have been, he at least took away with him one possession of value, to wit his doctorate of medicine, on the strength of which he began to practise as a country physician at Sacco. The record of his life during these years gives the impression that he must have been one of the most wretched of living mortals. The country was vexed by every sort of misfortune, by prolonged warfare, by raging pestilence, by famine, and by intolerable taxation;[46] but while he paints this picture of misery and desolation in one place, he goes on to declare in another that the time which he spent at Sacco was the happiest he ever knew.[47] No greater instance of inconsistency is to be found in his pages. He writes: "I gambled, I occupied myself with music, I walked abroad, I feasted, giving scant attention the while to my studies. I feared no hurt, I paid my respects to the Venetian gentlemen living in the town, and frequented their houses. I, too, was in the very flower of my age, and no time could have been more delightful than this which lasted for five years and a half."[48] But for almost the whole of this period Cardan was labouring under a physical misfortune concerning which he writes in another place in terms of almost savage bitterness. During ten years of his life, from his twenty-first to his thirty-first year, he suffered from the loss of virile power, a calamity which he laments in the following words: "And I maintain that this misfortune was to me the worst of evils. Compared with it neither the harsh servitude under my father, nor unkindness, nor the troubles of litigation, nor the wrongs done me by my fellow-townsmen, nor the scorn of my fellow-physicians, nor the ill things falsely spoken against me, nor all the measureless mass of possible evil, could have brought me to such despair, and hatred of life, and distaste of all pleasure, and lasting sorrow. I bitterly wept this misery, that I must needs be a laughing-stock, that marriage must be denied me, and that I must ever live in solitude. You ask for the cause of this misfortune, a matter which I am quite unable to explain. Because of the reasons just mentioned, and because I dreaded that men should know how grave was the ill afflicting me, I shunned the society of women; and, on account of this habit, the same miserable public scandal which I desired so earnestly to avoid, arose concerning me, and brought upon me the suspicion of still more nefarious practices: in sooth it seemed that there was no further calamity left for me to endure."[49] After reading these words, it is hard to believe that a man, afflicted with a misfortune which he characterizes in these terms, could have been even moderately happy; much less in that state of bliss which he sits down to describe forty years afterwards. But the end of his life at Sacco was fated to be happier than the beginning, and it is possible that memories of the last months he spent there may have helped to colour with rosy tint the picture of happiness recently referred to. In the first place he was suddenly freed from his physical infirmity, and shortly after his restoration he met and married the woman who, as long as she lived with him, did all that was possible to make him happy. Every momentous event of Cardan's life--and many a trifling one as well--was heralded by some manifestation of the powers lying beyond man's cognition. In writing about the signs and tokens which served as premonitions of his courtship and marriage, he glides easily into a description of the events themselves in terms which are worth producing. "In times past I had my home in Sacco, and there I led a joyful life, as if I were a man unvexed by misfortune (I recall this circumstance somewhat out of season, but the dream I am about to tell of seems only too appropriate to the occasion), or a mortal made free of the habitations of the blest, or rather of some region of delight. Then, on a certain night, I seemed to find myself in a pleasant garden, beautiful exceedingly, decked with flowers and filled with fruits of divers sorts, and a soft air breathed around. So lovely was it all that no painter nor our poet Pulci, nor any imagination of man could have figured the like. I was standing in the forecourt of this garden, the door whereof was open, and there was another door on the opposite side, when lo! I beheld before me a damsel clad in white. I embraced and kissed her; but before I could kiss her again, the gardener closed the door. I straightway begged him earnestly that he would open it again, but I begged in vain; wherefore, plunged in grief and clinging to the damsel, I seemed to be shut out of the garden. "A little time after this there was a rumour in the town of a house on fire, and I was roused from sleep to hurry to the spot. Then I learned that the house belonged to one Altobello Bandarini,[50] a captain of the Venetian levies in the district of Padua. I had no acquaintance with him, in sooth I scarcely knew him by sight. Now it chanced that after the fire he hired a house next door to my own, a step which displeased me somewhat, for such a neighbour was not to my taste; but what was I to do? After the lapse of a few days, when I was in the street, I perceived a young girl who, as to her face and her raiment, was the exact image of her whom I had beheld in my dream. But I said to myself, 'What is this girl to me? If I, poor wretch that I am, take to wife a girl dowered with naught, except a crowd of brothers and sisters, it will be all over with me; forasmuch as I can hardly keep myself as it is. If I should attempt to carry her off, or to have my will of her by stealth, there will of a surety be some tale-bearers about; and her father, being a fellow-townsman and a soldier to boot, would not sit down lightly under such an injury. In this case, or in that, it is hard to say what course I should follow, for if this affair should come to the issue I most desire, I must needs fly the place.' From that same hour these thoughts and others akin to them possessed my brain, which was only too ready to harbour them, and I felt it would be better to die than to live on in such perplexity. Thenceforth I was as one love-possessed, or even burnt up with passion, and I understood what meaning I might gather from the reading of my dream. Moreover I was by this time freed from the chain which had held me back from marriage. Thus I, a willing bridegroom, took a willing bride, her kinsfolk questioning us how this thing had been brought about, and offering us any help which might be of service; which help indeed proved of very substantial benefit. "But the interpretation of my dreams did not work itself out entirely in the after life of my wife; it made itself felt likewise in the lives of my children. My wife lived with me fifteen years, and alas! this ill-advised marriage was the cause of all the misfortunes which subsequently happened to me. These must have come about either by the working of the divine will, or as the recompense due for some ill deeds wrought by myself or by my forefathers."[51] The dream aforesaid was not the only portent having reference to his marriage. After describing shakings and tremblings of his bed, for which indeed a natural cause was not far to seek, he tells how in 1531 a certain dog, of gentle temper as a rule, and quiet, kept up a persistent howling for a long time; how some ravens perched on the house-top and began croaking in an unusual manner; and how, when his servant was breaking up a faggot, some sparks of fire flew out of the same; whereupon, "by an unlooked-for step I married a wife, and from that time divers misfortunes have attended me."[52] Lucia, the wife of his choice, was the eldest daughter of Altobello Bandarini, who had, besides her, three daughters and four sons. Jerome, as it has been already noted, was possessed with a fear lest he should be burdened by his brothers- and sisters-in-law after his marriage; but, considering that he was a young unknown physician, without either money or patients, and that Bandarini was a man of position and repute, with some wealth and more shrewdness, the chances were that the burden would lie on the other side. Cardan seems to have inherited Fazio's contempt for wealth, or at least to have made a profession thereof; for, in chronicling the event of his marriage, he sets down, with a certain degree of pomposity, that he took a wife without a dower on account of a certain vow he had sworn.[53] If the bride was penniless the father-in-law was wealthy, and the last-named fact might well have proved a powerful argument to induce Cardan to remain at Sacco, albeit he had little scope for his calling. That he soon determined to quit the place, is an evidence of his independence of spirit, and of his disinclination to sponge upon his well-to-do connections. Bandarini, when this scheme was proposed to him, vetoed it at once. He was unwilling to part with his daughter, and possibly he may have taken a fancy to his son-in-law, for Cardan has left it on record that Bandarini was greatly pleased with the match; he ended, however, by consenting to the migration, which was not made without the intervention of a warning portent. A short time before the young couple departed, it happened that a tile got mixed with the embers in Bandarini's bed-chamber; and, in the course of the night, exploded with a loud report, and the fragments thereof were scattered around. This event Bandarini regarded as an augury of evil, and indeed evil followed swiftly after. Before a year had passed he was dead, some holding that his death had been hastened by the ill conduct of his eldest son, and others whispering suspicions of poison. Jerome and his young wife betook themselves to Milan, but this visit seems to have been fully as unprofitable as the one he had paid in 1529. In that year he had to face his first rejection by the College of Physicians, when he made application for admission; and there is indirect evidence that he now made a second application with no better result.[54] In any case his affairs were in a very bad way. If he had money in his pocket he would not keep long away from the gaming-table; and, with the weight of trouble ever bearing him down more and more heavily, it is almost certain that his spirits must have suffered, and that poor Lucia must have passed many an unhappy hour on account of his nervous irritability. Then the gates of his profession remained closed to him by the action of the College. The pretext the authorities gave for their refusal to admit him was his illegitimate birth; but it is not unlikely that they may have mistrusted as a colleague the son of Fazio Cardano, and that stories of the profligate life and the intractable temper of the candidate may have been brought to them.[55] His health suffered from the bad air of the city almost as severely as before, and Lucia, who was at this time pregnant, miscarried at four months, and shortly afterwards had a second misfortune of the same kind. His mother's temper was not of the sweetest, and it is quite possible that between her and her daughter-in-law there may have been strained relations. Cardan at any rate found that he must once more beat a retreat from Milan, wherefore, at the end of April 1533, he made up his mind to remove to Gallarate. This town has already been mentioned as chief place of the district, from which the Cardan family took its origin. Before going thither Jerome had evidently weighed the matter well, and he has set down at some length the reasons which led him to make this choice. "Thus, acting under the reasons aforesaid (the family associations), I resolved to go to Gallarate, in order that I might have the enjoyment of four separate advantages which it offered. Firstly, that in the most healthy air of the place I might shake off entirely the distemper which I had contracted in Milan. Secondly, that I might earn something by my profession, seeing that then I should be free to practise. Thirdly, that there would be no need for me to pine away while I beheld those physicians, by whom I reckoned I had been despoiled, flourishing in wealth and in the high estimation of all men. Lastly, that by following a more frugal way of life, I might make what I possessed last the longer. For all things are cheaper in the country, since they have to be carried from the country into the town, and many necessaries may be had for the asking. Persuaded by these arguments, I went to this place, and I was not altogether deceived, seeing that I recovered my health, and the son--who was to be reft from me later on by the Senate--was born to me."[56] Employment at Gallarate was, however, almost as scarce as it had been at Sacco, wherefore Jerome found leisure in plenty for literary work. He began a treatise on Fate; but, even had this been completed, it would scarcely have filled the empty larder by the proceeds of its sale. More profitable was some chance employment which was given to him by Filippo Archinto,[57] a generous and accomplished young nobleman of Milan, who was ambitious to figure as a writer on Astronomy, and, it may be remarked, Archinto's benefactions were not confined to the payment for the hack work which Jerome did for him at this period. Had it not been for his subsequent patronage and support, it is quite possible that Cardan would have gone under in the sea of adversity. In spite of the cheapness of provisions at Gallarate, and of occasional meals taken gratis from the fields, complete destitution seemed to be only a matter of days, and just at this crisis, to add to his embarrassments--though he longed earnestly for the event--Lucia was brought to bed with her first-born living child on May 14, 1534. The child's birth was accompanied by divers omens, one of which the father describes, finding therein some premonition of future disaster. "I had great fear of his life until the fifteenth day of June, on which day, being a Sunday, he was baptized. The sun shone brightly into the bed-chamber: it was between the hours of eleven and twelve in the forenoon; and, according to custom, we were all gathered round the mother's bed except a young servant, the curtain was drawn away from the window and fastened to the wall, when suddenly a large wasp flew into the room, and circled round the infant. We were all greatly afeard for the child, but the wasp did him no hurt. The next moment it came against the curtain, making so great a noise that you would have said that a drum was being beaten, and all ran towards the place, but found no trace of the wasp. It could not have flown out of the room, because all eyes had been fixed upon it. Then all of us who were then present felt some foreboding of what subsequently came to pass, but did not deem that the end would be so bitter as it proved to be."[58] The impulse which drives men in desperate straits to seek shelter in the streets of a city was as strong in Cardan's time as it is to-day. At Gallarate the last coin was now spent, and there was an extra mouth to feed. There seemed to be no other course open but another retreat to Milan. Archinto was rich in literary ambitions, which might perchance stimulate him to find farther work for the starving scholar: and there was Chiara also who would scarcely let her grandchild die of want. The revelation which Cardan makes of himself and of his way of life at this time is not one to enlist sympathy for him entirely; but it is not wanting in a note of pathetic sincerity. "For a long time the College at Milan refused to admit me, and during these days I was assuredly a spendthrift and heedless. In body I was weakly, and in estate plundered by thieves on all sides, yet I never grudged money for the buying of books. My residence at Gallarate brought me no profit, for in the whole nineteen months I lived there, I did not receive more than twenty-five crowns towards the rent of the house I hired. I had such ill luck with the dice that I was forced to pawn all my wife's jewels, and our very bed. If it is a wonder that I found myself thus bereft of all my substance, it is still more wonderful that I did not take to begging on account of my poverty, and a wonder greater still that I harboured in my mind no unworthy thoughts against my forefathers, or against right living, or against those honours which I had won--honours which afterwards stood me in good stead--but bore my misfortunes with mind undisturbed."[59] Cardan's worldly fortunes were now at their lowest ebb. Burdened with a wife and child, he had found it necessary to return, after a second futile attempt to gain a living by his calling in a country town, to Milan, his "stony-hearted step-mother." If he had reckoned on his mother's bounty he was doomed to disappointment, for Chiara was an irritable woman, and as her son's temper was none of the sweetest, it is almost certain that they must have quarrelled occasionally. It is hard to believe that they could have been on good terms at this juncture, otherwise she would scarcely have allowed him to take his wife and child to what was then the public workhouse of the city;[60] but this place was his only refuge, and in October 1534 he was glad to shelter himself beneath its roof. There was in Cardan's nature a strong vein of melancholy, and up to the date now under consideration he had been the victim of a fortune calculated to deepen rather than disperse his morbid tendencies. A proof of his high courage and dauntless perseverance may be deduced from the fact that neither poverty, nor the sense of repeated failure, nor the flouts of the Milanese doctors, prevailed at any time to quench in his heart the love of fame,[61] or to disabuse him of the conviction that he, poverty-stricken wretch as he was, would before long bind Fortune to his chariot-wheels, and would force the adverse world to acknowledge him as one of its master minds. The dawn was now not far distant, but the last hours of his night of misfortune were very dark. The worst of the struggle, as far as the world was concerned, was over, and the sharpest sorrows and the heaviest disgrace reserved for Cardan in the future were to be those nourished in his own household. Writing of his way of life and of the vices and defects of his character, he says: "If a man shall fail in his carriage before the world as he fails in other things, who shall correct him? Thus I myself will do duty for that one leper who alone out of the ten who were healed came back to our Lord. By reasoning of this sort, Physicians and Astrologers trace back the origin of our natural habits to our primal qualities, to the training of our will, and to our occupations and conversation. In every man all these are found in proper ratio to the time of life of each individual; nevertheless it will be easy to discern marked variations in cases otherwise similar. Therefore it behoves us to hold fast to some guiding principle chosen out of these, and I on my part am inclined, as far as it may be allowed, to say with respect to all of them, [Greek: gnôthi seauton]. "My own nature in sooth was never a mystery to myself. I was ever hot-tempered, single-minded, and given to women. From these cardinal tendencies there proceeded truculence of temper, wrangling, obstinacy, rudeness of carriage, anger, and an inordinate desire, or rather a headstrong passion, for revenge in respect to any wrong done to me; so that this inclination, which is censured by many, became to me a delight. To put it briefly, I held _At vindicta bonum vita jucundius ipsa_. As a general rule I went astray but seldom, though it is a common saying, '_Natura nostra prona est ad malum_.' I am moreover truthful, mindful of benefits wrought to me, a lover of justice and of my own people, a despiser of money, a worshipper of that fame which defies death, prone to thrust aside what is commonplace, and still more disposed to treat mere trifles in the same way. Still, knowing well how great may be the power of little things at any moment during the course of an undertaking, I never make light of aught which may be useful. By nature I am prone to every vice and ill-doing except ambition, and I, if no one else does, know my own imperfections. But because of my veneration for God, and because I recognize the vanity and emptiness of all things of this sort, it often happens that, of my own free will, I forego certain opportunities for taking revenge which may be offered to me. I am timid, with a cold heart and a hot brain, given to reflection and the consideration of things many and mighty, and even of things which can never come to pass. I can even let my thoughts concern themselves with two distinct subjects at the same time. Those who throw out charges of garrulity and extravagance by way of contradicting any praise accorded to me, charge me with the faults of others rather than my own. I attack no man, I only defend myself. "And what reason is there why I should spend myself in this cause since I have so often borne witness of the emptiness of this life of ours? My excuse must be that certain men have praised me, wherefore they cannot deem me altogether wicked. I have always trained myself to let my face contradict my thoughts. Thus while I can simulate what is not, I cannot dissimulate what is. To accomplish this is no difficult task if a man cultivates likewise the habit of hoping for nothing. By striving for fifteen years to compass this end and by spending much trouble over the same I at last succeeded. Urged on by this humour I sometimes go forth in rags, sometimes finely dressed, sometimes silent, sometimes talkative, sometimes joyful, sometimes sad; and on this account my two-fold mood shows everything double. In my youth I rarely spent any care in keeping my hair in order, because of my inclination for other pursuits more to my taste. My gait is irregular. I move now quickly, now slowly. When I am at home I go with my legs naked as far as the ankles. I am slack in duty and reckless in speech, and specially prone to show irritation over anything which may disgust or irk me." The above-written self-description does not display a personality particularly attractive. Jerome Cardan was one of those men who experience a morbid gratification in cataloguing all their sinister points of character, and exaggerating them at the same time; and in this picture, as in many others scattered about the _De Vita Propria_, the shadows may have been put in too strongly. In the foregoing pages reference was made to certain acts of benevolence done to Cardan by the family of Archinto. It is not impossible that the promises and persuasions of his young patron Filippo may have had some weight in inducing Jerome to shift his home once more. Whatever befell he could hardly make his case worse; but whether Filippo had promised help or not, he showed himself now a true and valuable friend. There was in Milan a public lectureship in geometry and astronomy supported by a small endowment left by a certain Tommaso Plat, and to this post, which happened opportunely to be vacant, Cardan was appointed by the good offices of Filippo Archinto. Yet even when he was literally a pauper he seems to have felt some scruples about accepting this office, but fortunately in this instance his poverty overcame his pride. The salary was indeed a very small one,[62] and the lecturer was not suffered to handle the whole of it, but it was at least liberal enough to banish the dread of starvation, and his duties, which consisted solely in the preparation and delivery of his lectures, did not debar him from literary work on his own account. Wherefore in his leisure time he worked hard at his desk. Any differences which may have existed between him and his mother were now removed, for he took her to live with him, the household being made up of himself, his wife, his mother, a friend (a woman), a nurse, the little boy, a man- and maidservant, and a mule.[63] Possibly Chiara brought her own income with her, and thus allowed the establishment to be conducted on a more liberal scale. The Plat lectureship would scarcely have maintained three servants, and Jerome's gains from other sources must have been as yet very slender. His life at this time was a busy one, but he always contrived to portion out his days in such wise that certain hours were left for recreation. At such times as he was called upon to teach, the class-room, of course, had the first claims. After the lecture he would walk in the shade outside the city walls, then return to his dinner, then divert himself with music, and afterwards go fishing in the pools and streams hard by the town. In the course of time he obtained other employment, being appointed physician to the Augustinian friars. The Prior of this Order, Francesco Gaddi, was indeed his first patient of note. He tells how he cured this man of a biennial leprosy after treating him for six months;[64] adding that his labour was in vain, inasmuch as Gaddi died a violent death afterwards. The refusal of the College of Milan to admit him to membership did not forbid him to prescribe for whatever patients might like to consult him by virtue of his Paduan degree. He read voraciously everything which came in his way, and it must have been during these years that he stored his memory with that vast collection of facts out of which he subsequently compounded the row of tomes which form his legacy to posterity. Filippo Archinto was unfailing in his kindness, and Jerome at this time was fortunate enough to attract the attention of certain other Milanese citizens of repute who afterwards proved to be valuable friends; Ludovico Madio, Girolamo Guerrini a jeweller, Francesco Belloti, and Francesco della Croce. The last-named was a skilled jurisconsult, whose help proved of great service in a subsequent litigation between Jerome and the College of Physicians. All his life long Cardan was a dreamer of dreams, and he gives an account of one of his visions in this year, 1534, which, whether regarded as an allegory or as a portent, is somewhat remarkable. "In the year 1534, when I was as it were groping in the dark, when I had settled naught as to my future life, and when my case seemed to grow more desperate day by day, I beheld in a dream the figure of myself running towards the base of a mountain which stood upon my right hand, in company with a vast crowd of people of every station and age and sex--women, men, old men, boys, infants, poor men and rich men, clad in raiment of every sort. I inquired whither we were all running, whereupon one of the multitude answered that we were all hastening on to death. I was greatly terrified at these words, when I perceived a mountain on my left hand. Then, having turned myself round so that it stood on my right side, I grasped the vines (which, here in the midst of the mountains and as far as the place wherein I stood, were covered with dry leaves, and bare of grapes, as we commonly see them in autumn) and began to ascend. At first I found this difficult, for the reason that the mountain was very steep round the base, but having surmounted this I made my way upward easily. When I had come to the summit it seemed that I was like to pass beyond the dictates of my own will. Steep naked rocks appeared on every side, and I narrowly escaped falling down from a great height into a gloomy chasm. So dreadful is all this that now, what though forty years have rolled away, the memory thereof still saddens and terrifies me. Then, having turned towards the right where I could see naught but a plain covered with heath, I took that path out of fear, and, as I wended thither in reckless mood, I found that I had come to the entrance of a rude hut, thatched with straw and reeds and rushes, and that I held by my right hand a boy about twelve years of age and clad in a grey garment. Then at this very moment I was aroused from sleep, and my dream vanished. "In this vision was clearly displayed the deathless name which was to be mine, my life of heavy and ceaseless work, my imprisonment, my seasons of grievous terror and sadness, and my abiding-place foreshadowed as inhospitable, by the sharp stones I beheld: barren, by the want of trees and of all serviceable plants; but destined to be, nevertheless, in the end happy, and righteous, and easy. This dream told also of my lasting fame in the future, seeing that the vine yields a harvest every year. As to the boy, if he were indeed my good spirit, the omen was lucky, for I held him very close. If he were meant to foreshadow my grandson it would be less fortunate. That cottage in the desert was my hope of rest. That overwhelming horror and the sense of falling headlong may have had reference to the ruin of my son.[65] "My second dream occurred a short time after. It seemed to me that my soul was in the heaven of the moon, freed from the body and all alone, and when I was bewailing my fate I heard the voice of my father, saying: 'God has appointed me as a guardian to you. All this region is full of spirits, but these you cannot see, and you must not speak either to me or to them. In this part of heaven you will remain for seven thousand years, and for the same time in certain other stars, until you come to the eighth. After this you shall enter the kingdom of God.' I read this dream as follows. My father's soul is my tutelary spirit. What could be dearer or more delightful? The Moon signifies Grammar; Mercury Geometry and Arithmetic; Venus Music, the Art of Divination, and Poetry; the Sun the Moral, and Jupiter the Natural, World; Mars Medicine; Saturn Agriculture, the knowledge of plants, and other minor arts. The eighth star stands for a gleaning of all mundane things, natural science, and various other studies. After dealing with these I shall at last find my rest with the Prince of Heaven."[66] FOOTNOTES: [45] "Nec ullum mihi erat relictum auxilium nisi latrunculorum Ludus."--_Opera_, tom. i. p. 619. [46] From the formation of the League of Cambrai in 1508 to the establishment of the Imperial supremacy in Italy in 1530, the whole country was desolated by the marching and counter-marching of the contending forces. Milan, lying directly in the path of the French armies, suffered most of all. [47] Compare _De Vita Propria_, chaps. iv. and xxxi. pp. 13 and 92. [48] _De Vita Propria_, ch. xxxi. p. 92. In taking the other view he writes: "Vitam ducebam in Saccensi oppido, ut mihi videbar, infelicissime."--_Opera_, tom. i. p. 97. [49] _De Utilitate_, p. 235. [50] He gives a long and interesting sketch of his father-in-law in _De Utilitate_, p. 370. [51] _De Vita Propria_, ch. xxvi. p. 68; _Opera_, tom. i. p. 97. [52] _De Vita Propria_, ch. xli. p. 149. [53] _De Utilitate_, p. 350. [54] _De Utilitate_, p. 357: "Nam in urbe nec collegium recipere volebat nec cum aliquo ex illis artem exercere licebat et sine illis difficillimum erat." He writes thus while describing this particular visit to Milan. [55] Ill fortune seems to have pursued the whole family in their relations with learned societies. "Nam et pater meus ut ab eo accepi, diu in ingressu Collegii Jurisconsultorum laboravit, et ego, ut alias testatus sum, bis a medicorum Patavino, toties filius meus natu major, a Ticinensi, uterque a Mediolanensi rejecti sumus."--_Opera_, tom. i. p. 94. [56] _De Utilitate_, p. 358. [57] He became a priest, and died Archbishop of Milan in 1552. Cardan dedicated to him his first published book, _De Malo Medendi_. [58] _De Vita Propria_, ch. xxxvii. p. 119. [59] _De Vita Propria_, ch. xxv. p. 67. [60] The Xenodochium, which was originally a stranger's lodging-house. By this time places of this sort had become little else than _succursales_ of some religious house. The Governors of the Milanese Xenodochium were the patrons of the Plat endowment which Cardan afterwards enjoyed. [61] "Hoc unum sat scio, ab ineunte ætate me inextinguibili nominis immortalis cupiditate flagrasse."--_Opera_, tom. i. p. 61. [62] "Minimo tamen honorario, et illud etiam minimum suasu cujusdam amici egregii praefecti Xenodochii imminuerunt; ita cum hujus recordor in mentem venit fabellæ illius Apuleii de annonæ Praefecto."--_Opera_, tom. i. p. 64. [63] _De Utilitate_, p. 351. [64] The following gives a hint as to the treatment followed: "Referant leprosos balneo ejus aquae in qua cadaver ablutum sit, sanari."--_De Varietate_, p. 334. [65] _De Vita Propria_, ch. xxxvii. p. 121. This dream is also told in _De Libris Propriis_, Opera, tom. i. p. 64. [66] _De Vita Propria_, ch. xxxvii. p. 121. CHAPTER IV JEROME CARDAN is now standing on the brink of authorship. The very title of his first book, _De Malo Recentiorum Medicorum Medendi Usu_, gives plain indication of the humour which possessed him, when he formulated his subject and put it in writing. With his temper vexed by the persistent neglect and insult cast upon him by the Milanese doctors he would naturally sit down _con amore_ to compile a list of the errors perpetrated by the ignorance and bungling of the men who affected to despise him, and if his object was to sting the hides of these pundits and arouse them to hostility yet more vehement, he succeeded marvellously well. He was enabled to launch his book rather by the strength of private friendship than by the hope of any commercial success. Whilst at Pavia he had become intimate with Ottaviano Scoto, a fellow-student who came from Venice, and in after times he found Ottaviano's purse very useful to his needs. Since their college days Ottaviano's father had died and had left his son to carry on his calling of printing. In 1536 Jerome bethought him of his friend, and sent him the MS. of the treatise which was to let the world learn with what little wisdom it was being doctored.[67] Ottaviano seems to have expected no profit from this venture, which was manifestly undertaken out of a genuine desire to help his friend, and he generously bore all the costs. Cardan deemed that, whatever the result of the issue of the book might be, it would surely be to his benefit; he hazarded nothing, and the very publication of his work would give him at least notoriety. It would moreover give him the intense pleasure of knowing that he was repaying in some measure the debt of vengeance owing to his professional foes. The outcome was exactly the opposite of what printer and author had feared and hoped. The success of the book was rapid and great. Ottaviano must soon have recouped all the cost of publication; and, while he was counting his money, the doctors everywhere were reading Jerome's brochure, and preparing a ruthless attack upon the daring censor, who, with the impetuosity of youth, had laid himself open to attack by the careless fashion in which he had compiled his work. He took fifteen days to write it, and he confesses in his preface to the revised edition that he found therein over three hundred mistakes of one sort or another. The attack was naturally led by the Milanese doctors. They demanded to be told why this man, who was not good enough to practise by their sanction, was good enough to lay down the laws for the residue of the medical world. They heaped blunder upon blunder, and held him up to ridicule with all the wealth of invective characteristic of the learned controversy of the age. Cardan was deeply humbled and annoyed. "For my opponents, seizing the opportunity, took occasion to assail me through the reasoning of this book, and cried out: 'Who can doubt that this man is mad? and that he would teach a method and a practice of medicine differing from our own, since he has so many hard things to say of our procedure.' And, as Galen said, I must in truth have appeared crazy in my efforts to contradict this multitude raging against me. For, as it was absolutely certain that either I or they must be in the wrong, how could I hope to win? Who would take my word against the word of this band of doctors of approved standing, wealthy, for the most part full of years, well instructed, richly clad and cultivated in their bearing, well versed in speaking, supported by crowds of friends and kinsfolk, raised by popular approval to high position, and, what was more powerful than all else, skilled in every art of cunning and deceit?" Cardan had indeed prepared a bitter pill for his foes, but the draught they compelled him to swallow was hardly more palatable. The publication of the book naturally increased the difficulties of his position, and in this respect tended to make his final triumph all the more noteworthy. It was in 1536 that Cardan made his first essay as an author.[68] The next three years of his life at Milan were remarkable as years of preparation and accumulation, rather than as years of achievement. He had struck his first blow as a reformer, and, as is often the lot of reformers, his sword had broken in his hand, and there now rested upon him the sense of failure as a superadded torment. Yet now and again a gleam of consolation would disperse the gloom, and advise him that the world was beginning to recognize his existence, and in a way his merits. In this same year he received an offer from Pavia of the Professorship of Medicine, but this he refused because he did not see any prospect of being paid for his services. His friend Filippo Archinto was loyal still, and zealous in working for his success, and as he had been recently promoted to high office in the Imperial service, his good word might be very valuable indeed. He summoned his _protégé_ to join him at Piacenza, whither he had gone to meet Paul III., hoping to advance Cardan's interests with the Pope; but though Marshal Brissac, the French king's representative,[69] joined Archinto in advocating his cause, nothing was done, and Jerome returned disappointed to Milan. In these months Cardan, disgusted by the failure of his late attack upon the fortress of medical authority, turned his back, for a time, upon the study of medicine, and gave his attention almost entirely to mathematics, in which his reputation was high enough to attract pupils, and he always had one or more of them in his house, the most noteworthy of whom was Ludovico Ferrari of Bologna, who became afterwards a mathematician of repute, and a teacher both at Milan and Bologna. While he was working at the _De Malo Medendi_, he began a treatise upon Arithmetic, which he dedicated to his friend Prior Gaddi; but this work was not published till 1539. In 1536 he first heard a report of a fresh and important discovery in algebra, made by one Scipio Ferreo of Bologna; the prologue to one of the most dramatic incidents in his career, an incident which it will be necessary to treat at some length later on. Cardan was well aware that his excursions into astrology worked to his prejudice in public esteem, but in spite of this he could not refrain therefrom. It was during the plentiful leisure of this period that he cast the horoscope of Jesus Christ, a feat which subsequently brought upon him grave misfortune; a few patients came to him, moved no doubt by the spirit which still prompts people suffering from obscure diseases to consult professors of healing who are either in revolt or unqualified in preference to going to the orthodox physician. In connection with this irregular practice of his he gives a curious story about a certain Count Borromeo. "In 1536, while I was attending professionally in the house of the Borromei, it chanced that just about dawn I had a dream in which I beheld a serpent of enormous bulk, and I was seized with fear lest I should meet my death therefrom. Shortly afterwards there came a messenger to summon me to see the son of Count Carlo Borromeo. I went to the boy, who was about seven years old, and found him suffering from a slight distemper, but on feeling his pulse I perceived that it failed at every fourth beat. His mother, the Countess Corona, asked me how he fared, and I answered that there was not much fever about him; but that, because his pulse failed at every fourth beat, I was in fear of something, but what it might be I knew not rightly (but I had not then by me Galen's books on the indications of the pulse). Therefore, as the patient's state changed not, I determined on the third day to give him in small doses the drug called _Diarob: cum Turbit_: I had already written my prescription, and the messenger was just starting with it to the pharmacy, when I remembered my dream. 'How do I know,' said I to myself, 'that this boy may not be about to die as prefigured by the portent above written? and in that case these other physicians who hate me so bitterly, will maintain he died through taking this drug.' I called to the messenger, and said there was wanting in the prescription something which I desired to add. Then I privately tore up what I had written, and wrote out another made of pearls, of the horn of unicorn,[70] and certain gems. The powder was given, and was followed by vomiting. The bystanders perceived that the boy was indeed sick, whereupon they called in three of the chief physicians, one of whom was in a way friendly to me. They saw the description of the medicine, and demanded what I would do now. Now although two of these men hated me, it was not God's will that I should be farther attacked, and they not only praised the medicine, but ordered that it should be repeated. This was the saving of me. When I went again in the evening I understood the case completely. The following morning I was summoned at daybreak, and found the boy battling with death, and his father lying in tears. 'Behold him,' he cried, 'the boy whom you declared to ail nothing' (as if indeed I could have said such a thing); 'at least you will remain with him as long as he lives.' I promised that I would, and a little later the boy tried to rise, crying out the while. They held him down, and cast all the blame upon me. What more is there to say? If there had been found any trace of that drug _Diarob: cum Turbit_: (which in sooth was not safe) it would have been all over with me, since Borromeo all his life would either have launched against me complaints grave enough to make all men shun me, or another Canidia, more fatal than African serpents, would have breathed poison upon me."[71] In this same year, 1536, Lucia brought forth another child, a daughter, and it was about this time that Cardan first attracted the attention of Alfonso d'Avalos, the Governor of Milan, and an intimacy began which, albeit fruitless at first, was destined to be of no slight service to Jerome at the crisis of his fortunes.[72] In the following year, in 1537, he made a beginning of two of his books, which were subsequently found worthy of being finished, and which may still be read with a certain interest: the treatises _De Sapientia_ and _De Consolatione_. Of the last-named, he remarks that it pleased no one, forasmuch as it appealed not to those who were happy, and the wretched rejected it as entirely inadequate to give them solace in their evil case. In this year he made another attempt to gain admission to the College at Milan, and was again rejected; the issue of the _De Malo Medendi_ was too recent, and it needed other and more potent influences than those exercised by mere merit, to appease the fury of his rivals and to procure him due status. But it would appear that, in 1536 or 1537, he negotiated with the College to obtain a quasi-recognition on conditions which he afterwards describes as disgraceful to himself, and that this was granted to him.[73] Whatever his qualifications may have been, Cardan had no scruples in treating the few patients who came to him. The first case he notes is that of Donato Lanza,[74] a druggist, who had suffered for many years with blood-spitting, which ailment he treated successfully. Success of this sort was naturally helpful, but far more important than Lanza's cure was the introduction given by the grateful patient to the physician, commending him to Francesco Sfondrato, a noble Milanese, a senator, and a member of the Emperor's privy council. The eldest son of this gentleman had suffered many months from convulsions, and Cardan worked a cure in his case without difficulty. Shortly afterwards another child, only ten months old, was attacked by the same complaint, and was treated by Luca della Croce, the procurator of the College of Physicians, of which Sfondrato was a patron. As the attack threatened to be a serious one, Della Croce recommended that another physician, Ambrogio Cavenago, should be called in, but the father, remembering Cardan's cure of Lanza, wished for him as well. The description of the meeting of the doctors round the sick child's bed, of their quotations from Hippocrates, of the uncertainty and helplessness of the orthodox practitioners, and of the ready resource of the free-lance--who happens also to be the teller of the story--is a richly typical one.[75] "We, the physicians and the father of the child, met about seven in the morning, and Della Croce made a few general observations on death, for he knew that Sfondrato was a sensible man, and he himself was both honoured and learned. Cavenago kept silence at this stage, because the last word had been granted to him. Then I said, 'Do you not see that the child is suffering from Opisthotonos?' whereupon the first physician stood as one dazed, as if I were trying to trouble his wits by my hard words. But Della Croce at once swept aside all uncertainty by saying, 'He means the backward contraction of the muscles.' I confirmed his words, and added, 'I will show you what I mean.' Whereupon I raised the boy's head, which the doctors and all the rest believed was hanging down through weakness, and by its own weight, and bade them put it into its former position. Then Sfondrato turned to me, and said, 'As you have discovered what the disease is, tell us likewise what is the remedy therefor.' Since no one else spoke, I turned towards him and--careful lest I should do hurt to the credit I had gained already,--I said, 'You know what Hippocrates lays down in a case like this--_febrem convulsioni_'--and I recited the aphorism. Then I ordered a fomentation, and an application of lint moistened with linseed-oil and oil of lilies, and gave directions that the child should be gently handled until such time as the neck should be restored; that the nurse should eat no meat, and that the child should be nourished entirely by the milk of her breast, and not too much of that; that it should be kept in its cradle in a warm place, and rocked gently till it should fall asleep. After the other physicians had gone, I remember that the father of the child said to me, 'I give you this child for your own,' and that I answered, 'You are doing him an ill turn, in that you are supplanting his rich father by a poor one.' He answered, 'I am sure that you would care for him as if he were your own, fearing naught that you might thereby give offence to these others' (meaning the physicians). I said, 'It would please me well to work with them in everything, and to win their support.' I thus blended my words, so that he might understand I neither despaired of the child's cure, nor was quite confident thereanent. The cure came to a favourable end; for, after the fourteenth day of the fever--the weather being very warm--the child got well in four days' time. Now as I review the circumstances, I am of opinion that it was not because I perceived what the disease really was, for I might have done so much by reason of my special practice; nor because I healed the child, for that might have been attributed to chance; but because the child got well in four days, whereas his brother lay ill for six months, and was then left half dead, that his father was so much amazed at my skill, and afterwards preferred me to all others. That he thought well of me is certain, because Della Croce himself, during the time of his procuratorship, was full of spite and jealousy against me, and declared in the presence of Cavenago and of Sfondrato, that he would not, under compulsion, say a word in favour of a man like me, one whom the College regarded with disfavour. Whereupon Sfondrato saw that the envy and jealousy of the other physicians was what kept me out of the College, and not the circumstances of my birth. He told the whole story to the Senate, and brought such influence to bear upon the Governor of the Province and other men of worship, that at last the entrance to the College was opened to me." Up to the time of his admission to the College, Jerome had never felt that he could depend entirely upon medicine for his livelihood. He now determined to publish his _Practica Arithmeticæ_, the book which he had prepared _pari passu_ with the ill-starred _De Malo Medendi_. It seems to have been thoroughly revised and corrected, and was finally published in 1539, in Milan; Cardan only received ten crowns for his work, but the sudden fame he achieved as a mathematician ought to have set him on firm ground. His friends were still working to secure for him benefits yet more substantial. Alfonso d'Avalos, Francesco della Croce, the jurisconsult whose name has already been mentioned, and the senator Sfondrato, were doing their best to bring the physicians of the city into a more reasonable temper, and they finally succeeded in 1539; when, after having been denied admission for twelve years, Jerome Cardan became a member of the College, and a sharer in all the privileges appertaining thereto. Though Cardan was now a fully qualified physician, he spent his time for the next year or two rather with letters than with medicine. He worked hard at Greek, and as the result of his studies published somewhat prematurely a treatise, _De Immortalitate Animorum_, a collection of extracts from Greek writers which Julius Cæsar Scaliger with justice calls a confused farrago of other men's learning.[76] He published also about this period the treatise on Judicial Astrology, and the Essay _De Consolatione_, the only one of his books which has been found worthy of an English translation.[77] In 1541 he became Rector of the College of Physicians, but there is no record of any increase in the number of his patients by reason of this superadded dignity. A passage in the _De Vita Propria_, written with even more than his usual brutal candour, gives a graphic view of his manner of life at this period. "It was in the summer of the year 1543, a time when it was my custom to go every day to the house of Antonio Vicomercato, a gentleman of the city, and to play chess with him from morning till night. As we were wont to play for one real, or even three or four, on each game, I, seeing that I was generally the winner, would as a rule carry away with me a gold piece after each day's play, sometimes more and sometimes less. In the case of Vicomercato it was a pleasure and nothing else to spend money in this wise; but in my own there was an element of conflict as well; and in this manner I lost my self-respect so completely that, for two years and more, I took no thought of practising my art, nor considered that I was wasting all my substance--save what I made by play--that my good name and my studies as well would suffer shipwreck. But on a certain day towards the end of August, a new humour seized Vicomercato (either advisedly on account of the constant loss he suffered, or perhaps because he thought his decision would be for my benefit), a determination from which he was to be moved neither by arguments, nor adjurations, nor abuse. He forced me to swear that I would never again visit his house for the sake of gaming, and I, on my part, swore by all the gods as he wished. That day's play was our last, and thenceforth I gave myself up entirely to my studies."[78] But these studies unfortunately were not of a nature to keep the wolf from the door; and Jerome, albeit now a duly qualified physician, and known to fame as a writer on Mathematics far beyond the bounds of Italy, was well-nigh as poor as ever. His mother had died several years before, in 1537; but what little money she may have left would soon have been wasted in gratifying his extravagant taste for costly things,[79] and at the gaming-table. He found funds, however, for a journey to Florence, whither he went to see d'Avalos, who was a generous, open-handed man, and always ready to put his purse at the service of one whom he regarded as an honour to his city and country. There can be little doubt that he helped Cardan liberally at this juncture. The need for a loan was assuredly urgent enough. The recent resumption of hostilities between the French and the Imperialists had led to intolerable taxation throughout the Milanese provinces, and in consequence of dearth of funds in 1543, the Academy at Pavia was forced to close its class-rooms, and leave its teachers unpaid. The greater part of the professors migrated to Pisa; and the Faculty of Medicine, then vacant, was, _pro formâ_, transferred to Milan. This chair was now offered to Cardan. He was in desperate straits--a third child had been born this year--and, though there must have been even less chance of getting his salary paid than when he had refused it before, he accepted the post, explaining that he took this step because there was now no need for him to leave Milan, or danger that he would be rated as an itinerant teacher. It is not improbable that he may have been led to accept the office on account of the additional dignity it would give to him as a practising physician. When, a little later on, the authorities began to talk of returning to Pavia, he was in no mind to follow them, giving as a reason that, were he to leave Milan, he would lose his stipend for the Plat lectureship, and be put to great trouble in the transport of his household, and perhaps suffer in reputation as well. The Senate was evidently anxious to retain his services. They bade him consider the matter, promising to send on a certain date to learn his decision; and, as fate would have it, the question was conveniently decided for him by a portent. "On the night before the day upon which my answer was to be sent to the Senate to say what course I was going to take, the whole of the house fell down into a heap of ruins, and no single thing was left unwrecked, save the bed in which I and my wife and my children were sleeping. Thus the step, which I should never have taken of my own free will or without some sign, I was compelled to take by the course of events. This thing caused great wonder to all those who heard of it."[80] This was in 1544. Jerome hesitated no longer, and went forthwith to Pavia as Professor of Medicine at a salary of two hundred and forty gold crowns per annum; but, for the first year at least, this salary was not paid; and the new professor lectured for a time to empty benches; but, as he was at this time engaged in the final stage of his great work on Algebra, the leisure granted to him by the neglect of the students must have been most acceptable. He published at this time a treatise called _Contradicentium Medicorum_, and in 1545 his _Algebra_ or _Liber Artis Magnæ_ was issued from the press by Petreius of Nuremberg. The issue of this book, by which alone the name of Cardan holds a place in contemporary learning, is connected with an episode of his life important enough to demand special and detailed consideration in a separate place. His practice in medicine was now a fairly lucrative one, but his extravagant tastes and the many vices with which he charges himself would have made short work of the largest income he could possibly have earned, consequently poverty was never far removed from the household. Hitherto his reputation as a man of letters and a mathematician had exceeded his fame as a doctor; for, even after he had taken up his residence as Professor of Medicine at Padua, many applications were made to him for his services in other branches of learning. It was fortunate indeed that he had let his reading take a somewhat eclectic course, for medicine at this time seemed fated to play him false. At the end of 1544 no salary was forthcoming at Pavia, so he abandoned his class-room, and returned to Milan. During his residence there, in the summer of 1546, Cardinal Moroni, acting on behalf of Pope Paul III., made an offer for his services as a teacher of mathematics, accompanied by terms which, as he himself admits, were not to be despised; but, as was his wont, he found some reason for demur, and ultimately refused the offer. In his Harpocratic vein he argued, "This pope is an old man, a tottering wall, as it were. Why should I abandon a certainty for an uncertainty?"[81] The certainty he here alludes to must have been the salary for the Plat lectureship; and, as this emolument was a very small one, it would appear that he did not rate at a high figure any profits which might come to him in the future from his acceptance of the Pope's offer; but, as he admits subsequently, he did not then fully realize the benevolence of the Cardinal who approached him on the subject, or the magnificent patronage of the Farnesi.[82] It is quite possible that this refusal of his may have been caused by a reluctance to quit Milan, the city which had treated him in such cruel and inhospitable fashion, just at the time when he had become a man of mark. In the arrogance of success it was doubtless a keen pleasure to let his fellow-townsmen see that the man upon whom they had heaped insult after insult for so many years was one who could afford to let Popes and Cardinals pray for his services in vain. But whatever may have been his humour, he resolved to remain in Milan; and, as he had no other public duty to perform except the delivery of the Plat lectures, he had abundant leisure to spend upon the many and important works he had on hand at this season. Cardan had now achieved European fame, and was apparently on the high road to fortune, but on the very threshold of his triumph a great sorrow and misfortune befell him, the full effect of which he did not experience all at once. In the closing days of 1546 he lost his wife. There is very scant record of her life and character in any of her husband's writings,[83] although he wrote at great length concerning her father; and the few words that are to be found here and there favour the view that she was a good wife and mother. That Jerome could have been an easy husband to live with under any circumstances it is hard to believe. Lucia's life, had it been prolonged, might have been more free of trouble as the wife of a famous and wealthy physician; but it was her ill fortune to be the companion of her husband only in those dreary, terrible days at Sacco and Gallarate, and in the years of uncertainty which followed the final return to Milan. In the last-named period there was at least the Plat lectureship standing between them and starvation; but children increased the while in the nursery, and manuscripts in the desk of the physician without patients, and Lucia's short life was all consumed in this weary time of waiting for fame and fortune which, albeit hovering near, seemed destined to mock and delude the seeker to the end. Cardan was before all else a man of books and of the study, and it is not rare to find that one of this sort makes a harsh unsympathetic husband. The qualities which he attributes to himself in his autobiography suggest that to live with a man cursed with such a nature would have been difficult even in prosperity, and intolerable in trouble and privation. But fretful and irascible as Cardan shows himself to have been, there was a warm-hearted, affectionate side to his nature. He was capable of steadfast devotion to all those to whom his love had ever been given. His reverence for the memory of his tyrannical and irascible father had been noted already, and a still more remarkable instance of his fidelity and love will have to be considered when the time comes to deal with the crowning tragedy of his life. If Cardan had this tender side to his nature, if he could speak tolerant and even laudatory words concerning such a father as Fazio Cardano, and show evidences of a love strong as death in the fight he made for the life of his ill-starred and unworthy son, it may be hoped--in spite of his almost unnatural silence concerning her--that he gave Lucia some of that tenderness and sympathy which her life of hard toil and heavy sacrifice so richly deserved; and that even in the days when he sold her trinkets to pay his gambling losses, she was not destined to weep the bitter tears of a neglected wife. If her early married life had been full of care and travail, if she died when a better day seemed to be dawning, she was at least spared the supreme sorrow and disgrace which was destined to fall so soon upon the household. Judging by what subsequently happened, it will perhaps be held that fate, in cutting her thread of life, was kinder to her than to her husband, when it gave him a longer term of years under the sun. FOOTNOTES: [67] _De Libris Propriis_, Opera, tom. i. p. 102. [68] Besides the _De Malo Medendi Usu_, he published in 1536 a tract upon judicial astrology. This, in an enlarged form, was reprinted by Petreius at Nuremburg in 1542. [69] Cardan writes of Brissac: "Erat enim Brissacus Prorex singularis in studiosis amoris et humanitatis."--_De Vita Propria_, ch. iv. p. 14. [70] "Mirumque in modum venenis cornu ejus adversari creditur."--_De Subtilitate_, p. 315. Sir Thomas Browne (_Vulgar Errors_, Bk. iii. 23) deals at length with the pretended virtues of the horn, and in the Bestiary of Philip de Thaun (_Popular Treatises on Science during the Middle Ages_) is given an account of the many wonderful qualities of the beast. [71] _De Vita Propria_, ch. xxxiii. p. 105. He also alludes to this case in _De Libris Propriis_ (Opera, tom. i. p. 65), affirming that the other doctors concerned in the case raised a great prejudice against him on account of his reputation as an astrologer. "Ita tot modis et insanus paupertate, et Astrologus profitendo edendoque libros, et imperitus casu illustris pueri, et modum alium medendi observans ex titulo libri nuper edito, jam prope ab omnibus habebar. Atque hæc omnia in Urbe omnium nugacissima, et quæ calumniis maximè patet." [72] The founder of this family was Indico d'Avalos, a Spanish gentleman, who was chosen by Alfonso of Naples as a husband for Antonella, the daughter and heiress of the great Marchese Pescara of Aquino. This d'Avalos Marchese dal Guasto was the grandson of Indico. He commanded the advanced guard at the battle of Pavia, and took part in almost every battle between the French and Imperialists, and went with the Emperor to Tunis in 1535. Though he was a brave soldier and a skilful tactician, he was utterly defeated by d'Enghien at Cerisoles in 1544. He has been taxed with treachery in the case of the attack upon the messengers Rincon and Fregoso, who were carrying letters from Francis I. to the Sultan during a truce, but he did little more than imitate the tactics used by the French against himself; moreover, neither of the murdered men was a French subject, or had the status of an ambassador. D'Avalos was a liberal patron of letters and arts, and was very popular as Governor of Milan. He was a noted gallant and a great dandy. Brantôme writes of him--"qu'il était si dameret qu'il parfumait jusqu'aux selles de ses chevaux."--He died in 1546. [73] "Violentia quorundam Medicorum adactus sum anno MDXXXVI, seu XXXVII, turpi conditione pacisci cum Collegio, sed ut dixi, postmodum dissoluta est, anno MDXXXIX et restitutus sum integrè."--_De Vita Propria_, ch. xxxiii. p. 105. [74] _De Vita Propria_, ch. xl. p. 133.--He gives a long list of cases of his successful treatment in _Opera_, tom. i. p. 82. [75] There is a full account of this episode in _De Libris Propriis_, p. 128, and in _De Vita Propria_, ch. xl. p. 133. [76] Exotericarum exercitationum, p. 987. [77] _Cardanus Comforte, translated into Englishe_, 1573. It was the work of Thomas Bedingfield, a gentleman pensioner of Queen Elizabeth. [78] _De Vita Propria_, ch. xxxvii. p. 116. [79] "Delectant me gladii parvi, seu styli scriptorii, in quos plus viginti coronatis aureis impendi: multas etiam pecunias in varia pennarum genera, audeo dicere apparatum ad scribendum ducentis coronatis non potuisse emi."--_De Vita Propria_, ch. xviii. p. 57. [80] _De Vita Propria_, ch. iv. p. 15. [81] "At ego qui, ut dixi, Harpocraticus sum dicebam:--Summus Pont: decrepitus est: murus ruinosus, certa pro incertis derelinquam?"--_De Vita Propria_, ch. iv. p. 15. It is quite possible that Paul III. may have desired to have Cardan about him on account of his reputation as an astrologer, the Pope being a firm believer in the influence of the stars.--_Vide_ Ranke, _History of the Popes_ i. 166. [82] "Neque ego tum Moroni probitatem, nec Pharnesiorum splendorem intelligebam."--_De Vita Propria_, ch. iv. p. 15. [83] In writing of his own horoscope (_Geniturarum Exempla_, p. 461) he records that she miscarried thrice, brought forth three living children, and lived with him fifteen years. He dismisses his marriage as follows: "Duxi uxorem inexpectato, a quo tempore multa adversa concomitata sunt."--_De Vita Propria_, ch. xli. p. 149. But in _De Rerum Subtilitate_, p. 375, he records his grief at her death:--"Itaque cum a luctu dolor et vigilia invadere soleant, ut mihi anno vertente in morte uxoris Luciæ Bandarenæ quanquam institutis philosophiæ munitus essem, repugnante tamen natura, memorque vinculi c[o=]jugalis, suspiriis ac lachrymis et inedia quinque dierum, a periculo me vindicavi." CHAPTER V AT this point it may not be inopportune to make a break in the record of Cardan's life and work, and to treat in retrospect of that portion of his time which he spent in the composition of his treatises on Arithmetic and Algebra. Ever since 1535 he had been working intermittently at one or other of these, but it would have been impossible to deal coherently and effectively with the growth and completion of these two books--really the most important of all he left behind him--while chronicling the goings and comings of a life so adventurous as that of the author. The prime object of Cardan's ambition was eminence as a physician. But, during the long years of waiting, while the action of the Milanese doctors kept him outside the bounds of their College, and even after this had been opened to him without inducing ailing mortals to call for his services, he would now and again fall into a transport of rage against his persecutors, and of contempt for the public which refused to recognize him as a master of his art, and cast aside his medical books for months at a time, devoting himself diligently to Mathematics, the field of learning which, next to Medicine, attracted him most powerfully. His father Fazio was a geometrician of repute and a student of applied mathematics, and, though his first desire was to make his son a jurisconsult, he gave Jerome in early youth a fairly good grounding in arithmetic and geometry, deeming probably that such training would not prove a bad discipline for an intellect destined to attack those formidable tomes within which lurked the mysteries of the Canon and Civil Law. Mathematical learning has given to Cardan his surest title to immortality, and at the outset of his career he found in mathematics rather than in medicine the first support in the arduous battle he had to wage with fortune. His appointment to the Plat lectureship at Milan has already been noted. In the discharge of his new duties he was bound, according to the terms of the endowment of the Plat lecturer, to teach the sciences of geometry, arithmetic, and astronomy, and he began his course upon the lines laid down by the founder. Few listeners came, however, and at this juncture Cardan took a step which serves to show how real was his devotion to the cause of true learning, and how lightly he thought of an additional burden upon his own back, if this cause could be helped forward thereby. Keenly as he enjoyed his mathematical work, he laid a part of it aside when he perceived that the benches before him were empty, and, by way of making his lectures more attractive, he occasionally substituted geography for geometry, and architecture for arithmetic. The necessary research and the preparation of these lectures led naturally to the accumulation of a large mass of notes, and as these increased under his hand Jerome began to consider whether it might not be worth his while to use them in the composition of one or more volumes. In 1535 he delivered as Plat lecturer his address, the _Encomium Geometriæ_, which he followed up shortly after by the publication of a work, _Quindecim Libri Novæ Geometriæ_. But the most profitable labour of these years was that which produced his first important book, _The Practice of Arithmetic and Simple Mensuration_, which was published in 1539, a venture which brought to the author a reward of ten crowns.[84] It was a well-planned and well-arranged manual, giving proof of the wide erudition and sense of proportion possessed by the author. Besides dealing with Arithmetic as understood by the modern school-boy, it discusses certain astronomical operations, multiplication by memory, the mysteries of the Roman and Ecclesiastical Calendars, and gives rules for the solution of any problem arising from the terms of the same. It treats of partnership in agriculture, the Mezzadria system still prevalent in Tuscany and in other parts of Italy, of the value of money, of the strange properties of certain numbers, and gives the first simple rules of Algebra to serve as stepping-stones to the higher mathematics. It ends with information as to house-rent, letters of credit and exchange, tables of interest, games of chance, mensuration, and weights and measures. In an appendix Cardan examines critically the work of Fra Luca Pacioli da Borgo, an earlier writer on the subject, and points out numerous errors in the same. The book from beginning to end shows signs of careful study and compilation, and the fame which it brought to its author was well deserved. Cardan appended to the Arithmetic a printed notice which may be regarded as an early essay in advertising. He was fully convinced that his works were valuable and quite worth the sums of money he asked for them; the world was blind, perhaps wilfully, to their merits, therefore he now determined that it should no longer be able to quote ignorance of the author as an excuse for not buying the book. This appendix was a notification to the learned men of Europe that the writer of the _Practice of Arithmetic_ had in his press at home thirty-four other works in MS. which they might read with profit, and that of these only two had been printed, to wit the _De Malo Medendi Usu_ and a tract on _Simples_. This advertisement had something of the character of a legal document, for it invoked the authority of the Emperor to protect the copyright of Cardan's books within the Duchy of Milan for ten years, and to prevent the introduction of them from abroad. The Arithmetic proved far superior to any other treatise extant, and everywhere won the approval of the learned. It was from Nuremberg that its appearance brought the most valuable fruits. Andreas Osiander,[85] a learned humanist and a convert to Lutheranism, and Johannes Petreius, an eminent printer, were evidently impressed by the terms of Cardan's advertisement, for they wrote to him and offered in combination to edit and print any of the books awaiting publication in his study at Milan. The result of this offer was the reprinting of _De Malo Medendi_, and subsequently of the tract on Judicial Astrology, and of the treatise _De Consolatione_; the _Book of the Great Art_, the treatises _De Sapientia_ and _De Immortalitate Animorum_ were published in the first instance by these same patrons from the Nuremberg press. But Cardan, while he was hard at work on his Arithmetic, had not forgotten a certain report which had caused no slight stir in the world of Mathematics some three years before the issue of his book on Arithmetic, an episode which may be most fittingly told in his own words. "At this time[86] it happened that there came to Milan a certain Brescian named Giovanni Colla, a man of tall stature, and very thin, pale, swarthy, and hollow-eyed. He was of gentle manners, slow in gait, sparing of his words, full of talent, and skilled in mathematics. His business was to bring word to me that there had been recently discovered two new rules in Algebra for the solution of problems dealing with cubes and numbers. I asked him who had found them out, whereupon he told me the name of the discoverer was Scipio Ferreo of Bologna. 'And who else knows these rules?' I said. He answered, 'Niccolo Tartaglia and Antonio Maria Fiore.' And indeed some time later Tartaglia, when he came to Milan, explained them to me, though unwillingly; and afterwards I myself, when working with Ludovico Ferrari,[87] made a thorough study of the rules aforesaid. We devised certain others, heretofore unnoticed, after we had made trial of these new rules, and out of this material I put together my _Book of the Great Art_."[88] Before dealing with the events which led to the composition of the famous work above-named, it may be permitted to take a rapid survey of the condition of Algebra at the time when Cardan sat down to write. Up to the beginning of the sixteenth century the knowledge of Algebra in Italy, originally derived from Greek and Arabic sources, had made very little progress, and the science had been developed no farther than to provide for the solution of equations of the first or second degree.[89] In the preface to the _Liber Artis Magnæ_ Cardan writes:--"This art takes its origin from a certain Mahomet, the son of Moses, an Arabian, a fact to which Leonard the Pisan bears ample testimony. He left behind him four rules, with his demonstrations of the same, which I duly ascribe to him in their proper place. After a long interval of time, some student, whose identity is uncertain, deduced from the original four rules three others, which Luca Paciolus put with the original ones into his book. Then three more were discovered from the original rules, also by some one unknown, but these attracted very little notice though they were far more useful than the others, seeing that they taught how to arrive at the value of the _cubus_ and the _numerus_ and of the _cubus quadratus_.[90] But in recent times Scipio Ferreo of Bologna discovered the rule of the _cubus_ and the _res_ equal to the _numerus_ (_x^3 + px=q_), truly a beautiful and admirable discovery. For this Algebraic art outdoes all other subtlety of man, and outshines the clearest exposition mortal wit can achieve: a heavenly gift indeed, and a test of the powers of a man's mind. So excellent is it in itself that whosoever shall get possession thereof, will be assured that no problem exists too difficult for him to disentangle. As a rival of Ferreo, Niccolo Tartaglia of Brescia, my friend, at that time when he engaged in a contest with Antonio Maria Fiore, the pupil of Ferreo, made out this same rule to help secure the victory, and this rule he imparted to me after I had diligently besought him thereanent. I, indeed, had been deceived by the words of Luca Paciolus, who denied that there could be any general rule besides these which he had published, so I was not moved to seek that which I despaired of finding; but, having made myself master of Tartaglia's method of demonstration, I understood how many other results might be attained; and, having taken fresh courage, I worked these out, partly by myself and partly by the aid of Ludovico Ferrari, a former pupil of mine. Now all the discoveries made by the men aforesaid are here marked with their names. Those unsigned were found out by me; and the demonstrations are all mine, except three discovered by Mahomet and two by Ludovico."[91] This is Cardan's account of the scheme and origin of his book, and the succeeding pages will be mainly an amplification thereof. The earliest work on Algebra used in Italy was a translation of the MS. treatise of Mahommed ben Musa of Corasan, and next in order is a MS. written by a certain Leonardo da Pisa in 1202. Leonardo was a trader, who had learned the art during his voyages to Barbary, and his treatise and that of Mahommed were the sole literature on the subject up to the year 1494, when Fra Luca Pacioli da Borgo[92] brought out his volume treating of Arithmetic and Algebra as well. This was the first printed work on the subject. After the invention of printing the interest in Algebra grew rapidly. From the time of Leonardo to that of Fra Luca it had remained stationary. The important fact that the resolution of all the cases of a problem may be comprehended in a simple formula, which may be obtained from the solution of one of its cases merely by a change of the signs, was not known, but in 1505 the Scipio Ferreo alluded to by Cardan, a Bolognese professor, discovered the rule for the solution of one case of a compound cubic equation. This was the discovery that Giovanni Colla announced when he went to Milan in 1536. Cardan was then working hard at his Arithmetic--which dealt also with elementary Algebra--and he was naturally anxious to collect in its pages every item of fresh knowledge in the sphere of mathematics which might have been discovered since the publication of the last treatise. The fact that Algebra as a science had made such scant progress for so many years, gave to this new process, about which Giovanni Colla was talking, an extraordinary interest in the sight of all mathematical students; wherefore when Cardan heard the report that Antonio Maria Fiore, Ferreo's pupil, had been entrusted by his master with the secret of this new process, and was about to hold a public disputation at Venice with Niccolo Tartaglia, a mathematician of considerable repute, he fancied that possibly there would be game about well worth the hunting. Fiore had already challenged divers opponents of less weight in the other towns of Italy, but now that he ventured to attack the well-known Brescian student, mathematicians began to anticipate an encounter of more than common interest. According to the custom of the time, a wager was laid on the result of the contest, and it was settled as a preliminary that each one of the competitors should ask of the other thirty questions. For several weeks before the time fixed for the contest Tartaglia studied hard; and such good use did he make of his time that, when the day of the encounter came, he not only fathomed the formula upon which Fiore's hopes were based, but, over and beyond this, elaborated two other cases of his own which neither Fiore nor his master Ferreo had ever dreamt of. The case which Ferreo had solved by some unknown process was the equation _x^3 + px = q_, and the new forms of cubic equation which Tartaglia elaborated were as follows: _x^3 + px^2 = q_: and _x^3 - px^2 = q_. Before the date of the meeting, Tartaglia was assured that the victory would be his, and Fiore was probably just as confident. Fiore put his questions, all of which hinged upon the rule of Ferreo which Tartaglia had already mastered, and these questions his opponent answered without difficulty; but when the turn of the other side came, Tartaglia completely puzzled the unfortunate Fiore, who managed indeed to solve one of Tartaglia's questions, but not till after all his own had been answered. By this triumph the fame of Tartaglia spread far and wide, and Jerome Cardan, in consequence of the rumours of the Brescian's extraordinary skill, became more anxious than ever to become a sharer in the wonderful secret by means of which he had won his victory. Cardan was still engaged in working up his lecture notes on Arithmetic into the Treatise when this contest took place; but it was not till four years later, in 1539, that he took any steps towards the prosecution of his design. If he knew anything of Tartaglia's character, and it is reasonable to suppose that he did, he would naturally hesitate to make any personal appeal to him, and trust to chance to give him an opportunity of gaining possession of the knowledge aforesaid, rather than seek it at the fountain-head. Tartaglia was of very humble birth, and according to report almost entirely self-educated. Through a physical injury which he met with in childhood his speech was affected; and, according to the common Italian usage, a nickname[93] which pointed to this infirmity was given to him. The blow on the head, dealt to him by some French soldier at the sack of Brescia in 1512, may have made him a stutterer, but it assuredly did not muddle his wits; nevertheless, as the result of this knock, or for some other cause, he grew up into a churlish, uncouth, and ill-mannered man, and, if the report given of him by Papadopoli[94] at the end of his history be worthy of credit, one not to be entirely trusted as an autobiographer in the account he himself gives of his early days in the preface to one of his works. Papadopoli's notice of him states that he was in no sense the self-taught scholar he represented himself to be, but that he was indebted for some portion at least of his training to the beneficence of a gentleman named Balbisono,[95] who took him to Padua to study. From the passage quoted below he seems to have failed to win the goodwill of the Brescians, and to have found Venice a city more to his taste. It is probable that the contest with Fiore took place after his final withdrawal from his birthplace to Venice. In 1537 Tartaglia published a treatise on Artillery, but he gave no sign of making public to the world his discoveries in Algebra. Cardan waited on, but the morose Brescian would not speak, and at last he determined to make a request through a certain Messer Juan Antonio, a bookseller, that, in the interests of learning, he might be made a sharer of Tartaglia's secret. Tartaglia has given a version of this part of the transaction; and, according to what is there set down, Cardan's request, even when recorded in Tartaglia's own words, does not appear an unreasonable one, for up to this time Tartaglia had never announced that he had any intention of publishing his discoveries as part of a separate work on Mathematics. There was indeed a good reason why he should refrain from doing this in the fact that he could only speak and write Italian, and that in the Brescian dialect, being entirely ignorant of Latin, the only tongue which the writer of a mathematical work could use with any hope of success. Tartaglia's record of his conversation with Messer Juan Antonio, the emissary employed by Cardan, and of all the subsequent details of the controversy, is preserved in his principal work, _Quesiti et Inventioni Diverse de Nicolo Tartalea Brisciano_,[96] a record which furnishes abundant and striking instance of his jealous and suspicious temper. Much of it is given in the form of dialogue, the terms of which are perhaps a little too precise to carry conviction of its entire sincerity and spontaneity. It was probably written just after the final cause of quarrel in 1545, and its main object seems to be to set the author right in the sight of the world, and to exhibit Cardan as a meddlesome fellow not to be trusted, and one ignorant of the very elements of the art he professed to teach.[97] The inquiry begins with a courteously worded request from Messer Juan Antonio (speaking on behalf of Messer Hieronimo Cardano), that Messer Niccolo would make known to his principal the rule by means of which he had made such short work of Antonio Fiore's thirty questions. It had been told to Messer Hieronimo that Fiore's thirty questions had led up to a case of the _cosa_ and the _cubus_ equal to the _numerus_, and that Messer Niccolo had discovered a general rule for such case. Messer Hieronimo now especially desired to be taught this rule. If the inventor should be willing to let this rule be published, it should be published as his own discovery; but, if he were not disposed to let the same be made known to the world, it should be kept a profound secret. To this request Tartaglia replied that, if at any time he might publish his rule, he would give it to the world in a work of his own under his own name, whereupon Juan Antonio moderated his demand, and begged to be furnished merely with a copy of the thirty questions preferred by Fiore, and Tartaglia's solutions of the same; but Messer Niccolo was too wary a bird to be taken with such a lure as this. To grant so much, he replied, would be to tell everything, inasmuch as Cardan could easily find out the rule, if he should be furnished with a single question and its solution. Next Juan Antonio handed to Tartaglia eight algebraical questions which had been confided to him by Cardan, and asked for answers to them; but Tartaglia, having glanced at them, declared that they were not framed by Cardan at all, but by Giovanni Colla. Colla, he declared, had sent him one of these questions for solution some two years ago. Another, he (Tartaglia) had given to Colla, together with a solution thereof. Juan Antonio replied by way of contradiction--somewhat lamely--that the questions had been handed over to him by Cardan and no one else, wishing to maintain, apparently, that no one else could possibly have been concerned in them, whereupon Tartaglia replied that, supposing the questions had been given by Cardan to Juan Antonio his messenger, Cardan must have got the questions from Colla, and have sent them on to him (Tartaglia) for solution because he could not arrive at the meaning of them himself. He waved aside Juan Antonio's perfectly irrelevant and fatuous protests--that Cardan would not in any case have sent these questions if they had been framed by another person, or if he had been unable to solve them. Tartaglia, on the other hand, declared that Cardan certainly did not comprehend them. If he did not know the rule by which Fiore's questions had been answered (that of the _cosa_ and the _cubus_ equal to the _numerus_), how could he solve these questions which he now sent, seeing that certain of them involved operations much more complicated than that of the rule above written? If he understood the questions which he now sent for solution, he could not want to be taught this rule. Then Juan Antonio moderated his demand still farther, and said he would be satisfied with a copy of the questions which Fiore had put to Tartaglia, adding that the favour would be much greater if Tartaglia's own questions were also given. He probably felt that it would be mere waste of breath to beg again for Tartaglia's answers. The end of the matter was that Tartaglia handed over to the messenger the questions which Fiore had propounded in the Venetian contest, and authorized Juan Antonio to get a copy of his own from the notary who had drawn up the terms of the disputation with Fiore. The date of this communication is January 2, 1539, and on February 12 Cardan writes a long letter to Tartaglia, complaining in somewhat testy spirit of the reception given to his request. He is aggrieved that Tartaglia should have sent him nothing but the questions put to him by Fiore, thirty in number indeed, but only one in substance, and that he should have dared to hint that those which he (Cardan) had sent for solution were not his own, but the property of Giovanni Colla. Cardan had found Colla to be a conceited fool, and had dragged the conceit out of him--a process which he was now about to repeat for the benefit of Messer Niccolo Tartaglia. The letter goes on to contradict all Tartaglia's assertions by arguments which do not seem entirely convincing, and the case is not made better by the abusive passages interpolated here and there, and by the demonstration of certain errors in Tartaglia's book on Artillery. In short a more injudicious letter could not have been written by any man hoping to get a favour done to him by the person addressed. In the special matter of the problems which he sent to Tartaglia by the bookseller Juan Antonio, Cardan made a beginning of that tricky and crooked course which he followed too persistently all through this particular business. In his letter he maintains with a show of indignation that he had long known these questions, had known them in fact before Colla knew how to count ten, implying by these words that he knew how to solve them, while in reality all he knew about them was the fact that they existed. Tartaglia in his answer is not to be moved from his belief, and tells Cardan flatly that he is still convinced Giovanni Colla took the questions to Milan, where he found no one able to solve them, not even Messer Hieronimo Cardano, and that the mathematician last-named sent them on by the bookseller for solution, as has been already related. This letter of Tartaglia's bears the date of February 13, 1539, and after reading it and digesting its contents, Cardan seems to have come to the conclusion that he was not working in the right way to get possession of this secret which he felt he must needs master, if he wanted his forthcoming book to mark a new epoch in this History of Mathematics, and that a change of tactics was necessary. Alfonso d'Avalos, Cardan's friend and patron, was at this time the Governor of Milan. D'Avalos was a man of science, as well as a soldier, and Cardan had already sent to him a copy of Tartaglia's treatise on Artillery, deeming that a work of this kind would not fail to interest him. In his first letter to Tartaglia he mentions this fact, while picking holes in the writer's theories concerning transmitted force and views on gravitation. This mention of the name of D'Avalos, the master of many legions and of many cannons as well, to a man who had written a Treatise on the management of Artillery, and devised certain engines and instruments for the management of the same, was indeed a clever cast, and the fly was tempting enough to attract even so shy a fish as Niccolo Tartaglia. In his reply to Jerome's scolding letter of February 12, 1539, Tartaglia concludes with a description of the instruments which he was perfecting: a square to regulate the discharge of cannon, and to level and determine every elevation; and another instrument for the investigation of distances upon a plane surface. He ends with a request that Cardan will accept four copies of the engines aforesaid, two for himself and two for the Marchese d'Avalos. The tone of this letter shows that Cardan had at least begun to tame the bear, who now seemed disposed to dance _ad libitum_ to the pleasant music of words suggesting introductions to the governor, and possible patronage of these engines for the working of artillery. Cardan's reply of March 19, 1539, is friendly--too friendly indeed--and the wonder is that Tartaglia's suspicions were not aroused by its almost sugary politeness. It begins with an attempt to soften down the asperities of their former correspondence, some abuse of Giovanni Colla, and an apology for the rough words of his last epistle. Cardan then shows how their misunderstanding arose chiefly from a blunder made by Juan Antonio in delivering the message, and invites Tartaglia to come and visit him in his own house in Milan, so that they might deliberate together on mathematical questions; but the true significance of the letter appears in the closing lines. "I told the Marchese of the instruments which you had sent him, and he showed himself greatly pleased with all you had done. And he commanded me to write to you forthwith in pressing terms, and to tell you that, on the receipt of my letter, you should come to Milan without fail, for he desires to speak with you. And I, too, exhort you to come at once without further deliberation, seeing that this said Marchese is wonted to reward all men of worth in such noble and magnanimous and liberal fashion that none of them ever goes away dissatisfied." The receipt of this letter seems to have disquieted Tartaglia somewhat; for he has added a note to it, in which he says that Cardan has placed him in a position of embarrassment. He had evidently wished for an introduction to D'Avalos, but now it was offered to him it seemed a burden rather than a benefit. He disliked the notion of going to Milan; yet, if he did not go, the Marchese d'Avalos might take offence. But in the end he decided to undertake the journey; and, as D'Avalos happened then to be absent from Milan on a visit to his country villa at Vigevano, he stayed for three days in Cardan's house. As a recorder of conversations Tartaglia seems to have had something of Boswell's gift. He gives an abstract of an eventful dialogue with his host on March 25, 1539, which Cardan begins by a gentle reproach anent his guest's reticence in the matter of the rule of the _cosa_ and the _cubus_ equal to the _numerus_. Tartaglia's reply to this complaint seems reasonable enough (it must be borne in mind that he is his own reporter), and certainly helps to absolve him from the charge sometimes made against him that he was nothing more than a selfish curmudgeon who had resolved to let his knowledge die with him, rather than share it with other mathematicians of whom he was jealous. He told Cardan plainly that he kept his rules a secret because, for the present, it suited his purpose to do so. At this time he had not the leisure to elaborate farther the several rules in question, being engaged over a translation of Euclid into Italian; but, when this work should be completed, he proposed to publish a treatise on Algebra in which he would disclose to the world all the rules he already knew, as well as many others which he hoped to discover in the course of his present work. He concludes: "This is the cause of my seeming discourtesy towards your excellency. I have been all the ruder, perhaps, because you write to me that you are preparing a book similar to mine, and that you propose to publish my inventions, and to give me credit for the same. This I confess is not to my taste, forasmuch as I wish to set forth my discoveries in my own works, and not in those of others." In his reply to this, Cardan points out that he had promised, if Tartaglia so desired, that he would not publish the rules at all; but here Messer Niccolo's patience and good manners gave way, and he told Messer Hieronimo bluntly that he did not believe him. Then said Cardan: "I swear to you by the Sacred Evangel, and by myself as a gentleman, that I will not only abstain from publishing your discoveries--if you will make them known to me--but that I will promise and pledge my faith of a true Christian to set them down for my own use in cypher, so that after my death no one may be able to understand them. If you will believe this promise, believe it; if you will not, let us have done with the matter." "If I were not disposed to believe such oaths as these you now swear," said Tartaglia, "I might as well be set down as a man without any faith at all. I have determined to go forthwith to Vigevano to visit the Signor Marchese, as I have now been here for three days and am weary of the delay, but I promise when I return that I will show you all the rules." Cardan replied: "As you are bent on going to Vigevano, I will give you a letter of introduction to the Marchese, so that he may know who you are; but I would that, before you start, you show me the rule as you have promised." "I am willing to do this," said Tartaglia, "but I must tell you that, in order to be able to recall at any time my system of working, I have expressed it in rhyme; because, without this precaution, I must often have forgotten it. I care naught that my rhymes are clumsy, it has been enough for me that they have served to remind me of my rules. These I will write down with my own hand, so that you may be assured that my discovery is given to you correctly." Then follow Tartaglia's verses: "Quando chel cubo con le cose apresso Se agualia à qualche numero discreto Trouan dui altri differenti in esso Dapoi terrai questo per consueto Ch'el lor' produtto sempre sia eguale Al terzo cubo delle cose neto El residuo poi suo generale Delli lor lati cubi ben sottratti Varra la tua cosa principale. In el secondo de cotesti atti Quando chel cubo restasse lui solo Tu osseruarai quest' altri contratti Del numer farai due tal part 'a uolo Che luna in l'altra si produca schietto El terzo cubo delle cose in stolo Delle qual poi, per commun precetto Torrai li lati cubi insieme gionti Et cotal summa sara il tuo concetto Et terzo poi de questi nostri conti Se solve col recordo se ben guardi Che per natura son quasi congionti Questi trouai, et non con passi tardi Nel mille cinquecent' e quatro è trenta Con fondamenti ben sald' è gagliardi Nella citta del mar' intorno centa." Having handed over to his host these rhymes, with the precious rules enshrined therein, Tartaglia told him that, with so clear an exposition, he could not fail to understand them, ending with a warning hint to Cardan that, if he should publish the rules, either in the work he had in hand, or in any future one, either under the name of Tartaglia or of Cardan, he, the author, would put into print certain things which Messer Hieronimo would not find very pleasant reading. After all Tartaglia was destined to quit Milan without paying his respects to D'Avalos. There is not a word in his notes which gives the reason of this eccentric action on his part. He simply says that he is no longer inclined to go to Vigevano, but has made up his mind to return to Venice forthwith; and Cardan, probably, was not displeased at this exhibition of petulant impatience on the part of his guest, but was rather somewhat relieved to see Messer Niccolo ride away, now that he had extracted from him the coveted information. From the beginning to the end of this affair Cardan has been credited with an amount of subtle cunning which he assuredly did not manifest at other times when his wits were pitted for contest with those of other men. It has been advanced to his disparagement that he walked in deceitful ways from the very beginning; that he dangled before Tartaglia's eyes the prospect of gain and preferment simply for the purpose of enticing him to Milan, where he deemed he might use more efficaciously his arguments for the accomplishment of the purpose which was really in his mind; that he had no intention of advancing Tartaglia's fortunes when he suggested the introduction to D'Avalos, but that the Governor of Milan was brought into the business merely that he might be used as a potent ally in the attack upon Tartaglia's obstinate silence. Whether this may have been his line of action or not, the issue shows that he was fully able to fight his battle alone, and that his powers of persuasion and hard swearing were adequate when occasion arose for their exercise. It is quite possible that Tartaglia, when he began to reflect over what he had done by writing out and handing over to Cardan his mnemonic rhymes, fell into an access of suspicious anger--at Cardan for his wheedling persistency, and at himself for yielding thereto--and packed himself off in a rage with the determination to have done with Messer Hieronimo and all his works. Certainly his carriage towards Cardan in the weeks ensuing, as exhibited in his correspondence, does not picture him in an amiable temper. On April 9 Jerome wrote to him in a very friendly strain, expressing regret that his guest should have left Milan without seeing D'Avalos, and fear lest he might have prejudiced his fortunes by taking such a step. He then goes on to describe to Tartaglia the progress he is making in his work with the Practice of Arithmetic, and to ask him for help in solving one of the cases in Algebra, the rule for which was indeed contained in Tartaglia's verses, but expressed somewhat obscurely, for which reason Cardan had missed its meaning.[98] In his reply, Tartaglia ignores Jerome's courtesies altogether, and tells him that what he especially desires at the present moment is a sight of that volume on the Practice of Arithmetic, "for," says he, "if I do not see it soon, I shall begin to suspect that this work of yours will probably make manifest some breach of faith; in other words, that it will contain as interpolations certain of the rules I taught you." Niccolo then goes on to explain the difficulty which had puzzled Cardan, using terms which showed plainly that he had as poor an opinion of his correspondent's wit as of his veracity. Cardan was an irascible man, and it is a high tribute to his powers of restraint that he managed to keep his temper under the uncouth insults of such a letter as the foregoing. The more clearly Tartaglia's jealous, suspicious nature displays itself, the greater seems the wonder that a man of such a disposition should ever have disclosed such a secret. He did not believe Cardan when he promised that he would not publish the rules in question without his (the discoverer's) consent--why then did he believe him when he swore by the Gospel? The age was one in which the binding force of an oath was not regarded as an obligation of any particular sanctity if circumstances should arise which made the violation of the oath more convenient than its observance. However, the time was not yet come for Jerome to begin to quibble with his conscience. On May 12, 1539, he wrote another letter to Tartaglia, also in a very friendly tone, reproaching him gently for his suspicions, and sending a copy of the _Practice of Arithmetic_ to show him that they were groundless. He protested that Tartaglia might search from beginning to end without finding any trace of his jealously-guarded rules, inasmuch as, beyond correcting a few errors, the writer had only carried Algebra to the point where Fra Luca had left it. Tartaglia searched, and though he could not put his finger on any spot which showed that Messer Hieronimo had broken his oath, he found what must have been to him as a precious jewel, to wit a mistake in reckoning, which he reported to Cardan in these words: "In this process your excellency has made such a gross mistake that I am amazed thereat, forasmuch as any man with half an eye must have seen it--indeed, if you had not gone on to repeat it in divers examples, I should have set it down to a mistake of the printer." After pointing out to Cardan the blunders aforesaid, he concludes: "The whole of this work of yours is ridiculous and inaccurate, a performance which makes me tremble for your good name."[99] Every succeeding page of Tartaglia's notes shows more and more clearly that he was smarting under a sense of his own folly in having divulged his secret. Night and day he brooded over his excess of confidence, and as time went by he let his suspicions of Cardan grow into savage resentment. His ears were open to every rumour which might pass from one class-room to another. On July 10 a letter came to him from one Maphio of Bergamo, a former pupil, telling how Cardan was about to publish certain new mathematical rules in a book on Algebra, and hinting that in all probability these rules would prove to be Tartaglia's, whereupon he at once jumped to the conclusion that Maphio's gossip was the truth, and that this book would make public the secret which Cardan had sworn to keep. He left many of Cardan's letters unanswered; but at last he seems to have found too strong the temptation to say something disagreeable; so, in answer to a letter from Cardan containing a request for help in solving an equation which had baffled his skill, Tartaglia wrote telling Cardan that he had bungled in his application of the rule, and that he himself was now very sorry he had ever confided the rule aforesaid to such a man. He ends with further abuse of Cardan's _Practice of Arithmetic_, which he declares to be merely a confused farrago of other men's knowledge,[100] and with a remark which he probably intended to be a crowning insult. "I well remember when I was at your house in Milan, that you told me you had never tried to discover the rule of the _cosa_ and the _cubus_ equal to the _numerus_ which was found out by me, because Fra Luca had declared it to be impossible;[101] as if to say that, if you had set yourself to the task you could have accomplished it, a thing which sets me off laughing when I call to mind the fact that it is now two months since I informed you of the blunders you made in the extraction of the cube root, which process is one of the first to be taught to students who are beginning Algebra. Wherefore, if after the lapse of all this time you have not been able to find a remedy to set right this your mistake (which would have been an easy matter enough), just consider whether in any case your powers could have been equal to the discovery of the rule aforesaid."[102] In this quarrel Messer Giovanni Colla had appeared as the herald of the storm, when he carried to Milan in 1536 tidings of the discovery of the new rule which had put Cardan on the alert, and now, as the crisis approached, he again came upon the scene, figuring as unconscious and indirect cause of the final catastrophe. On January 5, 1540, Cardan wrote to Tartaglia, telling him that Colla had once more appeared in Milan, and was boasting that he had found out certain new rules in Algebra. He went on to suggest to his correspondent that they should unite their forces in an attempt to fathom this asserted discovery of Colla's, but to this letter Tartaglia vouchsafed no reply. In his diary it stands with a superadded note, in which he remarks that he thinks as badly of Cardan as of Colla, and that, as far as he is concerned, they may both of them go whithersoever they will.[103] Colla propounded divers questions to the Algebraists of Milan, and amongst them was one involving the equation _x^4 + 6x^2 + 36 = 60x_, one which he probably found in some Arabian treatise. Cardan tried all his ingenuity over this combination without success, but his brilliant pupil, Ludovico Ferrari, worked to better purpose, and succeeded at last in solving it by adding to each side of the equation, arranged in a certain fashion, some quadratic and simple quantities of which the square root could be extracted.[104] Cardan seems to have been baffled by the fact that the equation aforesaid could not be solved by the recently-discovered rules, because it produced a bi-quadratic. This difficulty Ferrari overcame, and, pursuing the subject, he discovered a general rule for the solution of all bi-quadratics by means of a cubic equation. Cardan's subsequent demonstration of this process is one of the masterpieces of the _Book of the Great Art_. It is an example of the use of assuming a new indeterminate quantity to introduce into an equation, thus anticipating by a considerable space of time Descartes, who subsequently made use of a like assumption in a like case. How far this discovery of Ferrari's covered the rules given by Tartaglia to Cardan, and how far it relieved Cardan of the obligation of secresy, is a problem fitted for the consideration of the mathematician and the casuist severally.[105] An apologist of Cardan might affirm that he cannot be held to have acted in bad faith in publishing the result of Ferrari's discovery. If this discovery included and even went beyond Tartaglia's, so much the worse for Tartaglia. The lesser discovery (Tartaglia's) Cardan never divulged before Ferrari unravelled Giovanni Colla's puzzle; but it was inevitable that it must be made known to the world as a part of the greater discovery (Ferrari's) which Cardan was in no way bound to keep a secret. The case might be said to run on all fours with that where a man confides a secret to a friend under a promise of silence, which promise the friend keeps religiously, until one day he finds that the secret, and even more than the secret, is common talk of the market-place. Is the obligation of silence, with which he was bound originally, still to lie upon the friend, even when he may have sworn to observe it by the Holy Evangel and the honour of a gentleman; and is the fact that great renown and profit would come to him by publishing the secret to be held as an additional reason for keeping silence, or as a justification for speech? In forming a judgment after a lapse of three and a half centuries as to Cardan's action, while having regard both to the sanctity of an oath at the time in question, and to the altered state of the case between him and Tartaglia consequent on Ludovico Ferrari's discovery, an hypothesis not overstrained in the direction of charity may be advanced to the effect that Cardan might well have deemed he was justified in revealing to the world the rules which Tartaglia had taught him, considering that these isolated rules had been developed by his own study and Ferrari's into a principle by which it would be possible to work a complete revolution in the science of Algebra. In any case, six years were allowed to elapse before Cardan, by publishing Tartaglia's rules in the _Book of the Great Art_, did the deed which, in the eyes of many, branded him as a liar and dishonest, and drove Tartaglia almost wild with rage. That his offence did not meet with universal reprobation is shown by negative testimony in the _Judicium de Cardano_, by Gabriel Naudé.[106] In the course of his essay Naudé lets it be seen how thoroughly he dislikes the character of the man about whom he writes. No evil disposition attributed to Cardan by himself or by his enemies is left unnoticed, and a lengthy catalogue of his offences is set down, but this list does not contain the particular sin of broken faith in the matter of Tartaglia's rules. On the contrary, after abusing and ridiculing a large portion of his work, Naudé breaks out into almost rhapsodical eulogy about Cardan's contributions to Mathematical science. "Quis negabit librum de Proportionibus dignum esse, qui cum pulcherrimis antiquorum inventis conferatur? Quis in Arithmetica non stupet, eum tot difficultates superasse, quibus explicandis Villafrancus, Lucas de Burgo, Stifelius, Tartalea, vix ac ne vix quidem pares esse potuissent?" It seems hard to believe, after reading elsewhere the bitter assaults of Naudé,[107] that he would have neglected so tempting an opportunity of darkening the shadows, if he himself had felt the slightest offence, or if public opinion in the learned world was in any perceptible degree scandalized by the disclosure made by the publication of the _Book of the Great Art_. This book was published at Nuremberg in 1545, and in its preface and dedication Cardan fully acknowledges his obligations to Tartaglia and Ferrari, with respect to the rules lately discussed, and gives a catalogue of the former students of the Art, and attributes to each his particular contribution to the mass of knowledge which he here presents to the world. Leonardo da Pisa,[108] Fra Luca da Borgo, and Scipio Ferreo all receive due credit for their work, and then Cardan goes on to speak of "my friend Niccolo Tartaglia of Brescia, who, in his contest with Antonio Maria Fiore, the pupil of Ferreo, elaborated this rule to assure him of victory, a rule which he made known to me in answer to my many prayers." He goes on to acknowledge other obligations to Tartaglia:[109] how the Brescian had first taught him that algebraical discovery could be most effectively advanced by geometrical demonstration, and how he himself had followed this counsel, and had been careful to give the demonstration aforesaid for every rule he laid down. The _Book of the Great Art_ was not published till six years after Cardan had become the sharer of Tartaglia's secret, which had thus had ample time to germinate and bear fruit in the fertile brain upon which it was cast. It is almost certain that the treatise as a whole--leaving out of account the special question of the solution of cubic equations--must have gained enormously in completeness and lucidity from the fresh knowledge revealed to the writer thereof by Tartaglia's reluctant disclosure, and, over and beyond this, it must be borne in mind that Cardan had been working for several years at Giovanni Colla's questions in conjunction with Ferrari, an algebraist as famous as Tartaglia or himself. The opening chapters of the book show that Cardan was well acquainted with the chief properties of the roots of equations of all sorts. He lays it down that all square numbers have two different kinds of root, one positive and one negative,[110] _vera_ and _ficta_: thus the root of 9 is either 3. or -3. He shows that when a case has all its roots, or when none are impossible, the number of its positive roots is the same as the number of changes in the signs of the terms when they are all brought to one side. In the case of _x^3 + 3bx = 2c_, he demonstrates his first resolution of a cubic equation, and gives his own version of his dealings with Tartaglia. His chief obligation to the Brescian was the information how to solve the three cases which follow, _i.e. x^3 + bx = c. x^3 = bx + c._ and _x^3 + c = bx_, and this he freely acknowledges, and furthermore admits the great service of the system of geometrical demonstration which Tartaglia had first suggested to him, and which he always employed hereafter. He claims originality for all processes in the book not ascribed to others, asserting that all the demonstrations of existing rules were his own except three which had been left by Mahommed ben Musa, and two invented by Ludovico Ferrari. With this vantage ground beneath his feet Cardan raised the study of Algebra to a point it had never reached before, and climbed himself to a height of fame to which Medicine had not yet brought him. His name as a mathematician was known throughout Europe, and the success of his book was remarkable. In the _De Libris Propriis_ there is a passage which indicates that he himself was not unconscious of the renown he had won, or disposed to underrate the value of his contribution to mathematical science. "And even if I were to claim this art (Algebra) as my own invention, I should perhaps be speaking only the truth, though Nicomachus, Ptolemæus, Paciolus, Boetius, have written much thereon. For men like these never came near to discover one-hundredth part of the things discovered by me. But with regard to this matter--as with divers others--I leave judgment to be given by those who shall come after me. Nevertheless I am constrained to call this work of mine a perfect one, seeing that it well-nigh transcends the bounds of human perception."[111] FOOTNOTES: [84] It was published at Milan by Bernardo Caluschio, with a dedication--dated 1537--to Francesco Gaddi, a descendant of the famous family of Florence. This man was Prior of the Augustinian Canons in Milan, and a great personage, but ill fortune seems to have overtaken him in his latter days. Cardan writes (_Opera_, tom. i. p. 107):--"qui cum mihi amicus esset dum floreret, Rexque cognomine ob potentiam appellaretur, conjectus in carcerem, miseré vitam ibi, ne dicam crudeliter, finivit: nam per quindecim dies in profundissima gorgyne fuit, ut vivus sepeliretur." [85] There is a reference to Osiander in _De Subtilitate_, p. 523. Cardan gives a full account of his relations with Osiander and Petreius in _Opera_, tom. i. p. 67. [86] November 1536. [87] Ferrari was one of Cardan's most distinguished pupils. "Ludovicus Ferrarius Bononiensis qui Mathematicas et Mediolani et in patria sua professus est, et singularis in illis eruditionis."--_De Vita Propria_, ch. xxxv. p. 111. There is a short memoir of Ferrari in _Opera_, tom. ix. [88] _Opera_, tom. i. p. 66. [89] Fra Luca's book, _Summa de Arithmetica Geometria Proportioni é Proportionalita_, extends as far as the solution of quadratic equations, of which only the positive roots were used. At this time letters were rarely used to express known quantities. [90] The early writers on Algebra used _numerus_ for the absolute or known term, _res_ or _cosa_ for the first power, _quadratum_ for the second, and _cubus_ for the third. The signs + and - first appear in the work of Stifelius, a German writer, who published a book of Arithmetic in 1544. Robert Recorde in his _Whetstone of Wit_ seems first to have used the sign of equality =. Vieta in France first applied letters as general symbols of quantity, though the earlier algebraists used them occasionally, chiefly as abbreviations. Aristotle also used them in the _Physics.--Libri. Hist. des Sciences Mathématiques_. i. 104. [91] _Opera_, tom. iv. p. 222. [92] In the conclusion of the Treatise on Arithmetic, Cardan points out certain errors in the work of Fra Luca. Fra Luca was a pupil of Piero della Francesca, who was highly skilled in Geometry, and who, according to Vasari, first applied perspective to the drawing of the human form. [93] Tartaglia, _i.e._ the stutterer. [94] Papadopoli, _Hist. Gymn. Pata._ (Ven. 1724). [95] "Balbisonem post relatam jurisprudentiæ lauream redeuntem Brixiam Nicolaus secutus est, cæpitque ex Mathematicis gloriam sibi ac divitias parare, æque paupertatis impatiens, ac fortunæ melioris cupidus, quam dum Brixiæ tuetur, homo morosæ, et inurbanæ rusticitatis prope omnium civium odia sibi conciliavit. Quamobrem alibi vivere coactus, varias Italiæ urbes incoluit, ac Ferrariæ, Parmæ, Mediolani, Romæ, Genuæ, arithmeticam, geometricam, ceteraque quæ ad Mathesim pertinent, docuit; depugnavitque scriptis accerrimis cum Cardano ac sibi ex illis quæsivit nomen et gloriam. Tandem domicilium posuit Venetiis, ubi non a Senatoribus modo, ut mos Venetus habet eruditorum hominum studiosissimus, maximi habitus est, at etiam a variis Magnatum ac Principum legatis præmiis ac muneribus auctus sortem, quam tamdiu expetierat visus sibi est conciliasse. Ergo ratus se majorem, quam ut a civibus suis contemneretur, Brixiam rediit, ubi spe privati stipendii Euclidis elementa explanare coepit; sed quæ illum olim a civitate sua austeritas, rustica, acerba, morosa, depulerat, eadem illum in eum apud omnes contemptum, et odium iterum dejicit, ut exinde horrendus ac detestabilis omnibus fugere, atque iterum Venetias confugere compulsus fuerit. Ibi persenex decessit."--Papadopoli, _Hist. Gymn. Pata.,_ ii. p. 210. [96] This work is the chief authority for the facts which follow. The edition referred to is that of Venice, 1546. There is also a full account of the same in Cossali, _Origine dell' Algebra_ (Parma, 1799). vol. ii. p. 96. [97] _Quesiti et Inventioni_, p. 115. [98] Cardan writes: "Vi supplico per l'amor che mi portati, et per l'amicitia ch'è tra noi, che spero durara fin che viveremo, che mi mandati sciolta questa questione. 1 cubo piu 3. cose egual à 10." Cardan had mistaken (1/3 _b_)^3 for 1/3 _b_^3, or the cube of 1/3 of the co-efficient for 1/3 of the cube of the co-efficient.--_Quesiti et Inventioni_ p. 124. [99] _Quesiti et Inventioni_, p. 125. [100] "Non ha datta fora tal opera come cose composto da sua testa ma come cose ellette raccolte e copiate de diverse libri a penna."--_Quesiti et Inventioni_, p. 127. [101] Cardan repeats the remark in the first chapter of the _Liber Artis Magnæ_ (_Opera_, tom. iv. p. 222). "Deceptus enim ego verbis Lucæ Paccioli, qui ultra sua capitula, generale ullum aliud esse posse negat (quanquam tot jam antea rebus a me inventis, sub manibus esset) desperabam tamen invenire, quod quærere non audebam." Perhaps he wrote them down as an apology or a defence against the storm which he anticipated as soon as Tartaglia should have seen the new Algebra. [102] Subsequently Tartaglia wrote very bitterly against Cardan, as the latter mentions in _De Libris Propriis_. "Nam etsi Nicolaus Tartalea libris materna lingua editis nos calumniatur, impudentiæ tamen ac stultitiæ suæ non aliud testimonium quæras, quam ipsos illius libros, in quibus nominatim splendidiorem unumquemque e civibus suis proscindit: adeò ut nemo dubitet insanisse hominem aliquo infortunio."--_Opera_, tom. i. p. 80. [103] _Quesiti et Inventioni_, p. 129. [104] Montucla, _Histoire de Math._ i. 596, gives a full account of Ferrari's process. [105] In the _De Vita Propria_, Cardan dismisses the matter briefly: "Ex hoc ad artem magnam, quam collegi, dum Jo. Colla certaret nobiscum, et Tartalea, à quo primum acceperam capitulum, qui maluit æmulum habere, et superiorem, quam amicum et beneficio devinctum, cum alterius fuisset inventum."--ch. xlv. p. 175. [106] Prefixed to the _De Vita Propria_. [107] In a question of broken faith, Cardan laid himself open especially to attack by reason of his constant self-glorification in the matter of veracity. [108] Leonardo knew that quadratic equations might have two positive roots, and Cardan pursued this farther by the discovery that they might also have negative roots. [109] "Caput xxviii. De capitulo generali cubi et rerum æqualium numero, Magistri Nicolai Tartagliæ, Brixiensis--Hoc capitulum habui à prefato viro ante considerationem demonstrationum secundi libri super Euclidem, et æquatio hæc cadit in [Symbol: Rx]. cu v binomii ex genere binomii secundi et qunti [m~]. [Symbol: Rx]. cuba universali recisi ejusdem binomii."--_Opera_, tom. iv. p. 341. [110] Montucla, who as a historian of Mathematics has a strong bias against Cardan, gives him credit for the discovery of the _fictæ radices_, but on the other hand he attributes to Vieta Cardan's discovery of the method of changing a complete cubic equation into one wanting the second term.--Ed. 1729, p. 595. [111] _Opera_, tom. i. p. 66. CHAPTER VI IT has been noted that Cardan quitted Pavia at the end of 1544 on account of the bankruptcy of the University, and that in 1546 a generous offer was made to him on condition of his entering the service of Pope Paul III.; an offer which after some hesitation he determined to refuse. In the autumn of this same year he resumed his teaching at Pavia, a fact which sanctions the assumption that this luckless seat of learning must have been once more in funds. In the year following, in 1547, there came to him another offer of employment accompanied by terms still more munificent than the Pope's, conveyed through Vesalius[112] and the ambassador of the King of Denmark. "The emolument was to be a salary of three hundred gold crowns per annum of the Hungarian currency, and in addition to these six hundred more to be paid out of the tax on skins of price. This last-named money differed in value by about an eighth from the royal coinage, and would be somewhat slower in coming in. Also the security for its payment was not so solid, and would in a measure be subject to risk. To this was farther added maintenance for myself and five servants and three horses. This offer I did not accept because the country was very cold and damp, and the people well-nigh barbarians; moreover the rites and doctrines of religion were quite foreign to those of the Roman Church."[113] Cardan was now forty-six years of age, a mathematician of European fame, and the holder of an honourable post at an ancient university, which he might have exchanged for other employment quite as dignified and far more lucrative. In dealing with a character as bizarre as his, it would be as a rule unprofitable to search deeply for motives of action, but in this instance it is no difficult matter to detect upon the surface several causes which may have swayed him in this decision to remain at Pavia. However firmly he may have set himself to win fame as a physician, he was in no way disposed to put aside those mathematical studies in which he had already made so distinguished a name, nor to abandon his astrology and chiromancy and discursive reading of all kinds. At Pavia he would find leisure for all these, and would in addition be able to make good any arrears of medical and magical knowledge into which he might have fallen during the years so largely devoted to the production of the _Book of the Great Art_. Moreover, the time in question was one of the prime epochs in the history of the healing art. A new light had just arisen in Vesalius, who had recently published his book, _Corporis Humani Fabrica_, and was lecturing in divers universities on the new method of Anatomy, the actual dissection of the human body. He went to Pavia in the course of his travels and left traces of his visit in the form of a revived and re-organized school of Anatomy. This fact alone would have been a powerful attraction to Cardan, ever greedy as he was of new knowledge, but there was another reason which probably swayed him more strongly still, to wit, the care of his eldest son's education and training. Gian Battista Cardano was now in his fourteenth year, and, according to the usages of the time, old enough to make a beginning of his training in Medicine, the profession he was destined to follow. It is not recorded whether or not he chose this calling for himself; but, taking into account the deep and tender affection Jerome always manifested towards his eldest son, it is not likely that undue compulsion was used in the matter. The youth, according to his father's description, strongly favoured in person his grandfather Fazio.[114] He had come into the world at a time when his parents' fortunes were at their lowest ebb, during those terrible months spent at Gallarate,[115] and in his adolescence he bore divers physical evidences of the ill nurture--it would be unjust to call it neglect--which he had received. At one time he was indeed put in charge of a good nurse, but he had to be withdrawn from her care almost immediately through her husband's jealousy, and he was next sent to a slattern, who fed him with old milk, and not enough of that; or more often with chewed bread. His body was swollen and unhealthy, he suffered greatly from an attack of fever, which ultimately left him deaf in one ear. He gave early evidence of a fine taste in music, an inheritance from his father, and was, according to Cardan's showing, upright and honest in his carriage, gifted with talents which must, under happier circumstances, have placed him in the first rank of men of learning, and in every respect a youth of the fairest promise. The father records that he himself, though well furnished by experience in the art of medicine, was now and again worsted by his son in disputation, and alludes in words of pathetic regret to divers problems, too deep for his own powers of solution, which Gian Battista would assuredly have mastered in the course of time. He does not forget to notice certain of the young man's failings; for he remarks that he was temperate of speech, except when he was angered, and then he would pour forth such a torrent of words that he scarce seemed in his right mind. Cardan professes to have discerned a cause for these failings, and the calamities flowing therefrom, in the fact that Gian Battista had the third and fourth toes of his right foot united by a membrane; he declares that, if he had known of this in time, he would have counteracted the evil by dividing the toes.[116] Gian Battista eventually gained the _baccalaureat_ in his twenty-second year, and two years after became a member of the College. The life which Cardan planned to lead at Pavia was unquestionably a full one. He had several young men under his care as pupils besides his son, amongst them being a kinsman of his, Gasparo Cardano, a youth of sterling virtue and a useful coadjutor in times to come. He was at this time engaged on his most important works in Medicine and Physical Science. He worked hard at his profession, practising occasionally and reading voraciously all books bearing on his studies. He wrote and published several small works during the four years--from 1547 to 1551--of his Professorship at Pavia; the most noteworthy of which were the Book of Precepts for the guidance of his children, and some Treatises on the Preservation of Health. He also wrote a book on Physiognomy, or as he called it Metoposcopy, an abstract of which appears as a chapter in _De Utilitate_ (lib. iii. c. 10), but the major part of his time must have been consumed in collecting and reducing to form the huge mass of facts out of which his two great works, _De Subtilitate_ and _De Varietate Rerum_, were built up. A mere abstract of the contents of these wonderful books would fill many pages, and prove as uninteresting and unsuggestive as abstracts must always be; and a commentary upon the same, honestly executed, would make a heavy draft on the working life-time of an industrious student. In reference to each book the author has left a statement of the reasons which impelled him to undertake his task, the most cogent of which were certain dreams.[117] Soon after he had begun to write the _De Astrorum Judiciis_ he dreamt one night that his soul, freed from his body, was ranging the vault of heaven near to the moon, and the soul of his father was there likewise. But he could not see this spirit, which spake to him saying, "Behold, I am given to you as a comrade." The spirit of the father then went on to tell the son how, after various stages of probation, he would attain the highest heaven, and in the terms of this discourse Cardan professed to discern the scheme of his more important works. The _De Subtilitate_ represents Cardan's original conception of a treatise dealing with the Cosmos, but during the course of its preparation a vast mass of subsidiary and contingent knowledge accumulated in his note-books, and rendered necessary the publication of a supplementary work, the _De Varietate_,[118] which, by the time it was finished, had grown to a bulk exceeding that of the original treatise. The seminal ideas which germinated and produced such a vast harvest of printed words, were substantially the same which had possessed the brains of Paracelsus and Agrippa. Cardan postulates in the beginning a certain sympathy between the celestial bodies and our own, not merely general, but distributive, the sun being in harmony with the heart, and the moon with the animal humours. He considers that all organized bodies are animated, so that what we call the Spirit of Nature is present everywhere. Beyond this everything is ruled by the properties of numbers.[119] Heat and moisture are the only real qualities in Nature, the first being the formal, and the second the material, cause of all things; these conceptions he gleaned probably from some criticisms of Aristotle on the archaic doctrines of Heraclitus and Thales as to the origin of the universe. It is no marvel that a writer, gifted with so bizarre and imaginative a temper, so restless and greedy of knowledge, sitting down to work with such a projection before him, should have produced the richest, and at the same time the most chaotic, collection of the facts of Natural Philosophy that had yet issued from the press. The erudition and the industry displayed in the gathering together of these vast masses of information, and in their verification by experiment, are indeed amazing; and, in turning over his pages, it is impossible to stifle regret that Cardan's confused method and incoherent system should have rendered his work comparatively useless for the spread of true knowledge, and qualified it only for a place among the _labores ineptiarum_. Cardan begins with a definition of Subtilty. "By subtilty I mean a certain faculty of the mind by which certain phenomena, discernible by the senses and comprehensible by the intellect, may be understood, albeit with difficulty." Subtilty, as he understood it, possesses a threefold character: substance, accident, and manifestation. With regard to the senses he admits but four to the first rank: touch, sight, smell, and hearing; the claims of taste, he affirms, are open to contention. He then passes on to discuss the properties of matter: fire, moisture, cold, dryness, and vacuum. The last-named furnishes him with a text for a discourse on a wonderful lamp which he invented by thinking out the principle of the vacuum. This digression on the very threshold of the work is a sample of what the reader may expect to encounter all through the twenty-one books of the _De Subtilitate_ and the seventeen of the _De Varietate_. Regardless of the claims of continuity, he jumps from principle to practice without the slightest warning. Intermingled with dissertations on abstract causes and the hidden forces of Nature are to be found descriptions of taps and pumps and syphons, and of the water-screw of Archimedes, the re-invention of which caused poor Galeazzo Rosso, Fazio's blacksmith friend, to go mad for joy. There are diagrams of furnaces, of machinery for raising sunken ships, and of the common steelyard. Cardan finds no problem of the universe too recondite to essay, and in like manner he sets down information as to the most trivial details of every-day economy: how to kill mice, why dogs bay the moon, how to make vinegar, why a donkey is stupid, why flint and steel produce fire, how to make the hands white, how to tell good mushrooms from bad, and how to mark household linen. He treats of the elements, Earth, Air, and Water, excluding Fire, because it produces nothing material; of the heavens and light: metals, stones, plants, and animals. Marvellous stories abound, and the most whimsical theories are advanced to account for the working of Nature. He tells how he once saw a man from Porto Maurizio, pallid, with little hair on his face, and fat in person, who had in his breasts milk enough to suckle a child. He was a soldier, and this strange property caused him no slight inconvenience. Sages, he affirms, on account of their studious lives, are little prone to sexual passion. With them the vital power is carried from the heart to a region remote from the genitals, _i.e._ to the brain, and for this reason such men as a rule beget children weak and unlike themselves. Diet has a valid effect on character, as the Germans, who subsist chiefly on the milk of wild cows, are fierce and bold and brutal. Again, the Corsicans, who eat young dogs, wild as well as domestic, are notably fierce, cruel, treacherous, fearless, nimble, and strong, following thus the nature of dogs. He argues at length to show that man is neither an animal nor a plant, but something between the two. A man is no more an animal than an animal is a plant. The animal has the _anima sensitiva_ which the plant lacks, and man transcends the animal through the gift of the _anima intellectiva_, which, as Aristotle testifies, differs from the _sensitiva_. Some maintain that man and the animals must be alike in nature and spirit, because it is possible for man to catch certain diseases from animals. But animals take certain properties from plants, and no one thinks of calling an animal a plant. Man's nature is threefold: the Divine, which neither deceives nor is deceived; the Human, which deceives, but is not deceived; the Brutish, which does not deceive, but is deceived. Dissertations on the various sciences, the senses, the soul and intellect, things marvellous, demons and angels, occupy the rest of the chapters of the _De Subtilitate_. At the end of the last book of _De Varietate_, Cardan gives a table showing the books of the two works arranged in parallel columns so as to exhibit the relation they bear to each other. A comparison of the treatment accorded to any particular branch of Natural Philosophy in the _De Subtilitate_ with that given in the _De Varietate_, will show that in the last-named work Cardan used his most discursive and anecdotic method. Mechanics are chiefly dealt with in the _De Subtilitate_, and all through this treatise he set himself to observe in a certain degree the laws of proportion, and kept more or less to the point with which he was dealing, a system of treatment which left him with a vast heap of materials on his hands, even after he had built up the heavy tome of the _De Subtilitate_. Perhaps when he began his work upon the fresh volume he found this _ingens acervus_ too intractable and heterogeneous to be susceptible of symmetrical arrangement, and was forced to let it remain in confusion. Few men would sit down with a light heart to frame a well-ordered treatise out of the _débris_ of a heap of note-books, and it would be unjust to censure Cardan's literary performance because he failed in this task. Probably no other man living in his day would have achieved a better result. It is certain that he expended a vast amount of labour in attempting to reduce his collected mass of facts even to the imperfect form it wears in the _De Varietate Rerum_.[120] Considering that this book covers to a great extent the same ground as its predecessor, Cardan must be credited with considerable ingenuity of treatment in presenting his supplementary work without an undue amount of repetition. In the _De Varietate_ he always contrives to bring forward some fresh fact or fancy to illustrate whatever section of the universe he may have under treatment, even though this section may have been already dealt with in the _De Subtilitate_. The characteristic most strongly marked in the later book is the increased eagerness with which he plunges into the investigation of certain forces, which he professes to appreciate as lying beyond Nature, and incapable of scientific verification in the modern sense, and the fabled manifestations of the same. He loses no opportunity of trying to peer behind the curtain, and of seeking--honestly enough--to formulate those various pseudo-sciences, politely called occult, which have now fallen into ridicule and disrepute with all except the charlatan and the dupe, who are always with us. Where he occupies in the _De Subtilitate_ one page in considering those things which lie outside Nature--demons, ghosts, incantations, succubi, incubi, divinations, and such like--he spends ten in the _De Varietate_ over kindred subjects. There is a wonderful story[121] told by his father of a ghost or demon which he saw in his youth while he was a scholar in the house of Giovanni Resta at Pavia. He searches the pages of Hector Boethius, Nicolaus Donis, Rugerus, Petrus Toletus, Leo Africanus, and other chroniclers of the marvellous, for tales of witchcraft, prodigies, and monstrous men and beasts, and devotes a whole chapter to chiromancy,[122] a subject with which he had occupied his plenteous leisure when he was waiting for patients at Sacco. The diagram of the human hand given by him does not differ greatly from that of the contemporary hand-books of the "Art," and the leading lines are just the same. The heavenly bodies are as potent here as in Horoscopy. The thumb is given to Mars, the index finger to Jupiter, the middle finger to Saturn, the ring finger to the Sun, and the little finger to Venus. Each finger-joint has its name, the lowest being called the procondyle, the middle the condyle, and the upper the metacondyle. He passes briefly over as lines of little import, the _via combusta_ and the _Cingulus Orionis_, but lays some stress on the character of the nails and the knitting together of the hand, declaring that hands which can be bent easily backward denote effeminacy or a rapacious spirit. He teaches that lines are most abundant in the hands of children, on account of the tenderness of the skin, and of old men on account of the dryness, a statement which might suggest the theory that lines come into existence through the opening and closing of the hand. But the adoption of this view would have proved more disastrous to chiromancy than ridicule or serious criticism; so he straightway finds an explanation for this fact in the postulate that lines in young people's hands speak as to the future, and in old men's as to the past. Later he goes on to affirm that lines in the hand cannot be treated as mere wrinkles arising from the folding of the skin, unless we are prepared to admit that wrinkled people are more humorous than others, alluding no doubt to the lines in the face caused by laughter, a proposition which does not seem altogether convincing or consequential, unless we also postulate that all humorous men laugh at every joke. There is a line in the hand which he calls the _linea jecoraria_, and the triangle formed by this and the _linea vitæ_ and the _linea cerebri_, rules the disposition of the subject, due consideration being given to the acuteness or obtuseness of the angles of this triangle. Cardan seems to have based his treatise on one written by a certain Ruffus Ephesius, and it is without doubt one of the dullest portions of his work.[123] It is almost certain that Cardan purposed to let the _De Varietate_ come forth from the press immediately after the _De Subtilitate_, but before the MS. was ready, it came to pass that he was called to make that memorable journey to Scotland in order to find a remedy for the ailment which was troubling the Archbishop of St. Andrews, a journey which has given to Britons a special interest in his life and work. In dealing with the Cosmos in the _De Subtilitate_ he had indeed made brief mention of Britain; but, writing then, he had no personal relations with either England or Scotland, or the people thereof; and, but for his subsequent visit, he would not have been able to set down in the pages of his second book so many interesting and suggestive notes of what he had seen and heard, and his ideas of the politics of the time. Again, if he had not been urged by the desire all men feel to read what others may have to say about places they have visited, it is not likely that he would have searched the volumes of Hector Boethius and other early writers for legends and stories of our island. Writing of Britain[124] in the _De Subtilitate_ he had praised its delicate wool and its freedom from poisonous beasts: a land where the wolf had been exterminated, and where the sheep might roam unvexed by any beast more formidable than the fox. The inordinate breeding of rooks seems even in those days[125] to have led to a war of extermination against them, carried on upon a system akin to that which was waged against the sparrow in the memory of men yet living. But besides this one, he records, in the _De Subtilitate_, few facts concerning Britain. He quotes the instances of Duns Scotus and Suisset in support of the view that the barbarians are equal to the Italians in intellect,[126] and he likewise notices the use of a fertilizing earth--presumably marl--in agriculture,[127] and the longevity of the people, some of whom have reached their hundred and twentieth year.[128] The first notice of us in the _De Varietate_ is in praise of our forestry, forasmuch as he remarked that the plane tree, which is almost unknown in Italy through neglect, thrives well in Scotland, he himself having seen specimens over thirty feet high growing in the garden of the Augustinian convent near Edinburgh. The lack of fruit in England he attributes rather to the violence of the wind than to the cold; but, in spite of our cruel skies, he was able to eat ripe plums in September, in a district close to the Scottish border. He bewails the absence of olives and nuts, and recommends the erection of garden-walls in order to help on the cultivation of the more delicate fruits. In a conversation with the Archbishop of St. Andrews he was told that the King of Scots ruled over one hundred and sixty-one islands, that the people of the Shetland Islands lived for the most part on fish prepared by freezing or sun-drying or fire, and had no other wealth than the skins of beasts. Cardan pictures the Shetlanders of that time as leading an ideal life, unvexed by discord, war, or ambition, labouring in the summer for the needs of winter, worshipping Christ, visited only once a year by a priest from Orkney, who came over to baptize the children born within the last twelve months, and was remunerated by a tenth of the catch of fish. He speaks of the men of Orkney as a very lively, robust, and open-hearted crew, furnished with heads strong enough to defy drunkenness, even after swallowing draughts of the most potent wine. The land swarms with birds, and the sheep bring forth two or even three lambs at a time. The horses are a mean breed, and resemble asses both as to their size and their patience. Some one told him of a fish, often seen round about the islands, as big or even bigger than a horse, with a hide of marvellous toughness, and useful for the abundance of oil yielded by its carcase. He attributes the bodily strength of these northerners to the absence of four deleterious influences--drunkenness, care, heat, and dry air. Cardan seems to have been astonished at the wealth of precious stones he found in Scotland--dark blue stones, diamonds, and carbuncles[129]--"maxime juxta academiam Glaguensis oppidi in Gludisdalia regione," and he casts about to explain how it is that England produces nothing of the kind, but only silver and lead. He solves the question by laying down an axiom that the harder the environment, the harder the stone produced. The mountains of Scotland are both higher and presumably harder than those of England, hence the carbuncles. He was evidently fascinated with the wealth of local legend and story which haunted the misty regions he visited. In dealing with demons and familiar spirits he cites the authority of Merlin, "whose fame is still great in England," and tells a story of a young woman living in the country of Mar.[130] This damsel was of noble family and very fair in person, but she displayed a great unwillingness to enter the marriage state. One day it was discovered that she was pregnant, and when the parents went to make inquisition for the seducer, the girl confessed that, both by day and night, a young man of surpassing beauty used to come and lie with her. Who he was and whence he came she knew not. They, though they gave little credit to her words, were informed by her handmaid, some three days afterwards, that the young man was once more with her; wherefore, having broken open the door, they entered, bearing lights and torches, and beheld, lying in their daughter's arms, a monster, fearsome and dreadful beyond human belief. All the neighbours ran quickly to behold the grisly sight, and amongst them a good priest, well acquainted with pagan rites. When he had come anear, and had said some verses of the Gospel of Saint John, the fiend vanished with a terrible noise, bearing away the roof of the chamber, and leaving the bed in flames. In three days' time the girl gave birth to a monstrous child, more hideous than anything heretofore seen in Scotland, wherefore the nurses, to keep off disgrace from the family, caused it to be burnt on a pile of wood. There is another story of a youth living about fourteen miles from Aberdeen, who was visited every night by a demon lady of wonderful loveliness, though he bolted and locked his chamber-door; but by fasting and praying and keeping his thoughts fixed on holy things he rid himself at last of the unclean spirit.[131] He quotes from Boethius the whole story of Macbeth,[132] and tells how "Duffus rex" languished and wasted under the malefic arts of certain witches who made an image of the king in wax and, by using various incantations, let the same melt slowly away before the fire. The unhappy king came near to die, but, as soon as these nefarious practices were discovered, the image was destroyed, whereupon the king was restored to health.[133] When Cardan received the first letter from Scotland the manuscript of the _De Varietate_ must have been ready or nearly ready for the printer; but, for some reason or other, he determined to postpone the publication of the work until he should have finished with the Archbishop, and took his manuscript with him when he set forth on his travels. In 1550 there came another break in Cardan's life as Professor at Pavia, the reason being the usual one of dearth of funds.[134] In 1551 he went back for a short time, but the storms of war were rising on all sides, and the luckless city of Pavia was in the very centre of the disturbance. The French once more crossed the Alps, pillaging and devastating the country, their ostensible mission being the vindication of the rights of Ottavio Farnese to the Duchy of Parma. Ottavio had quarrelled with Pope Julius III., who called upon the Emperor for assistance. War was declared, and Charles set to work to annex Parma and Piacenza as well to the Milanese. Cardan withdrew to Milan at the end of the year. Gian Battista had now completed his medical course, so there was now no reason why he should continue to live permanently at Pavia. Moreover at this juncture he seems to have been strongly moved to augment the fame which he had already won in Mathematics and Medicine by some great literary achievement, and he worked diligently with this object in view.[135] At the beginning of November 1551, a letter came to him from Cassanate,[136] a Franco-Spanish physician, who was at that time in attendance upon the Archbishop of St. Andrews, requesting him to make the journey to Paris, and there to meet the Archbishop, who was suffering from an affection of the lungs. The fame of Cardan as a physician had spread as far as Scotland, and the Archbishop had set his heart on consulting him. Cassanate's letter is of prodigious length. After a diffuse exordium he proceeds to praise in somewhat fulsome terms the _De Libris Propriis_ and the treatises _De Sapientia_[137] and _De Consolatione_, which had been given to him by a friend when he was studying at Toulouse in 1549. He had just read the _De Subtilitate_, and was inflamed with desire to become acquainted with everything which Cardan had ever written. But what struck Cassanate more than anything was a passage in the _De Sapientia_ on a medical question, which he extracts and incorporates in his epistle. Cardan writes there: "But if my profession itself will not give me a living, nor open out an avenue to some other career, I must needs set my brains to work, to find therein something unknown hitherto, for the charm of novelty is unfailing, something which would prove of the highest utility in a particular case. In Milan, while I was fighting the battle against hostile prejudice, and was unable to earn enough to pay my way (so much harder is the lot of manifest than of hidden merit, and no man is honoured as a prophet in his own country), I brought to light much fresh knowledge, and worked my hardest at my art, for outside my art there was naught to be done. At last I discovered a cure for phthisis, which is also known as Phthoe, a disease for many centuries deemed incurable, and I healed many who are alive to this day as easily as I have cured the _Gallicus morbus_. I also discovered a cure for intercutaneous water in many who still survive. But in the matter of invention, Reason will be the leader, but Experiment the Master, the stimulating cause of work in others. If in any experiment there should seem to be an element of danger, let it be performed gently, and little by little."[138] It is not wonderful that the Archbishop, who doubtless heard all about Cardan's asserted cure of phthisis from Cassanate, should have been eager to submit his asthma to Cardan's skill. After acknowledging the deep debt of gratitude which he, in common with the whole human race, owed to Cardan in respect to the two discoveries aforesaid, Cassanate comes to the business in hand, to wit, the Archbishop's asthma. Not content with giving a most minute description of the symptoms, he furnishes Cardan also with a theory of the operations of the distemper. He writes: "The disease at first took the form of a distillation from the brain into the lungs, accompanied with hoarseness, which, with the help of the physician in attendance, was cured for a time, but the temperature of the brain continued unfavourable, being too cold and too moist, so that certain unhealthy humours were collected in the head and there remained, because the brain could neither assimilate its own nutriment, nor disperse the humours which arose from below, being weakened through its nutriment of pituitous blood. After an attack of this nature it always happened that, whenever the body was filled with any particular matter, which, in the form of substance, or vapour, or quality, might invade the brain, a fresh attack would certainly arise, in the form of a fresh flow of the same humour down to the lungs. Moreover these attacks were found to agree almost exactly with the conjunctions and oppositions of the moon."[139] Cassanate goes on to say that his patient had proved somewhat intractable, refusing occasionally to have anything to do with his medical attendants, and that real danger was impending owing to the flow of humour having become chronic. Fortunately this humour was not acrid or salt; if it were, phthisis must at once supervene. But the Archbishop's lungs were becoming more and more clogged with phlegm, and a stronger effort of coughing was necessary to clear them. Latterly much of the thick phlegm had adhered to the lungs, and consequently the difficulty of breathing was great. Cassanate declares that he had been able to do no more than to keep the Archbishop alive, and he fears no one would be able to work a complete cure, seeing that the air of Scotland is so moist and salt, and that the Archbishop is almost worried to death by the affairs of State. He next urges Cardan to consent to meet the Archbishop in Paris, a city in which learning of all sorts flourishes exceedingly, the nurse of many great philosophers, and one in which Cardan would assuredly meet the honour and reverence which is his due. The Archbishop's offer was indeed magnificent in its terms. Funds would be provided generous enough to allow the physician to travel post the whole of the journey, and the goodwill of all the rulers of the states _en route_ would be enlisted in his favour. Cassanate finishes by fixing the end of January 1552 as a convenient date for the _rendezvous_ in Paris, and, as time and place accorded with Cardan's wishes, he wrote to Cassanate accepting the offer. The Archbishop of St. Andrews was John Hamilton, the illegitimate brother of James, Earl of Arran, who had been chosen Regent of the kingdom after the death of James V. at Flodden, and the bar sinister, in this case as in many others, was the ensign of a courage and talent and resource in which the lawful offspring was conspicuously wanting. Any student taking a cursory glance at the epoch of violence and complicated intrigue which marked the infancy of Mary of Scotland, may well be astonished that a man so weak and vain and incompetent as James Hamilton--albeit his footing was made more secure by his position as the Queen's heir-presumptive--should have held possession of his high dignities so long as he did. Alternately the tool of France and of England, he would one day cause his great rival Cardinal Beatoun to be proclaimed an enemy of his country, and the next would meet him amicably and adopt his policy. After becoming the partisan of Henry VIII. and the foe of Rome, he finally put the coping-stone to his inconsistencies by becoming a convert to Catholicism in 1543. But in spite of his indolence and weakness, he was still Regent of Scotland, when his brother, the Archbishop, was seized with that attack of periodic asthma which threatened to change vitally the course of Scottish politics. A very slight study of contemporary records will show that Arran had been largely, if not entirely, indebted to the distinguished talents and to the ambition of his brother for his continued tenure of the chief power of the State. If confirmation of this view be needed, it will be found in the fact that, as soon as the Archbishop was confined to a sick-room, Mary of Guise, the Queen Mother, supported by her brothers in France and by the Catholic party at home, began to undermine the Regent's position by intrigue, and ultimately, partly by coaxing, partly by threats, won from him a promise to surrender his power into her hands. In the meantime Cardan was waiting for further intelligence and directions as to his journey. The end of January had been fixed as the date of the meeting at Paris, and it was not until the middle of February that any further tidings came to him. Then he received a letter from Cassanate and a remittance to cover the expenses of his journey.[140] He set out at once on February 22, undaunted by the prospect of a winter crossing of the Simplon, and, having travelled by way of Sion and Geneva, arrived at Lyons on March 13. In Cassanate's first letter Paris had been named as the place of meeting; but, as a concession to Cardan's convenience, Lyons was added as an alternative, in case he should find it impossible to spare time for a longer journey. Cardan accordingly halted at Lyons, but neither Archbishop nor physician was there to meet him. After he had waited for more than a month, Cassanate appeared alone, and brought with him a heavy purse of money for the cost of the long journey to Scotland, which he now begged Cardan to undertake, and a letter from the Archbishop himself, who wrote word that, though he had fully determined in the first instance to repair to Paris, or even to Lyons, to meet Cardan, he found himself at present mastered by the turn of circumstances, and compelled to stay at home. He promised Cardan a generous reward, and a reception of a nature to convince him that the Scots are not such Scythians as they might perchance be deemed in Milan.[141] Cardan's temper was evidently upset by this turn of affairs, and his suspicions aroused; for he sets down his belief that patient and physician had from the first worked with the intention of dragging him all the way to Scotland, but that they had waited till he was across the Alps before showing their hand, fearing lest if the word Scotland should have been used at the outset, he would never have moved from Milan.[142] In describing his journey he writes:--"I tarried in Lyons forty-six days, seeing nothing of the Archbishop, nor of the physician whom I expected, nevertheless I gained more than I spent. I met there Ludovico Birago, a gentleman of Milan, and commander of the King's foot-soldiers, and with him I contracted a close friendship, so much so that, had I been minded to take service under Brissac, the King's lieutenant, I might have enjoyed a salary of one thousand crowns a year. Shortly afterwards Guglielmo Cassanate, the Archbishop's physician, arrived in Lyons and brought with him three hundred other golden crowns, which he handed to me, in order that I might make the journey with him to Scotland, offering in addition to pay the cost of travel, and promising me divers gifts in addition. Thus, making part of our journey down the Loire, I arrived at Paris. While I was there I met Orontius; but he for some reason or other refused to visit me. Under the escort of Magnienus[143] I inspected the treasury of the French Kings, and the Church of Saint Denis. I saw likewise something there, not so famous, but more interesting to my mind, and this was the horn of a unicorn, whole and uninjured. After this we met the King's physicians, and we all dined together, but I declined to hold forth to them during dinner, because before we sat down they were urgent that I should begin a discussion. I next set forth on my journey, my relations with Pharnelius and Silvius, and another of the King's physicians,[144] whom I left behind, being of a most friendly nature, and travelled to Boulogne in France, where, by the command of the Governor of Sarepont, an escort of fourteen armed horsemen and twenty foot-soldiers was assigned to me, and so to Calais. I saw the tower of Cæsar still standing. Then having crossed the narrow sea I went to London, and at last met the Archbishop at Edinburgh on the twenty-ninth of June. I remained there till the thirteenth of September. I received as a reward four hundred more gold crowns; a chain of gold worth a hundred and twenty crowns, a noble horse, and many other gifts, in order that no one of those who were with me should return empty-handed."[145] The Archbishop's illness might in itself have supplied a reason for his inability to travel abroad and meet Cardan as he had agreed to do; but the real cause of his change of plan was doubtless the condition of public affairs in Scotland at the beginning of 1552. In the interval of time between Cassanate's first letter to Cardan and the end of 1551, the Regent had half promised to surrender his office into the hands of the Guise party in Scotland, wherefore it was no wonder that the Primate, recognizing how grave was the danger which threatened the source of his power, should have resolved that, sick or sound, his proper place was at the Scottish Court. FOOTNOTES: [112] Vesalius had certainly lectured on anatomy at Pavia, but it would appear that Cardan did not know him personally, seeing that he writes in _De Libris Propriis_ (_Opera_, tom. i. p. 138): "Brasavolum ... nunquam vidi, ut neque Vesalium quamquam intimum mihi amicum." [113] _De Vita Propria_, ch. xxxii. p. 99. [114] In describing Fazio, Jerome writes: "Erat Euclidis operum studiosus, et humeris incurvis: et filius meus natu major ore, oculis, incessu, humeris, illi simillimus."--_De Vita Propria_, ch. iii. p. 8. In the same chapter Fazio is described as "Blæsus in loquendo; variorum studiorum amator: ruber, oculis albis et quibus noctu videret." [115] "At uxor mea imaginabatur assidue se videre calvariam patris, qui erat absens dum utero gereret Jo: Baptistam."--_Paralipomenon_, lib. iii. c. 21. [116] _De Utilitate_, p. 832. [117] "Post ex geminatis somniis, scripsi libros de Subtilitate quos impressos auxi et denuo superauctos tertio excudi curavi."--_De Vita Propria_, ch. xlv. p. 175. [118] "Libros de Rerum varietate anno MDLVIII edidi: erant enim reliquiæ librorum de subtilitate."--_De Vita Propria_, p. 176. "Reversus in patriam, perfeci libros XVII de Rerum varietate quos jampridem inchoaveram."--_Opera_, tom. i. p. 110. He had collected much material during his life at Gallarate. [119] Aristotle, _Metaphysics_, book I. ch. v., contains an examination of the Pythagorean doctrine which maintains Number to be the Substance of all things:--[Greek: all' auto to apeiron kai auto to hen ousian einai toutôn ôn katêgorountai.] [120] "Sed nullus major labor quam libri de Rerum Varietate quem cum sæpius mutassem, demum traductis quibuscunque insignioribus rebus in libros de Subtilitate, ita illum exhausi, ut totus denuo conscribendus fuerit atque ex integro restituendus."--_Opera_, tom. i. p. 74. He seems to have utilized the services of Ludovico Ferrari in compiling this work.--_Opera_, tom. i. p. 64. [121] _De Varietate_, p. 661. [122] Book XV. ch. lxxix. [123] He gives one example of his skill as a palmist in the _De Vita Propria_: "Memini me dum essem adolescens, persuasum fuisse cuidam Joanni Stephano Biffo, quod essem Chiromanticus, et tamen nil minus: rogat ille, ut prædicam ei aliquid de vita; dixi delusum esse a sociis, urget, veniam peto si quicquam gravius prædixero: dixi periculum imminere brevi de suspendio, intra hebdomadam capitur, admovetur tormentis: pertinaciter delictum negat, nihilominus tandem post sex menses laqueo vitam finivit."--ch. xlii. p. 156. [124] "Ergo nunc Britannia inclyta vellere est. Nec mirum cum null[u=] animal venenat[u=] mittat, imò nec infestum præter vulpem, olim et lupum: nunc vero exterminatis etiam lupis, tutò pecus vagat. Rore coeli sitim sedant greges, ab omni alio potu arcentur, quod aquæ ibi ovibus sint exitiales: quia tamen in pabulo humido vermes multi abundant, cornic[u=] adeo multitudo crevit, ut ob frugum damna nuper publico consilio illas perdentibus proposita præmia sint: ubi enim pabulum, ibi animalia sunt quæ eo vescuntur, atque immodicè tunc multiplicantur cum ubique abundaverit. Caret tamen ut dixi, serpentibus, tribus ex causis: nam pauci possunt generari ob frigus immensum."--_De Subtilitate_, p. 298. [125] Æneas Sylvius in describing his visit to Britain a century earlier says that rooks had been recently introduced, and that the trees on which they roosted and built belonged to the King's Exchequer. [126] "Ejusdem insulæ accola fuit Ioannes, ut dixi, Suisset [Richard Swineshead] cognom[e=]to Calculator; in cujus solius unius argumenti solutione, quod contra experiment[u=] est de actione mutua tota laboravit posteritas; quem senem admodum, nec inventa sua dum legeret intelligentem, flevisse referunt. Ex quo haud dubium esse reor, quod etiam in libro de animi immortalite scripsi, barbaros ingenio nobis haud esse inferiores: quandoquidem sub Brumæ cælo divisa toto orbe Britannia duos tam clari ingenii viros emiserit."--_De Subtilitate_, p. 444. [127] _Ibid.,_ p. 142. [128] p. 369. [129] The fame of Scots as judges of precious stones had spread to Italy before Cardan's time. In the _Novellino_ of Masuccio, which was first printed in 1476, there is a passage in the tenth novel of the first part, in which a rogue passes as "grandissimo cognoscitore" of gems because he had spent much time in Scotland. [130] _De Varietate_, p. 636. [131] _De Varietate_, p. 637. [132] _Ibid.,_ p. 637. [133] _Ibid.,_ p. 565. [134] "Peracto L anno quod stipendium non remuneraretur mansi Mediolani."--_De Vita Propria_, ch. iv. p. 15. [135] About this time he wrote the _Liber Decem Problematum_, and the treatise _Delle Burle Calde_, one of his few works written in Italian.--_Opera_, tom. i. p. 109. [136] Cassanate's letter is given in full (_Opera_, tom. i. p. 89). [137] The quotation from the _De Sapientia_ differs somewhat from the original passage which stands on p. 578 of the same volume. [138] _Opera_, tom. i. p. 89. [139] In a subsequent interview with Cardan, Cassanate modifies this statement.--_Opera_, tom. ix. p. 124. [140] "Accepique antequam discederem aureos coronatos Gallicos 500 et M.C.C. in reditu."--_De Vita Propria_, ch. iv. p. 16. [141] "Difficillimis causis victus venire non potui." The Archbishop's letter is given in _Opera_, tom. i. p. 137. [142] _Geniturarum Exempla_, p. 469. [143] He mentions this personage in _De Varietate_, p. 672: "Johannes Manienus medicus, vir egregius et mathematicaram studiosus." He was physician to the monks of Saint Denis. [144] The reception given to Cardan in Paris was a very friendly one. Orontius was a mechanician and mathematician; and jealousy of Cardan's great repute may have kept him away from the dinner, but the physicians were most hospitable. Pharnelius [Fernel] was Professor of Medicine at the University, and physician to the Court. Sylvius was an old man of a jocular nature, but as an anatomist bitterly opposed to the novel methods of Vesalius, who was one of Cardan's heroes. With this possibility of quarrelling over the merits of Vesalius, it speaks well for the temper of the doctors that they parted on good terms. Ranconet, another Parisian who welcomed Cardan heartily, was one of the Presidents of the Parliament of Paris. He seems to have been a man of worth and distinguished attainments, and Cardan gives an interesting account of him in _Geniturarum Exempla_, p. 423. [145] _De Vita Propria_, ch. xxix. p. 75. Cardan refers more than once to the generosity of the Archbishop. He computes (_Opera_, tom. i. p. 93) that his visit must have cost Hamilton four talents of gold; that is to say, two thousand golden crowns. CHAPTER VII CARDAN, as he has himself related, arrived at Edinburgh on June 29, 1552. The coming of such a man at such a time must have been an event of extraordinary interest. In England the Italy of the Renaissance had been in a measure realized by men of learning and intellect through the reports of the numerous scholars--John Tiptoft, Earl of Worcester, Henry Parker, Lord Morley, Howard Earl of Surrey, and Sir Thomas Wyat, may be taken as examples--who had wandered thither and come back with a stock of histories setting forth the beauty and charm, and also the terror and wickedness, of that wonderful land. Some echoes of this legend had doubtless drifted down to Scotland, and possibly still more may have been wafted over from France. Ascham had taken up his parable in the _Schoolmaster_, describing the devilish sins and corruptions of Italy, and now the good people of Edinburgh were to be given the sight of a man coming thence, one who was fabled to have gathered together more knowledge, both of this world and of that other hidden one which was to them just as real, than any mortal man alive. Under these circumstances it is not surprising that Cardan should have been regarded rather as a magician than as a doctor, and in the _Scotichronicon_[146] it is recorded that the Primate was cured of a lingering asthma by the incantations of an astrologer named Cardan, from Milan. Cardan in his narrative speaks of Edinburgh as the place where he met his patient, and does not mention any other place of sojourn, but the record just quoted goes on to say that he abode with the Primate for eleven weeks at his country residence at Monimail, near Cupar, Fife, where there is a well called to this day Cardan's Well. Cardan, as it has been noticed already, refused to commit himself to any opinion as to the character of the Archbishop's distemper over the dinner-table where he and Cassanate had been entertained by the French King's physicians. Cassanate had set forth his views in full as to the nature of the asthma which had to be dealt with in his letter to Cardan, and it is highly probable that he would again bring forward these views in the hearing of the Paris doctors. It is certain that some of the French physicians had, previous to this, prescribed a course of treatment for the Archbishop, probably without seeing him, and that the course was being tried when Cardan arrived in Edinburgh.[147] For the first six weeks of his stay he watched the case, and let the treatment aforesaid go on--whether it differed from that which Cassanate recommended or not there is no evidence to show. But no good result came of it. The Archbishop wasted in body and became fretful and disturbed in mind, and, at last, Cardan was obliged to let his opinion of the case be known; and, as this was entirely hostile to the treatment which was being pursued, the inevitable quarrel between the doctors burst forth with great violence. The Archbishop was irate with his ordinary medical attendant, probably the physician who was left in charge during Cassanate's absence--and this man retaliated upon Cardan for having thus stirred up strife. Cardan's position was certainly a very uneasy one. The other physicians were full of jealousy and malice, and the Archbishop began to accuse him of dilatory conduct of the case, redoubling his complaints as soon as he found himself getting better under the altered treatment. So weary did Cardan become of this bickering that he begged leave to depart at once, but this proposition the Primate took in very ill part. Cassanate in his first diagnosis had traced the Archbishop's illness to an excess of coldness and humidity in the brain. Now Cardan, on the other hand, maintained that the brain was too hot. He found Cassanate's treatment too closely fettered by his theory as to the causes of periodic asthma, but he did not venture to exhibit his own course of treatment till after he had gained some knowledge of the Archbishop's temper and habits. He came to the conclusion that his patient was overwrought with the cares of State, that he ate too freely, that he did not sleep enough, and that he was of a temper somewhat choleric. Cardan set forth this view of the case in a voluminous document, founding the course of treatment he proposed to pursue upon the aphorisms of Galen. He altogether rejected Cassanate's view as to the retention of the noxious humours in the head. The Archbishop had the ruddy complexion of a man in good health, a condition which could scarcely co-exist with the loading of the brain with matter which would certainly putrify if retained for any long time. Cardan maintained that the serous humour descended into the lungs, not by the passages, but by soaking through the membranes as through linen.[148] After describing the origin and the mode of descent of this humour, he goes on to search for an auxiliary cause of the mischief, and this he finds in the imperfect digestive powers of the stomach and liver. If the cause lay entirely in the brain, how was it that all the cerebral functions were not vitiated? In fine, the source of the disease lay, not in the weakness of the brain, but in an access of heat, caused possibly by exposure to the sun, by which the matter of the brain had become so rarefied that it showed unhealthy activity in absorbing moisture from the other parts. This heat, therefore, must be reduced. To accomplish this end three lines of treatment must be followed. First, a proper course of diet; second, drugs; and third, certain manual operations. As to diet, the Archbishop was ordered to take nothing but light and cooling food, two to four pints of asses' milk in the early morning, drawn from an ass fed on cooling herbs, and to use all such foods as had a fattening tendency; tortoise or turtle-soup,[149] distilled snails, barley-water and chicken-broth, and divers other rich edibles. The purging of the brain was a serious business; it was to be compassed by an application to the coronal suture of an ointment made of Greek pitch, ship's tar, white mustard, euphorbium, and honey of anathardus: the compound to be sharpened, if necessary, by the addition of blister fly, or rendered less searching by leaving out the euphorbium and mustard. Cardan adds, that, by the use of this persuasive application, he had sometimes brought out two pints of water in twenty-four hours. The use of the shower-bath and plentiful rubbing with dry cloths was also recommended. The purging of the body was largely a question of diet. To prevent generation of moisture, perfumes were to be used; the patient was to sleep on raw silk and not upon feathers, and to let an hour and a half come between supper and bed-time. Sleep, after all, was the great thing to be sought. The Archbishop was counselled to sleep from seven to ten hours, and to subtract time from his studies and his business and add the same to sleep.[150] Cardan's treatment, which seems to have been suggested as much by the man of common-sense as by the physician, soon began to tell favourably upon the Archbishop. He remained for thirty-five days in charge of his patient, during which time the distemper lost its virulence and the patient gained flesh. In the meantime the fame of his skill had spread abroad, and well-nigh the whole nobility of Scotland flocked to consult him,[151] and they paid him so liberally that on one day he made nineteen golden crowns. But when winter began to draw near, Cardan felt that it was time to move southward. He feared the cold; he longed to get back to his sons, and he was greatly troubled by the continued ill-behaviour of one of the servants he had brought with him--"maledicus, invidus, avarissimus, Dei contemptor;" but he found his patient very loth to let him depart. The Archbishop declared that his illness was alleviated but not cured, and only gave way unwillingly when Cardan brought forward arguments to show what dangers and inconveniences he would incur through a longer stay. Cardan had originally settled to return by way of Paris, but letters which he received from his young kinsman, Gasparo Cardano, and from Ranconet, led him to change his plans. The country was in a state of anarchy, the roads being infested with thieves, and Gasparo himself had the bad fortune to be taken by a gang of ruffians. In consequence of these things Cardan determined to return by way of Flanders and the Empire. It was not in reason that Cardan would quit Scotland and resign the care of his patient without taking the stars into his counsel as to the future. He cast the Archbishop's horoscope, and published it in the _Geniturarum Exempla_. It was not a successful feat. In his forty-eighth year, _i.e._ in 1560, the astrologer declared that Hamilton would be in danger of poison and of suffering from an affection of the heart. But the time of the greatest peril seemed to lie between July 30 and September 21, 1554. The stars gave no warning of the tragic fate which befell Archbishop Hamilton in the not very distant future. For the succeeding six years he governed the Church in Scotland with prudence and leniency, but in 1558 he began a persecution of the reformers which kindled a religious strife, highly embarrassing to the Catholic party then holding the reins of power. His cruelties were borne in mind by the reformers when they got the upper hand. In 1563 he was imprisoned for saying mass. In 1568 Mary, after her escape from Loch Leven, gave the chief direction of her affairs into the hands of the Archbishop, who was the bitter foe of the Regent Murray. Murray having defeated the Queen's forces at Langside, Hamilton took refuge in Dumbarton Castle, which was surprised and captured in 1571, when the Archbishop was taken to Stirling and hanged. In the words of the _Diurnal of Occurrants_: "as the bell struck six hours at even, he was hangit at the mercat cross of Stirling upon a jebat."[152] His enemies would not let him rest even there, for the next day, fixed to the tree, were found the following verses: "Cresce diu, felix arbor, semperque vireto Frondibus ut nobis talia poma feras." To return to Cardan. Having at last won from his patient leave to depart, he set forth laden with rich gifts. In Scotland, Cardan found the most generous paymasters he had ever met. In recording the niggard treatment which he subsequently experienced at the hands of Brissac, the French Viceroy, he contrasts it with the liberal rewards granted to him in what must then have been the poorest of the European kingdoms;[153] and in the Preface of the _De Astrorum Judiciis_ (Basel, 1554) he writes in sympathetic and grateful terms of the kind usage he had met in the North.[154] It must have been a severe disappointment to him that he was unable to revisit Paris on his way home, for letters from his friend Ranconet told him that a great number of illustrious men had proposed to repair to Paris for the sake of meeting him; and many of the nobles of France were anxious to consult him professionally, one of them offering a fee of a thousand gold crowns. But Cardan was so terrified by the report given by Gasparo of the state of France, that he made up his mind he would on no account touch its frontiers on his homeward journey. Before he quitted Scotland there had come to him letters from the English Court entreating him to tarry there some days on his way home to Italy, and give his opinion on the health of Edward VI., who was then slowly recovering from an attack of smallpox and measles. The young King's recovery was more apparent than real, for he was, in fact, slowly sinking under the constitutional derangement which killed him a few months later. Cardan could hardly refuse to comply with this request, nor is there any evidence to show that he made this visit to London unwillingly. But he soon found out that those about the Court were anxious to hear from him something more than a statement of his opinion as to Edward's health. They wanted, before all else, to learn what the stars had to say as to the probable duration of the sovereign's life. During his stay in Scotland Cardan would certainly have gained some intelligence of the existing state of affairs at the English Court; how in the struggle for the custody of the regal power, the Lord High Admiral and the Lord Protector, the King's uncles, had lost their heads; and how the Duke of Northumberland, the son of Dudley, the infamous minion of Henry VII. and the destroyer of the ill-fated Seymours, had now gathered all the powers and dignities of the kingdom into his own hands, and was waiting impatiently for the death of Edward, an event which would enable him to control yet more completely the supreme power, through the puppet queen whom he had ready at hand to place upon the throne. An Italian of the sixteenth century, steeped in the traditions of the bloody and insidious state-craft of Milan and the Lombard cities, Cardan would naturally shrink from committing himself to any such perilous utterance: all the more for the reason that he had already formed an estimate of the English as a fierce and cruel people. With his character as a magician to maintain he could scarcely keep entire silence, so he wrote down for the satisfaction of his interrogators a horoscope: a mere perfunctory piece of work, as we learn afterwards. He begins by reciting the extraordinary nature of the King's birth, repeating the legend that his mother was delivered of him by surgical aid, and only lived a few hours afterwards; and declares that, in his opinion, it would have been better had this boy never been born at all. "Nevertheless, seeing that he had come into this world and been duly trained and educated, it would be well for mankind were he to live long, for all the graces waited upon him. Boy as he was, he was skilled in divers tongues, Latin, English, and French, and not unversed in Greek, Italian, and Spanish; he had likewise knowledge of dialectics, natural philosophy, and music. His culture is the reflection of our mortal nature; his gravity that of kingly majesty, and his disposition is worthy of so illustrious a prince. Speaking generally, it was indeed a strange experience to realize that this boy of so great talent and promise was being educated in the knowledge of the affairs of men. I have not set forth his accomplishments, tricked out with rhetoric so as to exceed the truth; of which, in sooth, my relation falls short." Cardan next draws a figure of Edward's horoscope, and devotes several pages to the customary jargon of astrologers; and, under the heading "De animi qualitatibus," says: "There was something portentous about this boy. He had learnt, as I heard, seven languages, and certainly he knew thoroughly his own, French, and Latin. He was skilled in Dialectic, and eager to be instructed in all subjects. When I met him, he was in his fifteenth year, and he asked me (speaking Latin no less perfectly and fluently than myself), 'What is contained in those rare books of yours, _De rerum varietate_?' for I had dedicated these manuscripts to his name.[155] Whereupon I began by pointing out to him what I had written in the opening chapter on the cause of the comets which others had sought so long in vain. He was curious to hear more of this cause, so I went on to tell him that it was the collected light of the wandering stars. 'Then,' said he, 'how is it, since the stars are set going by various impulses, that this light is not scattered, or carried along with the stars in their courses?' I replied: 'It does indeed move with them, but at a speed vastly greater on account of the difference of our point of view; as, for instance, when the prism is cast upon the wall by the sun and the crystal, then the least motion of the crystal will shift the position of the reflection to a great distance.' The King said: 'But how can this be done when no _subjectum_ is provided? for in the case you quote the wall is the _subjectum_ to the reflection.' I replied: 'It is a similar effect to that which we observe in the Milky Way, and in the reflection of light when many candles are lighted in a mass; these always produce a certain clear and lucent medium. _Itaque ex ungue leonem_.' "This youth was the great hope of good and learned men everywhere, by reason of his frankness and the gentleness of his manners. He began to take an interest in the Arts before he understood them, and to understand them before he had full occasion to use them. The production of such a personality was an effort of humanity; and, should he be snatched away before his time, not only England, but all the world must mourn his loss. "When he was required to show the gravity of a king, he would appear to be an old man. He played upon the lyre; he took interest in public affairs; and was of a kingly mind, following thus the example of his father, who, while he was over-careful to do right, managed to exhibit himself to the world in an evil light. But the son was free from any suspicion of such a charge, and his intelligence was brought to maturity by the study of philosophy." Cardan next makes an attempt to gauge the duration of the King's life, and when it is considered that he was a skilled physician, and Edward a sickly boy, fast sinking into a decline, it is to be feared that he let sincerity give way to prudence when he proclaimed that, in his fifty-sixth year the King would be troubled with divers illnesses. "Speaking generally of the whole duration of his life he will be found to be steadfast, firm, severe, chaste, intelligent, an observer of righteousness, patient under trouble, mindful both of injuries and benefits, one demanding reverence and seeking his own. He would lust as a man, but would suffer the curse of impotence. He would be wise beyond measure, and thereby win the admiration of the world; very prudent and high-minded; fortunate, and indeed a second Solomon." Edward VI. died on July 6, 1553, about six months after Cardan had returned to Milan; and, before the publication of the _Geniturarum Exempla_ in 1554, the author added to the King's horoscope a supplementary note, explaining his conduct thereanent and shedding some light upon the tortuous and sinister intrigues which at that time engaged the ingenuity of the leaders about the English Court. Now that he was safe from the consequences of giving offence, he wrote in terms much less guarded as to the state of English affairs. It must be admitted that his calculations as to the King's length of days, published after death, have no special value as calculations; but his impressions of the probable drift of events in England are interesting as the view of a foreigner upon English politics, and moreover they exhibit in strong light the sinister designs of Northumberland. Cardan records his belief that, in the fourth month of his fifteenth year, the King had been in peril of his life from the plottings of those immediately about him. On one occasion a particular disposition of the sun and Mars denoted that he was in danger of plots woven by a wicked minister, nay, there were threatenings even of poison.[156] He does not shrink from affirming that this unfortunate boy met his death by the treachery of those about him. As an apology for the horoscope he drew when he was in England, he lays down the principle that it is inexpedient to give opinions as to the duration of life in dealing with the horoscopes of those in feeble health, unless you shall beforehand consult all the directions and processes and ingresses of the ruling planets, "and if I had not made this reservation in the prognostic I gave to the English courtiers, they might justly have found fault with me." He next remarks that he had spent much time in framing this horoscope--albeit it was imperfect--according to his usual practice, and that if he had gone on somewhat farther, and consulted the direction of the sun and moon, the danger of death in which the King stood would straightway have manifested itself. If he had still been distrustful as to the directions aforesaid, and had gone on to observe the processes and ingresses, the danger would have been made clear, but even then he would not have dared to predict an early death to one in such high position: he feared the treacheries and tumults and the transfer of power which must ensue, and drew a picture of all the evils which might befall himself, evils which he was in no mood to face. Where should he look for protection amongst a strange people, who had little mercy upon one another and would have still less for him, a foreigner, with their ruler a mere boy, who could protect neither himself nor his guest? It might easily come about that his return to Italy would be hindered; and, supposing the crisis to come to the most favourable issue, what would he get in return for all this danger and anxiety? He calls to mind the cases of two soothsayers who were foolish enough to predict the deaths of princes, Ascletarion, and a certain priest, who foretold the deaths of Domitian and Galeazzo Sforza; and describes their fate, which was one he did not desire to call down upon himself. Although his forecast as to Edward's future was incomplete and unsatisfactory, he foresaw what was coming upon the kingdom from the fact that all the powers thereof, the strong places, the treasury, the legislature, and the fleet, were gathered into the hands of one man (Northumberland). "And this man, forsooth, was one whose father[157] the King's father had beheaded; one who had plunged into confusion all the affairs of the realm; seeing that he had brought to the scaffold, one after the other, the two maternal uncles of the King. Wherefore he was driven on both by his evil disposition and by his dread of the future to conspire against his sovereign's life. Now in such a season as this, when all men held their tongues for fear (for he brought to trial whomsoever he would), when he had gained over the greater part of the nobles to his side by dividing amongst them the spoil of the Church; when he, the most bitter foe of the King's title and dignity, had so contrived that his own will was supreme in the business of the State, I became weary of the whole affair; and, being filled with pity for the young King, proved to be a better prophet on the score of my inborn common-sense, than through my skill in Astrology. I took my departure straightway, conscious of some evil hovering anigh, and full of tears."[158] The above is Cardan's view of the machinations of the statesmen in high places in the English Court during the last months of Edward's life. Judged by the subsequent action of Northumberland it is in the main correct; and, taking into consideration his associations and environment during his stay in London, this view bears evident traces of independent judgment. Sir John Cheke, the King's former preceptor, and afterwards Professor of Greek at Cambridge, had received him with all the courtesy due to a fellow-scholar, and probably introduced him at Court. Cheke was a Chamberlain of the Exchequer, and just about this time was appointed Clerk to the Privy Council, wherefore he must have been fully acquainted with the aims and methods of the opposing factions about the Court. His fellow-clerk, Cecil, was openly opposed to Northumberland's designs, and prudently advanced a plea of ill health to excuse his absence from his duties: but Cheke at this time was an avowed partisan of the Duke, and of the policy which professed to secure the ascendency of the anti-Papal party. Cardan, living in daily intercourse with Cheke, might reasonably have taken up the point of view of his kind and genial friend; but no,--he evidently rated Northumberland, from beginning to end, as a knave and a traitor, and a murderer at least in will. When he quitted England in the autumn of 1552 Cardan did not shake himself entirely free from English associations. In an ill-starred moment he determined to take back to Italy with him an English boy.[159] He was windbound for several days at Dover, and the man with whom he lodged seems to have offered to let him take his son, named William, aged twelve years, back to Italy. Cardan was pleased with the boy's manner and appearance, and at once consented; but the adventure proved a disastrous one. The boy and his new protector could not exchange a word, and only managed to make each other understand by signs, and that very imperfectly. The boy was resolute to go on while Cardan wanted to be rid of him; but his conscience would not allow him to send him home unless he should, of his own free will, ask to be sent, and by way of giving William a distaste for the life he had chosen, he records that he often beat him cruelly on the slightest pretext. But the boy was not to be shaken off. He persisted in following his venture to the end, and arrived in Cardan's train at Milan, where he was allowed to go his own way. The only care for his training Cardan took was to have him taught music. He chides the unhappy boy for his indifference to learning and for his love of the company of other youths. What with his literary work and the family troubles which so soon fell upon him, Cardan's hands were certainly full; but, all allowance being made, it is difficult to find a valid excuse for this neglect on his part. William grew up to be a young man, and was finally apprenticed to a tailor at Pavia, but his knavish master set him to work as a vinedresser, suspecting that Cardan cared little what happened so long as the young man was kept out of his sight. William seems to have been a merry, good-tempered fellow; but his life was a short one, for he took fever, and died in his twenty-second year.[160] Besides chronicling this strange and somewhat pathetic incident, Cardan sets down in the _Dialogus de Morte_ his general impressions of the English people. Alluding to the fear of death, he remarks that the English, so far as he has observed, were scarcely at all affected by it, and he commends their wisdom, seeing that death is the last ill we have to suffer, and is, moreover, inevitable. "And if an Englishman views his own death with composure, he is even less disturbed over that of a friend or kinsman: he will look forward to re-union in a future state of immortality. People like these, who stand up thus readily to face death and mourn not over their nearest ones, surely deserve sympathy, and this boy (William) was sprung from the same race. In stature the English resemble Italians, they are fairer in complexion, less ruddy, and broad in the chest. There are some very tall men amongst them: they are gentle in manner and friendly to travellers, but easily angered, and in this case are much to be dreaded. They are brave in battle, but wanting in caution; great eaters and drinkers, but in this respect the Germans exceed them, and they are prone rather than prompt to lust. Some amongst them are distinguished in talent, and of these Scotus and Suisset[161] may be given as examples. They dress like Italians, and are always fain to declare that they are more nearly allied to us than to any others, wherefore they try specially to imitate us in habit and manners as closely as they can. They are trustworthy, freehanded, and ambitious; but in speaking of bravery, nothing can be more marvellous than the conduct of the Highland Scots, who are wont to take with them, when they are led to execution, one playing upon the pipes, who, as often as not, is condemned likewise, and thus he leads the train dancing to death." Like as the English were to Italians in other respects, Cardan was struck with the difference between the two nations as soon as the islanders opened their mouths to speak. He could not understand a single word, but stood amazed, deeming them to be Italians who had lost their wits. "The tongue is curved upon the palate; they turn about their words in the mouth, and make a hissing sound with their teeth." He then goes on to say that all the time of his absence his mind was full of thoughts of his own people in Italy, wherefore he sought leave to return at once. FOOTNOTES: [146] _Scotichronicon_, vol. i. p. 286 [ed. G. F. S. Gordon, Glasgow, 1867]. Naudé, in his _Apologie pour les grands hommes soupçonnez_ de Magie, writes: "Ceux qui recherchoiant les Mathématiques et les Sciences les moins communes étoient soupçonnez d'être enchanteurs et Magiciens."--p. 15. [147] "Curam agebat Medicus ex constituto Medicorum Lutetianorum."--_De Vita Propria_, ch. xl. p. 137. Cardan makes no direct mention of any other physician in Scotland besides Cassanate; but the Archbishop would certainly have a body physician in attendance during Cassanate's absence. [148] "Per totam tunicam sicut in linteis."--_Opera_, tom. ix. p. 128. [149] "Accipe testudinem maximam et illam incoque in aqua, donec dissolvatur, deinde abjectis corticibus accipiantur caro, et ossa et viscera omnia mundata."--_Opera_, tom. ix. p. 140. [150] Another piece of advice runs as follows: "De venere certe non est bona, neque utilis, ubi tamen contingat necessitas, debet uti ea inter duos somnos, scilicet post mediam noctem, et melius est exercere eam ter in sex diebus pro exemplo ut singulis duobus diebus semel, quam bis in una die, etiam quod staret per decem dies."--_Opera_, tom. ix. p. 135. [151] "Interim autem concurrebant multi, imo pené tota nobilitas."--_Opera_, tom. l. p. 93. [152] _Scotichronicon_, vol. i. p. 234. Larrey in his _History of England_ seems to have given currency to the legend that Cardan foretold the Archbishop's death. "S'il en faut croire ce que l'Histoire nous dit de ce fameux Astrologe, il donna une terrible preuve de sa science à l'Archevêque qu'il avoit gueri, lorsque prenait congé de lire, il lui tint ce discours: 'Qu'il avoit bien pu le guerir de sa maladie; mais qu'il n'étoit pas en son pouvoir de changer sa destinée, ni d'empêcher qu'il ne fût pendu.'"--Larrey, _Hist. d'Angleterre_, vol. ii. p. 711. [153] _De Vita Propria_, ch. xxxii. p. 101. [154] "Scotic[u=] nomen antea horruer[a=], eorum exemplo qui prius coeperunt odisse quam cognoscere. Nunc cum ipsa gens per se humanissima sit atque supra existimationem civilis, tu tamen tantum illi addis ornamenti, ut longe nomine tuo jam nobilior evadat."--_De Astrorum Judiciis_, p. 3. [155] Cardan evidently carried the MS. with him, for he writes (_Opera_, tom. i. p. 72): "Hoc fuit quod Regi Angliæ Edoardo sexto admodum adolescenti dum redirem a Scotia ostendi." [156] "Cumque ibi esset nodus eti[a=] venenum, quod utin[a=] abfuerit."--_Geniturarum Exempla_, p. 411. [157] Edmund Dudley, the infamous minister of Henry VII. [158] _Geniturarum Exempla_, p. 412. [159] In the prologue to _Dialogus de Morte_, Opera, tom. i. p. 673, he gives a full account of this transaction. Of the boy himself he writes: "hospes ostendit mihi filium nomine Guglielmum, ætatis annorum duodecim, probum, scitulum, et parentibus obsequentem. Avus paternus nomine Gregorius adhuc vivebat, et erat Ligur: pater Laurentius, familia nobili Cataneorum." [160] _Opera_, tom. i. p. 119. Cardan here calls him "Gulielmus _Lataneus_ Anglus adolescens mihi charissimus." In the _De Morte_, however, he speaks of him as "ex familia Cataneorum" (see last page). [161] Cardan writes (_De Subtilitate_, p. 444) that Suisset [Richard Swineshead], who lived about 1350, was known as the Calculator; but Kästner [_Gesch. der Math._ I. 50] maintains that the title Calculator should be applied to the book rather than to the author, and hints that this misapprehension on Cardan's part shows that he knew of Suisset only by hearsay. The title of the copy of Suisset in the British Museum stands "Subtilissimi Doctoris Anglici Suiset. Calculationes Liber," Padue [1485]. Brunet gives one, "Opus aureum calculationum," Pavia, 1498. CHAPTER VIII CARDAN travelled southward by way of the Low Countries. He stayed some days at Antwerp, and during his visit he was pressed urgently to remain in the city and practise his art. A less pleasant experience was a fall into a ditch when he was coming out of a goldsmith's shop. He was cut and bruised about the left ear, but the damage was only skin-deep. He went on by Brussels and Cologne to Basel, where he once more tarried several days. He had a narrow escape here of falling into danger, for, had he not been forewarned by Guglielmo Gratarolo, a friend, he would have taken up his quarters in a house infected by the plague. He was received as a guest by Carlo Affaidato, a learned astronomer and physicist, who, on the day of departure, made him accept a valuable mule, worth a hundred crowns. Another generous offer of a similar kind was made to him shortly afterwards by a Genoese gentleman of the family of Ezzolino, who fell in with him accidentally on the road. This was the gift of a very fine horse (of the sort which the English call Obinum), but, greatly as Cardan desired to have the horse, his sense of propriety kept him back from accepting this gift.[162] He went next to Besançon, where he was received by Franciscus Bonvalutus, a scholar of some note, and then by Berne to Zurich. He must have crossed the Alps by the Splugen Pass, as Chur is named in his itinerary, and he also describes his voyage down the Lake of Como on the way to Milan, where he arrived on January 3, 1553. Cardan was a famous physician when he set out on his northward journey; but now on his return he stood firmly placed by the events of the last few months at the head of his profession. Writing of the material results of his mission to Scotland, he declares that he is ashamed to set down the terms upon which he was paid, so lavishly was he rewarded for his services. The offers made to him by so many exalted personages to secure his permanent and exclusive attention would indeed have turned the heads of most men. There was the offer from the King of Denmark; another, in 1552, from the King of France at a salary of thirteen hundred crowns a year; and yet another made by the agents of Charles V., who was then engaged in his disastrous attack upon Metz. All of them he refused: he had no inclination to share the perils of the leaguer of Metz, and his sense of loyalty forbad him to join himself to the power which was at that time warring against his sovereign. He speaks also of another offer made to him by the Queen of Scotland of a generous salary if he would settle in Scotland; but the country was too remote for his taste. There is no authority for this offer except the _De Vita Propria_, and it is there set down in terms which render it somewhat difficult to identify the Queen aforesaid.[163] As soon as he entered Milan, Ferrante Gonzaga, the Governor, desired to secure his services as physician to the Duke of Mantua, his brother, offering him thirty thousand gold crowns as honorarium; but, in spite of the Governor's persuasions and threats, he would not accept the office. When the news had come to Paris that Cardan was about to quit Britain, forty of the most illustrious scientists of France repaired to Paris in order to hear him expound the art of Medicine; but the disturbed state of the country deterred him from setting foot in France. He refers to a letter from his friend Ranconet as a testimony of the worship that was paid to him, and goes on to say that, in his journeying through France and Germany, he fared much as Plato fared at the Olympic games. In a passage which Cardan wrote shortly after his return from Britain, he lets it be seen that he was not ill-satisfied with the figure he then made in the world. He writes--"Therefore, since all those with whom I am intimate think well of me for my truth and probity, I can let my envious rivals indulge themselves as they list in the shameful habit of evil-speaking. With regard to folly, if I now utter, or ever have uttered, foolish words, let those who accuse me show their evidence. I, who was born poor, with a weakly body, in an age vexed almost incessantly by wars and tumults, helped on by no family influence, but forced to contend against the bitter opposition of the College at Milan, contrived to overcome all the plots woven against me, and open violence as well. All the honours which a physician can possess I either enjoy, or have refused when they were offered to me. I have raised the fortunes of my family, and have lived a blameless life. I am well known to all men of worship, and to the whole of Europe. What I have written has been lauded; in sooth, I have written of so many things and at such length, that a man could scarcely read my works if he spent his life therewith. I have taken good care of my domestic affairs, and by common consent I have come off victor in every contest I have tried. I have refused always to flatter the great; and over and beyond this I have often set myself in active opposition to them. My name will be found scattered about the pages of many writers. I shall deem my life long enough if I come in perfect health to the age of fifty-six. I have been most fortunate as the discoverer of many and important contributions to knowledge, as well as in the practice of my art and in the results attained; so much so that if my fame in the first instance has raised up envy against me, it has prevailed finally, and extinguished all ill-feeling."[164] These words were written before the publication of the _Geniturarum Exempla_ in 1554. Cardan's life for the six years which followed was busy and prosperous, but on the whole uneventful. The Archbishop of St. Andrews wrote to him according to promise at the end of two years to give an account of the results of his treatment. His letter is worthy of remark as showing that he, the person most interested, was well satisfied with Cardan's skill as a physician. Michael, the Archbishop's chief chamberlain, was the bearer thereof, and as Hamilton speaks of him as "epistolam vivam," it is probable that he bore likewise certain verbal messages which could be more safely carried thus than in writing. A sentence in the _De Vita Propria_,[165] mixed up with the account of Hamilton's cure, seems to refer to this embassy, and to suggest that Michael was authorized to promise Cardan a liberal salary if he would accept permanent office in the Primate's household. Moreover, Hamilton writes somewhat querulously about Cassanate's absence abroad on a visit to his family, a fact which would make him all the more eager to secure Cardan's services. His letter runs as follows--"Two of your most welcome letters, written some months ago, I received by the hand of an English merchant; others came by the care of the Lord Bishop of Dunkeld, together with the Indian balsam. The last were from Scoto, who sent at the same time your most scholarly comments on that difficult work of Ptolemy.[166] To all that you have written to me I have replied fully in three or four letters of my own, but I know not whether, out of all I have written, any letter of mine has reached you. But now I have directed that a servant of mine, who is known to you, and who is travelling to Rome, shall wait upon you and salute you in my name, and bear to you my gratitude, not only for the various gifts I have received from you, but likewise because my health is well-nigh restored, the ailment which vexed me is driven away, my strength increased, and my life renewed. Wherefore I rate myself debtor for all these benefits, as well as this very body of mine. For, from the time when I began to take these medicines of yours, selected and compounded with so great skill, my complaint has afflicted me less frequently and severely; indeed, now, as a rule, I am not troubled therewith more than once a month; sometimes I escape for two months."[167] In the following year (1555) Cardan's daughter Chiara, who seems to have been a virtuous and well-conducted girl, was married to Bartolomeo Sacco, a young Milanese gentleman of good family, a match which proved to be fortunate. Cardan had now reached that summit of fame against which the shafts of jealousy will always be directed. The literary manners of the age certainly lacked urbanity, and of all living controversialists there was none more truculent than Julius Cæsar Scaliger, who had begun his career as a man of letters by a fierce assault upon Erasmus with regard to his _Ciceronianus_, a leading case amongst the quarrels of authors. Erasmus he had attacked for venturing to throw doubts upon the suitability of Cicero's Latin as a vehicle of modern thought; this quarrel was over a question of form; and now Scaliger went a step farther, and, albeit he knew little of the subject in hand, published a book of _Esoteric Exercitations_ to show that the _De Subtilitate_ of Cardan was nothing but a tissue of nonsense.[168] The book was written with all the heavy-handed brutality he was accustomed to use, but it did no hurt to Cardan's reputation, and, irritable as he was by nature, it failed to provoke him to make an immediate rejoinder, a delay which was the cause of one of the most diverting incidents in the whole range of literary warfare. Scaliger sat in his study, eagerly expecting a reply, but Cardan took no notice of the attack. Then one day some tale-bearer, moved either by the spirit of tittle-tattle or the love of mischief, brought to Julius Cæsar the news that Jerome Cardan had sunk under his tremendous battery of abuse, and was dead. It is but bare charity to assume that Scaliger was touched by some stings of regret when he heard what had been the fatal result of his onslaught; still there can be little doubt that his mind was filled with a certain satisfaction when he reflected that he was in sooth a terrible assailant, and that his fist was heavier than any other man's. In any case, he felt that it behoved him to make some sign, wherefore he sat down and penned a funeral oration over his supposed victim, which is worth giving at length.[169] "At this season, when fate has dealt with me in a fashion so wretched and untoward that it has connected my name with a cruel public calamity, when a literary essay of mine, well known to the world, and undertaken at the call of duty, has ensued in dire misfortune, it seems to me that I am bound to bequeath to posterity a testimony that, sharp as may have been the vexation brought upon Jerome Cardan by my trifling censures, the grief which now afflicts me on account of his death is ten times sharper. For, even if Cardan living should have been a terror to me, I, who am but a single unit in the republic of letters, ought to have postponed my own and singular convenience to the common good, seeing how excellent were the merits of this man, in every sort of learning. For now the republic is bereft of a great and incomparable scholar, and must needs suffer a loss which, peradventure, none of the centuries to come will repair. What though I am a person of small account, I could count upon him as a supporter, a judge, and (immortal gods) even a laudator of my lucubrations; for he was so greatly impressed by their weighty merits, that he deemed he would best defend himself by avoiding all comment on the same, despairing of his own strength, and knowing not how great his powers really were. In this respect he was so skilful a master, that he could assuredly have fathomed the depths of every method and every device used against him, and would thereby have made his castigation of myself to serve as an augmentation of his own fame. He, in sooth, was a man of such quality that, if he had deemed it a thing demanded of him by equity, he would never have hesitated to point out to other students the truth of those words which I had written against him as an accusation, while, on the other hand, this same constancy of mind would have made him adhere to the opinions he might have put forth in the first instance, so far as these opinions were capable of proof. I, when I addressed my _Exercitations_ to him during his life--to him whom I knew by common report to be the most ingenious and learned of mortal men--was in good hope that I might issue from this conflict a conqueror; and is there living a man blind enough not to perceive that what I looked for was hard-earned credit, which I should certainly have won by finding my views confirmed by Cardan living, and not for inglorious peace brought about by his death? And indeed I might have been suffered to have share in the bounty and kindliness of this illustrious man, whom I have always heard described as a shrewd antagonist and one full of confidence in his own high position, for it was an easy task to win from him the ordinary rights of friendship by any trifling letter, seeing that he was the most courteous of mankind. It is scarcely likely that I, weary as I was, one who in fighting had long been used to perils of all sorts, should thus cast aside my courage; that I, worn out by incessant controversies and consumed by the daily wear and tear of writing, should care for an inglorious match with so distinguished an antagonist; or that I should have set my heart upon winning a bare victory in the midst of all this dust and tumult. For not only was the result which has ensued unlooked for in the nature of things and in the opinion of all men qualified to judge in such a case; it was also the last thing I could have desired to happen, for the sake of my good name. My judgment has ever been that all men (for in sooth all of us are, so to speak, little less than nothing) may so lose their heads in controversy that they may actually fight against their own interests. And if such a mischance as this may happen to any man of eminence--as has been my case, and the case of divers others I could recall--it shall not be written down in the list of his errors, unless in aftertimes he shall seek to justify the same. It is necessary to advance roughness in the place of refinement, and stubborn tenacity for steadfastness. No man can be pronounced guilty of offence on the score of some hasty word or other which may escape his lips; such a charge should rather be made when he defends himself by unworthy methods. Therefore if Cardan during his life, being well advised in the matter, should have kept silent over my attempts to correct him, what could have brought me greater credit than this? He would have bowed to my opinion in seemly fashion, and would have taken my censures as those of a father or a preceptor. But supposing that he had ventured to engage in a sharper controversy with me over this question, is there any one living who would fail to see that he might have gone near to lose his wits on account of the mental agitation which had afflicted him in the past? But as soon as his superhuman intellect had thoroughly grasped the question, it seemed to him that he must needs be called upon to bear what was intolerable. He could not pluck up courage enough to bear it by living, so he bore it by dying. Moreover, what he might well have borne, he could not bring himself to bear, to wit that he and I should come to an agreement and should formulate certain well-balanced decisions for the common good. For this reason I lament deeply my share in this affair, I who had most obvious reasons for engaging in this conflict, and the clearest ones for inventing a story as to the victory I hoped to gain; reasons which a man of sober temper could never anticipate, which a brave man would never desire. "Cardan's fame has its surest foundation in the praise of his adversaries. I lament greatly this misfortune of our republic: the causes of which the parliament of lettered men may estimate by its particular rules, but it cannot rate this calamity in relation to the excellences of this illustrious personality. For in a man of learning three properties ought to stand out pre-eminently--a spotless and gentle rule of life; manifold and varied learning; and consummate talent joined to the shrewdest capacity for forming a judgment. These three points Cardan attained so completely that he seemed to have been made entirely for himself, and at the same time to have been the only mortal made for mankind at large. No one could be more courteous to his inferiors or more ready to discuss the scheme of the universe with any man of mark with whom he might chance to foregather. He was a man of kingly courtesy, of sympathetic loftiness of mind, one fitted for all places, for all occasions, for all men and for all fortunes. In reference to learning itself, I beg you to look around upon the accomplished circle of the learned now living on the earth, in this most fortunate age of ours; here the combination of individual talent shows us a crowd of illustrious men, but each one of these displays himself as occupied with some special portion of Philosophy. But Cardan, in addition to his profound knowledge of the secrets of God and Nature, was a consummate master of the humaner letters, and was wont to expound the same with such eloquence that those who listened to him would have been justified in affirming that he could have studied nothing else all his life. A great man indeed! Great if he could lay claim to no other excellence than this; and forsooth, when we come to consider the quickness of his wit, his fiery energy in everything he undertook, whether of the least or the greatest moment, his laborious diligence and unconquerable steadfastness, I affirm that the man who shall venture to compare himself with Cardan may well be regarded as one lacking in all due modesty. I forsooth feel no hostility towards one whose path never crossed mine, nor envy of one whose shadow never touched mine; the numerous and weighty questions dealt with in his monumental work urged me on to undertake the task of gaining some knowledge of the same. After the completion of the Commentaries on Subtlety, he published as a kind of appendix to these that most learned work the _De Rerum Varietate_. And in this case, before news was brought to me of his death, I followed my customary practice, and in the course of three days compiled an Excursus in short chapters. When I heard that he was dead I brought them together into one little book, in order that I also might lend a hand in this great work of his, and this thing I did after a fashion which he himself would have approved, supposing that at some time or other he might have held discourse with me, or with some other yet more learned man, concerning his affairs."[170] It is a matter of regret that this cry of _peccavi_ was not published till all the chief literary contemporaries of Scaliger were in their graves. As it did not appear till 1621, the men of his own time were not able to enjoy the shout of laughter over his discomfiture which would surely have gone up from Paris and Strasburg and Basel and Zurich. Estienne and Gessner would hardly have felt acute sorrow at a flout put upon Julius Cæsar Scaliger. Crooked-tempered as he was, Cardan, compared with Scaliger, was as a rose to a thistle, but there were reasons altogether unconnected with the personalities of the disputants which swayed the balance to Cardan's advantage. The greater part of Scaliger's criticism was worthless, and the opinion of learned Europe weighed overwhelmingly on Cardan's side. Thuanus,[171] who assuredly did not love him, and Naudé, who positively disliked him, subsequently gave testimony in his favour. He did not follow the example of Erasmus, and let Scaliger's abuse go by in silence, but he took the next wisest course. He published a short and dignified reply, _Actio prima in Calumniatorem_, in which, from title-page to colophon, Scaliger's name never once occurs. The gist of the book may best be understood by quoting an extract from the criticism of Cardan by Naudé prefixed to the _De Vita Propria_. He writes: "This proposition of mine will best be comprehended by the man who shall set to work to compare Cardan with Julius Cæsar Scaliger, his rival, and a man endowed with an intellect almost superhuman. For Scaliger, although he came upon the stage with greater pomp and display, and brought with him a mind filled with daring speculation, and adequate to the highest flights, kept closely behind the lattices of the humaner letters and of medical philosophy, leaving to Cardan full liberty to occupy whatever ground of argument he might find most advantageous in any other of the fields of learning. Moreover, if any one shall give daily study to these celebrated _Exercitations_, he will find therein nothing to show that Cardan is branded by any mark of shame which may not be removed with the slightest trouble, if the task be undertaken in a spirit of justice. For, in the first place, who can maintain that Scaliger was justified in publishing his _Exercitations_ three years after the issue of the second edition of the _Libri de Subtilitate_, without ever having taken the trouble to read this edition, and without exempting from censure the errors which Cardan had diligently expunged from his book in the course of his latest revision, lest he (Scaliger) should find that all the mighty labour expended over his criticisms had been spent in vain? Besides, who does not know that Cardan, in his _Actio prima in Calumniatorem_, blunted the point of all his assailant's weapons, swept away all his objections, and broke in pieces all his accusations, in such wise that the very reason of their existence ceased to be? Cardan, in sooth, was a true man, and held all humanity as akin to him. There is small reason why we should marvel that he erred now and again; it is a marvel much greater that he should only have gone astray so seldom and in things of such trifling moment. Indeed I will dare to affirm, and back my opinion with a pledge, that the errors which Scaliger left behind him in these _Exercitations_ were more in number than those which he so wantonly laid to Cardan's charge, having sweated nine years over the task. And this he did not so much in the interests of true erudition as with the desire of coming to blows with all those whom he recognized as the chiefs of learning." During the whole dispute Cardan kept his temper admirably. Scaliger was a physician of repute; and it is not improbable that the spectacle of Cardan's triumphal progress back to Milan from the North may have aroused his jealousy and stimulated him to make his ill-judged attack. But even on the ground of medical science he was no match for Cardan, while in mathematics and philosophy he was immeasurably inferior. Cardan felt probably that the attack was nothing more than the buzzing of a gadfly, and that in any case it would make for his own advantage and credit, wherefore he saw no reason why he should disquiet himself; indeed his attitude of dignified indifference was admirably calculated to win for him the approval of the learned world by the contrast it furnished to the raging fury of his adversary.[172] After the heavy labour of editing and issuing to the world the _De Rerum Varietate_, and of re-editing the first issue of the _De Subtilitate_, Cardan might well have given himself a term of rest, but to a man of his temper, idleness, or even a relaxation of the strain, is usually irksome. The _De Varietate_ was first printed at Basel in 1553, and, as soon as it was out of the press, it brought a trouble--not indeed a very serious one--upon the author. The printer, Petrus of Basel (who must not be confused with Petreius of Nuremberg) took it upon him to add to Chapter LXXX of the work some disparaging remarks about the Dominican brotherhoods, making Cardan responsible for the assertion that they were rapacious wolves who hunted down reputed witches and despisers of God, not because of their offences, but because they chanced to be the possessors of much wealth. Cardan remonstrated at once--he always made it his practice to keep free from all theological wrangling,--but Petrus treated the whole question with ridicule,[173] and it does not seem that Cardan could have had any very strong feeling in the matter, for the obnoxious passage is retained in the editions of 1556 and 1557. The religious authorities were however fully justified in assuming that the presence of such a passage in the pages of a book so widely popular as the _De Varietate_ would necessarily prove a cause of scandal, and give cause to the enemy to blaspheme. For Reginald Scot, in the eighth chapter of _Discoverie of Witchcraft_, alludes to the passage in question in the following terms: "Cardanus writeth that the cause of such credulitie consisteth in three points: to wit in the imagination of the melancholike, in the constancie of them that are corrupt therewith, and in the deceipt of the Judges; who being inquisitors themselves against heretikes and witches, did both accuse and condemne them, having for their labour the spoile of their goods. So as these inquisitors added many fables hereunto, least they should seeme to have doone injurie to the poore wretches, in condemning and executing them for none offense. But sithens (said he) the springing up of Luther's sect, these priests have tended more diligentlie upon the execution of them; bicause more wealth is to be caught from them; insomuch as now they deale so looselie with witches (through distrust of gaines) that all is seene to be malice, follie, or avarice that hath beene practised against them. And whosoever shall search into this cause, or read the cheefe writers hereupon, shall find his words true." In 1554 Cardan published also with Petrus of Basel the _Ptolemæi de astrorum judiciis_ with the _Geniturarum Exempla_, bound in one volume, but he seems to have written nothing but a book of fables for the young, concerning which he subsequently remarks that, in his opinion, grown men might read the same with advantage. It is a matter of regret that this work should have disappeared, for it would have been interesting to note how far Cardan's intellect, acute and many-sided as it was, was capable of dealing with the literature of allegory and imagination. He has set down one fact concerning it, to wit that it contained "multa de futuris arcana." The next year he produced only a few medical trifles, but in 1557 he brought out two other scientific works which he characterizes as admirable--one the _Ars parva curandi_, and the other a treatise _De Urinis_. In the same year he published the book which, in forming a judgment of him as a man and a writer, is perhaps as valuable as the _De Vita Propria_ and the _De Utilitate_, to wit the _De Libriis Propriis_. This work exists in three forms: the first, a short treatise, "cui titulus est ephemerus," is dedicated to "Hieronymum Cardanum medicum, affinem suum," and has the date of 1543. The second has the date of 1554, and, according to Naudé, was first published "apud Gulielmum Rovillium sub scuto Veneto, Lugduni, 1557." The third was begun in 1560,[174] and contains comments written in subsequent years. The first is of slight interest, the second is a sort of register of his works, amplified from year to year, while the third has more the form of a treatise, and presents with some degree of symmetry the crude materials contained in the first. Having finished with his writings up to the year 1564, Cardan lapses into a philosophizing strain, and opens his discourse with the ominous words, "Sed jam ad institutum revertamur, déque ipso vitæ humanæ genere aliquo dicamus." He begins with a disquisition on the worthlessness of life, and repeats somewhat tediously the story of his visit to Scotland. He gives a synopsis of all the sciences he had ever studied--Theology, Dialectics, Arithmetic, Music, Optics, Astronomy, Astrology, Geometry, Chiromancy, Agriculture, Medicine, passing on to treat of Magic, portents and warnings, and of his own experience of the same at the crucial moments of his life. He ends by a reference to an incident already chronicled in the _De Vita Propria_,[175] how he escaped death or injury from a falling mass of masonry by crossing the street in obedience to an impulse he could not explain, and speculates why God, who was able to save him on this occasion with so little trouble, should have let him rush on and court the overwhelming stroke which ultimately laid him low. FOOTNOTES: [162] _De Vita Propria_, ch. xxxii. p. 100. [163] _De Vita Propria_, ch. iv. p. 16: "cum Scotorum Regina cujus levirum curaveram." Cardan had probably prescribed for a brother of the Duc de Longueville, the first husband of Mary of Guise, during his sojourn in Paris. [164] _Geniturarum Exempla_, p. 459. [165] _De Vita Propria_, ch. xl. p. 137. [166] _Commentaria in Ptolemæi de Astrorum Judiciis_ (Basil, 1554). He wrote these notes while going down the Loire in company with Cassanate on his way from Lyons to Paris in 1552.--_De Vita Propria_, ch. xlv. p. 175. He gives an interesting account (_Opera_, tom. i. p. 110) as to how the book first came under his notice. The day before he quitted Lyons with Cassanate, a school-master came to ask for advice, which Cardan gave gratis. Then the patient, knowing perhaps the physician's taste for the marvellous, related how there was a certain boy in the place who could see spirits by looking into an earthen vessel, but Cardan was little impressed by what he saw, and began to talk with the school-master about Archimedes. The school-master brought out a work of the Greek philosopher with which was bound up the _Ptolemæi Libri de Judiciis_. Cardan fastened upon it at once, and wanted to buy it, but the school-master insisted that he should take it as a gift. He declares that his Commentaries thereupon are the most perfect of all his writings. The book contains his famous Nativity of Christ. A remark in _De Libris Propriis_ (cf. _Opera_, tom. i. p. 67) indicates that there was an earlier edition of Ptolemy, printed at Milan at Cardan's own cost, because when he saw the numerous mistakes made by Ottaviano Scoto in printing the _De Malo Medendi_ and the _De Consolatione_, he determined to go to another printer. [167] _Opera_, tom. i. p. 93. [168] Cardan notices the attack in these words--"His diebus quidam conscripserat adversus nostrum de Subtilitate librum, Opus ingens. Adversus quem ego Apologiam scripsi."--_Opera_, tom. i. p. 117. Scaliger absurdly calls his work the _fifteenth_ book of _Exercitations_, and wished the world to believe that he had written, though not printed, the fourteen others. [169] It was not printed until many years after the deaths of both disputants, and appeared for the first time in a volume of Scaliger's letters and speeches published at Toulouse in 1621, and it was afterwards affixed to the _De Vita Propria_. [170] "Si Scaliger avoit eu un peu moins de démangeaison de contre dire, il auroit acquis plus de gloire, qu'il n'a fait dans ce combat: mais, ce que les Grecs ont apellé [Greek: ametria tês antholkês], une passion excessive de prendre le contrepied des autres, a fait grand tort à Scaliger. C'est par ce principe qu'il a soutenu que le perroquet est une très laide bête. Si Cardan l'eût dit, Scaliger lui eût opposé ce qu'on trouve dans les anciens Poètes touchant la beauté de cet oiseau. Vossius a fait une Critique très judicieuse de cette humeur contrariante de Scaliger, et a marqué en même temps en quoi ces deux Antagonistes étoient supérieurs et inférieures, l'un à l'autre."--(Scaliger, in _Exercitat.,_ 246.) "Quia Cardanus psittacum commendarat a colorum varietate ac præterea fulgore, quod et Appuleius facit in secundo Floridorum, contra contendit esse deformem, non modo ob foeditatem rostri, ac crurum, et linguæ, sed etiam quia sit coloris fusci ac cinericii, qui tristis. Quid faciamus summo Viro? Si Cardanus ea dixisset, provocasset ad judicia poëtarum, atque adeo omnium hominum. Nunc quia pulchri dixit coloris, ille deformis contendit. Hoc contradictionis studium, quod ubique in hisce exercitationibus se prodit, sophista dignius est, quamque philosopho."--Bayle: Article "Cardan." (Sir Thomas Browne, in one of his Commonplace Books, observes--"If Cardan saith a parrot is a beautiful bird, Scaliger will set his wits on work to prove it a deformed animal.") Naudé (_Apologie_, ch. xiii.) says that of the great men of modern times Scaliger and Cardan each claimed the possession of a guardian spirit, and hints that Scaliger may have been moved to make this claim in order not to be outdone by his great antagonist. It should, however, be remembered that Cardan did not seriously assert this belief till long after his controversy with Scaliger. Naudé sums up thus: "D'où l'on peut juger asseurement, que lui et Scaliger n'ont point eu d'autre Genie que la grande doctrine qu'ils s'étoient acquis par leurs veilles, par leurs travaux, et par l'expérience qu'ils avoient des choses sur lesquelles venant à élever leur jugement ils jugeoint pertinemment de toutes matières, et ne laissoient rien échapper qui ne leur fust conneu et manifeste." [171] Thuanus, ad Annum MDLXXVI, part of the Appendix to the _De Vita Propria_. [172] Cardan does not seem to have harboured animosity against Scaliger. In the _De Vita Propria_, ch. xlviii. p. 198, he writes: "Julius Cæsar Scaliger plures mihi titulos ascribit, quam ego mihi concedi postulassem, appellans _ingenium profundissimum, felicissimum, et incomparabile_." [173] "Quid tua interest quod quatuor verba adjecerim? an hoc tantum crimen est! quid facerem absens absenti?" Cardan writes on in meditative strain: "Coeterum cum non ignorem maculatos fuisse codices B. Hieronimi, atque aliorum patrum nostrorum, ab his qui aliter sentiebant, erroremque suum auctoritate viri tegere voluerunt: ut ne quis in nostris operibus hallucinetur vel ab aliis decipiatur, sciant omnes me nullibi Theologum agere, nec velle in alienam messem falcem ponere."--_Opera_, tom. i. p. 112. Johannes Wierus, one of the first rationalists on the subject of witchcraft, has quoted largely from Chapter LXXX of _De Varietate_ in his book _De Præstigiis Dæmonum_, in urging his case against the orthodox view. [174] _Opera_, tom. i. p. 96. "Annus hic est Salutis millesimus quingentesimus ac sexagesimus." [175] _De Vita Propria_, ch. xxx. p. 78. CHAPTER IX THE year 1555 may be held to mark the point of time at which Cardan reached the highest point of his fortunes. After a long and bitter struggle with an adverse world he had come out a conqueror, and his rise to fame and opulence, if somewhat slow, had been steady and secure. He longed for wealth, not that he might figure as a rich man, but so that he might win the golden independence which permits a student to prosecute the task which seems to subserve the highest purposes of true learning, and frees him from the irksome battle for daily bread. He loved, indeed, to spend money over beautiful things, and there are few more attractive touches in the picture he draws of himself than the confession of his passion for costly penholders, gems, rare books, vessels of brass and silver, and painted spheres.[176] In this brief season of ease and security, there were no flaming portents in the sky to foretell the cruel stroke of evil fortune which was destined so soon to fall upon him. Cardan has left a very pathetic sketch of his own miserable boyhood in the strangely ordered home in Milan, with his callous, tyrannical father, his quick-tempered mother, and the superadded torment of his Aunt Margaret's presence. Fazio Cardano was a man of rigorous sobriety, and he seems moreover to have atoned for his early irregularities by the practice of that austere piety which Jerome notices more than once as a characteristic of his old age.[177] The discipline was hard, and the life unlovely, but the home was at least decent and orderly, and no opportunities or provocations to loose manners or ill doing existed therein. In Cardan's own case it is to be feared that, after Lucia's death, the affairs of his household fell into dire confusion, in spite of the presence of his mother-in-law, Thadea, who had come to him as housekeeper--her husband, Altobello, having died soon after the marriage of his daughter with Cardan. He was an ardent lover of music, and, as a consequence, his house would be constantly filled with singing men and boys, a tribe of somewhat sinister reputation.[178] Then, when he was not engaged with music, he would be gambling in some fashion or other. After lamenting the vast amount of time he has wasted over the game of chess, he goes on: "But the play with the dice, an evil far more noxious, found its way into my house; and, after my sons had learned to play the same, my doors always stood open to dicers. I can find no excuse for this practice except the trivial one, that, what I did, I did in the hope of relieving the poverty of my children."[179] In a home of this sort, ruled by a father who was assuredly more careful of his work in the study and class-room than of his duties as paterfamilias, it is not wonderful that the two young men, Gian Battista and Aldo, should grow up into worthless profligates. It has been recorded how Cardan, during a journey to Genoa, wrote a Book of Precepts for his children,[180] a task the memory of which afterwards wrung from him a cry of despair. There never was compiled a more admirable collection of maxims; but, excellent as they were, it was not enough to write them down on paper; and the young men, if ever they took the trouble to read them, must have smiled as they called to mind the difference between their father's practices and the precepts he had composed for their guidance. Furthermore, he had written at length, in the _De Consolatione_, on the folly which parents for the most part display in the education of their children. "They show their affection in such foolish wise, that it would be nearer the mark to say they hate, rather than love, their offspring. They bring them up not to follow virtue, but to occupy themselves with all manner of hurtful things; not to learning, but to riot; not to the worship of God, but to foster in them the desire to drain the cup of lustful pleasure; not for the life eternal, but to the enticements of lechery."[181] At this time Gian Battista had gained the doctorate of medicine at Pavia, and had made his contribution to medical knowledge by the publication of an insignificant tract, _De cibis foetidis non edendis_. Cardan was evidently full of hope for his elder son's career, but Aldo seems to have been a trouble from the first. Yet, in casting Aldo's horoscope (probably at the time of his birth) Cardan predicts for him a flourishing future.[182] Never was there made a worse essay in prophecy. Aldo's childhood had been a sickly one. He had well-nigh died of convulsions, and later on he had been troubled with dysentery, abscesses of the brain, and a fever which lasted six months. Moreover, he could not walk till he was three years old. With a weakly body, his nature seems to have put forth all sorts of untoward growths. There is a story which Naudé brings forward as part of his indictment against Cardan, that the father being irritated beyond endurance by some ill conduct of his younger son during supper, cut off his ear by way of punishment. It was a most barbarous act; one going far beyond the range of any tradition of the early _patria potestas_, which may have yet lingered in Italy; and scarcely calculated to bring about reformation in the youth thus punished. In any case, Aldo went on from bad to worse; at one time his father found it necessary to place him under restraint, and the last record of him is that one in Cardan's testament, by which he was disinherited. Gian Battista's failings were doubtless grave and numerous, but he had at least sufficient industry to qualify himself as a physician. He was certainly his father's favourite child, and on this account the eulogies written of him in those dark hours when Cardan's reason was reeling under the accumulated blows of private grief and public disgrace, must be accepted with caution. There is no evidence to show he was in intellect anything like the budding genius his father deemed him; as to conduct and manner of life, his carriage was exactly what the majority of youths, brought up in a similar fashion, would have adopted. There must have been something in the young man's humours which from the first made his father apprehensive as to the future, for Cardan soon came to see that an early marriage would be the surest safeguard for Gian Battista's future. With his mind bent on this scheme, he pointed out to his son various damsels of suitable station, any one of whom he would be ready to welcome into his family, but Gian Battista always found some excuse for declining matrimony. He declared that he was too closely engaged with his work; and, over and beyond this, it would not be seemly to bring home a bride into a house like their own, full of young men, for Cardan, as usual, had several pupils living with him. It was at the end of 1557 that the first forebodings of misfortune appeared. To Cardan, according to custom, they came in the form of a portent, for he records how he lay awake at midnight on December 20, and was suddenly conscious that his bed was shaking. He at once attributed this to a shock of an earthquake, and in the morning he demanded of the servant, Simone Sosia, who occupied the truckle bed in the room, whether he had felt the same. Simone replied that he had, whereupon Cardan, as soon as he arose, went to the piazza and asked of divers persons he met there, whether they had also been disturbed, but no one had felt anything of the shock he alluded to. He went home, and while the family were at table, a messenger, sent, as he afterwards records, by a certain woman of the town,[183] entered the room, and told him that his son was going to be married immediately after breakfast. Cardan asked who the bride might be, but the messenger said he knew not, and departed. It is not quite clear whether Gian Battista was present or not, but as soon as ever the messenger had departed, Cardan let loose an indignant outburst over his son's misconduct, reproaching him with undutiful secresy, and setting forth how he had introduced to him four young ladies of good family, of whom two were certainly enamoured of him. Any one of the four would have been acceptable as a daughter-in-law, but he declared that now he would insist upon having full information as to the antecedents of any other bride his son might have selected, before admitting her to the shelter of his roof. Over and over again had he counselled Gian Battista that he must on no account marry in haste, or without his advice, or without making sure that his income would be sufficient to support the responsibilities of the married state; rather than this should happen, he would willingly allow the young man to keep a mistress in the house for the sake of offspring, for he desired beyond all else to rear grandchildren from Gian Battista, because he nursed the belief that, as the son resembled his grandfather Fazio, so the son's children would resemble their grandfather--himself. When he was questioned, Gian Battista declared he knew nothing about the report, and was fully as astonished as his father; but two days later Gian Battista's own servant came to the house, and announced that his master had been married that same morning,[184] but that he knew not the name of the bride. Cardan now ascertained that Gian Battista's disinclination for matrimony had arisen from the fact that he had been amusing himself with a girl who was nothing else than an attractive and finely-dressed harlot, named Brandonia Seroni, the last woman in all Milan whom he could with decency receive into his house. And the pitiful story was not yet complete. In marrying her the foolish youth had burdened himself with her mother, two or more sisters, and three brothers, the last-named being rough fellows without any calling but that of common soldiers. The character of the girl herself may be judged by the answer given by her father Evangelista Seroni to Cardan during the subsequent trial. When Seroni was asked whether he had given his daughter as a virgin in marriage, he answered frankly in the negative. Cardan at once made up his mind to shut his door upon the newly-married pair; but the unconquerable tenderness he felt for Gian Battista urged him on to send to the young man all the ready money he had saved. After two years of married life, two children, a boy and a girl, were born: husband and wife alike were in ill health, and every day brought its domestic quarrel. In the meantime sinister whispers were heard, set going in the first instance by the mother and sister of Brandonia, that Gian Battista was the father neither of the first nor of the second child. They even went so far as to designate the men to whom they rightly belonged, and contrived that this rumour should come to the ears of the injured husband. The consequence of their malignant tale-bearing was a quarrel more violent than ever, and the rise of a resolution in Gian Battista's mind to rid himself at all hazard of the accursed burden he had bound upon his shoulders. Until the end of 1559 Cardan continued to live in Milan, vexed no doubt by the ever-present spectacle of the wretched case into which his beloved son had fallen. He records how the young wife, unknown to her husband, handed over to her father the wedding-ring which he (Cardan) had given to his son, along with a piece of silken stuff, in order to pledge them for money. This outrage, joined to the certain conviction that his wife was false to him, proved a provocation beyond the limits of Gian Battista's patience, and finally incited him to make a criminal attempt upon Brandonia's life. Hitherto he had been earnest enough in his desire to rid himself of his wife so long as she raged against him; but, on the restoration of peace, his anger against her would vanish. Now he had lost all patience; he laid his plans advisedly, and set to work to execute them by enlisting the cooperation of the servant who had been with him ever since his marriage, and by taking to live with him in his own house Seroni, his wife, and son and daughter.[185] It cannot be said that the would-be murderer displayed at this juncture any of the traditional Italian craft in setting about his deadly task. The day before the attempt was made he took out of pawn the goods which Evangelista Seroni had pledged, and promised his servant a gift of clothes and money if he would compass the death of Brandonia, who was still ailing from the effects of her second confinement. To this suggestion the servant, who had also warned Gian Battista of his wife's misconduct, at once assented. But even on the very day when he had fully determined to make his essay in murder he vacillated again and again, and it seemed likely that Brandonia would once more be reprieved. When he entered her bed-chamber, full of his resolve to strike for freedom, he found her lying gravely ill with an attack of fever, shivering violently, and cold at the extremities. His anger forthwith vanished, and his hand was stayed; but as if urged on by ruthless fate, the mother-in-law, and the sister, and Brandonia herself, ill as she was, attacked Gian Battista with the foulest abuse and reproaches; this was the last straw. He went out and sought his servant, and told the fellow at once to make a cake and put a poison therein. The date of this fatal action was some day early in 1560. On October 1, 1559, Cardan had left Milan, and gone back to Pavia to resume his work as professor, taking Aldo with him. He threw himself into the discharge of his office and the life of the city with his customary ardour. Over and above his work of teaching he completed his treatise _De Secretis_, and likewise found time to hold a long disputation on the decisions of Galen with Andrea Camutio, one of the most illustrious physicians of the age. Concerning this episode he writes: "In disputation I showed myself so keen of wit that all men marvelled at the instances I brought forward, but for a long time no one ventured to put me to the proof. Thus I escaped the trouble of any such undertaking until two accidents both unforeseen involved me therein. At Pavia, Branda Porro, my whilom teacher in Philosophy, interrupted me one day when I was disputing with Camutio[186] on some matter of Philosophy, for, as I have said before, my colleagues were wont to lead me on to argue in philosophy because they were well assured that it would be vain to try to get the better of me in Medicine. Now Branda began by advancing Aristotle as an authority, whereupon I, when he brought out his citation, said, 'Take care, you have left out the "_non_" which should stand after "_album_."' Then Branda contradicted me, and I, spitting out the phlegm with which I am often troubled, told him quietly that he was in the wrong. He sent for the Codex in great rage, and when it was brought I asked that it might be given to me. I then read out the words just as they stood; but he, as if he suspected that I was reading falsely, snatched the volume out of my hands, and declared that I was puting a cheat upon my hearers. When he came to the word in dispute he held his tongue forthwith, and all the others looked at me in amazement."[187] It is certain that Cardan was still vexed in mind by the trouble he had left behind him at Milan. If he had not forgiven Gian Battista, he was full of kindly thought of him. He sent him from Pavia a new silk cloak, such as physicians wear, so that he might make a better show in his calling, and doubtless continued his supplies of money. Just a week before the quarrel last recorded, Aldo, against his father's wish, left Pavia and returned to Milan. Cardan used every argument he could bring forward to keep his younger son with him, but in vain; and, as he was unwilling to put constraint upon him, Aldo departed. Cardan says that he was within an ace of going with him, for the University was then in vacation: then the crowning catastrophe might have been averted, but the same fate which was driving on the son to destruction, kept the father at Pavia. Thus it happened that Aldo was an inmate of his brother's house when the poisoned cake was made. Cardan has written down a detailed account of the perpetration of this squalid tragedy, and no clearer presentation can be given than the one which his own words supply. He writes: "Thus my son and the servant went together to make the cake, and the servant put therein secretly some of the poison which had been given him. After the cake had been made, a small piece was given to my son's wife, who was very ill at the time, but her stomach rejected it at once. Her mother ate some of it, and likewise vomited after taking it. Though Gian Battista saw what happened he did not believe that the cake was really poisoned, for two reasons. First, because he had not, in truth, ordered that the poison should be mixed therewith; and second, because his brother-in-law (Bartolomeo Sacco) had said to him, before the cake was finished, 'See that you make it big enough, for I also am minded to taste it.' Next he gave some to his father-in-law, who straightway vomited, and complained of a pricking of the tongue. He warned my son; but he, still holding that the cake was harmless, ate thereof somewhat greedily; and, after having been sick, had to lie by for some time. On the second day after this Gian Battista, and his brother, and the servant as well were taken in hold: and on the Sunday following I, having been informed of what had happened, went to Milan in great anxiety as to what I should do." The news which had been brought to Cardan at Pavia told him, over and beyond what is written above, that his son's wife was dead, poisoned as every one believed through having eaten the cake, which had caused nausea and pain to every one else who had tasted it.[188] The catastrophe was accompanied by the usual portents. Some weeks previous to the attempt Gian Battista had chanced to walk out to the Porta Tonsa, clad in the smart silk gown which his father had recently given him, and as he was passing a butcher's shop, a certain pig, one of a drove which was there, rose up out of the mud and attacked the young physician and befouled his gown. The butcher and his men, to whom the thing seemed portentous, drove off the hog with staves, but this they could only do after the beast had wearied itself, and after Gian Battista had gone away. Again, at the beginning of February following, while Cardan was in residence as a Professor at Pavia, he chanced to look at the palm of his hand, and there, at the root of the third finger of the right hand, he beheld a mark like a bloody sword. That same evening a messenger arrived from Milan with the news of his son's arrest, and a letter from his son-in-law, begging him to come at once. The mark on his hand grew and grew for fifty-three days, gradually mounting up the finger, until the last fatal day, when it extended to the tip of the finger, and shone bright like fiery blood. The morning after Gian Battista's execution the mark had almost vanished, and in a day or two no sign of it remained. Cardan hurried to Milan to hear from Bartolomeo Sacco, his son-in-law, the full extent of the calamity. Probably there were few people in the city who did not regard Gian Battista as a worthless fellow, whose death would be a gain to the State and a very light loss to his immediate friends, but Cardan was not of this mind. He turned his back upon his professional engagements at Pavia, and threw himself, heart and soul, into the fight for his son's life. He could not make up his mind as to Gian Battista's recent conduct; if he ate of the cake, he surely could not have put in poison himself, or directed others to do so; if, on the other hand, he had poisoned the cake, Cardan feared greatly that, in the simplicity of his nature, he would assuredly let his accusers know what he had done. And his mind was greatly upset by the prodigies of which he had recently had experience. For some reason or other he did not visit the accused in prison, or give him any advice as to what course he should follow, a piece of neglect which he cites as a reproach against himself afterwards; but certain associates of Gian Battista, and his fellow-captives as well, urged him to assert his innocence, a course which Cardan recognized as the only safe one. At the first examination the accused followed this counsel; at the second he began to waver when the servant deposed that his master had given him a certain powder to mix with Brandonia's food in order to increase her flow of milk; and, later on, when confronted with the man from whom he had received the poison, he confessed all; and, simpleton as he was, admitted that for two months past his mind had been set upon the deed, and that on two previous occasions he had attempted to administer to her the noxious drug against the advice of his servant. From the first Cardan had placed his hopes of deliverance in the intervention of the Milanese Governor, the Duca di Sessa, who had not long ago consulted him as physician,[189] but the Duke refused to interfere. The intervention of an executive officer in the procedure of a Court of Justice was no rare occurrence at that period, and Cardan was deeply disappointed at the squeamishness or indolence of his whilom patient. He records afterwards how the Duke met his full share of the calamities which fell upon all those who were concerned in Gian Battista's condemnation;[190] and in the _Dialogus Tetim_, a work which he wrote immediately after the trial, he bewails afresh the inaction of this excellent ruler and the consequent loss of his son.[191] For twenty days and more, while Gian Battista lay in prison, Cardan, almost mad with apprehension and suspense, spent his time studying in the library at Milan. Sitting there one day, he heard a warning voice which told him that the thing he most feared had indeed come to pass. He felt that his heart was broken, and, springing up, he rushed out into the court, where he met certain of the Palavicini, the friends with whom he was staying, and cried out, "Alas, alas, he was indeed privy to the death of his wife, and now he has confessed it all, therefore he will be condemned to death and beheaded." Then having caught up a garment he went out to the piazza, and, before he had gone half-way he met his son-in-law, who asked him in sorrowful tones whither he was going. Cardan answered that he was troubled with apprehensions lest Gian Battista should have confessed his crime, whereupon Bartolomeo Sacco told him that what he feared had indeed come to pass. Gian Battista had admitted the truth of the charge against him: he was ultimately put on his trial before the Senate of Milan,[192] the President of the Court being one Rigone, a man whom Cardan afterwards accused of partiality and of a hostile bias towards the prisoner. Cardan himself stood up to defend his son; but with a full confession staring him in the face, he was sorely puzzled to fix upon a line of defence. This he perceived must of necessity be largely rhetorical; and, after he had grasped the entire situation, he set to work to convince the Court on two main points, first, that Gian Battista was a youth of simple guileless character; and, second, there was no proof that Brandonia had died of poison. A physician of good repute, Vincenzo Dinaldo, swore that she had died of fever (_lipyria_), and not from the effect of poison; and five others, men of the highest character, declared that she bore no signs of poison, either externally or internally. Her tongue and extremities and her body were not blackened, nor was the stomach swollen, nor did the hair and nails show any signs of falling, nor were the tissues eaten away. In the opening of his defence Cardan attempted to discredit the character of Brandonia. He showed how great were the injuries and provocations which Gian Battista had received from her, and that she was a dissolute wanton; her father himself, when under examination, having refused to say that she was a virgin when she left his house to be married. He claimed justification for the husband who should slay his wife convicted of adultery; and here, in this case, Brandonia was convicted by her own confession. He maintained that, if homicide is to be committed at all, poison is preferable to the knife, and then he went on to weave a web of ineffectual casuistry in support of his view, which moved the Court to pity and contempt. He cited the _Lex Cornelia_, which doomed the common people to the arena, and the patricians to exile, and claimed the penalty last-named as the one fitting to the present case.[193] Then he proceeded to show that the woman had really died from natural causes; for, even granting that she had swallowed arsenic in the cake, she had vomited at once, and the poison would have no time to do its work; moreover there was no proof that Gian Battista had given specific directions to anybody to mix poison with the ingredients of the cake. The most he had done was to utter some vague words thereanent to his servant, who forthwith took the matter into his own hands.[194] If Gian Battista had known, if he had merely been suspicious that the cake was poisoned, would he have let a crumb of it pass his lips; and if any large quantity of poison had been present, would he and the other persons who had eaten thereof have recovered so quickly? Cardan next went on to argue that, whatever motive may have swayed Gian Battista at this juncture, it could not have been the deliberate intent to kill his wife, because forsooth the wretched youth was incapable of deliberate action of any sort. He could never keep in the same mood for four-and-twenty hours at a stretch. He nursed alternately in his heart vengeance and forgiveness, changing as discord or peace ruled in his house. Cardan showed what a life of misery the wretched youth had passed since his marriage. Had this life continued, the finger of shame would have been pointed at him, he must have lost his status as a member of his profession, and have been cut off from the society of all decent people; nay, he would most likely have died by the hand of one or other of his wife's paramours. This was to show how powerful was the temptation to which the husband was exposed, and again he sang the praises of poison as an instrument of "removal"; because if effectively employed, it led to no open scandal. He next brought forward the simple and unsophisticated character of the accused, and the physical afflictions which had vexed him all his life, giving as illustrations of his son's folly the headlong haste with which he had rushed into a marriage, his folly in giving an ineffectual dose, if he really meant to poison his wife, in letting his plot be known to his servant, and in confessing. Lastly, Cardan had in readiness one of his favourite portents to lay before the Court. When Brandonia's brother had come into the house and found his father and sister sick through eating the cake, he suspected foul play and rushed at Gian Battista and at Aldo who was also there, and threatened them with his sword; but before he could harm them he fell down in a fit, his hand having been arrested by Providence. Providence had thus shown pity to this wretched youth, and now Cardan besought the Senate to be equally merciful. Cardan's pleas were all rejected; indeed such issue was inevitable from the first, if the Senate of Milan were not determined to abdicate the primary functions of a judicial tribunal. Gian Battista was condemned to death, but a strange condition was annexed to the sentence, to wit that his life would be spared, if the prosecutors, the Seroni family, could be induced to consent. But their consent was only to be gained by the payment of a sum of money entirely beyond Cardan's means, their demand having been stimulated through some foolish boasting of the family wealth by the condemned prisoner.[195] Cardan was powerless to arrest the course of the law, and Gian Battista was executed in prison on the night of April 7, 1560. In the whole world of biographic record it would be hard to find a figure more pathetic than that of Cardan fighting for the life of his unworthy son. No other episode of his career wins from the reader sympathy half so deep. The experience of these terrible days certainly shook still further off its balance a mind not over steady in its calmest moments. Cardan wrote voluminously and laboriously over Gian Battista's fate, but in his dirges and lamentations he never lets fall an expression of detestation or regret with regard to the crime itself: all his soul goes out in celebrating the charm and worth of his son, and in moaning over the ruin of mind, body, and estate which had fallen upon him through this cruel stroke of adverse fate. When he sat down to write the _De Vita Propria_, Cardan was strongly possessed with the belief that all through his career he had been subject to continuous and extraordinary persecution at the hands of his enemies. The entire thirtieth chapter is devoted to the description of these plots and assaults. In his earlier writings he attributes his calamities to evil fate and the influences of the stars; his wit was indeed great, and assuredly it was allied to madness, so it is not impossible that these personal foes who dogged his steps were largely the creatures of an old man's monomaniacal fancies. The persecution, he affirms, began to be so bitter as to be almost intolerable after the condemnation of Gian Battista. "Certain members of the Senate afterwards admitted (though I am sure they would be loth that men should hold them capable of such a wish) that they condemned my son to death in the hope that I might be killed likewise, or at least might lose my wits, and the powers above can bear witness how nearly one of these ills befell me. I would that you should know what these times were like, and what practices were in fashion. I am well assured that I never wrought offence to any of these men, even by my shadow. I took advice how I might put forward a defence of some kind on my son's behalf, but what arguments would have prevailed with minds so exasperated against me as were theirs?"[196] FOOTNOTES: [176] _De Vita Propria_, p. 57. [177] "In ore illud semper ei erat: Omnis spiritus laudet Dominum, qui ipse est fons omnium virtutum."--_De Vita Propria_, ch. iii. p. 7. Reginald Scot, in the _Discoverie of Witchcraft_, says that the aforesaid exclamation of Fazio was the Paracelsian charm to drive away spirits that haunt any house. There is a passage in _De Consolatione_ (_Opera_, tom. i. p. 600) which gives Fazio's view of happiness after death:--"Memineram patrem meum, Facium Cardanum, cum viveret, in ore semper habuisse, se mortem optare, quod nullum suavius tempus experiretur, qu[a=] id in quo profundissime dormiens omnium quæ in hac vita fiunt expers esset." [178] Cardan gives his impressions of musicians:--"Unde nostra ætate neminem ferine musicum invenias, qui non omni redundat vitiorum genere. Itaque hujusmodi musica maximo impedimento non solum pauperi et negotioso viro est, sed etiam omnibus generaliter. Quin etiam virorum egregiorum nostræ ætatis neminem musicum agnovimus, Erasmum, Alciatum, Budæum, Jasonem, Vesalium, Gesnerum. At vero quod domum everterit meam, si dicam, vera fatebor meo more. Nam et pecuniæ non levem jacturam feci, et quod majus est, filiorum mores corrupi. Sunt enim plerique ebrii, gulosi, procaces, inconstantes, impatientes, stolidi, inertes, omnisque libidinis genere coinquinati. Optimi quique inter illos stulti sunt."--_De Utilitate_, p. 362. [179] _De Vita Propria_, ch. xiii. p. 45. [180] "Quid profuit hæc tua industria, quis infelicior in filiis? quorum alter male periit: alter nec regi potest nec regere?"--_Opera_, tom. i. p. 109. [181] _Opera_, tom. i. p. 614. [182] "In cæteris erit elegans, splendidus, humanus, gravis et qui ab omnibus, potentioribusque, præsertim probetur."--_Geniturarum Exempla_, p. 464. [183] "A scorto nuntius venit."--_De Utilitate_, p. 833. [184] This incident is taken from the _De Utilitate_, which was written soon after the events chronicled. The account given in the _De Vita Propria_, written twenty years later, differs in some details. "Venio domum, accurrit famulus admodum tristis, nunciat Johannem Baptistam duxisse uxorem Brandoniam Seronam."--_De Vita Propria_, ch. xli. p. 147. [185] Cardan in describing this action of Gian Battista, who was then determined to murder his wife, says of him: "Erat enim natura clemens admodum et gratus."--_De Utilitate_, p. 834. [186] "Triduana illa disceptatio Papiæ cum Camutio instituta, publicata apud Senatum: ipse primo argumento primæ diei siluit."--_De Vita Propria_, ch. xii. p. 37. This does not exactly tally with Camutio's version. With regard to Cardan's assertion that his colleagues hesitated to meet him in medical discussion it may be noted that Camutio printed a book at Pavia in 1563, with the following title: "Andrææ Camutii disputationes quibus Hieronymi Cardani magni nominis viri conclusiones infirmantur, Galenus ab ejusdem injuria vindicatur, Hippocratis præterea aliquot loca diligentius multo quam unquam alias explicantur." In his version (_De Vita Propria_, ch. xii. p. 37) Cardan inquires sarcastically: "Habentur ejusdem imagines quædam typis excusæ in Camutii monumentis." [187] _De Vita Propria_, ch. xii. p. 39. The Third Book of the _Theonoston_ (_Opera_, tom. ii. p. 403) is in the form of a disputation, "De animi immortalite," with this same Branda. [188] In his defence at the trial Cardan affirmed that, while Brandonia was lying sick from eating the cake, her mother and the nurse quarrelled and fought, and finally fell down upon the sick woman. When the fight was over Brandonia was dead. In _Opera_, tom. ii. p. 311 (_Theonoston_, lib. i.) he writes: "Obiit illa non veneno, sed vi morbi atque Fato quo tam inclytus juvenis morte sua, omnia turbare debuerat." [189] "Vocatus sum enim ad Ducem Suessanum ex Ticinensi Academia accepique C. aureos coronatos et dona ex serico."--_De Vita Propria_, ch. xl. p. 138. [190] _De Vita Propria_, ch. xli. p. 153. [191] _Opera_, tom. i. p. 671. He cites the names of former Governors of Milan and other patrons, many of them harsh men, and not one as kind and beneficent as the Duca di Sessa; to wit Antonio Leva, Cardinal Caracio, Alfonso d'Avalos, Ferrante Gonzaga, the Cardinal of Trent, and the Duca d'Alba. Yet the rule of his best friend brought him his worst misfortune. [192] There is a full account of the trial in an appendix to the _De Utilitate ex Adversis Capienda_ (Basel, 1561). It is not included in the edition hitherto cited. [193] Laudabatur ejus benignitas aC simul factum Io. Petri Solarii tabellionis, qui cum filium spurium convictum haberet de veneficio, in duas sorores legitimas, solum hæreditatis consequendæ causa, satis habuit damnasse illum ad triremes."--_De Vita Propria_, ch. x. p. 33. [194] "Evasit nuper ob constantiam in tormentis famulus filii mei, qui pretio venenum dederat dominæ sine causa: periit filius meus, qui nec jusserat dari."--_De Utilitate_, p. 339. [195] Gian Battista seems to have boasted about the family wealth, and thus stirred up the Seroni to demand an excessive and impossible sum. "Hæc et alia hujusmodi cum protulissem, non valere, nisi eousque, ut decretum sit, si impetrare pacem potuissem vitæ parceretur. Sed non potuit filii stultitia, qui dum jactat opes quæ non sunt, illi quod non erat exigunt."--_De Vita Propria_, ch. x. p. 34. [196] _De Vita Propria_, ch. x. p. 33. CHAPTER X CARDAN had risen to high and well-deserved fame, and this fact alone might account for the existence of jealousy and ill-feeling amongst certain of those whom he had passed in the race. Some men, it is true, rise to eminence without making more than a few enemies, but Cardan was not one of these. His foes must have been numerous and truculent, the assault they delivered must have been deadly and overwhelming to have brought to such piteous wreck fortunes which seemed to rest upon the solid ground of desert. The public voice might accuse him of folly, but assuredly not of crime; he was the victim and not the culprit; his skill as a physician was as great as ever, but these considerations weighed little with the hounds who were close upon his traces. Now that the tide of his fortune seemed to be on the ebb they gathered around him. He writes: "And this, in sooth, was the chief, the culminating misfortune of my life: forasmuch as I could not with any show of decency be kept in my office, nor could I be dismissed without some more valid excuse, I could neither continue to reside in Milan with safety, nor could I depart therefrom. As I walked about the city men looked askance at me; and whenever I might be forced to exchange words with any one, I felt that I was a disgraced man. Thus, being conscious that my company was unacceptable, I shunned my friends. I had no notion what I should do, or whither I should go. I cannot say whether I was more wretched in myself than I was odious to my fellows."[197] Cardan gathered a certain amount of consolation from meditating over the ills which befell all those who were concerned in Gian Battista's fate. The Senator Falcutius, a man of the highest character in other respects, died about four months later, exclaiming with his dying breath that he was undone through the brutal ignorance of a certain man, who had been eager for the death sentence. One Hala shortly afterwards followed Falcutius to the grave, having fallen sick with phthisis immediately after the trial. Rigone, the President of the Court, lost his wife, and gave her burial bereft of the usual decencies of the last rite, a thing which Cardan says he could not have believed, had he not been assured of the same by the testimony of many witnesses. It was reported too, that Rigone himself, though a man of good reputation, was forced to feign death in order to escape accusation on some charge or other. His only son had died shortly before, so it might be said with reason that his house was as it were thrown under an evil spell by the avenging Furies of the youth whom he had sent to die in a dungeon. Again, within a few days the prosecutor himself, Evangelista Seroni, the man who was the direct cause of his son-in-law's death, was thrown into prison, and, having been deprived of his office of debt collector, became a beggar. Moreover, the son whom he specially loved was condemned to death in Sicily, and died on the gallows. Public and private calamity fell upon the Duca di Sessa,[198] the Governor of Milan, doubtless because he had allowed the law to take its course. Indeed every person great or small who had been concerned in Gian Battista's condemnation, was, by Cardan's showing, overtaken by grave misfortune. Cardan still held his Professorship at Pavia, and in spite of the difficulties and embarrassments of his position he went back to resume his work of teaching a few days after the fatal issue of his son's trial and condemnation. By the pathetic simplicity of its diction the following extract gives a vivid and piteous picture of the utter desolation and misery into which he was cast: it shows likewise that, after a lapse of fifteen years, the memory of his shame and sorrow was yet green, and that a powerful stimulus had been given to his superstitious fancies by the events lately chronicled. "In the month of May, in the year MDLX, a time when sleep had refused to come to me because of my grief for my son's death: when I could get no relief from fasting nor from the flagellation I inflicted upon my legs when I rode abroad, nor from the game of chess which I then played with Ercole Visconti, a youth very dear to me, and like myself troubled with sleeplessness, I prayed God to have pity upon me, because I felt that I must needs die, or lose my wits, or at least give up my work as Professor, unless I got some sleep, and that soon. Were I to resign my office, I could find no other means of earning my bread: if I should go mad I must become a laughing-stock to all. I must in any case lavish what still remained of my patrimony, for at my advanced age I could not hope to find fresh employment. Therefore I besought God that He would send me death, which is the lot of all men. I went to bed: it was already late, and, as I must needs rise at four in the morning, I should not have more than two hours' rest. Sleep, however, fell upon me at once, and meseemed that I heard a voice speaking to me out of the darkness. I could discern naught, so it was impossible to say what voice it was, or who was the speaker. It said, 'What would you have?' or 'What are you grieving over?' and added, 'Is it that you mourn for your son's death?' I replied, 'Can you doubt this?' Then the voice answered, 'Take the stone which is hanging round your neck and place it to your mouth, and so long as you hold it there you will not be troubled with thoughts of your son.' Here I awoke, and at once asked myself what this beryl stone could have to do with sleep, but after a little, when I found no other means of escape from my trouble, I called to mind the words spoken of a certain man: 'He hoped even beyond hope, and it was accounted to him as righteousness' (spoken of Abraham), and put the stone in my mouth, whereupon a thing beyond belief came to pass. In a moment all remembrance of my son faded from my mind, and the same thing happened when I fell asleep a second time after being aroused."[199] The record of Cardan's life for the next two years is a meagre one. His rest was constantly disturbed either by the machinations of his foes or by the dread thereof, the evil last-named being probably the more noxious of the two. As long ago as 1557 he had begun the treatise _De Utilitate ex Adversis Capienda_, a work giving evidence of careful construction, and one which, as a literary performance, takes the first rank.[200] This book had been put aside, either through pressure of other work or family troubles, but now the circumstances in which he found himself seemed perfectly congenial for the elaboration of a subject of this nature, so he set to work to finish it, concluding with the chapter _De Luctu_, which has been used largely as the authority for the foregoing narrative of Gian Battista's crime and death. At this period, when his mind was fully stored and his faculties adequately disciplined for the production of the best work, he seems to have realized with sharp regret that the time before him was so short, and that whatever fresh fruit of knowledge he might put forth would prove of very slight profit to him, as author. Writing of his replies given to certain mathematical professors, who had sent him problems for solution, he remarks that, although he may have a happy knack of dispatching with rapidity any work begun, he always begins too late. In his fifty-eighth year he answered one of these queries, involving three very difficult problems, within seven days; a feat which he judges to be a marvel: but what profit will it bring him now? If he had written this treatise when he was thirty he would straightway have risen to fame and fortune, in spite of his poverty, his rivals, and his enemies. Then, in ten years' space, he would have finished and brought out all those books which were now lying unfinished around him in his old age; and moreover would have won so great gain and glory, that no farther good fortune would have remained for him to ask for. Another work which he had begun about the same time (1558) was the treatise on _Dialectic_, illustrated by geometrical problems and theorems, and likewise by the well-known logical catch lines _Barbara Celarent_. During the summer vacation of 1561 he returned to Milan, and began a _Commentary on the Anatomy of Mundinus_, the recognized text-book of the schools up to the appearance of Vesalius. In the preface to this work he puts forward a vigorous plea for the extended use of anatomy in reaching a diagnosis.[201] He had likewise on hand the _Theonoston_, a set of essays on Moral subjects written something in the spirit of Seneca; and, after Gian Battista's death, he wrote the dialogue _Tetim, seu de Humanis Consiliis_. In the year following, 1561, a farther sorrow and trouble came upon him by the death of the English youth, William. If he was guilty of neglect in the case of this young man--and by his own confession he was--he was certainly profoundly grieved at his death. In the Argument to the _Dialogus de Morte_ he laments that he ever let the youth leave his house without sending him back to England, and tells how he was cozened by Daldo, the crafty tailor, out of a premium of thirty-one gold crowns, in return for which William was to be taught a trade. "But during the summer, Daldo, who had a little farm in the country, took the youth there and let him join in the village games, and by degrees made him into a vinedresser. But if at any time it chanced that William's services were also wanted at the tailor's shop, his master would force him to return thereto in the evening (for the farm was two miles distant), and sit sewing all the night. Besides this the boy would go dancing with the villagers, and in the course of their merry-making he fell in love with a girl. While I was living at Milan he was taken with fever, and came to me; but, for various reasons, I did not give proper attention to him, first, because he himself made light of his ailment; second, because I knew not that his sickness had been brought on by excessive toil and exposure to the sun; and third, because, when he had been seized with a similar distemper on two or three occasions before this, he had always got well within four or five days. Besides this I was then in trouble owing to the running away of my son Aldo and one of my servants. What more is there to tell? Four days after I had ordered him to be bled, messengers came to me in the night and begged me to go and see him, for he was apparently near his end. He was seized with convulsions and lost his senses, but I battled with the disease and brought him round. I was obliged to return to Pavia to resume my teaching, and William, when he was well enough to get up, was forced to sleep in the workshop by his master, who had been bidden to a wedding. There he suffered so much from cold and bad food that, when he was setting out for Pavia to seek me, he was again taken ill. His unfeeling master caused him to be removed to the poor-house, and there he died the following morning from the violence of the distemper, from agony of mind, and from the cold he had suffered. Indeed I was so heavily stricken by mischance that meseemed I had lost another son." It was partly as a consolation in his own grief, and partly as a monument to the ill-fated youth, that Cardan wrote the _Dialogus de Morte_, a work which contains little of interest beyond the record of Cardan's impressions of Englishmen already quoted. But it was beyond hope that he should find adequate solace for the gnawing grief and remorse which oppressed him in this, or any other literary work. He was ill looked upon at Milan, but his position at Pavia seems to have been still more irksome. He grew nervous as to his standing as a physician, for, with the powerful prejudice which had been raised against him both as to his public and his private affairs, he felt that a single slip in his treatment of any particular case would be fatal to him. In Milan he did meet with a certain amount of gratitude from the wealthier citizens for the services he had wrought them; but in Pavia, his birthplace, the public mind was strongly set against him; indeed in 1562 he was subjected to so much petty persecution at the hands of the authorities and of his colleagues, that he determined to give up his Professorship at all cost. He describes at great length one of the most notable intrigues against him. "Now in dealing with the deadly snares woven against my life, I will tell you of something strange which befell me. During my Professorship at Pavia I was in the habit of reading in my own house. I had in my household at that time a woman to do occasional work, the youth Ercole Visconti, two boys, and another servant. Of the two boys, one was my amanuensis and well skilled in music, and the other was a lackey. It was in 1562 that I made up my mind to resign my office of teaching and quit Pavia, a resolution which the Senate took in ill part, and dealt with me as with a man transported with rage. But there were two doctors of the city who strove with all their might to drive me away: one a crafty fellow who had formerly been a pupil of mine; the other was the teacher extraordinary in Medicine, a simple-minded man, and, as I take it, not evil by nature; but covetous and ambitious men will stop at nothing, especially when the prize to be won is an office held in high esteem. Thus, when they despaired of getting rid of me through the action of the Senate--what though I was petitioning to be relieved of my duties--they laid a plot to kill me, not by the dagger for fear of the Senate and of possible scandal, but by malignant craft. My opponent perceived that he could not be promoted to the post of principal teacher unless I should leave the place, and for this reason he and his allies spread their nets from a distance. In the first place, they caused to be written to me, in the name of my son-in-law[202] and of my daughter as well, a most vile and filthy letter telling how they were ashamed of their kinship with me; that they were ashamed likewise for the sake of the Senate, and of the College; and that the authorities ought to take cognizance of the matter and pronounce me unworthy of the office of teacher and cause me to be removed therefrom forthwith. Confounded at receiving such an impudent and audacious reproof at the hands of my own kindred, I knew not what to do or say, or what reply I should make; nor could I divine for what reason this unseemly and grievous affront had been put upon me. It afterwards came to light that the letter was written in order to serve as an occasion for fresh attacks; for, before many days had passed, another letter came to me bearing the name of one Fioravanti, written in the following strain. This man was likewise shocked for the sake of the city, the college, and the body of professors, seeing that a report had been spread abroad that I was guilty of abominable offences which cannot be named. He would call upon a number of his friends to take steps to compel me to consider the public scandal I was causing, and would see that the houses where these offences were committed should be pointed out. When I read this letter I was as one stupefied, nor could I believe it was the work of Fioravanti, whom I had hitherto regarded as a man of seemly carriage and a friend. But this letter and its purport remained fixed in my mind and prompted me to reply to my son-in-law; for I believed no longer that he had aught to do with the letter which professed to come from him; indeed I ought never to have harboured such a suspicion, seeing that both then and now he has always had the most kindly care for me; nor has he ever judged ill of me. "I called for my cloak at once and went to Fioravanti, whom I questioned about the letter. He admitted that he wrote it, whereupon I was more than ever astonished, for I was loth to suspect him of crooked dealing, much more of any premeditated treachery. I began to reason with him, and to inquire where all these wonderful plans had been concocted, and then he began to waver, and failed to find an answer. He could only put forward common report, and the utterances of the Rector of the Gymnasium, as the source of them."[203] Cardan goes on to connect the foregoing incident, by reasoning which is not very clear, with what he maintained to have been a veritable attempt against his life. "The first act of the tragedy having come to an end, the second began, and this threw certain light upon the first. My foes made it their special care that I, whom they held up as a disgrace to my country, to my family, to the Senate, to the Colleges of Milan and Pavia, to the Council of Professors, and to the students, should become a member of the Accademia degli Affidati, a society in which were enrolled divers illustrious theologians, two Cardinals, and two princes, the Duke of Mantua, and the Marquis Pescara. When they perceived how loth I was to take this step they began to threaten. What was I to do, broken down by the cruel fate of my son, and suffering every possible evil? Finally I agreed, induced by the promise they made me, that, in the course of a few days, I should be relieved of my duties as Professor; but I did not then perceive the snare, or consider how it was that they should now court the fellowship of one whom, less than fifteen days ago, all ranks of the College had declared to be a monster not to be tolerated. Alas for faith in heaven, for the barbarity of men, for the hatred of false friends, for that shamelessness and cruelty more fell than serpent's bite! What more is there to tell? The first time I entered the room of the Affidati I saw that a heavy beam had been poised above in such fashion that it might easily fall and kill whatsoever person might be passing underneath. Whether this had been done by accident or design I cannot say. But hereafter I attended as rarely as possible, making excuses for my absence; and, when I did go, I went when no one looked for me, and out of season, taking good heed of this trap the while. Wherefore no evil befell me thereby, either because my foes deemed it unwise to work such wickedness in public, or because they had not finally agreed to put their scheme in operation, or because they were plotting some fresh evil against me. Another attempt was made a few days later, when I was called to the ailing son of one Piero Trono, a surgeon; they placed high over the door a leaden weight which might easily be made to fall, pretending that it had been put there to hold up the curtain. This weight did fall; and, had it struck me, it would certainly have killed me: how near I was to death, God knows. Wherefore I began to be suspicious of something I could not define, so greatly was my mind upset. Then a third attempt was made, which was evident enough. A few days later, when they were about to sing a new Mass, the same rascally crew came to me, asking me whether I would lend them the services of my two singing boys, for my enemies knew well enough that these boys acted as my cup-bearers, and over and beyond this they made an agreement with my hired woman that she should give me poison. They first went to Ercole and tried to persuade him to go to the function; and he, suspecting nothing, at first promised his help; but when he heard that his fellow was to go likewise, he began to smell mischief and said, 'Only one of us knows music.' Then Fioravanti, a blunt fellow, was so wholly set on getting them out of the house that he said, 'Let us have both of you, for we know that the other is also a musician; and, though he may not be one of the best, still he will serve to swell the band of choristers.' Then Ercole said somewhat vaguely that he would ask his master. He came to me, having fathomed and laid bare the whole intention of the plot, so that, if I had not been stark mad and stupid, I might easily have seen through their design. Fifteen days or so had passed when the same men once more sought me out and begged me to let them have the two boys to help them in the performance of a comedy. Then Ercole came to me and said, 'Now in sooth the riddle is plain to read; they are planning to get all your people away from your table, so that they may kill you with poison; nor are they satisfied with plotting your death merely by tricks of this sort; they are determined to kill you by any chance which may offer."[204] How far these plots were real, and how far they sprang from monomania it is impossible to say. Cardan's relations with his brother physicians had never been of the happiest, and it is quite possible that a set may have been made in the Pavian Academy to get rid of a colleague, difficult to live with at the best, and now cankered still more in temper by misfortune, and likewise, in a measure, disgraced by the same. Surrounded by annoyances such as these, and tormented by the intolerable memories and associations of the last few years, it is not wonderful that he should seek a way out of his troubles by a change of scene and occupation. As early as 1536 Cardan had had professional relations with certain members of the Borromeo family, which was one of the most illustrious in Milan, and in 1560 Carlo Borromeo was appointed Archbishop of Milan. There is no record of the date when Cardan first made acquaintance with this generous patron, who was the nephew of the reigning Pope, Pius IV., himself a Milanese, but it is certain that Cardan had at an earlier date successfully treated the mother of the future Cardinal,[205] wherefore it is legitimate to assume that the physician was _persona grata_ to the whole family. As soon as Cardan had determined to withdraw from Pavia he applied to the Cardinal, who had just made a magnificent benefaction to Bologna in the form of the University buildings. He espoused Cardan's interests at once, and most opportunely, for the protection of a powerful personage was almost as needful at Bologna, as the sequel shows, as it would have been at Pavia. It was evident that Cardan had foes elsewhere than in Pavia; indeed the early stages of the negotiation, which went on in reference to his transfer to Bologna, suggest a doubt whether the change would bring him any advantage other than the substitution of one set of enemies for another. He writes: "When I was about to be summoned to teach at Bologna, some persons of that place who were envious of my reputation sent a certain officer (a getter-up of petitions) to Pavia. Now this fellow, who never once entered the class-room, nor had a word with any one of my pupils, wrote, on what authority I know not, a report in these words: 'Concerning Girolamo Cardano, I am told that he taught in this place, but got no pupils, always lecturing to empty benches: that he is a man of evil life, ill regarded by all, and little less than a fool, repulsive in his manners, and entirely unskilled in medicine. After he had promulgated certain of his opinions he found no one in the city who would employ him, nor did he practise his art.' "These words were read to the Senate by the messenger on his return in the presence of the illustrious Borromeo, the Pope's Legate to the city. The Senate were upon the point of breaking off all further negotiations, but while the man was reading his report, some one present heard the words in which he declared that I did not practise medicine. 'Hui!' he cried, 'I know that is not true, for I myself have seen divers men of the highest consideration going to him for help, and I--though I am not to be ranked with them--have often consulted him myself.' Then the Legate took up the parole and said, 'I too bear witness that he cured my own mother when she was given up by every one else.' Then the first speaker suggested that probably the rest of the tale was just as worthy of belief as this one statement, the Legate agreeing thereto; whereupon the messenger aforesaid held his tongue and blushed for shame. Ultimately the Senate determined to appoint me Professor for one year, 'for,' they said, 'if he should prove to be the sort of man the officer describes, or if his teaching should profit us nothing, we can let him go; but if it be otherwise, the contract may be ratified.' With regard to the salary, over which a dispute had already arisen, the Legate gave his consent, and the business came to an end. "But, disregarding this settlement, my opponents urged one of their number to wait upon me as a delegate from the Senate, and this man would fain have added to the terms already sanctioned by the Senate, others which I could not possibly accept. He offered me a smaller stipend, no teaching room was assigned to me, and no allowance for travelling expenses. I refused to treat with him, whereupon he was forced to depart, and to return to me later on with the terms of my engagement duly set forth."[206] It was in June 1562 that Cardan finally resigned his position at Pavia, but it was not until some months after this date that the final agreement with the Bolognese Senate, lately referred to, was concluded, and in the interim he was forced to suffer no slight annoyance and persecution at the hands of his adversaries in Pavia, in Bologna, and in Milan as well. Just before he resigned his Professorship he was warned by the portentous kindling of a fire, seemingly dead,[207] that fresh mischief was afoot, and he at once determined in his mind that his foes had planned destruction against him afresh. So impressed was he at this manifestation that he swore he would not leave home on the day following. "But early in the morning there came to my house four or five of my pupils bidding me to a feast, where all the chief Professors of the Gymnasium and the Academy proposed to be present. I replied that I could not come, whereupon they, knowing that it was not my wont to dine in the middle of the day, and deeming that it was on this score that I refused to join them, said, 'Then for your sake we will make the feast a supper.' I answered that I could not on any account make one of their party, and then they demanded to know the cause of my refusal. I replied it was because of a strange event which had befallen me, and of a vow I had made thereanent. At this they were greatly astonished, and two of them exchanged significant glances, and they urged me again and again that I should not be so firmly set upon marring so illustrious a gathering by my absence, but I gave back the same answer as before."[208] They came a second time, but Cardan was not to be moved. He records, however, that he did break his vow after all by going out after dusk to see a poor butcher who was seriously ill. It is hard to detect any evidence of deadly intent in what seems, by contemporary daylight, to have been a complimentary invitation to dinner; but to the old man, possessed as he was by hysterical terrors, this episode undoubtedly foreshadowed another assault against his life. He finds some compensation, however, in once more recording the fact that all these disturbers of his peace--like the men who were concerned in Gian Battista's condemnation--came to a bad end. His rival, who had taken his place as Professor, had not taught in the schools more than three or four times before he was seized with disease and died after three months' suffering. "Upon him there lay only the suspicion of the charge, but I heard afterwards that a friend of his was certainly privy to the deed of murder which they had resolved to work upon me by giving me a cup of poisoned wine at the supper. In the same year died Delfino, and a little while after Fioravanti."[209] In July Cardan withdrew to Milan, where, to add to his other troubles, he was seized with an attack of fever. He was now thoroughly alarmed at the look of his affairs. Many of his fears may have been imaginary, but the burden of real trouble which he had to carry was one which might easily bring him to the ground, and, when once a man is down, the crowd has little pity or scruple in trampling him to death. He set about to review his position, and to spy out all possible sources of danger. He writes: "I called to mind all the books I had written, and, seeing that in them there were many obscure passages upon which an unfavourable meaning might be put by the malice of my enemies, I wrote to the Council, submitting all my writings to its judgment and will and pleasure. By this action I saved myself from grave danger and disgrace in the future."[210] The Council to which Cardan here refers was probably the Congregation of the Index appointed by the Council at Trent for the authoritative examination of all books before allowing them to be read by the faithful. Before the close of the Council (1563) these duties had been handed over to the Pope (Pius IV.), who published the revised and definite Roman Index in 1564. FOOTNOTES: [197] _De Vita Propria_, ch. xxvii. p. 71. [198] "Quin etiam dominus ac Princeps alioquin generosus et humanus, cum ipsum ob invidiam meam et accusatorum multitudinem deseruisset, et ipse multis modis conflictatus est gravibus morbis, cæde propriæ neptis à conjuge suo, litibus gravibus: tum etiam subsecuta calamitas publica, Zotophagite insula amissa, classe regia dissipata."--_De Vita Propria_, ch. xli. p. 153. The island alluded to must have been _Lotophagites insula_, an island near the Syrtes Minor on the African coast, and the loss of the same probably refers to some disaster during the Imperialist wars against the Moors. [199] _De Vita Propria_, ch. xliii. p. 160. [200] Cardan rates it as his best work on an ethical subject.--_Opera_, tom i. p. 146. And on p. 115 he writes: "Utinam contigisset absolvere ante errorem filii; neque enim ille errasset, nec errandi causam aliquam habuisset: nec, etiamsi errasset, periisset." He also quotes a letter full of sound and loving counsels which he had sent to Gian Battista six months before he fell into the snare. [201] _Opera_, tom. x. p. 129. [202] Bartolomeo Sacco was evidently living at Pavia at this date. [203] _De Vita Propria_, ch. xxx. p. 83. [204] _De Vita Propria_, ch. xxx. p. 86. [205] _De Vita Propria_, ch. xvii. p. 55. [206] _De Vita Propria_, ch. xvii. p. 54. [207] _Ibid.,_ ch. xxx. p. 88. There is also a long account of this occurrence in _Opera_, tom. x. p. 459. [208] _De Vita Propria_, ch. xxx. p. 89. [209] _De Vita Propria_, ch. xxx. p. 90. [210] _Opera_, tom. x. p. 460. CHAPTER XI WHILE Cardan was lying sick at Milan, a messenger came from Pavia, begging him to hasten thither to see his infant grandson, who had been ailing when he left Pavia, and was now much worse. The journey under the burning sun of the hottest summer known for many years aggravated his malady, but he brought the child out of danger. He caught erysipelas in the face, and to this ailment succeeded severe trouble with the teeth. If it had not been for the fact that the time of the new moon had been near, he says that he must have submitted to blood-letting; but after the new moon his health mended, and thus he escaped the two-fold danger--that of the disease, and that of the lancet. He tells of an attempt made against his life by a servant for the sake of robbery, an attempt which came very near success; and of a severe attack of gout in the knee. After a month's confinement to his house he began to practise Medicine; and, finding patients in plenty, he nourished a hope that Fortune had done her worst, and that he might be allowed to repair his shattered fortunes by the exercise of his calling, but the activity of his adversaries--which may or may not have been provoked solely by malignity--was unsleeping. He hints at further attempts against his good name and his life, and gives at length some painful details of another charge made against him of an infamous character. It is almost certain that his way was made all the harder for him from the complaints which he had put in print about the indifference of the Duca di Sessa to his interests at the time of Gian Battista's trial. The Milanese doctors had no love for him, and every petulant word he might let fall would almost surely be brought to the Governor's ears. By Cardan's own admission it appears that utterances of this sort were both frequent and acrid. There was a certain physician of the city who wished to place his son gratis in Cardan's household. Cardan, however, refused, whereupon the physician in question called attention to a certain book in which Cardan had made some remarks to the effect that the friendship of the Duca di Sessa had been a fatal one to him, inasmuch as, having trusted too entirely to this friendship for his support, he had let go other interests which might have served him better. The physician aforesaid made a second application to Cardan to receive his son, offering this time to intercede with the Governor on his behalf. This proposition roused the old man's anger, and he exclaimed that he had no need of such friendship or protection; that in fact the interruption of their good understanding had come about more by his own act than the Governor's, who had been either unable or unwilling to save Gian Battista's life. The doctor replied, in the presence of divers persons, that Gian Battista had perished through his own foolishness: if he had not confessed he would never have been condemned; that the Senate had condemned him and not the Duca di Sessa, and that Cardan was now slandering this prince most unjustly. A lot of busy-bodies had by this time been attracted by the wrangle, and these heard the doctor's accusations in full, but gathered a very imperfect notion of Cardan's reply. He indignantly denied this charge, and in his own account of the scene he affirms that he won the approbation of all who listened, by the moderation of his bearing and speech. Four days after this occurrence he again met this physician, who declared he knew for certain that a kinsman of the Duca di Sessa, a hot-tempered man, had just read some slanders written by Cardan about the Duke, and had declared he would cut the writer in half and throw his remains into the jakes; the physician went on to say that he had appeased this gentleman's resentment, and that Cardan had now no cause for fear. Cardan at once saw through the dishonesty of the fellow, who was not content with bringing forward an unjust accusation, but must likewise subject him to these calumnies and the consequent dangers. After a bout of wrangling, in which the physician sought vainly to win from him an acknowledgment of the service he had wrought, the malicious fellow shouted out to the crowd which had gathered around them that Cardan persisted in his infamous slanders against the Governor. Wanton as the charge was, Cardan felt that with his present unpopularity it might easily grow into a fatal danger. Might was right in Milan as far as he was concerned, but he determined that he must make a stand against this pestilent fellow. By good luck he met some friends, to whom he told the adventure; and while he was speaking, the gentleman who was said to have threatened him, and the slanderous physician as well, joined the gathering; whereupon one of Cardan's friends repeated the whole story to the gentleman; who, as he was quite unversed in letters, was hugely diverted at hearing himself set down as a student, and told the physician that he was a fool, thereby delivering Cardan at least from this annoyance. He had refused the terms which the party opposed to him in the Senate at Bologna had sent for his acceptance, and was still waiting to hear whether they would carry out their original propositions. It was during this time of suspense that he was subjected to strange and inexplicable treatment at the hands of the Milanese Senate, treatment which, viewed by the light of his own report--the only one extant--seems very harsh and unjust. He writes: "At the time when I was greatly angered by the action of the Bolognese agent, four of the Senators persuaded me to seek practice once more in Milan, wherefore I, having altered my plans, began to try to earn an honest living, for I reckoned that the Senate of Milan knew that I had rejected the offers from Bologna, since these offers were unjust in themselves, and put before me in unjust fashion. But afterwards, although the same iniquitous terms were offered to me, I accepted them, not indeed because I was satisfied therewith, but because of my necessity, and so that I might be free from those dangers which, as I have before stated, pressed upon me in those days. The reason why I took this step was that the Senate, by most unexpected action, removed my name from the lists of those licensed to teach; nor was this all. They warned me by a message that they had recently given hearing to a double charge against me of very grave offences, and that nothing but my position, and the interests of the College, kept them back from laying me in hold. Nevertheless, influenced by these considerations, they had been moved to reduce my punishment to that of exile. But neither my good fortune nor God deserted me; for on the same day certain things came to pass by means of which I was able, with a single word, to free myself from all suspicion upon either charge, and to prove my innocence. Moreover, I forced them to admit that no mention of this affair had ever been made before the Senate, although two graduates had informed me that it had been discussed."[211] The Senate, however, was reluctant to stultify its late action, and refused to restore Cardan's name to the list of teachers. But he was put right in the sight of the world by the sharp censure pronounced by the Senate upon those busy-bodies who had ventured to speak in its name. Cardan's last days in Milan were cheered with a brief gleam of good fortune. His foes seem to have overshot the mark, and to have aroused sympathy for the old man, who, whatever his faults, was alike an honour to his country and the victim of fortune singularly cruel. The city took him under its protection, assured of his innocence as to the widespread charges against him, and pitying his misfortunes. His friend Borromeo had probably been forwarding his interests at the Papal Court, for he records that, just at this time, certain Cardinals and men of weight wrote to him from Rome in kindly and flattering terms. On November 16, 1562, the messenger from the Senate of Bologna arrived at Milan, bearing an offer of slightly more liberal terms. They were not so favourable as Cardan wished for; but, even had they been worse, he would probably have closed with them. In spite of the benevolent attitude of his well-wishers in Milan, it irked him to be there; the faces in the streets, the town gossip, all tended to recall to him the death of his son, so he departed at once to take up his duties. At Bologna Cardan went first to live in a hired house in the Via Gombru. Aldo was nominally a member of his household; but his presence must have been a plague rather than a comfort to his father, and he took with him likewise his orphan grandson, the son of Gian Battista and Brandonia, whom he destined to make his heir on account of Aldo's ill conduct.[212] This young man seems to have been a hopeless scoundrel from the first. The ratio in which fathers apportion their affection amongst their offspring is a very capricious one, and Cardan may have been fully as wide of the mark in chiding his younger as he was in lauding the talents and virtues of his elder son. But it is certain that on several occasions the authorities shared Cardan's view of Aldo's ill behaviour. More than once he alludes to the young reprobate's shameful conduct, and the intolerable annoyance caused by the same. Many of the ancient rights of parents over their children, which might to-day be deemed excessive, were still operative in the cities of Italy, and Cardan readily invoked the help of them in trying to work reformation of a sort upon Aldo, whom he caused to be imprisoned more than once, and finally to be banished.[213] The numerous hitches which delayed his final call to Bologna were probably due to the fact that a certain party amongst the teachers there were opposed to his appointment, and things did not run too smoothly after he had taken up his residence in his new home. It was not in Cardan's nature, however much he may have been cowed and broken down by misfortune, to mix with men inimical to himself without letting them have a taste of his quality. He records one skirmish which he had with Fracantiano, the Professor of the Practice of Medicine, a skirmish which, in its details, resembles so closely his encounter with Branda Porro, at Pavia, some time before, that it suggests a doubt whether it ever had a separate existence, and was not simply a variant of the Branda legend. "It happened that he (Fracantiano) was giving an account of the passage of the gall into the stomach, and was speaking in Greek before the whole Academy (he was making the while an anatomical dissection), when I cried out, 'There is an "[Greek: ou]" wanting in that sentence.' And as he delayed making any correction of his error, and I kept on repeating my remark in a low voice, the students cried out, 'Let the _Codex_ be sent for.' Fracantiano sent for it gladly. It was brought at once, and when he came to read the passage, he found that what I had affirmed was true to a hair. He spake not another word, being overwhelmed with confusion and astonishment. Moreover the students, who had almost compelled me to come to the lecture, were even more impressed by what had happened. But from that day forth my opponent avoided all meeting with me; nay, he even gave orders to his servants that they should warn him whenever they might see me approaching, and thus he contrived that we should never foregather. One day when he was teaching Anatomy, the students brought me, by a trick, into the room, whereupon he straightway fled, and having entangled his feet in his robe, he fell down headlong. This accident caused no little confusion, and shortly afterwards he left the place, being then a man well advanced in years."[214] He had not lived long in Bologna before he was fated to experience another repetition of one of the untoward episodes of his past life, to wit the fall of a house. It was not his own house this time, but it was sufficiently near to induce him to change his abode without delay. Next door to the house he had hired in the Via Gombru stood a palace belonging to a certain Gramigna. "The entire house fell, and was ruined in a single night, and together with the house perished the owner thereof." It was believed that this man had divers powerful enemies, and, in order that he might secure his position, he contrived to bring certain of his foes into his house, having first made a mine of gunpowder under the portico, and set a match thereto. But for some reason or other the plot miscarried the night when he destined to carry it out. Gramigna went to see what was amiss, and at that very moment the mine exploded and brought the house to the ground. After this explosion Cardan moved to a house in the Galera quarter, belonging to the family of Ranucci; but he did not find this dwelling perfect, as he was forced to vacate the rooms which were most to his taste on account of the bad state of the ceilings, the plaster of which, more than once, fell down upon his head. In his _Paralipomena_, "the last fruit off an old tree," which he put together about this time, there are numerous stories of prodigies and portents; of doors which would not close, and doors which opened of their own accord; of rappings on the walls, and of mysterious thunderings and noises during the night. He tells, at length, the story, already referred to, of the strange thing which happened to him, on the eve of his departure from Pavia in 1562, while he was awaiting tidings from Rome as to his appointment at Bologna. "I wore on the index finger of my right hand a selenite stone set in a ring, and on my left a jacinth, which I never took off my finger, this stone being large and hexagonal in shape. I took the selenite from my finger and put it beneath my pillow, for I fancied it kept off sleep, wearing still the jacinth because it appeared to have the opposite effect. I slept until midnight, when I awoke and missed the ring from my left hand. I called Jacopo Antonio, a boy of fifteen years of age who acted as my servant and slept in a truckle bed, and bade him look for my rings. He found the selenite at once where I had placed it; but though we both of us sought closely for the jacinth we could not find it. I was sorrowful to death on account of this omen, and despair seized upon my soul when I remembered the dire consequences of similar signs, all of which I had duly noted in my writings. I could scarcely believe this to be a thing happening in the order of nature. After a short delay I collected my thoughts, and told the servant to bring a light from the hearth. He replied that he would rather not do this, that he was afraid of the darkness, and that the fire was always extinguished in the evening. I bade him light a candle with the flint, when he told me that we had neither matches nor tinder nor sulphur. I persisted, and determined that a light should be got by one means or another, for I knew that, if I should go to sleep under so dire an omen, I must needs perish. So I ordered him to get a light as best he could. He went away and raked up the ashes, and found a bit of coal about the bigness of a cherry all alight, and caught hold of it with the tongs. At the same time I had little hope of getting a light, but he applied it to the wick of a lamp and blew thereon. The wick was lighted without any flame issuing from the live coal, which thing seemed to me a further marvel." After a search with the candle the ring was found on the floor under the middle of the bed, but the marvel was not yet worked out: the ring could not possibly have got into such a place unless it had been put there by hand. It could not have rolled there, on account of its shape, nor could it have fallen from the bed, because the pillow was closely joined to the head of the bed, round which ran a raised edge with no rift therein. Cardan concludes: "I know that much may be said over this matter, but nothing, forsooth, which will convince a man, ever so little inclined to superstition, that there was no boding sign manifested thereby, foretelling the ruin of my position and good name. Then, having soothed my mind, albeit I was well-nigh hopeless, I consoled myself with the belief that God still protected me." After pondering long and anxiously over the possible significance of this sign he took a more sanguine view of the future. He next put the jacinth ring on his finger and bade the boy try to pull it off, but he tried in vain, so well and closely did the ring fit the finger. From this time forth Cardan laid aside this ring, after having worn it for many years as a safeguard against lightning, plague, wakefulness, and palpitation of the heart.[215] Many other instances of a like character might be given from the _Paralipomena_; but the foregoing will suffice to show that the natural inclination of Cardan's temper towards the marvellous had been aggravated by his recent troubles. Also the belief that all men's hands were against him never slumbered, but for this disposition there may well have been some justification. Scarcely had he settled in Bologna before an intrigue was set in motion against him. "After the events aforesaid, and after I had gone to teach in Bologna, my adversaries, by a trick, managed to deprive me of the use of a class-room, that is to say they allotted to me an hour just about the time of dinner, or they gave the class-room at the very same hour, or a little earlier, to another teacher. When I perceived that the authorities were unwilling to accede to three distinct propositions which I made to them, namely, that this other teacher should begin his lecture sooner and leave off sooner: or that he should teach alternately with me: I so far got my own way at the next election that the other lecturer had to do his teaching elsewhere."[216] It would appear that the intrigues, of which Cardan gives so many instances, must have been the work of certain individuals, jealous of his fame and perhaps smarting under some caustic speech or downright insult, rather than of the authorities; the Senate of Bologna showed no hostility to him, but on the other hand procured for him the privileges of citizenship. While the negotiations were going on at Bologna for the further regulation of his position as a teacher, he tells a strange story how, on three or four different occasions, certain men came to him by night, in the name of the Senate and of the Judicial officers, and tried to induce him to recommend that a certain woman, who had been condemned for blasphemy, and for poisoning or witchcraft as well, should be pardoned, both by the temporal and spiritual authorities, bringing forward specially the argument that, in the sight of philosophers, such things as demons and spirits did not exist. They likewise urged him to procure the release from prison of another woman, who had not yet been condemned, because a certain sick man had died under the hands of some other doctors. They brought also a lot of nativities for him to read, as if he had been a soothsayer, and not a teacher of medicine, but he would have nothing to say to them.[217] It is somewhat strange that Cardan should have detected no trace of the snare of the enemy in this manoeuvre. Bearing in mind the character of the request made, and the fact that Cardan was by no means a _persona grata_ to the petitioners, it seems highly probable that they might have been more anxious to draw from Cardan a profession of his disbelief in witchcraft, than to procure the enlargement of the accused persons whose cause they had nominally espoused. At this period it was indeed dangerous to be a wizard, but it was perhaps still more dangerous to pose as an avowed sceptic of witchcraft. At the end of the fifteenth century the frequency of executions for sorcery in the north of Italy had provoked a strong outburst of popular feeling against this wanton bloodshed; but Spina, writing in the interest of orthodox religion, deplores that disbelief in the powers of Evil and their manifestations, always recognized by the Church, should have led men on to profess by their action any doubt as to the truth of witchcraft. But in spite of the fulminations of men of this sort, from this time onwards the more enlightened scholars of Europe began to modify their opinions on the subject of demoniac possession, and of witchcraft in general. The first book in which the new views were enunciated was the treatise _De Præstigiis Dæmonum_, by Johann Wier, a physician of Cleves, published in 1563. The step in advance taken by this reformer was not a revolutionary one. He simply denied that witches were willing and conscious instruments of the malefic powers, asserting that what evil they wrought came about by reason of the delusions with which the evil spirits infected the persons said to be possessed. The devil afflicted his victims directly, and then threw the suspicion of the evil deed upon some old woman. Wier's book was condemned and denounced by the clergy--he himself was a Protestant--but the most serious counterblast against it came from the pen of Jean Bodin, the illustrious French philosopher and jurist. He held up Wier to execration as an impious blasphemer, and asserted that the welfare of Christendom must needs suffer great injury through the dissemination of doctrines so detestable as those set forth in his book.[218] Seeing that such a spirit was dominant in the minds of men like Bodin, it will be evident that a charge of impiety or atheism might well follow a profession of disbelief, or even scepticism, as to the powers of witches or of evil spirits. A maxim familiar as an utterance of Sir Thomas Browne, "Ubi tres medici duo athei," was, no doubt, in common use in Cardan's time; and he, as a doctor, would consequently be ill-looked upon by the champions of orthodoxy, who would certainly not be conciliated by the fact that he was the friend of Cardinal Morone. This learned and enlightened prelate had been imprisoned by the savage and fanatical Paul IV., on a charge of favouring opinions analogous to Protestantism, but Pius IV., the easy-going Milanese jurisconsult, turned ecclesiastic, enlarged him by one of the first acts of his Papacy, and restored him to the charge of the diocese of Modena. Besides enjoying at Bologna the patronage of princes of the Church like Borromeo and Morone, Cardan found there an old friend in Ludovico Ferrari, who was at this time lecturing on mathematics. He also received into his house a new pupil, a Bolognese youth named Rodolfo Sylvestro, who was destined hereafter to bring as great credit to his teacher's name in Medicine as Ferrari had already brought thereto in Mathematics. Rodolfo proved to be one of the most faithful and devoted of friends; he remained at Bologna as long as Cardan continued to live there, sharing his master's ill-fortune, and ultimately accompanied him to Rome in 1571. He gives the names of two other Bolognese students, Giulio Pozzo and Camillo Zanolino, but of all his surviving pupils he rates Sylvestro as the most gifted. The records of Cardan's life at this period are scant and fragmentary, few events being chronicled except dreams and portents. In giving an account of one of these manifestations, which happened in September 1563, he incidentally lets light upon certain changes and vicissitudes in his own affairs. He was at this time living in an apartment in the house of the Ranucci, next door to a half-ruined palace of the Ghislieri. One night he awoke from sleep, and found that the neck-band of his shirt had become entangled with the cord by which he kept his precious emerald and a written charm suspended round his neck. He tried to disentangle the knot, but in vain, so he left the complication as it was, purposing to unravel it by daylight. He did not fall asleep; but, after lying quiet for a little, he determined to attempt once more whether he could undo the knot, when he found that everything was clear, and the stone under his armpit. "This sign showed me an unhoped-for solution of certain weighty difficulties, and at the same time proved, as I have often said elsewhere, that there must have been present something else unperceived by me. For my affairs were in this condition: my son-in-law at Milan had the administration of the scant remains of my property, and I received no rents therefrom for a whole year. My literary work was lying at the printer's, but it was not printed. Here, at Bologna, I was forced to lecture without having a fixed hour assigned to me. A crowd of enemies were intriguing against me. My son Aldo was in prison, and of little profit to me. But immediately after this portent I learned that my two chief opponents were either dying or about to retire. The question of the lecture-room was settled amicably, so that for the next year I was able to live in quiet. These two matters having come to an issue, I will next describe what came to pass with regard to the others. "During the next July (1564), through the help of Francesco Alciati,[219] the secretary of Pope Pius IV., a man to whom I am indebted for almost every benefit I have received since 1561, I began to enjoy my own again. On August 26 I received from the printer my books all printed with the greatest care, and by reason of the dispatch of this business my income was greatly increased. The next day my chief opponent resigned his office, and left vacant a salary of seven hundred gold crowns. The only manifestation of adverse fortune left to trouble me was the conspiracy of the doctors against me, but there were already signs that this would disappear before long, and in sooth it came to an end after the lapse of another year."[220] During this portion of his life at Bologna, Cardan seems to have lived comparatively alone, and to have spent his weary leisure in brooding over his sorrows. He began his long rambling epilogue to the _De Libris Propriis_, and, almost on the threshold, pours out his sorrow afresh over Gian Battista's unhappy fate. After affirming that Death must necessarily come as a friend to those whose lives are wretched, he begins to speculate whether, after all, he ought not to rejoice rather than mourn over his son's death. "Certes he is rid of this miserable life of danger and difficulty, vain, sorrowful, brief, and inconstant; these times in which the major part of the good things of the world fall to the trickster's share, and all may be enjoyed by those who are backed up by wealth or power or favour. Power is good when it is in the hands of those who use it well, but it is a great evil when murderers and poisoners are allowed to wield it. To the ill-starred, to the ungodly, and to the foolish, death is a boon, freeing them from numberless dangers, from heavy griefs, from fatal troubles, and from infamy; wherefore in such cases it ought not to be spoken of as something merely good or indifferent, but rated as the best of fortune. Shall I not declare to God (for He willed the deed), to myself, and to my surviving family, that my son's death was a thing to be desired, for God does all justly, wisely, and lovingly? He lets me stand as an example to show others that a good and upright man cannot be altogether wretched. I am poor, infirm, and old; bereaved by a cruel wrong of my best-loved son, a youth of the fairest promise, and left only with the faintest hope of any ray of future good fortune, or of seeing my race perpetuated after my death, for my daughter, who has been nine years married, is barren. "At one time I was prosperous in every relation of life: in my friendships, in my children, and in my health. In my youth I seemed to be one raised up to realize the highest hopes. I was accustomed to all the good things--nay, to all the luxuries of life. Now I am wretched, despised, with foes swarming around me; I not only count myself miserable, I feel I am far more miserable now than I was happy aforetime. Yet I neither lose my wits nor make any boast, as my actions prove. I do my work as a teacher with my mind closely set on the matter in question, and for this reason I attract a large number of hearers. I manage my affairs better than heretofore; and, if any man shall compare the book which I have lately published with those which I wrote some time ago, he will not fail to perceive how vastly my intellect has gained in richness, in vivacity, and in purity." Though the note of sorrow or even of despair is perceptible in these sentences, there is no sign that the virile and elastic spirit of the writer is broken. But there are manifest signs of an increasing tendency towards mental detachment from the world which had used him so ill. With the happiest of men the almost certain prospect of extinction at the end of a dozen years usually tends to foster the growth of a conviction that the world after all is a poor affair, and that to quit it is no great evil. How strongly therefore must reflections of a kindred nature have worked upon a man so cruelly tried as Cardan! FOOTNOTES: [211] _Opera_, tom. x. p. 462. [212] "Sed filius minor natu adeò malè se gessit, ut malim transire in nepotem ex primo filio."--_De Vita Propria_, ch. xxxvi. p. 112. [213] _De Vita Propria_, ch. xxvii. p. 71. [214] _De Vita Propria_, ch. xii. p. 40. [215] _Opera_, tom. x. p. 459. [216] _De Vita Propria_, ch. xvii. p. 56. [217] _De Vita Propria_, ch. xxiii. p. 104. [218] This opinion prevailed with men of learning far into the next century. Sir Thomas Browne writes: "They that doubt of these, do not only deny them, but spirits; and are obliquely and upon consequence a sect not of infidels, but atheists."--_Religio Medici, Works_, vol. ii. p. 89. [219] This was the Cardinal, the nephew of Andrea the great jurist, who was also a good friend of Cardan. [220] _Opera_, tom. x. p. 463. CHAPTER XII AT the beginning of the year 1565 Cardan had a narrow escape from death by burning, for his bed from some unknown cause caught fire twice in the same night while he was asleep. The servant was disturbed by the smoke, and having aroused his master, told him what was amiss, whereupon Cardan flew into a violent rage, for he deemed that the youth must be drunk. But he soon perceived the danger, and then they both set to work to extinguish the flames. His own description of the occurrence is highly characteristic. "Having put out the fire, I settled myself again to sleep, and, while I was dreaming of alarms, and that I was flying from some danger, it happened that either these terrifying dreams, or the fire and smoke again aroused me, and, looking around, I found that the bed was once more alight, and the greater part of it consumed. The vari-coloured coverlet, the leather hangings, and all the covering of the bed was unhurt. Thus this great alarm and danger and serious disturbance caused only a trifling loss; less than half of the bed-linen was burnt, but the blankets were entirely consumed. On the first alarm the flames burnt out twice or thrice with little smoke, and caused scarcely any damage. The second time the fire and the mishap forced me to rise just before dawn, the fire lasting altogether about seven hours." There was naturally a warning sign to be found in this accident.[221] The smoke, Cardan said, denoted disgrace; the fire, peril and fear; the flame, a grave and pressing danger to his life. The smouldering fire signified secret plots which were to be put into execution against him by his servants while he lay in bed. And the fact that he set fire to the bed himself, denoted that he would be able to meet any coming danger alone and without assistance. The indictment against him was foreshadowed by the fire and the flames and the smoke. Poison and assault were not to be feared. Men might indeed ask questions as to what kind of danger it could be which only arose from those about him, and fell short of poison and violence. The fire, he goes on to say, signifies the Magistrate. More than once it seemed to be extinct, but it always revived. Danger seemed to threaten him less from open hostility than from the cunning flattery of foes, and from over-confidence on his own part. His books, which he had lately caused to be printed, appeared to be in grave peril, but a graver one overhung his life. He deemed that he would quit the tribunal condemned by the empty scandal of the crowd, suffering no slight loss, and worsted chiefly through putting faith in false friends, and through his own instability. On the whole, the loss would prove inconsiderable; the danger moderate, but the vexation exceedingly heavy. These results might have sprung from causes other than natural ones; but, on the other hand, such things often come about through chance. They might prove to be a warning to him to keep clear of hostile prejudice, and to make friends of those in authority, care being taken not to let himself become involved in their private affairs, and not to seek too close an acquaintance.[222] Up to this date, Cardan, when he visited his patients, had either walked or ridden a mule. In 1562 he began to use a carriage, but this change of habit brought ill luck with it, for, in this same year, his horses ran away; he was thrown out of the vehicle, and sustained an injury to one of the fingers of his right hand, and to the right arm as well.[223] The finger soon healed, but the damage to the right arm shifted itself over to the left side, leaving the right arm sound. The foregoing details, taken chiefly from the _Paralipomena_ (Book III. ch. xii.), are somewhat significant in respect to the serious trouble which came upon him soon afterwards. Though he had now secured a class-room for himself, the malice of his enemies was not yet abated. Just before the end of his term, certain of them went to Cardinal Morone and told him that it would be inexpedient to allow Cardan to retain his Professorship any longer, seeing that scarcely any pupils went to listen to him. The terms Cardan used in describing this hostile movement against him,[224] rouse a suspicion that there may have been some ground for the assertion of his adversaries; but he declares that, at any rate, he had a good many pupils from the beginning of the session up to the time of Lent. He gives no clue whereby the date of this intrigue may be exactly ascertained, but it probably happened near the end of his sojourn at Bologna, because in his account of it he describes likewise the cessation of his public teaching, and makes no mention of any resumption of the same. He declares that he was at last overborne by the multitude of his foes, and their cunning plots. Under the pretence that, in seeking Cardan's removal, they were really acting for his benefit, they succeeded in bringing Cardinal Morone round to their views. Cardan's final words in dealing with this matter help to fix the date of this episode as some time in 1570. Speaking of his enemies, he writes: "Nay indeed they have given me greater leisure for the codification of my books, they have lengthened my days, they have increased my fame, and, by procuring my removal from the work which was too laborious for me, they secured for me the pleasure I now enjoy in the discovery and investigation of divers of the secrets of Nature. Therefore I constantly tell myself that I do not hate these men, nor deem them blameworthy, because they wrought me an ill turn, but because of the malignancy they had in their hearts."[225] It is almost certain that this removal of Cardan from his office of teacher was part and parcel of a carefully-devised plot against him, and a prelude to more serious trouble in the near future. Early in April 1570 he had occasion to put into writing a certain medical opinion which was to be sent to Cardinal Morone. He describes the episode: "It chanced that one of the sheets of my manuscript fell from the table down upon the floor, and then flew by itself up to the cornice of the room, where it hung, fixed to the woodwork. Greatly amazed, I called for Rodolfo, and pointed out to him this marvel. He did not indeed see it fly up, and at that time I was ignorant as to what it might foretell, for I had no foreboding of the many ills which were about to molest me. But now I see that the meaning of this portent must have been that, after the approaching shipwreck of my fortunes, my bark would be sped along with a more favouring breeze. It was during the month following, unless I am mistaken, that, when I was once more writing a letter to Cardinal Morone, I looked for a certain powder-box which had been missing for some long time, and, when I lifted up a sheet of paper in order to powder it with dust gathered up from the floor of the room, there was the powder-box, hidden beneath the sheet. How could it have come there on the level writing-desk? This sign confirmed the hope I had already conceived of the Cardinal's wisdom and humanity; that he would plead with the Pope, the best of men, in such wise that I should find a prosperous end to my toilsome life."[226] The blow thus foreshadowed fell on October 6, 1570, when he was suddenly arrested and put under restraint. He speaks of a bond which he gave for eighteen hundred gold crowns; and says that, while he was in hold, all his estate was administered by the civil authorities. Rodolfo Sylvestro was constantly with him during his incarceration, and on January 1, 1571, he was released, just at nightfall, and allowed to return to his own house. While he was in prison in the month of October some mysterious knockings at the door supplied him with a fulfilment and explanation of the portents lately chronicled. The knockings appeared furthermore to warn him of approaching death, and he began to bewail his misery; but, having gathered courage, he heartened himself to face his doom, which could be nothing worse than death. Young men, leaders of armies, courted death in battle to win the favour of their sovereigns; wherefore he, a decrepit old man, might surely await his end with calmness. He then wanders off into a long disquisition on the philosophy of Polybius, and forgets entirely to set down further details of his imprisonment, or to explain the cause thereof. Pius IV. had died at the end of 1565, and had been succeeded by Michele Ghislieri, the Cardinal of Alessandria, as Pius V. Like his predecessor, the new Pope was a Milanese by birth, but in character and aims the two Popes were entirely different. Pius. V. identified himself completely with the work of the Holy Office, and straightway set in operation all its powers for the extirpation of the heretical opinions which, on account of the easy-going character of the late Pope, had made much progress in Italy, and nowhere more than in Bologna. Von Ranke, in the _History of the Popes_, gives an extract (vol. i. p. 97) from the compendium of the Inquisitors, which sets forth that "Bologna was in a very perilous state, because there the heretics were especially numerous; amongst them was a certain Gian Battista Rotto, who enjoyed the friendship and support of many persons of weight, such as Morone, Pole, and the Marchesa Pescara (Vittoria Colonna). Rotto made himself very active in collecting money, which he distributed amongst the poor folk of Bologna who were heretics." It will be remembered that in 1562, while he was waiting in Milan for the appointment as Professor at Bologna, Cardan submitted his books to the Congregation of the Index for approval. He was known to be a fellow-citizen and friend of the reigning Pope: the _corpus_ of his work had by that time reached a portentous size, wherefore it is quite possible that the official readers may have been lenient, or cursory, over their work; but when Pius V., the strenuous ascetic foe of heresy, stepped into the place of the indolent Pius IV., jurist and politician rather than Churchman, it is more than probable that certain amateur inquisitors at Bologna, fully as anxious to work Cardan's ruin as to safeguard the faith, may have busied themselves in hunting through his various works for passages upon which to base a charge of unorthodoxy. Such passages were not hard to find. There was the horoscope of Jesus Christ, which subsequently affronted the piety of De Thou. There was the passage already noticed in which he said such hard things of the Dominicans (_De Varietate Rerum_, 1557, p. 572). He had indeed disclaimed it, but there it stood unexpunged in the subsequent editions of the book; and, while considering this detail, it may be remarked that Pius V. began his career as a member of the Dominican Order, the practices of which Cardan had impugned. In the first and second editions of the _De Subtilitate_ was another passage in which the tenets of Islam and the circumstances of the birth of Christ were handled in a way which caused grave scandal and offence.[227] This passage indeed was expunged in the edition of 1560. The _Paralipomena_ were not in print and available, but what can be read in them to-day doubtless reflects with accuracy the attitude of Cardan's mind towards religious matters in 1570. Though the _Paralipomena_ were locked in his desk, it is almost certain that the spirit with which they were inspired would have infected Cardan's brain, and prompted him to repeat in words the views on religion and a future state which he had already put on paper, for he rarely let discretion interfere with the enunciation of any opinion he favoured. In the _Paralipomena_ are many passages written in the spirit of universalism, and treating of the divine principle as something which animates wise men alone, wise men and philosophers of every age and every clime, Aristotle being the head and chief. Plato and Socrates and the Seven Sages adorn this illustrious circle, which includes likewise the philosophers of Chaldea and Egypt. Opinions like these were no longer the passport to Papal favour or even toleration. The age of the humanist Popes was past, and the Puritan movement, stimulated into life by the active competition of the Reformers, was beginning to show its strength, so that a man who spoke in terms of respect or reverence concerning Averroes or Plato would put himself in no light peril. Thus for those of Cardan's enemies who were minded to search and listen it must have been an easy task to formulate against him a charge of heresy, specious enough to carry conviction to such a burning zealot as Pius V. This Pope, in his new regulations for the maintenance of Church discipline, requisitioned the services of physicians in the detection of laxity of religious practices, or of unsoundness. "We forbid," he says in one of his bulls, "every physician, who may be called to the bedside of a patient, to visit for more than three days, unless he receives an attestation that the sick man has made fresh confession of his sins."[228] Cardan, with his irritable temper, may very likely have treated this regulation as an unwarrantable interference with his profession, and have paid no attention to it. Again, he evidently followed Hippocrates in rejecting the supernatural origin of disease; a position greatly in advance of that held by certain of the leading physiologists of the time.[229] Thus in more ways than one he may have laid himself open to some charge of disrespect shown to religion or to the spiritual powers. The absence of any other specific accusation and the circumstances of his incarceration, taken in conjunction with the foregoing considerations, almost compel the conclusion that his arrest and imprisonment in 1570 were brought about by a charge of impiety whispered by some envious tongue which will never now be identified. The sanction given by the authorities of the Church to his writings in 1562, operated without doubt to mitigate the punishment which fell upon him, and suffered him, after due purgation of his offences, to enjoy for the residue of his days a life comparatively quiet and prosperous under the patronage of Pius V. Though he was let out of prison he was not yet a free man. For some twelve weeks longer he remained a prisoner in his own house, the bond for eighteen hundred gold crowns having doubtless been given on this account. Almost his last reflection about his life at Bologna is one in which he records his satisfaction that all the men who plotted against him there met their death soon after their attempt, thus sharing the fate of his enemies at Milan and Pavia. If he is to be believed in this matter, the Fates, though they might not shield him from attack, proved themselves to be diligent and remorseless avengers of his wrongs. At the end of September he turned his back upon Bologna and the cold hospitality it had given him, and set forth on his last journey. He travelled by easy stages, and entered Rome on October 7, 1571, the day upon which Don John of Austria annihilated the Turkish fleet at Lepanto. There are evidences in his later writings beyond those already cited, that Cardan's views on religion had undergone change during his sojourn at Bologna. It was the custom, even with theologians of the time, to illustrate freely from the classics, wherefore the spectacle of the names of the great men of Greek and Roman letters, scattered thickly about the pages of any book, would not prove or even suggest unorthodoxy. Cardan quotes Plato or Aristotle or Plotinus twenty times for any saint in the Calendar. He does not mention the Virgin more than once or twice in the whole of the _De Vita Propria_; and, in discoursing on the immortality of the soul, he cites the opinion of Avicenna, but makes no mention of either saint or father.[230] The world of classic thought was immeasurably nearer and more real to Cardan than it can be to any modern dweller beyond the Alps: to him there had been no solution of continuity between classic times and his own. When he sat down to write in the _Theonoston_ his meditations on the death of his son, in the vain hope of reaping consolation therefrom, he invoked the golden rule of Plotinus, which lays down that the future is foreseen and arranged by the gods. Being thus arranged, it must needs be just, for God is the highest expression of justice. Against a fate thus settled for us we have no right to complain, lest we should seem to be setting ourselves into opposition to God's will. Here, although he writes in the spirit of a Christian, the authority cited is that of a heathen philosopher, and the form of his meditations is taken rather from Seneca than from father or schoolman. The devotional bias of Cardan's nature seems to have been strengthened temporarily by the terrible experiences of Gian Battista's trial and death; but in the course of his residence at Bologna a marked reaction set in, and the fervent religious outburst, in which he sought consolation during his intolerable sorrow, was succeeded by a calmer mood which regarded the necessary evils of life as transitory accidents, and death as the one and certain end of sorrow, and perhaps of consciousness as well. What he wrote during his residence in Rome he kept in manuscript; his recent experience at Bologna warned him that, living under the shadow of the Vatican with Pius V. as the ruler thereof, it behoved him to walk as an obedient son of the Church. Cardan went first to live in the Piazza di San Girolamo, not far from the Porto del Popolo, but subsequently he lived in a house in the Via Giulia near the church of Santa Maria di Monserrato, where probably he died. He had not long been settled in Rome before he was able to add a fresh supernatural experience to his already overburdened list. In the month of August 1572 he was lying awake one night with a lamp burning, when suddenly he heard a loud noise to the right of the chamber, as if a cart laden with planks was being unloaded. He looked up, and, the door being open at the time, he perceived a peasant entering the room. Just as he was on the threshold the intruder uttered the words, "_Te sin casa_," and straightway vanished. This apparition puzzled him greatly, and he alludes to it again in chapter xlvii. of the _De Vita Propria_. Ultimately he dismisses it with the remark that the explanation of such phenomena is rather the duty of theologians than of philosophers. With regard to matters of religious belief he seems to have taken as a rule of conduct the remark above written, and left them to the care of professional experts, for very few of his recorded opinions throw any light upon his views of the dogmas and doctrines of the Church. Whatever the tenor of these opinions may have been, he never proclaimed them definitely. Probably they interested him little, for he was not the man to keep silent over a subject which he had greatly at heart. He gave a general assent to the teaching of the Church, taking up the mental attitude of the vast majority of the learned men of his time, and expected that the Church would do all that was necessary for him in its own particular province. If he regarded Erasmus and Luther as disturbers of the faith and heretics, he did not say so, nor did he censure their activity. (Erasmus he praises highly in the opening words of the horoscope which he drew for him.--_Gen. Ex.,_ p. 496.) But he had certainly no desire to emulate them or give them his support. The world of letters and science was wide enough even for his active spirit; the world lying behind the veil he left to the exploration of those inquirers who might have a taste for such a venture. Still every page of his life's record shows how strong was his bent towards the supernatural; but the phase of the supernatural which he chose for study was one which Churchmen, as a rule, had let alone. Spirits wandering about this world were of greater moment to him than spirits fixed in beatitude or bane in the next; and accordingly, whenever he finds an opportunity, he discourses of apparitions, lamiæ, incubi, succubi, malignant and beneficent genii, and the methods of invoking them. Now that old age was pressing heavily upon him and he began to yearn for support, he sought consolation not in the ecstatic vision of the fervent Catholic, but in fostering the belief that he was in sooth under the protection of some guardian spirit like that which had attended his father and divers of the sages of old. Although he had in his earlier days treated his father's belief with a certain degree of respect and credence,[231] there is no evidence that he was possessed with the notion that any such supernatural guardian attended his own footsteps at the time when he put together the _De Varietate_; indeed it would seem that his belief was exactly the opposite. He writes as follows: "It is first of all necessary to know that there is one God, the Author of all good, by whose power all things were made, and in whose name all good things are brought to pass; also, that if a man shall err he need not be guilty of sin. That there is no other to whom we owe anything or whom we are bound to worship or serve. If we keep these sayings with a pure mind we shall be kept pure ourselves and free from sin. What a demon may be I know not, these beings I neither recognize nor love. I worship one God, and Him alone I serve. And in truth these things ought not to be published in the hearing of unlearned folk; for, if once this belief in spirits be taken up, it may easily come to pass that they who apply themselves to such arts will attribute God's work to the devil."[232] And in another place: "I of a truth know of no spirit or genius which attends me; but should one come to me, after being warned of the same in dreams, if it should be given to me by God, I will still reverence God alone; to Him alone will I give thanks, for any benefit which may befall me, as the bountiful source and principle of all good. And, in sooth, the spirit may rest untroubled if I repay my debt to our common Master. I know full well that He has given to me, for my good genius, reason, patience in trouble, a good disposition, a disregard of money and dignities, which gifts I use to the full, and deem them better and greater possessions than the Demon of Socrates."[233] About the Demon of Socrates Cardan has much to say in the _De Varietate_. He never even hints a doubt as to the veracity and sincerity of Socrates. He is quite sure that Socrates was fully persuaded of the reality of his attendant genius, and favours the view that this belief may have been well founded. He takes an agnostic position,[234] confining his positive statement to an assertion of his own inability to realize the presence of any ghostly minister attendant upon himself. In the _De Subtilitate_ he tells an experience of his own by way of suggesting that some of the demons spoken of by the retailers of marvels might be figments of the brain. In 1550 Cardan was called in to see a certain woman who had long been troubled with an obscure disease of the bladder. Every known remedy was tried in vain, when one day a certain Josephus Niger,[235] a distinguished Greek scholar, went to see the patient. Niger, according to Cardan's account, was quite ignorant of medicine, but he was reputed to be a skilled master of magic arts. The woman had a son, a boy about ten years old, and Josephus having handed him a three-cornered crystal, which he had with him, bade the youth secretly to look into it, and then declare, in his mother's hearing, that he could see in the crystal three very terrible demons going on foot. Then, after Josephus had whispered certain other words in the boy's ear, the boy went on to say that he beheld another demon, vastly bigger than the first, riding on horseback and bearing in his hand a three-tined fork. This monster overthrew the other demons, and led them away captive, bound with chains to his saddlebow. After listening to these words the woman rapidly got well, and Cardan, in commenting on the event, declares that she must have been cured either by the agency of the demons or by the force of the imagination, inasmuch as it would be difficult, if not impossible, to invent any other reason of her recovery.[236] In another passage of the _De Subtilitate_ he displays judicious reserve in writing of Demons in general.[237] During those terrible days, when his son had just died a felon's death, and when he himself was haunted by the real dangers which beset him, and almost maddened by the signs and tokens which seemed to tell of others to come, the belief which Fazio his father had nourished easily found a lodgment in his shaken and bewildered brain. In the _Dialogus de Humanis Consiliis_, one of the speakers tells of a certain man who is clearly meant to be Cardan himself. The speaker goes on to say that he is sure this man is attended by a genius, which manifested itself to him somewhat late in his life. "Aforetime, indeed, it had been wont to convey to him warnings in dreams and by certain noises. What greater proof of his power could there be than the cure of this man, without the use of drugs, of an intestinal rupture on the right side? If indeed it had not fared with him thus, after his son's death, he would at once have passed out of this life, whereby many and great evils might have come to pass. He was freed also from another troublesome ailment. In sooth, so many and so mighty are the wonderful things which had befallen him, that I, who am very intimate with him (and he himself thinks the same), am constrained to believe that he is attended by a genius, great and powerful and rare, and that he is not the master of his own actions. What he would have, he has not; and what he has, he would not have chosen, or even wished for. This thing causes him much trouble, but he submits when he reflects that all things are God's handiwork." The speaker ends by saying that he never heard of any others thus attended, save this man, and his father before him, and Socrates.[238] But it is in chapter xlvii. of the _De Vita Propria_, which must have been written shortly before his death, that he lets the reader see most plainly how strong was the hold which this belief in a guardian spirit of his own had taken upon him. "It is an admitted truth," he writes, "that attendant spirits have protected certain men, to wit, Socrates, Plotinus, Synesius, Dion, Flavius Josephus, and myself. All of these have enjoyed prosperous lives except Socrates and me, and I, as I have said before, was at one time offered many and favourable opportunities for the achievement of happiness. But C. Cæsar the dictator, Cicero, Antony, Brutus, and Cassius were also attended by mighty spirits, albeit malignant. For a long time I have been persuaded that I too had one, but by what method it gave me intelligence as to events about to happen, I could not exactly ascertain until I reached the seventy-fourth year of my age, the season when I began to write this record of my life. I now perceive that when I was in Milan in 1557, when my genius perceived what was hanging over me--how that my son on that same evening had promised to marry Brandonia Seroni, and that he would complete the nuptials the following day--it produced in me that palpitation of the heart of which I have already made mention, a weakness known to my genius alone, a manifestation which served to simulate a trembling of the bed." Cardan writes at length to show that the mysterious knocking which he and Rodolfo Sylvestro had heard during his imprisonment at Bologna, the peasant who entered his bed-chamber saying "_Te sin casa_," and divers other manifestations, going back as far as 1531--croaking of ravens, barking of dogs, and the ignition of fire-wood--must all have been brought about by the working of this powerful spirit. In 1570 there happened to him one of his everyday experiences of the presence of supernatural powers. In the middle of the night he was conscious of some presence walking about the room. It sat down beside him, and at the same time a loud noise arose from a chest which stood near. This phenomenon, he admits, might well have been the figment of a brain overburdened with thought; but suddenly his memory flies back to an experience of his twentieth year, upon which he proceeds to build a story, wild and fanciful even for his powers of imagination. "What man was it," he asks, "who sold me that copy of Apuleius when I was in my twentieth year, and forthwith went away? I indeed, at that time, had made only one essay in the literary arena, and had no knowledge of the Latin tongue; but in spite of this, and because the book had a gilded cover, I was imprudent enough to buy it. The very next day I found myself just as well versed in Latin as I am now. Moreover, almost at the same time I acquired knowledge of Greek and Spanish and French, sufficient for reading books written in these languages." Cardan was by this time completely possessed by the belief in his attendant genius, and the flash of memory which recalled the purchase of some book or other in his youth, suggested likewise the attribution of certain mystic powers to this guardian genius, and conjured up some fanciful explanation as to the way these powers had been exercised upon himself; he, the person most closely concerned, being entirely unconscious of their operation at the time when they first affected him. This recorded belief in a gift of tongues is one of the most convincing bits of evidence to be gleaned from Cardan's writings of the insanity which undoubtedly afflicted him, at least periodically, at this crisis of his life. FOOTNOTES: [221] He mentions this matter briefly in the _De Vita Propria_: "Bis arsisset lectus, prædixi me non permansurum Bononiæ, et prima vice restiti, secunda non potui."--ch. xli. p. 151. A fuller account of it is in _Opera_, tom. x. p. 464. [222] _Opera_, tom. x. p. 464. [223] _De Vita Propria_, ch. xxx. p. 80. He seems to have had many untoward experiences in driving. He tells of another mishap (_Opera_, tom. i. p. 472) in June 1570; how a fellow, some tipstaff of the courts, jumped into his carriage and frightened the mares Cardan was driving, jeering at them likewise because they were rather bare of flesh. [224] "Demum sub conductionis fine, voces sparserunt, et maxime apud Moronum Cardinalem, me exiguo auditorio profiteri, quod quanquam non omnino verum esset, quinimo ab initio Academiæ multos, et usque ad dies jejunii haberem auditores."--_De Vita Propria_, ch. xvii. p. 56. [225] _De Vita Propria_, ch. xvii. p. 57. [226] _De Vita Propria_, ch. xliii. p. 163. [227] "Alii multis diebus abstinent cibo, alii igne uruntur, ac ferro secantur, nullum doloris vestigium preferentes; multi sunt vocem e pectore mittentes, qui olim engastrimuthi dicebantur; hoc autem maxime eis contingit cum orgia quædam exercent, atque circumferuntur in orbem. Quæ tria ut verissima sunt et naturali ratione mira tamen constant, cujus superius mentionem fecimus, ita illud confictum nasci pueros e mulieribus absque concubitu."--_De Subtilitate_, p. 353. [228] Ranke, _History of the Popes_, vol. i. p. 246. [229] Mr. Stephen Paget in his life of Ambroise Paré, the great contemporary French surgeon, gives an interesting account of Paré's beliefs on the divine cause of the plague, p. 269. [230] _De Vita Propria_, ch. xxii. p. 63. [231] "Multa de dæmonibus narrabat, quæ quam vera essent nescio."--_De Utilitate_, p. 348. [232] _De Varietate_, p. 351. [233] _Ibid.,_ p. 658. [234] In his counsel to his children, he writes: "Do not believe that you hear demons speak to you, or that you behold the dead. Seek not to learn the truth of these things, for they are amongst the things which are hidden from us." [235] Cardan alludes to Niger in _De Varietate_, p. 641: "Referebat aliquando Josephus Niger harum rerum maximé peritus, dæmonem pueris se sub forma Christi ostendisse, petiisseque ut adoraretur." [236] _De Subtilitate_, p. 530. [237] "Nolim ego ad trutinam hæc sectari, velut Porphyrius, Psellus, Plotinus, Proclus, Jamblicus, qui copiose de his quæ non videre, velut historiam natæ rei scripserunt."--_De Subtilitate_, p. 540. [238] _Opera_, tom. i. 672. CHAPTER XIII AFTER the accusation brought against him at Milan in 1562, Cardan had been prohibited from teaching or lecturing in that city, and similar disabilities had followed his recent imprisonment at Bologna. At Rome no duties of this kind awaited him, so he had full time to follow his physician's calling after taking up his residence there. He records the cure of a noble matron, Clementina Massa, and of Cesare Buontempo, a jurisconsult, both of whom had been suffering for nearly two years. The circumstances of his retirement from Bologna would not affect his reputation as a physician, and he seems to have had in Rome as many or even more patients than he cared to treat; and in writing in general terms concerning his successes as a healer, he says: "In all, I restored to health more than a hundred patients, given up as incurable in Milan, in Bologna, and in Rome." Of all the friends Cardan had in this closing period of his life, none was more useful or benevolent than Cardinal Alciati, who, although he had been secretary to Pius IV., contrived to retain the favour of his successor. This piece of good fortune Alciati owed to the protection of Carlo Borromeo, who had been his pupil at Pavia, and had procured for him from Pius IV. a bishopric, a cardinal's hat, and the secretaryship of Dataria. Another of Cardan's powerful friends was the Prince of Matellica, of whom he speaks in terms of praise inflated enough to be ridiculous, were it not for the accompanying note of pathos. After celebrating the almost divine character of this nobleman, his munificence and his superhuman abilities, he goes on: "What could there be in me to win the kindly notice of such a patron? Certainly I had done him no service, nor could he hope I should ever do him any in the future, I, an old man, an outcast of fortune, and prostrated by calamity. In sooth, there was naught about me to attract him; if indeed he found any merit in me, it must have been my uprightness." Powerful friends are never superfluous, and Cardan seems to have needed them in Rome as much as in Bologna. In 1573 he again hints at plots against his life, but almost immediately after recording his suspicions he goes on to suggest that his danger had arisen chiefly from his ignorance of the streets of Rome, and from the uncouth manners of the populace. "Many physicians, more cautious than myself, and better versed in the customs of the place, have come by their death from similar cause." The danger, whatever its nature, seems to have threatened him as a member of the practising faculty at Rome rather than as the persecuted ex-teacher of Pavia and Bologna. Rodolfo Sylvestro was not the only one of his former associates near him in his old age, for he notes that Simone Sosia, who had been his _famulus_ at Pavia in 1562, was still in his service at Rome. In reviewing the machinations of his enemies to bring about his dismissal from the Professorship at Bologna, Cardan indulges in the reflection that these men unwillingly did him good service, that is, they procured him leisure which he might use in the completion of his unfinished works, and in the construction of fresh monuments which he proposed to build up out of the vast store of material accumulated in his industrious brain. The literary record of his life in Rome shows that this was no vain saying. He was at work on the later chapters of the _De Vita Propria_ up to the last weeks of his life; and, scattered about these, there are records of his work of correction and revising. While telling of the books he has lately been engaged with, he wanders off in the same sentence to talk of the dream which urged him to write the _De Subtilitate_, and of the execution of the _Commentarii in Ptolomæum_, during his voyage down the Loire. In 1573 he seems to have found the mass of undigested work more than he could bear to behold; for, after making extracts of such matter as he deemed worth keeping, he consigned to the flames no less than a hundred and twenty of his manuscripts.[239] Before leaving Bologna he had put into shape the _Proxenata_, a lengthy collection of hints, maxims, and reflections as to everyday life; he had re-edited the _Liber Artis Magnæ_, and had added thereto the treatise _De Proportionibus_, and the _Regula Aliza_. He also took in hand two books on Geometry, and one on Music, and this last he completed in 1574. On November 16, 1574, he records that he is at that moment writing an explanation of the more abstruse works of Hippocrates, but that he is yet far from the end of his task. In the _De Libris Propriis_ he gives a list of all his published works, and likewise a table of the same arranged in the order in which they ought to be read. He apologizes for the imperfect state in which some of them are left, and declares that the sight of his unfinished tasks never fails to awaken in his breast a bitter sense of resentment over that loss which he had never ceased to mourn. "At one time I hoped," he writes, "that these works would be corrected by my son, but this favour you see has been denied to me. The desire of my enemies was not to make an end of him, but of me; not by gentle means, in sooth, but by cruel open murder; to let me fall in the very blood of my son." It is somewhat remarkable that in this matter Cardan was destined to suffer a disappointment similar to that which he himself brought upon his own father by refusing to qualify himself to become the commentator on Archbishop Peckham's _Perspectiva_. He next gives the names of all those who had commended him in their works, and finds a special cause for gratification in the fact that, out of the long list set down, only four or five were known to him personally, and these not intimately. There is, however, another short list of censors; and of these he affirms that a certain Brodeus alone is worthy of respect. Of Buteon, who criticized the treatise on _Arithmetic_, he says: "_Est plane stultus et elleboro indiget._" Tartaglia's name is there, and he, according to Cardan, was forced to eat his words; "but he was ashamed to do what he promised, and unwilling to blot out what he had written. He went on in his wrong-headed course, living upon the labour of other men like a greedy crow, a manifest robber of other men's wealth of study; so impudent that he published as his own, in the Italian tongue, that invention for the raising of sunken ships which I had made known four years before. This he did, understanding the subject only imperfectly, and making no mention of my name. But men of real learning also attacked me: Rondeletius, and Julius Scaliger; and Fuchsius, in the proem of his book, says that my work _Medicinæ Contradictiones_ should be avoided like deadly poison. Julius Scaliger has been fully answered in the _Apologia_ in the Books on Subtlety."[240] There is a passage from De Thou's _History of his Own Times_, affixed to all editions of the _De Vita Propria_,[241] in which is given a contemporary sketch of Cardan during his residence at Rome. "His whole life," De Thou writes, "has been as strange as his present manners, and he, in sooth, out of singleness of mind or frankness, has written about himself certain statements, the like of which have never before been heard of a man of letters, and these I do not feel bound to unfold to any one, let him be ever so curious. I, myself, happening to be in Rome a few years before his death, often spoke to him and observed him with astonishment as he took his walks about the city clad in strange garb. When I considered the many writings of this famous man, I could perceive in him nothing to justify his great renown. Wherefore I am all the more inclined to turn to that very acute criticism of Julius Cæsar Scaliger, who exercised his extraordinary genius in making a special examination of the treatise _De Subtilitate Rerum_. He, having carefully noted everywhere the unequal powers of this writer, decided that he was one who, in certain subjects, knew more than a man could know, while in others he seemed more simple than a child. In the science of Arithmetic he worked hard and made many discoveries; but he was subject to strange and excessive aberration of mind, and was guilty of the most impudent blasphemy, in that he was minded to subject to the artificial laws of the stars the Ruler of the stars Himself, for this thing he did in the horoscope of our Saviour which he drew." Another witness of his life in Rome is François d'Amboise, a young French nobleman, who was engaged on his book _De Symbolis Heroicis_. He says that he saw Cardan, who was living in a spacious house, on the walls of which, in place of elegant paintings or vari-coloured tapestries, were written the words, "_Tempus mea possessio_." In his later writings there are farther indications that he was wont to conjure up omens and portents chiefly at those times when he was in danger and mental distress. In the case which is given below, the omen showed itself in a season of trouble, but Cardan, in describing it later, treats it as if he were a modern scientist. The distressing memories of the imprisonment had faded, and writing in ease and security at Rome he begins to rationalize. In the dialogue between himself and his father, written shortly before his death, Fazio calls his son's attention to certain of the omens and portents already noticed; and, after discussing these, Jerome goes on to tell for the first time of another boding event which, as he affirms, distressed him even more than the loss of his office and the prohibition to publish his books. On the day of his incarceration, on two different occasions, he met a cow being driven to the slaughter-house, with much shouting and beating with sticks and barking of dogs. The explanation of this event which he puts in Fazio's mouth is entirely conceived in the spirit of rationalism. What was there to wonder at? There was a butcher's shop in the street, and animals going to slaughter would naturally be met there. Why should a man fear to meet a cow? If it had been a bull there might have been something in it. Then with regard to the shaking of a window-casement; this might easily have been occasioned by the flight of a bird.[242] He was certainly less inclined to put faith in the warnings of the stars and in the lines of his hand. His line of life was very short and irregular, intersected and bifurcated, while the rest of the lines were little thicker than hairs. In his horoscope was a certain malefic influence which threatened that his life would be cut short before his forty-fifth year. "But," he writes in the year before his death, "here I am, living at the age of seventy-five."[243] The one supernatural idea which seems to have deepened with old age and remained undisturbed to the end was his belief in his attendant genius. In what he wrote during his last years his mood was almost entirely introspective, contemplative, and didactic, yet here and there he introduces a sentence which lets in a little light from his way of life and personal affairs, and helps to show how he occupied himself, and what his humour was. He tells how one day, in 1576, he was writing about the fennel plant in his treatise _De Tuenda Sanitate_, a plant which he praised highly because it pleased his palate. But shortly afterwards, when he was walking one day in the Roman vegetable market, an old man, shabbily dressed, met him and dissuaded him from the use of the plant aforesaid, saying: "In Galen's opinion you may as readily meet your death thereby as by eating hemlock." "I answered that I knew well enough the difference between hemlock and fennel, but the old man said, 'Take care, I know what I am saying,' and went on murmuring something about Galen. Whereupon I went home and found in Galen a passage I had not hitherto noticed, and, having changed my former views, I added many fresh excerpts to my treatise." Although his faith may have been shaken in the ability of the stars to govern his own fortunes, he records a case in which he himself filled the post of _vates_, and which came to a sudden and terrible issue. Cardan was present at a supper-party, and in the course of conversation let fall the remark, "I should like to say something, were I not afraid that my words would disturb the company," to which one of the guests replied, "You mean that you would prophesy death to one of us here present." Cardan replied, "Yes, within the present year," and in the next sentence he tells how on the first day of December in that same year a certain young man, named Virgilius, who had been present at the gathering aforesaid, died, and he sets down this event as a fulfilment of his prophecy. But in the same chapter he lets the reader into the secret of his system of prophecy, and displays it as simply an affair of common-sense, one recommended by Aristotle as the only trustworthy method of divining future events. Cardan writes: "I used to inquire what might be the exact nature of the business in hand, and began by making myself acquainted with the character of the locality, the ways of the people, and the quality of the chief actors. I unfolded a vast number of historical instances, leading events and secret transactions as well, and then, when I had confirmed the facts set forth by my method of art, I gave my judgment thereupon."[244] In his latter years Cardan must have been in easy circumstances. The pension from the Pope--no mention is made of its amount--and the fees he received from his patients allowed him to keep a carriage; and writing in his seventy-fifth year, he says that no fees would tempt him to join any consultation unless he should be well assured what sort of men he was expected to meet.[245] In the _Norma Vitæ Consarcinata_[246] he relates how in April 1576 there were two inmates of the Xenodochium at Rome, Troilus and Dominicus. It seemed that Troilus exercised some strange and malefic influence over his companion, who was taken with fever. He got well of this, but only to fall into a dropsy, which despatched him in a week. Shortly before his death, at the seventh hour, he cried out to two Spaniards who were standing by the bed that he had suffered such great torture from the working of Troilus, and that he was dying therefrom. "Therefore," he cried, "in your presence I summon him with my dying words to appear before God's tribunal, that he may give an account of all the evil he has wrought against me." On the following day there came a messenger from Corneto, a few miles from Rome, saying that Troilus, who was sojourning there, had fallen sick. The physician inquired at what hour, and the messenger said it was at seven o'clock, a day or two ago. He lay ill some days, an unfavourable case, but not a desperate one, and one night shortly afterwards at seven o'clock, the top of the mosquito curtains fell, and he died at exactly the same hour as Dominicus. He tells another long story of an adventure which befell him in May 1576. One day he was driving in his carriage in the Forum, when he remembered that he wanted to see a certain jeweller who lived in a narrow alley close by. Wherefore he told his coachman, a stupid fellow, to go to the Campo Altoviti, and await him there. The coachman drove off apparently understanding the order; but, instead of going to the place designated, went somewhere else; so Cardan, when he set about to find his carriage, sought in vain. He had a notion that the man had gone to a spot near the citadel, so he walked thither, encumbered with the thick garments he had put on as necessary for riding in the carriage. Just then he met a friend of his, Vincenzio, a Bolognese musician, who remarked that Cardan was not in his carriage as usual. The old man went on towards the citadel, but saw nothing of the carriage; and now he began to be seriously troubled, for there was naught else to be done but to go back over the bridge, and he was wearied with long fasting and his heavy clothes. He might indeed have asked for the loan of a carriage from the Governor of the castle; but he was unwilling to do this, so having commended himself to God, he resolved to use all his patience and prudence in finding his way back. He set out, and when he had crossed the bridge, he entered the banking-house of the Altoviti to inquire as to the alteration in the rate of exchange on Naples, and there sat down to rest. While the banker was giving him this information, the Governor entered the place, whereupon Cardan went out and there he found his carriage, the driver having been informed by Vincenzio, whom he had met, of the mistake he had made. Cardan got into the carriage, and while he was wondering whether or not he had better go home and break his fast, he found three raisins in his pocket, and thus made a fortunate ending of all his difficulties. All this reads like a commonplace chapter of accidents; but the events recorded did not present themselves to Cardan in this guise. He sits down to moralize over the succession of momentary events: his meeting with Vincenzio; Vincenzio's meeting with the driver, and directions given to the man to drive to the money-changers'; the presence of the Governor, his exit from the bank, his consequent meeting with the carriage, and his discovery of the raisins, seven occurrences in all, any one of which, if it had happened a little sooner or a little later, would have brought about great inconvenience, or even worse. He does not deny that other men may not now and then encounter like experiences, but the experiences of other men were not fraught with such momentous crises, nor did they foreshadow so many or grave dangers. The chronicling of this episode and the fanciful coincidence of the deaths of Dominicus and Troilus may be taken as evidence that his idiosyncrasies were becoming aggravated by the decay of his faculties. Writing on October 1, 1576, he makes mention of the various testaments he had already made, and goes on to say that he had resolved to make a new and final disposition of his goods. He would fain have let his property descend to his immediate offspring, but with a son like Aldo this was impossible, so he left all to Gian Battista's son, who would now be a youth about eighteen years of age, Aldo getting nothing. He desired, for reasons best known to himself, that all his descendants should remain _in curatela_ as long as possible, and that all his property should be held on trust; if the issue of his body should fail, then the succession should pass in perpetuity to his kinsfolk on the father's side. He desired that his works should be corrected and printed, and that, if heirs failed entirely, his house at Bologna should pass to the University, and be styled, after his family, _Collegium Cardanorum_. There is no authentic record of the exact date of Cardan's death. De Thou, in writing the record of 1576, says that if Cardan's life had been prolonged by three days he would have completed his seventy-fifth year. As Cardan's birthday was September 24, 1501, this would fix his death on September 21, 1576. The exact figures given by De Thou are: "eodem, quo prædixerat, anno et die, videlicet XI. Kalend. VIII.," and he adds by way of information that a belief was current at the time that Cardan, who had foretold how he would die on this day and in this year, had abstained from food for some days previous to his death in order to make the fatal day square with the prophecy. But the details which Cardan himself has set down concerning the last few weeks of his life are inconsistent with the facts chronicled by De Thou. In the _De Vita Propria_, chapter xxxvi., Cardan records how on October 1, 1576, he set to work to make his last will and testament, wherefore if credit is to be given to his version rather than to that of De Thou, he was alive and active some days after the date of his death as fixed by the chronicler. In cases where the record of an event of his early life given in the _De Vita Propria_ differs from an account of the same in some contemporary writing, the testimony of the _De Vita Propria_ may justly be put aside; but in this instance he was writing of something which could only have happened a few days past, and the balance of probability is that he was right and De Thou wrong. Bayle notices this discrepancy, and in the same paragraph taxes De Thou with a mistake of which he is innocent. He states that De Thou placed the date of Cardan's death in 1575, whereas the excerpt cited above runs: "Thuanus ad annum MDLXXVI., p. 136, lib. lxii. tom. 4. Romæ magni nominis sive Mathematicus, sive Medicus Hieronymus Cardanus Mediol. natus hoc anno itidem obiit." No mention is made of the disease to which Cardan finally succumbed. Had his frame not been of the strongest and most wiry, it must have gone to pieces long before through the havoc wrought by the severe and continuous series of ailments with which it was afflicted; so it seems permissible to assume that he died of natural decay. His body was interred in the church of Sant Andrea at Rome, and was subsequently transferred to Milan to be deposited finally under the stone which covered the bones of his father in the church of San Marco. This tomb, which Jerome had erected after Fazio's death, bore the following inscription: FACIO CARDANO 1.C. Mors fuit id quod vixi: vitam mors dedit ipsa, Mens æterna manet, gloria tuta quies. Obiit anno MDXXIV. IV. Kalend. Sept. anno Ætatis LXXX. Hieronymus Cardanus Medicus Parenti posterisque V.P.[247] FOOTNOTES: [239] "Qua causa permotus sim ad scribendum, superius intellexisse te existimo, quippe somnio monitus, inde bis, terque, ac quater, ac pluries, ut alias testatus sum; sed et desiderio perpetuandi nominis. Bis autem magnam copiam ac numerum eorum perdidi; primum circa XXXVII annum, cum circiter IX. libros exussi, quod vanos ac nullius utilitatis futuros esse intelligerem; anno autem MDLXXIII alios CXX libros, cum jam calamitas illa cessasset cremavi."--_De Vita Propria_, ch. xlv. pp. 174, 175. [240] _Opera_, tom. i. p. 122. [241] _De Vita Propria_, p. 232. [242] _Opera_, tom. i. p. 639. In the _De Varietate_ he says that natural causes may in most cases be found for seeming marvels. "Ecce auditur strepitus in domo, potest esse mus, felis, ericius, aut quod tigna subsidant blatta."--p. 624. [243] _De Vita Propria_, ch. xli. p. 152. [244] _De Vita Propria_, chapter xlii., _passim_. [245] _Ibid.,_ p. 66. [246] _Opera_, tom. i. p. 339. [247] Tomasinus, _Gymnasium Patavinum_. CHAPTER XIV THE estimates hitherto made concerning Cardan's character appear to have been influenced too completely, one way or the other, by the judgment pronounced upon him by Gabriel Naudé, and prefixed to all editions of the _De Vita Propria_. Some writers have been disposed to treat Naudé as a hide-bound pedant, insensible to the charm of genius, and the last man who ought to be trusted as the valuator of a nature so richly gifted, original, and erratic as was Cardan's. Such critics are content to regard as black anything which Naudé calls white and _vice versâ_. Others accept him as a witness entirely trustworthy, and adopt as a true description of Cardan the paragraphs made up of uncomplimentary adjectives--applied by Cardan to himself--which Naudé has transferred from the _De Vita Propria_ and the _Geniturarum Exempla_ to his _Judicium de Cardano_. It may be conceded at once that the impression received from a perusal of this criticism is in the main an unfavourable one of Cardan as a man, although Naudé shows himself no niggard of praise when he deals with Cardan's achievements in Medicine and Mathematics. But in appraising the qualifications of Naudé to act as a judge in this case, it will be necessary to bear in mind the fact that he was in his day a leading exponent of liberal opinions, the author of a treatise exposing the mummeries and sham mysteries of the Rosicrucians, and of an "Apologie pour les Grands Hommes soupçonnez de Magie," and a disbeliever in supernatural manifestations of every kind. With a mind thus attuned it is no matter of surprise that Naudé should have been led to speak somewhat severely when called upon to give judgment on a man saturated as Cardan was with the belief in sorcery, witches, and attendant demons. If Naudé indeed set to work with the intention of drawing a figure of Cardan which should stand out a sinister apparition in the eyes of posterity, his task was an easy one. All he had to do was to place Jerome Cardan himself in the witness-box. Reference to the passages already quoted will show that, in the whole _corpus_ of autobiographic literature, there does not exist a volume in which the work of self-dissection has been so ruthlessly and completely undertaken and executed as in Cardan's memoirs. It has all the vices of an old man's book; it is garrulous, vain-glorious, and full of needless repetition; but, whatever portion of his life may be under consideration, the author never shrinks from holding up to the world's gaze the result of his searches in the deepest abysses of his conscience. Autobiographers, as a rule, do not feel themselves subject to a responsibility so deep as this. Memory turns back to the contemplation of certain springs of action, certain achievements in the past, making a judicious selection from these, and excerpting only such as promise to furnish the possible reader with a pleasing impression of the personality of the subject. With material of this sort at hand, the autobiographer sets to work to construct a fair and gracious monument, being easily persuaded that it would be a barbarous act to mar its symmetry by the introduction of loathly and misshapen blocks like those which Cardan, had he been the artist, would have chosen first of all. Naudé, after he has recorded the fact that, from his first essay in letters, he had been a zealous and appreciative student of Cardan's works, sets down Cardan's picture of himself, taken from his own Horoscope in the _Geniturarum Exempla_, "nugacem, religionis contemptorem, injuriæ illatæ memorem, invidum, tristem, insidiatorem, proditorem, magum, incantatorem, frequentibus calamitatibus obnoxium, suorum osor[e=], turpi libidini deditum, solitarium, inamoenum, austerum, spontè etiam divinantem, zelotypum, lascivum, obscoenum, maledicum, obsequiosum, senum conversatione se delectantem, varium, ancipitem, impur[u=], et dolis mulierum obnoxium, calumniatorem, et omnino incognitum propter naturæ et morum repugnantiam, etiam his cum quibus assidue versor." The critic at once goes on to state that in his opinion this description, drawn by the person who ought to know best, is, in the main, a correct one. What better account could you expect, he asks, of a man who put faith in dreams and portents and auguries; who believed fully in the utterances of crazy beldames, who saw ghosts, and who believed he was attended by a familiar demon? Then follows a catalogue of moral offences and defects of character, all taken from Cardan's own confessions, and a pronunciation by Naudé that the man who says he never lies, must be of all liars the greatest; the charge of mendacity being driven home by references to Cardan's alleged miraculous comprehension of the classic tongues in a single night, and his pretended knowledge of a cure for phthisis. There is no need to follow Naudé farther in his diatribe against the faults and imperfections, real and apparent, of Cardan's character; these must be visible enough to the most cursory student. Passages like these arouse the suspicion that Naudé knew books better than men, that at any rate he did not realize that men are to be found, and not seldom, who take pleasure in magnifying their foibles into gigantic follies, and their peccadilloes into atrocious crimes; while the rarity is to come across one who will set down these details with the circumstantiality used by Cardan. There is one defect in the _De Vita Propria_--an artistic one--which Naudé does not notice, namely, that in his narrative of his early days Cardan often over-reaches himself. His show of extreme accuracy destroys the perspective of the story, and, in his anxiety to be minute over the sequence of his childish ailments, the most trivial details of his uneasy dreams, and the cuffs he got from his father and his Aunt Margaret, he confuses the reader with multitudinous particulars and ceases to be dramatic. But the hallucinations which he nourished about himself were not all the outcome of senility. In the _De Varietate_, the work upon which he spent the greatest care, and the product moreover of his golden prime, he gives an account of four marvellous properties with which he was gifted.[248] The first of these was the power to pass, whenever the whim seized him, from sense into a kind of ecstasy. While he was in this state he could hear but faintly the sound of voices, and could not distinguish spoken words. Whether he would be sensitive to any great pain he could not say, but twitchings and the sharpest attacks of gout affected him not. When he fell into this state he felt a certain separation about the heart, as if his soul were departing from that region and taking possession of his whole body, a door being opened for the passage of the same. The sensation would begin in the cerebellum, and thence would be diffused along the spine. The one thing of which he was fully conscious, was that he had passed out of himself. The second property was that, when he would, he could conjure up any images he liked before his eyes, real [Greek: eidôla], and not at all to be compared with the blurred processions of phantoms which he was wont to see when he was a child. At the time when he wrote, perhaps by reason of his busy life, he no longer saw them whensoever he would, nor so perfectly expressed, nor for so long at a time. These images constantly gave place one to another, and he would behold groves, and animals, and orbs, and whatever he was fain to see. This property he attributed to the force of his imaginative power, and his clearness of vision. The third property was that he never failed to be warned in dreams of things about to happen to him; and the fourth was that premonitory signs of coming events would display themselves in the form of spots on his nails. The signs of evil were black or livid, and appeared on the middle finger; white spots on the same nail portending good fortune. Honours were indicated on the thumb, riches on the fore-finger, matters relating to his studies and of grave import on the third finger, and minor affairs on the little finger. In putting together the record of his life, Cardan eschewed the narrative form and followed a method of his own. He collected the details of his qualities, habits, and adventures in separate chapters; his birth and lineage, his physical stature, his diet, his rule of life, his imperfections, his poverty, the misfortunes of his sons, his masters and pupils, his travels, his experiences of things beyond nature, his cures, the persecutions of his foes, and divers other categories being grouped together to make up the _De Vita Propria_, which, though it is the most interesting book he has left behind him, is certainly the most clumsy and chaotic from a literary point of view. The chapters for the most part begin with his early years, and end with some detail as to his life in Rome, each one being a categorical survey of a certain side of his life; but remarks as to his personal peculiarities are scattered about from beginning to end. He tells how he could always see the moon in broad daylight;[249] of his passion for wandering about the city by night carrying arms forbidden by the law; of his practice of self-torture, beating his legs with a switch, twisting his fingers, pinching his flesh, and biting his left arm; and of going about within doors with naked legs; how at one time he was possessed with the desire, _heroica passio_, of suicide; of his habit of filling his house with pets of all sorts--kids, lambs, hares, rabbits, and storks. The chapter in which he records all the maladies which afflicted him, puts upon the reader's credulity a burden almost as heavy as is the catalogue given by another philosopher of the number of authors he mastered before his twelfth year. Two attacks of the plague, agues, tertian and quotidian, malignant ulcers, hernia, hæmorrhoids, varicose veins, palpitation of the heart, gout, indigestion, the itch, and foulness of skin. Relief in the second attack of plague came from a sweat so copious that it soaked the bed and ran in streams down to the floor; and, in a case of continuous fever, from voiding a hundred and twenty ounces of urine. As a boy he was a sleep-walker, and he never became warm below the knees till he had been in bed six hours, a circumstance which led his mother to predict that his time on earth would be brief. Cardan lived an abstemious life. He broke his fast on bread-and-water and a few grapes. He sometimes dined off bread, the yolk of an egg, and a little wine, and would take for supper a mess of beetroot and rice and a chicory salad. The catalogue of his favourite dishes seems to exhaust every known edible, and it will suffice to remark that he was specially inclined to sound and well-stewed wild boar, the wings of young cockerels and the livers of pullets, oysters, mussels, fresh-water crayfish because his mother ate greedily thereof when she was pregnant with him; but of all dishes he rates the best a carp from three pounds weight to seven, taken from a good feeding-ground. He praises all sweet fruit, oil, olives, and finds in rue an antidote to poison. Ten o'clock was his hour for going to bed, and he allowed himself eight hours' sleep. When wakeful he would walk about the room and repeat the multiplication table. As a further remedy for sleeplessness he would reduce his food by half, and would anoint his thighs, the soles of his feet, the neck, the elbows, the carpal bones, the temples, the jugulars, the region of the heart and of the liver, and the upper lip with ointment of poplars, or the fat of bear, or the oil of water-lilies. These few extracts will show that an intelligible narrative could scarcely be produced by the methods Cardan used. The book is a collection of facts, classified as a scientific writer would arrange the sections and subsections of his subject. In gathering together and grouping the leading points of his life, a method somewhat similar to his own will suffice, but there will be no need to descend to a subdivision so minute as his own. A task of this sort is never an easy one, and in this instance the difficulties are increased by the diffuse and complicated nature of the subject matter; and because, owing to Cardan's wayward mental habit, there is no saying in what corner of the ten large folios which contain his writings some pregnant and characteristic sentence, picturing effectively some aspect of his nature or perhaps exhibiting the man at a glance, may not be hidden away. It must not be inferred, because Cardan himself and his critics after him, have laid such great stress upon his vices and imperfections, that he was devoid of virtues. The most striking and remarkable of his merits was his industry, but even in this particular instance, where his excellence is most clearly manifest, he is constantly lamenting his waste of time and idleness. Again and again he mourns over the precious hours he has spent over chess and dice and games of chance. In his counsels to his children, he compares a gambler to a sink of all the vices, and in writing of his early life at Sacco he describes himself as an idle profligate, and tells how he entirely neglected his profession. If indeed such monstrous cantles were cut out of his time through idleness he must, though his life proved a long one, have possessed extraordinary power of rapid production; for the huge mass of his published work, without taking any account of the many manuscripts he burned from time to time, would, in the case of most men, represent the ceaseless labour of a long life. And the _corpus_ is not great by reason of haste or want of finish. He has recorded more than once how it was ever his habit to let his work be polished to the utmost before putting it in type. The citations with which his pages bristle proclaim him to be a reader almost as voracious and catholic as Burton; and Naudé, with the watchfulness of the hostile critic in his heart and the bookworm's knowledge in his brain, would have been ready and able to convict him of quoting authors he had not read, if the least handle for this charge should have been given, but no accusation of the kind is preferred. The story of his life shows him to be full of rough candour and honesty, and unlikely to descend to subterfuge, while his great love of reading and his accurate retentive memory would make easy for him a task which ordinary mortals might well regard as hopeless. Those critics who pass judgment on Cardan, taken solely as a Physician or as a Mathematician, will give a presentment more fallacious than imperfect generalizations usually furnish, for in Cardan's case the man, taken as a whole, was incomparably greater than the sum of his parts. Naudé remarks that a man who knows a little of everything, and that little imperfectly, deserves small respect as a citizen of the republic of letters, but Cardan did not belong to this category, as Julius Cæsar Scaliger found to his cost. He was not like the bookmen of the revival of learning--Poliziano, Valla, or Alberti may stand as examples--who after putting on the armour of the learned language and saturating themselves with the _literæ humaniores_, made excursions into some domain of science for the sake of recreation. Cardan might rather be compared with Varro or Theophrastus in classic, and with Erasmus, Pico, Grotius, or Casaubon in modern times. On this point Naudé indulges in something approaching panegyric. He writes--"Investigation will show us that many excelled him in the humanities or in Theology, some even in Mathematics, some in Medicine and in the knowledge of Philosophy, some in Oriental tongues and in either side of Jurisprudence, but where shall we find any one who had mastered so many sciences by himself, who had plumbed so deeply the abysses of learning and had written such ample commentaries on the subjects he studied? Assuredly in Philosophy, in Metaphysics, in History, in Politics, in Morals, as well as in the more abstruse fields of learning, nothing that was worth consideration escaped his notice." The foregoing eulogy from the pen of an adverse critic gives eloquent testimony to Cardan's industry and the catholicity of his knowledge. As to his industry, the record of his literary production, chronicled incidentally in the course of the preceding pages, will be evidence enough, seeing that, from the time when he "commenced author," scarcely a year went by when he did not print a volume of some sort or other; to say nothing of the production of those multitudinous unpublished MSS., of which some went to build up the pile he burnt in his latter years in Rome, while others, perhaps, are still mouldering in the presses of university or city libraries of Italy. Frequent reference has been made to the more noteworthy of his works. Books like the _De Vita Propria_, the _De Libris Propriis_, the _De Utilitate ex Adversis Capienda_, the _Geniturarum Exempla_, the _Theonoston_, the _Consilia Medica_, the dialogues _Tetim_ and _De Morte_, have necessarily been drawn upon for biographical facts. The _De Subtilitate_ and the _De Varietate Rerum_; the _Liber Artis Magnæ_, the _Practica Arithmeticæ_, have been noticed as the most enduring portions of his legacy to posterity; wherefore, before saying the final word as to his literary achievement, it may not be superfluous to give a brief glance at those of his books which, although of minor importance to those already cited, engaged considerable attention in the lifetime of the writer. The work upon which Cardan founded his chief hope of immortality was his _Commentary on Hippocrates_. In bulk it ranks first easily, filling as it does one of the large folios of the edition of 1663. Curiously enough, in addition to a permanent place in the annals of medicine, Cardan anticipated for this forgotten mass of type a general and immediate popularity; wider than any which his technical works could possibly enjoy, seeing that it dealt with the preservation of health, the greatest mortal blessing, and must on this account be of interest to all men. It will be enough to remark of these commentaries that no portion of Cardan's work yields less information as to the author's life and personality; to dilate upon them, ever so superficially, from a scientific point of view, would be waste of time and paper. Another of his works, which he rated highly, was his treatise on Music. It was begun during his tenure of office at Pavia, _circa_ 1547, and he was still at work upon it two years before his death.[250] It is not difficult to realize, even at this interval of time, that this book at the date of its publication must have been welcomed by all musical students as a valuable contribution to the literature of their subject. It is strongly marked by Cardan's particular touch, that formative faculty by which he almost always succeeded in stimulating fresh interest in the reader, and exhibiting fresh aspects of whatever subject he might be treating. This work begins by laying down at length the general rules and principles of the art, and then goes on to treat of ancient music in all its forms; of music as Cardan knew and enjoyed it; of the system of counterpoint and composition, and of the construction of musical instruments. The Commentary on _Ptolemæi de Astrorum Judiciis_, the writing of which beguiled the tedium of his voyage down the Loire on his journey to Paris in 1552, is a book upon which he spent great care, and is certainly worthy of notice. Cardan's gratitude to Archbishop Hamilton for the liberal treatment and gracious reception he had recently encountered in Scotland, prompted him to dedicate this volume to his late patient. He writes in the preface how he had expected to find the Scots a pack of barbarians, but their country, he affirms, is cultivated and humanized beyond belief,--"and you yourself reflect such splendour upon your nation that now, by the very lustre of your name, it must needs appear to the world more noble and illustrious than at any time heretofore. What need is there for me to speak of the school founded by you at St. Andrews, of sedition quelled, of your country delivered, of the authority of your brother the Regent vindicated? These are merely the indications of your power, and not the source thereof." In the preface he also writes at length, concerning the horoscope of Christ,[251] in a strain of apology, as if he scented already the scandal which the publication of this injudicious performance was destined to raise. In estimating the influence of comets he sets down several instances which had evidently been brought to his notice during his sojourn in Scotland: how in 1165, within fourteen days of the appearance of a great comet, Malcolm IV., known on account of his continence as the virgin king, fell sick and died. Again, in 1214 two comets, one preceding and the other following the sun, appeared as fore-runners of the death of King William after a reign of forty-nine years. Perhaps the most interesting of his comments on Ptolemy's text are those which estimate the power of the stellar influences on the human frame, an aspect of the question which, by reason of his knowledge of medicine and surgery, would naturally engage his more serious attention. He tells of the birth of a monstrous child--a most loathsome malformation--at Middleton Stoney, near Oxford, during his stay in England,[252] and gives many other instances of the disastrous effects of untoward conjunction of the planets upon infants born under the influence of the same. He accuses monks and nuns of detestable vices in the plainest words, words which were probably read by the emissaries of the spiritual authority when the charge of impiety was being got up against him. In the _Geniturarum Exempla_ the horoscopes of Edward VI., Archbishop Hamilton, and Cardan himself have been already noticed; that of Sir John Cheke comes next in interest to these, and, it must be admitted, is no more trustworthy. It declares that Cheke would attain the age of sixty-one years, that he would be most fortunate in gathering wealth and friends around him, that he would die finally of lingering disease, and involve many in misfortune by his death--a faulty guess, indeed, as to the future of a man who died at forty-three, borne down by the weight of his misfortunes, neglected and forgotten by his former adherents, stripped of his wealth and covered with shame, in that he had abjured his faith to save a life which was so little worth preserving. Naudé does not neglect to censure Cardan for his maladroit attempts to read the future. He writes:--"This matter, forsooth, gave a ready handle to Cardan's rivals, and especially to those who were sworn foes of astrology; so that they were able to jibe at him freely because, neither in his own horoscope, nor in that of his son Giovanni Battista, nor in that of Aymer Ranconet, nor in that of Edward VI., king of England, nor in any other of the schemes that he drew, did he rightly foresee any of the events which followed. He did not divine that he himself was doomed to imprisonment, his son to the halter, Ranconet to a violent death, and Edward to a brief term of life, but predicted for each one of these some future directly contrary."[253] The treatise _De Consolatione_, probably the best known of Cardan's ethical works, was first published at Venice in 1542 by Girolamo Scoto, but it failed at first to please the public taste. It was not until 1544, when it was re-issued bound up with the _De Sapientia_ and the first version of the _De Libris Propriis_ from the press of Petreius at Nuremberg, that it met with any success. Perhaps the sober tone and didactic method of this treatise appealed more readily to the mood of the German than of the Italian reader. From internal evidence it is obvious that Cardan was urged to write it by the desire of making known to the world the bitter experience of his early literary and professional struggles. In the opening paragraph he lets it be seen that he intends to follow a Ciceronian model, and records his regret that the lament of Cicero over his daughter's death should have perished in the barbarian wars. The original title of the book was _The Accuser_, to wit, something which might censure the vain passions and erring tendencies of mankind, "at post mutato nomine, et in tres libellos diviso, de Consolatione eum inscripsimus, quod longe magis infelices consolatione, quam fortunati reprehensione, indigere viderentur." The subsequent success of the book was probably due to this change of name, though the author himself preferred to have discovered a special reason for its early failure.[254] The plan of the treatise is the same as that of a dozen others of the same nature: an effort to persuade men in evil case that they may find relief by regarding the misfortunes they suffer as transitory accidents in no way affecting the chief end of life, and by seeking happiness alone in trafficking with the riches of the mind. It is doubtful whether any of the books written with this object have ever served their purpose, save in the case of their originators. Cardan may have found the burden of his failure and poverty grow lighter as he set down his woes on paper, but the rest of the world must have read the book for some other reason than the hope of consolation. Read to-day in Bedingfield's quaint English, the book is full of charm and interest. It is filled with apt illustration from Greek philosophy and from Holy Writ as well, and lighted up by spaces of lively wit. It was accepted by the public taste for reasons akin to those which would secure popularity for a clever volume of essays at the present time, and was translated into more than one foreign language, Bedingfield's translation being published some thirty years after its first appearance. The _De Sapientia_, with which it is generally classed, is of far less interest. It is a series of ethical discourses, lengthy and discursive, which must have seemed dull enough to contemporary students: to read it through now would be a task almost impossible. It is only remembered because Cardan has inserted therein, somewhat incongruously, that account of his asserted cures of phthisis which Cassanate quoted when he wrote to Cardan about Archbishop Hamilton's asthma, and which were afterwards seized upon by hostile critics as evidence of his disregard of truth. Another of his minor works highly characteristic of the author is the _Somniorum Synesiorum_, a collection of all the remarkable dreams he ever dreamt, many of which have been already noticed. To judge from what specimens of his epistles are extant, Cardan seems to have been a good letter-writer. One of the most noteworthy is that which he addressed to Gian Battista after his marriage. It shows Cardan to have been a loving father and a master of sapient exhortation, while the son's fate gives melancholy testimony of the futility of good counsel unaided by direction and example. He tells of his grief at seeing the evil case into which his son had fallen, vexed by poverty, disgrace, and loss of health, how he would gladly even now receive the prodigal into his house (he says nothing about the wife), did he not fear that such a step would lead to his own ruin rather than to his son's restoration. After showing that any fresh misfortune to himself must needs cut away the last hope for Gian Battista, he sketches out a line of conduct for the ill-starred youth which he declared, if rightly pursued, might re-establish his fortunes. He begins by advising his son to read and lay to heart the contents of the _De Consolatione_ and the _De Utilitate_, and then, somewhat more to the purpose, promises him half his earnings of the present and the coming year. Beyond this Gian Battista should have half the salary of any office which his father might get for himself, and half of the piece of silk which he had received from the Venetian Ambassador, supposing that the young man should not be able to get a like piece for himself from the same source. He next cites the _De Consolatione_ to demonstrate the futility of lamentation over misfortune past or present, or indeed over any decree of fate. He bids Gian Battista reflect that he is human not a brute, a man not a woman, a Christian not a Moslem or Jew, an Italian not a barbarian, sprung from a worthy city and family, and from a father whose name by itself will prove a title to fame. His only real troubles are a weak body and infirm health--one a gift of heredity, the other aggravated by dissolute habits. It may be a vain thing for men to congratulate themselves over their happiness, but it is vainer for them to cry out for solace over past calamity. Contempt of money is foolish, but contempt of God is ten times worse. Cardan concludes this part of his letter by reciting two maxims given him by his father--one, to have daily remembrance of God and of His vast bounty, the other, to pursue with the utmost diligence any task taken in hand. Cardan then treats the scapegrace to a string of maxims from the _De Utilitate_, maxims which a model son might have read, but which Gian Battista would certainly put aside unnoticed, and finishes with some serviceable practical counsel: "Keep your mind calm, go early to bed, for ours is a hot-blooded race and predisposed to suffer from stone. Take nine hours' sleep, rise at six and visit your patients, being careful to use no speech unconnected with the case before you. Avoid heating your body to perspiration; go forth on horseback, come back on foot; and on your return put on warm clothes. Drink little, break your fast on bread, dried fish, and meat, and then give four hours to study, for studies bring pleasure, relief from care, and mental riches; they are the foundations of renown, and enable a man to do his duty with credit. See your patients again; and, before you sup, take exercise in the woods and fields adjacent. Should you become over-heated or wet with rain, cast off and dry your damp clothes, and don dry ones. Sup heartily, and go to bed at eight; and when, by the brevity of the night, this is not convenient, take a corresponding rest during the day. Abstain from summer fruit, from black wine, from vain overflow of talk, from falsehood and gaming, from trusting a woman or over-indulging her, for she is a foolish animal and full of deceit. Over-fondness towards a woman will surely bring evil upon you. Bleed and purge yourself as little as possible; learn by experience of other men's faults and misfortunes; live frugally; bear yourself suavely to all men; and let study be your main end. All this and more have I set forth in the books I have named. Trust neither promises nor hopes, for these may be vain and delusive; and reckon your own only that which you hold in your hand. Farewell." From the fact that Cardan took part in an unofficial medical conference in Paris, that he afterwards superseded Cassanate as the Archbishop of St. Andrews' physician, and did not find himself with a dozen or so quarrels on his hands, it may be assumed that he was laudably free from the jealousy attributed by tradition to his profession. This instance becomes all the more noteworthy when his natural irascibility, and the character of the learned controversy of the times comes to be considered. He does not spare his censure in remarking on the too frequent quarrels of men of letters,[255] albeit these quarrels must have lent no little gaiety to the literary world. No one who reads the account of Gian Battista's fate can doubt the sincerity of Cardan's remorse for that neglect of the boy's youthful training which helped to bring him to ruin, and the care which he bestowed upon his grandson Fazio proved that his regret was not of that sort which exhales itself in empty words. The zeal with which he threw himself into the struggle for his son's life, and his readiness to strip himself of his last coin as the fight went on, show that he was capable of warm-hearted affection, and afraid of no sacrifice in the cause of duty. The brutal candour which Cardan used in probing the weaknesses of his own nature and in displaying them to the world, he used likewise in his dealings with others. If he detected Branda Porro or Camutio in a blunder he would inform them they were blockheads without hesitation, and plume himself afterwards on the score of his blunt honesty. Veracity was not a common virtue in those days, but Cardan laid claim to it with a display of insistence which was not, perhaps, in the best taste. Over and over again he writes that he never told a lie;[256] a contention which seems to have roused especially the bile of Naudé, and to have spurred him on to make his somewhat clumsy assault on Cardan's veracity.[257] His citation of the case of the stranger who came with the volume of Apuleius for sale, and of the miraculous gift of classic tongues, has already been referred to; but these may surely be attributed to an exaggerated activity of that particular side of Cardan's imagination which was specially prone to seize upon some figment of the brain, and some imperfectly apprehended sensation of the optic nerve, and fashion from these materials a tale of marvel. Delusions of this sort were common in reputed witches, as Reginald Scot writes--"They learne strange toongs with small industrie (as Aristotle and others affirme)."[258] The other charge preferred by Naudé as to the pretended cure of consumption, and the consequent quibbling and tergiversation, is a more valid one. It has been noted how Cardan, previous to his journey to Scotland, had posed as the discoverer of a cure for this malady. In the list of his cures successfully treated he includes several in which he restored patients suffering from blood-spitting, fever, and extreme emaciation to sound health, the most noteworthy of these being that of Girolamo Tiboldo, a sea-captain. When the sick man had risen from his bed and had become fat and healthy, Cardan deemed that the occasion justified a certain amount of self-gratulation, but the physicians, out of envy, declared that Tiboldo had never suffered from true phthisis. In his account of the case Cardan says that he, and the physicians as well, were indeed untruthful over the matter, his own falsehood having been the result of over-sanguine hope, and theirs the outcome of spiteful envy. Tiboldo died after all of chest disease, but not till five years later, and then from a chill caught through sitting in wet garments.[259] The term consumption has always been applied somewhat loosely, and Cardan probably would have been allowed the benefit of this usage if he had not, in an excess of candour, set down the workings of his mind and conscience with regard to this matter. Writing of his treatment of Archbishop Hamilton, he says: "And in truth I cured scarcely any patients of phthisic disease, though I did find a remedy for many who were suffering from similar maladies, wherefore that boast of mine, that proclamation of merit to which I had no right, worked no small profit to me, a man very little given to lying. For the people about the Archbishop, urged on by these and other considerations, persuaded him that he had no chance of regaining his health except by putting himself under my care, and that he should fly to me as his last hope."[260] It has already been noted that Cardan's claim to some past knowledge in the successful treatment of chest diseases had weight with the Archbishop and Cassanate, and the result of his visit surely proved that their confidence was not ill-placed; his boasting may have been a trifle excessive, but it was based on hope rather than achievement; and if proof can be adduced that it was not prompted by any greed of illegitimate fame or profit, it may justly be ranked as a weakness rather than as a serious offence. To these two instances of falsehood Naudé adds a third, to wit, Cardan's claim to the guidance of a familiar spirit. He refuses to let this rank as a delusion; and, urged no doubt by righteous indignation against the ills springing from kindred superstitions, he writes down as a liar rather than a dupe the man who, after mastering the whole world of science, could profess such folly. Considering the catholicity of Cardan's achievements, and the eager spirit of inquiry he displayed in fields of learning remote from his own particular one, it is worthy of notice that he did not allow this discursive humour, which is not seldom a token of instability, to hold him back from pursuing the supreme aim of his life, that is, eminence in the art of Medicine. In his youth the threats and persuasions of his father could not induce him to take up Jurisprudence with an assured income and abandon Medicine. At Sacco, at Gallarate, and afterwards in Milan he was forced by the necessity of bread-winning to use his pen in all sorts of minor subjects that had no real fascination for him, but all his leisure was devoted to the acquisition of Medical knowledge. Prudence as well as inclination had a share in directing his energies into this channel, for a report, for which no doubt there was some warrant, was spread abroad that what skill he had lay entirely in the knowledge of Astrology; and, as this rumour operated greatly to his prejudice,[261] he resolved to perfect himself in Medicine and free his reputation from this aspersion. He had quarrelled violently with the physicians over the case of Count Borromeo's child which died, and with Borromeo himself, and, almost immediately after this, he published his book, _De Astrorum Judiciis_, a step which tended to identify him yet more closely with Astrology, and to raise a cry against him in Milan, which he declares to be the most scandal-mongering city in the Universe. But it is clear that in this instance scandal was not far wrong, and that Cardan himself was right in purging himself of the quasi science he ought never to have taken up. Medicine, when Cardan began his studies, was beginning to feel the effects of the revival of Greek learning. With the restored knowledge of the language of Greece there arose a desire to investigate the storehouses of science, as well as those of literature, and the extravagant assumption of the dogmatists, and the eccentricities of the Arabic school gave additional cogency to the cry for more light. The sects which Galen had endeavoured to unite sprang into new activity within a century after his death. The Arabian physicians, acute and curious as they were, had exercised but a very transient influence upon the real progress of the art, the chief cause of their non-success being their adhesion to arbitrary and empirical tradition. At the end of the fifteenth century, Leonicinus, a professor at Ferrara, recalled the allegiance of his pupils to the authority of Hippocrates by the ability and eloquence of his teaching; and, by his translation of Galen's works into Latin, he helped still farther to confirm the ascendency of the fathers of Medicine. The Arabians, sprung from the East, the storehouse of drugs and simples, and skilled in Chemistry, were the founders of the Pharmacopoeia,[262] but with this exception they did nothing to advance Medicine beyond the point where the Greeks had left it. The treatises of Haly, Avicenna, and Maimonides were little better than faint transcriptions of the writings of the great forerunners. Their teaching was random and spasmodic, whereas the system of Hippocrates was conceived in the spirit of Greek philosophy, moving on by select experience, always observant and cautious, and ascending by slow and certain steps to the generalities of Theory. Indeed the science of Medicine in the hands of Hippocrates and his school seems, more than any other, to have presented to the world a rudimentary essay, a faint foreshadowing of the great fabric of inductive process, subsequently formulated by the genius of Bacon. At various epochs Medicine had been specially stimulated by the vivifying spirit of Greek science; in the Roman school in the days of Celsus, and in the Arabian teaching likewise. Fuller acknowledgment of the authority of Greek Medicine came with the Renaissance,[263] but even this long step in advance did not immediately liberate the art from bondage. A new generation of professors arose who added fresh material to the storehouses, already overflowing, of pedantic erudition, and showed the utmost contempt for any fruit of other men's labour which might not square exactly with the utterances of the founders. This attitude rendered these professors of Medicine the legitimate objects of ridicule, as soon as the leaven of the revival began to work, and the darts of satire still fly, now and then, at the same quarry. Paracelsus, disfigured as his teaching was by mysticism, the arts of the charlatan, and by his ignorant repudiation of the service of Anatomy, struck the first damaging blows at this illegitimate ascendency, by the frequent success of his empirical treatment, by the contempt he heaped upon the scholastic authorities, and by the boldness with which he assailed every thesis which they maintained. Men of more sober intellect and weighty learning soon followed in his track. Fernelius, one of the physicians Cardan met in Paris, boldly rejected what he could not approve by experience in the writings of Hippocrates and Galen, and stood forth as the advocate for free inquiry, and Joubert of Montpelier, Argentier of Turin, and Botal of Asti subsequently took a similar course. When Cardan went to study at Pavia in 1519 this tradition was unshaken. It was not until the advent of Vesalius that the doom of the ancient system was sounded. Then, when Anatomy sprang to the front as the potent ally of Medicine, the science of healing entered upon a fresh stage, but this new force did not make itself felt soon enough to seduce Cardan from the altars of the ancients to the worship of new gods. As long as he lived he was a follower of the great masters, though at the same time his admiration of the teaching of Vesalius was enthusiastic and profound. His love of truth and sound learning forbade him to give unreflecting adhesion to the precepts of any man, however eminent, and when he found that Galen was a careless commentator on Hippocrates,[264] and failed to elucidate the difficulties with which he professed to deal, he did not spare his censure.[265] In the _De Subtilitate_ he speaks of him as "Verbosus et studio contradicendi tædulus ut alterum vix ferre queas, in reliquo gravis jactura artium posita sit, quam nostræ ætatis viri restituere conati sunt."[266] But as Galen's name is quoted as an authority on almost every page of the _Consilia Medica_, it may be assumed that Cardan's faith in his primary theories was unshaken. In his Commentaries on Hippocrates, Galen professes a profound respect for his master, but the two great men must be regarded as the leaders of rival schools; indeed it could hardly be otherwise, seeing how vast was the mass of knowledge which Galen added to the art during his lifetime. Hippocrates, by denying the supernatural origin of disease, by his method of diagnosis, by the importance he attached to air and diet, by his discriminating use of drugs, and by the simplicity of his system generally, had placed Medicine on a rational basis. In the six hundred years' space which elapsed before the appearance of Galen, Medicine was broken up into many rival schools. The Dogmatici and the Empirici for many years wrangled undisturbed, but shortly after the Christian era the Methodici entered the field, to be followed later on by the Eclectici and a troop of other sects, whose wranglings, and whose very names, are now forgotten. In his _History of Medicine_, Dr. Bostock gives a sketch of the attitude of Galen towards the rival schools. "In his general principles he may be considered as belonging to the Dogmatic sect, for his method was to reduce all his knowledge, as acquired by the observation of facts, to general theoretical principles. These principles he indeed professed to deduce from experience and observation,[267] and we have abundant proofs of his diligence in collecting experience and his accuracy in making observations; but still, in a certain sense at least, he regards individual facts and the details of experience as of little value, unconnected with the principles which he laid down as the basis of all medical reasoning. In this fundamental point, therefore, the method pursued by Galen appears to have been directly the reverse of that which we now consider as the correct method of scientific investigation; and yet, such is the force of actual genius, that in most instances he attained the ultimate object in view, although by an indirect path. He was an admirer of Hippocrates, and always speaks of him with the most profound respect, professing to act upon his principles, and to do little more than expound his doctrines and support them by new facts and observations. Yet in reality we have few writers whose works, both as to substance and manner, are more different from each other than those of Hippocrates and Galen, the simplicity of the former being strongly contrasted with the abstruseness and refinement of the latter." The antagonism between these two great men was not perhaps more marked than might have been expected, considering that an interval of six hundred years lay between them. However loyal he may have been to his master, Galen, with his keen, catholic, and subtle intellect, was bound to fall under the sway of Alexandrian influence while he studied in Alexandria as the pupil of Heraclianus. The methods of the contemporary school of philosophy fascinated him; and, in his endeavour to bring Medicine out of the chaotic welter in which he found it, he attempted--unhappily for the future of science--to use the hyper-idealistic Platonism then dominant in Alexandria, rather than the gradual and orderly induction of Hippocrates, as a bond of union between professional and scientific medicine; a false step for which not even his great services to anatomy and physiology can altogether atone. Yet most likely it was this same error, an error which practically led to the enslavement of Medicine till the seventeenth century, which caused Cardan to regard him, and not Hippocrates, as his master. The vastness and catholicity of Galen's scheme of Medicine must have been peculiarly attractive to a man of Cardan's temper; and that Galen attempted to reconcile the incongruous in the teleological system which he devised, would not have been rated as a fault by his Milanese disciple. Galen taught as a cardinal truth the doctrine of the Hippocratic elements, heat, cold, moisture, and dryness, and a glance at the Consilium which Cardan wrote out on Archbishop Hamilton's illness, will show how completely he was under the sway of this same teaching. The genius of Hippocrates was perhaps too sober and orderly to win his entire sympathy; the encyclopædic knowledge, the literary grace, and the more daring flights of Galen's intellect attracted him much more strongly. Hippocrates scoffed at charms and amulets, while Galen commended them, and is said to have invented the anodyne necklace which was long known and worn in England. There is no need to specify which of the masters Cardan would swear by in this matter. The choice which Cardan made, albeit it was exactly what might have been anticipated, was in every respect an unfortunate one. He put himself under a master whose teaching could have no other effect than to accentuate the failings of the pupil, whereas had he let his mind come under the more regular discipline of Hippocrates' method, it is almost certain that the mass of his work, now shut in dusty folios which stand undisturbed on the shelves for decade after decade, would have been immeasurably more fruitful of good. With all his industry in collecting, and his care in verifying, his medical work remains a heap of material, and nothing more valuable. Learning and science would have profited much had he put himself under the standard of the Father of Medicine, and still more if fate had sent him into being at some period after the world of letters had learned to realize the capabilities of the inductive system of Philosophy. It may readily be conceded that Cardan during his career turned to good account the medical knowledge which he had gathered from the best attainable sources, and that he was on the whole the most skilful physician of his age. He likewise foreshadowed the system of deaf mute instruction. A certain Georgius Agricola, a physician of Heidelberg who died in 1485, makes mention of a deaf mute who had learnt to read and write, but this statement was received with incredulity. Cardan, taking a more philosophic view, declared that people thus afflicted might easily be taught to hear by reading, and to speak by writing; writing was associated with speech, and speech with thought, but written characters and ideas might be connected without the intervention of sounds.[268] This view, put forward with all the authority of Cardan's name, would certainly rouse fresh interest in the question, and, whether stimulated by his words or not, an attempt to teach deaf mutes was made by Pedro de Ponce, a Spanish Dominican, about 1560. But it would not be permissible to claim for Cardan any share in the epoch-making discoveries in Medicine. Galen as an experimental physiologist had brought diagnosis to a level unattained before. His methods had been abandoned by his successors, and practice had in consequence suffered deterioration, but Cardan, studying under the revived Galenism, called into life by the teaching of Vesalius, went to deal with his cures under conditions more favourable than those offered by any previous period of the world's history. His cure of Archbishop Hamilton's asthma, over which Cassanate and the other doctors had failed, was due to a more careful diagnosis and a more judicious application of existing rules, rather than to the working of any new discoveries of his own. Viewed as a soldier in the service of Hygeia, how transient and slender is the fame of Cardan compared with that of Linacre, Vesalius, or Harvey! Were his claims to immortality to rest entirely on his contribution to Medicine, his name would have gone down to oblivion along with that of Cavenago, Camutio, Della Croce, and the multitude of jealous rivals who, according to his account, were ever plotting his downfall. But it was rescued from this fate by his excellence as a mathematician, by the interest clinging to his personality, by the enormous range of his learning, by his picturesque reputation as a dreamer of dreams, and a searcher into the secrets of the hidden world. In an age when books were few and ill-composed, his works became widely popular; because, although he dealt with abstruse subjects, he wrote--as even Naudé admits--in a passably good style, and handled his subject with a lightness of touch which was then very rare. This was the reason why men went on reading him long after his works had ceased to have any scientific value; which induced writers like Burton and Sir Thomas Browne to embroider their pages freely with quotations from his works, and thus make his name familiar to many who have never handled a single one of his volumes. It is somewhat strange to find running through the complex web of Cardan's character a well-defined thread of worldly wisdom and common-sense; to find that a man, described by almost every one who has dealt with his character as a credulous simpleton, one with disordered wits, or a down-right madman, should, when occasion demanded, prove himself to be a sharp man of business. When Fazio died he left his son with a number of unsettled law-suits on hand, concerning which he writes: "From my father's death until I was forty-six, that is to say for a space of twenty-three years, I was almost continually involved in law-suits. First with Alessandro Castillione, surnamed Gatico, with respect to certain plantations, and afterwards with his kinsfolk. Next with the Counts of Barbiani, next with the college, next with the heirs of Dominico de Tortis, who had held me in his arms when I was baptized. Out of all these suits I came victorious. It was indeed a matter for surprise that I should have got the better of Alessandro Castillione, seeing that his uncle sat as judge. Moreover, he had already got a decision against me, a decision which, as the jurisconsults declared, helped my case as the trial went on, and I was able to force him to pay me all the money which was in dispute. A like good fortune attended me while my claims were considered by the heads of the Milanese College, and finally rejected by several votes. Then afterwards, when they had decided to admit me, and when they tried to subject me to certain rules which placed me on a footing inferior to their own, I compelled them to grant me full membership. In the case of the Barbiani, after long litigation and many angry words and much trouble, I came to terms with them; and, having received the sum of money covenanted by agreement, I was entirely freed from vexation of the law."[269] Writing generally of his monetary dealings, Cardan says: "Whenever I may have incurred a loss, I have never been content merely to retrieve the same, I have always contrived to seize upon something extra."[270] Or again: "If at any time I have lost twenty crowns, I have never rested until I have succeeded in getting back these and twenty more in addition."[271] Cardan left in his _Dicta Familiaria_ and _Præceptorum ad filios Libellus_ a long list of aphorisms and counsels, many of which give evidence of keen insight and busy observation of mankind, while some are distinguished by a touch of humour rare in his other writings. He bids his children to be careful how they offend princes, and, offence being given, never to flatter themselves that it has been pardoned; to live joyfully as long as they can, for men are for the most part worn out by care; never to take a wife from a witless stock or one tainted with hereditary disease; to refrain from deliberating when the mind is disturbed; to learn how to be worsted and suffer loss; and to trust a school-master to teach children, but not to feed them. One of the dicta is a gem of quaint wisdom. "Before you begin to wash your face, see that you have a towel handy to dry the same." If all the instances of prodigies, portents, visions, and mysterious warnings which Cardan has left on record were set down in order, a perusal of this catalogue would justify, if it did not compel, the belief that he was little better than a credulous fool, and raise doubts whether such a man could have written such orderly and coherent works as the treatise on Arithmetic, or the book of the Great Art. But Cardan was beyond all else a man of moods, and it would be unfair to figure as his normal mental condition those periods of overwrought nervousness and the hallucinations they brought with them. In his old age the nearness of the inevitable stroke, and the severance of all earthly ties, led him to discipline his mind into a calmer mood, but early and late during his season of work his nature was singularly sensitive to the wearing assaults of cares and calamities. In crises of this kind his mind would be brought into so morbid a condition, that it would fall entirely under the sway of any single idea then dominant; such idea would master him entirely, or even haunt him like one of those unclean spectres he describes with such gusto in the _De Varietate_. What he may have uttered when these moods were upon him must not be taken seriously; these are the moments to which the major part of his experiences of things _supra naturam_ may be referred. But there are numerous instances in which he describes marvellous phenomena with philosophic calm, and examines them in the true spirit of scepticism. In his account of the trembling of the bed on which he lay the night before he heard of Gian Battista's marriage, he goes on to say that a few nights after the first manifestation, he was once more conscious of a strange movement; and, having put his hand to his breast, found that his heart was palpitating violently because he had been lying on his left side. Then he remembered that a similar physical trouble had accompanied the first trembling of the bed, and admits that this manifestation may be referred to a natural cause, _i.e._ the palpitation. He tells also how he found amongst his father's papers a record of a cure of the gout by a prayer offered to the Virgin at eight in the morning on the first of April, and how he duly put up the prayer and was cured of the gout, but he adds: "Sed in hoc, auxiliis etiam artis usus sum."[272] Again with regard to the episode of the ignition of his bed twice in the same night, without visible cause, he says that this portent may have come about by some supernatural working; but that, on the other hand, it may have been the result of mere chance. He tells another story of an experience which befell him when he was in Belgium.[273] He was aroused early in the morning by the noise made outside his door by a dog catching fleas. Having got out of bed to see to this, he heard the sound as of a key being softly put into the lock. He told this fact to the servants, who at once took up the tale, and persuaded themselves that they had heard many noises of the same kind, and others vastly more wonderful; in short, the whole house was swarming with apparitions. The next night the noise was repeated, and a second observation laid bare the real cause thereof. The scratching of the dog had caused the bolt to fall into the socket, and this produced the noise which had disquieted him. He writes in conclusion: "Thus many events which seem to defy all explanation have really come to pass by accident, or in the course of nature. Out of such manifestations as these the unlettered, the superstitious, the timorous, and the over-hasty make for themselves miracles."[274] Again, after telling a strange story of a boy who beheld the image of a thief in the neck of a phial, and of some incantations of Josephus Niger, he concludes: "Nevertheless I am of opinion that all these things were fables, and that no one could have had any real knowledge thereof, seeing that they were nothing else than vain triflings."[275] In a nature so complex and many-sided as Cardan's, strange resemblances may be sought for and discovered, and it certainly is an unexpected revelation to find a mental attitude common to Cardan, a man tied and bound by authority and the traditions of antiquity, and such a daring assailant of the schools and of Aristotle as Doctor Joseph Glanvil. The conclusions of Cardan as to certain obscure phenomena recently cited show that, in matters lying beyond sensual cognition, he kept an open mind. In summing up the case of the woman said to have been cured by the incantations of Josephus Niger, he says that she must have been cured either by the power of the imagination, or by the agency of the demons. Here he anticipates the arguments which Glanvil sets forth in _Sadducismus Triumphatus_. Writing on the belief in witchcraft Glanvil says, "We have the attestation of thousands of eye and ear witnesses, and these not of the easily-deceivable vulgar only, but of wise and grave discerners; and that when no interest could oblige them to agree together in a common Lye. I say, we have the light of all these circumstances to confirm us in the belief of things done by persons of despicable power and knowledge, beyond the reach of Art and ordinary Nature. Standing public Records have been kept of these well-attested Relations, and Epochas made of those unwonted events. Laws in many Nations have been enacted against those vile practices; those amongst the Jews and our own are notorious; such cases have often been determined near us by wise and reverend Judges, upon clear and convictive Evidence; and thousands of our own Nation have suffered death for their vile compacts with Apostate spirits. All these I might largely prove in their particular instances, but that 'tis not needful since these did deny the being of Witches, so it was not out of ignorance of these heads of Argument, of which probably they have heard a thousand times; but from an apprehension that such a belief is absurd, and the things impossible. And upon these presumptions they condemn all demonstrations of this nature, and are hardened against conviction. And I think those that can believe all Histories and Romances; That all the wiser would have agreed together to juggle mankind into a common belief of ungrounded fables, that the sound senses of multitudes together may deceive them, and Laws are built upon Chimeras; That the greatest and wisest Judges have been Murderers, and the sagest persons Fools, or designing Impostors; I say those that can believe this heap of absurdities, are either more credulous than those whose credulity they reprehend; or else have some extraordinary evidence of their perswasion, viz.: That it is absurd and impossible that there should be a Witch or Apparition."[276] Cardan's argument in the case of the sick woman, that it would be difficult if not impossible to invent cause for her cure, other than the power of imagination or Demoniac agency, if less emphatic and lengthy than Glanvil's, certainly runs upon parallel lines therewith, and suggests, if it does not proclaim, the existence of such a thing as the credulity of unbelief; in other words that those who were disposed to brush aside the alternative causes of the cure as set down by him, and search for others, and put faith in them, would be fully as credulous as those who held the belief which he recorded as his own. FOOTNOTES: [248] _De Varietate_, p. 314. [249] _De Vita Propria_, ch. xxxvii. p. 115. [250] "Musicam, sed hanc anno post VI. scilicet MDLXXIV. correxi et transcribi curavi."--_De Vita Propria_, ch. xlv. p. 176. [251] This is on p. 164. [252] Page 266. [253] _Judicium de Cardano_. [254] Page 57. [255] "Ita nostra ætate, lapsi sunt clarissimi alioqui viri in hoc genere. Budæus adversus Erasmum, Fuchsius adversus Cornarium, Silvius adversus Vesalium, Nizolius adversus Maioragium: non tam credo justis contentionum causis, quam vanitate quadam et spe augendæ opinionis in hominibus."--_Opera_, tom. i. p. 135. [256] He writes in this strain in _De Vita Propria_, ch. xiv. p. 49, in _De Varietate Rerum_, p. 626, and in _Geniturarum Exempla_, p. 431. [257] On the subject of dissimulation Cardan writes: "Assuevi vultum in contrarium semper efformare; ideo simulare possum, dissimulare nescio."--_De Vita Propria_, ch. xiii. p. 42. Again in _Libellus Præceptorum ad filios_ (_Opera_, tom. i. p. 481), "Nolite unquam mentiri, sed circumvenire [circumvenite?]." [258] _Discoverie of Witchcraft_, ch. xi. [259] Donato Lanza, the druggist, who had been his first introducer to Sfondrato, was equally perverse. After Cardan had cured him of phthisis, he jumped out of a window to avoid arrest, and fell into a fish-pond, and died of the cold he took.--_Opera_, tom. i. p. 83. [260] _Opera_, tom. i. p. 136. [261] _De Vita Propria_, ch. x. p. 32. [262] The Materia Medica of Mesua, dating from the eleventh century, was used by the London College of Physicians in framing their Pharmacopoeia in 1618. [263] In 1443 a copy of Celsus was found at Milan; Paulus Ægineta was discovered a little later. [264] _Opera_, tom. ix. p. 1. [265] _De Immortalitate Animorum_ (Lyons, 1545), p. 73. _De Varietate_, p. 77. _Opera_, tom. i. p. 135. [266] _De Subtilitate_, p. 445. [267] "Galen's great complaint against the Peripatetics or Aristotelians, was that while they discoursed about Anatomy they could not dissect. He met an argument with a dissection or an experiment. Come and see for yourselves, was his constant cry."--_Harveian Oration_, Dr. J.F. Payne, 1896. [268] _Opera_, tom. x. p. 462. [269] _De Vita Propria_, ch. xxviii. p. 73. [270] _Ibid.,_ ch. xxiii. p. 64. [271] _De Utilitate_, p. 309. He also writes at length in the Proxenata on Domestic Economy.--Chapter xxxvii. _et seq. Opera_, tom. i. p. 377. [272] _De Vita Propria_, ch. xxxvii. p. 118. [273] _De Varietate_, p. 589. [274] _De Varietate_, p. 589. [275] _Ibid.,_ p. 640. [276] _Sadducismus Triumphatus_ (Ed. 1682), p. 4. CHAPTER XV WHEN dealing with Cardan's sudden incarceration in 1570, in the chronicle of his life, it was assumed that his offence must have been some spoken or written words upon which a charge of impiety might have been fastened. Leaving out of consideration the fiery zeal of the reigning Pope Pius V., it is hard to determine what plea could have been found for a serious charge of this nature. Cardan's work had indeed passed the ecclesiastical censors in 1562; but in the estimation of Pius V. the smallest lapse from the letter of orthodoxy would have seemed grave enough to send to prison, and perhaps to death, a man as deeply penetrated with the spirit of religion as Cardan assuredly was. One of his chief reasons for refusing the King of Denmark's generous offer was the necessity involved of having to live amongst a people hostile to the Catholic religion; and, in writing of his visit to the English Court, he declares that he was unwilling to recognize the title of King Edward VI., inasmuch as by so doing he might seem to prejudice the rights of the Pope.[277] In spite of this positive testimony, and the absence of any utterances of manifest heresy, divers writers in the succeeding century classed him with the unbelievers. Dr. Samuel Parker in his _Tractatus de Deo_, published in 1678, includes him amongst the atheistical philosophers; but a perusal of the Doctor's remarks leaves the reader unconvinced as to the justice of such a charge. The term Atheism, however, was at this time used in the very loosest sense, and was even applied to disbelievers in the apostolical succession.[278] Dr. Parker writes, "Another cause which acted, together with the natural disposition of Cardan, to produce that odd mixture of folly and wisdom in him, was his habit of continual thinking by which the bile was absorbed and burnt up; he suffered neither eating, pleasure, nor pain to interrupt the course of his thoughts. He was well acquainted with the writings of all the ancients--nor did he just skim over the heads and contents of books as some do who ought not to be called learned men, but skilful bookmongers. Every author that Cardan read (and he read nearly all) he became intimately acquainted with, so that if any one disputing with him, quoted the authority of the ancients, and made any the least slip or mistake, he would instantly set them right." Dr. Parker is as greatly amazed at the mass of work he produced, as at his powers of accumulation, and maintains that Cardan believed he was endowed with a faculty which he calls _repræsentatio_, through which he was able to apprehend things without study, "by means of an interior light shining within him. From which you may learn the fact that he had studied with such enduring obstinacy that he began to persuade himself that the visions which appeared before him in these fits and transports of the mind, were the genuine inspirations of the Deity." This is evidently Dr. Parker's explanation of the attendant demon, and he ends by declaring that Cardan was rather fanatic than infidel. Mention has been made of the list of his vices and imperfections which Cardan wrote down with his own hand. Out of such a heap of self-accusation it would have been an easy task for some meddlesome enemy to gather up a plentiful selection of isolated facts which by artful combination might be so arranged as to justify a formal charge of impiety. The most definite of these charges were made by Martin del Rio,[279] who declares that Cardan once wrote a book on the Mortality of the Soul which he was wont to exhibit to his intimate friends. He did not think it prudent to print this work, but wrote another, taking a more orthodox view, called _De Immortalitate Animorum_. Another assailant, Theophile Raynaud, asserts that certain passages in this book suggest, if they do not prove, that Cardan did not set down his real opinions on the subject in hand. Raynaud ends by forbidding the faithful to read any of Cardan's books, and describes him as "Homo nullius religionis ac fidei, et inter clancularios atheos secundi ordinis ævo suo facile princeps." Of all Cardan's books the _De Immortalitate Animorum_ is the one in which materials for a charge of impiety might most easily be found. It was put together at a time when he had had very little practice in the Greek tongue, and it is possible that many of his conclusions may be drawn from premises only imperfectly apprehended. Scaliger in his Exercitations seizes upon one passage[280] which, according to his rendering, implied that Cardan reckoned the intelligence of men and beasts to be the same in essence, the variety of operation being produced by the fact that the apprehensive faculty was inherent in the one, and only operative upon the other from without. But all through this book it is very difficult to determine whether the propositions advanced are Cardan's own, or those of the Greek and Arabian writers he quotes so freely: and this charge of Scaliger, which is the best supported of all, goes very little way to convict him of impiety. In the _De Vita Propria_ there are several passages[281] which suggest a belief akin to that of the Anima Mundi; he had without doubt made up his mind that this work should not see the light till he was beyond the reach of Pope or Council. The origin of this charge of impiety may be referred with the best show of probability to his attempt to cast the horoscope of Jesus Christ.[282] This, together with a diagram, is given in the Commentaries on Ptolemy, and soon after it appeared it was made the occasion of a fierce attack by Julius Cæsar Scaliger, who declared that such a scheme must be flat blasphemy, inasmuch as the author proved that all the actions of Christ necessarily followed the position of the stars at the time of His nativity. If Scaliger had taken the trouble to glance at the Commentary he would have discovered that Cardan especially guarded himself against any accusation of this sort, by setting down that no one was to believe he had any intention of asserting that Christ's divinity, or His miracles, or His holy life, or the promulgation of His laws were in any way influenced by the stars.[283] Naudé, in recording the censures of De Thou, "Verum extremæ amentiæ fuit, imo impiæ audaciæ, astrorum commentitiis legibus verum astrorum dominum velle subjicere. Quod ille tamen exarata Servatoris nostri genitura fecit," and of Joseph Scaliger, "impiam dicam magis, an jocularem audaciam quæ et dominum stellarum stellis subjecerit, et natum eo tempore putarit, quod adhuc in lite positum est, ut vanitas cum impietate certaret,"[284] declares that it was chiefly from the publication of this horoscope that Cardan incurred the suspicion of blasphemy; but, with his free-thinking bias, abstains from adding his own censure. He rates Scaliger for ignorance because he was evidently under the impression that Cardan was the first to draw a horoscope of Christ, and attacks Cardan chiefly on the score of plagiary. He records how divers writers in past times had done the same thing. Albumasar, one of the most learned of the Arabs, whose _thema natalium_ is quoted by Roger Bacon in one of his epistles to Clement V., Albertus Magnus, Peter d'Ailly the Cardinal of Cambrai, and Tiberius Russilanus who lived in the time of Leo X., all constructed nativities of Christ, but Cardan makes no mention of these horoscopists, and, according to the view of Naudé, poses as the inventor of this form of impiety, and is consequently guilty of literary dishonesty, a worse sin, in his critics' eyes, than the framing of the horoscope itself. That there was in Cardan's practice enough of curiosity and independence to provoke suspicion of his orthodoxy in the minds of the leaders of the post-Tridentine revival, is abundantly possible; but there is nothing in all his life and works to show that he was, according to the standard of every age, anything else than a spiritually-minded man.[285] It would be hard to find words more instinct with the true feeling of piety, than the following taken from the fifty-third chapter of the _De Vita Propria_,--"I love solitude, for I never seem to be so entirely with those who are especially dear to me as when I am alone. I love God and the spirit of good, and when I am by myself I let my thoughts dwell on these, their immeasurable beneficence; the eternal wisdom, the source and origin of clearest light, that true joy within us which never fears that God will forsake us; that groundwork of truth; that willing love; and the Maker of us all, who is blessed in Himself, and likewise the desire and safeguard of all the blessed. Ah, what depth and what height of righteousness, mindful of the dead and not forgetting the living. He is the Spirit who protects me by His commands, my good and merciful counsellor, my helper and consoler in misfortune." Two or three of Cardan's treatises are in the _materna lingua_, but he wrote almost entirely in Latin, using a style which was emphatically literary.[286] His Latin is probably above the average excellence of the age, and if the classic writers held the first place in his estimation--as naturally they would--he assuredly did not neglect the firstfruits of modern literature. Pulci was his favourite poet. He evidently knew Dante and Boccaccio well, and his literary insight was clear enough to perceive that the future belonged to those who should write in the vulgar tongue of the lands which produced them.[287] Perhaps it was impossible that a man endowed with so catholic a spirit and with such earnest desire for knowledge, should sink into the mere pedant with whom later ages have been made acquainted through the farther specialization of science. At all events Cardan is an instance that the man of liberal education need not be killed by the man of science. For him the path of learning was not an easy one to tread, and, as it not seldom happens, opposition and coldness drove him on at a pace rarely attained by those for whom the royal road to learning is smoothed and prepared. For a long time his father refused to give him instruction in Latin, or to let him be taught by any one else, and up to his twentieth year he seems to have known next to nothing of this language which held the keys both of letters and science. He began to learn Greek when he was about thirty-five, but it was not till he had turned forty that he took up the study of it in real earnest;[288] and, writing some years later, he gives quotations from a Latin version of Aristotle.[289] In his commentaries on Hippocrates he used a Latin text, presumably the translation of Calvus printed in Rome in 1525, and quotes Epicurus in Latin in the _De Subtilitate_ (p. 347), but in works like the _De Sapientia_ and the _De Consolatione_ he quotes Greek freely, supplying in nearly every case a Latin version of the passages cited. These treatises bristle with quotations, Horace being his favourite author. "Vir in omni sapientiæ genere admirandus."[290] As with many moderns his love for Horace did not grow less as old age crept on, for the _De Vita Propria_ is perhaps fuller of Horatian tags than any other of his works. It would seem somewhat of a paradox that a sombre and earnest nature like Cardan's should find so great pleasure in reading the elegant _poco curante_ triflings of the Augustan singer, were it not a recognized fact that Horace has always been a greater favourite with serious practical Englishmen than with the descendants of those for whom he wrote his verses. It was a habit with Cardan to apologize in the prefaces of his scientific works for the want of elegance in his Latin, explaining that the baldness and simplicity of his periods arose from his determination to make his meaning plain, and to trouble nothing about style for the time being; but the following passage shows that he had a just and adequate conception of the necessary laws of literary art. "That book is perfect which goes straight to its point in one single line of argument, which neither leaves out aught that is necessary, nor brings in aught that is superfluous: which observes the rule of correct division; which explains what is obscure; and shows plainly the groundwork upon which it is based."[291] The _De Vita Propria_ from which this extract comes is in point of style one of his weakest books, but even in this volume passages may here and there be found of considerable merit, and Cardan was evidently studious to let his ideas be presented in intelligible form, for he records that in 1535 he read through the whole of Cicero, for the sake of improving his Latin. His style, according to Naudé, held a middle place between the high-flown and the pedestrian, and of all his books the _De Utilitate ex Adversis Capienda_, which was begun in 1557, shows the nearest approach to elegance, but even this is not free from diffuseness, the fault which Naudé finds in all his writings. Long dissertations entirely alien from the subject in hand are constantly interpolated. In the Practice of Arithmetic he turns aside to treat of the marvellous properties of certain numbers, of the motion of the planets, and of the Tower of Babel; and in the treatise on Dialectic he gives an estimate of the historians and letter-writers of the past. But here Cardan did not sin in ignorance; his poverty and not his will consented to these literary outrages. He was paid for his work by the sheet, and the thicker the volume the higher the pay.[292] When he made a beginning of the _De Utilitate_ Cardan was at the zenith of his fortunes. He had lately returned from his journey to Scotland, having made a triumphant progress through the cities of Western Europe. Thus, with his mind well stored with experience of divers lands, his wits sharpened by intercourse with the _élite_ of the learned world, and his hand nerved by the magnetic stimulant of success, he sat down to write as the philosopher and man of the world, rather than as the man of science. He was, in spite of his prosperity, inclined to deal with the more sombre side of life. He seems to have been specially drawn to write of death, disease, and of the peculiar physical misfortune which befell him in early manhood. Like Cicero he goes on to treat of Old Age, but in a spirit so widely different that a brief comparison of the conclusions of the two philosophers will not be without interest. Old age, Cardan declares to be the most cruel and irreparable evil with which man is cursed, and to talk of old age is to talk of the crowning misfortune of humanity. Old men are made wretched by avarice, by dejection, and by terror. He bids men not to be deceived by the flowery words of Cicero,[293] when he describes Cato as an old man, like to a fair statue of Polycleitus, with faculties unimpaired and memory fresh and green. He next goes on to catalogue the numerous vices and deformities of old age, and instances from Aristotle what he considers to be the worst of all its misfortunes, to wit that an old man is well-nigh cut off from hope; and by way of comment grimly adds, "If any man be plagued by the ills of old age he should blame no one but himself, for it is by his own choice that his life has run on so long." He vouchsafes a few words of counsel as to how this hateful season may be robbed of some of its horror. Our bodies grow old first, then our senses, then our minds. Therefore let us store our treasures in that part of us which will hold out longest, as men in a beleaguered city are wont to collect their resources in the citadel, which, albeit it must in the end be taken, will nevertheless be the last to fall into the foeman's hands. Old men should avoid society, seeing that they can bring nothing thereto worth having: whether they speak or keep silent they are in the way, and they are as irksome to themselves when they are silent, as they are to others when they speak. The old man should take a lesson from the lower animals, which are wont to defend themselves with the best arms given them by nature: bulls with their horns, horses with their hoofs, and cats with their claws; wherefore an old man should at least show himself to be as wise as the brutes and maintain his position by his wisdom and knowledge, seeing that all the grace and power of his manhood must needs have fled.[294] In another of his moral treatises he has formulated a long indictment against old age, that hateful state with its savourless joys and sleepless nights. Did not Zeno the philosopher strangle himself when he found that time refused to do its work. The happiest are those who earliest lay down the burden of existence, and the Law itself causes these offenders who are least guilty to die first, letting the more nefarious and hardened criminals stand by and witness the death of their fellows. There can be no evil worse than the daily expectation of the blow that is inevitable, and old age, when it comes, must make every man regret that he did not die in infancy. "When I was a boy," he writes, "I remember one day to have heard my mother, Chiara Micheria--herself a young woman--cry out that she wished it had been God's will to let her die when she was a child. I asked her why, and she answered: 'Because I know I must soon die, to the great peril of my soul, and besides this, if we shall diligently weigh and examine all our experiences of life, we shall not light upon a single one which will not have brought us more sorrow than joy. For afflictions when they come mar the recollection of our pleasures, and with just cause; for what is there in life worthy the name of delight, the ever-present burden of existence, the task of dressing and undressing every day, hunger, thirst, evil dreams? What more profit and ease have we than the dead? We must endure the heat of summer, the cold of winter, the confusion of the times, the dread of war, the stern rule of parents, the anxious care of our children, the weariness of domestic life, the ill carriage of servants, lawsuits, and, what is worst of all, the state of the public mind which holds probity as silliness; which practises deceit and calls it prudence. Craftsmen are counted excellent, not by their skill in their art, but by reason of their garish work and of the valueless approbation of the mob. Wherefore one must needs either incur God's displeasure or live in misery, despised and persecuted by men.'"[295] These words, though put into his mother's mouth, are manifestly an expression of Cardan's own feelings. Cardan was the product of an age to which there had recently been revealed the august sources from which knowledge, as we understand the term, has flowed without haste or rest since the unsealing of the fountain. He counts it rare fortune to have been born in such an age, and rhapsodizes over the flowery meadow of knowledge in which his generation rejoices, and over the vast Western world recently made known. Are not the artificial thunderbolts of man far more destructive than those of heaven? What praise is too high for the magnet which leads men safely over perilous seas, or for the art of printing? Indeed it needs but little more to enable man to scale the very heavens. With his mind thus set upon the exploration of these new fields of knowledge; with the full realization how vast was the treasure lying hid therein; it was only natural that a spirit so curious and greedy of fresh mental food should have fretted at the piteous brevity of the earthly term allowed to man, and have rated as a supreme evil that old age which brought with it decay of the faculties and foreshadowed the speedy and inevitable fall of the curtain. Cicero on the other hand had been nurtured in a creed and philosophy alike outworn. The blight of finality had fallen upon the moral world, and the physical universe still guarded jealously her mighty secrets. To the eyes of Cicero the mirror of nature was blank void and darkness, while Cardan, gazing into the same glass, must have been embarrassed with the number and variety of the subjects offered, and may well have felt that the longest life of man ten times prolonged would rank but as a moment in that Titanic spell of work necessary to bring to the birth the teeming burden with which the universe lay in travail. Here is one and perhaps the strongest reason of his hatred of old age; because through the shortness of his span of time he could only deal with a grain or two of the sand lying upon the shores of knowledge. Cicero, with his more limited vision, conscious that sixty years or so of life would exhaust every physical delight, and blunt and mar the intellectual; ignorant both of the world of new light lying beyond the void, and of the rapture which the conquering investigator of the same must feel in wringing forth its secrets, welcomed the gathering shades as friendly visitants, a mood which has asserted itself in later times with certain weary spirits, sated with knowledge as Vitellius was sated with his banquets of nightingales' tongues. Cardan with all his curiosity and restless mental activity was hampered and restrained in his explorations by the bonds which had been imposed upon thought during the rule of authority. These bonds held him back--acting imperceptibly--as they held back Abelard and many other daring spirits trained in the methods of the schoolmen, and allowed him to do little more than range at large over the fields of fresh knowledge which were destined to be reaped by later workers trained in other schools and under different masters. Learning was still subject to authority, though in milder degree, than when Thomas of Aquino dominated the mental outlook of Europe, and the great majority of the men who posed as Freethinkers, and sincerely believed themselves to be Freethinkers, were unconsciously swayed by the associations of the method of teaching they professed to despise. Their progress for the most part resembled the movement of a squirrel in a rotatory cage, but though their efforts to conquer the new world of knowledge were vain, it cannot be questioned that the restrictions placed around them, while nullifying the result of their investigations, stimulated enormously the activity of the brain and gave it a formal discipline which proved of the highest value when the real literary work of Modern Europe began. The futilities of the problems upon which the scholastic thinkers exercised themselves gave occasion for the satiric onslaught both of Rabelais and Erasmus. "Quæstio subtilissima, utrum Chimæra in vacuo bombinans possit comedere secundas intentiones; et fuit debatuta per decem hebdomadas in Consilio Constantiensi," and "Quid consecrasset Petrus, si consecrasset eo tempore, quo corpus Christi pendebet in cruce?" are samples which will be generally familiar, but the very absurdity of these exercitations serves to prove how strenuous must have been the temper of the times which preferred to exhaust itself over such banalities as are typified by the extracts above written, rather than remain inactive. The dogmas in learning were fixed as definitely as in religion, and the solution of every question was found and duly recorded. The Philosopher was allowed to strike out a new track, but if he valued his life or his ease, he would take care to arrive finally at the conclusion favoured by authority. Cardan may with justice be classed both with men of science and men of letters. In spite of the limitations just referred to it is certain that as he surveyed the broadening horizon of the world of knowledge, he must have felt the student's spasm of agony when he first realized the infinity of research and the awful brevity of time. His reflections on old age give proof enough of this. If he missed the labour in the full harvest-field, the glimpse of the distant mountain tops, suffused for the first time by the new light, he missed likewise the wearing labour which fell upon the shoulders of those who were compelled by the new philosophy to use new methods in presenting to the world the results of their midnight research. Such work as Cardan undertook in the composition of his moral essays, and in the Commentary on Hippocrates put no heavy tax on the brain or the vital energies; the Commentary was of portentous length, but it was not much more than a paraphrase with his own experiences added thereto. Mathematics were his pastime, to judge by the ease and rapidity with which he solved the problems sent to him by Francesco Sambo of Ravenna and others.[296] He worked hard no doubt, but as a rule mere labour inflicts no heavier penalty than healthy fatigue. The destroyer of vital power and spring is hard work, combined with that unsleeping diligence which must be exercised when a man sets himself to undertake something more complex than the mere accumulation of data, when he is forced to keep his mental powers on the strain through long hours of selection and co-ordination, and to fix and concentrate his energies upon the task of compelling into symmetry the heap of materials lying under his hand. The _De Subtilitate_ and the _De Varietate_ are standing proofs that Cardan did not overstrain his powers by exertion of this kind. Leaving out of the reckoning his mathematical treatises, the vogue enjoyed by Cardan's published works must have been a short one. They came to the birth only to be buried in the yawning graves which lie open in every library. At the time when Spon brought out his great edition in ten folio volumes in 1663, the mists of oblivion must have been gathering around the author's fame, and in a brief space his words ceased to have any weight in the teaching of that Art he had cultivated with so great zeal and affection. The mathematician who talked about "Cardan's rule" to his pupils was most likely ignorant both of his century and his birthplace. Had it not been for the references made by writers like Burton to his dabblings in occult learning, his claims to read the stars, and to the guidance of a peculiar spirit, his name would have been now unknown, save to a few algebraists; and his desire, expressed in one of the meditative passages of the _De Vita Propria_, would have been amply fulfilled: "Non tamen unquam concupivi gloriam aut honores: imo sprevi, cuperem notum esse quod sim, non opto ut sciatur qualis sim."[297] FOOTNOTES: [277] _De Vita Propria_, ch. xxix. p. 76. [278] Dugald Stewart, _Dissertations_, p. 378. [279] The writer, a Jesuit, says in _Disquisitionum Magicarum_ (Louvanii, 1599), tom. i.:--"In Cardani de Subtilitate et de Varietate libris passim latet anguis in herba et indiget expurgatione Ecclesiasticæ limæ." Del Rio was a violent assailant of Cornelius Agrippa. [280] "Quoniam intellectus intrinsecus est homini, belluis extrinsecus collucet: unus etiam satisfacere omnibus, quæ in una specie sunt potest, hominibus plures sunt necessarii: tertia est quod hominis anima tanquam speculum est levigata, splendida, solida, clara: belluarum autem tenebrosa nec levis; atque ideo in nostra anima lux mentis refulget multipliciter confracta, inde ipse Intellectus intelligit. Ceteris autem potentiis, ut diximus, nullus limes prescriptus est: at belluarum internis facultatibus tantum licet agnoscere, quantum per exteriores sensus accesserit."--_De Imm. Anim.,_ p. 283. [281] "Deum debere dici immensum: omnia quæ partes habent diversas ordinatas animam habere et vitam."--p. 167. [282] In the last edition of _De Libris Propriis_ he calls it "Christique nativitas admirabilis."--_Opera_, tom. i. p. 110. [283] _Ptolemæi de Astrorum Judiciis_, p. 163. [284] _Præfatio in Manilium_. [285] A proof of his liberal tone of mind is found in his appreciation of the fine qualities of Edward VI. as a man, although he resented his encroachments as a king upon the Pope's rights. [286] In the _De Vita Propria_, ch. xxxiii. p. 106, he fixes into his prose an entire line of Horace, "Canidia afflasset pejor serpentibus Afris." [287] "At Boccatii fabulæ nunc majus virent quam antea: et Dantis Petrarchæque ac Virgilii totque aliorum poemata sunt in maxima veneratione."--_Opera_, tom. i. p. 125. [288] _Ibid.,_ tom. i. p. 59. [289] _De Vita Propria_, ch. xii.-xiii. pp. 39, 44. [290] _Opera_, tom. i. p. 505. [291] _De Vita Propria_, ch. xxvii. p. 72. [292] "Eo tantum fine, quemadmodum alicubi fatetur, ut plura folia Typographis mitteret, quibuscum antea de illorum pretio pepigerat; atque hoc modo fami, non secus ac famæ scriberet."--Naudæus, _Judicium_. [293] In _De Consolatione_ (_Opera_, tom. i. p. 604) he writes:--"Quantum diligentiæ, quantum industriæ Cicero adjecit, quo conatu nixus est ut persuaderet senectutem esse tolerandam." [294] _De Utilitate_, book ii. ch. 4. [295] _De Consolatione_ (_Opera_, tom. i. p. 605). [296] _Opera_, tom. i. p. 113. On the same page he adds:--"Fui autem tam felix in cito absoluendo, quam infelicissimus in sero inchoando. Coepi enim illum anno ætatis meæ quinquagesimo octavo, absolvi intra septem dies; pene prodigio similis." [297] _De Vita Propria_, ch. ix. p. 30. INDEX Adda, battle, 7 Alberio, Antonio, 4 Alciati, Cardinal, 212, 233 Algebra, 65, 73, 98, 235 Appearance of Cardan, 19 Apuleius, 231, 256, 264 Archinto, Filippo, 40, 41, 46, 54 Aristotle, 16, 105, 108, 224, 240, 256, 288 Arithmetic, 54, 61, 71, 91, 290 Astrology, 5, 54, 259 Avicenna, 224, 268 Bandarini, Altobello, 35-38, 163 Bandarini, Lucia (Cardan's wife), 35, 37, 39, 40, 57, 67, 163 Bayle, 1, 154, 245 Bologna, 193, 195, 201-205, 207, 212, 220, 224 Borgo, Fra Luca da, 76, 92, 96, 97 Borromeo, Carlo, 193, 194, 202, 210, 233 Borromeo, Count, 55, 259 Browne, Sir T., 56, 154, 210, 267 Brissac, Marquis, 54, 122, 131 Camutio, 170, 171, 256, 264 Cantone, Otto, 9, 11 Cardano, Aldo, 164, 165, 170, 172, 203, 212, 243 Cardano, Fazio, 1, 2, 10, 15, 22, 68, 69, 162, 238, 245, 267 Cardano, Gasparo, 103, 130, 132 Cardano, Gian Battista, 40, 102, 103, 164-180, 199, 261 Cardano, Niccolo, 21 Cassanate, G., 117-122, 126, 225, 266 Cavenago, Ambrogio, 58, 59, 60, 266 Cheke, Sir J., 139, 258 Chiara (Cardan's daughter), 148, 213 Chiromancy, 110 Cicero, 259, 290-291, 294 Colla, Giovanni, 73, 76, 81, 83, 85, 93, 97 _Consolatione, De_, 57, 62, 117, 164, 288 Croce, Francesco della, 47, 61 Croce, Luca della, 58-60, 266 D'Avalos, Alfonso, 57, 61, 63, 84, 85, 88, 89 Deaf mutes, 274 Demons, 115, 147, 155, 229 Denmark, King of, 100, 144, 282 Diet, Cardan's, 251: for the Archbishop of St. Andrews, 128 Diseases, Cardan's, 5, 7, 31, 33, 251 Doctorate of Padua, 23, 30 Dreams, Cardan's, 20, 34, 48, 104, 235 Edinburgh, 113, 125, 126 Edward VI., 132-139, 282 English, the, 141 Erasmus, 148, 163, 226, 295 Familiar spirit of Cardan, 227, 229, 258 Familiar spirit of Fazio Cardano, 12, 227 Ferrari, Ludovico, 54, 73, 94-96, 98, 211 Ferreo, Scipio, 54, 73, 76, 77, 97 Fioravanti, 189, 190, 192, 197 Fiore Antonio, 73, 76, 77, 79, 80, 82, 97 Gaddi, Franc., 47 Galen, 55, 170, 239, 240, 260-268, 270-273 Gallarate, 1, 39, 102, 258 Gambling, 22, 27, 28, 32, 42, 62, 163 _Geniturarum Exempla_, 4, 136 Geometry, 70 Glanvil, Jos., 279-281 Greek, study of, 232, 288 Hamilton, James, Earl of Arran, 120, 121, 124 Hippocrates, 59, 223, 255, 260, 268, 270-273, 296 Horace, 287, 289 Horoscope of Cardan, 5, 248 Horoscope of Aldo Cardano, 165 Horoscope of Cheke, 258 Horoscope of Christ, 55, 221, 257, 285, 286 Horoscope of Edward VI., 133, 259 Horoscope of Gian Battista Cardano, 258 Horoscope of Ranconet, 259 Horoscope of the Archbishop of St. Andrews, 130, 258 _Immortalitate Animorum, De_, 61, 284 Imprisonment of Cardan, 219, 231 Index, Congregation of the, 197 Juan Antonio, 79, 81-83 Lanza, Donato, 58, 257 Latin, study of, 12, 279, 282 Lawsuits, 31, 48, 267, 275 Leonardo Pisano, 74-76, 97 _Libris Propriis, De_, 160, 213, 235 Lyons, 121 Mahomet the Algebraist, 74 Mahomet Ben Musa, 75, 98 Margarita, 6, 21, 163, 249 Medicine, state of, 267 Micheria Chiara (Cardan's mother), 1, 3, 27, 39, 41, 42, 46, 292 Milan, College of, 31, 38, 41, 47, 52, 57, 61, 62, 145, 276 Moroni, Cardinal, 65, 210, 217-220 Music, 163, 235, 256 Naudé, Gabriel, 96, 155, 156, 165, 246-249, 253, 254, 256, 264, 290 Niger, Josephus, 228, 279 Northumberland, Duke of, 133, 136, 138, 139 Orontius, 123 Osiander, A., 72 Paciolus, Luca, 74 Padua, University, 23-30 Paracelsus, 163, 269 Paris, 119, 121 Parker, Dr. S., 282, 283 Pavia, University, 18, 22, 53, 63, 65, 100, 116, 170, 183, 195, 269 Paul III., Pope, 54, 65, 100 Peckham, John, 16, 236 Petreius, 65 Petrus, 158, 159 Pharnelius [Fernel], 123, 260 Phthisis, cure of, 118, 256 Pius IV., 193, 197, 220, 221, 233 Pius V., 220-223, 225, 282 Plat Lectureship, 46, 64, 66, 70 Porro, Branda, 170, 171, 204, 256, 264 Portents, 38, 40, 64, 161, 166, 173, 175, 184, 205-207, 216, 219, 231, 238, 278 Precepts for Children, 164, 276 _Ptolemæi de Astrorum Judiciis_, 147, 154, 159, 235, 256, 285 Ranconet, A., 123, 130, 132, 145, 259 Ranke, Von, 220, 223 Rectorship at Padua, 23, 26-28 Rigone, 176, 182 Rome, 224, 233, 242 Rosso, Galeazzo, 14, 106 Sacco, 10, 30, 32, 67, 110, 258, 267 Sacco, Bartolomeo, 172, 174, 176 Saint Andrews, Abp. of, 112, 113, 118-122, 124, 126, 131, 146-148, 257, 265, 257 _Sapientia, De_, 57, 117, 260 Scaliger, J.C., 61, 148-157, 237 254, 284-286 Scot, Reginald, 159, 163, 256, 265 Scotland, 111-116, 141 Scoto, Ottaviano, 51, 147 Scotus, Duns, 113-141 Seroni, Brandonia, 168, 170, 172, 176-180, 231 Seroni, Evangelista, 168, 177, 182 Sessa, Duca di, 175, 182, 199, 200 Sfondrato, Francesco, 58, 59, 61 Shetlands, 113 Socrates, 228, 230 _Subtilitate, De_, 104-117, 149, 158, 221, 228 Suisset (Swineshead), 113, 141 Sylvestro, Rodolfo, 211, 219, 231, 234 Sylvius, 123 Tartaglia, Niccolo, 73, 75-99, 236 Thuanus [de Thou], 155, 221, 237 244, 278 Tiboldo, G., 265 Troilus and Dominicus, story of, 241, 243 _Utilitate, De_, 4, 184, 290 _Varietate, De_, 104-117, 154, 158, 159, 227, 249 Vesalius, 100, 101, 123, 261, 270 Vicomercato, Antonio, 62 Visconti, Ercole, 183, 188, 192 _Vita Propria, De_, 9, 45, 161, 235, 237, 244, 246, 249, 250, 284, 285, 289 Weir, Johann, 209, 210 William, the English boy, 139-141, 186, 187 Transcriber's notes Page 299 Faizo corrected to Fazio Typographical errors in equations corrected. a with macron [a=] e with macron [e=] u with macron [u=] o with macron [o=] m with tilde [m~] 
38536	----	[Transcriber's Note: This book includes extensive mathematical expressions and equations, which can not always be easily represented in plain text. The reader is encouraged to download the HTML version of the text, which represents the math more clearly. For the plain text version, the following conventions are used: Mixed fractions are represented by a dash with no spaces, while subtraction is represented by a dash with spaces on either side. For example: 1-1/2 is "one and one half." 1 - 1/2 is "one minus one half." The "sideways-8" symbol for infinity is represented as [infinity]. Square, cube, and other roots are shown by raising a quantity to the appropriate fractional power. For example: [4]^(1/2) is "the square root of 4." [x]^(1/n) is "the nth root of x." Extra parentheses have been added as needed to clarify the correct order of operations.] A REVIEW OF ALGEBRA BY ROMEYN HENRY RIVENBURG, A.M. HEAD OF THE DEPARTMENT OF MATHEMATICS THE PEDDIE INSTITUTE, HIGHTSTOWN, N.J. [Illustration] AMERICAN BOOK COMPANY NEW YORK CINCINNATI CHICAGO COPYRIGHT, 1914, BY ROMEYN H. RIVENBURG. COPYRIGHT, 1914, IN GREAT BRITAIN. A REVIEW OF ALGEBRA. E. P. 6 PREFACE In most high schools the course in Elementary Algebra is finished by the end of the second year. By the senior year, most students have forgotten many of the principles, and a thorough review is necessary in order to prepare college candidates for the entrance examinations and for effective work in the freshman year in college. Recognizing this need, many schools are devoting at least two periods a week for part of the senior year to a review of algebra. For such a review the regular textbook is inadequate. From an embarrassment of riches the teacher finds it laborious to select the proper examples, while the student wastes time in searching for scattered assignments. The object of this book is to conserve the time and effort of both teacher and student, by providing a thorough and effective review that can readily be completed, if need be, in two periods a week for a half year. Each student is expected to use his regular textbook in algebra for reference, as he would use a dictionary,--to recall a definition, a rule, or a process that he has forgotten. He should be encouraged to _think_ his way out wherever possible, however, and to refer to the textbook only when _forced_ to do so as a last resort. The definitions given in the General Outline should be reviewed as occasion arises for their use. The whole Outline can be profitably employed for rapid class reviews, by covering the part of the Outline that indicates the answer, the method, the example, or the formula, as the case may be. The whole scheme of the book is ordinarily to have a page of problems represent a day's work. This, of course, does not apply to the Outlines or the few pages of theory, which can be covered more rapidly. By this plan, making only a part of the omissions indicated in the next paragraph, the essentials of the algebra can be readily covered, if need be, in from thirty to thirty-two lessons, thus leaving time for tests, even if only eighteen weeks, of two periods each, are allotted to the course. If a brief course is desired, the Miscellaneous Examples (pp. 31 to 35, 50 to 52), many of the problems at the end of the book, and the College Entrance Examinations may be omitted without marring the continuity or the comprehensiveness of the review. ROMEYN H. RIVENBURG. CONTENTS PAGES OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA 7-13 ORDER OF OPERATIONS, EVALUATION, PARENTHESES 14 SPECIAL RULES OF MULTIPLICATION AND DIVISION 15 CASES IN FACTORING 16, 17 FACTORING 18 HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE 19 FRACTIONS 20 COMPLEX FRACTIONS AND FRACTIONAL EQUATIONS 21, 22 SIMULTANEOUS EQUATIONS AND INVOLUTION 23, 24 SQUARE ROOT 25 THEORY OF EXPONENTS 26-28 RADICALS 29, 30 MISCELLANEOUS EXAMPLES, ALGEBRA TO QUADRATICS 31-35 QUADRATIC EQUATIONS 36, 37 THE THEORY OF QUADRATIC EQUATIONS 38-41 OUTLINE OF SIMULTANEOUS QUADRATICS 42, 43 SIMULTANEOUS QUADRATICS 44 RATIO AND PROPORTION 45, 46 ARITHMETICAL PROGRESSION 47 GEOMETRICAL PROGRESSION 48 THE BINOMIAL THEOREM 49 MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND 50-52 PROBLEMS--LINEAR EQUATIONS, SIMULTANEOUS EQUATIONS, QUADRATIC EQUATIONS, SIMULTANEOUS QUADRATICS 53-57 COLLEGE ENTRANCE EXAMINATIONS 58-80 OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA ~Important Definitions~ Factors; coefficient; exponent; power; base; term; algebraic sum; similar terms; degree; homogeneous expression; linear equation; root of an equation; root of an expression; identity; conditional equation; prime quantity; highest common factor (H. C. F.); lowest common multiple (L. C. M.); involution; evolution; imaginary number; real number; rational; similar radicals; binomial surd; pure quadratic equation; affected quadratic equation; equation in the quadratic form; simultaneous linear equations; simultaneous quadratic equations; discriminant; symmetrical expression; ratio; proportion; fourth proportional; third proportional; mean proportional; arithmetic progression; geometric progression; S [infinity] ~Special Rules for Multiplication and Division~ 1. Square of the sum of two quantities. (x + y)^2. 2. Square of the difference of two quantities. (x - y)^2. 3. Product of the sum and difference of two quantities. (s + t)(s - t). 4. Product of two binomials having a common term. (x + r)(x + m). 5. Product of two binomials whose corresponding terms are similar. (3x + 2t)(2x - 5t). 6. Square of a polynomial. (m - n/3 + k)^2. 7. Sum of two cubes. (x^3 + y^3)/(x + y) = x^2 - xy + y^2. 8. Difference of two cubes. (x^3 - y^3)/(x - y) = x^2 + xy + y^2. 9. Sum or difference of two like powers. (x^7 + y^7)/(x + y), (x^5 - y^5)/(x - y), (x^4 - y^4)/(x - y), (x^4 - y^4)/(x + y). ~Cases in Factoring~ 1. Common monomial factor. mx + my - mz = m(x + y - z). 2. Trinomial that is a perfect square. x^2 ± 2xy + y^2 = (x ± y)^2. 3. The difference of two squares. (a) Two terms. x^2 - y^2 = (x + y)(x - y). (b) Four terms. x^2 + 2xy + y^2 - m^2 = (x + y + m)(x + y - m). (c) Six terms. x^2 + 2xy + y^2 - p^2 - 2pq - q^2 = [(x + y) + (p + q)][(x + y) - (p + q)]. (d) Incomplete square. x^4 + x^2 y^2 + y^4 = x^4 + 2x^2 y^2 + y^4 - x^2 y^2 = (x^2 + y^2 + xy)(x^2 + y^2 - xy). 4. Trinomial of the form x^2 + bx + c. x^2 - 5x + 6 = (x - 2)(x - 3). 5. Trinomial of the form ax^2 + bx + c. 20x^2 + 7x - 6 = (4x + 3)(5x - 2). 6. Sum or difference of two cubes. See "Special Rules," 7 and 8. two like powers. See "Special Rules," 9. 7. Common polynomial factor. Grouping. t^2 p + t^2 q - 2mp - 2mq = t^2(p + q) - 2m(p + q) = (p + q)(t^2 - 2m). 8. Factor Theorem. x^3 + 17x - 18 = (x - 1)(x^2 + x + 18). ~H. C. F. and L. C. M.~ a^2 + 2a - 3 = (a + 3)(a - 1). a^2 + 7a + 12 = (a + 3)(a + 4). a^4 + 27a = a(a + 3)(a^2 - 3a + 9). H. C. F. = a + 3. L. C. M. = (a + 3)(a - 1)(a + 4)a(a^2 - 3a + 9). ~Fractions~ Reduction to lowest terms. Reduction of a mixed number to an improper fraction. Reduction of an improper fraction to a mixed number. Addition and subtraction of fractions. Multiplication and division of fractions. Law of signs in division, changing signs of factors, etc. Complex fractions. ~Simultaneous Equations~ Solved by addition or subtraction. substitution. comparison. Graphical representation. ~Involution~ Law of signs. Binomial theorem laws. Expansion of monomials and fractions. binomials. trinomials. ~Evolution~ Law of signs. Evolution of monomials and fractions. Square root of algebraic expressions. Square root of arithmetical numbers. Optional Cube root of algebraic expressions. Cube root of arithmetical numbers. ~Theory of Exponents~ Proofs: a^m × a^n = a^(m + n); (a^m)/(a^n) = a^(m - n); (a^m)^n = a^(mn); [a^(mn)]^(1/n) = a^m; (a/b)^n = (a^n)/(b^n); (abc)^n = a^n b^n c^n. Meaning of fractional exponent. zero exponent. negative exponent. Four rules To multiply quantities having the same base, add exponents. To divide quantities having the same base, subtract exponents. To raise to a power, multiply exponents. To extract a root, divide the exponent of the power by the index of the root. ~Radicals~ Radical in its simplest form. Transformation of radicals Fraction under the radical sign. Reduction to an entire surd. Changing to surds of different order. Reduction to simplest form. Addition and subtraction of radicals. Multiplication and division of radicals a^(1/n) · b^(1/n) = [ab]^(1/n). ([ab]^(1/n))/(a^(1/n)) = b^(1/n). Rationalization Monomial denominator. Binomial denominator. Trinomial denominator. Square root of a binomial surd. Radical equations. _Always_ check results to avoid extraneous roots. ~Quadratic Equations~ Pure. x^2 = a. Affected. ax^2 + bx + c = 0. Methods of solving Completing the square. Formula. Developed from ax^2 + bx + c = 0. Factoring. Equations in the quadratic form. Properties of quadratics r_1 = -b/2a + ([b^2 - 4ac]^(1/2))/(2a). r_2 = -b/2a - ([b^2 - 4ac]^(1/2))/(2a). Then r_1 + r_2 = -b/a. r_1 · r_2 = c/a. Discriminant, b^2 - 4ac, and its discussion. Nature or character of the roots. ~Simultaneous Quadratics~ CASE I. One equation linear. The other quadratic. 3x - y = 12, x^2 - y^2 = 16. CASE II. Both equations homogeneous and of the second degree. x^2 - xy + y^2 = 21, y^2 - 2xy = -15. CASE III. Any two of the quantities x + y, x^2 + y^2, xy, x^3 + y^3, x^3 - y^3, x - y, x^2 ± xy + y^2, etc., given. x^2 + y^2 = 41, x + y = 9. CASE IV. Both equations symmetrical or symmetrical except for sign. Usually one equation of high degree, the other of the first degree. x^5 + y^5 = 242, x + y = 2. CASE V. Special Devices I. Solve for a compound unknown, like xy, x + y, (1)/(xy), etc., first. x^2y^2 + xy = 6, x + 2y = -5. II. Divide the equations, member by member. x^4 - y^4 = 20, x^2 - y^2 = 5. III. Eliminate the quadratic terms. 4x + 3y = 2xy, 7x - 5y = 5xy. ~Ratio and Proportion~ Proportionals mean, third, fourth. Theorems 1. Product of means equals product of extremes. 2. If the product of two numbers equals the product of two other numbers, either pair, etc. 3. Alternation. 4. Inversion. 5. Composition. 6. Division. 7. Composition and division. 8. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent, etc. Special method of proving four quantities in proportion. Let a/b = x, a = bx, etc. ~Progressions~ Development of formulas. { l = ar^(n - 1). { l = a + (n - 1)d. { S = (ar^n - a)/(r - 1). { S = (n/2)(a + l). { S = (rl - a)/(r - 1). { S = (n/2)[2a + (n - 1)d]. { S[infinity] = (a)/(1 - r). Insertion of means Arithmetical. Geometrical. ~Binomial Theorem~ Review of binomial theorem laws. See Involution. Expansion of (a + b)^n. Finding any term by key number method. r^(th) or (r + 1)^(th) term method. A REVIEW OF ALGEBRA ORDER OF OPERATIONS, EVALUATION, PARENTHESES Order of operations: First of all, raising to a power and extracting a root. Next, multiplication and division. Last of all, addition and subtraction. Find the value of: 1. 5 · 2^2 - 25^(1/2) ÷ 5 + 2^2 · 8 ÷ 4 - 2. 2. (3 × 6 ÷ 9)/2 - 2[100^(1/2)] ÷ 5 + 4 · 2^3 - (14 · 2)/28. 3. 9 · 2 ÷ 6 + 3 - 2 · 4^2 ÷ 8^(1/3) - 4 + (3 · 2^2)/6. Evaluate: 4. (a^4 - a^3 + b^3)/([a^2 b^2]^(1/2)) + (c[a^(1/2)] + a^3bc)/(abc), if a = 1, b = 2, c = 3. 5. t^(1/3) + [tm]^(1/3) + m^(1/3), if t = 8, m = 27. 6. (2[3 + 2d + a]^(1/2))/(3[a + b - cx - c]^(1/2)) + ((3c - d)x)/(7ad - [abc]^(1/2)), if a = 5, b = 3, c = -1, d = -2, x = 0. 7. a - {5b - [a - (3c - 3b) + 2c - 3(a - 2b - c)]}, if a = -3, b = 4, c = -5. (_Yale._) Simplify: 8. m - [2m - {3r - (4r - 2m)}]. 9. 2a - [5d + {3c - (a + [2d - 3a + 4c])}]. 10. 3c^2 + c(2a - [6c - {3a + c - 4a}]). SPECIAL RULES OF MULTIPLICATION AND DIVISION Give results by inspection: 1. (g + 1/2 k)^2. 2. (s - (2m)/3)^2. 3. (2v + 3w)(2v - 3w). 4. (x + 3ts)(x - 7ts). 5. (2l + 3g)(4l - 11g). 6. (a - (2b)/3 + c - d)^2. 7. (x^3 + 8m^3)/(x + 2m). 8. (y^3 - 27k^(3m))/(y - 3k^m). 9. (c^5 - d^5)/(c - d). 10. (e^5 + d^5)/(e + d). 11. (x^4 - y^4)/(x - y). 12. (x^4 - y^4)/(x + y). 13. (a - .03)(a - .0007). 14. (g^n - 1/2)(g^n + 3/4). 15. (t^7 - v^(7/2))/(t - v^(1/2)). 16. (k^32 + 1)(k^16 +1)(k^8 + 1)(k^4 +1)(k^2 +1)(k + 1)(k - 1). 17. [(a + b) + (c + d)][(a + b) - (c + d)]. 18. (p - q + r - s)(p - q - r + s). 19. (3m - n - l + 2r)(3m + n - l - 2r). 20. (x + 5)(x - 2)(x - 5)(x + 2). 21. (a^2 + b^2 - c - 2d + 3e)^2. 22. (s + t - 2v/5 + 3w/6 + z^2)^2. 23. (x^5 + 32)/(x + 2). ~References:~ The chapter on Special Rules of Multiplication and Division in any algebra. Special Rules of Multiplication and Division in the Outline in the front of the book. CASES IN FACTORING The number of terms in an expression usually gives the clue to the possible cases under which it may come. By applying the _test_ for each and eliminating the _possible_ cases one by one, the right case is readily found. Hence, the number of terms in the expression and a ready and accurate knowledge of the Cases in Factoring are the real keys to success in this vitally important part of algebra. CASE I. A common monomial factor. Applies to any number of terms. 5cx - 5ct + 5cv - 15 c^2 m + 25 c^3 m^2 = 5c(x - t + v - 3cm + 5c^2 m^2). CASE II. A trinomial that is a perfect square. Three terms. x^2 ± 2xm + m^2 = (x ± m)^2. CASE III. The difference of two squares. _A._ Two terms. x^2 - y^2 = (x + y)(x - y). _B._ Four terms. x^2 + 2xy + y^2 - m^2 = (x^2 + 2xy + y^2) - m^2 = (x + y + m)(x + y - m) _C._ Six terms. x^2 - 2xy + y^2 - m^2 - 2mn - n^2 = (x^2 - 2xy + y^2) - (m^2 + 2mn + n^2) = (x - y)^2 - (m + n)^2 = [(x - y) + (m + n)][(x - y) - (m + n)]. _D._ An incomplete square. Three terms, and 4th powers or multiples of 4. c^4 + c^2 d^2 + d^4 = c^4 + 2c^2 d^2 + d^4 - c^2 d^2 = (c^2 + d^2)^2 - c^2 d^2 = (c^2 + d^2 + cd)(c^2 + d^2 - cd). CASE IV. A trinomial of the form x^2 + bx + c. Three terms. x^2 + x - 30 = (x + 6)(x - 5). CASE V. A trinomial of the form ax^2 + bx + c. Three terms. 20x^2 + 7x - 6 = (4x + 3)(5x - 2). CASE VI. _A._ The sum or difference of two cubes. Two terms. x^3 + y^3 = (x + y)(x^2 - xy + y^2); x^3 - y^3 = (x - y)(x^2 + xy + y^2). _B._ The sum or difference of two like powers. Two terms. x^4 - y^4 = (x - y)(x^3 + x^2y + xy^2 + y^3); x^5 + y^5 = (x + y)(x^4 - x^3y + x^2 y^2 - xy^3 + y^4). CASE VII. A common polynomial factor. Any _composite_ number of terms. t^2 p + t^2 q - t^2 r - g^2 p - g^2 q + g^2 r = t^2 (p + q - r) - g^2 (p + q -r) = (p + q - r)(t^2 - g^2) = (p + q - r)(t + g)(t - g). CASE VIII. The Factor Theorem. Any number of terms. x^3 + 17x - 18 = (x - 1)(x^2 + x + 18). FACTORING Review the _Cases in Factoring_ (see Outline on preceding pages) and write out the prime factors of the following: 1. 8a^(13) + am^(12). 2. x^7 + y^7. 3. 4x^2 + 11x - 3. 4. m^2 + n^2 - (1 + 2mn). 5. -x^2 + 2x - 1 + x^4. 6. x^(16) - y^(16). (Five factors.) 7. (x + 1)^2 - 5x - 29. 8. x^4 + x^2 y^2 + y^4. 9. x^4 - 11x^2 + 1. 10. x^(2m) + 2 + 1/(x^(2m)). 11. x^(6m) + 13x^(3m) + 12. 12. 4a^2 b^2 - (a^2 + b^2 - c^2)^2. 13. (x^2 - x - 6)(x^2 - x - 20). 14. a^4 - 8a - a^3 + 8. 15. p^3 + 7p^2 + 14p + 8. 16. 18a^2 b + 60ab^2 + 50b^3. 17. x^3 - 7x + 6. 18. 24c^2 d^2 - 47cd - 75. 19. (a^2 - b^2)^2 - (a^2 - ab)^2. 20. a^2 x^3 - (8a^2)/(y^3) - x^3 + 8/(y^3). 21. gt - gk + gl^2 + xt - xk + xl^2. 22. (m - n)(2a^2 - 2ab) + (n - m)(2ab - 2b^2). 23. a^2 - x^2 - y^2 + b^2 + 2ab + 2xy. 24. (2c^2 + 3d^2)a + (2a^2 + 3c^2)d. 25. (n(n - 1))/(1 · 2) a^(n - 2) b^2 + (n(n - 1)(n - 2))/(1 · 2 · 3) a^(n - 3) b^3. 26. (x - x^2)^3 + (x^2 - 1)^3 + (1 - x)^3. (_M. I. T._) 27. (27y^3)^2 - 2(27y^3)(8b^3) + (8b^3)^2. (_Princeton._) 28. (a^3 + 8b^3)(a + b) - 6ab(a^2 - 2ab + 4b^2). (_M. I. T._) Solve by factoring: 29. x^3 = x. 30. z^2 - 4z - 45 = 0. 31. x^3 - x^2 = 4x - 4. ~Reference:~ The chapter on Factoring in any algebra. HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE Define H. C. F. and L. C. M. Find by factoring the H. C. F. and L. C. M.: 1. 3x^2 - 3x, 12x^2 (x^2 - 1), 18x^3 (x^3 - 1). 2. (x^2 - 1)(x^2 + 5x + 6), (x^2 + 3x)(x^2 - x - 6). (_Harvard._) 3. x^2 - y^2, x^2 + y^2, x^3 + y^3, x^6 + y^6, x^6 - y^6. (_College Entrance Board._) 4. x^3 + x^2 - 2, x^3 + 2x^2 - 3. (_Cornell._) 5. x^5 - 2x^4 + x^2, 2x^4 - 4x^3 - 4x + 6. (_Yale._) 6. x^2 + a^2 - b^2 + 2ax, x^2 - a^2 + b^2 + 2bx, x^2 - a^2 - b^2 - 2ab. (_Harvard._) 7. 2x^2 - x - 15, 3x^2 - 11x + 6, 2x^3 - x^2 - 13x - 6. (_College Entrance Board._) 8. (tv - v^2)^3, v^3 - t^2v, t^3 - v^3, v^2 - 2vt + t^2. Pick out the H. C. F. and the L. C. M. of the following: 9. 8(x^2 + y)^(32) (t^2 + z)^(19) (m - n^3)^(14), 12(x^2 + y)^(23) (t^2 + z)^(41) (m - n^3)^(17), 18(m - n^3)^(11) (x^2 + y)^(39) (t^2 + z)^(37). 10. 17ax^3 (y + z)^(10) (y - x)^(19) (x + z)^(27), 34a^2 x^4 (y + z)^(11) (y - x)^(21) (x + z)^(13), 51a^3 x^5 (y + z)^4 (x + z)^(32) (y - x)^(29). ~Reference:~ The chapter on H. C. F. and L. C. M. in any algebra. FRACTIONS Define: fraction, terms of a fraction, reciprocal of a number. Look up _the law of signs_ as it applies to fractions. Except for this, fractions in algebra are treated exactly the same as they are in arithmetic. 1. Reduce to lowest terms: (_a_) 32/24; (_b_) (a^6 - x^6)/(a^4 - x^4); (_c_) [(a + b)^2 - (c + d)^2]/[(a + c)^2 - (b + d)^2]. (_M. I. T._) 2. Reduce to a mixed expression: (_a_) 756/11; (_b_) (a^3 + b^3)/(a - b). 3. Reduce to an improper fraction: (_a_) 45-1/8; (_b_) 9-11/12 qt.; (_c_) a^2 - ab + b^2 - (b^3)/(a + b). Add: 4. 5/18 + 7/9 + 11/16 + 5/8. 5. 5/(1 + 2x) - (3x)/(1 - 2x) + (4 - 13x)/(4x^2 - 1). 6. 1/[x(x - a)(x - b)] + 1/[a(a - x)(a - b)] + 1/[b(b - x)(b - a)]. Multiply: 7. 72/121 × 55/56 × 77/90. 8. (b - y)/(a^3 + y^3) × (ca + cy)/(b^2 + by) × (b^6 + y^6)/(b^2 + y^2) × b/c. Divide: 9. (12/25) ÷ (6/50). 10. [1 - (ab)/(a^2 - ab + b^2)] [1 - (ab)/(a^2 + 2ab + b^2)] ÷ (a^3 - b^3)/(a^3 + b^3). (_Yale._) 11. [(x^4 - y^4)/(x^2 - y^2) ÷ (x + y)/(x^2 - xy)] ÷ [(x^2 + y^2)/(x - y) ÷ (x + y)/(xy - y^2)]. (_Sheffield._) Simplify: 12. [(4y)/x - (15y^2)/(x^2) + 4] ÷ [4 - (16y)/x + (15y^2)/(x^2)] × [3 - (4x + 20y)/(2x + 5y)]. ~Reference:~ The chapter on Fractions in any algebra. COMPLEX FRACTIONS AND FRACTIONAL EQUATIONS Define a complex fraction. Simplify: 1. (3/7 + 4/5)/(2 - 3/7 · 4/5). 2. (2 - 3/2 + 2/3)/(5 - 2/3 + 3/2). 3. 2 - 2/(1 - 1/[1 - 1/(1 + 1/2)]). 4. a/(b^2) - a/[b^2 + (cb)/(a - c/b)]. (_Harvard._) 5. If m = 1/(a + 1), n = 2/(a + 2), p = 3/(a + 3), what is the value of m/(1 - m) + n/(1 - n) + p/(1 - p)? (_Univ. of Penn._) 6. Simplify the expression {x + y - 1/[x + y - xy/(x + y)]}(x^3 - y^3)/(x^2 - y^2). (_Cornell._) 7. Simplify [1 - (2xy)/((x + y)^2)]/[1 + (2xy)/((x - y)^2)] ÷ {(1 - y/x)/(1 + y/x)}^2. 8. Solve (7y + 9)/4 - [y - (2y - 1)/9] = 7. 9. Solve 2-1/3 - (2/5)(x^2 + 3) = (10x)/3 + 1 - (2x^2)/5. 10. How much water must be added to 80 pounds of a 5 per cent salt solution to obtain a 4 per cent solution? (_Yale._) ~Reference:~ See Complex Fractions, and the first part of the chapter on Fractional Equations in any algebra. FRACTIONAL EQUATIONS 1. Solve for each letter in turn 1/b = 1/p + 1/q. 2. Solve and check: (5x + 2)/3 - (3 - (3x - 1)/2) = (3x + 19)/2 - ((x + 1)/6 + 3). 3. Solve and check: (1/2)(x - a/3) - (1/3)(x - a/4) + (1/4)(x - a/5) = 0. 4. Solve (after looking up the special _short_ method): (3x - 1)/30 + (4x - 7)/15 = x/4 - (2x - 3)/(12x - 11) + (7x - 15)/60. 5. Solve by the special _short_ method: 1/(x - 2) - 1/(x - 3) = 1/(x - 4) - 1/(x - 5). 6. At what time between 8 and 9 o'clock are the hands of a watch (_a_) opposite each other? (_b_) at right angles? (_c_) together? Work out (_a_) and state the equations for (_b_) and (_c_). 7. The formula for converting a temperature of F degrees Fahrenheit into its equivalent temperature of C degrees Centigrade is C = (5/9)(F - 32). Express F in terms of C, and compute F for the values C = 30 and C = 28. (_College Entrance Exam. Board._) 8. What is the price of eggs when 2 less for 24 cents raises the price 2 cents a dozen? (_Yale._) 9. Solve 2/(x - 2) + 2/(4 - x^2) = 5/(x + 2). ~Reference:~ The Chapter on Fractional Equations in any algebra. Note particularly the special _short_ methods, usually given about the middle of the chapter. SIMULTANEOUS EQUATIONS NOTE. Up to this point each topic presented has reviewed to some extent the preceding topics. For example, factoring reviews the special rules of multiplication and division; H. C. F. and L. C. M. review factoring; addition and subtraction of fractions and fractional equations review H. C. F. and L. C. M., etc. From this point on, however, the interdependence is not so marked, and miscellaneous examples illustrating the work already covered will be given very frequently in order to keep the whole subject fresh in mind. 1. Solve by three methods--addition and subtraction, substitution, and comparison: { 5x + y = 11, { 3x + 2y = 1. Solve and check: 2. { 12R_1 - 11R_2 = b + 12c, { R_1 + R_2 = 2b + c. 3. { (r - s)/2 = 25/6 - (r + s)/3, { (r + s - 9)/2 - (s - r - 6)/3 = 0. 4. One half of A's marbles exceeds one half of B's and C's together by 2; twice B's marbles falls short of A's and C's together by 16; if C had four more marbles, he would have one fourth as many as A and B together. How many has each? (_College Entrance Board._) 5. The sides of a triangle are a, b, c. Calculate the radii of the three circles having the vertices as centers, each being tangent externally to the other two. (_Harvard._) 6. Solve { 2x + 3y = 7, x - y = 1 } graphically; then solve algebraically and compare results. (Use coördinate or squared paper.) Factor: 7. x^4 + 4. 8. 2d^(10) - 1024d. 9. 2(x^3 - 1) - 7(x^2 - 1). ~References:~ The chapters on Simultaneous Equations and Graphs in any algebra. SIMULTANEOUS EQUATIONS AND INVOLUTION 1. Solve (3/4)x - (5/3)y = 11-1/2, (5/8)x - (3/2)y = 10-1/4. Look up the method of solving when the unknowns are in the denominator. Should you clear of fractions? 2. Solve 1/x - 1/y - 1/z = 1/a, 1/y - 1/z - 1/x = 1/b, 1/z - 1/x - 1/y = 1/c. 3. Solve graphically and algebraically 2x - y = 4, 2x + 3y = 12. 4. Solve graphically and algebraically 3x + 7y = 5, 8x + 3y = -18. Review: 5. The squares of the numbers from 1 to 25. 6. The cubes of the numbers from 1 to 12. 7. The fourth powers of the numbers from 1 to 5. 8. The fifth powers of the numbers from 1 to 3. 9. The binomial theorem laws. (See Involution.) Expand: (Indicate first, then reduce.) 10. (b + y)^7. 11. [(2a)/3 - 1]^5. 12. (x^2 + 2a)^5. 13. (x - y + 2z)^3. 14. A train lost one sixth of its passengers at the first stop, 25 at the second stop, 20% of the remainder at the third stop, three quarters of the remainder at the fourth stop; 25 remain. What was the original number? (_M. I. T._) ~References:~ The chapter on Involution in any algebra. Also the references on the preceding page. SQUARE ROOT Find the square root of: 1. 1 + 16m^6 - 40m^4 + 10m - 8m^3 + 25m^2. 2. (a^2)/(x^2) + (6a)/x + 11 + (6x)/a + (x^2)/(a^2). 3. Find the square root to three terms of x^2 + 5. 4. Find the square root of 337,561. 5. Find the square root of 1823.29. 6. Find to four decimal places the square root of 1.672. (_Princeton._) 7. Add 2/[(x - 1)^3] + 1/[(1 - x)^2] - 2/(1 - x) - 1/x. 8. Find the value of: (64^(1/3) · 12)/24 ÷ 2 × 3 - (2 · 7^2)/(14) ÷ 7 × 1 + (1^(1/3) · 1^7)/(1 · 1^2) - 4 · 0. 9. Simplify [(x + y)^5 + (x - y)^5][(x + y)^5 - (x - y)^5]. 10. Solve by the short method: 5/(7 - x) - [(2-1/4)x - 3]/4 - (x + 11)/8 + (11x + 5)/16 = 0. 11. It takes 3/4 of a second for a ball to go from the pitcher to the catcher, and 1/2 of a second for the catcher to handle it and get off a throw to second base. It is 90 feet from first base to second, and 130 feet from the catcher's position to second. A runner stealing second has a start of 13 feet when the ball leaves the pitcher's hand, and beats the throw to the base by 1/8 of a second. The next time he tries it, he gets a start of only 3-1/2 feet, and is caught by 6 feet. What is his rate of running, and the velocity of the catcher's throw? (_Cornell._) ~Reference:~ The chapter on Square Root in any algebra. THEORY OF EXPONENTS Review the proofs, for positive integral exponents, of: I. a^m × a^n = a^(m + n). II. (a^m)/(a^n) = a^(m - n). III. (a^m)^n = a^(mn). IV. [a^(mn)]^(1/n) = a^m. V. [a/b]^n = (a^n)/(b^n). VI. (abc)^n = a^n b^n c^n. ~To find the meaning of a fractional exponent.~ Assume that Law I holds for _all_ exponents. If so, a^(2/3) · a^(2/3) · a^(2/3) = a^(6/3) = a^2. Hence, a^(2/3) is _one of the three equal factors_ (hence the cube root) of a^2. Therefore a^(2/3) = [a^2]^(1/3). In the same way, a^(4/5) · a^(4/5) · a^(4/5) · a^(4/5) · a^(4/5) = a^(20/5) = a^4. Hence, a^(4/5) is _one of the five equal factors_ (hence the fifth root) of a^4. Therefore a^(4/5) = [a^4]^(1/5). In the same way, in general, a^(p/q) = [a^p]^(1/q). Hence, _the numerator of a fractional exponent indicates the power, the denominator indicates the root_. ~To find the meaning of a zero exponent.~ Assume that Law II holds for _all_ exponents. If so, (a^m)/(a^m) = a^(m - m) = a^0. But by division, (a^m)/(a^m) = 1. Therefore a^0 = 1. Axiom I. ~To find the meaning of a negative exponent.~ Assume that Law I holds for _all_ exponents. If so, a^m × a^(-m) = a^(m - m) = a^0 = 1. Hence, a^m × a^(-m) = 1. Therefore a^(-m) = 1/(a^m). Rules: _To multiply quantities having the same base, add exponents._ _To divide quantities having the same base, subtract exponents._ _To raise a quantity to a power, multiply exponents._ _To extract a root, divide the exponent of the power by the index of the root._ 1. Find the value of 3^2 - 5 × 4^0 + 8^(-2/3) + 1^(2/5). 2. Find the value of 8^(-2/3) + 9^(3/2) - 2^(-2) + 1^(-2/5) - 7^0. Give the value of each of the following: 3. (3^0)/5, 3/(5^0), (3^0)/(5^0), 3^0 × 5, 3 × 5^0, 3^0 × 5^0, 3^0 + 5^0, 3^0 - 5^0. 4. Express 7^0 as some power of 7 divided by itself. Simplify: 5. 16^(1/3) · 2^(1/2) · 32^(5/6). (Change to the same base first.) 6. [2/(8^(-3))]^(1/5). 7. [(x^n)^(n + 2)]/[(x^(n + 1))(x^(n - 1))]. 8. (x + 3x^(2/3) - 2x^(1/3))(3 - 2x^(-1/3) + 4x^(-2/3)). 9. [(a^2b)/(c^2d)]^(1/2) × [(c^3d)/(ab^3)]^(1/3) × [(a^(1/3)c)/(b^(1/4)d^(5/12))]^2. 10. [(a^(-4))/(b^(-2)c)]^(-3/4) × [(a^(-1)b[c^(-3)]^(1/2))/(ab^(-1))]^(1/2). 11. [([a^2]^(1/3))/([b^(-1)]^(1/4)) · ([c^(-3)]^(1/2))/(a^(1/3)) · (b^(-1/4)a^(1/3))/(c^(-1))]^(-6). ~Reference:~ The chapter on Theory of Exponents in any algebra. Solve for x: 1. x^(2/3) = 4. 2. x^(-3/4) = 8. Factor: 3. x^(2/3) - 9. 4. x^(3/5) + 27. 5. x^(2a) - y^(-6). 6. a^(1/3) x^(1/2) - 3a^(1/3) + 5x^(1/2) - 15. 7. Find the H. C. F. and L. C. M. of a^2 + a^(3/2) b^(1/2) + a^(1/2) b^(3/2) - b^2, a^2 - a^(3/2) b^(1/2) - a^(1/2) b^(3/2) - b^2. 8. Simplify the product of: (ayx^(-1))^(1/2), (bxy^(-2))^(1/3), and (y^2a^(-2)b^(-2))^(1/4). (_Princeton._) 9. Find the square root of: 25a^(4/3)b^(-3) - 10a^(2/3)b^(-3/2) - 49 + 10a^(-2/3)b^(3/2) + 25a^(-4/3)b^3. 10. Simplify [(2^(n + 2))/(4^(-n)) ÷ (8^n)/(2^3)]^(1/5). 11. Find the value of (7 · 13^0 ÷ 7)/(21^0) + 3^0 × (4^0 · 7^0)/[(7a + b)^0] + 8^(-2/3). 12. Express as a power of 2: 8^3; 4^5; 4^3 · 8^(2/3) · 16^(3/4). 13. Simplify {[(x^(a + 1))/(x^(1 - a))]^a ÷ [(x^a)/(x^(1 - a))]^(a - 1)}^(1/(3a - 1)). 14. Simplify [(x^(5/2) y^(4/3))/(z^(-5/4)) · (z^4)/(x^(-3) y^(-5/3)) ÷ (y^(-2) z^(1/4))/(x^(-1/2))]^(1/5). 15. Expand (a^(1/2) + b^(1/3))^4, writing the result with fractional exponents. ~Reference:~ The chapter on Theory of Exponents in any algebra. RADICALS 1. Review all definitions in Radicals, also the methods of transforming and simplifying radicals. When is _a radical in its simplest form_? 2. Simplify (to simplest form): [2/3]^(1/2); [1/11]^(1/2); [3/5]^(1/3); 3[5/6]^(1/2); (2a/b)[(8b^2)/(27a)]^(1/2); [5/(x^n)]^(1/2n); (a + b)^2 [(-a^4)/((a + b)^5)]^(1/3); 27^(1/2); [54]^(1/3); -5[125^(1/2)]. 3. Reduce to entire surds: 2[3^(1/2)]; 2[3^(1/4)]; 6[2^(1/3)]; a[[b^2]^(1/n)]; -3[2^(1/3)]; 3a[[(a + 2)/(6a^2)]^(1/3)]; (a + 2y)[(a - 2y)/(a + 2y)]^(1/2). 4. Reduce to radicals of lower order (or simplify indices): [a^2]^(1/4); [a^3]^(1/6); [27a^3]^(1/6); [81 a^4 x^8]^(1/12); [9x^2 y^4 z^10]^(1/2n). 5. Reduce to radicals of the same degree (order, or index): 7^(1/2) and [11]^(1/3); 5^(1/3) and 3^(1/4); 7^(1/6) and 3^(1/2); [x^m]^(1/n) and [x^n]^(1/m); [c^y]^(1/x), [c^z]^(1/y), and [c^x]^(1/z). 6. Which is greater, 3^(1/2) or 4^(1/3)? [23]^(1/3) or 2[2^(1/2)]? 7. Which is greatest, 3^(1/2), 5^(1/3), or 7^(1/4)? Give work and arrange in descending order of magnitude. Collect: 8. 128^(1/2) - 2[50^(1/2)] + 72^(1/2) - 18^(1/2). 9. 2[5/3]^(1/2) + (1/6)60^(1/2) + 15^(1/2) + [3/5]^(1/2). 10. [(m - n)^2a]^(1/2) + [(m + n)^2a]^(1/2) - [am^2]^(1/2) + [a(n - m)^2]^(1/2) - a^(1/2). 11. A and B each shoot thirty arrows at a target. B makes twice as many hits as A, and A makes three times as many misses as B. Find the number of hits and misses of each. (_Univ. of Cal._) ~Reference:~ The chapter on Radicals in any algebra (first part of the chapter). The most important principle in Radicals is the following: (ab)^(1/n) = a^(1/n) b^(1/n). Hence [ab]^(1/n) = a^(1/n) · b^(1/n). Or, a^(1/n) · b^(1/n) = [ab]^(1/n). From this also ([ab]^(1/n))/(a^(1/n)) = b^(1/n). Multiply: 1. 2[4^(1/3)] by 3[6^(1/3)]. 2. 2^(1/2) by 3^(1/3). 3. 2^(1/4) by 4^(1/6). 4. [a + x^(1/2)]^(1/2) by [a - x^(1/2)]^(1/2). 5. 2^(1/2) + 3^(1/2) - 5^(1/2) by 2^(1/2) - 3^(1/2) + 5^(1/2). 6. -p/2 + ([p^2 - 4q]^(1/2))/2 by -p/2 - ([p^2 - 4q]^(1/2))/2. Divide: 7. 27^(1/2) by 3^(1/2). 8. 4[18^(1/2)] by 5[32^(1/2)]. 9. 3[12]^(1/3) by 6^(1/2). 10. 3^(1/2) by 3^(1/4). 11. 6[105^(1/2)] + 18[40^(1/2)] - 45[12^(1/2)] by 3[15^(1/2)]. (_Short division._) 12. 10[18]^(1/3) - 4[60]^(1/3) + 5[100]^(1/3) by 3[30]^(1/3). Rationalize the denominator: 13. 2/(3^(1/2)); 7/(7^(1/2)); 5/(2[5^(1/2)]); 3/([a^2]^(1/5)); 4/([a^3]^(1/7)). 14. 2/(2^(1/2)) + 3^(1/2)); (a^(1/2) + b^(1/2))/(a^(1/2) - b^(1/2)); 3/(3 - 3^(1/2)). 15. [3^(1/2) + 2^(1/2)]/[6^(1/2) + 3^(1/2) - 2^(1/2)]. Review the method of finding the square root of a binomial surd. (By inspection preferably.) Then find square root of: 16. 5 + 2[6^(1/2)]. 17. 17 - 12[2^(1/2)]. 18. 7 - 33^(1/2). ~Reference:~ The chapter on Radicals in any algebra, beginning at Addition and Subtraction of Radicals. MISCELLANEOUS EXAMPLES, ALGEBRA TO QUADRATICS Results by inspection, examples 1-10. Divide: 1. (x^(5/17) + y^(5/17))/(x^(1/17) + y^(1/17)). 2. (x - y)/(x^(1/3) - y^(1/3)). 3. (m^2 + n^2)/(m^(2/3) + n^(2/3)). 4. (x - y^2)/(x^(1/3) - [y^2]^(1/3)). Multiply: 5. [a^(-3/4) + 2/(m^(1/2))]^2. 6. (K^(-2/7) - g^(-11/25))^2. 7. (r^(2s) + l^(-3m))(r^(2s) - l^(-3m)). 8. [a^(-2) + b^(-3) - 1/(c^2)]^2. 9. (3K^x + 4t^(-3))(3K^x - 7t^(-3)). 10. (2y^(2/7) - 40K^3)(3y^(2/7) + 55K^3). Factor: 11. x^(2/3) - 64. 12. y^(3/5) + 27. 13. b^(3/2) - 8m^(-1). 14. 3p - 8p^(1/2) - 35. Factor, using radicals instead of exponents: 15. 60 - 7[3b^(1/2)] - 6b. 16. 15m - 2[[mn]^(1/2)] - 24n. 17. a - b (factor as difference of two squares). 18. a - b (factor as difference of two cubes). 19. a - b (factor as difference of two fourth powers). 20. Find the H. C. F. and L. C. M. of x^2 + xy^(1/2) - 2y, 2x^2 + 5xy^(1/2) + 2y, 2x^2 - xy^(1/2) - y. 21. Solve (short method) (x - 7)/(x - 8) - (x - 8)/(x - 9) = (x - 4)/(x - 5) - (x - 5)/(x - 6). 22. Simplify (ab/c + bc/a + ca/b)/(a/bc + b/ca + c/ab) × [((a + b + c)^2)/(ab + bc + ca) - 2]. (_Princeton._) 1. Solve for p: 2^(p - 3) = 128. 2. Solve for t: t^(3/2) = -27. 3. Find the square root of 8114.4064. What, then, is the square root of .0081144064? of 811440.64? From any of the above can you determine the square root of .081144064? 4. The H. C. F. of two expressions is a(a - b), and their L. C. M. is a^2b(a + b)(a - b). If one expression is ab(a^2 - b^2), what is the other? 5. Solve (short method): 5/(7 - x) - [(2-1/4)x - 3]/4 - (x + 11)/8 + (11x + 5)/16 = 0. 6. Solve 2/m - 3/n + 10/p = -3, 4/m + 5/p + 6/n = 15, 1/m - 1/n + 5/p = -1/2. 7. Simplify 21[2/3]^(1/2) - 5[4/5]^(1/2) + 6[4-1/6]^(1/2) - 10[3-1/5]^(1/2) + (40/3)[11-1/4]^(1/2). 8. Does [16 × 25]^(1/2) = 4 × 5? Does [16 + 25]^(1/2) = 4 + 5? 9. Write the fraction 5/(4 + 2[3^(1/2)]) with rational denominator, and find its value correct to two decimal places. 10. Simplify [{([p + [p^2 - q]^(1/2)]/2)^(1/2) + ([p - [p^2 - q]^(1/2)]/2)^(1/2)}^2]/[p + q^(1/2)]. (_Princeton._) 1. Rationalize the denominator of {6^(1/2) + 3^(1/2) - 3[2^(1/2)]}/{6^(1/2) - 3^(1/2) + 3[2^(1/2)]}. (_Univ. of Cal._) 2. Simplify [2^(n + 4) - 2(2^n)]/[2(2^(n + 3))]. (_Univ. of Penn._) 3. Find the value of [1 + 8^(-x/3)]/[(8x)^(1/2) + 10^(x - 2)], when x = 2. (_Cornell._) 4. Find the value of x if x^(6/5) = y^4, y^(2/3) = 9. (_M. I. T._) 5. A fisherman told a yarn about a fish he had caught. If the fish were half as long as he said it was, it would be 10 inches more than twice as long as it is. If it were 4 inches longer than it is, and he had further exaggerated its length by adding 4 inches, it would be 1/5 as long as he now said it was. How long is the fish, and how long did he first say it was? (_M. I. T._) 6. The force _P_ necessary to lift a weight _W_ by means of a certain machine is given by the formula P = a + bW, where _a_ and _b_ are constants depending on the amount of friction in the machine. If a force of 7 pounds will raise a weight of 20 pounds, and a force of 13 pounds will raise a weight of 50 pounds, what force is necessary to raise a weight of 40 pounds? (First determine the constants _a_ and _b_.) (_Harvard._) 7. Reduce to the simplest form: [[4/[2^(n + 2)]]^(1/n); [ax(a^(-1)x - ax^(-1))]/[x^(2/3) - a^(2/3)]. 8. Determine the H. C. F. and L. C. M. of (xy - y^2)^3 and y^3 - x^2y. (_College Entrance Board._) 1. Simplify (a - 8m)/(a^(1/3) - 2m^(1/3)) - 2a^(1/3)m^(1/3). 2. Simplify, writing the result with rational denominator: ([a^(1/2) + (1)/(x^(-1/2))]^2 - [(1)/(a^(-1/2)) - x^(1/2)]^2) / (x + [a^2 + x^2]^(1/2)). (_M. I. T._) 3. Find [7 - 48^(1/2)]^(1/2). 4. Expand ([a^3]^(1/2) - [b^5]^(1/2))^5. 5. Expand and simplify (1 - 2[3^(1/2)] + 3[2^(1/2)])^2. 6. Solve the simultaneous equations x ^(-1/2) + 2y^(-1/2) = 7/6, 2x^(-1/2) - y^(-1/2) = 2/3. (_Yale._) 7. Find to three places of decimals the value of {[(a + b)^(-1/3)]/[(11a + b^2)^(1/6)] · [({a^3 - b^3)^(-1/2)]/[(a - b)^(1/2)]}^(1/2), when a = 5 and b = 3. (_Columbia._) 8. Show that (10 - 4[5^(1/2)])/(5 + 3[5^(1/2)]) is the negative of the reciprocal of (10 + 4[5^(1/2)])/(5 - 3[5^(1/2)]). (_Columbia._) 9. Solve and check {5}/{[3x + 2]^(1/2)} = [3x + 2]^(1/2) + [3x - 1]^(1/2). 10. Assuming that when an apple falls from a tree the distance (S meters) through which it falls in any time (t seconds) is given by the formula S = (1/2)gt^2 (where g = 9.8), find to two decimal places the time taken by an apple in falling 15 meters. (_College Entrance Board._) Excellent practice may be obtained by solving the ordinary formulas used in arithmetic, geometry, and physics _orally, for each letter in turn_. ARITHMETIC p = br i = prt a = p + prt GEOMETRY K = (1/2) bh K = bh K = (a^2)/4 3^(1/2) K = (1/2) (b + b') h K = [pi] R^2 C = 2 [pi] R K = [pi] R L S = 4 [pi] R^2 V = [pi] R^2 H V = (1/3) [pi] R^2 H V = (4/3) [pi] R^3 S = ([pi] R^2 E)/(180) C/(C') = R/(R') K/(K') = (R^2)/(R'^2) PHYSICS v = gt s = (1/2) gt^2 s = (v^2)/(2g) C = E/R E = (wv^2)/(2g) e = (4Pl^3)/(bh^3 m) E = (mv^2)/(2) t = [pi] [l/g]^(1/2) F = (mV^2)/(r) mh = (mv^2)/(2g) R = gs/(g + s) E = (4n^2l^2w)/(g) C = (5/9)(F - 32) QUADRATIC EQUATIONS 1. Define a quadratic equation; a pure quadratic; an affected (or complete) quadratic; an equation in the quadratic form. 2. Solve the pure quadratic (7)/(3S^2) - (11)/(9S^2) = 5/6. Review the first (or usual) method of completing the square. Solve by it the following: 3. x^2 + 10x = 24. 4. 2x^2 - 5x = 7. 5. (x - 1)/2 + 2/(x - 1) = 2-1/2. 6. ax^2 + bx + c = 0. Review the solution by factoring. Solve by it the following: 7. x^2 + 8x + 7 = 0. 8. 24x^2 = 2x + 15. 9. 3 = 10x - 3x^2. 10. -7 = 6x - x^2. Solve, by factoring, these equations, which are not quadratics: 11. x^4 = 16. 12. x^3 = 8. 13. x^3 = x. Review the solution by formula. Solve by it the following: 14. 5x^2 - 6x = 8. 15. (1/2)(x + 1) - (x/3)(2x - 1) = -12. 16. x^2 + 4ax = 12a^2. 17. 3x^2 = 2rx + 2r^2. Solve graphically: 18. x^2 - 2x - 8 = 0. 19. x^2 + x - 2 = 0. ~Reference:~ The chapter on Quadratic Equations in any algebra (first part of the chapter). 1. Solve by three methods--formula, factoring, and completing the square: x^2 + 10x = 24. Review equations in the quadratic form and solve: 2. x^4 - 5x^2 = -4. 3. 2[x^(-2)]^(1/3) - 3[x^(-1)]^(1/3) = 2. 4. (x + 3)/(x - 3) + 6 = 5[(x + 3)/(x - 3)]^(1/2). (Let y = [(x + 3)/(x - 3)]^(1/2) and substitute.) 5. 3x^2 - 4x + 2[3x^2 - 4x - 6]^(1/2) = 21. 6. x^2 + 5x - 5 = (6)/(x^2 + 5x). Solve and check: 7. [x + 7]^(1/2) + [3x - 2]^(1/2) = (4x + 9)/([3x - 2]^(1/2)). 8. [x^2 - 5]^(1/2) + 6/[[x^2 - 5]^(1/2)] = 5. 9. (10w)/([10w - 9]^(1/2)) - [10w + 2]^(1/2) = 2/([10w - 9]^(1/2)). Give results by inspection: 10. (a^(1/2) + b^(1/2))(a^(1/2) - b^(1/2)). 11. ([10 + 19^(1/2)]^(1/2))([10 - 19^(1/2)]^(1/2)). 12. How many gallons each of cream containing 33% butter fat and milk containing 6% butter fat must be mixed to produce 10 gallons of cream containing 25% butter fat? 13. I have $6 in dimes, quarters, and half-dollars, there being 33 coins in all. The number of dimes and quarters together is ten times the number of half-dollars. How many coins of each kind are there? (_College Entrance Board._) ~Reference:~ The last part of the chapter on Quadratic Equations in any algebra. THE THEORY OF QUADRATIC EQUATIONS ~I. To find the sum and the product of the roots.~ The general quadratic equation is ax^2 + bx + c = 0. (1) Or, x^2 + (b/a)x + c/a = 0. (2) To derive the formula, we have by transposing x^2 + (b/a)x = -c/a. Completing the square, x^2 + (b/a)x + [b/2a]^2 = (b^2)/(4a^2) - c/a = (b^2 - 4ac)/(4a^2). Extracting square root, x + b/2a = [±[b^2 - 4ac]^(1/2)]/(2a). Transposing, x = -b/2a ± [[b^2 - 4ac]^(1/2)]/(2a). Hence, x = [-b ± [b^2 - 4ac]^(1/2)]/(2a). These two values of x we call _roots_. For convenience represent them by r_1 and r_2. Hence, r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a). r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a). --------------------------------------------- Adding, r_1 + r_2 = -(2b)/(2a) = -b/a. (3) Also, r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a). r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a). ------------------------------------------- Multiplying, r_1 r_2 = (b^2)/(4a^2) - (b^2 - 4ac)/(4a^2) = (b^2 - b^2 + 4ac)/(4a^2) = (4ac)/(4a^2) = c/a. (4) Hence we have shown that r_1 + r_2 = -b/a, and r_1 r_2 = c/a. Or, referring to equation (2) above, we have the following rule: _When the coefficient of x^2 is unity, the sum of the roots is the coefficient of x with the sign changed; the product of the roots is the independent term._ EXAMPLES: 1. x^2 - 9x + 21 = 0. Sum of the roots = 9. Products of the roots = 21. 2. 3x^2 - 7x - 18 = 0. Sum of the roots = 7/3. Product of the roots = -6. 3. -21x = 17 - 4x^2. Sum of the roots = 21/4. Product of the roots = -17/4. ~II. To find the nature or character of the roots.~ As before, r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a), r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a). The [b^2 - 4ac]^(1/2) determines the _nature_ or _character_ of the roots; hence it is called the _discriminant_. ~If b^2 - 4ac is positive, the roots are real, unequal, and either rational or irrational.~ ~If b^2 - 4ac is negative, the roots are imaginary and unequal.~ ~If b^2 - 4ac is zero, the roots are real, equal, and rational.~ EXAMPLES: 1. x^2 - 4x + 2 = 0. [b^2 - 4ac]^(1/2) = [16 - 8]^(1/2) = 8^(1/2). Therefore: The roots are real, unequal, and irrational. 2. x^2 - 4x + 6 = 0. [b^2 - 4ac]^(1/2) = [16 - 24]^(1/2) = -8^(1/2). Therefore: The roots are imaginary and unequal. 3. x^2 - 4x + 4 = 0. [b^2 - 4ac]^(1/2) = [16 - 16]^(1/2) = 0^(1/2). Therefore: The roots are real, equal, and rational. ~III. To form the quadratic equation when the roots are given.~ Suppose the roots are 3, -7. Then, x = 3, Or, x - 3 = 0, x = -7. x + 7 = 0. ------------------- Multiplying to get a quadratic, (x - 3)(x + 7) = 0. Or, x^2 + 4x - 21 = 0. _Or_, use the sum and product idea developed on the preceding page. The coefficient of x^2 must be unity. Add the roots and change the sign to get the coefficient of x. Multiply the roots to get the independent term. Therefore: The equation is x^2 + 4x - 21 = 0. In the same way, if the roots are [2 + 3^(1/2)]/7, [2 - 3^(1/2)]/7, the equation is x^2 - (4/7)x + 1/49 = 0. Find the sum, the product, and the nature or character of the roots of the following: 1. x^2 - 7x + 12 = 0. 2. 9x^2 - 6x + 1 = 0. 3. x^2 + 2x + 9734 = 0. 4. 16 + 5/x = 17/(x^2). 5. (x - 8)/(x - 3) = x. 6. (x + 7)(x - 6) = 70. 7. x^2 - x(2)^(1/2) = 3. 8. pr^2 + qr + s = 0. Form the equations whose roots are: 9. 5, -3. 10. 2/3, 5/3. 11. c + d, c - d. 12. -3, -5. 13. [2 ± -3^(1/2)]/5. 14. 8/3 + (2/3)37^(1/2), 8/3 - (2/3)37^(1/2). 15. [-2 ± -2^(1/2)]/2. 16. Solve x^2 - 3x + 4 = 0. Check by substituting the values of x; then check by finding the sum and the product of the roots. Compare the amount of labor required in each case. 17. Solve (x - 3)(x + 2)(x^2 + 3x - 4) = 0. 18. Is e^(4z) + 2e^(3z) + e^(2z) + 2e^z + 2 + e^(-2z) a perfect square? 19. Find the square root (short method): (x^2 - 1)(x^2 - 3x + 2)(x^2 - x - 2). 20. Solve (1.2x - 1.5)/(1.5) + (.4x + 1)/(.2x - .2) = (.4x + 1)/(.5). 21. The glass of a mirror is 18 inches by 12 inches, and it has a frame of uniform width whose area is equal to that of the glass. Find the width of the frame. OUTLINE OF SIMULTANEOUS QUADRATICS ~Simultaneous Quadratics~ CASE I. One equation linear. The other quadratic. 2x + y = 7, x^2 + 2y^2 = 22. METHOD: Solve for x as in terms of y, or _vice versa_, in the linear and substitute in the quadratic. CASE II. Both equations homogeneous and of the second degree. x^2 - xy + y^2 = 39, 2x^2 - 3xy + 2y^2 = 43. METHOD: Let y = vx, and substitute in both equations. ALTERNATE METHOD: Solve for x in terms of y in one equation and substitute in the other. CASE III. Any two of the quantities x + y x^2 + y^2 xy x - y x^3 + y^3 x^3 - y^3 x^2 + xy + y^2 x^2 - xy + y^2 given. x + y = 5, x^2 - xy + y^2 = 7. METHOD: Solve for x + y and x - y; then add to get x, subtract to get y. CASE IV. Both equations symmetrical or symmetrical except for sign. Usually one equation of high degree, the other of the first degree. x^5 + y^5 = 242, x + y = 2. METHOD: Let x = u + v and y = u - v, and substitute in both equations. ~Special Devices~ I. Consider some compound quantity like xy, [x - y]^(1/2), [xy]^(1/2), x/y, etc., as the unknown, at first. Solve for the compound unknown, and combine the resulting equation with the simpler original equation. x^2 y^2 + xy = 6, x + 2y = -5. II. Divide the equations member by member. Then solve by Case I, II, or III. x^3 - y^3 = 152, x - y = 2. III. Eliminate the quadratic terms. Then solve by Case I, II, or III. xy + x = 15, xy + y = 16. SIMULTANEOUS QUADRATICS Solve: 1. x + y = 7, x^2 + 4xy = 57. 2. 2x^2 = 46 + y^2, xy + y^2 = 14. 3. x^2 + y^2 = 25, x + y = 1. 4. x^4 + y^4 = 2, x - y = 2. 5. x^3 + y^3 = 28, x + y = 4. 6. x^2 y^2 + xy - 12 = 0, x + y = 4. 7. 2xy - x + 2y = 16, 3xy + 2x - 4y = 10. 8. (3x - 2y)(2x - 3y) = 26, x + 1 = 2y. 9. 4x^2 + 3xy + 2y^2 = 18, 3x^2 + 2xy - y^2 = 3. 10. x^5 + y^5 = 242, x + y = 2. 11. x - y + [x - y]^(1/2) = 6, xy = 5. 12. 4x^2 - x + y = 67, 3x^2 - 3y = 27. 13. x - y - [x - y]^(1/2) = 2, x^3 - y^3 = 2044. (_Yale._) 14. x^2 + xy + x = 14, y^2 + xy + y = 28. (_Princeton._) 15. x^2 + y^2 = 13, y^2 = 4(x - 2). Plot the graph of each equation. (_Cornell._) 16. x^2 + y^2 = xy + 37, x + y = xy - 17. (_Columbia._) _In grouping the answers, be sure to associate each value of x with the corresponding value of y._ 17. The course of a yacht is 30 miles in length and is in the shape of a right triangle one arm of which is 2 miles longer than the other. What is the distance along each side? ~Reference:~ The chapter on Simultaneous Quadratics in any algebra. RATIO AND PROPORTION 1. Define ratio, proportion, mean proportional, third proportional, fourth proportional. 2. Find a mean proportional between 4 and 16; 18 and 50; 12m^2n and 3mn^2. 3. Find a third proportional to 4 and 7; 5 and 10; a^2 - 9 and a - 3. 4. Find a fourth proportional to 2, 5, and 4; 35, 20, and 14. 5. Write out the proofs for the following, stating the theorem in full in each case: (_a_) The product of the extremes equals etc. (_b_) If the product of two numbers equals the product of two other numbers, either pair etc. (_c_) Alternation. (_d_) Inversion. (_e_) Composition. (_f_) Division. (_g_) Composition and division. (_h_) In a series of equal ratios, the sum of the antecedents is to the sum of the consequents etc. (_i_) Like powers or like roots of the terms of a proportion etc. 6. If x : m :: 13 : 7, write all the possible proportions that can be derived from it. [See (5) above.] 7. Given rs = 161m; write the eight proportions that may be derived from it, and quote your authority. 8. (_a_) What theorem allows you to change any proportion into an equation? (_b_) What theorem allows you to change any equation into a proportion? 9. If xy = rg, what is the ratio of x to g? of y to r? of y to g? 10. Find two numbers such that their sum, difference, and the sum of their squares are in the ratio 5 : 3 : 51. (_Yale._) ~Reference:~ The chapter on Ratio and Proportion in any algebra. An easy and powerful method of proving four expressions in proportion is illustrated by the following example: Given a : b = c : d; prove that 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2. Let a/b = r. Therefore a = br. Also c/d = r. Therefore c = dr. Substitute the value of a in the first ratio, and c in the second: Then (3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3b^3r^3 + 5b^3r)/(3b^3r^3 - 5b^3r) = [b^3r(3r^2 + 5)]/[b^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5). Also (3c^3 + 5cd^2)/(3c^3 - 5cd^2) = (3d^3r^3 + 5d^3r)/(3d^3r^3 - 5d^3r) = [d^3r(3r^2 + 5)]/[d^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5). Therefore (3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3c^3 + 5cd^2)/(3c^3 - 5cd^2). Axiom 1. Or, 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2. If a : b = c : d, prove: 1. a^2 + b^2 : a^2 = c^2 + d^2 : c^2. 2. a^2 + 3b^2 : a^2 - 3b^2 = c^2 + 3d^2 : c^2 - 3d^2. 3. a^2 + 2b^2 : 2b^2 = ac + 2bd : 2bd. 4. 2a + 3c : 2a - 3c = 8b + 12d : 8b - 12d. 5. a^2 - ab + b^2 : (a^3 - b^3)/a = c^2 - cd + d^2 : (c^3 - d^3)/c. 6. The second of three numbers is a mean proportional between the other two. The third number exceeds the sum of the other two by 20; and the sum of the first and third exceeds three times the second by 4. Find the numbers. 7. Three numbers are proportional to 5, 7, and 9; and their sum is 14. Find the numbers. (_College Entrance Board._) 8. A triangular field has the sides 15, 18, and 27 rods, respectively. Find the dimensions of a similar field having 4 times the area. ~ARITHMETICAL PROGRESSION~ 1. Define an arithmetical progression. Learn to derive the three formulas in arithmetical progression: l = a + (n - 1)d, S = (n/2)(a + l), S = (n/2)[2a + (n - 1)d]. 2. Find the sum of the first 50 odd numbers. 3. In the series 2, 5, 8, ···, which term is 92? 4. How many terms must be taken from the series 3, 5, 7, ···, to make a total of 255? 5. Insert 5 arithmetical means between 11 and 32. 6. Insert 9 arithmetical means between 7-1/2 and 30. 7. Find x, if 3 + 2x, 5 + 6x, 9 + 5x are in A. P. 8. The 7th term of an arithmetical progression is 17, and the 13th term is 59. Find the 4th term. 9. How can you turn an A. P. into an equation? 10. Given a = -5/3, n = 20, S = -5/3, find d and l. 11. Find the sum of the first n odd numbers. 12. An arithmetical progression consists of 21 terms. The sum of the three terms in the middle is 129; the sum of the last three terms is 237. Find the series. (Look up the short method for such problems.) (_Mass. Inst. of Technology._) 13. B travels 3 miles the first day, 7 miles the second day, 11 miles the third day, etc. In how many days will B overtake A who started from the same point 8 days in advance and who travels uniformly 15 miles a day? ~Reference:~ The chapter on Arithmetical Progression in any algebra. ~GEOMETRICAL PROGRESSION~ 1. Define a geometrical progression. Learn to derive the four formulas in geometrical progression: { I. l = ar^(n - 1). {II. S = (ar^n - a)/(r - 1). {III. S = (rl - a)/(r - 1). { IV. S_{[infinity]} = (a)/(1 - r). 2. How many terms must be taken from the series 9, 18, 36, ··· to make a total of 567? 3. In the G. P. 2, 6, 18, ···, which term is 486? 4. Find x, if 2x - 4, 5x - 7, 10x + 4 are in geometrical progression. 5. How can you turn a G. P. into an equation? 6. Insert 4 geometrical means between 4 and 972. 7. Insert 6 geometrical means between 5/16 and 5120. 8. Given a = -2, n = 5, l = -32; find r and S. 9. If the first term of a geometrical progression is 12 and the sum to infinity is 36, find the 4th term. 10. If the series 3-1/3, 2-1/2, ··· be an A. P., find the 97th term. If a G. P., find the sum to infinity. 11. The third term of a geometrical progression is 36; the 6th term is 972. Find the first and second terms. 12. Insert between 6 and 16 two numbers, such that the first three of the four shall be in arithmetical progression, and the last three in geometrical progression. 13. A rubber ball falls from a height of 40 inches and on each rebound rises 40% of the previous height. Find by formula how far it falls on its eighth descent. (_Yale._) ~Reference:~ The chapter on Geometrical Progression in any algebra. ~THE BINOMIAL THEOREM~ 1. Review the Binomial Theorem laws. (See Involution.) Expand: 2. (b - n)^7. 3. (x + x^(-1))^5. 4. [a/x - x/a]^6. 5. [x/2y - [xy]^(1/2)]^5. 6. (x^2 - x + 2)^3. 7. [(2[b^2]^(1/3))/(y) + (3[y^(1/2)])/(b^3)]^4. 8. (a + b)^n = a^n + na^(n - 1)b + [n(n - 1)]/(1·2) a^(n - 2)b^2 + [n(n - 1)(n - 2)]/(1·2·3) a^(n - 3)b^3 + [n(n - 1)(n - 2)(n - 3)]/(1·2·3·4) a^(n - 4) b^4 + ···. Show by observation that the formula for the (r + 1)th term = [n(n - 1)(n - 2)···(n - r + 1)]/[1·2·3·4 ··· r] a^(n - r)b^r. 9. Indicate what the 97th term of (a + b)^n would be. 10. Using the expansion of (a + b)^n in (8), derive a formula for the rth term by observing how each term is made up, then generalizing. Using either the formula in (8) or (10), whichever you are familiar with, find: 11. The 4th term of [a + 1/a]^(30). 12. The 8th term of (1 + x[y^(1/2)])^(13). 13. The middle term of (2a^(3/4) - y[a^(1/3)])^(10). 14. The term not containing x in [x^3 - 2/x]^(12). 15. The term containing x^(18) in [x^2 - a/x]^(15). ~Reference:~ The chapter on The Binomial Theorem in any algebra. ~MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND~ 1. Solve the equation x^2 - 1.6x - .23 = 0, obtaining the values of the roots correct to three significant figures. (_Harvard._) 2. Write the roots of (x^2 + 2x)(x^2 - 2x - 3)(x^2 - x + 1) = 0. (_Sheffield Scientific School._) 3. Solve 2[2x + 2]^(1/2) + [2x + 1]^(1/2) = (12x + 4)/([8x + 8]^{1/2}). (_Yale._) 4. Solve the equation V = (H/3)(B + x + [Bx]^(1/2)) for x, taking H = 6, B = 8, and V = 28; and verify your result. (_Harvard._) 5. Solve { x : y = 2 : 3, { x^2 + y^2 = 5(x + y) + 2. 6. Solve 2x^2 - 4x + 3[x^2 - 2x + 6]^(1/2) = 15. (_Coll. Ent. Board._) 7. Find all values of x and y which satisfy the equations: { x^(1/2) + y^(1/2) = 4, { 1/[[x + 1]^(1/2) - x^(1/2)] - 1/[[x + 1]^(1/2) + x^(1/2)] = y. (_Mass. Inst. of Technology._) 8. If [alpha] and [beta] represent the roots of px^2 + qx + r = 0, find [alpha] + [beta], [alpha] - [beta], and [alpha][beta] in terms of p, q, and r. (_Princeton._) 9. Form the equation whose roots are 2 + [3]^(1/2) and 2 - [-3]^(1/2). 10. Determine, without solving, the character of the roots of 9x^2 - 24x + 16 = 0. (_College Entrance Board._) 11. If a : b = c : d, prove that a + b : c + d = [a^2 + b^2]^(1/2) : [c^2 + d^2]^(1/2). (_College Entrance Board._) 12. Given a : b = c : d. Prove that a^2 + b^2 : (a^3)/(a + b) = c^2 + d^2 : (c^3)/(c + d). (_Sheffield._) 13. The 9th term of an arithmetical progression is 1/6; the 16th term is 5/2. Find the first term. (_Regents._) Solve graphically: 1. x^2 - x - 6 = 0. 2. x^2 + 3x - 10 = 0. 3. Find four numbers in arithmetical progression, such that the sum of the first two is 1, and the sum of the last two is -19. 4. What number added to 2, 20, 9, 34, will make the results proportional? 5. Find the middle term of [3a^5 + (b^(3/4))/(2)]^8. 6. Solve (x + 1)/(3x + 2) = (2x - 3)/(3x - 2) - 1 - 36/(4 - 9x^2). (_Princeton._) 7. A strip of carpet one half inch thick and 29-6/7 feet long is rolled on a roller four inches in diameter. Find how many turns there will be, remembering that each turn increases the diameter by one inch, and that the circumference of a circle equals (approximately) 22/7 times the diameter. (_Harvard._) 8. The sum of the first three terms of a geometrical progression is 21, and the sum of their squares is 189. What is the first term? (_Yale._) 9. Find the geometrical progression whose sum to infinity is 4, and whose second term is 3/4. 10. Solve 4x + 4[3x^2 - 7x + 3]^(1/2) = 3x^2 - 3x + 6. 11. Solve { 2x^2 + 3xy - 5y^2 = 4, { 2xy + 3y^2 = -3. 12. Two hundred stones are placed on the ground 3 feet apart, the first being 3 feet from a basket. If the basket and all the stones are in a straight line, how far does a person travel who starts from the basket and brings the stones to it one by one? Solve graphically; and check by solving algebraically: 1. { x^2 + y^2 = 25, { x + y = 1. 2. x^2 - 3x - 18 = 0. 3. x^2 + 3x - 10 = 0. Determine the value of m for which the roots of the equation will be equal: (HINT: See page 40. To have the roots equal, b^2 - 4ac must equal 0.) 4. 2x^2 - mx + 12-1/2 = 0. 5. (m - 1)x^2 + mx + 2m - 3 = 0. 6. If 2a + 3b is a root of x^2 - 6bx - 4a^2 + 9b^2 = 0, find the other root without solving the equation. (_Univ. of Penn._) 7. How many times does a common clock strike in 12 hours? 8. Find the sum to infinity of 2/(2^(1/2)), 1/(2^(1/2)), 1/(2[2]^(1/2)), ···. 9. Solve [x/2 + 6/x]^2 - 6[x/2 + 6/x] + 8 = 0. 10. Find the value of the recurring decimal 2.214214···. 11. A man purchases a $500 piano by paying monthly installments of $10 and interest on the debt. If the yearly rate is 6%, what is the total amount of interest? 12. The arithmetical mean between two numbers is 42-1/2, and their geometrical mean is 42. Find the numbers. (_College Entrance Exam. Board._) 13. If the middle term of [3x - (1)/(2[x^(1/2)])]^4 is equal to the fourth term of [2[x^(1/2)] + 1/2x]^7, find the value of x. (_M. I. T._) ~PROBLEMS~ ~Linear Equations, One Unknown~ 1. A train running 30 miles an hour requires 21 minutes longer to go a certain distance than does a train running 36 miles an hour. How great is the distance? (_Cornell._) 2. A man can walk 2-1/2 miles an hour up hill and 3-1/2 miles an hour down hill. He walks 56 miles in 20 hours on a road no part of which is level. How much of it is up hill? (_Yale._) 3. A physician having 100 cubic centimeters of a 6% solution of a certain medicine wishes to dilute it to a 3-1/2% solution. How much water must he add? (A 6% solution contains 6% of medicine and 94% of water.) (_Case._) 4. A clerk earned $504 in a certain number of months. His salary was increased 25%, and he then earned $450 in two months less time than it had previously taken him to earn $504. What was his original salary per month? (_College Entrance Board._) 5. A person who possesses $15,000 employs a part of the money in building a house. He invests one third of the money which remains at 6%, and the other two thirds at 9%, and from these investments he obtains an annual income of $500. What was the cost of the house? (_M. I. T._) 6. Two travelers have together 400 pounds of baggage. One pays $1.20 and the other $1.80 for excess above the weight carried free. If all had belonged to one person, he would have had to pay $4.50. How much baggage is allowed to go free? (_Yale._) 7. A man who can row 4-1/3 miles an hour in still water rows downstream and returns. The rate of the current is 2-1/4 miles per hour, and the time required for the trip is 13 hours. How many hours does he require to return? ~Simultaneous Equations, Two and Three Unknowns~ 1. A manual training student in making a bookcase finds that the distance from the top of the lowest shelf to the under side of the top shelf is 4 ft. 6 in. He desires to put between these four other shelves of inch boards in such a way that the book space will diminish one inch for each shelf from the bottom to the top. What will be the several spaces between the shelves? 2. A quantity of water, sufficient to fill three jars of different sizes, will fill the smallest jar 4 times, or the largest jar twice with 4 gallons to spare, or the second jar three times with 2 gallons to spare. What is the capacity of each jar? (_Case._) 3. A policeman is chasing a pickpocket. When the policeman is 80 yards behind him, the pickpocket turns up an alley; but coming to the end, he finds there is no outlet, turns back, and is caught just as he comes out of the alley. If he had discovered that the alley had no outlet when he had run halfway up and had then turned back, the policeman would have had to pursue the thief 120 yards beyond the alley before catching him. How long is the alley? (_Harvard._) 4. A and B together can do a piece of work in 14 days. After they have worked 6 days on it, they are joined by C who works twice as fast as A. The three finish the work in 4 days. How long would it take each man alone to do it? (_Columbia._) 5. In a certain mill some of the workmen receive $1.50 a day, others more. The total paid in wages each day is $350. An assessment made by a labor union to raise $200 requires $1.00 from each man receiving $1.50 a day, and half of one day's pay from every man receiving more. How many men receive $1.50 a day? (_Harvard._) 6. There are two alloys of silver and copper, of which one contains twice as much copper as silver, and the other three times as much silver as copper. How much must be taken from each to obtain a kilogram of an alloy to contain equal quantities of silver and copper? (_M. I. T._) 7. Two automobiles travel toward each other over a distance of 120 miles. A leaves at 9 A.M., 1 hour before B starts to meet him, and they meet at 12:00 M. If each had started at 9:15 A.M., they would have met at 12:00 M. also. Find the rate at which each traveled. (_M. I. T._) ~Quadratic Equations~ 1. Telegraph poles are set at equal distances apart. In order to have two less to the mile, it will be necessary to set them 20 feet farther apart. Find how far apart they are now. (_Yale._) 2. The distance S that a body falls from rest in t seconds is given by the formula S = 16t^2. A man drops a stone into a well and hears the splash after 3 seconds. If the velocity of sound in air is 1086 feet a second, what is the depth of the well? (_Yale._) 3. It requires 2000 square tiles of a certain size to pave a hall, or 3125 square tiles whose dimensions are one inch less. Find the area of the hall. How many solutions has the equation of this problem? How many has the problem itself? Explain the apparent discrepancy. (_Cornell._) 4. A rectangular tract of land, 800 feet long by 600 feet broad, is divided into four rectangular blocks by two streets of equal width running through it at right angles. Find the width of the streets, if together they cover an area of 77,500 square feet. (_M. I. T._) 5. (_a_) The height y to which a ball thrown vertically upward with a velocity of 100 feet per second rises in x seconds is given by the formula, y = 100x - 16x^2. In how many seconds will the ball rise to a height of 144 feet? (_b_) Draw the graph of the equation y = 100x - 16x^2. (_College Entrance Board._) 6. Two launches race over a course of 12 miles. The first steams 7-1/2 miles an hour. The other has a start of 10 minutes, runs over the first half of the course with a certain speed, but increases its speed over the second half of the course by 2 miles per hour, winning the race by a minute. What is the speed of the second launch? Explain the meaning of the negative answer. (_Sheffield Scientific School._) 7. The circumference of a rear wheel of a certain wagon is 3 feet more than the circumference of a front wheel. The rear wheel performs 100 fewer revolutions than the front wheel in traveling a distance of 6000 feet. How large are the wheels? (_Harvard._) 8. A man starts from home to catch a train, walking at the rate of 1 yard in 1 second, and arrives 2 minutes late. If he had walked at the rate of 4 yards in 3 seconds, he would have arrived 2-1/2 minutes early. Find the distance from his home to the station. (_College Entrance Board._) ~Simultaneous Quadratics~ 1. Two cubical coal bins together hold 280 cubic feet of coal, and the sum of their lengths is 10 feet. Find the length of each bin. 2. The sum of the radii of two circles is 25 inches, and the difference of their areas is 125[pi] square inches. Find the radii. 3. The area of a right triangle is 150 square feet, and its hypotenuse is 25 feet. Find the arms of the triangle. 4. The combined capacity of two cubical tanks is 637 cubic feet, and the sum of an edge of one and an edge of the other is 13 feet. (_a_) Find the length of a diagonal of any face of each cube. (_b_) Find the distance from upper left-hand corner to lower right-hand corner in either cube. 5. A and B run a mile. In the first heat A gives B a start of 20 yards and beats him by 30 seconds. In the second heat A gives B a start of 32 seconds and beats him by 9-5/11 yards. Find the rate at which each runs. (_Sheffield._) 6. After street improvement it is found that a certain corner rectangular lot has lost 1/10 of its length and 1/15 of its width. Its perimeter has been decreased by 28 feet, and the new area is 3024 square feet. Find the reduced dimensions of the lot. (_College Entrance Board._) 7. A man spends $539 for sheep. He keeps 14 of the flock that he buys, and sells the remainder at an advance of $2 per head, gaining $28 by the transaction. How many sheep did he buy, and what was the cost of each? (_Yale._) 8. A boat's crew, rowing at half their usual speed, row 3 miles downstream and back again in 2 hours and 40 minutes. At full speed they can go over the same course in 1 hour and 4 minutes. Find the rate of the crew, and the rate of the current in miles per hour. (_College Entrance Board._) 9. Find the sides of a rectangle whose area is unchanged if its length is increased by 4 feet and its breadth decreased by 3 feet, but which loses one third of its area if the length is increased by 16 feet and the breadth decreased by 10 feet. (_M. I. T._) COLLEGE ENTRANCE EXAMINATIONS ~UNIVERSITY OF CALIFORNIA~ ELEMENTARY ALGEBRA 1. If a = 4, b = -3, c = 2, and d = -4, find the value of: (_a_) ab^3 - 3cd^2 + 2(3a - b)(c - 2d). (_b_) 2a^3 - 3b^4 + (4c^3 + d^3)(4c^2 + d^2). 2. Reduce to a mixed number: (3a^4 - 4a^3 - 10a^2 + 41a - 28)/(a^2 - 3a + 4). Simplify: 3. (a + 2)/(a^2 + 3a - 40) - (b - 2)/(ab - 5b + 3a - 15). 4. [1 - (2 - 3b - 2c)/(a + 2)] ÷ (a^2 - 4c^2 + 9b^2 + 6ab)/(2a^2 + a - 6). 5. A's age 10 years hence will be 4 times what B's age was 11 years ago, and the amount that A's age exceeds B's age is one third of the sum of their ages 8 years ago. Find their present ages. 6. Draw the lines represented by the equations 3x - 2y = 13 and 2x + 5y = -4, and find by algebra the coördinates of the point where they intersect. 7. Solve the equations { bx - ay = b^2 - ab, { y - b = 2(x - 2a). 8. Solve (2x + 1)(3x - 2) - (5x - 7)(x - 2) = 41. ~COLORADO SCHOOL OF MINES~ ELEMENTARY ALGEBRA 1. Solve by factoring: x^3 + 30x = 11x^2. 2. Show that 1 - [(a^2 + b^2 - c^2)/(2ab)]^2 = (a + b + c)(a + b - c)(a - b + c)(b + c - a) ÷ 4a^2b^2. 3. How many pairs of numbers will satisfy simultaneously the two equations { 3x + 2y = 7, { x + y = 3? Show by means of a graph that your answer is correct. What is meant by eliminating x in the above equations by substitution? by comparison? by subtraction? 4. Find the square root of 223,728. 5. Simplify: (_a_) [1/3]^(1/2) + [12]^(1/2) - [3/4]^(1/2). (_b_) (-[-3[-4]^(1/2)]^(1/2))^4. 6. Solve the equation .03x^2 - 2.23x + 1.1075 = 0. 7. How far must a boy run in a potato race if there are n potatoes in a straight line at a distance d feet apart, the first being at a distance a feet from the basket? ~COLUMBIA UNIVERSITY~ ELEMENTARY ALGEBRA COMPLETE TIME: THREE HOURS Six questions are required; two from Group _A_, two from Group _B_, and both questions of Group _C_. No extra credit will be given for more than six questions. _Group A_ 1. (_a_) Resolve the following into their prime factors: (1) (x^2 - y^2)^2 - y^4. (2) 10x^2 - 7x - 6. (_b_) Find the H. C. F. and the L. C. M. of x^3 - 3x^2 + x - 3, x^3 - 3x^2 - x + 3. 2. (_a_) Simplify [x/y + y/x - 2]/[1/x + 1/y] + [x/y + y/x + 2]/[1/x - 1/y]. (_b_) If x : y = (x - z)^2 : (y - z)^2, prove that z is a mean proportional between x and y. 3. A crew can row 10 miles in 50 minutes downstream, and 12 miles in an hour and a half upstream. Find the rate of the current and of the crew in still water. _Group B_ 4. (_a_) Determine the values of k so that the equation (2 + k)x^2 + 2kx + 1 = 0 shall have equal roots. (_b_) Solve the equations x^2 - xy + y^2 = 7, 2x - 3y = 0. (_c_) Plot the following two equations, and find from the graphs the approximate values of their common solutions: x^2 + y^2 = 25, 4x^2 + 9y^2 = 144. 5. Two integers are in the ratio 4 : 5. Increase each by 15, and the difference of their squares is 999. What are the integers? 6. A man has $539 to spend for sheep. He wishes to keep 14 of the flock that he buys, but to sell the remainder at a gain of $2 per head. This he does and gains $28. How many sheep did he buy, and at what price each? _Group C_ 7. (_a_) Find the seventh term of [a + 1/a]^(13). (_b_) Derive the formula for the sum of n terms of an arithmetic progression. 8. A ball falling from a height of 60 feet rebounds after each fall one third of its last descent. What distance has it passed over when it strikes the ground for the eighth time? ~CORNELL UNIVERSITY~ ELEMENTARY ALGEBRA 1. Find the H. C. F.: x^4 - y^4, x^3 - xy^2 + x^2y - y^3, x^4 + 2x^2y^2 - 3y^4. 2. Solve the following set of equations: x + y = -1, x + 3y + 2z = -4, x - y + 4z = 5. 3. Expand and simplify: [2x^3 - 1/x]^7. 4. An automobile goes 80 miles and back in 9 hours. The rate of speed returning was 4 miles per hour faster than the rate going. Find the rate each way. 5. Simplify: {[(x + 1)/(x - 1)]^2 - 2 + [(x - 1)/(x + 1)]^2} /{[(x + 1)/(x - 1)]^2 - [(x - 1)/(x + 1)]^2}. 6. Solve for x: (2x + 3)/(x - 1) - 6 = 5/(x^2 + 2x - 3). 7. A, B, and C, all working together, can do a piece of work in 2-2/3 days. A works twice as fast as C, and A and C together could do the work in 4 days. How long would it take each one of the three to do the work alone? ~CORNELL UNIVERSITY~ INTERMEDIATE ALGEBRA 1. Solve the following set of equations: x + y = -1, 2z + 5w = 1, x + 3y + 2z = -4, x - y + 4z + 4w = 5. 2. Simplify: (_a_) [6 - 20^(1/2)]^(1/2). (_b_) [1 + [x^2 + 1]^(1/2)]/[1 + [x^2 + 1]^(1/2) + x^2]. 3. Find, and simplify, the 23d term in the expansion of [(2x^2)/(3) - 3/4]^(28). 4. The weight of an object varies directly as its distance from the center of the earth when it is below the earth's surface, and inversely as the square of its distance from the center when it is above the surface. If an object weighs 10 pounds at the surface, how far above, and how far below the surface will it weigh 9 pounds? (The radius of the earth may be taken as 4000 miles.) 5. Solve the following pair of equations for x and y: x^2 + y^2 = 4, x = (1 + 2^(1/2))y - 2. 6. Find the value of [1 + 8^(-x/3)]/[(8x)^(1/2) + 10^(x - 2)], when x = 2. 7. From a square of pasteboard, 12 inches on a side, square corners are cut, and the sides are turned up to form a rectangular box. If the squares cut out from the corners had been 1 inch larger on a side, the volume of the box would have been increased 28 cubic inches. What is the size of the square corners cut out? (See the figure on the blackboard.) ~HARVARD UNIVERSITY~ ELEMENTARY ALGEBRA TIME: ONE HOUR AND A HALF Arrange your work neatly and clearly, beginning each question on a separate page. 1. Simplify the following expression: [[1/a + 1/(b + c)]/[1/a - 1/(b + c)] [1 + (b^2 + c^2 - a^2)/(2bc)]. 2. (_a_) Write the middle term of the expansion of (a - b)^14 by the binomial theorem. (_b_) Find the value of a^7b^7, if a = x^(2/7)y^(-3/2) and b = (1/2) x^(-1/7)y^(1/2), and reduce the result to a form having only positive exponents. 3. Find correct to three significant figures the negative root of the equation 1 - 2/(x + 1) + 4x/{(x + 1)^2} = 0. 4. Prove the rule for finding the sum of n terms of a geometrical progression of which the first term is a and the constant ratio is r. Find the sum of 8 terms of the progression 5 + 3-1/3 + 2-2/9 + ···. 5. A goldsmith has two alloys of gold, the first being 3/4 pure gold, the second 5/12 pure gold. How much of each must he take to produce 100 ounces of an alloy which shall be 2/3 pure gold? ~HARVARD UNIVERSITY~ ELEMENTARY ALGEBRA TIME: ONE HOUR AND A HALF 1. Solve the simultaneous equations x + y = a + b, (y + b)/(x + a) = a/b, and verify your results. 2. Solve the equation x^2 - 1.6x - 0.23 = 0, obtaining the values of the roots correct to three significant figures. 3. Write out the first four terms of (a - b)^7. Find the fourth term of this expansion when a = [x^(-1) y^(1/2)]^(1/3), b = [9xy^(-4)]^(1/6), expressing the result in terms of a single radical, and without fractional or negative exponents. 4. Reduce the following expression to a polynomial in a and b: (6a^3 + 7ab^2 + 12b^3)/(3a^2 - 5ab - 4b^2) - 1/[3/19b - (5a + 4b)/(19a^2)]. 5. The cost of publishing a book consists of two main items: first, the fixed expense of setting up the type; and, second, the running expenses of presswork, binding, etc., which may be assumed to be proportional to the number of copies. A certain book costs 35 cents a copy if 1000 copies are published at one time, but only 19 cents a copy if 5000 copies are published at one time. Find (_a_) the cost of setting up the type for the book, and (_b_) the cost of presswork, binding, etc., per thousand copies. ~HARVARD UNIVERSITY~ ELEMENTARY ALGEBRA TIME: ONE HOUR AND A HALF 1. Find the highest common factor and the lowest common multiple of the three expressions a^4 - b^4; a^3 + b^3; a^3 + 2a^2 b + 2ab^2 + b^3. 2. Solve the quadratic equation x^2 - 1.6x + 0.3 = 0, computing the value of the larger root correct to three significant figures. 3. In the expression x^2 - 2xy + y^2 - 4[2^(1/2)](x + y) + 8, substitute for x and y the values x = (u + v + 1)/[2^(1/2)], y = (u - v + 1)/[2^(1/2)], and reduce the resulting expression to its simplest form. 4. State and prove the formula for the sum of the first n terms of a geometric progression in which a is the first term and r the constant ratio. 5. A state legislature is to elect a United States senator, a majority of all the votes cast being necessary for a choice. There are three candidates, A, B, and C, and 100 members vote. On the first ballot A has the largest number of votes, receiving 9 more votes than his nearest competitor, B; but he fails of the necessary majority. On the second ballot C's name is withdrawn, and all the members who voted for C now vote for B, whereupon B is elected by a majority of 2. How many votes were cast for each candidate on the first ballot? ~MASSACHUSETTS INSTITUTE OF TECHNOLOGY~ ALGEBRA A TIME: ONE HOUR AND THREE QUARTERS 1. Factor the expressions: x^3 + x^2 = 2x. x^3 + x^2 - 4x - 4. 2. Simplify the expression: [1 - (b^2)/(a^2)][1 - (ab - b^2)/(a^2)](a^4)/(a^3 + b^3) · (a - b)/(a^2 + b^2). 3. Find the value of x + [1 + x^2]^(1/2), when x = (1/2)[[a/b]^(1/2) - [b/a]^(1/2)]. 4. Solve the equations: (7x + 6)/11 + y - 16 = (5x - 13)/2 - (8y - x)/5, 3(3x + 4) = 10y - 15. 5. Solve the equations: A + C = 2, -A + B + C + D = 1, 2A - B + 2C + D = 5, B + D = 1. 6. Two squares are formed with a combined perimeter of 16 inches. One square contains 4 square inches more than the other. Find the area of each. 7. A man walked to a railway station at the rate of 4 miles an hour and traveled by train at the rate of 30 miles an hour, reaching his destination in 20 hours. If he had walked 3 miles an hour and ridden 35 miles an hour, he would have made the journey in 18 hours. Required the total distance traveled. ~MASSACHUSETTS INSTITUTE OF TECHNOLOGY~ ALGEBRA B TIME: ONE HOUR AND THREE QUARTERS 1. How many terms must be taken in the series 2, 5, 8, 11, ··· so that the sum shall be 345? 2. Prove the formula x = [-b ± [b^2 - 4ac]^(1/2)]/(2a) for solving the quadratic equation ax^2 + bx + c = 0. 3. Find all values of a for which [\sq]a is a root of x^2 + x + 20 = 2a, and check your results. 4. Solve {x^2 + 3y^2 = 10, x - y = 2,} and sketch the graphs. 5. The sum of two numbers x and y is 5, and the sum of the two middle terms in the expansion of (x + y)^3 is equal to the sum of the first and last terms. Find the numbers. 6. Solve x^4 - 2x^3 + 3x^2 - 2x + 1 = 0. (HINT: Divide by x^2 and substitute x + 1/x = z.) 7. In anticipation of a holiday a merchant makes an outlay of $50, which will be a total loss in case of rain, but which will bring him a clear profit of $150 above the outlay if the day is pleasant. To insure against loss he takes out an insurance policy against rain for a certain sum of money for which he has to pay a certain percentage. He then finds that whether the day be rainy or pleasant he will make $80 clear. What is the amount of the policy, and what rate did the company charge him? ~MASSACHUSETTS INSTITUTE OF TECHNOLOGY~ ALGEBRA A TIME: TWO HOURS 1. Simplify [m + 1/m]^2 + [n + 1/n]^2 + [mn + 1/mn]^2 - [m + 1/m][n + 1/n][mn + 1/mn]. 2. Find the prime factors of (_a_) (x - x^2)^3 + (x^2 - 1)^3 + (1 - x)^3. (_b_) (2x + a - b)^4 - (x - a + b)^4. 3. (_a_) Simplify [(x^q)/(x^r)]^(q + r) [(x^r)/(x^p)]^(r + p)[{x^p/{x^q}]^(p + q). (_b_) Show that ([[x]^[1/(n+1)]]^(1/n))/([[x]^[1/(n+2)]]^[1/(n+1)]) = {x^(1/n) · [x]^[1/(n+2)]}/{[x^2]^[1/(n+1)]}. 4. Define _homogeneous terms_. For what value of n is x^n y^(5 - n/2) + x^(n + 1) y^(2n - 6) a homogeneous binomial? 5. Extract the square root of x(x - 2^(1/2))(x - 8^(1/2))(x - 18^(1/2)) + 4. 6. Two vessels contain each a mixture of wine and water. In the first vessel the quantity of wine is to the quantity of water as 1 : 3, and in the second as 3 : 5. What quantity must be taken from each, so as to form a third mixture which shall contain 5 gallons of wine and 9 gallons of water? 7. Find a quantity such that by adding it to each of the quantities a, b, c, d, we obtain four quantities in proportion. 8. What values must be given to a and b, so that (3a + 2b + 17)/2, (2a - 3b + 25)/3, 4 - 5a - 13b may be equal? ~MOUNT HOLYOKE COLLEGE~ ELEMENTARY ALGEBRA TIME: TWO HOURS 1. Factor the following expressions: (_a_) a^(3/4) - b^(3/4). (_b_) x^2 y^2 z^2 - x^2 z - y^2 z + 1. (_c_) 16(x + y)^4 - (2x - y)^4. 2. (_a_) Simplify (a^2 + b^2){(b^4)/(b^2 - a^2) - a^2}/{a/(a + b) + b/(a - b)}}. (_b_) Extract the square root of x^4 - 2x^3 + 5x^2 - 4x + 4. 3. Solve the following equations: (_a_) 1/x + 1/y = 5, 1/(x^2) + 1/(y^2) = 13. (_b_) x^2 - 5x + 2 = 0. (_c_) [27x + 1]^(1/2) = 2 - 3[3x^(1/2)]. 4. Simplify: (_a_) 7[54]^(1/3) + 256^(1/6) + [432/(-250)]^(1/3). (_b_) 1/[(a - b)(b - c)] + 1/[(c - a)(b - a)]. (_c_) Find [19 - 8[3^(1/2)]]^(1/2). 5. Plot the graphs of the following system, and determine the solution from the point of intersection: { x - 2y = 0, { 2x - 3y = 4. 6. (_a_) Derive the formula for the solution of ax^2 + bx + c = 0. (_b_) Determine the value of m for which the roots of 2x^2 + 4x + m = 0 are (i) equal, (ii) real, (iii) imaginary. (_c_) Form the quadratic equation whose roots are 2 + 3^(1/2) and 2 - 3^(1/2). 7. A page is to have a margin of 1 inch, and is to contain 35 square inches of printing. How large must the page be, if the length is to exceed the width by 2 inches? 8. (_a_) In an arithmetical progression the sum of the first six terms is 261, and the sum of the first nine terms is 297. Find the common difference. (_b_) Three numbers whose sum is 27 are in arithmetical progression. If 1 is added to the first, 3 to the second, and 11 to the third, the sums will be in geometrical progression. Find the numbers. (_c_) Derive the formula for the sum of _n_ terms of a geometrical progression. 9. (_a_) Expand and simplify (2a^2 - 3x^3)^7. (_b_) For what value of x will the ratio 7 + x : 12 + x be equal to the ratio 5 : 6? ~UNIVERSITY OF PENNSYLVANIA~ ELEMENTARY ALGEBRA TIME: THREE HOURS 1. Simplify: [(a + x)/(a - x) - (a - x)/(a + x)] ÷ (4ax)/(a^2 - x^2). 2. Find the H. C. F. and L. C. M. of 10ab^2(x^2 - 2ax), 15a^3b(x^2 - ax - 2a^2), 25b^3(x^2 - a^2)^2. 3. A grocer buys eggs at 4 for 7¢. He sells 1/4 of them at 5 for 12¢, and the rest at 6 for 11¢, making 27¢ by the transaction. How many eggs does he buy? 4. Solve for t: (t + 4a + b)/(t + a + b) - (4t - a - 2b)/(t + a - b) = -3. 5. Find the square root of a^2 - (3/2)a^(3/2) - (3/2)a^(1/2) + (41/16)a + 1. 6. (_a_) For what values of m will the roots of 2x^2 + 3mx = -2 be equal? (_b_) If 2a + 3b is a root of x^2 - 6bx - 4a^2 + 9b^2 = 0, find the other root without solving the equation. 7. (_a_) Solve for x: [2x - 3a]^(1/2) + [3x - 2a]^(1/2) = 3[a^(1/2)]. (_b_) Solve for m: 1 - (1)/(2 - m) = 1/(m + 2) + (m - 6)/(4 - m^2). 8. Solve the system: x^2 + 2y^2 = 17; xy - y^2 = 2. 9. Two boats leave simultaneously opposite shores of a river 2-1/4 mi. wide and pass each other in 15 min. The faster boat completes the trip 6-3/4 min. before the other reaches the opposite shore. Find the rates of the boats in miles per hour. 10. Write the sixth term of [x/(2[y^2]^(1/3)) - (y^(1/2))/x]^9 without writing the preceding terms. 11. The sum of the 2d and 20th terms of an A. P. is 10, and their product is 23-47/64. What is the sum of sixteen terms? ~PRINCETON UNIVERSITY~ ALGEBRA A TIME: TWO HOURS Candidates who are at this time taking _both_ Algebra A and Algebra B may omit from Algebra A questions 4, 5, and 6, and from Algebra B questions 1 (_a_), 3, and 4. 1. Simplify (a^3 + a^2b + ab^2)/(a^2 - 3ab - 4b^2) ÷ {(a^2 + 6ab - 7b^2)/(a^2 + 8ab - 9b^2) · (a^3 - b^3)/(a^2 - 7ab + 12b^2)}. 2. (_a_) Divide a^(5/2) + ab^(3/2) + b^(5/2) - 2a^(1/2)b^2 - a^(3/2)b by a^(3/2) - b^(3/2) + a^(1/2)b - ab^(1/2). (_b_) Simplify (1)/(x^(-1) + y^(-1)} · (x^(1/4)y^(1/2))^3 + 1. 3. Factor: (_a_) (x^2 + 3x)^2 - (2x - 6)^2. (_b_) a^2 + ac - 4b^2 - 2bc. 4. Solve 1/(x + 1) - (1)/(x - 1) - (1)/(x - 3) + (1)/(x - 5) = 0. 5. Solve for x and y: mx + ax = my - by, x - y = a + b. 6. The road from A to B is uphill for 5 mi., level for 4 mi., and then downhill for 6 mi. A man walks from B to A in 4 hr.; later he walks halfway from A to B and back again to A in 3 hr. and 55 min.; and later he walks from A to B in 3 hr. and 52 min. What are his rates of walking uphill, downhill, and on the level, if these do not vary? ALGEBRA B 1. Solve (_a_) (x + 1)/(x - 2) + (2x + 1)/(x + 1) + (3x + 3)/(1 - x) = 0. (_b_) [2x + 7]^(1/2) + [3x - 18]^(1/2) - [7x + 1]^(1/2) = 0. (_c_) 6/(x^2 + 2x) = 5 - 2x - x^2. 2. Solve for x and y, checking one solution in each problem: (_a_) 2x + 3y = 1, 6/x + 1/y = 2. (_b_) x^2 = x + y, y^2 = 3y - x. 3. A man arranges to pay a debt of $3600 in 40 monthly payments which form an A. P. After paying 30 of them he still owes 1/3 of his debt. What was his first payment? 4. If 4 quantities are in proportion and the second is a mean proportional between the third and fourth, prove that the third will be a mean prop. between the first and second. 5. In the expansion of [2x + 1/3x]^6 the ratio of the fourth term to the fifth is 2 : 1. Find x. 6. Two men A and B can together do a piece of work in 12 days; B would need 10 days more than A to do the whole work. How many days would it take A alone to do the work? ALGEBRA TO QUADRATICS 1. Simplify (ab^(-2)c^2)^(1/2) · (a^3b^2c^(-3))^(1/3) + [(a^6)/(b)]^(1/3). 2. Simplify a/[(a - b)(a - c)] + b/[(b - c)(b - a)] + c/[(c - a)(c - b)]. 3. Factor (_a_) x^4 - 10x^2 + 9. (_b_) x^2 + 2xy - a^2 - 2ay. (_c_) (a + b)^2 + (a + c)^2 - (c + d)^2 - (b + d)^2. 4. Find H. C. F. of x^4 - x^3 + 2x^2 + x + 3 and (x + 2)(x^3 - 1). 5. Solve x/(x - 2) + (x - 9)/(x - 7) = (x + 1)/(x - 1) + (x - 8)/(x - 6). 6. The sum of three numbers is 51; if the first number be divided by the second, the quotient is 2 and the remainder 5; if the second number be divided by the third, the quotient is 3 and the remainder 2. What are the numbers? ~SMITH COLLEGE~ ELEMENTARY ALGEBRA 1. Factor e^(2x) - 2 + e^(-2x), x^(12) - 8, x^2 - x - y^2 - y, 18a^2x^2 -24axy - 10y^2. 2. Solve [7 + 4x + 3[2x^2 + 5x + 7]^(1/2)]^(1/2) - 3 = 0. 3. The second term of a geometrical progression is 3[2^(1/2)], and the fifth term is 3/16. Find the first term and the ratio. 4. Solve the following equations and check your results by plotting: { x^2 + y^2 - xy = 7, { x + y = 4. 5. Solve 1/(x^3) + 1/(y^3) = 243/8, 1/x + 1/y = 9/2. 6. In an arithmetical progression d = -11, n = 13, s = 0. Find a and l. 7. Expand by the binomial theorem and simplify: [(2x)/(y^3) - (y^4)/(x^5 [-6]^(1/2))]^5. 8. The diagonal of a rectangle is 13 ft. long. If each side were longer by 2 ft., the area would be increased by 38 sq. ft. Find the lengths of the sides. ~SMITH COLLEGE~ ELEMENTARY ALGEBRA 1. Find the H. C. F. of 8x^3 - 27, 32x^5 - 243, and 6x^3 - 9x^2 + 4x - 6. 2. Solve: (_a_) (2x + 5)^(-5) + 31(2x + 5)^(-5/2) = 32. (_b_) (x - 1)^(1/2) + (3x + 1)^(1/2) = 4. 3. A farmer sold a horse at $75 for which he had paid x dollars. He realized x per cent profit by his sale. Find x. 4. Find the 13th term and the sum of 13 terms of the arithmetical progression (2^(1/2) - 1)/2, (2^(1/2))/2, (1)/[2([2]^(1/2) - 1)], ···. 5. The difference between two numbers is 48. Their arithmetical mean exceeds their geometrical mean by 18. Find the numbers. 6. Expand by the binomial theorem and simplify [3a^(-2) - a/[-2]^(1/2)]^5. 7. Solve: 1/x + 1/y = 3/2, 1/(x^2) + 1/(y^2) = 5/4. 8. Solve the following equations and check the results by finding the intersections of the graphs of the two equations: { x^2 = 4y, { x + 2y = 4. ~VASSAR COLLEGE~ ELEMENTARY AND INTERMEDIATE ALGEBRA Answer any six questions. 1. Find the product of [1 + 2a/3 - (5a^2)/(6)] and [2 - 3a/4 + (a^2)/(3)]. 2. Resolve into linear factors: (_a_) 4x^2 - 25; (_b_) 6x^2 - x - 12; (_c_) a^2b^2 + 1 - a^2 - b^2; (_d_) y^3 + (x - 3)y^2 - (3x - 2)y + 2x. 3. Reduce to simplest form: (_a_) z/(1/x - 1/y) + y/(1 - y/x) - x/(1 - x/y). (_b_) [-(x^3)^(1/2)]^(1/3) × (4y^(-3))^(1/2). 4. (_a_) Divide x^(3/2) - x^(-3/2) by x^(1/2) - x^(-1/2). (_b_) Find correct to one place of decimals the value of [5^(1/2) + 7^(1/2)]/[2 - 3^(1/2)]. 5. (_a_) If a/b = c/d, show that (a^2 + c^2)/(b^2 + d^2) = ac/bd. (_b_) Two numbers are in the ratio 3 : 4, and if 7 be subtracted from each the remainders are in the ratio 2 : 3. Find the numbers. 6. Solve the equations: (_a_) (x + 1)/(2) - 3/x = x/3 - (5 - x)/(6). (_b_) 11x^2 - 11-1/4 = 9x. (_c_) { x^2 - 2y^2 = 71, { x + y = 20. 7. A field could be made into a square by diminishing the length by 10 feet and increasing the breadth by 5 feet, but its area would then be diminished by 210 square feet. Find the length and the breadth of the field. ~VASSAR COLLEGE~ ELEMENTARY AND INTERMEDIATE ALGEBRA Answer six questions, including No. 5 and No. 7 or 8. Candidates in Intermediate Algebra will answer Nos. 5-9. 1. Find two numbers whose ratio is 3 and such that two sevenths of the larger is 15 more than one half the smaller. 2. Determine the factors of the lowest common multiple of 3x^4 (x^3 - y^3), 15 (x^4 - 2x^2y^2 + y^4), and 10y (x^4 + x^2y^2 + y^4). 3. Find to two decimal places the value of [4a^(-2/5) + b^0[ab^(-1)]^(1/2)]^(1/2), when a = -32 and b = -8. 4. Solve the equations: 2x + 5y = 85, 2y + 5z = 103, 2z + 5x = 57. 5. Solve any 3 of these equations: (_a_) x^2 + 44 - 15x = 0. (_b_) 2/x - x/5 = x/20 - 223/30. (_c_) x^2 + 8x - [4x^2 + 32x + 12]^(1/2) = 21. (_d_) 5/(x + 1) + 8/(x - 2) = 12/(40 - 2x). 6. The sum of two numbers is 13, and the sum of their cubes is 910. Find the smaller number, correct to the second decimal place. 7. The sum of 9 terms of an arithmetical progression is 46; the sum of the first 5 terms is 25. Find the common difference. 8. Explain the terms, and prove that if four numbers are in proportion, they are in proportion by _alternation_, by _inversion_, and by _composition_. Find x when (3 + x)/(3 - x) = (40 + x^3)/(40 - x^3). 9. Find the value of x in each of these equations: (_a_) 7x^(1/4) - 3x^(1/2) = 2. (_b_) (x^2 + 2)^(5/2) + 3/{[x^2 + 2]^(1/2)} = 4x^2 + 8. ~YALE UNIVERSITY~ ALGEBRA A TIME: ONE HOUR Omit one question in Group II and one in Group III. Credit will be given for _six_ questions only. _Group I_ 1. Resolve into prime factors: (_a_) 6x^2 - 7x - 20; (_b_) (x^2 - 5x)^2 - 2(x^2 - 5x) - 24; (_c_) a^4 + 4a^2 + 16. 2. Simplify [5 - (a^2 - 19x^2)/(a^2 - 4x^2)] ÷ [3 - (a - 5x)/(a - 2x)]. 3. Solve [2(x - 7)]/(x^2 + 3x - 28) + (2 - x)/(4 - x) - (x + 3)/(x + 7) = 0. _Group II_ 4. Simplify [2^(1/2) + 2[3^(1/2)]]/[2^(1/2) - 12^(1/2)], and compute the value of the fraction to two decimal places. 5. Solve the simultaneous equations { x^(-1/2) + 2y^(-1/2) = 7/6, { 2x^(-1/2) - y^(-1/2) = 2/3. _Group III_ 6. Two numbers are in the ratio of c : d. If a be added to the first and subtracted from the second, the results will be in the ratio of 3 : 2. Find the numbers. 7. A dealer has two kinds of coffee, worth 30 and 40 cents per pound. How many pounds of each must be taken to make a mixture of 70 pounds, worth 36 cents per pound? 8. A, B, and C can do a piece of work in 30 hours. A can do half as much again as B, and B two thirds as much again as C. How long would each require to do the work alone? ~YALE UNIVERSITY~ ALGEBRA B TIME: ONE HOUR Omit one question in Group I and one in Group II. Credit will be given for _five_ questions only. _Group I_ 1. Solve (x + a)/(x + b) + (x + b)/(x + a) = 5/2. 2. Solve the simultaneous equations { x^2y^2 + 28xy - 480 = 0, { 2x + y = 11. Arrange the roots in corresponding pairs. 3. Solve 3x^(-3/2) + 20x^(-3/4) = 32. _Group II_ 4. In going 7500 yd. a front wheel of a wagon makes 1000 more revolutions than a rear one. If the wheels were each 1 yd. greater in circumference, a front wheel would make 625 more revolutions than a rear one. Find the circumference of each. 5. Two cars of equal speed leave A and B, 20 mi. apart, at different times. Just as the cars pass each other an accident reduces the power and their speed is decreased 10 mi. per hour. One car makes the journey from A to B in 56 min., and the other from B to A in 72 min. What is their common speed? _Group III_ 6. Write in the simplest form the last three terms of the expansion of (4a^(3/2) - a^(1/2) x^(1/3))^8. 7. (_a_) Derive the formula for the sum of an A. P. (_b_) Find the sum to infinity of the series 1, -1/2, 1/4, -1/8, ···. Also find the sum of the positive terms. 
729	----	Hackers, Heroes of the Computer Revolution, by Steven Levy (C)1984 by Steven Levy Chapters 1 and 2 of Hackers, Heroes of the Computer Revolution by Steven Levy Who's Who The Wizards and their Machines Bob Albrecht Found of People's Computer Company who took visceral pleasure in exposing youngsters to computers. Altair 8800 The pioneering microcomputer that galvanized hardware hackers. Building this kit made you learn hacking. Then you tried to figure out what to DO with it. Apple II ][ Steve Wozniak's friendly, flaky, good-looking computer, wildly successful and the spark and soul of a thriving industry. Atari 800 This home computer gave great graphics to game hackers like John Harris, though the company that made it was loath to tell you how it worked. Bob and Carolyn Box World-record-holding gold prospectors turned software stars, working for Sierra On-Line. Doug Carlston Corporate lawyer who chucked it all to form the Broderbund software company. Bob Davis Left job in liquor store to become best-selling author of Sierra On-Line computer game "Ulysses and the Golden Fleece." Success was his downfall. Peter Deutsch Bad in sports, brilliant at math, Peter was still in short pants when he stubled on the TX-0 at MIT--and hacked it along with the masters. Steve Dompier Homebrew member who first made the Altair sing, and later wrote the "Targe" game on the Sol which entranced Tom Snyder. John Draper The notorious "Captain Crunch" who fearlessly explored the phone systems, got jailed, hacked microprocessors. Cigarettes made his violent. Mark Duchaineau The young Dungeonmaster who copy-protected On-Lines disks at his whim. Chris Esponosa Fourteen-year-old follower of Steve Wozniak and early Apple employee. Lee Felsenstein Former "military editor" of Berkeley Barb, and hero of an imaginary science-fiction novel, he designed computers with "junkyard" approach and was central figure in Bay Area hardware hacking in the seventies. Ed Fredkin Gentle founder of Information International, thought himself world's greates programmer until he met Stew Nelson. Father figure to hackers. Gordon French Silver-haired hardware hacker whose garage held not cars but his homebrewed Chicken Hawk comptuer, then held the first Homebrew Computer Club meeting. Richard Garriott Astronaut's son who, as Lord British, created Ultima world on computer disks. Bill Gates Cocky wizard, Harvard dropout who wrote Altair BASIC, and complained when hackers copied it. Bill Gosper Horwitz of computer keyboards, master math and LIFE hacker at MIT AI lab, guru of the Hacker Ethic and student of Chinese restaurant menus. Richard Greenblatt Single-minded, unkempt, prolific, and canonical MIT hacker who went into night phase so often that he zorched his academic career. The hacker's hacker. John Harris The young Atari 800 game hacker who became Sierra On-Line's star programmer, but yearned for female companionship. IBM-PC IBM's entry into the personal computer market which amazingly included a bit of the Hacker Ethic, and took over. [H.E. as open architecture.] IBM 704 IBM was The Enemy, and this was its machine, the Hulking Giant computer in MIT's Building 26. Later modified into the IBM 709, then the IBM 7090. Batch-processed and intolerable. Jerry Jewell Vietnam vet turned programmer who founded Sirius Software. Steven Jobs Visionary, beaded, non-hacking youngster who took Wozniak's Apple II ][, made a lot of deals, and formed a company that would make a billion dollars. Tom Knight At sixteen, an MIT hacker who would name the Incompatible Time-sharing System. Later a Greenblatt nemesis over the LISP machine schism. Alan Kotok The chubby MIT student from Jersey who worked under the rail layout at TMRC, learned the phone system at Western Electric, and became a legendary TX-0 and PDP-1 hacker. Effrem Lipkin Hacker-activist from New York who loved machines but hated their uses. Co-Founded Community Memory; friend of Felsenstein. LISP Machine The ultimate hacker computer, invented mosly by Greenblatt and subject of a bitter dispute at MIT. "Uncle" John McCarthy Absent-minded but brilliant MIT [later Stanford] professor who helped pioneer computer chess, artificial intelligence, LISP. Bob Marsh Berkeley-ite and Homebrewer who shared garage with Felsenstein and founded Processor Technology, which made the Sol computer. Roger Melen Homebrewer who co-founded Cromemco company to make circuit boards for Altair. His "Dazzler" played LIFE programs on his kitchen table. Louis Merton Pseudonym for the AI chess hacker whose tendency to go catatonic brought the hacker community together. Jude Milhon Met Lee Felsenstein through a classified ad in the Berkeley Barb, and became more than a friend-- a member of the Community Memory collective. Marvin Minsky Playful and brilliant MIT prof who headed the AI lave and allowed the hackers to run free. Fred Moore Vagabond pacifist who hated money, loved technology, and co-founded Homebrew Club. Stewart Nelson Buck-toothed, diminutive, but fiery AI lab hacker who connected the PDP-1 comptuer to hack the phone system. Later co-founded the Systems Concepts company. Ted Nelson Self-described "innovator" and noted curmudgeon who self-published the influential Computer Lib book. Russel Noftsker Harried administrator of MIT AI lab in the late sixties; later president of Symbolics company. Adam Osborne Bangkok-born publisher-turned-computer-manufacturer who considered himself a philsopher. Founded Osborne Computer Company to make "adequate" machines. PDP-1 Digital Equipment's first minicomputer, and in 1961 an interactive godsend to the MIT hackers and a slap in the face to IBM fascism. PDP-6 Designed in part by Kotok, this mainframe computer was cornerstone of AI lab, with its gorgeious instruction set and sixteen sexy registers. Tom Pittman The religious Homebrew hacker who lost his wife but kept the faith with his Tiny Basic. Ed Roberts Enigmatic founder of MITS company who shook the world with his Altair computer. He wanted to help people build mental pyramids. Steve [Slug] Russell McCarthy's "coolie," who hacked the Spacewar program, first videogame, on the PDP-1. Never made a dime from it. Peter Samson MIT hacker, one of the first, who loved systems, trains, TX-0, music, parliamentary procedure, pranks, and hacking. Bob Saunders Jolly, balding TMRC hacker who married early, hacked till late at night eating "lemon gunkies," and mastered the "CBS Strategy on Spacewar. Warren Schwader Big blond hacker from rural Wisconsin who went from the assembly line to software stardom but couldn't reconcile the shift with his devotion to Jehovah's Witnesses. David Silver Left school at fourteen to be mascot of AI lab; maker of illicit keys and builder of a tiny robot that did the impossible. Dan Sokol Long-haired prankster who reveled in revealing technological secrets at Homebrew Club. Helped "liberate" Alair BASIC on paper tape. Les Solomon Editor of Popular Electroics, the puller of strings who set the computer revolution into motion. Marty Spergel The Junk Man, the Homebrew member who supplied circuits and cables and could make you a deal for anything. Richard Stallman The Last of the Hackers, who vowed to defend the principles of Hackerism to the bitter end. Remained at MIT until there was no one to eat Chinese food with. Jeff Stephenson Thirty-year-old martial arts veteran and hacker who was astounded that joining Sierra On-Line meant enrolling in Summer Camp. Jay Sullivan MAddeningly clam wizard-level programmer at Informatics who impressed Ken Williams by knowing the meaning of the word "any." Dick Sunderland Chalk-complexioned MBA who believed that firm managerial bureaucracy was a worth goal, but as president of Sierra On-Line found that hackers didn't think that way. Gerry Sussman Young MIT hacker branded "loser" because he smoked a pipe and "munged" his programs; later became "winner" by algorithmic magic. Margot Tommervik With her husband Al, long-haired Margot parlayed her game show winnings into a magazine that deified the Apple Computer. Tom Swift Terminal Lee Felsenstein's legendary, never-to-be-built computer terminal which would give the user ultimate leave to get his hands on the world. TX-0 Filled a small room, but in the late fifties this $3 million machine was the world's first personal computer--for the community of MIT hackers that formed around it. Jim Warren Portly purveyor of "techno-gossip" at Homebrew, he was first editor of hippie-styled Dr. Dobbs Journal, later started the lucrative Computer Faire. Randy Wigginton Fifteen-year-old member of Steve Wozniak's kiddie corps, he help Woz trundle the Apple II to Homebrew. Still in high school when he became Apple's first software employee. Ken Williams Arrogant and brilliant young programmer who saw the writing on the CRT and started Sierra On-Line to make a killing and improve society by selling games for the Apple computer. Roberta Williams Ken Williams' timid wife who rediscovered her own creativity by writing "Mystery House," the first of her many bestselling computer games. Steven "Woz" Wozniak Openhearted, technologically daring hardware hacker from San Jose suburbs. Woz built the Apple Computer for the pleasure of himself and friends. PART ONE True Hackers CAMBRIDGE: The Fifties and Sixties CHAPTER 1 THE TECH MODEL RAILROAD CLUB Just why Peter Samson was wandering around in Building 26 in the middle of the night is a matter that he would find difficult to explain. Some things are not spoken. If you were like the people whom Peter Samson was coming to know and befriend in this, his freshman year at the Massachusetts Institute of Technology in the winter of 1958-59, no explanation would be required. Wandering around the labyrinth of laboratories and storerooms, searching for the secrets of telephone switching in machine rooms, tracing paths of wires or relays in subterranean steam tunnels . . . for some, it was common behavior, and there was no need to justify the impulse, when confronted with a closed door with an unbearably intriguing noise behind it, to open the door uninvited. And then, if there was no one to physically bar access to whatever was making that intriguing noise, to touch the machine, start flicking switches and noting responses, and eventually to loosen a screw, unhook a template, jiggle some diodes and tweak a few connections. Peter Samson and his friends had grown up with a specific relationship to the world, wherein things had meaning only if you found out how they worked. And how would you go about that if not by getting your hands on them? It was in the basement of Building 26 that Samson and his friends discovered the EAM room. Building 26 was a long glass-and-steel structure, one of MIT's newer buildings, contrasting with the venerable pillared structures that fronted the Institute on Massachusetts Avenue. In the basement of this building void of personality, the EAM room. Electronic Accounting Machinery. A room that housed machines which ran like computers. Not many people in 1959 had even seen a computer, let alone touched one. Samson, a wiry, curly-haired redhead with a way of extending his vowels so that it would seem he was racing through lists of possible meanings of statements in mid-word, had viewed computers on his visits to MIT from his hometown of Lowell, Massachusetts, less than thirty miles from campus. This made him a "Cambridge urchin," one of dozens of science-crazy high schoolers in the region who were drawn, as if by gravitational pull, to the Cambridge campus. He had even tried to rig up his own computer with discarded parts of old pinball machines: they were the best source of logic elements he could find. LOGIC ELEMENTS: the term seems to encapsulate what drew Peter Samson, son of a mill machinery repairman, to electronics. The subject made sense. When you grow up with an insatiable curiosity as to how things work, the delight you find upon discovering something as elegant as circuit logic, where all connections have to complete their loops, is profoundly thrilling. Peter Samson, who early on appreciated the mathematical simplicity of these things, could recall seeing a television show on Boston's public TV channel, WGBH, which gave a rudimentary introduction to programming a computer in its own language. It fired his imagination: to Peter Samson, a computer was surely like Aladdin's lamp--rub it, and it would do your bidding. So he tried to learn more about the field, built machines of his own, entered science project competitions and contests, and went to the place that people of his ilk aspired to: MIT. The repository of the very brightest of those weird high school kids with owl-like glasses and underdeveloped pectorals who dazzled math teachers and flunked PE, who dreamed not of scoring on prom night, but of getting to the finals of the General Electric Science Fair competition. MIT, where he would wander the hallways at two o'clock in the morning, looking for something interesting, and where he would indeed discover something that would help draw him deeply into a new form of creative process, and a new life-style, and would put him into the forefront of a society envisioned only by a few science-fiction writers of mild disrepute. He would discover a computer that he could play with. The EAM room which Samson had chanced on was loaded with large keypunch machines the size of squat file cabinets. No one was protecting them: the room was staffed only by day, when a select group who had attained official clearance were privileged enough to submit long manila cards to operators who would then use these machines to punch holes in them according to what data the privileged ones wanted entered on the cards. A hole in the card would represent some instruction to the computer, telling it to put a piece of data somewhere, or perform a function on a piece of data, or move a piece of data from one place to another. An entire stack of these cards made one computer program, a program being a series of instructions which yield some expected result, just as the instructions in a recipe, when precisely followed, lead to a cake. Those cards would be taken to yet another operator upstairs who would feed the cards into a "reader" that would note where the holes were and dispatch this information to the IBM 704 computer on the first floor of Building 26. The Hulking Giant. The IBM 704 cost several million dollars, took up an entire room, needed constant attention from a cadre of professional machine operators, and required special air-conditioning so that the glowing vacuum tubes inside it would not heat up to data-destroying temperatures. When the air-conditioning broke down--a fairly common occurrences--a loud gong would sound, and three engineers would spring from a nearby office to frantically take covers off the machine so its innards wouldn't melt. All these people in charge of punching cards, feeding them into readers, and pressing buttons and switches on the machine were what was commonly called a Priesthood, and those privileged enough to submit data to those most holy priests were the official acolytes. It was an almost ritualistic exchange. ACOLYTE: Oh machine, would you accept my offer of information so you may run my program and perhaps give me a computation? PRIEST (on behalf of the machine): We will try. We promise nothing. As a general rule, even these most privileged of acolytes were not allowed direct access to the machine itself, and they would not be able to see for hours, sometimes for days, the results of the machine's ingestion of their "batch" of cards. This was something Samson knew, and of course it frustrated the hell out of Samson, who wanted to get at the damn machine. For this was what life was all about. What Samson did not know, and was delighted to discover, was that the EAM room also had a particular keypunch machine called the 407. Not only could it punch cards, but it could also read cards, sort them, and print them on listings. No one seemed to be guarding these machines, which were computers, sort of. Of course, using them would be no picnic: one needed to actually wire up what was called a plug board, a two-inch-by-two-inch plastic square with a mass of holes in it. If you put hundreds of wires through the holes in a certain order, you would get something that looked like a rat's nest but would fit into this electromechanical machine and alter its personality. It could do what you wanted it to do. So, without any authorization whatsoever, that is what Peter Samson set out to do, along with a few friends of his from an MIT organization with a special interest in model railroading. It was a casual, unthinking step into a science-fiction future, but that was typical of the way that an odd subculture was pulling itself up by its bootstraps and growing to underground prominence--to become a culture that would be the impolite, unsanctioned soul of computerdom. It was among the first computer hacker escapades of the Tech Model Railroad Club, or TMRC. * * * Peter Samson had been a member of the Tech Model Railroad Club since his first week at MIT in the fall of 1958. The first event that entering MIT freshmen attended was a traditional welcoming lecture, the same one that had been given for as long as anyone at MIT could remember. LOOK AT THE PERSON TO YOUR LEFT . . . LOOK AT THE PERSON TO YOUR RIGHT . . . ONE OF YOU THREE WILL NOT GRADUATE FROM THE INSTITUTE. The intended effect of the speech was to create that horrid feeling in the back of the collective freshman throat that signaled unprecedented dread. All their lives, these freshmen had been almost exempt from academic pressure. The exemption had been earned by virtue of brilliance. Now each of them had a person to the right and a person to the left who was just as smart. Maybe even smarter. But to certain students this was no challenge at all. To these youngsters, classmates were perceived in a sort of friendly haze: maybe they would be of assistance in the consuming quest to find out how things worked, and then to master them. There were enough obstacles to learning already--why bother with stupid things like brown-nosing teachers and striving for grades? To students like Peter Samson, the quest meant more than the degree. Sometime after the lecture came Freshman Midway. All the campus organizations--special-interest groups, fraternities, and such-- set up booths in a large gymnasium to try to recruit new members. The group that snagged Peter was the Tech Model Railroad Club. Its members, bright-eyed and crew-cutted upperclassmen who spoke with the spasmodic cadences of people who want words out of the way in a hurry, boasted a spectacular display of HO gauge trains they had in a permanent clubroom in Building 20. Peter Samson had long been fascinated by trains, especially subways. So he went along on the walking tour to the building, a shingle-clad temporary structure built during World War II. The hallways were cavernous, and even though the clubroom was on the second floor it had the dank, dimly lit feel of a basement. The clubroom was dominated by the huge train layout. It just about filled the room, and if you stood in the little control area called "the notch" you could see a little town, a little industrial area, a tiny working trolley line, a papier-mache mountain, and of course a lot of trains and tracks. The trains were meticulously crafted to resemble their full-scale counterparts, and they chugged along the twists and turns of track with picture-book perfection. And then Peter Samson looked underneath the chest-high boards which held the layout. It took his breath away. Underneath this layout was a more massive matrix of wires and relays,and crossbar switches than Peter Samson had ever dreamed existed. There were neat regimental lines of switches, and achingly regular rows of dull bronze relays, and a long, rambling tangle of red, blue, and yellow wires--twisting and twirling like a rainbow-colored explosion of Einstein's hair. It was an incredibly complicated system, and Peter Samson vowed to find out how it worked. The Tech Model Railroad Club awarded its members a key to the clubroom after they logged forty hours of work on the layout. Freshman Midway had been on a Friday. By Monday, Peter Samson had his key. * * * There were two factions of TMRC. Some members loved the idea of spending their time building and painting replicas of certain trains with historical and emotional value, or creating realistic scenery for the layout. This was the knife-and-paintbrush contingent, and it subscribed to railroad magazines and booked the club for trips on aging train lines. The other faction centered on the Signals and Power Subcommittee of the club, and it cared far more about what went on under the layout. This was The System, which worked something like a collaboration between Rube Goldberg and Wernher von Braun, and it was constantly being improved, revamped, perfected, and sometimes "gronked"--in club jargon, screwed up. S&P people were obsessed with the way The System worked, its increasing complexities, how any change you made would affect other parts, and how you could put those relationships between the parts to optimal use. Many of the parts for The System had been donated by the Western Electric College Gift Plan, directly from the phone company. The club's faculty advisor was also in charge of the campus phone system, and had seen to it that sophisticated phone equipment was available for the model railroaders. Using that equipment as a starting point, the Railroaders had devised a scheme which enabled several people to control trains at once, even if the trains were at different parts of the same track. Using dials appropriated from telephones, the TMRC "engineers" could specify which block of track they wanted control of, and run a train from there. This was done by using several types of phone company relays, including crossbar executors and step switches which let you actually hear the power being transferred from one block to another by an other-worldly chunka-chunka-chunka sound. It was the S&P group who devised this fiendishly ingenious scheme, and it was the S&P group who harbored the kind of restless curiosity which led them to root around campus buildings in search of ways to get their hands on computers. They were lifelong disciples of a Hands-On Imperative. Head of S&P was an upperclassman named Bob Saunders, with ruddy, bulbous features, an infectious laugh, and a talent for switch gear. As a child in Chicago, he had built a high-frequency transformer for a high school project; it was his six-foot-high version of a Tesla coil, something devised by an engineer in the 1800s which was supposed to send out furious waves of electrical power. Saunders said his coil project managed to blow out television reception for blocks around. Another person who gravitated to S&P was Alan Kotok, a plump, chinless, thick-spectacled New Jerseyite in Samson's class. Kotok's family could recall him, at age three, prying a plug out of a wall with a screwdriver and causing a hissing shower of sparks to erupt. When he was six, he was building and wiring lamps. In high school he had once gone on a tour of the Mobil Research Lab in nearby Haddonfield, and saw his first computer--the exhilaration of that experience helped him decide to enter MIT. In his freshman year, he earned a reputation as one of TMRC's most capable S&P people. The S&P people were the ones who spent Saturdays going to Eli Heffron's junkyard in Somerville scrounging for parts, who would spend hours on their backs resting on little rolling chairs they called "bunkies" to get underneath tight spots in the switching system, who would work through the night making the wholly unauthorized connection between the TMRC phone and the East Campus. Technology was their playground. The core members hung out at the club for hours; constantly improving The System, arguing about what could be done next, developing a jargon of their own that seemed incomprehensible to outsiders who might chance on these teen-aged fanatics, with their checked short-sleeve shirts, pencils in their pockets, chino pants, and, always, a bottle of Coca-Cola by their side. (TMRC purchased its own Coke machine for the then forbidding sum of $165; at a tariff of five cents a bottle, the outlay was replaced in three months; to facilitate sales, Saunders built a change machine for Coke buyers that was still in use a decade later.) When a piece of equipment wasn't working, it was "losing"; when a piece of equipment was ruined, it was "munged" (Mash Until No Good); the two desks in the corner of the room were not called the office, but the "orifice"; one who insisted on studying for courses was a "tool"; garbage was called "cruft"; and a project undertaken or a product built not solely to fulfill some constructive goal, but with some wild pleasure taken in mere involvement, was called a "hack." This latter term may have been suggested by ancient MIT lingo-- the word "hack" had long been used to describe the elaborate college pranks that MIT students would regularly devise, such as covering the dome that overlooked the campus with reflecting foil. But as the TMRC people used the word, there was serious respect implied. While someone might call a clever connection between relays a "mere hack," it would be understood that, to qualify as a hack, the feat must be imbued with innovation, style, and technical virtuosity. Even though one might self-deprecatingly say he was "hacking away at The System" (much as an axe-wielder hacks at logs), the artistry with which one hacked was recognized to be considerable. The most productive people working on Signals and Power called themselves "hackers" with great pride. Within the confines of the clubroom in Building 20, and of the "Tool Room" (where some study and many techno bull sessions took place), they had unilaterally endowed themselves with the heroic attributes of Icelandic legend. This is how Peter Samson saw himself and his friends in a Sandburg-esque poem in the club newsletter: Switch Thrower for the World, Fuze Tester, Maker of Routes, Player with the Railroads and the System's Advance Chopper; Grungy, hairy, sprawling, Machine of the Point-Function Line-o-lite: They tell me you are wicked and I believe them; for I have seen your painted light bulbs under the lucite luring the system coolies . . . Under the tower, dust all over the place, hacking with bifur- cated springs . . . Hacking even as an ignorant freshman acts who has never lost occupancy and has dropped out Hacking the M-Boards, for under its locks are the switches, and under its control the advance around the layout, Hacking! Hacking the grungy, hairy, sprawling hacks of youth; uncabled, frying diodes, proud to be Switch-thrower, Fuze- tester, Maker of Routes, Player with Railroads, and Advance Chopper to the System. Whenever they could, Samson and the others would slip off to the EAM room with their plug boards, trying to use the machine to keep track of the switches underneath the layout. Just as important, they were seeing what the electromechanical counter could do, taking it to its limit. That spring of 1959, a new course was offered at MIT. It was the first course in programming a computer that freshmen could take. The teacher was a distant man with a wild shock of hair and an equally unruly beard--John McCarthy. A master mathematician, McCarthy was a classically absent-minded professor; stories abounded about his habit of suddenly answering a question hours, sometimes even days after it was first posed to him. He would approach you in the hallway, and with no salutation would begin speaking in his robotically precise diction, as if the pause in conversation had been only a fraction of a second, and not a week. Most likely, his belated response would be brilliant. McCarthy was one of a very few people working in an entirely new form of scientific inquiry with computers. The volatile and controversial nature of his field of study was obvious from the very arrogance of the name that McCarthy had bestowed upon it: Artificial Intelligence. This man actually thought that computers could be SMART. Even at such a science-intensive place as MIT, most people considered the thought ridiculous: they considered computers to be useful, if somewhat absurdly expensive, tools for number-crunching huge calculations and for devising missile defense systems (as MIT's largest computer, the Whirlwind, had done for the early-warning SAGE system), but scoffed at the thought that computers themselves could actually be a scientific field of study, Computer Science did not officially exist at MIT in the late fifties, and McCarthy and his fellow computer specialists worked in the Electrical Engineering Department, which offered the course, No. 641, that Kotok, Samson, and a few other TRMC members took that spring. McCarthy had started a mammoth program on the IBM 704--the Hulking Giant--that would give it the extraordinary ability to play chess. To critics of the budding field of Artificial Intelligence, this was just one example of the boneheaded optimism of people like John McCarthy. But McCarthy had a certain vision of what computers could do, and playing chess was only the beginning. All fascinating stuff, but not the vision that was driving Kotok and Samson and the others. They wanted to learn how to WORK the damn machines, and while this new programming language called LISP that McCarthy was talking about in 641 was interesting, it was not nearly as interesting as the act of programming, or that fantastic moment when you got your printout back from the Priesthood--word from the source itself!--and could then spend hours poring over the results of the program, what had gone wrong with it, how it could be improved. The TMRC hackers were devising ways to get into closer contact with the IBM 704, which soon was upgraded to a newer model called the 709. By hanging out at the computation center in the wee hours of the morning, and by getting to know the Priesthood, and by bowing and scraping the requisite number of times, people like Kotok were eventually allowed to push a few buttons on the machine, and watch the lights as it worked. There were secrets to those IBM machines that had been painstakingly learned by some of the older people at MIT with access to the 704 and friends among the Priesthood. Amazingly, a few of these programmers, grad students working with McCarthy, had even written a program that utilized one of the rows of tiny lights: the lights would be lit in such an order that it looked like a little ball was being passed from right to left: if an operator hit a switch at just the right time, the motion of the lights could be reversed--Computer Ping-Pong! This obviously was the kind of thing that you'd show off to impress your peers, who would then take a look at the actual program you had written and see how it was done. To top the program, someone else might try to do the same thing with fewer instructions--a worthy endeavor, since there was so little room in the small "memory" of the computers of those days that not many instructions could fit into them, John McCarthy had once noticed how his graduate students who loitered around the 704 would work over their computer programs to get the most out of the fewest instructions, and get the program compressed so that fewer cards would need to be fed to the machine. Shaving off an instruction or two was almost an obsession with them. McCarthy compared these students to ski bums. They got the same kind of primal thrill from "maximizing code" as fanatic skiers got from swooshing frantically down a hill. So the practice of taking a computer program and trying to cut off instructions without affecting the outcome came to be called "program bumming," and you would often hear people mumbling things like "Maybe I can bum a few instructions out and get the octal correction card loader down to three cards instead of four." McCarthy in 1959 was turning his interest from chess to a new way of talking to the computer, the whole new "language" called LISP. Alan Kotok and his friends were more than eager to take over the chess project. Working on the batch-processed IBM, they embarked on the gargantuan project of teaching the 704, and later the 709, and even after that its replacement the 7090, how to play the game of kings. Eventually Kotok's group became the largest users of computer time in the entire MIT computation center. Still, working with the IBM machine was frustrating. There was nothing worse than the long wait between the time you handed in your cards and the time your results were handed back to you. If you had misplaced as much as one letter in one instruction, the program would crash, and you would have to start the whole process over again. It went hand in hand with the stifling proliferation of goddamn RULES that permeated the atmosphere of the computation center. Most of the rules were designed to keep crazy young computer fans like Samson and Kotok and Saunders physically distant from the machine itself. The most rigid rule of all was that no one should be able to actually touch or tamper with the machine itself. This, of course, was what those Signals and Power people were dying to do more than anything else in the world, and the restrictions drove them mad. One priest--a low-level sub-priest, really--on the late-night shift was particularly nasty in enforcing this rule, so Samson devised a suitable revenge. While poking around at Eli's electronic junk shop one day, he chanced upon an electrical board precisely like the kind of board holding the clunky vacuum tubes which resided inside the IBM. One night, sometime before 4 A.M., this particular sub-priest stepped out for a minute; when he returned, Samson told him that the machine wasn't working, but they'd found the trouble--and held up the totally smashed module from the old 704 he'd gotten at Eli's. The sub-priest could hardly get the words out. "W-where did you get that?" Samson, who had wide green eyes that could easily look maniacal, slowly pointed to an open place on the machine rack where, of course, no board had ever been, but the space still looked sadly bare. The sub-priest gasped. He made faces that indicated his bowels were about to give out. He whimpered exhortations to the deity. Visions, no doubt, of a million-dollar deduction from his paycheck began flashing before him. Only after his supervisor, a high priest with some understanding of the mentality of these young wiseguys from the Model Railroad Club, came and explained the situation did he calm down. He was not the last administrator to feel the wrath of a hacker thwarted in the quest for access. * * * One day a former TMRC member who was now on the MIT faculty paid a visit to the clubroom. His name was Jack Dennis. When he had been an undergraduate in the early 1950s, he had worked furiously underneath the layout. Dennis lately had been working a computer which MIT had just received from Lincoln Lab, a military development laboratory affiliated with the Institute. The computer was called the TX-0, and it was one of the first transistor-run computers in the world. Lincoln Lab had used it specifically to test a giant computer called the TX-2, which had a memory so complex that only with this specially built little brother could its ills be capably diagnosed. Now that its original job was over, the three-million-dollar TX-0 had been shipped over to the Institute on "long-term loan," and apparently no one at Lincoln Lab had marked a calendar with a return date. Dennis asked the S&P people at TMRC whether they would like to see it. Hey you nuns! Would you like to meet the Pope? The TX-0 was in Building 26, in the second-floor Radio Laboratory of Electronics (RLE), directly above the first-floor Computation Center which housed the hulking IBM 704. The RLE lab resembled the control room of an antique spaceship. The TX-0, or Tixo, as it was sometimes called, was for its time a midget machine, since it was one of the first computers to use finger-size transistors instead of hand-size vacuum tubes. Still, it took up much of the room, along with its fifteen tons of supporting air-conditioning equipment. The TX-O's workings were mounted on several tall, thin chassis, like rugged metal bookshelves, with tangled wires and neat little rows of tiny, bottle-like containers in which the transistors were inserted. Another rack had a solid metal front speckled with grim-looking gauges. Facing the racks was an L-shaped console, the control panel of this H. G. Wells spaceship, with a blue countertop for your elbows and papers. On the short arm of the L stood a Flexowriter, which resembled a typewriter converted for tank warfare, its bottom anchored in a military gray housing. Above the top were the control panels, boxlike protrusions painted an institutional yellow. On the sides of the boxes which faced the user were a few gauges, several lines of quarter-inch blinking lights, a matrix of steel toggle switches the size of large grains of rice, and, best of all, an actual cathode ray tube display, round and smoke-gray. The TMRC people were awed. THIS MACHINE DID NOT USE CARDS. The user would first punch in a program onto a long, thin paper tape with a Flexowriter (there were a few extra Flexowriters in an adjoining room), then sit at the console, feed in the program by running the tape through a reader, and be able to sit there while the program ran. If something went wrong with the program, you knew immediately, and you could diagnose the problem by using some of the switches, or checking out which of the lights were blinking or lit. The computer even had an audio output: while the program ran, a speaker underneath the console would make a sort of music, like a poorly tuned electric organ whose notes would vibrate with a fuzzy, ethereal din. The chords on this "organ" would change, depending on what data the machine was reading at any given microsecond; after you were familiar with the tones, you could actually HEAR what part of your program the computer was working on. You would have to discern this, though, over the clacking of the Flexowriter, which could make you think you were in the middle of a machine-gun battle. Even more amazing was that, because of these "interactive" capabilities, and also because users seemed to be allowed blocks of time to use the TX-0 all by themselves, you could even modify a program WHILE SITTING AT THE COMPUTER. A miracle! There was no way in hell that Kotok, Saunders, Samson, and the others were going to be kept away from that machine. Fortunately, there didn't seem to be the kind of bureaucracy surrounding the TX-0 that there was around the IBM 704. No cadre of officious priests. The technician in charge was a canny white-haired Scotsman named John McKenzie. While he made sure that graduate students and those working on funded projects-- Officially Sanctioned Users--maintained access to the machine, McKenzie tolerated the crew of TMRC madmen who began to hang out in the RLE lab, where the TX-0 stood. Samson, Kotok, Saunders, and a freshman named Bob Wagner soon figured out that the best time of all to hang out in Building 26 was at night, when no person in his right mind would have signed up for an hour-long session on the piece of paper posted every Friday beside the air conditioner in the RLE lab. The TX-0 as a rule was kept running twenty-four hours a day--computers back then were too expensive for their time to be wasted by leaving them idle through the night, and besides, it was a hairy procedure to get the thing up and running once it was turned off. So the TMRC hackers, who soon were referring to themselves as TX-0 hackers, changed their life-style to accommodate the computer. They laid claim to what blocks of time they could, and would "vulture time" with nocturnal visits to the lab on the off chance that someone who was scheduled for a 3 A.M. session might not show up. "Oh!" Samson would say delightedly, a minute or so after someone failed to show up at the time designated in the logbook. "Make sure it doesn't go to waste!" It never seemed to, because the hackers were there almost all the time. If they weren't in the RLE lab waiting for an opening to occur, they were in the classroom next to the TMRC clubroom, the Tool Room, playing a "hangman"-style word game that Samson had devised called "Come Next Door," waiting for a call from someone who was near the TX-0, monitoring it to see if someone had not shown up for a session. The hackers recruited a network of informers to give advance notice of potential openings at the computer--if a research project was not ready with its program in time, or a professor was sick, the word would be passed to TMRC and the hackers would appear at the TX-0, breathless and ready to jam into the space behind the console. Though Jack Dennis was theoretically in charge of the operation, Dennis was teaching courses at the time, and preferred to spend the rest of his time actually writing code for the machine. Dennis played the role of benevolent godfather to the hackers: he would give them a brief hands-on introduction to the machine, point them in certain directions, be amused at their wild programming ventures. He had little taste for administration, though, and was just as happy to let John McKenzie run things. McKenzie early on recognized that the interactive nature of the TX-0 was inspiring a new form of computer programming, and the hackers were its pioneers. So he did not lay down too many edicts. The atmosphere was loose enough in 1959 to accommodate the strays--science-mad people whose curiosity burned like a hunger, who like Peter Samson would be exploring the uncharted maze of laboratories at MIT. The noise of the air-conditioning, the audio output, and the drill-hammer Flexowriter would lure these wanderers, who'd poke their heads into the lab like kittens peering into baskets of yarn. One of those wanderers was an outsider named Peter Deutsch. Even before discovering the TX-0, Deutsch had developed a fascination for computers. It began one day when he picked up a manual that someone had discarded, a manual for an obscure form of computer language for doing calculations. Something about the orderliness of the computer instructions appealed to him: he would later describe the feeling as the same kind of eerily transcendent recognition that an artist experiences when he discovers the medium that is absolutely right for him. THIS IS WHERE I BELONG. Deutsch tried writing a small program, and, signing up for time under the name of one of the priests, ran it on a computer. Within weeks, he had attained a striking proficiency in programming. He was only twelve years old. He was a shy kid, strong in math and unsure of most everything else. He was uncomfortably overweight, deficient in sports, but an intellectual star performer. His father was a professor at MIT, and Peter used that as his entree to explore the labs. It was inevitable that he would be drawn to the TX-0. He first wandered into the small "Kluge Room" (a "kluge" is a piece of inelegantly constructed equipment that seems to defy logic by working properly), where three off-line Flexowriters were available for punching programs onto paper tape which would later be fed into the TX-0. Someone was busy punching in a tape. Peter watched for a while, then began bombarding the poor soul with questions about that weird-looking little computer in the next room. Then Peter went up to the TX-0 itself, examined it closely, noting how it differed from other computers: it was smaller, had a CRT display, and other neat toys. He decided right then to act as if he had a perfect right to be there. He got hold of a manual and soon was startling people by spouting actual make-sense computer talk, and eventually was allowed to sign up for night and weekend sessions, and to write his own programs. McKenzie worried that someone might accuse him of running some sort of summer camp, with this short-pants little kid, barely tall enough to stick his head over the TX-O's console, staring at the code that an Officially Sanctioned User, perhaps some self-important graduate student, would be hammering into the Flexowriter, and saying in his squeaky, preadolescent voice something like "Your problem is that this credit is wrong over here . . . you need this other instruction over there," and the self-important grad student would go crazy--WHO IS THIS LITTLE WORM?--and start screaming at him to go out and play somewhere. Invariably, though, Peter Deutsch's comments would turn out to be correct. Deutsch would also brazenly announce that he was going to write better programs than the ones currently available, and he would go and do it. Samson, Kotok, and the other hackers accepted Peter Deutsch: by virtue of his computer knowledge he was worthy of equal treatment. Deutsch was not such a favorite with the Officially Sanctioned Users, especially when he sat behind them ready to spring into action when they made a mistake on the Flexowriter. These Officially Sanctioned Users appeared at the TX-0 with the regularity of commuters. The programs they ran were statistical analyses, cross correlations, simulations of an interior of the nucleus of a cell. Applications. That was fine for Users, but it was sort of a waste in the minds of the hackers. What hackers had in mind was getting behind the console of the TX-0 much in the same way as getting in behind the throttle of a plane, Or, as Peter Samson, a classical music fan, put it, computing with the TX-0 was like playing a musical instrument: an absurdly expensive musical instrument upon which you could improvise, compose, and, like the beatniks in Harvard Square a mile away, wail like a banshee with total creative abandon. One thing that enabled them to do this was the programming system devised by Jack Dennis and another professor, Tom Stockman. When the TX-0 arrived at MIT, it had been stripped down since its days at Lincoln Lab: the memory had been reduced considerably, to 4,096 "words" of eighteen bits each. (A "bit" is a BInary digiT, either a one or zero. These binary numbers are the only thing computers understand. A series of binary numbers is called a "word.") And the TX-0 had almost no software. So Jack Dennis, even before he introduced the TMRC people to the TX-0, had been writing "systems programs"--the software to help users utilize the machine. The first thing Dennis worked on was an assembler. This was something that translated assembly language--which used three- letter symbolic abbreviations that represented instructions to the machine--into machine language, which consisted of the binary numbers 0 and 1. The TX-0 had a rather limited assembly language: since its design allowed only two bits of each eighteen-bit word to be used for instructions to the computer, only four instructions could be used (each possible two-bit variation--00, 0 1, 10, and 11--represented an instruction). Everything the computer did could be broken down to the execution of one of those four instructions: it took one instruction to add two numbers, but a series of perhaps twenty instructions to multiply two numbers. Staring at a long list of computer commands written as binary numbers--for example, 10011001100001-- could make you into a babbling mental case in a matter of minutes. But the same command in assembly language might look like this: ADD Y. After loading the computer with the assembler that Dennis wrote, you could write programs in this simpler symbolic form, and wait smugly while the computer did the translation into binary for you, Then you'd feed that binary "object" code back into the computer. The value of this was incalculable: it enabled programmers to write in something that LOOKED like code, rather than an endless, dizzying series of ones and zeros. The other program that Dennis worked on with Stockman was something even newer--a debugger. The TX-0 came with a debugging program called UT-3, which enabled you to talk to the computer while it was running by typing commands directly into the Flexowriter, But it had terrible problems-for one thing, it only accepted typed-in code that used the octal numeric system. "Octal" is a base-eight number system (as opposed to binary, which is base two, and Arabic--ours-which is base ten), and it is a difficult system to use. So Dennis and Stockman decided to write something better than UT-3 which would enable users to use the symbolic, easier-to-work-with assembly language. This came to be called FLIT, and it allowed users to actually find program bugs during a session, fix them, and keep the program running. (Dennis would explain that "FLIT" stood for FLexowriter Interrogation Tape, but clearly the name's real origin was the insect spray with that brand name.) FLIT was a quantum leap forward, since it liberated programmers to actually do original composing on the machine--just like musicians composing on their musical instruments. With the use of the debugger, which took up one third of the 4,096 words of the TX-O's memory, hackers were free to create a new, more daring style of programming. And what did these hacker programs DO? Well, sometimes, it didn't matter much at all what they did. Peter Samson hacked the night away on a program that would instantly convert Arabic numbers to Roman numerals, and Jack Dennis, after admiring the skill with which Samson had accomplished this feat, said, "My God, why would anyone want to do such a thing?" But Dennis knew why. There was ample justification in the feeling of power and accomplishment Samson got when he fed in the paper tape, monitored the lights and switches, and saw what were once plain old blackboard Arabic numbers coming back as the numerals the Romans had hacked with. In fact it was Jack Dennis who suggested to Samson that there were considerable uses for the TX-O's ability to send noise to the audio speaker. While there were no built-in controls for pitch, amplitude, or tone character, there was a way to control the speaker--sounds would be emitted depending on the state of the fourteenth bit in the eighteen-bit words the TX-0 had in its accumulator in a given microsecond. The sound was on or off depending on whether bit fourteen was a one or zero. So Samson set about writing programs that varied the binary numbers in that slot in different ways to produce different pitches. At that time, only a few people in the country had been experimenting with using a computer to output any kind of music, and the methods they had been using required massive computations before the machine would so much as utter a note, Samson, who reacted with impatience to those who warned he was attempting the impossible, wanted a computer playing music right away. So he learned to control that one bit in the accumulator so adeptly that he could command it with the authority of Charlie Parker on the saxophone. In a later version of this music compiler, Samson rigged it so that if you made an error in your programming syntax, the Flexowriter would switch to a red ribbon and print "To err is human to forgive divine." When outsiders heard the melodies of Johann Sebastian Bach in a single-voice, monophonic square wave, no harmony, they were universally unfazed. Big deal! Three million dollars for this giant hunk of machinery, and why shouldn't it do at least as much as a five-dollar toy piano? It was no use to explain to these outsiders that Peter Samson had virtually bypassed the process by which music had been made for eons. Music had always been made by directly creating vibrations that were sound. What happened in Samson's program was that a load of numbers, bits of information fed into a computer, comprised a code in which the music resided. You could spend hours staring at the code, and not be able to divine where the music was. It only became music while millions of blindingly brief exchanges of data were taking place in the accumulator sitting in one of the metal, wire, and silicon racks that comprised the TX-0. Samson had asked the computer, which had no apparent knowledge of how to use a voice, to lift itself in song--and the TX-0 had complied. So it was that a computer program was not only metaphorically a musical composition--it was LITERALLY a musical composition! It looked like--and was--the same kind of program which yielded complex arithmetical computations and statistical analyses. These digits that Samson had jammed into the computer were a universal language which could produce ANYTHING--a Bach fugue or an anti-aircraft system. Samson did not say any of this to the outsiders who were unimpressed by his feat. Nor did the hackers themselves discuss this--it is not even clear that they analyzed the phenomenon in such cosmic terms. Peter Samson did it, and his colleagues appreciated it, because it was obviously a neat hack. That was justification enough. * * * To hackers like Bob Saunders--balding, plump, and merry disciple of the TX-0, president of TMRC's S&P group, student of systems-- it was a perfect existence. Saunders had grown up in the suburbs of Chicago, and for as long as he could remember the workings of electricity and telephone circuitry had fascinated him. Before beginning MIT, Saunders had landed a dream summer job, working for the phone company installing central office equipment, He would spend eight blissful hours with soldering iron and pliers in hand, working in the bowels of various systems, an idyll broken by lunch hours spent in deep study of phone company manuals. It was the phone company equipment underneath the TMRC layout that had convinced Saunders to become active in the Model Railroad Club. Saunders, being an upperclassman, had come to the TX-0 later in his college career than Kotok and Samson: he had used the breathing space to actually lay the foundation for a social life, which included courtship of and eventual marriage to Marge French, who had done some non-hacking computer work for a research project. Still, the TX-0 was the center of his college career, and he shared the common hacker experience of seeing his grades suffer from missed classes. It didn't bother him much, because he knew that his real education was occurring in Room 240 of Building 26, behind the Tixo console. Years later he would describe himself and the others as "an elite group. Other people were off studying, spending their days up on four-floor buildings making obnoxious vapors or off in the physics lab throwing particles at things or whatever it is they do. And we were simply not paying attention to what other folks were doing because we had no interest in it. They were studying what they were studying and we were studying what we were studying. And the fact that much of it was not on the officially approved curriculum was by and large immaterial." The hackers came out at night. It was the only way to take full advantage of the crucial "off-hours" of the TX-0. During the day, Saunders would usually manage to make an appearance in a class or two. Then some time spent performing "basic maintenance"--things like eating and going to the bathroom. He might see Marge for a while. But eventually he would filter over to Building 26. He would go over some of the programs of the night before, printed on the nine-and-a-half-inch-wide paper that the Flexowriter used. He would annotate and modify the listing to update the code to whatever he considered the next stage of operation. Maybe then he would move over to the Model Railroad Club, and he'd swap his program with someone, checking simultaneously for good ideas and potential bugs. Then back to Building 26, to the Kluge Room next to the TX-0, to find an off-line Flexowriter on which to update his code. All the while he'd be checking to see if someone had canceled a one-hour session on the machine; his own session was scheduled at something like two or three in the morning. He'd wait in the Kluge Room, or play some bridge back at the Railroad Club, until the time came. Sitting at the console, facing the metal racks that held the computer's transistors, each transistor representing a location that either held or did not hold a bit of memory, Saunders would set up the Flexowriter, which would greet him with the word "WALRUS." This was something Samson had hacked, in honor of Lewis Carroll's poem with the line "The time has come, the Walrus said . . ." Saunders might chuckle at that as he went into the drawer for the paper tape which held the assembler program and fed that into the tape reader. Now the computer would be ready to assemble his program, so he'd take the Flexowriter tape he'd been working on and send that into the computer. He'd watch the lights go on as the computer switched his code from "source" (the symbolic assembly language) to "object" code (binary), which the computer would punch out into another paper tape. Since that tape was in the object code that the TX-0 understood, he'd feed it in, hoping that the program would run magnificently. There would most probably be a few fellow hackers kibitzing behind him, laughing and joking and drinking Cokes and eating some junk food they'd extracted from the machine downstairs. Saunders preferred the lemon jelly wedges that the others called "lemon gunkies." But at four in the morning, anything tasted good. They would all watch as the program began to run, the lights going on, the whine from the speaker humming in high or low register depending on what was in Bit 14 in the accumulator, and the first thing he'd see on the CRT display after the program had been assembled and run was that the program had crashed. So he'd reach into the drawer for the tape with the FLIT debugger and feed THAT into the computer. The computer would then be a debugging machine, and he'd send the program back in. Now he could start trying to find out where things had gone wrong, and maybe if he was lucky he'd find out, and change things by putting in some commands by flicking some of the switches on the console in precise order, or hammering in some code on the Flexowriter. Once things got running--and it was always incredibly satisfying when something worked, when he'd made that roomful of transistors and wires and metal and electricity all meld together to create a precise output that he'd devised--he'd try to add the next advance to it. When the hour was over--someone already itching to get on the machine after him--Saunders would be ready to spend the next few hours figuring out what the heck had made the program go belly-up. The peak hour itself was tremendously intense, but during the hours before, and even during the hours afterward, a hacker attained a state of pure concentration. When you programmed a computer, you had to be aware of where all the thousands of bits of information were going from one instruction to the next, and be able to predict--and exploit--the effect of all that movement. When you had all that information glued to your cerebral being, it was almost as if your own mind had merged into the environment of the computer. Sometimes it took hours to build up to the point where your thoughts could contain that total picture, and when you did get to that point, it was such a shame to waste it that you tried to sustain it by marathon bursts, alternatively working on the computer or poring over the code that you wrote on one of the off-line Flexowriters in the Kluge Room. You would sustain that concentration by "wrapping around" to the next day. Inevitably, that frame of mind spilled over to what random shards of existence the hackers had outside of computing. The knife-and-paintbrush contingent at TMRC were not pleased at all by the infiltration of Tixo-mania into the club: they saw it as a sort of Trojan horse for a switch in the club focus, from railroading to computing. And if you attended one of the club meetings held every Tuesday at five-fifteen, you could see the concern: the hackers would exploit every possible thread of parliamentary procedure to create a meeting as convoluted as the programs they were hacking on the TX-0. Motions were made to make motions to make motions, and objections ruled out of order as if they were so many computer errors. A note in the minutes of the meeting on November 24, 1959, suggests that "we frown on certain members who would do the club a lot more good by doing more S&P-ing and less reading Robert's Rules of Order." Samson was one of the worst offenders, and at one point, an exasperated TMRC member made a motion "to purchase a cork for Samson's oral diarrhea." Hacking parliamentary procedure was one thing, but the logical mind-frame required for programming spilled over into more commonplace activities. You could ask a hacker a question and sense his mental accumulator processing bits until he came up with a precise answer to the question you asked. Marge Saunders would drive to the Safeway every Saturday morning in the Volkswagen and upon her return ask her husband, "Would you like to help me bring in the groceries?" Bob Saunders would reply, "No." Stunned, Marge would drag in the groceries herself. After the same thing occurred a few times, she exploded, hurling curses at him and demanding to know why he said no to her question. "That's a stupid question to ask," he said. "Of course I won't LIKE to help you bring in the groceries. If you ask me if I'll help you bring them in, that's another matter." It was as if Marge had submitted a program into the TX-0, and the program, as programs do when the syntax is improper, had crashed. It was not until she debugged her question that Bob Saunders would allow it to run successfully on his own mental computer. CHAPTER 2 THE HACKER ETHIC Something new was coalescing around the TX-0: a new way of life, with a philosophy, an ethic, and a dream. There was no one moment when it started to dawn on the TX-0 hackers that by devoting their technical abilities to computing with a devotion rarely seen outside of monasteries they were the vanguard of a daring symbiosis between man and machine. With a fervor like that of young hot-rodders fixated on souping up engines, they came to take their almost unique surroundings for granted, Even as the elements of a culture were forming, as legends began to accrue, as their mastery of programming started to surpass any previous recorded levels of skill, the dozen or so hackers were reluctant to acknowledge that their tiny society, on intimate terms with the TX-0, had been slowly and implicitly piecing together a body of concepts, beliefs, and mores. The precepts of this revolutionary Hacker Ethic were not so much debated and discussed as silently agreed upon. No manifestos were issued. No missionaries tried to gather converts. The computer did the converting, and those who seemed to follow the Hacker Ethic most faithfully were people like Samson, Saunders, and Kotok, whose lives before MIT seemed to be mere preludes to that moment when they fulfilled themselves behind the console of the TX-0. Later there would come hackers who took the implicit Ethic even more seriously than the TX-0 hackers did, hackers like the legendary Greenblatt or Gosper, though it would be some years yet before the tenets of hackerism would be explicitly delineated. Still, even in the days of the TX-0, the planks of the platform were in place. The Hacker Ethic: ACCESS TO COMPUTERS--AND ANYTHING WHICH MIGHT TEACH YOU SOMETHING ABOUT THE WAY THE WORLD WORKS--SHOULD BE UNLIMITED AND TOTAL. ALWAYS YIELD TO THE HANDS-ON IMPERATIVE! Hackers believe that essential lessons can be learned about the systems--about the world--from taking things apart, seeing how they work, and using this knowledge to create new and even more interesting things. They resent any person, physical barrier, or law that tries to keep them from doing this. This is especially true when a hacker wants to fix something that (from his point of view) is broken or needs improvement. Imperfect systems infuriate hackers, whose primal instinct is to debug them. This is one reason why hackers generally hate driving cars--the system of randomly programmed red lights and oddly laid out one-way streets causes delays which are so goddamned UNNECESSARY that the impulse is to rearrange signs, open up traffic-light control boxes . . .redesign the entire system. In a perfect hacker world, anyone pissed off enough to open up a control box near a traffic light and take it apart to make it work better should be perfectly welcome to make the attempt. Rules which prevent you from taking matters like that into your own hands are too ridiculous to even consider abiding by. This attitude helped the Model Railroad Club start, on an extremely informal basis, something called the Midnight Requisitioning Committee. When TMRC needed a set of diodes, or some extra relays, to build some new feature into The System, a few S&P people would wait until dark and find their way into the places where those things were kept. None of the hackers, who were as a rule scrupulously honest in other matters, seemed to equate this with "stealing." A willful blindness. ALL INFORMATION SHOULD BE FREE. If you don't have access to the information you need to improve things, how can you fix them? A free exchange of information particularly when the information was in the form of a computer program, allowed for greater overall creativity. When you were working on a machine like the TX-0, which came with almost no software, everyone would furiously write systems programs to make programming easier--Tools to Make Tools, kept in the drawer by the console for easy access by anyone using the machine. This prevented the dread, time-wasting ritual of reinventing the wheel: instead of everybody writing his own version of the same program, the best version would be available to everyone, and everyone would be free to delve into the code and improve on THAT. A world studded with feature-full programs, bummed to the minimum, debugged to perfection. The belief, sometimes taken unconditionally, that information should be free was a direct tribute to the way a splendid computer, or computer program, works--the binary bits moving in the most straightforward, logical path necessary to do their complex job, What was a computer but something which benefited from a free flow of information? If, say, the accumulator found itself unable to get information from the input/output (i/o) devices like the tape reader or the switches, the whole system would collapse. In the hacker viewpoint, any system could benefit from that easy flow of information. MISTRUST AUTHORITY--PROMOTE DECENTRALIZATION. The best way to promote this free exchange of information is to have an open system, something which presents no boundaries between a hacker and a piece of information or an item of equipment that he needs in his quest for knowledge, improvement, and time on-line. The last thing you need is a bureaucracy. Bureaucracies, whether corporate, government, or university, are flawed systems, dangerous in that they cannot accommodate the exploratory impulse of true hackers. Bureaucrats hide behind arbitrary rules (as opposed to the logical algorithms by which machines and computer programs operate): they invoke those rules to consolidate power, and perceive the constructive impulse of hackers as a threat. The epitome of the bureaucratic world was to be found at a very large company called International Business Machines--IBM. The reason its computers were batch-processed Hulking Giants was only partially because of vacuum tube technology, The real reason was that IBM was a clumsy, hulking company which did not understand the hacking impulse. If IBM had its way (so the TMRC hackers thought), the world would be batch-processed, laid out on those annoying little punch cards, and only the most privileged of priests would be permitted to actually interact with the computer. All you had to do was look at someone in the IBM world, and note the button-down white shirt, the neatly pinned black tie, the hair carefully held in place, and the tray of punch cards in hand. You could wander into the Computation Center, where the 704, the 709, and later the 7090 were stored--the best IBM had to offer--and see the stifling orderliness, down to the roped-off areas beyond which non-authorized people could not venture. And you could compare that to the extremely informal atmosphere around the TX-0, where grungy clothes were the norm and almost anyone could wander in. Now, IBM had done and would continue to do many things to advance computing. By its sheer size and mighty influence, it had made computers a permanent part of life in America. To many people, the words IBM and computer were virtually synonymous. IBM's machines were reliable workhorses, worthy of the trust that businessmen and scientists invested in them. This was due in part to IBM's conservative approach: it would not make the most technologically advanced machines, but would rely on proven concepts and careful, aggressive marketing. As IBM's dominance of the computer field was established, the company became an empire unto itself, secretive and smug. What really drove the hackers crazy was the attitude of the IBM priests and sub-priests, who seemed to think that IBM had the only "real" computers, and the rest were all trash. You couldn't talk to those people--they were beyond convincing. They were batch-processed people, and it showed not only in their preference of machines, but in their idea about the way a computation center, and a world, should be run. Those people could never understand the obvious superiority of a decentralized system, with no one giving orders: a system where people could follow their interests, and if along the way they discovered a flaw in the system, they could embark on ambitious surgery. No need to get a requisition form. just a need to get something done. This antibureaucratic bent coincided neatly with the personalities of many of the hackers, who since childhood had grown accustomed to building science projects while the rest of their classmates were banging their heads together and learning social skills on the field of sport. These young adults who were once outcasts found the computer a fantastic equalizer, experiencing a feeling, according to Peter Samson, "like you opened the door and walked through this grand new universe . . ." Once they passed through that door and sat behind the console of a million-dollar computer, hackers had power. So it was natural to distrust any force which might try to limit the extent of that power. HACKERS SHOULD BE JUDGED BY THEIR HACKING, NOT BOGUS CRITERIA SUCH AS DEGREES, AGE, RACE, OR POSITION. The ready acceptance of twelve-year-old Peter Deutsch in the TX-0 community (though not by non-hacker graduate students) was a good example. Likewise, people who trotted in with seemingly impressive credentials were not taken seriously until they proved themselves at the console of a computer. This meritocratic trait was not necessarily rooted in the inherent goodness of hacker hearts--it was mainly that hackers cared less about someone's superficial characteristics than they did about his potential to advance the general state of hacking, to create new programs to admire, to talk about that new feature in the system. YOU CAN CREATE ART AND BEAUTY ON A COMPUTER. Samson's music program was an example. But to hackers, the art of the program did not reside in the pleasing sounds emanating from the on-line speaker. The code of the program held a beauty of its own. (Samson, though, was particularly obscure in refusing to add comments to his source code explaining what he was doing at a given time. One well-distributed program Samson wrote went on for hundreds of assembly language instructions, with only one comment beside an instruction which contained the number 1750. The comment was RIPJSB, and people racked their brains about its meaning until someone figured out that 1750 was the year Bach died, and that Samson had written an abbreviation for Rest In Peace Johann Sebastian Bach.) A certain esthetic of programming style had emerged. Because of the limited memory space of the TX-0 (a handicap that extended to all computers of that era), hackers came to deeply appreciate innovative techniques which allowed programs to do complicated tasks with very few instructions. The shorter a program was, the more space you had left for other programs, and the faster a program ran. Sometimes when you didn't need speed or space much, and you weren't thinking about art and beauty, you'd hack together an ugly program, attacking the problem with "brute force" methods. "Well, we can do this by adding twenty numbers," Samson might say to himself, "and it's quicker to write instructions to do that than to think out a loop in the beginning and the end to do the same job in seven or eight instructions." But the latter program might be admired by fellow hackers, and some programs were bummed to the fewest lines so artfully that the author's peers would look at it and almost melt with awe. Sometimes program bumming became competitive, a macho contest to prove oneself so much in command of the system that one could recognize elegant shortcuts to shave off an instruction or two, or, better yet, rethink the whole problem and devise a new algorithm which would save a whole block of instructions. (An algorithm is a specific procedure which one can apply to solve a complex computer problem; it is sort of a mathematical skeleton key.) This could most emphatically be done by approaching the problem from an offbeat angle that no one had ever thought of before but that in retrospect made total sense. There was definitely an artistic impulse residing in those who could utilize this genius-from-Mars techniques black-magic, visionary quality which enabled them to discard the stale outlook of the best minds on earth and come up with a totally unexpected new algorithm. This happened with the decimal print routine program. This was a subroutines program within a program that you could sometimes integrate into many different programs--to translate binary numbers that the computer gave you into regular decimal numbers. In Saunders' words, this problem became the "pawn's ass of programming--if you could write a decimal print routine which worked you knew enough about the computer to call yourself a programmer of sorts." And if you wrote a GREAT decimal print routine, you might be able to call yourself a hacker. More than a competition, the ultimate bumming of the decimal print routine became a sort of hacker Holy Grail. Various versions of decimal print routines had been around for some months. If you were being deliberately stupid about it, or if you were a genuine moron--an out-and-out "loser"--it might take you a hundred instructions to get the computer to convert machine language to decimal. But any hacker worth his salt could do it in less, and finally, by taking the best of the programs, bumming an instruction here and there, the routine was diminished to about fifty instructions. After that, things got serious. People would work for hours, seeking a way to do the same thing in fewer lines of code. It became more than a competition; it was a quest. For all the effort expended, no one seemed to be able to crack the fifty-line barrier. The question arose whether it was even possible to do it in less. Was there a point beyond which a program could not be bummed? Among the people puzzling with this dilemma was a fellow named Jenson, a tall, silent hacker from Maine who would sit quietly in the Kluge Room and scribble on printouts with the calm demeanor of a backwoodsman whittling. Jenson was always looking for ways to compress his programs in time and space--his code was a completely bizarre sequence of intermingled Boolean and arithmetic functions, often causing several different computations to occur in different sections of the same eighteen-bit "word." Amazing things, magical stunts. Before Jenson, there had been general agreement that the only logical algorithm for a decimal print routine would have the machine repeatedly subtracting, using a table of the powers of ten to keep the numbers in proper digital columns. Jenson somehow figured that a powers-of-ten table wasn't necessary; he came up with an algorithm that was able to convert the digits in a reverse order but, by some digital sleight of hand, print them out in the proper order. There was a complex mathematical justification to it that was clear to the other hackers only when they saw Jenson's program posted on a bulletin board, his way of telling them that he had taken the decimal print routine to its limit. FORTY-SIX INSTRUCTIONS. People would stare at the code and their jaws would drop. Marge Saunders remembers the hackers being unusually quiet for days afterward. "We knew that was the end of it," Bob Saunders later said. "That was Nirvana." COMPUTERS CAN CHANGE YOUR LIFE FOR THE BETTER. This belief was subtly manifest. Rarely would a hacker try to impose a view of the myriad advantages of the computer way of knowledge to an outsider. Yet this premise dominated the everyday behavior of the TX-0 hackers, as well as the generations of hackers that came after them. Surely the computer had changed THEIR lives, enriched their lives, given their lives focus, made their lives adventurous. It had made them masters of a certain slice of fate. Peter Samson later said, "We did it twenty-five to thirty percent for the sake of doing it because it was something we could do and do well, and sixty percent for the sake of having something which was in its metaphorical way alive, our offspring, which would do things on its own when we were finished. That's the great thing about programming, the magical appeal it has . . . Once you fix a behavioral problem [a computer or program] has, it's fixed forever, and it is exactly an image of what you meant." LIKE ALADDIN'S LAMP, YOU COULD GET IT TO DO YOUR BIDDING. Surely everyone could benefit from experiencing this power. Surely everyone could benefit from a world based on the Hacker Ethic. This was the implicit belief of the hackers, and the hackers irreverently extended the conventional point of view of what computers could and should do--leading the world to a new way of looking and interacting with computers. This was not easily done. Even at such an advanced institution as MIT, some professors considered a manic affinity for computers as frivolous, even demented. TMRC hacker Bob Wagner once had to explain to an engineering professor what a computer was. Wagner experienced this clash of computer versus anti-computer even more vividly when he took a Numerical Analysis class in which the professor required each student to do homework using rattling, clunky electromechanical calculators. Kotok was in the same class, and both of them were appalled at the prospect of working with those lo-tech machines. "Why should we," they asked, "when we've got this computer?" So Wagner began working on a computer program that would emulate the behavior of a calculator. The idea was outrageous. To some, it was a misappropriation of valuable machine time. According to the standard thinking on computers, their time was too precious that one should only attempt things which took maximum advantage of the computer, things that otherwise would take roomfuls of mathematicians days of mindless calculating. Hackers felt otherwise: anything that seemed interesting or fun was fodder for computing--and using interactive computers, with no one looking over your shoulder and demanding clearance for your specific project, you could act on that belief. After two or three months of tangling with intricacies of floating-point arithmetic (necessary to allow the program to know where to place the decimal point) on a machine that had no simple method to perform elementary multiplication, Wagner had written three thousand lines of code that did the job. He had made a ridiculously expensive computer perform the function of a calculator that cost a thousand times less. To honor this irony, he called the program Expensive Desk Calculator, and proudly did the homework for his class on it. His grade--zero. "You used a computer!" the professor told him. "This CAN'T be right." Wagner didn't even bother to explain. How could he convey to his teacher that the computer was making realities out of what were once incredible possibilities? Or that another hacker had even written a program called Expensive Typewriter that converted the TX-0 to something you could write text on, could process your writing in strings of characters and print it out on the Flexowriter--could you imagine a professor accepting a classwork report WRITTEN BY THE COMPUTER? How could that professor--how could, in fact, anyone who hadn't been immersed in this uncharted man-machine universe--understand how Wagner and his fellow hackers were routinely using the computer to simulate, according to Wagner, "strange situations which one could scarcely envision otherwise"? The professor would learn in time, as would everyone, that the world opened up by the computer was a limitless one. If anyone needed further proof, you could cite the project that Kotok was working on in the Computation Center, the chess program that bearded Al professor "Uncle" John McCarthy, as he was becoming known to his hacker students, had begun on the IBM 704. Even though Kotok and the several other hackers helping him on the program had only contempt for the IBM batch-processing mentality that pervaded the machine and the people around it, they had managed to scrounge some late-night time to use it interactively, and had been engaging in an informal battle with the systems programmers on the 704 to see which group would be known as the biggest consumer of computer time. The lead would bounce back and forth, and the white-shirt-and-black-tie 704 people were impressed enough to actually let Kotok and his group touch the buttons and switches on the 704: rare sensual contact with a vaunted IBM beast. Kotok's role in bringing the chess program to life was indicative of what was to become the hacker role in Artificial Intelligence: a Heavy Head like McCarthy or like his colleague Marvin Minsky would begin a project or wonder aloud whether something might be possible, and the hackers, if it interested them, would set about doing it. The chess program had been started using FORTRAN, one of the early computer languages. Computer languages look more like English than assembly language, are easier to write with, and do more things with fewer instructions; however, each time an instruction is given in a computer language like FORTRAN, the computer must first translate that command into its own binary language. A program called a compiler does this, and the compiler takes up time to do its job, as well as occupying valuable space within the computer. In effect, using a computer language puts you an extra step away from direct contact with the computer, and hackers generally preferred assembly or, as they called it, "machine" language to less elegant, "higher-level" languages like FORTRAN. Kotok, though, recognized that because of the huge amounts of numbers that would have to be crunched in a chess program, part of the program would have to be done in FORTRAN, and part in assembly. They hacked it part by part, with "move generators," basic data structures, and all kinds of innovative algorithms for strategy. After feeding the machine the rules for moving each piece, they gave it some parameters by which to evaluate its position, consider various moves, and make the move which would advance it to the most advantageous situation. Kotok kept at it for years, the program growing as MIT kept upgrading its IBM computers, and one memorable night a few hackers gathered to see the program make some of its first moves in a real game. Its opener was quite respectable, but after eight or so exchanges there was real trouble, with the computer about to be checkmated. Everybody wondered how the computer would react. It too a while (everyone knew that during those pauses the computer was actually "thinking," if your idea of thinking included mechanically considering various moves, evaluating them, rejecting most, and using a predefined set of parameters to ultimately make a choice). Finally, the computer moved a pawn two squares forward--illegally jumping over another piece. A bug! But a clever one--it got the computer out of check. Maybe the program was figuring out some new algorithm with which to conquer chess. At other universities, professors were making public proclamations that computers would never be able to beat a human being in chess. Hackers knew better. They would be the ones who would guide computers to greater heights than anyone expected. And the hackers, by fruitful, meaningful association with the computer, would be foremost among the beneficiaries. But they would not be the only beneficiaries. Everyone could gain something by the use of thinking computers in an intellectually automated world. And wouldn't everyone benefit even more by approaching the world with the same inquisitive intensity, skepticism toward bureaucracy, openness to creativity, unselfishness in sharing accomplishments, urge to make improvements, and desire to build as those who followed the Hacker Ethic? By accepting others on the same unprejudiced basis by which computers accepted anyone who entered code into a Flexowriter? Wouldn't we benefit if we learned from computers the means of creating a perfect system? If EVERYONE could interact with computers with the same innocent, productive, creative impulse that hackers did, the Hacker Ethic might spread through society like a benevolent ripple, and computers would indeed change the world for the better. In the monastic confines of the Massachusetts Institute of Technology, people had the freedom to live out this dream--the hacker dream. No one dared suggest that the dream might spread. Instead, people set about building, right there at MIT, a hacker Xanadu the likes of which might never be duplicated. Hackers, Heroes of the Computer Revolution, by Steven Levy (C)1984 by Steven Levy 
929	----	The Real Cyberpunk Fakebook, by St. Jude, R.U. Sirius and Bart Nagel (C)1995 Ken Goffman and Jude Milhon This file contains the first three and half chapters. From Michael >>I changed a few spaces here and there to make it look better onscreen, >>let me know if you have any suggestions, corrections, additions, etc. From Jude orright, michael: i played with the formatting. it's HELL to make things look good in ASCII, but it looks bettah. *** okay, michael, stand back... here it comes.... i'm sending cybpunk fakebook as a MIME-encoded attachment AND a paste-in... look out.... it's terribly silly.... *** Dear Michael Hart and Project Gutenberg: This text comes over a little odd in ASCII. Like MONDO2000-- the zine we made infamous-- this book relies on its wacked layout and bizarre illustrations for much of its meaning, not to say charm. And it was difficult to figure what should be considered the first chapter, for obvious reasons. I think the first chapter really includes Section II, but never mind. Here it is, the beginning of... ****************************************************** ****************************************************** ****************************************************** ***** ***** THE ***** ***** REAL ***** ***** CYBERPUNK ***** ***** FAKEBOOK ***** ***** ***** ***** By St. Jude, R.U.Sirius, and Bart Nagel ***** ****************************************************** ****************************************************** ****************************************************** Dedication: For all our parents and lovers and housemates and children and friends, for the Cypherpunks, for Kevin Crow, Nesta Stubbs, The Omega, Phiber, and hackers everywhere. ======================================================== | | | INTRODUCTION to The Real Cyberpunk Fakebook | | by Bruce Sterling, | | A Renowned Cyberpunk Writer | | | ======================================================== I like this book so much that I'm thinking of changing my name to St.Erling. You couldn't ask for better guides to faking cyberpunk than these two utterly accomplished Bay Area fraudsters. These two characters are such consummate boho hustlers that they make Aleister Crowley look like Rebecca of Sunnybrook Farm. I don't believe in smart drugs, and I've never believed in smart drugs, but I do believe the following. It's genuinely useful to society to have some small, contained fraction of reckless fools who are willing to consume untested and unknown devices and substances. Sure, most of them will have their hearts explode or break out into great purple bleeding thalidomide warts. But who knows, maybe someday one of these jaspers will be eating handfuls of psychoactive crap out of some hippie pharmacy and he or she will suddenly learn to read Japanese in the original in six days. That's not at all likely, but it could happen-- grant me the possibility. The only drawback to this decentralized, libertarian, free-market regime of biomedical research is that you have to be ruthlessly prepared to sacrifice certain people-- just write 'em off, basically, like a cageful of control hamsters down at the NIMH. And if I ever met a man uniquely suited to this particular cutting-edge role in life, it is R.U. Sirius. R. U. Sirius basically resembles Gomez Addams in a purple fedora with an Andy Warhol badge pinned to the brim. The moment I met R.U., I felt a strong need to pith him and examine his viscera. I'm sure there are many other freelance biomedical researchers who will feel the same intellectual impulse. Read this book and you'll see what I mean. Then there's this saint person. Never draw to an inside straight. Never eat at a place called Mom's. And never eat a bag of ephedrine and a pumpkin pie ("the *whip* of vegetables!") from a California blonde who doesn't even have a real name. This female personage is so appallingly cagey that even her main squeeze delights in cryptographically baffling the NSA. If Pat Buchanan ever gets his not-so- secret wish and sets up a domestic American gulag for counterculture thought-criminals, the Judester's gonna be way, *way* up on the list-- maybe even number two, right after Bob Dobbs. Her trial's likely to prove rather interesting, however, as she only commits "crimes" in areas of social activity that haven't even been defined yet, much less successfully criminalized. A serious legal study of this woman's spectrum of activities would be like a CAT-scan of the American unconscious. There's also Bart Nagel, who is too nice a guy to be in the company of these people. Almost everything in this swell book is completely true. Except for everything about me. And my closest co- conspirators. We actual cyberpunks-- by this I mean *science fiction writers*, dammit, the people for whom the c-word was invented, the people who were professionally ahead of our time and were cyberpunks *twelve years ago*-- we never sneer and we never dress like, God forbid, Tom Wolfe. We just laugh at inappropriate times (like when testifying in Congress) and we dress and act just like industrial design professors. I hope this brief intro clears up any confusion. If you have any trouble at all with this book, take full advantage of your online d00dship and send email. Don't be afraid to ask "stupid" questions-- that's what the Internet is for! Ask nice, big, broad, open-ended questions. Stuff like "I'm doing a term paper so please tell me everything you know about cyberspace" or "I'm cyberpunk fan from Bulgaria and Enlgish not too good, but please say more what is about Virtual Reality?" Just don't send the email to me, of course. Send email to them. After this book, they deserve it! I feel sure that you'll get prompt answers that will surprise you. ............................................................. ............................................................. ======================================================== | | | The | | Authors | | Explain: | | | | A Technical Guide To This Technical Guide | | | ======================================================== WORDS IN BOLDFACE (enclosed in double <<angle braces>> for the ASCII version) These are terms that are defined in *Building Your Cyber Word Power*. Check there for anything that baffles you. Sometimes there's a double-anglebrace-enclosed term in the text that refers to a chapter subheading, and then you must practice your <<haqr smarts>> in order to find it. If all else fails, you could ask Bruce Sterling at his secret email address-- bruces@well.com. He will know. THE SHURIKEN AWARDS We may sometimes succumb to the temptation to rate things the way snotty critics do, by awarding stars. However, we will award them as *shuriken*, a cyber kinda star: ^ ^ ^ < X > < X > < X > v v v A shuriken is a throwing star-- a shiny-steel, sharp-edged, sharp-pointed weapon from Japan (which is cyberpunk's original home in certain misty urban legends). The shuriken itself as an assault weapon would rate one-half shuriken on a scale of four. A hydrogen bomb would rate five shuriken. You get the idea. Occasionally we may add Propeller Beanies to the Shuriken: <<<o>>> <<<o>>> <<<o>>> __|__ __|__ __|__ /_____\ /_____\ /_____\ This indicates nerdly interest over and above a cyberpunk rating. Propeller head is an ancient term for <<nerd>>. The real name for that key on the Macintosh is not COMMAND, but PROPELLER, and this is why. _________________ /| |\ | | || | | _/_ O || | | ( C O/ \O || | | \__/ O || | /\_________________/\| \/____________________\/ ========================================================= ======================================================= ===================================================== CONTENTS CONTENTS CONTENTS CONTENTS CONTENTS CONTENTS CONTENTS CONTENTS CONTENTS CONTENTS CONTENTS CONTENTS ===================================================== ======================================================= ========================================================= ****************************************************** *** SECTION I: CYBERPUNK... WHY?? OKAY-- HOW??? *** ****************************************************** Chapter 1/ CYBERPUNK: A CHALLENGING POSTMODERN LIFESTYLE! /Why Bother? Big Wins! (and Unexpected Smallstuff)/ Chapter 2/ ACHIEVING CYBERPUNK /Being It or Faking It/ Chapter 3/ A STYLE GUIDE TO THE CYBERTYPES /Recognizing Them and Fitting In/ ****************************************************** ***** SECTION II: CYBERPUNK... KNOWING ABOUT IT! ***** ****************************************************** Chapter 4/ BUILDING YOUR CYBER WORD POWER + A Dictionary of Terminally Hip Jargon and Useful Expressions + A Cyberpunk Phrasebook, with Hip Conversational Ploys for Winning Without a Clue Chapter 5/ CHEATCARDS FOR BOOKS YOU SHOULD HAVE READ /But Didn't/ Chapter 6/ CHEATCARDS FOR MOVIES/TV YOU DIDN'T SEE /But Should Know About/ Chapter 7/ ONLINE THINGS YOU SHOULD KNOW ABOUT /Even if You Never Go Online/ ****************************************************** ***** SECTION III: CYBERPUNK... DOING IT! ***** ****************************************************** Chapter 8/ ART OF THE HACK FOR BEGINNERS /A Child's First Book of Piracy, Intrusion & Espionage/ + Advice to Newbies + Haqr Mind, Haqr Smarts + Social Engineering for Fun and Profit Chapter 9/ THE HARDWARE/SOFTWARE YOU ACTUALLY NEED /Or How to Fake It/ + Computers, Modems, Encryption Programs + Plus Terminally Hip Extras: Laptops, Heads-up Displays, Personal Communicators, Pagers... Or + Realistic Balsa Mock-ups to Please Your Budget Chapter 10/ YOUR ONLINE PERSONA /How to Win Friends, Score Information, and Intrigue the Apposite Sex/ + Starting Out Right + Writing a Kewl dot.plan, + Designing a Non-lame dot.sig + Location, Location, Location-- What Your Eddress Says About YOU. + Beyond Attitude-- What??? + Netiquette + Art of the Flame + Online Poise: Cool in a MUD, Uncowed in a MOO Chapter 11/ HOW TO AVOID BANKRUPTCY /Sorry, that's just a little joke/ Chapter 12/ WHERE TO HANG /Finding the Cool Places in Cyberspace/ + Will the Net Kill Hacking? An Introductory Rant + IRCs, BBSes, MUDs, MOOs and MUSEs, Special Interest Groups, With a Special Word About alt.sex.bestiality ****************************************************** ***** SECTION IV: CYBERPUNK... THE SCENE ***** ****************************************************** Chapter 13/ FACE TIME /Pleased to Meat You?/ + Hacking Your Face2face IRL Persona + The Mandatory Black Leather Jacket + Leather Trousers? + Boots, Hair + Wearable Electronics: What's Chic, What's Rancid? And Buttons/Badges/Insignia, With a Special Warning About StarFleet Gear + Street Cred and Martial Arts Chapter 14/ TERMINALLY HIP WIDGETS /And High-Tech ToyZ/ + Fun With Your Cellular Phone + One Hundred Uses for Your Laser Pointer + Laminator 2: Identity Hacking + Why NOT to Buy a Stun Gun or a Nerve-Gas Dispenser or a TASER Chapter 15/ GAMES! + Video Games & Computer Games Fast-twitch Muscle Games, Exploration Games, Weird or X-Rated Games, Slacker Computer Games + Offline Games Magic, Hacker, The Glass Bead Game, DD&D Chapter 16/ CYBERPUNK LIFESTYLE HINTS /Trends, Faves and Hates/ + Interior Decorating Tips and Stylin Furnishings, Amusing Potted Plants, Stickers, Posters and Logos + What to Put on Top of Your Computer Monitor and Why + Nerd Comic Strips + Haqr Basic Diet, Stunt Foods & Intimidating Soft Drinks + Music That Doesn't Suck + Squeaky/Cuddly Toys With Really Good Rationalizations + Rubik's Hypercubes or Rubik's Dodecahedrons or Rubik's Other Strange Shapes and Hi-Tech Intellectual Adult Transformers In the Shape of Interlocked Rings, Chains, Blocks, Helices, and Platonic Solids That Shapeshift into Other Configurations of Rings, Chains, Etc Etc But Only If You Do Them Exactly Right, Which Is Very Difficult Or Impossible, but Which Gather Dust, Take Up Lots of Room On Your Monitor, and Taunt and Sneer at You Every Time You Look at Them. ****************************************************** ***** SECTION V: CYBERPUNK... THE INNER SCENE ***** ****************************************************** Chapter 17/ CYBERPUNK SECRETS REVEALED! (Yes, Just as We Promised-- REVEALED!) + Why Cyberpunks Seldom Have Their Organs Pierced + The Real Reason Why Cyberpunks Need to Encrypt Their Email + What Cyberpunks Are Doing at 3 AM in That Dumpster + Why Cyberpunks Avoid Altered States + Coping With Neurotoxins + Why Some Cyberpunks Love Star Trek Even Though It Sucks, + When Cyberpunks Always Diss What is Lame and Useless + What Cyberpunks Actually Admire, and Why + Secret Cyberpunk Handshaking, Signals and Head Motions Chapter 18/ CYBERPUNK: THE INNER GAME /The Tao of Punk; The Secret Dancing Masters of Cyber; And Everything You Wanted to Know About Cyber But Were Too Lame To Ask/ + The Hidden Hierarchy of Cyberpunk Revealed, from Bottom to Top Chapter 19/ CYBERPUNK: THE PARENTAL-DISCRETION SPECIAL /Sects and Politics... and Recipes/ + Disclaimer and Waiving of All Rights + Declaration of Age >> 21 and An Anti-Suicide Pact ******************************************************** *****SECTION VI: CYBERPUNK... ARE YOU CYBER ENOUGH?***** ******************************************************** Chapter 20/ IT *IS* AN INTELLIGENCE TEST! /Cyberpunk Skull-Tweakers and Fun Fare/ + The All-Cyber Cryptic Crossword Puzzle + Name That Nym! + Three-Letter-Acronyms From H.E.C.K. Cryptic Crossword Puzzles, Twisters and Max Headroom Memorial Rebuses Chapters 21/ 22/, and Of Course 23/ BOTTOM LINE TIME /Making It or Faking It/ A Cyberpunk Review to Prepare you, and then *****************THE FINAL EXAM ***************** It's Not True/False, We Don't Grade on the Curve, Stop Sniveling. ...................................................... ***** THE OFFICIAL CYBERPUNK HIPNESS CHECKLIST ***** You Won't Like This Either But It's for Your Own Good, Punk. ...................................................... ...................................................... ...................................................... ................-..That Is All..-..................... ...................................................... ++++++++++++++++++++++++++++++++++++++++++++++++++++++ APPENDIX A: Cyberpunk Valorized: Careers Under Deconstruction The Semiosis of Black Leather, Chrome, Mirrorshades and Modems ++++++++++++++++++++++++++++++++++++++++++++++++++++++ APPENDIX B: ASCII Charts ++++++++++++++++++++++++++++++++++++++++++++++++++++++ ...................................................... ...................................................... ...................................................... ...................................................... Now, Welcome to.the Text.............................. ****************************************************** ****************************************************** ****************************************************** ***** ***** THE ***** ***** REAL ***** ***** CYBERPUNK ***** ***** FAKEBOOK ***** ***** ***** ***** By St. Jude, R.U.Sirius, and Bart Nagel ***** ****************************************************** ****************************************************** ****************************************************** ++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++ + SECTION I: + + CYBERPUNK...WHY?? + + Okay... HOW??? + + + ++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++ ====================================================== Chapter 1: CYBERPUNK: A CHALLENGING POSTMODERN LIFESTYLE CHOICE Why Bother? Big Wins! (and Unexpected Smallstuff) ====================================================== Cyberpunk is extremely hip. Being extremely hip is the last hope for people with no money and no power. Being hip gets you big wins in the status game. Hipness can crush your enemies and attract the apposite sex. Best of all, cyberpunk is the next big thing AFTER the next big thing. You can hop on the cyberpunk bandwagon and coast for a long, long time. Think of the money you'll save on wardrobe updates! The worry you'll save on lifestyle decisions! Cyberpunk has not yet been co-opted. In fact, this handbook is the very first exploitation of this hip new underground trend. This is the ground floor. Get in on it! ====================================================== Chapter 2: ACHIEVING CYBERPUNK Being It or Faking It ====================================================== What is there to know about being a cyberpunk? Leather jacket, mirrorshades-- that just about does it, right? This kind of patronizing shirt must farking DIE.* *Since we can't afford to offend any parental units who might *purchase this book for their family circle, all chancy verbs *and nouns have been cleverly encyphered. This is in the *spirit of true cyberpunkhood, see <<Encryption>>. You think cyberpunk is just a leather jacket, some chrome studs, and fully reflective sunglasses? You think that's all there is? Hah! You can find those on Kansas City bikers and the whole California Highway Patrol. The true cyberpunk might tuck a *cellular-modem laptop* under a spiked leather arm, and a *laser pointer* in the upper zip pocket. Or, a true cyberpunk may look just like YOU. But sHe** who knows doesn't tell, and **hirm who tells doesn't know. **Pronoun disclaimer: **All pronouns in this book started life as intact males-- **he, his, and him. If anything bad happened to them **afterwards, blame it on the Riot Grrrls Bobbitt Squad. The lifestyle and goals of the true cyberpunk are carefully guarded secrets in a life *totally devoted* to coolness and secrecy. We will PIERCE THE VEIL, and REVEAL those SECRETS. We will display for you the INNER CYBERPUNK. We will give you everything you need to know about embarking on this challenging lifestyle. When you have read to the end of this EASY handbook, if you DON'T pass the hipness quiz... well, just read it again. But turn your TV up louder. ====================================================== Chapter 3: A STYLE GUIDE TO THE CYBERTYPES Recognizing Them and Fitting In ====================================================== While a cyberpunk is commonly a middleclass white male with way too many electrons, there are varieties of cyberpunk. Underlying all the types and genres is Basic Cyber Style, which breaks down to physical gear and mental attitude: --->Basic Cyberpunk Gear is simple. Black leather jacket. Boots. Mirrorshades. Laser pointer. (We don't know why all cyberpunks need a laser pointer, but it's mandatory.) We'll give you a more elaborate guide to basic cyberpunk gear. Later. --->The Basic Cyberpunk Attitude is quiet assurance. Subdued swashbuckling. Maybe a little menace. With these cyberpunk basics you can navigate through all the sub-genres. But if you want to pass as a native in a particular cyber sub-scene without getting jeered at or beaten up, you gotta accessorize, and pay close attention to detail. >>>>Motorpsycho Maniacs Cyberbikers pack the mystique of both worlds-- high tech, and big greasy loud engines. Standard cyberpunk costume is ideal for riding motorcycles, and a mirrorshades helmet is a big plus for the cyber look-- mega robotic coolness. Motorcycles are dangerous and can kill you. This is also cool. >>>>Goths, Deathcore, and Vampire-Wannabes Ideally, for this sub-scene, you should know about The Cure, which is a band. To fit in, grow your hair big and dye it blueblack. Spray it with <<Aquanet White>> to make it stick out, medusa-like. Go to a kidshop and buy plastic fangs. (The kind that glow in the dark are funny. Funny is NOT the object here.) All sexes should wear a Victorian shirt-- blouse-- white or black only--- that gapes to show flesh. You must practice looking tormented, tall and thin. The ideal is chalk-white face makeup with blueblack eyesockets. Blueblack makeup with white eyesockets is untested, but might work very well, if you avoid a minstrel look. At all times think intensity and torment. Torment...and ironic bitterness. No giggling or snickering, no kidding. >>>>Riot Grrrls! These are fierce girls who like tech. This is a sexist category, but there we are: girls only. A grrrl can be called "d00d" and "guy" at all times, but a non-female guy is not a grrrl. This is just the way things are. If you're a grrrl, you can wear anything you want to, because you're there to defend it. This is true for anybody, really-- look as tough as you wanta be, and be ready to back it up. Fierce is good. Grrrls with tech expertise are irresistible. NOTHING is more attractive than a fierce, blazing, ninja-type grrrl right now, and if she knows UNIX or phone-freeking, the world is hers. Hrrrs. >>>>Technopagans/Ravers/Neohippies Don't worry about this one. This scene is free, loving, noncomforming, spontaneous. You can dress any old way and fit right in... Unless you don't look cool. Maybe you should stick to basic cyberpunk. Dancing in leather is hot as h*ck, but sweating is better than not looking cool. Non-cyber ravers favor floppy hats, five kinds of plaid 'n' paisley, and multiple organ piercings. They sometimes take raver drugs. These drugs make you fonder of other people than you really want to be. (The morning- after Revulsion hangovers can be nasty.) In this scene, pretending to be on raver drugs is recommended, and easy, too. Unfocus your eyes and smile lovingly. In black leather you won't have to worry so much about getting hugged. >>>>Academic Cyber-Wannabes Students, teachers, whatever, dress down. Like you're always en route to a garage sale...maybe to donate what you're wearing. Casual. Jeans, black leather jacket, laser pointer. No tweed, notice, and no Birkenshtocken. If you flash paperbacks by Arthur Kroker, Paul Virilio and Jean Baudrillard, it means you're serious. Paperbacks by Mark Leyner and Kathy Acker means you're *way past* serious. >>>>Cybercowboys/grrrls Some of these people come from Texas or Oklahoma. In this crew, to yr cyberbasics you add a cowboy hat, cowboy boots, and grow any hair you've got really LONG. Males should try to get hair somewhere on their faces. >>>>Science Fiction Writers Full-steam straight-ahead hard edge, with a permanent sneer. Just to twist heads, some males writers go for the Tom Wolfe effete look-- blue blazer and wing-tips. Still they sneer. >>>>Web Crawlers and Other Bourgeois Types You don't really care about this one, do you? You do? Subscribe to WIRED. Next. >>>>Deep Geek: Supernerds, Hackers, Wizards, Phone Phreakers Things get difficult here. Deep geekware is unstandard. Very heavy Wizards can look like accountants, or like streetpeople. Facial hair and Goodwill Casual happen a lot. Chubby happens too, since these guys don't do enough dancing in leather pants. To get along in this scene, you really need to be very smart, very funny, or very sexy. To work yourself up to smart at least, learn UNIX. Or carry the 2600 zine in your back pocket and read that. Practice being technical. But until you get good, wear your cyberbasics and never leave home without your laser pointer. This will draw the admiration of people who don't know any better, which has its own rewards. Leading us inevitably to the final category... >>>>Phonies, Poseurs and Pretenders: Taking the Easy Way In Don't think: scheme! Forget about reading books, buy no computers or widgets. Don't do or buy anything. Save all your money for clothes and art materials. Make your girl/boyfriend help you assemble your hi-tek models-- you're gonna need mockups of a laptop computer, a personal communicator, a beeper, maybe even a fake stun-gun. Realism is key. Then wear them all with *attitude*. You're better than real. Strut. Sneer. Remember the 3 disses: distrust, disrespect, distroy. Wait, that's not right, is it? We know there are going to be mutterings about this category. Grumblings that being a poseur is not as easy as we think. A poseur has a lot of overhead-- in worry, just for starts-- what if you're exposed as <<a clueless>>? And staying locked to the HOTWIRED Website to catch what you should be imitating? Dang. ___ ___ | | | | [Photo of Billy Idol Goes Here] | | |__ __| ++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++ + SECTION II: + + CYBERPUNK... + + + + KNOWING ABOUT IT! + ++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++ ====================================================== Chapter 4: Building Your Cyber Word Power ====================================================== Part 1: A Dictionary of Terminally Hip Jargon and Useful Expressions XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX <<acronym>>: A word made from the initials of a name or phrase. Such as TLA. Three-Letter Acronym. Right. <<AFAIK>>: As far as I know. An <<acronym>>, in <<haqrese>>. <<AI>>: Artificial Intelligence. The next best thing to real. <<aka>>: Also known as. An acronym coined by the FBI in its popular Most Wanted lists. <<alliance>>: Among <<phreakers>>, a former AT&T trademark which refers to teleconferencing systems. <<anarchist>>: Somebody who feels that governments are an unreasonable restraint on free humans' being. <<anarcho-cryptographer>>: An anarchist who hopes to bring down the established order by persuading everybody to <<encrypt>> their email. <<anonymity>>: There's no handle like NO handle. Being completely unknown means you can't be traced. Maybe. You can be anonymous online by bouncing your email or postings through <<remailers>>. Who are you? Only penet.fi knows for sure. <<Aquanet White>>: This is the most intense hairspray on the planet, for that BIG <<goth>> hair. Since you're being so attentive, here's a bonus goth haiku: Sun! Hide white skin, run-- Burning, cloaked, I run... day sky!... Must... find... Aquanet <<ASCII>>: An <<acronym>> for... well, nobody remembers what it's an acronym for, but it means just plain keyboard characters, like your <<dot.sig>> is made of. This is a portrait of R.U. Sirius rendered in ascii art: � ################ # ____ # # __/=$==\_) # # //-OO-\\ # # >>( _ )<L # # J>>>\ /<<<L # � ################ So he comes out looking something like the cartoon character Cathy-- yeah, but that's the nature of the medium. St.Jude would look exactly the same, only no hat. Being subtle or elegant in ascii is a real challenge. <<attitude>>: Strutting. Sneering. Being BAD. Attitude is what all primates do to make their enemies feel inadequate. Keep it in mind. <<bah>>: Expresses the whole range of haqr negative emotions, from dysgruntlement up through horrible contempt, as in response to <<lameness>>. <<bahaha>>: A haqr evil laugh. Other common evil laughs are BYaa-hahah and pchtkwaaahahahaha. <<bang>>: Old haqr term for exclamation point. Sometimes bangs are a series of characters to add emphasis: w00t@%$%$@! <<BBS>>: A computerized bulletin-board system. Imagine a bulletin board in the sky. It's subdivided with topic labels. The cards displayed under each topic are email postings. You read them to follow the conversations. You can add your own comments or rebuttals. Some boards have a chat area where you can talk real-time, sort of like ham radio. The underground BBS chat areas are hangout places where bored hacker/phreaker types exchange quips and insults. Good H/P boards have libraries of up-to-date info on tools of the trade. <<beta release, in beta test>>: Not ready for prime time. This comes from the beta phase of program testing, when bugs are collected from patient users up for major <<lossage>>. "In beta" can describe anything unpleasing or forked up. If it's really <<FUBARed>>, it can be called ALPHA-release, which is software still being tested in-house, by programmers and unlucky affiliates. <<Big Room, The>>: Used to refer to the place you went OUT to, with one big bright light up there or else many small ones, you know? Now means the place you go INTO, the new Big Room-- Cyberspace. <<bogus>>: Untrue. Unreal. A spoof. Also, bogosity, which is the state of being bogus, and bogon, a unit of bogosity. Then there's the bogometer... <<boho>>: Bohemian. Means like, counter-cultural. Underground. Alternative, with people in black clothes. <<boxing>>: Using a gadget to get free phone calls. The Red Box plays the tones of coins registering in a pay phone. The Rainbow Box incorporates many previous boxes in one diabolical widget, thanks to our Dutch buddies. <<bridge>>: A <<hack>> into the phone company itself, allowing multiple <<phreakers>> to cross-talk, like a high-tech, illegal party line. Appropriating the phone company's own <<PBX>> systems is considered good <<phun>>. <<btw>>: By the way, in <<hackerese>>. <<carding>>: Making purchases on a phony or stolen credit account. The card as a physical chunk of plastic has become more or less irrelevant. <<celibate>>: Non-hacking hacker. Sometimes this is a haqr who has been <<newtered>>. <<clueless, a>>: (by analogy from "a homeless"??) One who doesn't get it and is doomed. <<codez>>: Phone numbers and authorization codes that allow you to make phree fonecalls. <<codez kid>>: A person whose purpose in life is finding ways to make phree fone calls. This is a terrible thing to call someone. Much worse than <<punk>>. <<coldlist>>: Shortlist for oblivion. By analogy to hotlist. <<Con>>: Convention, or maybe it was Conference-- nobody cares any more. A Con is a gathering of haqrz. There are several every year. The most famous is Hacking at the End of the Universe, held by the former Hac-tic in Amsterdam. Next is Hacking On Planet Earth, HOPE, and two infamous Cons are in Austin TX-- SummerCon, in the summer, and HoHoCon, figure it out. <<cookbook guyz>>: Haqr wannabes who don't figure out how to do things for th emselves. They copy down procedures for hacking computers or fones and follow the instructions, like using a cookbook. Everybody has to start out this way. Get over it. <<culture hacker>>: A pirate whose raw material is the society itself. A <<detournement>> specialist, who takes the <<memes>> of the culture on wild detours. <<cracker, kraqr>>: Somebody who breaks the copy protection on computer games or intrudes into other people's computers. Or invades cyberspace in strange ways. Or pirates any of the media. See <<pirating>> and <<hack>> and <<spoof>>. <<cryonics>>: Freezing your body (or just your head, in the budget plan) so that you can be revived (or provided with a whole-body transplant) at some time in the future when 1. they can do that sort of thing, and 2. they really WANT to do that sort of thing for frozen heads like you. See <<futurespoofing>>. <<cryptography, cryptology>>: Cryptology is the study of <<encryption>>. Cryptography is doing it. See <<public key encryption>> for a full rundown. <<cyberpunk>>: 1. A citizen of cyberspace. 2. A citizen of cyberspace who wears mirrorshades indoors, at night. <<cyberspace>>: The planetary Net linked by phonelines and satellites, whose nodes are computers and human beings. An online metaverse that's now realler than what's outside your window. <<cyber-yup>>: A tourist on the info highway. A <<Netcrawler>>, a Web Browser. If artists and nerds are sort of squatting or homesteading their homepages, cyber-yups create theirs as investment property. <<cypherpunk>>: A guerrilla in the war for privacy and lots more encryption>>. <<cypherpunks>>: A <<list>> of people interested in cryptology and cryptography. <<darqside>>: Antisocial. Evil. Weird. Someone who dares things you wouldn't, or couldn't. <<data>>: This is supposed to be plural. These data. If you don't wanna deal with that, see <<information>>. <<deck>>: What cyberpunks in NEUROMANCER typed on and jacked in through... like a keyboard with phonejacks that plug into YOU. <<deep geek>>: For the specs for deep geek, see Chapter Three. <<defect>>: <<Roll over>> to the <<narqside>>. <<detournement>>: Cultural hijacking. Taking something that has a usual meaning and making it play your way. A detournement is a cultural <<hack>>. <<dharma combat>>: Wrangling over standards and protocols, as in the <<IETF>>. Dharma is Buddhist for the principles of operation for the universe. Lots of <<deep geeks>> are Buddhists, Buddhist-wannabes, or jack-Buddhists. <<digital cash>>: The Philosopher's Stone of the nineties. Or maybe the Brooklyn Bridge. And good luck with it. <<diy>>: Do it yourself. A part of haqr mind, see Chapter Eight. <<domain name>>: This is part of the eddress that humans use. If you subscribe to an online service, like US Online, your domain eddress is theirs. Your whole eddress is whatever your handle is-- say skulldrool-- plus the server's domain name, like so: skulldrool@usol.com. Top level domain names are countries, like .au for australia, or categories, like .com, for company, .org for organization, .gov for the government, .mil for the military, etc. Domain names can be bought, and maybe they can be hacked. For example, if I had a military-industrial complex, I might hack an eddress like dark.satanic.mil. <<dot.plan>>: This is a file in your home directory within UNIX that people can read when they <<finger>> you. Your dot.plan file (actually it's just .plan) is where you put your advertisements for yourself. A typical dot.plan might start with a motto or a fave quote, such as, "In theory, there's no difference between theory and practice. But in practice, there is." Followed by as intriguing and flattering a profile of yourself as you can whomp together. <<dot.sig>>: This is your online signature, your digigraph, which you can tack onto the end of all your online appearances. A dot.sig is usually made up of thought-provoking quotes and ascii graphics. While somebody has to <<finger>> you to get your <<dot.plan>>, everybody is forced to see your dot.sig every time they read your postings or get email from you. Think of your dot.sig as a billboard advertising yourself. 
34268	----	Transcriber's note: A few typographical errors have been corrected: they are listed at the end of the text. * * * * * THE EARL OF PEMBROKE TO THE ABBESS OF WILTON. "Go spin, you jade! go spin!" [Illustration: MAGNETISM, LIGHT, AND MOLECULAR SPINNING TOPS. _Page 122._ _THE ROMANCE OF SCIENCE._ SPINNING TOPS. _THE "OPERATIVES' LECTURE"_ OF THE BRITISH ASSOCIATION MEETING AT LEEDS, 6th SEPTEMBER, 1890. BY PROFESSOR JOHN PERRY, M.E., D.Sc, LL.D., F.R.S. With Numerous Illustrations. _REPRINT OF NEW AND REVISED EDITION,_ _With an Illustrated Appendix on the Use of Gyrostats._ LONDON SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE, Northumberland Avenue, W.C.; 43, Queen Victoria Street, E.C. BRIGHTON: 129, North Street. NEW YORK: E. S. GORHAM. 1910 PUBLISHED UNDER THE DIRECTION OF THE GENERAL LITERATURE COMMITTEE [_Date of last impression, April 1908_] This Report of an Experimental Lecture WAS INSCRIBED TO THE LATE LORD KELVIN, BY HIS AFFECTIONATE PUPIL, THE LECTURER, WHO HEREBY TOOK A CONVENIENT METHOD OF ACKNOWLEDGING THE REAL AUTHOR OF WHATEVER IS WORTH PUBLICATION IN THE FOLLOWING PAGES. * * * * * PREFACE. This is not the lecture as it was delivered. Instead of two pages of letterpress and a woodcut, the reader may imagine that for half a minute the lecturer played with a spinning top or gyrostat, and occasionally ejaculated words of warning, admonition, and explanation towards his audience. A verbatim report would make rather uninteresting reading, and I have taken the liberty of trying, by greater fullness of explanation, to make up to the reader for his not having seen the moving apparatus. It has also been necessary in a treatise intended for general readers to simplify the reasoning, the lecture having been delivered to persons whose life experiences peculiarly fitted them for understanding scientific things. An "argument" has been added at the end to make the steps of the reasoning clearer. JOHN PERRY. * * * * * {9} SPINNING TOPS. -------- At a Leeds Board School last week, the master said to his class, "There is to be a meeting of the British Association in Leeds. What is it all about? Who are the members of the British Association? What do they do?" There was a long pause. At length it was broken by an intelligent shy boy: "Please, sir, I know--they spin tops!"[1] Now I am sorry to say that this answer was wrong. The members of the British Association and the Operatives of Leeds have neglected top-spinning since they were ten years of age. If more attention were paid to the intelligent examination of the behaviour of tops, there would be greater advances in mechanical engineering and a great many industries. There would be a better general knowledge of astronomy. Geologists would not make mistakes by millions of years, and our knowledge of Light, and Radiant Heat, and other {10} Electro-magnetic Phenomena would extend much more rapidly than it does. I shall try to show you towards the end of the lecture that the fact of our earth's being a spinning body is one which would make itself known to us even if we lived in subterranean regions like the coming race of an ingenious novelist.[2] It is the greatest and most persistent cause of many of the phenomena which occur around us and beneath us, and it is probable that even Terrestrial Magnetism is almost altogether due to it. Indeed there is only one possible explanation of the _Vril-ya_ ignorance about the earth's rotation. Their knowledge of mechanics and dynamics was immense; no member attending the meeting of the British Association can approach them in their knowledge of, I will not say, _Vril_, but even of quite vulgar electricity and magnetism; and yet this great race which expresses so strongly its contempt for Anglo-Saxon _Koom-Poshery_ was actually ignorant of the fact that it had existed for untold generations inside an object that spins about an axis. Can we imagine for one instant that the children of that race had never spun a top or trundled a hoop, and so had had no chance of being led to the greatest study of nature? No; the only possible explanation lies in the great novelist's never {11} having done these things himself. He had probably as a child a contempt for the study of nature, he was a baby Pelham, and as a man he was condemned to remain in ignorance even of the powers of the new race that he had created. The _Vril-ya_ ignorance of the behaviour of spinning bodies existing as it does side by side with their deep knowledge of magnetism, becomes even more remarkable when it comes home to us that the phenomena of magnetism and of light are certainly closely connected with the behaviour of spinning bodies, and indeed that a familiar knowledge of the behaviour of such bodies is absolutely necessary for a proper comprehension of most of the phenomena occurring in nature. The instinctive craving to investigate these phenomena seems to manifest itself soon after we are able to talk, and who knows how much of the intellectual inferiority of woman is due to her neglect of the study of spinning tops; but alas, even for boys in the pursuit of top-spinning, the youthful mind and muscle are left with no other guidance than that which is supplied by the experience of young and not very scientific companions. I remember distinctly that there were many puzzling problems presented to me every day. There were tops which nobody seemed able to spin, and there were others, well {12} prized objects, often studied in their behaviour and coveted as supremely valuable, that behaved well under the most unscientific treatment. And yet nobody, even the makers, seemed to know why one behaved badly and the other well. I do not disguise from myself the fact that it is rather a difficult task to talk of spinning tops to men who have long lost that skill which they wonder at in their children; that knowingness of touch and handling which gave them once so much power over what I fear to call inanimate nature. A problem which the child gives up as hopeless of solution, is seldom attacked again in maturer years; he drives his desire for knowledge into the obscure lumber-closets of his mind, and there it lies, with the accumulating dust of his life, a neglected and almost forgotten instinct. Some of you may think that this instinct only remains with those minds so many of which are childish even to the limit of life's span; and probably none of you have had the opportunity of seeing how the old dust rubs off from the life of the ordinary man, and the old desire comes back to him to understand the mysteries that surround him. But I have not only felt this desire myself, I have seen it in the excited eyes of the crowd of people who stand by the hour under the dropping cherry-blossoms beside the red-pillared temple of {13} Asakusa in the Eastern capital of Japan, watching the _tedzu-mashi_ directing the evolutions of his heavily rimmed _Koma_. First he throws away from him his great top obliquely into the air and catches it spinning on the end of a stick, or the point of a sword, or any other convenient implement; he now sends it about quite carelessly, catching it as it comes back to him from all sorts of directions; he makes it run up the hand-rail of a staircase into a house by the door and out again by the window; he makes it travel up a great corkscrew. Now he seizes it in his hands, and with a few dexterous twists gives it a new stock of spinning energy. He makes it travel along a stretched string or the edge of a sword; he does all sorts of other curious things with his tops, and suddenly sinks from his masterful position to beg for a few coppers at the end of his performance. How tame all this must seem to you who more than half forget your childish initiation into the mysteries of nature; but trust me, if I could only make that old top-spinner perform those magical operations of his on this platform, the delight of the enjoyment of beautiful motion would come back. Perhaps it is only in Japan that such an exhibition is possible; the land where the waving bamboo, and the circling hawk, and the undulating summer sea, and every beautiful motion of nature {14} are looked upon with tenderness; and perhaps it is from Japan that we shall learn the development of our childish enthusiasm. The devotees of the new emotional art of beautiful motion and changing colour are still in the main beggars like Homer, and they live in garrets like Johnson and Savage; but the dawn of a new era is heralded, or rather the dawn has already come, for Sir William Thomson's achievements in the study of spinning tops rank already as by no means the meanest of his great career. If you will only think of it, the behaviour of the commonest spinning top is very wonderful. When not spinning you see that it falls down at once, I find it impossible to balance it on its peg; but what a very different object it is when spinning; you see that it not only does not fall down, it offers a strange resistance when I strike it, and actually lifts itself more and more to an upright position. Once started on scientific observation, nature gives us facts of an analogous kind in great plenty. Those of you who have observed a rapidly moving heavy belt or rope, know that rapid motion gives a peculiar quasi-rigidity to flexible and even to fluid things. Here, for example, is a disc of quite thin paper (Fig. 1), and when I set it in rapid rotation you observe that it resists the force exerted by my {15} hand, the blow of my fist, as if it were a disc of steel. Hear how it resounds when I strike it with a stick. Where has its flexibility gone? [Illustration: FIG. 1.] Here again is a ring of chain which is quite flexible. It seems ridiculous to imagine that this {16} could be made to stand up like a stiff hoop, and yet you observe that when I give it a rapid rotation on this mandril and let it slide off upon the table, it runs over the table just as if it were a rigid ring, and when it drops on the floor it rebounds like a boy's hoop (Fig. 2). [Illustration: FIG. 2.] Here again is a very soft hat, specially made for this sort of experiment. You will note that it collapses to the table in a shapeless mass when I lay it down, and seems quite incapable of resisting forces which tend to alter its shape. In fact, there is almost a complete absence of rigidity; but when this is spun on the end of a stick, first note {17} how it has taken a very easily defined shape; secondly, note how it runs along the table as if it were made of steel; thirdly, note how all at once it collapses again into a shapeless heap of soft material when its rapid motion has ceased. Even so you will see that when a drunken man is not leaning against a wall or lamp-post, he feels that his only chance of escape from ignominious collapse is to get up a decent rate of speed, to obtain a quasi-sobriety of demeanour by rapidity of motion. The water inside this glass vessel (Fig. 3) is in a state of rapid motion, revolving with the vessel itself. Now observe the piece of paraffin wax A immersed in the water, and you will see when I push at it with a rod that it vibrates just as if it were surrounded with a thick jelly. Let us now apply Prof. Fitzgerald's improvement on this experiment of Sir William Thomson's. Here is a disc B stuck on the end of the rod; observe that when I introduce it, although it does not touch A, A is repelled from the disc. Now observe that when I twirl the disc it seems to attract A. [Illustration: FIG. 3.[3]] At the round hole in front of this box a rapid motion is given to a small quantity of air which is mixed with smoke that you may see it. That smoke-ring moves through the air almost like a solid body for a considerable distance unchanged, and I am not sure that it may not be possible yet {18} to send as a projectile a huge poisoned smoke-ring, so that it may destroy or stupefy an army miles away. Remember that it is really the same air all the time. You will observe that two smoke rings sent from two boxes have curious actions {19} upon one another, and the study of these actions has given rise to Thomson's smoke-ring or vortex theory of the constitution of matter (Fig. 4). [Illustration: FIG. 4.] It was Rankine, the great guide of all engineers, who first suggested the idea of molecular vortices in his explanations of heat phenomena and the phenomena of elasticity--the idea that every particle of matter is like a little spinning top; but I am now speaking of Thomson's theory. To imagine that an atom of matter is merely a {20} curiously shaped smoke-ring formed miraculously in a perfect fluid, and which can never undergo permanent alteration, looks to be a very curious and far-fetched hypothesis. But in spite of certain difficulties, it is the foundation of the theory which will best explain most of the molecular phenomena observed by philosophers. Whatever be the value of the theory, you see from these experiments that motion does give to small quantities of fluid curious properties of elasticity, attraction and repulsion; that each of these entities refuses to be cut in two; that you cannot bring a knife even near the smoke-ring; and that what may be called a collision between two of them is not very different in any way from the collision between two rings of india-rubber. Another example of the rigidity given to a fluid by rapid motion, is the feeling of utter helplessness which even the strongest swimmers sometimes experience when they get caught in an eddy underneath the water. I could, if I liked, multiply these instances of the quasi-rigidity which mere motion gives to flexible or fluid bodies. In Nevada a jet of water like the jet from a fireman's hose, except that it is much more rapid, which is nearly as easily projected in different directions, is used in mining, and huge masses of earth and rock are rapidly disintegrated {21} by the running water, which seems to be rather like a bar of steel than a jet of water in its rigidity. It is, however, probable that you will take more interest in this box of brass which I hold in my hands. You see nothing moving, but really, inside this case there is a fly-wheel revolving rapidly. Observe that I rest this case on the table on its sharp edge, a sort of skate, and it does not tumble down as an ordinary box would do, or as this box will do after a while, when its contents come to rest. Observe that I can strike it violent blows, and it does not seem to budge from its vertical position; it turns itself just a little round, but does not get tilted, however hard I strike it. Observe that if I do get it tilted a little it does not fall down, but slowly turns with what is called a precessional motion (Fig. 5). You will, I hope, allow me, all through this lecture, to use the term _precessional_ for any motion of this kind. Probably you will object more strongly to the great liberty I shall take presently, of saying that the case _precesses_ when it has this kind of motion; but I really have almost no option in the matter, as I must use some verb, and I have no time to invent a less barbarous one. [Illustration: FIG. 5.] When I hold this box in my hands (Fig. 6), I find that if I move it with a motion of mere translation in any direction, it feels just as it would do {22} if its contents were at rest, but if I try to turn it in my hands I find the most curious great resistance to such a motion. The result is that when you hold this in your hands, its readiness to move so long as it is not turned round, and its great resistance to turning round, and its unexpected tendency to turn in a different way from that in which you try to turn it, give one the most uncanny sensations. It seems almost as if an invisible being had hold of the box and exercised forces capriciously. And {23} indeed there is a spiritual being inside, what the algebraic people call an impossible quantity, what other mathematicians call "an operator." [Illustration: FIG. 6.] Nearly all the experiments, even the tops and other apparatus you have seen or will see to-night, have been arranged and made by my enthusiastic assistant, Mr. Shepherd. The following experiment is not only his in arrangement; even the idea of it is his. He said, you may grin and contort your body with that large gyrostat in your hands, but many of your audience will simply say to {24} themselves that you only _pretend_ to find a difficulty in turning the gyrostat. So he arranged this pivoted table for me to stand upon, and you will observe that when I now try to turn the gyrostat, it will not turn; however I may exert myself, it keeps pointing to that particular corner of the room, and all my efforts only result in turning round my own body and the table, but not the gyrostat. Now you will find that in every case this box only resists having the axis of revolution of its hidden flywheel turned round, and if you are interested in the matter and make a few observations, you will soon see that every spinning body like the fly-wheel inside this case resists more or less the change of direction of its spinning axis. When the fly-wheels of steam-engines and dynamo machines and other quick speed machines are rotating on board ship, you may be quite sure that they offer a greater resistance to the pitching or rolling or turning of the ship, or any other motion which tends to turn their axes in direction, than when they are not rotating. Here is a top lying on a plate, and I throw it up into the air; you will observe that its motion is very difficult to follow, and nobody could predict, before it falls, exactly how it will alight on the plate; it may come down peg-end foremost, or hindmost, or sideways. But when I spin it (Fig. 7), and now throw it up into the air, there is no doubt whatever {25} as to how it will come down. The spinning axis keeps parallel to itself, and I can throw the top up time after time, without disturbing much the spinning motion. [Illustration: FIG. 7.] [Illustration: FIG. 8.] If I pitch up this biscuit, you will observe that I can have no certainty as to how it will come down, but if I give it a spin before it leaves my hand there is no doubt whatever (Fig. 8). Here is a hat. I throw it up, and I cannot be sure as to how it will move, but if I give it a spin, you see that, as {26} with the top and the biscuit, the axis about which the spinning takes place keeps parallel to itself, and we have perfect certainty as to the hat's alighting on the ground brim downwards (Fig. 9). [Illustration: FIG. 9.] I need not again bring before you the very soft hat to which we gave a quasi-rigidity a few minutes ago; but you will remember that my assistant sent that off like a projectile through the air when it was spinning, and that it kept its spinning axis parallel to itself just like this more rigid hat and the biscuit. [Illustration: FIG. 10.] [Illustration: FIG. 11.] I once showed some experiments on spinning tops to a coffee-drinking, tobacco-smoking audience in that most excellent institution, the Victoria Music Hall in London. In that music hall, things are not very different from what they are at any other {27} music hall except in beer, wine, and spirits being unobtainable, and in short scientific addresses being occasionally given. Now, I impressed my audience as strongly as I could with the above fact, that if one wants to throw a quoit with certainty as to how it will alight, one gives it a spin; if one wants to throw a hoop or a hat to somebody to catch upon a stick, one gives the hoop or hat a spin; the disinclination of a spinning body to let its axis get altered in direction can always be depended upon. I told them that this was why smooth-bore guns cannot be depended upon for accuracy;[4] that the spin which an ordinary bullet took depended greatly on how it chanced to touch the muzzle as it just left the gun, whereas barrels are now rifled, that is, spiral grooves are now cut inside the barrel of a gun, and excrescences from the bullet or projectile fit into these grooves, so that as it is forced along the barrel of the gun by the explosive force of the powder, it must also spin about its axis. Hence it leaves the gun with a perfectly well-known spinning motion about which there can be no doubt, and we know too that Fig. 10 shows the {28} kind of motion which it has afterwards, for, just like the hat or the biscuit, its spinning axis keeps nearly parallel to itself. Well, this was all I could do, for I am not skilful in throwing hats or quoits. But after my address was finished, and after a young lady in a spangled dress had sung a comic song, two jugglers came upon the stage, and I could not have had better illustrations of the above principle than were given in almost every trick performed by this lady and gentleman. They sent hats, and hoops, and plates, and umbrellas spinning from one to the other. One of them threw a stream of knives into the air, catching them and throwing them up again with perfect precision and my now educated audience shouted with delight, and showed in other unmistakable {29} ways that they observed the spin which that juggler gave to every knife as it left his hand, so that he might have a perfect knowledge as to how it would come back to him again (Fig. 11). {30} It struck me with astonishment at the time that, almost without exception, every juggling trick performed that evening was an illustration of the above principle. And now, if you doubt my statement, just ask a child whether its hoop is more likely to tumble down when it is rapidly rolling along, or when it is going very slowly; ask a man on a bicycle to go more and more slowly to see if he keeps his balance better; ask a ballet-dancer how long she could stand on one toe without balancing herself with her arms or a pole, if she were not spinning; ask astronomers how many months would elapse before the earth would point ever so far away from the pole star if it were not spinning; and above all, ask a boy whether his top is as likely to stand upright upon its peg when it is not spinning as when it is spinning. [Illustration: FIG. 12.] We will now examine more carefully the behaviour of this common top (Fig. 12). It is not {31} spinning, and you observe that it tumbles down at once; it is quite unstable if I leave it resting upright on its peg. But now note that when it is spinning, it not only will remain upright resting on its peg, but if I give it a blow and so disturb its state, it goes circling round with a precessional motion which grows gradually less and less as time goes on, and the top lifts itself to the upright position again. I hope you do not think that time spent in careful observation of a phenomenon of this kind is wasted. Educated observation of the commonest phenomena occurring in our everyday life is never wasted, and I often feel that if workmen, who are the persons most familiar with inorganic nature, could only observe and apply simple scientific laws to their observations, instead of a great discovery every century we should have a great discovery every year. Well, to return to our top; there are two very curious observations to make. Please neglect for a short time the slight wobbling motions that occur. One observation we make is, that the top does not at first bow down in the direction of the blow. If I strike towards the south, the top bows towards the west; if I strike towards the west, the top bows down towards the north. Now the reason of this is known to all scientific men, and the principle underlying the top's behaviour is of very great {32} importance in many ways, and I hope to make it clear to you. The second fact, that the top gradually reaches its upright position again, is one known to everybody, but the reason for it is not by any means well known, although I think that you will have no great difficulty in understanding it. The first phenomenon will be observed in this case which I have already shown you. This case (Fig. 5), {33} with the fly-wheel inside it, is called a _gyrostat_. When I push the case it does not bow down, but slowly turns round. This gyrostat will not exhibit the second phenomenon; it will not rise up again if I manage to get it out of its upright position, but, on the contrary, will go precessing in wider and wider circles, getting further and further away from its upright position. [Illustration: FIG. 13.] [Illustration: FIG. 14.] The first phenomenon is most easily studied in this balanced gyrostat (Fig. 13). You here see the fly-wheel G in a strong brass frame F, which is supported so that it is free to move about the vertical axis A B, or about the horizontal axis C D. The gyrostat is balanced by a weight W. Observe that I can increase the leverage of W or diminish it by shifting the position of the sleeve at A so that it will tend to either lift or lower the gyrostat, or exactly balance it as it does now. You must observe exactly what it is that we wish to study. If I endeavour to push F downwards, with the end of this stick (Fig. 14), it really moves horizontally to the right; now I push it to the right (Fig. 15), and it only rises; now push it up, and you see that it goes to the left; push it to the left, and it only goes downwards. You will notice that if I clamp the instrument so that it cannot move vertically, it moves at once horizontally; if I prevent mere horizontal motion it readily moves vertically when I push it. Leaving it free as {34} before, I will now shift the position of the weight W, so that it tends continually to lift the gyrostat, and of course the instrument does not lift, it moves horizontally with a slow precessional motion. I now again shift the weight W, so that the gyrostat would fall if it were not spinning (Fig. 16), and it now moves horizontally with a slow precessional motion which is in a direction opposed to the last. These phenomena are easily explained, but, {35} as I said before, it is necessary first to observe them carefully. You all know now, vaguely, the fundamental fact. It is that if I try to make a very quickly spinning body change the direction of its axis, the direction of the axis will change, but not in the way I intended. It is even more curious than my countryman's pig, for when he wanted the pig to go to Cork, he had to pretend that he was driving the pig home. His rule was a very {36} simple one, and we must find a rule for our spinning body, which is rather like a crab, that will only go along the road when you push it sidewise. [Illustration: FIG. 15.] [Illustration: FIG. 16.[5]] [Illustration: FIG. 10.] As an illustration of this, consider the spinning projectile of Fig. 10. The spin tends to keep its axis always in the same direction. But there is a defect in the arrangement, which you are now in a {37} position to understand. You see that at A the air must be pressing upon the undersurface A A, and I have to explain that this pressure tends to make the projectile turn itself broadside on to the air. A boat in a current not allowed to move as a whole, but tied at its middle, sets itself broadside on to the current. Observe this disc of cardboard which I drop through the air edgewise, and note how quickly it sets itself broadside on and falls more slowly; and some of you may have thrown over into the water at Aden small pieces of silver for the diving boys, and you are aware that if it were not for this slow falling of the coins with a wobbling motion broadside on, it would be nearly impossible for any diving boy to get possession of them. Now all this is a parenthesis. The {38} pressure of the air tends to make the projectile turn broadside on, but as the projectile is spinning it does not tilt up, no more than this gyrostat does when I try to tilt it up, it really tilts out of the plane of the diagram, out of the plane of its flight; and only that artillerymen know exactly what it will do, this kind of _windage_ of the projectile would give them great trouble. You will notice that an experienced child when it wants to change the direction of a hoop, just exerts a tilting pressure with its hoop-stick. A man on a bicycle changes his direction by leaning over so as to be out of balance. It is well to remind you, however, that the motion of a bicycle and its rider is not all rotational, so that it is not altogether the analogue of a top or gyrostat. The explanation of the swerving from a straight path when the rider tilts his body, ultimately comes to the same simple principle, Newton's second law of motion, but it is arrived at more readily. It is for the same reason--put briefly, the exercise of a centripetal force--that when one is riding he can materially assist his horse to turn a corner quickly, if he does not mind appearances, by inclining his body towards the side to which he wants to turn; and the more slowly the horse is going the greater is the tendency to turn for a given amount of tilting of one's body. Circus-riders, when galloping in a circle, assist their horses greatly by the position of their bodies; it is {39} not to save themselves from falling by centrifugal force that they take a position on a horse's back which no riding-master would allow his pupil to imitate; and the respectable riders of this country would not scorn to help their horses in this way to quick turning movements, if they had to chase and collect cattle like American cowboys. Very good illustrations of change of direction are obtained in playing _bowls_. You know that a bowl, if it had no _bias_, that is, if it had no little weight inside it tending to tilt it, would roll along the level bowling-green in a straight path, its speed getting less and less till it stopped. As a matter of fact, however, you know that at the beginning, when it is moving fast, its path is pretty straight, but because it always has bias the path is never quite straight, and it bends more and more rapidly as the speed diminishes. In all our examples the slower the spin the quicker is the precession produced by given tilting forces. Now close observation will give you a simple rule about the behaviour of a gyrostat. As a matter of fact, all that has been incomprehensible or curious disappears at once, if instead of speaking of this gyrostat as moving up or down, or to the right or left, I speak of its motions about its various axes. It offers no resistance to mere motion of translation. But when I spoke of its moving {40} horizontally, I ought to have said that it moved about the vertical axis A B (Fig. 13). Again, what I referred to as up and down motion of F is really motion in a vertical plane about the horizontal axis C D. In future, when I speak of trying to give motion to F, think only of the axis about which I try to turn it, and then a little observation will clear the ground. [Illustration: FIG. 17.] [Illustration: FIG. 18.] Here is a gyrostat (Fig. 17), suspended in gymbals so carefully that neither gravity nor any frictional forces at the pivots constrain it; nothing that I can do to this frame which I hold in my hand will affect the direction of the axis E F of the gyrostat. Observe that I whirl round on my toes like a ballet-dancer while this is in my hand. I move it about in all sorts of ways, but if it was pointing to the pole star at the beginning it remains pointing to the pole star; if it pointed towards the moon at the beginning it still points {41} towards the moon. The fact is, that as there is almost no frictional constraint at the pivots there are almost no forces tending to turn the axis of rotation of the gyrostat, and I can only give it motions of translation. But now I will clamp this vertical spindle by means of a screw and repeat my ballet-dance whirl; you will note that I need not whirl round, a very small portion of a whirl is enough to cause this gyrostat (Fig. 18) to set its spinning axis vertical, to set its axis parallel to the vertical axis of rotation which I give it. Now I whirl in the opposite direction, the gyrostat at once turns a somersault, turns completely round and remains again with its axis vertical, and if you were to carefully note the direction of the spinning of the {42} gyrostat, you would find the following rule to be generally true:--Pay no attention to mere translational motion, think only of rotation about axes, and just remember that when you constrain the axis of a spinning body to rotate, it will endeavour to set its own axis parallel to the new axis about which you rotate it; and not only is this the case, but it will endeavour to have the direction of its own spin the same as the direction of the new rotation. I again twirl on my toes, holding this frame, and now I know that to a person looking down upon the gyrostat and me from the ceiling, as I revolved in the direction of the hands of a clock, the gyrostat is spinning in the direction of the hands of a clock; but if I revolve against the clock direction (Fig. 19) the gyrostat tumbles over so as again to be revolving in the same direction as that in which I revolve. [Illustration: FIG. 19.] This then is the simple rule which will enable you to tell beforehand how a gyrostat will move {43} when you try to turn it in any particular direction. You have only to remember that if you continued your effort long enough, the spinning axis would become parallel to your new axis of motion, and the direction of spinning would be the same as the direction of your new turning motion. Now let me apply my rule to this balanced gyrostat. I shove it, or give it an impulse downwards, but observe that this really means a rotation about the horizontal axis C D (Fig. 13), and hence the gyrostat turns its axis as if it wanted to become parallel to C D. Thus, looking down from above (as shown by Fig. 20), O E was the direction of the spinning axis, O D was the axis about which I endeavoured to move it, and the instantaneous effect was that O E altered to the position O G. A greater impulse of the same kind would have caused the spinning axis instantly to go to O H or O J, whereas an upward opposite impulse would have instantly made the spinning axis point in the direction O K, O L or O M, depending on how great the impulse was and the rate of spinning. When one observes these phenomena for the first time, one says, "I shoved it down, and it moved to the right; I shoved it up, and it moved to the left;" but if the direction of the spin were opposite to what it is, one would say, "I shoved it down, and it moved to the left; I shoved it up, and it moved to the right." The simple {44} statement in all cases ought to be, "I wanted to rotate it about a new axis, and the effect was to send its spinning axis towards the direction of the new axis." And now if you play with this balanced gyrostat as I am doing, shoving it about in all sorts of ways, you will find the rule to be a correct one, and there is no difficulty in predicting what will happen. [Illustration: FIG. 20.] {45} If this rule is right, we see at once why precession takes place. I put this gyrostat (Fig. 13) out of balance, and if it were not rotating it would fall downwards; but a force acting downwards really causes the gyrostat to move to the right, and so you see that it is continually moving in this way, for the force is always acting downwards, and the spinning axis is continually chasing the new axes about which gravity tends continually to make it revolve. We see also why it is that if the want of balance is the other way, if gravity tends to lift the gyrostat, the precession is in the opposite direction. And in playing with this gyrostat as I do now, giving it all sorts of pushes, one makes other observations and sees that the above rule simplifies them all; that is, it enables us to remember them. For example, if I use this stick to hurry on the precession, the gyrostat moves in opposition to the force which causes the precession. I am particularly anxious that you should remember this. At present the balance-weight is so placed that the gyrostat would fall if it were not spinning. But it is spinning, and so it precesses. If gravity were greater it would precess faster, and it comes home to us that it is this precession which enables the force of gravity to be inoperative in mere downward motion. You see that if the precession is hurried, it is more than sufficient to balance gravity, {46} and the gyrostat rises. If I retard the precession, it is unable to balance gravity, and the gyrostat falls. If I clamp this vertical axis so that precession is impossible, you will notice that the gyrostat falls just as if it were not spinning. If I clamp the instrument so that it cannot move vertically, you notice how readily I can make it move horizontally; I can set it rotating horizontally like any ordinary body. In applying our rule to this top, observe that the axis of spinning is the axis E F of the top (Fig. 12). As seen in the figure, gravity is tending to make the top rotate about the axis F D, and the spinning axis in its chase of the axis F D describes a cone in space as it precesses. This gyrostat, which is top-heavy, rotates and precesses in much the same way as the top; that is, if you apply our rule, or use your observation, you will find that to an observer above the table the spinning and precession occur in the same direction, that is, either both with the hands of a watch, or both against the hands of a watch. Whereas, a top like this before you (Fig. 21), supported above its centre of gravity, or the gyrostat here (Fig. 22), which is also supported above its centre of gravity, or the gyrostat shown in Fig. 56, or any other gyrostat supported in such a way that it would be in stable equilibrium if it were not spinning; in all these {47} cases, to an observer placed above the table, the precession is in a direction opposite to that of the spinning. [Illustration: FIG. 21.] [Illustration: FIG. 22.] {48} If an impulse be given to a top or gyrostat in the direction of the precession, it will rise in opposition to the force of gravity, and should at any instant the precessional velocity be greater than what it ought to be for the balance of the force of gravity, the top or gyrostat will rise, its precessional velocity diminishing. If the precessional velocity is too small, the top will fall, and as it falls the precessional velocity increases. Now I say that all these facts, which are mere facts of observation, agree with our rule. I wish I dare ask you to remember them all. You will observe that in this wall sheet I have made a list of them. I speak of gravity as causing the precession, but the forces may be any others than such as are due to gravity. WALL SHEET. I. RULE. When forces act upon a spinning body, tending to cause rotation about any other axis than the spinning axis, the spinning axis sets itself in better agreement with the new axis of rotation. Perfect agreement would mean perfect parallelism, the directions of rotation being the same. II. Hurry on the precession, and the body rises in opposition to gravity. {49} III. Delay the precession and the body falls, as gravity would make it do if it were not spinning. IV. A common top precesses in the same direction as that in which it spins. V. A top supported above its centre of gravity, or a body which would be in stable equilibrium if not spinning, precesses in the opposite direction to that of its spinning. VI. The last two statements come to this:--When the forces acting on a spinning body tend to make the _angle_ of precession greater, the precession is in the same direction as the spinning, and _vice versâ_. Having by observation obtained a rule, every natural philosopher tries to make his rule a rational one; tries to explain it. I hope you know what we mean when we say that we explain a phenomenon; we really mean that we show the phenomenon to be consistent with other better known phenomena. Thus when you unmask a spiritualist and show that the phenomena exhibited by him are due to mere sleight-of-hand and trickery, you explain the phenomena. When you show that they are all consistent with well-observed and established mesmeric influences, you are also said to explain the phenomena. When you show that they can be effected by means of telegraphic messages, or by reflection of light from mirrors, you explain the {50} phenomena, although in all these cases you do not really know the nature of mesmerism, electricity, light, or moral obliquity. The meanest kind of criticism is that of the man who cheapens a scientific explanation by saying that the very simplest facts of nature are unexplainable. Such a man prefers the chaotic and indiscriminate wonder of the savage to the reverence of a Sir Isaac Newton. [Illustration: FIG. 23.] The explanation of our rule is easy. Here is a gyrostat (Fig. 23) something like the earth in shape, and it is at rest. I am sorry to say that I am compelled to support this globe in a very visible manner by gymbal rings. If this globe were just floating in the air, if it had no tendency to fall, my explanation would be easier to understand, and I could illustrate it better experimentally. Observe the point P. If I move the globe slightly about the axis A, the point P moves to Q. But suppose instead of this that the globe and inner gymbal {51} ring had been moved about the axis B; the point P would have moved to R. Well, suppose both those rotations took place simultaneously. You all know that the point P would move neither to Q nor to R, but it would move to S; P S being the diagonal of the little parallelogram. The resultant motion then is neither about the axis O A in space, nor about the axis O B, but it is about some such axis as O C. To this globe I have given two rotations simultaneously. Suppose a little being to exist on this globe which could not see the gymbals, but was able to observe other objects in the room. It would say that the direction of rotation is neither about O A nor about O B, but that the real axis of its earth is some line intermediate, O C in fact. If then a ball is suddenly struck in two different directions at the same instant, to understand how it will spin we must first find how much spin each blow would produce if it acted alone, and about what axis. A spin of three turns per second about the axis O A (Fig. 24), and a spin of two turns per second about the axis O B, really mean that the ball will spin about the axis O C with a spin of three and a half turns per second. To arrive at this result, I made O A, 3 feet long (any other scale of representation would have been right) {52} and O B, 2 feet long, and I found the diagonal O C of the parallelogram shown on the figure to be 3½ feet long. Observe that if the rotation about the axis O A is _with_ the hands of a watch looking from O to A, the rotation about the axis O B looking from O to B, must also be with the hands of a watch, and the resultant rotation about the axis O C is also in a direction with the hands of a watch looking from O to C. Fig. 25 shows in two diagrams how necessary it is that on looking from O along either O A or O B, the rotation should be in the same direction as regards the hands of a watch. These constructions are well known to all who have studied elementary mechanical principles. Obviously if the rotation about O A is very much greater than the rotation about O B, then the position of the new axis O C must be much nearer O A than O B. [Illustration: FIG. 24.] [Illustration: FIG. 25.] We see then that if a body is spinning about an axis O A, and we apply forces to it which {53} would, if it were at rest, turn it about the axis O B; the effect is to cause the spinning axis to be altered to O C; that is, the spinning axis sets itself in better agreement with the new axis of rotation. This is the first statement on our wall sheet, the rule from which all our other statements are derived, assuming that they were not really derived from observation. Now I do not say that I have here given a complete proof for all cases, for the fly-wheels in these gyrostats are running in bearings, and the bearings constrain the axes to take the new positions, whereas there is no such {54} constraint in this top; but in the limited time of a popular lecture like this it is not possible, even if it were desirable, to give an exhaustive proof of such a universal rule as ours is. That I have not exhausted all that might be said on this subject will be evident from what follows. If we have a spinning ball and we give to it a new kind of rotation, what will happen? Suppose, for example, that the earth were a homogeneous sphere, and that there were suddenly impressed upon it a new rotatory motion tending to send Africa southwards; the axis of this new spin would have its pole at Java, and this spin combined with the old one would cause the earth to have its true pole somewhere between the present pole and Java. It would no longer rotate about its present axis. In fact the axis of rotation would be altered, and there would be no tendency for anything further to occur, because a homogeneous sphere will as readily rotate about one axis as another. But if such a thing were to happen to this earth of ours, which is not a sphere but a flattened spheroid like an orange, its polar diameter being the one-third of one per cent. shorter than the equatorial diameter; then as soon as the new axis was established, the axis of symmetry would resent the change and would try to become again the axis of rotation, and a great wobbling motion would ensue. {55} I put the matter in popular language when I speak of the resentment of an axis; perhaps it is better to explain more exactly what I mean. I am going to use the expression Centrifugal Force. Now there are captious critics who object to this term, but all engineers use it, and I like to use it, and our captious critics submit to all sorts of ignominious involution of language in evading the use of it. It means the force with which any body acts upon its constraints when it is constrained to move in a curved path. The force is always directed away from the centre of the curve. When a ball is whirled round in a curve at the end of a string its centrifugal force tends to break the string. When any body keyed to a shaft is revolving with the shaft, it may be that the centrifugal forces of all the parts just balance one another; but sometimes they do not, and then we say that the shaft is out of balance. Here, for example, is a disc of wood rotating. It is in balance. But I stop its motion and fix this piece of lead, A, to it, and you observe when it rotates that it is so much out of balance that the bearings of the shaft and the frame that holds them, and even the lecture-table, are shaking. Now I will put things in balance again by placing another piece of lead, B, on the side of the spindle remote from A, and when I again rotate the disc (Fig. 26) there {56} is no longer any shaking of the framework. When the crank-shaft of a locomotive has not been put in balance by means of weights suitably placed on the driving-wheels, there is nobody in the train who does not feel the effects. Yes, and the coal-bill shows the effects, for an unbalanced engine tugs the train spasmodically instead of exerting an efficient steady pull. My friend Professor Milne, of Japan, places earthquake measuring instruments on engines and in trains for measuring this and other wants of balance, and he has shown unmistakably that two engines of nearly the same general design, one balanced properly and the other not, consume very different amounts of coal in making the same journey at the same speed. [Illustration: FIG. 26.] If a rotating body is in balance, not only does the axis of rotation pass through the centre of gravity (or rather centre of mass) of the body, but {57} the axis of rotation must be one of the three principal axes through the centre of mass of the body. Here, for example, is an ellipsoid of wood; A A, B B, and C C (Fig. 27) are its three principal axes, and it would be in balance if it rotated about any one of these three axes, and it would not be in balance if it rotated about any other axis, unless, indeed, it were like a homogeneous sphere, every diameter of which is a principal axis. [Illustration: FIG. 27.] Every body has three such principal axes through its centre of mass, and this body (Fig. 27) has them; but I have here constrained it to rotate about the axis D D, and you all observe the effect of the unbalanced centrifugal forces, which is nearly great enough to tear the framework in pieces. The higher the speed the more important this want of balance is. If the speed is doubled, the centrifugal forces become four times as great; and modern mechanical engineers with their quick speed engines, some of which revolve, like the fan-engines of torpedo-boats, at 1700 revolutions per minute, require to pay great attention to this subject, which the older engineers never troubled their {58} heads about. You must remember that even when want of balance does not actually fracture the framework of an engine, it will shake everything, so that nuts and keys and other fastenings are pretty sure to get loose. I have seen, on a badly-balanced machine, a securely-fastened pair of nuts, one supposed to be locking the other, quietly revolving on their bolt at the same time, and gently lifting themselves at a regular but fairly rapid rate, until they both tumbled from the end of the bolt into my hand. If my hand had not been there, the bolts would have tumbled into a receptacle in which they would have produced interesting but most destructive phenomena. You would have somebody else lecturing to you to-night if that event had come off. Suppose, then, that our earth were spinning about any other axis than its present axis, the axis of figure. If spun about any diameter of the equator for example, centrifugal forces would just keep things in a state of unstable equilibrium, and no great change might be produced until some accidental cause effected a slight alteration in the spinning axis, and after that the earth would wobble very greatly. How long and how violently it would wobble, would depend on a number of circumstances about which I will not now venture to guess. If you {59} tell me that on the whole, in spite of the violence of the wobbling, it would not get shaken into a new form altogether, then I know that in consequence of tidal and other friction it would eventually come to a quiet state of spinning about its present axis. You see, then, that although every body has three axes about which it will rotate in a balanced fashion without any tendency to wobble, this balance of the centrifugal forces is really an unstable balance in two out of the three cases, and there is only one axis about which a perfectly stable balanced kind of rotation will take place, and a spinning body generally comes to rotate about this axis in the long run if left to itself, and if there is friction to still the wobbling. To illustrate this, I have here a method of spinning bodies which enables them to choose as their spinning axis that one principal axis about which their rotation is most stable. The various bodies can be hung at the end of this string, and I cause the pulley from which the string hangs to rotate. Observe that at first the disc (Fig. 28 _a_) rotates soberly about the axis A A, but you note the small beginning of the wobble; now it gets quite violent, and now the disc is stably and smoothly rotating about the axis B B, which is the most important of its principal axes. {60} [Illustration: FIG. 28.] Again, this cone (Fig. 28 _b_) rotates smoothly at first about the axis A A, but the wobble begins and gets very great, and eventually the cone rotates smoothly about the axis B B, which is the most important of its principal axes. Here again is a rod hung from one end (Fig. 28 _d_). See also this anchor ring. But you may be more interested in this limp ring of chain (Fig. 28 _c_). See how at first it hangs from the cord vertically, and how the wobbles and vibrations end in its becoming a perfectly circular ring lying all in a horizontal plane. This experiment illustrates also the quasi-rigidity given to a flexible body by rapid motion. To return to this balanced gyrostat of ours (Fig. 13). It is not precessing, so you know that the weight W just balances the gyrostat F. Now if I leave the instrument to itself after I give a downward impulse to F, not exerting merely a steady pressure, you will notice that F swings to the right for the reason already given; but it swings too fast and too far, just like any other swinging body, and it is easy from what I have already said, to see that this wobbling motion (Fig. 29) should be the result, and that it should continue until friction stills it, and F takes its permanent new position only after some time elapses. You see that I can impose this wobble or nodding {62} motion upon the gyrostat whether it has a motion of precession or not. It is now nodding as it processes round and round--that is, it is rising and falling as it precesses. [Illustration: FIG. 29.] Perhaps I had better put the matter a little more clearly. You see the same phenomenon in this top. If the top is precessing too fast for the force of gravity the top rises, and the precession diminishes in consequence; the precession being now too slow to balance gravity, the top falls a little and the {63} precession increases again, and this sort of vibration about a mean position goes on just as the vibration of a pendulum goes on till friction destroys it, and the top precesses more regularly in the mean position. This nodding is more evident in the nearly horizontal balanced gyrostat than in a top, because in a top the turning effect of gravity is less in the higher positions. When scientific men try to popularize their discoveries, for the sake of making some fact very plain they will often tell slight untruths, making statements which become rather misleading when their students reach the higher levels. Thus astronomers tell the public that the earth goes round the sun in an elliptic path, whereas the attractions of the planets cause the path to be only approximately elliptic; and electricians tell the public that electric energy is conveyed through wires, whereas it is really conveyed by all other space than that occupied by the wires. In this lecture I have to some small extent taken advantage of you in this way; for example, at first you will remember, I neglected the nodding or wobbling produced when an impulse is given to a top or gyrostat, and, all through, I neglect the fact that the instantaneous axis of rotation is only nearly coincident with the axis of figure of a precessing gyrostat or top. And indeed you may generally {64} take it that if all one's statements were absolutely accurate, it would be necessary to use hundreds of technical terms and involved sentences with explanatory, police-like parentheses; and to listen to many such statements would be absolutely impossible, even for a scientific man. You would hardly expect, however, that so great a scientific man as the late Professor Rankine, when he was seized with the poetic fervour, would err even more than the popular lecturer in making his accuracy of statement subservient to the exigencies of the rhyme as well as to the necessity for simplicity of statement. He in his poem, _The Mathematician in Love_, has the following lines-- "The lady loved dancing;--he therefore applied To the polka and waltz, an equation; But when to rotate on his axis he tried, His centre of gravity swayed to one side, And he fell by the earth's gravitation." Now I have no doubt that this is as good "dropping into poetry" as can be expected in a scientific man, and ----'s science is as good as can be expected in a man who calls himself a poet; but in both cases we have illustrations of the incompatibility of science and rhyming. [Illustration: FIG. 17.] The motion of this gyrostat can be made even more complicated than it was when we had {65} nutation and precession, but there is really nothing in it which is not readily explainable by the simple principles I have put before you. Look, for example, at this well-balanced gyrostat (Fig. 17). When I strike this inner gymbal ring in any way you see that it wriggles quickly just as if it were a lump of jelly, its rapid vibrations dying away just like the rapid vibrations of any yielding elastic body. This strange elasticity is of very great interest when we consider it in relation to the molecular properties of matter. Here again (Fig. 30) we have an example which is even more interesting. I have supported the cased {66} gyrostat of Figs. 5 and 6 upon a pair of stilts, and you will observe that it is moving about a perfectly stable position with a very curious staggering kind of vibratory motion; but there is nothing in these motions, however curious, that you cannot easily explain if you have followed me so far. [Illustration: FIG. 30.] Some of you who are more observant than the others, will have remarked that all these precessing gyrostats gradually fall lower and lower, just as they would do, only more quickly, if they were not spinning. And if you cast your eye upon the third statement of our wall sheet (p. 49) you will readily understand why it is so. "Delay the precession and the body falls, as gravity would make it do if it were not spinning." {67} Well, the precession of every one of these is resisted by friction, and so they fall lower and lower. I wonder if any of you have followed me so well as to know already why a spinning top rises. Perhaps you have not yet had time to think it out, but I have accentuated several times the particular fact which explains this phenomenon. Friction makes the gyrostats fall, what is it that causes a top to rise? Rapid rising to the upright position is the invariable sign of rapid rotation in a top, and I recollect that when quite vertical we used to say, "She sleeps!" Such was the endearing way in which the youthful experimenter thought of the beautiful object of his tender regard. All so well known as this rising tendency of a top has been ever since tops were first spun, I question if any person in this hall knows the explanation, and I question its being known to more than a few persons anywhere. Any great mathematician will tell you that the explanation is surely to be found published in _Routh_, or that at all events he knows men at Cambridge who surely know it, and he thinks that he himself must have known it, although he has now forgotten those elaborate mathematical demonstrations which he once exercised his mind upon. I believe that all such statements are made in error, but I cannot {68} be sure.[6] A partial theory of the phenomenon was given by Mr. Archibald Smith in the _Cambridge Mathematical Journal_ many years ago, but the problem was solved by Sir William Thomson and Professor Blackburn when they stayed together one year at the seaside, reading for the great Cambridge mathematical examination. It must have alarmed a person interested in Thomson's success to notice that the seaside holiday was really spent by him and his friend in spinning all sorts of rounded stones which they picked up on the beach. And I will now show you the curious phenomenon that puzzled him that year. This ellipsoid (Fig. 31) will represent a waterworn stone. It is lying in its most stable state on the table, and I give it a spin. You see that for a second or two it was inclined to go on spinning about the axis A A, but it began to wobble violently, and after a while, when these wobbles stilled, you saw that it was spinning nicely with its axis B B vertical; but then a new series of wobblings began and became more violent, and when they ceased you saw that the object had at length reached a settled state of {69} spinning, standing upright upon its longest axis. This is an extraordinary phenomenon to any person who knows about the great inclination of this body to spin in the very way in which I first started it spinning. You will find that nearly any rounded stone when spun will get up in this way upon its longest axis, if the spin is only vigorous enough, and in the very same way this spinning top tends to get more and more upright. [Illustration: FIG. 31.] I believe that there are very few mathematical explanations of phenomena which may not be given in quite ordinary language to people who have an ordinary amount of experience. In most cases the symbolical algebraic explanation must be given first by somebody, and then comes the time for its translation into ordinary language. This is the foundation of the new thing called Technical Education, which assumes that a {70} workman may be taught the principles underlying the operations which go on in his trade, if we base our explanations on the experience which the man has acquired already, without tiring him with a four years' course of study in elementary things such as is most suitable for inexperienced children and youths at public schools and the universities. [Illustration: FIG. 32.] [Illustration: FIG. 33.] With your present experience the explanation of the rising of the top becomes ridiculously simple. If you look at statement _two_ on this wall sheet (p. 48) and reflect a little, some of you will be able, without any elaborate mathematics, to give the simple reason for this that Thomson gave me sixteen years ago. "Hurry on the precession, and the body rises in opposition to gravity." Well, as I am not touching the top, and as the body does rise, we look at once for something that is hurrying on the precession, and we naturally look to the way in which its peg is rubbing on the table, for, with the exception of the atmosphere this top is touching nothing else than the table. Observe carefully how any of these objects precesses. Fig. 32 shows the way in which a top spins. Looked at from above, if the top is spinning in the direction of the hands of a watch, we know from the fourth statement of our wall sheet, or by mere observation, that it also precesses in the direction of the hands {71} of a watch; that is, its precession is such as to make the peg roll at B into the paper. For you will observe that the peg is rolling round a circular path on the table, G being nearly motionless, and the axis A G A describing nearly a cone in space whose vertex is G, above the table. Fig. 33 {72} shows the peg enlarged, and it is evident that the point B touching the table is really like the bottom of a wheel B B', and as this wheel is rotating, the rotation causes it to roll _into_ the paper, away from us. But observe that its mere precession is making it roll _into_ the paper, and that the spin if great enough wants to roll the top faster than the precession lets it roll, so that it hurries on the precession, and therefore the top rises. That is the simple explanation; the spin, so long as it is {73} great enough, is always hurrying on the precession, and if you will cast your recollection back to the days of your youth, when a top was supported on your hand as this is now on mine (Fig. 34), and the spin had grown to be quite small, and was unable to keep the top upright, you will remember that you dexterously helped the precession by giving your hand a circling motion so as to get from your top the advantages as to uprightness of a slightly longer spin. [Illustration: FIG. 34.] I must ask you now by observation, and the application of exactly the same argument, to explain the struggle for uprightness on its longer axis of any rounded stone when it spins on a table. I may tell you that some of these large rounded-looking objects which I now spin before you in illustration, are made hollow, and they are either of wood or zinc, because I have not the skill necessary to spin large solid objects, and yet I wanted to have objects which you would be able to see. This small one (Fig. 31) is the largest solid one to which my fingers are able to give sufficient spin. Here is a very interesting object (Fig. 35), spherical {74} in shape, but its centre of gravity is not exactly at its centre of figure, so when I lay it on the table it always gets to its position of stable equilibrium, the white spot touching the table as at A. Some of you know that if this sphere is thrown into the air it seems to have very curious motions, because one is so apt to forget that it is the motion of its centre of gravity which follows a simple path, and the boundary is eccentric to the centre of gravity. Its motions when set to roll upon a carpet are also extremely curious. [Illustration: FIG. 35.] Now for the very reasons that I have already given, when this sphere is made to spin on the table, it always endeavours to get its white spot uppermost, as in C, Fig. 35; to get into the position in which when not spinning it would be unstable. [Illustration: FIG. 36.] The precession of a top or gyrostat leads us at once to think of the precession of the great spinning body on which we live. You know that the earth {75} spins on its axis a little more than once every twenty-four hours, as this orange is revolving, and that it goes round the sun once in a year, as this orange is now going round a model sun, or as is shown in the diagram (Fig. 36). Its spinning axis points in the direction shown, very nearly to the star which is called the pole star, almost infinitely far away. In the figure and model I have greatly exaggerated the elliptic nature of the earth's path, as is quite usual, although it may be a little misleading, because the earth's path is much more nearly circular than many people imagine. As a matter of fact the earth is about three million miles nearer the sun in winter than it is in summer. This seems at first paradoxical, but we get to understand it when we reflect that, because of the slope of the earth's axis to the ecliptic, we people who live in the northern hemisphere have the sun less vertically above us, and have a shorter day in the winter, and hence each square foot of our part of the earth's surface receives much less heat every day, and so we feel colder. Now in about 13,000 years the earth will have precessed just half a revolution (_see_ Fig. 38); the axis will then be sloped towards the sun when it is nearest, instead of away from it as it is now; consequently we shall be much warmer in summer and colder in winter than we are now. Indeed we shall then be much worse off than the southern {77} hemisphere people are now, for they have plenty of oceanic water to temper their climate. It is easy to see the nature of the change from figures 36, 37, and 38, or from the model as I carry the orange and its symbolic knitting-needle round the model sun. Let us imagine an observer placed above this model, far above the north pole of the earth. He sees the earth rotating against the direction of the hands of a watch, and he finds that it precesses with the hands of a watch, so that spin and precession are in opposite directions. Indeed it is because of this that we have the word "precession," which we now apply to the motion of a top, although the precession of a top is in the same direction as that of the spin. [Illustration: FIG. 37.] [Illustration: FIG. 38.] The practical astronomer, in explaining the _luni-solar precession of the equinoxes_ to you, will not probably refer to tops or gyrostats. He will tell you that the _longitude_ and _right ascension_ of a star seem to alter; in fact that the point on the ecliptic from which he makes his measurements, namely, the spring equinox, is slowly travelling round the ecliptic in a direction opposite to that of the earth in its orbit, or to the apparent path of the sun. The spring equinox is to him for heavenly measurements what the longitude of Greenwich is to the navigator. He will tell you that aberration of light, and parallax of the stars, {80} but more than both, this precession of the equinoxes, are the three most important things which prevent us from seeing in an observatory by transit observations of the stars, that the earth is revolving with perfect uniformity. But his way of describing the precession must not disguise for you the physical fact that his phenomenon and ours are identical, and that to us who are acquainted with spinning tops, the slow conical motion of a spinning axis is more readily understood than those details of his measurements in which an astronomer's mind is bound up, and which so often condemn a man of great intellectual power to the life of drudgery which we generally associate with the idea of the pound-a-week cheap clerk. [Illustration: FIG. 22.] The precession of the earth is then of the same nature as that of a gyrostat suspended above its centre of gravity, of a body which would be stable and not top-heavy if it were not spinning. In fact the precession of the earth is of the same nature as that of this large gyrostat (Fig. 22), which is suspended in gymbals, so that it has a vibration like a pendulum when not spinning. I will now spin it, so that looked at from above it goes against the hands of a watch, and you observe that it precesses with the hands of a watch. Here again is a hemispherical wooden ship, in which there is a gyrostat with its axis vertical. It is in stable {81} equilibrium. When the gyrostat is not spinning, the ship vibrates slowly when put out of equilibrium; when the gyrostat is spinning the ship gets a motion of precession which is opposite in direction to that of the spinning. Astronomers, beginning with Hipparchus, have made observations of the earth's motion for us, and we have observed the motions of gyrostats, and we naturally seek for an explanation of the precessional motion of the earth. The equator of the earth makes an angle of 23½° with the ecliptic, which is the plane of the earth's orbit. Or the spinning axis of the earth is always at angle of 23½° with a perpendicular to the ecliptic, and makes a complete revolution in 26,000 years. The surface of the water on which this wooden ship is floating represents the ecliptic. The axis {82} of spinning of the gyrostat is about 23½° to the vertical; the precession is in two minutes instead of 26,000 years; and only that this ship does not revolve in a great circular path, we should have in its precession a pretty exact illustration of the earth's precession. The precessional motion of the ship, or of the gyrostat (Fig. 22), is explainable, and in the same way the earth's precession is at once explained if we find that there are forces from external bodies tending to put its spinning axis at right angles to the ecliptic. The earth is a nearly spherical body. If it were exactly spherical and homogeneous, the resultant force of attraction upon it, of a distant body, would be in a line through its centre. And again, if it were spherical and non-homogeneous, but if its mass were arranged in uniformly dense, spherical layers, like the coats of an onion. But the earth is not spherical, and to find what is the nature of the attraction of a distant body, it has been necessary to make pendulum observations all over the earth. You know that if a pendulum does not alter in length as we take it about to various places, its time of vibration at each place enables the force of gravity at each place to be determined; and Mr. Green proved that if we know the force of gravity at all places on the surface of the earth, although we may know nothing about the {83} state of the inside of the earth, we can calculate with absolute accuracy the force exerted by the earth on matter placed anywhere outside the earth; for instance, at any part of the moon's orbit, or at the sun. And hence we know the equal and opposite force with which such matter will act on the earth. Now pendulum observations have been made at a great many places on the earth, and we know, although of course not with absolute accuracy, the attraction on the earth, of matter outside the earth. For instance, we know that the resultant attraction of the sun on the earth is a force which does not pass through the centre of the earth's mass. You may comprehend the result better if I refer to this diagram of the earth at midwinter (Fig. 39), and use a popular method of description. A and B may roughly be called the protuberant parts of the earth--that protuberant belt of matter which makes the {84} earth orange-shaped instead of spherical. On the spherical portion inside, assumed roughly to be homogeneous, the resultant attraction is a force through the centre. [Illustration: FIG. 39.] I will now consider the attraction on the protuberant equatorial belt indicated by A and B. The sun attracts a pound of matter at B more than it attracts a pound of matter at A, because B is nearer than A, and hence the total resultant force is in the direction M N rather than O O, through the centre of the earth's mass. But we know that a force in the direction M N is equivalent to a force O O parallel to M N, together with a tilting couple of forces tending to turn the equator edge on to the sun. You will get the true result as to the tilting tendency by imagining the earth to be motionless, and the sun's mass to be distributed as a circular ring of matter 184 millions of miles in diameter, inclined to the equator at 23½°. Under the influence of the attraction of this ring the earth would heave like a great ship on a calm sea, rolling very slowly; in fact, making one complete swing in about three years. But the earth is spinning, and the tilting couple or torque acts upon it just like the forces which are always tending to cause this ship-model to stand upright, and hence it has a precessional motion whose complete period is 26,000 years. When there is no spin in the ship, its complete oscillation takes place in three seconds, and {85} when I spin the gyrostat on board the ship, the complete period of its precession is two minutes. In both cases the effect of the spin is to convert what would be an oscillation into a very much slower precession. There is, however, a great difference between the earth and the gyrostat. The forces acting on the top are always the same, but the forces acting on the earth are continually altering. At midwinter and midsummer the tilting forces are greatest, and at the equinoxes in spring and autumn there are no such forces. So that the precessional motion changes its rate every quarter year from a maximum to nothing, or from nothing to a maximum. It is, however, always in the same direction--the direction opposed to the earth's spin. When we speak then of the precessional motion of the earth, we usually think of the mean or average motion, since the motion gets quicker and slower every quarter year. Further, the moon is like the sun in its action. It tries to tilt the equatorial part of the earth into the plane of the moon's orbit. The plane of the moon's orbit is nearly the same as that of the ecliptic, and hence the average precession of the earth is of much the same kind as if only one of the two, the moon or the sun, alone acted. That is, the general phenomenon of precession of the {86} earth's axis in a conical path in 26,000 years is the effect of the combined tilting actions of the sun and moon. You will observe here an instance of the sort of untruth which it is almost imperative to tell in explaining natural phenomena. Hitherto I had spoken only of the sun as producing precession of the earth. This was convenient, because the plane of the ecliptic makes always almost exactly 23½° with the earth's equator, and although on the whole the moon's action is nearly identical with that of the sun, and about twice as great, yet it varies considerably. The superior tilting action of the moon, just like its tide-producing action, is due to its being so much nearer us than the sun, and exists in spite of the very small mass of the moon as compared with that of the sun. As the ecliptic makes an angle of 23½° with the earth's equator, and the moon's orbit makes an angle 5½° with the ecliptic, we see that the moon's orbit sometimes makes an angle of 29° with the earth's equator, and sometimes only 18°, changing from 29° to 18°, and back to 29° again in about nineteen years. This causes what is called "Nutation," or the nodding of the earth, for the tilting action due to the sun is greatly helped and greatly modified by it. The result of the variable nature of the moon's action is then that the earth's axis {87} rotates in an elliptic conical path round what might be called its mean position. We have also to remember that twice in every lunar month the moon's tilting action on the earth is greater, and twice it is zero, and that it continually varies in value. On the whole, then, the moon and sun, and to a small extent the planets, produce the general effect of a precession, which repeats itself in a period of about 25,695 years. It is not perfectly uniform, being performed at a speed which is a maximum in summer and winter; that is, there is a change of speed whose period is half a year; and there is a change of speed whose period is half a lunar month, the precession being quicker to-night than it will be next Saturday, when it will increase for about another week, and diminish the next. Besides this, because of 5½° of angularity of the orbits, we have something like the nodding of our precessing gyrostat, and the inclination of the earth's axis to the ecliptic is not constant at 23½°, but is changing, its periodic time being nineteen years. Regarding the earth's centre as fixed at O we see then, as illustrated in this model and in Fig. 40, the axis of the earth describes almost a perfect circle on the celestial sphere once in 25,866 years, its speed fluctuating every half year and every half month. But it is not a perfect circle, it is really a wavy {88} line, there being a complete wave every nineteen years, and there are smaller ripples in it, corresponding to the half-yearly and fortnightly periods. But the very cause of the nutation, the nineteen-yearly period of retrogression of the moon's nodes, as it is called, is itself really produced as the precession of a gyrostat is produced, that is, by tilting forces acting on a spinning body. [Illustration: FIG. 40.] Imagine the earth to be stationary, and the sun and moon revolving round it. It was Gauss who found that the present action is the same as if the masses of the moon and sun were distributed all {89} round their orbits. For instance, imagine the moon's mass distributed over her orbit in the form of a rigid ring of 480,000 miles diameter, and imagine less of it to exist where the present speed is greater, so that the ring would be thicker at the moon's apogee, and thinner at the perigee. Such a ring round the earth would be similar to Saturn's rings, which have also a precession of nodes, only Saturn's rings are not rigid, else there would be no equilibrium. Now if we leave out of account the earth and imagine this ring to exist by itself, and that its centre simply had a motion round the sun in a year, since it makes an angle of 5½° with the ecliptic it would vibrate into the ecliptic till it made the same angle on the other side and back again. But it revolves once about its centre in twenty-seven solar days, eight hours, and it will no longer swing like a ship in a ground-swell, but will get a motion of precession opposed in direction to its own revolution. As the ring's motion is against the hands of a watch, looking from the north down on the ecliptic, this retrogression of the moon's nodes is in the direction of the hands of a watch. It is exactly the same sort of phenomenon as the precession of the equinoxes, only with a much shorter period of 6798 days instead of 25,866 years. I told you how, if we knew the moon's mass or the sun's, we could tell the amount of the forces, or {90} the torque as it is more properly called, with which it tries to tilt the earth. We know the rate at which the earth is spinning, and we have observed the precessional motion. Now when we follow up the method which I have sketched already, we find that the precessional velocity of a spinning body ought to be equal to the torque divided by the spinning velocity and by the moment of inertia[7] of the body about the polar axis. Hence the greater the tilting forces, and the less the spin and the less the moment of inertia, the greater is the precessional speed. Given all of these elements except one, it is easy to calculate that unknown element. Usually what we aim at in such a calculation is the determination of the moon's mass, as this phenomenon of precession and the action of the tides are the only two natural phenomena which have as yet enabled the moon's mass to be calculated. I do not mean to apologize to you for the introduction of such terms as _Moment of Inertia_, nor do I mean to explain them. In this lecture I have avoided, as much as I could, the introduction of mathematical expressions and the use of technical terms. But I want you to {91} understand that I am not afraid to introduce technical terms when giving a popular lecture. If there is any offence in such a practice, it must, in my opinion, be greatly aggravated by the addition of explanations of the precise meanings of such terms. The use of a correct technical term serves several useful purposes. First, it gives some satisfaction to the lecturer, as it enables him to state, very concisely, something which satisfies his own weak inclination to have his reasoning complete, but which he luckily has not time to trouble his audience with. Second, it corrects the universal belief of all popular audiences that they know everything now that can be said on the subject. Third, it teaches everybody, including the lecturer, that there is nothing lost and often a great deal gained by the adoption of a casual method of skipping when one is working up a new subject. Some years ago it was argued that if the earth were a shell filled with liquid, if this liquid were quite frictionless, then the moment of inertia of the shell is all that we should have to take into account in considering precession, and that if it were viscous the precession would very soon disappear altogether. To illustrate the effect of the moment of inertia, I have hung up here a number of glasses--one _a_ filled with sand, another _b_ with treacle, a third _c_ with oil, the fourth _d_ with water, {92} [Illustration: FIG. 41.] {93} and the fifth _e_ is empty (Fig. 41). You see that if I twist these suspending wires and release them, a vibratory motion is set up, just like that of the balance of a watch. Observe that the glass with water vibrates quickly, its effective moment of inertia being merely that of the glass itself, and you see that the time of swing is pretty much the same as that of the empty glass; that is, the water does not seem to move with the glass. Observe that the vibration goes on for a fairly long time. The glass with sand vibrates slowly; here there is great moment of inertia, as the sand and glass behave like one rigid body, and again the vibration goes on for a long time. In the oil and treacle, however, there are longer periods of vibration than in the case of the water or empty glass, and less than would be the case if the vibrating bodies were all rigid, but the vibrations are stilled more rapidly because of friction. Boiled (_f_) and unboiled (_g_) eggs suspended from wires in the same way will exhibit the same differences in the behaviour of bodies, one of which is rigid and the other liquid inside; you see how much slower an oscillation the boiled has than the unboiled. Even on the table here it is easy to show the difference between boiled and unboiled eggs. {94} Roll them both; you see that one of them stops much sooner than the other; it is the unboiled one that stops sooner, because of its internal friction. I must ask you to observe carefully the following very distinctive test of whether an egg is boiled or not. I roll the egg or spin it, and then place my finger on it just for an instant; long enough to stop the motion of the shell. You see that the boiled egg had quite finished its motion, but the unboiled egg's shell alone was stopped; the liquid inside goes on moving, and now renews the motion of the shell when I take my finger away. It was argued that if the earth were fluid inside, the effective moment of inertia of the shell being comparatively small, and having, as we see in these examples, nothing whatever to do with the moment of inertia of the liquid, the precessional motion of the earth ought to be enormously quicker than it is. This was used as an argument against the idea of the earth's being fluid inside. We know that the observed half-yearly and half-monthly changes of the precession of the earth would be much greater than they are if the earth were a rigid shell containing much liquid, and if the shell were not nearly infinitely rigid the phenomena of the tides would not occur, but in regard to the general precession of the earth there is now {95} no doubt that the old line of argument was wrong. Even if the earth were liquid inside, it spins so rapidly that it would behave like a rigid body in regard to such a slow phenomenon as precession of the equinoxes. In fact, in the older line of argument the important fact was lost sight of, that rapid rotation can give to even liquids a quasi-rigidity. Now here (Fig. 42 _a_) is a hollow brass top filled with water. The frame is light, and the water inside has much more mass than the outside frame, and if you test this carefully you will find that the top spins in almost exactly the same way as if the water were quite rigid; in fact, as if the whole top were rigid. Here you see it spinning and precessing just like any rigid top. This top, I know, is not filled with water, it is only partially filled; but whether partially or wholly filled it spins very much like a rigid top. [Illustration: FIG. 42.] {96} This is not the case with a long hollow brass top with water inside. I told you that all bodies have one axis about which they prefer to rotate. The outside metal part of a top behaves in a way that is now well known to you; the friction of its peg on the table compels it to get up on its longer axis. But the fluid inside a top is not constrained to spin on its longer axis of figure, and as it prefers its shorter axis like all these bodies I showed you, it spins in its own way, and by friction and pressure against the case constrains the case to spin about the shorter axis, annulling completely the tendency of the outside part to rise or keep up on its long axis. Hence it is found to be simply impossible to spin a long hollow top when filled with water. [Illustration: FIG. 43.] [Illustration: FIG. 44.] Here, for example, is one (Fig. 42 _b_) that only differs from the last in being longer. It is filled, or partially filled, with water, and you observe that if {97} I slowly get up a great spin when it is mounted in this frame, and I let it out on the table as I did the other one, this one lies down at once and refuses to spin on its peg. This difference of behaviour is most remarkable in the two hollow tops you see before you (Fig. 43). They are both nearly spherical, both filled with water. They look so nearly alike that few persons among the audience are able to detect any difference in their shape. But one of them (_a_) is really slightly oblate like an orange, and the other (_b_) is slightly prolate like a lemon. I will give them both a gradually increasing rotation in this frame {98} (Fig. 44) for a time sufficient to insure the rotation of the water inside. When just about to be set free to move like ordinary tops on the table, water and brass are moving like the parts of a rigid top. You see that the orange-shaped one continues to spin and precess, and gets itself upright when disturbed, like an ordinary rigid top; indeed I have seldom seen a better behaved top; whereas the lemon-shaped one lies down on its side at once, and quickly ceases to move in any way. [Illustration: FIG. 45.] And now you will be able to appreciate a fourth test of a boiled egg, which is much more easily seen by a large audience than the last. Here is the unboiled one (Fig. 45 _b_). I try my best to spin it as it lies on the table, but you see that I cannot give it much spin, and so there is nothing of any importance to look at. But you observe that it is quite easy to spin the boiled {99} egg, and that for reasons now well known to you it behaves like the stones that Thomson spun on the sea-beach; it gets up on its longer axis, a very pretty object for our educated eyes to look at (Fig. 45 _a_). You are all aware, from the behaviour of the lemon-shaped top, that even if, by the use of a whirling table suddenly stopped, or by any other contrivance, I could get up a spin in this unboiled egg, it would never make the slightest effort to rise on its end and spin about its longer axis. I hope you don't think that I have been speaking too long about astronomical matters, for there is one other important thing connected with astronomy that I must speak of. You see, I have had almost nothing practically to do with astronomy, and hence I have a strong interest in the subject. It is very curious, but quite true, that men practically engaged in any pursuit are almost unable to see the romance of it. This is what the imaginative outsider sees. But the overworked astronomer has a different point of view. As soon as it becomes one's duty to do a thing, and it is part of one's every-day work, the thing loses a great deal of its interest. We have been told by a great American philosopher that the only coachmen who ever saw the romance of coach-driving are those titled individuals who pay nowadays so largely for the {100} privilege. In almost any branch of engineering you will find that if any invention is made it is made by an outsider; by some one who comes to the study of the subject with a fresh mind. Who ever heard of an old inhabitant of Japan or Peru writing an interesting book about those countries? At the end of two years' residence he sees only the most familiar things when he takes his walks abroad, and he feels unmitigated contempt for the ingenuous globe-trotter who writes a book about the country after a month's travel over the most beaten tracks in it. Now the experienced astronomer has forgotten the difficulties of his predecessors and the doubts of outsiders. It is a long time since he felt that awe in gazing at a starry sky that we outsiders feel when we learn of the sizes and distances apart of the hosts of heaven. He speaks quite coolly of millions of years, and is nearly as callous when he refers to the ancient history of humanity on our planet as a weather-beaten geologist. The reason is obvious. Most of you know that the _Nautical Almanac_ is as a literary production one of the most uninteresting works of reference in existence. It is even more disconnected than a dictionary, and I should think that preparing census-tables must be ever so much more romantic as an occupation than preparing the tables of the _Nautical Almanac_. And yet {101} a particular figure, one of millions set down by an overworked calculator, may have all the tragic importance of life or death to the crew and passengers of a ship, when it is heading for safety or heading for the rocks under the mandate of that single printed character. But this may not be a fair sort of criticism. I so seldom deal with astronomical matters, I know so little of the wear and tear and monotony of the every-day life of the astronomer, that I do not even know that the above facts are specially true about astronomers. I only know that they are very likely to be true because they are true of other professional men. I am happy to say that I come in contact with all sorts and conditions of men, and among others, with some men who deny many of the things taught in our earliest school-books. For example, that the earth is round, or that the earth revolves, or that Frenchmen speak a language different from ours. Now no man who has been to sea will deny the roundness of the earth, however greatly he may wonder at it; and no man who has been to France will deny that the French language is different from ours; but many men who learnt about the rotation of the earth in their school-days, and have had a plentiful opportunity of observing the heavenly bodies, deny the rotation of the earth. {102} They tell you that the stars and moon are revolving about the earth, for they see them revolving night after night, and the sun revolves about the earth, for they see it do so every day. And really if you think of it, it is not so easy to prove the revolution of the earth. By the help of good telescopes and the electric telegraph or good chronometers, it is easy to show from the want of parallax in stars that they must be very far away; but after all, we only know that either the earth revolves or else the sky revolves.[8] Of course, it seems infinitely more likely that the small earth should revolve than that the whole heavenly host should turn about the earth as a centre, and infinite likelihood is really absolute proof. Yet there is nobody who does not welcome an independent kind of proof. The phenomena of the tides, and nearly every new astronomical fact, may be said to be an addition to the proof. Still there is the absence of perfect certainty, and when we are told that these spinning-top phenomena give us a real proof of the rotation of the earth without our leaving the room, we welcome {103} it, even although we may sneer at it as unnecessary after we have obtained it. [Illustration: FIG. 17.] You know that a gyrostat suspended with perfect freedom about axes, which all pass through its centre of gravity, maintains a constant direction in space however its support may be carried. Its axis is not forced to alter its direction in any way. Now this gyrostat (Fig. 17) has not the perfect absence of friction at its axes of which I speak, and even the slightest friction will produce some constraint which is injurious to the experiment I am about to describe. It must be remembered, that if there were absolutely no constraint, then, even if the {104} gyrostat were _not_ spinning, its axis would keep a constant direction in space. But the spinning gyrostat shows its superiority in this, that any constraint due to friction is less powerful in altering the axis. The greater the spin, then, the better able are we to disregard effects due to friction. You have seen for yourselves the effect of carrying this gyrostat about in all sorts of ways--first, when it is not spinning and friction causes quite a large departure from constancy of direction of the axis; second, when it is spinning, and you see that although there is now the same friction as before, and I try to disturb the instrument more than before, the axis remains sensibly parallel to itself all the time. Now when this instrument is supported by the table it is really being carried round by the earth in its daily rotation. If the axis kept its direction perfectly, and it were now pointing horizontally due east, six hours after this it will point towards the north, but inclining downwards, six hours afterwards it will point due west horizontally, and after one revolution of the earth it will again point as it does now. Suppose I try the experiment, and I see that it points due east now in this room, and after a time it points due west, and yet I know that the gyrostat is constantly pointing in the same direction in space all the time, surely it is obvious that the room must {105} be turning round in space. Suppose it points to the pole star now, in six hours, or twelve, or eighteen, or twenty-four, it will still point to the pole star. Now it is not easy to obtain so frictionless a gyrostat that it will maintain a good spin for such a length of time as will enable the rotation of the room to be made visible to an audience. But I will describe to you how forty years ago it was proved in a laboratory that the earth turns on its axis. This experiment is usually connected with the name of Foucault, the same philosopher who with Fizeau showed how in a laboratory we can measure the velocity of light, and therefore measure the distance of the sun. It was suggested by Mr. Lang of Edinburgh in 1836, although only carried out in 1852 by Foucault. By these experiments, if you were placed on a body from which you could see no stars or other outside objects, say that you were living in underground regions, you could discover--first, whether there is a motion of rotation, and the amount of it; second, the meridian line or the direction of the true north; third, your latitude. Obtain a gyrostat like this (Fig. 46) but much larger, and far more frictionlessly suspended, so that it is free to move vertically or horizontally. For the vertical motion your gymbal pivots ought to be hard steel knife-edges. {106} [Illustration: FIG. 46.] As for the horizontal freedom, Foucault used a fine steel wire. Let there be a fine scale engraved crosswise on the outer gymbal ring, and try to discover if it moves horizontally by means of a microscope with cross wires. When this is carefully done we find that there is a motion, {107} but this is not the motion of the gyrostat, it is the motion of the microscope. In fact, the microscope and all other objects in the room are going round the gyrostat frame. Now let us consider what occurs. The room is rotating about the earth's axis, and we know the rate of rotation; but we only want to know for our present purpose how much of the total rotation is about a vertical line in the room. If the room were at the North Pole, the whole rotation would be about the vertical line. If the room were at the equator, none of its rotation would be about a vertical line. In our latitude now, the horizontal rate of rotation about a vertical axis is about four-fifths of the whole rate of rotation of the earth on its axis, and this is the amount that would be measured by our microscope. This experiment would give no result at a place on the equator, but in our latitude you would have a laboratory proof of the rotation of the earth. Foucault made the measurements with great accuracy. If you now clamp the frame, and allow the spinning axis to have no motion except in a horizontal plane, the motion which the earth tends to give it about a vertical axis cannot now affect the gyrostat, but the earth constrains it to move about an axis due north and south, and consequently the spinning axis tries to put itself parallel {108} to the north and south direction (Fig. 47). Hence with such an instrument it is easy to find the true north. If there were absolutely no friction the instrument would vibrate about the true north position like the compass needle (Fig. 50), although with an exceedingly slow swing. [Illustration: FIG. 47.] It is with a curious mixture of feelings that one first recognizes the fact that all rotating bodies, fly-wheels of steam-engines and the like, are always tending to turn themselves towards the pole star; gently and vainly tugging at their foundations {109} to get round towards the object of their adoration all the time they are in motion. [Illustration: FIG. 48.] Now we have found the meridian as in Fig. 47, we can begin a third experiment. Prevent motion horizontally, that is, about a vertical axis, but give the instrument freedom to move vertically in the meridian, like a transit instrument in an observatory {110} about its horizontal axis. Its revolution with the earth will tend to make it change its angular position, and therefore it places itself parallel to the earth's axis; when in this position the daily rotation no longer causes any change in its direction in space, so it continues to point to the pole star (Fig. 48). It would be an interesting experiment to measure with a delicate chemical balance the force with which the axis raises itself, and in this way _weigh_ the rotational motion of the earth.[9] Now let us turn the frame of the instrument G B round a right angle, so that the spinning axis can only move in a plane at right angles to the meridian; obviously it is constrained by the vertical component of the earth's rotation, and points vertically downwards. [Illustration: FIG. 49.] [Illustration: FIG. 50.] This last as well as the other phenomena of which I have spoken is very suggestive. Here is a magnetic needle (Fig. 49), sometimes called a dipping needle from the way in which it is suspended. If I turn its {111} frame so that it can only move at right angles to the meridian, you see that it points vertically. You may reflect upon the analogous properties of this magnetic needle (Fig. 50) and of the gyrostat (Fig. 47); they both, when only capable of moving horizontally, point to the north; and you see that a very frictionless gyrostat might be used as a compass, or at all events as a corrector of compasses.[10] I have just put before you another analogy, and I want you to understand that, although these are only analogies, they are not mere chance analogies, for there is undoubtedly a dynamical connection between the magnetic and the gyrostatic phenomena. Magnetism depends on rotatory motion. The molecules of matter are in actual rotation, and a certain allineation of the axes of the rotations produces what we call magnetism. In a steel bar not magnetized the little axes of rotation are all in different directions. The process {112} of magnetization is simply bringing these rotations to be more or less round parallel axes, an allineation of the axes. A honey-combed mass with a spinning gyrostat in every cell, with all the spinning axes parallel, and the spins in the same direction, would--I was about to say, would be a magnet, but it would not be a magnet in all its properties, and yet it would resemble a magnet in many ways.[11] [Illustration: FIG. 51.] [Illustration: FIG. 52.] Some of you, seeing electromotors and other electric contrivances near this table, may think that they have to do with our theories and explanations of magnetic phenomena. But I must explain that this electromotor which I hold in my hand (Fig. 51) is used by me merely as the {113} most convenient means I could find for the spinning of my tops and gyrostats. On the spindle of the motor is fastened a circular piece of wood; by touching this key I can supply the motor with electric energy, and the wooden disc is now rotating very rapidly. I have only to bring its rim in contact with any of these tops or gyrostats to set them spinning, and you see that I can set half a dozen gyrostats a-spinning in a few seconds; this chain of gyrostats, for instance. Again, this larger motor (Fig. 52), too large to move about in my hand, is fastened to the table, and I have used {114} it to drive my larger contrivances; but you understand that I use these just as a barber might use them to brush your hair, or Sarah Jane to clean the knives, or just as I would use a little steam-engine if it were more convenient for my purpose. It was more convenient for me to bring from London this battery of accumulators and these motors than to bring sacks of coals, and boilers, and steam-engines. But, indeed, all this has the deeper meaning that we can give to it if we like. Love is as old as the hills, and every day Love's messages are carried by the latest servant of man, the telegraph. These spinning tops were known probably to primeval man, and yet we have not learnt from them more than the most fractional portion of the lesson that they are always sending out to an unobservant world. Toys like these were spun probably by the builders of the Pyramids when they were boys, and here you see them side by side with the very latest of man's contrivances. I feel almost as Mr. Stanley might feel if, with the help of the electric light and a magic-lantern, he described his experiences in that dreadful African forest to the usual company of a London drawing-room. The phenomena I have been describing to you play such a very important part in nature, that if time admitted I might go on expounding and {115} explaining without finding any great reason to stop at one place rather than another. The time at my disposal allows me to refer to only one other matter, namely, the connection between light and magnetism and the behaviour of spinning tops. You are all aware that sound takes time to travel. This is a matter of common observation, as one can see a distant woodchopper lift his axe again before one hears the sound of his last stroke. A destructive sea wave is produced on the coast of Japan many hours after an earthquake occurs off the coast of America, the wave motion having taken time to travel across the Pacific. But although light travels more quickly than sound or wave motion in the sea, it does not travel with infinite rapidity, and the appearance of the eclipse of one of Jupiter's satellites is delayed by an observable number of minutes because light takes time to travel. The velocity has been measured by means of such observations, and we know that light travels at the rate of about 187,000 miles per second, or thirty thousand millions of centimetres per second. There is no doubt about this figure being nearly correct, for the velocity of light has been measured in the laboratory by a perfectly independent method. Now the most interesting physical work done since Newton's time is the outcome of the experiments of Faraday and the theoretical deductions of {116} Thomson and Maxwell. It is the theory that light and radiant heat are simply electro-magnetic disturbances propagated through space. I dare not do more than just refer to this matter, although it is of enormous importance. I can only say, that of all the observed facts in the sciences of light, electricity, and magnetism, we know of none that is in opposition to Maxwell's theory, and we know of many that support it. The greatest and earliest support that it had was this. If the theory is correct, then a certain electro-magnetic measurement ought to result in exactly the same quantity as the velocity of light. Now I want you to understand that the electric measurement is one of quantities that seem to have nothing whatever to do with light, except that one uses one's eyes in making the measurement; it requires the use of a two-foot rule and a magnetic needle, and coils of wire and currents of electricity. It seemed to bear a relationship to the velocity of light, which was not very unlike the fabled connection between Tenterden Steeple and the Goodwin Sands. It is a measurement which it is very difficult to make accurately. A number of skilful experimenters, working independently, and using quite different methods, arrived at results only one of which is as much as five per cent. different from the observed velocity of light, and some of them, {117} on which the best dependence may be placed, agree exactly with the average value of the measurements of the velocity of light. There is then a wonderful agreement of the two measurements, but without more explanation than I can give you now, you cannot perhaps understand the importance of this agreement between two seemingly unconnected magnitudes. At all events we now know, from the work of Professor Hertz in the last two years, that Maxwell's theory is correct, and that light is an electro-magnetic disturbance; and what is more, we know that electro-magnetic disturbances, incomparably slower than red-light or heat, are passing now through our bodies; that this now recognized kind of radiation may be reflected and refracted, and yet will pass through brick and stone walls and foggy atmospheres where light cannot pass, and that possibly all military and marine and lighthouse signalling may be conducted in the future through the agency of this new and wonderful kind of radiation, of which what we call light is merely one form. Why at this moment, for all I know, two citizens of Leeds may be signalling to each other in this way through half a mile of houses, including this hall in which we are present.[12] {118} I mention this, the greatest modern philosophical discovery, because the germ of it, which was published by Thomson in 1856, makes direct reference to the analogy between the behaviour of our spinning-tops and magnetic and electrical phenomena. It will be easier, however, for us to consider here a mechanical illustration of the rotation of the plane of polarized light by magnetism which Thomson elaborated in 1874. This phenomenon may, I think, be regarded as the most important of all Faraday's discoveries. It was of enormous scientific importance, because it was made in a direction where a new phenomenon was not even suspected. Of his discovery of induced currents of electricity, to which all electric-lighting companies and transmission of power companies of the present day owe their being, Faraday himself said that it was a natural consequence of the discoveries of an earlier experimenter, Oersted. But this magneto-optic discovery was quite unexpected. I will now describe the phenomenon. Some of you are aware that when a beam of light is sent through this implement, called a Nichol's Prism, it becomes polarized, or one-sided--that is, all the light that comes through is known to be propagated by vibrations which occur all in one plane. This rope (Fig. 53) hanging from the ceiling {119} illustrates the nature of plane polarized light. All points in the rope are vibrating in the same plane. Well, this prism A, Fig. 54, only lets through it light that is polarized in a vertical plane. And here at B I have a similar implement, and I place it so that it also will only allow light to pass through it which is polarized in a vertical plane. Hence most of the light coming through the polarizer, as the first prism is called, will pass readily through the analyzer, as the second is called, and I am now letting this light enter my eye. But when I turn the analyzer round through a right angle, I find that I see no light; there was a gradual darkening as I rotated the analyzer. The analyzer will now only allow light to pass through which is polarized in a horizontal plane, and it receives no such light. [Illustration: FIG. 53.] [Illustration: FIG. 54.] You will see in this model (Fig. 55) a good illustration of polarized light. The white, brilliantly illuminated thread M N is {120} pulled by a weight beyond the pulley M, and its end N is fastened to one limb of a tuning-fork. Some ragged-looking pieces of thread round the portion N A prevent its vibrating in any very determinate way, but from A to M the thread is free from all encumbrance. A vertical slot at A, through which the thread passes, determines the nature of the vibration of the part A B; every part of the thread between A and B is vibrating in up and down directions only. A vertical slot in B allows the vertical vibration to be communicated through it, and so we see the part B M vibrating in the same way as A B. I might point out quite a lot of ways in which this is not a perfect illustration of what occurs with light in Fig. 54. But it is quite good enough for my present purpose. A is a polarizer of vibration; it only allows up and down motion to pass through it, and B also allows up and down motion to pass through. But now, as B is turned round, it lets less and less of the up and down motion pass through it, until when it is in the second position shown in the lower part of the figure, it allows no up and down motion to pass through, and there is no visible motion of the thread between B and M. You will observe that if we did not know in what plane (in the present case the plane is vertical) the vibrations of the thread between A and B occurred, we should only have to turn B round until we found no vibration {122} passing through, to obtain the information. Hence, as in the light case, we may call A a polarizer of vibrations, and B an analyzer. [Illustration: FIG. 55.] Now if polarized light is passing from A to B (Fig. 54) through the air, say, and we have the analyzer placed so that there is darkness, we find that if we place in the path of the ray some solution of sugar we shall no longer have darkness at B; we must turn B round to get things dark again; this is evidence of the sugar solution having twisted round the plane of polarization of the light. I will now assume that you know something about what is meant by twisting the plane of polarization of light. You know that sugar solution will do it, and the longer the path of the ray through the sugar, the more twist it gets. This phenomenon is taken advantage of in the sugar industries, to find the strengths of sugar solutions. For the thread illustration I am indebted to Professor Silvanus Thomson, and the next piece of apparatus which I shall show also belongs to him. I have here (_see_ Frontispiece) a powerful armour-clad coil, or electro-magnet. There is a central hole through it, through which a beam of light may be passed from an electric lamp, and I have a piece of Faraday's heavy glass nearly filling this hole. I have a polarizer at one end, and an analyzer at the other. You see now that the {123} polarized light passes through the heavy glass and the analyzer, and enters the eye of an observer. I will now turn B until the light no longer passes. Until now there has been no magnetism, but I have the means here of producing a most intense magnetic field in the direction in which the ray passes, and if your eye were here you would see that there is light passing through the analyzer. The magnetism has done something to the light, it has made it capable of passing where it could not pass before. When I turn the analyzer a little I stop the light again, and now I know that what the magnetism did was to convert the glass into a medium like the sugar, a medium which rotates the plane of polarization of light. In this experiment you have had to rely upon my personal measurement of the actual rotation produced. But if I insert between the polarizer and analyzer this disc of Professor Silvanus Thomson's, built up of twenty-four radial pieces of mica, I shall have a means of showing to this audience the actual rotation of the plane of polarization of light. You see now on the screen the light which has passed through the analyzer in the form of a cross, and if the cross rotates it is a sign of the rotation of the plane of polarization of the light. By means of this electric key I can create, destroy, and reverse the magnetic {124} field in the glass. As I create magnetism you see the twisting of the cross; I destroy the magnetism, and it returns to its old position; I create the opposite kind of magnetism, and you see that the cross twists in the opposite way. I hope it is now known to you that magnetism rotates the plane of polarization of light as the solution of sugar did. [Illustration: FIG. 56.] [Illustration: FIG. 57.] As an illustration of what occurs between polarizer and analyzer, look again at this rope (Fig. 53) fastened to the ceiling. I move the bottom end sharply from east to west, and you see that every part of the rope moves from east to west. Can you imagine a rope such that when the bottom end was moved from east to west, a point some yards up moved from east-north-east to west-sou'-west, that a higher point moved from north-east to south-west, and so on, the direction gradually changing for higher and higher points? Some of you, knowing what I have done, may be able to imagine it. We should have what we want if this rope were a chain of gyrostats such as you see figured in the diagram; gyrostats all spinning in the same way looked at from below, with frictionless hinges between them. Here is such a chain (Fig. 56), one of many that I have tried to use in this way for several years. But although I have often believed that I saw the phenomenon occur in {126} such a chain, I must now confess to repeated failures. The difficulties I have met with are almost altogether mechanical ones. You see that by touching all the gyrostats in succession with this rapidly revolving disc driven by the little electromotor, I can get them all to spin at the same time; but you will notice that what with bad mechanism and bad calculation on my part, and want of skill, the phenomenon is completely masked by wild movements of the gyrostats, the causes of which are better known than capable of rectification. The principle of the action is very visible in this gyrostat suspended as the bob of a pendulum (Fig. 57). You may imagine this to represent a particle of the {127} substance which transmits light in the magnetic field, and you see by the trickling thin stream of sand which falls from it on the paper that it is continually changing the plane of polarization. But I am happy to say that I can show you to-night a really successful illustration of Thomson's principle; it is the very first time that this most suggestive experiment has been shown to an audience. I have a number of double gyrostats (Fig. 58) placed on the same line, joined end to end by short pieces of elastic. Each instrument is supported at its centre of gravity, and it can rotate both in horizontal and in vertical planes. [Illustration: FIG. 58.] The end of the vibrating lever A can only get a horizontal motion from my hand, and the motion is transmitted from one gyrostat to the next, until it has travelled to the very end one. Observe that when the gyrostats are not spinning, the motion is {128} everywhere horizontal. Now it is very important not to have any illustration here of a reflected ray of light, and so I have introduced a good deal of friction at all the supports. I will now spin all the gyrostats, and you will observe that when A moves nearly straight horizontally, the next gyrostat moves straight but in a slightly different plane, the second gyrostat moves in another plane, and so on, each gyrostat slightly twisting the plane in which the motion occurs; and you see that the end one does not by any means receive the horizontal motion of A, but a motion nearly vertical. This is a mechanical illustration, the first successful one I have made after many trials, of the effect on light of magnetism. The reason for the action that occurs in this model must be known to everybody who has tried to follow me from the beginning of the lecture. And you can all see that we have only to imagine that many particles of the glass are rotating like gyrostats, and that magnetism has partially caused an allineation of their axes, to have a dynamical theory of Faraday's discovery. The magnet twists the plane of polarization, and so does the solution of sugar; but it is found by experiment that the magnet does it indifferently for coming and going, whereas the sugar does it in a way that corresponds with a spiral structure of molecules. You see that in this important {129} particular the gyrostat analogue must follow the magnetic method, and not the sugar method. We must regard this model, then, the analogue to Faraday's experiment, as giving great support to the idea that magnetism consists of rotation. I have already exceeded the limits of time usually allowed to a popular lecturer, but you see that I am very far from having exhausted our subject. I am not quite sure that I have accomplished the object with which I set out. My object was, starting from the very different behaviour of a top when spinning and when not spinning, to show you that the observation of that very common phenomenon, and a determination to understand it, might lead us to understand very much more complex-looking things. There is no lesson which it is more important to learn than this--That it is in the study of every-day facts that all the great discoveries of the future lie. Three thousand years ago spinning tops were common, but people never studied them. Three thousand years ago people boiled water and made steam, but the steam-engine was unknown to them. They had charcoal and saltpetre and sulphur, but they knew nothing of gunpowder. They saw fossils in rocks, but the wonders of geology were unstudied by them. They had bits of iron and copper, but not one of them thought of any one of the fifty simple {130} ways that are now known to us of combining those known things into a telephone. Why, even the simplest kind of signalling by flags or lanterns was unknown to them, and yet a knowledge of this might have changed the fate of the world on one of the great days of battle that we read about. We look on Nature now in an utterly different way, with a great deal more knowledge, with a great deal more reverence, and with much less unreasoning superstitious fear. And what we are to the people of three thousand years ago, so will be the people of one hundred years hence to us; for indeed the acceleration of the rate of progress in science is itself accelerating. The army of scientific workers gets larger and larger every day, and it is my belief that every unit of the population will be a scientific worker before long. And so we are gradually making time and space yield to us and obey us. But just think of it! Of all the discoveries of the next hundred years; the things that are unknown to us, but which will be so well known to our descendants that they will sneer at us as utterly ignorant, because these things will seem to them such self-evident facts; I say, of all these things, if one of us to-morrow discovered one of them, he would be regarded as a great discoverer. And yet the children of a hundred years hence will know it: it will be brought home to {131} them perhaps at every footfall, at the flapping of every coat-tail. Imagine the following question set in a school examination paper of 2090 A.D.--"Can you account for the crass ignorance of our forefathers in not being able to see from England what their friends were doing in Australia?"[13] Or this--"Messages are being received every minute from our friends on the planet Mars, and are now being answered: how do you account for our ancestors being utterly ignorant that these messages were occasionally sent to them?" Or this--"What metal is as strong compared with steel as steel is compared with lead? and explain why the discovery of it was not made in Sheffield." But there is one question that our descendants will never ask in accents of jocularity, for to their bitter sorrow every man, woman, and child of them will know the answer, and that question is this--"If our ancestors in the matter of coal economy were not quite as ignorant as a baby who takes a penny {132} as equivalent for a half-crown, why did they waste our coal? Why did they destroy what never can be replaced?" My friends, let me conclude by impressing upon you the value of knowledge, and the importance of using every opportunity within your reach to increase your own store of it. Many are the glittering things that seem to compete successfully with it, and to exercise a stronger fascination over human hearts. Wealth and rank, fashion and luxury, power and fame--these fire the ambitions of men, and attract myriads of eager worshippers; but, believe it, they are but poor things in comparison with knowledge, and have no such pure satisfactions to give as those which it is able to bestow. There is no evil thing under the sun which knowledge, when wielded by an earnest and rightly directed will, may not help to purge out and destroy; and there is no man or woman born into this world who has not been given the capacity, not merely to gather in knowledge for his own improvement and delight, but even to add something, however little, to that general stock of knowledge which is the world's best wealth. * * * * * {133} ARGUMENT. 1. _Introduction_, pages 9-14, showing the importance of the study of spinning-top behaviour. 2. _Quasi-rigidity induced even in flexible and fluid bodies by rapid motion_, 14-21. Illustrations: Top, 14; belt or rope, 14; disc of thin paper, 14; ring of chain, 15; soft hat, 16; drunken man, 16; rotating water, 16; smoke rings, 17; Thomson's Molecular Theory, 19; swimmer caught in an eddy, 20; mining water jet, 20; cased gyrostat, 21. 3. _The nature of this quasi-rigidity in spinning bodies is a resistance to change of direction of the axis of spinning_, 21-30. Illustrations: Cased gyrostat, 21-24; tops, biscuits, hats, thrown into the air, 24-26; quoits, hoops, projectiles from guns, 27; jugglers at the Victoria Music Hall, 26-30; child trundling hoop, man on bicycle, ballet-dancer, the earth pointing to pole star, boy's top, 30. 4. _Study of the crab-like behaviour of a spinning body_, 30-49. Illustrations: Spinning top, 31; cased gyrostat, 32; balanced gyrostat, 33-36; windage of projectiles from {134} rifled guns, 36-38; tilting a hoop or bicycle, turning quickly on horseback, 38; bowls, 39; how to simplify one's observations, 39, 40; the illustration which gives us our simple universal rule, 40-42; testing the rule, 42-44; explanation of precession of gyrostat, 44, 45; precession of common top, 46; precession of overhung top, 46; list of our results given in a wall sheet, 48, 49. 5. _Proof or explanation of our simple universal rule_, 50-54. Giving two independent rotations to a body, 50, 51; composition of rotations, 52, 53. 6. _Warning that the rule is not, after all, so simple_, 54-66. Two independent spins given to the earth, 54; centrifugal force, 55; balancing of quick speed machinery, 56, 57; the possible wobbling of the earth, 58; the three principal axes of a body, 59; the free spinning of discs, cones, rods, rings of chain, 60; nodding motion of a gyrostat, 62; of a top, 63; parenthesis about inaccuracy of statement and Rankine's rhyme, 63, 64; further complications in gyrostatic behaviour, 64; strange elastic, jelly-like behaviour, 65; gyrostat on stilts, 66. 7. _Why a gyrostat falls_, 66, 67. 8. _Why a top rises_, 67-74. General ignorance, 67; Thomson preparing for the mathematical tripos, 68; behaviour of a water-worn stone when spun on a table, 68, 69; parenthesis on technical education, 70; simple explanation of why a top rises, 70-73; behaviour of heterogeneous sphere when spun, 74. 9. _Precessional motion of the earth_, 74-91. Its nature and effects on climate, 75-80; resemblance of the precessing earth to certain models, 80-82; tilting forces exerted by the sun and moon on the {135} earth, 82-84; how the earth's precessional motion is always altering, 85-88; the retrogression of the moon's nodes is itself another example, 88, 89; an exact statement made and a sort of apology for making it, 90, 91. 10. _Influence of possible internal fluidity of the earth on its precessional motion_, 91-98. Effect of fluids and sand in tumblers, 91-93; three tests of the internal rigidity of an egg, that is, of its being a boiled egg, 93, 94; quasi-rigidity of fluids due to rapid motion, forgotten in original argument, 95; beautiful behaviour of hollow top filled with water, 95; striking contrasts in the behaviour of two tops which are very much alike, 97, 98; fourth test of a boiled egg, 98. 11. Apology for dwelling further upon astronomical matters, and impertinent remarks about astronomers, 99-101. 12. How a gyrostat would enable a person living in subterranean regions to know, _1st, that the earth rotates_; _2nd, the amount of rotation_; _3rd, the direction of true north_; _4th, the latitude_, 101-111. Some men's want of faith, 101; disbelief in the earth's rotation, 102; how a free gyrostat behaves, 103, 104; Foucault's laboratory measurement of the earth's rotation, 105-107; to find the true north, 108; all rotating bodies vainly endeavouring to point to the pole star, 108; to find the latitude, 110; analogies between the gyrostat and the mariner's compass and the dipping needle, 110, 111; dynamical connection between magnetism and gyrostatic phenomena, 111. 13. How the lecturer spun his tops, using electro-motors, 112-114. 14. _Light_, _magnetism_, _and molecular spinning tops_, 115-128. Light takes time to travel, 115; the electro-magnetic {136} theory of light, 116, 117; signalling through fogs and buildings by means of a new kind of radiation, 117; Faraday's rotation of the plane of polarization by magnetism, with illustrations and models, 118-124; chain of gyrostats, 124; gyrostat as a pendulum bob, 126; Thomson's mechanical illustration of Faraday's experiment, 127, 128. 15. _Conclusion_, 129-132. The necessity for cultivating the observation, 129; future discovery, 130; questions to be asked one hundred years hence, 131; knowledge the thing most to be wished for, 132. * * * * * {137} APPENDIX I. THE USE OF GYROSTATS. In 1874 two famous men made a great mistake in endeavouring to prevent or diminish the rolling motion of the saloon of a vessel by using a rapidly rotating wheel. Mr. Macfarlane Gray pointed out their mistake. It is only when the wheel is allowed to _precess_ that it can exercise a steadying effect; the moment which it then exerts is equal to the angular speed of the precession multiplied by the moment of momentum of the spinning wheel. It is astonishing how many engineers who know the laws of motion of mere translation, are ignorant of angular motion, and yet the analogies between the two sets of laws are perfectly simple. I have set out these analogies in my book on _Applied Mechanics_. The last of them between centripetal force on a body moving in a curved path, and torque or moment on a rotating body is the simple key to all gyrostatic or top calculation. When the spin of a top is greatly reduced it is necessary to remember that the total moment of momentum is not about the spinning axis (see my _Applied Mechanics_, page 594); correction for this is, I suppose, what introduces the complexity which scares students from studying the vagaries of tops; but in all cases that are likely to come before an engineer it would be absurd to study {138} such a small correction, and consequently calculation is exceedingly simple. Inventors using gyrostats have succeeded in doing the following things-- (1) Keeping the platform of a gun level on board ship, however the ship may roll or pitch. Keeping a submarine vessel or a flying machine with any plane exactly horizontal or inclined in any specified way.[14] It is easy to effect such objects without the use of a gyrostat, as by means of spirit levels it is possible to command powerful electric or other motors to keep anything always level. The actual methods employed by Mr. Beauchamp Tower (an hydraulic method), and by myself (an electric method), depend upon the use of a gyrostat, which is really a pendulum, the axis being vertical. (2) Greatly reducing the rolling (or pitching) of a ship, or the saloon of a ship. This is the problem which Mr. Schlick has solved with great success, at any rate in the case of torpedo boats. (3) In Mr. Brennan's Mono-rail railway, keeping the resultant force due to weight, wind pressure, centrifugal force, etc., exactly in line with the rail, so that, however the load on a wagon may alter in position, and although the wagon may be going round a curve, it is quickly brought to a position such that there are no forces tending to alter its angular position. The wagon leans over towards the wind or towards the centre of the curve of the rail so as to be in equilibrium. (4) I need not refer to such matters as the use of gyrostats for the correction of compasses on board ship, referred to in page 111. {139} [Illustration: FIG. 1.] {140} Problems (2) and (3) are those to which I wish to refer. For a ship of 6,000 tons Mr. Schlick would use a large wheel of 10 to 20 tons, revolving about an axis E F (fig. 1) whose mean position is vertical. Its bearings are in a frame E C F D which can move about a thwart-ship axis C D with a precessional motion. Its centre of gravity is below this axis, so that like the ship itself the frame is in stable equilibrium. Let the ship have rolled through an angle R from its upright position, and suppose the axis E F to have precessed through the angle P from a vertical position. Let the angular velocity of rolling be called [.R], and the angular velocity of precession [.P]; let the moment of momentum of the wheel be m. For any vibrating body like a ship it is easy to write out the equation of motion; into this equation we have merely to introduce the moment m [.P] diminishing R; into the equation for P we merely introduce the moment m [.R] increasing P. As usual we introduce frictional terms; in the first place F [.R] (F being a constant co-efficient) stilling the roll of the ship; in the second case f [.P] a fluid friction introduced by a pair of dash pots applied at the pins A and B to still the precessional vibrations of the frame. It will be found that the angular motion P is very much greater than the roll R. Indeed, so great is P that there are stops to prevent its exceeding a certain amount. Of course so long as a stop acts, preventing precession, the roll of the ship proceeds as if the gyrostat wheel were not rotating. Mr. Schlick drives his wheels by steam; he will probably in future do as Mr. Brennan does, drive them by electromotors, and keep them in air-tight cases in good vacuums, because the loss of energy by friction against an atmosphere is proportional to the density of the atmosphere. The solution of the equations to find the nature of the R and P motions is sometimes tedious, but requires no great amount of mathematical knowledge. In a case considered by me of {141} a 6,000 ton ship, the period of a roll was increased from 14 to 20 seconds by the use of the gyrostat, and the roll rapidly diminished in amount. There was accompanying this slow periodic motion, one of a two seconds' period, but if it did appear it was damped out with great rapidity. Of course it is assumed that, by the use of bilge keels and rolling chambers, and as low a metacentre as is allowable, we have already lengthened the time of vibration, and damped the roll R as much as possible, before applying the gyrostat. I take it that everybody knows the importance of lengthening the period of the natural roll of a ship, although he may not know the reason. The reason why modern ships of great tonnage are so steady is because their natural periodic times of rolling vibration are so much greater than the probable periods of any waves of the sea, for if a series of waves acts upon a ship tending to make it roll, if the periodic time of each wave is not very different from the natural periodic time of vibration of the ship, the rolling motion may become dangerously great. If we try to apply Mr. Schlick's method to Mr. Brennan's car it is easy to show that there is instability of motion, whether there is or is not friction. If there is no friction, and we make the gyrostat frame unstable by keeping its centre of gravity above the axis C D, there will be vibrations, but the smallest amount of friction will cause these vibrations to get greater and greater. Even without friction there will be instability if m, the moment of momentum of the wheel, is less than a certain amount. We see, then, that no form of the Schlick method, or modification of it, can be applied to solve the Brennan problem. {142} [Illustration: FIG. 2.] {143} Mr. Brennan's method of working is quite different from that of Mr. Schlick. Fig. 2 shows his model car (about six feet long); it is driven by electric accumulators carried by the car. His gyrostat wheels are driven by electromotors (not shown in fig. 3); as they are revolving in nearly vacuous spaces they consume but little power, and even if the current were stopped they would continue running at sufficiently high speeds to be effective for a length of time. Still it must not be forgotten that energy is wasted in friction, and work has to be done in bringing the car to a new position of equilibrium, and this energy is supplied by the electromotors. Should the gyrostats really stop, or fall to a certain low speed, two supports are automatically dropped, one on either side of the car; each of them drops till it reaches the ground; one of them dropping, perhaps, much farther than the other. The real full-size car, which he is now constructing, may be pulled with other cars by any kind of locomotive using electricity or petrol or steam, or each of the wheels may be a driving wheel. He would prefer to generate electropower on his train, and to drive every wheel with an electric motor. His wheels are so independent of one another that they can take very quick curves and vertical inequalities of the rail. The rail is fastened to sleepers lying on ground that may have sidelong slope. The model car is supported by a mono-rail bogie at each end; each bogie has two wheels pivoted both vertically and horizontally; it runs on a round iron gas pipe, and sometimes on steel wire rope; the ground is nowhere levelled or cut, and at one place the rail is a steel wire rope spanning a gorge, as shown in fig. 2. It is interesting to stop the car in the middle of this rope and to swing the rope sideways to see the automatic balancing of the car. The car may be left here or elsewhere balancing itself with nobody in charge of it. If the load on the car--great lead weights--be dumped about into new positions, the car adjusts itself to the new conditions with great {144} quickness. When the car is stopped, if a person standing on the ground pushes the car sidewise, the car of course pushes in opposition, like an indignant animal, and by judicious pushing and yielding it is possible to cause a considerable tilt. Left now to itself the car rights itself very quickly. [Illustration: FIG. 3.] {145} [Illustration: FIG. _3^b_ (showing the ground-plan of Fig. 3).] {146} Fig. 3 is a diagrammatic representation of Mr. Brennan's pair of gyrostats in sectional elevation and plan. The cases G and G', inside which the wheels F and F' are rotating _in vacuo_ at the same speed and in opposite directions (driven by electromotors not shown in the figure), are pivoted about vertical axes E J and E' J'. They are connected by spur-toothed segments J J and J' J', so that their precessional motions are equal and opposite. The whole system is pivoted about C, a longitudinal axis. Thus when precessing so that H comes out of the paper, so will H', and when H goes into the paper, so does H'. When the car is in equilibrium the axes K H and K' H' are in line N O O' N' across the car in the plane of the paper. They are also in a line which is at right angles to the total resultant (vertical or nearly vertical) force on the car. I will call N O O' N' the mid position. Let ½m be the moment of momentum of either wheel. Let us suppose that suddenly the car finds that it is not in equilibrium because of a gust of wind, or centrifugal force, or an alteration of loading, so that the shelf D comes up against H, the spinning axis (or a roller revolving with the spinning axis) of the gyrostat. H begins to roll away from me, and if no slipping occurred (but there always is slipping, and, indeed, slipping is a necessary condition) it would roll, that is, the gyrostats would precess with a constant angular velocity [alpha], and exert the moment m[alpha] upon the shelf D, and therefore on the car. It is to be observed that this is greater as the diameter of the rolling part is greater. This precession continues until the roller and the shelf cease to touch. At first H lifts with the shelf, and afterwards the shelf moving downwards is followed for some distance by the roller. If the tilt had been in the opposite direction the shelf D' would have acted upwards upon the roller H', and caused just the opposite kind of precession, and a moment of the opposite kind. We now have the spindles out of their mid position; how are they brought back from O Q and O' Q' to O N and O' N', {147} but with H permanently lowered just the right amount? It is the essence of Mr. Brennan's invention that after a restoring moment has been applied to the car the spindles shall go back to the position N O O' N' (with H permanently lowered), so as to be ready to act again. He effects this object in various ways. Some ways described in his patents are quite different from what is used on the model, and the method to be used on the full-size wagon will again be quite different. I will describe one of the methods. Mr. Brennan tells me that he considers this old method to be crude, but he is naturally unwilling to allow me to publish his latest method. D' is a circular shelf extending from the mid position in my direction; D is a similar shelf extending from the mid position into the paper, or away from me. It is on these shelves that H' and H roll, causing precession away from N O O' N', as I have just described. When H' is inside the paper, or when H is outside the paper, they find no shelf to roll upon. There are, however, two other shelves L and L', for two other rollers M and M', which are attached to the frames concentric with the spindles; they are free to rotate, but are not rotated by the spindles. When they are pressed by their shelves L or L' this causes negative precession, and they roll towards the N O O' N' position. There is, of course, friction at their supports, retarding their rotation, and therefore the precession. The important thing to remember is that H and H', when they touch their shelves (when one is touching the other is not touching) cause a precession away from the mid position N O O' N' at a rate [alpha], which produces a restoring moment m[alpha] of nearly constant amount (except for slipping), whereas where M or M' touches its shelf L or L' (when one is touching the other is not touching) the pressure on the shelf and friction determine the rate of the precession towards the mid position N O O' N', {148} as well as the small vertical motion. The friction at the supports of M and M' is necessary. Suppose that the tilt from the equilibrium position to be corrected is R, when D presses H upward. The moment m[alpha], and its time of action (the total momental impulse) are too great, and R is over-corrected; this causes the roller M' to act on L', and the spindles return to the mid position; they go beyond the mid position, and now the roller H' acts on D', and there is a return to the mid position, and beyond it a little, and so it goes on, the swings of the gyrostats out of and into the mid position, and the vibrations of the car about its position of equilibrium getting rapidly less and less until again neither H nor H', nor M nor M' is touching a shelf. It is indeed marvellous to see how rapidly the swings decay. Friction accelerates the precession away from N O O' N'. Friction retards the precession towards the middle position. It will be seen that by using the two gyrostats instead of one when there is a curve on the line, although the plane N O O' N' rotates, and we may say that the gyrostats precess, the tilting couples which they might exercise are equal and opposite. I do not know if Mr. Brennan has tried a single gyrostat, the mid position of the axis of the wheel being vertical, but even in this case a change of slope, or inequalities in the line, might make it necessary to have a pair. It is evident that this method of Mr. Brennan is altogether different in character from that of Mr. Schlick. Work is here actually done which must be supplied by the electromotors. One of the most important things to know is this: the Brennan model is wonderfully successful; the weight of the apparatus is not a large fraction of the weight of the wagon; will this also be the case with a car weighing 1,000 times as {149} much? The calculation is not difficult, but I may not give it here. If we assume that suddenly the wagon finds itself at the angle R from its position of equilibrium, it may be taken that if the size of each dimension of the wagon be multiplied by n, and the size of each dimension of the apparatus be multiplied by p, then for a sudden gust of wind, or suddenly coming on a curve, or a sudden shift of position of part of the cargo, R may be taken as inversely proportional to n. I need not state the reasonable assumption which underlies this calculation, but the result is that if n is 10, p is 7.5. That is, if the weight of the wagon is multiplied by 1,000, the weight of the apparatus is only multiplied by 420. In fact, if, in the model, the weight of the apparatus is 10 per cent. of that of the wagon, in the large wagon the weight of the apparatus is only about 4 per cent. of that of the wagon. This is a very satisfactory result.[15] My calculations seem to show that Mr. Schlick's apparatus will form a larger fraction of the whole weight of a ship, as the ship is larger, but in the present experimental stage of the subject it is unfair to say more than that this seems probable. My own opinion is that large ships are sufficiently steady already. In both cases it has to be remembered that if the _diameter_ of the wheel can be increased in greater proportion than the dimensions of ship or wagon, the proportional weight of the apparatus may be diminished. A wheel of twice the diameter, but of the same weight, may have twice the moment of momentum, and may therefore be twice as effective. I assume the stresses in the material to be the same. * * * * * {150} APPENDIX II. Page 23; note at line 3. Prof. Osborne Reynolds made the interesting remark (_Collected Papers_, Vol. ii., p. 154), "That if solid matter had certain kinds of internal motions, such as the box has, pears differing, say, from apples, the laws of motion would not have been discovered; if discovered for pears they would not have applied to apples." Page 38; note at line 8. The motion of a rifle bullet is therefore one of precession about the tangent to the path. The mathematical solution is difficult, but Prof. Greenhill has satisfied himself mathematically that air friction damps the precession, and causes the axis of the shot to get nearer the tangential direction, so that fig. 10 illustrates what would occur in a vacuum, but not in air. It is probable that this statement applies only to certain proportions of length to diameter. Page 129; note at line 5. Many men wonder how the ether can have the enormous rigidity necessary for light transmission, and yet behave like a frictionless fluid. 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[4] In 1746 Benjamin Robins taught the principles of rifling as we know them now. He showed that the _spin_ of the round bullet was the most important thing to consider. He showed that even the bent barrel of a gun did not deflect the bullet to anything like the extent that the spin of the bullet made it deflect in the opposite direction. [5] NOTE.--In Fig. 16 the axis is shown inclined, but, only that it would have been more troublesome to illustrate, I should have preferred to show the precession occurring when the axis keeps horizontal. [6] When this lecture containing the above statement was in the hands of the printers, I was directed by Prof. Fitzgerald to the late Prof. Jellet's _Treatise on the Theory of Friction_, published in 1872, and there at page 18 I found the mathematical explanation of the rising of a top. [7] Roughly, the _Inertia_ or _Mass_ of a body expresses its resistance to change of mere translational velocity, whereas, the _Moment of Inertia_ of a body expresses its resistance to change of rotational velocity. [8] It is a very unlikely, and certainly absurd-looking, hypothesis, but it seems that it is not contradicted by any fact in spectrum analysis, or even by any probable theory of the constitution of the interstellar ether, that the stars are merely images of our own sun formed by reflection at the boundaries of the ether. [9] Sir William Thomson has performed this. [10] It must be remembered that in one case I speak of the true north, and in the other of the magnetic north. [11] Rotating a large mass of iron rapidly in one direction and then in the other in the neighbourhood of a delicately-suspended magnetic needle, well protected from air currents, ought, I think, to give rise to magnetic phenomena of very great interest in the theory of magnetism. I have hitherto failed to obtain any trace of magnetic action, but I attribute my failure to the comparatively slow speed of rotation which I have employed, and to the want of delicacy of my magnetometer. [12] I had applied for a patent for this system of signalling some time before the above words were spoken, but although it was valid I allowed it to lapse in pure shame that I should have so unblushingly patented the use of the work of Fitzgerald, Hertz, and Lodge. [13] How to see by electricity is perfectly well known, but no rich man seems willing to sacrifice the few thousands of pounds which are necessary for making the apparatus. If I could spare the money and time I would spend them in doing this thing--that is, I think so--but it is just possible that if I could afford to throw away three thousand pounds, I might feel greater pleasure in the growth of a great fortune than in any other natural process. [14] Probably first described by Mr. Brennan. [15] The weight of Mr. Brennan's loaded wagon is 313 lb., including gyrostats and storage cells. His two wheels weigh 13 lb. If made of nickel steel and run at their highest safe speed they would weigh much less. * * * * * Changes made against printed original. Page 91. "all that we should have to take into account": duplicated 'that' in original. Page 150. "applied to apples": 'applied to applies' in original. Advertisements. "Persia ... by the Rev. Professor Sayce": 'Professsor' in original. 
634	----	This is a collection of mathematical constants... These numbers have been downloaded from: "http://www.cecm.sfu.ca/projects/ISC/I_d.html" An index of high precision tables of functions can be found at: "http://www.cecm.sfu.ca/projects/ISC/rindex.html" You can find information about some of the constants below at: "http://www.mathsof.com/asolve/constant/constant.html" Thank you to Simon Plouffe (from Simon Fraser University) for his kind permission to distribute this collection of constants. ----------------------------------------------------------------------------- Contents -------- 1-6/(Pi^2) to 5000 digits. 1/log(2) the inverse of the natural logarithm of 2 to 2000 places. 1/sqrt(2*Pi) to 1024 digits. sum(1/2^(2^n),n=0..infinity). to 1024 digits. 3/(Pi*Pi) to 2000 digits. arctan(1/2) to 1000 digits. The Artin's Constant = product(1-1/(p**2-p),p=prime) The Backhouse constant The Berstein Constant The Catalan Constant The Champernowne Constant Copeland-Erdos constant cos(1) to 15000 digits. The cube root of 3 to 2000 places. 2**(1/3) to 2000 places Zeta(1,2) ot the derivative of Zeta function at 2. The Dubois-Raymond constant exp(1/e) to 2000 places. Gompertz (1825) constant exp(2) to 5000 digits. exp(E) to 2000 places. exp(-1)**exp(-1) to 2000 digits. The exp(gamma) to 1024 places. exp(-exp(1)) to 1024 digits. exp(-gamma) to 500 digits. exp(-1) = exp(Pi) to 5000 digits. exp(-Pi/2) also i**i to 2000 digits. exp(Pi/4) to 2000 digits. exp(Pi)-Pi to 2000 digits. exp(Pi)/Pi**E to 1100 places. Feigenbaum reduction parameter Feigenbaum bifurcation velocity constant Fransen-Robinson constant. gamma or Euler constant GAMMA(1/3) to 256 digits. GAMMA(1/4) to 512 digits. The Euler constant squared to 2000 digits. GAMMA(2/3) to 256 places gamma cubed. to 1024 digits. GAMMA(3/4) to 256 places. gamma**(exp(1) to 1024 digits. 2**sqrt(2) a transcendental number to 2000 digits. Si(Pi) or the Gibbs Constant to 1024 places. The Gauss-Kuzmin-Wirsing constant. The golden ratio : (1+sqrt(5))/2 to 20000 places. The Golomb constant. Grothendieck's majorant. 1/W(1), the inverse of the omega number : W(1). Khinchin constant to 1024 digits. Landau-Ramanujan constant The Lehmer constant to 1000 digits. Lemniscate constant or Gauss constant. The Lengyel constant. The Levy constant. log(10) the natural logarithm of 10 to 2000 digits. The log10 of 2 to 2000 digits. log(2), natural logarithm of 2 to 2000 places. log(2) squared to 2000 digits. log(2*Pi) to 2000 places. log(3), natural logarithm of 3 to 2000 places. log(4)/log(3) to 1024 places. -log(gamma) to 1024 digits. The log of the log of 2 to 2000 digits, absolute value. log(Pi) natural logarithm of Pi to 2000 places. The Madelung constant Minimal y of GAMMA(x) BesselI(1,2)/BesselI(0,2); The omega constant or W(1). 1/(one-ninth constant) The Parking or Renyi constant. Pi/2*sqrt(3) to 2000 digits. Pi**exp(1) to 2000 digits. Pi^2 to 10000 digits. The Smallest Pisot-Vijayaraghavan number. arctan(1/2)/Pi, to 1024 digits. product(1+1/n**3,n=1..infinity) exp(Pi*sqrt(163)), the Ramanujan number The Robbins constant Salem Constant sin(1) to 1024 digits. 2**(1/4) to 1024 places. sqrt(3)/2 to 5000 digits. sum(1/binomial(2*n,n),n=1..infinity) to 1024 digits. sum(1/(n*binomial(2*n,n)),,n=1..infinity); to 1024 digits. sum(1/n^n,n=1..infinity); to 1024 places. The Traveling Salesman Constant The Tribonacci constant The twin primes constant. The Varga constant, the one/ninth constant -Zeta(1,1/2). -Zeta(-1/2) to 256 digits. Zeta(2) or Pi**2/6 to 10000 places. Zeta(3) or Apery constant to 2000 places. Zeta(4) or Pi**4/90 to 10000 places. Zeta(5), the sum(1/n**5,n=1..infinity) to 512 digits. Zeta(7) to 512 places : sum(1/n**7,n=1..infinity) Zeta(9) or sum(1/n**9,n=1..infinity) The Hard hexagons Entropy Constant ----------------------------------------------------------------------------- 1-6/(Pi^2) to 5000 digits. .39207289814597337133672322074163416657384735196652070692634580863496127422658 735285274435644626897431826653343088568250991562839408348952558397869101910044 168287418837630346090344128142725232280081846950544588945104349233845519860235 478013752874888452546923326181835771108778185425297888417868576864617275811561 330630424192103992371844063005810729791367810232917738723885386964431826453535 905907614449167288215917896721626280528275896067038147627421438102874420209114 283031089287791823583188720457836037724958727540937325971240235006933941887088 652273182790886585142931926559181988974866244340862951315812052809204750474816 430247023973718215153491786148753491003381673460786833208291818530068999090721 752421441534903029493841963810349129854816275432069261689883499042672794563279 299504180713102088765758949225794484407306891253577533262758052911265557952815 325040663628650312916901015777561782819610508727218752638400753963946901892734 396711153225803445533941568858632445301649742519165316441371609711531245089243 290549824649975134158044128818527386726565538183303018146350709277119694372345 677582608647163425438890427150410024157713718860965862131327245429890180475113 153411263994036956927450905854836195277537880204828534118620902663388920837997 660386215683412323571455281034788094296469957634407205979637839396999291268859 280494867831202839632408231414702965284181311318387323905136101845230649191328 344204506538210488338362999418725024491290968463024341230939260937210637763357 668716325043532540720756824043914962647749839154837035616512309032638541576246 512363428759766225539481944983492434326527204170645681513760558107716849614234 624284323701601285720556600781803702070830269262536977533958130472783157895527 099648524055663579209506965406389148701201411165643257462862545248916282535924 283109135878831217758425399659926807364022613100715042102603188631532662678255 793368462608650127902461290448248933845382593062932405288099147085163337644259 096942457982869681884492751291945213055219225791268428646737404748762908271223 988080461936745870026987077963833251743802479327783763199318341165695354688986 587709006638984740347519367402758489989916610040443071767511540635748990264849 985865097486689959900054636548278168659769020552203441195594619095883719967595 163286233850666913354175920848129816950224785210602307170200324097923815543904 765622453721166092941083477472617302559945103931826892133402269758301852813673 313787284287044516786005234330589325533869618136662526023138681759816054564830 823941376406346235393059115570371588897152889961892481410619643955709600104785 676501470053957334404492263310332087541957463774082958556187073996705969238130 327560015852814044634211981886674723986988897022825327742090060873707979236631 087584065217349162647213909628975630351127856180937849171897544173187997735431 685164552213725723547887766893999809160919964767617090344204462604438931997737 794915491499507617367123976245445662884386972100089268492901081935107944719414 842581272481248389212828409389631643367179863342024797779288701814583298838958 832929265318994914512229305037934174323166686217001570566648749237816190371530 970670094366863915185878559044766538509033560898561258893529669960565355241845 298083885988208630792383965443493189702162463545680223954782323399990578055375 238166359760380063033268621526458667579176419424938930517625097922755311183710 745112135686482997935258127774601766702374701246949854388893425578843578779948 388764843816364323561857066550454768564160400372163688443710008619746963248721 285450733227692713183294357334410215067068643812289378210321931889489656820466 809967506206366896603638875961668977227433190924290041768209356873325152340791 912813035141325564405189779991290242963653040502971303969510916052321346803263 616347582473895485425915642466980587305549090607684017625337525040913199423035 386079297039620191288580069298373556249429733144977260490424072181862404526826 817071944122527238086545093279183706840284479537951297285943981771954473076657 685349498593661734118944882704643158420248935512451236217550768155753514781095 976468804752093019662179762466247347751258878263063530932485519204198416357559 668554659240563023943272791577074043369103540249505902292249986184531429207320 441603873665542536935000660592213839517613747530936270214955498346033948852217 917507874321865944958743538264769258134043919235895761280482175277831086617230 368023430246344754243125747354652746626371109702030400946709013790121636923479 334262138445002114553966856917269467351180089470344256454746771666622342456492 176453740878253161674182945911059921426724376964460732328571172726217308529523 262825426126910937230270053544839512546829497880117246462299113726750444859334 558341040251310724340825881906883649884796840752694488592880986955465404606887 058715891120910975896486172581109538650183092274820509139397244697423368852508 154738304143183735570326011149855299682867699250414750565458319892944377315536 391971718447000833094110391910495202247093032743184900344414039480499297560832 897901104 ----------------------------------------------------------------------------- 1/log(2) the inverse of the natural logarithm of 2. to 2000 places. 1.4426950408889634073599246810018921374266459541529859341354494069311092191811 850798855266228935063444969975183096525442555931016871683596427206621582234793 362745373698847184936307013876635320155338943189166648376431286154240474784222 894979047950915303513385880549688658930969963680361105110756308441454272158283 449418919339085777157900441712802468483413745226951823690112390940344599685399 061134217228862780291580106300619767624456526059950737532406256558154759381783 052397255107248130771562675458075781713301935730061687619373729826758974156238 179835671034434897506807055180884865613868329177321829349139684310593454022025 186369345262692150955971910022196792243214334244941790714551184993859212216753 653113007746327672064612337411082119137944333984805793109128776096702003757589 981588518061267880997609562525078410248470569007687680584613278654747820278086 594620609107490153248199697305790152723247872987409812541000334486875738223647 164945447537067167595899428099818267834901316666335348036789869446887091166604 973537292586072129486973545407080983067489383412371863140083597961886597586874 525330546892129766415704206212592463136924216805908774083358139286665415849711 625870695565785887476996312969525004593726273890268056693551287294338372191311 166508810015878626559156379540559056778223681400309688439348086228481847913456 331411930238402640972748436449621954492244652220471763586074796585566605340982 860985740278837433126885633544343069787018964358261391181002525990207661844329 848831847239159127013904570477357648310102119282970853289609316803539196498695 732643937914903084854706164337898563482389000045642618556224969309139603125202 237673760741538621162455511650864367991293893712255727528553585053886275469281 675504073039189843896410520398990210789077410746707154871874459278264803257453 294068365525441034657373203151382251293614376241422022507143703697307346094148 501086031893236041133111157449377024914688145536097228616724252720888890615174 510525315591783162470294301780959342523719751256123 ----------------------------------------------------------------------------- 1/sqrt(2*Pi) to 1024 digits. .39894228040143267793994605993438186847585863116493465766592582967065792589930 183850125233390730693643030255886263518268551099195455583724299621273062550770 634527058272049931756451634580753059725364273208366959347827170299918641906345 603280893338860670465365279671686934195477117721206532537536913347875056042405 570488425818048231790377280499717633857536399283914031869328369477175485823977 505444792776115507041270396967248504733760381481392390130056467602335630557008 570072664110001572156395357782312341095260906926908924456724555467210574392891 525673510930385068078318351980655196468743818998016595978188772145886161745990 050171296094036631329384620186504530996681431649143242106041745529453928221968 879979271810612541370164453636765287464840612259774030275763201370942219451172 546547075844214142250283806186859413525755477454980153057834914761302200742289 202782109330263327658274294341361264338498005796358789443727517115501354585988 939374551889434073832049151982961930707176175080332908654736428226919459067537 99881712938 ----------------------------------------------------------------------------- sum(1/2^(2^n),n=0..infinity). to 1024 digits. 0.8164215090218931437080797375305252217033113759205528043412109038 4305561419455530006048531324839726561755884354820793393249334253 1385023703470168591803162501641378819505539721136213701923284523 4283123411030157746618769850665609087759577356088592708255670961 1511603255836101453412728095225302660486164829592085247749725419 1191271500533834073674513177454416699480215530972684390616972105 9958065039379297587005270471610028297428995734644505701701103082 6930529896276673940020997391153902511692115693331856436193281886 7356259335520938127016626541645397371801227949921479099121251589 7719252957621869994522193843748736289511599560877623254242109788 8031249582337843804332880240487467096566555049952788767180351255 3443784826960014018156912683901006125559846031156431128801995466 7849660214879231535089640098219689014895803216854654610987884309 3375147537123678256705617554490069667937389945110543099411044968 8572271298811057185720835831609174885658074423123956455857403738 8490440331108074066818018534205109244035940825937632942762395325 ----------------------------------------------------------------------------- 3/(Pi*Pi) to 2000 digits. .30396355092701331433163838962918291671307632401673964653682709568251936288670 632357362782177686551284086673328455715874504218580295825523720801065449044977 915856290581184826954827935928637383859959076524727705527447825383077240069882 260993123562555773726538336909082114445610907287351055791065711567691362094219 334684787903948003814077968497094635104316094883541130638057306517784086773232 047046192775416355892041051639186859735862051966480926186289280948562789895442 858484455356104088208405639771081981137520636229531337014379882496533029056455 673863408604556707428534036720409005512566877829568524342093973595397624762591 784876488013140892423254106925623254498309163269606583395854090734965500454639 123789279232548485253079018094825435072591862283965369155058250478663602718360 350247909643448955617120525387102757796346554373211233368620973544367221023592 337479668185674843541549492111219108590194745636390623680799623018026549053632 801644423387098277233029215570683777349175128740417341779314195144234377455378 354725087675012432920977935590736306636717230908348490926824645361440152813827 161208695676418287280554786424794987921143140569517068934336377285054909762443 423294368002981521536274547072581902361231059897585732940689548668305539581001 169806892158293838214272359482605952851765021182796397010181080301500354365570 359752566084398580183795884292648517357909344340806338047431949077384675404335 827897746730894755830818500290637487754354515768487829384530369531394681118321 165641837478233729639621587978042518676125080422581482191743845483680729211876 743818285620116887230259027508253782836736397914677159243119720946141575192882 687857838149199357139721699609098148964584865368731511233020934763608421052236 450175737972168210395246517296805425649399294417178371268568727375541858732037 858445432060584391120787300170036596317988693449642478948698405684233668660872 103315768695674936048769354775875533077308703468533797355950426457418331177870 451528771008565159057753624354027393472390387104365 ----------------------------------------------------------------------------- arctan(1/2) to 1000 digits. 0.46364760900080611621425623146121440202853705428612026381093308872019786416574 170530060028398488789255652985225119083751350581818162501115547153056994410562 071933626616488010153250275598792580551685388916747823728653879391801251719948 401395583818511509502163330649387215460973207855555720860146322756524267305218 045746400869745058389736389648900264868778537801282363312171645781468369009933 405288824862445623881190901589497679971970114967760016450062530168121256093353 041349396630129319242748402931611194920616208441593723612731668769816870275931 895103339733259290385128925459459224632156097836380095374993209486073394918643 251602748279304503733177255465049960867577062275441628502227372371197447336697 731851069401381126995777925627482566009621167267481152728272252072259726842157 101958775620917015577687098665426689034493518054728900537078381242128547943030 243678452646699376838088771904127673115937480616288330320288044652395896189241 30515270876726439400070443923542442569122697771151892771722644634 ----------------------------------------------------------------------------- The Artin's Constant. = product(1-1/(p**2-p),p=prime) Reference : Wrench, John W., Jr. Evaluation of Artin's constant and the twin-prime constant. (English) Math. Comp. 15 1961 396--398. 0.373955813619202288054728054346516415111629249 ----------------------------------------------------------------------------- The Backhouse constant calculated by Philippe Flajolet INRIA Paris to 1300 places. 1.4560749485826896713995953511165435576531783748471315402707024 374140015062653898955996453194018603091099251436196347135486077 516491312123142920351770128317405369527499880254869230705808528 451124053000179297856106749197085005775005438769180068803215980 620273634173560481682324390971937912897855009041182006889374170 524605523103968123415765255124331292772157858632005469569315813 246500040902370666667117547152236564044351398169338973930393708 455830836636739542046997815299374792625225091766965656321726658 531118262706074545210728644758644231717911597527697966195100532 506679370361749364973096351160887145901201340918694999972951200 319685565787957715446072017436793132019277084608142589327171752 140350669471255826551253135545512621599175432491768704927031066 824955171959773604447488530521694205264813827872679158267956816 962042960183918841576453649251600489240011190224567845202131844 607922804066771020946499003937697924293579076067914951599294437 906214030884143685764890949235109954378252651983684848569010117 463899184591527039774046676767289711551013271321745464437503346 595005227041415954600886072536255114520109115277724099455296613 699531850998749774202185343255771313121423357927183815991681750 625176199614095578995402529309491627747326701699807286418966752 89794974645089663963739786981613361814875; ----------------------------------------------------------------------------- The Berstein Constant. 0.28016949902386913303643649123067200004248213981236 ----------------------------------------------------------------------------- The Catalan Constant. As calculated by Greg Fee using Maple Release 3 standard Catalan evaluation. This implementation uses 1 bit/term series of Ramanujan. Calculated on April 25 1996 in approx. 10 hours of CPU on a SGI R4000 machine. To do the same on your machine just type this. > catalan := evalf(Catalan,50100): bytes used=37569782748, alloc=5372968, time=38078.95 here are the 50000 digits (1000 lines of 50 digits each). it comes from formula 34.1 of page 293 of Ramanujan Notebooks,part I, the series used is by putting x--> -1/2 . in other words the formula used is : the ordinary formula for Catalan sum((-1)**(n+1)/(2*n+1)**2,n=0..infinity) and then you apply the Euler Transform to it : ref : Abramowitz & Stegun page , page 16. the article of Greg Fee that took those formulas appear in Computation of Catalan's constant using Ramanujan's Formula, by Greg Fee, ACM 1990, Proceedings of the ISAAC conference, 1990 (MAYBE 1989), held in Tokyo. catalan := 0. 91596559417721901505460351493238411077414937428167 21342664981196217630197762547694793565129261151062 48574422619196199579035898803325859059431594737481 15840699533202877331946051903872747816408786590902 47064841521630002287276409423882599577415088163974 70252482011560707644883807873370489900864775113225 99713434074854075532307685653357680958352602193823 23950800720680355761048235733942319149829836189977 06903640418086217941101917532743149978233976105512 24779530324875371878665828082360570225594194818097 53509711315712615804242723636439850017382875977976 53068370092980873887495610893659771940968726844441 66804621624339864838916280448281506273022742073884 31172218272190472255870531908685735423498539498309 91911596738846450861515249962423704374517773723517 75440708538464401321748392999947572446199754961975 87064007474870701490937678873045869979860644874974 64387206238513712392736304998503539223928787979063 36440323547845358519277777872709060830319943013323 16712476158709792455479119092126201854803963934243 49565375967394943547300143851807050512507488613285 64129344959502298722983162894816461622573989476231 81954200660718814275949755995898363730376753385338 13545031276817240118140721534688316835681686393272 93677586673925839540618033387830687064901433486017 29810699217995653095818715791155395603668903699049 39667538437758104931899553855162621962533168040162 73752130120940604538795076053827123197467900882369 17861557338912441722383393814812077599429849172439 76685756327180688082799829793788494327249346576074 90543874819526813074437046294635892810276531705076 54797449483994895947709278859119584872412786608408 85545978238124922605056100945844866989585768716111 71786662336847409949385541321093755281815525881591 50222824445444171860994658815176649607822367897051 92697113125713754543701243296730572468450158193130 16087766215650957554679666786617082347682558133518 68193774565001456526170409607468895393023479198060 00842455621751084234717363878793695778784409337922 19894575340961647424554622478788002922914803690711 52707955455054147826884981852460058144665178681423 15411487855409966516738539727614697016904391511490 08933307918457465762099677548123138201543601098852 72162977010876157478173564163698570355340672649351 96316955476721150777231590044833826051611638343086 51397972251617413853812932480119463625188008403981 94553905518210424606292185217560246548601929767239 74051103952645692429786421242403751892678729602717 73378738379978326676208611952067912151263821192523 29404069205994386427469321533885667117330827142408 33265920326075316592804231023099735840039594034263 22276880701186819617678090563158159784537637578356 37359027716488313102887693795053507320801807581022 38230803176250432942472226839122971295535135510431 47618866554743676921841201887716179922856205635220 54703200691808688066121174204060992412348760515406 82022625595048124858941187358346822904230836155547 69477770831940874812491674892900659369616416623436 83707543963838945144011955648738134292122982001302 10799619224249244930519992358581580826035249799850 59186697220123164897104830701793528112228966355128 31743735239301140279238980874456964830901320787765 87853623013542800016290558772950067958761782473748 71378060042208445346045064702443258085164777173903 19602865553832828141591524873526330715051314788284 49992386632431981063365152433113214639009333621591 60744482923457177454817169580181688900175285645046 48913909042035602983604565242526579727013858675765 38993029584492586921897886443888193581145267705631 60609737684654083694230203816826392458579107404870 87987785242614086871517857580100602368170349179773 36221966295377189138531167399655658859121646280155 82629873541376336076073020045591202946657347571852 74531163384777648683824850411630160522708694444270 36442512423639718149992349608389591682580361647498 81042639483890042940550431502193126864230059992926 36154064926266418658359490424937152362206840394037 01086807400984400015124653435350672338454694635760 21186762114341424761178341043127306116782248833969 91553909131097323106678111748553767902723184507654 57756998874113956861466315813615736740618811259146 20397423401125882131569075175754979658229689846231 32925727317533830231353323287005659568853417520457 39327581835139823476780092614265210747104566687631 34325667275929891952548849037809046546488268575204 45469505381349830902146048971831938778086340901416 82854845242480931043432177247887782487394860618002 33415225914146138782700545170971410457656614928953 10867248608048420437663793623021364581779802272088 27380717367112998222890691257630277791626510357625 77038104288680376054636303337940367377696744757171 91871280395437096641387722662688983731111160200451 85939731747646215428384601621445265537202925520515 04941828003032550267579038252786139633572720650890 36782017625857363660245964491453352814103725168382 20900971019436802783367089633146724973295039192592 98514966414498521873384370124517467421871213110205 72617434013405687655510418786654451890276500538217 86094121053538997849059821800230678908216061413670 18393687028304544346780536499566495053180837980207 95036583522762200650678617717109567200562970302355 35933738697718328353375572623444156649160057626666 04199085276789703504193295554568745338842121304879 86200092870617800767859273517538652367734853505306 61253960255362808093505625628213474323943992224427 39711562755985244339104126180433506987134104280978 45686951897766888265050375616759153547317366813568 33531685884402667262031966007851949052618190161355 40883210564405409027216204498851041761292787884227 85183520070443946096157166554344839280259250115630 62276507400503123514176565264499430425705315022305 52233576634208943102385867060630430297719853224212 04329861952863316219947980302165117007185321676809 50619341672862846747533072110055118542257586292926 81406381602461376952043278677852351940897487799588 26265101885716752644896425951624560816468058666260 58443282815376692095017001316910938643914700333459 06701868799246483109181855848104631118954767258303 66892265711699056543175998868028673114587345754977 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38553435988563869996601567186058612474588988257925 98304946331331136242504115036752660594397646535926 45323276332233198607510185476341931299728219728129 36917765373540523212220572346832475001299207035100 66576238472740236038469788599572252590845226606269 18632822114580758281992501723085071150021220576439 43620707427834155896022955746495351914508567512874 92979405670865653984987174441248881733453560855937 02374789351023494268216534496793285005822181596890 50573591689721439132211505274663264207286280411226 78255908457235177947866947938347915770490640391229 35503307041121029245168359103359865202320918829972 02408484458254568473159485768244652855633966093047 98144783823574995239064329696064042759459507981505 03221094697842432880685911089418989378961575508205 16744316407054439807581169313640269129171642991630 38596531630082280670694949115602661521273858451736 94340594841447759036208225564793415527194168680049 21836995551781485477483975205304202788655061495348 48105727803004141698374905968616401793413366933407 37345271784449129478438443104629357399390409029801 48998649493484704163930702722474652200796312165756 04462268735963249448716420919976809061371188594152 14366605526985916462635332444559966262728408025592 42054257589343986404452836081389978002591330301930 93582251756346790483545106853860486128669311660258 88432830039855698508556441332625423163629132362854 71166779905984501926940735617488269912585186782940 70796775115093101315160682682243483785857958808069 97227699256020374628822496781188176087645649137817 18121890224717594255690193502795232086999862955596 57990279144342367587666229831447445776059827318326 53932717548424075514061550891204842759303457326183 74685191590684516417544141463803609413979685388345 04168589246412716863234468225891856862945460220492 05659177971313582851542004419646353313442057363431 07130843155119302950777358076785333838806043196777 19550487613134989090019183928534780642078161985608 96774726014391749964703357741155341423161256208698 33514415443255666118206765463782722520773025214000 38108633059439276064496150103551519568294079685261 51191909731143149176354644669634322908982092314769 31916894038051233876504421677622559098506844870085 77381968193957459793916689118775200397548489475385 39847315390269771017865672090280679962749320721414 02973814580750409956165906615444836653859373403172 31348133655412308677306318216674596984243147763414 07140932740194826905648649093547830919668818468944 97861616837624118800499595576365339220832942263909 49054342120242302426912815202997098196510921544888 98180669000336073714710914841624083239490212673412 85654658027260921634828085880748855012536477024921 74541839712418712410842156966959758974826513535044 37982030839657599975548143650418053605131208618004 06348121131180290885645073485818852991283396147525 21640216417955848619478841441578374306774876011942 08469984692176661306609708544603789809627058924794 14613607001351406000098067552937395225099086837806 61764887114148968509385101497346290646398063873553 68380248229619709531076781259153037015822410213235 54465039653412311688557103292746579314045489850780 76051289621795112003265679656234838520070099452379 18695729171609920238426836033108773538343242978867 07684630403325940847966101796013645441455749556065 36761978256989928946226389427459461002145101062354 77445678675488424709769433165660365947851310628432 08112125111647545970759513170677192499928085724418 67867305645591104919069756026700970794392706246478 39137725862464455497974027505115816922954166718861 94831700328217545033892173871635914930862586695047 04809726604781614328725017472926883696305912276595 46927251633942317041790658723755282806001149141441 40325473525153569834996837500424912583198617305817 91446340864364651701438450560014619878957790034105 89939589114733114277340118162769916073566580688375 02801835671154187418527402547729677386094377127429 15582396842125410318943394469958870596228984319822 68837213198707867215620424190485607993582980429002 08606457935791525970125373231053163710610926563166 16850358757133468064030767994044406078978454509915 05874986938133580989255741557666221681594459902604 77765693393931216548249783317133410958497891874254 87783051254572174506572434355188159438775385461518 24292385902689734112299296844119619653204683588576 34687757780554977983897659194982931496779168443842 63436213698536280259738570349190955603823139917175 79105670108820269184785095523585275759268941178974 13809526891406292117233997044116162427192030976316 14664558448956035337521227987537282934170242070690 50592326179086426593228379050128544733177583153438 48132091117419463412138432286513401793430140727156 81887273228425536422698861469651935316064823119168 91574802307354674601013114597237063465514190575230 81807153666054778103118230543782434755626986526098 92909448327541354533273092186667305992077021187375 40302103124712502314359041674773979293471696975033 55720613802110189216451282476075054123773831714851 63446369964774003835027888440415544760432532075853 62250505275088067613459409267406975020479325528897 01113035082680958516000417523028203574885929493253 89303611184860456100947735717047080004090167214500 74405270358729608455072721446485581513498780806082 13521297937327044332259010629049066537039666180672 78240656203048997863778312230428209901808403796954 37718741840217161945870446401022627335998181585933 65452326877598181927503979385709516221261073350423 77698946837217187578804105526004849460164920354544 08746275385207322932703703480867578363547313180633 40788128602694851019869015427702324096815909854656 61836614932918923280531289745404950799828306646102 72694678323691114430825641845172134059094679468473 73074046195707787671207246870191107888630055547642 08188077181196184119833968574252698831654452851644 37064743601527934924356493257649035957243845506599 96092411284327238971798826825420250457224063981923 00955638310388236549807489180826624669570850430767 40017784465459119334279913895972967107031614058954 17906583737939519497303263305349199089276731952401 30243876745516359128426615887576337300834774026935 84062948601284113052065328339779358936107811863551 45975692621671550684936292327964473984063457314475 23182370855613685165285374795634877615121423095852 37503810250309527421573220614620839530850321406895 72374844879111418175913457973488676586418509863128 33977719479190274515412025099252587581579966827349 68448163562248680287787366054173416999163765665385 81095634919642246410021617453036166646767784493191 68768803760434245568232540632588889051025512587238 70200945074599656822076820223724012402082094281368 31620711042665728552343755203994535506870498582151 88144479894668176359043572476748738703131386727366 64237695726203461305176208696122708343562712933227 54244814441539762720328603234137383411832281787718 95907561204744686092884436650706532640140593743156 17021250121528978294584639139195216712554159112036 62617446988989738953558976129096935643101760553413 09761620077831766379383917299182221777169985552763 18764153621590502956600654056094291266739105832587 57789975918507663262661937192596555297915514056319 13209645903926242144949179598676274187517065656445 26851958812965483428820356484099912746214581149876 59408744652471538986596487203211007939581684178653 88761775030424132556818376436243089902251005891866 32528680409897695265152076541938056429617775037838 42477034169745357141688754466614766850847279648458 46570622777433415111929975121204331040549229806244 30619707801248858339640108442089549726785366033283 48370824846333642251242034255438554747923490542487 36973241215506297769504957694032259884052732383383 87007735474942066252923855687547595283673618969505 03254975105475685582904264280288223133583017113458 61975019784420232041785756047105885909576234677056 50430778775853587154899385008551889511670055906314 64963846456224224185199851310612230676179904894428 48831782517506544346538425133271300230240438306247 26219830610476018589074943812150970837649929489684 83123978975240059120343706965306631739923470616819 68172026390046232808553407601396429236442012304767 82267038336232588525467626853636782615075340531593 37313071457898910856926960637345974288402225005290 15520998884042809345391721847580917037513060813642 09848557290766958284218557360999719764035404970757 03445820150158257028881134452044496425524101031249 67662635529318143494937932601048277243418470639942 89534462222004129934514220744814190333789315550685 22756309684544363912187670081612847742123338998836 14812382079042635267202161427485062261887583694841 91172479483300041582873400512340189492090287674178 65440211256360975020769150161031730685245191308694 91594569139279624592810899540762589954635485223854 48757994147093065077267684512505887349957817689055 28225674785956202175853620241580733897536259435693 19362751420235075212169619136721223793548343218975 75221937495115660983627436716746887603997456876267 78025082784567540835581457350675069310164155301861 75479366277294083492223487189377371193343564738314 47218094113538389434288346905494847215674948971663 57418853498986877125372894967217015374967204028920 ----------------------------------------------------------------------------- Champernowne constant, the natural integers concatenated. this is a NORMAL number in base 10, Ref: D.G. Champernowne, The Construction of decimals normal in the scale 10, Journal of the London Math. Soc, 8, (1933). 0.123456789101112131415161718192021222324252627282930313233343536373839404142434 445464748495051525354555657585960616263646566676869707172737475767778798081828 384858687888990919293949596979899100101102103104105106107108109110111112113114 115116117118119120121122123124125126127128129130131132133134135136137138139140 141142143144145146147148149150151152153154155156157158159160161162163164165166 167168169170171172173174175176177178179180181182183184185186187188189190191192 193194195196197198199200201202203204205206207208209210211212213214215216217218 219220221222223224225226227228229230231232233234235236237238239240241242243244 245246247248249250251252253254255256257258259260261262263264265266267268269270 271272273274275276277278279280281282283284285286287288289290291292293294295296 297298299300301302303304305306307308309310311312313314315316317318319320321322 323324325326327328329330331332333334335336337338339340341342343344345346347348 349350351352353354355356357358359360361362363364365366367368369370371372373374 375376377378379380381382383384385386387388389390391392393394395396397398399400 401402403404405406407408409410411412413414415416417418419420421422423424425426 427428429430431432433434435436437438439440441442443444445446447448449450451452 453454455456457458459460461462463464465466467468469470471472473474475476477478 479480481482483484485486487488489490491492493494495496497498499500501502503504 505506507508509510511512513514515516517518519520521522523524525526527528529530 531532533534535536537538539540541542543544545546547548549550551552553554555556 557558559560561562563564565566567568569570571572573574575576577578579580581582 583584585586587588589590591592593594595596597598599600601602603604605606607608 609610611612613614615616617618619620621622623624625626627628629630631632633634 635636637638639640641642643644645646647648649650651652653654655656657658659660 661662663664665666667668669670671672673674675676677678679680681682683684685686 687688689690691692693694695696697698699700701702703704705706707708709710711712 713714715716717718719720721722723724725726727728729730731732733734735736737738 739740741742743744745746747748749750751752753754755756757758759760761762763764 765766767768769770771772773774775776777778779780781782783784785786787788789790 791792793794795796797798799800801802803804805806807808809810811812813814815816 817818819820821822823824825826827828829830831832833834835836837838839840841842 843844845846847848849850851852853854855856857858859860861862863864865866867868 869870871872873874875876877878879880881882883884885886887888889890891892893894 895896897898899900901902903904905906907908909910911912913914915916917918919920 921922923924925926927928929930931932933934935936937938939940941942943944945946 947948949950951952953954955956957958959960961962963964965966967968969970971972 973974975976977978979980981982983984985986987988989990991992993994995996997998 999 ----------------------------------------------------------------------------- Copeland-Erdos constant, the primes concatenated. Proved to be normal in base 10. 0.2357111317192329313741434753596167717379838997101103107109113127 1311371391491511571631671731791811911931971992112232272292332392 4125125726326927127728128329330731131331733133734734935335936737 3379383389397401409419421431433439443449457461463467479487491499 5035095215235415475575635695715775875935996016076136176196316416 4364765365966167367768369170170971972773373974375175776176977378 7797809811821823827829839853857859863877881883887907911919929937 9419479539679719779839919971009101310191021103110331039104910511 0611063106910871091109310971103110911171123112911511153116311711 1811187119312011213121712231229123112371249125912771279128312891 2911297130113031307131913211327136113671373138113991409142314271 4291433143914471451145314591471148114831487148914931499151115231 5311543154915531559156715711579158315971601160716091613161916211 6271637165716631667166916931697169917091721172317331741174717531 7591777178317871789180118111823183118471861186718711873187718791 8891901190719131931193319491951197319791987199319971999200320112 ----------------------------------------------------------------------------- cos(1) to 15000 digits. 0.5403023058681397174009366074429766037323104206179222276700972553811003947744717645179518560871830893 4357173116003008909786063376002166345640651226541731858471797116447447949423311792455139325433594351 7756702892596375736154327549641754491775115131222730100631357078232236771401517468995936678730674227 6202450776374406758749816178427202164558511156329688905710812427293316986852471456894904342375433094 4230240935962395831824547281736640780712434336217481003220271297578822917644683598726994264913443918 2656945351575076278251380499160730638031721445034986129488336335655779909793015287927884038980097454 8251049924537987740061453776371387833594234524168164283618828482374896327390556260912017589827502528 5999174385806924855842322178268582710882915646830067968759551300361081203367474727491810336735150934 5888830420321759659405270393476250248737075266131336984241605971059560659997869138441557441446642001 2839398870926323453338868626299654709768054836830358211823411732418465771864116514294188326444690783 8591321108965751039607059607592213323660177863514921082050310656541987948453420365753840298795631616 8157011981970355887547815277419962869363110268688811893191682997748017318703517866980706474458344000 3768601699062304869395907940670264065673462011834932716365990618313789462149345364297139262013102950 3484162582051809930594462107749878028801458089615943548951028274796450601572557509376953338025102336 5030138382522433851551922228134572839271134035925173441390906576186661661637799926895849611005089594 5140910111905132580259501544309513723985720939476474894288034135662757457443388341239875567257987784 5443398063565886106331085355741817317569001387196133456957598071765341625573943667036221555294034598 5708552810738513036224294909456716235233888778877525962168278584356199904722631795896992400744103515 2563369237496356968599770161805363912298556403527947611479952229720779961723024644092490022153525720 7426991775642129269196664606916524778858295045724845910860227611446165516523259778623762195275197559 7598738278690423089893199791243715706181235518117184633349798535115459342404647904621036119576141429 3607466912075634548915883761375863620912312830135706516519846929100660094332304358827069548826263823 1908488770766956401176948849786470380435504004834753756960400671057548254539507002983348819151735861 4598734086692434493677329587736577119246163840055471129847462892407837128650380459376193967554150580 5242328019089138696374948476035575783228204016278497018040069903724903967958484359627941479578415877 0059041001066118092143548344986183970630569497169890893982349122692316286614650145992158947638310980 0797876571632169111826159448759879455321376378672761156967784144663783839712182009726088613272017012 0465450291990583358093028318847784287222583386407249992092360939461258866104220723568750923605392971 9690140303017820623981506289149913655563065078796987157189416004614638686601943664192926349869382011 8367278959225031614593393680899199351432005207154630766350418318288779940961759164097048303106708922 7868810262114672053971388626984761678725059736800220184938164847589615401376053109380865786857293460 6446264014317775731125515946714533807639221825836208065247097397190379193401778465625365802148471292 5245855137234055963272782685593223878715505884114546568427257201206044832502508595311251215022123475 0565784594703542664668012540811642015956293906035075247838515558710172609971148468632280952657055929 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5336063197566145422977978387805759165018009510388762731827013910355064650917936734553911059085987183 8474714967975368300187292308759333056247924687766081047688661301030951912236351508961181078628460174 6317561183488864078759849515525252210598076279112301082060449369148953989659556686669095895222031292 2247099198021385946385202480175329390049463742660172601131679254108574893981180111674360094113454449 6561092296908376918617251121144163223810536416946182200648670079401613137290910905631112429092629138 7469357265932577133168692741215388274951528236712712985229762098724517787934331704093026627540885694 9635772470283342578624806814145409781466905678883792571174234769147649940717624167346662498155738636 6170997352045510172552092984219711132526402551226668709111582104937058561718195661764579884366283689 5782210185453779169233792836327935545176827650907272797357551069931918281992872743169427343221879815 6380324435315715122102255411134138827061163888283444771565654270641613841802707165618873578425615734 0896082649936539154499051208712770079669807394066339666681142116060475284927613013278800368621074281 6485801951065221886202823242769507156366297066314339191075047566808437713326685874985026006698837860 5490807311895236607024162189671087295591401922628126137796818133256687443390496347988433894093365319 7840651165810906297611563695607636357419851920766209004913543614374031254339290689365989894906243695 6329333551454275322212292047557997859637305333812442035914395035326412253212048492518456353118852804 4644039882443310678720901924694287085375850091236822779737725063768807304781593886990860497739100428 9094720796718842051372299836556581976609296825773500325758304877900076508488660891594536252139022987 7366341412916839361087932865573598073982026926689146529549860982265129715051500808099128226695073979 6296231545601901769613913810734881970842757767551370782121969873384880530784536199241835648915249807 5651637001879405757252711276047167297227351935486278511084971325605362578545227606716160514735472802 1559305445808976037394030180351703851313974929014930856294395901174546422713075035127359084525861055 1663625969072812189665811254023596921469135922356119305964997644101955631928798313392451833441080729 5151474802320060350178673297834022050692361070679608533825725410194723139833062716361423210800328214 0763716104004838570045668360276529171031656811459316892659908796125962894013684340627873702863822857 3159714638088533613408087440167275825036603355726628898365889022364877583126689935258803495556042347 6365924129268270009310026508983170470953639285441969408678653243072357995779574901523148836967975920 8916130428523847588358268550101838744600482369322692351842867439119163238031505991038538232326377290 7355908584704884737800537653294903151874817708861289170090456776487414505910024657629973399914709258 0728500417336423885746741151212660242088276320894448669205647632349138953872577888425931316525973386 1330125361204793689292526218999916795388241566933849148221081973542548244095146163924762042135961403 5283589131672876613482538060647300691064831939782309510987177158695369824034147159354009528699304286 2145749353979787828947113996662072044923377951775561953808185761431448184306837772454389287995888305 4046888026467803506570316840709807329505811211541949789420301432128396804171686341845978556469339569 3309926519591429322887537778331597947128464919652888478841428841224650796673801944551934185103185774 8270204541605437716608016371622261527605859151033157889614919071480229936992309185221033434932929170 2452343549234875995120044233227130130630614445349413844344928490379744013103659952332151414197461172 9188694989599674000923016440603028877722762112049552527595941978307601860473456233766661602286525514 1094325208861609153087169625742242721169394891054448748340500333149919904421934895728419020818601102 6358444204107454583676144890783589680131865414611695679924118021744436160180121408810987238386753127 7518543713480998227348175255915413805338733982929847648341175515978003794412518453854629169423494969 3170485005542955080363986556564560554661998334818913801398855474992307152934393238859411132969566511 0521027185818533638585723777481322050330470616223769372634150378820640746617326058706465633257965592 4883740287577890604153887249089796425699571344159299457726721179343402518949282679684725160132630305 9926791714520456658528782238921680358430768860748251253271990002272112766752130126967482118280471349 9672716389778850104108677513763396272548054629155895156622647345865443303650613883819687941586076002 7772561131648270433061018399406530898778857275033208278449945565388505586721108772595348181288371811 1213531246976786027645972839665516058700679866503529467956543175217951743866367529285010070398582959 7376252018358907162251379978133392078712900107970208199450389685489835835850535772860810309908855704 8896958811415137233711121841783357504725737825586792649467253381782073022576751219226927055746823835 4609779062469475288180630875763847006523901501950624203771077141776487766141819062538835511984723484 1365698247385456103615295875714992201335559776231276384491580381675113432162388219976537728339236383 8971144846523115448268392483777878489049218454997467138815081831742116230476810227475258474836072856 0400004218411716152406847513828577261484652039771764987834043642551675364807054892812661677590143056 7139493846198268485559403544070062941788160134960923986084729740421270302782017545326213881121910873 9892071895692396462599300992191804085768106711456870791372278086007442993885580024404251907911154191 9599418123236249397126320722960004489066335817271090186662299369493755516737530797515629546620979643 9595880022858106641123010564333062136591842846362057376367427650054610903947220797073684685778987863 7453212930971322271971638729415169949319111225907074096492162362841197052447312171305009284159027937 1673871163286106791834911427829074889014092046593791098182079637843053927263307512677716445095570510 2673749544883386541068666919052422559724257265851535793804492603758767755068235166363431881940769643 7275524896096218579284975959973091419123562503717162844579100027977065063207153616223418952083422771 0232153328825904504702535468466623705332205244093832199515476876453820547600393166337614261522004650 3436032239382026060799331810394757043853879984969306976983669777060829866614947891216973209468359607 7714018443650596738067533492451239901332947467565989782466602711105988071348524311497453679793892542 2021224636591601099286442129208540787617358953877252999007009916214823192161907277946477613728693414 6457804497173093293775074080470723025171724299968687741876456336692906959175126659810363104213334370 6196390027587125561241648817555383262783157450316299263685446724300314602910960920955564375533765226 8488998773723012618944967930085992474791869488929599602220008063918262772648667187241930103351969932 9904981522117018386694273437973306883556952927190868681491098567648297831271032350555420237638763562 6092984476986028684600632005848985545801930766136959623431087608411901722106093313214957611260789224 9513758511064500856183401890402008696644402926613355639447515617278047457485052561408550199589033946 5118029779258941842519885383936880805222659746128314306402864282936845918317752996575670062571279638 9929958923404985132438252553686071768775777965022983731446356818212933779654518336413620171859134793 9428122496344663847601430567896722988147143919892358005903111671247504250941751077049218301321668024 6816643299763365033390142213718478203341174624504088208028105687359133967930740660483856058362643727 5391577543212581364416676868475177869827451923655499811226014963411927181403704904684268137163824635 8974325748635488088850028360133079900954739589596839945986778897141183479537153849618943018174078169 1718173548455942117705095196318467034320123717168475204256009173461652239421457017700454008297648282 1573781871727607259105176096308556318543855842510602502318224862570382437292211202753818474723747276 7544279605121850550356036351520502984262945541686452738345085171243705484284299695146687741256260039 4727894956753227892317987733463157348070565171574954751186636773394077608826543509527280954215026104 9393182511497172482245979984810503852702018268964651043243547882199912652043123623947990774024951476 8816735900902991125382266550132889464735914282884301585580057807355477667933540323084638999170486737 3849101043594487607927429026768214058937278400877420414012106427278583500525979448663055003823012386 0826963596969757097047096595263087374628636880550540227211211898089157980460110792128672304131030321 2757924945797025724775970426689239757915101693962962316818337720681207836855321803748145020506122795 7055169620224118187232342842941855214836828678447034243041774195247377492016509528506732184017871844 2054184202186245066994608997180272716078325187282876751503969212111547614269904419953075509599966668 2756736720454716000971148096540079854927948400989725639332796183380138027576271912266246372346491917 1829535114450948903581163479938155719672670260324456303705811019905615560087263014279461430550807418 8125495775299514579387605296429340254637736350268016795765573224677683048739211364842570484399462469 7396908514490761555852348782419105422645337751767636251988884307837957593755070877435158125664491677 0738420134746511208806478235385658390869469427566797914835562648247490783773535388411561524329636705 4542069797419371486008658288240657655553071336412347072409230820371616094924793334135413395450732677 1841020782629497238192201626860364503999856729939265715452350019428629301695574560851487773467551619 8524098404536212563836271367540806891475361048960792539597116530199968999656452095176148720119784695 7964136293457268784971413249966188495307283087143598562177098710622338104029234071790414272657382673 5675155211811796033307001820743007529459393550614593503187517840907423957744894293355711064000326176 9959553330659098271589415325029778646454346312382274755979431730314883212696954701994047486471331930 ----------------------------------------------------------------------------- The cube root of 3 to 2000 places. 1.4422495703074083823216383107801095883918692534993505775464161945416875968299 973398547554797056452566868350808544895499664254239461102597148689501571852372 270903320238475984450610855400272600881454988727513673553524678660747156884392 233189182017038998238223321296166355085262673491335016654548957881758552741755 933631318741467200604638466647569374364197555749424906820810942671235906265763 689646373616178216558425874823856595235871903196104071395306028102853508443638 035194550133809152223907849897509193948036531196743457062338119411183556576924 832001231070159153329300428270666394443820480019012241818057851180278635499201 489352352796818010900623683532797037372461456517341535339099046710530415693769 030514949589952161665911663338019542272664828143118184417165535766881832140589 503272799127928026983572135676304667631409826930968622476494140464484288713308 799468418700020456187690275033046203665644407179091196980397474788838026707228 447481594820872396116012271067171066612781813201108139530097227226661910705939 909901191206759999972839234010665112017261908678382645549306351311165772467118 013661987390610857244224988944560935283818057572315979194877433541573356226311 595461038207967346941503135025476073551453433424461165314322615014485349866683 602832980563626925050343176583678023394977405157024328732956587653028673542262 319136728400150061488547762972908589372758076656574126999169140342169228720667 989164601581812852922185632265852061482709669693761466494213131347663694812007 554641393149115472708170804166986154374140450535686496436942865468268050323625 576976340202400189774509847282093076387561768341485471708922787978088375199630 922230750920799585424333055600530141398002959265610286745209039670707051933220 670000650335164775534326792392443529016468407192109040375770365934296127947126 325201326851675245529357969957045653909018613602450501385677092328413554183401 307615891413662731269785442444444893271976321424301679394389049166741038484553 173092098892100644535591061999694839444092529599082 ----------------------------------------------------------------------------- 2**(1/3) to 2000 places 1.259921049894873164767210607278228350570251464701507980081975112155299676513959483729396562436 2550941543102560356156652593990240406137372284591103042693552469606426166250009774745265654803068671 8540551868924587251676419937370969509838278316139915512931369536618394746344857657030311909589598474 1105981162907053590816478011473521325484771297880242208582053257972526662202669005665608199471562817 6405060664826773572670419486207621442965694205079319172441480920448232840127470321964282081201905714 1889964599983175038018886895942020559220211547299738488026073636974178877921579846750995396300782609 5962420348323866013985736343390973712652799599196996837791316816815442885027965152927810767971400204 0605674803938561251718357006907984996341976291474044834540269715476228513178020643878047649322579052 8984670858052862581300054293885607206097472230406313572349364584065759169169167270601244028967000010 6908103531385290270041508423233623988938649678219414983802707295717681287900144574622714770234835715 190550 ----------------------------------------------------------------------------- Zeta(1,2) ot the derivative of Zeta function at 2. -0.9375482543158437537025740945678649778978602886148299258854334803 6044381131270752279368941514115151749311382116241638535059404171 5961733247197185174912402688214443700163931015045107160373574873 1352956057133552593318050514872534799984717397570317550302619073 ----------------------------------------------------------------------------- This number is (exp(2)-7)/2 Ref: Francois le Lionnais, Les nombres remarquables, Paris, Hermann 1983, pp. 23. 0.1945280494653251136152137302875039065901577852759236620435639112 6128689803952888169215624253956089738687658063273943306194230184 6390636687239196106699038887450061447803705376851195665473775341 0432909101348239341042021104911276174378712312707073399640646659 4403538165050966894987036499348004765165375766040941184234739651 4956779385722841561961636382301294169998230606424642604839452569 4123319935614068634305323678131896475911139214742172930676438469 3349287600077498007403753598564668470942599861444131812798597054 7933095739935752164198846632305117558156194995005256891703382249 3319463428079109321077886242460055967658105859758658736348984146 7259992527092431598567842973511456278699178055257489684072513882 2403821492552091058527972095893841735642638248904856731252070117 6210793704693341271357851963226482022753143890006555463250692416 7265132318157078023594405882897139317429953835226355968647936199 7993536655407480626554885296765049525164840537710545438813154286 2425019139361380724333725282493692938578755281217194719835697214 The Dubois-Raymond constant to 1024 digits here is an expression for it. 1/2*(exp(2)-7); ----------------------------------------------------------------------------- exp(1/e) to 2000 places. 1.4446678610097661336583391085964302230585954532422531658205226643038549377186 145055735829230470988511429523184485575419803227050644507431903824515434045323 348299589354600565058450155229216855356115763888979191269527071068690455441925 402327632452829115155531294480526395191779440816753200019244730489909867275405 109516334654321860031956702982909430158801267338033175228207912837451102704873 260814978988983190463351143854405447262816274799749460481270356697908366707436 286857745469285242239557660491219676478966504162461997038023839327097318593376 749353786144331818222839808355940598080149837715987730914221376657228893083038 653373796085635927418524207501700955503473378672633650172532071964633094039805 791181211321822715792798249869313151257645431256672690785430973013197954007349 759915411284195117795095690802080276708206224917028910504992550858838301530990 168243094412628246769234504036301625290742722268679939821759708209959163409668 272661675291393988743978086274424216929030067674616098609214653187573187305086 126232965254339049074676664819105807348446759446087127244979017242483460404179 341747714250050989931616597470081561793341963839062094592757409458111774150123 401090738866584637703401804698629579215080920954993340355597764858183174482353 473455817208540672671046670074599189679769953032351553703420171175453974166247 809827198143203926183829504168693184240081340014797368333638242826144594773532 895315646529427993252140944435001058155602470222707034315433467824667390320672 298668340755575720341029504672999558891432765402324099908360099906068254994962 205614450346406873954438690905208353252717530240673528702084400924827736996425 665458538869892677127917308507971472097901815236540498385634815319237163639823 489127657214172616278624432703958190123012591459364636457700684303609959162148 471764293090265464352172321399050793239542414519952632715835083532242388928148 062704080236468208418817964014007539984012140940632398552002059931845792571919 870579761842699809457608403137282578292506081725353 ----------------------------------------------------------------------------- -exp(1)*Ei(-1) to 1024 digits. also called Gompertz (1825) constant = int(exp(-x)/(1+x),x=0..infinity); 0.5963473623231940743410784993692793760741778601525487815734849104 8232721911487441747043049709361276034423703474842862368981207829 9529057196617369222665894024318513514368293763296254771187974025 2432302052117885737856177283652365137855948674253562181300812083 3784238448595980666983593217826489686047231099964510855581415383 5206162575008318874187017581518579310050611604355294567103401503 6663635029755807141964659205370602563858754392239763839327096186 3555954208141117245933865465249552771087829990958035092991791621 6389635691355069731255489979569371930717843870146967280775178170 0499106605448472254946244137072561379284901975499830037495298303 8426547682453111389665104606160569870635068347161893124491230526 4149918184343827745648804281946265691438208018677444460174831369 8959152675647833695487186740099259602213107786153781858902163226 2956642078512987325163348487588340256844389750747943861531479299 3932807784399881769589219826357740623772168228057169916069633006 6837801738278339632544426209799414229337385628490796642900584404 ----------------------------------------------------------------------------- exp(2) to 5000 digits. 7.3890560989306502272304274605750078131803155705518473240871278225225737960790 577633843124850791217947737531612654788661238846036927812733744783922133980777 749001228956074107537023913309475506820865818202696478682084042209822552348757 424625414146799281293318880707633010193378997407299869600953033075153208188236 846947930299135587714456831239232727646025883399964612128492852096789051388246 639871228137268610647356263792951822278429484345861352876938669857520015499601 480750719712933694188519972288826362559719410958661914798715043283976932646102 351163123899900105137834067644986638926856158218642155772484920111935316211719 517317472697968293451998505418486319713568594702291255739835611051497936814502 776448076429851041821170559441917876834712852764978097134625041402352421587409 386682542715703926452964045506287780013110926501384833453026463631415604718881 176579427863485990767045271193729587239959870733108149612531097705935300990503 296810754210908776263085724850038278722761448667450564987385877157510562434389 439671394429509260066782961819652860639659716093395833353128273752767615716807 321951690196420724578844775506966145437379667573871682823798557571921419903428 286722894917809656472723245360455183273688341979349581145497709879594970647587 774356404788420783840317200610794937498514342848507542818121827763847107008928 430013343347406041207901096205079600966957256606617065204899682275744332229689 868778141305679410714248778230278035697065436930658102212059365744148111717372 322044376207885775691770653530977435291385579208636931367256934994119537433668 031809658792549783249819431959484459119080723629468718812314342471079573188059 411629168665877899187594284046874776602916796426275176173176029678431264875228 449532410503891160486326309580712207062800802613988163730123961812436859697073 687897512695426891917718851472182330500260721398680077113065104042744612432418 636538005821724437716331477461032168192992238880422697070054846741573798917399 664629667243783725544694984451303822508007659860365418755296638140706405978149 113966107310825202174733604913793172676414416872070100939381007659363791360656 378830743436073339549024125737991091522518587344681286529833287221910116921836 703851224586473924509470051806105676912398353017129631531317928130670873675512 293989672856993558195595942576194069903301750916817051960686947952596778782116 930308596118038967961203891800225090375471269275160190537092166047922404708309 416240552805821234037825405031635884767385305129541900147968059426298242777017 734996745786002038575746677162070901902201756573668953012000610370394745226690 333490222329674847018388022672641358086323396965584421695809648659885883078785 964787945274277326991008474611773087285290817112400532828353522787286349298011 466754586407723191590006556417367299581161660855749873433075517030969167280695 653202663209047583374547360436268123508018544017504043756080906361771198625470 149048104379275794921114783538875029113899376475005617514738376106868341618440 164912495798964383808812921962561491051901122911823612065514311598245021110875 712471517593218049994701618450176846899022261812635774014019853978932640323786 385253493641551644097115088418258263013563879614423017571852398499892146067182 638553689729688520649156395658919954114782075511831857426860122373498674870269 377784042456203383448711943399122554241567388165664833168685491971027077696622 998133080121145563012933654211395517371249780456303105901619652899751196234842 033074924963187644045591906539014423515625166139224777674555779001461298083240 670561570473915800706245618102188658585974040456456638626219452067339034885937 978146951625086693511616724093429504969947085637338958001476853062546109730748 376302212066849030245002245041158352623596964842952259953997748754773060375131 432377444155669094253632264252866815633646570176854539043225160772328374142235 489458806852028337790028376352092746419761677099346887904937940221571672439522 740132691815493434328136627024336195728813658138947689751364355698740198730204 510532190210150011714590951429011319612385212642480325179599473126617375628942 845133870702270951959022387235109134927515454530142550893141002994767608509974 099495643092380868481746445893150659622485102898867337568536562053113285278280 351437135381193985324853666970630897369008926778074784358432808168592231610023 937935862567524585182213177412486056516553217031350024159764870112911003745379 441637967667571824311172167030273585143500332343181952864044962265047451317936 372089394350861238019804894013124903695750759791221648513922781016702444000581 199558265295370743383107986866095266236801287979244682017813400494270034827947 193733181042241620822455794255265865304353863376071807362149344444252372011622 157846843555534601131362486554035416289136470782000305212931508564116297615265 979785699352599457163263071573219381166066389951148022466666367079637253197900 654223597796112430223855369266514443574027195687544688956958110007986602432617 897216285227051503020130341916578568404607717493009597456553690821784854445871 826786386 ----------------------------------------------------------------------------- exp(E) to 2000 places. 15.154262241479264189760430272629911905528548536856139769140746405914830973730 934432608456968357873460511587268852852295841083492664266576491187794797041548 104617616229388368454821943265188236980675811312322990354613338335185965954216 525072048711316948412488370282981016309404957477919913724532172853873219106809 779147336581876999676941747786490381633905056120497761253480544666296079402019 529877275185530879677728180527535931123975906005188808804151764154263227653969 369419281681418048811050162285713125125736860841705024753725516254728475141045 799649334649258377732997799526746207088566625779404589544900951646188503245155 543276102551379333718085468414791771323547050692212614636013851810485295066335 920575541400093728813275661177976041869730169672487165342920993670102150408829 499597506617616719842615571111326294774123810315151613066360242301079967098082 562692041094465730378984833988726011897854428408453570101750798918965140392429 845444888600384060710574361350926972543596160929674886120979106708943840028535 135440871391881852367895585832284186976550273436642950593016859041634964856093 336420483954820650384924580193949950779821150964872368373040063870083168095363 027825895373270231030783784706559323671105702265188720741128349888185678057390 161918502016416029651453196228069975763255414887223141197308260993680995553640 876987559152741737571115875654758656802173784981639101880003202229444914999791 121833614814370224661729299739195320853192431221206621702154741529067513733743 105281967108077994259019879889556684699959506479418340819142655482832234968003 081972689575466058206320833033553171990434965352214108774173325396334580400786 128927180361486215530618389677147746814551739039888122296264079342155258477423 050894548531430753273706246888147420871112695796318805808940094545849750800038 454605680063772534810291135198832505594830473508310601555491417396748201669626 466472692816532277039854341738387257233855372353460965326995319953544360415756 641401352111499181015985628664112142501524055296479 ----------------------------------------------------------------------------- exp(-1)**exp(-1) to 2000 digits. .69220062755534635386542199718278976149067802929754473593891489996517155902908\ 536212301238764935309834760400492151361541546843908266574243371525714656551784\ 111012118004581325684010925893771483618960486085537343879946652796899853787040\ 400275355096588887233775143501460016780690036801308541630111513237847051421431\ 735003757575027827657890910901800419868861476858068127728546724106522866167178\ 533394203986699512684407638296761887691144096768970964770952744814137508760262\ 347108202096328364345869620334620993862514484308367560881003972238303450654011\ 999302660669583712575390228335963343223317598290298325296741580034479426809617\ 916562716602703751876293992348406456168072272178626090821133759976972454939676\ 233705471764886959909661343202583435002924051282973829360881319733550247152310\ 617878169931510859670977308319692497459453778843104776531627139117951829952777\ 984784748083100509514689180507644575396893067426609748327310210350348865097934\ 847108079588231375369143094903983519917287211557028561309044418018748727861720\ 049246655346024866193001094782643551068607687413779478375026080693792371745667\ 326920931738054051554242249897838297144489837734713137322284350535618752976286\ 093895989639032722130480678499715176926503482305130453922095850818101953235988\ 815595737184281626484649685339288973565445822350279492095445940680566081003259\ 268179169113276938186323188870256490208853136983606712443143803762922403641831\ 707414719079593143314102646592545748237030101570952500100174998657495674530130\ 976736155424204383428711496304162222379650795775368562210611162645256123252749\ 309026738236443657254192886318471249495662113016226965924032587660041596226858\ 871842100005227499634563119014388646764794648285810011294558172412582269896636\ 148795687517513422803093683960921543788798700779038740080583786722919943702536\ 866284814961649540249074875351523999305921211941060037993733571079368644010896\ 558441568043291595432404954354108447990195383717938254334408912627182163556098\ 692787518571950661351407700606788750607311437028557 ----------------------------------------------------------------------------- The exp(gamma) to 1024 places. 1.781072417990197985236504103107179549169645214303430205357665876 5128410768135882937075742164884182803348222452251457420010557945 7424819650088156857512645001158459572674035828196794290950691578 4452444104950624749464673954422493920612973667189929611817817165 2864420491963881484122168597921107933464249196247355882269791909 6702915015433548608693369337054694559016232743529532598372576605 7036185991522439177800246866066358611728792783719231136775739394 1040997516402036473484352386382021226506642476962500214726344491 4484348856424178974964672272861347388162990813399863760165095122 5930470345744755950618891448569923966073975162156342628654955137 3909258141962310785511202080191889749032762449465399591732002370 2346972595868267504864024051730817333687405009498105976458687029 3896367065248715697709906547424211801204145061066518815890360582 3941798519634236370134771763646188686856689181798821908601430001 9066725574605712722169911418933648461722485988806975526316560646 4885402822786400170714829524143677069517465752235987166746273979 ----------------------------------------------------------------------------- exp(-exp(1)) to 1024 digits. 0.06598803584531253707679018759684642493857704825279643640247354156 6736330030756308104088242453371467745675265361417385915268129777 7682952099474319550375315816728478627294085441342517362404028137 3075319679080836893339721611007132085986867373329450125117846511 1997371033841820124908788824103402505629682279991950522007168769 9223409940814097787626953707932174652296941447728408313475779368 7286487969837972121942114360126409716777495118467526894899864788 8735031400612104221274029913304793894630640959962842551117400217 1224301523765801548364486266084996073901969183044451312867501330 6711338167540042368298531300473719224313617955264178491864330278 0756133449166065238609080229467962852341826426558255601122364081 7499678015473971397679826375733102162922394423779621713888837163 6549577694533983489312197194643498576817745878661766236458466956 5030064688400052046935488158098374252714019582660708187777614388 4410669718849179506625309950551125603873283026187944985821129679 3977208840286917829299168355913757001865422780775332862089897061 ----------------------------------------------------------------------------- exp(-gamma) to 500 digits. .56145948356688516982414321479088078676571038692515316815415907604508796707428 563713287115893421435876731913100954504183815294964765104385205667809151313057 747958292870260031414870646544898536405328477211580543159583446376788374801032 626348256998850040565178360339726237476948553637188597328437148468927020966712 340240735717852742632696218798915114734105359643954396180929458839301817871074 725053658507888643323283317624778561426262665910695346779694244638109730255111 461860331438877726770166639418389 ----------------------------------------------------------------------------- exp(-1) = .3678794411714423215955237701614608674458111310317678345078368016974614957 448998033571472743459196437466273252768439952082469757927901290086266535894940 987830921943673773381150486389911251456163449877199786844759579397473025498924 954532393662079648105146475206122942230891649265666003650745772837055328537383 881068047876119568298934544973507393185992166174330035699372082071022775180215 849942337816907156676717623366082303761229156237572094700070405097334256775762 525280303768861651570936537995427406370717878445419467490931306980560163702111 389774228214017380232832465287291389004660986659512444097699851459164287803720 202510224578732111059537776807437112206240005167965280975444780286486006838564 200433684662484349386918262062518994821970992423425207510492093445285124486022 451380986417421061219536368310078209224804653079806562854154786061793155705987 170215999699188228265397927803747127438635156296714511943986702682452679716814 389772141359579690542529103548859731078233269414118579235695949376986012657588 031279984679484673513468022653024462770569824438638729700298758880953411267542 378902616433104091860701225717581666034510079098588808084286840740013403838132 000405677583140619926558491845127806870837819198202812845022642081730224354601 095412338871575982537376825937442618108261974718650463417452873248272637661583 763933421278629358444618677226500353035131097140226114592021137432096713055954 505147204649927608239102334732395201495304468001908089101037412807847539359160 611641079862219291665624157286971019636369443758363500845526495936073271833152 931149024594615904547697593101389084399751933133348433477200696285948743528556 297391651507574098728381915385505736133180286124059186909373875909560442712511 863220530973179836403160420097491442842555227369917972099757563782455600807325 247976721458498517103721404406674873157772120399469746760971682571017742702091 976625319582603930922294102726724544711344475350616228877691947616866950710887 116942994673390516341643567379611607810782080925309343160677308435649505282277 143899798591755972954046342095672245369518957308579598643134793385812415355675 291899157522533720196926954880276460574076971653429884431763463713203880902338 189564902322438404549924474378588959906250499901257825080600147842732190812817 711934108024868894521447880640085291145580218572590239108929234950834560385598 264413050007085479362046499862563501578061804405417874248135200069738483165900 819542105115641466271702656052848039460906996074183387446906226245533123070003 746761999924821965359200292985623376468387329994047893933509321888579079558108 532106742280666761853263614256495835101702025138181821268264778420097841702272 074934499757852233420292950330729579362706266864030148742016735810561115119140 933150264112453530386361812864420158160056929083560649726579198229762814780621 626924189506016999966646523657255267820360115441201900912876474365515347254068 977126822504734351104238450420844021885018609482213474955937715411986797869406 901368616991961290260081527978111345593390116335390332063036239926050930337510 487387825129323182867557291501635809350054883222675608212720372702183721063618 766032448320860058510873972259420402021732839200930702666581026758380968319782 710140669865321655651908876853501690204501646055425696274997201851061994392484 716368827281255930651110357011903772384664149179502647766324344903110604933597 369307751543693916375069636837908788841175187618545795892326444959750662095009 337771076527052948187501487954618625510971793928190375637306344615327443788644 260965771903798281228597580348470380178567344461205480216157156816957565718704 604822270628628211141265346305533954300255062929387759528640885004979653887409 976058761847972051881407449924090939831494676302707547452061023683299266122338 036652047950535395587261542739207773632052723475401754881701586714569894018921 519193880329457862667201533527926629924358587337382557753575994189776992699036 929853342843832745594503529817089980657070431882652011567093710443866443096324 458514806188167500898899970525822778993842811875596117969870217991703566620053 881920936924983304499189729814016476423672781258837208970232443016192994987189 906657867441348308691569923239751117705773024216418915493813151331872222104876 018630449436875253249433513525449837483704514494029181449878168061359701292352 504191891313458792365601660548342810621469801384824090810669846170767085685899 785564857924282402194928690482144323229415123402340487067353488086617921820698 260187756567785955297236202072848114260810165814251172356630887346667214405755 064751632951709237696888433710461851626922336997430989108175292901274456729026 897398267322412205484307901349886509154522805054949542325635216190553843704547 497658046774943354976886666574975969604657936559634358260375014325936534014108 536530940933071023603703810210960333504418033446500026828172522656629218733068 103308604187028255043031128085804875636638726987353344641500894285398583468466 572923501425356405540074529179915839876321144568767670217198946211327367433659 212215796547779840135013939581575029998309788085684387785690874319389225831713 407375943246998104726893945574983977419048817505719440381194482726365055416394 529385023976318425828473726178514373156770369601906483482160845652447862127420 814646131468650282625925503737538301804016207199490810106475192285238033922767 863780524249883251503330102246571151978692652708838689354711650504855367013048 084537612063181884849418069060721506913296880645361700199073464209465192900024 373138841982650560792714648102609555071446761017633568355591661311936567379998 436093917299622324589324267935569164760481519092162867463547970866441954615923 829401162179371649318968810505190602085391506118242697656983735012095200971806 014373878126407313565412293474558497048135615861036141413184366745627830305377 197190556576066236193466219660303933611457032880555031901448584661557878795716 990989636747333893063473430706501036761944341196758158460737266520712984247289 322558483581239022075320148434047341543656233413013871387019594991544773609045 558244750536091386998883475017517860658931731223068290680819643240133271960944 730318954264562465337993541288743914044685265869684888013287084014418377776931 719549231983169199982843889112520808510341795692623364043981983963128160281444 600897621700699245717417420107664358501274174184771572661194748311584838869196 703144393823089342508678764977152401140220776085446190456414446822433789704093 292031860360113829694166642478717205882024222578801145523141536683669229311529 734786217344654179702616836864863339499713072862900573734515716534500666501962 351552329457403769234646005495122072991406185988955812305841818749655509711812 164543287171712717639249401742476715154545264238620185457154620973848007521650 628352934201627608761476938851309233389278954976749196541501500800284852120826 701884012419691634946140519381479099717211384451024901914563151171091631189614 222248052072388655676422044169833916995459050698200059848808825332052302218534 382103663153041434587301609749912935834330571898367063986023976570217860878414 399715908345499184027778983111390459768714416741846642660172741310016875498297 410066094152906536438942875283746741261371155681245232355988839735644298320927 261630410887181712445718308922356530754900264063982532184324189337770688146136 469873277333754186312213046603872637915650390392459392171779937754560843042554 167540337834368899663366605897005292579693730765130328561614211663515792543265 743245646456301233601498554986931800033991654339912025960260402340634343065644 117712233580250617770436239967580640636524315180852668140884585066781941957307 028574270412326416602581494620189216854239550746132753155458357904647968854618 482046924284150539290983398961908188174709206134955882747787667435877548764171 453603726061839812978696543203407802589408474474526353267969259940574749829054 355001368592055747391188426980337249951840967652222312074638032974733469322866 571753903353231632201537145455076280510816376149035959720954073093067293076431 105396958718486486181477738102263426286427890561116940082753118540387550686775 104955868138703817541913276236513990807247455941952135529925767114457620374347 405652435045786714555239113081761034315771605220693027381372065984152743262346 053755843062933114620558864444294215661431312036756247136531337923247716559613 319270116963909315587070545209824226313904823004243217381435487562567643449813 317561489366273578901166780930354747302380244656920293969727516115277217482733 819524443185463690049952764021583166958713996637548234358873298790304442980309 967910869066387633681839553674580396292731226636194488274575125951310709789961 546888223657749118137726673562886144544488289564782915082998840659546460369998 787409331138300923904936424648736791307531873114843250545090086024806659917866 772387465882527713630378473106752869425207983298333904366175303531589593361853 231916154783274333104573807030708131747959428905816723942666659380494632761157 865094987731550280301387035363099032720287388437348505820139554016845607652336 654416593433311201059163792230262781476839798832471237348530813571257336258058 491045323346674112018333063340423940883488103429445618775227198352955237778320 950290309444876990102358281329480666655243523306381638509004002637273446207840 058441681830543036373741793134797211406249680438187965078105764870809074600533 145409574510012912228419823080029355388675170041662957727277212067407292584894 995567066917869209368017562716594382397079150413885781563621199522416497895560 731774735975490495266556352501054308741257348949871882883387067924254585239093 685841449669579601997430953719386495529260719211200317078896918760584530630460 675501744307815292144268897830865763347229825902168455564492655607099118224300 215434134831116671819417609539030193859233380065226859210550582988448601276928 147371258710429899330978273714735482793847053318253743445022198568969304266497 876880582024520780116444982926341685617690723125638924173460190172487516320006 633910111535525361499052165232218983180225324935247412610260518462192680550463 043170478972296873819 > quit bytes used=248388, alloc=262096, time=34.57 ----------------------------------------------------------------------------- exp(Pi) to 5000 digits. 23.140692632779269005729086367948547380266106242600211993445046409524342350690 452783516971997067549219675952704801087773144428044414693835844717445879609849 365327965863669242230268991013741764684401410395183868477243068059588162449844 491430966778413671631963414784038216511287637731470347353833162821294047891936 224820221006032065443362736557271823744989618858059591684872645479013397834026 595101499643792422968160799565381423536206957600770590460899883002254304871211 791300849327379580729427301931042601691939325853203428968661895283290521711157 185185506802254197204566370865568386830544799278170407497768540367556534957218 867882563994384718224585889428535247260568210271076018491534518468064887386774 439630514005169440540665265430968869063937315359837311042174433023967896690035 041181486053390287203759918586886897487324321721585596074334676426167856117353 336421265631915665454892289692245773889570905361803836197510326567943624088359 906422347128465334373148717065178946374273412694796804321041476668230286429344 467874583036218963523921329324375520233126347824414750800627831282927151936516 683092962522682971552125741093828633003411897142507039161880015550503803553843 718861603288886144250899881003197448339122303772529361727307574134362759044269 954062892257908757705291101487630260102052478004206277235139719507830448876238 398163941174446827974730802486588543696271962723880546618553419729373637368419 148691027035328153724386748755117214258310847410169225398434038980853491740384 450721511408761212581190832133264923835593491754167962282322191228011153278155 204219018025908891765383953674502890902247970409692405785082751036913560354420 871706508620407109807604764966449837316163603450827937016534044097441642023703 410806122861535305138746183274011530452801071709010490216621635377646499033264 932280662787028761132223268803342739479758297830428171940689683261869189792204 372346952679542510781571693303751286098474711457649993852939553172993185224036 358516664323460206691753893686912422507033520673358633547358875825685074153842 033094308893131345705716033088381538800504452907863703981598171970921717902511 887696724028632841703625672494791651320238895154596088408050186290838580481035 843085189594358532168161396393517807454063446339968749650173057777744258116535 464257730771420029454123404787847316248466745152361050882905703604498230376703 340240226504235389465900223099783347649614910356582290356550731766853975911910 746202688908388745135615403239288022221725455920512691043974473013791321643862 047011511401952252331456616095730115275198105794921389057581899604069596312018 400348181169493900500632762923475243277921938328216385021270164263681502896970 692397118675914204346025209567370587484939686481736835309890886178757055350098 190233382599980455727852434399271283305687364708639771893621760100124175297705 664141792179364855677396908563589465417348490768969923568661640518224884603094 790569591022495866012011445605013870930489092384114472366038957507636688016240 927874382373744814916650614964417835499690440578359227244931236515185766806020 869550253766446694082251835120777892599001327222957119361829143530846402091415 726345692213517571401822146205028170456034371403687269158576340545739829098352 283077029287667093225004263913090050739682847703779856625416842792432550203402 413428541294789909000898098693461858186688607060762090953500339464606516734501 811483628242252585632808731673256257434598532767486270795829193495190539036088 759411369318223477773161684525680481157015658081193600273562249181657931076574 049762471337149153488285055374984402971807265324598661527818241276524939991452 293745346670743725231341876484456395225698215724941630362119767194560138326404 067453543645450177729297740351386800694550777684571313282941145462456613557751 403966004715191533236797106859367653313087705155824966589154335605275410365300 611426129360904442954486182159443158639568286689752024409215312802108583728336 953496882031714728135451194625391231730268114209967817073910495844777913706021 293115530486281301382266625229458162718456028463865879004483202230406831681996 316039536428156758520482690103982742029454980047903034146149921793383363678235 981221060893534718083901897228789865384560569608078870320677896324906867639823 735574035375558481160809838619270551688923514046990660511140072896133742208655 234971189424922834230143224068158856994814823887995135233753229973447855032760 252884773932621046682174269338931761295250984747504138165356384062608334290422 434220900128127653923205925952466762258512241619318556989790744826403802399650 526424941280974472624060256977463441304053665109452426288751383309082936134624 741933237922957302399406551590580236766203903859557822284868400314720637637680 811871278511142546865006754768893058619320291677608475718377710744737574994822 452359332074670151386680646665352175620384480229864105082880688043389335307661 299677728379350077524855876789194128395883222709514431750576748884395123359680 860153899816120343861701640009949542826862293622241916029918186565601791273991 417827293 ----------------------------------------------------------------------------- exp(-Pi/2) also i**i to 2000 digits. .20787957635076190854695561983497877003387784163176960807513588305541987728548 213978860027786542603534052177330723502180819061973037466398699991126317864120 573171777952006743376649542246381929737430538703760051890663033049700519005556 200475866205294351834431843455027479745344769934714172383230815271481800760921 074192047151878353489584821890186029582331295662952070823409567696363742039451 439394183861901080820897771751705004348176454751714529894341134142017562215488 095419920914735851528567953452697630499372957729482599702847752403248082077702 918719721753834752086086485875347786554698383255367901383517221186415195959120 390444802266967367943596502055843602956960655824943133694017295242896108616198 249990451356900573640511026643913735174062790749688490122755719177620377303584 528775757603495038129915398658737653591686400515998897106379906160863003099013 645709498138143803664034891345628757167799263377000749589344423980292093268230 632524978561696934908340259472484771680946553547691686005521521017215168296115 515373720404230774755428762255360151297094806897761624632936487322995579946223 667565364759901788242408718951443672395014444320374069000448514561029691104910 878057252357609122417789180933458448669890751102411596252119830838350644499993 773014283041837616232015551287486272843303258079418180873861157171614886718338 731911292453181830750086234591887636828944331510421374747327980503328795725867 541880427942460082555893105915997000395650625441364588266466031127067422310305 535187902531388974405712179856104006613759583217665543810066431723725472964991 961856292812999539439235771809766062492663191457900110787465809844280408451804 032780672618042041264065962778906740989424460902454836714117721549884956940452 112485912787321134061766389636298386755420256587872016951195708368559827864276 680718881231914244391811217652023500760896564348636759308347063169843396342762 380801002427823283187715222253641595206878156248356262571460314665809232684903 574697050158967802352025989063622982180850344064759 ----------------------------------------------------------------------------- exp(Pi/4) to 2000 digits. 2.1932800507380154565597696592787382234616376419942723348580159186570268641892 369341265228125781694047116775935790761569464704160085076260537732149882576637 180529757138045659787272477676744761875531295935525305656121941938325771933074 806085313682138961756598766510966436553091512057045356609836539616553346459064 373502076478360495913617354215297823304378002288296230580232637860784233742745 735702022885199809060605517273394101492864140169829651753522368650118179667647 457626718601024618354031140185283877957644241987211531869403181361164328095230 730024398040705662226389469435719611444017096638835693523253428765682612234595 951708986390752407413028766603575990876954471836670122447802557320011165920485 737487446430693887117667166587423547762822550179704551822757013641772397484169 934294428941022343604987484707337177835418960634659111004443583106161213962732 869557059035275368574238683663081722030128312529362175545918716754993797947622 129173925575936122032955821750739144790967485344605357664225364299538765324136 949018482774292694843822530262226861372921802584863940760156109818309384570069 470795640209894190345349837819076500001006116847165098191110703818397723847636 086648056388275991255550426835352067430644024117869749680856585873571170743657 136786044315394750064576180785147254885957951336271665283759692513277243122946 517550048565187973568254014390248480460064061788843773774508689443629914689070 863974320918429013628422650310544292255494696518842812225079252175007239925551 029582635039092486273352486912798309911063734307779598316684424998539795548125 268033997130011499846042483697015844414014841982307460020489537183068407159309 517983225320936363638581565633474911041859194338512125351318270196627438605518 304515781785368472522587318378751154292711692985750641614337759718281173294064 791775848005896300566492309884311173439857933604637590011752364247809250024131 668204754717613231485795970425851046070793232499371364443547727560313245357546 907886186064507648208682600737612333200339822268899 ----------------------------------------------------------------------------- exp(Pi)-Pi to 2000 digits. 19.999099979189475767266442984669044496068936843225106172470101817216525944404 243784888937171725432151693804618287805466497334199805143253612992086471481368 247877681760967303709163431369118815729471028430755057501577134613459686801610 704647801507211762486314847860577867900833311083256953746572913680020323304929 618504632831150544522399907303180108380621726267699580354342096658546876449879 643159988034359365697795039973428331350089575668158797355781334927791924908462 223948963574654689501489118919093471858265963412546785882640500336895296973966 483005645858551426665349194572391634445869980810501002365767972240411271396391 082111221236595109050948710707066806359343256840929468906163467675785198127850 897610557893040418579801231012809055434162544049876792334963083023969523711985 090121754320574190885164894127431550579021679199277342729649641164236667946343 333283426879029077921683908271628596220423601763550345768754857836784061224477 552634753376507552515368184893952139761271484818185608411825056469292191172143 402388719757353027370539501356145218281091045971445755027005231869035902214741 399613831007108114309671590397877803470295035414651150086781633795866338614524 463801202361184473111890032179184589977518740064869257017125631178766560097502 116613444002528982978443997440095613894005793745136785941826042478841296771486 236106975150388789824537291233206113660684322476384073354639220456769367445622 366336245399318812002745528830530899228024664435613518648595533486264904813388 759794300611251909625869666788277721079633255273502463162440356430257586908347 777676490163357073347809280783525118108447888762632389639833558863696345582070 730264535051858948446447412414316361574314135012304613109139900762893879606840 891822553375973205919523999001461276195913399918515888563154830389019266241478 846496278959060784450769167849458953118807617766203046735572290276973105663715 745401348437577225759361032117444541819852672262704946729225766212036820852119 071048899665886244277863027854266426693643042645769 ----------------------------------------------------------------------------- exp(Pi)/Pi**E to 1100 places. 1.0303455242162108324415524375441423913311674535426350477520603769436858333367 078466536634299653186541372113411215861485309267528306708178141431148217377434 464491473535305791217064585171952378312515789548509946623397488705415787396598 914128956695347553752512638550318082771091427083769596910701526504657102657014 692869502510623838492054960512997771472559153485184037328476999471131102482175 108766705405357550641075673536209065070065612083371548796051824396699408865713 070119453591522563130261505375573780321442206315118412633701828205392108525782 413195330127295606671997427108097591179860083444244927443504416473570457716741 027361944790276285858904376391561055460513844056484484786473059281875288705999 618242118516344206637486889073335672784807640819659793662267947301826094178286 628556298718293181640871018794887107215120378358047902368736163774600113536888 571530806116406546769959374670822838831591995246739397760825519219044581209189 563299741163333901285277924920149254250155930276721158235118621942060338299354 365607743394171754382061635835272405348932946679933596759506206130017828475418 918307145 ----------------------------------------------------------------------------- Feigenbaum reduction parameter 2.502907875095892822283902873218215786381271376727149977336192056 Feigenbaum bifurcation velocity constant 4.669201609102990671853203820466201617258185577475768632745651343 00413433021131473 References: Briggs, Keith A precise calculation of the Feigenbaum constants. (English) Math. Comp. 57 (1991), no. 195, 435--439. Briggs, Keith How to calculate the Feigenbaum constants on your PC. Austral. Math. Soc. Gaz. 16 (1989), no. 4, 89--92.58F14 ----------------------------------------------------------------------------- Fransen-Robinson constant. 2.80777024202851936522150118655777293230808592093019829122005 ref : Math of Computation, vol 34 1980 pp 553-566 Math of Computation vol 37 1981 pp 233-235 ----------------------------------------------------------------------------- 170000 digits of gamma, as calculated from a value furnished by Jon Borwein. gamma or Euler constant is Lim(n->infinity) {sum(1/k,k=1..n) - log(n)} .57721566490153286060651209008240243104215933593992359880576723488486772677766 467093694706329174674951463144724980708248096050401448654283622417399764492353 625350033374293733773767394279259525824709491600873520394816567085323315177661 152862119950150798479374508570574002992135478614669402960432542151905877553526 733139925401296742051375413954911168510280798423487758720503843109399736137255 306088933126760017247953783675927135157722610273492913940798430103417771778088 154957066107501016191663340152278935867965497252036212879226555953669628176388 792726801324310104765059637039473949576389065729679296010090151251959509222435 014093498712282479497471956469763185066761290638110518241974448678363808617494 551698927923018773910729457815543160050021828440960537724342032854783670151773 943987003023703395183286900015581939880427074115422278197165230110735658339673 487176504919418123000406546931429992977795693031005030863034185698032310836916 400258929708909854868257773642882539549258736295961332985747393023734388470703 702844129201664178502487333790805627549984345907616431671031467107223700218107 450444186647591348036690255324586254422253451813879124345735013612977822782881 489459098638460062931694718871495875254923664935204732436410972682761608775950 880951262084045444779922991572482925162512784276596570832146102982146179519579 590959227042089896279712553632179488737642106606070659825619901028807561251991 375116782176436190570584407835735015800560774579342131449885007864151716151945 657061704324507500816870523078909370461430668481791649684254915049672431218378 387535648949508684541023406016225085155838672349441878804409407701068837951113 078720234263952269209716088569083825113787128368204911789259447848619911852939 102930990592552669172744689204438697111471745715745732039352091223160850868275 588901094516811810168749754709693666712102063048271658950493273148608749402070 067425909182487596213738423114426531350292303175172257221628324883811245895743 862398703757662855130331439299954018531341415862127886480761100301521196578006 811777376350168183897338966398689579329914563886443103706080781744899579583245 794189620260498410439225078604603625277260229196829958609883390137871714226917 883819529844560791605197279736047591025109957791335157917722515025492932463250 287476779484215840507599290401855764599018626926776437266057117681336559088155 481074700006233637252889495546369714330120079130855526395954978230231440391497 404947468259473208461852460587766948828795301040634917229218580087067706904279 267432844469685149718256780958416544918514575331964063311993738215734508749883 255608888735280190191550896885546825924544452772817305730108060617701136377318 246292466008127716210186774468495951428179014511194893422883448253075311870186 097612246231767497755641246198385640148412358717724955422482016151765799408062 968342428905725947392696386338387438054713196764292683724907608750737852837023 046865034905120342272174366897928486297290889267897770326246239122618887653005 778627436060944436039280977081338369342355085839411267092187344145121878032761 505094780554663005868455631524546053151132528188910792314913110323443024509334 500030765586487422297177003317845391505669401599884929160911400294869020884853 816970095515663470554452217640358629398286581312387013253588006256866269269977 677377306832269009160851045150022610718025546592849389492775958975407615599337 826482419795064186814378817185088540803679963142395400919643887500789000006279 979428098863729925919777650404099220379404276168178371566865306693983091652432 270595530417667366401167929590129305374497183080042758486350838080424667350935 598323241169692148606498927636244329588548737897014897133435384480028904666509 028453768962239830488140627305408795911896705749385443247869148085337702640677 580812754587311176364787874307392066420112513527274996175450530855823566830683 229176766770410352315350325101246563861567064498471326959693301678661383333334 416579006058674971036468951745695971815537640783776501842783459918420159954314 490477255523061476701659934163906609120540053221589020913408027822515338528995 116654522458691859936712201321501448014242309862546044886725693431488704915930 446401891645020224054953862918475862930778893506437715966069096046812437023054 657031606799925871666752472194097779801863626256335825262794223932548601326935 307013889374369238428789385127647408565486502815630677404422030644037568263091 029175145722344410503693177114521708889074464160486887010838623114261284414259 609563704006192005793350341552426240262064656935430612585265834521921214977718 780695866085163349221048367379945925943403795600021927854183794177602033655946 730788798380848163146782414923546491488766833684074928938652818630485898203548 186243838481759976358490751807914806349439162847054822007549453489861338272357 309221900307400968003376668449325055676549375303181125164105524923840776451498 423957620127815523229449288545578538202489189424418570959195582081000715783840 396274799858178808888657168306994360607359904210685114279131696995967923008289 881560975383380591093603412529986567903895687956734550833629078238626385634907 473192752787401665575311901115434700181862569712611201268529231299371614039069 651122248166150823536439823966205326333222485051915936826907150043155898718027 833538454483091072494980578809617179963371670365541800414646675387195869484833 315435833306419359294874209514788323477484814181497768716944136400566451569361 165241615557341419354247213730674683338490544266260383727882175527099309581410 261369795007864658767716086308044607498028015769626759138977947722143375154708 293458791238984330550672234749699849424867067215025692735295850658695889974865 355621869580439971251689766541698626538628919775421877219396058170011042364141 587808103861721015575519237111600498806822916180977324219583289748692271839791 904677165426681388933792960368154579396113396219222454301515806317437084056085 364160313849829695185669526128221237169393681303212965619397187102070980079488 339101975351043074418234488333317969782773320911433245143050865734575006873914 754707775775599184671183085836601594371937184490390617702325365679775967444757 475115841957467009973450024544284065850245085856463927912461198790936930720198 040293036037388384307421628212016353864662260971989584367994305720301496380508 322323658255577245342371877374398183333064546629069933111259737219502746468990 654571554403039178354197564343157390348838667505427421618310500605504642235457 084273935493590517627174792994723989086329701019056101077426909264752357403046 301592434424649008341886308593206855225077909101958588953143287998175709819168 293159404530056325433144885173573026982569372534699640134408715801081452878657 904086636379450711085051042417976919112926151320103163634980866069486244078006 684006716962214637181147772683418466463642427340530031380773496119981468617685 854631208163164798937964263738356618938313710983289564905211488134029742388868 631543132978765799125454243338563472002681290489949550426980882130267263581532 480675387903230574210403301497887867523778607054688614721009926329425108878019 702841179224025910914665848092578571927862821476670740878635197142562924278670 284077032414375699318832433315590024333047691110092479791180062862022137078006 217257329047359943988831392799279693970635676281166940541288590819820238382770 354834968797340488882930167367709415846544009548624651461013539134968559120402 363618721509929806519058616828153028750427545258605331963432595777478813437239 394991243806143754498590686075185631427255255642593967014980414259818237852576 829436395965624388520656548071038845463944537701917845718741011862232278025251 943626574382422560935676925823877491160737759451401447031902241535591125061381 782974212649826416187246063133408919267023597958023658416317556792335662101231 335845494590590069984200672260251167743847364824385715407146265945642391127170 780306371416926386440100571310958960632649637552956769364689410517952000616452 021884353404730182439305148819845930762964044456877624165287162072767318606325 408014288745711986573074717018866036879703647708548528716700036229285288374682 466058814117540474460616763543037399237565965936967087923167744685693108382107 830483159196430021441259702289063203174101149366480952903011716334531917922939 242428772837872349569929232136092234947226458243755094515335520117612897517339 513717829332871586094386627011791841554587264898251392555943795197317867448769 925326179423381299941279398602644245196006054368186646709865944365930154376291 486979597694996533527210020020967910439482547244113342244870054637656840866762 153373627461591205470086290571698257353705239761231238412564349411789498615828 597020971097039196352168122582247562719519382728285209147182375343655254020746 203066730476952470094413814561782826663196139673593672572646033884946424894724 489784815967153061538467122369871282319784769311057166236373262075959711804015 143896231706019585709823813924666135279136956376559048616762051297409983149655 978602534101294543998672883006244084401861815117506870887646716097992920177696 249633015758292596188494858720022922480606288121778733831458825512939536510881 918651200449231549839477314735786889731420473109453814648226321023310794395974 628523541295751910635587923001956181312946120301575763401597856817517498423778 824473753982474575996867707408545574329424026781938482203540962106060721924990 825104854003184963199863221569089097614053107165113129326853498524402864482934 705916085869959819889039599559181107641344665881452542565881535450288847399753 243278090215225695218441452986508355129839802262382649748818115815250345997499 159664009832014520070480355568589973099815031041922385089745373276129471712681 653088670054728865779224670178265948272609722903471574140316697310705050782961 082607287042912608231197415101473578478101072797112427976028481411516338868969 078671757725938152479612378999903666175603588218132546756344830914872962669372 198850270018829817301702494421406317265197139506221082764507181663910436683295 663807307194543211255053620896783230310851714811492858899362870648875584389130 047186846798581655521590139256032061376427156280118971096114002391303934060617 784821102217165992659596829588905449896485740412204749735425646718402365046790 582748590695047355535311838358089652886158894174416219886804230349243895814973 672227495181127145343869749924324213868414848375770331087617671331994426349880 672166702983717230571304972563819884436539981947970223275432911444433463252114 793188464973594694484809517347709407261091493510518550431888516380180024268176 317264589577881465894731621991354217970696392959125603838048585831567272541846 146996603519007670111782305279998313186980724782168173389846912490557910867065 430256293808109482360911182960668949738197631510515384935175523971928535671639 294213365269580491039503518402421871935461814477959700303498159579720430677759 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350937000463861006031635526626081975615057281743286197573113312439953870280147 398646481951044043706171345291375172436515853579700413959268525998127206830937 440974827426549100448972845133920884290100392354174550892017405721341477968861 838150118728784484229782013346708946882973957166350009120024170327964084767582 879474715637448850019020912066997357415500918386337344316273385635359478187486 231891538682917624634816247070727154605269563573473480946626498924617401296277 765764623728025927779480657766730119544692743950126338678444030100649194269469 957667185565363382615662324210286732044396948485770363843327964356245450681615 513771880359763392935479951262966854820586892757707754826468364824774254559020 807145925453009353453225487341641107976962272723879767243025919066038955071383 136691818152342825205396838698720024030663475975605134609154469548162387425378 158302653098104742760607748290171664036336150871497476588172192978853908464446 528705641619046836523612432367732528674613688684282872712218847056520735270032 365972755740237383131307183137426933651450657979976777726799881447914358583652 734628844838133347320253032562039218550423745071246201702474289175965495898199 655426861179406034823110616497789765805879019925874069554366793037911629180282 626336505128834652109745963364668600722087891966477672374488583609530207665510 278220140742685179742262555195530120108977140157868272506731574046078508917622 773625815367966375862818839152555955251613018703144216909288827431719952956153 271512085651116775294542856160716086003685818294277547141318914905593218301506 196236101121808775264666512306459999615933951176549704193364674478497559809435 239357666243284437023988696401177364692704574611809528748877957078110501719666 214766418626949735746138893917281555165412641863779432045863818098406043470354 870027768855685015445395272991659315976801943009541965850172319561870163232017 714943145752454182789724144200910922235877784762143982533049141480551884762764 035237769794587133262129421332780056882892714132334279085669544052271630773613 222223624476492604444925157608167205125791983401414314117462430350958588366449 885635241216735096498110886480454556121486157144398897012515745641666786086974 599235092306913787822489885741010152307835427302686702643076573361527388044793 980964034533893414269988016666251433188572227755985299273106746685733619787048 404116734845406324070562038620208792937068210443308574090272915222256690268914 154418055934840966397852111013396317334823271359536191364785487924262667118227 265433909164704496304973642336756093387686017097896893746341790095852105996698 311184947184187211471898960323813655945633958099118010918898042657125950534933 432590632358450814434844929283495242487228534842158559858227162257611210612276 943587121299259340778966149794293635945828339925529844938205488768791335589322 378202474504684456545675207302749752781246295549575961201910351796377991332194 760924906812854053093940777922686627236493385363777797775841839985573147572508 278278133424401285982580466473015242013987134640272855003454864134949658500406 325010032627834898309246405253515128757565780041454196947196829194129125382322 323040242780772044913500813181044059147935301275937249608282170902630073205555 942055092241584936796162684784919911479278169140343465155973137653133769095915 408557353221970821520200012878529148624180724372726554026068919977669095929312 884966544823351307602582047220179815182981537115752145138942540406776414150378 261542887352743572109269918298733784934925067944884528014286190663864893203832 262475585291450877811812606609287425948652977367927474274345444494299917920010 592825788588113902413233669323988364616559366434889317217046632974698473209548 716538766597259787417373930873958472547273737620357931162811189212024790071245 606469357417284761827083359348188921482659253659222451433399363951099458994696 946356256322943123704715769176584334385021496772976038499059194240009998943838 797082608197390942977149427159059509449833904644963958202273423071851709842599 767135871155365317731794555314716713826977065239562572301017029691252626488197 191233757103889948396553423171412191470103447526027865609798213571692749883016 370886297798018562692270340541981638788082598785178596072450292199365464328545 845537098344079433001582732447215368560726924499924496560890592414895661596875 510475437876313196604588788445159012313418329514858528887605451799863092365858 284379784336048329938764878904605287054107746539136015280698825897759306969126 743288071527820619474346504546690173027794183372811515486678727546215387379115 550722143800612230582630841490858633187051228683676576287111524137827187928358 183112019706088688696349390736783165658883202173568093943999985504687912622999 501403710446795445902947321277458468364057512251147621689571851825122939713687 690034964094187762776565918785051166359770911611077709056893042232586633880423 332641476369701384662384172105006157634901704929732854275295654961204015190472 324601956711162863712879382278936420806227252391446338575146990663577833960269 166968095423399522369570065424450232261096844773919834646255336098360100685095 774964533666208297222159707073618423201453895130160080791291134744405932067551 763295733958650556239133883104100736158904717574756529573433505140903826999699 603607039261188627921921196874050802238921046483609662032780823943745532697338 678333874962639215512698373936532607451211556697184835766320687895642282136622 715867419516911165836626866744045455083127447812394910099323884790018944384080 848875050517056179744159971295571413750822633140320665846277239471306970143937 789246958168275660874369056961391946014632458426105667029785928612327249477805 277259332199024776783298212540205132110141164448518460584779710688655848291095 318621147278146482111054771441916191172 bytes used=3001004, alloc=2555436, time=4026.65 > quit bytes used=3001164, alloc=2555436, time=4026.65 ----------------------------------------------------------------------------- GAMMA(1/3) to 256 digits. 2.678938534707747633655692940974677644128689377957301100950428327 5904176101677438195409828890411887894191590492000722633357190845 6950447225997771336770846976816728982305000321834255032224715694 1817555449952728784394779441305765828401612319141596466526033727 ----------------------------------------------------------------------------- GAMMA(1/4) to 512 digits, 3.625609908221908311930685155867672002995167682880065467433377999 5699192435387291216183601367233843003614717513924207199658915240 9402255997742645889036145060641374489685419499920192677303799463 0892212412318323707992084397369907093905620929232342870274191448 6039571368350368654879959683684764758514890904041663407630339718 0668059577342379085590807145783129763563688255879288111906351681 5850849474881502788673107310524879825166366128793164184417443827 6457548009199147768019228150926119943229978378353634595543419474 ----------------------------------------------------------------------------- The Euler constant squared to 2000 digits. .33317792380771867431837613635524422665941714024962974315083333800226579369575 666966126326863171597730303956560340239859445266992699598365527983313786046034 268780373558371128830975908549888552751106825808280078552787375606684200974130 617588691178694315050324483702760663422216993421703577153253600060741015276339 995955943381572867437578342434224655682393243470405850320136057784655662305806 336598738504567052000302014780312845447665262628245409059368111793718690365391 787374644136628232478821575207439195083023003582529592911049416405497804079735 906414541071473018833423728505726964936987752237826402067444337399595520475012 288526287304305672083807917820314483332377181367931160843498881621249228779170 420363188448139000587782356093541658303906200570835173454721355359315749772478 567570824550788178469027530815855753394890223992432182646851222376597944082250 651535911750795209885106735426727042416802208197237186718051362364187225776476 470637129109290590422937316034959310316334478259163645260184821130314905132228 424515350718917437096729789431293384160628313117664130173704083301200438871961 377714361119697224389714966794999085492494451623705625139833485599878731684805 399265239678318576488579463179229662150505567200784546393706439420057318174508 184490004029944723412825377473700716194720015430484223559786301861306368734881 313371882248317461331182208906680249105105177437367317953837332537628210518958 414150659594781665278992239381206263980968759308421024899927860428325296015100 254083327019038389413937463567795591719222262167354922700124199888590086508057 007138340244455586691192098911631143303827436130349373813083644429999583728353 369352017064252566989786567296158684564092320306564449034531271427626316480736 098201513997004646088266563584289225772605733841988691095309995814940916745770 933938681533882829140446609531584136533622731814962566534978326918981782082993 418034436561490940423020354026303602299616110979002333613709999437631985392429 351407961444985983615358391278333538703785385676286 ----------------------------------------------------------------------------- GAMMA(2/3) to 256 places 1.354117939426400416945288028154513785519327266056793698394022467 9637829654017425416758341479529729111064348236100330588541422615 5258621182660719114811432283343415591562091750568259236652338521 1910858011501770153617023853945368317754599736504155930691384228 ----------------------------------------------------------------------------- gamma cubed. to 1024 digits. .19231551682118458966319237441963590712167826133337523867325291253917884491613 793593739097123785566051165094965780249772919084922981643738599345213543428957 742288546576032444206688457334614669849809568086840871644035741608308542723092 543816627167731079380200399942426915166058779273537957134087816599737672826305 617066052156709273249321401290067231051721766282999888936851124722927435845590 319942645201785383892893590850587178162063257541072513978189837556112379150636 148760910104390259034417469090937012971326354602687664132843476901741631775777 081234656412363217863473461261574743283715998060224863081938734918646849347145 033237076361895247637420595507164456702675975151779741772629529669464407992337 123086115363644441929623938101426279598514787173428909140466696339146115142823 325480256471561795210904195108554856936575846775437708628098851574803182659483 351246996032005245642253177026204332161514379092596972555376134272694772206998 440842283175674621131512645887533783859717598289275645296301371173946908237273 97365598808 ----------------------------------------------------------------------------- GAMMA(3/4) to 256 places. 1.225416702465177645129098303362890526851239248108070611230118938 2898228884267983572371723762149150665821733802375880331630166590 3296103947930471025505998382277791927689007765101690145533165791 5948759445277305159342900375786380960492388345759811873070193570 ----------------------------------------------------------------------------- gamma**(exp(1) to 1024 digits. .22451725198323206266512829374391428680958174657315887299976447489059275846479 851251681928362572708209730381722857833380421173934573990970329852312435340918 723516863644528124322010037083830953350325119406626169159798580553819189019644 525366844056133526380503994361412314960214996628220370770969562579508342794572 702991261240338976437164936112074563081171190911120222275138363004074757488349 445070892014203631363507636089439980519187261359111374500534809964325181302302 877148105978575777706778407410040095779608096178596573497347651697007658000109 073098455646714573376452062410999493231083774116020607012527713362204649452762 377532893260193987058525344732248680390433602987907816010399245176769654887875 249879130584301906395707120983550329899506386856496243934190271238088447545277 448372057783873211254473602348442474028053511549176413730565419185556166032672 995606386281785677423142512912153638567058976070288094249292809205132638547323 922070498702820218743762166666374086233986793871920852717856457523739869944430 40623078909 ----------------------------------------------------------------------------- 2**sqrt(2) a transcendental number to 2000 digits. is 1/2 (2 ) 2 2.6651441426902251886502972498731398482742113137146594928359795933649204461787 059548676091800051964169419893638542353875146742420314383674078186985054875748 950831147839628583561836083461266431794091489100534014373950342870833119045271 169737315956529056576328457297981774346372848330862819349528549927583773563188 830693383234459611805080976879081261274910728976742978426637632502369601695624 881711639702926903859903555628460115605232024465006631806391529947959280102745 500352847408628685697748491775145744996588372975745725899906388280003508036903 733879090467182805383659676795228294681896018049344615328808423789947168382540 568693719973773623543726326713082085669350247813249584441266170431183380998258 190363113378957685095232730797115780607677737672559957796299769439384975413846 210683714037918940572046075515482487463115813270055277740580117488694405652197 317909350581370671620544575999859693906949876978769247287723037903391158915566 420267606097113568152490853124555456599355741761228151551680899339284142161770 232062626842896876281108354897870567478353447994083334164280221005939665814268 099506656731699682085700857612904437186645383675046896232582721370112050740502 386252630389533268325697347198943234550477641417747497473252987559073026395876 017811879634413615303763505385762659815196734847570797396637608468720047344119 580770957940634323627053077957093296260478199325554149573588752687081621624236 506589930746778556987975753446508559720699750219983091077608980268862489801452 899911236684871179874279954411637105880602110171485781070632949664966383211861 495485877240590608092504748657763832877253295432068289256776649101883658978959 997849718819652644631951910159879232861820961763086731968735658298762079274371 471703151308725777185995258140397579936758760782486324489438492126898947226536 445638549209472367778232932556878676532449691522697937686003006888480147654028 470526446832672538365906814675431912844507558089096325837811183599592847614815 274128432280444427943751270444589333595911610552128 ----------------------------------------------------------------------------- Si(Pi) or the Gibbs Constant to 1024 places. 1.8519370519824661703610533701579913633458097289811549098047837818769818901663 483585327103365029547577016843616480071570093724507999019639342272322414165036 365074788027757760407005425387045947037548070012549126196000327078575312602462 781280151598692712625156658037819170657049819111714215383017286869095002766891 969837835648786933759294319175361858839873281361537111741600533650285988928906 414670095488877382247112955736673406636533206353917604135172039112403028911351 451318386134929257744182407526476030905279207782148560221871814904254471501463 635842777947117746613775605839980813601589774035700341407559120370214113987005 974964457642432794571720297914619514587500552129836800839402275440787337189077 600233378591748197346154415354013755202065349536370774797232235307627711101354 680926841172462714308267187960091741576168508046447756294559627846381809450570 206315108346086296761158384244642331395026518568824439952885040681806714182600 926258083237153223244690004091924289785349238396416174935955727240494968269552 94684580908 ----------------------------------------------------------------------------- The Gauss-Kuzmin-Wirsing constant. 0.303663002898732658597448121901 ----------------------------------------------------------------------------- The golden ratio: (1+sqrt(5))/2 to 20000 places. 1.61803398874989484820458683436563811772030917980576286213544862270526046281890 244970720720418939113748475408807538689175212663386222353693179318006076672635 443338908659593958290563832266131992829026788067520876689250171169620703222104 321626954862629631361443814975870122034080588795445474924618569536486444924104 432077134494704956584678850987433944221254487706647809158846074998871240076521 705751797883416625624940758906970400028121042762177111777805315317141011704666 599146697987317613560067087480710131795236894275219484353056783002287856997829 778347845878228911097625003026961561700250464338243776486102838312683303724292 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099793662043449569683547930754311327558643189731515171064432189249793277801264 964764475467078165807406131259375271847408816115479818307816751047809291413954 564631160581269051753953556915775580410671981231638405277556052272223764711883 233223099585068971018717504781906533494858423259762256575841898529144717833517 322602985786292943465056366932162627673816245957417932698892327220666636081992 490988831468529940991386734446049670842442978243630232938910355965601739942201 988690257245471401633009612146187208365108688185334060622017099515827070442337 042180176696349133695996064322005328873494893135966030424380804565944743335678 31672703729636367594216999379522 ----------------------------------------------------------------------------- The Golomb constant. 0.624329988543550870992936383100837244179642620180529286 ----------------------------------------------------------------------------- 1.782213978191369111774413452972549340791731909773239381024959956 8851541287637840802431676635782553089344691654390590242832007237 1609466241224698183193304440691580844300831623521485059225129118 5032205757091260620528266462450802768645378426168771943578383034 5978943604654179333058167030255047978147740633984429355512597392 7131851484715199464267093555750824146784832149267419783105716722 8768468972668325299481120507392284163942673878597000697791104289 8487803162999699269304351197324998710178766727173045863648514760 4701659316888114930720650660711405336160026631760170680854538265 9150318190903060193515801842051584567626640372089564659879643577 2488389325434670915813658089829180199037286782797617046758172337 5507340757273943007007796125518210897841661588530594349694531781 6864584734664771038367942228085531134516968772013659085159554971 0033709286835312292255286502952743356554669481909219776476206717 6817891575544489011908491514331205109901351470453441453432787641 6367319236317348475787753626711408551283923240403824950905050799 Grothendieck's majorant. Pi/(2*log(1+sqrt(2))); ----------------------------------------------------------------------------- 1/W(1), the inverse of the omega number : W(1). 1.763222834351896710225201776951707080436017986667473634570456905 5472758471869957367890838910506811055619330020274054680467376400 2401379520573801043392673302307036497529675447164274374303901741 6565384005522095243453556698266942639558302053548854145913508208 3104393643656736618587043331573171809287097789810954168363751211 5774719105876483128311371433944522684813012018209804037944042568 7558913817470781415827410676176997180106117658115344871122490403 8394819485117511829843123792540192533487442618499553352029977896 7912880511965457695181197786947920843282297623994619882094844581 4099680627550392429004489387836181076871507788319467266809965539 2277112528255933430969581697789255189648432700787154001865296672 8398733904857956041043333256191892446199131893146501992769492797 0415858853237673525171809006097427444281054965744837481066511899 5095678265475563359222255403243408821197885652730518924925961740 4375740897180239698196120787218493119629216937297365667602427646 8323643137024181077777851246850400311319023111832315787739592779 ----------------------------------------------------------------------------- Khinchin constant to 1024 digits. 2.685452001065306445309714835481795693820382293994462953051152345 5572188595371520028011411749318476979951534659052880900828976777 1641096305179253348325966838185231542133211949962603932852204481 9409618068664166428930847788062036073705350103367263357728904990 4270702723451702625237023545810686318501032374655803775026442524 8528694682341899491573066189872079941372355000579357366989339508 7902124464207528974145914769301844905060179349938522547040420337 7985639831015709022233910000220772509651332460444439191691460859 6823482128324622829271012690697418234847767545734898625420339266 2351862086778136650969658314699527183744805401219536666604964826 9890827548115254721177330319675947383719393578106059230401890711 3496246737068412217946810740608918276695667117166837405904739368 8095345048999704717639045134323237715103219651503824698888324870 9353994696082647818120566349467125784366645797409778483662049777 7486827656970871631929385128993141995186116737926546205635059513 8571376169712687229980532767327871051376395637190231452890030581 ----------------------------------------------------------------------------- Landau-Ramanujan constant calculated by Philippe Flajolet INRIA Paris and Paul Zimmermann .76422365358922066299069873125009232811679054139340951472168667374961464165873 285883840150501313123372193726912079259263418742064678084323063315434629380531 605171169636177508819961243824994277683469051623513921871962056905329564467041 917634977065956990571293866028938589982961051662960890991779298360729736972006 403169851286365173473921065768550978681981674707359066921830288751501689624646 710918081710618090086517493799082420450570666204898612757713333895484325083035 682950407721597524121430942470953115765559404064229125772724071563491218723272 555640889999512705135849728552347645942418505999635800934732669411548076911671 455813028066898593167493626295259560163215843892463887558347193993864581698751 045893518777945872755226448709943505595943671299977780669880564555921300690852 242867691102264527531455816088116296997029876937094388422089495290791626363527 791432286156863284215944899347183748322904155863814951281527102068249218645827 978145098870379211809629840943604891233924014852514327407923660178532707078811 584944045092539519718157085780907690772192962552262890529967200510669638584207 655081660527132551761150093619010182152039541621744474356571314026496051480322 439134457528009739604967190734667398621127034770623094786463721777245551191609 693349580116501538146897732947400254272699518373881294004390465050310091210361 980535760952228835847669743267507757379848939356645406017251962513826671863828 822629657399438626453078913514555113206475947913245582423662405126070382560901 984614575152951511943211356814416716008974384391847402590826495013602834007260 634108659796382596784136373377680857831279147106417370573337040146024737648200 768231118490558678994106995743922457089666910491534089500139419890965785853368 531985664042350494746329804481593573838687414276915611134778612290893976432134 279879206472381493290546264824907766030881348705331723336407298994245656611424 036824812873959790915799781062723446426357233234127834780836022424212901203199 698485951429216878840715626887034517436895639117072657935407050794141100343395 582796409783891583020407548189623248280478295465239223872194333981851251004747 915658247782096645428132405620504841629632689157664149594957463505689486587289 243413739100608393347108455293656982935338521814358746992934863313565820307192 052961665775757266627408455837127946799180904710259452519968016372631267038023 447298309515688684101430849594108797807013561524049847909714362331059569179431 431691402111049142985963053516771600084867260895318575293282183754558954666446 191468252314874744996401074402664686009572448671687582605311706563448494800841 013235616345298355661883397936163440983329415351662473668696589017509927510426 179402465228834883071274736258665127650610546028964911287287862738263596129333 646532191462193644375375986052224103348422135461338128126772131245890101073371 833507228401529641174179963903413664423919027895258486882959689695414733771140 886608735962775876801352400029217923528552903064508042269791458998101532140852 906522851155926562268843097382297291671643980514954868377297117263777565889196 871160509021211371651902341089659293730249465771175004282806372542711805610086 582199353671242849838516132864990427711284071698787833006397284922820344349455 347600218003759357103205064406756731817833717035278315824041187154686581171864 952708996224348497793884832819497863569163437333174207210017054301866996336546 345356688803248528083754492545177937909480719571292537184111634137303745351158 808536058387947244006076177381212102789234193301297654667548505996663777643498 372821340898092902889278088428356657996352849776141925529724571724867587136418 201398244839217272327643398650709245008820337499415470402244860516920495403808 927989711051717768667304408545186666225235192118021635892919167850976801426649 352465241315279351720847405377357578042399840807239581517870192914436593403426 041472365945332028684204462087017959678507105044335899885299980564606846787225 732214957985497309246622145607355305369122171822980735330797038775292460178947 026111522360709501432667888113281990837513268845551510892190791002610859910028 841896531906865518597376800549155279239015601038178005539072818968126480022740 306581014459893635059496241529118447243731448964790876170326036539290133430684 781818206966837601732903016602762498422827578841548456314713744987325573405808 059688168787542109613848415947550297583968310995807378723577778742169704341423 897424468452868147350993296144684543688264968869192161649670355650366847261236 344317060911028964507813116327249011793664610722979820574075779145800908301060 913216200897249373201032777744785128764388089981656932715040612090221827393598 823187046383122896921216104846546948319792790320515076726732119163218284371172 036046713301772401534216569368578887782115998981797541546286000230716000159982 248277212628384883047943066328546836223944311977655842794918993522521563237555 650145286211059365868171399984063710196404159462648235183782162356589166387664 599049458627927377620486160161495004228978098646407047162007645445619312738742 470000474 ----------------------------------------------------------------------------- The Lehmer constant to 1000 digits. .59263271820163619710407860499570146908427540719716107109956260815824735236416 000851066478429710125705118718346542386963492602972067606827856079871979437487 251403403000583692486915538640614660348566823412154845636446642830937804980068 991183184965696172012113759470623436986514577152028677342172295805168234444776 889445080541632061558512323623064379170223520781603669639276342712524885456742 628397234194861141555441118656607347923877980434663425531692753922997364863485 871218333767125332765552976528666126066185788713622949403089031984313907958405 478989509947007489250053656329690017664408563598385209478301821868952137848984 967705787356537385497593520672298603936207528576177510978795686629351912784453 933568454874614667993688671835743937379165908437616570415794750776200309560115 740427603024939939214964193674171720654296953903175526951535843439567271915190 168880981717597249067581873559990538542690575895690162307174847972701860908496 99968957675960828498381969270404195036511309800972643309599823665 ----------------------------------------------------------------------------- Lemniscate constant or Gauss constant. also known under this formula. 1/2*Pi^(3/2)/GAMMA(3/4)^2*2^(1/2); also known under the same number divided by sqrt(2)--> 1.854... see D.H. Lehmer The Lemniscate Constant : MTAC (or now MOC) vol 3. pp 550-551 (1948-49). or Abramowitz and Stegun, Handbook of Mathematical Functions, p 658 formula 18.14.7 Dover Publication, New York 1964. 2.6220575542921198104648395898911194136827549514316231628168217038007905870704 142502302955329614290934461357526717832180556089569013939356947011194347752358 404226414971649069519368999799321460723831213908102062218974296008565545397723 05369549710288888325525 ----------------------------------------------------------------------------- The Lengyel constant. 1.09868580552518701 ----------------------------------------------------------------------------- The Levy constant. 3.275822918721811159787681882453843863608475525982374149405198924 1907232156449603551812775404791745294926985262434016333281898085 1150341709970823046646564670370807129022418613959423772012981792 4251087697614930028806824926170594041290808697054441234922379888 4427089726916409835535854804837478582876252691844500764338370387 6741884459420351700003732122230624193601340916574804354470221732 5993822298382263233915155285779861204684944725214872290148093612 0890454342092099519978162400515039051381538771475429139307394617 8045527838277727414847221513498826672911215448270452018633487650 0197461804182629653652880116815110575135087115230163029427488336 4626099362896606336377631721650883480193611175632669841114675716 4673788536609062436703325968873753578142654195992507552782054563 7159407017761632115357055923392379438429270901774392301637440604 1267849616074783099029968056083369047909232336111034927147379005 2538954916571473419972293167664630764119832764653881696204634629 7332994286181358520050377154266892424005731518557308952871781294 ----------------------------------------------------------------------------- log(10) the natural logarithm of 10 to 2000 digits. 2.3025850929940456840179914546843642076011014886287729760333279009675726096773 524802359972050895982983419677840422862486334095254650828067566662873690987816 894829072083255546808437998948262331985283935053089653777326288461633662222876 982198867465436674744042432743651550489343149393914796194044002221051017141748 003688084012647080685567743216228355220114804663715659121373450747856947683463 616792101806445070648000277502684916746550586856935673420670581136429224554405 758925724208241314695689016758940256776311356919292033376587141660230105703089 634572075440370847469940168269282808481184289314848524948644871927809676271275 775397027668605952496716674183485704422507197965004714951050492214776567636938 662976979522110718264549734772662425709429322582798502585509785265383207606726 317164309505995087807523710333101197857547331541421808427543863591778117054309 827482385045648019095610299291824318237525357709750539565187697510374970888692 180205189339507238539205144634197265287286965110862571492198849978748873771345 686209167058498078280597511938544450099781311469159346662410718466923101075984 383191912922307925037472986509290098803919417026544168163357275557031515961135 648465461908970428197633658369837163289821744073660091621778505417792763677311 450417821376601110107310423978325218948988175979217986663943195239368559164471 182467532456309125287783309636042629821530408745609277607266413547875766162629 265682987049579549139549180492090694385807900327630179415031178668620924085379 498612649334793548717374516758095370882810674524401058924449764796860751202757 241818749893959716431055188481952883307466993178146349300003212003277656541304 726218839705967944579434683432183953044148448037013057536742621536755798147704 580314136377932362915601281853364984669422614652064599420729171193706024449293 580370077189810973625332245483669885055282859661928050984471751985036666808749 704969822732202448233430971691111368135884186965493237149969419796878030088504 089796185987565798948364452120436982164152929878117 ----------------------------------------------------------------------------- The log10 of 2 to 2000 digits. .30102999566398119521373889472449302676818988146210854131042746112710818927442 450948692725211818617204068447719143099537909476788113352350599969233370469557 506450296425419340266181973431160294350118390289817858261715443953186192904635 388469952023931084961246254040026331259462147884584731828267268398232619654279 350763131754835092713896494691778576891805079000759954808781545971458503196487 762612249229082911819095149899717161986047767650006782051791255732862866834200 040292050983708457222489549429756214970724465970861368960922190948276121439149 652823516782649231480402774624324416331153873825930388303938063321613023905188 058213191568546169290530150513192698537848841871832006575356946839297174213201 090589689085058562464098721839687664853985623516127730263892787826084983668103 030843141556081394361767454885666342453812373393242246959434906021204450429682 746068847854611568476841064379795004659699177456575408640184640794565295443410 774082939997454007372170168019488905548569106940037541168996341575929721806443 038102815203392388085633198685453987393548560657842896848982613944260846632782 952602876621276230434192202628912112083612600558368625489999909279487843197474 433888686291177131574131432228241690729958547252661570168378653248437724845014 942310709810575476442391111669469145546531582130875457148591552640646694593973 872746626264815563731353272693379596968024623637358037017027865278713823682667 495198288846233675574623064477933647769803714706831332588818731312138647402960 387841835706778409896729322309228363640902016770371618273369284540872180801447 717626255069534761608867969624937665753204434486879532892939253551114683172522 672690275744806780237681755348374057043821812232253331678962079755990322930597 596747208666484230417392379259986253497978309395579390585310379752521430687788 055906173448921911090260258267733075735592578884228777929210367534078634908553 047948919541274191849959984720028965124825229007476444632358842089065039549599 585584910351150484927218240498074544155997149894779 ----------------------------------------------------------------------------- log(2), natural logarithm of 2 to 2000 places. .69314718055994530941723212145817656807550013436025525412068000949339362196969 471560586332699641868754200148102057068573368552023575813055703267075163507596 193072757082837143519030703862389167347112335011536449795523912047517268157493 206515552473413952588295045300709532636664265410423915781495204374043038550080 194417064167151864471283996817178454695702627163106454615025720740248163777338 963855069526066834113727387372292895649354702576265209885969320196505855476470 330679365443254763274495125040606943814710468994650622016772042452452961268794 654619316517468139267250410380254625965686914419287160829380317271436778265487 756648508567407764845146443994046142260319309673540257444607030809608504748663 852313818167675143866747664789088143714198549423151997354880375165861275352916 610007105355824987941472950929311389715599820565439287170007218085761025236889 213244971389320378439353088774825970171559107088236836275898425891853530243634 214367061189236789192372314672321720534016492568727477823445353476481149418642 386776774406069562657379600867076257199184734022651462837904883062033061144630 073719489002743643965002580936519443041191150608094879306786515887090060520346 842973619384128965255653968602219412292420757432175748909770675268711581705113 700915894266547859596489065305846025866838294002283300538207400567705304678700 184162404418833232798386349001563121889560650553151272199398332030751408426091 479001265168243443893572472788205486271552741877243002489794540196187233980860 831664811490930667519339312890431641370681397776498176974868903887789991296503 619270710889264105230924783917373501229842420499568935992206602204654941510613 918788574424557751020683703086661948089641218680779020818158858000168811597305 618667619918739520076671921459223672060253959543654165531129517598994005600036 651356756905124592682574394648316833262490180382424082423145230614096380570070 255138770268178516306902551370323405380214501901537402950994226299577964742713 815736380172987394070424217997226696297993931270694 ----------------------------------------------------------------------------- log(2) squared to 2000 digits. .48045301391820142466710252632666497173055295159454558686686413362366538225983 447219994826344392699093271559766135889748125512841335826850317755529488084429 083918466479889640433525242367364365809288123088602963911280715303182661763796 098673082702453105925226656312820024956976451435307963064082905548298567572314 978510155867910849608390915193311084870624052845434182445492796725716952952577 112359657358801041335486474970435268692388583338828166487908857540975096649723 158050786731973450614471200509341451651211862109203508748202985578691271609236 429867173301844624563281759641081926635865778233923697428001426552796882397958 699081971291893820699930825839343233464154340966319873368980954169096883844669 475455105834078055480402621723577305762889765443344080058548975962214119995192 270879454946813913700234932649350986356999460108860298900645465619378210410794 978760457745388343582681206074499441569171338158390742524789445815173948614268 602975364968994533565229182173888716905266128207010601191677397255402480756248 841017533750114667793807441522051507737288756300106495195566095982819853197718 427233204358653582673030819925273550282782475835975021051202156925695807293114 719561323480377325289297137716733902980240850280661658200161208902474534456408 020708391564466394074251782258237392673066393349088630941472195142023389586745 687155804449158357836503410995389549790766768832250176230597517393677471159719 441479083953466637902057135006471738221771003656262750524725740125181133520413 927493114167990581194158219634131737550551497740716861951132459145167430824226 892920380483557544751404050461029915092336083146207824608201177405271530386414 047510596432138801973441847956220910403492441981420917492865466428695218727873 069462991782596535703111962746013890662738693072095637923459636963328544864722 713944629064010714484517585123118685587273025311551324838859175961721429578102 705178803700774249482144350081547902724003849735372234169034789932632165255953 580079452806705485149346920455794828652839542437604 ----------------------------------------------------------------------------- log(2*Pi) to 2000 places. 1.8378770664093454835606594728112352797227949472755668256343030809655313918545 207953894865972719083952440112932492686748927337257636815871443117518304453627 872071214850947173380927918119827616112603264697461892547492510365033899089548 201917187027839632231962611480106953907721299179844624279113855486999422005670 391966389850627885412925913729488231249524260974736305689987586887646607970258 953093145638634759757061713788462725643079461672052950585309829800787111999992 074126943705144047152430700687247592054316975009722719076849626583582485399922 753679280302789575459100202066417683936712388159514332525411750507649724518605 059042160990362403936104519600917610771497670658882278136156555534754445076266 765187901482804052386787426337408944137118915686982655208159082601536796094035 051774961877174911446465066877848938559655749937054225161751623317487505801769 689661835077881525919088198969357960783242618144657028735729075124759420708690 852634755752923440722283452753593767913238054014882609582282799976925761217812 723574091548090088859200013721780671774949241617759590438569372865738534554510 858290166156189544297285501617489057171251457966376452423264234211827830275279 345774101074566235939829931461103920384721043500747453198570298026622864955882 036406811561405812376973825437118859959735664628545106931132183001138639465392 306050795351816792533819663298534779884036037020478135606436496477299027604002 092012506123353425852902749694014191995559279413339875867033134479231884084453 309463607997148584790173555347026302571024022261206634314825908584287923797432 404825950473549492476599560915436415666900271726522243108707762907191899187406 281836671397296579799946974032226225842483012034488444326956286621496907231624 486368839777509650818360425810279746687118323736301682533727156029993798858368 398903598191817954714232819502479300845331670333100137070663902974828337021050 508970968577278481543549869408464596915721244519043263369140886159423909787913 265766212905488060203614413244662072164682621389882 ----------------------------------------------------------------------------- log(3), natural logarithm of 3 to 2000 places. 1.0986122886681096913952452369225257046474905578227494517346943336374942932186 089668736157548137320887879700290659578657423680042259305198210528018707672774 106031627691833813671793736988443609599037425703167959115211455919177506713470 549401667755802222031702529468975606901065215056428681380363173732985777823669 916547921318181490200301038236301222486527481982259910974524908964580534670088 459650857484441190188570876474948670796130858294116021661211840014098255143919 487688936798494302255731535329685345295251459213876494685932562794416556941578 272310355168866102118469890439943063138255285736466882824988136822800634143910 786893251456437510204451627561934973982116941585740535361758900975122233797736 969687754354795135712982177017581242122351405810163272465588937249564919185242 960796684234647069377237252655082032078333928055892853146873095132606458309184 397496822230325765467533311823019649275257599132217851353390237482964339502546 074245824934666866121881436526565429542767610505477795422933973323401173743193 974579847018559548494059478353943841010602930762292228131207489306344534025277 732685627148001681871547243978207187803444678021617815841904282007672124325573 801436417887682616104101681872424068790890992987420815218323752894275273253407 100283575069506240396546275224430846258845085978625308322477453888506800348832 434049008399005808094356528212237038870203680454860077621424408869725941358436 599922621173967080495095279271436315464044462308915818536711960837030485352090 967262958241504035599512135545033224174847410033198148783245256933470494993730 165633666099190395712282284488167431215062856999387403881901274483956479103477 288597211985064942279698579166995641855126504150219155471966585692972660652357 329373683002783092177660538703046200766158494670022601175679751800393479176327 784493514263496836003755785716070049818151918437343829093474666045775065927367 012111537058249647984793040420582396475385785096062609338991470612013024310826 051826295864007600305949432116688044610613468453398 ----------------------------------------------------------------------------- log(4)/log(3) to 1024 places. 1.2618595071429148741990542286855217085991712802637608557413098876773704027618 296101223453770989034911227080318766274303898468982938729508273723927866999000 719332811694866233549044312251923997037373455857086816990621624176838752185803 683719187644374061640579715851375818026262655154375649795097952287600939872473 230715403031591651644062384321705384972156977118844019083272884878941881844307 919345990258940632328035983263301113215653893982156786724339409664856033938681 661787242581989937448641376211170665290364479393502434022916027382720472752374 808965684893618694361814346465683854082033596995240401209016306158827847405499 261082972675109261175005020650983049155299442886378651470617671771509903573992 209277680173191791043413795237434922731457393130552595836144508236780593504772 405451361420889870883520714648836891577020310086195388598795555979960374899465 159159118712371167108503708699743976431753685554512678490816826280850570570090 294634207695926608500623581465176344913641844063185272050909348889287724579617 19746496097 ----------------------------------------------------------------------------- -log(gamma) to 1024 digits. 0.5495393129816448223376617688029077883306989812630647910901513045 7663142005575304756261898911276140684146692757919040495526318590 5450417734549848078207129395256287725094703876281688270975610026 1156408875886889419139970426537839323510832374953687141792987794 3000698803152906006836392355186819384773558380559416839086586250 2841553060753907344342607909650625913030645934230065412361060041 1186074892174667506461526100843421125988247546108823882428539431 6454281589506290937285564063171286057328260161429150263449874790 9691567676023896506619892254286017265556137668232879979106763396 2776903826327089219839457230041344361865407953362338720245711646 0691442538786831806598144179106273232873298589435840446451356990 6835067495862965072477055694509983292056288856482328938870143806 4548296318506275719460957159631960983323959080438147279242020449 5163099676146527251947570818748536000180541287063016105886560109 7015592439767669989094255929681870729160076789008393992781794303 4763903514203857308736435378434747170357841069573597602072345417 ----------------------------------------------------------------------------- The log of the log of 2 to 2000 digits, absolute value. .36651292058166432701243915823266946945426344783710526305367771367056161531935 273854945582285669890835830252304536483476556634251719406466348146550305627921 387302556189226997176722860418008326130199421895328554633938904615831328063710 408093693844171143967478468991705264226741840273910350046311904221893236804840 957722768754358035596648711491884558132023331639659768082499364529335846684623 574263437565253698610956952159177357201065013422224499941535070781422978330296 377743119702939743503626331562446876550886089941354907099089430394945825668449 468115227170631943254890446975649362234698192704667228041009029618868802182017 491019923610554905353248775853399980970743779162782181934419963190210504293301 242415059279268097540561402920991761582248023420901884127368122693212840349330 525077289091675757913935886266719087864681159566395944299795802063954813959230 279378674693155920271933093224151240077317162470589936907038275335137392318007 010419557221304563325866289473260701848823391100394107328066534014994695730210 827482741897152875253930306881389072913952011284420830343238504866857365878601 190465284347941981959538527516075714884787745270712517536149184711555374580933 501476344855689822357324218830557986891108083551035175426573698361264322026343 115299594227880362178756321039238401475311641419391888550299063392194019786380 699715464067242520949120024266549954436249466858160040981594217436885386512021 604072269424181483000915687305164387694737281901935916941408480035782907353488 307611185501000617441320718653608681267977410426445808179494847004384504009330 405535140614775145038048303614763630318479359673814223230902118107800177274264 210818656122969187369118689382050536042394269815560177525349171142132855949034 100553883795898327918010260094014333325283859699408123711860074284226847041345 049345789235874120085949073626871893573109110848363179526917077695085066923713 696243058862086805613028965326048016127350910020944965629549560018447387238456 668466998830683373478492430931052375224007892721665 ----------------------------------------------------------------------------- 1/2 ln(1/2 + 1/2 5 ) and here is 2000 digits of it .48121182505960344749775891342436842313518433438566051966101816884016386760822 177441200942912272347499723183995829365641127256832372673762275305924186440975 418241700721183715022382393746918727524327919301879707900356172679694454575230 534543418876528553256490207399693496618755630102123996367930820635997798850998 015682579785264932866665111624171380827259278847902609653311324722751493140649 850889321763660025666619532106796817576618473073515986039848457545412056323413 570047800639487224315261789680045093639052503490478543352197865370437193903357 677241670370417641767978031965232099656758795421613175997885741759883069252399 717590046453960557551254692968807903367049621356294492555120383931774697654826 977541909002148287591795010410315009172040285181976301883343507599305507581267 421313032934991077388766751780351352387576508756665097521115192509805325161772 335414969051191031376000829815753239644609931361117839554965237333780624451158 972538538125625324467105392756233692811966537796197589176667110958463736359845 597437135943592053489075726261345821454327659167799080629243972731468565536314 092311391895063108539696637577527511079577705177124842749437409823450126791364 958219681888381115601710967123335538333939270275009967204943917336889395799674 130763597728326095101756364120387680017229574545727080162869726194293372934966 548187257084227073667108151143278740184115332931445134747747519570047997464676 573641483610494903280902249776225702986230973608580845275935793406227236160751 971088461984502310559538051316421863947239859494645234070234046955383565216388 882407729962183950603296536936010628310600983980249842603128194607404195468678 698754289980250108751726295370872829184831419012990151954975813462049581615570 765362754084555825696012212173496741080404051978451915071512814910182254675129 624858581222722255988561112107929352729069072128826890998825768857977503070665 560540491091499109395002416249253602970873298576618011862110489015088164984777 470696739474116453343444463404706900064984082377102 ----------------------------------------------------------------------------- log(Pi) natural logarithm of Pi to 2000 places. 1.1447298858494001741434273513530587116472948129153115715136230714721377698848 260797836232702754897077020098122286979891590482055279234565872790810788102868 252763939142663459029024847733588699377892031196308247567940119160282172273798 881265631780498236973133106950036000644054872638802232700964335049595118150662 372524683433912698965797514047770385779953998258425660228485014813621791592525 056707638686028076345688975051233436078143991414426429596712897781136526452345 041059007160818570824981188183186897672845928110257656875172422338337189273043 288217348651042761532375161028392221340143696717585616442473718780506046692056 283377310133621627451589875201512996545465739691528252391695852453793594601400 379956519666036538000112659858500129765699060744667455472671045084950668558743 390774251341592412652317771784917799588095767880510296444750901508911403278080 768337337938949488075152890091875363766086707435833345108139232535574067684327 431198049633999761803046221286361595859836404758009861799938264629277646275948 484896414107483132593462053635073046055030768215494444154778884559535228440047 850918217255915179900785243523837112867132342905566964492585582623118824223244 661476739136153339414264534600881979155478967757529878307593230499751706785370 666315222134751026417324918906534257373051835228316776877311442944368108997522 287634554909933469253981028398378467695079971965163008386496663274223886761392 944112379606529081463545502415193643368404005225615575618053680459613160686367 226297126848055518038239624057983138433955882483556816617339018195508924667782 042898879384623081953507082523699065543916029676565349509487102686726405036344 889957813954840804697878603723560031033518890166410542245140400821480026071893 924502077785635698810693233664357379481092927781936265980614204270094398298364 733767922501305495445975380037647617519082652294857728828349379913418698964043 483457091550460629912859614271432256377699794328889523074041463529466113313641 884192574888189320796571991444939402534883228262813 ----------------------------------------------------------------------------- The Madelung constant (in absolute value). (for the NaCl) References : Richard E. Crandall, Topics in Advanced Scientific Computation, Springer , Telos books, 1996. pages 73-79. Andre Hautot, New applications of Poisson's summation formula, J of Phys, A vol. 8 #6, 1975 pp 853-862. David H. Bailey, personal communication, 1.7475645946331821906362120355443974034851614366247417581528 253507650406235327611798907583626946078899308325815387537105932820299441838280 130369330021565993632823766071722975686592380371672038104106034214556064382777 786832173132243697558773426250474787821285086056791668167573992447684129703678 251857628109371313372076707193197424971581157230969923096692739496577811072226 715205474090115068915716583082820050184892117803134673122964985828828184357133 159143170054956325334887536302670425627486948438002800259270026847557436497550 492246136239920400157506303972146648111512373640102950660119390467194373312530 445102911514639759331918047977946099333746429426562908969344779296885419044079 142558327219971840906746802376153893544565503602730285440849344302806267044182 412004397418676617724475639534442306853849527943580751895490309305073843954464 206438717926390780392074428209795791773699230408221437464566804310569266319755 045922443248074894080624749361070936309149224368986933140903796823240790046284 487394 ----------------------------------------------------------------------------- The gamma function has a minumum at this point. 1.461632144968362341262659542325721328468196204006446351295988409 is the solution of the equation : Psi(x)*GAMMA(x)=0 the point y of that function is 0.8856031944108887002788159005825887332079515336699034488712001659 ----------------------------------------------------------------------------- Minimal y of GAMMA(x), The gamma function has a minumum at this point. 1.461632144968362341262659542325721328468196204006446351295988409 is the solution of the exquation : Psi(x)*GAMMA(x)=0 the point y of that function is 0.8856031944108887002788159005825887332079515336699034488712001659 ----------------------------------------------------------------------------- BesselI(1,2)/BesselI(0,2); 0.6977746579640079820067905925517525994866582629980212323686300828 1653085276464111299696565418267656872398282187739641339311319229 6119532583948267154023368572077084687931653259676802609699344773 5279134807392866925472877889269341631325163541360922351694910777 6671270197989917890435512998227488474178151185828274743128000168 8397357503158963055814845672281277378531389353796457494911144399 5739496545408641490244407439658462383405191698214657075454152356 1619789277021570199808441532569484324720553204382546010895369539 5675614108617595161315382073293136443905115788991399794118453170 7255433214244317404753282387468232949778600917592531885601921774 7449173024758275105885300397998919614286772988729202691184797255 4158448910383265324626150602695958039517132533551829053986426116 0122414588261244351825022559389263137501312747096674905266754096 5360402524508385488358940164115563648340734549714076368827296266 5134246462433258344593103771627997645494162129529126660817978430 0381978775455761063199246771331988090510227282881824758312010616 ----------------------------------------------------------------------------- The omega constant or W(1). 0.5671432904097838729999686622103555497538157871865125081351310792 2304579308668456669321944696175229455763802497286678978545235846 5940072995608516439289994614311571492959803594376698474635606134 2268461356989570453977624855707865877337063566333012384304556354 2978608509015429081920856055752374819658465950807273089050157336 1831596070667108039283918360149499646349348448317465915933636893 3680971490856983717510093546792166747552889731475588925030572822 4604865124854109688318448770433467727016574464765200627013360494 8057883875774914635983034808686985627342099151198306130250270223 7292838727216542426572698430693890685874829642167823425504200307 1966795220895590936913343950051154949542716768789494702443830337 8784002606637609098563645828787818795338304237475555696975428665 6135480540090110477123732473016808842009334259193374301935466235 3076567270975761218841385994442828002635250684447525596225061138 7848128978427693880472920268889238516484753423844953902789609971 1060547842212061361983111973376227976096491011771088137407049732 ----------------------------------------------------------------------------- 1/(one-ninth constant) 0.10765391922648457661532344509094719058797 ----------------------------------------------------------------------------- The Parking or Renyi constant. 0.74759792025341143517873094363652421026172 ----------------------------------------------------------------------------- Pi/2*sqrt(3) to 2000 digits. 2.7206990463513267758911173864632335984260993721391108633548274030821847716895 308255261874823180902532843336217215997883561940787410516122246973780471945928 074272151105071606069926407320631215050260199436490411738611994573104634564097 276251425400592337791384867496792738531503755908688294159392930817350006553434 105356116752769892544162138120764758932680400960448160574491309596803480585543 534970781437557797530573104924990842174305955365341922005452803161699025970771 025521173408012537984272114454694894037585991508686216831576021038048514103191 563504020401188548314412282035224115988825835545717429691325531600376513919561 019710037879836953583879901983392966634781510062500750756944690533333707348100 039222314716346524885715551854825024485691037579685679693038369707885725177104 783247588661153444463427387859212832429279247448410169983512035228487431086946 359808564858159787685162879186164433753334576814659887581309409558887064246861 421259503383020983023174436454225936586618158636979284521950970630746419121908 923081201542756378309298996229447185580881264591539469858458740661651509142538 794755404627240173080387592375699778205320713962567715219470572493857701378055 651612330089104621562535503293796318749220344258295905168829438348497792425069 965232798395101160887116725143361470643578026381152996572241488892468259982047 719368012173188900662946509655974598048346528979686976958888804076209361166540 704371609471365658085340968039713604386497802875757765485851456902692930178553 526336770293386901221522831898204546414170768297808208403723684204215427401316 902549740082060396161935441001000042327260835241627100691542183693970238796767 421706031793419565345410469549027373119213813026406577379804785073767429451804 666337697136947502388991025107668294693700271592474273898844052457144153342167 385800911617768074294556500450374678259995000280226355161352143259600644716783 216865021466013908629559836007620301423861196436445664580994458272741704107342 793660030280520213415008028445591430514626020556462 ----------------------------------------------------------------------------- 1/2 1/6 Pi 3 to a precision of 5000 digits. .90689968211710892529703912882107786614203312404637028778494246769406159056317 694184206249410603008442811120724053326278539802624701720407489912601573153093 580907170350238686899754691068770716834200664788301372462039981910348781880324 254171418001974459304616224989309128438345853028960980531309769391166688511447 017853722509232975147207127069215863108934669868160535248304365322678268618478 449902604791859325101910349749969473914353184551139740018176010538996753235903 418403911360041793280907048182316313458619971695620722771920070126828380343971 878346734670628494381374273450747053296086118485724765637751772001255046398536 732366792932789845279599673277976555449271700208335835856482301777779024493666 797407715721155082952385172849416748285636791932285598976794565692952417257015 944158628870511481544757959530709441430930824828033899945040117428291436956487 866028549527199292283876263953881445844448589382199625271031365196290214156204 737531677943403276743914788180753121955393862123264281739836568769154730406363 076937338475854594364329987431490618602937548638464899528195802205505030475129 315851348757467243601291974585665927351069046541892384064901908312859004593518 838707766963682071875118344312654395830734480860986350562764794494992641416899 884109327983670536290389083811204902145260087937176655240804962974894199940159 064560040577296335543155032186581993494488429932289923196296013587364537221802 347905364904552193617803226799045347954992676252525884952838189675643100595178 421122567644623004071742772994015154713902560992694028012412280680718091337723 008499133606867987206451470003333474424202784138757002305140612313234129322558 072353439311398551151368231830091243730712710088021924599349283579224764839348 887792323789825007963303417025560982312334238641580912996146841523813844473891 286003038725893580981855001501248927533316667600754517204507144198668815722610 722883404886713028765199453358734338079537321454818881936648194242472347024475 978866767601734044716693428151971435048753401854871978498171578797862574705055 586118480974544127839248838014963419252344251830539387345286023806694607504484 080699475624505200207655323792600572889555014667273281155797489419762824871884 385791643672348212893972728665478979118115392675063032572810519042832281433674 735643323719280793738316082322206416519190295943581796522481636008924890101073 666519661501290429787243208904403012876626194449044016157378609712734956779582 031482137001916370540383857568261585661103574743525068448269778512883738443557 674524034077728878998702772569096453262689728629595424298723115594455484988359 010996281935717965626127656826745836674377041052621490122494450710658034350390 431689408929848679398272748137079245045045352548317805547044624078231091684741 214551839785876185742803899708541173837262223533010862959657624385190705542791 062996139957266756355730572410532673631201130374518368873305443371849574899901 355948479321460897476868132910430034376703324787393202743638664455860101309881 764610136862267602037066772046106877964951317277861046920902714653888182807814 211213671222758284028943874586162736956875149461136669984139029006187320003023 603149621185767235947463797853592787185161453556788800249111109656608976028424 163662393594652233107578878380185674326918648747303549997207146531236431569535 514976826794247564611217912173100469724161417810355646977690411235395124173047 735617068484130728553131996986496032025652271018367018304070329314662192787524 250815852548957073875543590846436778174981977871990477635089689845408160906027 428588816122021189171750762664766966248165008305097122278220744089713381058759 034675750002441516101207527950026880912289356909667956760984198265010450061816 712805215933533403503064411352281629175957132823423118163582966751223518682190 426023391469836041317483992586163095745244902060262616676518788160617561276810 288662404852029213590791859971956608582285614197257237879064017826369197652915 393301883781294192630721563157800432396160460581288794934892643018107634786319 789508201469398876369446495040257822648044684428828823858111165183209987470961 598253969205023701557453003267966995272063889368816075123662092981757722714868 536171903801585121645471566102595699431873637886388075821329235378499067725385 652153956681815060912640455178789386570759994447393989304270939277910530904936 223570528841084571875793004902564358037853197231847877219137872737686044355208 905974233521071758132943419441361661663712116370387547622103164339601887104133 091142788987634285896311863484505113811056267796267398961137929634371383758288 012470538401837922115034863779118548544172814751594556642469518173170686208163 621752594966990780958005390460080252530314021376527471457740983685683475225373 972279766587932996788971006143471503590204712318787919809849211378056472569967 209648491698544346609469779020329096977538214167731836506277664471251247141518 596158833853805286990113176619956161666106388384993161422475470958917955383964 028680736703122374121895175286549688306643381126803222735828420690127416301092 516148748 ----------------------------------------------------------------------------- Pi**exp(1) to 2000 digits. 22.459157718361045473427152204543735027589315133996692249203002554066926040399 117912318519752727143031531450073148896372716654162727200036841245878483825780 197399927516270911185238671352940834892162337692496730536751662599601668725547 775888060873742920118171661161372246197209044896331314599273279140914840576764 889753784885344102006492593490357594763469165286294007847395407755298019829026 131224029511379990688652442931146335939285716073329541491532530137722767555183 068793043622842319088286797578297967264718164000704357181058364260641458021223 969069744749885161613318405106216364559561084133094289928312637557836158743046 929164246018335319342336420036127263824523624703549571835049073419630023545342 564129097921194303116908001292522770494403698852612861911839515968731591698101 911513070221704547646617959225224684510983208759621594421483998747447264175934 025693668265035580476642374237061708505644664324457496738884193688683115925200 555602645705059320117009362760553595826641201998120042761370192174862031910519 905506816571425870811340528853808629975276085614460926314249761823361686869406 515266685962413229106796479209792454561359400328316215872353548736449613197898 651307023062935703537082846360598423157484945238146522934841939651220934032766 631221216839837299037626473546428923658982018887142263051892244637286989688952 629932221082170328499564646586884098400854477924636623980341662609379906902596 890092791699406503118363879200217974924860319354120458999797010669295391213233 516672591218324888062076891463762345298253525351404898329971337436517350003961 883125449684954203283005442012232052307716980668470945979870907851778445856606 483290848480861820758755919609039050305827913278716879189191452266911950355455 869776770198226974316621958022717872653726612167154548315173292985803868104995 405043526449458038171403377697461873928491691112281324245397651802020663132788 881315151831992999654261917159535830028527474699812892854833574079721772874399 120441167663005232606009645213664966955702763587607 ----------------------------------------------------------------------------- Pi^2 to 10000 digits. 9.8696044010893586188344909998761511353136994072407906264133493762200448224192 052430017734037185522318240259137740231440777723481220300467276106176779851976 609903998562065756305715060412328403287808693527693421649396665715190445387352 617794138202582605816934125155920483098188732700330762666711043589508715041003 257885365952763577528379226833187450864045463541250269737295669583342278581500 063652270954724908597560726692647527790052853364522066698082641589687710573278 892917469015455100692544324570364496561725379286076060081459725892292324142400 442959813618144137067777781947396583031708566327895707534079917145231589263721 144638282644328528037928503480952338995039685746094853460090177429322057990359 173578204657580419316868230021961468992704206142969634660057998403516421365430 499845337217365572404636768488762615122990270599380102994468861817162609801308 765300370601583691986762860050799364683226697315683671755589711987529752963949 163155394491954838770687211307898665755909865363656307636308806115008353739182 611017808879024933772955782165794152474429338040786774075917093911979674301710 197011933994782022551476370229160802356358689156949579810351725151158286387293 277492004738462303294161944944950482112274446835725813185686259164943537639200 245003154821872041892476434436456337341364878452945127591637200743026136518682 117133046089777848572443256426333072365045231672315616051773692148498509071244 808367251079619884021248924514013965314763445957717647448759069546304375317353 067681435847515688509468429655050941311966099761993928593528770616690680969158 020553597781851324066690861770052217484091351102484588263306531907440319802006 689728609754185245016232046990294306437167133128246527441455128482197747519442 522936754874415353118001181393254334638307722867084758425378966015720137099654 767906621280174009756998567044437250552077804965634117081429051429437956774749 247521494382427232511802876279805621393263109127872561635360942478452947871754 251615265191330283407473028263587424337973810856876863949300992557237204901226 980769528397502168128621385708117824179109883401397518132046411884555584811300 206496819201030573698664416231870521502551644845016434425822921150995221122676 823169859477070880760028354214951270260667879890831705251698037149755032075546 739353790664991275286270904351343457716507473168210816011762804948451576594635 941758812614520495389456565236901499040341581369495434370994611114770114180337 050875014299446011339694109996842858347703643457588349286012312250872202731757 514199334733626583889229748495856193888834239544151667502218841230922834931531 503775433323872764004739676719079165248824115121900359781976053423694171525906 351893263258110365757154987316965790422671497821821284531161080960760023664011 827060294290645986837706910492665967709726606634015487578207063844618783057522 774798544124848269691600014288030197114786935973447885997415639814201684668181 882453828142845515271665634338154613488269646125634010584242331806504888119459 980651714678470201077451172779441620203150356378529272583772670625627052800726 116946586284541156754927922407382178743899639944159298538349243240264821879440 597603535529046967850494820470624984189963921518775371814632179244111897323352 839886048258467496260696262848480117741453428149333135564153426050193665392112 384051703068106821484250914035428865434001067320722503552390254325516377456702 947138170075412331140584055904531054235091647701437495157641156469104116895519 221992591244125063537949157434477187976072985039892995607262182578151496011546 909949578789545156511095639499194568048477769601461169809150924199241906522025 339701857041320060709228676570693818249464327191041456120875394184172772414392 110267801128456003123905634312916802334461082596817610850472185984702471200309 971671192940623139270921696035930280269549671539347265684386338695994645240994 471804698508079695777259521677282120192269861787653812010897171876028968343668 025142519223060489946243443954825503928003122061817520882009392142310894621512 211423807480575296276217095800659572136159922884813095433558008409125472476132 290881205397015024091803245154148016132534541725485713796739568584903674764429 535993311300473635976666216084381171675628270488139893315400869323564627133406 930743416199955760250917711645599956551079791124389790050154383508353503409012 653757497459442025015614536124068511932237930722224390121306235462149331535250 806742359498690027774683108247136713900862616838832025913804292479525424185038 598737935675632047911580061446327106945653496434004629296041565703410921150180 388713090151481484456257860298433768511545289245512381421908647636226357819661 260013256717832904035501212429188810223443306144790831047437322174168533405671 749501512322346857491187209961786713844922691096325621722834656032811535771833 253505872662653011477320420032510948036958841218791538139868571441163362660042 019634508563519704433212502577110522585551558587825701537326327953694001760140 392522666190496071808022741631843306990214649060274520397675040135563194555278 775565944348404810311717493335976206894434123008582716918937188582150761698735 854801584605757588122686593591270603979782687256675034141898862749859661309026 781052599007356287636651833150395757814863130606929676042251310159297301685410 893621915794385043460099031725894642379951900071193600960700189047527767130723 855875587594980435537140931558314891144168752979659924605268664286476331829258 848992194100932372812701269103132192322533579777118184154690422226025485662242 642594901587020729350928495181373695709237647044325946794269231525191684594551 568990337797540830358361016409940805482161595594298673822704933204143535957901 090020222129003677967699896364043293980926913358336989329718916743077667797680 410377730715003105378193079556031226371741931595349576932598162176270747640947 455833218359905261184491284194296679690596362983718269880776678011530918098146 217543091543352017427952306834517444813423217623555885825270426898634151203252 331908819795579460339196043017991086746005861096866493000934332624191392832643 922255530499205354191313235845300818669478029806126392202530272431058224244860 709165972161313820278169580753101510085738642173313142007746836222132310677030 168052025322957847600863375424540193077116839169556469220616336216867972857749 208617582790814920657902305527938111963666671857886255954946278286075154753041 262781743279613674991134362608644791777699304312242980064473459896580467724159 963586748722844213040907517668770551924273526221983098292641175818137693773764 400674411776739196946345386398335031018712273945550681968588924594926402073694 116117931135228569415135542368508111542298684547669171806693292123069086089544 985978263750717485668458407718537841810503168698812392372532335935832192065550 437741667556818583136219098274096743656398552671710855861413443447722468810524 734170336494079049424010692543996380818678491502361782954524767478143593895381 808766230716183766845428565243266806297269062637897717695366732052527175938383 912650823794375977365073928564585603050109079601158738614860701382431415267692 774218651718647063080895712849599805325468580261808680812235865761576147195665 908236576594455276365825013464807100629837136533610059255977577712359171893300 426220696882053354979437676184846342379482668816672878082887827672491680706425 638359990485336743555195895057377299026105027262168350606230744351237149165761 418422876475885687387617066744495596056760002934390299725665069339665063650367 659535178388869210715798225289233530602862034069794578375621193301968034844930 968494598682396040747419807100557700359990785042645939820294135402706632794640 528609463900020393569546564092360817372837219399394658720878896059728151605838 238676541669512380743569758358701858377371117259888780536957895421806929878196 991123260704012747968888503858935881847399437111883577118797335213663519006363 356336125505919740919820037511002378906574695952324863553936871927846576208586 659175511594471580765089597051217652603550309904868454249557969037813956563594 482997493334897293282633417267926516663109973557959402873171439643843894266101 751268487927216306390042424669277974465714476788026712039482962173280070047308 974243669764635657817636051997502458648819197813655943095237185243708701150925 610336416014224978861669080856275475946113601937783054747912989907442756546070 783913011885888210156954343664032303812052220928325612326849381275792340760255 561787352596337566427966665968131741929910022208359169435684336368930565778945 156969476036071760433712670283717373191319728575044618098510831085917410929491 361006316306318037199798426917940818267874624258234546696489522670407596570606 074754839461201167427814970723460982831541234941429656565121067056600254878783 589237565130410781059911782769167059138430437519791960874541928055005869289049 291711556603993189126847523121546044523719971012837695087188160206756547228936 005765563250153700224626372601737813818519283311322944958977254117270525467715 366914234154445476128923480244820684772189434024072170529882335814557261465276 850628492831415315034847076740387368315795950051529411191540069193790282274974 554746032136668642137987537954572644155142175814395255351744600789503877148525 901463839595437596225083991287076356810294267674644597758070825982365915676008 864631501415090090222950956132949315171627373349722803781005530107769116745640 844361556397634129833196411910842900824162283699981680258860575114760215441640 134678237039170365961310870366156461120962501261417426042415389452011622574832 257443295821554286346941411897838223641951632621469937958300218207427481975486 367007243424715627715638444890091326426839557457280383877985268576286970497755 875267571231273572469185254307162096355144039737539258545864701731623345381553 702609774600372866228706286479026267328365967355621390701142715285968765440480 413999519423326295528747110104881861313937258368711586199138423542005340917940 029029608428514521398919508640364199254963336863203248008422891547180641933006 809271563435528027313370857776489971194369981939115604037133620154489683868865 89931759486282677 ----------------------------------------------------------------------------- The Smallest Pisot-Vijayaraghavan number. 1.324717957244746025960908854478097340734404056901733364534015050 3028278512455475940546993479817872803299109209947422074251089026 3904589779559431475709672347175416683903886741875173693158425354 9908246622354533727350458987990956815062774550980248621301216989 4157524574548625075626524610368938904839932269952074975962828868 5569081507045136961098533525772815860334411419278282737652960329 9358467423102848324169523900610854333821850839810180895735387047 3931343967313767646021031652768893963935325943992483103109583953 7751942602887740927186203389282016152555321827094706130567612398 8920463730657196297771688630876153324800111768073116684532277431 5662899607266383572210363470709838371598022337102130982468490863 1296936634439244500715415042900081903067058984533905346887287406 6195775626167061764288919391230837918311716229603886147635880730 6315097483767582459270289013195095515560122800385957615401784215 1761874421595586099669924711478012082373365413973711912926405796 2484832322634420095923073636101515091300390033271919208565844628 ----------------------------------------------------------------------------- arctan(1/2)/Pi, to 1024 digits. 0.1475836176504332741754010762247405259511345238869178945999223128 6271147678602633673171429894778980400728170225841282815020514562 1521257493949857822174541459994018799541759109327949871559160403 1349271099342015425615134736832575251590871914549016863000485145 0655748506991483594199692099780283990946970320377371042160788150 0515580090470643113961197326187224085791750089161218300305450802 0736904639543575260558948426244398071837799927125671333584571034 6725722237801422969315704088646303590972236098731619147633079382 5469232528467804542055856389692426230479993325789292960386625904 2429298823134246665023844512493698971815322280687865131446724736 2358152669150209742367008522568183250062234492658383966328619506 9626204974008292187552899967087920922016160309839732183814608938 5342288815877356876363720519379780946756702578438637088625279626 1519018681561216852475138834954270945514499387170085692896342697 6347308117732513304560528686378658169845690944192418796963769141 9540758225444374125323403758176172251088269388603393288997880015 ----------------------------------------------------------------------------- product(1+1/n**3,n=1..infinity); this number is also 1/2 cosh(1/2 3 Pi) ----------------- Pi to 2000 digits it is... 2.4281897920988703287360414361791463581183629447833904976327499747264447341208 683681238055015720590438813806801377705872956847589966936033836187324105300392 865808669244394657890892491651371066377104252323673607675147979646688639836904 850393739332252145805640347256592540290028830867001979398074213872522790089323 154228502236515521885815194297560892363347908641995792622259984250909040378684 263470763030768385636842185608142679764460596043796303811421041621275844292057 700163007280916980100703596621893466128982343192673735582936526960268252695827 969447527973522788556399884636296118681899499078622248798356109363043725581948 681540712668089039110046272735513927716078337416320928815428889994397208487845 323415270800430803554866401575762091559108205822661443142552172670570426310007 280323994887051306540159374175370062983819312615469974069317481395354238438130 325673813846292110990298812971632204025757373026637787021404782680466464668362 246757474415547510332404401129732859289172057826361202415591341320691414286355 066207235839140402947798916145909194748511067491857528988108472600419916917960 275893578887621848777804200666414640094066666366659061695962487433212942135659 098862138218072682527574354007044692378416377618789192673645148831322767020796 403995170593560502621672218884712643146275368343297780459511975061235555673521 375875748035778776659777816235811663079155556113809261875545662039223374633395 136872880693684118840597415337352831152304828824841286248567801036556995482649 553616364949385709861118707654681305841219015174685662159709787730991016345808 078455565925980891003288634435743709825026053099020264256367104788436657955995 570619349714104886667745264632292843536272232451933428675315001115884283005963 548074076280385891472617273743006391427828655498922164766406576688662750419251 884672348578621105429233053556502012952422035910770779156223960943510643593924 861383856922056804894136302733137543295955164256954576711200844560050492902481 435945529529155148304619158429612546849411015726741 ----------------------------------------------------------------------------- exp(Pi*sqrt(163)), the Ramanujan number, to a precision of 2000 digits. 262537412640768743.99999999999925007259719818568887935385633733699086270753741 037821064791011860731295118134618606450419308388794975386404490572871447719681 485232243203911647829148864228272013117831706501045222687801444841770346969463 355707681723887681000923706539519386506362757657888558223948114276912100830886 651107284710623465811298183012459132836100064982665923651726178830863710786452 195528154274665109611001472502097904639381778712575009803657792230643121651131 087380599298242335584945612399567699978435964864096003266482443521306491599303 270530753256568618388265483309802846696242873884751844436838530734115044469478 840059464469131682120592946054542163754891890060150356872862933140063632268146 351612163764864131429342351600214180513528287731960179813917884407150662994919 093496277396207234135302557578180281180210206340974993923837290330361739816633 600322612620886664117180538328558970002735722645233287010649586367726698687384 859165698266261741988551156844303327351231032433075727331649536152620482684798 306053981003157759802511144595774183596489094220203477196778483082245007019118 206108478776225735878584402319091953216420763414005680399431546526673794350216 992134747713261128519133178491606658068403489787814431322679410839519360265028 960726537291276226938242717551278279653750700784001190019241713358327134701518 756952318950577522896149682821650782166855605218622283761511045290704651981350 624064015699555055607723527235898359267993820905324184058912744801439474570950 647586555194756066347107978366612927647920909687903131865554282732062606593248 413261523705890098275370715373630772580812755826920872591581902005039751192726 281420515295848284628604840714806749933756897548169897911661250320738399632947 197475066080743912282251610298715312153928673289056455168511094510850241868813 357753938319988751316257344799941108118740096770682577450950592795177900534229 227625135157671393352553508698193649538153388239870759679764768250913442427211 537562946093572780028074511889735844312259940735819 ----------------------------------------------------------------------------- The Robbins constant. ref: D. Robbins, Average distance between two points in a box, Amer. Mathematical Monthly, 85.4 1978 p.278 0.6617071822671762351558311332484135817464001357909536048089442294 7958464613859763130665248076810712015170977531075941097247868058 1643721687453324207229824442327640922920607860008648053326693895 1526942028215425692085403456100394606163834472771107263924054689 7434592322069695104571767853038748238911194887130919810475594295 3120545589150326753940164393320790294473473479010132900154516660 0642731445463113650395856252896443964373900626507351434749911653 3540376378675705958829699270063500978386289740462915842777306955 7430187885803716470017544601967121335982623876512065551505953382 8281442492815931568016481658129911912468681742538796067114408338 5962036245968755328720698995275209149543768315871982607183656932 7991821337185639447759795860031495377302353537591681976432088663 8761213723743456544539160466691236289725645485547899749367949903 6787454198087305903903975046429880243733984398127096523272663805 7779297187173093916715126325857845393789259694776116795702854531 2570851202124609101739182422265676598386800760949577826879652854 4/105+17/105*2^(1/2)-2/35*3^(1/2)+1/5*ln(1+2^(1/2))+2/5*ln(2+3^(1/2))-1/15*Pi; ----------------------------------------------------------------------------- Salem Constant ref : D.H. Lehmer , factorization of certain cyclotomic functions, Annals of Math, Serie 2, vol 34, 1933, pp. 461-479. David Boyd, Small Salem numbers, Duke Math Journal, vol 44, 1977, pp. 315-328 this number is an algebraic number of the 10th degree. 1.176280818259917506544070338474035050693415806564695259830106347 0296883765485499620968301155818153946592071813793476817656271429 9390469080189480252316007759657054606241887504896232590717733457 1567548096997559812677289401128791972456983735177677402547018406 6278603009315383369626077626819915970468346466323231071265612414 2230084750982757531788114948316855868535248394324346506941148983 5604855670999941131248924651646199928894650701513975703312904628 5965316234036730870359350603811812061902043009241085523839830214 9953872876195952056739715886750661129345807575743980651247047412 2134188106798291251486337803701296891625290465195911765657939458 5147548608924166974891816070204188007795273821303291763399098187 4464693191554220975967586181179145555664298356496556596386045043 4719067256426322958012208664666341022433004123110637753690615489 2804267030782226373027706825877145786773674445329003975537521344 3969594529828030667432608207317189943453247528925058415973944040 8254461851691378006656323698981137666295496727278749361016763363 ----------------------------------------------------------------------------- sin(1) to 1024 digits. 0.8414709848078965066525023216302989996225630607983710656727517099 9191040439123966894863974354305269585434903790792067429325911892 0991898881193410327729212409480791955826766606999907764011978408 7827325663474848028702986561570179624553948935729246701270864862 8105338203056137721820386844966776167426623901338275339795676425 5565477963989764824328690275696429120630058303651523031278255289 8532648513981934521359709559620621721148144417810576010756741366 4805500891672660580414007806239307037187795626128880463608173452 4656391420252404187763420749206952007713347809814279021452682556 6320823352154416091644209058929870224733844604489723713979912740 8192472504885548731193103506819081515326074573929111833196282150 8973486881142145283822986512570166738407445519237561432212906059 2482739703681801585630905432667846431075312638121732567019856011 0683602890189501942151616655191791451720046686595971691072197805 8854064600199401370140530958085520528052531711332305461638363601 8169947971500485150793983830395678167948161221402208916987109743 ----------------------------------------------------------------------------- 2**(1/4) to 1024 places. 1.1892071150027210667174999705604759152929720924638174130190022247194666682269 171598707813445381376737160373947747692131860637263617898477567853608625380177 750701515114035570922731623428688899241754460719087105038499725591050098371044 920154845735674580904839940930900034977959080384896588430050411987170093790798 209846252353739812817408181137808285520148422100609589324124459310350575191963 029413832634742802798244080228008217292720586153666393704002382073085456530674 477148598887334576271867838116547045872761271112699886784349301758614249701700 541314551438919987437667621785161783177987307048236318734734842180537156986842 636482761056228477995862896332939281687874758656034737919964594007561544437157 418903039869712943062486253517341291535975311215446746159086477606517445957055 930979119465756398917686972170262497475333629918606531157083493680769804948170 607437684746785586528255014184649792489099515633782998595087643532396621477896 547910454186934661861396145218563917026341604354229856108549326870868151717454 04554548532 ----------------------------------------------------------------------------- sqrt(3)/2 to 5000 digits. .86602540378443864676372317075293618347140262690519031402790348972596650845440 001854057309337862428783781307070770335151498497254749947623940582775604718682 426404661595115279103398741005054233746163250765617163345166144332533612733446 091898561352356583018393079400952499326868992969473382517375328802537830917406 480305047380109359516254157291476197991649889491225414435723191645867361208199 229392769883397903190917683305542158689044718915805104415276245083501176035557 214434799547818289854358424903644974664824214151039320430199436934876879115865 891569799649150391935143852695668478165605185363200962455338411559964418782057 071100837137605118649713541552994922973799383214444889807391897919511442742645 178801692640403219098617233052984486143643263207691133234921001059774207763922 059064326725351759582500834464720774042303563857199988146341731478871918094755 506357431937348827299122589427548768950694033248095598111147855527762146186159 609886913128081573442101642685834146932480595852486941819774796907287883592668 681656295544982771231241739359880261799888459616178511015265142019295770748553 621477960335310125476008798159293638317998764183171554007533292685532366426931 296113029111025520184013514875239936403973082905020852634097000954786673108797 194683512466021134551718490623186005559263054213445514986015560105003175358818 729120260192377759863996689880745305394749277211166300200942565181578057244342 364079464408162259363253332269243879958128832143605562042103400838175855005147 159035775759548082123045351970406460845175874648068200206983521552062681616351 546128866398014618829887276854773455787107021211539099616380870095321225622743 875843134805266684710810680269730212282707006426650390681672492836820335198867 111490598052146276725080070297023977357726727420363586882813118327458332011650 300663287203505391842923422615658023387724025020112031995598518110930146011943 357535550858470014843437983175002044765810711671261397841703350673509295101418 035838107387174672478179790404106522129323473426130545413167650437830630173033 609770202799206445648799740500038602872011502383662940004575718574473772243957 859564732954178543698075776889882013103418542402303648469135979284487987981305 207957632878889539116749028392001145076602606946768688776828321352341343714498 172069787183303687222279154323894660649265107409869767073908525830747627588164 599684978287226131955625954677069349468340871546911321236846310103649548391557 706597324218895772995796196414385734757463701320460682282702082229074510097287 470965263450130698632304053253571980160303875529709389914239699309762498208260 656985764679971094870832353759361789431473305428008521443480289919702645320371 540591669433894078131793357800419838012267461497196943352987715772147171547862 923549410773155563038338703393228578903032372374987517727227965664327459492466 828637381314870736911784345741891568168064181395191242008190333580358992436427 792146567461304662028297877682556837732219391714165673332227709019541094949164 731317250808556100846480973008466031051651987243311582800178339090672155044828 435337148282908872278634250781579720625368059903173644323312066359683471335647 572519875946415674309642065162759421558450733571189718736416717232624102189802 986643530813745547079843131265127944488423520875100157013114234552402349201621 895330336887481355125635530103043133142183302509644544691916000304561434594091 842118084695176156346518430689840576347349894596331764219496350825991275112646 178968314081728405445653705633918909972488638272414746021940065089177324960678 701639578875389300114821175213799849881345936920742369738692578177604561048165 791951728359470764717852032653804264508712488409275144263817247478872455521125 122686157803180905175243587957901639589918375503356805003310878302478688338855 646264365123996336007949820946888100805705914494984147795125582819571118249753 142501534840648675362586395929915055763877114942594265588179739300156153186304 918492515415542607279787417977785786967831684043241053939562352551581578479080 217437441480744707729267778098749964843897754964276371179512590516188343080479 627627215997745371033986957853630712266447521404867551898386833006795876131353 061615629472534881740658887722803330178811024150345211269474743386476066285988 462296383533038290827914933752828644732943045951482219064252892987971875320122 303106072804280427262462568416437742171936686010084615581521635282403006879396 842836076754390842203901164603231158190564588614887636276553758069184106730228 889992731281104098031581800960316548425810811583034173443155831075066192714369 565224756157760818281064759373776548828531997751494021282367027889259567476651 994432153571589252600399312123424069148817040108814852822860658109915341468231 434111113996190820817658629913660653985709633814345784059091998325755015350541 335156433200795915635980201946177034129575229788245918427630740173255464494827 998091485820424906785146136481107013187529775997375080905292723700988248579688 317119428485099110702270510482145419412443049360943824740041797224596723946566 554038885 ----------------------------------------------------------------------------- sum(1/binomial(2*n,n),n=1..infinity) to 1024 digits. 0.7363998587187150779097951683649234960631258329094979056821966523 0847181802807864081869444182490225974582720321801478346017690055 4229868477732944895880680415915142979334394163998909738083425408 1520029546146727664979554751571056972458855740951911198864857982 9433328581834861487045790649324680582119729407417116198674601654 4485479889543142786974292724928598532747380156659130512545236749 4154597773449101860414448973793322220865507304585980050655111918 9338017331890327068185957293937796352569292021414362805981608876 3091647656764089200563681690417652792652154091682197250552326447 7646813159383043809989583900078755611335395490521438524130346215 3457599854790211802421898533425927038158436578567901788663851909 9589847649578146455045212664074436825052408587935995452420291968 6774100311819740383505356065846433687090253752952485814436801506 6642240052190351749758439051568244234310796570611672868391753708 6438175031998334537917178650178729583132166807457526236785528510 1696922360975795282761033968077326069723073543573616136752770598 ----------------------------------------------------------------------------- sum(1/(n*binomial(2*n,n)),,n=1..infinity); to 1024 digits. 0.6045997880780726168646927525473852440946887493642468585232949784 6270772704211796122804166273735338961874080482702217519026535083 1344802716599417343821020623872714469001591245998364607125138112 2280044319220091497469332127356585458688283611427866798297286974 4149992872752292230568685973987020873179594111125674298011902481 6728219834314714180461439087392897799121070234988695768817855124 1231896660173652790621673460689983331298260956878970075982667878 4007025997835490602278935940906694528853938032121544208972413314 4637471485146133800845522535626479188978231137523295875828489671 6470219739074565714984375850118133417003093235782157786195519323 0186399782185317703632847800138890557237654867851852682995777864 9384771474367219682567818996111655237578612881903993178630437953 0161150467729610575258034098769650530635380629428728721655202259 9963360078285527624637658577352366351466194855917509302587630562 9657262547997501806875767975268094374698250211186289355178292765 2545383541463692924141550952115989104584610315360424205129155898 ----------------------------------------------------------------------------- sum(1/n^n,n=1..infinity); to 1024 places. 1.291285997062663540407282590595600541498619368274522317310002445 1369445387652344555588170411294297089849950709248154305484104874 1928486419757916355594791369649697415687802079972917794827300902 5649230550720966638128467012053685745978703001277894129288253551 7702223833753193457492599677796483008495491110669649755010519757 4291162109702156166953289768924278900580939081478809403679930558 9535200633716110465094638606808864998606531021853412479159737305 2710686824652246770336860469870234201965831431339687388172956893 5536851798521420666264165438061224569940966356043885239969381304 4840101532338556989547899226146597068180753342912289091004995136 4103584723741679660994037428872280908239472403012423375069665874 3147683502983470096596930198071220594154742391888495488920431478 4037389693592832744937301860181757952468190913559650620576842700 8907326547137233834847185623248044173423385652705113744822086069 8381169706447896315548031108686846807807010570342300009547766282 9927022264266182213029160934485049255679995121281765081062180734 ----------------------------------------------------------------------------- The Traveling Salesman Constant, conjectured to be is equal to 4/153*(1+2*sqrt(2))*sqrt(51) to 1000 digits. .71478270079129427201898487962108409673134559709443031939645700411546117738335\ 879706770213413096294533561547227555717895434127457058654186783324525211448435\ 423370160734747472156550615029635220251467885538763575736849440141040232425552\ 364704664879061099570515393895856312208463669793487083110116620844381148478166\ 953397235099760820248716126335472464734965931893615249427223312525010786175723\ 903850094286618856777573472030439593602004416562703436281430743460123517870481\ 605658651710683396096658326275655282564938079930443149087689479702230621110332\ 425071472991466740480185001283536160284031917506648494911514005453049419741227\ 682161417117934301981301137112382110439175900888848785626934265741110708345544\ 731999904108101036079296059394893034776038533840976912765053467151339515952296\ 425034733122079333744376059531233173573812633038639781766805813536012423214277\ 007401299039458343003042376467569131088941308597225474822014342730622766746260\ 22472480156659330677754354367566446245619515011589704068286465445 ----------------------------------------------------------------------------- The Tribonacci constant, is such that 1/(1-x-x^2-x^3) once expanded into a series will give coefficients proportional to approx. c**n and c = (to 1000 digits). 1.8392867552141611325518525646532866004241787460975922467787586394042032220819\ 664257384354194283070141419798268592409741641784507465074369438315458204995137\ 962496555396446136661215402779726781189410412116092232821559560718167121823659\ 866522733785378156969892521173957914132287210618789840852549569311453491349853\ 459576175035965221323814247272722417358187700069790551025490449657107425265477\ 228110065989375556363093330528262357538519719942991453008254663977472900587005\ 974481391931672825848839626332970700687236831127837750250557122275153259578946\ 560570686422283918659698294691356239220443192476147068811451726766712743964146\ 212571843342662340390218352494591033227231061513286997030808036302223324997105\ 243107472354231399744381826565607351940357874911762680524537079221110849710806\ 876410050156541475662235008885665949715821834184868714802901255436993480513679\ 165025853053878276666126224317766358200942985505387325991651787730184472388604\ 26222324857820792721049160181783725613203439814302274533997621231 / 19 1/2\1/3 4 |---- + 1/9 33 | + ----------------------- + 1/3 \ 27 / / 19 1/2\1/3 9 |---- + 1/9 33 | \ 27 / in fact the n'th Tribonacci number is given by this EXACT formula. --------------------------------------------------------------------- See : http://www.labri.u-bordeaux.fr/~loeb/book/92pl.html Comment calculer le nieme nombre de Tribonacci Resume of a conference given in 1993 (Universite Bordeaux I, LaBRI). 1/2 1/3 1/2 1/3 n 1/2 1/3 (1/3 (19 + 3 33 ) + 1/3 (19 - 3 33 ) + 1/3) (586 + 102 33 ) 3 --------------------------------------------------------------------------- 1/2 2/3 1/2 1/3 (586 + 102 33 ) + 4 - 2 (586 + 102 33 ) To get the actual n'th Tribonacci number just round the result to the nearest integer. Here is the formula 'lprinted'... 3*(1/3*(19+3*33^(1/2))^(1/3)+1/3*(19-3*33^(1/2))^(1/3)+1/3)^n/((586+102*33^(1 /2))^(2/3)+4-2*(586+102*33^(1/2))^(1/3))*(586+102*33^(1/2))^(1/3); This formula has 2 parts, first the numerator is the root of (x^3-x^2-x-1) no surprise here, but the denominator was obtained using LLL (Pari-Gp) algorithm. The thing is, if you try to get a closed formula by doing the Z-transform or anything classical, it won't work very well since the actual symbolic expression will be huge and won't simplify. The numerical values of Tribonacci numbers are c**n essentially and the c here is one of the roots of (x^3-x^2-x-1), then there is another constant c2. So the exact formula is c**n/c2. Another way of doing 'exact formulas' are given by using [ ] function the n'th term of the series expansion of 1/(1+x+x**2) is 1 - 2 floor(1/3 n + 2/3) + floor(1/3 n + 1/3) + floor(1/3 n) ----------------------------------------------------------------------------- The twin primes constant. 0.660161815846869573927812110014555778432623 ----------------------------------------------------------------------------- The Varga constant, also known to be the 1/(one-ninth constant). 9.2890254919208189187554494359517450610317 One-ninth constant is 0.1076539192264845766153234450909471905879765038 ----------------------------------------------------------------------------- 0.4749493799879206503325046363279829685595493732172029822833310248 6455792917488386027427564125050214441890378494262395464775250455 2099778523950882780814821592082565202912193041770281959987798787 6404342380353179170625016170252803841553681975679189489592083858 to 256 digits is also this closed expression. 2**(5/4)*sqrt(Pi)*exp(Pi/8)*GAMMA(1/4)**(-2); ----------------------------------------------------------------------------- -Zeta(1,1/2). is also equal to -Zeta(1/2)*(1/2*gamma+1/2*ln(8*Pi)+1/4*Pi). 3.922646139209151727471531446714599513730323971506505209568298485 2547208031503382848806505231041456914038034379886764996843321856 0187370796648866325531877003002927708284792679262934379740474743 4560678349258709176744625306684542186046544092107149397014020908 ----------------------------------------------------------------------------- -Zeta(-1/2) to 256 digits. 0.2078862249773545660173067253970493022262685312876725376101135571 0614729193229234048754326694073321564310997561412868956566132691 4694458311965705623294109531061640017807007041375078320755666248 7877869206615046914282912338325693716136777293836109459387888090 ----------------------------------------------------------------------------- Zeta(2) or Pi**2/6 to 10000 places. 1.6449340668482264364724151666460251892189499012067984377355582293700074704032 008738336289006197587053040043189623371906796287246870050077879351029463308662 768317333093677626050952510068721400547968115587948903608232777619198407564558 769632356367097100969489020859320080516364788783388460444451840598251452506833 876314227658793929588063204472197908477340910590208378289549278263890379763583 343942045159120818099593454448774587965008808894087011116347106931614618428879 815486244835909183448757387428394082760287563214346010013576620982048720690400 073826635603024022844629630324566097171951427721315951255679986190871931543953 524106380440721421339654750580158723165839947624349142243348362904887009665059 862263034109596736552811371670326911498784034357161605776676333067252736894238 416640889536227595400772794748127102520498378433230017165744810302860434966884 794216728433597281997793810008466560780537782885947278625931618664588292160658 193859232415325806461781201884649777625984977560609384606051467685834725623197 101836301479837488962159297027632358745738223006797795679319515651996612383618 366168655665797003758579395038193467059393114859491596635058620858526381064548 879582000789743717215693657490825080352045741139287635530947709860823922939866 707500525803645340315412739072742722890227479742157521265272866790504356086447 019522174348296308095407209404388845394174205278719269341962282024749751511874 134727875179936647336874820752335660885793907659619607908126511591050729219558 844613572641252614751578071609175156885327683293665654765588128436115113494859 670092266296975220677781810295008702914015225183747431377217755317906719967001 114954768292364207502705341165049051072861188854707754573575854747032957919907 087156125812402558853000196898875722439717953811180793070896494335953356183275 794651103546695668292833094507406208425346300827605686180238175238239659462458 207920249063737872085300479379967603565543851521312093605893490413075491311959 041935877531888380567912171377264570722995635142812810658216832092872867483537 830128254732917028021436897618019637363184980566899586355341068647425930801883 367749469866838428949777402705311753583758607474169405737637153525165870187112 803861643246178480126671392369158545043444646648471950875283006191625838679257 789892298444165212547711817391890576286084578861368469335293800824741929432439 323626468769086749231576094206150249840056930228249239061832435185795019030056 175145835716574335223282351666140476391283940576264724881002052041812033788626 252366555788937763981538291415976032314805706590691944583703140205153805821921 917295905553978794000789946119846527541470685853650059963662675570615695254317 725315543876351727626192497886160965070445249636970214088526846826793337277335 304510049048440997806284485082110994618287767772335914596367843974103130509587 129133090687474711615266669048005032852464489328907980999569273302366947444696 980408971357140919211944272389692435581378274354272335097373721967750814686576 663441952446411700179575195463240270033858392729754878763962111770937842133454 352824431047423526125821320401230363123983273324026549756391540540044136979906 766267255921507827975082470078437497364993986919795895302438696540685316220558 806647674709744582710116043808080019623575571358222189260692237675032277565352 064008617178017803580708485672571477572333511220120417258731709054252729576117 157856361679235388523430675984088509039181941283572915859606859411517352815919 870332098540687510589658192905746197996012164173315499267877030429691916001924 484991596464924192751849273249865761341412961600243528301525154033206984420337 556616976173553343451538112761782303041577387865173576020145899030695462069065 351711300188076000520650939052152800389076847099469601808412030997450411866718 328611865490103856545153616005988380044924945256557877614064389782665774206832 411967449751346615962876586946213686698711643631275635335149528646004828057278 004190419870510081657707240659137583988000520343636253480334898690385149103585 368570634580095882712702849300109928689359987147468849238926334734854245412688 715146867566169170681967207525691336022089090287580952299456594764150612460738 255998885216745605996111036014063528612604711748023315552566811553927437855567 821790569366659293375152951940933326091846631854064965008359063918058917234835 442292916243240337502602422687344751988706321787037398353551039243691555255875 134457059916448337962447184707856118983477102806472004318967382079920904030839 766456322612605341318596676907721184490942249405667438216006927617235153525030 064785515025246914076042976716405628085257548207585396903651441272704392969943 543335542786305484005916868738198135037240551024131805174572887029028088900945 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848336703688167279661283316060673882330154485559722831554953152790512944632946 735062955119167184229698846592671871061956988599224929488766360362711791273491 242638869726650876864081880699049446615099393830619711646796113001921819683024 369590515257225336237992051139086240802237202937259314304211737816439025200542 055318136632596576723199340502998514457667643516144415500155722104031898805440 653709255083200892365218872640883469778246338301021065367088378738509704040810 118194328693552303379694930125516918347623107028885523667957806037022051779505 028008670887159641266810562570756698846186139861592744870102722702811328809624 868102930465135820109650384254656351993944445309647709325824379714345859125506 877130290546602279165189060434774131962949884052040496677412243316096744620693 327264458120474035506817919611461758654045587703663849715440195969689615628960 733445735296123199491057564399722505169785378990925113661431487432487733678949 019352988522538094902522590394751351923716447424611528634448882020511514348257 497663043958452914278076401286422973635083861449802065395422055989305365344258 406290277926136430522703183045682790609399758778618475976902240574620411468420 789028389415679841570668448757332730136446415250393630492420794579690598982563 634794371786030627807571427540544467716211510439649619615894455342087862656397 318775137299062662894178988094097600508351513266859789769143450230405235877948 795703108619774510513482618808266634220911430043634780135372644293596024532610 984706096099075879394304168910801183438306189422268343209329596285393195315550 071036782813675559163239612697474390396580444802778813013814637945415280117737 606393331747556123925865982509562883171017504543694725101038457391872858194293 569737146079314281231269511124082599342793333822398383287610844889944177275061 276589196398144868452633037548205588433810339011632429729270198883661339140821 828082433113732673457903301183426283393331797507107656636715689233784438799106 754768243983336732261591094015393469562139536566565776453479816009954691934306 373112756944918730123928293059783643062895186209981463422826315903634488313032 831853876784002124661481417309822646974566572851980596186466222535610586501060 559389354250986623486636672918500396484429115992054143925656145321307762701431 109862585265745263460848266175202942100591718317478075708259661506302326093932 413832915555816215547105569544654419443851662259659900478861906607307315711016 958544747987869384398340404111546329077619079464671118673247160362213345007884 829040611627439276302939341999583743108136532968942657182539530873951450191820 935056069335704163143611513476045912657685600322963842457985498317907126091011 797318835314314701692825723944005383968675370154720935387808230212632056793375 926964558766056261071327777661355290321651670368059861572614056061488427629824 192828246006011960072285445047286228865219954762507436349751805180986235154915 226834386051053006199966404486323469711312437376372424449414920445067932761767 679125806576866861237969161787243497138590205823571609427520177842766709146463 931539594188401796843318630461527843189738406253298660145756988009167644881508 215285259433998864854474587186924340753953328502139615847864693367792757871489 334294260541692283370771062100289635636419880551887157493162875686211754244619 227819039025740912688153913374136780795364905670678695088313722635759543577546 141771415471902552505807846123397894719299325008588235198590011532298380379162 425791005356111440356331256325762107359190362635732542558624100131583979524754 316910639932572932704180665214512726135049044612440766293011804330394319279334 810771916902515015037158492688824885861937895558287133963500921684628186124273 474060259399605688305532735318473816804027047283330280043143429185793369240273 355779706173195060993551811727692743520160416876902904340402564908668603762472 042907215970259047724490235316306370606991938770244989659716703034571246995914 394501207237452604619273074148348554404473259576213397312997544762714495082959 312544595205212262078197542384527016059190673289589876424310783621937224230258 950434962433395477704784381079837711221394327892603565116857119214328127573413 402333253237221049254791185017480310218989543061451931033189737257000890152990 004838268071419086899819918106727366542493889477200541334737148591196773655501 134878593905921337885561809629414995199061663656519267339522270025748280644810 98321959914380446 ----------------------------------------------------------------------------- Zeta(3) or Apery constant to 2000 places. 1.2020569031595942853997381615114499907649862923404988817922715553418382057863 130901864558736093352581461991577952607194184919959986732832137763968372079001 614539417829493600667191915755222424942439615639096641032911590957809655146512 799184051057152559880154371097811020398275325667876035223369849416618110570147 157786394997375237852779370309560257018531827900030765471075630488433208697115 737423807934450316076253177145354444118311781822497185263570918244899879620350 833575617202260339378587032813126780799005417734869115253706562370574409662217 129026273207323614922429130405285553723410330775777980642420243048828152100091 460265382206962715520208227433500101529480119869011762595167636699817183557523 488070371955574234729408359520886166620257285375581307928258648728217370556619 689895266201877681062920081779233813587682842641243243148028217367450672069350 762689530434593937503296636377575062473323992348288310773390527680200757984356 793711505090050273660471140085335034364672248565315181177661810922279191022488 396800266606568705190627597387735357444478775379164142738132256957319602018748 847471046993365661400806930325618537188600727185359482884788624504185554640857 155630071250902713863468937416826654665772926111718246036305660465300475221703 265136391058698857884245041340007617472791371842774108750867905018896539635695 864308196137299023274934970241622645433923929267278367865571555817773966377191 281418224664126866345281105514013167325366841827929537266050341518527048802890 268315833479592038755984988617867005963731015727172000114334767351541882552524 663262972025386614259375933490112495445188844587988365323760500686216425928461 880113716666635035656010025131275200124346538178852251664505673955057386315263 765954302814622423017747501167684457149670488034402130730241278731540290425115 091994087834862014280140407162144654788748177582604206667340250532107702583018 381329938669733199458406232903960570319092726406838808560840747389568335052094 151491733048363304771434582553921221820451656004278 ----------------------------------------------------------------------------- Zeta(4) or Pi**4/90 to 10000 places. 1.0823232337111381915160036965411679027747509519187269076829762154441206161869 688465569096359416999172329908139080427424145840715745700453492820035147162192 070877834809108370293261887348261752736042355062193737506171117453492968677507 330760668693411890586283379527951203344958904688626269482208350329836321490205 321239557248466462255011566604558826867876535044954351371974951488631328974725 885751455324761892324749088343183216559962899648054020498855660906710813145472 438251775250469502552514132207698095596477686277529297400362468833633531227758 668098332337740208149807142410957545732327968829227632494222596768214873164210 130083383048578880296049867785925835977212171325665188800281027638581269931387 545523778137133635462656850510299889287023951692086070339616088259169901546588 829589456014838452545931150701738279902071569195340951585273588301542879433793 746390084034611646372042627028266824368992792302461651111441102686560899069254 234916963890635275530295265490294558420276762570241101886780740855794520052537 619233699975159280962799577357881560429246227564596438489387074199902375906750 607371161632214547578450995034633065293054464210084330411049428823987182779472 210717957162013973497506273354919182726537315801429070644112238984462888749346 123621785684422095337765665063123393418309331590366489436671353950052108795316 627967870123212606511753317383291994351943063709445458703075841875966315880470 976209778331910566278373749222678619336932243629212937680214776276988227275161 168195674581326992671514767615743996645318562807522634712756683205732676873043 690872655267444910730887471434771768813238590527416830613773207408761753459968 131671551610024798988217912278432609940552842489853554822285939017206671602845 270243345579339355738139745080825577027485254916354927551049265465602698556315 339949578895606778338186500146965223230533534669952369976343379670422876496446 956754049652503997706078434631929556382871737860047504977021567642914163208355 134805792566068258029946197161166626296206873985365333182877235965645025559136 257276543969869988263262252546952360187873782368673354821387004235792861595706 529762634076158431188855993259791932713520010274020791811758790707479416547489 106183607617773678547349113072916281762908896372877205362179853486880174699470 107383795934201535381924332932601995742371149445554137644649833874733600917286 939054275258964969221373438159681513527365152368160859347449314348075900345248 022371248340709893964541153866142119927965830143459708005575640219601213663806 527389078736847155824812085033248336785809277461345965241697530456781539113888 521821359486098822884971197488419528446391075495066353252263979396256780568023 412994361020882530460315369615010615907640388449002875649111956761004893444788 262667636891482868798355412860868083881677258693652904007497955167213959088437 014275117694129854270050771940027704855368527442476609058963047355537726975352 606857483276704063949198474837601584854183457622643688691068255182852613072480 280116483639755956169869839328933485751218575154980335069046376361009455721878 935484849344137476171947848917049046209355129479618398660809902190592737601620 169040890810646508898460380317543721534054169065012444317105131721986309172378 537608940683232656099437288581269590045631303620508424266923250471846659288150 681335483184307726970195422317477929132443869159416282481643778054583836828018 127995153507811603759817449284836263090201220997756637752944272923039050840588 526655478542632456194649397920321886086276148959318204123715442572261905064051 748829412167006730489244277871224198449924807997912473097994195519502920052843 575035251296805230980217455834950259915190120471703736893969162481741729962970 670053192448016331645644501333390769048925825620404903143689985910667455231442 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515494214492670669082127457130974283485983092914088726492118497432266371226824 875948389614981716856752378501606845324359404776151498977944763288087490189431 567980871984467577345685128211910689436861931973467442977900191686105709960704 247139640024918766861671955618581917473309178753765735212385607298227106480106 537149722093254436283421847474576282179725315520039011067876317406749486747639 347982896766183337576896202860401071552989371016008309060190992412809088614441 646654105893505762407387440923795927678677314054708985208520611435535408016654 229067285430682554469717607581231080604156967903901761908112936876560748896341 920294370713216767793388638761930280085425060461906222473604145351687985760903 215643034996835766181131997746531597187699717610564527952652882093884115991577 451698549942629678093993472726354309939190357942645720968543107431197468809980 627437570862144295710924616576802276874075744221432380876748250155470605742281 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618266368031324230774053892443623086218378228126439958968424445897200361459364 463623172532572596661183895027638353423444389643114391115334388645902941449500 732571763151162628164649829973364651802997793189287950022681105360372194839533 016176403858484554564108507065133121480345105324438099040318190955849197497061 904745526328509255928560324798988508452008254659458580751965574079575664172151 429283457530334433832247339205550340081934271407875523588239038928688231031116 685774070902353399406220553935353359705991585294396065635832205296490274719954 954545635571104827608342293449851043206010425769596017548513029751046669457729 911002771224625850764192851256486039760777545031797863829805430625454242197192 650845549179102957624514082855678903328840344325001075325771190441903524679102 916635927653682510365850466208370557558260830505848475444490942055517025514061 236295517217126857903003771312835776356693150237134797866633953858174574710649 136579953232455487107124841923945710055281481281172026560468060409693818161392 120593310600816179615501558311898550270948482669262547402586210264886603224423 444239787177354946164616170997318446945587112298937701381742794064800317168436 849421744820503433814670428017724395345703671119167117585728943065604163190547 839533264443118121740628530720157344406090463915631451724267831928308834119239 736248383292471178282470624518063900179921030228782015138996734766295351374675 554673622002323542920078983611532099684121749186225665063965240769129740102540 785260559694885346373325632780322499928187657550733044222657648029891583042976 473511950585855477705357784614191197986186803349909050399051684277447168462645 827309613058038524324779844038106483284405352269783143279061442328225443426863 267646966313161361597434286041079509870675820652906643205446812078165087198608 410452351966497715341000213896226639542237898664203168559646819644328848841845 270367785915074494145826117421318834712891456565891957492382282189903366592406 972810028353566198885128802997885995081975922287980943864508926875597853178261 854102911427737034499520706208814113273004255217324799402676464990116279901559 293602974751606325065922379687952954853542015714664761686279842547745463992391 558076861160339246996760129883338283079538922436438622983346118907620620672087 053503624129784260426616778509719545312833367099778911819532601228794907920967 614092904418446876172334172096949688116212054509841294897097252924033729833667 006043967308172363020394063078521448894823418466897974356016880504076300639363 41762674388635891 ----------------------------------------------------------------------------- Zeta(5), the sum(1/n**5,n=1..infinity) to 512 digits. 1.036927755143369926331365486457034168057080919501912811974192677 9038035897862814845600431065571333363796203414665566090428009617 7915597084183511072180087644866286337180353598363962365128888981 3352767752398275032022436845766444665958115993917977745039244643 9196666159664016205325205021519226713512567859748692860197447984 3200672681297530919900774656558601526573730037561532683149897971 9350398378581319922884886425335104251602510849904346402941172432 7576341508162332245618649927144272264614113007580868316916497918 ----------------------------------------------------------------------------- Zeta(7) to 512 places : sum(1/n**7,n=1..infinity); 1.008349277381922826839797549849796759599863560565238706417283136 5716014783173557353460969689138513239689614536514910748872867774 1984033544031579830103398456212106946358524390658335396467699756 7696691427804314333947495215378902800259045551979353108370084210 7329399046107085641235605890622599776098694754076320000481632951 2586769250630734413632555601360305007373302413187037951026624779 3954650225467042015510405582224239250510868837727077426002177100 0195455778989836046745406121952650765461161356548679150080858554 ----------------------------------------------------------------------------- Zeta(9) or sum(1/n**9,n=1..infinity); 1.002008392826082214417852769232412060485605851394888756548596615 9097850533902583989503930691271695861574086047658470602614253739 7072243015306913249876425109092948687676545396979415407826022964 1544836250668629056707364521601531424421326337598815558052591454 0848901539527747456133451028740613274660692763390016294270864220 1123162209241265753326205462293215454665179945038662778223564776 1660330281492364570399301119383985017167926002064923069795850945 8457966548540026945118759481561430375776154443343398399851419383 ----------------------------------------------------------------------------- This number, the Product[Cos[Pi/n], {n,3,infinity}] is the limit of an interesting figure in geometry.: If we take a circle, inscribe a triangle, then incribe another circle inside the triangle, then inscribe a square inside the inner circle, then inscribe another circle inside the square, then inscribe a pentagon... The radius of this figure (the number of sides of the polygon increase with every step:triangle 3, square 4, pentagon 5, ...) approaches a limit: Product[Cos[Pi/n], {n,3,infinity}] Is there any way to get an analytic solution to this? Like this would be the square root of Pi or some combination of radicals and irrational numbers? Anyway, Thanks, Mounitra Chatterji mounitra@seas.ucla.edu mentioned in december 1995. By Mounitra Chatterji .1149420448532962007010401574695987428307953372008635168440233965; maple routine --> product(cos(Pi/n),n=3..infinity);evalf(",64); ------------------------------------------------------------------ The request was sent by achim flammenkamp on Tue Feb 27 09:05:13 PST 1996 The email address is: achim@mathematik.uni-jena.de The number is 1.60140224354988761393325 (to 24 digits of precision). -int(sqrt(x)/log(1-x),x=0..1); ------------------------------------------------------------------- .283265121310307732587685540450858868452123075913479495609303244760289207466703551200728343246718266 1721794706326872389237418265273196389116929121819750888062495294277256191719424273967384545908106616 5124702322513598413388920213387535350692362866707758376138858482266928332718882186473891252470626193 1134162075403008037881499615240658150936661712754874529120769279078826146925069339158824377250780006 81691683658433538480533518043146405030754456294577975558177142447872562829157 There is a pattern in the binary expansion of this number. The request was sent by B.J. Mares on Sun Dec 3 15:20:18 PST 1995 The email address is: bjmares@teleport.com ------------------------------------------------------- The request was sent by Joe Keane on Sun Sep 10 05:02:26 PDT 1995 The email address is: jgk@netcom.com The number to be tested is: 1.38432969165678691636600070469187275993602894672280031682863878069088210808356345 The number of correct digits in the number: 79 The hints given by the user: It's log((3+sqrt(7))/sqrt(2)) or 1/2*arccosh(8). -------------------------------------------------------- The request was sent by (Mr.) B.J. Mares on Sat Dec 9 19:10:27 PST 1995 The email address is: bjmares@teleport.com The number to be tested is: .86224012586805457155779028324939457856576474276829909451607121455730674059051645804203844143861813$ 451257229030330958513908111490904372705631904836799517334609935566864203581911199877725969528883243$ Another binary pattern. --------------------------------------------------------- The request was sent by Jon Borwein on Sun Nov 5 06:09:28 GMT 1995 The email address is: jborwein@cecm.sfu.ca The number to be tested is: .01118680003287710787004681 The number of correct digits in the number: 20 The test(s) to be performed on the number: algebraic -------------------------------------------------------- 1.456791031046907 The number of correct digits in the number: 16 The test(s) to be performed on the number: algebraic gamma_multiplicative gamma_additve zeta_multiplicative zeta_additive psi_digamma linear_dependence_salvage The hints given by the user: p(0)=1 q(0)=2 p(i+1)=sqrt(p(i)*q(i)) i = 0,1,2,.. q(i+1)=(p(i) + q(i))/2 i = 0,1,2,.. x = lim p(i) = lim q(i) i->+inf i->+inf -------------------------------------------------------- The request was sent by Olivier Gerard on Mon Jan 29 18:48:42 PST 1996 The email address is: quadrature@onco.techlink.fr The number to be tested is: 1.062550805496255938 This number arises in the study of generalized Zeta functions on non associative sets. -------------------------------------------------------- The request was sent by Michael Mossinghoff on Fri Feb 9 14:40:28 PST 1996 The email address is: mjm@math.appstate.edu The number to be tested is: 1.296210659593309 (see below for 2500 digits of it). As I mentioned in the original note, it would be interesting to see if this number satisfies a simple polynomial of degree > 34. The simplest polynomial I know of that it satisfies is x^38-x^36-x^34-x^29+x^28-x^24-x^14+x^10-x^9-x^4-x^2+1 I found this during a search for polynomials with height 1, degree 38, and Mahler measure < 1.3. I also have a second new Salem number that would be interesting to try. Thanks for running this! Best regards, Mike Mossinghoff mjm@math.appstate.edu 1.2962106595933092168517831791253754042307237363926176836463419715400357507663\ 555372700460810162259842255138960885075885472138523375229647035948031308222869\ 213377761985420998401465270339786283142588526265385851765349326219909024384324\ 298668143261669279113959085262729367911041451897621484638159134108808507417558\ 371227480609429111967509190900525542468572422201267290352457473788303514632978\ 531591219560940258062757424400763572149784569551257493407108061275808255266204\ 988526404732083078237046586577078037338486088388181584983281574252897177808263\ 147692481736785688370028996889741999268557158363474402864561998038209817582814\ 010732290535268946721928114002527443568020359790313185377702725896115435126307\ 841519785171242185997657977732689357703555840184684554577244752237497568339160\ 938205575175811976414747122955198011255949965359970687280700475477368518212756\ 924749820065045209604606889253335548989681523027453599219856774850675170030081\ 340461412329460883636590018878175768282781839837697211776636498168350816554156\ 904601023147786817236407289883278093415918634119620218433047846657184261144649\ 040715513536648841284787099601551612909626813632800691067564404454541790010887\ 679088108728482285977923782153457884089162309486388513634809308430291906873755\ 353865787785568433558148544650806363798445573460997103012214477139122206697676\ 151378710063572151250547043624062114013563819037462333697027524356258777528864\ 271328965733484293667236211401267719087175146826163754038706366216877272628132\ 296182344392845125506127123945469182368918766036231606918375224969603018840277\ 778903237698826183111400261578682603995590568903906955569848314084496482503972\ 906016618276547328327517227379822958377122743985938689837061722495995392321936\ 345285971817821600170724492417762482659737742843585759061520292400466743607983\ 593438732628413114256276767063139352552076489085606199932942061150333621663624\ 667294211959583161911171198313494502505440901133068426838051637173543721800267\ 607050254597479936347302850855318828765200608121163125879643065717811879123723\ 939826702878343201235748915166745912187493987556824139288848294746007488299743\ 663817162198495190194616103659925459932420514340386336983265209362290719538034\ 616103846861918706369114431911997889483119661422295458652413962075819025018423\ 406629086461013112957825351840936858715307617702746177132615020866202346765384\ 189199689332174745118809280247719860161398327812075021357273956644275172873038\ 687900608249173662145494837168975704911668609774430992557238265593517876057742\ 2513 ------------------------------------------------------------- Reference Philippe Flajolet and Andrew Odlyzko in Random Mapping Statistics you can have the article at ftp://netlib.att.com/netlib/att/math/odlyzko/index.html 1 - ln(1 - 1/E) > evalf(",1024); 1.4586751453870818910216436450673297018769779066921941448349981657928142090774\ 201612200442809516952542077265289812147224950456505217508488257192318776903978\ 283958471454981649855439295026537053597338520354935148025543820985296873219986\ 302608076828991375664708977028227357407155020168390466081440332929613402809962\ 987761600422067245386552208829277426092542078462258992350164685882837621214882\ 780180315165656808973787662538495808236640442271087689278355793100958663124347\ 608912549488795731777070799343730722066801620056545869945636645492898791927486\ 575158188313946857834776772734408679626984363705284330037652725380287794676349\ 373789251316549424606319247455867160631085208147788915528328222030175460874293\ 072958579419651653681072447431245769874928136318703222181432813223236987618651\ 560035148342838332185451812183617068075562954967559891795834498316055598164437\ 208325189384466039982301475617199617179127040273935951240040637361969048372804\ 683416371677229307327020903657448359390542480371335759362920019292630614667717\ 96831954446 1 + exp(-1) ----------- exp(-1) - 1 -2.163953413738652848770004010218023117093738602150792272533574120 ----------------------------------------------------------------------------- The Hard hexagons Entropy Constant The hard-hexagons entropy constant is algebraic (see below z number). The value is : 1.3954859724793027352295006635668880689541037281446611908174721561357608803586 977746898378730852754279026689685607685657184842212457119511639349818266947083 252547173794947534862281229126187281554340126162747356973585709823756812898414 948800016934903723995652094568253572538633572005211925074739811015138086289661 268136787831885630404682747107477204686894756657580905530270066675404962427719 060854536142216836296933016900330937276956621269398726823104923047442882514781 702966107270054292812280795061336321550953581179745072336957434963259935073449 490894249329307540816210555328068610619705545037955077580725537613858033619505 210958967729699416630942601615566925218549336476968551824281894615092855649748 501359906929152571833851080212811049755339847366927914398892041851355831303575 673710465224807454744982583885183287167357146092090743402851746571565499082292 999884612996137479952358336507860770516087879631202738350102895965881076822440 14681214726789035888008851819053742866660552775722734105313225337 Taken from The Favorite mathematical constants of Steven Finch, Mathsoft Inc. The constant is given by this (see z below)... 124 1/3 a := - --- 11 363 2501 1/2 b := ----- 33 11979 / 31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\\1/3 c := |1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| || \ 242 \\11979 / \11979 / // 1/4 7/12 3 11 z1 := 3/44 ------------------------------------------------------------------ / 31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\\2/3 |1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| || \ 242 \\11979 / \11979 / // 1/3 1/2 1/3 1/3 2/3 1/2 1/2 2 z2 := (1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) ) 31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\ %1 := 1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| | 242 \\11979 / \11979 / / 1/3 1/2 1/3 1/3 2/3 1/2 1/2 2 z3 := (- 1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) ) 31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\ %1 := 1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| | 242 \\11979 / \11979 / / 1/3 1/2 z4 := 1/(1/33 (1089 + 372 11 ) / 124 1/3 / 124 1/3 15376 2/3\1/2\1/2 + |2 - --- 11 + 2 |1 - --- 11 + ------ 11 | | )^1/2 \ 363 \ 363 131769 / / 1/4 7/12 z := 3/44 3 11 1/3 1/2 1/3 1/3 2/3 1/2 1/2 2 (1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) ) 1/3 1/2 1/3 1/3 2/3 1/2 1/2 2 / (- 1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) ) / ( / 2/3 1/3 1/2 %1 (1/33 (1089 + 372 11 ) / 124 1/3 / 124 1/3 15376 2/3\1/2\1/2 + |2 - --- 11 + 2 |1 - --- 11 + ------ 11 | | )^1/2) \ 363 \ 363 131769 / / 31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\ %1 := 1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| | 242 \\11979 / \11979 / / evalf(z); 1.395485972479302735229500663566888068954103728144661190817472165 ----------------------------------------------------------------------------- 
35550	----	Transcriber's notes: (1) Numbers following letters (without space) like C2 were originally printed in subscript. Letter subscripts are preceded by an underscore, like C_n. (2) Characters following a carat (^) were originally printed in superscript. (3) [root] stands for the root symbol; [alpha], [beta], etc. for greek letters. [Illustration: _Aristippus Philosophus Socraticus, nausragio cum ejectus ad Rhodiensium litus animadvertisses Geometrica Schemata descripta, exclamavisse ad comites ita dicitur_, Bene Speremus, Hominum enim vestigia video. _Vitruv. Architect lib. 6. Prief_. delin MBurghers sculptUniv. Oxon.] PIONEERS OF PROGRESS MEN OF SCIENCE EDITED BY S. CHAPMAN, M.A., D.SC., F.R.S. ARCHIMEDES BY SIR THOMAS HEATH K.C.B., K.C.V.O., F.R.S.; SC.D., CAMB. HON. D.SC., OXFORD [Greek: Dos moi pou stô, kai kinô tên gên] LONDON: SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE NEW YORK: THE MACMILLAN CO. 1920 CONTENTS. CHAP. PAGE I. ARCHIMEDES 1 II. GREEK GEOMETRY TO ARCHIMEDES 7 III. THE WORKS OF ARCHIMEDES 24 IV. GEOMETRY IN ARCHIMEDES 29 V. THE SANDRECKONER 45 VI. MECHANICS 50 VII. HYDROSTATICS 53 BIBLIOGRAPHY 57 CHRONOLOGY 58 CHAPTER I. ARCHIMEDES. If the ordinary person were asked to say off-hand what he knew of Archimedes, he would probably, at the most, be able to quote one or other of the well-known stories about him: how, after discovering the solution of some problem in the bath, he was so overjoyed that he ran naked to his house, shouting [Greek: eurêka, eurêka] (or, as we might say, "I've got it, I've got it"); or how he said "Give me a place to stand on and I will move the earth"; or again how he was killed, at the capture of Syracuse in the Second Punic War, by a Roman soldier who resented being told to get away from a diagram drawn on the ground which he was studying. And it is to be feared that few who are not experts in the history of mathematics have any acquaintance with the details of the original discoveries in mathematics of the greatest mathematician of antiquity, perhaps the greatest mathematical genius that the world has ever seen. History and tradition know Archimedes almost exclusively as the inventor of a number of ingenious mechanical appliances, things which naturally appeal more to the popular imagination than the subtleties of pure mathematics. Almost all that is told of Archimedes reaches us through the accounts by Polybius and Plutarch of the siege of Syracuse by Marcellus. He perished in the sack of that city in 212 B.C., and, as he was then an old man (perhaps 75 years old), he must have been born about 287 B.C. He was the son of Phidias, an astronomer, and was a friend and kinsman of King Hieron of Syracuse and his son Gelon. He spent some time at Alexandria studying with the successors of Euclid (Euclid who flourished about 300 B.C. was then no longer living). It was doubtless at Alexandria that he made the acquaintance of Conon of Samos, whom he admired as a mathematician and cherished as a friend, as well as of Eratosthenes; to the former, and to the latter during his early period he was in the habit of communicating his discoveries before their publication. It was also probably in Egypt that he invented the water-screw known by his name, the immediate purpose being the drawing of water for irrigating fields. After his return to Syracuse he lived a life entirely devoted to mathematical research. Incidentally he became famous through his clever mechanical inventions. These things were, however, in his case the "diversions of geometry at play," and he attached no importance to them. In the words of Plutarch, "he possessed so lofty a spirit, so profound a soul, and such a wealth of scientific knowledge that, although these inventions had won for him the renown of more than human sagacity, yet he would not consent to leave behind him any written work on such subjects, but, regarding as ignoble and sordid the business of mechanics and every sort of art which is directed to practical utility, he placed his whole ambition in those speculations in the beauty and subtlety of which there is no admixture of the common needs of life". During the siege of Syracuse Archimedes contrived all sorts of engines against the Roman besiegers. There were catapults so ingeniously constructed as to be equally serviceable at long or short range, and machines for discharging showers of missiles through holes made in the walls. Other machines consisted of long movable poles projecting beyond the walls; some of these dropped heavy weights upon the enemy's ships and on the constructions which they called _sambuca_, from their resemblance to a musical instrument of that name, and which consisted of a protected ladder with one end resting on two quinqueremes lashed together side by side as base, and capable of being raised by a windlass; others were fitted with an iron hand or a beak like that of a crane, which grappled the prows of ships, then lifted them into the air and let them fall again. Marcellus is said to have derided his own engineers and artificers with the words, "Shall we not make an end of fighting with this geometrical Briareus who uses our ships like cups to ladle water from the sea, drives our _sambuca_ off ignominiously with cudgel-blows, and, by the multitude of missiles that he hurls at us all at once, outdoes the hundred-handed giants of mythology?" But the exhortation had no effect, the Romans being in such abject terror that, "if they did but see a piece of rope or wood projecting above the wall they would cry 'there it is,' declaring that Archimedes was setting some engine in motion against them, and would turn their backs and run away, insomuch that Marcellus desisted from all fighting and assault, putting all his hope in a long siege". Archimedes died, as he had lived, absorbed in mathematical contemplation. The accounts of the circumstances of his death differ in some details. Plutarch gives more than one version in the following passage: "Marcellus was most of all afflicted at the death of Archimedes, for, as fate would have it, he was intent on working out some problem with a diagram, and, his mind and his eyes being alike fixed on his investigation, he never noticed the incursion of the Romans nor the capture of the city. And when a soldier came up to him suddenly and bade him follow to Marcellus, he refused to do so until he had worked out his problem to a demonstration; whereat the soldier was so enraged that he drew his sword and slew him. Others say that the Roman ran up to him with a drawn sword, threatening to kill him; and, when Archimedes saw him, he begged him earnestly to wait a little while in order that he might not leave his problem incomplete and unsolved, but the other took no notice and killed him. Again, there is a third account to the effect that, as he was carrying to Marcellus some of his mathematical instruments, sundials, spheres, and angles adjusted to the apparent size of the sun to the sight, some soldiers met him and, being under the impression that he carried gold in the vessel, killed him." The most picturesque version of the story is that which represents him as saying to a Roman soldier who came too close, "Stand away, fellow, from my diagram," whereat the man was so enraged that he killed him. Archimedes is said to have requested his friends and relatives to place upon his tomb a representation of a cylinder circumscribing a sphere within it, together with an inscription giving the ratio (3/2) which the cylinder bears to the sphere; from which we may infer that he himself regarded the discovery of this ratio as his greatest achievement. Cicero, when quaestor in Sicily, found the tomb in a neglected state and restored it. In modern times not the slightest trace of it has been found. Beyond the above particulars of the life of Archimedes, we have nothing but a number of stories which, if perhaps not literally accurate, yet help us to a conception of the personality of the man which we would not willingly have altered. Thus, in illustration of his entire preoccupation by his abstract studies, we are told that he would forget all about his food and such necessities of life, and would be drawing geometrical figures in the ashes of the fire, or, when anointing himself, in the oil on his body. Of the same kind is the story mentioned above, that, having discovered while in a bath the solution of the question referred to him by Hieron as to whether a certain crown supposed to have been made of gold did not in fact contain a certain proportion of silver, he ran naked through the street to his home shouting [Greek: eurêka, eurêka]. It was in connexion with his discovery of the solution of the problem _To move a given weight by a given force_ that Archimedes uttered the famous saying, "Give me a place to stand on, and I can move the earth" ([Greek: dos moi pou stô kai kinô tên gên], or in his broad Doric, as one version has it, [Greek: pa bô kai kinô tan gan]). Plutarch represents him as declaring to Hieron that any given weight could be moved by a given force, and boasting, in reliance on the cogency of his demonstration, that, if he were given another earth, he would cross over to it and move this one. "And when Hieron was struck with amazement and asked him to reduce the problem to practice and to show him some great weight moved by a small force, he fixed on a ship of burden with three masts from the king's arsenal which had only been drawn up by the great labour of many men; and loading her with many passengers and a full freight, sitting himself the while afar off, with no great effort but quietly setting in motion with his hand a compound pulley, he drew the ship towards him smoothly and safely as if she were moving through the sea." Hieron, we are told elsewhere, was so much astonished that he declared that, from that day forth, Archimedes's word was to be accepted on every subject! Another version of the story describes the machine used as a _helix_; this term must be supposed to refer to a screw in the shape of a cylindrical helix turned by a handle and acting on a cog-wheel with oblique teeth fitting on the screw. Another invention was that of a sphere constructed so as to imitate the motions of the sun, the moon, and the five planets in the heavens. Cicero actually saw this contrivance, and he gives a description of it, stating that it represented the periods of the moon and the apparent motion of the sun with such accuracy that it would even (over a short period) show the eclipses of the sun and moon. It may have been moved by water, for Pappus speaks in one place of "those who understand the making of spheres and produce a model of the heavens by means of the regular circular motion of water". In any case it is certain that Archimedes was much occupied with astronomy. Livy calls him "unicus spectator caeli siderumque". Hipparchus says, "From these observations it is clear that the differences in the years are altogether small, but, as to the solstices, I almost think that both I and Archimedes have erred to the extent of a quarter of a day both in observation and in the deduction therefrom." It appears, therefore, that Archimedes had considered the question of the length of the year. Macrobius says that he discovered the distances of the planets. Archimedes himself describes in the _Sandreckoner_ the apparatus by which he measured the apparent diameter of the sun, i.e. the angle subtended by it at the eye. The story that he set the Roman ships on fire by an arrangement of burning-glasses or concave mirrors is not found in any authority earlier than Lucian (second century A.D.); but there is no improbability in the idea that he discovered some form of burning-mirror, e.g. a paraboloid of revolution, which would reflect to one point all rays falling on its concave surface in a direction parallel to its axis. CHAPTER II. GREEK GEOMETRY TO ARCHIMEDES. In order to enable the reader to arrive at a correct understanding of the place of Archimedes and of the significance of his work it is necessary to pass in review the course of development of Greek geometry from its first beginnings down to the time of Euclid and Archimedes. Greek authors from Herodotus downwards agree in saying that geometry was invented by the Egyptians and that it came into Greece from Egypt. One account says:-- "Geometry is said by many to have been invented among the Egyptians, its origin being due to the measurement of plots of land. This was necessary there because of the rising of the Nile, which obliterated the boundaries appertaining to separate owners. Nor is it marvellous that the discovery of this and the other sciences should have arisen from such an occasion, since everything which moves in the sense of development will advance from the imperfect to the perfect. From sense-perception to reasoning, and from reasoning to understanding, is a natural transition. Just as among the Phoenicians, through commerce and exchange, an accurate knowledge of numbers was originated, so also among the Egyptians geometry was invented for the reason above stated. "Thales first went to Egypt and thence introduced this study into Greece." But it is clear that the geometry of the Egyptians was almost entirely practical and did not go beyond the requirements of the land-surveyor, farmer or merchant. They did indeed know, as far back as 2000 B.C., that in a triangle which has its sides proportional to 3, 4, 5 the angle contained by the two smaller sides is a right angle, and they used such a triangle as a practical means of drawing right angles. They had formulæ, more or less inaccurate, for certain measurements, e.g. for the areas of certain triangles, parallel-trapezia, and circles. They had, further, in their construction of pyramids, to use the notion of similar right-angled triangles; they even had a name, _se-qet_, for the ratio of the half of the side of the base to the height, that is, for what we should call the _co-tangent_ of the angle of slope. But not a single general theorem in geometry can be traced to the Egyptians. Their knowledge that the triangle (3, 4, 5) is right angled is far from implying any knowledge of the general proposition (Eucl. I., 47) known by the name of Pythagoras. The science of geometry, in fact, remained to be discovered; and this required the genius for pure speculation which the Greeks possessed in the largest measure among all the nations of the world. Thales, who had travelled in Egypt and there learnt what the priests could teach him on the subject, introduced geometry into Greece. Almost the whole of Greek science and philosophy begins with Thales. His date was about 624-547 B.C. First of the Ionian philosophers, and declared one of the Seven Wise Men in 582-581, he shone in all fields, as astronomer, mathematician, engineer, statesman and man of business. In astronomy he predicted the solar eclipse of 28 May, 585, discovered the inequality of the four astronomical seasons, and counselled the use of the Little Bear instead of the Great Bear as a means of finding the pole. In geometry the following theorems are attributed to him--and their character shows how the Greeks had to begin at the very beginning of the theory--(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground (this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids). After Thales come the Pythagoreans. We are told that the Pythagoreans were the first to use the term [Greek: mathêmata] (literally "subjects of instruction") in the specialised sense of "mathematics"; they, too, first advanced mathematics as a study pursued for its own sake and made it a part of a liberal education. Pythagoras, son of Mnesarchus, was born in Samos about 572 B.C., and died at a great age (75 or 80) at Metapontum. His interests were as various as those of Thales; his travels, all undertaken in pursuit of knowledge, were probably even more extended. Like Thales, and perhaps at his suggestion, he visited Egypt and studied there for a long period (22 years, some say). It is difficult to disentangle from the body of Pythagorean doctrines the portions which are due to Pythagoras himself because of the habit which the members of the school had of attributing everything to the Master ([Greek: autos epha], _ipse dixit_). In astronomy two things at least may safely be attributed to him; he held that the earth is spherical in shape, and he recognised that the sun, moon and planets have an independent motion of their own in a direction contrary to that of the daily rotation; he seems, however, to have adhered to the geocentric view of the universe, and it was his successors who evolved the theory that the earth does not remain at the centre but revolves, like the other planets and the sun and moon, about the "central fire". Perhaps his most remarkable discovery was the dependence of the musical intervals on the lengths of vibrating strings, the proportion for the octave being 2 : 1, for the fifth 3 : 2 and for the fourth 4 : 3. In arithmetic he was the first to expound the theory of _means_ and of proportion as applied to commensurable quantities. He laid the foundation of the theory of numbers by considering the properties of numbers as such, namely, prime numbers, odd and even numbers, etc. By means of _figured_ numbers, square, oblong, triangular, etc. (represented by dots arranged in the form of the various figures) he showed the connexion between numbers and geometry. In view of all these properties of numbers, we can easily understand how the Pythagoreans came to "liken all things to numbers" and to find in the principles of numbers the principles of all things ("all things are numbers"). We come now to Pythagoras's achievements in geometry. There is a story that, when he came home from Egypt and tried to found a school at Samos, he found the Samians indifferent, so that he had to take special measures to ensure that his geometry might not perish with him. Going to the gymnasium, he sought out a well-favoured youth who seemed likely to suit his purpose, and was withal poor, and bribed him to learn geometry by promising him sixpence for every proposition that he mastered. Very soon the youth got fascinated by the subject for its own sake, and Pythagoras rightly judged that he would gladly go on without the sixpence. He hinted, therefore, that he himself was poor and must try to earn his living instead of doing mathematics; whereupon the youth, rather than give up the study, volunteered to pay sixpence to Pythagoras for each proposition. In geometry Pythagoras set himself to lay the foundations of the subject, beginning with certain important definitions and investigating the fundamental principles. Of propositions attributed to him the most famous is, of course, the theorem that in a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the sides about the right angle (Eucl. I., 47); and, seeing that Greek tradition universally credits him with the proof of this theorem, we prefer to believe that tradition is right. This is to some extent confirmed by another tradition that Pythagoras discovered a general formula for finding two numbers such that the sum of their squares is a square number. This depends on the theory of the _gnomon_, which at first had an arithmetical signification corresponding to the geometrical use of it in Euclid, Book II. A figure in the shape of a _gnomon_ put round two sides of a square makes it into a larger square. Now consider the number 1 represented by a dot. Round this place three other dots so that the four dots form a square (1 + 3 = 2²). Round the four dots (on two adjacent sides of the square) place five dots at regular and equal distances, and we have another square (1 + 3 + 5 = 3²); and so on. The successive odd numbers 1, 3, 5 ... were called _gnomons_, and the general formula is 1 + 3 + 5 + ... + (2n - 1) = n². Add the next odd number, i.e. 2n + 1, and we have n² + (2n + 1) = (n + 1)². In order, then, to get two square numbers such that their sum is a square we have only to see that 2n + 1 is a square. Suppose that 2n + 1 = m²; then n = ½(m² - 1), and we have {½(m² - 1)}² + m² = {½(m² + 1)}², where m is any odd number; and this is the general formula attributed to Pythagoras. Proclus also attributes to Pythagoras the theory of proportionals and the construction of the five "cosmic figures," the five regular solids. One of the said solids, the dodecahedron, has twelve pentagonal faces, and the construction of a regular pentagon involves the cutting of a straight line "in extreme and mean ratio" (Eucl. II., 11, and VI., 30), which is a particular case of the method known as the _application of areas_. How much of this was due to Pythagoras himself we do not know; but the whole method was at all events fully worked out by the Pythagoreans and proved one of the most powerful of geometrical methods. The most elementary case appears in Euclid, I., 44, 45, where it is shown how to apply to a given straight line as base a parallelogram having a given angle (say a rectangle) and equal in area to any rectilineal figure; this construction is the geometrical equivalent of arithmetical _division_. The general case is that in which the parallelogram, though _applied_ to the straight line, overlaps it or falls short of it in such a way that the part of the parallelogram which extends beyond, or falls short of, the parallelogram of the same angle and breadth on the given straight line itself (exactly) as base is similar to another given parallelogram (Eucl. VI., 28, 29). This is the geometrical equivalent of the most general form of quadratic equation ax ± mx² = C, so far as it has real roots; while the condition that the roots may be real was also worked out (= Eucl. VI., 27). It is important to note that this method of _application of areas_ was directly used by Apollonius of Perga in formulating the fundamental properties of the three conic sections, which properties correspond to the equations of the conics in Cartesian co-ordinates; and the names given by Apollonius (for the first time) to the respective conics are taken from the theory, _parabola_ ([Greek: parabolê]) meaning "application" (i.e. in this case the parallelogram is applied to the straight line exactly), _hyperbola_ ([Greek: hyperbolê]), "exceeding" (i.e. in this case the parallelogram exceeds or overlaps the straight line), _ellipse_ ([Greek: elleipsis]), "falling short" (i.e. the parallelogram falls short of the straight line). Another problem solved by the Pythagoreans is that of drawing a rectilineal figure equal in area to one given rectilineal figure and similar to another. Plutarch mentions a doubt as to whether it was this problem or the proposition of Euclid I., 47, on the strength of which Pythagoras was said to have sacrificed an ox. The main particular applications of the theorem of the square on the hypotenuse (e.g. those in Euclid, Book II.) were also Pythagorean; the construction of a square equal to a given rectangle (Eucl. II., 14) is one of them and corresponds to the solution of the pure quadratic equation x² = ab. The Pythagoreans proved the theorem that the sum of the angles of any triangle is equal to two right angles (Eucl. I., 32). Speaking generally, we may say that the Pythagorean geometry covered the bulk of the subject-matter of Books I., II., IV., and VI. of Euclid (with the qualification, as regards Book VI., that the Pythagorean theory of proportion applied only to commensurable magnitudes). Our information about the origin of the propositions of Euclid, Book III., is not so complete; but it is certain that the most important of them were well known to Hippocrates of Chios (who flourished in the second half of the fifth century, and lived perhaps from about 470 to 400 B.C.), whence we conclude that the main propositions of Book III. were also included in the Pythagorean geometry. Lastly, the Pythagoreans discovered the existence of incommensurable lines, or of _irrationals_. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of [root]2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a _reductio ad absurdum_ proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable was bound to cause geometers a great shock, because it showed that the theory of proportion invented by Pythagoras was not of universal application, and therefore that propositions proved by means of it were not really established. Hence the stories that the discovery of the irrational was for a time kept secret, and that the first person who divulged it perished by shipwreck. The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus (408-355 B.C.) discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes. By the time of Hippocrates of Chios the scope of Greek geometry was no longer even limited to the Elements; certain special problems were also attacked which were beyond the power of the geometry of the straight line and circle, and which were destined to play a great part in determining the direction taken by Greek geometry in its highest flights. The main problems in question were three: (1) the doubling of the cube, (2) the trisection of any angle, (3) the squaring of the circle; and from the time of Hippocrates onwards the investigation of these problems proceeded _pari passu_ with the completion of the body of the Elements. Hippocrates himself is an example of the concurrent study of the two departments. On the one hand, he was the first of the Greeks who is known to have compiled a book of Elements. This book, we may be sure, contained in particular the most important propositions about the circle included in Euclid, Book III. But a much more important proposition is attributed to Hippocrates; he is said to have been the first to prove that circles are to one another as the squares on their diameters (= Eucl. XII., 2), with the deduction that similar segments of circles are to one another as the squares on their bases. These propositions were used by him in his tract on the squaring of _lunes_, which was intended to lead up to the squaring of the circle. The latter problem is one which must have exercised practical geometers from time immemorial. Anaxagoras for instance (about 500-428 B.C.) is said to have worked at the problem while in prison. The essential portions of Hippocrates's tract are preserved in a passage of Simplicius (on Aristotle's _Physics_), which contains substantial fragments from Eudemus's _History of Geometry_. Hippocrates showed how to square three particular lunes of different forms, and then, lastly, he squared the sum of a certain circle and a certain lune. Unfortunately, however, the last-mentioned lune was not one of those which can be squared, and so the attempt to square the circle in this way failed after all. Hippocrates also attacked the problem of doubling the cube. There are two versions of the origin of this famous problem. According to one of them, an old tragic poet represented Minos as having been dissatisfied with the size of a tomb erected for his son Glaucus, and having told the architect to make it double the size, retaining, however, the cubical form. According to the other, the Delians, suffering from a pestilence, were told by the oracle to double a certain cubical altar as a means of staying the plague. Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in continued proportion between two given straight lines, i.e. finding x, y such that a : x = x : y = y : b, where a, b are the two given straight lines. It is easy to see that, if a : x = x : y = y : b, then b/a = (x/a)³, and, as a particular case, if b = 2a, x³ = 2a³, so that the side of the cube which is double of the cube of side a is found. The problem of doubling the cube was henceforth tried exclusively in the form of the problem of the two mean proportionals. Two significant early solutions are on record. (1) Archytas of Tarentum (who flourished in first half of fourth century B.C.) found the two mean proportionals by a very striking construction in three dimensions, which shows that solid geometry, in the hands of Archytas at least, was already well advanced. The construction was usually called mechanical, which it no doubt was in form, though in reality it was in the highest degree theoretical. It consisted in determining a point in space as the intersection of three surfaces: (a) a cylinder, (b) a cone, (c) an "anchor-ring" with internal radius = 0. (2) Menæchmus, a pupil of Eudoxus, and a contemporary of Plato, found the two mean proportionals by means of conic sections, in two ways, ([alpha]) by the intersection of two parabolas, the equations of which in Cartesian co-ordinates would be x² = ay, y² = bx, and ([beta]) by the intersection of a parabola and a rectangular hyperbola, the corresponding equations being x² = ay, and xy = ab respectively. It would appear that it was in the effort to solve this problem that Menæchmus discovered the conic sections, which are called, in an epigram by Eratosthenes, "the triads of Menæchmus". The trisection of an angle was effected by means of a curve discovered by Hippias of Elis, the sophist, a contemporary of Hippocrates as well as of Democritus and Socrates (470-399 B.C.). The curve was called the _quadratrix_ because it also served (in the hands, as we are told, of Dinostratus, brother of Menæchmus, and of Nicomedes) for squaring the circle. It was theoretically constructed as the locus of the point of intersection of two straight lines moving at uniform speeds and in the same time, one motion being angular and the other rectilinear. Suppose OA, OB are two radii of a circle at right angles to one another. Tangents to the circle at A and B, meeting at C, form with the two radii the square OACB. The radius OA is made to move uniformly about O, the centre, so as to describe the angle AOB in a certain time. Simultaneously AC moves parallel to itself at uniform speed such that A just describes the line AO in the same length of time. The intersection of the moving radius and AC in their various positions traces out the _quadratrix_. The rest of the geometry which concerns us was mostly the work of a few men, Democritus of Abdera, Theodorus of Cyrene (the mathematical teacher of Plato), Theætetus, Eudoxus, and Euclid. The actual writers of Elements of whom we hear were the following. Leon, a little younger than Eudoxus (408-355 B.C.), was the author of a collection of propositions more numerous and more serviceable than those collected by Hippocrates. Theudius of Magnesia, a contemporary of Menæchmus and Dinostratus, "put together the elements admirably, making many partial or limited propositions more general". Theudius's book was no doubt the geometrical text-book of the Academy and that used by Aristotle. Theodorus of Cyrene and Theætetus generalised the theory of irrationals, and we may safely conclude that a great part of the substance of Euclid's Book X. (on irrationals) was due to Theætetus. Theætetus also wrote on the five regular solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron), and Euclid was therefore no doubt equally indebted to Theætetus for the contents of his Book XIII. In the matter of Book XII. Eudoxus was the pioneer. These facts are confirmed by the remark of Proclus that Euclid, in compiling his Elements, collected many of the theorems of Eudoxus, perfected many others by Theætetus, and brought to irrefragable demonstration the propositions which had only been somewhat loosely proved by his predecessors. Eudoxus (about 408-355 B.C.) was perhaps the greatest of all Archimedes's predecessors, and it is his achievements, especially the discovery of the _method of exhaustion_, which interest us in connexion with Archimedes. In astronomy Eudoxus is famous for the beautiful theory of concentric spheres which he invented to explain the apparent motions of the planets, and, particularly, their apparent stationary points and retrogradations. The theory applied also to the sun and moon, for which Eudoxus required only three spheres in each case. He represented the motion of each planet as compounded of the rotations of four interconnected spheres about diameters, all of which pass through the centre of the earth. The outermost sphere represents the daily rotation, the second a motion along the zodiac circle or ecliptic; the poles of the third sphere, about which that sphere revolves, are fixed at two opposite points on the zodiac circle, and are carried round in the motion of the second sphere; and on the surface of the third sphere the poles of the fourth sphere are fixed; the fourth sphere, revolving about the diameter joining its two poles, carries the planet which is fixed at a point on its equator. The poles and the speeds and directions of rotation are so chosen that the planet actually describes a _hippopede_, or _horse-fetter_, as it was called (i.e. a figure of eight), which lies along and is longitudinally bisected by the zodiac circle, and is carried round that circle. As a _tour de force_ of geometrical imagination it would be difficult to parallel this hypothesis. In geometry Eudoxus discovered the great theory of proportion, applicable to incommensurable as well as commensurable magnitudes, which is expounded in Euclid, Book V., and which still holds its own and will do so for all time. He also solved the problem of the two mean proportionals by means of certain curves, the nature of which, in the absence of any description of them in our sources, can only be conjectured. Last of all, and most important for our purpose, is his use of the famous _method of exhaustion_ for the measurement of the areas of curves and the volumes of solids. The example of this method which will be most familiar to the reader is the proof in Euclid XII., 2, of the theorem that the areas of circles are to one another as the squares on their diameters. The proof in this and in all cases depends on a lemma which forms Prop. 1 of Euclid's Book X. to the effect that, if there are two unequal magnitudes of the same kind and from the greater you subtract not less than its half, then from the remainder not less than its half, and so on continually, you will at length have remaining a magnitude less than the lesser of the two magnitudes set out, however small it is. Archimedes says that the theorem of Euclid XII., 2, was proved by means of a certain lemma to the effect that, if we have two unequal magnitudes (i.e. lines, surfaces, or solids respectively), the greater exceeds the lesser by such a magnitude as is capable, if added continually to itself, of exceeding any magnitude of the same kind as the original magnitudes. This assumption is known as the Axiom or Postulate of Archimedes, though, as he states, it was assumed before his time by those who used the method of exhaustion. It is in reality used in Euclid's lemma (Eucl. X., 1) on which Euclid XII., 2, depends, and only differs in statement from Def. 4 of Euclid, Book V., which is no doubt due to Eudoxus. The method of exhaustion was not discovered all at once; we find traces of gropings after such a method before it was actually evolved. It was perhaps Antiphon, the sophist, of Athens, a contemporary of Socrates (470-399 B.C.), who took the first step. He inscribed a square (or, according to another account, an equilateral triangle) in a circle, then bisected the arcs subtended by the sides, and so inscribed a polygon of double the number of sides; he then repeated the process, and maintained that, by continuing it, we should at last arrive at a polygon with sides so small as to make the polygon coincident with the circle. Though this was formally incorrect, it nevertheless contained the germ of the method of exhaustion. Hippocrates, as we have seen, is said to have proved the theorem that circles are to one another as the squares on their diameters, and it is difficult to see how he could have done this except by some form, or anticipation, of the method. There is, however, no doubt about the part taken by Eudoxus; he not only based the method on rigorous demonstration by means of the lemma or lemmas aforesaid, but he actually applied the method to find the volumes (1) of any pyramid, (2) of the cone, proving (1) that any pyramid is one third part of the prism which has the same base and equal height, and (2) that any cone is one third part of the cylinder which has the same base and equal height. Archimedes, however, tells us the remarkable fact that these two theorems were first discovered by Democritus (who flourished towards the end of the fifth century B.C.), though he was not able to prove them (which no doubt means, not that he gave no sort of proof, but that he was not able to establish the propositions by the rigorous method of Eudoxus). Archimedes adds that we must give no small share of the credit for these theorems to Democritus; and this is another testimony to the marvellous powers, in mathematics as well as in other subjects, of the great man who, in the words of Aristotle, "seems to have thought of everything". We know from other sources that Democritus wrote on irrationals; he is also said to have discussed the question of two parallel sections of a cone (which were evidently supposed to be indefinitely close together), asking whether we are to regard them as unequal or equal: "for if they are unequal they will make the cone irregular as having many indentations, like steps, and unevennesses, but, if they are equal, the cone will appear to have the property of the cylinder and to be made up of equal, not unequal, circles, which is very absurd". This explanation shows that Democritus was already close on the track of infinitesimals. Archimedes says further that the theorem that spheres are in the triplicate ratio of their diameters was proved by means of the same lemma. The proofs of the propositions about the volumes of pyramids, cones and spheres are, of course, contained in Euclid, Book XII. (Props. 3-7 Cor., 10, 16-18 respectively). It is no doubt desirable to illustrate Eudoxus's method by one example. We will take one of the simplest, the proposition (Eucl. XII., 10) about the cone. Given ABCD, the circular base of the cylinder which has the same base as the cone and equal height, we inscribe the square ABCD; we then bisect the arcs subtended by the sides, and draw the regular inscribed polygon of eight sides, then similarly we draw the regular inscribed polygon of sixteen sides, and so on. We erect on each regular polygon the prism which has the polygon for base, thereby obtaining successive prisms inscribed in the cylinder, and of the same height with it. Each time we double the number of sides in the base of the prism we take away more than half of the volume by which the cylinder exceeds the prism (since we take away more than half of the excess of the area of the circular base over that of the inscribed polygon, as in Euclid XII., 2). Suppose now that V is the volume of the cone, C that of the cylinder. We have to prove that C = 3V. If C is not equal to 3V, it is either greater or less than 3V. Suppose (1) that C > 3V, and that C = 3V + E. Continue the construction of prisms inscribed in the cylinder until the parts of the cylinder left over outside the final prism (of volume P) are together less than E. Then C - P < E. But C - 3V = E; Therefore P > 3V. But it has been proved in earlier propositions that P is equal to three times the pyramid with the same base as the prism and equal height. Therefore that pyramid is greater than V, the volume of the cone: which is impossible, since the cone encloses the pyramid. Therefore C is not greater than 3V. Next (2) suppose that C < 3V, so that, inversely, V > 1/3 C. This time we inscribe successive pyramids in the cone until we arrive at a pyramid such that the portions of the cone left over outside it are together less than the excess of V over 1/3 C. It follows that the pyramid is greater than 1/3 C. Hence the prism on the same base as the pyramid and inscribed in the cylinder (which prism is three times the pyramid) is greater than C: which is impossible, since the prism is enclosed by the cylinder, and is therefore less than it. Therefore V is not greater than 1/3 C, or C is not less than 3V. Accordingly C, being neither greater nor less than 3V, must be equal to it; that is, V = 1/3 C. It only remains to add that Archimedes is fully acquainted with the main properties of the conic sections. These had already been proved in earlier treatises, which Archimedes refers to as the "Elements of Conics". We know of two such treatises, (1) Euclid's four Books on Conics, (2) a work by one Aristæus called "Solid Loci," probably a treatise on conics regarded as loci. Both these treatises are lost; the former was, of course, superseded by Apollonius's great work on Conics in eight Books. CHAPTER III. THE WORKS OF ARCHIMEDES. The range of Archimedes's writings will be gathered from the list of his various treatises. An extraordinarily large proportion of their contents represents entirely new discoveries of his own. He was no compiler or writer of text-books, and in this respect he differs from Euclid and Apollonius, whose work largely consisted in systematising and generalising the methods used and the results obtained by earlier geometers. There is in Archimedes no mere working-up of existing material; his objective is always something new, some definite addition to the sum of knowledge. Confirmation of this is found in the introductory letters prefixed to most of his treatises. In them we see the directness, simplicity and humanity of the man. There is full and generous recognition of the work of predecessors and contemporaries; his estimate of the relation of his own discoveries to theirs is obviously just and free from any shade of egoism. His manner is to state what particular discoveries made by his predecessors had suggested to him the possibility of extending them in new directions; thus he says that, in connexion with the efforts of earlier geometers to square the circle, it occurred to him that no one had tried to square a parabolic segment; he accordingly attempted the problem and finally solved it. Similarly he describes his discoveries about the volumes and surfaces of spheres and cylinders as supplementing the theorems of Eudoxus about the pyramid, the cone and the cylinder. He does not hesitate to say that certain problems baffled him for a long time; in one place he positively insists, for the purpose of pointing a moral, on specifying two propositions which he had enunciated but which on further investigation proved to be wrong. The ordinary MSS. of the Greek text of Archimedes give his works in the following order:-- 1. _On the Sphere and Cylinder_ (two books). 2. _Measurement of a Circle._ 3. _On Conoids and Spheroids._ 4. _On Spirals._ 5. _On Plane Equilibriums_ (two books). 6. _The Sandreckoner._ 7. _Quadrature of a Parabola._ A most important addition to this list has been made in recent years through an extraordinary piece of good fortune. In 1906 J. L. Heiberg, the most recent editor of the text of Archimedes, discovered a palimpsest of mathematical content in the "Jerusalemic Library" of one Papadopoulos Kerameus at Constantinople. This proved to contain writings of Archimedes copied in a good hand of the tenth century. An attempt had been made (fortunately with only partial success) to wash out the old writing, and then the parchment was used again to write a Euchologion upon. However, on most of the leaves the earlier writing remains more or less legible. The important fact about the MS. is that it contains, besides substantial portions of the treatises previously known, (1) a considerable portion of the work, in two books, _On Floating Bodies_, which was formerly supposed to have been lost in Greek and only to have survived in the translation by Wilhelm of Mörbeke, and (2) most precious of all, the greater part of the book called _The Method, treating of Mechanical Problems_ and addressed to Eratosthenes. The important treatise so happily recovered is now included in Heiberg's new (second) edition of the Greek text of Archimedes (Teubner, 1910-15), and some account of it will be given in the next chapter. The order in which the treatises appear in the MSS. was not the order of composition; but from the various prefaces and from internal evidence generally we are able to establish the following as being approximately the chronological sequence:-- 1. _On Plane Equilibriums_, I. 2. _Quadrature of a Parabola._ 3. _On Plane Equilibriums_, II. 4. _The Method._ 5. _On the Sphere and Cylinder_, I, II. 6. _On Spirals._ 7. _On Conoids and Spheroids._ 8. _On Floating Bodies_, I, II. 9. _Measurement of a Circle._ 10. _The Sandreckoner._ In addition to the above we have a collection of geometrical propositions which has reached us through the Arabic with the title "Liber assumptorum Archimedis". They were not written by Archimedes in their present form, but were probably collected by some later Greek writer for the purpose of illustrating some ancient work. It is, however, quite likely that some of the propositions, which are remarkably elegant, were of Archimedean origin, notably those concerning the geometrical figures made with three and four semicircles respectively and called (from their shape) (1) the _shoemaker's knife_ and (2) the _Salinon_ or _salt-cellar_, and another theorem which bears on the trisection of an angle. An interesting fact which we now know from Arabian sources is that the formula for the area of any triangle in terms of its sides which we write in the form [Delta] = [root]{s(s - a)(s - b)(s - c)}, and which was supposed to be Heron's because Heron gives the geometrical proof of it, was really due to Archimedes. Archimedes is further credited with the authorship of the famous Cattle-Problem enunciated in a Greek epigram edited by Lessing in 1773. According to its heading the problem was communicated by Archimedes to the mathematicians at Alexandria in a letter to Eratosthenes; and a scholium to Plato's _Charmides_ speaks of the problem "called by Archimedes the Cattle-Problem". It is an extraordinarily difficult problem in indeterminate analysis, the solution of which involves enormous figures. Of lost works of Archimedes the following can be identified:-- 1. Investigations relating to _polyhedra_ are referred to by Pappus, who, after speaking of the five regular solids, gives a description of thirteen other polyhedra discovered by Archimedes which are semi-regular, being contained by polygons equilateral and equiangular but not similar. One at least of these semi-regular solids was, however, already known to Plato. 2. A book of arithmetical content entitled _Principles_ dealt, as we learn from Archimedes himself, with the _naming of numbers_, and expounded a system of expressing large numbers which could not be written in the ordinary Greek notation. In setting out the same system in the _Sandreckoner_ (see Chapter V. below), Archimedes explains that he does so for the benefit of those who had not seen the earlier work. 3. _On Balances_ (or perhaps _levers_). Pappus says that in this work Archimedes proved that "greater circles overpower lesser circles when they rotate about the same centre". 4. A book _On Centres of Gravity_ is alluded to by Simplicius. It is not, however, certain that this and the last-mentioned work were separate treatises, Possibly Book I. _On Plane Equilibriums_ may have been part of a larger work (called perhaps _Elements of Mechanics_), and _On Balances_ may have been an alternative title. The title _On Centres of Gravity_ may be a loose way of referring to the same treatise. 5. _Catoptrica_, an optical work from which Theon of Alexandria quotes a remark about refraction. 6. _On Sphere-making_, a mechanical work on the construction of a sphere to represent the motions of the heavenly bodies (cf. pp. 5-6 above). Arabian writers attribute yet further works to Archimedes, (1) On the circle, (2) On a heptagon in a circle, (3) On circles touching one another, (4) On parallel lines, (5) On triangles, (6) On the properties of right-angled triangles, (7) a book of _Data_; but we have no confirmation of these statements. CHAPTER IV. GEOMETRY IN ARCHIMEDES. The famous French geometer, Chasles, drew an instructive distinction between the predominant features of the geometry of the two great successors of Euclid, namely, Archimedes and Apollonius of Perga (the "great geometer," and author of the classical treatise on Conics). The works of these two men may, says Chasles, be regarded as the origin and basis of two great inquiries which seem to share between them the domain of geometry. Apollonius is concerned with the _Geometry of Forms and Situations_, while in Archimedes we find the _Geometry of Measurements_, dealing with the quadrature of curvilinear plane figures and with the quadrature and cubature of curved surfaces, investigations which gave birth to the calculus of the infinite conceived and brought to perfection by Kepler, Cavalieri, Fermat, Leibniz and Newton. In geometry Archimedes stands, as it were, on the shoulders of Eudoxus in that he applied the method of exhaustion to new and more difficult cases of quadrature and cubature. Further, in his use of the method he introduced an interesting variation of the procedure as we know it from Euclid. Euclid (and presumably Eudoxus also) only used _inscribed_ figures, "exhausting" the figure to be measured, and had to invert the second half of the _reductio ad absurdum_ to enable approximation from below (so to speak) to be applied in that case also. Archimedes, on the other hand, approximates from above as well as from below; he approaches the area or volume to be measured by taking closer and closer _circumscribed_ figures, as well as inscribed, and thereby _compressing_, as it were, the inscribed and circumscribed figure into one, so that they ultimately coincide with one another and with the figure to be measured. But he follows the cautious method to which the Greeks always adhered; he never says that a given curve or surface is the _limiting form_ of the inscribed or circumscribed figure; all that he asserts is that we can approach the curve or surface _as nearly as we please_. The deductive form of proof by the method of exhaustion is apt to obscure not only the way in which the results were arrived at but also the real character of the procedure followed. What Archimedes actually does in certain cases is to perform what are seen, when the analytical equivalents are set down, to be real _integrations_; this remark applies to his investigation of the areas of a parabolic segment and a spiral respectively, the surface and volume respectively of a sphere and a segment of a sphere, and the volume of any segments of the solids of revolution of the second degree. The result is, as a rule, only obtained after a long series of preliminary propositions, all of which are links in a chain of argument elaborately forged for the one purpose. The method suggests the tactics of some master of strategy who foresees everything, eliminates everything not immediately conducive to the execution of his plan, masters every position in its order, and then suddenly (when the very elaboration of the scheme has almost obscured, in the mind of the onlooker, its ultimate object) strikes the final blow. Thus we read in Archimedes proposition after proposition the bearing of which is not immediately obvious but which we find infallibly used later on; and we are led on by such easy stages that the difficulty of the original problem, as presented at the outset, is scarcely appreciated. As Plutarch says, "It is not possible to find in geometry more difficult and troublesome questions, or more simple and lucid explanations". But it is decidedly a rhetorical exaggeration when Plutarch goes on to say that we are deceived by the easiness of the successive steps into the belief that any one could have discovered them for himself. On the contrary, the studied simplicity and the perfect finish of the treatises involve at the same time an element of mystery. Although each step depends upon the preceding ones, we are left in the dark as to how they were suggested to Archimedes. There is, in fact, much truth in a remark of Wallis to the effect that he seems "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results". A partial exception is now furnished by the _Method_; for here we have (as it were) a lifting of the veil and a glimpse of the interior of Archimedes's workshop. He tells us how he discovered certain theorems in quadrature and cubature, and he is at the same time careful to insist on the difference between (1) the means which may serve to suggest the truth of theorems, although not furnishing scientific proofs of them, and (2) the rigorous demonstrations of them by approved geometrical methods which must follow before they can be finally accepted as established. Writing to Eratosthenes he says: "Seeing in you, as I say, an earnest student, a man of considerable eminence in philosophy and an admirer of mathematical inquiry when it comes your way, I have thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, which, when you see it, will put you in possession of a means whereby you can investigate some of the problems of mathematics by mechanics. This procedure is, I am persuaded, no less useful for the proofs of the actual theorems as well. For certain things which first became clear to me by a mechanical method had afterwards to be demonstrated by geometry, because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired by the method some knowledge of the questions, to supply the proof than it is to find the proof without any previous knowledge. This is a reason why, in the case of the theorems the proof of which Eudoxus was the first to discover, namely, that the cone is a third part of the cylinder, and the pyramid a third part of the prism, having the same base and equal height, we should give no small share of the credit to Democritus, who was the first to assert this truth with regard to the said figures, though he did not prove it. I am myself in the position of having made the discovery of the theorem now to be published in the same way as I made my earlier discoveries; and I thought it desirable now to write out and publish the method, partly because I have already spoken of it and I do not want to be thought to have uttered vain words, but partly also because I am persuaded that it will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not occurred to me. "First then I will set out the very first theorem which became known to me by means of mechanics, namely, that _Any segment of a section of a right-angled cone_ [_i.e. a parabola_] _is four-thirds of the triangle which has the same base and equal height_; and after this I will give each of the other theorems investigated by the same method. Then, at the end of the book, I will give the geometrical proofs of the propositions." The following description will, I hope, give an idea of the general features of the mechanical method employed by Archimedes. Suppose that X is the plane or solid figure the area or content of which is to be found. The method in the simplest case is to weigh infinitesimal elements of X against the corresponding elements of another figure, B say, being such a figure that its area or content and the position of its centre of gravity are already known. The diameter or axis of the figure X being drawn, the infinitesimal elements taken are parallel sections of X in general, but not always, at right angles to the axis or diameter, so that the centres of gravity of all the sections lie at one point or other of the axis or diameter and their weights can therefore be taken as acting at the several points of the diameter or axis. In the case of a plane figure the infinitesimal sections are spoken of as parallel _straight lines_ and in the case of a solid figure as parallel _planes_, and the aggregate of the infinite number of sections is said to _make up_ the whole figure X. (Although the sections are so spoken of as straight lines or planes, they are really indefinitely narrow plane strips or indefinitely thin laminae respectively.) The diameter or axis is produced in the direction away from the figure to be measured, and the diameter or axis as produced is imagined to be the bar or lever of a balance. The object is now to apply all the separate elements of X at _one point_ on the lever, while the corresponding elements of the known figure B operate at different points, namely, _where they actually are_ in the first instance. Archimedes contrives, therefore, to move the elements of X away from their original position and to concentrate them at one point on the lever, such that each of the elements balances, about the point of suspension of the lever, the corresponding element of B acting at its centre of gravity. The elements of X and B respectively balance about the point of suspension in accordance with the property of the lever that the weights are inversely proportional to the distances from the fulcrum or point of suspension. Now the centre of gravity of B as a whole is known, and it may then be supposed to act as one mass at its centre of gravity. (Archimedes assumes as known that the sum of the "moments," as we call them, of all the elements of the figure B, acting severally at the points where they actually are, is equal to the moment of the whole figure applied as one mass at one point, its centre of gravity.) Moreover all the elements of X are concentrated at the one fixed point on the bar or lever. If this fixed point is H, and G is the centre of gravity of the figure B, while C is the point of suspension, X : B = CG : CH. Thus the area or content of X is found. Conversely, the method can be used to find the centre of gravity of X when its area or volume is known beforehand. In this case the elements of X, and X itself, have to be applied where they are, and the elements of the known figure or figures have to be applied at the one fixed point H on the other side of C, and since X, B and CH are known, the proportion B : X = CG : CH determines CG, where G is the centre of gravity of X. The mechanical method is used for finding (1) the area of any parabolic segment, (2) the volume of a sphere and a spheroid, (3) the volume of a segment of a sphere and the volume of a right segment of each of the three conicoids of revolution, (4) the centre of gravity (a) of a hemisphere, (b) of any segment of a sphere, (c) of any right segment of a spheroid and a paraboloid of revolution, and (d) of a half-cylinder, or, in other words, of a semicircle. Archimedes then proceeds to find the volumes of two solid figures, which are the special subject of the treatise. The solids arise as follows:-- (1) Given a cylinder inscribed in a rectangular parallelepiped on a square base in such a way that the two bases of the cylinder are circles inscribed in the opposite square faces, suppose a plane drawn through one side of the square containing one base of the cylinder and through the parallel diameter of the opposite base of the cylinder. The plane cuts off a solid with a surface resembling that of a horse's hoof. Archimedes proves that the volume of the solid so cut off is one sixth part of the volume of the parallelepiped. (2) A cylinder is inscribed in a cube in such a way that the bases of the cylinder are circles inscribed in two opposite square faces. Another cylinder is inscribed which is similarly related to another pair of opposite faces. The two cylinders include between them a solid with all its angles rounded off; and Archimedes proves that the volume of this solid is two-thirds of that of the cube. Having proved these facts by the mechanical method, Archimedes concluded the treatise with a rigorous geometrical proof of both propositions by the method of exhaustion. The MS. is unfortunately somewhat mutilated at the end, so that a certain amount of restoration is necessary. I shall now attempt to give a short account of the other treatises of Archimedes in the order in which they appear in the editions. The first is-- _On the Sphere and Cylinder._ Book I. begins with a preface addressed to Dositheus (a pupil of Conon), which reminds him that on a former occasion he had communicated to him the treatise proving that any segment of a "section of a right-angled cone" (i.e. a parabola) is four-thirds of the triangle with the same base and height, and adds that he is now sending the proofs of certain theorems which he has since discovered, and which seem to him to be worthy of comparison with Eudoxus's propositions about the volumes of a pyramid and a cone. The theorems are (1) that the surface of a sphere is equal to four times its greatest circle (i.e. what we call a "great circle" of the sphere); (2) that the surface of any segment of a sphere is equal to a circle with radius equal to the straight line drawn from the vertex of the segment to a point on the circle which is the base of the segment; (3) that, if we have a cylinder circumscribed to a sphere and with height equal to the diameter, then (a) the volume of the cylinder is 1½ times that of the sphere and (b) the surface of the cylinder, including its bases, is 1½ times the surface of the sphere. Next come a few definitions, followed by certain _Assumptions_, two of which are well known, namely:-- 1. _Of all lines which have the same extremities the straight line is the least_ (this has been made the basis of an alternative definition of a straight line). 2. _Of unequal lines, unequal surfaces and unequal solids the greater exceeds the less by such a magnitude as, when (continually) added to itself, can be made to exceed any assigned magnitude among those which are comparable_ [_with it and_] _with one another_ (i.e. are of the same kind). This is the _Postulate of Archimedes_. He also assumes that, of pairs of lines (including broken lines) and pairs of surfaces, concave in the same direction and bounded by the same extremities, the outer is greater than the inner. These assumptions are fundamental to his investigation, which proceeds throughout by means of figures inscribed and circumscribed to the curved lines or surfaces that have to be measured. After some preliminary propositions Archimedes finds (Props. 13, 14) the area of the surfaces (1) of a right cylinder, (2) of a right cone. Then, after quoting certain Euclidean propositions about cones and cylinders, he passes to the main business of the book, the measurement of the volume and surface of a sphere and a segment of a sphere. By circumscribing and inscribing to a great circle a regular polygon of an even number of sides and making it revolve about a diameter connecting two opposite angular points he obtains solids of revolution greater and less respectively than the sphere. In a series of propositions he finds expressions for (a) the surfaces, (b) the volumes, of the figures so inscribed and circumscribed to the sphere. Next he proves (Prop. 32) that, if the inscribed and circumscribed polygons which, by their revolution, generate the figures are similar, the surfaces of the figures are in the duplicate ratio, and their volumes in the triplicate ratio, of their sides. Then he proves that the surfaces and volumes of the inscribed and circumscribed figures respectively are less and greater than the surface and volume respectively to which the main propositions declare the surface and volume of the sphere to be equal (Props. 25, 27, 30, 31 Cor.). He has now all the material for applying the method of exhaustion and so proves the main propositions about the surface and volume of the sphere. The rest of the book applies the same procedure to a segment of the sphere. Surfaces of revolution are inscribed and circumscribed to a segment less than a hemisphere, and the theorem about the surface of the segment is finally proved in Prop. 42. Prop. 43 deduces the surface of a segment greater than a hemisphere. Prop. 44 gives the volume of the sector of the sphere which includes any segment. Book II begins with the problem of finding a sphere equal in volume to a given cone or cylinder; this requires the solution of the problem of the two mean proportionals, which is accordingly assumed. Prop. 2 deduces, by means of 1., 44, an expression for the volume of a segment of a sphere, and Props. 3, 4 solve the important problems of cutting a given sphere by a plane so that (a) the surfaces, (b) the volumes, of the segments may have to one another a given ratio. The solution of the second problem (Prop. 4) is difficult. Archimedes reduces it to the problem of dividing a straight line AB into two parts at a point M such that MB : (a given length) = (a given area) : AM². The solution of this problem with a determination of the limits of possibility are given in a fragment by Archimedes, discovered and preserved for us by Eutocius in his commentary on the book; they are effected by means of the points of intersection of two conics, a parabola and a rectangular hyperbola. Three problems of construction follow, the first two of which are to construct a segment of a sphere similar to one given segment, and having (a) its volume, (b) its surface, equal to that of another given segment of a sphere. The last two propositions are interesting. Prop. 8 proves that, if V, V' be the volumes, and S, S' the surfaces, of two segments into which a sphere is divided by a plane, V and S belonging to the greater segment, then S² : S'² > V : V' > S^(3/2) : S'^(3/2). Prop. 9 proves that, of all segments of spheres which have equal surfaces, the hemisphere is the greatest in volume. _The Measurement of a Circle._ This treatise, in the form in which it has come down to us, contains only three propositions; the second, being an easy deduction from Props. 1 and 3, is out of place in so far as it uses the result of Prop. 3. In Prop. 1 Archimedes inscribes and circumscribes to a circle a series of successive regular polygons, beginning with a square, and continually doubling the number of sides; he then proves in the orthodox manner by the method of exhaustion that the area of the circle is equal to that of a right-angled triangle, in which the perpendicular is equal to the radius, and the base equal to the circumference, of the circle. Prop. 3 is the famous proposition in which Archimedes finds by sheer calculation upper and lower arithmetical limits to the ratio of the circumference of a circle to its diameter, or what we call [pi]; the result obtained is 3-1/7> [pi] > 3-10/71. Archimedes inscribes and circumscribes successive regular polygons, beginning with hexagons, and doubling the number of sides continually, until he arrives at inscribed and circumscribed regular polygons with 96 sides; seeing then that the length of the circumference of the circle is intermediate between the perimeters of the two polygons, he calculates the two perimeters in terms of the diameter of the circle. His calculation is based on two close approximations (an upper and a lower) to the value of [root]3, that being the cotangent of the angle of 30°, from which he begins to work. He assumes as known that 265/153 < [root]3 < 1351/780. In the text, as we have it, only the results of the steps in the calculation are given, but they involve the finding of approximations to the square roots of several large numbers: thus 1172-1/8 is given as the approximate value of [root](1373943-33/64), 3013¾ as that of [root](9082321) and 1838-9/11 as that of [root](3380929). In this way Archimedes arrives at 14688/(4673½) as the ratio of the perimeter of the circumscribed polygon of 96 sides to the diameter of the circle; this is the figure which he rounds up into 3-1/7. The corresponding figure for the inscribed polygon is 6336/(2017¼), which, he says, is > 3-10/71. This example shows how little the Greeks were embarrassed in arithmetical calculations by their alphabetical system of numerals. _On Conoids and Spheroids._ The preface addressed to Dositheus shows, as we may also infer from internal evidence, that the whole of this book also was original. Archimedes first explains what his conoids and spheroids are, and then, after each description, states the main results which it is the aim of the treatise to prove. The conoids are two. The first is the _right-angled conoid_, a name adapted from the old name ("section of a right-angled cone") for a parabola; this conoid is therefore a paraboloid of revolution. The second is the _obtuse-angled conoid_, which is a hyperboloid of revolution described by the revolution of a hyperbola (a "section of an obtuse-angled cone") about its transverse axis. The spheroids are two, being the solids of revolution described by the revolution of an ellipse (a "section of an acute-angled cone") about (1) its major axis and (2) its minor axis; the first is called the "oblong" (or oblate) spheroid, the second the "flat" (or prolate) spheroid. As the volumes of oblique segments of conoids and spheroids are afterwards found in terms of the volume of the conical figure with the base of the segment as base and the vertex of the segment as vertex, and as the said base is thus an elliptic section of an oblique circular cone, Archimedes calls the conical figure with an elliptic base a "segment of a cone" as distinct from a "cone". As usual, a series of preliminary propositions is required. Archimedes first sums, in geometrical form, certain series, including the arithmetical progression, a, 2a, 3a, ... na, and the series formed by the squares of these terms (in other words the series 1², 2², 3², ... n²); these summations are required for the final addition of an indefinite number of elements of each figure, which amounts to an _integration_. Next come two properties of conics (Prop. 3), then the determination by the method of exhaustion of the area of an ellipse (Prop. 4). Three propositions follow, the first two of which (Props. 7, 8) show that the conical figure above referred to is really a segment of an oblique _circular_ cone; this is done by actually finding the circular sections. Prop. 9 gives a similar proof that each elliptic section of a conoid or spheroid is a section of a certain oblique _circular_ cylinder (with axis parallel to the axis of the segment of the conoid or spheroid cut off by the said elliptic section). Props. 11-18 show the nature of the various sections which cut off segments of each conoid and spheroid and which are circles or ellipses according as the section is perpendicular or obliquely inclined to the axis of the solid; they include also certain properties of tangent planes, etc. The real business of the treatise begins with Props. 19, 20; here it is shown how, by drawing many plane sections equidistant from one another and all parallel to the base of the segment of the solid, and describing cylinders (in general oblique) through each plane section with generators parallel to the axis of the segment and terminated by the contiguous sections on either side, we can make figures circumscribed and inscribed to the segment, made up of segments of cylinders with parallel faces and presenting the appearance of the steps of a staircase. Adding the elements of the inscribed and circumscribed figures respectively and using the method of exhaustion, Archimedes finds the volumes of the respective segments of the solids in the approved manner (Props. 21, 22 for the paraboloid, Props. 25, 26 for the hyperboloid, and Props. 27-30 for the spheroids). The results are stated in this form: (1) Any segment of a paraboloid of revolution is half as large again as the cone or segment of a cone which has the same base and axis; (2) Any segment of a hyperboloid of revolution or of a spheroid is to the cone or segment of a cone with the same base and axis in the ratio of AD + 3CA to AD + 2CA in the case of the hyperboloid, and of 3CA - AD to 2CA - AD in the case of the spheroid, where C is the centre, A the vertex of the segment, and AD the axis of the segment (supposed in the case of the spheroid to be not greater than half the spheroid). _On Spirals._ The preface addressed to Dositheus is of some length and contains, first, a tribute to the memory of Conon, and next a summary of the theorems about the sphere and the conoids and spheroids included in the above two treatises. Archimedes then passes to the spiral which, he says, presents another sort of problem, having nothing in common with the foregoing. After a definition of the spiral he enunciates the main propositions about it which are to be proved in the treatise. The spiral (now known as the Spiral of Archimedes) is defined as the locus of a point starting from a given point (called the "origin") on a given straight line and moving along the straight line at uniform speed, while the line itself revolves at uniform speed about the origin as a fixed point. Props. 1-11 are preliminary, the last two amounting to the summation of certain series required for the final addition of an indefinite number of element-areas, which again amounts to integration, in order to find the area of the figure cut off between any portion of the curve and the two radii vectores drawn to its extremities. Props. 13-20 are interesting and difficult propositions establishing the properties of tangents to the spiral. Props. 21-23 show how to inscribe and circumscribe to any portion of the spiral figures consisting of a multitude of elements which are narrow sectors of circles with the origin as centre; the area of the spiral is intermediate between the areas of the inscribed and circumscribed figures, and by the usual method of exhaustion Archimedes finds the areas required. Prop. 24 gives the area of the first complete turn of the spiral (= 1/3[pi](2[pi]a)², where the spiral is r = a[theta]), and of any portion of it up to OP where P is any point on the first turn. Props. 25, 26 deal similarly with the second turn of the spiral and with the area subtended by any arc (not being greater than a complete turn) on any turn. Prop. 27 proves the interesting property that, if R1 be the area of the first turn of the spiral bounded by the initial line, R2 the area of the ring added by the second complete turn, R3 the area of the ring added by the third turn, and so on, then R3 = 2R2, R4 = 3R2, R5 = 4R2, and so on to R_n = (n - 1)R2, while R2, = 6R1. _Quadrature of the Parabola._ The title of this work seems originally to have been _On the Section of a Right-angled Cone_ and to have been changed after the time of Apollonius, who was the first to call a parabola by that name. The preface addressed to Dositheus was evidently the first communication from Archimedes to him after the death of Conon. It begins with a feeling allusion to his lost friend, to whom the treatise was originally to have been sent. It is in this preface that Archimedes alludes to the lemma used by earlier geometers as the basis of the method of exhaustion (the Postulate of Archimedes, or the theorem of Euclid X., 1). He mentions as having been proved by means of it (1) the theorems that the areas of circles are to one another in the duplicate ratio of their diameters, and that the volumes of spheres are in the triplicate ratio of their diameters, and (2) the propositions proved by Eudoxus about the volumes of a cone and a pyramid. No one, he says, so far as he is aware, has yet tried to square the segment bounded by a straight line and a section of a right-angled cone (a parabola); but he has succeeded in proving, by means of the same lemma, that the parabolic segment is equal to four-thirds of the triangle on the same base and of equal height, and he sends the proofs, first as "investigated" by means of mechanics and secondly as "demonstrated" by geometry. The phraseology shows that here, as in the _Method_, Archimedes regarded the mechanical investigation as furnishing evidence rather than proof of the truth of the proposition, pure geometry alone furnishing the absolute proof required. The mechanical proof with the necessary preliminary propositions about the parabola (some of which are merely quoted, while two, evidently original, are proved, Props. 4, 5) extends down to Prop. 17; the geometrical proof with other auxiliary propositions completes the book (Props. 18-24). The mechanical proof recalls that of the _Method_ in some respects, but is more elaborate in that the elements of the area of the parabola to be measured are not straight lines but narrow strips. The figures inscribed and circumscribed to the segment are made up of such narrow strips and have a saw-like edge; all the elements are trapezia except two, which are triangles, one in each figure. Each trapezium (or triangle) is weighed where it is against another area hung at a fixed point of an assumed lever; thus the whole of the inscribed and circumscribed figures respectively are weighed against the sum of an indefinite number of areas all suspended from one point on the lever. The result is obtained by a real _integration_, confirmed as usual by a proof by the method of exhaustion. The geometrical proof proceeds thus. Drawing in the segment the inscribed triangle with the same base and height as the segment, Archimedes next inscribes triangles in precisely the same way in each of the segments left over, and proves that the sum of the two new triangles is ¼ of the original inscribed triangle. Again, drawing triangles inscribed in the same way in the four segments left over, he proves that their sum is ¼ of the sum of the preceding pair of triangles and therefore (¼)² of the original inscribed triangle. Proceeding thus, we have a series of areas exhausting the parabolic segment. Their sum, if we denote the first inscribed triangle by [Delta], is [Delta]{1 + ¼ + (¼)² + (¼)³ + . . . .} Archimedes proves geometrically in Prop. 23 that the sum of this infinite series is 4/3[Delta], and then confirms by _reductio ad absurdum_ the equality of the area of the parabolic segment to this area. CHAPTER V. THE SANDRECKONER. The _Sandreckoner_ deserves a place by itself. It is not mathematically very important; but it is an arithmetical curiosity which illustrates the versatility and genius of Archimedes, and it contains some precious details of the history of Greek astronomy which, coming from such a source and at first hand, possess unique authority. We will begin with the astronomical data. They are contained in the preface addressed to King Gelon of Syracuse, which begins as follows:-- "There are some, King Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again, there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the earth, including in it all the seas and the hollows of the earth filled up to a height equal to that of the highest of the mountains, would be many times further still from recognising that any number could be expressed which exceeded the multitude of the sand so taken. But I will try to show you, by means of geometrical proofs which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in size to the earth filled up in the way described, but also that of a mass equal in size to the universe. "Now you are aware that 'universe' is the name given by most astronomers to the sphere the centre of which is the centre of the earth, while the radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account, as you have heard from astronomers. But Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premises lead to the conclusion that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the centre of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a ratio to the distance of the fixed stars as the centre of the sphere bears to its surface." Here then is absolute and practically contemporary evidence that the Greeks, in the person of Aristarchus of Samos (about 310-230 B.C.), had anticipated Copernicus. By the last words quoted Aristarchus only meant to say that the size of the earth is negligible in comparison with the immensity of the universe. This, however, does not suit Archimedes's purpose, because he has to assume a definite size, however large, for the universe. Consequently he takes a liberty with Aristarchus. He says that the centre (a mathematical point) can have no ratio whatever to the surface of the sphere, and that we must therefore take Aristarchus to mean that the size of the earth is to that of the so-called "universe" as the size of the so-called "universe" is to that of the real universe in the new sense. Next, he has to assume certain dimensions for the earth, the moon and the sun, and to estimate the angle subtended at the centre of the earth by the sun's diameter; and in each case he has to exaggerate the probable figures so as to be on the safe side. While therefore (he says) some have tried to prove that the perimeter of the earth is 300,000 stadia (Eratosthenes, his contemporary, made it 252,000 stadia, say 24,662 miles, giving a diameter of about 7,850 miles), he will assume it to be ten times as great or 3,000,000 stadia. The diameter of the earth, he continues, is greater than that of the moon and that of the sun is greater than that of the earth. Of the diameter of the sun he observes that Eudoxus had declared it to be nine times that of the moon, and his own father, Phidias, had made it twelve times, while Aristarchus had tried to prove that the diameter of the sun is greater than eighteen times but less than twenty times the diameter of the moon (this was in the treatise of Aristarchus _On the Sizes and Distances of the Sun and Moon_, which is still extant, and is an admirable piece of geometry, proving rigorously, on the basis of certain assumptions, the result stated). Archimedes again intends to be on the safe side, so he takes the diameter of the sun to be thirty times that of the moon and not greater. Lastly, he says that Aristarchus discovered that the diameter of the sun appeared to be about 1/720th part of the zodiac circle, i.e. to subtend an angle of about half a degree; and he describes a simple instrument by which he himself found that the angle subtended by the diameter of the sun at the time when it had just risen was less than 1/164th part and greater than 1/200th part of a right angle. Taking this as the size of the angle subtended at the eye of the observer on the surface of the earth, he works out, by an interesting geometrical proposition, the size of the angle subtended at the centre of the earth, which he finds to be > 1/203rd part of a right angle. Consequently the diameter of the sun is greater than the side of a regular polygon of 812 sides inscribed in a great circle of the so-called "universe," and _a fortiori_ greater than the side of a regular _chiliagon_ (polygon of 1000 sides) inscribed in that circle. On these assumptions, and seeing that the perimeter of a regular chiliagon (as of any other regular polygon of more than six sides) inscribed in a circle is more than 3 times the length of the diameter of the circle, it easily follows that, while the diameter of the earth is less than 1,000,000 stadia, the diameter of the so-called "universe" is less than 10,000 times the diameter of the earth, and therefore less than 10,000,000,000 stadia. Lastly, Archimedes assumes that a quantity of sand not greater than a poppy-seed contains not more than 10,000 grains, and that the diameter of a poppy-seed is not less than 1/40th of a _dactylus_ (while a stadium is less than 10,000 _dactyli_). Archimedes is now ready to work out his calculation, but for the inadequacy of the alphabetic system of numerals to express such large numbers as are required. He, therefore, develops his remarkable terminology for expressing large numbers. The Greek has names for all numbers up to a myriad (10,000); there was, therefore, no difficulty in expressing with the ordinary numerals all numbers up to a myriad myriads (100,000,000). Let us, says Archimedes, call all these numbers numbers of the _first order_. Let the _second order_ of numbers begin with 100,000,000, and end with 100,000,000². Let 100,000,000² be the first number of the _third order_, and let this extend to 100,000,000³; and so on, to the _myriad-myriadth_ order, beginning with 100,000,000^(99,999,999) and ending with 100,000,000^(100,000,000), which for brevity we will call P. Let all the numbers of all the orders up to P form the _first period_, and let the _first order_ of the _second period_ begin with P and end with 100,000,000 P; let the _second order_ begin with this, the _third order_ with 100,000,000² P, and so on up to the _100,000,000th order_ of the _second period_, ending with 1,000,000,000^(100,000,000) P or P². The _first order_ of the _third period_ begins with P², and the _orders_ proceed as before. Continuing the series of _periods_ and _orders_ of each _period_, we finally arrive at the _100,000,000th period_ ending with P^(100,000,000). The prodigious extent of this scheme is seen when it is considered that the last number of the first period would now be represented by 1 followed by 800,000,000 ciphers, while the last number of the 100,000,000th period would require 100,000,000 times as many ciphers, i.e. 80,000 million million ciphers. As a matter of fact, Archimedes does not need, in order to express the "number of the sand," to go beyond the _eighth order_ of the _first period_. The orders of the _first period_ begin respectively with 1, 10^8, 10^16, 10^24, ... (10^8)^(99,999,999); and we can express all the numbers required in powers of 10. Since the diameter of a poppy-seed is not less than 1/40th of a dactylus, and spheres are to one another in the triplicate ratio of their diameters, a sphere of diameter 1 _dactylus_ is not greater than 64,000 poppy-seeds, and, therefore, contains not more than 64,000 × 10,000 grains of sand, and _a fortiori_ not more than 1,000,000,000, or 10^9 grains of sand. Archimedes multiplies the diameter of the sphere continually by 100, and states the corresponding number of grains of sand. A sphere of diameter 10,000 _dactyli_ and _a fortiori_ of one stadium contains less than 10^21 grains; and proceeding in this way to spheres of diameter 100 stadia, 10,000 stadia and so on, he arrives at the number of grains of sand in a sphere of diameter 10,000,000,000 stadia, which is the size of the so-called universe; the corresponding number of grains of sand is 10^51. The diameter of the real universe being 10,000 times that of the so-called universe, the final number of grains of sand in the real universe is found to be 10^63, which in Archimedes's terminology is a myriad-myriad units of the _eighth order_ of numbers. CHAPTER VI. MECHANICS. It is said that Archytas was the first to treat mechanics in a systematic way by the aid of mathematical principles; but no trace survives of any such work by him. In practical mechanics he is said to have constructed a mechanical dove which would fly, and also a rattle to amuse children and "keep them from breaking things about the house" (so says Aristotle, adding "for it is impossible for children to keep still"). In the Aristotelian _Mechanica_ we find a remark on the marvel of a great weight being moved by a small force, and the problems discussed bring in the lever in various forms as a means of doing this. We are told also that practically all movements in mechanics reduce to the lever and the principle of the lever (that the weight and the force are in inverse proportion to the distances from the point of suspension or fulcrum of the points at which they act, it being assumed that they act in directions perpendicular to the lever). But the lever is merely "referred to the circle"; the force which acts at the greater distance from the fulcrum is said to move a weight more easily because it describes a greater circle. There is, therefore, no proof here. It was reserved for Archimedes to prove the property of the lever or balance mathematically, on the basis of certain postulates precisely formulated and making no large demand on the faith of the learner. The treatise _On Plane Equilibriums_ in two books is, as the title implies, a work on statics only; and, after the principle of the lever or balance has been established in Props. 6, 7 of Book I., the rest of the treatise is devoted to finding the centre of gravity of certain figures. There is no dynamics in the work and therefore no room for the parallelogram of velocities, which is given with a fairly adequate proof in the Aristotelian _Mechanica_. Archimedes's postulates include assumptions to the following effect: (1) Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium, but the system in that case "inclines towards the weight which is at the greater distance," in other words, the action of the weight which is at the greater distance produces motion in the direction in which it acts; (2) and (3) If when weights are in equilibrium something is added to or subtracted from one of the weights, the system will "incline" towards the weight which is added to or the weight from which nothing is taken respectively; (4) and (5) If equal and similar figures be applied to one another so as to coincide throughout, their centres of gravity also coincide; if figures be unequal but similar, their centres of gravity are similarly situated with regard to the figures. The main proposition, that two magnitudes balance at distances reciprocally proportional to the magnitudes, is proved first for commensurable and then for incommensurable magnitudes. Preliminary propositions have dealt with equal magnitudes disposed at equal distances on a straight line and odd or even in number, and have shown where the centre of gravity of the whole system lies. Take first the case of commensurable magnitudes. If A, B be the weights acting at E, D on the straight line ED respectively, and ED be divided at C so that A : B = DC : CE, Archimedes has to prove that the system is in equilibrium about C. He produces ED to K, so that DK = EC, and DE to L so that EL = CD; LK is then a straight line bisected at C. Again, let H be taken on LK such that LH = 2LE or 2CD, and it follows that the remainder HK = 2DK or 2EC. Since A, B are commensurable, so are EC, CD. Let x be a common measure of EC, CD. Take a weight w such that w is the same part of A that x is of LH. It follows that w is the same part of B that x is of HK. Archimedes now divides LH, HK into parts equal to x, and A B into parts equal to w, and places the w's at the middle points of the x's respectively. All the w's are then in equilibrium about C. But all the w's acting at the several points along LH are equivalent to A acting as a whole at the point E. Similarly the w's acting at the several points on HK are equivalent to B acting at D. Therefore A, B placed at E, D respectively balance about C. Prop. 7 deduces by _reductio ad absurdum_ the same result in the case where A, B are incommensurable. Prop. 8 shows how to find the centre of gravity of the remainder of a magnitude when the centre of gravity of the whole and of a part respectively are known. Props. 9-15 find the centres of gravity of a parallelogram, a triangle and a parallel-trapezium respectively. Book II., in ten propositions, is entirely devoted to finding the centre of gravity of a parabolic segment, an elegant but difficult piece of geometrical work which is as usual confirmed by the method of exhaustion. CHAPTER VII. HYDROSTATICS. The science of hydrostatics is, even more than that of statics, the original creation of Archimedes. In hydrostatics he seems to have had no predecessors. Only one of the facts proved in his work _On Floating Bodies_, in two books, is given with a sort of proof in Aristotle. This is the proposition that the surface of a fluid at rest is that of a sphere with its centre at the centre of the earth. Archimedes founds his whole theory on two postulates, one of which comes at the beginning and the other after Prop. 7 of Book I. Postulate 1 is as follows:-- "Let us assume that a fluid has the property that, if its parts lie evenly and are continuous, the part which is less compressed is expelled by that which is more compressed, and each of its parts is compressed by the fluid above it perpendicularly, unless the fluid is shut up in something and compressed by something else." Postulate 2 is: "Let us assume that any body which is borne upwards in water is carried along the perpendicular [to the surface] which passes through the centre of gravity of the body". In Prop. 2 Archimedes proves that the surface of any fluid at rest is the surface of a sphere the centre of which is the centre of the earth. Props. 3-7 deal with the behaviour, when placed in fluids, of solids (1) just as heavy as the fluid, (2) lighter than the fluid, (3) heavier than the fluid. It is proved (Props. 5, 6) that, if the solid is lighter than the fluid, it will not be completely immersed but only so far that the weight of the solid will be equal to that of the fluid displaced, and, if it be forcibly immersed, the solid will be driven upwards by a force equal to the difference between the weight of the solid and that of the fluid displaced. If the solid is heavier than the fluid, it will, if placed in the fluid, descend to the bottom and, if weighed in the fluid, the solid will be lighter than its true weight by the weight of the fluid displaced (Prop. 7). The last-mentioned theorem naturally connects itself with the story of the crown made for Hieron. It was suspected that this was not wholly of gold but contained an admixture of silver, and Hieron put to Archimedes the problem of determining the proportions in which the metals were mixed. It was the discovery of the solution of this problem when in the bath that made Archimedes run home naked, shouting [Greek: eurêka, eurêka]. One account of the solution makes Archimedes use the proposition last quoted; but on the whole it seems more likely that the actual discovery was made by a more elementary method described by Vitruvius. Observing, as he is said to have done, that, if he stepped into the bath when it was full, a volume of water was spilt equal to the volume of his body, he thought of applying the same idea to the case of the crown and measuring the volumes of water displaced respectively (1) by the crown itself, (2) by the same weight of pure gold, and (3) by the same weight of pure silver. This gives an easy means of solution. Suppose that the weight of the crown is W, and that it contains weights w1 and w2, of gold and silver respectively. Now experiment shows (1) that the crown itself displaces a certain volume of water, V say, (2) that a weight W of gold displaces a certain other volume of water, V1 say, and (3) that a weight W of silver displaces a volume V2. From (2) it follows, by proportion, that a weight w1 of gold will displace w1/W · V1 of the fluid, and from (3) it follows that a weight w2 of silver displaces w2/W · V2 of the fluid. Hence V = w1/W · V1 + w2/W · V2; therefore WV = w1V1 + w2V2, that is, (w1 + w2)V = w1V1 + w2V2, so that w1/w2 = (V2 - V)/(V - V1), which gives the required ratio of the weights of gold and silver contained in the crown. The last two propositions of Book I. investigate the case of a segment of a sphere floating in a fluid when the base of the segment is (1) entirely above and (2) entirely below the surface of the fluid; and it is shown that the segment will in either case be in equilibrium in the position in which the axis is vertical, the equilibrium being in the first case stable. Book II. is a geometrical _tour de force_. Here, by the methods of pure geometry, Archimedes investigates the positions of rest and stability of a right segment of a paraboloid of revolution floating with its base upwards or downwards (but completely above or completely below the surface) for a number of cases differing (1) according to the relation between the length of the axis of the paraboloid and the principal parameter of the generating parabola, and (2) according to the specific gravity of the solid in relation to the fluid; where the position of rest and stability is such that the axis of the solid is not vertical, the angle at which it is inclined to the vertical is fully determined. The idea of specific gravity appears all through, though this actual term is not used. Archimedes speaks of the solid being lighter or heavier than the fluid or equally heavy with it, or when a ratio has to be expressed, he speaks of a solid the weight of which (for an equal volume) has a certain ratio to that of the fluid. BIBLIOGRAPHY. The _editio princeps_ of the works of Archimedes with the commentaries of Eutocius was brought out by Hervagius (Herwagen) at Basel in 1544. D. Rivault (Paris, 1615) gave the enunciations in Greek and the proofs in Latin somewhat retouched. The _Arenarius_ (_Sandreckoner_) and the _Dimensio circuli_ with Eutocius's commentary were edited with Latin translation and notes by Wallis in 1678 (Oxford). Torelli's monumental edition (Oxford, 1792) of the Greek text of the complete works and of the commentaries of Eutocius, with a new Latin translation, remained the standard text until recent years; it is now superseded by the definitive text with Latin translation of the complete works, Eutocius's commentaries, the fragments, scholia, etc., edited by Heiberg in three volumes (Teubner, Leipzig, first edition, 1880-1; second edition, including the newly discovered _Method_, etc., 1910-15). Of translations the following may be mentioned. The Aldine edition of 1558, 4to, contains the Latin translation by Commandinus of the _Measurement of a Circle_, _On Spirals_, _Quadrature of the Parabola_, _On Conoids and Spheroids_, _The Sandreckoner_. Isaac Barrow's version was contained in _Opera Archimedis_, _Apollonii Pergoei conicorum libri_, _Theodosii Sphoerica_, _methodo novo illustrata et demonstrata_ (London, 1675). The first French version of the works was by Peyrard in two volumes (second edition, 1808). A valuable German translation, with notes, by E. Nizze, was published at Stralsund in 1824. There is a complete edition in modern notation by T. L. Heath (_The Works of Archimedes_, Cambridge, 1897, supplemented by _The Method of Archimedes_, Cambridge, 1912). CHRONOLOGY. (APPROXIMATE IN SOME CASES.) B.C. 624-547 Thales 572-497 Pythagoras 500-428 Anaxagoras 470-400 / Hippocrates of Chios \ Hippias of Elis 470-380 Democritus 460-385 Theodorus of Cyrene 430-360 Archytas of Taras (Tarentum) 427-347 Plato 415-369 Theætetus 408-355 Eudoxus of Cnidos / Leon fl. about 350 < Menæchmus | Dinostratus \ Theudius fl. 300 Euclid 310-230 Aristarchus of Samos 287-212 Archimedes 284-203 Eratosthenes 265-190 Apollonius of Perga 
63	----	The Number "e" Below is the value of 'e' to about 100,000 places, computed on the NCSA Cray Y-MP using the Brent multiple precision routines (published as Algorithm 524 in the March 1978 issue of Transactions on Mathematical Software). The method used was to compute first the alternating series for 1/e, then to invert this result. The time to compute 1/e was about 594 seconds, and the time to invert was about 97 seconds. No special optimization was attempted on the code, other than the default vectorization that the cft77 compiler attempts to do. 2.718281828459045235360287471352662497 7572470936999595749669676277240766303535 4759457138217852516642742746639193200305 9921817413596629043572900334295260595630 7381323286279434907632338298807531952510 1901157383418793070215408914993488416750 9244761460668082264800168477411853742345 4424371075390777449920695517027618386062 6133138458300075204493382656029760673711 3200709328709127443747047230696977209310 1416928368190255151086574637721112523897 8442505695369677078544996996794686445490 5987931636889230098793127736178215424999 2295763514822082698951936680331825288693 9849646510582093923982948879332036250944 3117301238197068416140397019837679320683 2823764648042953118023287825098194558153 0175671736133206981125099618188159304169 0351598888519345807273866738589422879228 4998920868058257492796104841984443634632 4496848756023362482704197862320900216099 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2490933606999809477625379297359703775906 6145994638578378211017122446355845171941 6703447321627224432659148585957978237529 7632344291124231136860372451443876580127 1594060878788638511089680883165505046309 0061488325454528199082562388058720428439 4183468786514254137768605429107972100427 
69	----	The 32nd Mersenne Prime Found by David Slowinski In honor of Andrew Wiles' proof of the theorem known as "Fermat's Last Theorem" stated 350 years ago--but unproven until this week (February, 1993). [Fermat's thoughts on primes did not fare so well, however. A prime number is an integer, which is evenly integer divisible only by itself and 1] Took 26.562767 minutes to calculate using Maple 4.0 on a 512-MW 4 CPU Cray 2 17413590682008709732516359924590332789077936369050 70309746547355383827215620662576319147974364224616 10635130071368293660728159709054586772369049491142 93477202089620405024218873003497567737597556640892 78997985072561905731032163710847069465291689885445 30722380248547797941846968948877581472117196096521 07130138147783655536756743589920967534065512007429 20360681239094095454312630905781679734461358821352 20353524610720279709899877492086995072413691418815 60320836818547291247759862604646096021625722838827 38939891890104617219312155545932570137995324568811 85759966782043075143856381987226504677899075388614 68405916279603561174627301118740917331778061439712 23252614922823735925880038798335047860320508646260 66228106030298325285057443402167635765093406904801 96180095803280184094325223694430165424132888797257 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97	----	Flatland by Edwin A. Abbott 1884 To The Inhabitance of SPACE IN GENERAL And H.C. IN PARTICULAR This Work is Dedicated By a Humble Native of Flatland In the Hope that Even as he was Initiated into the Mysteries Of THREE DIMENSIONS Having been previously conversant With ONLY TWO So the Citizens of that Celestial Region May aspire yet higher and higher To the Secrets of FOUR FIVE or EVEN SIX Dimensions Thereby contributing To the Enlargment of THE IMAGINATION And the possible Development Of that most and excellent Gift of MODESTY Among the Superior Races Of SOLID HUMANITY *** FLATLAND PART 1 THIS WORLD SECTION 1 Of the Nature of Flatland I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space. Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows--only hard with luminous edges--and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said "my universe:" but now my mind has been opened to higher views of things. In such a country, you will perceive at once that it is impossible that there should be anything of what you call a "solid" kind; but I dare say you will suppose that we could at least distinguish by sight the Triangles, Squares, and other figures, moving about as I have described them. On the contrary, we could see nothing of the kind, not at least so as to distinguish one figure from another. Nothing was visible, nor could be visible, to us, except Straight Lines; and the necessity of this I will speedily demonstrate. Place a penny on the middle of one of your tables in Space; and leaning over it, look down upon it. It will appear a circle. But now, drawing back to the edge of the table, gradually lower your eye (thus bringing yourself more and more into the condition of the inhabitants of Flatland), and you will find the penny becoming more and more oval to your view, and at last when you have placed your eye exactly on the edge of the table (so that you are, as it were, actually a Flatlander) the penny will then have ceased to appear oval at all, and will have become, so far as you can see, a straight line. The same thing would happen if you were to treat in the same way a Triangle, or a Square, or any other figure cut out from pasteboard. As soon as you look at it with your eye on the edge of the table, you will find that it ceases to appear to you as a figure, and that it becomes in appearance a straight line. Take for example an equilateral Triangle--who represents with us a Tradesman of the respectable class. Figure 1 represents the Tradesman as you would see him while you were bending over him from above; figures 2 and 3 represent the Tradesman, as you would see him if your eye were close to the level, or all but on the level of the table; and if your eye were quite on the level of the table (and that is how we see him in Flatland) you would see nothing but a straight line. When I was in Spaceland I heard that your sailors have very similar experiences while they traverse your seas and discern some distant island or coast lying on the horizon. The far-off land may have bays, forelands, angles in and out to any number and extent; yet at a distance you see none of these (unless indeed your sun shines bright upon them revealing the projections and retirements by means of light and shade), nothing but a grey unbroken line upon the water. Well, that is just what we see when one of our triangular or other acquaintances comes towards us in Flatland. As there is neither sun with us, nor any light of such a kind as to make shadows, we have none of the helps to the sight that you have in Spaceland. If our friend comes closer to us we see his line becomes larger; if he leaves us it becomes smaller; but still he looks like a straight line; be he a Triangle, Square, Pentagon, Hexagon, Circle, what you will--a straight Line he looks and nothing else. You may perhaps ask how under these disadvantagous circumstances we are able to distinguish our friends from one another: but the answer to this very natural question will be more fitly and easily given when I come to describe the inhabitants of Flatland. For the present let me defer this subject, and say a word or two about the climate and houses in our country. SECTION 2 Of the Climate and Houses in Flatland As with you, so also with us, there are four points of the compass North, South, East, and West. There being no sun nor other heavenly bodies, it is impossible for us to determine the North in the usual way; but we have a method of our own. By a Law of Nature with us, there is a constant attraction to the South; and, although in temperate climates this is very slight--so that even a Woman in reasonable health can journey several furlongs northward without much difficulty--yet the hampering effort of the southward attraction is quite sufficient to serve as a compass in most parts of our earth. Moreover, the rain (which falls at stated intervals) coming always from the North, is an additional assistance; and in the towns we have the guidance of the houses, which of course have their side-walls running for the most part North and South, so that the roofs may keep off the rain from the North. In the country, where there are no houses, the trunks of the trees serve as some sort of guide. Altogether, we have not so much difficulty as might be expected in determining our bearings. Yet in our more temperate regions, in which the southward attraction is hardly felt, walking sometimes in a perfectly desolate plain where there have been no houses nor trees to guide me, I have been occasionally compelled to remain stationary for hours together, waiting till the rain came before continuing my journey. On the weak and aged, and especially on delicate Females, the force of attraction tells much more heavily than on the robust of the Male Sex, so that it is a point of breeding, if you meet a Lady on the street, always to give her the North side of the way--by no means an easy thing to do always at short notice when you are in rude health and in a climate where it is difficult to tell your North from your South. Windows there are none in our houses: for the light comes to us alike in our homes and out of them, by day and by night, equally at all times and in all places, whence we know not. It was in old days, with our learned men, an interesting and oft-investigate question, "What is the origin of light?" and the solution of it has been repeatedly attempted, with no other result than to crowd our lunatic asylums with the would-be solvers. Hence, after fruitless attempts to suppress such investigations indirectly by making them liable to a heavy tax, the Legislature, in comparatively recent times, absolutely prohibited them. I--alas, I alone in Flatland--know now only too well the true solution of this mysterious problem; but my knowledge cannot be made intelligible to a single one of my countrymen; and I am mocked at--I, the sole possessor of the truths of Space and of the theory of the introduction of Light from the world of three Dimensions--as if I were the maddest of the mad! But a truce to these painful digressions: let me return to our homes. The most common form for the construction of a house is five-sided or pentagonal, as in the annexed figure. The two Northern sides RO, OF, constitute the roof, and for the most part have no doors; on the East is a small door for the Women; on the West a much larger one for the Men; the South side or floor is usually doorless. Square and triangular houses are not allowed, and for this reason. The angles of a Square (and still more those of an equilateral Triangle,) being much more pointed than those of a Pentagon, and the lines of inanimate objects (such as houses) being dimmer than the lines of Men and Women, it follows that there is no little danger lest the points of a square or triangular house residence might do serious injury to an inconsiderate or perhaps absentminded traveller suddenly running against them: and therefore, as early as the eleventh century of our era, triangular houses were universally forbidden by Law, the only exceptions being fortifications, powder-magazines, barracks, and other state buildings, which is not desirable that the general public should approach without circumspection. At this period, square houses were still everywhere permitted, though discouraged by a special tax. But, about three centuries afterwards, the Law decided that in all towns containing a population above ten thousand, the angle of a Pentagon was the smallest house-angle that could be allowed consistently with the public safety. The good sense of the community has seconded the efforts of the Legislature; and now, even in the country, the pentagonal construction has superseded every other. It is only now and then in some very remote and backward agricultural district that an antiquarian may still discover a square house. SECTION 3 Concerning the Inhabitants of Flatland The greatest length or breadth of a full grown inhabitant of Flatland may be estimated at about eleven of your inches. Twelve inches may be regarded as a maximum. Our Women are Straight Lines. Our Soldiers and Lowest Class of Workmen are Triangles with two equal sides, each about eleven inches long, and a base or third side so short (often not exceeding half an inch) that they form at their vertices a very sharp and formidable angle. Indeed when their bases are of the most degraded type (not more than the eighth part of an inch in size), they can hardly be distinguished from Straight lines or Women; so extremely pointed are their vertices. With us, as with you, these Triangles are distinguished from others by being called Isosceles; and by this name I shall refer to them in the following pages. Our Middle Class consists of Equilateral or Equal-Sided Triangles. Our Professional Men and Gentlemen are Squares (to which class I myself belong) and Five-Sided Figures or Pentagons. Next above these come the Nobility, of whom there are several degrees, beginning at Six-Sided Figures, or Hexagons, and from thence rising in the number of their sides till they receive the honourable title of Polygonal, or many-Sided. Finally when the number of the sides becomes so numerous, and the sides themselves so small, that the figure cannot be distinguished from a circle, he is included in the Circular or Priestly order; and this is the highest class of all. It is a Law of Nature with us that a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility. Thus the son of a Square is a Pentagon; the son of a Pentagon, a Hexagon; and so on. But this rule applies not always to the Tradesman, and still less often to the Soldiers, and to the Workmen; who indeed can hardly be said to deserve the name of human Figures, since they have not all their sides equal. With them therefore the Law of Nature does not hold; and the son of an Isosceles (i.e. a Triangle with two sides equal) remains Isosceles still. Nevertheless, all hope is not such out, even from the Isosceles, that his posterity may ultimately rise above his degraded condition. For, after a long series of military successes, or diligent and skillful labours, it is generally found that the more intelligent among the Artisan and Soldier classes manifest a slight increase of their third side or base, and a shrinkage of the two other sides. Intermarriages (arranged by the Priests) between the sons and daughters of these more intellectual members of the lower classes generally result in an offspring approximating still more to the type of the Equal-Sided Triangle. Rarely--in proportion to the vast numbers of Isosceles births--is a genuine and certifiable Equal-Sided Triangle produced from Isosceles parents (footnote 1). Such a birth requires, as its antecedents, not only a series of carefully arranged intermarriages, but also a long-continued exercise of frugality and self-control on the part of the would-be ancestors of the coming Equilateral, and a patient, systematic, and continuous development of the Isosceles intellect through many generations. The birth of a True Equilateral Triangle from Isosceles parents is the subject of rejoicing in our country for many furlongs round. After a strict examination conducted by the Sanitary and Social Board, the infant, if certified as Regular, is with solemn ceremonial admitted into the class of Equilaterals. He is then immediately taken from his proud yet sorrowing parents and adopted by some childless Equilateral, who is bound by oath never to permit the child henceforth to enter his former home or so much as to look upon his relations again, for fear lest the freshly developed organism may, by force of unconscious imitation, fall back again into his hereditary level. The occasional emergence of an Equilateral from the ranks of his serf-born ancestors is welcomed, not only by the poor serfs themselves, as a gleam of light and hope shed upon the monotonous squalor of their existence, but also by the Aristocracy at large; for all the higher classes are well aware that these rare phenomena, while they do little or nothing to vulgarize their own privileges, serve as almost useful barrier against revolution from below. Had the acute-angled rabble been all, without exception, absolutely destitute of hope and of ambition, they might have found leaders in some of their many seditious outbreaks, so able as to render their superior numbers and strength too much even for the wisdom of the Circles. But a wise ordinance of Nature has decreed that in proportion as the working-classes increase in intelligence, knowledge, and all virtue, in that same proportion their acute angle (which makes them physically terrible) shall increase also and approximate to their comparatively harmless angle of the Equilateral Triangle. Thus, in the most brutal and formidable off the soldier class--creatures almost on a level with women in their lack of intelligence--it is found that, as they wax in the mental ability necessary to employ their tremendous penetrating power to advantage, so do they wane in the power of penetration itself. How admirable is the Law of Compensation! And how perfect a proof of the natural fitness and, I may almost say, the divine origin of the aristocratic constitution of the States of Flatland! By a judicious use of this Law of Nature, the Polygons and Circles are almost always able to stifle sedition in its very cradle, taking advantage of the irrepressible and boundless hopefulness of the human mind. Art also comes to the aid of Law and Order. It is generally found possible--by a little artificial compression or expansion on the part of the State physicians--to make some of the more intelligent leaders of a rebellion perfectly Regular, and to admit them at once into the privileged classes; a much larger number, who are still below the standard, allured by the prospect of being ultimately ennobled, are induced to enter the State Hospitals, where they are kept in honourable confinement for life; one or two alone of the most obstinate, foolish, and hopelessly irregular are led to execution. Then the wretched rabble of the Isosceles, planless and leaderless, are either transfixed without resistance by the small body of their brethren whom the Chief Circle keeps in pay for emergencies of this kind; or else more often, by means of jealousies and suspicious skillfully fomented among them by the Circular party, they are stirred to mutual warfare, and perish by one another's angles. No less than one hundred and twenty rebellions are recorded in our annals, besides minor outbreaks numbered at two hundred and thirty-five; and they have all ended thus. Footnote 1. "What need of a certificate?" a Spaceland critic may ask: "Is not the procreation of a Square Son a certificate from Nature herself, proving the Equal-sidedness of the Father?" I reply that no Lady of any position will mary an uncertified Triangle. Square offspring has sometimes resulted from a slightly Irregular Triangle; but in almost every such case the Irregularity of the first generation is visited on the third; which either fails to attain the Pentagonal rank, or relapses to the Triangular. SECTION 4 Concerning the Women If our highly pointed Triangles of the Soldier class are formidable, it may be readily inferred that far more formidable are our Women. For, if a Soldier is a wedge, a Woman is a needle; being, so to speak, ALL point, at least at the two extremities. Add to this the power of making herself practically invisible at will, and you will perceive that a Female, in Flatland, is a creature by no means to be trifled with. But here, perhaps, some of my younger Readers may ask HOW a woman in Flatland can make herself invisible. This ought, I think, to be apparent without any explanation. However, a few words will make it clear to the most unreflecting. Place a needle on the table. Then, with your eye on the level of the table, look at it side-ways, and you see the whole length of it; but look at it end-ways, and you see nothing but a point, it has become practically invisible. Just so is it with one of our Women. When her side is turned towards us, we see her as a straight line; when the end containing her eye or mouth--for with us these two organs are identical--is the part that meets our eye, then we see nothing but a highly lustrous point; but when the back is presented to our view, then--being only sub-lustrous, and, indeed, almost as dim as an inanimate object--her hinder extremity serves her as a kind of Invisible Cap. The dangers to which we are exposed from our Women must now be manifest to the meanest capacity of Spaceland. If even the angle of a respectable Triangle in the middle class is not without its dangers; if to run against a Working Man involves a gash; if collision with an Officer of the military class necessitates a serious wound; if a mere touch from the vertex of a Private Soldier brings with it danger of death;--what can it be to run against a woman, except absolute and immediate destruction? And when a Woman is invisible, or visible only as a dim sub-lustrous point, how difficult must it be, even for the most cautious, always to avoid collision! Many are the enactments made at different times in the different States of Flatland, in order to minimize this peril; and in the Southern and less temperate climates, where the force of gravitation is greater, and human beings more liable to casual and involuntary motions, the Laws concerning Women are naturally much more stringent. But a general view of the Code may be obtained from the following summary:-- 1. Every house shall have one entrance on the Eastern side, for the use of Females only; by which all females shall enter "in a becoming and respectful manner" (footnote 1) and not by the Men's or Western door. 2. No Female shall walk in any public place without continually keeping up her Peace-cry, under penalty of death. 3. Any Female, duly certified to be suffering from St. Vitus's Dance, fits, chronic cold accompanied by violent sneezing, or any disease necessitating involuntary motions, shall be instantly destroyed. In some of the States there is an additional Law forbidding Females, under penalty of death, from walking or standing in any public place without moving their backs constantly from right to left so as to indicate their presence to those behind them; others oblige a Woman, when travelling, to be followed by one of her sons, or servants, or by her husband; others confine Women altogether in their houses except during the religious festivals. But it has been found by the wisest of our Circles or Statesmen that the multiplication of restrictions on Females tends not only to the debilitation and diminution of the race, but also to the increase of domestic murders to such an extent that a State loses more than it gains by a too prohibitive Code. For whenever the temper of the Women is thus exasperated by confinement at home or hampering regulations abroad, they are apt to vent their spleen upon their husbands and children; and in the less temperate climates the whole male population of a village has been sometimes destroyed in one or two hours of a simultaneous female outbreak. Hence the Three Laws, mentioned above, suffice for the better regulated States, and may be accepted as a rough exemplification of our Female Code. After all, our principal safeguard is found, not in Legislature, but in the interests of the Women themselves. For, although they can inflict instantaneous death by a retrograde movement, yet unless they can at once disengage their stinging extremity from the struggling body of their victim, their own frail bodies are liable to be shattered. The power of Fashion is also on our side. I pointed out that in some less civilized States no female is suffered to stand in any public place without swaying her back from right to left. This practice has been universal among ladies of any pretensions to breeding in all well-governed States, as far back as the memory of Figures can reach. It is considered a disgrace to any state that legislation should have to enforce what ought to be, and is in every respectable female, a natural instinct. The rhythmical and, if I may so say, well-modulated undulation of the back in our ladies of Circular rank is envied and imitated by the wife of a common Equilateral, who can achieve nothing beyond a mere monotonous swing, like the ticking of a pendulum; and the regular tick of the Equilateral is no less admired and copied by the wife of the progressive and aspiring Isosceles, in the females of whose family no "back-motion" of any kind has become as yet a necessity of life. Hence, in every family of position and consideration, "back motion" is as prevalent as time itself; and the husbands and sons in these households enjoy immunity at least from invisible attacks. Not that it must be for a moment supposed that our Women are destitute of affection. But unfortunately the passion of the moment predominates, in the Frail Sex, over every other consideration. This is, of course, a necessity arising from their unfortunate conformation. For as they have no pretensions to an angle, being inferior in this respect to the very lowest of the Isosceles, they are consequently wholly devoid of brainpower, and have neither reflection, judgment nor forethought, and hardly any memory. Hence, in their fits of fury, they remember no claims and recognize no distinctions. I have actually known a case where a Woman has exterminated her whole household, and half an hour afterwards, when her rage was over and the fragments swept away, has asked what has become of her husband and children. Obviously then a Woman is not to be irritated as long as she is in a position where she can turn round. When you have them in their apartments--which are constructed with a view to denying them that power--you can say and do what you like; for they are then wholly impotent for mischief, and will not remember a few minutes hence the incident for which they may be at this moment threatening you with death, nor the promises which you may have found it necessary to make in order to pacify their fury. On the whole we got on pretty smoothly in our domestic relations, except in the lower strata of the Military Classes. There the want of tact and discretion on the part of the husbands produces at times indescribable disasters. Relying too much on the offensive weapons of their acute angles instead of the defensive organs of good sense and seasonable simulations, these reckless creatures too often neglect the prescribed construction of the women's apartments, or irritate their wives by ill-advised expressions out of doors, which they refuse immediately to retract. Moreover a blunt and stolid regard for literal truth indisposes them to make those lavish promises by which the more judicious Circle can in a moment pacify his consort. The result is massacre; not, however, without its advantages, as it eliminates the more brutal and troublesome of the Isosceles; and by many of our Circles the destructiveness of the Thinner Sex is regarded as one among many providential arrangements for suppressing redundant population, and nipping Revolution in the bud. Yet even in our best regulated and most approximately Circular families I cannot say that the ideal of family life is so high as with you in Spaceland. There is peace, in so far as the absence of slaughter may be called by that name, but there is necessarily little harmony of tastes or pursuits; and the cautious wisdom of the Circles has ensured safety at the cost of domestic comfort. In every Circular or Polygonal household it has been a habit from time immemorial--and now has become a kind of instinct among the women of our higher classes--that the mothers and daughters should constantly keep their eyes and mouths towards their husband and his male friends; and for a lady in a family of distinction to turn her back upon her husband would be regarded as a kind of portent, involving loss of STATUS. But, as I shall soon shew, this custom, though it has the advantage of safety, is not without disadvantages. In the house of the Working Man or respectable Tradesman--where the wife is allowed to turn her back upon her husband, while pursuing her household avocations--there are at least intervals of quiet, when the wife is neither seen nor heard, except for the humming sound of the continuous Peace-cry; but in the homes of the upper classes there is too often no peace. There the voluble mouth and bright penetrating eye are ever directed toward the Master of the household; and light itself is not more persistent than the stream of Feminine discourse. The tact and skill which suffice to avert a Woman's sting are unequal to the task of stopping a Woman's mouth; and as the wife has absolutely nothing to say, and absolutely no constraint of wit, sense, or conscience to prevent her from saying it, not a few cynics have been found to aver that they prefer the danger of the death-dealing but inaudible sting to the safe sonorousness of a Woman's other end. To my readers in Spaceland the condition of our Women may seen truly deplorable, and so indeed it is. A Male of the lowest type of the Isosceles may look forward to some improvement of his angle, and to the ultimate elevation of the whole of his degraded caste; but no Woman can entertain such hopes for her sex. "Once a Woman, always a Woman" is a Decree of Nature; and the very Laws of Evolution seem suspended in her disfavour. Yet at least we can admire the wise Prearrangement which has ordained that, as they have no hopes, so they shall have no memory to recall, and no forethought to anticipate, the miseries and humiliations which are at once a necessity of their existence and the basis of the constitution of Flatland. SECTION 5 Of our Methods of Recognizing one another You, who are blessed with shade as well as light, you, who are gifted with two eyes, endowed with a knowledge of perspective, and charmed with the enjoyment of various colours, you, who can actually SEE an angle, and contemplate the complete circumference of a Circle in the happy region of the Three Dimensions--how shall I make it clear to you the extreme difficulty which we in Flatland experience in recognizing one another's configuration? Recall what I told you above. All beings in Flatland, animate and inanimate, no matter what their form, present TO OUR VIEW the same, or nearly the same, appearance, viz. that of a straight Line. How then can one be distinguished from another, where all appear the same? The answer is threefold. The first means of recognition is the sense of hearing; which with us is far more highly developed than with you, and which enables us not only to distinguish by the voice of our personal friends, but even to discriminate between different classes, at least so far as concerns the three lowest orders, the Equilateral, the Square, and the Pentagon--for the Isosceles I take no account. But as we ascend the social scale, the process of discriminating and being discriminated by hearing increases in difficulty, partly because voices are assimilated, partly because the faculty of voice-discrimination is a plebeian virtue not much developed among the Aristocracy. And wherever there is any danger of imposture we cannot trust to this method. Amongst our lowest orders, the vocal organs are developed to a degree more than correspondent with those of hearing, so that an Isosceles can easily feign the voice of a Polygon, and, with some training, that of a Circle himself. A second method is therefore more commonly resorted to. FEELING is, among our Women and lower classes--about our upper classes I shall speak presently--the principal test of recognition, at all events between strangers, and when the question is, not as to the individual, but as to the class. What therefore "introduction" is among the higher classes in Spaceland, that the process of "feeling" is with us. "Permit me to ask you to feel and be felt by my friend Mr. So-and-so"--is still, among the more old-fashioned of our country gentlemen in districts remote from towns, the customary formula for a Flatland introduction. But in the towns, and among men of business, the words "be felt by" are omitted and the sentence is abbreviated to, "Let me ask you to feel Mr. So-and-so"; although it is assumed, of course, that the "feeling" is to be reciprocal. Among our still more modern and dashing young gentlemen--who are extremely averse to superfluous effort and supremely indifferent to the purity of their native language--the formula is still further curtailed by the use of "to feel" in a technical sense, meaning, "to recommend-for-the-purposes-of-feeling-and-being-felt"; and at this moment the "slang" of polite or fast society in the upper classes sanctions such a barbarism as "Mr. Smith, permit me to feel Mr. Jones." Let not my Reader however suppose that "feeling" is with us the tedious process that it would be with you, or that we find it necessary to feel right round all the sides of every individual before we determine the class to which he belongs. Long practice and training, begun in the schools and continued in the experience of daily life, enable us to discriminate at once by the sense of touch, between the angles of an equal-sided Triangle, Square, and Pentagon; and I need not say that the brainless vertex of an acute-angled Isosceles is obvious to the dullest touch. It is therefore not necessary, as a rule, to do more than feel a single angle of an individual; and this, once ascertained, tells us the class of the person whom we are addressing, unless indeed he belongs to the higher sections of the nobility. There the difficulty is much greater. Even a Master of Arts in our University of Wentbridge has been known to confuse a ten-sided with a twelve-sided Polygon; and there is hardly a Doctor of Science in or out of that famous University who could pretend to decide promptly and unhesitatingly between a twenty-sided and a twenty-four sided member of the Aristocracy. Those of my readers who recall the extracts I gave above from the Legislative code concerning Women, will readily perceive that the process of introduction by contact requires some care and discretion. Otherwise the angles might inflict on the unwary Feeling irreparable injury. It is essential for the safety of the Feeler that the Felt should stand perfectly still. A start, a fidgety shifting of the position, yes, even a violent sneeze, has been known before now to prove fatal to the incautious, and to nip in the bud many a promising friendship. Especially is this true among the lower classes of the Triangles. With them, the eye is situated so far from their vertex that they can scarcely take cognizance of what goes on at that extremity of their frame. They are, moreover, of a rough coarse nature, not sensitive to the delicate touch of the highly organized Polygon. What wonder then if an involuntary toss of the head has ere now deprived the State of a valuable life! I have heard that my excellent Grandfather--one of the least irregular of his unhappy Isosceles class, who indeed obtained, shortly before his decease, four out of seven votes from the Sanitary and Social Board for passing him into the class of the Equal-sided--often deplored, with a tear in his venerable eye, a miscarriage of this kind, which had occurred to his great-great-great-Grandfather, a respectable Working Man with an angle or brain of 59 degrees 30 minutes. According to his account, my unfortunately Ancestor, being afflicted with rheumatism, and in the act of being felt by a Polygon, by one sudden start accidentally transfixed the Great Man through the diagonal and thereby, partly in consequence of his long imprisonment and degradation, and partly because of the moral shock which pervaded the whole of my Ancestor's relations, threw back our family a degree and a half in their ascent towards better things. The result was that in the next generation the family brain was registered at only 58 degrees, and not till the lapse of five generations was the lost ground recovered, the full 60 degrees attained, and the Ascent from the Isosceles finally achieved. And all this series of calamities from one little accident in the process of Feeling. As this point I think I hear some of my better educated readers exclaim, "How could you in Flatland know anything about angles and degrees, or minutes? We SEE an angle, because we, in the region of Space, can see two straight lines inclined to one another; but you, who can see nothing but on straight line at a time, or at all events only a number of bits of straight lines all in one straight line,--how can you ever discern an angle, and much less register angles of different sizes?" I answer that though we cannot SEE angles, we can INFER them, and this with great precision. Our sense of touch, stimulated by necessity, and developed by long training, enables us to distinguish angles far more accurately than your sense of sight, when unaided by a rule or measure of angles. Nor must I omit to explain that we have great natural helps. It is with us a Law of Nature that the brain of the Isosceles class shall begin at half a degree, or thirty minutes, and shall increase (if it increases at all) by half a degree in every generation until the goal of 60 degrees is reached, when the condition of serfdom is quitted, and the freeman enters the class of Regulars. Consequently, Nature herself supplies us with an ascending scale or Alphabet of angles for half a degree up to 60 degrees, Specimen of which are placed in every Elementary School throughout the land. Owing to occasional retrogressions, to still more frequent moral and intellectual stagnation, and to the extraordinary fecundity of the Criminal and Vagabond classes, there is always a vast superfluity of individuals of the half degree and single degree class, and a fair abundance of Specimens up to 10 degrees. These are absolutely destitute of civil rights; and a great number of them, not having even intelligence enough for the purposes of warfare, are devoted by the States to the service of education. Fettered immovably so as to remove all possibility of danger, they are placed in the classrooms of our Infant Schools, and there they are utilized by the Board of Education for the purpose of imparting to the offspring of the Middle Classes the tact and intelligence which these wretched creatures themselves are utterly devoid. In some States the Specimens are occasionally fed and suffered to exist for several years; but in the more temperate and better regulated regions, it is found in the long run more advantageous for the educational interests of the young, to dispense with food, and to renew the Specimens every month--which is about the average duration of the foodless existence of the Criminal class. In the cheaper schools, what is gained by the longer existence of the Specimen is lost, partly in the expenditure for food, and partly in the diminished accuracy of the angles, which are impaired after a few weeks of constant "feeling." Nor must we forget to add, in enumerating the advantages of the more expensive system, that it tends, though slightly yet perceptibly, to the diminution of the redundant Isosceles population--an object which every statesman in Flatland constantly keeps in view. On the whole therefore--although I am not ignorant that, in many popularly elected School Boards, there is a reaction in favour of "the cheap system" as it is called--I am myself disposed to think that this is one of the many cases in which expense is the truest economy. But I must not allow questions of School Board politics to divert me from my subject. Enough has been said, I trust, to shew that Recognition by Feeling is not so tedious or indecisive a process as might have been supposed; and it is obviously more trustworthy than Recognition by hearing. Still there remains, as has been pointed out above, the objection that this method is not without danger. For this reason many in the Middle and Lower classes, and all without exception in the Polygonal and Circular orders, prefer a third method, the description of which shall be reserved for the next section. SECTION 6 Of Recognition by Sight I am about to appear very inconsistent. In the previous sections I have said that all figures in Flatland present the appearance of a straight line; and it was added or implied, that it is consequently impossible to distinguish by the visual organ between individuals of different classes: yet now I am about to explain to my Spaceland critics how we are able to recognize one another by the sense of sight. If however the Reader will take the trouble to refer to the passage in which Recognition by Feeling is stated to be universal, he will find this qualification--"among the lower classes." It is only among the higher classes and in our more temperate climates that Sight Recognition is practised. That this power exists in any regions and for any classes is the result of Fog; which prevails during the greater part of the year in all parts save the torrid zones. That which is with you in Spaceland an unmixed evil, blotting out the landscape, depressing the spirits, and enfeebling the health, is by us recognized as a blessing scarcely inferior to air itself, and as the Nurse of arts and Parent of sciences. But let me explain my meaning, without further eulogies on this beneficent Element. If Fog were non-existent, all lines would appear equally and indistinguishably clear; and this is actually the case in those unhappy countries in which the atmosphere is perfectly dry and transparent. But wherever there is a rich supply of Fog, objects that are at a distance, say of three feet, are appreciably dimmer than those at the distance of two feet eleven inches; and the result is that by careful and constant experimental observation of comparative dimness and clearness, we are enabled to infer with great exactness the configuration of the object observed. An instance will do more than a volume of generalities to make my meaning clear. Suppose I see two individuals approaching whose rank I wish to ascertain. They are, we will suppose, a Merchant and a Physician, or in other words, an Equilateral Triangle and a Pentagon; how am I to distinguish them? It will be obvious, to every child in Spaceland who has touched the threshold of Geometrical Studies, that, if I can bring my eye so that its glance may bisect an angle (A) of the approaching stranger, my view will lie as it were evenly between the two sides that are next to me (viz. CA and AB), so that I shall contemplate the two impartially, and both will appear of the same size. Now in the case of (1) the Merchant, what shall I see? I shall see a straight line DAE, in which the middle point (A) will be very bright because it is nearest to me; but on either side the line will shade away RAPIDLY TO DIMNESS, because the sides AC and AB RECEDE RAPIDLY INTO THE FOG and what appear to me as the Merchant's extremities, viz. D and E, will be VERY DIM INDEED. On the other hand in the case of (2) the Physician, though I shall here also see a line (D'A'E') with a bright centre (A'), yet it will shade away LESS RAPIDLY to dimness, because the sides (A'C', A'B') RECEDE LESS RAPIDLY INTO THE FOG: and what appear to me the Physician's extremities, viz. D' and E', will not be NOT SO DIM as the extremities of the Merchant. The Reader will probably understand from these two instances how--after a very long training supplemented by constant experience--it is possible for the well-educated classes among us to discriminate with fair accuracy between the middle and lowest orders, by the sense of sight. If my Spaceland Patrons have grasped this general conception, so far as to conceive the possibility of it and not to reject my account as altogether incredible--I shall have attained all I can reasonably expect. Were I to attempt further details I should only perplex. Yet for the sake of the young and inexperienced, who may perchance infer--from the two simple instances I have given above, of the manner in which I should recognize my Father and my Sons--that Recognition by sight is an easy affair, it may be needful to point out that in actual life most of the problems of Sight Recognition are far more subtle and complex. If for example, when my Father, the Triangle, approaches me, he happens to present his side to me instead of his angle, then, until I have asked him to rotate, or until I have edged my eye around him, I am for the moment doubtful whether he may not be a Straight Line, or, in other words, a Woman. Again, when I am in the company of one of my two hexagonal Grandsons, contemplating one of his sides (AB) full front, it will be evident from the accompanying diagram that I shall see one whole line (AB) in comparative brightness (shading off hardly at all at the ends) and two smaller lines (CA and BD) dim throughout and shading away into greater dimness towards the extremities C and D. But I must not give way to the temptation of enlarging on these topics. The meanest mathematician in Spaceland will readily believe me when I assert that the problems of life, which present themselves to the well-educated--when they are themselves in motion, rotating, advancing or retreating, and at the same time attempting to discriminate by the sense of sight between a number of Polygons of high rank moving in different directions, as for example in a ball-room or conversazione--must be of a nature to task the angularity of the most intellectual, and amply justify the rich endowments of the Learned Professors of Geometry, both Static and Kinetic, in the illustrious University of Wentbridge, where the Science and Art of Sight Recognition are regularly taught to large classes of the ELITE of the States. It is only a few of the scions of our noblest and wealthiest houses, who are able to give the time and money necessary for the thorough prosecution of this noble and valuable Art. Even to me, a Mathematician of no mean standing, and the Grandfather of two most hopeful and perfectly regular Hexagons, to find myself in the midst of a crowd of rotating Polygons of the higher classes, is occasionally very perplexing. And of course to a common Tradesman, or Serf, such a sight is almost as unintelligible as it would be to you, my Reader, were you suddenly transported to my country. In such a crowd you could see on all sides of you nothing but a Line, apparently straight, but of which the parts would vary irregularly and perpetually in brightness or dimness. Even if you had completed your third year in the Pentagonal and Hexagonal classes in the University, and were perfect in the theory of the subject, you would still find there was need of many years of experience, before you could move in a fashionable crowd without jostling against your betters, whom it is against etiquette to ask to "feel," and who, by their superior culture and breeding, know all about your movements, while you know very little or nothing about theirs. In a word, to comport oneself with perfect propriety in Polygonal society, one ought to be a Polygon oneself. Such at least is the painful teaching of my experience. It is astonishing how much the Art--or I may almost call it instinct--of Sight Recognition is developed by the habitual practice of it and by the avoidance of the custom of "Feeling." Just as, with you, the deaf and dumb, if once allowed to gesticulate and to use the hand-alphabet, will never acquire the more difficult but far more valuable art of lip-speech and lip-reading, so it is with us as regards "Seeing" and "Feeling." None who in early life resort to "Feeling" will ever learn "Seeing" in perfection. For this reason, among our Higher Classes, "Feeling" is discouraged or absolutely forbidden. From the cradle their children, instead of going to the Public Elementary schools (where the art of Feeling is taught,) are sent to higher Seminaries of an exclusive character; and at our illustrious University, to "feel" is regarded as a most serious fault, involving Rustication for the first offence, and Expulsion for the second. But among the lower classes the art of Sight Recognition is regarded as an unattainable luxury. A common Tradesman cannot afford to let his son spend a third of his life in abstract studies. The children of the poor are therefore allowed to "feel" from their earliest years, and they gain thereby a precocity and an early vivacity which contrast at first most favourably with the inert, undeveloped, and listless behaviour of the half-instructed youths of the Polygonal class; but when the latter have at last completed their University course, and are prepared to put their theory into practice, the change that comes over them may almost be described as a new birth, and in every art, science, and social pursuit they rapidly overtake and distance their Triangular competitors. Only a few of the Polygonal Class fail to pass the Final Test or Leaving Examination at the University. The condition of the unsuccessful minority is truly pitiable. Rejected from the higher class, they are also despised by the lower. They have neither the matured and systematically trained powers of the Polygonal Bachelors and Masters of Arts, nor yet the native precocity and mercurial versatility of the youthful Tradesman. The professions, the public services, are closed against them, and though in most States they are not actually debarred from marriage, yet they have the greatest difficulty in forming suitable alliances, as experience shews that the offspring of such unfortunate and ill-endowed parents is generally itself unfortunate, if not positively Irregular. It is from these specimens of the refuse of our Nobility that the great Tumults and Seditions of past ages have generally derived their leaders; and so great is the mischief thence arising that an increasing minority of our more progressive Statesmen are of opinion that true mercy would dictate their entire suppression, by enacting that all who fail to pass the Final Examination of the University should be either imprisoned for life, or extinguished by a painless death. But I find myself digressing into the subject of Irregularities, a matter of such vital interest that it demands a separate section. SECTION 7 Concerning Irregular Figures Throughout the previous pages I have been assuming--what perhaps should have been laid down at the beginning as a distinct and fundamental proposition--that every human being in Flatland is a Regular Figure, that is to say of regular construction. By this I mean that a Woman must not only be a line, but a straight line; that an Artisan or Soldier must have two of his sides equal; that Tradesmen must have three sides equal; Lawyers (of which class I am a humble member), four sides equal, and, generally, that in every Polygon, all the sides must be equal. The sizes of the sides would of course depend upon the age of the individual. A Female at birth would be about an inch long, while a tall adult Woman might extend to a foot. As to the Males of every class, it may be roughly said that the length of an adult's size, when added together, is two feet or a little more. But the size of our sides is not under consideration. I am speaking of the EQUALITY of sides, and it does not need much reflection to see that the whole of the social life in Flatland rests upon the fundamental fact that Nature wills all Figures to have their sides equal. If our sides were unequal our angles might be unequal. Instead of its being sufficient to feel, or estimate by sight, a single angle in order to determine the form of an individual, it would be necessary to ascertain each angle by the experiment of Feeling. But life would be too short for such a tedious groping. The whole science and art of Sight Recognition would at once perish; Feeling, so far as it is an art, would not long survive; intercourse would become perilous or impossible; there would be an end to all confidence, all forethought; no one would be safe in making the most simple social arrangements; in a word, civilization might relapse into barbarism. Am I going too fast to carry my Readers with me to these obvious conclusions? Surely a moment's reflection, and a single instance from common life, must convince every one that our social system is based upon Regularity, or Equality of Angles. You meet, for example, two or three Tradesmen in the street, whom your recognize at once to be Tradesman by a glance at their angles and rapidly bedimmed sides, and you ask them to step into your house to lunch. This you do at present with perfect confidence, because everyone knows to an inch or two the area occupied by an adult Triangle: but imagine that your Tradesman drags behind his regular and respectable vertex, a parallelogram of twelve or thirteen inches in diagonal:--what are you to do with such a monster sticking fast in your house door? But I am insulting the intelligence of my Readers by accumulating details which must be patent to everyone who enjoys the advantages of a Residence in Spaceland. Obviously the measurements of a single angle would no longer be sufficient under such portentous circumstances; one's whole life would be taken up in feeling or surveying the perimeter of one's acquaintances. Already the difficulties of avoiding a collision in a crowd are enough to tax the sagacity of even a well-educated Square; but if no one could calculate the Regularity of a single figure in the company, all would be chaos and confusion, and the slightest panic would cause serious injuries, or--if there happened to be any Women or Soldiers present--perhaps considerable loss of life. Expediency therefore concurs with Nature in stamping the seal of its approval upon Regularity of conformation: nor has the Law been backward in seconding their efforts. "Irregularity of Figure" means with us the same as, or more than, a combination of moral obliquity and criminality with you, and is treated accordingly. There are not wanting, it is true, some promulgators of paradoxes who maintain that there is no necessary connection between geometrical and moral Irregularity. "The Irregular," they say, "is from his birth scouted by his own parents, derided by his brothers and sisters, neglected by the domestics, scorned and suspected by society, and excluded from all posts of responsibility, trust, and useful activity. His every movement is jealously watched by the police till he comes of age and presents himself for inspection; then he is either destroyed, if he is found to exceed the fixed margin of deviation, at an uninteresting occupation for a miserable stipend; obliged to live and board at the office, and to take even his vacation under close supervision; what wonder that human nature, even in the best and purest, is embittered and perverted by such surroundings!" All this very plausible reasoning does not convince me, as it has not convinced the wisest of our Statesmen, that our ancestors erred in laying it down as an axiom of policy that the toleration of Irregularity is incompatible with the safety of the State. Doubtless, the life of an Irregular is hard; but the interests of the Greater Number require that it shall be hard. If a man with a triangular front and a polygonal back were allowed to exist and to propagate a still more Irregular posterity, what would become of the arts of life? Are the houses and doors and churches in Flatland to be altered in order to accommodate such monsters? Are our ticket-collectors to be required to measure every man's perimeter before they allow him to enter a theatre, or to take his place in a lecture room? Is an Irregular to be exempted from the militia? And if not, how is he to be prevented from carrying desolation into the ranks of his comrades? Again, what irresistible temptations to fraudulent impostures must needs beset such a creature! How easy for him to enter a shop with his polygonal front foremost, and to order goods to any extent from a confiding tradesman! Let the advocates of a falsely called Philanthropy plead as they may for the abrogation of the Irregular Penal Laws, I for my part have never known an Irregular who was not also what Nature evidently intended him to be--a hypocrite, a misanthropist, and, up to the limits of his power, a perpetrator of all manner of mischief. Not that I should be disposed to recommend (at present) the extreme measures adopted by some States, where an infant whose angle deviates by half a degree from the correct angularity is summarily destroyed at birth. Some of our highest and ablest men, men of real genius, have during their earliest days laboured under deviations as great as, or even greater than forty-five minutes: and the loss of their precious lives would have been an irreparable injury to the State. The art of healing also has achieved some of its most glorious triumphs in the compressions, extensions, trepannings, colligations, and other surgical or diaetetic operations by which Irregularity has been partly or wholly cured. Advocating therefore a VIA MEDIA, I would lay down no fixed or absolute line of demarcation; but at the period when the frame is just beginning to set, and when the Medical Board has reported that recovery is improbably, I would suggest that the Irregular offspring be painlessly and mercifully consumed. SECTION 8 Of the Ancient Practice of Painting If my Readers have followed me with any attention up to this point, they will not be surprised to hear that life is somewhat dull in Flatland. I do not, of course, mean that there are not battles, conspiracies, tumults, factions, and all those other phenomena which are supposed to make History interesting; nor would I deny that the strange mixture of the problems of life and the problems of Mathematics, continually inducing conjecture and giving an opportunity of immediate verification, imparts to our existence a zest which you in Spaceland can hardly comprehend. I speak now from the aesthetic and artistic point of view when I say that life with us is dull; aesthetically and artistically, very dull indeed. How can it be otherwise, when all one's prospect, all one's landscapes, historical pieces, portraits, flowers, still life, are nothing but a single line, with no varieties except degrees of brightness and obscurity? It was not always thus. Colour, if Tradition speaks the truth, once for the space of half a dozen centuries or more, threw a transient splendour over the lives of our ancestors in the remotest ages. Some private individual--a Pentagon whose name is variously reported--having casually discovered the constituents of the simpler colours and a rudimentary method of painting, is said to have begun by decorating first his house, then his slaves, then his Father, his Sons, and Grandsons, lastly himself. The convenience as well as the beauty of the results commended themselves to all. Wherever Chromatistes,--for by that name the most trustworthy authorities concur in calling him,--turned his variegated frame, there he at once excited attention, and attracted respect. No one now needed to "feel" him; no one mistook his front for his back; all his movements were readily ascertained by his neighbours without the slightest strain on their powers of calculation; no one jostled him, or failed to make way for him; his voice was saved the labour of that exhausting utterance by which we colourless Squares and Pentagons are often forced to proclaim our individuality when we move amid a crowd of ignorant Isosceles. The fashion spread like wildfire. Before a week was over, every Square and Triangle in the district had copied the example of Chromatistes, and only a few of the more conservative Pentagons still held out. A month or two found even the Dodecagons infected with the innovation. A year had not elapsed before the habit had spread to all but the very highest of the Nobility. Needless to say, the custom soon made its way from the district of Chromatistes to surrounding regions; and within two generations no one in all Flatland was colourless except the Women and the Priests. Here Nature herself appeared to erect a barrier, and to plead against extending the innovations to these two classes. Many-sidedness was almost essential as a pretext for the Innovators. "Distinction of sides is intended by Nature to imply distinction of colours"--such was the sophism which in those days flew from mouth to mouth, converting whole towns at a time to a new culture. But manifestly to our Priests and Women this adage did not apply. The latter had only one side, and therefore--plurally and pedantically speaking--NO SIDES. The former--if at least they would assert their claim to be readily and truly Circles, and not mere high-class Polygons, with an infinitely large number of infinitesimally small sides--were in the habit of boasting (what Women confessed and deplored) that they also had no sides, being blessed with a perimeter of only one line, or, in other words, a Circumference. Hence it came to pass that these two Classes could see no force in the so-called axiom about "Distinction of Sides implying Distinction of Colour;" and when all others had succumbed to the fascinations of corporal decoration, the Priests and the Women alone still remained pure from the pollution of paint. Immoral, licentious, anarchical, unscientific--call them by what names you will--yet, from an aesthetic point of view, those ancient days of the Colour Revolt were the glorious childhood of Art in Flatland--a childhood, alas, that never ripened into manhood, nor even reached the blossom of youth. To live then in itself a delight, because living implied seeing. Even at a small party, the company was a pleasure to behold; the richly varied hues of the assembly in a church or theatre are said to have more than once proved too distracting from our greatest teachers and actors; but most ravishing of all is said to have been the unspeakable magnificence of a military review. The sight of a line of battle of twenty thousand Isosceles suddenly facing about, and exchanging the sombre black of their bases for the orange of the two sides including their acute angle; the militia of the Equilateral Triangles tricoloured in red, white, and blue; the mauve, ultra-marine, gamboge, and burnt umber of the Square artillerymen rapidly rotating near their vermillion guns; the dashing and flashing of the five-coloured and six-coloured Pentagons and Hexagons careering across the field in their offices of surgeons, geometricians and aides-de-camp--all these may well have been sufficient to render credible the famous story how an illustrious Circle, overcome by the artistic beauty of the forces under his command, threw aside his marshal's baton and his royal crown, exclaiming that he henceforth exchanged them for the artist's pencil. How great and glorious the sensuous development of these days must have been is in part indicated by the very language and vocabulary of the period. The commonest utterances of the commonest citizens in the time of the Colour Revolt seem to have been suffused with a richer tinge of word or thought; and to that era we are even now indebted for our finest poetry and for whatever rhythm still remains in the more scientific utterance of those modern days. SECTION 9 Of the Universal Colour Bill But meanwhile the intellectual Arts were fast decaying. The Art of Sight Recognition, being no longer needed, was no longer practised; and the studies of Geometry, Statics, Kinetics, and other kindred subjects, came soon to be considered superfluous, and fell into disrespect and neglect even at our University. The inferior Art of Feeling speedily experienced the same fate at our Elementary Schools. Then the Isosceles classes, asserting that the Specimens were no longer used nor needed, and refusing to pay the customary tribute from the Criminal classes to the service of Education, waxed daily more numerous and more insolent on the strength of their immunity from the old burden which had formerly exercised the twofold wholesome effect of at once taming their brutal nature and thinning their excessive numbers. Year by year the Soldiers and Artisans began more vehemently to assert--and with increasing truth--that there was no great difference between them and the very highest class of Polygons, now that they were raised to an equality with the latter, and enabled to grapple with all the difficulties and solve all the problems of life, whether Statical or Kinetical, by the simple process of Colour Recognition. Not content with the natural neglect into which Sight Recognition was falling, they began boldly to demand the legal prohibition of all "monopolizing and aristocratic Arts" and the consequent abolition of all endowments for the studies of Sight Recognition, Mathematics, and Feeling. Soon, they began to insist that inasmuch as Colour, which was a second Nature, had destroyed the need of aristocratic distinctions, the Law should follow in the same path, and that henceforth all individuals and all classes should be recognized as absolutely equal and entitled to equal rights. Finding the higher Orders wavering and undecided, the leaders of the Revolution advanced still further in their requirements, and at last demanded that all classes alike, the Priests and the Women not excepted, should do homage to Colour by submitting to be painted. When it was objected that Priests and Women had no sides, they retorted that Nature and Expediency concurred in dictating that the front half of every human being (that is to say, the half containing his eye and mouth) should be distinguishable from his hinder half. They therefore brought before a general and extraordinary Assembly of all the States of Flatland a Bill proposing that in every Woman the half containing the eye and mouth should be coloured red, and the other half green. The Priests were to be painted in the same way, red being applied to that semicircle in which the eye and mouth formed the middle point; while the other or hinder semicircle was to be coloured green. There was no little cunning in this proposal, which indeed emanated not from any Isosceles--for no being so degraded would have angularity enough to appreciate, much less to devise, such a model of state-craft--but from an Irregular Circle who, instead of being destroyed in his childhood, was reserved by a foolish indulgence to bring desolation on his country and destruction on myriads of followers. On the one hand the proposition was calculated to bring the Women in all classes over to the side of the Chromatic Innovation. For by assigning to the Women the same two colours as were assigned to the Priests, the Revolutionists thereby ensured that, in certain positions, every Woman would appear as a Priest, and be treated with corresponding respect and deference--a prospect that could not fail to attract the Female Sex in a mass. But by some of my Readers the possibility of the identical appearance of Priests and Women, under a new Legislation, may not be recognized; if so, a word or two will make it obvious. Imagine a woman duly decorated, according to the new Code; with the front half (i.e., the half containing the eye and mouth) red, and with the hinder half green. Look at her from one side. Obviously you will see a straight line, HALF RED, HALF GREEN. Now imagine a Priest, whose mouth is at M, and whose front semicircle (AMB) is consequently coloured red, while his hinder semicircle is green; so that the diameter AB divides the green from the red. If you contemplate the Great Man so as to have your eye in the same straight line as his dividing diameter (AB), what you will see will be a straight line (CBD), of which ONE HALF (CB) WILL BE RED, AND THE OTHER (BD) GREEN. The whole line (CD) will be rather shorter perhaps than that of a full-sized Woman, and will shade off more rapidly towards its extremities; but the identity of the colours would give you an immediate impression of identity in Class, making you neglectful of other details. Bear in mind the decay of Sight Recognition which threatened society at the time of the Colour revolt; add too the certainty that Woman would speedily learn to shade off their extremities so as to imitate the Circles; it must then be surely obvious to you, my dear Reader, that the Colour Bill placed us under a great danger of confounding a Priest with a young Woman. How attractive this prospect must have been to the Frail Sex may readily be imagined. They anticipated with delight the confusion that would ensue. At home they might hear political and ecclesiastical secrets intended not for them but for their husbands and brothers, and might even issue some commands in the name of a priestly Circle; out of doors the striking combination of red and green without addition of any other colours, would be sure to lead the common people into endless mistakes, and the Woman would gain whatever the Circles lost, in the deference of the passers by. As for the scandal that would befall the Circular Class if the frivolous and unseemly conduct of the Women were imputed to them, and as to the consequent subversion of the Constitution, the Female Sex could not be expected to give a thought to these considerations. Even in the households of the Circles, the Women were all in favour of the Universal Colour Bill. The second object aimed at by the Bill was the gradual demoralization of the Circles themselves. In the general intellectual decay they still preserved their pristine clearness and strength of understanding. From their earliest childhood, familiarized in their Circular households with the total absence of Colour, the Nobles alone preserved the Sacred Art of Sight Recognition, with all the advantages that result from that admirable training of the intellect. Hence, up to the date of the introduction of the Universal Colour Bill, the Circles had not only held their own, but even increased their lead of the other classes by abstinence from the popular fashion. Now therefore the artful Irregular whom I described above as the real author of this diabolical Bill, determined at one blow to lower the status of the Hierarchy by forcing them to submit to the pollution of Colour, and at the same time to destroy their domestic opportunities of training in the Art of Sight Recognition, so as to enfeeble their intellects by depriving them of their pure and colourless homes. Once subjected to the chromatic taint, every parental and every childish Circle would demoralize each other. Only in discerning between the Father and the Mother would the Circular infant find problems for the exercise of his understanding--problems too often likely to be corrupted by maternal impostures with the result of shaking the child's faith in all logical conclusions. Thus by degrees the intellectual lustre of the Priestly Order would wane, and the road would then lie open for a total destruction of all Aristocratic Legislature and for the subversion of our Privileged Classes. SECTION 10 Of the Suppression of the Chromatic Sedition The agitation for the Universal Colour Bill continued for three years; and up to the last moment of that period it seemed as though Anarchy were destined to triumph. A whole army of Polygons, who turned out to fight as private soldiers, was utterly annihilated by a superior force of Isosceles Triangles--the Squares and Pentagons meanwhile remaining neutral. Worse than all, some of the ablest Circles fell a prey to conjugal fury. Infuriated by political animosity, the wives in many a noble household wearied their lords with prayers to give up their opposition to the Colour Bill; and some, finding their entreaties fruitless, fell on and slaughtered their innocent children and husband, perishing themselves in the act of carnage. It is recorded that during that triennial agitation no less than twenty-three Circles perished in domestic discord. Great indeed was the peril. It seemed as though the Priests had no choice between submission and extermination; when suddenly the course of events was completely changed by one of those picturesque incidents which Statesmen ought never to neglect, often to anticipate, and sometimes perhaps to originate, because of the absurdly disproportionate power with which they appeal to the sympathies of the populace. It happened that an Isosceles of a low type, with a brain little if at all above four degrees--accidentally dabbling in the colours of some Tradesman whose shop he had plundered--painted himself, or caused himself to be painted (for the story varies) with the twelve colours of a Dodecagon. Going into the Market Place he accosted in a feigned voice a maiden, the orphan daughter of a noble Polygon, whose affection in former days he had sought in vain; and by a series of deceptions--aided, on the one side, by a string of lucky accidents too long to relate, and, on the other, by an almost inconceivable fatuity and neglect of ordinary precautions on the part of the relations of the bride--he succeeded in consummating the marriage. The unhappy girl committed suicide on discovering the fraud to which she had been subjected. When the news of this catastrophe spread from State to State the minds of the Women were violently agitated. Sympathy with the miserable victim and anticipations of similar deceptions for themselves, their sisters, and their daughters, made them now regard the Colour Bill in an entirely new aspect. Not a few openly avowed themselves converted to antagonism; the rest needed only a slight stimulus to make a similar avowal. Seizing this favourable opportunity, the Circles hastily convened an extraordinary Assembly of the States; and besides the usual guard of Convicts, they secured the attendance of a large number of reactionary Women. Amidst an unprecedented concourse, the Chief Circle of those days--by name Pantocyclus--arose to find himself hissed and hooted by a hundred and twenty thousand Isosceles. But he secured silence by declaring that henceforth the Circles would enter on a policy of Concession; yielding to the wishes of the majority, they would accept the Colour Bill. The uproar being at once converted to applause, he invited Chromatistes, the leader of the Sedition, into the centre of the hall, to receive in the name of his followers the submission of the Hierarchy. Then followed a speech, a masterpiece of rhetoric, which occupied nearly a day in the delivery, and to which no summary can do justice. With a grave appearance of impartiality he declared that as they were now finally committing themselves to Reform or Innovation, it was desirable that they should take one last view of the perimeter of the whole subject, its defects as well as its advantages. Gradually introduction the mention of the dangers to the Tradesmen, the Professional Classes and the Gentlemen, he silenced the rising murmurs of the Isosceles by reminding them that, in spite of all these defects, he was willing to accept the Bill if it was approved by the majority. But it was manifest that all, except the Isosceles, were moved by his words and were either neutral or averse to the Bill. Turning now to the Workmen he asserted that their interests must not be neglected, and that, if they intended to accept the Colour Bill, they ought at least to do so with full view of the consequences. Many of them, he said, were on the point of being admitted to the class of the Regular Triangles; others anticipated for their children a distinction they could not hope for themselves. That honourable ambition would not have to be sacrificed. With the universal adoption of Colour, all distinctions would cease; Regularity would be confused with Irregularity; development would give place to retrogression; the Workman would in a few generations be degraded to the level of the Military, or even the Convict Class; political power would be in the hands of the greatest number, that is to say the Criminal Classes, who were already more numerous than the Workmen, and would soon out-number all the other Classes put together when the usual Compensative Laws of Nature were violated. A subdued murmur of assent ran through the ranks of the Artisans, and Chromatistes, in alarm, attempted to step forward and address them. But he found himself encompassed with guards and forced to remain silent while the Chief Circle in a few impassioned words made a final appeal to the Women, exclaiming that, if the Colour Bill passed, no marriage would henceforth be safe, no woman's honour secure; fraud, deception, hypocrisy would pervade every household; domestic bliss would share the fate of the Constitution and pass to speedy perdition. "Sooner than this," he cried, "come death." At these words, which were the preconcerted signal for action, the Isosceles Convicts fell on and transfixed the wretched Chromatistes; the Regular Classes, opening their ranks, made way for a band of Women who, under direction of the Circles, moved back foremost, invisibly and unerringly upon the unconscious soldiers; the Artisans, imitating the example of their betters, also opened their ranks. Meantime bands of Convicts occupied every entrance with an impenetrable phalanx. The battle, or rather carnage, was of short duration. Under the skillful generalship of the Circles almost every Woman's charge was fatal and very many extracted their sting uninjured, ready for a second slaughter. But no second blow was needed; the rabble of the Isosceles did the rest of the business for themselves. Surprised, leader-less, attacked in front by invisible foes, and finding egress cut off by the Convicts behind them, they at once--after their manner--lost all presence of mind, and raised the cry of "treachery." This sealed their fate. Every Isosceles now saw and felt a foe in every other. In half an hour not one of that vast multitude was living; and the fragments of seven score thousand of the Criminal Class slain by one another's angles attested the triumph of Order. The Circles delayed not to push their victory to the uttermost. The Working Men they spared but decimated. The Militia of the Equilaterals was at once called out, and every Triangle suspected of Irregularity on reasonable grounds, was destroyed by Court Martial, without the formality of exact measurement by the Social Board. The homes of the Military and Artisan classes were inspected in a course of visitation extending through upwards of a year; and during that period every town, village, and hamlet was systematically purged of that excess of the lower orders which had been brought about by the neglect to pay the tribute of Criminals to the Schools and University, and by the violation of other natural Laws of the Constitution of Flatland. Thus the balance of classes was again restored. Needless to say that henceforth the use of Colour was abolished, and its possession prohibited. Even the utterance of any word denoting Colour, except by the Circles or by qualified scientific teachers, was punished by a severe penalty. Only at our University in some of the very highest and most esoteric classes--which I myself have never been privileged to attend--it is understood that the sparing use of Colour is still sanctioned for the purpose of illustrating some of the deeper problems of mathematics. But of this I can only speak from hearsay. Elsewhere in Flatland, Colour is now non-existent. The art of making it is known to only one living person, the Chief Circle for the time being; and by him it is handed down on his death-bed to none but his Successor. One manufactory alone produces it; and, lest the secret should be betrayed, the Workmen are annually consumed, and fresh ones introduced. So great is the terror with which even now our Aristocracy looks back to the far-distant days of the agitation for the Universal Colour Bill. SECTION 11 Concerning our Priests It is high time that I should pass from these brief and discursive notes about things in Flatland to the central event of this book, my initiation into the mysteries of Space. THAT is my subject; all that has gone before is merely preface. For this reason I must omit many matters of which the explanation would not, I flatter myself, be without interest for my Readers: as for example, our method of propelling and stopping ourselves, although destitute of feet; the means by which we give fixity to structures of wood, stone, or brick, although of course we have no hands, nor can we lay foundations as you can, nor avail ourselves of the lateral pressure of the earth; the manner in which the rain originates in the intervals between our various zones, so that the northern regions do not intercept the moisture falling on the southern; the nature of our hills and mines, our trees and vegetables, our seasons and harvests; our Alphabet and method of writing, adapted to our linear tablets; these and a hundred other details of our physical existence I must pass over, nor do I mention them now except to indicate to my readers that their omission proceeds not from forgetfulness on the part of the author, but from his regard for the time of the Reader. Yet before I proceed to my legitimate subject some few final remarks will no doubt be expected by my Readers upon these pillars and mainstays of the Constitution of Flatland, the controllers of our conduct and shapers of our destiny, the objects of universal homage and almost of adoration: need I say that I mean our Circles or Priests? When I call them Priests, let me not be understood as meaning no more than the term denotes with you. With us, our Priests are Administrators of all Business, Art, and Science; Directors of Trade, Commerce, Generalship, Architecture, Engineering, Education, Statesmanship, Legislature, Morality, Theology; doing nothing themselves, they are the Causes of everything worth doing, that is done by others. Although popularly everyone called a Circle is deemed a Circle, yet among the better educated Classes it is known that no Circle is really a Circle, but only a Polygon with a very large number of very small sides. As the number of the sides increases, a Polygon approximates to a Circle; and, when the number is very great indeed, say for example three or four hundred, it is extremely difficult for the most delicate touch to feel any polygonal angles. Let me say rather it WOULD be difficult: for, as I have shown above, Recognition by Feeling is unknown among the highest society, and to FEEL a Circle would be considered a most audacious insult. This habit of abstention from Feeling in the best society enables a Circle the more easily to sustain the veil of mystery in which, from his earliest years, he is wont to enwrap the exact nature of his Perimeter or Circumference. Three feet being the average Perimeter it follows that, in a Polygon of three hundred sides each side will be no more than the hundredth part of a foot in length, or little more than the tenth part of an inch; and in a Polygon of six or seven hundred sides the sides are little larger than the diameter of a Spaceland pin-head. It is always assumed, by courtesy, that the Chief Circle for the time being has ten thousand sides. The ascent of the posterity of the Circles in the social scale is not restricted, as it is among the lower Regular classes, by the Law of Nature which limits the increase of sides to one in each generation. If it were so, the number of sides in the Circle would be a mere question of pedigree and arithmetic, and the four hundred and ninety-seventh descendant of an Equilateral Triangle would necessarily be a polygon with five hundred sides. But this is not the case. Nature's Law prescribes two antagonistic decrees affecting Circular propagation; first, that as the race climbs higher in the scale of development, so development shall proceed at an accelerated pace; second, that in the same proportion, the race shall become less fertile. Consequently in the home of a Polygon of four or five hundred sides it is rare to find a son; more than one is never seen. On the other hand the son of a five-hundred-sided Polygon has been known to possess five hundred and fifty, or even six hundred sides. Art also steps in to help the process of higher Evolution. Our physicians have discovered that the small and tender sides of an infant Polygon of the higher class can be fractured, and his whole frame re-set, with such exactness that a Polygon of two or three hundred sides sometimes--by no means always, for the process is attended with serious risk--but sometimes overleaps two or three hundred generations, and as it were double at a stroke, the number of his progenitors and the nobility of his descent. Many a promising child is sacrificed in this way. Scarcely one out of ten survives. Yet so strong is the parental ambition among those Polygons who are, as it were, on the fringe of the Circular class, that it is very rare to find the Nobleman of that position in society, who has neglected to place his first-born in the Circular Neo-Therapeutic Gymnasium before he has attained the age of a month. One year determines success or failure. At the end of that time the child has, in all probability, added one more to the tombstones that crowd the Neo-Therapeutic Cemetery; but on rare occasional a glad procession bears back the little one to his exultant parents, no longer a Polygon, but a Circle, at least by courtesy: and a single instance of so blessed a result induces multitudes of Polygonal parents to submit to similar domestic sacrifice, which have a dissimilar issue. SECTION 12 Of the Doctrine of our Priests As to the doctrine of the Circles it may briefly be summed up in a single maxim, "Attend to your Configuration." Whether political, ecclesiastical, or moral, all their teaching has for its object the improvement of individual and collective Configuration--with special reference of course to the Configuration of the Circles, to which all other objects are subordinated. It is the merit of the Circles that they have effectually suppressed those ancient heresies which led men to waste energy and sympathy in the vain belief that conduct depends upon will, effort, training, encouragement, praise, or anything else but Configuration. It was Pantocyclus--the illustrious Circle mentioned above, as the queller of the Colour Revolt--who first convinced mankind that Configuration makes the man; that if, for example, you are born an Isosceles with two uneven sides, you will assuredly go wrong unless you have them made even--for which purpose you must go to the Isosceles Hospital; similarly, if you are a Triangle, or Square, or even a Polygon, born with any Irregularity, you must be taken to one of the Regular Hospitals to have your disease cured; otherwise you will end your days in the State Prison or by the angle of the State Executioner. All faults or defects, from the slightest misconduct to the most flagitious crime, Pantocyclus attributed to some deviation from perfect Regularity in the bodily figure, caused perhaps (if not congenital) by some collision in a crowd; by neglect to take exercise, or by taking too much of it; or even by a sudden change of temperature, resulting in a shrinkage or expansion in some too susceptible part of the frame. Therefore, concluded that illustrious Philosopher, neither good conduct nor bad conduct is a fit subject, in any sober estimation, for either praise or blame. For why should you praise, for example, the integrity of a Square who faithfully defends the interests of his client, when you ought in reality rather to admire the exact precision of his right angles? Or again, why blame a lying, thievish Isosceles, when you ought rather to deplore the incurable inequality of his sides? Theoretically, this doctrine is unquestionable; but it has practical drawbacks. In dealing with an Isosceles, if a rascal pleads that he cannot help stealing because of his unevenness, you reply that for that very reason, because he cannot help being a nuisance to his neighbours, you, the Magistrate, cannot help sentencing him to be consumed--and there's an end of the matter. But in little domestic difficulties, when the penalty of consumption, or death, is out of the question, this theory of Configuration sometimes comes in awkwardly; and I must confess that occasionally when one of my own Hexagonal Grandsons pleads as an excuse for his disobedience that a sudden change of temperature has been too much for his Perimeter, and that I ought to lay the blame not on him but on his Configuration, which can only be strengthened by abundance of the choicest sweetmeats, I neither see my way logically to reject, nor practically to accept, his conclusions. For my own part, I find it best to assume that a good sound scolding or castigation has some latent and strengthening influence on my Grandson's Configuration; though I own that I have no grounds for thinking so. At all events I am not alone in my way of extricating myself from this dilemma; for I find that many of the highest Circles, sitting as Judges in law courts, use praise and blame towards Regular and Irregular Figures; and in their homes I know by experience that, when scolding their children, they speak about "right" and "wrong" as vehemently and passionately as if they believe that these names represented real existence, and that a human Figure is really capable of choosing between them. Constantly carrying out their policy of making Configuration the leading idea in every mind, the Circles reverse the nature of that Commandment which in Spaceland regulates the relations between parents and children. With you, children are taught to honour their parents; with us--next to the Circles, who are the chief object of universal homage--a man is taught to honour his Grandson, if he has one; or, if not, his Son. By "honour," however, is by no means mean "indulgence," but a reverent regard for their highest interests: and the Circles teach that the duty of fathers is to subordinate their own interests to those of posterity, thereby advancing the welfare of the whole State as well as that of their own immediate descendants. The weak point in the system of the Circles--if a humble Square may venture to speak of anything Circular as containing any element of weakness--appears to me to be found in their relations with Women. As it is of the utmost importance for Society that Irregular births should be discouraged, it follows that no Woman who has any Irregularities in her ancestry is a fit partner for one who desires that his posterity should rise by regular degrees in the social scale. Now the Irregularity of a Male is a matter of measurement; but as all Women are straight, and therefore visibly Regular so to speak, one has to devise some other means of ascertaining what I may call their invisible Irregularity, that is to say their potential Irregularities as regards possible offspring. This is effected by carefully-kept pedigrees, which are preserved and supervised by the State; and without a certified pedigree no Woman is allowed to marry. Now it might have been supposed the a Circle--proud of his ancestry and regardful for a posterity which might possibly issue hereafter in a Chief Circle--would be more careful than any other to choose a wife who had no blot on her escutcheon. But it is not so. The care in choosing a Regular wife appears to diminish as one rises in the social scale. Nothing would induce an aspiring Isosceles, who has hopes of generating an Equilateral Son, to take a wife who reckoned a single Irregularity among her Ancestors; a Square or Pentagon, who is confident that his family is steadily on the rise, does not inquire above the five-hundredth generation; a Hexagon or Dodecagon is even more careless of the wife's pedigree; but a Circle has been known deliberately to take a wife who has had an Irregular Great-Grandfather, and all because of some slight superiority of lustre, or because of the charms of a low voice--which, with us, even more than with you, is thought "an excellent thing in a Woman." Such ill-judged marriages are, as might be expected, barren, if they do not result in positive Irregularity or in diminution of sides; but none of these evils have hitherto provided sufficiently deterrent. The loss of a few sides in a highly-developed Polygon is not easily noticed, and is sometimes compensated by a successful operation in the Neo-Therapeutic Gymnasium, as I have described above; and the Circles are too much disposed to acquiesce in infecundity as a law of the superior development. Yet, if this evil be not arrested, the gradual diminution of the Circular class may soon become more rapid, and the time may not be far distant when, the race being no longer able to produce a Chief Circle, the Constitution of Flatland must fall. One other word of warning suggest itself to me, though I cannot so easily mention a remedy; and this also refers to our relations with Women. About three hundred years ago, it was decreed by the Chief Circle that, since women are deficient in Reason but abundant in Emotion, they ought no longer to be treated as rational, nor receive any mental education. The consequence was that they were no longer taught to read, nor even to master Arithmetic enough to enable them to count the angles of their husband or children; and hence they sensibly declined during each generation in intellectual power. And this system of female non-education or quietism still prevails. My fear is that, with the best intentions, this policy has been carried so far as to react injuriously on the Male Sex. For the consequence is that, as things now are, we Males have to lead a kind of bi-lingual, and I may almost say bimental, existence. With Women, we speak of "love," "duty," "right," "wrong," "pity," "hope," and other irrational and emotional conceptions, which have no existence, and the fiction of which has no object except to control feminine exuberances; but among ourselves, and in our books, we have an entirely different vocabulary and I may also say, idiom. "Love" them becomes "the anticipation of benefits"; "duty" becomes "necessity" or "fitness"; and other words are correspondingly transmuted. Moreover, among Women, we use language implying the utmost deference for their Sex; and they fully believe that the Chief Circle Himself is not more devoutly adored by us than they are: but behind their backs they are both regarded and spoken of--by all but the very young--as being little better than "mindless organisms." Our Theology also in the Women's chambers is entirely different from our Theology elsewhere. Now my humble fear is that this double training, in language as well as in thought, imposes somewhat too heavy a burden upon the young, especially when, at the age of three years old, they are taken from the maternal care and taught to unlearn the old language--except for the purpose of repeating it in the presence of the Mothers and Nurses--and to learn the vocabulary and idiom of science. Already methinks I discern a weakness in the grasp of mathematical truth at the present time as compared with the more robust intellect of our ancestors three hundred years ago. I say nothing of the possible danger if a Woman should ever surreptitiously learn to read and convey to her Sex the result of her perusal of a single popular volume; nor of the possibility that the indiscretion or disobedience of some infant Male might reveal to a Mother the secrets of the logical dialect. On the simple ground of the enfeebling of the male intellect, I rest this humble appeal to the highest Authorities to reconsider the regulations of Female education. PART II OTHER WORLDS "O brave new worlds, That have such people in them!" SECTION 13 How I had a Vision of Lineland It was the last day but one of the 1999th year of our era, and the first day of the Long Vacation. Having amused myself till a late hour with my favourite recreation of Geometry, I had retired to rest with an unsolved problem in my mind. In the night I had a dream. I saw before me a vast multitude of small Straight Lines (which I naturally assumed to be Women) interspersed with other Beings still smaller and of the nature of lustrous points--all moving to and fro in one and the same Straight Line, and, as nearly as I could judge, with the same velocity. A noise of confused, multitudinous chirping or twittering issued from them at intervals as long as they were moving; but sometimes they ceased from motion, and then all was silence. Approaching one of the largest of what I thought to be Women, I accosted her, but received no answer. A second and third appeal on my part were equally ineffectual. Losing patience at what appeared to me intolerable rudeness, I brought my mouth to a position full in front of her mouth so as to intercept her motion, and loudly repeated my question, "Woman, what signifies this concourse, and this strange and confused chirping, and this monotonous motion to and fro in one and the same Straight Line?" "I am no Woman," replied the small Line: "I am the Monarch of the world. But thou, whence intrudest thou into my realm of Lineland?" Receiving this abrupt reply, I begged pardon if I had in any way startled or molested his Royal Highness; and describing myself as a stranger I besought the King to give me some account of his dominions. But I had the greatest possible difficulty in obtaining any information on points that really interested me; for the Monarch could not refrain from constantly assuming that whatever was familiar to him must also be known to me and that I was simulating ignorance in jest. However, by preserving questions I elicited the following facts: It seemed that this poor ignorant Monarch--as he called himself--was persuaded that the Straight Line which he called his Kingdom, and in which he passed his existence, constituted the whole of the world, and indeed the whole of Space. Not being able either to move or to see, save in his Straight Line, he had no conception of anything out of it. Though he had heard my voice when I first addressed him, the sounds had come to him in a manner so contrary to his experience that he had made no answer, "seeing no man," as he expressed it, "and hearing a voice as it were from my own intestines." Until the moment when I placed my mouth in his World, he had neither seen me, nor heard anything except confused sounds beating against, what I called his side, but what he called his INSIDE or STOMACH; nor had he even now the least conception of the region from which I had come. Outside his World, or Line, all was a blank to him; nay, not even a blank, for a blank implies Space; say, rather, all was non-existent. His subjects--of whom the small Lines were men and the Points Women--were all alike confined in motion and eyesight to that single Straight Line, which was their World. It need scarcely be added that the whole of their horizon was limited to a Point; nor could any one ever see anything but a Point. Man, woman, child, thing--each as a Point to the eye of a Linelander. Only by the sound of the voice could sex or age be distinguished. Moreover, as each individual occupied the whole of the narrow path, so to speak, which constituted his Universe, and no one could move to the right or left to make way for passers by, it followed that no Linelander could ever pass another. Once neighbours, always neighbours. Neighbourhood with them was like marriage with us. Neighbours remained neighbours till death did them part. Such a life, with all vision limited to a Point, and all motion to a Straight Line, seemed to me inexpressibly dreary; and I was surprised to note that vivacity and cheerfulness of the King. Wondering whether it was possible, amid circumstances so unfavourable to domestic relations, to enjoy the pleasures of conjugal union, I hesitated for some time to question his Royal Highness on so delicate a subject; but at last I plunged into it by abruptly inquiring as to the health of his family. "My wives and children," he replied, "are well and happy." Staggered at this answer--for in the immediate proximity of the Monarch (as I had noted in my dream before I entered Lineland) there were none but Men--I ventured to reply, "Pardon me, but I cannot imagine how your Royal Highness can at any time either see or approach their Majesties, when there at least half a dozen intervening individuals, whom you can neither see through, nor pass by? Is it possible that in Lineland proximity is not necessary for marriage and for the generation of children?" "How can you ask so absurd a question?" replied the Monarch. "If it were indeed as you suggest, the Universe would soon be depopulated. No, no; neighbourhood is needless for the union of hearts; and the birth of children is too important a matter to have been allowed to depend upon such an accident as proximity. You cannot be ignorant of this. Yet since you are pleased to affect ignorance, I will instruct you as if you were the veriest baby in Lineland. Know, then, that marriages are consummated by means of the faculty of sound and the sense of hearing. "You are of course aware that every Man has two mouths or voices--as well as two eyes--a bass at one and a tenor at the other of his extremities. I should not mention this, but that I have been unable to distinguish your tenor in the course of our conversation." I replied that I had but one voice, and that I had not been aware that his Royal Highness had two. "That confirms my impression," said the King, "that you are not a Man, but a feminine Monstrosity with a bass voice, and an utterly uneducated ear. But to continue. "Nature having herself ordained that every Man should wed two wives--" "Why two?" asked I. "You carry your affected simplicity too far," he cried. "How can there be a completely harmonious union without the combination of the Four in One, viz. the Bass and Tenor of the Man and the Soprano and Contralto of the two Women?" "But supposing," said I, "that a man should prefer one wife or three?" "It is impossible," he said; "it is as inconceivable as that two and one should make five, or that the human eye should see a Straight Line." I would have interrupted him; but he proceeded as follows: "Once in the middle of each week a Law of Nature compels us to move to and fro with a rhythmic motion of more than usual violence, which continues for the time you would take to count a hundred and one. In the midst of this choral dance, at the fifty-first pulsation, the inhabitants of the Universe pause in full career, and each individual sends forth his richest, fullest, sweetest strain. It is in this decisive moment that all our marriages are made. So exquisite is the adaptation of Bass and Treble, of Tenor to Contralto, that oftentimes the Loved Ones, though twenty thousand leagues away, recognize at once the responsive note of their destined Lover; and, penetrating the paltry obstacles of distance, Love unites the three. The marriage in that instance consummated results in a threefold Male and Female offspring which takes its place in Lineland." "What! Always threefold?" said I. "Must one wife then always have twins?" "Bass-voice Monstrosity! yes," replied the King. "How else could the balance of the Sexes be maintained, if two girls were not born for every boy? Would you ignore the very Alphabet of Nature?" He ceased, speechless for fury; and some time elapsed before I could induce him to resume his narrative. "You will not, of course, suppose that every bachelor among us finds his mates at the first wooing in this universal Marriage Chorus. On the contrary, the process is by most of us many times repeated. Few are the hearts whose happy lot is at once to recognize in each other's voice the partner intended for them by Providence, and to fly into a reciprocal and perfectly harmonious embrace. With most of us the courtship is of long duration. The Wooer's voices may perhaps accord with one of the future wives, but not with both; or not, at first, with either; or the Soprano and Contralto may not quite harmonize. In such cases Nature has provided that every weekly Chorus shall bring the three Lovers into closer harmony. Each trial of voice, each fresh discovery of discord, almost imperceptibly induces the less perfect to modify his or her vocal utterance so as to approximate to the more perfect. And after many trials and many approximations, the result is at last achieved. There comes a day at last when, while the wonted Marriage Chorus goes forth from universal Lineland, the three far-off Lovers suddenly find themselves in exact harmony, and, before they are aware, the wedded Triplet is rapt vocally into a duplicate embrace; and Nature rejoices over one more marriage and over three more births." SECTION 14 How I vainly tried to explain the nature of Flatland Thinking that it was time to bring down the Monarch from his raptures to the level of common sense, I determined to endeavour to open up to him some glimpses of the truth, that is to say of the nature of things in Flatland. So I began thus: "How does your Royal Highness distinguish the shapes and positions of his subjects? I for my part noticed by the sense of sight, before I entered your Kingdom, that some of your people are lines and others Points; and that some of the lines are larger--" "You speak of an impossibility," interrupted the King; "you must have seen a vision; for to detect the difference between a Line and a Point by the sense of sight is, as every one knows, in the nature of things, impossible; but it can be detected by the sense of hearing, and by the same means my shape can be exactly ascertained. Behold me--I am a Line, the longest in Lineland, over six inches of Space--" "Of Length," I ventured to suggest. "Fool," said he, "Space is Length. Interrupt me again, and I have done." I apologized; but he continued scornfully, "Since you are impervious to argument, you shall hear with your ears how by means of my two voices I reveal my shape to my Wives, who are at this moment six thousand miles seventy yards two feet eight inches away, the one to the North, the other to the South. Listen, I call to them." He chirruped, and then complacently continued: "My wives at this moment receiving the sound of one of my voice, closely followed by the other, and perceiving that the latter reaches them after an interval in which sound can traverse 6.457 inches, infer that one of my mouths is 6.457 inches further from them than the other, and accordingly know my shape to be 6.457 inches. But you will of course understand that my wives do not make this calculation every time they hear my two voices. They made it, once for all, before we were married. But they COULD make it at any time. And in the same way I can estimate the shape of any of my Male subjects by the sense of sound." "But how," said I, "if a Man feigns a Woman's voice with one of his two voices, or so disguises his Southern voice that it cannot be recognized as the echo of the Northern? May not such deceptions cause great inconvenience? And have you no means of checking frauds of this kind by commanding your neighbouring subjects to feel one another?" This of course was a very stupid question, for feeling could not have answered the purpose; but I asked with the view of irritating the Monarch, and I succeeded perfectly. "What!" cried he in horror, "explain your meaning." "Feel, touch, come into contact," I replied. "If you mean by FEELING," said the King, "approaching so close as to leave no space between two individuals, know, Stranger, that this offence is punishable in my dominions by death. And the reason is obvious. The frail form of a Woman, being liable to be shattered by such an approximation, must be preserved by the State; but since Women cannot be distinguished by the sense of sight from Men, the Law ordains universally that neither Man nor Woman shall be approached so closely as to destroy the interval between the approximator and the approximated. "And indeed what possible purpose would be served by this illegal and unnatural excess of approximation which you call TOUCHING, when all the ends of so brutal and course a process are attained at once more easily and more exactly by the sense of hearing? As to your suggested danger of deception, it is non-existent: for the Voice, being the essence of one's Being, cannot be thus changed at will. But come, suppose that I had the power of passing through solid things, so that I could penetrate my subjects, one after another, even to the number of a billion, verifying the size and distance of each by the sense of FEELING: How much time and energy would be wasted in this clumsy and inaccurate method! Whereas now, in one moment of audition, I take as it were the census and statistics, local, corporeal, mental and spiritual, of every living being in Lineland. Hark, only hark!" So saying he paused and listened, as if in an ecstasy, to a sound which seemed to me no better than a tiny chirping from an innumerable multitude of lilliputian grasshoppers. "Truly," replied I, "your sense of hearing serves you in good stead, and fills up many of your deficiencies. But permit me to point out that your life in Lineland must be deplorably dull. To see nothing but a Point! Not even to be able to contemplate a Straight Line! Nay, not even to know what a Straight Line is! To see, yet to be cut off from those Linear prospects which are vouchsafed to us in Flatland! Better surely to have no sense of sight at all than to see so little! I grant you I have not your discriminative faculty of hearing; for the concert of all Lineland which gives you such intense pleasure, is to me no better than a multitudinous twittering or chirping. But at least I can discern, by sight, a Line from a Point. And let me prove it. Just before I came into your kingdom, I saw you dancing from left to right, and then from right to left, with Seven Men and a Woman in your immediate proximity on the left, and eight Men and two Women on your right. Is not this correct?" "It is correct," said the King, "so far as the numbers and sexes are concerned, though I know not what you mean by 'right' and 'left.' But I deny that you saw these things. For how could you see the Line, that is to say the inside, of any Man? But you must have heard these things, and then dreamed that you saw them. And let me ask what you mean by those words 'left' and 'right.' I suppose it is your way of saying Northward and Southward." "Not so," replied I; "besides your motion of Northward and Southward, there is another motion which I call from right to left." King. Exhibit to me, if you please, this motion from left to right. I. Nay, that I cannot do, unless you could step out of your Line altogether. King. Out of my Line? Do you mean out of the world? Out of Space? I. Well, yes. Out of YOUR world. Out of YOUR Space. For your Space is not the true Space. True Space is a Plane; but your Space is only a Line. King. If you cannot indicate this motion from left to right by yourself moving in it, then I beg you to describe it to me in words. I. If you cannot tell your right side from your left, I fear that no words of mine can make my meaning clearer to you. But surely you cannot be ignorant of so simple a distinction. King. I do not in the least understand you. I. Alas! How shall I make it clear? When you move straight on, does it not sometimes occur to you that you COULD move in some other way, turning your eye round so as to look in the direction towards which your side is now fronting? In other words, instead of always moving in the direction of one of your extremities, do you never feel a desire to move in the direction, so to speak, of your side? King. Never. And what do you mean? How can a man's inside "front" in any direction? Or how can a man move in the direction of his inside? I. Well then, since words cannot explain the matter, I will try deeds, and will move gradually out of Lineland in the direction which I desire to indicate to you. At the word I began to move my body out of Lineland. As long as any part of me remained in his dominion and in his view, the King kept exclaiming, "I see you, I see you still; you are not moving." But when I had at last moved myself out of his Line, he cried in his shrillest voice, "She is vanished; she is dead." "I am not dead," replied I; "I am simply out of Lineland, that is to say, out of the Straight Line which you call Space, and in the true Space, where I can see things as they are. And at this moment I can see your Line, or side--or inside as you are pleased to call it; and I can see also the Men and Women on the North and South of you, whom I will now enumerate, describing their order, their size, and the interval between each." When I had done this at great length, I cried triumphantly, "Does that at last convince you?" And, with that, I once more entered Lineland, taking up the same position as before. But the Monarch replied, "If you were a Man of sense--though, as you appear to have only one voice I have little doubt you are not a Man but a Woman--but, if you had a particle of sense, you would listen to reason. You ask me to believe that there is another Line besides that which my senses indicate, and another motion besides that of which I am daily conscious. I, in return, ask you to describe in words or indicate by motion that other Line of which you speak. Instead of moving, you merely exercise some magic art of vanishing and returning to sight; and instead of any lucid description of your new World, you simply tell me the numbers and sizes of some forty of my retinue, facts known to any child in my capital. Can anything be more irrational or audacious? Acknowledge your folly or depart from my dominions." Furious at his perversity, and especially indignant that he professed to be ignorant of my sex, I retorted in no measured terms, "Besotted Being! You think yourself the perfection of existence, while you are in reality the most imperfect and imbecile. You profess to see, whereas you see nothing but a Point! You plume yourself on inferring the existence of a Straight Line; but I CAN SEE Straight Lines, and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. Why waste more words? Suffice it that I am the completion of your incomplete self. You are a Line, but I am a Line of Lines called in my country a Square: and even I, infinitely superior though I am to you, am of little account among the great nobles of Flatland, whence I have come to visit you, in the hope of enlightening your ignorance." Hearing these words the King advanced towards me with a menacing cry as if to pierce me through the diagonal; and in that same movement there arose from myriads of his subjects a multitudinous war-cry, increasing in vehemence till at last methought it rivalled the roar of an army of a hundred thousand Isosceles, and the artillery of a thousand Pentagons. Spell-bound and motionless, I could neither speak nor move to avert the impending destruction; and still the noise grew louder, and the King came closer, when I awoke to find the breakfast-bell recalling me to the realities of Flatland. SECTION 15 Concerning a Stranger from Spaceland From dreams I proceed to facts. It was the last day of our 1999th year of our era. The patterning of the rain had long ago announced nightfall; and I was sitting (footnote 3) in the company of my wife, musing on the events of the past and the prospects of the coming year, the coming century, the coming Millennium. My four Sons and two orphan Grandchildren had retired to their several apartments; and my wife alone remained with me to see the old Millennium out and the new one in. I was rapt in thought, pondering in my mind some words that had casually issued from the mouth of my youngest Grandson, a most promising young Hexagon of unusual brilliancy and perfect angularity. His uncles and I had been giving him his usual practical lesson in Sight Recognition, turning ourselves upon our centres, now rapidly, now more slowly, and questioning him as to our positions; and his answers had been so satisfactory that I had been induced to reward him by giving him a few hints on Arithmetic, as applied to Geometry. Taking nine Squares, each an inch every way, I had put them together so as to make one large Square, with a side of three inches, and I had hence proved to my little Grandson that--though it was impossible for us to SEE the inside of the Square--yet we might ascertain the number of square inches in a Square by simply squaring the number of inches in the side: "and thus," said I, "we know that three-to-the-second, or nine, represents the number of square inches in a Square whose side is three inches long." The little Hexagon meditated on this a while and then said to me; "But you have been teaching me to raise numbers to the third power: I suppose three-to-the-third must mean something in Geometry; what does it mean?" "Nothing at all," replied I, "not at least in Geometry; for Geometry has only Two Dimensions." And then I began to shew the boy how a Point by moving through a length of three inches makes a Line of three inches, which may be represented by three; and how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches every way, which may be represented by three-to-the-second. xxx Upon this, my Grandson, again returning to his former suggestion, took me up rather suddenly and exclaimed, "Well, then, if a Point by moving three inches, makes a Line of three inches represented by three; and if a straight Line of three inches, moving parallel to itself, makes a Square of three inches every way, represented by three-to-the-second; it must be that a Square of three inches every way, moving somehow parallel to itself (but I don't see how) must make Something else (but I don't see what) of three inches every way--and this must be represented by three-to-the-third." "Go to bed," said I, a little ruffled by this interruption: "if you would talk less nonsense, you would remember more sense." So my Grandson had disappeared in disgrace; and there I sat by my Wife's side, endeavouring to form a retrospect of the year 1999 and of the possibilities of the year 2000; but not quite able to shake of the thoughts suggested by the prattle of my bright little Hexagon. Only a few sands now remained in the half-hour glass. Rousing myself from my reverie I turned the glass Northward for the last time in the old Millennium; and in the act, I exclaimed aloud, "The boy is a fool." Straightway I became conscious of a Presence in the room, and a chilling breath thrilled through my very being. "He is no such thing," cried my Wife, "and you are breaking the Commandments in thus dishonouring your own Grandson." But I took no notice of her. Looking around in every direction I could see nothing; yet still I FELT a Presence, and shivered as the cold whisper came again. I started up. "What is the matter?" said my Wife, "there is no draught; what are you looking for? There is nothing." There was nothing; and I resumed my seat, again exclaiming, "The boy is a fool, I say; three-to-the-third can have no meaning in Geometry." At once there came a distinctly audible reply, "The boy is not a fool; and three-to-the-third has an obvious Geometrical meaning." My Wife as well as myself heard the words, although she did not understand their meaning, and both of us sprang forward in the direction of the sound. What was our horror when we saw before us a Figure! At the first glance it appeared to be a Woman, seen sideways; but a moment's observation shewed me that the extremities passed into dimness too rapidly to represent one of the Female Sex; and I should have thought it a Circle, only that it seemed to change its size in a manner impossible for a Circle or for any regular Figure of which I had had experience. But my Wife had not my experience, nor the coolness necessary to note these characteristics. With the usual hastiness and unreasoning jealousy of her Sex, she flew at once to the conclusion that a Woman had entered the house through some small aperture. "How comes this person here?" she exclaimed, "you promised me, my dear, that there should be no ventilators in our new house." "Nor are they any," said I; "but what makes you think that the stranger is a Woman? I see by my power of Sight Recognition--" "Oh, I have no patience with your Sight Recognition," replied she, "'Feeling is believing' and 'A Straight Line to the touch is worth a Circle to the sight'"--two Proverbs, very common with the Frailer Sex in Flatland. "Well," said I, for I was afraid of irritating her, "if it must be so, demand an introduction." Assuming her most gracious manner, my Wife advanced towards the Stranger, "Permit me, Madam to feel and be felt by--" then, suddenly recoiling, "Oh! it is not a Woman, and there are no angles either, not a trace of one. Can it be that I have so misbehaved to a perfect Circle?" "I am indeed, in a certain sense a Circle," replied the Voice, "and a more perfect Circle than any in Flatland; but to speak more accurately, I am many Circles in one." Then he added more mildly, "I have a message, dear Madam, to your husband, which I must not deliver in your presence; and, if you would suffer us to retire for a few minutes--" But my wife would not listen to the proposal that our august Visitor should so incommode himself, and assuring the Circle that the hour of her own retirement had long passed, with many reiterated apologies for her recent indiscretion, she at last retreated to her apartment. I glanced at the half-hour glass. The last sands had fallen. The third Millennium had begun. Footnote 3. When I say "sitting," of course I do not mean any change of attitude such as you in Spaceland signify by that word; for as we have no feet, we can no more "sit" nor "stand" (in your sense of the word) than one of your soles or flounders. Nevertheless, we perfectly well recognize the different mental states of volition implied by "lying," "sitting," and "standing," which are to some extent indicated to a beholder by a slight increase of lustre corresponding to the increase of volition. But on this, and a thousand other kindred subjects, time forbids me to dwell. SECTION 16 How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland As soon as the sound of the Peace-cry of my departing Wife had died away, I began to approach the Stranger with the intention of taking a nearer view and of bidding him be seated: but his appearance struck me dumb and motionless with astonishment. Without the slightest symptoms of angularity he nevertheless varied every instant with graduations of size and brightness scarcely possible for any Figure within the scope of my experience. The thought flashed across me that I might have before me a burglar or cut-throat, some monstrous Irregular Isosceles, who, by feigning the voice of a Circle, had obtained admission somehow into the house, and was now preparing to stab me with his acute angle. In a sitting-room, the absence of Fog (and the season happened to be remarkably dry), made it difficult for me to trust to Sight Recognition, especially at the short distance at which I was standing. Desperate with fear, I rushed forward with an unceremonious, "You must permit me, Sir--" and felt him. My Wife was right. There was not the trace of an angle, not the slightest roughness or inequality: never in my life had I met with a more perfect Circle. He remained motionless while I walked around him, beginning from his eye and returning to it again. Circular he was throughout, a perfectly satisfactory Circle; there could not be a doubt of it. Then followed a dialogue, which I will endeavour to set down as near as I can recollect it, omitting only some of my profuse apologies--for I was covered with shame and humiliation that I, a Square, should have been guilty of the impertinence of feeling a Circle. It was commenced by the Stranger with some impatience at the lengthiness of my introductory process. Stranger. Have you felt me enough by this time? Are you not introduced to me yet? I. Most illustrious Sir, excuse my awkwardness, which arises not from ignorance of the usages of polite society, but from a little surprise and nervousness, consequent on this somewhat unexpected visit. And I beseech you to reveal my indiscretion to no one, and especially not to my Wife. But before your Lordship enters into further communications, would he deign to satisfy the curiosity of one who would gladly know whence his visitor came? Stranger. From Space, from Space, Sir: whence else? I. Pardon me, my Lord, but is not your Lordship already in Space, your Lordship and his humble servant, even at this moment? Stranger. Pooh! what do you know of Space? Define Space. I. Space, my Lord, is height and breadth indefinitely prolonged. Stranger. Exactly: you see you do not even know what Space is. You think it is of Two Dimensions only; but I have come to announce to you a Third--height, breadth, and length. I. Your Lordship is pleased to be merry. We also speak of length and height, or breadth and thickness, thus denoting Two Dimensions by four names. Stranger. But I mean not only three names, but Three Dimensions. I. Would your Lordship indicate or explain to me in what direction is the Third Dimension, unknown to me? Stranger. I came from it. It is up above and down below. I. My Lord means seemingly that it is Northward and Southward. Stranger. I mean nothing of the kind. I mean a direction in which you cannot look, because you have no eye in your side. I. Pardon me, my Lord, a moment's inspection will convince your Lordship that I have a perfectly luminary at the juncture of my two sides. Stranger: Yes: but in order to see into Space you ought to have an eye, not on your Perimeter, but on your side, that is, on what you would probably call your inside; but we in Spaceland should call it your side. I. An eye in my inside! An eye in my stomach! Your Lordship jests. Stranger. I am in no jesting humour. I tell you that I come from Space, or, since you will not understand what Space means, from the Land of Three Dimensions whence I but lately looked down upon your Plane which you call Space forsooth. From that position of advantage I discerned all that you speak of as SOLID (by which you mean "enclosed on four sides"), your houses, your churches, your very chests and safes, yes even your insides and stomachs, all lying open and exposed to my view. I. Such assertions are easily made, my Lord. Stranger. But not easily proved, you mean. But I mean to prove mine. When I descended here, I saw your four Sons, the Pentagons, each in his apartment, and your two Grandsons the Hexagons; I saw your youngest Hexagon remain a while with you and then retire to his room, leaving you and your Wife alone. I saw your Isosceles servants, three in number, in the kitchen at supper, and the little Page in the scullery. Then I came here, and how do you think I came? I. Through the roof, I suppose. Strange. Not so. Your roof, as you know very well, has been recently repaired, and has no aperture by which even a Woman could penetrate. I tell you I come from Space. Are you not convinced by what I have told you of your children and household? I. Your Lordship must be aware that such facts touching the belongings of his humble servant might be easily ascertained by any one of the neighbourhood possessing your Lordship's ample means of information. Stranger. (TO HIMSELF.) What must I do? Stay; one more argument suggests itself to me. When you see a Straight Line-- your wife, for example--how many Dimensions do you attribute to her? I. Your Lordship would treat me as if I were one of the vulgar who, being ignorant of Mathematics, suppose that a Woman is really a Straight Line, and only of One Dimension. No, no, my Lord; we Squares are better advised, and are as well aware of your Lordship that a Woman, though popularly called a Straight Line, is, really and scientifically, a very thin Parallelogram, possessing Two Dimensions, like the rest of us, viz., length and breadth (or thickness). Stranger. But the very fact that a Line is visible implies that it possesses yet another Dimension. I. My Lord, I have just acknowledged that a Woman is broad as well as long. We see her length, we infer her breadth; which, though very slight, is capable of measurement. Stranger. You do not understand me. I mean that when you see a Woman, you ought--besides inferring her breadth--to see her length, and to SEE what we call her HEIGHT; although the last Dimension is infinitesimal in your country. If a Line were mere length without "height," it would cease to occupy Space and would become invisible. Surely you must recognize this? I. I must indeed confess that I do not in the least understand your Lordship. When we in Flatland see a Line, we see length and BRIGHTNESS. If the brightness disappears, the Line is extinguished, and, as you say, ceases to occupy Space. But am I to suppose that your Lordship gives the brightness the title of a Dimension, and that what we call "bright" you call "high"? Stranger. No, indeed. By "height" I mean a Dimension like your length: only, with you, "height" is not so easily perceptible, being extremely small. I. My Lord, your assertion is easily put to the test. You say I have a Third Dimension, which you call "height." Now, Dimension implies direction and measurement. Do but measure my "height," or merely indicate to me the direction in which my "height" extends, and I will become your convert. Otherwise, your Lordship's own understand must hold me excused. Stranger. (TO HIMSELF.) I can do neither. How shall I convince him? Surely a plain statement of facts followed by ocular demonstration ought to suffice. --Now, Sir; listen to me. You are living on a Plane. What you style Flatland is the vast level surface of what I may call a fluid, or in, the top of which you and your countrymen move about, without rising above or falling below it. I am not a plane Figure, but a Solid. You call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other. When I cut through your plane as I am now doing, I make in your plane a section which you, very rightly, call a Circle. For even a Sphere--which is my proper name in my own country--if he manifest himself at all to an inhabitant of Flatland--must needs manifest himself as a Circle. Do you not remember--for I, who see all things, discerned last night the phantasmal vision of Lineland written upon your brain--do you not remember, I say, how when you entered the realm of Lineland, you were compelled to manifest yourself to the King, not as a Square, but as a Line, because that Linear Realm had not Dimensions enough to represent the whole of you, but only a slice or section of you? In precisely the same way, your country of Two Dimensions is not spacious enough to represent me, a being of Three, but can only exhibit a slice or section of me, which is what you call a Circle. The diminished brightness of your eye indicates incredulity. But now prepare to receive proof positive of the truth of my assertions. You cannot indeed see more than one of my sections, or Circles, at a time; for you have no power to raise your eye out of the plane of Flatland; but you can at least see that, as I rise in Space, so my sections become smaller. See now, I will rise; and the effect upon your eye will be that my Circle will become smaller and smaller till it dwindles to a point and finally vanishes. There was no "rising" that I could see; but he diminished and finally vanished. I winked once or twice to make sure that I was not dreaming. But it was no dream. For from the depths of nowhere came forth a hollow voice--close to my heart it seemed--"Am I quite gone? Are you convinced now? Well, now I will gradually return to Flatland and you shall see my section become larger and larger." Every reader in Spaceland will easily understand that my mysterious Guest was speaking the language of truth and even of simplicity. But to me, proficient though I was in Flatland Mathematics, it was by no means a simple matter. The rough diagram given above will make it clear to any Spaceland child that the Sphere, ascending in the three positions indicated there, must needs have manifested himself to me, or to any Flatlander, as a Circle, at first of full size, then small, and at last very small indeed, approaching to a Point. But to me, although I saw the facts before me, the causes were as dark as ever. All that I could comprehend was, that the Circle had made himself smaller and vanished, and that he had now re-appeared and was rapidly making himself larger. When he regained his original size, he heaved a deep sigh; for he perceived by my silence that I had altogether failed to comprehend him. And indeed I was now inclining to the belief that he must be no Circle at all, but some extremely clever juggler; or else that the old wives' tales were true, and that after all there were such people as Enchanters and Magicians. After a long pause he muttered to himself, "One resource alone remains, if I am not to resort to action. I must try the method of Analogy." Then followed a still longer silence, after which he continued our dialogue. Sphere. Tell me, Mr. Mathematician; if a Point moves Northward, and leaves a luminous wake, what name would you give to the wake? I. A straight Line. Sphere. And a straight Line has how many extremities? I. Two. Sphere. Now conceive the Northward straight Line moving parallel to itself, East and West, so that every point in it leaves behind it the wake of a straight Line. What name will you give to the Figure thereby formed? We will suppose that it moves through a distance equal to the original straight line. --What name, I say? I. A square. Sphere. And how many sides has a Square? How many angles? I. Four sides and four angles. Sphere. Now stretch your imagination a little, and conceive a Square in Flatland, moving parallel to itself upward. I. What? Northward? Sphere. No, not Northward; upward; out of Flatland altogether. If it moved Northward, the Southern points in the Square would have to move through the positions previously occupied by the Northern points. But that is not my meaning. I mean that every Point in you--for you are a Square and will serve the purpose of my illustration--every Point in you, that is to say in what you call your inside, is to pass upwards through Space in such a way that no Point shall pass through the position previously occupied by any other Point; but each Point shall describe a straight Line of its own. This is all in accordance with Analogy; surely it must be clear to you. Restraining my impatience--for I was now under a strong temptation to rush blindly at my Visitor and to precipitate him into Space, or out of Flatland, anywhere, so that I could get rid of him--I replied:-- "And what may be the nature of the Figure which I am to shape out by this motion which you are pleased to denote by the word 'upward'? I presume it is describable in the language of Flatland." Sphere. Oh, certainly. It is all plain and simple, and in strict accordance with Analogy--only, by the way, you must not speak of the result as being a Figure, but as a Solid. But I will describe it to you. Or rather not I, but Analogy. We began with a single Point, which of course--being itself a Point--has only ONE terminal Point. One Point produces a Line with TWO terminal Points. One Line produces a Square with FOUR terminal Points. Now you can give yourself the answer to your own question: 1, 2, 4, are evidently in Geometrical Progression. What is the next number? I. Eight. Sphere. Exactly. The one Square produces a SOMETHING-WHICH-YOU- DO-NOT-AS-YET-KNOW-A-NAME-FOR-BUT-WHICH-WE-CALL-A-CUBE with EIGHT terminal Points. Now are you convinced? I. And has this Creature sides, as well as Angles or what you call "terminal Points"? Sphere. Of course; and all according to Analogy. But, by the way, not what YOU call sides, but what WE call sides. You would call them SOLIDS. I. And how many solids or sides will appertain to this Being whom I am to generate by the motion of my inside in an "upward" direction, and whom you call a Cube? Sphere. How can you ask? And you a mathematician! The side of anything is always, if I may so say, one Dimension behind the thing. Consequently, as there is no Dimension behind a Point, a Point has 0 sides; a Line, if I may so say, has 2 sides (for the points of a Line may be called by courtesy, its sides); a Square has 4 sides; 0, 2, 4; what Progression do you call that? I. Arithmetical. Sphere. And what is the next number? I. Six. Sphere. Exactly. Then you see you have answered your own question. The Cube which you will generate will be bounded by six sides, that is to say, six of your insides. You see it all now, eh? "Monster," I shrieked, "be thou juggler, enchanter, dream, or devil, no more will I endure thy mockeries. Either thou or I must perish." And saying these words I precipitated myself upon him. SECTION 17 How the Sphere, having in vain tried words, resorted to deeds It was in vain. I brought my hardest right angle into violent collision with the Stranger, pressing on him with a force sufficient to have destroyed any ordinary Circle: but I could feel him slowly and unarrestably slipping from my contact; not edging to the right nor to the left, but moving somehow out of the world, and vanishing into nothing. Soon there was a blank. But still I heard the Intruder's voice. Sphere. Why will you refuse to listen to reason? I had hoped to find in you--as being a man of sense and an accomplished mathematician--a fit apostle for the Gospel of the Three Dimensions, which I am allowed to preach once only in a thousand years: but now I know not how to convince you. Stay, I have it. Deeds, and not words, shall proclaim the truth. Listen, my friend. I have told you I can see from my position in Space the inside of all things that you consider closed. For example, I see in yonder cupboard near which you are standing, several of what you call boxes (but like everything else in Flatland, they have no tops or bottom) full of money; I see also two tablets of accounts. I am about to descend into that cupboard and to bring you one of those tablets. I saw you lock the cupboard half an hour ago, and I know you have the key in your possession. But I descend from Space; the doors, you see, remain unmoved. Now I am in the cupboard and am taking the tablet. Now I have it. Now I ascend with it. I rushed to the closet and dashed the door open. One of the tablets was gone. With a mocking laugh, the Stranger appeared in the other corner of the room, and at the same time the tablet appeared upon the floor. I took it up. There could be no doubt--it was the missing tablet. I groaned with horror, doubting whether I was not out of my sense; but the Stranger continued: "Surely you must now see that my explanation, and no other, suits the phenomena. What you call Solid things are really superficial; what you call Space is really nothing but a great Plane. I am in Space, and look down upon the insides of the things of which you only see the outsides. You could leave the Plane yourself, if you could but summon up the necessary volition. A slight upward or downward motion would enable you to see all that I can see. "The higher I mount, and the further I go from your Plane, the more I can see, though of course I see it on a smaller scale. For example, I am ascending; now I can see your neighbour the Hexagon and his family in their several apartments; now I see the inside of the Theatre, ten doors off, from which the audience is only just departing; and on the other side a Circle in his study, sitting at his books. Now I shall come back to you. And, as a crowning proof, what do you say to my giving you a touch, just the least touch, in your stomach? It will not seriously injure you, and the slight pain you may suffer cannot be compared with the mental benefit you will receive." Before I could utter a word of remonstrance, I felt a shooting pain in my inside, and a demoniacal laugh seemed to issue from within me. A moment afterwards the sharp agony had ceased, leaving nothing but a dull ache behind, and the Stranger began to reappear, saying, as he gradually increased in size, "There, I have not hurt you much, have I? If you are not convinced now, I don't know what will convince you. What say you?" My resolution was taken. It seemed intolerable that I should endure existence subject to the arbitrary visitations of a Magician who could thus play tricks with one's very stomach. If only I could in any way manage to pin him against the wall till help came! Once more I dashed my hardest angle against him, at the same time alarming the whole household by my cries for aid. I believe, at the moment of my onset, the Stranger had sunk below our Plane, and really found difficulty in rising. In any case he remained motionless, while I, hearing, as I thought, the sound of some help approaching, pressed against him with redoubled vigor, and continued to shout for assistance. A convulsive shudder ran through the Sphere. "This must not be," I thought I heard him say: "either he must listen to reason, or I must have recourse to the last resource of civilization." Then, addressing me in a louder tone, he hurriedly exclaimed, "Listen: no stranger must witness what you have witnessed. Send your Wife back at once, before she enters the apartment. The Gospel of Three Dimensions must not be thus frustrated. Not thus must the fruits of one thousand years of waiting be thrown away. I hear her coming. Back! back! Away from me, or you must go with me--wither you know not--into the Land of Three Dimensions!" "Fool! Madman! Irregular!" I exclaimed; "never will I release thee; thou shalt pay the penalty of thine impostures." "Ha! Is it come to this?" thundered the Stranger: "then meet your fate: out of your Plane you go. Once, twice, thrice! 'Tis done!" SECTION 18 How I came to Spaceland, and what I saw there An unspeakable horror seized me. There was a darkness; then a dizzy, sickening sensation of sight that was not like seeing; I saw a Line that was no Line; Space that was not Space: I was myself, and not myself. When I could find voice, I shrieked loud in agony, "Either this is madness or it is Hell." "It is neither," calmly replied the voice of the Sphere, "it is Knowledge; it is Three Dimensions: open your eye once again and try to look steadily." I looked, and, behold, a new world! There stood before me, visibly incorporate, all that I had before inferred, conjectured, dreamed, of perfect Circular beauty. What seemed the centre of the Stranger's form lay open to my view: yet I could see no heart, lungs, nor arteries, only a beautiful harmonious Something--for which I had no words; but you, my Readers in Spaceland, would call it the surface of the Sphere. Prostrating myself mentally before my Guide, I cried, "How is it, O divine ideal of consummate loveliness and wisdom that I see thy inside, and yet cannot discern thy heart, thy lungs, thy arteries, thy liver?" "What you think you see, you see not," he replied; "it is not giving to you, nor to any other Being, to behold my internal parts. I am of a different order of Beings from those in Flatland. Were I a Circle, you could discern my intestines, but I am a Being, composed as I told you before, of many Circles, the Many in the One, called in this country a Sphere. And, just as the outside of a Cube is a Square, so the outside of a Sphere represents the appearance of a Circle." Bewildered though I was by my Teacher's enigmatic utterance, I no longer chafed against it, but worshipped him in silent adoration. He continued, with more mildness in his voice. "Distress not yourself if you cannot at first understand the deeper mysteries of Spaceland. By degrees they will dawn upon you. Let us begin by casting back a glance at the region whence you came. Return with me a while to the plains of Flatland and I will shew you that which you have often reasoned and thought about, but never seen with the sense of sight--a visible angle." "Impossible!" I cried; but, the Sphere leading the way, I followed as if in a dream, till once more his voice arrested me: "Look yonder, and behold your own Pentagonal house, and all its inmates." I looked below, and saw with my physical eye all that domestic individuality which I had hitherto merely inferred with the understanding. And how poor and shadowy was the inferred conjecture in comparison with the reality which I now behold! My four Sons calmly asleep in the North-Western rooms, my two orphan Grandsons to the South; the Servants, the Butler, my Daughter, all in their several apartments. Only my affectionate Wife, alarmed by my continued absence, had quitted her room and was roving up and down in the Hall, anxiously awaiting my return. Also the Page, aroused by my cries, had left his room, and under pretext of ascertaining whether I had fallen somewhere in a faint, was prying into the cabinet in my study. All this I could now SEE, not merely infer; and as we came nearer and nearer, I could discern even the contents of my cabinet, and the two chests of gold, and the tablets of which the Sphere had made mention. Touched by my Wife's distress, I would have sprung downward to reassure her, but I found myself incapable of motion. "Trouble not yourself about your Wife," said my Guide: "she will not be long left in anxiety; meantime, let us take a survey of Flatland." Once more I felt myself rising through space. It was even as the Sphere had said. The further we receded from the object we beheld, the larger became the field of vision. My native city, with the interior of every house and every creature therein, lay open to my view in miniature. We mounted higher, and lo, the secrets of the earth, the depths of the mines and inmost caverns of the hills, were bared before me. Awestruck at the sight of the mysteries of the earth, thus unveiled before my unworthy eye, I said to my Companion, "Behold, I am become as a God. For the wise men in our country say that to see all things, or as they express it, OMNIVIDENCE, is the attribute of God alone." There was something of scorn in the voice of my Teacher as he made answer: "is it so indeed? Then the very pick-pockets and cut-throats of my country are to be worshipped by your wise men as being Gods: for there is not one of them that does not see as much as you see now. But trust me, your wise men are wrong." I. Then is omnividence the attribute of others besides Gods? Sphere. I do not know. But, if a pick-pocket or a cut-throat of our country can see everything that is in your country, surely that is no reason why the pick-pocket or cut-throat should be accepted by you as a God. This omnividence, as you call it--it is not a common word in Spaceland--does it make you more just, more merciful, less selfish, more loving? Not in the least. Then how does it make you more divine? I. "More merciful, more loving!" But these are the qualities of women! And we know that a Circle is a higher Being than a Straight Line, in so far as knowledge and wisdom are more to be esteemed than mere affection. Sphere. It is not for me to classify human faculties according to merit. Yet many of the best and wisest in Spaceland think more of the affections than of the understanding, more of your despised Straight Lines than of your belauded Circles. But enough of this. Look yonder. Do you know that building? I looked, and afar off I saw an immense Polygonal structure, in which I recognized the General Assembly Hall of the States of Flatland, surrounded by dense lines of Pentagonal buildings at right angles to each other, which I knew to be streets; and I perceived that I was approaching the great Metropolis. "Here we descend," said my Guide. It was now morning, the first hour of the first day of the two thousandth year of our era. Acting, as was their wont, in strict accordance with precedent, the highest Circles of the realm were meeting in solemn conclave, as they had met on the first hour of the first day of the year 1000, and also on the first hour of the first day of the year 0. The minutes of the previous meetings were now read by one whom I at once recognized as my brother, a perfectly Symmetrical Square, and the Chief Clerk of the High Council. It was found recorded on each occasion that: "Whereas the States had been troubled by divers ill-intentioned persons pretending to have received revelations from another World, and professing to produce demonstrations whereby they had instigated to frenzy both themselves and others, it had been for this cause unanimously resolved by the Grand Council that on the first day of each millenary, special injunctions be sent to the Prefects in the several districts of Flatland, to make strict search for such misguided persons, and without formality of mathematical examination, to destroy all such as were Isosceles of any degree, to scourge and imprison any regular Triangle, to cause any Square or Pentagon to be sent to the district Asylum, and to arrest any one of higher rank, sending him straightway to the Capital to be examined and judged by the Council." "You hear your fate," said the Sphere to me, while the Council was passing for the third time the formal resolution. "Death or imprisonment awaits the Apostle of the Gospel of Three Dimensions." "Not so," replied I, "the matter is now so clear to me, the nature of real space so palpable, that methinks I could make a child understand it. Permit me but to descend at this moment and enlighten them." "Not yet," said my Guide, "the time will come for that. Meantime I must perform my mission. Stay thou there in thy place." Saying these words, he leaped with great dexterity into the sea (if I may so call it) of Flatland, right in the midst of the ring of Counsellors. "I come," said he, "to proclaim that there is a land of Three Dimensions." I could see many of the younger Counsellors start back in manifest horror, as the Sphere's circular section widened before them. But on a sign from the presiding Circle--who shewed not the slightest alarm or surprise--six Isosceles of a low type from six different quarters rushed upon the Sphere. "We have him," they cried; "No; yes; we have him still! he's going! he's gone!" "My Lords," said the President to the Junior Circles of the Council, "there is not the slightest need for surprise; the secret archives, to which I alone have access, tell me that a similar occurrence happened on the last two millennial commencements. You will, of course, say nothing of these trifles outside the Cabinet." Raising his voice, he now summoned the guards. "Arrest the policemen; gag them. You know your duty." After he had consigned to their fate the wretched policemen--ill-fated and unwilling witnesses of a State-secret which they were not to be permitted to reveal--he again addressed the Counsellors. "My Lords, the business of the Council being concluded, I have only to wish you a happy New Year." Before departing, he expressed, at some length, to the Clerk, my excellent but most unfortunate brother, his sincere regret that, in accordance with precedent and for the sake of secrecy, he must condemn him to perpetual imprisonment, but added his satisfaction that, unless some mention were made by him of that day's incident, his life would be spared. SECTION 19 How, though the Sphere shewed me other mysteries of Spaceland, I still desire more; and what came of it When I saw my poor brother led away to imprisonment, I attempted to leap down into the Council Chamber, desiring to intercede on his behalf, or at least bid him farewell. But I found that I had no motion of my own. I absolutely depended on the volition of my Guide, who said in gloomy tones, "Heed not thy brother; haply thou shalt have ample time hereafter to condole with him. Follow me." Once more we ascended into space. "Hitherto," said the Sphere, "I have shewn you naught save Plane Figures and their interiors. Now I must introduce you to Solids, and reveal to you the plan upon which they are constructed. Behold this multitude of moveable square cards. See, I put one on another, not, as you supposed, Northward of the other, but ON the other. Now a second, now a third. See, I am building up a Solid by a multitude of Squares parallel to one another. Now the Solid is complete, being as high as it is long and broad, and we call it a Cube." "Pardon me, my Lord," replied I; "but to my eye the appearance is as of an Irregular Figure whose inside is laid open to view; in other words, methinks I see no Solid, but a Plane such as we infer in Flatland; only of an Irregularity which betokens some monstrous criminal, so that the very sight of it is painful to my eyes." "True," said the Sphere; "it appears to you a Plane, because you are not accustomed to light and shade and perspective; just as in Flatland a Hexagon would appear a Straight Line to one who has not the Art of Sight Recognition. But in reality it is a Solid, as you shall learn by the sense of Feeling." He then introduced me to the Cube, and I found that this marvellous Being was indeed no Plane, but a Solid; and that he was endowed with six plane sides and eight terminal points called solid angles; and I remembered the saying of the Sphere that just such a Creature as this would be formed by the Square moving, in Space, parallel to himself: and I rejoiced to think that so insignificant a Creature as I could in some sense be called the Progenitor of so illustrious an offspring. But still I could not fully understand the meaning of what my Teacher had told me concerning "light" and "shade" and "perspective"; and I did not hesitate to put my difficulties before him. Were I to give the Sphere's explanation of these matters, succinct and clear though it was, it would be tedious to an inhabitant of Space, who knows these things already. Suffice it, that by his lucid statements, and by changing the position of objects and lights, and by allowing me to feel the several objects and even his own sacred Person, he at last made all things clear to me, so that I could now readily distinguish between a Circle and a Sphere, a Plane Figure and a Solid. This was the Climax, the Paradise, of my strange eventful History. Henceforth I have to relate the story of my miserable Fall:--most miserable, yet surely most undeserved! For why should the thirst for knowledge be aroused, only to be disappointed and punished? My volition shrinks from the painful task of recalling my humiliation; yet, like a second Prometheus, I will endure this and worse, if by any means I may arouse in the interiors of Plane and Solid Humanity a spirit of rebellion against the Conceit which would limit our Dimensions to Two or Three or any number short of Infinity. Away then with all personal considerations! Let me continue to the end, as I began, without further digressions or anticipations, pursuing the plain path of dispassionate History. The exact facts, the exact words,--and they are burnt in upon my brain,--shall be set down without alteration of an iota; and let my Readers judge between me and Destiny. The Sphere would willingly have continued his lessons by indoctrinating me in the conformation of all regular Solids, Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured to interrupt him. Not that I was wearied of knowledge. On the contrary, I thirsted for yet deeper and fuller draughts than he was offering to me. "Pardon me," said I, "O Thou Whom I must no longer address as the Perfection of all Beauty; but let me beg thee to vouchsafe thy servant a sight of thine interior." Sphere. My what? I. Thine interior: thy stomach, thy intestines. Sphere. Whence this ill-timed impertinent request? And what mean you by saying that I am no longer the Perfection of all Beauty? I. My Lord, your own wisdom has taught me to aspire to One even more great, more beautiful, and more closely approximate to Perfection than yourself. As you yourself, superior to all Flatland forms, combine many Circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And even as we, who are now in Space, look down on Flatland and see the insides of all things, so of a certainty there is yet above us some higher, purer region, whither thou dost surely purpose to lead me--O Thou Whom I shall always call, everywhere and in all Dimensions, my Priest, Philosopher, and Friend--some yet more spacious Space, some more dimensionable Dimensionality, from the vantage-ground of which we shall look down together upon the revealed insides of Solid things, and where thine own intestines, and those of thy kindred Spheres, will lie exposed to the view of the poor wandering exile from Flatland, to whom so much has already been vouchsafed. Sphere. Pooh! Stuff! Enough of this trifling! The time is short, and much remains to be done before you are fit to proclaim the Gospel of Three Dimensions to your blind benighted countrymen in Flatland. I. Nay, gracious Teacher, deny me not what I know it is in thy power to perform. Grant me but one glimpse of thine interior, and I am satisfied for ever, remaining henceforth thy docile pupil, thy unemancipable slave, ready to receive all thy teachings and to feed upon the words that fall from thy lips. Sphere. Well, then, to content and silence you, let me say at once, I would shew you what you wish if I could; but I cannot. Would you have me turn my stomach inside out to oblige you? I. But my Lord has shewn me the intestines of all my countrymen in the Land of Two Dimensions by taking me with him into the Land of Three. What therefore more easy than now to take his servant on a second journey into the blessed region of the Fourth Dimension, where I shall look down with him once more upon this land of Three Dimensions, and see the inside of every three-dimensioned house, the secrets of the solid earth, the treasures of the mines of Spaceland, and the intestines of every solid living creature, even the noble and adorable Spheres. Sphere. But where is this land of Four Dimensions? I. I know not: but doubtless my Teacher knows. Sphere. Not I. There is no such land. The very idea of it is utterly inconceivable. I. Not inconceivable, my Lord, to me, and therefore still less inconceivable to my Master. Nay, I despair not that, even here, in this region of Three Dimensions, your Lordship's art may make the Fourth Dimension visible to me; just as in the Land of Two Dimensions my Teacher's skill would fain have opened the eyes of his blind servant to the invisible presence of a Third Dimension, though I saw it not. Let me recall the past. Was I not taught below that when I saw a Line and inferred a Plane, I in reality saw a Third unrecognized Dimension, not the same as brightness, called "height"? And does it not now follow that, in this region, when I see a Plane and infer a Solid, I really see a Fourth unrecognized Dimension, not the same as colour, but existent, though infinitesimal and incapable of measurement? And besides this, there is the Argument from Analogy of Figures. Sphere. Analogy! Nonsense: what analogy? I. Your Lordship tempts his servant to see whether he remembers the revelations imparted to him. Trifle not with me, my Lord; I crave, I thirst, for more knowledge. Doubtless we cannot SEE that other higher Spaceland now, because we have no eye in our stomachs. But, just as there WAS the realm of Flatland, though that poor puny Lineland Monarch could neither turn to left nor right to discern it, and just as there WAS close at hand, and touching my frame, the land of Three Dimensions, though I, blind senseless wretch, had no power to touch it, no eye in my interior to discern it, so of a surety there is a Fourth Dimension, which my Lord perceives with the inner eye of thought. And that it must exist my Lord himself has taught me. Or can he have forgotten what he himself imparted to his servant? In One Dimension, did not a moving Point produce a Line with TWO terminal points? In Two Dimensions, did not a moving Line produce a Square with FOUR terminal points? In Three Dimensions, did not a moving Square produce--did not this eye of mine behold it--that blessed Being, a Cube, with EIGHT terminal points? And in Four Dimensions shall not a moving Cube--alas, for Analogy, and alas for the Progress of Truth, if it be not so--shall not, I say, the motion of a divine Cube result in a still more divine Organization with SIXTEEN terminal points? Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this--if I might quote my Lord's own words--"strictly according to Analogy"? Again, was I not taught by my Lord that as in a Line there are TWO bounding Points, and in a Square there are FOUR bounding Lines, so in a Cube there must be SIX bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, "strictly according to Analogy"? O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not knowing the facts; and I appeal to your Lordship to confirm or deny my logical anticipations. If I am wrong, I yield, and will no longer demand a Fourth Dimension; but, if I am right, my Lord will listen to reason. I ask therefore, is it, or is it not, the fact, that ere now your countrymen also have witnessed the descent of Beings of a higher order than their own, entering closed rooms, even as your Lordship entered mine, without the opening of doors or windows, and appearing and vanishing at will? On the reply to this question I am ready to stake everything. Deny it, and I am henceforth silent. Only vouchsafe an answer. Sphere (AFTER A PAUSE). It is reported so. But men are divided in opinion as to the facts. And even granting the facts, they explain them in different ways. And in any case, however great may be the number of different explanations, no one has adopted or suggested the theory of a Fourth Dimension. Therefore, pray have done with this trifling, and let us return to business. I. I was certain of it. I was certain that my anticipations would be fulfilled. And now have patience with me and answer me yet one more question, best of Teachers! Those who have thus appeared--no one knows whence--and have returned--no one knows whither--have they also contracted their sections and vanished somehow into that more Spacious Space, whither I now entreat you to conduct me? Sphere (MOODILY). They have vanished, certainly--if they ever appeared. But most people say that these visions arose from the thought--you will not understand me--from the brain; from the perturbed angularity of the Seer. I. Say they so? Oh, believe them not. Or if it indeed be so, that this other Space is really Thoughtland, then take me to that blessed Region where I in Thought shall see the insides of all solid things. There, before my ravished eye, a Cube moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space, with a wake of its own--shall create a still more perfect perfection than himself, with sixteen terminal Extra-solid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger at the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Six Dimension shall fly open; after that a Seventh, and then an Eighth-- How long I should have continued I know not. In vain did the Sphere, in his voice of thunder, reiterate his command of silence, and threaten me with the direst penalties if I persisted. Nothing could stem the flood of my ecstatic aspirations. Perhaps I was to blame; but indeed I was intoxicated with the recent draughts of Truth to which he himself had introduced me. However, the end was not long in coming. My words were cut short by a crash outside, and a simultaneous crash inside me, which impelled me through space with a velocity that precluded speech. Down! down! down! I was rapidly descending; and I knew that return to Flatland was my doom. One glimpse, one last and never-to-be-forgotten glimpse I had of that dull level wilderness--which was now to become my Universe again--spread out before my eye. Then a darkness. Then a final, all-consummating thunder-peal; and, when I came to myself, I was once more a common creeping Square, in my Study at home, listening to the Peace-Cry of my approaching Wife. SECTION 20 How the Sphere encouraged me in a Vision. Although I had less than a minute for reflection, I felt, by a kind of instinct, that I must conceal my experiences from my Wife. Not that I apprehended, at the moment, any danger from her divulging my secret, but I knew that to any Woman in Flatland the narrative of my adventures must needs be unintelligible. So I endeavoured to reassure her by some story, invented for the occasion, that I had accidentally fallen through the trap-door of the cellar, and had there lain stunned. The Southward attraction in our country is so slight that even to a Woman my tale necessarily appeared extraordinary and well-nigh incredible; but my Wife, whose good sense far exceeds that of the average of her Sex, and who perceived that I was unusually excited, did not argue with me on the subject, but insisted that I was ill and required repose. I was glad of an excuse for retiring to my chamber to think quietly over what had happened. When I was at last by myself, a drowsy sensation fell on me; but before my eyes closed I endeavoured to reproduce the Third Dimension, and especially the process by which a Cube is constructed through the motion of a Square. It was not so clear as I could have wished; but I remembered that it must be "Upward, and yet not Northward," and I determined steadfastly to retain these words as the clue which, if firmly grasped, could not fail to guide me to the solution. So mechanically repeating, like a charm, the words, "Upward, yet not Northward," I fell into a sound refreshing sleep. During my slumber I had a dream. I thought I was once more by the side of the Sphere, whose lustrous hue betokened that he had exchanged his wrath against me for perfectly placability. We were moving together towards a bright but infinitesimally small Point, to which my Master directed my attention. As we approached, methought there issued from it a slight humming noise as from one of your Spaceland bluebottles, only less resonant by far, so slight indeed that even in the perfect stillness of the Vacuum through which we soared, the sound reached not our ears till we checked our flight at a distance from it of something under twenty human diagonals. "Look yonder," said my Guide, "in Flatland thou hast lived; of Lineland thou hast received a vision; thou hast soared with me to the heights of Spaceland; now, in order to complete the range of thy experience, I conduct thee downward to the lowest depth of existence, even to the realm of Pointland, the Abyss of No dimensions. "Behold yon miserable creature. That Point is a Being like ourselves, but confined to the non-dimensional Gulf. He is himself his own World, his own Universe; of any other than himself he can form no conception; he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality; for he is himself his One and All, being really Nothing. Yet mark his perfect self-contentment, and hence learn his lesson, that to be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy. Now listen." He ceased; and there arose from the little buzzing creature a tiny, low, monotonous, but distinct tinkling, as from one of your Spaceland phonographs, from which I caught these words, "Infinite beatitude of existence! It is; and there is nothing else beside It." "What," said I, "does the puny creature mean by 'it'?" "He means himself," said the Sphere: "have you not noticed before now, that babies and babyish people who cannot distinguish themselves from the world, speak of themselves in the Third Person? But hush!" "It fills all Space," continued the little soliloquizing Creature, "and what It fills, It is. What It thinks, that It utters; and what It utters, that It hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All. Ah, the happiness, ah, the happiness of Being!" "Can you not startle the little thing out of its complacency?" said I. "Tell it what it really is, as you told me; reveal to it the narrow limitations of Pointland, and lead it up to something higher." "That is no easy task," said my Master; "try you." Hereon, raising by voice to the uttermost, I addressed the Point as follows: "Silence, silence, contemptible Creature. You call yourself the All in All, but you are the Nothing: your so-called Universe is a mere speck in a Line, and a Line is a mere shadow as compared with--" "Hush, hush, you have said enough," interrupted the Sphere, "now listen, and mark the effect of your harangue on the King of Pointland." The lustre of the Monarch, who beamed more brightly than ever upon hearing my words, shewed clearly that he retained his complacency; and I had hardly ceased when he took up his strain again. "Ah, the joy, ah, the joy of Thought! What can It not achieve by thinking! Its own Thought coming to Itself, suggestive of its disparagement, thereby to enhance Its happiness! Sweet rebellion stirred up to result in triumph! Ah, the divine creative power of the All in One! Ah, the joy, the joy of Being!" "You see," said my Teacher, "how little your words have done. So far as the Monarch understand them at all, he accepts them as his own--for he cannot conceive of any other except himself--and plumes himself upon the variety of 'Its Thought' as an instance of creative Power. Let us leave this God of Pointland to the ignorant fruition of his omnipresence and omniscience: nothing that you or I can do can rescue him from his self-satisfaction." After this, as we floated gently back to Flatland, I could hear the mild voice of my Companion pointing the moral of my vision, and stimulating me to aspire, and to teach others to aspire. He had been angered at first--he confessed--by my ambition to soar to Dimensions above the Third; but, since then, he had received fresh insight, and he was not too proud to acknowledge his error to a Pupil. Then he proceeded to initiate me into mysteries yet higher than those I had witnessed, shewing me how to construct Extra-Solids by the motion of Solids, and Double Extra-Solids by the motion of Extra-Solids, and all "strictly according to Analogy," all by methods so simple, so easy, as to be patent even to the Female Sex. SECTION 21 How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success I awoke rejoicing, and began to reflect on the glorious career before me. I would go forth, methought, at once, and evangelize the whole of Flatland. Even to Women and Soldiers should the Gospel of Three Dimensions be proclaimed. I would begin with my Wife. Just as I had decided on the plan of my operations, I heard the sound of many voices in the street commanding silence. Then followed a louder voice. It was a herald's proclamation. Listening attentively, I recognized the words of the Resolution of the Council, enjoining the arrest, imprisonment, or execution of any one who should pervert the minds of people by delusions, and by professing to have received revelations from another World. I reflected. This danger was not to be trifled with. It would be better to avoid it by omitting all mention of my Revelation, and by proceeding on the path of Demonstration--which after all, seemed so simple and so conclusive that nothing would be lost by discarding the former means. "Upward, not Northward"--was the clue to the whole proof. It had seemed to me fairly clear before I fell asleep; and when I first awoke, fresh from my dream, it had appeared as patent as Arithmetic; but somehow it did not seem to me quite so obvious now. Though my Wife entered the room opportunely at just that moment, I decided, after we had exchanged a few words of commonplace conversation, not to begin with her. My Pentagonal Sons were men of character and standing, and physicians of no mean reputation, but not great in mathematics, and, in that respect, unfit for my purpose. But it occurred to me that a young and docile Hexagon, with a mathematical turn, would be a most suitable pupil. Why therefore not make my first experiment with my little precocious Grandson, whose casual remarks on the meaning of three-to-the-third had met with the approval of the Sphere? Discussing the matter with him, a mere boy, I should be in perfect safety; for he would know nothing of the Proclamation of the Council; whereas I could not feel sure that my Sons--so greatly did their patriotism and reverence for the Circles predominate over mere blind affection--might not feel compelled to hand me over to the Prefect, if they found me seriously maintaining the seditious heresy of the Third Dimension. But the first thing to be done was to satisfy in some way the curiosity of my Wife, who naturally wished to know something of the reasons for which the Circle had desired that mysterious interview, and of the means by which he had entered the house. Without entering into the details of the elaborate account I gave her,--an account, I fear, not quite so consistent with truth as my Readers in Spaceland might desire,--I must be content with saying that I succeeded at last in persuading her to return quietly to her household duties without eliciting from me any reference to the World of Three Dimensions. This done, I immediately sent for my Grandson; for, to confess the truth, I felt that all that I had seen and heard was in some strange way slipping away from me, like the image of a half-grasped, tantalizing dream, and I longed to essay my skill in making a first disciple. When my Grandson entered the room I carefully secured the door. Then, sitting down by his side and taking our mathematical tablets,--or, as you would call them, Lines--I told him we would resume the lesson of yesterday. I taught him once more how a Point by motion in One Dimension produces a Line, and how a straight Line in Two Dimensions produces a Square. After this, forcing a laugh, I said, "And now, you scamp, you wanted to make believe that a Square may in the same way by motion 'Upward, not Northward' produce another figure, a sort of extra square in Three Dimensions. Say that again, you young rascal." At this moment we heard once more the herald's "O yes! O yes!" outside in the street proclaiming the Resolution of the Council. Young though he was, my Grandson--who was unusually intelligent for his age, and bred up in perfect reverence for the authority of the Circles--took in the situation with an acuteness for which I was quite unprepared. He remained silent till the last words of the Proclamation had died away, and then, bursting into tears, "Dear Grandpapa," he said, "that was only my fun, and of course I meant nothing at all by it; and we did not know anything then about the new Law; and I don't think I said anything about the Third Dimension; and I am sure I did not say one word about 'Upward, not Northward,' for that would be such nonsense, you know. How could a thing move Upward, and not Northward? Upward and not Northward! Even if I were a baby, I could not be so absurd as that. How silly it is! Ha! ha! ha!" "Not at all silly," said I, losing my temper; "here for example, I take this Square," and, at the word, I grasped a moveable Square, which was lying at hand--"and I move it, you see, not Northward but--yes, I move it Upward--that is to say, Northward but I move it somewhere--not exactly like this, but somehow--" Here I brought my sentence to an inane conclusion, shaking the Square about in a purposeless manner, much to the amusement of my Grandson, who burst out laughing louder than ever, and declared that I was not teaching him, but joking with him; and so saying he unlocked the door and ran out of the room. Thus ended my first attempt to convert a pupil to the Gospel of Three Dimensions. SECTION 22 How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result My failure with my Grandson did not encourage me to communicate my secret to others of my household; yet neither was I led by it to despair of success. Only I saw that I must not wholly rely on the catch-phrase, "Upward, not Northward," but must rather endeavour to seek a demonstration by setting before the public a clear view of the whole subject; and for this purpose it seemed necessary to resort to writing. So I devoted several months in privacy to the composition of a treatise on the mysteries of Three Dimensions. Only, with the view of evading the Law, if possible, I spoke not of a physical Dimension, but of a Thoughtland whence, in theory, a Figure could look down upon Flatland and see simultaneously the insides of all things, and where it was possible that there might be supposed to exist a Figure environed, as it were, with six Squares, and containing eight terminal Points. But in writing this book I found myself sadly hampered by the impossibility of drawing such diagrams as were necessary for my purpose: for of course, in our country of Flatland, there are no tablets but Lines, and no diagrams but Lines, all in one straight Line and only distinguishable by difference of size and brightness; so that, when I had finished my treatise (which I entitled, "Through Flatland to Thoughtland") I could not feel certain that many would understand my meaning. Meanwhile my wife was under a cloud. All pleasures palled upon me; all sights tantalized and tempted me to outspoken treason, because I could not compare what I saw in Two Dimensions with what it really was if seen in Three, and could hardly refrain from making my comparisons aloud. I neglected my clients and my own business to give myself to the contemplation of the mysteries which I had once beheld, yet which I could impart to no one, and found daily more difficult to reproduce even before my own mental vision. One day, about eleven months after my return from Spaceland, I tried to see a Cube with my eye closed, but failed; and though I succeeded afterwards, I was not then quite certain (nor have I been ever afterwards) that I had exactly realized the original. This made me more melancholy than before, and determined me to take some step; yet what, I knew not. I felt that I would have been willing to sacrifice my life for the Cause, if thereby I could have produced conviction. But if I could not convince my Grandson, how could I convince the highest and most developed Circles in the land? And yet at times my spirit was too strong for me, and I gave vent to dangerous utterances. Already I was considered heterodox if not treasonable, and I was keenly alive to the danger of my position; nevertheless I could not at times refrain from bursting out into suspicious or half-seditious utterances, even among the highest Polygonal or Circular society. When, for example, the question arose about the treatment of those lunatics who said that they had received the power of seeing the insides of things, I would quote the saying of an ancient Circle, who declared that prophets and inspired people are always considered by the majority to be mad; and I could not help occasionally dropping such expressions as "the eye that discerns the interiors of things," and "the all-seeing land"; once or twice I even let fall the forbidden terms "the Third and Fourth Dimensions." At last, to complete a series of minor indiscretions, at a meeting of our Local Speculative Society held at the palace of the Prefect himself,--some extremely silly person having read an elaborate paper exhibiting the precise reasons why Providence has limited the number of Dimensions to Two, and why the attribute of omnividence is assigned to the Supreme alone--I so far forgot myself as to give an exact account of the whole of my voyage with the Sphere into Space, and to the Assembly Hall in our Metropolis, and then to Space again, and of my return home, and of everything that I had seen and heard in fact or vision. At first, indeed, I pretended that I was describing the imaginary experiences of a fictitious person; but my enthusiasm soon forced me to throw off all disguise, and finally, in a fervent peroration, I exhorted all my hearers to divest themselves of prejudice and to become believers in the Third Dimension. Need I say that I was at once arrested and taken before the Council? Next morning, standing in the very place where but a very few months ago the Sphere had stood in my company, I was allowed to begin and to continue my narration unquestioned and uninterrupted. But from the first I foresaw my fate; for the President, noting that a guard of the better sort of Policemen was in attendance, of angularity little, if at all, under 55 degrees, ordered them to be relieved before I began my defence, by an inferior class of 2 or 3 degrees. I knew only too well what that meant. I was to be executed or imprisoned, and my story was to be kept secret from the world by the simultaneous destruction of the officials who had heard it; and, this being the case, the President desired to substitute the cheaper for the more expensive victims. After I had concluded my defence, the President, perhaps perceiving that some of the junior Circles had been moved by evident earnestness, asked me two questions:-- 1. Whether I could indicate the direction which I meant when I used the words "Upward, not Northward"? 2. Whether I could by any diagrams or descriptions (other than the enumeration of imaginary sides and angles) indicate the Figure I was pleased to call a Cube? I declared that I could say nothing more, and that I must commit myself to the Truth, whose cause would surely prevail in the end. The President replied that he quite concurred in my sentiment, and that I could not do better. I must be sentenced to perpetual imprisonment; but if the Truth intended that I should emerge from prison and evangelize the world, the Truth might be trusted to bring that result to pass. Meanwhile I should be subjected to no discomfort that was not necessary to preclude escape, and, unless I forfeited the privilege by misconduct, I should be occasionally permitted to see my brother who had preceded me to my prison. Seven years have elapsed and I am still a prisoner, and--if I except the occasional visits of my brother--debarred from all companionship save that of my jailers. My brother is one of the best of Squares, just, sensible, cheerful, and not without fraternal affection; yet I confess that my weekly interviews, at least in one respect, cause me the bitterest pain. He was present when the Sphere manifested himself in the Council Chamber; he saw the Sphere's changing sections; he heard the explanation of the phenomena then give to the Circles. Since that time, scarcely a week has passed during seven whole years, without his hearing from me a repetition of the part I played in that manifestation, together with ample descriptions of all the phenomena in Spaceland, and the arguments for the existence of Solid things derivable from Analogy. Yet--I take shame to be forced to confess it--my brother has not yet grasped the nature of Three Dimensions, and frankly avows his disbelief in the existence of a Sphere. Hence I am absolutely destitute of converts, and, for aught that I can see, the millennial Revelation has been made to me for nothing. Prometheus up in Spaceland was bound for bringing down fire for mortals, but I--poor Flatland Prometheus--lie here in prison for bringing down nothing to my countrymen. Yet I existing the hope that these memoirs, in some manner, I know not how, may find their way to the minds of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality. That is the hope of my brighter moments. Alas, it is not always so. Heavily weights on me at times the burdensome reflection that I cannot honestly say I am confident as to the exact shape of the once-seen, oft-regretted Cube; and in my nightly visions the mysterious precept, "Upward, not Northward," haunts me like a soul-devouring Sphinx. It is part of the martyrdom which I endure for the cause of Truth that there are seasons of mental weakness, when Cubes and Spheres flit away into the background of scarce-possible existences; when the Land of Three Dimensions seems almost as visionary as the Land of One or None; nay, when even this hard wall that bars me from my freedom, these very tablets on which I am writing, and all the substantial realities of Flatland itself, appear no better than the offspring of a diseased imagination, or the baseless fabric of a dream. *** PREFACE TO THE SECOND AND REVISED EDITION, 1884. BY THE EDITOR If my poor Flatland friend retained the vigour of mind which he enjoyed when he began to compose these Memoirs, I should not now need to represent him in this preface, in which he desires, fully, to return his thanks to his readers and critics in Spaceland, whose appreciation has, with unexpected celerity, required a second edition of this work; secondly, to apologize for certain errors and misprints (for which, however, he is not entirely responsible); and, thirdly, to explain one or two misconceptions. But he is not the Square he once was. Years of imprisonment, and the still heavier burden of general incredulity and mockery, have combined with the thoughts and notions, and much also of the terminology, which he acquired during his short stay in spaceland. He has, therefore, requested me to reply in his behalf to two special objections, one of an intellectual, the other of a moral nature. The first objection is, that a Flatlander, seeing a Line, sees something that must be THICK to the eye as well as LONG to the eye (otherwise it would not be visible, if it had not some thickness); and consequently he ought (it is argued) to acknowledge that his countrymen are not only long and broad, but also (though doubtless to a very slight degree) THICK or HIGH. This objection is plausible, and, to Spacelanders, almost irresistible, so that, I confess, when I first heard it, I knew not what to reply. But my poor old friend's answer appears to me completely to meet it. "I admit," said he--when I mentioned to him this objection--"I admit the truth of your critic's facts, but I deny his conclusions. It is true that we have really in Flatland a Third unrecognized Dimension called 'height,' just as it also is true that you have really in Spaceland a Fourth unrecognized Dimension, called by no name at present, but which I will call 'extra-height.' But we can no more take cognizance of our 'height' than you can of your 'extra-height.' Even I--who have been in Spaceland, and have had the privilege of understanding for twenty-four hours the meaning of 'height'--even I cannot now comprehend it, nor realize it by the sense of sight or by any process of reason; I can but apprehend it by faith. "The reason is obvious. Dimension implies direction, implies measurement, implies the more and the less. Now, all our lines are EQUALLY and INFINITESIMALLY thick (or high, whichever you like); consequently, there is nothing in them to lead our minds to the conception of that Dimension. No 'delicate micrometer'--as has been suggested by one too hasty Spaceland critic--would in the least avail us; for we should not know WHAT TO MEASURE, NOR IN WHAT DIRECTION. When we see a Line, we see something that is long and BRIGHT; BRIGHTNESS, as well as length, is necessary to the existence of a Line; if the brightness vanishes, the Line is extinguished. Hence, all my Flatland friends--when I talk to them about the unrecognized Dimension which is somehow visible in a Line--say, 'Ah, you mean BRIGHTNESS': and when I reply, 'No, I mean a real Dimension,' they at once retort, 'Then measure it, or tell us in what direction it extends'; and this silences me, for I can do neither. Only yesterday, when the Chief Circle (in other words our High Priest) came to inspect the State Prison and paid me his seventh annual visit, and when for the seventh time he put me the question, 'Was I any better?' I tried to prove to him that he was 'high,' as well as long and broad, although he did not know it. But what was his reply? 'You say I am "high"; measure my "high-ness" and I will believe you.' What could I do? How could I meet his challenge? I was crushed; and he left the room triumphant. "Does this still seem strange to you? Then put yourself in a similar position. Suppose a person of the Fourth Dimension, condescending to visit you, were to say, 'Whenever you open your eyes, you see a Plane (which is of Two Dimensions) and you INFER a Solid (which is of Three); but in reality you also see (though you do not recognize) a Fourth Dimension, which is not colour nor brightness nor anything of the kind, but a true Dimension, although I cannot point out to you its direction, nor can you possibly measure it.' What would you say to such a visitor? Would not you have him locked up? Well, that is my fate: and it is as natural for us Flatlanders to lock up a Square for preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube for preaching the Fourth. Alas, how strong a family likeness runs through blind and persecuting humanity in all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes--we are all liable to the same errors, all alike the Slaves of our respective Dimensional prejudices, as one of our Spaceland poets has said-- 'One touch of Nature makes all worlds akin.'" (footnote 1) On this point the defence of the Square seems to me to be impregnable. I wish I could say that his answer to the second (or moral) objection was equally clear and cogent. It has been objected that he is a woman-hater; and as this objection has been vehemently urged by those whom Nature's decree has constituted the somewhat larger half of the Spaceland race, I should like to remove it, so far as I can honestly do so. But the Square is so unaccustomed to the use of the moral terminology of Spaceland that I should be doing him an injustice if I were literally to transcribe his defence against this charge. Acting, therefore, as his interpreter and summarizer, I gather that in the course of an imprisonment of seven years he has himself modified his own personal views, both as regards Women and as regards the Isosceles or Lower Classes. Personally, he now inclines to the opinion of the Sphere (see page 86) that the Straight Lines are in many important respects superior to the Circles. But, writing as a Historian, he has identified himself (perhaps too closely) with the views generally adopted by Flatland, and (as he has been informed) even by Spaceland, Historians; in whose pages (until very recent times) the destinies of Women and of the masses of mankind have seldom been deemed worthy of mention and never of careful consideration. In a still more obscure passage he now desires to disavow the Circular or aristocratic tendencies with which some critics have naturally credited him. While doing justice to the intellectual power with which a few Circles have for many generations maintained their supremacy over immense multitudes of their countrymen, he believes that the facts of Flatland, speaking for themselves without comment on his part, declare that Revolutions cannot always be suppressed by slaughter, and that Nature, in sentencing the Circles to infecundity, has condemned them to ultimate failure--"and herein," he says, "I see a fulfilment of the great Law of all worlds, that while the wisdom of Man thinks it is working one thing, the wisdom of Nature constrains it to work another, and quite a different and far better thing." For the rest, he begs his readers not to suppose that every minute detail in the daily life of Flatland must needs correspond to some other detail in Spaceland; and yet he hopes that, taken as a whole, his work may prove suggestive as well as amusing, to those Spacelanders of moderate and modest minds who--speaking of that which is of the highest importance, but lies beyond experience--decline to say on the one hand, "This can never be," and on the other hand, "It must needs be precisely thus, and we know all about it." Footnote 1. The Author desires me to add, that the misconceptions of some of his critics on this matter has induced him to insert (on pp. 74 and 92) in his dialogue with the Sphere, certain remarks which have a bearing on the point in question and which he had previously omitted as being tedious and unnecessary. 
682	----	Catalan's Constant [Ramanujan's Formula] Catalan constant to 300000 digits computed on September 29, 1996 by using a Sun Ultra-Sparc in 1 day 8 hour 15 min 15 sec 55 hsec. The algorithm used is the standard series for Catalan, accelerated by an Euler transform. The algorithm was implemented using the LiDIA library for computational number theory and it is part of the multiprecision floating-point arithmetic of the package. LiDIA is available from ftp://crypt1.cs.uni-sb.de/pub/systems/LiDIA/LiDIA-1.2.1.tgz http://www-jb.cs.uni-sb.de/LiDIA/linkhtml/lidia/lidia.html The implementation of the algorithm is: inline void const_catalan (bigfloat & y) { bigfloat p; bigfloat t; int i = 1, j = 3; // j = 2*i+1 // y = t = p = 1/2 divide (y, 1, 2); t.assign (y); p.assign (y); // while t is greater than the desired accuracy while (!t.is_approx_zero ()) { // do // p = p * (i/j); // t = (t * i + p) / j; // y = y + t; // i++; j+=2; multiply (p, p, i); divide (p, p, j); multiply (t, t, i); add (t, t, p); divide (t, t, j); add (y, y, t); i++; j += 2; } } Here is the output of the program: Calculating Catalan's constant to 300000 decimals Time required: 1 day 8 hour 15 min 15 sec 55 hsec -------------------------------------------------------------------------- Additional REFERENCES: Catalan constant is: sum((-1)**(n+1)/(2*n-1)**2,n=1..infinity) also known under the name beta(2), see ?catalan in Maple for more details. The previous record was 200000 digits, also from Thomas Papanikolaou and before that: 100000 digits was due to Greg Fee and Simon Plouffe on August 14, 1996, by using a SGI r10000 Power Challenge with 194 Mhz in 5.63 hours and the standard implementation of Catalan on MapleV, Release 4. (which uses Greg's idea). Euler Tranform: References, Abramowitz and Stegun, formula 3.6.27 page 16 in Handbook of Mathematical Functions and Tables, Dover 1964. Ramanujan Notebooks, part I formula 34.1 of page 293. The series used is by putting x--> -1/2 . In other words the formula used is: the ordinary formula for Catalan sum((-1)**(n+1)/(2*n+1)**2,n=0..infinity) and then you apply the Euler Transform to it. Computation of Catalan's constant using Ramanujan's Formula, by Greg Fee, ACM 1990, Proceedings of the ISAAC conference, 1990, p. 157. Catalan constant to 300000 digits ------------------------------------------------------------------------ .91596559417721901505460351493238411077414937428167213426649811962176301977625 476947935651292611510624857442261919619957903589880332585905943159473748115840 699533202877331946051903872747816408786590902470648415216300022872764094238825 995774150881639747025248201156070764488380787337048990086477511322599713434074 854075532307685653357680958352602193823239508007206803557610482357339423191498 298361899770690364041808621794110191753274314997823397610551224779530324875371 878665828082360570225594194818097535097113157126158042427236364398500173828759 779765306837009298087388749561089365977194096872684444166804621624339864838916 280448281506273022742073884311722182721904722558705319086857354234985394983099 191159673884645086151524996242370437451777372351775440708538464401321748392999 947572446199754961975870640074748707014909376788730458699798606448749746438720 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432568999414277671149482737656695972837510170120731021202853085289585580262839 009038589909633411329463088278787461042577068922850898191537791785946970991510 951691571196251071871059641004166660572567197085658059074859369510151513239134 701046753988956130952781573873277268397439036130671226266274315817155149983369 255232611835020702025334380328028447809317987826912290482102307728881699091027 238217624823814141166942609035417856997584043316823744116630923803965016670561 130179842030708619496906015870141452658258605005599198260333820012987567795243 3428285033676 
633	----	The Golden Mean [Golden Ratio] (1+sqrt(5))/2 to 20000 places. This etext also contains these mathematical items: Catalan evaluation 170000 digits of gamma or Euler constant The Artin's Constant The Backhouse Constant Zeta(3) or Apery Constant Zeta(1,2) or the derivative of Zeta function at 2 Feigenbaum reduction parameter Feigenbaum bifurcation velocity constant Franson-Robinson constant The Gauss-Kusmin-Wirsing constant Khinchin constant Landau-Ramanujan constant The twin primes constant The Lengyel constant The Levy constant The golden ratio: (1+sqrt(5))/2 to 20000 places. 1.61803398874989484820458683436563811772030917980576286213544862270526046281890 244970720720418939113748475408807538689175212663386222353693179318006076672635 443338908659593958290563832266131992829026788067520876689250171169620703222104 321626954862629631361443814975870122034080588795445474924618569536486444924104 432077134494704956584678850987433944221254487706647809158846074998871240076521 705751797883416625624940758906970400028121042762177111777805315317141011704666 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327197652607107643153112391526077219362144346096089758726934223674331613718574 577608117751518069662104795585140130069701845007026290479492570837120175279378 554957627391245587148332010170361840521636818017341425089806160634676330850504 184585816629334093479199103685913053789482158651701181210113330006695775232786 685518078256752836149494920745837336845813691407977595925267273966423478746614 399819648081036705066005238269165055144634711116867428177319502560642951637959 659475644987891461446925936629309364804816174059808214254340525211371332408113 913579971622858101419103410460569290782498956214560041045692221416830893236662 517618696271719453854998551484275173369241202680159928083201458300754484742331 264387808478085056104304909999364345905195187494843696772757473359670883349609 157447435750398602016397666114276536952670441155200193914842934601015129531174 458876483070371677396154265591399083037577663021309908712719887069032930470124 105861506399852998141757804303480803588203202011047607004755710169423412034108 915643947825303164593730437558194686752534953230130276782353560116641311177996 099793662043449569683547930754311327558643189731515171064432189249793277801264 964764475467078165807406131259375271847408816115479818307816751047809291413954 564631160581269051753953556915775580410671981231638405277556052272223764711883 233223099585068971018717504781906533494858423259762256575841898529144717833517 322602985786292943465056366932162627673816245957417932698892327220666636081992 490988831468529940991386734446049670842442978243630232938910355965601739942201 988690257245471401633009612146187208365108688185334060622017099515827070442337 042180176696349133695996064322005328873494893135966030424380804565944743335678 31672703729636367594216999379522 ------------------------------------------------------------------------------ As calculated by Greg Fee using Maple Release 3 standard Catalan evaluation. This implementation uses 1 bit/term series of Ramanujan. Calculated on April 25 1996 in approx. 10 hours of CPU on a SGI R4000 machine. To do the same on your machine just type this. > catalan := evalf(Catalan,50100): bytes used=37569782748, alloc=5372968, time=38078.95 here are the 50000 digits (1000 lines of 50 digits each). It comes from formula 34.1 of page 293 of Ramanujan Notebooks, part I, the series used is by putting x--> -1/2 . In other words the formula used is: the ordinary formula for Catalan sum((-1)**(n+1)/(2*n+1)**2,n=0..infinity) and then you apply the Euler Transform to it: ref: Abramowitz & Stegun page 16. The article of Greg Fee that took those formulas appear in Computation of Catalan's constant using Ramanujan's Formula, by Greg Fee, ACM 1990, Proceedings of the ISAAC conference, 1990 (MAYBE 1989), held in Tokyo. catalan := 0. 91596559417721901505460351493238411077414937428167 21342664981196217630197762547694793565129261151062 48574422619196199579035898803325859059431594737481 15840699533202877331946051903872747816408786590902 47064841521630002287276409423882599577415088163974 70252482011560707644883807873370489900864775113225 99713434074854075532307685653357680958352602193823 23950800720680355761048235733942319149829836189977 06903640418086217941101917532743149978233976105512 24779530324875371878665828082360570225594194818097 53509711315712615804242723636439850017382875977976 53068370092980873887495610893659771940968726844441 66804621624339864838916280448281506273022742073884 31172218272190472255870531908685735423498539498309 91911596738846450861515249962423704374517773723517 75440708538464401321748392999947572446199754961975 87064007474870701490937678873045869979860644874974 64387206238513712392736304998503539223928787979063 36440323547845358519277777872709060830319943013323 16712476158709792455479119092126201854803963934243 49565375967394943547300143851807050512507488613285 64129344959502298722983162894816461622573989476231 81954200660718814275949755995898363730376753385338 13545031276817240118140721534688316835681686393272 93677586673925839540618033387830687064901433486017 29810699217995653095818715791155395603668903699049 39667538437758104931899553855162621962533168040162 73752130120940604538795076053827123197467900882369 17861557338912441722383393814812077599429849172439 76685756327180688082799829793788494327249346576074 90543874819526813074437046294635892810276531705076 54797449483994895947709278859119584872412786608408 85545978238124922605056100945844866989585768716111 71786662336847409949385541321093755281815525881591 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33977719479190274515412025099252587581579966827349 68448163562248680287787366054173416999163765665385 81095634919642246410021617453036166646767784493191 68768803760434245568232540632588889051025512587238 70200945074599656822076820223724012402082094281368 31620711042665728552343755203994535506870498582151 88144479894668176359043572476748738703131386727366 64237695726203461305176208696122708343562712933227 54244814441539762720328603234137383411832281787718 95907561204744686092884436650706532640140593743156 17021250121528978294584639139195216712554159112036 62617446988989738953558976129096935643101760553413 09761620077831766379383917299182221777169985552763 18764153621590502956600654056094291266739105832587 57789975918507663262661937192596555297915514056319 13209645903926242144949179598676274187517065656445 26851958812965483428820356484099912746214581149876 59408744652471538986596487203211007939581684178653 88761775030424132556818376436243089902251005891866 32528680409897695265152076541938056429617775037838 42477034169745357141688754466614766850847279648458 46570622777433415111929975121204331040549229806244 30619707801248858339640108442089549726785366033283 48370824846333642251242034255438554747923490542487 36973241215506297769504957694032259884052732383383 87007735474942066252923855687547595283673618969505 03254975105475685582904264280288223133583017113458 61975019784420232041785756047105885909576234677056 50430778775853587154899385008551889511670055906314 64963846456224224185199851310612230676179904894428 48831782517506544346538425133271300230240438306247 26219830610476018589074943812150970837649929489684 83123978975240059120343706965306631739923470616819 68172026390046232808553407601396429236442012304767 82267038336232588525467626853636782615075340531593 37313071457898910856926960637345974288402225005290 15520998884042809345391721847580917037513060813642 09848557290766958284218557360999719764035404970757 03445820150158257028881134452044496425524101031249 67662635529318143494937932601048277243418470639942 89534462222004129934514220744814190333789315550685 22756309684544363912187670081612847742123338998836 14812382079042635267202161427485062261887583694841 91172479483300041582873400512340189492090287674178 65440211256360975020769150161031730685245191308694 91594569139279624592810899540762589954635485223854 48757994147093065077267684512505887349957817689055 28225674785956202175853620241580733897536259435693 19362751420235075212169619136721223793548343218975 75221937495115660983627436716746887603997456876267 78025082784567540835581457350675069310164155301861 75479366277294083492223487189377371193343564738314 47218094113538389434288346905494847215674948971663 57418853498986877125372894967217015374967204028920 ------------------------------------------------------------------------------ 170000 digits of gamma, as calculated from a value furnished by Jon Borwein. gamma or Euler constant is Lim(n->infinity) {sum(1/k,k=1..n) - log(n)} .57721566490153286060651209008240243104215933593992359880576723488486772677766 467093694706329174674951463144724980708248096050401448654283622417399764492353 625350033374293733773767394279259525824709491600873520394816567085323315177661 152862119950150798479374508570574002992135478614669402960432542151905877553526 733139925401296742051375413954911168510280798423487758720503843109399736137255 306088933126760017247953783675927135157722610273492913940798430103417771778088 154957066107501016191663340152278935867965497252036212879226555953669628176388 792726801324310104765059637039473949576389065729679296010090151251959509222435 014093498712282479497471956469763185066761290638110518241974448678363808617494 551698927923018773910729457815543160050021828440960537724342032854783670151773 943987003023703395183286900015581939880427074115422278197165230110735658339673 487176504919418123000406546931429992977795693031005030863034185698032310836916 400258929708909854868257773642882539549258736295961332985747393023734388470703 702844129201664178502487333790805627549984345907616431671031467107223700218107 450444186647591348036690255324586254422253451813879124345735013612977822782881 489459098638460062931694718871495875254923664935204732436410972682761608775950 880951262084045444779922991572482925162512784276596570832146102982146179519579 590959227042089896279712553632179488737642106606070659825619901028807561251991 375116782176436190570584407835735015800560774579342131449885007864151716151945 657061704324507500816870523078909370461430668481791649684254915049672431218378 387535648949508684541023406016225085155838672349441878804409407701068837951113 078720234263952269209716088569083825113787128368204911789259447848619911852939 102930990592552669172744689204438697111471745715745732039352091223160850868275 588901094516811810168749754709693666712102063048271658950493273148608749402070 067425909182487596213738423114426531350292303175172257221628324883811245895743 862398703757662855130331439299954018531341415862127886480761100301521196578006 811777376350168183897338966398689579329914563886443103706080781744899579583245 794189620260498410439225078604603625277260229196829958609883390137871714226917 883819529844560791605197279736047591025109957791335157917722515025492932463250 287476779484215840507599290401855764599018626926776437266057117681336559088155 481074700006233637252889495546369714330120079130855526395954978230231440391497 404947468259473208461852460587766948828795301040634917229218580087067706904279 267432844469685149718256780958416544918514575331964063311993738215734508749883 255608888735280190191550896885546825924544452772817305730108060617701136377318 246292466008127716210186774468495951428179014511194893422883448253075311870186 097612246231767497755641246198385640148412358717724955422482016151765799408062 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486979597694996533527210020020967910439482547244113342244870054637656840866762 153373627461591205470086290571698257353705239761231238412564349411789498615828 597020971097039196352168122582247562719519382728285209147182375343655254020746 203066730476952470094413814561782826663196139673593672572646033884946424894724 489784815967153061538467122369871282319784769311057166236373262075959711804015 143896231706019585709823813924666135279136956376559048616762051297409983149655 978602534101294543998672883006244084401861815117506870887646716097992920177696 249633015758292596188494858720022922480606288121778733831458825512939536510881 918651200449231549839477314735786889731420473109453814648226321023310794395974 628523541295751910635587923001956181312946120301575763401597856817517498423778 824473753982474575996867707408545574329424026781938482203540962106060721924990 825104854003184963199863221569089097614053107165113129326853498524402864482934 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325830253832126211335913998147254731088561103502523150232260764659143479577070 212255923857619651541130675902020771818471255444048094317583914104182857184684 254356428021148612817114059931241855097230094836718158216635032352144301276029 408110483083684033797276910734947127639485242365436010121611472601302712302114 936394618640713323418155883170291132471155668517400914663308941679541388606459 041308602475814809253881136334315220213957367287804727138375085893419354244464 347169486851505717320834190054678315225496047820058249682459499423860216615636 221511785277723734839254123719912398519291223166073953380408600317855535616697 152666159714298093829640817484067595658252229511494206329027779249773802025863 049002338914339161519730475496173930034909831850312164452992323900730390224074 960970238606174088433933033116621413766889370324384494435952138468815104628978 355361643207570463255409224335023740400655662384598817253627657820515918887137 428224533698114375777755070470831999858049255399615577688353196603888411154532 510323408417668685656064533645932582928561534870050036225742135675646177049353 802796329267220164454445475026639009734156461408161237802378693542006719308703 239489020605194625691681219163898897868246791575908260118074941898215409019644 400609858251539514525840505405733419715524119332247162481715869590459607132227 120270750038073415057285273051413461724364324350759083064442511793105821732556 350937000463861006031635526626081975615057281743286197573113312439953870280147 398646481951044043706171345291375172436515853579700413959268525998127206830937 440974827426549100448972845133920884290100392354174550892017405721341477968861 838150118728784484229782013346708946882973957166350009120024170327964084767582 879474715637448850019020912066997357415500918386337344316273385635359478187486 231891538682917624634816247070727154605269563573473480946626498924617401296277 765764623728025927779480657766730119544692743950126338678444030100649194269469 957667185565363382615662324210286732044396948485770363843327964356245450681615 513771880359763392935479951262966854820586892757707754826468364824774254559020 807145925453009353453225487341641107976962272723879767243025919066038955071383 136691818152342825205396838698720024030663475975605134609154469548162387425378 158302653098104742760607748290171664036336150871497476588172192978853908464446 528705641619046836523612432367732528674613688684282872712218847056520735270032 365972755740237383131307183137426933651450657979976777726799881447914358583652 734628844838133347320253032562039218550423745071246201702474289175965495898199 655426861179406034823110616497789765805879019925874069554366793037911629180282 626336505128834652109745963364668600722087891966477672374488583609530207665510 278220140742685179742262555195530120108977140157868272506731574046078508917622 773625815367966375862818839152555955251613018703144216909288827431719952956153 271512085651116775294542856160716086003685818294277547141318914905593218301506 196236101121808775264666512306459999615933951176549704193364674478497559809435 239357666243284437023988696401177364692704574611809528748877957078110501719666 214766418626949735746138893917281555165412641863779432045863818098406043470354 870027768855685015445395272991659315976801943009541965850172319561870163232017 714943145752454182789724144200910922235877784762143982533049141480551884762764 035237769794587133262129421332780056882892714132334279085669544052271630773613 222223624476492604444925157608167205125791983401414314117462430350958588366449 885635241216735096498110886480454556121486157144398897012515745641666786086974 599235092306913787822489885741010152307835427302686702643076573361527388044793 980964034533893414269988016666251433188572227755985299273106746685733619787048 404116734845406324070562038620208792937068210443308574090272915222256690268914 154418055934840966397852111013396317334823271359536191364785487924262667118227 265433909164704496304973642336756093387686017097896893746341790095852105996698 311184947184187211471898960323813655945633958099118010918898042657125950534933 432590632358450814434844929283495242487228534842158559858227162257611210612276 943587121299259340778966149794293635945828339925529844938205488768791335589322 378202474504684456545675207302749752781246295549575961201910351796377991332194 760924906812854053093940777922686627236493385363777797775841839985573147572508 278278133424401285982580466473015242013987134640272855003454864134949658500406 325010032627834898309246405253515128757565780041454196947196829194129125382322 323040242780772044913500813181044059147935301275937249608282170902630073205555 942055092241584936796162684784919911479278169140343465155973137653133769095915 408557353221970821520200012878529148624180724372726554026068919977669095929312 884966544823351307602582047220179815182981537115752145138942540406776414150378 261542887352743572109269918298733784934925067944884528014286190663864893203832 262475585291450877811812606609287425948652977367927474274345444494299917920010 592825788588113902413233669323988364616559366434889317217046632974698473209548 716538766597259787417373930873958472547273737620357931162811189212024790071245 606469357417284761827083359348188921482659253659222451433399363951099458994696 946356256322943123704715769176584334385021496772976038499059194240009998943838 797082608197390942977149427159059509449833904644963958202273423071851709842599 767135871155365317731794555314716713826977065239562572301017029691252626488197 191233757103889948396553423171412191470103447526027865609798213571692749883016 370886297798018562692270340541981638788082598785178596072450292199365464328545 845537098344079433001582732447215368560726924499924496560890592414895661596875 510475437876313196604588788445159012313418329514858528887605451799863092365858 284379784336048329938764878904605287054107746539136015280698825897759306969126 743288071527820619474346504546690173027794183372811515486678727546215387379115 550722143800612230582630841490858633187051228683676576287111524137827187928358 183112019706088688696349390736783165658883202173568093943999985504687912622999 501403710446795445902947321277458468364057512251147621689571851825122939713687 690034964094187762776565918785051166359770911611077709056893042232586633880423 332641476369701384662384172105006157634901704929732854275295654961204015190472 324601956711162863712879382278936420806227252391446338575146990663577833960269 166968095423399522369570065424450232261096844773919834646255336098360100685095 774964533666208297222159707073618423201453895130160080791291134744405932067551 763295733958650556239133883104100736158904717574756529573433505140903826999699 603607039261188627921921196874050802238921046483609662032780823943745532697338 678333874962639215512698373936532607451211556697184835766320687895642282136622 715867419516911165836626866744045455083127447812394910099323884790018944384080 848875050517056179744159971295571413750822633140320665846277239471306970143937 789246958168275660874369056961391946014632458426105667029785928612327249477805 277259332199024776783298212540205132110141164448518460584779710688655848291095 318621147278146482111054771441916191172 ------------------------------------------------------------------------------ The Artin's Constant. = product(1-1/(p**2-p),p=prime) 0.373955813619202288054728054346516415111629249 ------------------------------------------------------------------------------ The Backhouse constant calculated by Philippe Flajolet INRIA Paris to 1300 places. 1.4560749485826896713995953511165435576531783748471315402707024 374140015062653898955996453194018603091099251436196347135486077 516491312123142920351770128317405369527499880254869230705808528 451124053000179297856106749197085005775005438769180068803215980 620273634173560481682324390971937912897855009041182006889374170 524605523103968123415765255124331292772157858632005469569315813 246500040902370666667117547152236564044351398169338973930393708 455830836636739542046997815299374792625225091766965656321726658 531118262706074545210728644758644231717911597527697966195100532 506679370361749364973096351160887145901201340918694999972951200 319685565787957715446072017436793132019277084608142589327171752 140350669471255826551253135545512621599175432491768704927031066 824955171959773604447488530521694205264813827872679158267956816 962042960183918841576453649251600489240011190224567845202131844 607922804066771020946499003937697924293579076067914951599294437 906214030884143685764890949235109954378252651983684848569010117 463899184591527039774046676767289711551013271321745464437503346 595005227041415954600886072536255114520109115277724099455296613 699531850998749774202185343255771313121423357927183815991681750 625176199614095578995402529309491627747326701699807286418966752 89794974645089663963739786981613361814875; ------------------------------------------------------------------------------ Zeta(3) or Apery constant to 2000 places. 1.2020569031595942853997381615114499907649862923404988817922715553418382057863 130901864558736093352581461991577952607194184919959986732832137763968372079001 614539417829493600667191915755222424942439615639096641032911590957809655146512 799184051057152559880154371097811020398275325667876035223369849416618110570147 157786394997375237852779370309560257018531827900030765471075630488433208697115 737423807934450316076253177145354444118311781822497185263570918244899879620350 833575617202260339378587032813126780799005417734869115253706562370574409662217 129026273207323614922429130405285553723410330775777980642420243048828152100091 460265382206962715520208227433500101529480119869011762595167636699817183557523 488070371955574234729408359520886166620257285375581307928258648728217370556619 689895266201877681062920081779233813587682842641243243148028217367450672069350 762689530434593937503296636377575062473323992348288310773390527680200757984356 793711505090050273660471140085335034364672248565315181177661810922279191022488 396800266606568705190627597387735357444478775379164142738132256957319602018748 847471046993365661400806930325618537188600727185359482884788624504185554640857 155630071250902713863468937416826654665772926111718246036305660465300475221703 265136391058698857884245041340007617472791371842774108750867905018896539635695 864308196137299023274934970241622645433923929267278367865571555817773966377191 281418224664126866345281105514013167325366841827929537266050341518527048802890 268315833479592038755984988617867005963731015727172000114334767351541882552524 663262972025386614259375933490112495445188844587988365323760500686216425928461 880113716666635035656010025131275200124346538178852251664505673955057386315263 765954302814622423017747501167684457149670488034402130730241278731540290425115 091994087834862014280140407162144654788748177582604206667340250532107702583018 381329938669733199458406232903960570319092726406838808560840747389568335052094 151491733048363304771434582553921221820451656004278 ------------------------------------------------------------------------------ Zeta(1,2) ot the derivative of Zeta function at 2. -0.9375482543158437537025740945678649778978602886148299258854334803 6044381131270752279368941514115151749311382116241638535059404171 5961733247197185174912402688214443700163931015045107160373574873 1352956057133552593318050514872534799984717397570317550302619073 ------------------------------------------------------------------------------ Feigenbaum reduction parameter 2.502907875095892822283902873218215786381271376727149977336192056 Feigenbaum bifurcation velocity constant 4.669201609102990671853203820466201617258185577475768632745651343 00413433021131473 ------------------------------------------------------------------------------ Fransen-Robinson constant. 2.80777024202851936522150118655777293230808592093019829122005 ref: Math of Computation, vol 34 1980 pp 553-566 Math of Computation vol 37 1981 pp 233-235 ------------------------------------------------------------------------------ The Gauss-Kuzmin-Wirsing constant. 0.303663002898732658597448121901 ------------------------------------------------------------------------------ Khinchin constant to 1024 digits. 2.685452001065306445309714835481795693820382293994462953051152345 5572188595371520028011411749318476979951534659052880900828976777 1641096305179253348325966838185231542133211949962603932852204481 9409618068664166428930847788062036073705350103367263357728904990 4270702723451702625237023545810686318501032374655803775026442524 8528694682341899491573066189872079941372355000579357366989339508 7902124464207528974145914769301844905060179349938522547040420337 7985639831015709022233910000220772509651332460444439191691460859 6823482128324622829271012690697418234847767545734898625420339266 2351862086778136650969658314699527183744805401219536666604964826 9890827548115254721177330319675947383719393578106059230401890711 3496246737068412217946810740608918276695667117166837405904739368 8095345048999704717639045134323237715103219651503824698888324870 9353994696082647818120566349467125784366645797409778483662049777 7486827656970871631929385128993141995186116737926546205635059513 8571376169712687229980532767327871051376395637190231452890030581 ------------------------------------------------------------------------------ Landau-Ramanujan constant calculated by Philippe Flajolet INRIA Paris and Paul Zimmermann .76422365358922066299069873125009232811679054139340951472168667374961464165873 285883840150501313123372193726912079259263418742064678084323063315434629380531 605171169636177508819961243824994277683469051623513921871962056905329564467041 917634977065956990571293866028938589982961051662960890991779298360729736972006 403169851286365173473921065768550978681981674707359066921830288751501689624646 710918081710618090086517493799082420450570666204898612757713333895484325083035 682950407721597524121430942470953115765559404064229125772724071563491218723272 555640889999512705135849728552347645942418505999635800934732669411548076911671 455813028066898593167493626295259560163215843892463887558347193993864581698751 045893518777945872755226448709943505595943671299977780669880564555921300690852 242867691102264527531455816088116296997029876937094388422089495290791626363527 791432286156863284215944899347183748322904155863814951281527102068249218645827 978145098870379211809629840943604891233924014852514327407923660178532707078811 584944045092539519718157085780907690772192962552262890529967200510669638584207 655081660527132551761150093619010182152039541621744474356571314026496051480322 439134457528009739604967190734667398621127034770623094786463721777245551191609 693349580116501538146897732947400254272699518373881294004390465050310091210361 980535760952228835847669743267507757379848939356645406017251962513826671863828 822629657399438626453078913514555113206475947913245582423662405126070382560901 984614575152951511943211356814416716008974384391847402590826495013602834007260 634108659796382596784136373377680857831279147106417370573337040146024737648200 768231118490558678994106995743922457089666910491534089500139419890965785853368 531985664042350494746329804481593573838687414276915611134778612290893976432134 279879206472381493290546264824907766030881348705331723336407298994245656611424 036824812873959790915799781062723446426357233234127834780836022424212901203199 698485951429216878840715626887034517436895639117072657935407050794141100343395 582796409783891583020407548189623248280478295465239223872194333981851251004747 915658247782096645428132405620504841629632689157664149594957463505689486587289 243413739100608393347108455293656982935338521814358746992934863313565820307192 052961665775757266627408455837127946799180904710259452519968016372631267038023 447298309515688684101430849594108797807013561524049847909714362331059569179431 431691402111049142985963053516771600084867260895318575293282183754558954666446 191468252314874744996401074402664686009572448671687582605311706563448494800841 013235616345298355661883397936163440983329415351662473668696589017509927510426 179402465228834883071274736258665127650610546028964911287287862738263596129333 646532191462193644375375986052224103348422135461338128126772131245890101073371 833507228401529641174179963903413664423919027895258486882959689695414733771140 886608735962775876801352400029217923528552903064508042269791458998101532140852 906522851155926562268843097382297291671643980514954868377297117263777565889196 871160509021211371651902341089659293730249465771175004282806372542711805610086 582199353671242849838516132864990427711284071698787833006397284922820344349455 347600218003759357103205064406756731817833717035278315824041187154686581171864 952708996224348497793884832819497863569163437333174207210017054301866996336546 345356688803248528083754492545177937909480719571292537184111634137303745351158 808536058387947244006076177381212102789234193301297654667548505996663777643498 372821340898092902889278088428356657996352849776141925529724571724867587136418 201398244839217272327643398650709245008820337499415470402244860516920495403808 927989711051717768667304408545186666225235192118021635892919167850976801426649 352465241315279351720847405377357578042399840807239581517870192914436593403426 041472365945332028684204462087017959678507105044335899885299980564606846787225 732214957985497309246622145607355305369122171822980735330797038775292460178947 026111522360709501432667888113281990837513268845551510892190791002610859910028 841896531906865518597376800549155279239015601038178005539072818968126480022740 306581014459893635059496241529118447243731448964790876170326036539290133430684 781818206966837601732903016602762498422827578841548456314713744987325573405808 059688168787542109613848415947550297583968310995807378723577778742169704341423 897424468452868147350993296144684543688264968869192161649670355650366847261236 344317060911028964507813116327249011793664610722979820574075779145800908301060 913216200897249373201032777744785128764388089981656932715040612090221827393598 823187046383122896921216104846546948319792790320515076726732119163218284371172 036046713301772401534216569368578887782115998981797541546286000230716000159982 248277212628384883047943066328546836223944311977655842794918993522521563237555 650145286211059365868171399984063710196404159462648235183782162356589166387664 599049458627927377620486160161495004228978098646407047162007645445619312738742 470000474 ------------------------------------------------------------------------------ The twin primes constant. 0.660161815846869573927812110014555778432623 ------------------------------------------------------------------------------ The Lengyel constant. 1.09868580552518701 ------------------------------------------------------------------------------ The Levy constant. 3.275822918721811159787681882453843863608475525982374149405198924 1907232156449603551812775404791745294926985262434016333281898085 1150341709970823046646564670370807129022418613959423772012981792 4251087697614930028806824926170594041290808697054441234922379888 4427089726916409835535854804837478582876252691844500764338370387 6741884459420351700003732122230624193601340916574804354470221732 5993822298382263233915155285779861204684944725214872290148093612 0890454342092099519978162400515039051381538771475429139307394617 8045527838277727414847221513498826672911215448270452018633487650 0197461804182629653652880116815110575135087115230163029427488336 4626099362896606336377631721650883480193611175632669841114675716 4673788536609062436703325968873753578142654195992507552782054563 7159407017761632115357055923392379438429270901774392301637440604 1267849616074783099029968056083369047909232336111034927147379005 2538954916571473419972293167664630764119832764653881696204634629 7332994286181358520050377154266892424005731518557308952871781294 
16713	----	[Transcribers note: Many of the puzzles in this book assume a familiarity with the currency of Great Britain in the early 1900s. As this is likely not common knowledge for those outside Britain (and possibly many within,) I am including a chart of relative values. The most common units used were: the Penny, abbreviated: d. (from the Roman penny, denarius) the Shilling, abbreviated: s. the Pound, abbreviated: £ There was 12 Pennies to a Shilling and 20 Shillings to a Pound, so there was 240 Pennies in a Pound. To further complicate things, there were many coins which were various fractional values of Pennies, Shillings or Pounds. Farthing ¼d. Half-penny ½d. Penny 1d. Three-penny 3d. Sixpence (or tanner) 6d. Shilling (or bob) 1s. Florin or two shilling piece 2s. Half-crown (or half-dollar) 2s. 6d. Double-florin 4s. Crown (or dollar) 5s. Half-Sovereign 10s. Sovereign (or Pound) £1 or 20s. This is by no means a comprehensive list, but it should be adequate to solve the puzzles in this book. Exponents are represented in this text by ^, e.g. '3 squared' is 3^2. Numbers with fractional components (other than ¼, ½ and ¾) have a + symbol separating the whole number component from the fraction. It makes the fraction look odd, but yeilds correct solutions no matter how it is interpreted. E.G., 4 and eleven twenty-thirds is 4+11/23, not 411/23 or 4-11/23. ] AMUSEMENTS IN MATHEMATICS by HENRY ERNEST DUDENEY In Mathematicks he was greater Than Tycho Brahe or Erra Pater: For he, by geometrick scale, Could take the size of pots of ale; Resolve, by sines and tangents, straight, If bread or butter wanted weight; And wisely tell what hour o' th' day The clock does strike by algebra. BUTLER'S _Hudibras_. 1917 PREFACE In issuing this volume of my Mathematical Puzzles, of which some have appeared in periodicals and others are given here for the first time, I must acknowledge the encouragement that I have received from many unknown correspondents, at home and abroad, who have expressed a desire to have the problems in a collected form, with some of the solutions given at greater length than is possible in magazines and newspapers. Though I have included a few old puzzles that have interested the world for generations, where I felt that there was something new to be said about them, the problems are in the main original. It is true that some of these have become widely known through the press, and it is possible that the reader may be glad to know their source. On the question of Mathematical Puzzles in general there is, perhaps, little more to be said than I have written elsewhere. The history of the subject entails nothing short of the actual story of the beginnings and development of exact thinking in man. The historian must start from the time when man first succeeded in counting his ten fingers and in dividing an apple into two approximately equal parts. Every puzzle that is worthy of consideration can be referred to mathematics and logic. Every man, woman, and child who tries to "reason out" the answer to the simplest puzzle is working, though not of necessity consciously, on mathematical lines. Even those puzzles that we have no way of attacking except by haphazard attempts can be brought under a method of what has been called "glorified trial"--a system of shortening our labours by avoiding or eliminating what our reason tells us is useless. It is, in fact, not easy to say sometimes where the "empirical" begins and where it ends. When a man says, "I have never solved a puzzle in my life," it is difficult to know exactly what he means, for every intelligent individual is doing it every day. The unfortunate inmates of our lunatic asylums are sent there expressly because they cannot solve puzzles--because they have lost their powers of reason. If there were no puzzles to solve, there would be no questions to ask; and if there were no questions to be asked, what a world it would be! We should all be equally omniscient, and conversation would be useless and idle. It is possible that some few exceedingly sober-minded mathematicians, who are impatient of any terminology in their favourite science but the academic, and who object to the elusive x and y appearing under any other names, will have wished that various problems had been presented in a less popular dress and introduced with a less flippant phraseology. I can only refer them to the first word of my title and remind them that we are primarily out to be amused--not, it is true, without some hope of picking up morsels of knowledge by the way. If the manner is light, I can only say, in the words of Touchstone, that it is "an ill-favoured thing, sir, but my own; a poor humour of mine, sir." As for the question of difficulty, some of the puzzles, especially in the Arithmetical and Algebraical category, are quite easy. Yet some of those examples that look the simplest should not be passed over without a little consideration, for now and again it will be found that there is some more or less subtle pitfall or trap into which the reader may be apt to fall. It is good exercise to cultivate the habit of being very wary over the exact wording of a puzzle. It teaches exactitude and caution. But some of the problems are very hard nuts indeed, and not unworthy of the attention of the advanced mathematician. Readers will doubtless select according to their individual tastes. In many cases only the mere answers are given. This leaves the beginner something to do on his own behalf in working out the method of solution, and saves space that would be wasted from the point of view of the advanced student. On the other hand, in particular cases where it seemed likely to interest, I have given rather extensive solutions and treated problems in a general manner. It will often be found that the notes on one problem will serve to elucidate a good many others in the book; so that the reader's difficulties will sometimes be found cleared up as he advances. Where it is possible to say a thing in a manner that may be "understanded of the people" generally, I prefer to use this simple phraseology, and so engage the attention and interest of a larger public. The mathematician will in such cases have no difficulty in expressing the matter under consideration in terms of his familiar symbols. I have taken the greatest care in reading the proofs, and trust that any errors that may have crept in are very few. If any such should occur, I can only plead, in the words of Horace, that "good Homer sometimes nods," or, as the bishop put it, "Not even the youngest curate in my diocese is infallible." I have to express my thanks in particular to the proprietors of _The Strand Magazine_, _Cassell's Magazine_, _The Queen_, _Tit-Bits_, and _The Weekly Dispatch_ for their courtesy in allowing me to reprint some of the puzzles that have appeared in their pages. THE AUTHORS' CLUB _March_ 25, 1917 CONTENTS PREFACE v ARITHMETICAL AND ALGEBRAICAL PROBLEMS 1 Money Puzzles 1 Age and Kinship Puzzles 6 Clock Puzzles 9 Locomotion and Speed Puzzles 11 Digital Puzzles 13 Various Arithmetical and Algebraical Problems 17 GEOMETRICAL PROBLEMS 27 Dissection Puzzles 27 Greek Cross Puzzles 28 Various Dissection Puzzles 35 Patchwork Puzzles 46 Various Geometrical Puzzles 49 POINTS AND LINES PROBLEMS 56 MOVING COUNTER PROBLEMS 58 UNICURSAL AND ROUTE PROBLEMS 68 COMBINATION AND GROUP PROBLEMS 76 CHESSBOARD PROBLEMS 85 The Chessboard 85 Statical Chess Puzzles 88 The Guarded Chessboard 95 Dynamical Chess Puzzles 96 Various Chess Puzzles 105 MEASURING, WEIGHING, AND PACKING PUZZLES 109 CROSSING RIVER PROBLEMS 112 PROBLEMS CONCERNING GAMES 114 PUZZLE GAMES 117 MAGIC SQUARE PROBLEMS 119 Subtracting, Multiplying, and Dividing Magics 124 Magic Squares of Primes 125 MAZES AND HOW TO THREAD THEM 127 THE PARADOX PARTY 137 UNCLASSIFIED PROBLEMS 142 SOLUTIONS 148 INDEX 253 AMUSEMENTS IN MATHEMATICS. ARITHMETICAL AND ALGEBRAICAL PROBLEMS. "And what was he? Forsooth, a great arithmetician." _Othello_, I. i. The puzzles in this department are roughly thrown together in classes for the convenience of the reader. Some are very easy, others quite difficult. But they are not arranged in any order of difficulty--and this is intentional, for it is well that the solver should not be warned that a puzzle is just what it seems to be. It may, therefore, prove to be quite as simple as it looks, or it may contain some pitfall into which, through want of care or over-confidence, we may stumble. Also, the arithmetical and algebraical puzzles are not separated in the manner adopted by some authors, who arbitrarily require certain problems to be solved by one method or the other. The reader is left to make his own choice and determine which puzzles are capable of being solved by him on purely arithmetical lines. MONEY PUZZLES. "Put not your trust in money, but put your money in trust." OLIVER WENDELL HOLMES. 1.--A POST-OFFICE PERPLEXITY. In every business of life we are occasionally perplexed by some chance question that for the moment staggers us. I quite pitied a young lady in a branch post-office when a gentleman entered and deposited a crown on the counter with this request: "Please give me some twopenny stamps, six times as many penny stamps, and make up the rest of the money in twopence-halfpenny stamps." For a moment she seemed bewildered, then her brain cleared, and with a smile she handed over stamps in exact fulfilment of the order. How long would it have taken you to think it out? 2.--YOUTHFUL PRECOCITY. The precocity of some youths is surprising. One is disposed to say on occasion, "That boy of yours is a genius, and he is certain to do great things when he grows up;" but past experience has taught us that he invariably becomes quite an ordinary citizen. It is so often the case, on the contrary, that the dull boy becomes a great man. You never can tell. Nature loves to present to us these queer paradoxes. It is well known that those wonderful "lightning calculators," who now and again surprise the world by their feats, lose all their mysterious powers directly they are taught the elementary rules of arithmetic. A boy who was demolishing a choice banana was approached by a young friend, who, regarding him with envious eyes, asked, "How much did you pay for that banana, Fred?" The prompt answer was quite remarkable in its way: "The man what I bought it of receives just half as many sixpences for sixteen dozen dozen bananas as he gives bananas for a fiver." Now, how long will it take the reader to say correctly just how much Fred paid for his rare and refreshing fruit? 3.--AT A CATTLE MARKET. Three countrymen met at a cattle market. "Look here," said Hodge to Jakes, "I'll give you six of my pigs for one of your horses, and then you'll have twice as many animals here as I've got." "If that's your way of doing business," said Durrant to Hodge, "I'll give you fourteen of my sheep for a horse, and then you'll have three times as many animals as I." "Well, I'll go better than that," said Jakes to Durrant; "I'll give you four cows for a horse, and then you'll have six times as many animals as I've got here." No doubt this was a very primitive way of bartering animals, but it is an interesting little puzzle to discover just how many animals Jakes, Hodge, and Durrant must have taken to the cattle market. 4.--THE BEANFEAST PUZZLE. A number of men went out together on a bean-feast. There were four parties invited--namely, 25 cobblers, 20 tailors, 18 hatters, and 12 glovers. They spent altogether £6, 13s. It was found that five cobblers spent as much as four tailors; that twelve tailors spent as much as nine hatters; and that six hatters spent as much as eight glovers. The puzzle is to find out how much each of the four parties spent. 5.--A QUEER COINCIDENCE. Seven men, whose names were Adams, Baker, Carter, Dobson, Edwards, Francis, and Gudgeon, were recently engaged in play. The name of the particular game is of no consequence. They had agreed that whenever a player won a game he should double the money of each of the other players--that is, he was to give the players just as much money as they had already in their pockets. They played seven games, and, strange to say, each won a game in turn, in the order in which their names are given. But a more curious coincidence is this--that when they had finished play each of the seven men had exactly the same amount--two shillings and eightpence--in his pocket. The puzzle is to find out how much money each man had with him before he sat down to play. 6.--A CHARITABLE BEQUEST. A man left instructions to his executors to distribute once a year exactly fifty-five shillings among the poor of his parish; but they were only to continue the gift so long as they could make it in different ways, always giving eighteenpence each to a number of women and half a crown each to men. During how many years could the charity be administered? Of course, by "different ways" is meant a different number of men and women every time. 7.--THE WIDOW'S LEGACY. A gentleman who recently died left the sum of £8,000 to be divided among his widow, five sons, and four daughters. He directed that every son should receive three times as much as a daughter, and that every daughter should have twice as much as their mother. What was the widow's share? 8.--INDISCRIMINATE CHARITY. A charitable gentleman, on his way home one night, was appealed to by three needy persons in succession for assistance. To the first person he gave one penny more than half the money he had in his pocket; to the second person he gave twopence more than half the money he then had in his pocket; and to the third person he handed over threepence more than half of what he had left. On entering his house he had only one penny in his pocket. Now, can you say exactly how much money that gentleman had on him when he started for home? 9.--THE TWO AEROPLANES. A man recently bought two aeroplanes, but afterwards found that they would not answer the purpose for which he wanted them. So he sold them for £600 each, making a loss of 20 per cent. on one machine and a profit of 20 per cent. on the other. Did he make a profit on the whole transaction, or a loss? And how much? 10.--BUYING PRESENTS. "Whom do you think I met in town last week, Brother William?" said Uncle Benjamin. "That old skinflint Jorkins. His family had been taking him around buying Christmas presents. He said to me, 'Why cannot the government abolish Christmas, and make the giving of presents punishable by law? I came out this morning with a certain amount of money in my pocket, and I find I have spent just half of it. In fact, if you will believe me, I take home just as many shillings as I had pounds, and half as many pounds as I had shillings. It is monstrous!'" Can you say exactly how much money Jorkins had spent on those presents? 11.--THE CYCLISTS' FEAST. 'Twas last Bank Holiday, so I've been told, Some cyclists rode abroad in glorious weather. Resting at noon within a tavern old, They all agreed to have a feast together. "Put it all in one bill, mine host," they said, "For every man an equal share will pay." The bill was promptly on the table laid, And four pounds was the reckoning that day. But, sad to state, when they prepared to square, 'Twas found that two had sneaked outside and fled. So, for two shillings more than his due share Each honest man who had remained was bled. They settled later with those rogues, no doubt. How many were they when they first set out? 12.--A QUEER THING IN MONEY. It will be found that £66, 6s. 6d. equals 15,918 pence. Now, the four 6's added together make 24, and the figures in 15,918 also add to 24. It is a curious fact that there is only one other sum of money, in pounds, shillings, and pence (all similarly repetitions of one figure), of which the digits shall add up the same as the digits of the amount in pence. What is the other sum of money? 13.--A NEW MONEY PUZZLE. The largest sum of money that can be written in pounds, shillings, pence, and farthings, using each of the nine digits once and only once, is £98,765, 4s. 3½d. Now, try to discover the smallest sum of money that can be written down under precisely the same conditions. There must be some value given for each denomination--pounds, shillings, pence, and farthings--and the nought may not be used. It requires just a little judgment and thought. 14.--SQUARE MONEY. "This is queer," said McCrank to his friend. "Twopence added to twopence is fourpence, and twopence multiplied by twopence is also fourpence." Of course, he was wrong in thinking you can multiply money by money. The multiplier must be regarded as an abstract number. It is true that two feet multiplied by two feet will make four square feet. Similarly, two pence multiplied by two pence will produce four square pence! And it will perplex the reader to say what a "square penny" is. But we will assume for the purposes of our puzzle that twopence multiplied by twopence is fourpence. Now, what two amounts of money will produce the next smallest possible result, the same in both cases, when added or multiplied in this manner? The two amounts need not be alike, but they must be those that can be paid in current coins of the realm. 15.--POCKET MONEY. What is the largest sum of money--all in current silver coins and no four-shilling piece--that I could have in my pocket without being able to give change for a half-sovereign? 16.--THE MILLIONAIRE'S PERPLEXITY. Mr. Morgan G. Bloomgarten, the millionaire, known in the States as the Clam King, had, for his sins, more money than he knew what to do with. It bored him. So he determined to persecute some of his poor but happy friends with it. They had never done him any harm, but he resolved to inoculate them with the "source of all evil." He therefore proposed to distribute a million dollars among them and watch them go rapidly to the bad. But he was a man of strange fancies and superstitions, and it was an inviolable rule with him never to make a gift that was not either one dollar or some power of seven--such as 7, 49, 343, 2,401, which numbers of dollars are produced by simply multiplying sevens together. Another rule of his was that he would never give more than six persons exactly the same sum. Now, how was he to distribute the 1,000,000 dollars? You may distribute the money among as many people as you like, under the conditions given. 17.--THE PUZZLING MONEY-BOXES. Four brothers--named John, William, Charles, and Thomas--had each a money-box. The boxes were all given to them on the same day, and they at once put what money they had into them; only, as the boxes were not very large, they first changed the money into as few coins as possible. After they had done this, they told one another how much money they had saved, and it was found that if John had had 2s. more in his box than at present, if William had had 2s. less, if Charles had had twice as much, and if Thomas had had half as much, they would all have had exactly the same amount. Now, when I add that all four boxes together contained 45s., and that there were only six coins in all in them, it becomes an entertaining puzzle to discover just what coins were in each box. 18.--THE MARKET WOMEN. A number of market women sold their various products at a certain price per pound (different in every case), and each received the same amount--2s. 2½d. What is the greatest number of women there could have been? The price per pound in every case must be such as could be paid in current money. 19.--THE NEW YEAR'S EVE SUPPERS. The proprietor of a small London café has given me some interesting figures. He says that the ladies who come alone to his place for refreshment spend each on an average eighteenpence, that the unaccompanied men spend half a crown each, and that when a gentleman brings in a lady he spends half a guinea. On New Year's Eve he supplied suppers to twenty-five persons, and took five pounds in all. Now, assuming his averages to have held good in every case, how was his company made up on that occasion? Of course, only single gentlemen, single ladies, and pairs (a lady and gentleman) can be supposed to have been present, as we are not considering larger parties. 20.--BEEF AND SAUSAGES. "A neighbour of mine," said Aunt Jane, "bought a certain quantity of beef at two shillings a pound, and the same quantity of sausages at eighteenpence a pound. I pointed out to her that if she had divided the same money equally between beef and sausages she would have gained two pounds in the total weight. Can you tell me exactly how much she spent?" "Of course, it is no business of mine," said Mrs. Sunniborne; "but a lady who could pay such prices must be somewhat inexperienced in domestic economy." "I quite agree, my dear," Aunt Jane replied, "but you see that is not the precise point under discussion, any more than the name and morals of the tradesman." 21.--A DEAL IN APPLES. I paid a man a shilling for some apples, but they were so small that I made him throw in two extra apples. I find that made them cost just a penny a dozen less than the first price he asked. How many apples did I get for my shilling? 22.--A DEAL IN EGGS. A man went recently into a dairyman's shop to buy eggs. He wanted them of various qualities. The salesman had new-laid eggs at the high price of fivepence each, fresh eggs at one penny each, eggs at a halfpenny each, and eggs for electioneering purposes at a greatly reduced figure, but as there was no election on at the time the buyer had no use for the last. However, he bought some of each of the three other kinds and obtained exactly one hundred eggs for eight and fourpence. Now, as he brought away exactly the same number of eggs of two of the three qualities, it is an interesting puzzle to determine just how many he bought at each price. 23.--THE CHRISTMAS-BOXES. Some years ago a man told me he had spent one hundred English silver coins in Christmas-boxes, giving every person the same amount, and it cost him exactly £1, 10s. 1d. Can you tell just how many persons received the present, and how he could have managed the distribution? That odd penny looks queer, but it is all right. 24.--A SHOPPING PERPLEXITY. Two ladies went into a shop where, through some curious eccentricity, no change was given, and made purchases amounting together to less than five shillings. "Do you know," said one lady, "I find I shall require no fewer than six current coins of the realm to pay for what I have bought." The other lady considered a moment, and then exclaimed: "By a peculiar coincidence, I am exactly in the same dilemma." "Then we will pay the two bills together." But, to their astonishment, they still required six coins. What is the smallest possible amount of their purchases--both different? 25.--CHINESE MONEY. The Chinese are a curious people, and have strange inverted ways of doing things. It is said that they use a saw with an upward pressure instead of a downward one, that they plane a deal board by pulling the tool toward them instead of pushing it, and that in building a house they first construct the roof and, having raised that into position, proceed to work downwards. In money the currency of the country consists of taels of fluctuating value. The tael became thinner and thinner until 2,000 of them piled together made less than three inches in height. The common cash consists of brass coins of varying thicknesses, with a round, square, or triangular hole in the centre, as in our illustration. [Illustration] These are strung on wires like buttons. Supposing that eleven coins with round holes are worth fifteen ching-changs, that eleven with square holes are worth sixteen ching-changs, and that eleven with triangular holes are worth seventeen ching-changs, how can a Chinaman give me change for half a crown, using no coins other than the three mentioned? A ching-chang is worth exactly twopence and four-fifteenths of a ching-chang. 26.--THE JUNIOR CLERK'S PUZZLE. Two youths, bearing the pleasant names of Moggs and Snoggs, were employed as junior clerks by a merchant in Mincing Lane. They were both engaged at the same salary--that is, commencing at the rate of £50 a year, payable half-yearly. Moggs had a yearly rise of £10, and Snoggs was offered the same, only he asked, for reasons that do not concern our puzzle, that he might take his rise at £2, 10s. half-yearly, to which his employer (not, perhaps, unnaturally!) had no objection. Now we come to the real point of the puzzle. Moggs put regularly into the Post Office Savings Bank a certain proportion of his salary, while Snoggs saved twice as great a proportion of his, and at the end of five years they had together saved £268, 15s. How much had each saved? The question of interest can be ignored. 27.--GIVING CHANGE. Every one is familiar with the difficulties that frequently arise over the giving of change, and how the assistance of a third person with a few coins in his pocket will sometimes help us to set the matter right. Here is an example. An Englishman went into a shop in New York and bought goods at a cost of thirty-four cents. The only money he had was a dollar, a three-cent piece, and a two-cent piece. The tradesman had only a half-dollar and a quarter-dollar. But another customer happened to be present, and when asked to help produced two dimes, a five-cent piece, a two-cent piece, and a one-cent piece. How did the tradesman manage to give change? For the benefit of those readers who are not familiar with the American coinage, it is only necessary to say that a dollar is a hundred cents and a dime ten cents. A puzzle of this kind should rarely cause any difficulty if attacked in a proper manner. 28.--DEFECTIVE OBSERVATION. Our observation of little things is frequently defective, and our memories very liable to lapse. A certain judge recently remarked in a case that he had no recollection whatever of putting the wedding-ring on his wife's finger. Can you correctly answer these questions without having the coins in sight? On which side of a penny is the date given? Some people are so unobservant that, although they are handling the coin nearly every day of their lives, they are at a loss to answer this simple question. If I lay a penny flat on the table, how many other pennies can I place around it, every one also lying flat on the table, so that they all touch the first one? The geometrician will, of course, give the answer at once, and not need to make any experiment. He will also know that, since all circles are similar, the same answer will necessarily apply to any coin. The next question is a most interesting one to ask a company, each person writing down his answer on a slip of paper, so that no one shall be helped by the answers of others. What is the greatest number of three-penny-pieces that may be laid flat on the surface of a half-crown, so that no piece lies on another or overlaps the surface of the half-crown? It is amazing what a variety of different answers one gets to this question. Very few people will be found to give the correct number. Of course the answer must be given without looking at the coins. 29.--THE BROKEN COINS. A man had three coins--a sovereign, a shilling, and a penny--and he found that exactly the same fraction of each coin had been broken away. Now, assuming that the original intrinsic value of these coins was the same as their nominal value--that is, that the sovereign was worth a pound, the shilling worth a shilling, and the penny worth a penny--what proportion of each coin has been lost if the value of the three remaining fragments is exactly one pound? 30.--TWO QUESTIONS IN PROBABILITIES. There is perhaps no class of puzzle over which people so frequently blunder as that which involves what is called the theory of probabilities. I will give two simple examples of the sort of puzzle I mean. They are really quite easy, and yet many persons are tripped up by them. A friend recently produced five pennies and said to me: "In throwing these five pennies at the same time, what are the chances that at least four of the coins will turn up either all heads or all tails?" His own solution was quite wrong, but the correct answer ought not to be hard to discover. Another person got a wrong answer to the following little puzzle which I heard him propound: "A man placed three sovereigns and one shilling in a bag. How much should be paid for permission to draw one coin from it?" It is, of course, understood that you are as likely to draw any one of the four coins as another. 31.--DOMESTIC ECONOMY. Young Mrs. Perkins, of Putney, writes to me as follows: "I should be very glad if you could give me the answer to a little sum that has been worrying me a good deal lately. Here it is: We have only been married a short time, and now, at the end of two years from the time when we set up housekeeping, my husband tells me that he finds we have spent a third of his yearly income in rent, rates, and taxes, one-half in domestic expenses, and one-ninth in other ways. He has a balance of £190 remaining in the bank. I know this last, because he accidentally left out his pass-book the other day, and I peeped into it. Don't you think that a husband ought to give his wife his entire confidence in his money matters? Well, I do; and--will you believe it?--he has never told me what his income really is, and I want, very naturally, to find out. Can you tell me what it is from the figures I have given you?" Yes; the answer can certainly be given from the figures contained in Mrs. Perkins's letter. And my readers, if not warned, will be practically unanimous in declaring the income to be--something absurdly in excess of the correct answer! 32.--THE EXCURSION TICKET PUZZLE. When the big flaming placards were exhibited at the little provincial railway station, announcing that the Great ---- Company would run cheap excursion trains to London for the Christmas holidays, the inhabitants of Mudley-cum-Turmits were in quite a flutter of excitement. Half an hour before the train came in the little booking office was crowded with country passengers, all bent on visiting their friends in the great Metropolis. The booking clerk was unaccustomed to dealing with crowds of such a dimension, and he told me afterwards, while wiping his manly brow, that what caused him so much trouble was the fact that these rustics paid their fares in such a lot of small money. He said that he had enough farthings to supply a West End draper with change for a week, and a sufficient number of threepenny pieces for the congregations of three parish churches. "That excursion fare," said he, "is nineteen shillings and ninepence, and I should like to know in just how many different ways it is possible for such an amount to be paid in the current coin of this realm." Here, then, is a puzzle: In how many different ways may nineteen shillings and ninepence be paid in our current coin? Remember that the fourpenny-piece is not now current. 33.--PUZZLE IN REVERSALS. Most people know that if you take any sum of money in pounds, shillings, and pence, in which the number of pounds (less than £12) exceeds that of the pence, reverse it (calling the pounds pence and the pence pounds), find the difference, then reverse and add this difference, the result is always £12, 18s. 11d. But if we omit the condition, "less than £12," and allow nought to represent shillings or pence--(1) What is the lowest amount to which the rule will not apply? (2) What is the highest amount to which it will apply? Of course, when reversing such a sum as £14, 15s. 3d. it may be written £3, 16s. 2d., which is the same as £3, 15s. 14d. 34.--THE GROCER AND DRAPER. A country "grocer and draper" had two rival assistants, who prided themselves on their rapidity in serving customers. The young man on the grocery side could weigh up two one-pound parcels of sugar per minute, while the drapery assistant could cut three one-yard lengths of cloth in the same time. Their employer, one slack day, set them a race, giving the grocer a barrel of sugar and telling him to weigh up forty-eight one-pound parcels of sugar While the draper divided a roll of forty-eight yards of cloth into yard pieces. The two men were interrupted together by customers for nine minutes, but the draper was disturbed seventeen times as long as the grocer. What was the result of the race? 35.--JUDKINS'S CATTLE. Hiram B. Judkins, a cattle-dealer of Texas, had five droves of animals, consisting of oxen, pigs, and sheep, with the same number of animals in each drove. One morning he sold all that he had to eight dealers. Each dealer bought the same number of animals, paying seventeen dollars for each ox, four dollars for each pig, and two dollars for each sheep; and Hiram received in all three hundred and one dollars. What is the greatest number of animals he could have had? And how many would there be of each kind? 36.--BUYING APPLES. As the purchase of apples in small quantities has always presented considerable difficulties, I think it well to offer a few remarks on this subject. We all know the story of the smart boy who, on being told by the old woman that she was selling her apples at four for threepence, said: "Let me see! Four for threepence; that's three for twopence, two for a penny, one for nothing--I'll take _one_!" There are similar cases of perplexity. For example, a boy once picked up a penny apple from a stall, but when he learnt that the woman's pears were the same price he exchanged it, and was about to walk off. "Stop!" said the woman. "You haven't paid me for the pear!" "No," said the boy, "of course not. I gave you the apple for it." "But you didn't pay for the apple!" "Bless the woman! You don't expect me to pay for the apple and the pear too!" And before the poor creature could get out of the tangle the boy had disappeared. Then, again, we have the case of the man who gave a boy sixpence and promised to repeat the gift as soon as the youngster had made it into ninepence. Five minutes later the boy returned. "I have made it into ninepence," he said, at the same time handing his benefactor threepence. "How do you make that out?" he was asked. "I bought threepennyworth of apples." "But that does not make it into ninepence!" "I should rather think it did," was the boy's reply. "The apple woman has threepence, hasn't she? Very well, I have threepennyworth of apples, and I have just given you the other threepence. What's that but ninepence?" I cite these cases just to show that the small boy really stands in need of a little instruction in the art of buying apples. So I will give a simple poser dealing with this branch of commerce. An old woman had apples of three sizes for sale--one a penny, two a penny, and three a penny. Of course two of the second size and three of the third size were respectively equal to one apple of the largest size. Now, a gentleman who had an equal number of boys and girls gave his children sevenpence to be spent amongst them all on these apples. The puzzle is to give each child an equal distribution of apples. How was the sevenpence spent, and how many children were there? 37.--BUYING CHESTNUTS. Though the following little puzzle deals with the purchase of chestnuts, it is not itself of the "chestnut" type. It is quite new. At first sight it has certainly the appearance of being of the "nonsense puzzle" character, but it is all right when properly considered. A man went to a shop to buy chestnuts. He said he wanted a pennyworth, and was given five chestnuts. "It is not enough; I ought to have a sixth," he remarked! "But if I give you one chestnut more." the shopman replied, "you will have five too many." Now, strange to say, they were both right. How many chestnuts should the buyer receive for half a crown? 38.--THE BICYCLE THIEF. Here is a little tangle that is perpetually cropping up in various guises. A cyclist bought a bicycle for £15 and gave in payment a cheque for £25. The seller went to a neighbouring shopkeeper and got him to change the cheque for him, and the cyclist, having received his £10 change, mounted the machine and disappeared. The cheque proved to be valueless, and the salesman was requested by his neighbour to refund the amount he had received. To do this, he was compelled to borrow the £25 from a friend, as the cyclist forgot to leave his address, and could not be found. Now, as the bicycle cost the salesman £11, how much money did he lose altogether? 39.--THE COSTERMONGER'S PUZZLE. "How much did yer pay for them oranges, Bill?" "I ain't a-goin' to tell yer, Jim. But I beat the old cove down fourpence a hundred." "What good did that do yer?" "Well, it meant five more oranges on every ten shillin's-worth." Now, what price did Bill actually pay for the oranges? There is only one rate that will fit in with his statements. AGE AND KINSHIP PUZZLES. "The days of our years are threescore years and ten." --_Psalm_ xc. 10. For centuries it has been a favourite method of propounding arithmetical puzzles to pose them in the form of questions as to the age of an individual. They generally lend themselves to very easy solution by the use of algebra, though often the difficulty lies in stating them correctly. They may be made very complex and may demand considerable ingenuity, but no general laws can well be laid down for their solution. The solver must use his own sagacity. As for puzzles in relationship or kinship, it is quite curious how bewildering many people find these things. Even in ordinary conversation, some statement as to relationship, which is quite clear in the mind of the speaker, will immediately tie the brains of other people into knots. Such expressions as "He is my uncle's son-in-law's sister" convey absolutely nothing to some people without a detailed and laboured explanation. In such cases the best course is to sketch a brief genealogical table, when the eye comes immediately to the assistance of the brain. In these days, when we have a growing lack of respect for pedigrees, most people have got out of the habit of rapidly drawing such tables, which is to be regretted, as they would save a lot of time and brain racking on occasions. 40.--MAMMA'S AGE. Tommy: "How old are you, mamma?" Mamma: "Let me think, Tommy. Well, our three ages add up to exactly seventy years." Tommy: "That's a lot, isn't it? And how old are you, papa?" Papa: "Just six times as old as you, my son." Tommy: "Shall I ever be half as old as you, papa?" Papa: "Yes, Tommy; and when that happens our three ages will add up to exactly twice as much as to-day." Tommy: "And supposing I was born before you, papa; and supposing mamma had forgot all about it, and hadn't been at home when I came; and supposing--" Mamma: "Supposing, Tommy, we talk about bed. Come along, darling. You'll have a headache." Now, if Tommy had been some years older he might have calculated the exact ages of his parents from the information they had given him. Can you find out the exact age of mamma? 41.--THEIR AGES. "My husband's age," remarked a lady the other day, "is represented by the figures of my own age reversed. He is my senior, and the difference between our ages is one-eleventh of their sum." 42.--THE FAMILY AGES. When the Smileys recently received a visit from the favourite uncle, the fond parents had all the five children brought into his presence. First came Billie and little Gertrude, and the uncle was informed that the boy was exactly twice as old as the girl. Then Henrietta arrived, and it was pointed out that the combined ages of herself and Gertrude equalled twice the age of Billie. Then Charlie came running in, and somebody remarked that now the combined ages of the two boys were exactly twice the combined ages of the two girls. The uncle was expressing his astonishment at these coincidences when Janet came in. "Ah! uncle," she exclaimed, "you have actually arrived on my twenty-first birthday!" To this Mr. Smiley added the final staggerer: "Yes, and now the combined ages of the three girls are exactly equal to twice the combined ages of the two boys." Can you give the age of each child? 43.--MRS. TIMPKINS'S AGE. Edwin: "Do you know, when the Timpkinses married eighteen years ago Timpkins was three times as old as his wife, and to-day he is just twice as old as she?" Angelina: "Then how old was Mrs. Timpkins on the wedding day?" Can you answer Angelina's question? 44--A CENSUS PUZZLE. Mr. and Mrs. Jorkins have fifteen children, all born at intervals of one year and a half. Miss Ada Jorkins, the eldest, had an objection to state her age to the census man, but she admitted that she was just seven times older than little Johnnie, the youngest of all. What was Ada's age? Do not too hastily assume that you have solved this little poser. You may find that you have made a bad blunder! 45.--MOTHER AND DAUGHTER. "Mother, I wish you would give me a bicycle," said a girl of twelve the other day. "I do not think you are old enough yet, my dear," was the reply. "When I am only three times as old as you are you shall have one." Now, the mother's age is forty-five years. When may the young lady expect to receive her present? 46.--MARY AND MARMADUKE. Marmaduke: "Do you know, dear, that in seven years' time our combined ages will be sixty-three years?" Mary: "Is that really so? And yet it is a fact that when you were my present age you were twice as old as I was then. I worked it out last night." Now, what are the ages of Mary and Marmaduke? 47--ROVER'S AGE. "Now, then, Tommy, how old is Rover?" Mildred's young man asked her brother. "Well, five years ago," was the youngster's reply, "sister was four times older than the dog, but now she is only three times as old." Can you tell Rover's age? 48.--CONCERNING TOMMY'S AGE. Tommy Smart was recently sent to a new school. On the first day of his arrival the teacher asked him his age, and this was his curious reply: "Well, you see, it is like this. At the time I was born--I forget the year--my only sister, Ann, happened to be just one-quarter the age of mother, and she is now one-third the age of father." "That's all very well," said the teacher, "but what I want is not the age of your sister Ann, but your own age." "I was just coming to that," Tommy answered; "I am just a quarter of mother's present age, and in four years' time I shall be a quarter the age of father. Isn't that funny?" This was all the information that the teacher could get out of Tommy Smart. Could you have told, from these facts, what was his precise age? It is certainly a little puzzling. 49.--NEXT-DOOR NEIGHBOURS. There were two families living next door to one another at Tooting Bec--the Jupps and the Simkins. The united ages of the four Jupps amounted to one hundred years, and the united ages of the Simkins also amounted to the same. It was found in the case of each family that the sum obtained by adding the squares of each of the children's ages to the square of the mother's age equalled the square of the father's age. In the case of the Jupps, however, Julia was one year older than her brother Joe, whereas Sophy Simkin was two years older than her brother Sammy. What was the age of each of the eight individuals? 50.--THE BAG OF NUTS. Three boys were given a bag of nuts as a Christmas present, and it was agreed that they should be divided in proportion to their ages, which together amounted to 17½ years. Now the bag contained 770 nuts, and as often as Herbert took four Robert took three, and as often as Herbert took six Christopher took seven. The puzzle is to find out how many nuts each had, and what were the boys' respective ages. 51.--HOW OLD WAS MARY? Here is a funny little age problem, by the late Sam Loyd, which has been very popular in the United States. Can you unravel the mystery? The combined ages of Mary and Ann are forty-four years, and Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann. How old is Mary? That is all, but can you work it out? If not, ask your friends to help you, and watch the shadow of bewilderment creep over their faces as they attempt to grip the intricacies of the question. 52.--QUEER RELATIONSHIPS. "Speaking of relationships," said the Parson at a certain dinner-party, "our legislators are getting the marriage law into a frightful tangle, Here, for example, is a puzzling case that has come under my notice. Two brothers married two sisters. One man died and the other man's wife also died. Then the survivors married." "The man married his deceased wife's sister under the recent Act?" put in the Lawyer. "Exactly. And therefore, under the civil law, he is legally married and his child is legitimate. But, you see, the man is the woman's deceased husband's brother, and therefore, also under the civil law, she is not married to him and her child is illegitimate." "He is married to her and she is not married to him!" said the Doctor. "Quite so. And the child is the legitimate son of his father, but the illegitimate son of his mother." "Undoubtedly 'the law is a hass,'" the Artist exclaimed, "if I may be permitted to say so," he added, with a bow to the Lawyer. "Certainly," was the reply. "We lawyers try our best to break in the beast to the service of man. Our legislators are responsible for the breed." "And this reminds me," went on the Parson, "of a man in my parish who married the sister of his widow. This man--" "Stop a moment, sir," said the Professor. "Married the sister of his widow? Do you marry dead men in your parish?" "No; but I will explain that later. Well, this man has a sister of his own. Their names are Stephen Brown and Jane Brown. Last week a young fellow turned up whom Stephen introduced to me as his nephew. Naturally, I spoke of Jane as his aunt, but, to my astonishment, the youth corrected me, assuring me that, though he was the nephew of Stephen, he was not the nephew of Jane, the sister of Stephen. This perplexed me a good deal, but it is quite correct." The Lawyer was the first to get at the heart of the mystery. What was his solution? 53.--HEARD ON THE TUBE RAILWAY. First Lady: "And was he related to you, dear?" Second Lady: "Oh, yes. You see, that gentleman's mother was my mother's mother-in-law, but he is not on speaking terms with my papa." First Lady: "Oh, indeed!" (But you could see that she was not much wiser.) How was the gentleman related to the Second Lady? 54.--A FAMILY PARTY. A certain family party consisted of 1 grandfather, 1 grandmother, 2 fathers, 2 mothers, 4 children, 3 grandchildren, 1 brother, 2 sisters, 2 sons, 2 daughters, 1 father-in-law, 1 mother-in-law, and 1 daughter-in-law. Twenty-three people, you will say. No; there were only seven persons present. Can you show how this might be? 55.--A MIXED PEDIGREE. Joseph Bloggs: "I can't follow it, my dear boy. It makes me dizzy!" John Snoggs: "It's very simple. Listen again! You happen to be my father's brother-in-law, my brother's father-in-law, and also my father-in-law's brother. You see, my father was--" But Mr. Bloggs refused to hear any more. Can the reader show how this extraordinary triple relationship might have come about? 56.--WILSON'S POSER. "Speaking of perplexities--" said Mr. Wilson, throwing down a magazine on the table in the commercial room of the Railway Hotel. "Who was speaking of perplexities?" inquired Mr. Stubbs. "Well, then, reading about them, if you want to be exact--it just occurred to me that perhaps you three men may be interested in a little matter connected with myself." It was Christmas Eve, and the four commercial travellers were spending the holiday at Grassminster. Probably each suspected that the others had no homes, and perhaps each was conscious of the fact that he was in that predicament himself. In any case they seemed to be perfectly comfortable, and as they drew round the cheerful fire the conversation became general. "What is the difficulty?" asked Mr. Packhurst. "There's no difficulty in the matter, when you rightly understand it. It is like this. A man named Parker had a flying-machine that would carry two. He was a venturesome sort of chap--reckless, I should call him--and he had some bother in finding a man willing to risk his life in making an ascent with him. However, an uncle of mine thought he would chance it, and one fine morning he took his seat in the machine and she started off well. When they were up about a thousand feet, my nephew suddenly--" "Here, stop, Wilson! What was your nephew doing there? You said your uncle," interrupted Mr. Stubbs. "Did I? Well, it does not matter. My nephew suddenly turned to Parker and said that the engine wasn't running well, so Parker called out to my uncle--" "Look here," broke in Mr. Waterson, "we are getting mixed. Was it your uncle or your nephew? Let's have it one way or the other." "What I said is quite right. Parker called out to my uncle to do something or other, when my nephew--" "There you are again, Wilson," cried Mr. Stubbs; "once for all, are we to understand that both your uncle and your nephew were on the machine?" "Certainly. I thought I made that clear. Where was I? Well, my nephew shouted back to Parker--" "Phew! I'm sorry to interrupt you again, Wilson, but we can't get on like this. Is it true that the machine would only carry two?" "Of course. I said at the start that it only carried two." "Then what in the name of aerostation do you mean by saying that there were three persons on board?" shouted Mr. Stubbs. "Who said there were three?" "You have told us that Parker, your uncle, and your nephew went up on this blessed flying-machine." "That's right." "And the thing would only carry two!" "Right again." "Wilson, I have known you for some time as a truthful man and a temperate man," said Mr. Stubbs, solemnly. "But I am afraid since you took up that new line of goods you have overworked yourself." "Half a minute, Stubbs," interposed Mr. Waterson. "I see clearly where we all slipped a cog. Of course, Wilson, you meant us to understand that Parker is either your uncle or your nephew. Now we shall be all right if you will just tell us whether Parker is your uncle or nephew." "He is no relation to me whatever." The three men sighed and looked anxiously at one another. Mr. Stubbs got up from his chair to reach the matches, Mr. Packhurst proceeded to wind up his watch, and Mr. Waterson took up the poker to attend to the fire. It was an awkward moment, for at the season of goodwill nobody wished to tell Mr. Wilson exactly what was in his mind. "It's curious," said Mr. Wilson, very deliberately, "and it's rather sad, how thick-headed some people are. You don't seem to grip the facts. It never seems to have occurred to either of you that my uncle and my nephew are one and the same man." "What!" exclaimed all three together. "Yes; David George Linklater is my uncle, and he is also my nephew. Consequently, I am both his uncle and nephew. Queer, isn't it? I'll explain how it comes about." Mr. Wilson put the case so very simply that the three men saw how it might happen without any marriage within the prohibited degrees. Perhaps the reader can work it out for himself. CLOCK PUZZLES. "Look at the clock!" _Ingoldsby Legends_. In considering a few puzzles concerning clocks and watches, and the times recorded by their hands under given conditions, it is well that a particular convention should always be kept in mind. It is frequently the case that a solution requires the assumption that the hands can actually record a time involving a minute fraction of a second. Such a time, of course, cannot be really indicated. Is the puzzle, therefore, impossible of solution? The conclusion deduced from a logical syllogism depends for its truth on the two premises assumed, and it is the same in mathematics. Certain things are antecedently assumed, and the answer depends entirely on the truth of those assumptions. "If two horses," says Lagrange, "can pull a load of a certain weight, it is natural to suppose that four horses could pull a load of double that weight, six horses a load of three times that weight. Yet, strictly speaking, such is not the case. For the inference is based on the assumption that the four horses pull alike in amount and direction, which in practice can scarcely ever be the case. It so happens that we are frequently led in our reckonings to results which diverge widely from reality. But the fault is not the fault of mathematics; for mathematics always gives back to us exactly what we have put into it. The ratio was constant according to that supposition. The result is founded upon that supposition. If the supposition is false the result is necessarily false." If one man can reap a field in six days, we say two men will reap it in three days, and three men will do the work in two days. We here assume, as in the case of Lagrange's horses, that all the men are exactly equally capable of work. But we assume even more than this. For when three men get together they may waste time in gossip or play; or, on the other hand, a spirit of rivalry may spur them on to greater diligence. We may assume any conditions we like in a problem, provided they be clearly expressed and understood, and the answer will be in accordance with those conditions. 57.--WHAT WAS THE TIME? "I say, Rackbrane, what is the time?" an acquaintance asked our friend the professor the other day. The answer was certainly curious. "If you add one quarter of the time from noon till now to half the time from now till noon to-morrow, you will get the time exactly." What was the time of day when the professor spoke? 58.--A TIME PUZZLE. How many minutes is it until six o'clock if fifty minutes ago it was four times as many minutes past three o'clock? 59.--A PUZZLING WATCH. A friend pulled out his watch and said, "This watch of mine does not keep perfect time; I must have it seen to. I have noticed that the minute hand and the hour hand are exactly together every sixty-five minutes." Does that watch gain or lose, and how much per hour? 60.--THE WAPSHAW'S WHARF MYSTERY. There was a great commotion in Lower Thames Street on the morning of January 12, 1887. When the early members of the staff arrived at Wapshaw's Wharf they found that the safe had been broken open, a considerable sum of money removed, and the offices left in great disorder. The night watchman was nowhere to be found, but nobody who had been acquainted with him for one moment suspected him to be guilty of the robbery. In this belief the proprietors were confirmed when, later in the day, they were informed that the poor fellow's body had been picked up by the River Police. Certain marks of violence pointed to the fact that he had been brutally attacked and thrown into the river. A watch found in his pocket had stopped, as is invariably the case in such circumstances, and this was a valuable clue to the time of the outrage. But a very stupid officer (and we invariably find one or two stupid individuals in the most intelligent bodies of men) had actually amused himself by turning the hands round and round, trying to set the watch going again. After he had been severely reprimanded for this serious indiscretion, he was asked whether he could remember the time that was indicated by the watch when found. He replied that he could not, but he recollected that the hour hand and minute hand were exactly together, one above the other, and the second hand had just passed the forty-ninth second. More than this he could not remember. What was the exact time at which the watchman's watch stopped? The watch is, of course, assumed to have been an accurate one. 61.--CHANGING PLACES. [Illustration] The above clock face indicates a little before 42 minutes past 4. The hands will again point at exactly the same spots a little after 23 minutes past 8. In fact, the hands will have changed places. How many times do the hands of a clock change places between three o'clock p.m. and midnight? And out of all the pairs of times indicated by these changes, what is the exact time when the minute hand will be nearest to the point IX? 62.--THE CLUB CLOCK. One of the big clocks in the Cogitators' Club was found the other night to have stopped just when, as will be seen in the illustration, the second hand was exactly midway between the other two hands. One of the members proposed to some of his friends that they should tell him the exact time when (if the clock had not stopped) the second hand would next again have been midway between the minute hand and the hour hand. Can you find the correct time that it would happen? [Illustration] 63.--THE STOP-WATCH. [Illustration] We have here a stop-watch with three hands. The second hand, which travels once round the face in a minute, is the one with the little ring at its end near the centre. Our dial indicates the exact time when its owner stopped the watch. You will notice that the three hands are nearly equidistant. The hour and minute hands point to spots that are exactly a third of the circumference apart, but the second hand is a little too advanced. An exact equidistance for the three hands is not possible. Now, we want to know what the time will be when the three hands are next at exactly the same distances as shown from one another. Can you state the time? 64.--THE THREE CLOCKS. On Friday, April 1, 1898, three new clocks were all set going precisely at the same time--twelve noon. At noon on the following day it was found that clock A had kept perfect time, that clock B had gained exactly one minute, and that clock C had lost exactly one minute. Now, supposing that the clocks B and C had not been regulated, but all three allowed to go on as they had begun, and that they maintained the same rates of progress without stopping, on what date and at what time of day would all three pairs of hands again point at the same moment at twelve o'clock? 65.--THE RAILWAY STATION CLOCK. A clock hangs on the wall of a railway station, 71 ft. 9 in. long and 10 ft. 4 in. high. Those are the dimensions of the wall, not of the clock! While waiting for a train we noticed that the hands of the clock were pointing in opposite directions, and were parallel to one of the diagonals of the wall. What was the exact time? 66.--THE VILLAGE SIMPLETON. A facetious individual who was taking a long walk in the country came upon a yokel sitting on a stile. As the gentleman was not quite sure of his road, he thought he would make inquiries of the local inhabitant; but at the first glance he jumped too hastily to the conclusion that he had dropped on the village idiot. He therefore decided to test the fellow's intelligence by first putting to him the simplest question he could think of, which was, "What day of the week is this, my good man?" The following is the smart answer that he received:-- "When the day after to-morrow is yesterday, to-day will be as far from Sunday as to-day was from Sunday when the day before yesterday was to-morrow." Can the reader say what day of the week it was? It is pretty evident that the countryman was not such a fool as he looked. The gentleman went on his road a puzzled but a wiser man. LOCOMOTION AND SPEED PUZZLES. "The race is not to the swift."--_Ecclesiastes_ ix. II. 67.--AVERAGE SPEED. In a recent motor ride it was found that we had gone at the rate of ten miles an hour, but we did the return journey over the same route, owing to the roads being more clear of traffic, at fifteen miles an hour. What was our average speed? Do not be too hasty in your answer to this simple little question, or it is pretty certain that you will be wrong. 68.--THE TWO TRAINS. I put this little question to a stationmaster, and his correct answer was so prompt that I am convinced there is no necessity to seek talented railway officials in America or elsewhere. Two trains start at the same time, one from London to Liverpool, the other from Liverpool to London. If they arrive at their destinations one hour and four hours respectively after passing one another, how much faster is one train running than the other? 69.--THE THREE VILLAGES. I set out the other day to ride in a motor-car from Acrefield to Butterford, but by mistake I took the road going _via_ Cheesebury, which is nearer Acrefield than Butterford, and is twelve miles to the left of the direct road I should have travelled. After arriving at Butterford I found that I had gone thirty-five miles. What are the three distances between these villages, each being a whole number of miles? I may mention that the three roads are quite straight. 70.--DRAWING HER PENSION. "Speaking of odd figures," said a gentleman who occupies some post in a Government office, "one of the queerest characters I know is an old lame widow who climbs up a hill every week to draw her pension at the village post office. She crawls up at the rate of a mile and a half an hour and comes down at the rate of four and a half miles an hour, so that it takes her just six hours to make the double journey. Can any of you tell me how far it is from the bottom of the hill to the top?" [Illustration] 71.--SIR EDWYN DE TUDOR. In the illustration we have a sketch of Sir Edwyn de Tudor going to rescue his lady-love, the fair Isabella, who was held a captive by a neighbouring wicked baron. Sir Edwyn calculated that if he rode fifteen miles an hour he would arrive at the castle an hour too soon, while if he rode ten miles an hour he would get there just an hour too late. Now, it was of the first importance that he should arrive at the exact time appointed, in order that the rescue that he had planned should be a success, and the time of the tryst was five o'clock, when the captive lady would be taking her afternoon tea. The puzzle is to discover exactly how far Sir Edwyn de Tudor had to ride. 72.--THE HYDROPLANE QUESTION. The inhabitants of Slocomb-on-Sea were greatly excited over the visit of a certain flying man. All the town turned out to see the flight of the wonderful hydroplane, and, of course, Dobson and his family were there. Master Tommy was in good form, and informed his father that Englishmen made better airmen than Scotsmen and Irishmen because they are not so heavy. "How do you make that out?" asked Mr. Dobson. "Well, you see," Tommy replied, "it is true that in Ireland there are men of Cork and in Scotland men of Ayr, which is better still, but in England there are lightermen." Unfortunately it had to be explained to Mrs. Dobson, and this took the edge off the thing. The hydroplane flight was from Slocomb to the neighbouring watering-place Poodleville--five miles distant. But there was a strong wind, which so helped the airman that he made the outward journey in the short time of ten minutes, though it took him an hour to get back to the starting point at Slocomb, with the wind dead against him. Now, how long would the ten miles have taken him if there had been a perfect calm? Of course, the hydroplane's engine worked uniformly throughout. 73.--DONKEY RIDING. During a visit to the seaside Tommy and Evangeline insisted on having a donkey race over the mile course on the sands. Mr. Dobson and some of his friends whom he had met on the beach acted as judges, but, as the donkeys were familiar acquaintances and declined to part company the whole way, a dead heat was unavoidable. However, the judges, being stationed at different points on the course, which was marked off in quarter-miles, noted the following results:--The first three-quarters were run in six and three-quarter minutes, the first half-mile took the same time as the second half, and the third quarter was run in exactly the same time as the last quarter. From these results Mr. Dobson amused himself in discovering just how long it took those two donkeys to run the whole mile. Can you give the answer? 74.--THE BASKET OF POTATOES. A man had a basket containing fifty potatoes. He proposed to his son, as a little recreation, that he should place these potatoes on the ground in a straight line. The distance between the first and second potatoes was to be one yard, between the second and third three yards, between the third and fourth five yards, between the fourth and fifth seven yards, and so on--an increase of two yards for every successive potato laid down. Then the boy was to pick them up and put them in the basket one at a time, the basket being placed beside the first potato. How far would the boy have to travel to accomplish the feat of picking them all up? We will not consider the journey involved in placing the potatoes, so that he starts from the basket with them all laid out. 75.--THE PASSENGER'S FARE. At first sight you would hardly think there was matter for dispute in the question involved in the following little incident, yet it took the two persons concerned some little time to come to an agreement. Mr. Smithers hired a motor-car to take him from Addleford to Clinkerville and back again for £3. At Bakenham, just midway, he picked up an acquaintance, Mr. Tompkins, and agreed to take him on to Clinkerville and bring him back to Bakenham on the return journey. How much should he have charged the passenger? That is the question. What was a reasonable fare for Mr. Tompkins? DIGITAL PUZZLES. "Nine worthies were they called." DRYDEN: _The Flower and the Leaf._ I give these puzzles, dealing with the nine digits, a class to themselves, because I have always thought that they deserve more consideration than they usually receive. Beyond the mere trick of "casting out nines," very little seems to be generally known of the laws involved in these problems, and yet an acquaintance with the properties of the digits often supplies, among other uses, a certain number of arithmetical checks that are of real value in the saving of labour. Let me give just one example--the first that occurs to me. If the reader were required to determine whether or not 15,763,530,163,289 is a square number, how would he proceed? If the number had ended with a 2, 3, 7, or 8 in the digits place, of course he would know that it could not be a square, but there is nothing in its apparent form to prevent its being one. I suspect that in such a case he would set to work, with a sigh or a groan, at the laborious task of extracting the square root. Yet if he had given a little attention to the study of the digital properties of numbers, he would settle the question in this simple way. The sum of the digits is 59, the sum of which is 14, the sum of which is 5 (which I call the "digital root"), and therefore I know that the number cannot be a square, and for this reason. The digital root of successive square numbers from 1 upwards is always 1, 4, 7, or 9, and can never be anything else. In fact, the series, 1, 4, 9, 7, 7, 9, 4, 1, 9, is repeated into infinity. The analogous series for triangular numbers is 1, 3, 6, 1, 6, 3, 1, 9, 9. So here we have a similar negative check, for a number cannot be triangular (that is, (n² + n)/2) if its digital root be 2, 4, 5, 7, or 8. 76.--THE BARREL OF BEER. A man bought an odd lot of wine in barrels and one barrel containing beer. These are shown in the illustration, marked with the number of gallons that each barrel contained. He sold a quantity of the wine to one man and twice the quantity to another, but kept the beer to himself. The puzzle is to point out which barrel contains beer. Can you say which one it is? Of course, the man sold the barrels just as he bought them, without manipulating in any way the contents. [Illustration: ( 15 Gals ) (31 Gals) (19 Gals) (20 Gals) (16 Gals) (18 Gals) ] 77.--DIGITS AND SQUARES. [Illustration: +---+---+---+ | 1 | 9 | 2 | +---+---+---+ | 3 | 8 | 4 | +---+---+---+ | 5 | 7 | 6 | +---+---+---+ ] It will be seen in the diagram that we have so arranged the nine digits in a square that the number in the second row is twice that in the first row, and the number in the bottom row three times that in the top row. There are three other ways of arranging the digits so as to produce the same result. Can you find them? 78.--ODD AND EVEN DIGITS. The odd digits, 1, 3, 5, 7, and 9, add up 25, while the even figures, 2, 4, 6, and 8, only add up 20. Arrange these figures so that the odd ones and the even ones add up alike. Complex and improper fractions and recurring decimals are not allowed. 79.--THE LOCKERS PUZZLE. [Illustration: A B C ================== ================== ================== | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ | | | | | | | | | | | | | | | | | | | | | | | | | | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ | | | | | | | | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ | | | | | | | | | | | | | | | | | | | | | | | | | | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ | | | | | | | ================== ================== ================== | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ | | | | | | | | | | | | | | | | | | | | | | | | | | +--+ +--+ +--+ | | +--+ +--+ +--+ | | +--+ +--+ +--+ | ------------------ ------------------ ------------------ ] A man had in his office three cupboards, each containing nine lockers, as shown in the diagram. He told his clerk to place a different one-figure number on each locker of cupboard A, and to do the same in the case of B, and of C. As we are here allowed to call nought a digit, and he was not prohibited from using nought as a number, he clearly had the option of omitting any one of ten digits from each cupboard. Now, the employer did not say the lockers were to be numbered in any numerical order, and he was surprised to find, when the work was done, that the figures had apparently been mixed up indiscriminately. Calling upon his clerk for an explanation, the eccentric lad stated that the notion had occurred to him so to arrange the figures that in each case they formed a simple addition sum, the two upper rows of figures producing the sum in the lowest row. But the most surprising point was this: that he had so arranged them that the addition in A gave the smallest possible sum, that the addition in C gave the largest possible sum, and that all the nine digits in the three totals were different. The puzzle is to show how this could be done. No decimals are allowed and the nought may not appear in the hundreds place. 80.--THE THREE GROUPS. There appeared in "Nouvelles Annales de Mathématiques" the following puzzle as a modification of one of my "Canterbury Puzzles." Arrange the nine digits in three groups of two, three, and four digits, so that the first two numbers when multiplied together make the third. Thus, 12 × 483 = 5,796. I now also propose to include the cases where there are one, four, and four digits, such as 4 × 1,738 = 6,952. Can you find all the possible solutions in both cases? 81.--THE NINE COUNTERS. [Illustration: (1)(5)(8) (7)(9) (2)(3) (4)(6) ] I have nine counters, each bearing one of the nine digits, 1, 2, 3, 4, 5, 6, 7, 8 and 9. I arranged them on the table in two groups, as shown in the illustration, so as to form two multiplication sums, and found that both sums gave the same product. You will find that 158 multiplied by 23 is 3,634, and that 79 multiplied by 46 is also 3,634. Now, the puzzle I propose is to rearrange the counters so as to get as large a product as possible. What is the best way of placing them? Remember both groups must multiply to the same amount, and there must be three counters multiplied by two in one case, and two multiplied by two counters in the other, just as at present. 82.--THE TEN COUNTERS. In this case we use the nought in addition to the 1, 2, 3, 4, 5, 6, 7, 8, 9. The puzzle is, as in the last case, so to arrange the ten counters that the products of the two multiplications shall be the same, and you may here have one or more figures in the multiplier, as you choose. The above is a very easy feat; but it is also required to find the two arrangements giving pairs of the highest and lowest products possible. Of course every counter must be used, and the cipher may not be placed to the left of a row of figures where it would have no effect. Vulgar fractions or decimals are not allowed. 83.--DIGITAL MULTIPLICATION. Here is another entertaining problem with the nine digits, the nought being excluded. Using each figure once, and only once, we can form two multiplication sums that have the same product, and this may be done in many ways. For example, 7 × 658 and 14 × 329 contain all the digits once, and the product in each case is the same--4,606. Now, it will be seen that the sum of the digits in the product is 16, which is neither the highest nor the lowest sum so obtainable. Can you find the solution of the problem that gives the lowest possible sum of digits in the common product? Also that which gives the highest possible sum? 84.--THE PIERROT'S PUZZLE. [Illustration] The Pierrot in the illustration is standing in a posture that represents the sign of multiplication. He is indicating the peculiar fact that 15 multiplied by 93 produces exactly the same figures (1,395), differently arranged. The puzzle is to take any four digits you like (all different) and similarly arrange them so that the number formed on one side of the Pierrot when multiplied by the number on the other side shall produce the same figures. There are very few ways of doing it, and I shall give all the cases possible. Can you find them all? You are allowed to put two figures on each side of the Pierrot as in the example shown, or to place a single figure on one side and three figures on the other. If we only used three digits instead of four, the only possible ways are these: 3 multiplied by 51 equals 153, and 6 multiplied by 21 equals 126. 85.--THE CAB NUMBERS. A London policeman one night saw two cabs drive off in opposite directions under suspicious circumstances. This officer was a particularly careful and wide-awake man, and he took out his pocket-book to make an entry of the numbers of the cabs, but discovered that he had lost his pencil. Luckily, however, he found a small piece of chalk, with which he marked the two numbers on the gateway of a wharf close by. When he returned to the same spot on his beat he stood and looked again at the numbers, and noticed this peculiarity, that all the nine digits (no nought) were used and that no figure was repeated, but that if he multiplied the two numbers together they again produced the nine digits, all once, and once only. When one of the clerks arrived at the wharf in the early morning, he observed the chalk marks and carefully rubbed them out. As the policeman could not remember them, certain mathematicians were then consulted as to whether there was any known method for discovering all the pairs of numbers that have the peculiarity that the officer had noticed; but they knew of none. The investigation, however, was interesting, and the following question out of many was proposed: What two numbers, containing together all the nine digits, will, when multiplied together, produce another number (the _highest possible_) containing also all the nine digits? The nought is not allowed anywhere. 86.--QUEER MULTIPLICATION. If I multiply 51,249,876 by 3 (thus using all the nine digits once, and once only), I get 153,749,628 (which again contains all the nine digits once). Similarly, if I multiply 16,583,742 by 9 the result is 149,253,678, where in each case all the nine digits are used. Now, take 6 as your multiplier and try to arrange the remaining eight digits so as to produce by multiplication a number containing all nine once, and once only. You will find it far from easy, but it can be done. 87.--THE NUMBER-CHECKS PUZZLE. [Illustration] Where a large number of workmen are employed on a building it is customary to provide every man with a little disc bearing his number. These are hung on a board by the men as they arrive, and serve as a check on punctuality. Now, I once noticed a foreman remove a number of these checks from his board and place them on a split-ring which he carried in his pocket. This at once gave me the idea for a good puzzle. In fact, I will confide to my readers that this is just how ideas for puzzles arise. You cannot really create an idea: it happens--and you have to be on the alert to seize it when it does so happen. It will be seen from the illustration that there are ten of these checks on a ring, numbered 1 to 9 and 0. The puzzle is to divide them into three groups without taking any off the ring, so that the first group multiplied by the second makes the third group. For example, we can divide them into the three groups, 2--8 9 0 7--1 5 4 6 3, by bringing the 6 and the 3 round to the 4, but unfortunately the first two when multiplied together do not make the third. Can you separate them correctly? Of course you may have as many of the checks as you like in any group. The puzzle calls for some ingenuity, unless you have the luck to hit on the answer by chance. 88.--DIGITAL DIVISION. It is another good puzzle so to arrange the nine digits (the nought excluded) into two groups so that one group when divided by the other produces a given number without remainder. For example, 1 3 4 5 8 divided by 6 7 2 9 gives 2. Can the reader find similar arrangements producing 3, 4, 5, 6, 7, 8, and 9 respectively? Also, can he find the pairs of smallest possible numbers in each case? Thus, 1 4 6 5 8 divided by 7 3 2 9 is just as correct for 2 as the other example we have given, but the numbers are higher. 89.--ADDING THE DIGITS. If I write the sum of money, £987, 5s. 4½d., and add up the digits, they sum to 36. No digit has thus been used a second time in the amount or addition. This is the largest amount possible under the conditions. Now find the smallest possible amount, pounds, shillings, pence, and farthings being all represented. You need not use more of the nine digits than you choose, but no digit may be repeated throughout. The nought is not allowed. 90.--THE CENTURY PUZZLE. Can you write 100 in the form of a mixed number, using all the nine digits once, and only once? The late distinguished French mathematician, Edouard Lucas, found seven different ways of doing it, and expressed his doubts as to there being any other ways. As a matter of fact there are just eleven ways and no more. Here is one of them, 91+5742/638. Nine of the other ways have similarly two figures in the integral part of the number, but the eleventh expression has only one figure there. Can the reader find this last form? 91.--MORE MIXED FRACTIONS. When I first published my solution to the last puzzle, I was led to attempt the expression of all numbers in turn up to 100 by a mixed fraction containing all the nine digits. Here are twelve numbers for the reader to try his hand at: 13, 14, 15, 16, 18, 20, 27, 36, 40, 69, 72, 94. Use every one of the nine digits once, and only once, in every case. 92.--DIGITAL SQUARE NUMBERS. Here are the nine digits so arranged that they form four square numbers: 9, 81, 324, 576. Now, can you put them all together so as to form a single square number--(I) the smallest possible, and (II) the largest possible? 93.--THE MYSTIC ELEVEN. Can you find the largest possible number containing any nine of the ten digits (calling nought a digit) that can be divided by 11 without a remainder? Can you also find the smallest possible number produced in the same way that is divisible by 11? Here is an example, where the digit 5 has been omitted: 896743012. This number contains nine of the digits and is divisible by 11, but it is neither the largest nor the smallest number that will work. 94.--THE DIGITAL CENTURY. 1 2 3 4 5 6 7 8 9 = 100. It is required to place arithmetical signs between the nine figures so that they shall equal 100. Of course, you must not alter the present numerical arrangement of the figures. Can you give a correct solution that employs (1) the fewest possible signs, and (2) the fewest possible separate strokes or dots of the pen? That is, it is necessary to use as few signs as possible, and those signs should be of the simplest form. The signs of addition and multiplication (+ and ×) will thus count as two strokes, the sign of subtraction (-) as one stroke, the sign of division (÷) as three, and so on. 95.--THE FOUR SEVENS. [Illustration] In the illustration Professor Rackbrane is seen demonstrating one of the little posers with which he is accustomed to entertain his class. He believes that by taking his pupils off the beaten tracks he is the better able to secure their attention, and to induce original and ingenious methods of thought. He has, it will be seen, just shown how four 5's may be written with simple arithmetical signs so as to represent 100. Every juvenile reader will see at a glance that his example is quite correct. Now, what he wants you to do is this: Arrange four 7's (neither more nor less) with arithmetical signs so that they shall represent 100. If he had said we were to use four 9's we might at once have written 99+9/9, but the four 7's call for rather more ingenuity. Can you discover the little trick? 96.--THE DICE NUMBERS. [Illustration] I have a set of four dice, not marked with spots in the ordinary way, but with Arabic figures, as shown in the illustration. Each die, of course, bears the numbers 1 to 6. When put together they will form a good many, different numbers. As represented they make the number 1246. Now, if I make all the different four-figure numbers that are possible with these dice (never putting the same figure more than once in any number), what will they all add up to? You are allowed to turn the 6 upside down, so as to represent a 9. I do not ask, or expect, the reader to go to all the labour of writing out the full list of numbers and then adding them up. Life is not long enough for such wasted energy. Can you get at the answer in any other way? VARIOUS ARITHMETICAL AND ALGEBRAICAL PROBLEMS. "Variety's the very spice of life, That gives it all its flavour." COWPER: _The Task._ 97.--THE SPOT ON THE TABLE. A boy, recently home from school, wished to give his father an exhibition of his precocity. He pushed a large circular table into the corner of the room, as shown in the illustration, so that it touched both walls, and he then pointed to a spot of ink on the extreme edge. [Illustration] "Here is a little puzzle for you, pater," said the youth. "That spot is exactly eight inches from one wall and nine inches from the other. Can you tell me the diameter of the table without measuring it?" The boy was overheard to tell a friend, "It fairly beat the guv'nor;" but his father is known to have remarked to a City acquaintance that he solved the thing in his head in a minute. I often wonder which spoke the truth. 98.--ACADEMIC COURTESIES. In a certain mixed school, where a special feature was made of the inculcation of good manners, they had a curious rule on assembling every morning. There were twice as many girls as boys. Every girl made a bow to every other girl, to every boy, and to the teacher. Every boy made a bow to every other boy, to every girl, and to the teacher. In all there were nine hundred bows made in that model academy every morning. Now, can you say exactly how many boys there were in the school? If you are not very careful, you are likely to get a good deal out in your calculation. 99.--THE THIRTY-THREE PEARLS. [Illustration] "A man I know," said Teddy Nicholson at a certain family party, "possesses a string of thirty-three pearls. The middle pearl is the largest and best of all, and the others are so selected and arranged that, starting from one end, each successive pearl is worth £100 more than the preceding one, right up to the big pearl. From the other end the pearls increase in value by £150 up to the large pearl. The whole string is worth £65,000. What is the value of that large pearl?" "Pearls and other articles of clothing," said Uncle Walter, when the price of the precious gem had been discovered, "remind me of Adam and Eve. Authorities, you may not know, differ as to the number of apples that were eaten by Adam and Eve. It is the opinion of some that Eve 8 (ate) and Adam 2 (too), a total of 10 only. But certain mathematicians have figured it out differently, and hold that Eve 8 and Adam a total of 16. Yet the most recent investigators think the above figures entirely wrong, for if Eve 8 and Adam 8 2, the total must be 90." "Well," said Harry, "it seems to me that if there were giants in those days, probably Eve 8 1 and Adam 8 2, which would give a total of 163." "I am not at all satisfied," said Maud. "It seems to me that if Eve 8 1 and Adam 8 1 2, they together consumed 893." "I am sure you are all wrong," insisted Mr. Wilson, "for I consider that Eve 8 1 4 Adam, and Adam 8 1 2 4 Eve, so we get a total of 8,938." "But, look here," broke in Herbert. "If Eve 8 1 4 Adam and Adam 8 1 2 4 2 oblige Eve, surely the total must have been 82,056!" At this point Uncle Walter suggested that they might let the matter rest. He declared it to be clearly what mathematicians call an indeterminate problem. 100.--THE LABOURER'S PUZZLE. Professor Rackbrane, during one of his rambles, chanced to come upon a man digging a deep hole. "Good morning," he said. "How deep is that hole?" "Guess," replied the labourer. "My height is exactly five feet ten inches." "How much deeper are you going?" said the professor. "I am going twice as deep," was the answer, "and then my head will be twice as far below ground as it is now above ground." Rackbrane now asks if you could tell how deep that hole would be when finished. 101.--THE TRUSSES OF HAY. Farmer Tompkins had five trusses of hay, which he told his man Hodge to weigh before delivering them to a customer. The stupid fellow weighed them two at a time in all possible ways, and informed his master that the weights in pounds were 110, 112, 113, 114, 115, 116, 117, 118, 120, and 121. Now, how was Farmer Tompkins to find out from these figures how much every one of the five trusses weighed singly? The reader may at first think that he ought to be told "which pair is which pair," or something of that sort, but it is quite unnecessary. Can you give the five correct weights? 102.--MR. GUBBINS IN A FOG. Mr. Gubbins, a diligent man of business, was much inconvenienced by a London fog. The electric light happened to be out of order and he had to manage as best he could with two candles. His clerk assured him that though both were of the same length one candle would burn for four hours and the other for five hours. After he had been working some time he put the candles out as the fog had lifted, and he then noticed that what remained of one candle was exactly four times the length of what was left of the other. When he got home that night Mr. Gubbins, who liked a good puzzle, said to himself, "Of course it is possible to work out just how long those two candles were burning to-day. I'll have a shot at it." But he soon found himself in a worse fog than the atmospheric one. Could you have assisted him in his dilemma? How long were the candles burning? 103.--PAINTING THE LAMP-POSTS. Tim Murphy and Pat Donovan were engaged by the local authorities to paint the lamp-posts in a certain street. Tim, who was an early riser, arrived first on the job, and had painted three on the south side when Pat turned up and pointed out that Tim's contract was for the north side. So Tim started afresh on the north side and Pat continued on the south. When Pat had finished his side he went across the street and painted six posts for Tim, and then the job was finished. As there was an equal number of lamp-posts on each side of the street, the simple question is: Which man painted the more lamp-posts, and just how many more? 104.--CATCHING THE THIEF. "Now, constable," said the defendant's counsel in cross-examination," you say that the prisoner was exactly twenty-seven steps ahead of you when you started to run after him?" "Yes, sir." "And you swear that he takes eight steps to your five?" "That is so." "Then I ask you, constable, as an intelligent man, to explain how you ever caught him, if that is the case?" "Well, you see, I have got a longer stride. In fact, two of my steps are equal in length to five of the prisoner's. If you work it out, you will find that the number of steps I required would bring me exactly to the spot where I captured him." Here the foreman of the jury asked for a few minutes to figure out the number of steps the constable must have taken. Can you also say how many steps the officer needed to catch the thief? 105.--THE PARISH COUNCIL ELECTION. Here is an easy problem for the novice. At the last election of the parish council of Tittlebury-in-the-Marsh there were twenty-three candidates for nine seats. Each voter was qualified to vote for nine of these candidates or for any less number. One of the electors wants to know in just how many different ways it was possible for him to vote. 106.--THE MUDDLETOWN ELECTION. At the last Parliamentary election at Muddletown 5,473 votes were polled. The Liberal was elected by a majority of 18 over the Conservative, by 146 over the Independent, and by 575 over the Socialist. Can you give a simple rule for figuring out how many votes were polled for each candidate? 107.--THE SUFFRAGISTS' MEETING. At a recent secret meeting of Suffragists a serious difference of opinion arose. This led to a split, and a certain number left the meeting. "I had half a mind to go myself," said the chair-woman, "and if I had done so, two-thirds of us would have retired." "True," said another member; "but if I had persuaded my friends Mrs. Wild and Christine Armstrong to remain we should only have lost half our number." Can you tell how many were present at the meeting at the start? 108.--THE LEAP-YEAR LADIES. Last leap-year ladies lost no time in exercising the privilege of making proposals of marriage. If the figures that reached me from an occult source are correct, the following represents the state of affairs in this country. A number of women proposed once each, of whom one-eighth were widows. In consequence, a number of men were to be married of whom one-eleventh were widowers. Of the proposals made to widowers, one-fifth were declined. All the widows were accepted. Thirty-five forty-fourths of the widows married bachelors. One thousand two hundred and twenty-one spinsters were declined by bachelors. The number of spinsters accepted by bachelors was seven times the number of widows accepted by bachelors. Those are all the particulars that I was able to obtain. Now, how many women proposed? 109.--THE GREAT SCRAMBLE. After dinner, the five boys of a household happened to find a parcel of sugar-plums. It was quite unexpected loot, and an exciting scramble ensued, the full details of which I will recount with accuracy, as it forms an interesting puzzle. You see, Andrew managed to get possession of just two-thirds of the parcel of sugar-plums. Bob at once grabbed three-eighths of these, and Charlie managed to seize three-tenths also. Then young David dashed upon the scene, and captured all that Andrew had left, except one-seventh, which Edgar artfully secured for himself by a cunning trick. Now the fun began in real earnest, for Andrew and Charlie jointly set upon Bob, who stumbled against the fender and dropped half of all that he had, which were equally picked up by David and Edgar, who had crawled under a table and were waiting. Next, Bob sprang on Charlie from a chair, and upset all the latter's collection on to the floor. Of this prize Andrew got just a quarter, Bob gathered up one-third, David got two-sevenths, while Charlie and Edgar divided equally what was left of that stock. [Illustration] They were just thinking the fray was over when David suddenly struck out in two directions at once, upsetting three-quarters of what Bob and Andrew had last acquired. The two latter, with the greatest difficulty, recovered five-eighths of it in equal shares, but the three others each carried off one-fifth of the same. Every sugar-plum was now accounted for, and they called a truce, and divided equally amongst them the remainder of the parcel. What is the smallest number of sugar-plums there could have been at the start, and what proportion did each boy obtain? 110.--THE ABBOT'S PUZZLE. The first English puzzlist whose name has come down to us was a Yorkshireman--no other than Alcuin, Abbot of Canterbury (A.D. 735-804). Here is a little puzzle from his works, which is at least interesting on account of its antiquity. "If 100 bushels of corn were distributed among 100 people in such a manner that each man received three bushels, each woman two, and each child half a bushel, how many men, women, and children were there?" Now, there are six different correct answers, if we exclude a case where there would be no women. But let us say that there were just five times as many women as men, then what is the correct solution? 111.--REAPING THE CORN. A farmer had a square cornfield. The corn was all ripe for reaping, and, as he was short of men, it was arranged that he and his son should share the work between them. The farmer first cut one rod wide all round the square, thus leaving a smaller square of standing corn in the middle of the field. "Now," he said to his son, "I have cut my half of the field, and you can do your share." The son was not quite satisfied as to the proposed division of labour, and as the village schoolmaster happened to be passing, he appealed to that person to decide the matter. He found the farmer was quite correct, provided there was no dispute as to the size of the field, and on this point they were agreed. Can you tell the area of the field, as that ingenious schoolmaster succeeded in doing? 112.--A PUZZLING LEGACY. A man left a hundred acres of land to be divided among his three sons--Alfred, Benjamin, and Charles--in the proportion of one-third, one-fourth, and one-fifth respectively. But Charles died. How was the land to be divided fairly between Alfred and Benjamin? 113.--THE TORN NUMBER. [Illustration] I had the other day in my possession a label bearing the number 3 0 2 5 in large figures. This got accidentally torn in half, so that 3 0 was on one piece and 2 5 on the other, as shown on the illustration. On looking at these pieces I began to make a calculation, scarcely conscious of what I was doing, when I discovered this little peculiarity. If we add the 3 0 and the 2 5 together and square the sum we get as the result the complete original number on the label! Thus, 30 added to 25 is 55, and 55 multiplied by 55 is 3025. Curious, is it not? Now, the puzzle is to find another number, composed of four figures, all different, which may be divided in the middle and produce the same result. 114.--CURIOUS NUMBERS. The number 48 has this peculiarity, that if you add 1 to it the result is a square number (49, the square of 7), and if you add 1 to its half, you also get a square number (25, the square of 5). Now, there is no limit to the numbers that have this peculiarity, and it is an interesting puzzle to find three more of them--the smallest possible numbers. What are they? 115.--A PRINTER'S ERROR. In a certain article a printer had to set up the figures 5^4×2^3, which, of course, means that the fourth power of 5 (625) is to be multiplied by the cube of 2 (8), the product of which is 5,000. But he printed 5^4×2^3 as 5 4 2 3, which is not correct. Can you place four digits in the manner shown, so that it will be equally correct if the printer sets it up aright or makes the same blunder? 116.--THE CONVERTED MISER. Mr. Jasper Bullyon was one of the very few misers who have ever been converted to a sense of their duty towards their less fortunate fellow-men. One eventful night he counted out his accumulated wealth, and resolved to distribute it amongst the deserving poor. He found that if he gave away the same number of pounds every day in the year, he could exactly spread it over a twelvemonth without there being anything left over; but if he rested on the Sundays, and only gave away a fixed number of pounds every weekday, there would be one sovereign left over on New Year's Eve. Now, putting it at the lowest possible, what was the exact number of pounds that he had to distribute? Could any question be simpler? A sum of pounds divided by one number of days leaves no remainder, but divided by another number of days leaves a sovereign over. That is all; and yet, when you come to tackle this little question, you will be surprised that it can become so puzzling. 117.--A FENCE PROBLEM. [Illustration] The practical usefulness of puzzles is a point that we are liable to overlook. Yet, as a matter of fact, I have from time to time received quite a large number of letters from individuals who have found that the mastering of some little principle upon which a puzzle was built has proved of considerable value to them in a most unexpected way. Indeed, it may be accepted as a good maxim that a puzzle is of little real value unless, as well as being amusing and perplexing, it conceals some instructive and possibly useful feature. It is, however, very curious how these little bits of acquired knowledge dovetail into the occasional requirements of everyday life, and equally curious to what strange and mysterious uses some of our readers seem to apply them. What, for example, can be the object of Mr. Wm. Oxley, who writes to me all the way from Iowa, in wishing to ascertain the dimensions of a field that he proposes to enclose, containing just as many acres as there shall be rails in the fence? The man wishes to fence in a perfectly square field which is to contain just as many acres as there are rails in the required fence. Each hurdle, or portion of fence, is seven rails high, and two lengths would extend one pole (16½ ft.): that is to say, there are fourteen rails to the pole, lineal measure. Now, what must be the size of the field? 118.--CIRCLING THE SQUARES. [Illustration] The puzzle is to place a different number in each of the ten squares so that the sum of the squares of any two adjacent numbers shall be equal to the sum of the squares of the two numbers diametrically opposite to them. The four numbers placed, as examples, must stand as they are. The square of 16 is 256, and the square of 2 is 4. Add these together, and the result is 260. Also--the square of 14 is 196, and the square of 8 is 64. These together also make 260. Now, in precisely the same way, B and C should be equal to G and H (the sum will not necessarily be 260), A and K to F and E, H and I to C and D, and so on, with any two adjoining squares in the circle. All you have to do is to fill in the remaining six numbers. Fractions are not allowed, and I shall show that no number need contain more than two figures. 119.--RACKBRANE'S LITTLE LOSS. Professor Rackbrane was spending an evening with his old friends, Mr. and Mrs. Potts, and they engaged in some game (he does not say what game) of cards. The professor lost the first game, which resulted in doubling the money that both Mr. and Mrs. Potts had laid on the table. The second game was lost by Mrs. Potts, which doubled the money then held by her husband and the professor. Curiously enough, the third game was lost by Mr. Potts, and had the effect of doubling the money then held by his wife and the professor. It was then found that each person had exactly the same money, but the professor had lost five shillings in the course of play. Now, the professor asks, what was the sum of money with which he sat down at the table? Can you tell him? 120.--THE FARMER AND HIS SHEEP. [Illustration] Farmer Longmore had a curious aptitude for arithmetic, and was known in his district as the "mathematical farmer." The new vicar was not aware of this fact when, meeting his worthy parishioner one day in the lane, he asked him in the course of a short conversation, "Now, how many sheep have you altogether?" He was therefore rather surprised at Longmore's answer, which was as follows: "You can divide my sheep into two different parts, so that the difference between the two numbers is the same as the difference between their squares. Maybe, Mr. Parson, you will like to work out the little sum for yourself." Can the reader say just how many sheep the farmer had? Supposing he had possessed only twenty sheep, and he divided them into the two parts 12 and 8. Now, the difference between their squares, 144 and 64, is 80. So that will not do, for 4 and 80 are certainly not the same. If you can find numbers that work out correctly, you will know exactly how many sheep Farmer Longmore owned. 121.--HEADS OR TAILS. Crooks, an inveterate gambler, at Goodwood recently said to a friend, "I'll bet you half the money in my pocket on the toss of a coin--heads I win, tails I lose." The coin was tossed and the money handed over. He repeated the offer again and again, each time betting half the money then in his possession. We are not told how long the game went on, or how many times the coin was tossed, but this we know, that the number of times that Crooks lost was exactly equal to the number of times that he won. Now, did he gain or lose by this little venture? 122.--THE SEE-SAW PUZZLE. Necessity is, indeed, the mother of invention. I was amused the other day in watching a boy who wanted to play see-saw and, in his failure to find another child to share the sport with him, had been driven back upon the ingenious resort of tying a number of bricks to one end of the plank to balance his weight at the other. As a matter of fact, he just balanced against sixteen bricks, when these were fixed to the short end of plank, but if he fixed them to the long end of plank he only needed eleven as balance. Now, what was that boy's weight, if a brick weighs equal to a three-quarter brick and three-quarters of a pound? 123.--A LEGAL DIFFICULTY. "A client of mine," said a lawyer, "was on the point of death when his wife was about to present him with a child. I drew up his will, in which he settled two-thirds of his estate upon his son (if it should happen to be a boy) and one-third on the mother. But if the child should be a girl, then two-thirds of the estate should go to the mother and one-third to the daughter. As a matter of fact, after his death twins were born--a boy and a girl. A very nice point then arose. How was the estate to be equitably divided among the three in the closest possible accordance with the spirit of the dead man's will?" 124.--A QUESTION OF DEFINITION. "My property is exactly a mile square," said one landowner to another. "Curiously enough, mine is a square mile," was the reply. "Then there is no difference?" Is this last statement correct? 125.--THE MINERS' HOLIDAY. Seven coal-miners took a holiday at the seaside during a big strike. Six of the party spent exactly half a sovereign each, but Bill Harris was more extravagant. Bill spent three shillings more than the average of the party. What was the actual amount of Bill's expenditure? 126.--SIMPLE MULTIPLICATION. If we number six cards 1, 2, 4, 5, 7, and 8, and arrange them on the table in this order:-- 1 4 2 8 5 7 We can demonstrate that in order to multiply by 3 all that is necessary is to remove the 1 to the other end of the row, and the thing is done. The answer is 428571. Can you find a number that, when multiplied by 3 and divided by 2, the answer will be the same as if we removed the first card (which in this case is to be a 3) From the beginning of the row to the end? 127.--SIMPLE DIVISION. Sometimes a very simple question in elementary arithmetic will cause a good deal of perplexity. For example, I want to divide the four numbers, 701, 1,059, 1,417, and 2,312, by the largest number possible that will leave the same remainder in every case. How am I to set to work Of course, by a laborious system of trial one can in time discover the answer, but there is quite a simple method of doing it if you can only find it. 128.--A PROBLEM IN SQUARES. We possess three square boards. The surface of the first contains five square feet more than the second, and the second contains five square feet more than the third. Can you give exact measurements for the sides of the boards? If you can solve this little puzzle, then try to find three squares in arithmetical progression, with a common difference of 7 and also of 13. 129.--THE BATTLE OF HASTINGS. All historians know that there is a great deal of mystery and uncertainty concerning the details of the ever-memorable battle on that fatal day, October 14, 1066. My puzzle deals with a curious passage in an ancient monkish chronicle that may never receive the attention that it deserves, and if I am unable to vouch for the authenticity of the document it will none the less serve to furnish us with a problem that can hardly fail to interest those of my readers who have arithmetical predilections. Here is the passage in question. "The men of Harold stood well together, as their wont was, and formed sixty and one squares, with a like number of men in every square thereof, and woe to the hardy Norman who ventured to enter their redoubts; for a single blow of a Saxon war-hatchet would break his lance and cut through his coat of mail.... When Harold threw himself into the fray the Saxons were one mighty square of men, shouting the battle-cries, 'Ut!' 'Olicrosse!' 'Godemitè!'" Now, I find that all the contemporary authorities agree that the Saxons did actually fight in this solid order. For example, in the "Carmen de Bello Hastingensi," a poem attributed to Guy, Bishop of Amiens, living at the time of the battle, we are told that "the Saxons stood fixed in a dense mass," and Henry of Huntingdon records that "they were like unto a castle, impenetrable to the Normans;" while Robert Wace, a century after, tells us the same thing. So in this respect my newly-discovered chronicle may not be greatly in error. But I have reason to believe that there is something wrong with the actual figures. Let the reader see what he can make of them. The number of men would be sixty-one times a square number; but when Harold himself joined in the fray they were then able to form one large square. What is the smallest possible number of men there could have been? In order to make clear to the reader the simplicity of the question, I will give the lowest solutions in the case of 60 and 62, the numbers immediately preceding and following 61. They are 60 × 4² + 1 = 31², and 62 × 8² + 1 = 63². That is, 60 squares of 16 men each would be 960 men, and when Harold joined them they would be 961 in number, and so form a square with 31 men on every side. Similarly in the case of the figures I have given for 62. Now, find the lowest answer for 61. 130.--THE SCULPTOR'S PROBLEM. An ancient sculptor was commissioned to supply two statues, each on a cubical pedestal. It is with these pedestals that we are concerned. They were of unequal sizes, as will be seen in the illustration, and when the time arrived for payment a dispute arose as to whether the agreement was based on lineal or cubical measurement. But as soon as they came to measure the two pedestals the matter was at once settled, because, curiously enough, the number of lineal feet was exactly the same as the number of cubical feet. The puzzle is to find the dimensions for two pedestals having this peculiarity, in the smallest possible figures. You see, if the two pedestals, for example, measure respectively 3 ft. and 1 ft. on every side, then the lineal measurement would be 4 ft. and the cubical contents 28 ft., which are not the same, so these measurements will not do. [Illustration] 131.--THE SPANISH MISER. There once lived in a small town in New Castile a noted miser named Don Manuel Rodriguez. His love of money was only equalled by a strong passion for arithmetical problems. These puzzles usually dealt in some way or other with his accumulated treasure, and were propounded by him solely in order that he might have the pleasure of solving them himself. Unfortunately very few of them have survived, and when travelling through Spain, collecting material for a proposed work on "The Spanish Onion as a Cause of National Decadence," I only discovered a very few. One of these concerns the three boxes that appear in the accompanying authentic portrait. [Illustration] Each box contained a different number of golden doubloons. The difference between the number of doubloons in the upper box and the number in the middle box was the same as the difference between the number in the middle box and the number in the bottom box. And if the contents of any two of the boxes were united they would form a square number. What is the smallest number of doubloons that there could have been in any one of the boxes? 132.--THE NINE TREASURE BOXES. The following puzzle will illustrate the importance on occasions of being able to fix the minimum and maximum limits of a required number. This can very frequently be done. For example, it has not yet been ascertained in how many different ways the knight's tour can be performed on the chess board; but we know that it is fewer than the number of combinations of 168 things taken 63 at a time and is greater than 31,054,144--for the latter is the number of routes of a particular type. Or, to take a more familiar case, if you ask a man how many coins he has in his pocket, he may tell you that he has not the slightest idea. But on further questioning you will get out of him some such statement as the following: "Yes, I am positive that I have more than three coins, and equally certain that there are not so many as twenty-five." Now, the knowledge that a certain number lies between 2 and 12 in my puzzle will enable the solver to find the exact answer; without that information there would be an infinite number of answers, from which it would be impossible to select the correct one. This is another puzzle received from my friend Don Manuel Rodriguez, the cranky miser of New Castile. On New Year's Eve in 1879 he showed me nine treasure boxes, and after informing me that every box contained a square number of golden doubloons, and that the difference between the contents of A and B was the same as between B and C, D and E, E and F, G and H, or H and I, he requested me to tell him the number of coins in every one of the boxes. At first I thought this was impossible, as there would be an infinite number of different answers, but on consideration I found that this was not the case. I discovered that while every box contained coins, the contents of A, B, C increased in weight in alphabetical order; so did D, E, F; and so did G, H, I; but D or E need not be heavier than C, nor G or H heavier than F. It was also perfectly certain that box A could not contain more than a dozen coins at the outside; there might not be half that number, but I was positive that there were not more than twelve. With this knowledge I was able to arrive at the correct answer. In short, we have to discover nine square numbers such that A, B, C; and D, E, F; and G, H, I are three groups in arithmetical progression, the common difference being the same in each group, and A being less than 12. How many doubloons were there in every one of the nine boxes? 133.--THE FIVE BRIGANDS. The five Spanish brigands, Alfonso, Benito, Carlos, Diego, and Esteban, were counting their spoils after a raid, when it was found that they had captured altogether exactly 200 doubloons. One of the band pointed out that if Alfonso had twelve times as much, Benito three times as much, Carlos the same amount, Diego half as much, and Esteban one-third as much, they would still have altogether just 200 doubloons. How many doubloons had each? There are a good many equally correct answers to this question. Here is one of them: A 6 × 12 = 72 B 12 × 3 = 36 C 17 × 1 = 17 D 120 × ½ = 60 E 45 × 1/3 = 15 ___ ___ 200 200 The puzzle is to discover exactly how many different answers there are, it being understood that every man had something and that there is to be no fractional money--only doubloons in every case. This problem, worded somewhat differently, was propounded by Tartaglia (died 1559), and he flattered himself that he had found one solution; but a French mathematician of note (M.A. Labosne), in a recent work, says that his readers will be astonished when he assures them that there are 6,639 different correct answers to the question. Is this so? How many answers are there? 134.--THE BANKER'S PUZZLE. A banker had a sporting customer who was always anxious to wager on anything. Hoping to cure him of his bad habit, he proposed as a wager that the customer would not be able to divide up the contents of a box containing only sixpences into an exact number of equal piles of sixpences. The banker was first to put in one or more sixpences (as many as he liked); then the customer was to put in one or more (but in his case not more than a pound in value), neither knowing what the other put in. Lastly, the customer was to transfer from the banker's counter to the box as many sixpences as the banker desired him to put in. The puzzle is to find how many sixpences the banker should first put in and how many he should ask the customer to transfer, so that he may have the best chance of winning. 135.--THE STONEMASON'S PROBLEM. A stonemason once had a large number of cubic blocks of stone in his yard, all of exactly the same size. He had some very fanciful little ways, and one of his queer notions was to keep these blocks piled in cubical heaps, no two heaps containing the same number of blocks. He had discovered for himself (a fact that is well known to mathematicians) that if he took all the blocks contained in any number of heaps in regular order, beginning with the single cube, he could always arrange those on the ground so as to form a perfect square. This will be clear to the reader, because one block is a square, 1 + 8 = 9 is a square, 1 + 8 + 27 = 36 is a square, 1 + 8 + 27 + 64 = 100 is a square, and so on. In fact, the sum of any number of consecutive cubes, beginning always with 1, is in every case a square number. One day a gentleman entered the mason's yard and offered him a certain price if he would supply him with a consecutive number of these cubical heaps which should contain altogether a number of blocks that could be laid out to form a square, but the buyer insisted on more than three heaps and _declined to take the single block_ because it contained a flaw. What was the smallest possible number of blocks of stone that the mason had to supply? 136.--THE SULTAN'S ARMY. A certain Sultan wished to send into battle an army that could be formed into two perfect squares in twelve different ways. What is the smallest number of men of which that army could be composed? To make it clear to the novice, I will explain that if there were 130 men, they could be formed into two squares in only two different ways--81 and 49, or 121 and 9. Of course, all the men must be used on every occasion. 137.--A STUDY IN THRIFT. Certain numbers are called triangular, because if they are taken to represent counters or coins they may be laid out on the table so as to form triangles. The number 1 is always regarded as triangular, just as 1 is a square and a cube number. Place one counter on the table--that is, the first triangular number. Now place two more counters beneath it, and you have a triangle of three counters; therefore 3 is triangular. Next place a row of three more counters, and you have a triangle of six counters; therefore 6 is triangular. We see that every row of counters that we add, containing just one more counter than the row above it, makes a larger triangle. Now, half the sum of any number and its square is always a triangular number. Thus half of 2 + 2² = 3; half of 3 + 3² = 6; half of 4 + 4² = 10; half of 5 + 5²= 15; and so on. So if we want to form a triangle with 8 counters on each side we shall require half of 8 + 8², or 36 counters. This is a pretty little property of numbers. Before going further, I will here say that if the reader refers to the "Stonemason's Problem" (No. 135) he will remember that the sum of any number of consecutive cubes beginning with 1 is always a square, and these form the series 1², 3², 6², 10², etc. It will now be understood when I say that one of the keys to the puzzle was the fact that these are always the squares of triangular numbers--that is, the squares of 1, 3, 6, 10, 15, 21, 28, etc., any of which numbers we have seen will form a triangle. Every whole number is either triangular, or the sum of two triangular numbers or the sum of three triangular numbers. That is, if we take any number we choose we can always form one, two, or three triangles with them. The number 1 will obviously, and uniquely, only form one triangle; some numbers will only form two triangles (as 2, 4, 11, etc.); some numbers will only form three triangles (as 5, 8, 14, etc.). Then, again, some numbers will form both one and two triangles (as 6), others both one and three triangles (as 3 and 10), others both two and three triangles (as 7 and 9), while some numbers (like 21) will form one, two, or three triangles, as we desire. Now for a little puzzle in triangular numbers. Sandy McAllister, of Aberdeen, practised strict domestic economy, and was anxious to train his good wife in his own habits of thrift. He told her last New Year's Eve that when she had saved so many sovereigns that she could lay them all out on the table so as to form a perfect square, or a perfect triangle, or two triangles, or three triangles, just as he might choose to ask he would add five pounds to her treasure. Soon she went to her husband with a little bag of £36 in sovereigns and claimed her reward. It will be found that the thirty-six coins will form a square (with side 6), that they will form a single triangle (with side 8), that they will form two triangles (with sides 5 and 6), and that they will form three triangles (with sides 3, 5, and 5). In each of the four cases all the thirty-six coins are used, as required, and Sandy therefore made his wife the promised present like an honest man. The Scotsman then undertook to extend his promise for five more years, so that if next year the increased number of sovereigns that she has saved can be laid out in the same four different ways she will receive a second present; if she succeeds in the following year she will get a third present, and so on until she has earned six presents in all. Now, how many sovereigns must she put together before she can win the sixth present? What you have to do is to find five numbers, the smallest possible, higher than 36, that can be displayed in the four ways--to form a square, to form a triangle, to form two triangles, and to form three triangles. The highest of your five numbers will be your answer. 138.--THE ARTILLERYMEN'S DILEMMA. [Illustration: [Pyramid of cannon-balls]] MMMMMMMr MM MM: M 0 rWZX M : MWM aX ,BM M 0M M aMMMM2MW 02 MMWMMr ZM. M@M 8MM 7 XM2 MS2 M.MMMWMMMM MM M MX iMM M7W 8 . M r W M@ Z;M M 0r ; M M M W 22 W M @ M M M.M2WMMMMZ ;MM@X:7MMMB; MMM ZM M:MM0;8: ,MS Ma 8 MMMMMMMi rM 2MMMMMM MB M 7 XM, ,: BMM: r7S .,MM MM MB M i ,M , 2 ; aMMMMMMMMM XM; MZM M . M 7 M . Z M M M8 M M S M .0 M 8MM aMi: MMMM7M ,7 .iM X M @ aZ M M 8 ,@MMMMBMMMa SMW 7M,XZ@MM M 8M M .M MMMM@X MMr Ma MMMMMMMMM@ M .WM M @WM7WMM .WX MZS M 8M :MMMWMMMM 8X MMMBMMM7 7aM 2MM r, 8r ZM2 Mr2 aMM; Mai :MS :iM ZiM @MX M M . M Wr.MMMaBMMMB M M MZ. ,M MMZ Mr M M B0 Z 2S iM S XM 7 WMM MM @.M M M W M. M M 0;M2M;MMMM: WW8aMM M S@ M M M : MaMMMMMM MM0W;MZM: M i M M MM MMMZMBZa0ar B20rMMM Si i BW MMM02 7MM0 2MMM MMMMMMMM . M SM@aiMM20BWM XM 0ZMMM:MMMMW; r. 0WMBM2XrB: . "All cannon-balls are to be piled in square pyramids," was the order issued to the regiment. This was done. Then came the further order, "All pyramids are to contain a square number of balls." Whereupon the trouble arose. "It can't be done," said the major. "Look at this pyramid, for example; there are sixteen balls at the base, then nine, then four, then one at the top, making thirty balls in all. But there must be six more balls, or five fewer, to make a square number." "It _must_ be done," insisted the general. "All you have to do is to put the right number of balls in your pyramids." "I've got it!" said a lieutenant, the mathematical genius of the regiment. "Lay the balls out singly." "Bosh!" exclaimed the general. "You can't _pile_ one ball into a pyramid!" Is it really possible to obey both orders? 139.--THE DUTCHMEN'S WIVES. I wonder how many of my readers are acquainted with the puzzle of the "Dutchmen's Wives"--in which you have to determine the names of three men's wives, or, rather, which wife belongs to each husband. Some thirty years ago it was "going the rounds," as something quite new, but I recently discovered it in the _Ladies' Diary_ for 1739-40, so it was clearly familiar to the fair sex over one hundred and seventy years ago. How many of our mothers, wives, sisters, daughters, and aunts could solve the puzzle to-day? A far greater proportion than then, let us hope. Three Dutchmen, named Hendrick, Elas, and Cornelius, and their wives, Gurtrün, Katrün, and Anna, purchase hogs. Each buys as many as he (or she) gives shillings for one. Each husband pays altogether three guineas more than his wife. Hendrick buys twenty-three more hogs than Katrün, and Elas eleven more than Gurtrün. Now, what was the name of each man's wife? [Illustration] 140.--FIND ADA'S SURNAME. This puzzle closely resembles the last one, my remarks on the solution of which the reader may like to apply in another case. It was recently submitted to a Sydney evening newspaper that indulges in "intellect sharpeners," but was rejected with the remark that it is childish and that they only published problems capable of solution! Five ladies, accompanied by their daughters, bought cloth at the same shop. Each of the ten paid as many farthings per foot as she bought feet, and each mother spent 8s. 5¼d. more than her daughter. Mrs. Robinson spent 6s. more than Mrs. Evans, who spent about a quarter as much as Mrs. Jones. Mrs. Smith spent most of all. Mrs. Brown bought 21 yards more than Bessie--one of the girls. Annie bought 16 yards more than Mary and spent £3, 0s. 8d. more than Emily. The Christian name of the other girl was Ada. Now, what was her surname? 141.--SATURDAY MARKETING. Here is an amusing little case of marketing which, although it deals with a good many items of money, leads up to a question of a totally different character. Four married couples went into their village on a recent Saturday night to do a little marketing. They had to be very economical, for among them they only possessed forty shilling coins. The fact is, Ann spent 1s., Mary spent 2s., Jane spent 3s., and Kate spent 4s. The men were rather more extravagant than their wives, for Ned Smith spent as much as his wife, Tom Brown twice as much as his wife, Bill Jones three times as much as his wife, and Jack Robinson four times as much as his wife. On the way home somebody suggested that they should divide what coin they had left equally among them. This was done, and the puzzling question is simply this: What was the surname of each woman? Can you pair off the four couples? GEOMETRICAL PROBLEMS. "God geometrizes continually." PLATO. "There is no study," said Augustus de Morgan, "which presents so simple a beginning as that of geometry; there is none in which difficulties grow more rapidly as we proceed." This will be found when the reader comes to consider the following puzzles, though they are not arranged in strict order of difficulty. And the fact that they have interested and given pleasure to man for untold ages is no doubt due in some measure to the appeal they make to the eye as well as to the brain. Sometimes an algebraical formula or theorem seems to give pleasure to the mathematician's eye, but it is probably only an intellectual pleasure. But there can be no doubt that in the case of certain geometrical problems, notably dissection or superposition puzzles, the æsthetic faculty in man contributes to the delight. For example, there are probably few readers who will examine the various cuttings of the Greek cross in the following pages without being in some degree stirred by a sense of beauty. Law and order in Nature are always pleasing to contemplate, but when they come under the very eye they seem to make a specially strong appeal. Even the person with no geometrical knowledge whatever is induced after the inspection of such things to exclaim, "How very pretty!" In fact, I have known more than one person led on to a study of geometry by the fascination of cutting-out puzzles. I have, therefore, thought it well to keep these dissection puzzles distinct from the geometrical problems on more general lines. DISSECTION PUZZLES. "Take him and cut him out in little stars." _Romeo and Juliet_, iii. 2. Puzzles have infinite variety, but perhaps there is no class more ancient than dissection, cutting-out, or superposition puzzles. They were certainly known to the Chinese several thousand years before the Christian era. And they are just as fascinating to-day as they can have been at any period of their history. It is supposed by those who have investigated the matter that the ancient Chinese philosophers used these puzzles as a sort of kindergarten method of imparting the principles of geometry. Whether this was so or not, it is certain that all good dissection puzzles (for the nursery type of jig-saw puzzle, which merely consists in cutting up a picture into pieces to be put together again, is not worthy of serious consideration) are really based on geometrical laws. This statement need not, however, frighten off the novice, for it means little more than this, that geometry will give us the "reason why," if we are interested in knowing it, though the solutions may often be discovered by any intelligent person after the exercise of patience, ingenuity, and common sagacity. If we want to cut one plane figure into parts that by readjustment will form another figure, the first thing is to find a way of doing it at all, and then to discover how to do it in the fewest possible pieces. Often a dissection problem is quite easy apart from this limitation of pieces. At the time of the publication in the _Weekly Dispatch_, in 1902, of a method of cutting an equilateral triangle into four parts that will form a square (see No. 26, "Canterbury Puzzles"), no geometrician would have had any difficulty in doing what is required in five pieces: the whole point of the discovery lay in performing the little feat in four pieces only. Mere approximations in the case of these problems are valueless; the solution must be geometrically exact, or it is not a solution at all. Fallacies are cropping up now and again, and I shall have occasion to refer to one or two of these. They are interesting merely as fallacies. But I want to say something on two little points that are always arising in cutting-out puzzles--the questions of "hanging by a thread" and "turning over." These points can best be illustrated by a puzzle that is frequently to be found in the old books, but invariably with a false solution. The puzzle is to cut the figure shown in Fig. 1 into three pieces that will fit together and form a half-square triangle. The answer that is invariably given is that shown in Figs. 1 and 2. Now, it is claimed that the four pieces marked C are really only one piece, because they may be so cut that they are left "hanging together by a mere thread." But no serious puzzle lover will ever admit this. If the cut is made so as to leave the four pieces joined in one, then it cannot result in a perfectly exact solution. If, on the other hand, the solution is to be exact, then there will be four pieces--or six pieces in all. It is, therefore, not a solution in three pieces. [Illustration: Fig. 1] [Illustration: Fig. 2] If, however, the reader will look at the solution in Figs. 3 and 4, he will see that no such fault can be found with it. There is no question whatever that there are three pieces, and the solution is in this respect quite satisfactory. But another question arises. It will be found on inspection that the piece marked F, in Fig. 3, is turned over in Fig. 4--that is to say, a different side has necessarily to be presented. If the puzzle were merely to be cut out of cardboard or wood, there might be no objection to this reversal, but it is quite possible that the material would not admit of being reversed. There might be a pattern, a polish, a difference of texture, that prevents it. But it is generally understood that in dissection puzzles you are allowed to turn pieces over unless it is distinctly stated that you may not do so. And very often a puzzle is greatly improved by the added condition, "no piece may be turned over." I have often made puzzles, too, in which the diagram has a small repeated pattern, and the pieces have then so to be cut that not only is there no turning over, but the pattern has to be matched, which cannot be done if the pieces are turned round, even with the proper side uppermost. [Illustration: Fig. 3] [Illustration: Fig. 4] Before presenting a varied series of cutting-out puzzles, some very easy and others difficult, I propose to consider one family alone--those problems involving what is known as the Greek cross with the square. This will exhibit a great variety of curious transpositions, and, by having the solutions as we go along, the reader will be saved the trouble of perpetually turning to another part of the book, and will have everything under his eye. It is hoped that in this way the article may prove somewhat instructive to the novice and interesting to others. GREEK CROSS PUZZLES. "To fret thy soul with crosses." SPENSER. "But, for my part, it was Greek to me." _Julius Cæsar_, i. 2. Many people are accustomed to consider the cross as a wholly Christian symbol. This is erroneous: it is of very great antiquity. The ancient Egyptians employed it as a sacred symbol, and on Greek sculptures we find representations of a cake (the supposed real origin of our hot cross buns) bearing a cross. Two such cakes were discovered at Herculaneum. Cecrops offered to Jupiter Olympus a sacred cake or _boun_ of this kind. The cross and ball, so frequently found on Egyptian figures, is a circle and the _tau_ cross. The circle signified the eternal preserver of the world, and the T, named from the Greek letter _tau_, is the monogram of Thoth, the Egyptian Mercury, meaning wisdom. This _tau_ cross is also called by Christians the cross of St. Anthony, and is borne on a badge in the bishop's palace at Exeter. As for the Greek or mundane cross, the cross with four equal arms, we are told by competent antiquaries that it was regarded by ancient occultists for thousands of years as a sign of the dual forces of Nature--the male and female spirit of everything that was everlasting. [Illustration: Fig. 5.] The Greek cross, as shown in Fig. 5, is formed by the assembling together of five equal squares. We will start with what is known as the Hindu problem, supposed to be upwards of three thousand years old. It appears in the seal of Harvard College, and is often given in old works as symbolical of mathematical science and exactitude. Cut the cross into five pieces to form a square. Figs. 6 and 7 show how this is done. It was not until the middle of the nineteenth century that we found that the cross might be transformed into a square in only four pieces. Figs. 8 and 9 will show how to do it, if we further require the four pieces to be all of the same size and shape. This Fig. 9 is remarkable because, according to Dr. Le Plongeon and others, as expounded in a work by Professor Wilson of the Smithsonian Institute, here we have the great Swastika, or sign, of "good luck to you "--the most ancient symbol of the human race of which there is any record. Professor Wilson's work gives some four hundred illustrations of this curious sign as found in the Aztec mounds of Mexico, the pyramids of Egypt, the ruins of Troy, and the ancient lore of India and China. One might almost say there is a curious affinity between the Greek cross and Swastika! If, however, we require that the four pieces shall be produced by only two clips of the scissors (assuming the puzzle is in paper form), then we must cut as in Fig. 10 to form Fig. 11, the first clip of the scissors being from a to b. Of course folding the paper, or holding the pieces together after the first cut, would not in this case be allowed. But there is an infinite number of different ways of making the cuts to solve the puzzle in four pieces. To this point I propose to return. [Illustration: Fig. 6] [Illustration: Fig. 7] [Illustration: Fig. 8] [Illustration: Fig. 9] [Illustration: Fig. 10] [Illustration: Fig. 11] It will be seen that every one of these puzzles has its reverse puzzle--to cut a square into pieces to form a Greek cross. But as a square has not so many angles as the cross, it is not always equally easy to discover the true directions of the cuts. Yet in the case of the examples given, I will leave the reader to determine their direction for himself, as they are rather obvious from the diagrams. Cut a square into five pieces that will form two separate Greek crosses of _different sizes_. This is quite an easy puzzle. As will be seen in Fig. 12, we have only to divide our square into 25 little squares and then cut as shown. The cross A is cut out entire, and the pieces B, C, D, and E form the larger cross in Fig. 13. The reader may here like to cut the single piece, B, into four pieces all similar in shape to itself, and form a cross with them in the manner shown in Fig. 13. I hardly need give the solution. [Illustration: FIG. 12.] [Illustration: FIG. 13.] Cut a square into five pieces that will form two separate Greek crosses of exactly the _same size_. This is more difficult. We make the cuts as in Fig. 14, where the cross A comes out entire and the other four pieces form the cross in Fig. 15. The direction of the cuts is pretty obvious. It will be seen that the sides of the square in Fig. 14 are marked off into six equal parts. The sides of the cross are found by ruling lines from certain of these points to others. [Illustration: FIG. 14.] [Illustration: FIG. 15.] I will now explain, as I promised, why a Greek cross may be cut into four pieces in an infinite number of different ways to make a square. Draw a cross, as in Fig. 16. Then draw on transparent paper the square shown in Fig. 17, taking care that the distance c to d is exactly the same as the distance a to b in the cross. Now place the transparent paper over the cross and slide it about into different positions, only be very careful always to keep the square at the same angle to the cross as shown, where a b is parallel to c d. If you place the point c exactly over a the lines will indicate the solution (Figs. 10 and 11). If you place c in the very centre of the dotted square, it will give the solution in Figs. 8 and 9. You will now see that by sliding the square about so that the point c is always within the dotted square you may get as many different solutions as you like; because, since an infinite number of different points may theoretically be placed within this square, there must be an infinite number of different solutions. But the point c need not necessarily be placed within the dotted square. It may be placed, for example, at point e to give a solution in four pieces. Here the joins at a and f may be as slender as you like. Yet if you once get over the edge at a or f you no longer have a solution in four pieces. This proof will be found both entertaining and instructive. If you do not happen to have any transparent paper at hand, any thin paper will of course do if you hold the two sheets against a pane of glass in the window. [Illustration: FIG. 16.] [Illustration: FIG. 17.] It may have been noticed from the solutions of the puzzles that I have given that the side of the square formed from the cross is always equal to the distance a to b in Fig. 16. This must necessarily be so, and I will presently try to make the point quite clear. We will now go one step further. I have already said that the ideal solution to a cutting-out puzzle is always that which requires the fewest possible pieces. We have just seen that two crosses of the same size may be cut out of a square in five pieces. The reader who succeeded in solving this perhaps asked himself: "Can it be done in fewer pieces?" This is just the sort of question that the true puzzle lover is always asking, and it is the right attitude for him to adopt. The answer to the question is that the puzzle may be solved in four pieces--the fewest possible. This, then, is a new puzzle. Cut a square into four pieces that will form two Greek crosses of the same size. [Illustration: FIG. 18.] [Illustration: FIG. 19.] [Illustration: FIG. 20.] The solution is very beautiful. If you divide by points the sides of the square into three equal parts, the directions of the lines in Fig. 18 will be quite obvious. If you cut along these lines, the pieces A and B will form the cross in Fig. 19 and the pieces C and D the similar cross in Fig. 20. In this square we have another form of Swastika. The reader will here appreciate the truth of my remark to the effect that it is easier to find the directions of the cuts when transforming a cross to a square than when converting a square into a cross. Thus, in Figs. 6, 8, and 10 the directions of the cuts are more obvious than in Fig. 14, where we had first to divide the sides of the square into six equal parts, and in Fig. 18, where we divide them into three equal parts. Then, supposing you were required to cut two equal Greek crosses, each into two pieces, to form a square, a glance at Figs. 19 and 20 will show how absurdly more easy this is than the reverse puzzle of cutting the square to make two crosses. Referring to my remarks on "fallacies," I will now give a little example of these "solutions" that are not solutions. Some years ago a young correspondent sent me what he evidently thought was a brilliant new discovery--the transforming of a square into a Greek cross in four pieces by cuts all parallel to the sides of the square. I give his attempt in Figs. 21 and 22, where it will be seen that the four pieces do not form a symmetrical Greek cross, because the four arms are not really squares but oblongs. To make it a true Greek cross we should require the additions that I have indicated with dotted lines. Of course his solution produces a cross, but it is not the symmetrical Greek variety required by the conditions of the puzzle. My young friend thought his attempt was "near enough" to be correct; but if he bought a penny apple with a sixpence he probably would not have thought it "near enough" if he had been given only fourpence change. As the reader advances he will realize the importance of this question of exactitude. [Illustration: FIG. 21.] [Illustration: FIG. 22.] In these cutting-out puzzles it is necessary not only to get the directions of the cutting lines as correct as possible, but to remember that these lines have no width. If after cutting up one of the crosses in a manner indicated in these articles you find that the pieces do not exactly fit to form a square, you may be certain that the fault is entirely your own. Either your cross was not exactly drawn, or your cuts were not made quite in the right directions, or (if you used wood and a fret-saw) your saw was not sufficiently fine. If you cut out the puzzles in paper with scissors, or in cardboard with a penknife, no material is lost; but with a saw, however fine, there is a certain loss. In the case of most puzzles this slight loss is not sufficient to be appreciable, if the puzzle is cut out on a large scale, but there have been instances where I have found it desirable to draw and cut out each part separately--not from one diagram--in order to produce a perfect result. [Illustration: FIG. 23.] [Illustration: FIG. 24.] Now for another puzzle. If you have cut out the five pieces indicated in Fig. 14, you will find that these can be put together so as to form the curious cross shown in Fig. 23. So if I asked you to cut Fig. 24 into five pieces to form either a square or two equal Greek crosses you would know how to do it. You would make the cuts as in Fig. 23, and place them together as in Figs. 14 and 15. But I want something better than that, and it is this. Cut Fig. 24 into only four pieces that will fit together and form a square. [Illustration: FIG. 25.] [Illustration: FIG. 26.] The solution to the puzzle is shown in Figs. 25 and 26. The direction of the cut dividing A and C in the first diagram is very obvious, and the second cut is made at right angles to it. That the four pieces should fit together and form a square will surprise the novice, who will do well to study the puzzle with some care, as it is most instructive. I will now explain the beautiful rule by which we determine the size of a square that shall have the same area as a Greek cross, for it is applicable, and necessary, to the solution of almost every dissection puzzle that we meet with. It was first discovered by the philosopher Pythagoras, who died 500 B.C., and is the 47th proposition of Euclid. The young reader who knows nothing of the elements of geometry will get some idea of the fascinating character of that science. The triangle ABC in Fig. 27 is what we call a right-angled triangle, because the side BC is at right angles to the side AB. Now if we build up a square on each side of the triangle, the squares on AB and BC will together be exactly equal to the square on the long side AC, which we call the hypotenuse. This is proved in the case I have given by subdividing the three squares into cells of equal dimensions. [Illustration: FIG. 27.] [Illustration: FIG. 28.] It will be seen that 9 added to 16 equals 25, the number of cells in the large square. If you make triangles with the sides 5, 12 and 13, or with 8, 15 and 17, you will get similar arithmetical proofs, for these are all "rational" right-angled triangles, but the law is equally true for all cases. Supposing we cut off the lower arm of a Greek cross and place it to the left of the upper arm, as in Fig. 28, then the square on EF added to the square on DE exactly equals a square on DF. Therefore we know that the square of DF will contain the same area as the cross. This fact we have proved practically by the solutions of the earlier puzzles of this series. But whatever length we give to DE and EF, we can never give the exact length of DF in numbers, because the triangle is not a "rational" one. But the law is none the less geometrically true. [Illustration: FIG. 29.] [Illustration: FIG. 30.] Now look at Fig. 29, and you will see an elegant method for cutting a piece of wood of the shape of two squares (of any relative dimensions) into three pieces that will fit together and form a single square. If you mark off the distance _ab_ equal to the side _cd_ the directions of the cuts are very evident. From what we have just been considering, you will at once see why _bc_ must be the length of the side of the new square. Make the experiment as often as you like, taking different relative proportions for the two squares, and you will find the rule always come true. If you make the two squares of exactly the same size, you will see that the diagonal of any square is always the side of a square that is twice the size. All this, which is so simple that anybody can understand it, is very essential to the solving of cutting-out puzzles. It is in fact the key to most of them. And it is all so beautiful that it seems a pity that it should not be familiar to everybody. We will now go one step further and deal with the half-square. Take a square and cut it in half diagonally. Now try to discover how to cut this triangle into four pieces that will form a Greek cross. The solution is shown in Figs. 31 and 32. In this case it will be seen that we divide two of the sides of the triangle into three equal parts and the long side into four equal parts. Then the direction of the cuts will be easily found. It is a pretty puzzle, and a little more difficult than some of the others that I have given. It should be noted again that it would have been much easier to locate the cuts in the reverse puzzle of cutting the cross to form a half-square triangle. [Illustration: FIG. 31.] [Illustration: FIG. 32.] [Illustration: FIG. 33.] [Illustration: FIG. 34.] Another ideal that the puzzle maker always keeps in mind is to contrive that there shall, if possible, be only one correct solution. Thus, in the case of the first puzzle, if we only require that a Greek cross shall be cut into four pieces to form a square, there is, as I have shown, an infinite number of different solutions. It makes a better puzzle to add the condition that all the four pieces shall be of the same size and shape, because it can then be solved in only one way, as in Figs. 8 and 9. In this way, too, a puzzle that is too easy to be interesting may be improved by such an addition. Let us take an example. We have seen in Fig. 28 that Fig. 33 can be cut into two pieces to form a Greek cross. I suppose an intelligent child would do it in five minutes. But suppose we say that the puzzle has to be solved with a piece of wood that has a bad knot in the position shown in Fig. 33--a knot that we must not attempt to cut through--then a solution in two pieces is barred out, and it becomes a more interesting puzzle to solve it in three pieces. I have shown in Figs. 33 and 34 one way of doing this, and it will be found entertaining to discover other ways of doing it. Of course I could bar out all these other ways by introducing more knots, and so reduce the puzzle to a single solution, but it would then be overloaded with conditions. And this brings us to another point in seeking the ideal. Do not overload your conditions, or you will make your puzzle too complex to be interesting. The simpler the conditions of a puzzle are, the better. The solution may be as complex and difficult as you like, or as happens, but the conditions ought to be easily understood, or people will not attempt a solution. If the reader were now asked "to cut a half-square into as few pieces as possible to form a Greek cross," he would probably produce our solution, Figs. 31-32, and confidently claim that he had solved the puzzle correctly. In this way he would be wrong, because it is not now stated that the square is to be divided diagonally. Although we should always observe the exact conditions of a puzzle we must not read into it conditions that are not there. Many puzzles are based entirely on the tendency that people have to do this. The very first essential in solving a puzzle is to be sure that you understand the exact conditions. Now, if you divided your square in half so as to produce Fig. 35 it is possible to cut it into as few as three pieces to form a Greek cross. We thus save a piece. I give another puzzle in Fig. 36. The dotted lines are added merely to show the correct proportions of the figure--a square of 25 cells with the four corner cells cut out. The puzzle is to cut this figure into five pieces that will form a Greek cross (entire) and a square. [Illustration: FIG. 35.] [Illustration: FIG. 36.] The solution to the first of the two puzzles last given--to cut a rectangle of the shape of a half-square into three pieces that will form a Greek cross--is shown in Figs. 37 and 38. It will be seen that we divide the long sides of the oblong into six equal parts and the short sides into three equal parts, in order to get the points that will indicate the direction of the cuts. The reader should compare this solution with some of the previous illustrations. He will see, for example, that if we continue the cut that divides B and C in the cross, we get Fig. 15. [Illustration: FIG. 37.] [Illustration: FIG. 38.] The other puzzle, like the one illustrated in Figs. 12 and 13, will show how useful a little arithmetic may sometimes prove to be in the solution of dissection puzzles. There are twenty-one of those little square cells into which our figure is subdivided, from which we have to form both a square and a Greek cross. Now, as the cross is built up of five squares, and 5 from 21 leaves 16--a square number--we ought easily to be led to the solution shown in Fig. 39. It will be seen that the cross is cut out entire, while the four remaining pieces form the square in Fig. 40. [Illustration: FIG. 39] [Illustration: FIG. 40] Of course a half-square rectangle is the same as a double square, or two equal squares joined together. Therefore, if you want to solve the puzzle of cutting a Greek cross into four pieces to form two separate squares of the same size, all you have to do is to continue the short cut in Fig. 38 right across the cross, and you will have four pieces of the same size and shape. Now divide Fig. 37 into two equal squares by a horizontal cut midway and you will see the four pieces forming the two squares. [Illustration: FIG. 41] Cut a Greek cross into five pieces that will form two separate squares, one of which shall contain half the area of one of the arms of the cross. In further illustration of what I have already written, if the two squares of the same size A B C D and B C F E, in Fig. 41, are cut in the manner indicated by the dotted lines, the four pieces will form the large square A G E C. We thus see that the diagonal A C is the side of a square twice the size of A B C D. It is also clear that half the diagonal of any square is equal to the side of a square of half the area. Therefore, if the large square in the diagram is one of the arms of your cross, the small square is the size of one of the squares required in the puzzle. The solution is shown in Figs. 42 and 43. It will be seen that the small square is cut out whole and the large square composed of the four pieces B, C, D, and E. After what I have written, the reader will have no difficulty in seeing that the square A is half the size of one of the arms of the cross, because the length of the diagonal of the former is clearly the same as the side of the latter. The thing is now self-evident. I have thus tried to show that some of these puzzles that many people are apt to regard as quite wonderful and bewildering, are really not difficult if only we use a little thought and judgment. In conclusion of this particular subject I will give four Greek cross puzzles, with detached solutions. 142.--THE SILK PATCHWORK. The lady members of the Wilkinson family had made a simple patchwork quilt, as a small Christmas present, all composed of square pieces of the same size, as shown in the illustration. It only lacked the four corner pieces to make it complete. Somebody pointed out to them that if you unpicked the Greek cross in the middle and then cut the stitches along the dark joins, the four pieces all of the same size and shape would fit together and form a square. This the reader knows, from the solution in Fig. 39, is quite easily done. But George Wilkinson suddenly suggested to them this poser. He said, "Instead of picking out the cross entire, and forming the square from four equal pieces, can you cut out a square entire and four equal pieces that will form a perfect Greek cross?" The puzzle is, of course, now quite easy. 143.--TWO CROSSES FROM ONE. Cut a Greek cross into five pieces that will form two such crosses, both of the same size. The solution of this puzzle is very beautiful. 144.--THE CROSS AND THE TRIANGLE. Cut a Greek cross into six pieces that will form an equilateral triangle. This is another hard problem, and I will state here that a solution is practically impossible without a previous knowledge of my method of transforming an equilateral triangle into a square (see No. 26, "Canterbury Puzzles"). 145.--THE FOLDED CROSS. Cut out of paper a Greek cross; then so fold it that with a single straight cut of the scissors the four pieces produced will form a square. VARIOUS DISSECTION PUZZLES. We will now consider a small miscellaneous selection of cutting-out puzzles, varying in degrees of difficulty. 146.--AN EASY DISSECTION PUZZLE. First, cut out a piece of paper or cardboard of the shape shown in the illustration. It will be seen at once that the proportions are simply those of a square attached to half of another similar square, divided diagonally. The puzzle is to cut it into four pieces all of precisely the same size and shape. 147.--AN EASY SQUARE PUZZLE. If you take a rectangular piece of cardboard, twice as long as it is broad, and cut it in half diagonally, you will get two of the pieces shown in the illustration. The puzzle is with five such pieces of equal size to form a square. One of the pieces may be cut in two, but the others must be used intact. 148.--THE BUN PUZZLE. THE three circles represent three buns, and it is simply required to show how these may be equally divided among four boys. The buns must be regarded as of equal thickness throughout and of equal thickness to each other. Of course, they must be cut into as few pieces as possible. To simplify it I will state the rather surprising fact that only five pieces are necessary, from which it will be seen that one boy gets his share in two pieces and the other three receive theirs in a single piece. I am aware that this statement "gives away" the puzzle, but it should not destroy its interest to those who like to discover the "reason why." 149.--THE CHOCOLATE SQUARES. Here is a slab of chocolate, indented at the dotted lines so that the twenty squares can be easily separated. Make a copy of the slab in paper or cardboard and then try to cut it into nine pieces so that they will form four perfect squares all of exactly the same size. 150.--DISSECTING A MITRE. The figure that is perplexing the carpenter in the illustration represents a mitre. It will be seen that its proportions are those of a square with one quarter removed. The puzzle is to cut it into five pieces that will fit together and form a perfect square. I show an attempt, published in America, to perform the feat in four pieces, based on what is known as the "step principle," but it is a fallacy. [Illustration] We are told first to cut oft the pieces 1 and 2 and pack them into the triangular space marked off by the dotted line, and so form a rectangle. So far, so good. Now, we are directed to apply the old step principle, as shown, and, by moving down the piece 4 one step, form the required square. But, unfortunately, it does _not_ produce a square: only an oblong. Call the three long sides of the mitre 84 in. each. Then, before cutting the steps, our rectangle in three pieces will be 84 × 63. The steps must be 10½ in. in height and 12 in. in breadth. Therefore, by moving down a step we reduce by 12 in. the side 84 in. and increase by 10½ in. the side 63 in. Hence our final rectangle must be 72 in. × 73½ in., which certainly is not a square! The fact is, the step principle can only be applied to rectangles with sides of particular relative lengths. For example, if the shorter side in this case were 61+5/7 (instead of 63), then the step method would apply. For the steps would then be 10+2/7 in. in height and 12 in. in breadth. Note that 61+5/7 × 84 = the square of 72. At present no solution has been found in four pieces, and I do not believe one possible. 151.--THE JOINER'S PROBLEM. I have often had occasion to remark on the practical utility of puzzles, arising out of an application to the ordinary affairs of life of the little tricks and "wrinkles" that we learn while solving recreation problems. [Illustration] The joiner, in the illustration, wants to cut the piece of wood into as few pieces as possible to form a square table-top, without any waste of material. How should he go to work? How many pieces would you require? 152.--ANOTHER JOINER'S PROBLEM. [Illustration] A joiner had two pieces of wood of the shapes and relative proportions shown in the diagram. He wished to cut them into as few pieces as possible so that they could be fitted together, without waste, to form a perfectly square table-top. How should he have done it? There is no necessity to give measurements, for if the smaller piece (which is half a square) be made a little too large or a little too small it will not affect the method of solution. 153--A CUTTING-OUT PUZZLE. Here is a little cutting-out poser. I take a strip of paper, measuring five inches by one inch, and, by cutting it into five pieces, the parts fit together and form a square, as shown in the illustration. Now, it is quite an interesting puzzle to discover how we can do this in only four pieces. [Illustration] 154.--MRS. HOBSON'S HEARTHRUG. [Illustration] Mrs. Hobson's boy had an accident when playing with the fire, and burnt two of the corners of a pretty hearthrug. The damaged corners have been cut away, and it now has the appearance and proportions shown in my diagram. How is Mrs. Hobson to cut the rug into the fewest possible pieces that will fit together and form a perfectly square rug? It will be seen that the rug is in the proportions 36 × 27 (it does not matter whether we say inches or yards), and each piece cut away measured 12 and 6 on the outside. 155.--THE PENTAGON AND SQUARE. I wonder how many of my readers, amongst those who have not given any close attention to the elements of geometry, could draw a regular pentagon, or five-sided figure, if they suddenly required to do so. A regular hexagon, or six-sided figure, is easy enough, for everybody knows that all you have to do is to describe a circle and then, taking the radius as the length of one of the sides, mark off the six points round the circumference. But a pentagon is quite another matter. So, as my puzzle has to do with the cutting up of a regular pentagon, it will perhaps be well if I first show my less experienced readers how this figure is to be correctly drawn. Describe a circle and draw the two lines H B and D G, in the diagram, through the centre at right angles. Now find the point A, midway between C and B. Next place the point of your compasses at A and with the distance A D describe the arc cutting H B at E. Then place the point of your compasses at D and with the distance D E describe the arc cutting the circumference at F. Now, D F is one of the sides of your pentagon, and you have simply to mark off the other sides round the circle. Quite simple when you know how, but otherwise somewhat of a poser. [Illustration] Having formed your pentagon, the puzzle is to cut it into the fewest possible pieces that will fit together and form a perfect square. [Illustration] 156.--THE DISSECTED TRIANGLE. A good puzzle is that which the gentleman in the illustration is showing to his friends. He has simply cut out of paper an equilateral triangle--that is, a triangle with all its three sides of the same length. He proposes that it shall be cut into five pieces in such a way that they will fit together and form either two or three smaller equilateral triangles, using all the material in each case. Can you discover how the cuts should be made? Remember that when you have made your five pieces, you must be able, as desired, to put them together to form either the single original triangle or to form two triangles or to form three triangles--all equilateral. 157.--THE TABLE-TOP AND STOOLS. I have frequently had occasion to show that the published answers to a great many of the oldest and most widely known puzzles are either quite incorrect or capable of improvement. I propose to consider the old poser of the table-top and stools that most of my readers have probably seen in some form or another in books compiled for the recreation of childhood. The story is told that an economical and ingenious schoolmaster once wished to convert a circular table-top, for which he had no use, into seats for two oval stools, each with a hand-hole in the centre. He instructed the carpenter to make the cuts as in the illustration and then join the eight pieces together in the manner shown. So impressed was he with the ingenuity of his performance that he set the puzzle to his geometry class as a little study in dissection. But the remainder of the story has never been published, because, so it is said, it was a characteristic of the principals of academies that they would never admit that they could err. I get my information from a descendant of the original boy who had most reason to be interested in the matter. The clever youth suggested modestly to the master that the hand-holes were too big, and that a small boy might perhaps fall through them. He therefore proposed another way of making the cuts that would get over this objection. For his impertinence he received such severe chastisement that he became convinced that the larger the hand-hole in the stools the more comfortable might they be. [Illustration] Now what was the method the boy proposed? Can you show how the circular table-top may be cut into eight pieces that will fit together and form two oval seats for stools (each of exactly the same size and shape) and each having similar hand-holes of smaller dimensions than in the case shown above? Of course, all the wood must be used. 158.--THE GREAT MONAD. [Illustration] Here is a symbol of tremendous antiquity which is worthy of notice. It is borne on the Korean ensign and merchant flag, and has been adopted as a trade sign by the Northern Pacific Railroad Company, though probably few are aware that it is the Great Monad, as shown in the sketch below. This sign is to the Chinaman what the cross is to the Christian. It is the sign of Deity and eternity, while the two parts into which the circle is divided are called the Yin and the Yan--the male and female forces of nature. A writer on the subject more than three thousand years ago is reported to have said in reference to it: "The illimitable produces the great extreme. The great extreme produces the two principles. The two principles produce the four quarters, and from the four quarters we develop the quadrature of the eight diagrams of Feuh-hi." I hope readers will not ask me to explain this, for I have not the slightest idea what it means. Yet I am persuaded that for ages the symbol has had occult and probably mathematical meanings for the esoteric student. I will introduce the Monad in its elementary form. Here are three easy questions respecting this great symbol:-- (I.) Which has the greater area, the inner circle containing the Yin and the Yan, or the outer ring? (II.) Divide the Yin and the Yan into four pieces of the same size and shape by one cut. (III.) Divide the Yin and the Yan into four pieces of the same size, but different shape, by one straight cut. 159.--THE SQUARE OF VENEER. The following represents a piece of wood in my possession, 5 in. square. By markings on the surface it is divided into twenty-five square inches. I want to discover a way of cutting this piece of wood into the fewest possible pieces that will fit together and form two perfect squares of different sizes and of known dimensions. But, unfortunately, at every one of the sixteen intersections of the cross lines a small nail has been driven in at some time or other, and my fret-saw will be injured if it comes in contact with any of these. I have therefore to find a method of doing the work that will not necessitate my cutting through any of those sixteen points. How is it to be done? Remember, the exact dimensions of the two squares must be given. [Illustration] 160.--THE TWO HORSESHOES. [Illustration] Why horseshoes should be considered "lucky" is one of those things which no man can understand. It is a very old superstition, and John Aubrey (1626-1700) says, "Most houses at the West End of London have a horseshoe on the threshold." In Monmouth Street there were seventeen in 1813 and seven so late as 1855. Even Lord Nelson had one nailed to the mast of the ship _Victory_. To-day we find it more conducive to "good luck" to see that they are securely nailed on the feet of the horse we are about to drive. Nevertheless, so far as the horseshoe, like the Swastika and other emblems that I have had occasion at times to deal with, has served to symbolize health, prosperity, and goodwill towards men, we may well treat it with a certain amount of respectful interest. May there not, moreover, be some esoteric or lost mathematical mystery concealed in the form of a horseshoe? I have been looking into this matter, and I wish to draw my readers' attention to the very remarkable fact that the pair of horseshoes shown in my illustration are related in a striking and beautiful manner to the circle, which is the symbol of eternity. I present this fact in the form of a simple problem, so that it may be seen how subtly this relation has been concealed for ages and ages. My readers will, I know, be pleased when they find the key to the mystery. Cut out the two horseshoes carefully round the outline and then cut them into four pieces, all different in shape, that will fit together and form a perfect circle. Each shoe must be cut into two pieces and all the part of the horse's hoof contained within the outline is to be used and regarded as part of the area. 161.--THE BETSY ROSS PUZZLE. A correspondent asked me to supply him with the solution to an old puzzle that is attributed to a certain Betsy Ross, of Philadelphia, who showed it to George Washington. It consists in so folding a piece of paper that with one clip of the scissors a five-pointed star of Freedom may be produced. Whether the story of the puzzle's origin is a true one or not I cannot say, but I have a print of the old house in Philadelphia where the lady is said to have lived, and I believe it still stands there. But my readers will doubtless be interested in the little poser. Take a circular piece of paper and so fold it that with one cut of the scissors you can produce a perfect five-pointed star. 162.--THE CARDBOARD CHAIN. [Illustration] Can you cut this chain out of a piece of cardboard without any join whatever? Every link is solid; without its having been split and afterwards joined at any place. It is an interesting old puzzle that I learnt as a child, but I have no knowledge as to its inventor. 163.--THE PAPER BOX. It may be interesting to introduce here, though it is not strictly a puzzle, an ingenious method for making a paper box. Take a square of stout paper and by successive foldings make all the creases indicated by the dotted lines in the illustration. Then cut away the eight little triangular pieces that are shaded, and cut through the paper along the dark lines. The second illustration shows the box half folded up, and the reader will have no difficulty in effecting its completion. Before folding up, the reader might cut out the circular piece indicated in the diagram, for a purpose I will now explain. This box will be found to serve excellently for the production of vortex rings. These rings, which were discussed by Von Helmholtz in 1858, are most interesting, and the box (with the hole cut out) will produce them to perfection. Fill the box with tobacco smoke by blowing it gently through the hole. Now, if you hold it horizontally, and softly tap the side that is opposite to the hole, an immense number of perfect rings can be produced from one mouthful of smoke. It is best that there should be no currents of air in the room. People often do not realise that these rings are formed in the air when no smoke is used. The smoke only makes them visible. Now, one of these rings, if properly directed on its course, will travel across the room and put out the flame of a candle, and this feat is much more striking if you can manage to do it without the smoke. Of course, with a little practice, the rings may be blown from the mouth, but the box produces them in much greater perfection, and no skill whatever is required. Lord Kelvin propounded the theory that matter may consist of vortex rings in a fluid that fills all space, and by a development of the hypothesis he was able to explain chemical combination. [Illustration: ·-----------·-----------·-----------·-----------· | · · ·|||||||· ·|||||||· · · | | · ·|||· ·|||· · · | | · · · · | | · · · · · · · · | | · · · · · · · · | · · · · · ||· · · · · /|\ · ·|| ||||· · · · · \|/ · ·|||| ||||||· · · ·|||||| ||||· · · · · · · ·|||| ||· · · · · · · ·|| · · · · · ||· · · · · · · ·|| ||||· · · · · · · ·|||| ||||||· · · ·|||||| ||||· · · · · · · ·|||| ||· · · · · · · ·|| · · · · · | · · · · · · · · | | · · · · · · · · | | · · · · | | · ·|||· ·|||· · · | | · · ·|||||||· ·|||||||· · · | ·-----------·-----------·-----------·-----------· ] [Illustration] 164.--THE POTATO PUZZLE. Take a circular slice of potato, place it on the table, and see into how large a number of pieces you can divide it with six cuts of a knife. Of course you must not readjust the pieces or pile them after a cut. What is the greatest number of pieces you can make? [Illustration: -------- / \ 1/ \ / \ 2 \/ 3 / \ / \ /\ / \ / \ 4 \/ 5\/ 6 / \ | \ /\ /\ / | \ 7\/ 8\/ 9\/10 / \ /\ /\ /\ / \/11\/12\/13\/ \ /\ /\ / \/14\/15\/ \ /\ / \/16\/ ----- ] The illustration shows how to make sixteen pieces. This can, of course, be easily beaten. 165.--THE SEVEN PIGS. [Illustration] +------------------------------+ | | | P | | | | P | | P | | P | | P | | P | | P | | | +------------------------------+ Here is a little puzzle that was put to one of the sons of Erin the other day and perplexed him unduly, for it is really quite easy. It will be seen from the illustration that he was shown a sketch of a square pen containing seven pigs. He was asked how he would intersect the pen with three straight fences so as to enclose every pig in a separate sty. In other words, all you have to do is to take your pencil and, with three straight strokes across the square, enclose each pig separately. Nothing could be simpler. [Illustration] The Irishman complained that the pigs would not keep still while he was putting up the fences. He said that they would all flock together, or one obstinate beast would go into a corner and flock all by himself. It was pointed out to him that for the purposes of the puzzle the pigs were stationary. He answered that Irish pigs are not stationery--they are pork. Being persuaded to make the attempt, he drew three lines, one of which cut through a pig. When it was explained that this is not allowed, he protested that a pig was no use until you cut its throat. "Begorra, if it's bacon ye want without cutting your pig, it will be all gammon." We will not do the Irishman the injustice of suggesting that the miserable pun was intentional. However, he failed to solve the puzzle. Can you do it? 166.--THE LANDOWNER'S FENCES. The landowner in the illustration is consulting with his bailiff over a rather puzzling little question. He has a large plan of one of his fields, in which there are eleven trees. Now, he wants to divide the field into just eleven enclosures by means of straight fences, so that every enclosure shall contain one tree as a shelter for his cattle. How is he to do it with as few fences as possible? Take your pencil and draw straight lines across the field until you have marked off the eleven enclosures (and no more), and then see how many fences you require. Of course the fences may cross one another. 167.--THE WIZARD'S CATS. [Illustration] A wizard placed ten cats inside a magic circle as shown in our illustration, and hypnotized them so that they should remain stationary during his pleasure. He then proposed to draw three circles inside the large one, so that no cat could approach another cat without crossing a magic circle. Try to draw the three circles so that every cat has its own enclosure and cannot reach another cat without crossing a line. 168.--THE CHRISTMAS PUDDING. [Illustration] "Speaking of Christmas puddings," said the host, as he glanced at the imposing delicacy at the other end of the table. "I am reminded of the fact that a friend gave me a new puzzle the other day respecting one. Here it is," he added, diving into his breast pocket. "'Problem: To find the contents,' I suppose," said the Eton boy. "No; the proof of that is in the eating. I will read you the conditions." "'Cut the pudding into two parts, each of exactly the same size and shape, without touching any of the plums. The pudding is to be regarded as a flat disc, not as a sphere.'" "Why should you regard a Christmas pudding as a disc? And why should any reasonable person ever wish to make such an accurate division?" asked the cynic. "It is just a puzzle--a problem in dissection." All in turn had a look at the puzzle, but nobody succeeded in solving it. It is a little difficult unless you are acquainted with the principle involved in the making of such puddings, but easy enough when you know how it is done. 169.--A TANGRAM PARADOX. Many pastimes of great antiquity, such as chess, have so developed and changed down the centuries that their original inventors would scarcely recognize them. This is not the case with Tangrams, a recreation that appears to be at least four thousand years old, that has apparently never been dormant, and that has not been altered or "improved upon" since the legendary Chinaman Tan first cut out the seven pieces shown in Diagram I. If you mark the point B, midway between A and C, on one side of a square of any size, and D, midway between C and E, on an adjoining side, the direction of the cuts is too obvious to need further explanation. Every design in this article is built up from the seven pieces of blackened cardboard. It will at once be understood that the possible combinations are infinite. [Illustration] The late Mr. Sam Loyd, of New York, who published a small book of very ingenious designs, possessed the manuscripts of the late Mr. Challenor, who made a long and close study of Tangrams. This gentleman, it is said, records that there were originally seven books of Tangrams, compiled in China two thousand years before the Christian era. These books are so rare that, after forty years' residence in the country, he only succeeded in seeing perfect copies of the first and seventh volumes with fragments of the second. Portions of one of the books, printed in gold leaf upon parchment, were found in Peking by an English soldier and sold for three hundred pounds. A few years ago a little book came into my possession, from the library of the late Lewis Carroll, entitled _The Fashionable Chinese Puzzle_. It contains three hundred and twenty-three Tangram designs, mostly nondescript geometrical figures, to be constructed from the seven pieces. It was "Published by J. and E. Wallis, 42 Skinner Street, and J. Wallis, Jun., Marine Library, Sidmouth" (South Devon). There is no date, but the following note fixes the time of publication pretty closely: "This ingenious contrivance has for some time past been the favourite amusement of the ex-Emperor Napoleon, who, being now in a debilitated state and living very retired, passes many hours a day in thus exercising his patience and ingenuity." The reader will find, as did the great exile, that much amusement, not wholly uninstructive, may be derived from forming the designs of others. He will find many of the illustrations to this article quite easy to build up, and some rather difficult. Every picture may thus be regarded as a puzzle. But it is another pastime altogether to create new and original designs of a pictorial character, and it is surprising what extraordinary scope the Tangrams afford for producing pictures of real life--angular and often grotesque, it is true, but full of character. I give an example of a recumbent figure (2) that is particularly graceful, and only needs some slight reduction of its angularities to produce an entirely satisfactory outline. As I have referred to the author of _Alice in Wonderland_, I give also my designs of the March Hare (3) and the Hatter (4). I also give an attempt at Napoleon (5), and a very excellent Red Indian with his Squaw by Mr. Loyd (6 and 7). A large number of other designs will be found in an article by me in _The Strand Magazine_ for November, 1908. [Illustration: 2] [Illustration: 3] [Illustration: 4] On the appearance of this magazine article, the late Sir James Murray, the eminent philologist, tried, with that amazing industry that characterized all his work, to trace the word "tangram" to its source. At length he wrote as follows:--"One of my sons is a professor in the Anglo-Chinese college at Tientsin. Through him, his colleagues, and his students, I was able to make inquiries as to the alleged Tan among Chinese scholars. Our Chinese professor here (Oxford) also took an interest in the matter and obtained information from the secretary of the Chinese Legation in London, who is a very eminent representative of the Chinese literati." [Illustration: 5] "The result has been to show that the man Tan, the god Tan, and the 'Book of Tan' are entirely unknown to Chinese literature, history, or tradition. By most of the learned men the name, or allegation of the existence, of these had never been heard of. The puzzle is, of course, well known. It is called in Chinese _ch'i ch'iao t'u_; literally, 'seven-ingenious-plan' or 'ingenious-puzzle figure of seven pieces.' No name approaching 'tangram,' or even 'tan,' occurs in Chinese, and the only suggestions for the latter were the Chinese _t'an_, 'to extend'; or _t'ang_, Cantonese dialect for 'Chinese.' It was suggested that probably some American or Englishman who knew a little Chinese or Cantonese, wanting a name for the puzzle, might concoct one out of one of these words and the European ending 'gram.' I should say the name 'tangram' was probably invented by an American some little time before 1864 and after 1847, but I cannot find it in print before the 1864 edition of Webster. I have therefore had to deal very shortly with the word in the dictionary, telling what it is applied to and what conjectures or guesses have been made at the name, and giving a few quotations, one from your own article, which has enabled me to make more of the subject than I could otherwise have done." [Illustration: 6] [Illustration: 7] Several correspondents have informed me that they possess, or had possessed, specimens of the old Chinese books. An American gentleman writes to me as follows:--"I have in my possession a book made of tissue paper, printed in black (with a Chinese inscription on the front page), containing over three hundred designs, which belongs to the box of 'tangrams,' which I also own. The blocks are seven in number, made of mother-of-pearl, highly polished and finely engraved on either side. These are contained in a rosewood box 2+1/8 in. square. My great uncle, ----, was one of the first missionaries to visit China. This box and book, along with quite a collection of other relics, were sent to my grandfather and descended to myself." My correspondent kindly supplied me with rubbings of the Tangrams, from which it is clear that they are cut in the exact proportions that I have indicated. I reproduce the Chinese inscription (8) for this reason. The owner of the book informs me that he has submitted it to a number of Chinamen in the United States and offered as much as a dollar for a translation. But they all steadfastly refused to read the words, offering the lame excuse that the inscription is Japanese. Natives of Japan, however, insist that it is Chinese. Is there something occult and esoteric about Tangrams, that it is so difficult to lift the veil? Perhaps this page will come under the eye of some reader acquainted with the Chinese language, who will supply the required translation, which may, or may not, throw a little light on this curious question. [Illustration: 8] By using several sets of Tangrams at the same time we may construct more ambitious pictures. I was advised by a friend not to send my picture, "A Game of Billiards" (9), to the Academy. He assured me that it would not be accepted because the "judges are so hide-bound by convention." Perhaps he was right, and it will be more appreciated by Post-impressionists and Cubists. The players are considering a very delicate stroke at the top of the table. Of course, the two men, the table, and the clock are formed from four sets of Tangrams. My second picture is named "The Orchestra" (10), and it was designed for the decoration of a large hall of music. Here we have the conductor, the pianist, the fat little cornet-player, the left-handed player of the double-bass, whose attitude is life-like, though he does stand at an unusual distance from his instrument, and the drummer-boy, with his imposing music-stand. The dog at the back of the pianoforte is not howling: he is an appreciative listener. [Illustration: 9] [Illustration: 10] One remarkable thing about these Tangram pictures is that they suggest to the imagination such a lot that is not really there. Who, for example, can look for a few minutes at Lady Belinda (11) and the Dutch girl (12) without soon feeling the haughty expression in the one case and the arch look in the other? Then look again at the stork (13), and see how it is suggested to the mind that the leg is actually much more slender than any one of the pieces employed. It is really an optical illusion. Again, notice in the case of the yacht (14) how, by leaving that little angular point at the top, a complete mast is suggested. If you place your Tangrams together on white paper so that they do not quite touch one another, in some cases the effect is improved by the white lines; in other cases it is almost destroyed. [Illustration: 11] [Illustration: 12] Finally, I give an example from the many curious paradoxes that one happens upon in manipulating Tangrams. I show designs of two dignified individuals (15 and 16) who appear to be exactly alike, except for the fact that one has a foot and the other has not. Now, both of these figures are made from the same seven Tangrams. Where does the second man get his foot from? [Illustration: 13] [Illustration: 14] [Illustration: 15] [Illustration: 16] PATCHWORK PUZZLES. "Of shreds and patches."--_Hamlet_, iii. 4. 170.--THE CUSHION COVERS. [Illustration] The above represents a square of brocade. A lady wishes to cut it in four pieces so that two pieces will form one perfectly square cushion top, and the remaining two pieces another square cushion top. How is she to do it? Of course, she can only cut along the lines that divide the twenty-five squares, and the pattern must "match" properly without any irregularity whatever in the design of the material. There is only one way of doing it. Can you find it? 171.--THE BANNER PUZZLE. [Illustration] A Lady had a square piece of bunting with two lions on it, of which the illustration is an exactly reproduced reduction. She wished to cut the stuff into pieces that would fit together and form two square banners with a lion on each banner. She discovered that this could be done in as few as four pieces. How did she manage it? Of course, to cut the British Lion would be an unpardonable offence, so you must be careful that no cut passes through any portion of either of them. Ladies are informed that no allowance whatever has to be made for "turnings," and no part of the material may be wasted. It is quite a simple little dissection puzzle if rightly attacked. Remember that the banners have to be perfect squares, though they need not be both of the same size. 172.--MRS. SMILEY'S CHRISTMAS PRESENT. Mrs. Smiley's expression of pleasure was sincere when her six granddaughters sent to her, as a Christmas present, a very pretty patchwork quilt, which they had made with their own hands. It was constructed of square pieces of silk material, all of one size, and as they made a large quilt with fourteen of these little squares on each side, it is obvious that just 196 pieces had been stitched into it. Now, the six granddaughters each contributed a part of the work in the form of a perfect square (all six portions being different in size), but in order to join them up to form the square quilt it was necessary that the work of one girl should be unpicked into three separate pieces. Can you show how the joins might have been made? Of course, no portion can be turned over. [Illustration] 173.--MRS. PERKINS'S QUILT. [Illustration] It will be seen that in this case the square patchwork quilt is built up of 169 pieces. The puzzle is to find the smallest possible number of square portions of which the quilt could be composed and show how they might be joined together. Or, to put it the reverse way, divide the quilt into as few square portions as possible by merely cutting the stitches. 174.--THE SQUARES OF BROCADE. [Illustration] I happened to be paying a call at the house of a lady, when I took up from a table two lovely squares of brocade. They were beautiful specimens of Eastern workmanship--both of the same design, a delicate chequered pattern. "Are they not exquisite?" said my friend. "They were brought to me by a cousin who has just returned from India. Now, I want you to give me a little assistance. You see, I have decided to join them together so as to make one large square cushion-cover. How should I do this so as to mutilate the material as little as possible? Of course I propose to make my cuts only along the lines that divide the little chequers." [Illustration] I cut the two squares in the manner desired into four pieces that would fit together and form another larger square, taking care that the pattern should match properly, and when I had finished I noticed that two of the pieces were of exactly the same area; that is, each of the two contained the same number of chequers. Can you show how the cuts were made in accordance with these conditions? 175--ANOTHER PATCHWORK PUZZLE. [Illustration] A lady was presented, by two of her girl friends, with the pretty pieces of silk patchwork shown in our illustration. It will be seen that both pieces are made up of squares all of the same size--one 12 × 12 and the other 5 × 5. She proposes to join them together and make one square patchwork quilt, 13 × 13, but, of course, she will not cut any of the material--merely cut the stitches where necessary and join together again. What perplexes her is this. A friend assures her that there need be no more than four pieces in all to join up for the new quilt. Could you show her how this little needlework puzzle is to be solved in so few pieces? 176.--LINOLEUM CUTTING. [Illustration] The diagram herewith represents two separate pieces of linoleum. The chequered pattern is not repeated at the back, so that the pieces cannot be turned over. The puzzle is to cut the two squares into four pieces so that they shall fit together and form one perfect square 10 × 10, so that the pattern shall properly match, and so that the larger piece shall have as small a portion as possible cut from it. 177.--ANOTHER LINOLEUM PUZZLE. [Illustration] Can you cut this piece of linoleum into four pieces that will fit together and form a perfect square? Of course the cuts may only be made along the lines. VARIOUS GEOMETRICAL PUZZLES. "So various are the tastes of men." MARK AKENSIDE. 178.--THE CARDBOARD BOX. This puzzle is not difficult, but it will be found entertaining to discover the simple rule for its solution. I have a rectangular cardboard box. The top has an area of 120 square inches, the side 96 square inches, and the end 80 square inches. What are the exact dimensions of the box? 179.--STEALING THE BELL-ROPES. Two men broke into a church tower one night to steal the bell-ropes. The two ropes passed through holes in the wooden ceiling high above them, and they lost no time in climbing to the top. Then one man drew his knife and cut the rope above his head, in consequence of which he fell to the floor and was badly injured. His fellow-thief called out that it served him right for being such a fool. He said that he should have done as he was doing, upon which he cut the rope below the place at which he held on. Then, to his dismay, he found that he was in no better plight, for, after hanging on as long as his strength lasted, he was compelled to let go and fall beside his comrade. Here they were both found the next morning with their limbs broken. How far did they fall? One of the ropes when they found it was just touching the floor, and when you pulled the end to the wall, keeping the rope taut, it touched a point just three inches above the floor, and the wall was four feet from the rope when it hung at rest. How long was the rope from floor to ceiling? 180.--THE FOUR SONS. Readers will recognize the diagram as a familiar friend of their youth. A man possessed a square-shaped estate. He bequeathed to his widow the quarter of it that is shaded off. The remainder was to be divided equitably amongst his four sons, so that each should receive land of exactly the same area and exactly similar in shape. We are shown how this was done. But the remainder of the story is not so generally known. In the centre of the estate was a well, indicated by the dark spot, and Benjamin, Charles, and David complained that the division was not "equitable," since Alfred had access to this well, while they could not reach it without trespassing on somebody else's land. The puzzle is to show how the estate is to be apportioned so that each son shall have land of the same shape and area, and each have access to the well without going off his own land. [Illustration] 181.--THE THREE RAILWAY STATIONS. As I sat in a railway carriage I noticed at the other end of the compartment a worthy squire, whom I knew by sight, engaged in conversation with another passenger, who was evidently a friend of his. "How far have you to drive to your place from the railway station?" asked the stranger. "Well," replied the squire, "if I get out at Appleford, it is just the same distance as if I go to Bridgefield, another fifteen miles farther on; and if I changed at Appleford and went thirteen miles from there to Carterton, it would still be the same distance. You see, I am equidistant from the three stations, so I get a good choice of trains." Now I happened to know that Bridgefield is just fourteen miles from Carterton, so I amused myself in working out the exact distance that the squire had to drive home whichever station he got out at. What was the distance? 182.--THE GARDEN PUZZLE. Professor Rackbrain tells me that he was recently smoking a friendly pipe under a tree in the garden of a country acquaintance. The garden was enclosed by four straight walls, and his friend informed him that he had measured these and found the lengths to be 80, 45, 100, and 63 yards respectively. "Then," said the professor, "we can calculate the exact area of the garden." "Impossible," his host replied, "because you can get an infinite number of different shapes with those four sides." "But you forget," Rackbrane said, with a twinkle in his eye, "that you told me once you had planted this tree equidistant from all the four corners of the garden." Can you work out the garden's area? 183.--DRAWING A SPIRAL. If you hold the page horizontally and give it a quick rotary motion while looking at the centre of the spiral, it will appear to revolve. Perhaps a good many readers are acquainted with this little optical illusion. But the puzzle is to show how I was able to draw this spiral with so much exactitude without using anything but a pair of compasses and the sheet of paper on which the diagram was made. How would you proceed in such circumstances? [Illustration] 184.--HOW TO DRAW AN OVAL. Can you draw a perfect oval on a sheet of paper with one sweep of the compasses? It is one of the easiest things in the world when you know how. 185.--ST. GEORGE'S BANNER. At a celebration of the national festival of St. George's Day I was contemplating the familiar banner of the patron saint of our country. We all know the red cross on a white ground, shown in our illustration. This is the banner of St. George. The banner of St. Andrew (Scotland) is a white "St. Andrew's Cross" on a blue ground. That of St. Patrick (Ireland) is a similar cross in red on a white ground. These three are united in one to form our Union Jack. Now on looking at St. George's banner it occurred to me that the following question would make a simple but pretty little puzzle. Supposing the flag measures four feet by three feet, how wide must the arm of the cross be if it is required that there shall be used just the same quantity of red and of white bunting? [Illustration] 186.--THE CLOTHES LINE PUZZLE. A boy tied a clothes line from the top of each of two poles to the base of the other. He then proposed to his father the following question. As one pole was exactly seven feet above the ground and the other exactly five feet, what was the height from the ground where the two cords crossed one another? 187.--THE MILKMAID PUZZLE. [Illustration] Here is a little pastoral puzzle that the reader may, at first sight, be led into supposing is very profound, involving deep calculations. He may even say that it is quite impossible to give any answer unless we are told something definite as to the distances. And yet it is really quite "childlike and bland." In the corner of a field is seen a milkmaid milking a cow, and on the other side of the field is the dairy where the extract has to be deposited. But it has been noticed that the young woman always goes down to the river with her pail before returning to the dairy. Here the suspicious reader will perhaps ask why she pays these visits to the river. I can only reply that it is no business of ours. The alleged milk is entirely for local consumption. "Where are you going to, my pretty maid?" "Down to the river, sir," she said. "I'll _not_ choose your dairy, my pretty maid." "Nobody axed you, sir," she said. If one had any curiosity in the matter, such an independent spirit would entirely disarm one. So we will pass from the point of commercial morality to the subject of the puzzle. Draw a line from the milking-stool down to the river and thence to the door of the dairy, which shall indicate the shortest possible route for the milkmaid. That is all. It is quite easy to indicate the exact spot on the bank of the river to which she should direct her steps if she wants as short a walk as possible. Can you find that spot? 188.--THE BALL PROBLEM. [Illustration] A stonemason was engaged the other day in cutting out a round ball for the purpose of some architectural decoration, when a smart schoolboy came upon the scene. "Look here," said the mason, "you seem to be a sharp youngster, can you tell me this? If I placed this ball on the level ground, how many other balls of the same size could I lay around it (also on the ground) so that every ball should touch this one?" The boy at once gave the correct answer, and then put this little question to the mason:-- "If the surface of that ball contained just as many square feet as its volume contained cubic feet, what would be the length of its diameter?" The stonemason could not give an answer. Could you have replied correctly to the mason's and the boy's questions? 189.--THE YORKSHIRE ESTATES. [Illustration] I was on a visit to one of the large towns of Yorkshire. While walking to the railway station on the day of my departure a man thrust a hand-bill upon me, and I took this into the railway carriage and read it at my leisure. It informed me that three Yorkshire neighbouring estates were to be offered for sale. Each estate was square in shape, and they joined one another at their corners, just as shown in the diagram. Estate A contains exactly 370 acres, B contains 116 acres, and C 74 acres. Now, the little triangular bit of land enclosed by the three square estates was not offered for sale, and, for no reason in particular, I became curious as to the area of that piece. How many acres did it contain? 190.--FARMER WURZEL'S ESTATE. [Illustration] I will now present another land problem. The demonstration of the answer that I shall give will, I think, be found both interesting and easy of comprehension. Farmer Wurzel owned the three square fields shown in the annexed plan, containing respectively 18, 20, and 26 acres. In order to get a ring-fence round his property he bought the four intervening triangular fields. The puzzle is to discover what was then the whole area of his estate. 191.--THE CRESCENT PUZZLE. [Illustration] Here is an easy geometrical puzzle. The crescent is formed by two circles, and C is the centre of the larger circle. The width of the crescent between B and D is 9 inches, and between E and F 5 inches. What are the diameters of the two circles? 192.--THE PUZZLE WALL. [Illustration] There was a small lake, around which four poor men built their cottages. Four rich men afterwards built their mansions, as shown in the illustration, and they wished to have the lake to themselves, so they instructed a builder to put up the shortest possible wall that would exclude the cottagers, but give themselves free access to the lake. How was the wall to be built? 193.--THE SHEEPFOLD. It is a curious fact that the answers always given to some of the best-known puzzles that appear in every little book of fireside recreations that has been published for the last fifty or a hundred years are either quite unsatisfactory or clearly wrong. Yet nobody ever seems to detect their faults. Here is an example:--A farmer had a pen made of fifty hurdles, capable of holding a hundred sheep only. Supposing he wanted to make it sufficiently large to hold double that number, how many additional hurdles must he have? 194.--THE GARDEN WALLS. [Illustration] A speculative country builder has a circular field, on which he has erected four cottages, as shown in the illustration. The field is surrounded by a brick wall, and the owner undertook to put up three other brick walls, so that the neighbours should not be overlooked by each other, but the four tenants insist that there shall be no favouritism, and that each shall have exactly the same length of wall space for his wall fruit trees. The puzzle is to show how the three walls may be built so that each tenant shall have the same area of ground, and precisely the same length of wall. Of course, each garden must be entirely enclosed by its walls, and it must be possible to prove that each garden has exactly the same length of wall. If the puzzle is properly solved no figures are necessary. 195.--LADY BELINDA'S GARDEN. Lady Belinda is an enthusiastic gardener. In the illustration she is depicted in the act of worrying out a pleasant little problem which I will relate. One of her gardens is oblong in shape, enclosed by a high holly hedge, and she is turning it into a rosary for the cultivation of some of her choicest roses. She wants to devote exactly half of the area of the garden to the flowers, in one large bed, and the other half to be a path going all round it of equal breadth throughout. Such a garden is shown in the diagram at the foot of the picture. How is she to mark out the garden under these simple conditions? She has only a tape, the length of the garden, to do it with, and, as the holly hedge is so thick and dense, she must make all her measurements inside. Lady Belinda did not know the exact dimensions of the garden, and, as it was not necessary for her to know, I also give no dimensions. It is quite a simple task no matter what the size or proportions of the garden may be. Yet how many lady gardeners would know just how to proceed? The tape may be quite plain--that is, it need not be a graduated measure. [Illustration] 196.--THE TETHERED GOAT. [Illustration] Here is a little problem that everybody should know how to solve. The goat is placed in a half-acre meadow, that is in shape an equilateral triangle. It is tethered to a post at one corner of the field. What should be the length of the tether (to the nearest inch) in order that the goat shall be able to eat just half the grass in the field? It is assumed that the goat can feed to the end of the tether. 197.--THE COMPASSES PUZZLE. It is curious how an added condition or restriction will sometimes convert an absurdly easy puzzle into an interesting and perhaps difficult one. I remember buying in the street many years ago a little mechanical puzzle that had a tremendous sale at the time. It consisted of a medal with holes in it, and the puzzle was to work a ring with a gap in it from hole to hole until it was finally detached. As I was walking along the street I very soon acquired the trick of taking off the ring with one hand while holding the puzzle in my pocket. A friend to whom I showed the little feat set about accomplishing it himself, and when I met him some days afterwards he exhibited his proficiency in the art. But he was a little taken aback when I then took the puzzle from him and, while simply holding the medal between the finger and thumb of one hand, by a series of little shakes and jerks caused the ring, without my even touching it, to fall off upon the floor. The following little poser will probably prove a rather tough nut for a great many readers, simply on account of the restricted conditions:-- Show how to find exactly the middle of any straight line by means of the compasses only. You are not allowed to use any ruler, pencil, or other article--only the compasses; and no trick or dodge, such as folding the paper, will be permitted. You must simply use the compasses in the ordinary legitimate way. 198.--THE EIGHT STICKS. I have eight sticks, four of them being exactly half the length of the others. I lay every one of these on the table, so that they enclose three squares, all of the same size. How do I do it? There must be no loose ends hanging over. 199.--PAPA'S PUZZLE. Here is a puzzle by Pappus, who lived at Alexandria about the end of the third century. It is the fifth proposition in the eighth book of his _Mathematical Collections_. I give it in the form that I presented it some years ago under the title "Papa's Puzzle," just to see how many readers would discover that it was by Pappus himself. "The little maid's papa has taken two different-sized rectangular pieces of cardboard, and has clipped off a triangular piece from one of them, so that when it is suspended by a thread from the point A it hangs with the long side perfectly horizontal, as shown in the illustration. He has perplexed the child by asking her to find the point A on the other card, so as to produce a similar result when cut and suspended by a thread." Of course, the point must not be found by trial clippings. A curious and pretty point is involved in this setting of the puzzle. Can the reader discover it? [Illustration] 200.--A KITE-FLYING PUZZLE. While accompanying my friend Professor Highflite during a scientific kite-flying competition on the South Downs of Sussex I was led into a little calculation that ought to interest my readers. The Professor was paying out the wire to which his kite was attached from a winch on which it had been rolled into a perfectly spherical form. This ball of wire was just two feet in diameter, and the wire had a diameter of one-hundredth of an inch. What was the length of the wire? Now, a simple little question like this that everybody can perfectly understand will puzzle many people to answer in any way. Let us see whether, without going into any profound mathematical calculations, we can get the answer roughly--say, within a mile of what is correct! We will assume that when the wire is all wound up the ball is perfectly solid throughout, and that no allowance has to be made for the axle that passes through it. With that simplification, I wonder how many readers can state within even a mile of the correct answer the length of that wire. 201.--HOW TO MAKE CISTERNS. [Illustration] Our friend in the illustration has a large sheet of zinc, measuring (before cutting) eight feet by three feet, and he has cut out square pieces (all of the same size) from the four corners and now proposes to fold up the sides, solder the edges, and make a cistern. But the point that puzzles him is this: Has he cut out those square pieces of the correct size in order that the cistern may hold the greatest possible quantity of water? You see, if you cut them very small you get a very shallow cistern; if you cut them large you get a tall and slender one. It is all a question of finding a way of cutting put these four square pieces exactly the right size. How are we to avoid making them too small or too large? 202.--THE CONE PUZZLE. [Illustration] I have a wooden cone, as shown in Fig. 1. How am I to cut out of it the greatest possible cylinder? It will be seen that I can cut out one that is long and slender, like Fig. 2, or short and thick, like Fig. 3. But neither is the largest possible. A child could tell you where to cut, if he knew the rule. Can you find this simple rule? 203.--CONCERNING WHEELS. [Illustration] There are some curious facts concerning the movements of wheels that are apt to perplex the novice. For example: when a railway train is travelling from London to Crewe certain parts of the train at any given moment are actually moving from Crewe towards London. Can you indicate those parts? It seems absurd that parts of the same train can at any time travel in opposite directions, but such is the case. In the accompanying illustration we have two wheels. The lower one is supposed to be fixed and the upper one running round it in the direction of the arrows. Now, how many times does the upper wheel turn on its own axis in making a complete revolution of the other wheel? Do not be in a hurry with your answer, or you are almost certain to be wrong. Experiment with two pennies on the table and the correct answer will surprise you, when you succeed in seeing it. 204.--A NEW MATCH PUZZLE. [Illustration] In the illustration eighteen matches are shown arranged so that they enclose two spaces, one just twice as large as the other. Can you rearrange them (1) so as to enclose two four-sided spaces, one exactly three times as large as the other, and (2) so as to enclose two five-sided spaces, one exactly three times as large as the other? All the eighteen matches must be fairly used in each case; the two spaces must be quite detached, and there must be no loose ends or duplicated matches. 205.--THE SIX SHEEP-PENS. [Illustration] Here is a new little puzzle with matches. It will be seen in the illustration that thirteen matches, representing a farmer's hurdles, have been so placed that they enclose six sheep-pens all of the same size. Now, one of these hurdles was stolen, and the farmer wanted still to enclose six pens of equal size with the remaining twelve. How was he to do it? All the twelve matches must be fairly used, and there must be no duplicated matches or loose ends. POINTS AND LINES PROBLEMS. "Line upon line, line upon line; here a little and there a little."--_Isa_. xxviii. 10. What are known as "Points and Lines" puzzles are found very interesting by many people. The most familiar example, here given, to plant nine trees so that they shall form ten straight rows with three trees in every row, is attributed to Sir Isaac Newton, but the earliest collection of such puzzles is, I believe, in a rare little book that I possess--published in 1821--_Rational Amusement for Winter Evenings_, by John Jackson. The author gives ten examples of "Trees planted in Rows." These tree-planting puzzles have always been a matter of great perplexity. They are real "puzzles," in the truest sense of the word, because nobody has yet succeeded in finding a direct and certain way of solving them. They demand the exercise of sagacity, ingenuity, and patience, and what we call "luck" is also sometimes of service. Perhaps some day a genius will discover the key to the whole mystery. Remember that the trees must be regarded as mere points, for if we were allowed to make our trees big enough we might easily "fudge" our diagrams and get in a few extra straight rows that were more apparent than real. [Illustration] 206.--THE KING AND THE CASTLES. There was once, in ancient times, a powerful king, who had eccentric ideas on the subject of military architecture. He held that there was great strength and economy in symmetrical forms, and always cited the example of the bees, who construct their combs in perfect hexagonal cells, to prove that he had nature to support him. He resolved to build ten new castles in his country all to be connected by fortified walls, which should form five lines with four castles in every line. The royal architect presented his preliminary plan in the form I have shown. But the monarch pointed out that every castle could be approached from the outside, and commanded that the plan should be so modified that as many castles as possible should be free from attack from the outside, and could only be reached by crossing the fortified walls. The architect replied that he thought it impossible so to arrange them that even one castle, which the king proposed to use as a royal residence, could be so protected, but his majesty soon enlightened him by pointing out how it might be done. How would you have built the ten castles and fortifications so as best to fulfil the king's requirements? Remember that they must form five straight lines with four castles in every line. [Illustration] 207.--CHERRIES AND PLUMS. [Illustration] The illustration is a plan of a cottage as it stands surrounded by an orchard of fifty-five trees. Ten of these trees are cherries, ten are plums, and the remainder apples. The cherries are so planted as to form five straight lines, with four cherry trees in every line. The plum trees are also planted so as to form five straight lines with four plum trees in every line. The puzzle is to show which are the ten cherry trees and which are the ten plums. In order that the cherries and plums should have the most favourable aspect, as few as possible (under the conditions) are planted on the north and east sides of the orchard. Of course in picking out a group of ten trees (cherry or plum, as the case may be) you ignore all intervening trees. That is to say, four trees may be in a straight line irrespective of other trees (or the house) being in between. After the last puzzle this will be quite easy. 208.--A PLANTATION PUZZLE. [Illustration] A man had a square plantation of forty-nine trees, but, as will be seen by the omissions in the illustration, four trees were blown down and removed. He now wants to cut down all the remainder except ten trees, which are to be so left that they shall form five straight rows with four trees in every row. Which are the ten trees that he must leave? 209.--THE TWENTY-ONE TREES. A gentleman wished to plant twenty-one trees in his park so that they should form twelve straight rows with five trees in every row. Could you have supplied him with a pretty symmetrical arrangement that would satisfy these conditions? 210.--THE TEN COINS. Place ten pennies on a large sheet of paper or cardboard, as shown in the diagram, five on each edge. Now remove four of the coins, without disturbing the others, and replace them on the paper so that the ten shall form five straight lines with four coins in every line. This in itself is not difficult, but you should try to discover in how many different ways the puzzle may be solved, assuming that in every case the two rows at starting are exactly the same. [Illustration] 211.--THE TWELVE MINCE-PIES. It will be seen in our illustration how twelve mince-pies may be placed on the table so as to form six straight rows with four pies in every row. The puzzle is to remove only four of them to new positions so that there shall be _seven_ straight rows with four in every row. Which four would you remove, and where would you replace them? [Illustration] 212.--THE BURMESE PLANTATION. [Illustration] A short time ago I received an interesting communication from the British chaplain at Meiktila, Upper Burma, in which my correspondent informed me that he had found some amusement on board ship on his way out in trying to solve this little poser. If he has a plantation of forty-nine trees, planted in the form of a square as shown in the accompanying illustration, he wishes to know how he may cut down twenty-seven of the trees so that the twenty-two left standing shall form as many rows as possible with four trees in every row. Of course there may not be more than four trees in any row. 213.--TURKS AND RUSSIANS. This puzzle is on the lines of the Afridi problem published by me in _Tit-Bits_ some years ago. On an open level tract of country a party of Russian infantry, no two of whom were stationed at the same spot, were suddenly surprised by thirty-two Turks, who opened fire on the Russians from all directions. Each of the Turks simultaneously fired a bullet, and each bullet passed immediately over the heads of three Russian soldiers. As each of these bullets when fired killed a different man, the puzzle is to discover what is the smallest possible number of soldiers of which the Russian party could have consisted and what were the casualties on each side. MOVING COUNTER PROBLEMS. "I cannot do't without counters." _Winter's Tale_, iv. 3. Puzzles of this class, except so far as they occur in connection with actual games, such as chess, seem to be a comparatively modern introduction. Mathematicians in recent times, notably Vandermonde and Reiss, have devoted some attention to them, but they do not appear to have been considered by the old writers. So far as games with counters are concerned, perhaps the most ancient and widely known in old times is "Nine Men's Morris" (known also, as I shall show, under a great many other names), unless the simpler game, distinctly mentioned in the works of Ovid (No. 110, "Ovid's Game," in _The Canterbury Puzzles_), from which "Noughts and Crosses" seems to be derived, is still more ancient. In France the game is called Marelle, in Poland Siegen Wulf Myll (She-goat Wolf Mill, or Fight), in Germany and Austria it is called Muhle (the Mill), in Iceland it goes by the name of Mylla, while the Bogas (or native bargees) of South America are said to play it, and on the Amazon it is called Trique, and held to be of Indian origin. In our own country it has different names in different districts, such as Meg Merrylegs, Peg Meryll, Nine Peg o'Merryal, Nine-Pin Miracle, Merry Peg, and Merry Hole. Shakespeare refers to it in "Midsummer Night's Dream" (Act ii., scene 1):-- "The nine-men's morris is filled up with mud; And the quaint mazes in the wanton green, For lack of tread, are undistinguishable." It was played by the shepherds with stones in holes cut in the turf. John Clare, the peasant poet of Northamptonshire, in "The Shepherd Boy" (1835) says:--"Oft we track his haunts .... By nine-peg-morris nicked upon the green." It is also mentioned by Drayton in his "Polyolbion." It was found on an old Roman tile discovered during the excavations at Silchester, and cut upon the steps of the Acropolis at Athens. When visiting the Christiania Museum a few years ago I was shown the great Viking ship that was discovered at Gokstad in 1880. On the oak planks forming the deck of the vessel were found boles and lines marking out the game, the holes being made to receive pegs. While inspecting the ancient oak furniture in the Rijks Museum at Amsterdam I became interested in an old catechumen's settle, and was surprised to find the game diagram cut in the centre of the seat--quite conveniently for surreptitious play. It has been discovered cut in the choir stalls of several of our English cathedrals. In the early eighties it was found scratched upon a stone built into a wall (probably about the date 1200), during the restoration of Hargrave church in Northamptonshire. This stone is now in the Northampton Museum. A similar stone has since been found at Sempringham, Lincolnshire. It is to be seen on an ancient tombstone in the Isle of Man, and painted on old Dutch tiles. And in 1901 a stone was dug out of a gravel pit near Oswestry bearing an undoubted diagram of the game. The game has been played with different rules at different periods and places. I give a copy of the board. Sometimes the diagonal lines are omitted, but this evidently was not intended to affect the play: it simply meant that the angles alone were thought sufficient to indicate the points. This is how Strutt, in _Sports and Pastimes_, describes the game, and it agrees with the way I played it as a boy:--"Two persons, having each of them nine pieces, or men, lay them down alternately, one by one, upon the spots; and the business of either party is to prevent his antagonist from placing three of his pieces so as to form a row of three, without the intervention of an opponent piece. If a row be formed, he that made it is at liberty to take up one of his competitor's pieces from any part he thinks most to his advantage; excepting he has made a row, which must not be touched if he have another piece upon the board that is not a component part of that row. When all the pieces are laid down, they are played backwards and forwards, in any direction that the lines run, but only can move from one spot to another (next to it) at one time. He that takes off all his antagonist's pieces is the conqueror." [Illustration] 214.--THE SIX FROGS. [Illustration] The six educated frogs in the illustration are trained to reverse their order, so that their numbers shall read 6, 5, 4, 3, 2, 1, with the blank square in its present position. They can jump to the next square (if vacant) or leap over one frog to the next square beyond (if vacant), just as we move in the game of draughts, and can go backwards or forwards at pleasure. Can you show how they perform their feat in the fewest possible moves? It is quite easy, so when you have done it add a seventh frog to the right and try again. Then add more frogs until you are able to give the shortest solution for any number. For it can always be done, with that single vacant square, no matter how many frogs there are. 215.--THE GRASSHOPPER PUZZLE. It has been suggested that this puzzle was a great favourite among the young apprentices of the City of London in the sixteenth and seventeenth centuries. Readers will have noticed the curious brass grasshopper on the Royal Exchange. This long-lived creature escaped the fires of 1666 and 1838. The grasshopper, after his kind, was the crest of Sir Thomas Gresham, merchant grocer, who died in 1579, and from this cause it has been used as a sign by grocers in general. Unfortunately for the legend as to its origin, the puzzle was only produced by myself so late as the year 1900. On twelve of the thirteen black discs are placed numbered counters or grasshoppers. The puzzle is to reverse their order, so that they shall read, 1, 2, 3, 4, etc., in the opposite direction, with the vacant disc left in the same position as at present. Move one at a time in any order, either to the adjoining vacant disc or by jumping over one grasshopper, like the moves in draughts. The moves or leaps may be made in either direction that is at any time possible. What are the fewest possible moves in which it can be done? [Illustration] 216.--THE EDUCATED FROGS. [Illustration] Our six educated frogs have learnt a new and pretty feat. When placed on glass tumblers, as shown in the illustration, they change sides so that the three black ones are to the left and the white frogs to the right, with the unoccupied tumbler at the opposite end--No. 7. They can jump to the next tumbler (if unoccupied), or over one, or two, frogs to an unoccupied tumbler. The jumps can be made in either direction, and a frog may jump over his own or the opposite colour, or both colours. Four successive specimen jumps will make everything quite plain: 4 to 1, 5 to 4, 3 to 5, 6 to 3. Can you show how they do it in ten jumps? 217.--THE TWICKENHAM PUZZLE. [Illustration: ( I ) ((N)) ( M ) ((A)) ( H ) ((T)) ( E ) ((W)) ( C ) ((K)) ( ) ] In the illustration we have eleven discs in a circle. On five of the discs we place white counters with black letters--as shown--and on five other discs the black counters with white letters. The bottom disc is left vacant. Starting thus, it is required to get the counters into order so that they spell the word "Twickenham" in a clockwise direction, leaving the vacant disc in the original position. The black counters move in the direction that a clock-hand revolves, and the white counters go the opposite way. A counter may jump over one of the opposite colour if the vacant disc is next beyond. Thus, if your first move is with K, then C can jump over K. If then K moves towards E, you may next jump W over C, and so on. The puzzle may be solved in twenty-six moves. Remember a counter cannot jump over one of its own colour. 218.--THE VICTORIA CROSS PUZZLE. [Illustration: +---------------------+ | \... A .../ | | (I) |.......| (V) | |\_____|_______|_____/| |......| |------| |.. R .| |. I ..| |......| |......| | _____|_______|_____ | |/ |.......| \| | (O) |.. T ..| (C) | | /.........\ | +---------------------+ ] The puzzle-maker is peculiarly a "snapper-up of unconsidered trifles," and his productions are often built up with the slenderest materials. Trivialities that might entirely escape the observation of others, or, if they were observed, would be regarded as of no possible moment, often supply the man who is in quest of posers with a pretty theme or an idea that he thinks possesses some "basal value." When seated opposite to a lady in a railway carriage at the time of Queen Victoria's Diamond Jubilee, my attention was attracted to a brooch that she was wearing. It was in the form of a Maltese or Victoria Cross, and bore the letters of the word VICTORIA. The number and arrangement of the letters immediately gave me the suggestion for the puzzle which I now present. The diagram, it will be seen, is composed of nine divisions. The puzzle is to place eight counters, bearing the letters of the word VICTORIA, exactly in the manner shown, and then slide one letter at a time from black to white and white to black alternately, until the word reads round in the same direction, only with the initial letter V on one of the black arms of the cross. At no time may two letters be in the same division. It is required to find the shortest method. Leaping moves are, of course, not permitted. The first move must obviously be made with A, I, T, or R. Supposing you move T to the centre, the next counter played will be O or C, since I or R cannot be moved. There is something a little remarkable in the solution of this puzzle which I will explain. 219.--THE LETTER BLOCK PUZZLE. [Illustration: +-----+-----+-----+\ | | | | | | G | E | F | | | | | | | +-----+-----+-----+\| | | | | | | H | C | B | | | | | | | +-----+-----+-----+\| | |\____| | | | D || | A | | | || | | | +-----+-----+-----+ | \_________________\| ] Here is a little reminiscence of our old friend the Fifteen Block Puzzle. Eight wooden blocks are lettered, and are placed in a box, as shown in the illustration. It will be seen that you can only move one block at a time to the place vacant for the time being, as no block may be lifted out of the box. The puzzle is to shift them about until you get them in the order-- A B C D E F G H This you will find by no means difficult if you are allowed as many moves as you like. But the puzzle is to do it in the fewest possible moves. I will not say what this smallest number of moves is, because the reader may like to discover it for himself. In writing down your moves you will find it necessary to record no more than the letters in the order that they are shifted. Thus, your first five moves might be C, H, G, E, F; and this notation can have no possible ambiguity. In practice you only need eight counters and a simple diagram on a sheet of paper. 220.--A LODGING-HOUSE DIFFICULTY. [Illustration] The Dobsons secured apartments at Slocomb-on-Sea. There were six rooms on the same floor, all communicating, as shown in the diagram. The rooms they took were numbers 4, 5, and 6, all facing the sea. But a little difficulty arose. Mr. Dobson insisted that the piano and the bookcase should change rooms. This was wily, for the Dobsons were not musical, but they wanted to prevent any one else playing the instrument. Now, the rooms were very small and the pieces of furniture indicated were very big, so that no two of these articles could be got into any room at the same time. How was the exchange to be made with the least possible labour? Suppose, for example, you first move the wardrobe into No. 2; then you can move the bookcase to No. 5 and the piano to No. 6, and so on. It is a fascinating puzzle, but the landlady had reasons for not appreciating it. Try to solve her difficulty in the fewest possible removals with counters on a sheet of paper. 221.--THE EIGHT ENGINES. The diagram represents the engine-yard of a railway company under eccentric management. The engines are allowed to be stationary only at the nine points indicated, one of which is at present vacant. It is required to move the engines, one at a time, from point to point, in seventeen moves, so that their numbers shall be in numerical order round the circle, with the central point left vacant. But one of the engines has had its fire drawn, and therefore cannot move. How is the thing to be done? And which engine remains stationary throughout? [Illustration] 222.--A RAILWAY PUZZLE. [Illustration] Make a diagram, on a large sheet of paper, like the illustration, and have three counters marked A, three marked B, and three marked C. It will be seen that at the intersection of lines there are nine stopping-places, and a tenth stopping-place is attached to the outer circle like the tail of a Q. Place the three counters or engines marked A, the three marked B, and the three marked C at the places indicated. The puzzle is to move the engines, one at a time, along the lines, from stopping-place to stopping-place, until you succeed in getting an A, a B, and a C on each circle, and also A, B, and C on each straight line. You are required to do this in as few moves as possible. How many moves do you need? 223.--A RAILWAY MUDDLE. The plan represents a portion of the line of the London, Clodville, and Mudford Railway Company. It is a single line with a loop. There is only room for eight wagons, or seven wagons and an engine, between B and C on either the left line or the right line of the loop. It happened that two goods trains (each consisting of an engine and sixteen wagons) got into the position shown in the illustration. It looked like a hopeless deadlock, and each engine-driver wanted the other to go back to the next station and take off nine wagons. But an ingenious stoker undertook to pass the trains and send them on their respective journeys with their engines properly in front. He also contrived to reverse the engines the fewest times possible. Could you have performed the feat? And how many times would you require to reverse the engines? A "reversal" means a change of direction, backward or forward. No rope-shunting, fly-shunting, or other trick is allowed. All the work must be done legitimately by the two engines. It is a simple but interesting puzzle if attempted with counters. [Illustration] 224.--THE MOTOR-GARAGE PUZZLE. [Illustration] The difficulties of the proprietor of a motor garage are converted into a little pastime of a kind that has a peculiar fascination. All you need is to make a simple plan or diagram on a sheet of paper or cardboard and number eight counters, 1 to 8. Then a whole family can enter into an amusing competition to find the best possible solution of the difficulty. The illustration represents the plan of a motor garage, with accommodation for twelve cars. But the premises are so inconveniently restricted that the proprietor is often caused considerable perplexity. Suppose, for example, that the eight cars numbered 1 to 8 are in the positions shown, how are they to be shifted in the quickest possible way so that 1, 2, 3, and 4 shall change places with 5, 6, 7, and 8--that is, with the numbers still running from left to right, as at present, but the top row exchanged with the bottom row? What are the fewest possible moves? One car moves at a time, and any distance counts as one move. To prevent misunderstanding, the stopping-places are marked in squares, and only one car can be in a square at the same time. 225.--THE TEN PRISONERS. If prisons had no other use, they might still be preserved for the special benefit of puzzle-makers. They appear to be an inexhaustible mine of perplexing ideas. Here is a little poser that will perhaps interest the reader for a short period. We have in the illustration a prison of sixteen cells. The locations of the ten prisoners will be seen. The jailer has queer superstitions about odd and even numbers, and he wants to rearrange the ten prisoners so that there shall be as many even rows of men, vertically, horizontally, and diagonally, as possible. At present it will be seen, as indicated by the arrows, that there are only twelve such rows of 2 and 4. I will state at once that the greatest number of such rows that is possible is sixteen. But the jailer only allows four men to be removed to other cells, and informs me that, as the man who is seated in the bottom right-hand corner is infirm, he must not be moved. Now, how are we to get those sixteen rows of even numbers under such conditions? [Illustration] 226.--ROUND THE COAST. [Illustration] Here is a puzzle that will, I think, be found as amusing as instructive. We are given a ring of eight circles. Leaving circle 8 blank, we are required to write in the name of a seven-lettered port in the United Kingdom in this manner. Touch a blank circle with your pencil, then jump over two circles in either direction round the ring, and write down the first letter. Then touch another vacant circle, jump over two circles, and write down your second letter. Proceed similarly with the other letters in their proper order until you have completed the word. Thus, suppose we select "Glasgow," and proceed as follows: 6--1, 7--2, 8--3, 7--4, 8--5, which means that we touch 6, jump over 7 and and write down "G" on 1; then touch 7, jump over 8 and 1, and write down "l" on 2; and so on. It will be found that after we have written down the first five letters--"Glasg"--as above, we cannot go any further. Either there is something wrong with "Glasgow," or we have not managed our jumps properly. Can you get to the bottom of the mystery? 227.--CENTRAL SOLITAIRE. [Illustration] This ancient puzzle was a great favourite with our grandmothers, and most of us, I imagine, have on occasions come across a "Solitaire" board--a round polished board with holes cut in it in a geometrical pattern, and a glass marble in every hole. Sometimes I have noticed one on a side table in a suburban front parlour, or found one on a shelf in a country cottage, or had one brought under my notice at a wayside inn. Sometimes they are of the form shown above, but it is equally common for the board to have four more holes, at the points indicated by dots. I select the simpler form. Though "Solitaire" boards are still sold at the toy shops, it will be sufficient if the reader will make an enlarged copy of the above on a sheet of cardboard or paper, number the "holes," and provide himself with 33 counters, buttons, or beans. Now place a counter in every hole except the central one, No. 17, and the puzzle is to take off all the counters in a series of jumps, except the last counter, which must be left in that central hole. You are allowed to jump one counter over the next one to a vacant hole beyond, just as in the game of draughts, and the counter jumped over is immediately taken off the board. Only remember every move must be a jump; consequently you will take off a counter at each move, and thirty-one single jumps will of course remove all the thirty-one counters. But compound moves are allowed (as in draughts, again), for so long as one counter continues to jump, the jumps all count as one move. Here is the beginning of an imaginary solution which will serve to make the manner of moving perfectly plain, and show how the solver should write out his attempts: 5-17, 12-10, 26-12, 24-26 (13-11, 11-25), 9-11 (26-24, 24-10, 10-12), etc., etc. The jumps contained within brackets count as one move, because they are made with the same counter. Find the fewest possible moves. Of course, no diagonal jumps are permitted; you can only jump in the direction of the lines. 228.--THE TEN APPLES. [Illustration] The family represented in the illustration are amusing themselves with this little puzzle, which is not very difficult but quite interesting. They have, it will be seen, placed sixteen plates on the table in the form of a square, and put an apple in each of ten plates. They want to find a way of removing all the apples except one by jumping over one at a time to the next vacant square, as in draughts; or, better, as in solitaire, for you are not allowed to make any diagonal moves--only moves parallel to the sides of the square. It is obvious that as the apples stand no move can be made, but you are permitted to transfer any single apple you like to a vacant plate before starting. Then the moves must be all leaps, taking off the apples leaped over. 229.--THE NINE ALMONDS. "Here is a little puzzle," said a Parson, "that I have found peculiarly fascinating. It is so simple, and yet it keeps you interested indefinitely." The reverend gentleman took a sheet of paper and divided it off into twenty-five squares, like a square portion of a chessboard. Then he placed nine almonds on the central squares, as shown in the illustration, where we have represented numbered counters for convenience in giving the solution. "Now, the puzzle is," continued the Parson, "to remove eight of the almonds and leave the ninth in the central square. You make the removals by jumping one almond over another to the vacant square beyond and taking off the one jumped over--just as in draughts, only here you can jump in any direction, and not diagonally only. The point is to do the thing in the fewest possible moves." The following specimen attempt will make everything clear. Jump 4 over 1, 5 over 9, 3 over 6, 5 over 3, 7 over 5 and 2, 4 over 7, 8 over 4. But 8 is not left in the central square, as it should be. Remember to remove those you jump over. Any number of jumps in succession with the same almond count as one move. [Illustration] 230.--THE TWELVE PENNIES. Here is a pretty little puzzle that only requires twelve pennies or counters. Arrange them in a circle, as shown in the illustration. Now take up one penny at a time and, passing it over two pennies, place it on the third penny. Then take up another single penny and do the same thing, and so on, until, in six such moves, you have the coins in six pairs in the positions 1, 2, 3, 4, 5, 6. You can move in either direction round the circle at every play, and it does not matter whether the two jumped over are separate or a pair. This is quite easy if you use just a little thought. [Illustration] 231.--PLATES AND COINS. Place twelve plates, as shown, on a round table, with a penny or orange in every plate. Start from any plate you like and, always going in one direction round the table, take up one penny, pass it over two other pennies, and place it in the next plate. Go on again; take up another penny and, having passed it over two pennies, place it in a plate; and so continue your journey. Six coins only are to be removed, and when these have been placed there should be two coins in each of six plates and six plates empty. An important point of the puzzle is to go round the table as few times as possible. It does not matter whether the two coins passed over are in one or two plates, nor how many empty plates you pass a coin over. But you must always go in one direction round the table and end at the point from which you set out. Your hand, that is to say, goes steadily forward in one direction, without ever moving backwards. [Illustration] 232.--CATCHING THE MICE. [Illustration] "Play fair!" said the mice. "You know the rules of the game." "Yes, I know the rules," said the cat. "I've got to go round and round the circle, in the direction that you are looking, and eat every thirteenth mouse, but I must keep the white mouse for a tit-bit at the finish. Thirteen is an unlucky number, but I will do my best to oblige you." "Hurry up, then!" shouted the mice. "Give a fellow time to think," said the cat. "I don't know which of you to start at. I must figure it out." While the cat was working out the puzzle he fell asleep, and, the spell being thus broken, the mice returned home in safety. At which mouse should the cat have started the count in order that the white mouse should be the last eaten? When the reader has solved that little puzzle, here is a second one for him. What is the smallest number that the cat can count round and round the circle, if he must start at the white mouse (calling that "one" in the count) and still eat the white mouse last of all? And as a third puzzle try to discover what is the smallest number that the cat can count round and round if she must start at the white mouse (calling that "one") and make the white mouse the third eaten. 233.--THE ECCENTRIC CHEESEMONGER. [Illustration] The cheesemonger depicted in the illustration is an inveterate puzzle lover. One of his favourite puzzles is the piling of cheeses in his warehouse, an amusement that he finds good exercise for the body as well as for the mind. He places sixteen cheeses on the floor in a straight row and then makes them into four piles, with four cheeses in every pile, by always passing a cheese over four others. If you use sixteen counters and number them in order from 1 to 16, then you may place 1 on 6, 11 on 1, 7 on 4, and so on, until there are four in every pile. It will be seen that it does not matter whether the four passed over are standing alone or piled; they count just the same, and you can always carry a cheese in either direction. There are a great many different ways of doing it in twelve moves, so it makes a good game of "patience" to try to solve it so that the four piles shall be left in different stipulated places. For example, try to leave the piles at the extreme ends of the row, on Nos. 1, 2, 15 and 16; this is quite easy. Then try to leave three piles together, on Nos. 13, 14, and 15. Then again play so that they shall be left on Nos. 3, 5, 12, and 14. 234.--THE EXCHANGE PUZZLE. Here is a rather entertaining little puzzle with moving counters. You only need twelve counters--six of one colour, marked A, C, E, G, I, and K, and the other six marked B, D, F, H, J, and L. You first place them on the diagram, as shown in the illustration, and the puzzle is to get them into regular alphabetical order, as follows:-- A B C D E F G H I J K L The moves are made by exchanges of opposite colours standing on the same line. Thus, G and J may exchange places, or F and A, but you cannot exchange G and C, or F and D, because in one case they are both white and in the other case both black. Can you bring about the required arrangement in seventeen exchanges? [Illustration] It cannot be done in fewer moves. The puzzle is really much easier than it looks, if properly attacked. 235.--TORPEDO PRACTICE. [Illustration] If a fleet of sixteen men-of-war were lying at anchor and surrounded by the enemy, how many ships might be sunk if every torpedo, projected in a straight line, passed under three vessels and sank the fourth? In the diagram we have arranged the fleet in square formation, where it will be seen that as many as seven ships may be sunk (those in the top row and first column) by firing the torpedoes indicated by arrows. Anchoring the fleet as we like, to what extent can we increase this number? Remember that each successive ship is sunk before another torpedo is launched, and that every torpedo proceeds in a different direction; otherwise, by placing the ships in a straight line, we might sink as many as thirteen! It is an interesting little study in naval warfare, and eminently practical--provided the enemy will allow you to arrange his fleet for your convenience and promise to lie still and do nothing! 236.--THE HAT PUZZLE. Ten hats were hung on pegs as shown in the illustration--five silk hats and five felt "bowlers," alternately silk and felt. The two pegs at the end of the row were empty. [Illustration] The puzzle is to remove two contiguous hats to the vacant pegs, then two other adjoining hats to the pegs now unoccupied, and so on until five pairs have been moved and the hats again hang in an unbroken row, but with all the silk ones together and all the felt hats together. Remember, the two hats removed must always be contiguous ones, and you must take one in each hand and place them on their new pegs without reversing their relative position. You are not allowed to cross your hands, nor to hang up one at a time. Can you solve this old puzzle, which I give as introductory to the next? Try it with counters of two colours or with coins, and remember that the two empty pegs must be left at one end of the row. 237.--BOYS AND GIRLS. If you mark off ten divisions on a sheet of paper to represent the chairs, and use eight numbered counters for the children, you will have a fascinating pastime. Let the odd numbers represent boys and even numbers girls, or you can use counters of two colours, or coins. The puzzle is to remove two children who are occupying adjoining chairs and place them in two empty chairs, _making them first change sides_; then remove a second pair of children from adjoining chairs and place them in the two now vacant, making them change sides; and so on, until all the boys are together and all the girls together, with the two vacant chairs at one end as at present. To solve the puzzle you must do this in five moves. The two children must always be taken from chairs that are next to one another; and remember the important point of making the two children change sides, as this latter is the distinctive feature of the puzzle. By "change sides" I simply mean that if, for example, you first move 1 and 2 to the vacant chairs, then the first (the outside) chair will be occupied by 2 and the second one by 1. [Illustration] 238.--ARRANGING THE JAMPOTS. I happened to see a little girl sorting out some jam in a cupboard for her mother. She was putting each different kind of preserve apart on the shelves. I noticed that she took a pot of damson in one hand and a pot of gooseberry in the other and made them change places; then she changed a strawberry with a raspberry, and so on. It was interesting to observe what a lot of unnecessary trouble she gave herself by making more interchanges than there was any need for, and I thought it would work into a good puzzle. It will be seen in the illustration that little Dorothy has to manipulate twenty-four large jampots in as many pigeon-holes. She wants to get them in correct numerical order--that is, 1, 2, 3, 4, 5, 6 on the top shelf, 7, 8, 9, 10, 11, 12 on the next shelf, and so on. Now, if she always takes one pot in the right hand and another in the left and makes them change places, how many of these interchanges will be necessary to get all the jampots in proper order? She would naturally first change the 1 and the 3, then the 2 and the 3, when she would have the first three pots in their places. How would you advise her to go on then? Place some numbered counters on a sheet of paper divided into squares for the pigeon-holes, and you will find it an amusing puzzle. [Illustration] UNICURSAL AND ROUTE PROBLEMS. "I see them on their winding way." REGINALD HEBER. It is reasonable to suppose that from the earliest ages one man has asked another such questions as these: "Which is the nearest way home?" "Which is the easiest or pleasantest way?" "How can we find a way that will enable us to dodge the mastodon and the plesiosaurus?" "How can we get there without ever crossing the track of the enemy?" All these are elementary route problems, and they can be turned into good puzzles by the introduction of some conditions that complicate matters. A variety of such complications will be found in the following examples. I have also included some enumerations of more or less difficulty. These afford excellent practice for the reasoning faculties, and enable one to generalize in the case of symmetrical forms in a manner that is most instructive. 239.--A JUVENILE PUZZLE. For years I have been perpetually consulted by my juvenile friends about this little puzzle. Most children seem to know it, and yet, curiously enough, they are invariably unacquainted with the answer. The question they always ask is, "Do, please, tell me whether it is really possible." I believe Houdin the conjurer used to be very fond of giving it to his child friends, but I cannot say whether he invented the little puzzle or not. No doubt a large number of my readers will be glad to have the mystery of the solution cleared up, so I make no apology for introducing this old "teaser." The puzzle is to draw with three strokes of the pencil the diagram that the little girl is exhibiting in the illustration. Of course, you must not remove your pencil from the paper during a stroke or go over the same line a second time. You will find that you can get in a good deal of the figure with one continuous stroke, but it will always appear as if four strokes are necessary. [Illustration] Another form of the puzzle is to draw the diagram on a slate and then rub it out in three rubs. 240.--THE UNION JACK. [Illustration] The illustration is a rough sketch somewhat resembling the British flag, the Union Jack. It is not possible to draw the whole of it without lifting the pencil from the paper or going over the same line twice. The puzzle is to find out just _how much_ of the drawing it is possible to make without lifting your pencil or going twice over the same line. Take your pencil and see what is the best you can do. 241.--THE DISSECTED CIRCLE. How many continuous strokes, without lifting your pencil from the paper, do you require to draw the design shown in our illustration? Directly you change the direction of your pencil it begins a new stroke. You may go over the same line more than once if you like. It requires just a little care, or you may find yourself beaten by one stroke. [Illustration] 242.--THE TUBE INSPECTOR'S PUZZLE. The man in our illustration is in a little dilemma. He has just been appointed inspector of a certain system of tube railways, and it is his duty to inspect regularly, within a stated period, all the company's seventeen lines connecting twelve stations, as shown on the big poster plan that he is contemplating. Now he wants to arrange his route so that it shall take him over all the lines with as little travelling as possible. He may begin where he likes and end where he likes. What is his shortest route? Could anything be simpler? But the reader will soon find that, however he decides to proceed, the inspector must go over some of the lines more than once. In other words, if we say that the stations are a mile apart, he will have to travel more than seventeen miles to inspect every line. There is the little difficulty. How far is he compelled to travel, and which route do you recommend? [Illustration] 243.--VISITING THE TOWNS. [Illustration] A traveller, starting from town No. 1, wishes to visit every one of the towns once, and once only, going only by roads indicated by straight lines. How many different routes are there from which he can select? Of course, he must end his journey at No. 1, from which he started, and must take no notice of cross roads, but go straight from town to town. This is an absurdly easy puzzle, if you go the right way to work. 244.--THE FIFTEEN TURNINGS. Here is another queer travelling puzzle, the solution of which calls for ingenuity. In this case the traveller starts from the black town and wishes to go as far as possible while making only fifteen turnings and never going along the same road twice. The towns are supposed to be a mile apart. Supposing, for example, that he went straight to A, then straight to B, then to C, D, E, and F, you will then find that he has travelled thirty-seven miles in five turnings. Now, how far can he go in fifteen turnings? [Illustration] 245.--THE FLY ON THE OCTAHEDRON. "Look here," said the professor to his colleague, "I have been watching that fly on the octahedron, and it confines its walks entirely to the edges. What can be its reason for avoiding the sides?" "Perhaps it is trying to solve some route problem," suggested the other. "Supposing it to start from the top point, how many different routes are there by which it may walk over all the edges, without ever going twice along the same edge in any route?" [Illustration] The problem was a harder one than they expected, and after working at it during leisure moments for several days their results did not agree--in fact, they were both wrong. If the reader is surprised at their failure, let him attempt the little puzzle himself. I will just explain that the octahedron is one of the five regular, or Platonic, bodies, and is contained under eight equal and equilateral triangles. If you cut out the two pieces of cardboard of the shape shown in the margin of the illustration, cut half through along the dotted lines and then bend them and put them together, you will have a perfect octahedron. In any route over all the edges it will be found that the fly must end at the point of departure at the top. 246.--THE ICOSAHEDRON PUZZLE. The icosahedron is another of the five regular, or Platonic, bodies having all their sides, angles, and planes similar and equal. It is bounded by twenty similar equilateral triangles. If you cut out a piece of cardboard of the form shown in the smaller diagram, and cut half through along the dotted lines, it will fold up and form a perfect icosahedron. Now, a Platonic body does not mean a heavenly body; but it will suit the purpose of our puzzle if we suppose there to be a habitable planet of this shape. We will also suppose that, owing to a superfluity of water, the only dry land is along the edges, and that the inhabitants have no knowledge of navigation. If every one of those edges is 10,000 miles long and a solitary traveller is placed at the North Pole (the highest point shown), how far will he have to travel before he will have visited every habitable part of the planet--that is, have traversed every one of the edges? [Illustration] 247.--INSPECTING A MINE. The diagram is supposed to represent the passages or galleries in a mine. We will assume that every passage, A to B, B to C, C to H, H to I, and so on, is one furlong in length. It will be seen that there are thirty-one of these passages. Now, an official has to inspect all of them, and he descends by the shaft to the point A. How far must he travel, and what route do you recommend? The reader may at first say, "As there are thirty-one passages, each a furlong in length, he will have to travel just thirty-one furlongs." But this is assuming that he need never go along a passage more than once, which is not the case. Take your pencil and try to find the shortest route. You will soon discover that there is room for considerable judgment. In fact, it is a perplexing puzzle. [Illustration] 248.--THE CYCLISTS' TOUR. Two cyclists were consulting a road map in preparation for a little tour together. The circles represent towns, and all the good roads are represented by lines. They are starting from the town with a star, and must complete their tour at E. But before arriving there they want to visit every other town once, and only once. That is the difficulty. Mr. Spicer said, "I am certain we can find a way of doing it;" but Mr. Maggs replied, "No way, I'm sure." Now, which of them was correct? Take your pencil and see if you can find any way of doing it. Of course you must keep to the roads indicated. [Illustration] 249.--THE SAILOR'S PUZZLE. The sailor depicted in the illustration stated that he had since his boyhood been engaged in trading with a small vessel among some twenty little islands in the Pacific. He supplied the rough chart of which I have given a copy, and explained that the lines from island to island represented the only routes that he ever adopted. He always started from island A at the beginning of the season, and then visited every island once, and once only, finishing up his tour at the starting-point A. But he always put off his visit to C as long as possible, for trade reasons that I need not enter into. The puzzle is to discover his exact route, and this can be done with certainty. Take your pencil and, starting at A, try to trace it out. If you write down the islands in the order in which you visit them--thus, for example, A, I, O, L, G, etc.--you can at once see if you have visited an island twice or omitted any. Of course, the crossings of the lines must be ignored--that is, you must continue your route direct, and you are not allowed to switch off at a crossing and proceed in another direction. There is no trick of this kind in the puzzle. The sailor knew the best route. Can you find it? [Illustration] 250.--THE GRAND TOUR. One of the everyday puzzles of life is the working out of routes. If you are taking a holiday on your bicycle, or a motor tour, there always arises the question of how you are to make the best of your time and other resources. You have determined to get as far as some particular place, to include visits to such-and-such a town, to try to see something of special interest elsewhere, and perhaps to try to look up an old friend at a spot that will not take you much out of your way. Then you have to plan your route so as to avoid bad roads, uninteresting country, and, if possible, the necessity of a return by the same way that you went. With a map before you, the interesting puzzle is attacked and solved. I will present a little poser based on these lines. I give a rough map of a country--it is not necessary to say what particular country--the circles representing towns and the dotted lines the railways connecting them. Now there lived in the town marked A a man who was born there, and during the whole of his life had never once left his native place. From his youth upwards he had been very industrious, sticking incessantly to his trade, and had no desire whatever to roam abroad. However, on attaining his fiftieth birthday he decided to see something of his country, and especially to pay a visit to a very old friend living at the town marked Z. What he proposed was this: that he would start from his home, enter every town once and only once, and finish his journey at Z. As he made up his mind to perform this grand tour by rail only, he found it rather a puzzle to work out his route, but he at length succeeded in doing so. How did he manage it? Do not forget that every town has to be visited once, and not more than once. [Illustration] 251.--WATER, GAS, AND ELECTRICITY. There are some half-dozen puzzles, as old as the hills, that are perpetually cropping up, and there is hardly a month in the year that does not bring inquiries as to their solution. Occasionally one of these, that one had thought was an extinct volcano, bursts into eruption in a surprising manner. I have received an extraordinary number of letters respecting the ancient puzzle that I have called "Water, Gas, and Electricity." It is much older than electric lighting, or even gas, but the new dress brings it up to date. The puzzle is to lay on water, gas, and electricity, from W, G, and E, to each of the three houses, A, B, and C, without any pipe crossing another. Take your pencil and draw lines showing how this should be done. You will soon find yourself landed in difficulties. [Illustration] 252.--A PUZZLE FOR MOTORISTS. [Illustration] Eight motorists drove to church one morning. Their respective houses and churches, together with the only roads available (the dotted lines), are shown. One went from his house A to his church A, another from his house B to his church B, another from C to C, and so on, but it was afterwards found that no driver ever crossed the track of another car. Take your pencil and try to trace out their various routes. 253.--A BANK HOLIDAY PUZZLE. Two friends were spending their bank holiday on a cycling trip. Stopping for a rest at a village inn, they consulted a route map, which is represented in our illustration in an exceedingly simplified form, for the puzzle is interesting enough without all the original complexities. They started from the town in the top left-hand corner marked A. It will be seen that there are one hundred and twenty such towns, all connected by straight roads. Now they discovered that there are exactly 1,365 different routes by which they may reach their destination, always travelling either due south or due east. The puzzle is to discover which town is their destination. [Illustration] Of course, if you find that there are more than 1,365 different routes to a town it cannot be the right one. 254.--THE MOTOR-CAR TOUR. [Illustration] In the above diagram the circles represent towns and the lines good roads. In just how many different ways can a motorist, starting from London (marked with an L), make a tour of all these towns, visiting every town once, and only once, on a tour, and always coming back to London on the last ride? The exact reverse of any route is not counted as different. 255.--THE LEVEL PUZZLE. [Illustration] This is a simple counting puzzle. In how many different ways can you spell out the word LEVEL by placing the point of your pencil on an L and then passing along the lines from letter to letter. You may go in any direction, backwards or forwards. Of course you are not allowed to miss letters--that is to say, if you come to a letter you must use it. 256.--THE DIAMOND PUZZLE. IN how many different ways may the word DIAMOND be read in the arrangement shown? You may start wherever you like at a D and go up or down, backwards or forwards, in and out, in any direction you like, so long as you always pass from one letter to another that adjoins it. How many ways are there? [Illustration] 257.--THE DEIFIED PUZZLE. In how many different ways may the word DEIFIED be read in this arrangement under the same conditions as in the last puzzle, with the addition that you can use any letters twice in the same reading? [Illustration] 258.--THE VOTERS' PUZZLE. [Illustration] Here we have, perhaps, the most interesting form of the puzzle. In how many different ways can you read the political injunction, "RISE TO VOTE, SIR," under the same conditions as before? In this case every reading of the palindrome requires the use of the central V as the middle letter. 259.--HANNAH'S PUZZLE. A man was in love with a young lady whose Christian name was Hannah. When he asked her to be his wife she wrote down the letters of her name in this manner:-- H H H H H H H A A A A H H A N N A H H A N N A H H A A A A H H H H H H H and promised that she would be his if he could tell her correctly in how many different ways it was possible to spell out her name, always passing from one letter to another that was adjacent. Diagonal steps are here allowed. Whether she did this merely to tease him or to test his cleverness is not recorded, but it is satisfactory to know that he succeeded. Would you have been equally successful? Take your pencil and try. You may start from any of the H's and go backwards or forwards and in any direction, so long as all the letters in a spelling are adjoining one another. How many ways are there, no two exactly alike? 260.--THE HONEYCOMB PUZZLE. [Illustration] Here is a little puzzle with the simplest possible conditions. Place the point of your pencil on a letter in one of the cells of the honeycomb, and trace out a very familiar proverb by passing always from a cell to one that is contiguous to it. If you take the right route you will have visited every cell once, and only once. The puzzle is much easier than it looks. 261.--THE MONK AND THE BRIDGES. In this case I give a rough plan of a river with an island and five bridges. On one side of the river is a monastery, and on the other side is seen a monk in the foreground. Now, the monk has decided that he will cross every bridge once, and only once, on his return to the monastery. This is, of course, quite easy to do, but on the way he thought to himself, "I wonder how many different routes there are from which I might have selected." Could you have told him? That is the puzzle. Take your pencil and trace out a route that will take you once over all the five bridges. Then trace out a second route, then a third, and see if you can count all the variations. You will find that the difficulty is twofold: you have to avoid dropping routes on the one hand and counting the same routes more than once on the other. [Illustration] COMBINATION AND GROUP PROBLEMS. "A combination and a form indeed." _Hamlet_, iii. 4. Various puzzles in this class might be termed problems in the "geometry of situation," but their solution really depends on the theory of combinations which, in its turn, is derived directly from the theory of permutations. It has seemed convenient to include here certain group puzzles and enumerations that might, perhaps, with equal reason have been placed elsewhere; but readers are again asked not to be too critical about the classification, which is very difficult and arbitrary. As I have included my problem of "The Round Table" (No. 273), perhaps a few remarks on another well-known problem of the same class, known by the French as La Problême des Ménages, may be interesting. If n married ladies are seated at a round table in any determined order, in how many different ways may their n husbands be placed so that every man is between two ladies but never next to his own wife? This difficult problem was first solved by Laisant, and the method shown in the following table is due to Moreau:-- 4 0 2 5 3 13 6 13 80 7 83 579 8 592 4738 9 4821 43387 10 43979 439792 The first column shows the number of married couples. The numbers in the second column are obtained in this way: 5 × 3 + 0 - 2 = 13; 6 × 13 + 3 + 2 = 83; 7 × 83 + 13 - 2 = 592; 8 × 592 + 83 + 2 = 4821; and so on. Find all the numbers, except 2, in the table, and the method will be evident. It will be noted that the 2 is subtracted when the first number (the number of couples) is odd, and added when that number is even. The numbers in the third column are obtained thus: 13 - 0 = 13; 83 - 3 = 80; 592 - 13 = 579; 4821 - 83 = 4738; and so on. The numbers in this last column give the required solutions. Thus, four husbands may be seated in two ways, five husbands may be placed in thirteen ways, and six husbands in eighty ways. The following method, by Lucas, will show the remarkable way in which chessboard analysis may be applied to the solution of a circular problem of this kind. Divide a square into thirty-six cells, six by six, and strike out all the cells in the long diagonal from the bottom left-hand corner to the top right-hand corner, also the five cells in the diagonal next above it and the cell in the bottom right-hand corner. The answer for six couples will be the same as the number of ways in which you can place six rooks (not using the cancelled cells) so that no rook shall ever attack another rook. It will be found that the six rooks may be placed in eighty different ways, which agrees with the above table. 262.--THOSE FIFTEEN SHEEP. A certain cyclopædia has the following curious problem, I am told: "Place fifteen sheep in four pens so that there shall be the same number of sheep in each pen." No answer whatever is vouchsafed, so I thought I would investigate the matter. I saw that in dealing with apples or bricks the thing would appear to be quite impossible, since four times any number must be an even number, while fifteen is an odd number. I thought, therefore, that there must be some quality peculiar to the sheep that was not generally known. So I decided to interview some farmers on the subject. The first one pointed out that if we put one pen inside another, like the rings of a target, and placed all sheep in the smallest pen, it would be all right. But I objected to this, because you admittedly place all the sheep in one pen, not in four pens. The second man said that if I placed four sheep in each of three pens and three sheep in the last pen (that is fifteen sheep in all), and one of the ewes in the last pen had a lamb during the night, there would be the same number in each pen in the morning. This also failed to satisfy me. [Illustration] The third farmer said, "I've got four hurdle pens down in one of my fields, and a small flock of wethers, so if you will just step down with me I will show you how it is done." The illustration depicts my friend as he is about to demonstrate the matter to me. His lucid explanation was evidently that which was in the mind of the writer of the article in the cyclopædia. What was it? Can you place those fifteen sheep? 263.--KING ARTHUR'S KNIGHTS. King Arthur sat at the Round Table on three successive evenings with his knights--Beleobus, Caradoc, Driam, Eric, Floll, and Galahad--but on no occasion did any person have as his neighbour one who had before sat next to him. On the first evening they sat in alphabetical order round the table. But afterwards King Arthur arranged the two next sittings so that he might have Beleobus as near to him as possible and Galahad as far away from him as could be managed. How did he seat the knights to the best advantage, remembering that rule that no knight may have the same neighbour twice? 264.--THE CITY LUNCHEONS. Twelve men connected with a large firm in the City of London sit down to luncheon together every day in the same room. The tables are small ones that only accommodate two persons at the same time. Can you show how these twelve men may lunch together on eleven days in pairs, so that no two of them shall ever sit twice together? We will represent the men by the first twelve letters of the alphabet, and suppose the first day's pairing to be as follows-- (A B) (C D) (E F) (G H) (I J) (K L). Then give any pairing you like for the next day, say-- (A C) (B D) (E G) (F H) (I K) (J L), and so on, until you have completed your eleven lines, with no pair ever occurring twice. There are a good many different arrangements possible. Try to find one of them. 265.--A PUZZLE FOR CARD-PLAYERS. Twelve members of a club arranged to play bridge together on eleven evenings, but no player was ever to have the same partner more than once, or the same opponent more than twice. Can you draw up a scheme showing how they may all sit down at three tables every evening? Call the twelve players by the first twelve letters of the alphabet and try to group them. 266.--A TENNIS TOURNAMENT. Four married couples played a "mixed double" tennis tournament, a man and a lady always playing against a man and a lady. But no person ever played with or against any other person more than once. Can you show how they all could have played together in the two courts on three successive days? This is a little puzzle of a quite practical kind, and it is just perplexing enough to be interesting. 267.--THE WRONG HATS. "One of the most perplexing things I have come across lately," said Mr. Wilson, "is this. Eight men had been dining not wisely but too well at a certain London restaurant. They were the last to leave, but not one man was in a condition to identify his own hat. Now, considering that they took their hats at random, what are the chances that every man took a hat that did not belong to him?" "The first thing," said Mr. Waterson, "is to see in how many different ways the eight hats could be taken." "That is quite easy," Mr. Stubbs explained. "Multiply together the numbers, 1, 2, 3, 4, 5, 6, 7, and 8. Let me see--half a minute--yes; there are 40,320 different ways." "Now all you've got to do is to see in how many of these cases no man has his own hat," said Mr. Waterson. "Thank you, I'm not taking any," said Mr. Packhurst. "I don't envy the man who attempts the task of writing out all those forty-thousand-odd cases and then picking out the ones he wants." They all agreed that life is not long enough for that sort of amusement; and as nobody saw any other way of getting at the answer, the matter was postponed indefinitely. Can you solve the puzzle? 268.--THE PEAL OF BELLS. A correspondent, who is apparently much interested in campanology, asks me how he is to construct what he calls a "true and correct" peal for four bells. He says that every possible permutation of the four bells must be rung once, and once only. He adds that no bell must move more than one place at a time, that no bell must make more than two successive strokes in either the first or the last place, and that the last change must be able to pass into the first. These fantastic conditions will be found to be observed in the little peal for three bells, as follows:-- 1 2 3 2 1 3 2 3 1 3 2 1 3 1 2 1 3 2 How are we to give him a correct solution for his four bells? 269.--THREE MEN IN A BOAT. A certain generous London manufacturer gives his workmen every year a week's holiday at the seaside at his own expense. One year fifteen of his men paid a visit to Herne Bay. On the morning of their departure from London they were addressed by their employer, who expressed the hope that they would have a very pleasant time. "I have been given to understand," he added, "that some of you fellows are very fond of rowing, so I propose on this occasion to provide you with this recreation, and at the same time give you an amusing little puzzle to solve. During the seven days that you are at Herne Bay every one of you will go out every day at the same time for a row, but there must always be three men in a boat and no more. No two men may ever go out in a boat together more than once, and no man is allowed to go out twice in the same boat. If you can manage to do this, and use as few different boats as possible, you may charge the firm with the expense." One of the men tells me that the experience he has gained in such matters soon enabled him to work out the answer to the entire satisfaction of themselves and their employer. But the amusing part of the thing is that they never really solved the little mystery. I find their method to have been quite incorrect, and I think it will amuse my readers to discover how the men should have been placed in the boats. As their names happen to have been Andrews, Baker, Carter, Danby, Edwards, Frith, Gay, Hart, Isaacs, Jackson, Kent, Lang, Mason, Napper, and Onslow, we can call them by their initials and write out the five groups for each of the seven days in the following simple way: 1 2 3 4 5 First Day: (ABC) (DEF) (GHI) (JKL) (MNO). The men within each pair of brackets are here seen to be in the same boat, and therefore A can never go out with B or with C again, and C can never go out again with B. The same applies to the other four boats. The figures show the number on the boat, so that A, B, or C, for example, can never go out in boat No. 1 again. 270.--THE GLASS BALLS. A number of clever marksmen were staying at a country house, and the host, to provide a little amusement, suspended strings of glass balls, as shown in the illustration, to be fired at. After they had all put their skill to a sufficient test, somebody asked the following question: "What is the total number of different ways in which these sixteen balls may be broken, if we must always break the lowest ball that remains on any string?" Thus, one way would be to break all the four balls on each string in succession, taking the strings from left to right. Another would be to break all the fourth balls on the four strings first, then break the three remaining on the first string, then take the balls on the three other strings alternately from right to left, and so on. There is such a vast number of different ways (since every little variation of order makes a different way) that one is apt to be at first impressed by the great difficulty of the problem. Yet it is really quite simple when once you have hit on the proper method of attacking it. How many different ways are there? [Illustration] 271.--FIFTEEN LETTER PUZZLE. ALE FOE HOD BGN CAB HEN JOG KFM HAG GEM MOB BFH FAN KIN JEK DFL JAM HIM GCL LJH AID JIB FCJ NJD OAK FIG HCK MLN BED OIL MCD BLK ICE CON DGK The above is the solution of a puzzle I gave in _Tit-bits_ in the summer of 1896. It was required to take the letters, A, B, C, D, E, F, G, H, I, J, K, L, M, N, and O, and with them form thirty-five groups of three letters so that the combinations should include the greatest number possible of common English words. No two letters may appear together in a group more than once. Thus, A and L having been together in ALE, must never be found together again; nor may A appear again in a group with E, nor L with E. These conditions will be found complied with in the above solution, and the number of words formed is twenty-one. Many persons have since tried hard to beat this number, but so far have not succeeded. More than thirty-five combinations of the fifteen letters cannot be formed within the conditions. Theoretically, there cannot possibly be more than twenty-three words formed, because only this number of combinations is possible with a vowel or vowels in each. And as no English word can be formed from three of the given vowels (A, E, I, and O), we must reduce the number of possible words to twenty-two. This is correct theoretically, but practically that twenty-second word cannot be got in. If JEK, shown above, were a word it would be all right; but it is not, and no amount of juggling with the other letters has resulted in a better answer than the one shown. I should, say that proper nouns and abbreviations, such as Joe, Jim, Alf, Hal, Flo, Ike, etc., are disallowed. Now, the present puzzle is a variation of the above. It is simply this: Instead of using the fifteen letters given, the reader is allowed to select any fifteen different letters of the alphabet that he may prefer. Then construct thirty-five groups in accordance with the conditions, and show as many good English words as possible. 272.--THE NINE SCHOOLBOYS. This is a new and interesting companion puzzle to the "Fifteen Schoolgirls" (see solution of No. 269), and even in the simplest possible form in which I present it there are unquestionable difficulties. Nine schoolboys walk out in triplets on the six week days so that no boy ever walks _side by side_ with any other boy more than once. How would you arrange them? If we represent them by the first nine letters of the alphabet, they might be grouped on the first day as follows:-- A B C D E F G H I Then A can never walk again side by side with B, or B with C, or D with E, and so on. But A can, of course, walk side by side with C. It is here not a question of being together in the same triplet, but of walking side by side in a triplet. Under these conditions they can walk out on six days; under the "Schoolgirls" conditions they can only walk on four days. 273.--THE ROUND TABLE. Seat the same n persons at a round table on (n - 1)(n - 2) -------------- 2 occasions so that no person shall ever have the same two neighbours twice. This is, of course, equivalent to saying that every person must sit once, and once only, between every possible pair. 274.--THE MOUSE-TRAP PUZZLE. [Illustration 6 20 2 19 13 21 7 5 3 18 17 8 15 11 14 16 1 9 10 4 12 ] This is a modern version, with a difference, of an old puzzle of the same name. Number twenty-one cards, 1, 2, 3, etc., up to 21, and place them in a circle in the particular order shown in the illustration. These cards represent mice. You start from any card, calling that card "one," and count, "one, two, three," etc., in a clockwise direction, and when your count agrees with the number on the card, you have made a "catch," and you remove the card. Then start at the next card, calling that "one," and try again to make another "catch." And so on. Supposing you start at 18, calling that card "one," your first "catch" will be 19. Remove 19 and your next "catch" is 10. Remove 10 and your next "catch" is 1. Remove the 1, and if you count up to 21 (you must never go beyond), you cannot make another "catch." Now, the ideal is to "catch" all the twenty-one mice, but this is not here possible, and if it were it would merely require twenty-one different trials, at the most, to succeed. But the reader may make any two cards change places before he begins. Thus, you can change the 6 with the 2, or the 7 with the 11, or any other pair. This can be done in several ways so as to enable you to "catch" all the twenty-one mice, if you then start at the right place. You may never pass over a "catch"; you must always remove the card and start afresh. 275.--THE SIXTEEN SHEEP. [Illustration: +========================+ || | | | || || 0 | 0 | 0 | 0 || +-----+-----+-----+------+ || | | | || || 0 | 0 | 0 | 0 || +========================+ || || | || || || 0 || 0 | 0 || 0 || +-----+=====+=====+------+ || | || | || || 0 | 0 || 0 | 0 || +========================+ ] Here is a new puzzle with matches and counters or coins. In the illustration the matches represent hurdles and the counters sheep. The sixteen hurdles on the outside, and the sheep, must be regarded as immovable; the puzzle has to do entirely with the nine hurdles on the inside. It will be seen that at present these nine hurdles enclose four groups of 8, 3, 3, and 2 sheep. The farmer requires to readjust some of the hurdles so as to enclose 6, 6, and 4 sheep. Can you do it by only replacing two hurdles? When you have succeeded, then try to do it by replacing three hurdles; then four, five, six, and seven in succession. Of course, the hurdles must be legitimately laid on the dotted lines, and no such tricks are allowed as leaving unconnected ends of hurdles, or two hurdles placed side by side, or merely making hurdles change places. In fact, the conditions are so simple that any farm labourer will understand it directly. 276.--THE EIGHT VILLAS. In one of the outlying suburbs of London a man had a square plot of ground on which he decided to build eight villas, as shown in the illustration, with a common recreation ground in the middle. After the houses were completed, and all or some of them let, he discovered that the number of occupants in the three houses forming a side of the square was in every case nine. He did not state how the occupants were distributed, but I have shown by the numbers on the sides of the houses one way in which it might have happened. The puzzle is to discover the total number of ways in which all or any of the houses might be occupied, so that there should be nine persons on each side. In order that there may be no misunderstanding, I will explain that although B is what we call a reflection of A, these would count as two different arrangements, while C, if it is turned round, will give four arrangements; and if turned round in front of a mirror, four other arrangements. All eight must be counted. [Illustration: /\ /\ /\ |2 | |5 | |2 | /\ /\ |5 | |5 | /\ /\ /\ |2 | |5 | |2 | +---+---+---+ +---+---+---+ +---+---+---+ | 1 | 6 | 2 | | 2 | 6 | 1 | | 1 | 6 | 2 | +---+---+---+ +---+---+---+ +---+---+---+ | 6 | | 6 | | 6 | | 6 | | 4 | | 4 | +---+---+---+ +---+---+---+ +---+---+---+ | 2 | 6 | 1 | | 1 | 6 | 2 | | 4 | 2 | 3 | +---+---+---+ +---+---+---+ +---+---+---+ A B C ] 277.--COUNTER CROSSES. All that we need for this puzzle is nine counters, numbered 1, 2, 3, 4, 5, 6, 7, 8, and 9. It will be seen that in the illustration A these are arranged so as to form a Greek cross, while in the case of B they form a Latin cross. In both cases the reader will find that the sum of the numbers in the upright of the cross is the same as the sum of the numbers in the horizontal arm. It is quite easy to hit on such an arrangement by trial, but the problem is to discover in exactly how many different ways it may be done in each case. Remember that reversals and reflections do not count as different. That is to say, if you turn this page round you get four arrangements of the Greek cross, and if you turn it round again in front of a mirror you will get four more. But these eight are all regarded as one and the same. Now, how many different ways are there in each case? [Illustration: (1) (2) (2) (4) (5) (1) (6) (7) (3) (4) (9) (5) (6) (3) (7) (8) A (8) B (9) ] 278.--A DORMITORY PUZZLE. In a certain convent there were eight large dormitories on one floor, approached by a spiral staircase in the centre, as shown in our plan. On an inspection one Monday by the abbess it was found that the south aspect was so much preferred that six times as many nuns slept on the south side as on each of the other three sides. She objected to this overcrowding, and ordered that it should be reduced. On Tuesday she found that five times as many slept on the south side as on each of the other sides. Again she complained. On Wednesday she found four times as many on the south side, on Thursday three times as many, and on Friday twice as many. Urging the nuns to further efforts, she was pleased to find on Saturday that an equal number slept on each of the four sides of the house. What is the smallest number of nuns there could have been, and how might they have arranged themselves on each of the six nights? No room may ever be unoccupied. [Illustration +---+---+---+ | | | | | | | | | | | | +---+---+---+ | |\|/| | | |-*-| | | |/|\| | +---+---+---+ | | | | | | | | | | | | +---+---+---+ ] 279.--THE BARRELS OF BALSAM. A merchant of Bagdad had ten barrels of precious balsam for sale. They were numbered, and were arranged in two rows, one on top of the other, as shown in the picture. The smaller the number on the barrel, the greater was its value. So that the best quality was numbered "1" and the worst numbered "10," and all the other numbers of graduating values. Now, the rule of Ahmed Assan, the merchant, was that he never put a barrel either beneath or to the right of one of less value. The arrangement shown is, of course, the simplest way of complying with this condition. But there are many other ways--such, for example, as this:-- 1 2 5 7 8 3 4 6 9 10 Here, again, no barrel has a smaller number than itself on its right or beneath it. The puzzle is to discover in how many different ways the merchant of Bagdad might have arranged his barrels in the two rows without breaking his rule. Can you count the number of ways? 280.--BUILDING THE TETRAHEDRON. I possess a tetrahedron, or triangular pyramid, formed of six sticks glued together, as shown in the illustration. Can you count correctly the number of different ways in which these six sticks might have been stuck together so as to form the pyramid? Some friends worked at it together one evening, each person providing himself with six lucifer matches to aid his thoughts; but it was found that no two results were the same. You see, if we remove one of the sticks and turn it round the other way, that will be a different pyramid. If we make two of the sticks change places the result will again be different. But remember that every pyramid may be made to stand on either of its four sides without being a different one. How many ways are there altogether? [Illustration] 281.--PAINTING A PYRAMID. This puzzle concerns the painting of the four sides of a tetrahedron, or triangular pyramid. If you cut out a piece of cardboard of the triangular shape shown in Fig. 1, and then cut half through along the dotted lines, it will fold up and form a perfect triangular pyramid. And I would first remind my readers that the primary colours of the solar spectrum are seven--violet, indigo, blue, green, yellow, orange, and red. When I was a child I was taught to remember these by the ungainly word formed by the initials of the colours, "Vibgyor." [Illustration] In how many different ways may the triangular pyramid be coloured, using in every case one, two, three, or four colours of the solar spectrum? Of course a side can only receive a single colour, and no side can be left uncoloured. But there is one point that I must make quite clear. The four sides are not to be regarded as individually distinct. That is to say, if you paint your pyramid as shown in Fig. 2 (where the bottom side is green and the other side that is out of view is yellow), and then paint another in the order shown in Fig. 3, these are really both the same and count as one way. For if you tilt over No. 2 to the right it will so fall as to represent No. 3. The avoidance of repetitions of this kind is the real puzzle of the thing. If a coloured pyramid cannot be placed so that it exactly resembles in its colours and their relative order another pyramid, then they are different. Remember that one way would be to colour all the four sides red, another to colour two sides green, and the remaining sides yellow and blue; and so on. 282.--THE ANTIQUARY'S CHAIN. An antiquary possessed a number of curious old links, which he took to a blacksmith, and told him to join together to form one straight piece of chain, with the sole condition that the two circular links were not to be together. The following illustration shows the appearance of the chain and the form of each link. Now, supposing the owner should separate the links again, and then take them to another smith and repeat his former instructions exactly, what are the chances against the links being put together exactly as they were by the first man? Remember that every successive link can be joined on to another in one of two ways, just as you can put a ring on your finger in two ways, or link your forefingers and thumbs in two ways. [Illustration] 283.--THE FIFTEEN DOMINOES. In this case we do not use the complete set of twenty-eight dominoes to be found in the ordinary box. We dispense with all those dominoes that have a five or a six on them and limit ourselves to the fifteen that remain, where the double-four is the highest. In how many different ways may the fifteen dominoes be arranged in a straight line in accordance with the simple rule of the game that a number must always be placed against a similar number--that is, a four against a four, a blank against a blank, and so on? Left to right and right to left of the same arrangement are to be counted as two different ways. 384.--THE CROSS TARGET. +-+-+ |*|*| +-+-+ |*|*| +-+-+-+-+-+-+ | | | |*| | | +-+-+-+-+-+-+ | | |*| |*| | +-+-+-+-+-+-+ | |*| +-+-+ | | | +-+-+ In the illustration we have a somewhat curious target designed by an eccentric sharpshooter. His idea was that in order to score you must hit four circles in as many shots so that those four shots shall form a square. It will be seen by the results recorded on the target that two attempts have been successful. The first man hit the four circles at the top of the cross, and thus formed his square. The second man intended to hit the four in the bottom arm, but his second shot, on the left, went too high. This compelled him to complete his four in a different way than he intended. It will thus be seen that though it is immaterial which circle you hit at the first shot, the second shot may commit you to a definite procedure if you are to get your square. Now, the puzzle is to say in just how many different ways it is possible to form a square on the target with four shots. 285.--THE FOUR POSTAGE STAMPS. +---+----+----+----+ | 1 | 2 | 3 | 4 | +---+----+----+----+ | 5 | 6 | 7 | 8 | +---+----+----+----+ | 9 | 10 | 11 | 12 | +---+----+----+----+ "It is as easy as counting," is an expression one sometimes hears. But mere counting may be puzzling at times. Take the following simple example. Suppose you have just bought twelve postage stamps, in this form--three by four--and a friend asks you to oblige him with four stamps, all joined together--no stamp hanging on by a mere corner. In how many different ways is it possible for you to tear off those four stamps? You see, you can give him 1, 2, 3, 4, or 2, 3, 6, 7, or 1, 2, 3, 6, or 1, 2, 3, 7, or 2, 3, 4, 8, and so on. Can you count the number of different ways in which those four stamps might be delivered? There are not many more than fifty ways, so it is not a big count. Can you get the exact number? 286.--PAINTING THE DIE. In how many different ways may the numbers on a single die be marked, with the only condition that the 1 and 6, the 2 and 5, and the 3 and 4 must be on opposite sides? It is a simple enough question, and yet it will puzzle a good many people. 287.--AN ACROSTIC PUZZLE. In the making or solving of double acrostics, has it ever occurred to you to consider the variety and limitation of the pair of initial and final letters available for cross words? You may have to find a word beginning with A and ending with B, or A and C, or A and D, and so on. Some combinations are obviously impossible--such, for example, as those with Q at the end. But let us assume that a good English word can be found for every case. Then how many possible pairs of letters are available? CHESSBOARD PROBLEMS. "You and I will goe to the chesse." GREENE'S _Groatsworth of Wit._ During a heavy gale a chimney-pot was hurled through the air, and crashed upon the pavement just in front of a pedestrian. He quite calmly said, "I have no use for it: I do not smoke." Some readers, when they happen to see a puzzle represented on a chessboard with chess pieces, are apt to make the equally inconsequent remark, "I have no use for it: I do not play chess." This is largely a result of the common, but erroneous, notion that the ordinary chess puzzle with which we are familiar in the press (dignified, for some reason, with the name "problem") has a vital connection with the game of chess itself. But there is no condition in the game that you shall checkmate your opponent in two moves, in three moves, or in four moves, while the majority of the positions given in these puzzles are such that one player would have so great a superiority in pieces that the other would have resigned before the situations were reached. And the solving of them helps you but little, and that quite indirectly, in playing the game, it being well known that, as a rule, the best "chess problemists" are indifferent players, and _vice versa_. Occasionally a man will be found strong on both subjects, but he is the exception to the rule. Yet the simple chequered board and the characteristic moves of the pieces lend themselves in a very remarkable manner to the devising of the most entertaining puzzles. There is room for such infinite variety that the true puzzle lover cannot afford to neglect them. It was with a view to securing the interest of readers who are frightened off by the mere presentation of a chessboard that so many puzzles of this class were originally published by me in various fanciful dresses. Some of these posers I still retain in their disguised form; others I have translated into terms of the chessboard. In the majority of cases the reader will not need any knowledge whatever of chess, but I have thought it best to assume throughout that he is acquainted with the terminology, the moves, and the notation of the game. I first deal with a few questions affecting the chessboard itself; then with certain statical puzzles relating to the Rook, the Bishop, the Queen, and the Knight in turn; then dynamical puzzles with the pieces in the same order; and, finally, with some miscellaneous puzzles on the chessboard. It is hoped that the formulæ and tables given at the end of the statical puzzles will be of interest, as they are, for the most part, published for the first time. THE CHESSBOARD. "Good company's a chessboard." BYRON'S _Don Juan_, xiii. 89. A chessboard is essentially a square plane divided into sixty-four smaller squares by straight lines at right angles. Originally it was not chequered (that is, made with its rows and columns alternately black and white, or of any other two colours), and this improvement was introduced merely to help the eye in actual play. The utility of the chequers is unquestionable. For example, it facilitates the operation of the bishops, enabling us to see at the merest glance that our king or pawns on black squares are not open to attack from an opponent's bishop running on the white diagonals. Yet the chequering of the board is not essential to the game of chess. Also, when we are propounding puzzles on the chessboard, it is often well to remember that additional interest may result from "generalizing" for boards containing any number of squares, or from limiting ourselves to some particular chequered arrangement, not necessarily a square. We will give a few puzzles dealing with chequered boards in this general way. 288.--CHEQUERED BOARD DIVISIONS. I recently asked myself the question: In how many different ways may a chessboard be divided into two parts of the same size and shape by cuts along the lines dividing the squares? The problem soon proved to be both fascinating and bristling with difficulties. I present it in a simplified form, taking a board of smaller dimensions. [Illustration: +---+---*---+---+ +---+---+---*---+ +---+---+---*---+ | | H | | | | | H | | | | H | +---+---*---+---+ +---+---*===*---+ +---*===*---*---+ | | H | | | | H | | | H H H | +---+---*---+---+ +---+---*---+---+ +---*---*---*---+ | | H | | | | H | | | H H H | +---+---*---+---+ +---*===*---+---+ +---*---*===*---+ | | H | | | H | | | | H | | | +---+---*---+---+ +---*---+---+---+ +---*---+---+---+ +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ +---+---*---+---+ +---+---+---*---+ +---+---+---*---+ | | H | | | | | H | | | | H | +---*===*---+---+ +---*===*===*---+ +---+---*===*---+ | H | | | | H | | | | | H | | +---*===*===*---+ +---*===*===*---+ +---+---*---+---+ | | | H | | | | H | | | H | | +---+---*===*---+ +---*===*===*---+ +---*===*---+---+ | | H | | | H | | | | H | | | +---+---*---+---+ +---*---+---+---+ +---*---+---+---+ ] It is obvious that a board of four squares can only be so divided in one way--by a straight cut down the centre--because we shall not count reversals and reflections as different. In the case of a board of sixteen squares--four by four--there are just six different ways. I have given all these in the diagram, and the reader will not find any others. Now, take the larger board of thirty-six squares, and try to discover in how many ways it may be cut into two parts of the same size and shape. 289.--LIONS AND CROWNS. The young lady in the illustration is confronted with a little cutting-out difficulty in which the reader may be glad to assist her. She wishes, for some reason that she has not communicated to me, to cut that square piece of valuable material into four parts, all of exactly the same size and shape, but it is important that every piece shall contain a lion and a crown. As she insists that the cuts can only be made along the lines dividing the squares, she is considerably perplexed to find out how it is to be done. Can you show her the way? There is only one possible method of cutting the stuff. [Illustration: +-+-+-+-+-+-+ | | | | | | | +-+-+-+-+-+-+ | |L|L|L| | | +-+-+-+-+-+-+ | | |C|C| | | +-+-+-+-+-+-+ | | |C|C| | | +-+-+-+-+-+-+ |L| | | | | | +-+-+-+-+-+-+ | | | | | | | +-+-+-+-+-+-+ ] 290.--BOARDS WITH AN ODD NUMBER OF SQUARES. We will here consider the question of those boards that contain an odd number of squares. We will suppose that the central square is first cut out, so as to leave an even number of squares for division. Now, it is obvious that a square three by three can only be divided in one way, as shown in Fig. 1. It will be seen that the pieces A and B are of the same size and shape, and that any other way of cutting would only produce the same shaped pieces, so remember that these variations are not counted as different ways. The puzzle I propose is to cut the board five by five (Fig. 2) into two pieces of the same size and shape in as many different ways as possible. I have shown in the illustration one way of doing it. How many different ways are there altogether? A piece which when turned over resembles another piece is not considered to be of a different shape. [Illustration: +---*---+---+ | H | | +---*===*---+ | HHHHH | +---*===*---+ | | H | +---+---*---+ Fig 1] [Illustration: +---+---+---+---+---+ | | | | | | *===*===*===*---+---+ | | | H | | +---+---*===*---+---+ | | HHHHH | | +---+---*===*---+---+ | | H | | | +---+---*===*===*===* | H | | | | +---*---+---+---+---+ Fig 2] 291.--THE GRAND LAMA'S PROBLEM. Once upon a time there was a Grand Lama who had a chessboard made of pure gold, magnificently engraved, and, of course, of great value. Every year a tournament was held at Lhassa among the priests, and whenever any one beat the Grand Lama it was considered a great honour, and his name was inscribed on the back of the board, and a costly jewel set in the particular square on which the checkmate had been given. After this sovereign pontiff had been defeated on four occasions he died--possibly of chagrin. [Illustration: +---+---+---+---+---+---+---+---+ | * | | | | | | | | +---+---+---+---+---+---+---+---+ | | * | | | | | | | +---+---+---+---+---+---+---+---+ | | | * | | | | | | +---+---+---+---+---+---+---+---+ | | | | * | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ ] Now the new Grand Lama was an inferior chess-player, and preferred other forms of innocent amusement, such as cutting off people's heads. So he discouraged chess as a degrading game, that did not improve either the mind or the morals, and abolished the tournament summarily. Then he sent for the four priests who had had the effrontery to play better than a Grand Lama, and addressed them as follows: "Miserable and heathenish men, calling yourselves priests! Know ye not that to lay claim to a capacity to do anything better than my predecessor is a capital offence? Take that chessboard and, before day dawns upon the torture chamber, cut it into four equal parts of the same shape, each containing sixteen perfect squares, with one of the gems in each part! If in this you fail, then shall other sports be devised for your special delectation. Go!" The four priests succeeded in their apparently hopeless task. Can you show how the board may be divided into four equal parts, each of exactly the same shape, by cuts along the lines dividing the squares, each part to contain one of the gems? 292.--THE ABBOT'S WINDOW. [Illustration] Once upon a time the Lord Abbot of St. Edmondsbury, in consequence of "devotions too strong for his head," fell sick and was unable to leave his bed. As he lay awake, tossing his head restlessly from side to side, the attentive monks noticed that something was disturbing his mind; but nobody dared ask what it might be, for the abbot was of a stern disposition, and never would brook inquisitiveness. Suddenly he called for Father John, and that venerable monk was soon at the bedside. "Father John," said the Abbot, "dost thou know that I came into this wicked world on a Christmas Even?" The monk nodded assent. "And have I not often told thee that, having been born on Christmas Even, I have no love for the things that are odd? Look there!" The Abbot pointed to the large dormitory window, of which I give a sketch. The monk looked, and was perplexed. "Dost thou not see that the sixty-four lights add up an even number vertically and horizontally, but that all the _diagonal_ lines, except fourteen are of a number that is odd? Why is this?" "Of a truth, my Lord Abbot, it is of the very nature of things, and cannot be changed." "Nay, but it _shall_ be changed. I command thee that certain of the lights be closed this day, so that every line shall have an even number of lights. See thou that this be done without delay, lest the cellars be locked up for a month and other grievous troubles befall thee." Father John was at his wits' end, but after consultation with one who was learned in strange mysteries, a way was found to satisfy the whim of the Lord Abbot. Which lights were blocked up, so that those which remained added up an even number in every line horizontally, vertically, and diagonally, while the least possible obstruction of light was caused? 293.--THE CHINESE CHESSBOARD. Into how large a number of different pieces may the chessboard be cut (by cuts along the lines only), no two pieces being exactly alike? Remember that the arrangement of black and white constitutes a difference. Thus, a single black square will be different from a single white square, a row of three containing two white squares will differ from a row of three containing two black, and so on. If two pieces cannot be placed on the table so as to be exactly alike, they count as different. And as the back of the board is plain, the pieces cannot be turned over. 294.--THE CHESSBOARD SENTENCE. [Illustration] I once set myself the amusing task of so dissecting an ordinary chessboard into letters of the alphabet that they would form a complete sentence. It will be seen from the illustration that the pieces assembled give the sentence, "CUT THY LIFE," with the stops between. The ideal sentence would, of course, have only one full stop, but that I did not succeed in obtaining. The sentence is an appeal to the transgressor to cut himself adrift from the evil life he is living. Can you fit these pieces together to form a perfect chessboard? STATICAL CHESS PUZZLES. "They also serve who only stand and wait." MILTON. 295.--THE EIGHT ROOKS. [Illustration: +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | R | R | R | R | R | R | R | R | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ FIG. 1.] [Illustration: +---+---+---+---+---+---+---+---+ | R | | | | | | | | +---+---+---+---+---+---+---+---+ | | R | | | | | | | +---+---+---+---+---+---+---+---+ | | | R | | | | | | +---+---+---+---+---+---+---+---+ | | | | R | | | | | +---+---+---+---+---+---+---+---+ | | | | | R | | | | +---+---+---+---+---+---+---+---+ | | | | | | R | | | +---+---+---+---+---+---+---+---+ | | | | | | | R | | +---+---+---+---+---+---+---+---+ | | | | | | | | R | +---+---+---+---+---+---+---+---+ FIG. 2.] It will be seen in the first diagram that every square on the board is either occupied or attacked by a rook, and that every rook is "guarded" (if they were alternately black and white rooks we should say "attacked") by another rook. Placing the eight rooks on any row or file obviously will have the same effect. In diagram 2 every square is again either occupied or attacked, but in this case every rook is unguarded. Now, in how many different ways can you so place the eight rooks on the board that every square shall be occupied or attacked and no rook ever guarded by another? I do not wish to go into the question of reversals and reflections on this occasion, so that placing the rooks on the other diagonal will count as different, and similarly with other repetitions obtained by turning the board round. 296.--THE FOUR LIONS. The puzzle is to find in how many different ways the four lions may be placed so that there shall never be more than one lion in any row or column. Mere reversals and reflections will not count as different. Thus, regarding the example given, if we place the lions in the other diagonal, it will be considered the same arrangement. For if you hold the second arrangement in front of a mirror or give it a quarter turn, you merely get the first arrangement. It is a simple little puzzle, but requires a certain amount of careful consideration. [Illustration +---+---+---+---+ | L | | | | +---+---+---+---+ | | L | | | +---+---+---+---+ | | | L | | +---+---+---+---+ | | | | L | +---+---+---+---+ ] 297.--BISHOPS--UNGUARDED. Place as few bishops as possible on an ordinary chessboard so that every square of the board shall be either occupied or attacked. It will be seen that the rook has more scope than the bishop: for wherever you place the former, it will always attack fourteen other squares; whereas the latter will attack seven, nine, eleven, or thirteen squares, according to the position of the diagonal on which it is placed. And it is well here to state that when we speak of "diagonals" in connection with the chessboard, we do not limit ourselves to the two long diagonals from corner to corner, but include all the shorter lines that are parallel to these. To prevent misunderstanding on future occasions, it will be well for the reader to note carefully this fact. 298.--BISHOPS--GUARDED. Now, how many bishops are necessary in order that every square shall be either occupied or attacked, and every bishop guarded by another bishop? And how may they be placed? 299.--BISHOPS IN CONVOCATION. [Illustration: +---+---+---+---+---+---+---+---+ | B | B | B | B | B | B | B | B | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | B | B | B | B | B | B | | +---+---+---+---+---+---+---+---+ ] The greatest number of bishops that can be placed at the same time on the chessboard, without any bishop attacking another, is fourteen. I show, in diagram, the simplest way of doing this. In fact, on a square chequered board of any number of squares the greatest number of bishops that can be placed without attack is always two less than twice the number of squares on the side. It is an interesting puzzle to discover in just how many different ways the fourteen bishops may be so placed without mutual attack. I shall give an exceedingly simple rule for determining the number of ways for a square chequered board of any number of squares. 300.--THE EIGHT QUEENS. [Illustration: +---+---+---+---+---+---+---+---+ | | | | ..Q | | | | +---+---+---+...+---+---+---+---+ | | ..Q.. | | | | | +---+...+---+---+---+---+---+---+ | Q.. | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | Q | | +---+---+---+---+---+---+---+---+ | | Q | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | ..Q | +---+---+---+---+---+---+...+---+ | | | | | ..Q.. | | +---+---+---+---+...+---+---+---+ | | | | Q.. | | | | +---+---+---+---+---+---+---+---+ ] The queen is by far the strongest piece on the chessboard. If you place her on one of the four squares in the centre of the board, she attacks no fewer than twenty-seven other squares; and if you try to hide her in a corner, she still attacks twenty-one squares. Eight queens may be placed on the board so that no queen attacks another, and it is an old puzzle (first proposed by Nauck in 1850, and it has quite a little literature of its own) to discover in just how many different ways this may be done. I show one way in the diagram, and there are in all twelve of these fundamentally different ways. These twelve produce ninety-two ways if we regard reversals and reflections as different. The diagram is in a way a symmetrical arrangement. If you turn the page upside down, it will reproduce itself exactly; but if you look at it with one of the other sides at the bottom, you get another way that is not identical. Then if you reflect these two ways in a mirror you get two more ways. Now, all the other eleven solutions are non-symmetrical, and therefore each of them may be presented in eight ways by these reversals and reflections. It will thus be seen why the twelve fundamentally different solutions produce only ninety-two arrangements, as I have said, and not ninety-six, as would happen if all twelve were non-symmetrical. It is well to have a clear understanding on the matter of reversals and reflections when dealing with puzzles on the chessboard. Can the reader place the eight queens on the board so that no queen shall attack another and so that no three queens shall be in a straight line in any oblique direction? Another glance at the diagram will show that this arrangement will not answer the conditions, for in the two directions indicated by the dotted lines there are three queens in a straight line. There is only one of the twelve fundamental ways that will solve the puzzle. Can you find it? 301.--THE EIGHT STARS. [Illustration: +---+---+---+---+---+---+---+---+ |///| | | | | | |///| +---+---+---+---+---+---+---+---+ | |///| | | | |///| * | +---+---+---+---+---+---+---+---+ | | |///| | |///| | | +---+---+---+---+---+---+---+---+ | | | |///|///| | | | +---+---+---+---+---+---+---+---+ | | | |///|///| | | | +---+---+---+---+---+---+---+---+ | | |///| | |///| | | +---+---+---+---+---+---+---+---+ | |///| | | | |///| | +---+---+---+---+---+---+---+---+ |///| | | | | | |///| +---+---+---+---+---+---+---+---+ ] The puzzle in this case is to place eight stars in the diagram so that no star shall be in line with another star horizontally, vertically, or diagonally. One star is already placed, and that must not be moved, so there are only seven for the reader now to place. But you must not place a star on any one of the shaded squares. There is only one way of solving this little puzzle. 302.--A PROBLEM IN MOSAICS. The art of producing pictures or designs by means of joining together pieces of hard substances, either naturally or artificially coloured, is of very great antiquity. It was certainly known in the time of the Pharaohs, and we find a reference in the Book of Esther to "a pavement of red, and blue, and white, and black marble." Some of this ancient work that has come down to us, especially some of the Roman mosaics, would seem to show clearly, even where design is not at first evident, that much thought was bestowed upon apparently disorderly arrangements. Where, for example, the work has been produced with a very limited number of colours, there are evidences of great ingenuity in preventing the same tints coming in close proximity. Lady readers who are familiar with the construction of patchwork quilts will know how desirable it is sometimes, when they are limited in the choice of material, to prevent pieces of the same stuff coming too near together. Now, this puzzle will apply equally to patchwork quilts or tesselated pavements. It will be seen from the diagram how a square piece of flooring may be paved with sixty-two square tiles of the eight colours violet, red, yellow, green, orange, purple, white, and blue (indicated by the initial letters), so that no tile is in line with a similarly coloured tile, vertically, horizontally, or diagonally. Sixty-four such tiles could not possibly be placed under these conditions, but the two shaded squares happen to be occupied by iron ventilators. [Illustration: +---+---+---+---+---+---+---+---+ | V | R | Y | G | O | P | W | B | +---+---+---+---+---+---+---+---+ | W | B | O | P | Y | G | V | R | +---+---*===*---+---*===*---+---+ | G | P H W H V | B H R H Y | O | +---+---*===*---+---*===*---+---+ | R | Y | B | O | G | V | P | W | +---+---+---+---+---+---+---+---+ | B | G | R | Y | P | W | O | V | +---+---+---+---+---+---+---+---+ | O | V | P | W | R | Y | B | G | +---+---+---+---+---+---+---+---+ | P | W | G | B | V | O | R | Y | +---+---+---+---+---+---+---+---+ |///| O | V | R | W | B | G |///| +---+---+---+---+---+---+---+---+ ] The puzzle is this. These two ventilators have to be removed to the positions indicated by the darkly bordered tiles, and two tiles placed in those bottom corner squares. Can you readjust the thirty-two tiles so that no two of the same colour shall still be in line? 303.--UNDER THE VEIL. [Illustration: +---+---+---+---+---+---+---+---+ | | | V | E | I | L | | | +---+---+---+---+---+---+---+---+ | | | I | L | V | E | | | +---+---+---+---+---+---+---+---+ | I | V | | | | | L | E | +---+---+---+---+---+---+---+---+ | L | E | | | | | I | V | +---+---+---+---+---+---+---+---+ | V | I | | | | | E | L | +---+---+---+---+---+---+---+---+ | E | L | | | | | V | I | +---+---+---+---+---+---+---+---+ | | | E | V | L | I | | | +---+---+---+---+---+---+---+---+ | | | L | I | E | V | | | +---+---+---+---+---+---+---+---+ ] If the reader will examine the above diagram, he will see that I have so placed eight V's, eight E's, eight I's, and eight L's in the diagram that no letter is in line with a similar one horizontally, vertically, or diagonally. Thus, no V is in line with another V, no E with another E, and so on. There are a great many different ways of arranging the letters under this condition. The puzzle is to find an arrangement that produces the greatest possible number of four-letter words, reading upwards and downwards, backwards and forwards, or diagonally. All repetitions count as different words, and the five variations that may be used are: VEIL, VILE, LEVI, LIVE, and EVIL. This will be made perfectly clear when I say that the above arrangement scores eight, because the top and bottom row both give VEIL; the second and seventh columns both give VEIL; and the two diagonals, starting from the L in the 5th row and E in the 8th row, both give LIVE and EVIL. There are therefore eight different readings of the words in all. This difficult word puzzle is given as an example of the use of chessboard analysis in solving such things. Only a person who is familiar with the "Eight Queens" problem could hope to solve it. 304.--BACHET'S SQUARE. One of the oldest card puzzles is by Claude Caspar Bachet de Méziriac, first published, I believe, in the 1624 edition of his work. Rearrange the sixteen court cards (including the aces) in a square so that in no row of four cards, horizontal, vertical, or diagonal, shall be found two cards of the same suit or the same value. This in itself is easy enough, but a point of the puzzle is to find in how many different ways this may be done. The eminent French mathematician A. Labosne, in his modern edition of Bachet, gives the answer incorrectly. And yet the puzzle is really quite easy. Any arrangement produces seven more by turning the square round and reflecting it in a mirror. These are counted as different by Bachet. Note "row of four cards," so that the only diagonals we have here to consider are the two long ones. 305.--THE THIRTY-SIX LETTER-BLOCKS. [Illustration] The illustration represents a box containing thirty-six letter-blocks. The puzzle is to rearrange these blocks so that no A shall be in a line vertically, horizontally, or diagonally with another A, no B with another B, no C with another C, and so on. You will find it impossible to get all the letters into the box under these conditions, but the point is to place as many as possible. Of course no letters other than those shown may be used. 306.--THE CROWDED CHESSBOARD. [Illustration] The puzzle is to rearrange the fifty-one pieces on the chessboard so that no queen shall attack another queen, no rook attack another rook, no bishop attack another bishop, and no knight attack another knight. No notice is to be taken of the intervention of pieces of another type from that under consideration--that is, two queens will be considered to attack one another although there may be, say, a rook, a bishop, and a knight between them. And so with the rooks and bishops. It is not difficult to dispose of each type of piece separately; the difficulty comes in when you have to find room for all the arrangements on the board simultaneously. 307.--THE COLOURED COUNTERS. [Illustration] The diagram represents twenty-five coloured counters, Red, Blue, Yellow, Orange, and Green (indicated by their initials), and there are five of each colour, numbered 1, 2, 3, 4, and 5. The problem is so to place them in a square that neither colour nor number shall be found repeated in any one of the five rows, five columns, and two diagonals. Can you so rearrange them? 308.--THE GENTLE ART OF STAMP-LICKING. The Insurance Act is a most prolific source of entertaining puzzles, particularly entertaining if you happen to be among the exempt. One's initiation into the gentle art of stamp-licking suggests the following little poser: If you have a card divided into sixteen spaces (4 × 4), and are provided with plenty of stamps of the values 1d., 2d., 3d., 4d., and 5d., what is the greatest value that you can stick on the card if the Chancellor of the Exchequer forbids you to place any stamp in a straight line (that is, horizontally, vertically, or diagonally) with another stamp of similar value? Of course, only one stamp can be affixed in a space. The reader will probably find, when he sees the solution, that, like the stamps themselves, he is licked He will most likely be twopence short of the maximum. A friend asked the Post Office how it was to be done; but they sent him to the Customs and Excise officer, who sent him to the Insurance Commissioners, who sent him to an approved society, who profanely sent him--but no matter. 309.--THE FORTY-NINE COUNTERS. [Illustration] Can you rearrange the above forty-nine counters in a square so that no letter, and also no number, shall be in line with a similar one, vertically, horizontally, or diagonally? Here I, of course, mean in the lines parallel with the diagonals, in the chessboard sense. 310.--THE THREE SHEEP. [Illustration] A farmer had three sheep and an arrangement of sixteen pens, divided off by hurdles in the manner indicated in the illustration. In how many different ways could he place those sheep, each in a separate pen, so that every pen should be either occupied or in line (horizontally, vertically, or diagonally) with at least one sheep? I have given one arrangement that fulfils the conditions. How many others can you find? Mere reversals and reflections must not be counted as different. The reader may regard the sheep as queens. The problem is then to place the three queens so that every square shall be either occupied or attacked by at least one queen--in the maximum number of different ways. 311.--THE FIVE DOGS PUZZLE. In 1863, C.F. de Jaenisch first discussed the "Five Queens Puzzle"--to place five queens on the chessboard so that every square shall be attacked or occupied--which was propounded by his friend, a "Mr. de R." Jaenisch showed that if no queen may attack another there are ninety-one different ways of placing the five queens, reversals and reflections not counting as different. If the queens may attack one another, I have recorded hundreds of ways, but it is not practicable to enumerate them exactly. [Illustration] The illustration is supposed to represent an arrangement of sixty-four kennels. It will be seen that five kennels each contain a dog, and on further examination it will be seen that every one of the sixty-four kennels is in a straight line with at least one dog--either horizontally, vertically, or diagonally. Take any kennel you like, and you will find that you can draw a straight line to a dog in one or other of the three ways mentioned. The puzzle is to replace the five dogs and discover in just how many different ways they may be placed in five kennels _in a straight row_, so that every kennel shall always be in line with at least one dog. Reversals and reflections are here counted as different. 312.--THE FIVE CRESCENTS OF BYZANTIUM. When Philip of Macedon, the father of Alexander the Great, found himself confronted with great difficulties in the siege of Byzantium, he set his men to undermine the walls. His desires, however, miscarried, for no sooner had the operations been begun than a crescent moon suddenly appeared in the heavens and discovered his plans to his adversaries. The Byzantines were naturally elated, and in order to show their gratitude they erected a statue to Diana, and the crescent became thenceforward a symbol of the state. In the temple that contained the statue was a square pavement composed of sixty-four large and costly tiles. These were all plain, with the exception of five, which bore the symbol of the crescent. These five were for occult reasons so placed that every tile should be watched over by (that is, in a straight line, vertically, horizontally, or diagonally with) at least one of the crescents. The arrangement adopted by the Byzantine architect was as follows:-- [Illustration] Now, to cover up one of these five crescents was a capital offence, the death being something very painful and lingering. But on a certain occasion of festivity it was necessary to lay down on this pavement a square carpet of the largest dimensions possible, and I have shown in the illustration by dark shading the largest dimensions that would be available. The puzzle is to show how the architect, if he had foreseen this question of the carpet, might have so arranged his five crescent tiles in accordance with the required conditions, and yet have allowed for the largest possible square carpet to be laid down without any one of the five crescent tiles being covered, or any portion of them. 313.--QUEENS AND BISHOP PUZZLE. It will be seen that every square of the board is either occupied or attacked. The puzzle is to substitute a bishop for the rook on the same square, and then place the four queens on other squares so that every square shall again be either occupied or attacked. [Illustration] 314.--THE SOUTHERN CROSS. [Illustration] In the above illustration we have five Planets and eighty-one Fixed Stars, five of the latter being hidden by the Planets. It will be found that every Star, with the exception of the ten that have a black spot in their centres, is in a straight line, vertically, horizontally, or diagonally, with at least one of the Planets. The puzzle is so to rearrange the Planets that all the Stars shall be in line with one or more of them. In rearranging the Planets, each of the five may be moved once in a straight line, in either of the three directions mentioned. They will, of course, obscure five other Stars in place of those at present covered. 315.--THE HAT-PEG PUZZLE. Here is a five-queen puzzle that I gave in a fanciful dress in 1897. As the queens were there represented as hats on sixty-four pegs, I will keep to the title, "The Hat-Peg Puzzle." It will be seen that every square is occupied or attacked. The puzzle is to remove one queen to a different square so that still every square is occupied or attacked, then move a second queen under a similar condition, then a third queen, and finally a fourth queen. After the fourth move every square must be attacked or occupied, but no queen must then attack another. Of course, the moves need not be "queen moves;" you can move a queen to any part of the board. [Illustration] 316.--THE AMAZONS. [Illustration] This puzzle is based on one by Captain Turton. Remove three of the queens to other squares so that there shall be eleven squares on the board that are not attacked. The removal of the three queens need not be by "queen moves." You may take them up and place them anywhere. There is only one solution. 317.--A PUZZLE WITH PAWNS. Place two pawns in the middle of the chessboard, one at Q 4 and the other at K 5. Now, place the remaining fourteen pawns (sixteen in all) so that no three shall be in a straight line in any possible direction. Note that I purposely do not say queens, because by the words "any possible direction" I go beyond attacks on diagonals. The pawns must be regarded as mere points in space--at the centres of the squares. See dotted lines in the case of No. 300, "The Eight Queens." 318.--LION-HUNTING. [Illustration] My friend Captain Potham Hall, the renowned hunter of big game, says there is nothing more exhilarating than a brush with a herd--a pack--a team--a flock--a swarm (it has taken me a full quarter of an hour to recall the right word, but I have it at last)--a _pride_ of lions. Why a number of lions are called a "pride," a number of whales a "school," and a number of foxes a "skulk" are mysteries of philology into which I will not enter. Well, the captain says that if a spirited lion crosses your path in the desert it becomes lively, for the lion has generally been looking for the man just as much as the man has sought the king of the forest. And yet when they meet they always quarrel and fight it out. A little contemplation of this unfortunate and long-standing feud between two estimable families has led me to figure out a few calculations as to the probability of the man and the lion crossing one another's path in the jungle. In all these cases one has to start on certain more or less arbitrary assumptions. That is why in the above illustration I have thought it necessary to represent the paths in the desert with such rigid regularity. Though the captain assures me that the tracks of the lions usually run much in this way, I have doubts. The puzzle is simply to find out in how many different ways the man and the lion may be placed on two different spots that are not on the same path. By "paths" it must be understood that I only refer to the ruled lines. Thus, with the exception of the four corner spots, each combatant is always on two paths and no more. It will be seen that there is a lot of scope for evading one another in the desert, which is just what one has always understood. 319.--THE KNIGHT-GUARDS. [Illustration] The knight is the irresponsible low comedian of the chessboard. "He is a very uncertain, sneaking, and demoralizing rascal," says an American writer. "He can only move two squares, but makes up in the quality of his locomotion for its quantity, for he can spring one square sideways and one forward simultaneously, like a cat; can stand on one leg in the middle of the board and jump to any one of eight squares he chooses; can get on one side of a fence and blackguard three or four men on the other; has an objectionable way of inserting himself in safe places where he can scare the king and compel him to move, and then gobble a queen. For pure cussedness the knight has no equal, and when you chase him out of one hole he skips into another." Attempts have been made over and over again to obtain a short, simple, and exact definition of the move of the knight--without success. It really consists in moving one square like a rook, and then another square like a bishop--the two operations being done in one leap, so that it does not matter whether the first square passed over is occupied by another piece or not. It is, in fact, the only leaping move in chess. But difficult as it is to define, a child can learn it by inspection in a few minutes. I have shown in the diagram how twelve knights (the fewest possible that will perform the feat) may be placed on the chessboard so that every square is either occupied or attacked by a knight. Examine every square in turn, and you will find that this is so. Now, the puzzle in this case is to discover what is the smallest possible number of knights that is required in order that every square shall be either occupied or attacked, and every knight protected by another knight. And how would you arrange them? It will be found that of the twelve shown in the diagram only four are thus protected by being a knight's move from another knight. THE GUARDED CHESSBOARD. On an ordinary chessboard, 8 by 8, every square can be guarded--that is, either occupied or attacked--by 5 queens, the fewest possible. There are exactly 91 fundamentally different arrangements in which no queen attacks another queen. If every queen must attack (or be protected by) another queen, there are at fewest 41 arrangements, and I have recorded some 150 ways in which some of the queens are attacked and some not, but this last case is very difficult to enumerate exactly. On an ordinary chessboard every square can be guarded by 8 rooks (the fewest possible) in 40,320 ways, if no rook may attack another rook, but it is not known how many of these are fundamentally different. (See solution to No. 295, "The Eight Rooks.") I have not enumerated the ways in which every rook shall be protected by another rook. On an ordinary chessboard every square can be guarded by 8 bishops (the fewest possible), if no bishop may attack another bishop. Ten bishops are necessary if every bishop is to be protected. (See Nos. 297 and 298, "Bishops unguarded" and "Bishops guarded.") On an ordinary chessboard every square can be guarded by 12 knights if all but 4 are unprotected. But if every knight must be protected, 14 are necessary. (See No. 319, "The Knight-Guards.") Dealing with the queen on n² boards generally, where n is less than 8, the following results will be of interest:-- 1 queen guards 2² board in 1 fundamental way. 1 queen guards 3² board in 1 fundamental way. 2 queens guard 4² board in 3 fundamental ways (protected). 3 queens guard 4² board in 2 fundamental ways (not protected). 3 queens guard 5² board in 37 fundamental ways (protected). 3 queens guard 5² board in 2 fundamental ways (not protected). 3 queens guard 6² board in 1 fundamental way (protected). 4 queens guard 6² board in 17 fundamental ways (not protected). 4 queens guard 7² board in 5 fundamental ways (protected). 4 queens guard 7² board in 1 fundamental way (not protected). NON-ATTACKING CHESSBOARD ARRANGEMENTS. We know that n queens may always be placed on a square board of n² squares (if n be greater than 3) without any queen attacking another queen. But no general formula for enumerating the number of different ways in which it may be done has yet been discovered; probably it is undiscoverable. The known results are as follows:-- Where n = 4 there is 1 fundamental solution and 2 in all. Where n = 5 there are 2 fundamental solutions and 10 in all. Where n = 6 there is 1 fundamental solution and 4 in all. Where n = 7 there are 6 fundamental solutions and 40 in all. Where n = 8 there are 12 fundamental solutions and 92 in all. Where n = 9 there are 46 fundamental solutions. Where n = 10 there are 92 fundamental solutions. Where n = 11 there are 341 fundamental solutions. Obviously n rooks may be placed without attack on an n² board in n! ways, but how many of these are fundamentally different I have only worked out in the four cases where n equals 2, 3, 4, and 5. The answers here are respectively 1, 2, 7, and 23. (See No. 296, "The Four Lions.") We can place 2n-2 bishops on an n² board in 2^{n} ways. (See No. 299, "Bishops in Convocation.") For boards containing 2, 3, 4, 5, 6, 7, 8 squares, on a side there are respectively 1, 2, 3, 6, 10, 20, 36 fundamentally different arrangements. Where n is odd there are 2^{½(n-1)} such arrangements, each giving 4 by reversals and reflections, and 2^{n-3} - 2^{½(n-3)} giving 8. Where n is even there are 2^{½(n-2)}, each giving 4 by reversals and reflections, and 2^{n-3} - 2^{½(n-4)}, each giving 8. We can place ½(n²+1) knights on an n² board without attack, when n is odd, in 1 fundamental way; and ½n² knights on an n² board, when n is even, in 1 fundamental way. In the first case we place all the knights on the same colour as the central square; in the second case we place them all on black, or all on white, squares. THE TWO PIECES PROBLEM. On a board of n² squares, two queens, two rooks, two bishops, or two knights can always be placed, irrespective of attack or not, in ½(n^{4} - n²) ways. The following formulæ will show in how many of these ways the two pieces may be placed with attack and without:-- With Attack. Without Attack. 2 Queens 5n³ - 6n² + n 3n^{4} - 10n³ + 9n² - 2n ------------------- ------------------------------ 3 6 2 Rooks n³ - n² n^{4} - 2n³ + n² ---------------------- 2 2 Bishops 4n³ - 6n² + 2n 3n^{4} - 4n³ + 3n² - 2n -------------------- ----------------------------- 6 6 2 Knights 4n² - 12n + 8 n^{4} - 9n² + 24n -------------------- 2 (See No. 318, " Lion Hunting.") DYNAMICAL CHESS PUZZLES. "Push on--keep moving." THOS. MORTON: _Cure for the Heartache_. 320.--THE ROOK'S TOUR. [Illustration: +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | R | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ ] The puzzle is to move the single rook over the whole board, so that it shall visit every square of the board once, and only once, and end its tour on the square from which it starts. You have to do this in as few moves as possible, and unless you are very careful you will take just one move too many. Of course, a square is regarded equally as "visited" whether you merely pass over it or make it a stopping-place, and we will not quibble over the point whether the original square is actually visited twice. We will assume that it is not. 321.--THE ROOK'S JOURNEY. This puzzle I call "The Rook's Journey," because the word "tour" (derived from a turner's wheel) implies that we return to the point from which we set out, and we do not do this in the present case. We should not be satisfied with a personally conducted holiday tour that ended by leaving us, say, in the middle of the Sahara. The rook here makes twenty-one moves, in the course of which journey it visits every square of the board once and only once, stopping at the square marked 10 at the end of its tenth move, and ending at the square marked 21. Two consecutive moves cannot be made in the same direction--that is to say, you must make a turn after every move. [Illustration: +---+---+---+---+---+---+---+---+ | | | | | | | | R | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | 21| | 10| | | | | +---+---+---+---+---+---+---+---+ ] 322.--THE LANGUISHING MAIDEN. [Illustration: --+-----+-----+-----+-----+-----+-----+-----+ | | | | | | | | | Kt | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | M | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ ] A wicked baron in the good old days imprisoned an innocent maiden in one of the deepest dungeons beneath the castle moat. It will be seen from our illustration that there were sixty-three cells in the dungeon, all connected by open doors, and the maiden was chained in the cell in which she is shown. Now, a valiant knight, who loved the damsel, succeeded in rescuing her from the enemy. Having gained an entrance to the dungeon at the point where he is seen, he succeeded in reaching the maiden after entering every cell once and only once. Take your pencil and try to trace out such a route. When you have succeeded, then try to discover a route in twenty-two straight paths through the cells. It can be done in this number without entering any cell a second time. 323.--A DUNGEON PUZZLE. [Illustration: +-----+-----+-----+-----+-----+-----+-----+-----+ | | | | | | | | | | ............. ....... ............. | | . | | . | . | . | . | | . | +--.--+-- --+--.--+--.--+--.--+--.--+-- --+--.--+ | . | | . | . | . | . | | . | | ....... ....... ....... ....... | | | . | | | | | . | | +-- --+--.--+-- --+-- --+-- --+-- --+--.--+-- --+ | | . | | | | | . | | | ....... ....... ....... ....... | | . | | . | . | . | . | | . | +--.--+-- --+--.--+--.--+--.--+--.--+-- --+--.--+ | . | | . | . | . | . | | . | | ............. ....... . ....... | | | | | | | . | . | | +-- --+-- --+-- --+-- --+-- --+--.--+--.--+-- --+ | | | | | | . | . | | | ............. ....... . ....... | | . | | . | . | . | . | | . | +--.--+-- --+--.--+--.--+--.--+--.--+-- --+--.--+ | . | | . | . | . | . | | . | | ....... ....... ....... ....... | | | . | | | | | . | | +-- --+--.--+-- --+-- --+-- --+-- --+--.--+-- --+ | | . | | | | | . | | | ....... ....... ....... ....... | | . | | . | . | . | . | | . | +--.--+-- --+--.--+--.--+--.--+--.--+-- --+--.--+ | . | | . | . | . | . | | . | | ............. . P ............. | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ ] A French prisoner, for his sins (or other people's), was confined in an underground dungeon containing sixty-four cells, all communicating with open doorways, as shown in our illustration. In order to reduce the tedium of his restricted life, he set himself various puzzles, and this is one of them. Starting from the cell in which he is shown, how could he visit every cell once, and only once, and make as many turnings as possible? His first attempt is shown by the dotted track. It will be found that there are as many as fifty-five straight lines in his path, but after many attempts he improved upon this. Can you get more than fifty-five? You may end your path in any cell you like. Try the puzzle with a pencil on chessboard diagrams, or you may regard them as rooks' moves on a board. 324.--THE LION AND THE MAN. In a public place in Rome there once stood a prison divided into sixty-four cells, all open to the sky and all communicating with one another, as shown in the illustration. The sports that here took place were watched from a high tower. The favourite game was to place a Christian in one corner cell and a lion in the diagonally opposite corner and then leave them with all the inner doors open. The consequent effect was sometimes most laughable. On one occasion the man was given a sword. He was no coward, and was as anxious to find the lion as the lion undoubtedly was to find him. [Illustration: +-----+-----+-----+-----+-----+-----+-----+-----+ | | | | | | | | | | L | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | C | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ ] The man visited every cell once and only once in the fewest possible straight lines until he reached the lion's cell. The lion, curiously enough, also visited every cell once and only once in the fewest possible straight lines until he finally reached the man's cell. They started together and went at the same speed; yet, although they occasionally got glimpses of one another, they never once met. The puzzle is to show the route that each happened to take. 325.--AN EPISCOPAL VISITATION. The white squares on the chessboard represent the parishes of a diocese. Place the bishop on any square you like, and so contrive that (using the ordinary bishop's move of chess) he shall visit every one of his parishes in the fewest possible moves. Of course, all the parishes passed through on any move are regarded as "visited." You can visit any squares more than once, but you are not allowed to move twice between the same two adjoining squares. What are the fewest possible moves? The bishop need not end his visitation at the parish from which he first set out. 326.--A NEW COUNTER PUZZLE. Here is a new puzzle with moving counters, or coins, that at first glance looks as if it must be absurdly simple. But it will be found quite a little perplexity. I give it in this place for a reason that I will explain when we come to the next puzzle. Copy the simple diagram, enlarged, on a sheet of paper; then place two white counters on the points 1 and 2, and two red counters on 9 and 10, The puzzle is to make the red and white change places. You may move the counters one at a time in any order you like, along the lines from point to point, with the only restriction that a red and a white counter may never stand at once on the same straight line. Thus the first move can only be from 1 or 2 to 3, or from 9 or 10 to 7. [Illustration: 4 8 / \ / \ 2 6 10 \ / \ / 3 7 / \ / \ 1 5 9 ] 327.--A NEW BISHOP'S PUZZLE. [Illustration: +---+---+---+---+ | b | b | b | b | +---+---+---+---+ | | | | | +---+---+---+---+ | | | | | +---+---+---+---+ | B | B | B | B | +---+---+---+---+ ] This is quite a fascinating little puzzle. Place eight bishops (four black and four white) on the reduced chessboard, as shown in the illustration. The problem is to make the black bishops change places with the white ones, no bishop ever attacking another of the opposite colour. They must move alternately--first a white, then a black, then a white, and so on. When you have succeeded in doing it at all, try to find the fewest possible moves. If you leave out the bishops standing on black squares, and only play on the white squares, you will discover my last puzzle turned on its side. 328.--THE QUEEN'S TOUR. The puzzle of making a complete tour of the chessboard with the queen in the fewest possible moves (in which squares may be visited more than once) was first given by the late Sam Loyd in his _Chess Strategy_. But the solution shown below is the one he gave in _American Chess-Nuts_ in 1868. I have recorded at least six different solutions in the minimum number of moves--fourteen--but this one is the best of all, for reasons I will explain. [Illustration: +---+---+---+---+---+---+---+---+ | | | | | | | | | | ............................. | | . | | | | | | | . | +-.-+---+---+---+---+---+---+-.-+ | . | | | | | | | . | | . | ..........................| | . | .| | | | | | . | +-.-+---.---+---+---+---+---+..-+ | . | |. | | | | . . | | . | ................. | .| . | | . | .| .| | |. | . | . | +-.-+---.---.---+---.---+.--+-.-+ | . | |. |. | .| . | . | | . | . | . | . | . | .| . | . | | . | ..| .| .|. | . |.. | . | +-.-+-.-.---.---.---+.--.-.-+-.-+ | . | . |. |. .|. . .| . | . | | . | . | . | . | ..| . | . | . | | . | . | .|. .| ..|. | . | . | +-.-+-.-+---.---..--.---+-.-+-.-+ | . | . | .|. .. .|. | . | . | | . | . | . | ..| . | . | . | . | | . | . |. | ..|. .| .| . | . | +-.-+-.-.---+.--.---.---.-.-+-.-+ | . | ..| . .|. |. |.. | . | | . | . | .| . | . | . | . | . | | . |.. | . |. | .| .| ..| . | +-.-.-.-+.--.---+---.---.-.-.-.-+ | ..| . . .| | |. |.. |.. | | . | ..| ............. | . | . | | | . | | | | | | | +---+---+---+---+---+---+---+---+ ] If you will look at the lettered square you will understand that there are only ten really differently placed squares on a chessboard--those enclosed by a dark line--all the others are mere reversals or reflections. For example, every A is a corner square, and every J a central square. Consequently, as the solution shown has a turning-point at the enclosed D square, we can obtain a solution starting from and ending at any square marked D--by just turning the board about. Now, this scheme will give you a tour starting from any A, B, C, D, E, F, or H, while no other route that I know can be adapted to more than five different starting-points. There is no Queen's Tour in fourteen moves (remember a tour must be re-entrant) that may start from a G, I, or J. But we can have a non-re-entrant path over the whole board in fourteen moves, starting from any given square. Hence the following puzzle:-- [Illustration: +---+---+---+---*---+---+---+---+ | A | B | C | G " G | C | B | A | *===*---+---+---*---+---+---+---+ | B " D | E | H " H | E | D | B | +---*===*---+---*---+---+---+---+ | C | E " F | I " I | F | E | C | +---+---*===*---*---+---+---+---+ | G | H | I " J " J | I | H | G | +---+---+---*===*---+---+---+---+ | G | H | I | J | J | I | H | G | +---+---+---+---+---+---+---+---+ | C | E | F | I | I | F | E | C | +---+---+---+---+---+---+---+---+ | B | D | E | H | H | E | D | B | +---+---+---+---+---+---+---+---+ | A | B | C | G | G | C | B | A | +---+---+---+---+---+---+---+---+ ] Start from the J in the enclosed part of the lettered diagram and visit every square of the board in fourteen moves, ending wherever you like. 329.--THE STAR PUZZLE. [Illustration: +---+---+---+---+---+---+---+---+ | * | * | * | * | * | * | * | * | +---+---+---+---+---+---+---+---+ | * | * | * | * | * | * | * | * | +---+---+---+---+---+---+---+---+ | * | * | * | * | * | * | * | * | +---+---+---+---+---+---+---+---+ | * | * | ¤ | * | * | * | * | * | +---+---+---+---+---+---+---+---+ | * | * | * | ¤ | * | * | * | * | +---+---+---+---+---+---+---+---+ | * | * | * | * | * | * | * | * | +---+---+---+---+---+---+---+---+ | * | * | * | * | * | * | * | * | +---+---+---+---+---+---+---+---+ | * | * | * | * | * | * | * | * | +---+---+---+---+---+---+---+---+ ] Put the point of your pencil on one of the white stars and (without ever lifting your pencil from the paper) strike out all the stars in fourteen continuous straight strokes, ending at the second white star. Your straight strokes may be in any direction you like, only every turning must be made on a star. There is no objection to striking out any star more than once. In this case, where both your starting and ending squares are fixed inconveniently, you cannot obtain a solution by breaking a Queen's Tour, or in any other way by queen moves alone. But you are allowed to use oblique straight lines--such as from the upper white star direct to a corner star. 330.--THE YACHT RACE. Now then, ye land-lubbers, hoist your baby-jib-topsails, break out your spinnakers, ease off your balloon sheets, and get your head-sails set! Our race consists in starting from the point at which the yacht is lying in the illustration and touching every one of the sixty-four buoys in fourteen straight courses, returning in the final tack to the buoy from which we start. The seventh course must finish at the buoy from which a flag is flying. This puzzle will call for a lot of skilful seamanship on account of the sharp angles at which it will occasionally be necessary to tack. The point of a lead pencil and a good nautical eye are all the outfit that we require. [Illustration] This is difficult, because of the condition as to the flag-buoy, and because it is a re-entrant tour. But again we are allowed those oblique lines. 331.--THE SCIENTIFIC SKATER. [Illustration] It will be seen that this skater has marked on the ice sixty-four points or stars, and he proposes to start _from his present position_ near the corner and enter every one of the points in fourteen straight lines. How will he do it? Of course there is no objection to his passing over any point more than once, but his last straight stroke must bring him back to the position from which he started. It is merely a matter of taking your pencil and starting from the spot on which the skater's foot is at present resting, and striking out all the stars in fourteen continuous straight lines, returning to the point from which you set out. 332.--THE FORTY-NINE STARS. [Illustration] The puzzle in this case is simply to take your pencil and, starting from one black star, strike out all the stars in twelve straight strokes, ending at the other black star. It will be seen that the attempt shown in the illustration requires fifteen strokes. Can you do it in twelve? Every turning must be made on a star, and the lines must be parallel to the sides and diagonals of the square, as shown. In this case we are dealing with a chessboard of reduced dimensions, but only queen moves (without going outside the boundary as in the last case) are required. 333.--THE QUEEN'S JOURNEY. [Illustration] Place the queen on her own square, as shown in the illustration, and then try to discover the greatest distance that she can travel over the board in five queen's moves without passing over any square a second time. Mark the queen's path on the board, and note carefully also that she must never cross her own track. It seems simple enough, but the reader may find that he has tripped. 334.--ST. GEORGE AND THE DRAGON. [Illustration] Here is a little puzzle on a reduced chessboard of forty-nine squares. St. George wishes to kill the dragon. Killing dragons was a well-known pastime of his, and, being a knight, it was only natural that he should desire to perform the feat in a series of knight's moves. Can you show how, starting from that central square, he may visit once, and only once, every square of the board in a chain of chess knight's moves, and end by capturing the dragon on his last move? Of course a variety of different ways are open to him, so try to discover a route that forms some pretty design when you have marked each successive leap by a straight line from square to square. 335.--FARMER LAWRENCE'S CORNFIELDS. One of the most beautiful districts within easy distance of London for a summer ramble is that part of Buckinghamshire known as the Valley of the Chess--at least, it was a few years ago, before it was discovered by the speculative builder. At the beginning of the present century there lived, not far from Latimers, a worthy but eccentric farmer named Lawrence. One of his queer notions was that every person who lived near the banks of the river Chess ought to be in some way acquainted with the noble game of the same name, and in order to impress this fact on his men and his neighbours he adopted at times strange terminology. For example, when one of his ewes presented him with a lamb, he would say that it had "queened a pawn"; when he put up a new barn against the highway, he called it "castling on the king's side"; and when he sent a man with a gun to keep his neighbour's birds off his fields, he spoke of it as "attacking his opponent's rooks." Everybody in the neighbourhood used to be amused at Farmer Lawrence's little jokes, and one boy (the wag of the village) who got his ears pulled by the old gentleman for stealing his "chestnuts" went so far as to call him "a silly old chess-protector!" One year he had a large square field divided into forty-nine square plots, as shown in the illustration. The white squares were sown with wheat and the black squares with barley. When the harvest time came round he gave orders that his men were first to cut the corn in the patch marked 1, and that each successive cutting should be exactly a knight's move from the last one, the thirteenth cutting being in the patch marked 13, the twenty-fifth in the patch marked 25, the thirty-seventh in the one marked 37, and the last, or forty-ninth cutting, in the patch marked 49. This was too much for poor Hodge, and each day Farmer Lawrence had to go down to the field and show which piece had to be operated upon. But the problem will perhaps present no difficulty to my readers. [Illustration] 336.--THE GREYHOUND PUZZLE. In this puzzle the twenty kennels do not communicate with one another by doors, but are divided off by a low wall. The solitary occupant is the greyhound which lives in the kennel in the top left-hand corner. When he is allowed his liberty he has to obtain it by visiting every kennel once and only once in a series of knight's moves, ending at the bottom right-hand corner, which is open to the world. The lines in the above diagram show one solution. The puzzle is to discover in how many different ways the greyhound may thus make his exit from his corner kennel. [Illustration] 337.--THE FOUR KANGAROOS. [Illustration] In introducing a little Commonwealth problem, I must first explain that the diagram represents the sixty-four fields, all properly fenced off from one another, of an Australian settlement, though I need hardly say that our kith and kin "down under" always _do_ set out their land in this methodical and exact manner. It will be seen that in every one of the four corners is a kangaroo. Why kangaroos have a marked preference for corner plots has never been satisfactorily explained, and it would be out of place to discuss the point here. I should also add that kangaroos, as is well known, always leap in what we call "knight's moves." In fact, chess players would probably have adopted the better term "kangaroo's move" had not chess been invented before kangaroos. The puzzle is simply this. One morning each kangaroo went for his morning hop, and in sixteen consecutive knight's leaps visited just fifteen different fields and jumped back to his corner. No field was visited by more than one of the kangaroos. The diagram shows how they arranged matters. What you are asked to do is to show how they might have performed the feat without any kangaroo ever crossing the horizontal line in the middle of the square that divides the board into two equal parts. 338.--THE BOARD IN COMPARTMENTS. [Illustration] We cannot divide the ordinary chessboard into four equal square compartments, and describe a complete tour, or even path, in each compartment. But we may divide it into four compartments, as in the illustration, two containing each twenty squares, and the other two each twelve squares, and so obtain an interesting puzzle. You are asked to describe a complete re-entrant tour on this board, starting where you like, but visiting every square in each successive compartment before passing into another one, and making the final leap back to the square from which the knight set out. It is not difficult, but will be found very entertaining and not uninstructive. Whether a re-entrant "tour" or a complete knight's "path" is possible or not on a rectangular board of given dimensions depends not only on its dimensions, but also on its shape. A tour is obviously not possible on a board containing an odd number of cells, such as 5 by 5 or 7 by 7, for this reason: Every successive leap of the knight must be from a white square to a black and a black to a white alternately. But if there be an odd number of cells or squares there must be one more square of one colour than of the other, therefore the path must begin from a square of the colour that is in excess, and end on a similar colour, and as a knight's move from one colour to a similar colour is impossible the path cannot be re-entrant. But a perfect tour may be made on a rectangular board of any dimensions provided the number of squares be even, and that the number of squares on one side be not less than 6 and on the other not less than 5. In other words, the smallest rectangular board on which a re-entrant tour is possible is one that is 6 by 5. A complete knight's path (not re-entrant) over all the squares of a board is never possible if there be only two squares on one side; nor is it possible on a square board of smaller dimensions than 5 by 5. So that on a board 4 by 4 we can neither describe a knight's tour nor a complete knight's path; we must leave one square unvisited. Yet on a board 4 by 3 (containing four squares fewer) a complete path may be described in sixteen different ways. It may interest the reader to discover all these. Every path that starts from and ends at different squares is here counted as a different solution, and even reverse routes are called different. 339.--THE FOUR KNIGHTS' TOURS. [Illustration] I will repeat that if a chessboard be cut into four equal parts, as indicated by the dark lines in the illustration, it is not possible to perform a knight's tour, either re-entrant or not, on one of the parts. The best re-entrant attempt is shown, in which each knight has to trespass twice on other parts. The puzzle is to cut the board differently into four parts, each of the same size and shape, so that a re-entrant knight's tour may be made on each part. Cuts along the dotted lines will not do, as the four central squares of the board would be either detached or hanging on by a mere thread. 340.--THE CUBIC KNIGHT'S TOUR. Some few years ago I happened to read somewhere that Abnit Vandermonde, a clever mathematician, who was born in 1736 and died in 1793, had devoted a good deal of study to the question of knight's tours. Beyond what may be gathered from a few fragmentary references, I am not aware of the exact nature or results of his investigations, but one thing attracted my attention, and that was the statement that he had proposed the question of a tour of the knight over the six surfaces of a cube, each surface being a chessboard. Whether he obtained a solution or not I do not know, but I have never seen one published. So I at once set to work to master this interesting problem. Perhaps the reader may like to attempt it. 341.--THE FOUR FROGS. [Illustration] In the illustration we have eight toadstools, with white frogs on 1 and 3 and black frogs on 6 and 8. The puzzle is to move one frog at a time, in any order, along one of the straight lines from toadstool to toadstool, until they have exchanged places, the white frogs being left on 6 and 8 and the black ones on 1 and 3. If you use four counters on a simple diagram, you will find this quite easy, but it is a little more puzzling to do it in only seven plays, any number of successive moves by one frog counting as one play. Of course, more than one frog cannot be on a toadstool at the same time. 342.--THE MANDARIN'S PUZZLE. The following puzzle has an added interest from the circumstance that a correct solution of it secured for a certain young Chinaman the hand of his charming bride. The wealthiest mandarin within a radius of a hundred miles of Peking was Hi-Chum-Chop, and his beautiful daughter, Peeky-Bo, had innumerable admirers. One of her most ardent lovers was Winky-Hi, and when he asked the old mandarin for his consent to their marriage, Hi-Chum-Chop presented him with the following puzzle and promised his consent if the youth brought him the correct answer within a week. Winky-Hi, following a habit which obtains among certain solvers to this day, gave it to all his friends, and when he had compared their solutions he handed in the best one as his own. Luckily it was quite right. The mandarin thereupon fulfilled his promise. The fatted pup was killed for the wedding feast, and when Hi-Chum-Chop passed Winky-Hi the liver wing all present knew that it was a token of eternal goodwill, in accordance with Chinese custom from time immemorial. The mandarin had a table divided into twenty-five squares, as shown in the diagram. On each of twenty-four of these squares was placed a numbered counter, just as I have indicated. The puzzle is to get the counters in numerical order by moving them one at a time in what we call "knight's moves." Counter 1 should be where 16 is, 2 where 11 is, 4 where 13 now is, and so on. It will be seen that all the counters on shaded squares are in their proper positions. Of course, two counters may never be on a square at the same time. Can you perform the feat in the fewest possible moves? [Illustration] In order to make the manner of moving perfectly clear I will point out that the first knight's move can only be made by 1 or by 2 or by 10. Supposing 1 moves, then the next move must be by 23, 4, 8, or 21. As there is never more than one square vacant, the order in which the counters move may be written out as follows: 1--21--14--18--22, etc. A rough diagram should be made on a larger scale for practice, and numbered counters or pieces of cardboard used. 343.--EXERCISE FOR PRISONERS. The following is the plan of the north wing of a certain gaol, showing the sixteen cells all communicating by open doorways. Fifteen prisoners were numbered and arranged in the cells as shown. They were allowed to change their cells as much as they liked, but if two prisoners were ever in the same cell together there was a severe punishment promised them. [Illustration] Now, in order to reduce their growing obesity, and to combine physical exercise with mental recreation, the prisoners decided, on the suggestion of one of their number who was interested in knight's tours, to try to form themselves into a perfect knight's path without breaking the prison regulations, and leaving the bottom right-hand corner cell vacant, as originally. The joke of the matter is that the arrangement at which they arrived was as follows:-- 8 3 12 1 11 14 9 6 4 7 2 13 15 10 5 The warders failed to detect the important fact that the men could not possibly get into this position without two of them having been at some time in the same cell together. Make the attempt with counters on a ruled diagram, and you will find that this is so. Otherwise the solution is correct enough, each member being, as required, a knight's move from the preceding number, and the original corner cell vacant. The puzzle is to start with the men placed as in the illustration and show how it might have been done in the fewest moves, while giving a complete rest to as many prisoners as possible. As there is never more than one vacant cell for a man to enter, it is only necessary to write down the numbers of the men in the order in which they move. It is clear that very few men can be left throughout in their cells undisturbed, but I will leave the solver to discover just how many, as this is a very essential part of the puzzle. 344.--THE KENNEL PUZZLE. [Illustration] A man has twenty-five dog kennels all communicating with each other by doorways, as shown in the illustration. He wishes to arrange his twenty dogs so that they shall form a knight's string from dog No. 1 to dog No. 20, the bottom row of five kennels to be left empty, as at present. This is to be done by moving one dog at a time into a vacant kennel. The dogs are well trained to obedience, and may be trusted to remain in the kennels in which they are placed, except that if two are placed in the same kennel together they will fight it out to the death. How is the puzzle to be solved in the fewest possible moves without two dogs ever being together? 345.--THE TWO PAWNS. [Illustration] Here is a neat little puzzle in counting. In how many different ways may the two pawns advance to the eighth square? You may move them in any order you like to form a different sequence. For example, you may move the Q R P (one or two squares) first, or the K R P first, or one pawn as far as you like before touching the other. Any sequence is permissible, only in this puzzle as soon as a pawn reaches the eighth square it is dead, and remains there unconverted. Can you count the number of different sequences? At first it will strike you as being very difficult, but I will show that it is really quite simple when properly attacked. VARIOUS CHESS PUZZLES. "Chesse-play is a good and wittie exercise of the minde for some kinde of men." Burton's _Anatomy of Melancholy_. 346.--SETTING THE BOARD. I have a single chessboard and a single set of chessmen. In how many different ways may the men be correctly set up for the beginning of a game? I find that most people slip at a particular point in making the calculation. 347.--COUNTING THE RECTANGLES. Can you say correctly just how many squares and other rectangles the chessboard contains? In other words, in how great a number of different ways is it possible to indicate a square or other rectangle enclosed by lines that separate the squares of the board? 348.--THE ROOKERY. [Illustration] The White rooks cannot move outside the little square in which they are enclosed except on the final move, in giving checkmate. The puzzle is how to checkmate Black in the fewest possible moves with No. 8 rook, the other rooks being left in numerical order round the sides of their square with the break between 1 and 7. 349.--STALEMATE. Some years ago the puzzle was proposed to construct an imaginary game of chess, in which White shall be stalemated in the fewest possible moves with all the thirty-two pieces on the board. Can you build up such a position in fewer than twenty moves? 350.--THE FORSAKEN KING. [Illustration] Set up the position shown in the diagram. Then the condition of the puzzle is--White to play and checkmate in six moves. Notwithstanding the complexities, I will show how the manner of play may be condensed into quite a few lines, merely stating here that the first two moves of White cannot be varied. 351.--THE CRUSADER. The following is a prize puzzle propounded by me some years ago. Produce a game of chess which, after sixteen moves, shall leave White with all his sixteen men on their original squares and Black in possession of his king alone (not necessarily on his own square). White is then to _force_ mate in three moves. 352.--IMMOVABLE PAWNS. Starting from the ordinary arrangement of the pieces as for a game, what is the smallest possible number of moves necessary in order to arrive at the following position? The moves for both sides must, of course, be played strictly in accordance with the rules of the game, though the result will necessarily be a very weird kind of chess. [Illustration] 353.--THIRTY-SIX MATES. [Illustration] Place the remaining eight White pieces in such a position that White shall have the choice of thirty-six different mates on the move. Every move that checkmates and leaves a different position is a different mate. The pieces already placed must not be moved. 354.--AN AMAZING DILEMMA. In a game of chess between Mr. Black and Mr. White, Black was in difficulties, and as usual was obliged to catch a train. So he proposed that White should complete the game in his absence on condition that no moves whatever should be made for Black, but only with the White pieces. Mr. White accepted, but to his dismay found it utterly impossible to win the game under such conditions. Try as he would, he could not checkmate his opponent. On which square did Mr. Black leave his king? The other pieces are in their proper positions in the diagram. White may leave Black in check as often as he likes, for it makes no difference, as he can never arrive at a checkmate position. [Illustration] 355.--CHECKMATE! [Illustration] Strolling into one of the rooms of a London club, I noticed a position left by two players who had gone. This position is shown in the diagram. It is evident that White has checkmated Black. But how did he do it? That is the puzzle. 356.--QUEER CHESS. Can you place two White rooks and a White knight on the board so that the Black king (who must be on one of the four squares in the middle of the board) shall be in check with no possible move open to him? "In other words," the reader will say, "the king is to be shown checkmated." Well, you can use the term if you wish, though I intentionally do not employ it myself. The mere fact that there is no White king on the board would be a sufficient reason for my not doing so. 357.--ANCIENT CHINESE PUZZLE. [Illustration] My next puzzle is supposed to be Chinese, many hundreds of years old, and never fails to interest. White to play and mate, moving each of the three pieces once, and once only. 358.--THE SIX PAWNS. In how many different ways may I place six pawns on the chessboard so that there shall be an even number of unoccupied squares in every row and every column? We are not here considering the diagonals at all, and every different six squares occupied makes a different solution, so we have not to exclude reversals or reflections. 359.--COUNTER SOLITAIRE. Here is a little game of solitaire that is quite easy, but not so easy as to be uninteresting. You can either rule out the squares on a sheet of cardboard or paper, or you can use a portion of your chessboard. I have shown numbered counters in the illustration so as to make the solution easy and intelligible to all, but chess pawns or draughts will serve just as well in practice. [Illustration] The puzzle is to remove all the counters except one, and this one that is left must be No. 1. You remove a counter by jumping over another counter to the next space beyond, if that square is vacant, but you cannot make a leap in a diagonal direction. The following moves will make the play quite clear: 1-9, 2-10, 1-2, and so on. Here 1 jumps over 9, and you remove 9 from the board; then 2 jumps over 10, and you remove 10; then 1 jumps over 2, and you remove 2. Every move is thus a capture, until the last capture of all is made by No. 1. 360.--CHESSBOARD SOLITAIRE. [Illustration] Here is an extension of the last game of solitaire. All you need is a chessboard and the thirty-two pieces, or the same number of draughts or counters. In the illustration numbered counters are used. The puzzle is to remove all the counters except two, and these two must have originally been on the same side of the board; that is, the two left must either belong to the group 1 to 16 or to the other group, 17 to 32. You remove a counter by jumping over it with another counter to the next square beyond, if that square is vacant, but you cannot make a leap in a diagonal direction. The following moves will make the play quite clear: 3-11, 4-12, 3-4, 13-3. Here 3 jumps over 11, and you remove 11; 4 jumps over 12, and you remove 12; and so on. It will be found a fascinating little game of patience, and the solution requires the exercise of some ingenuity. 361.--THE MONSTROSITY. One Christmas Eve I was travelling by rail to a little place in one of the southern counties. The compartment was very full, and the passengers were wedged in very tightly. My neighbour in one of the corner seats was closely studying a position set up on one of those little folding chessboards that can be carried conveniently in the pocket, and I could scarcely avoid looking at it myself. Here is the position:-- [Illustration] My fellow-passenger suddenly turned his head and caught the look of bewilderment on my face. "Do you play chess?" he asked. "Yes, a little. What is that? A problem?" "Problem? No; a game." "Impossible!" I exclaimed rather rudely. "The position is a perfect monstrosity!" He took from his pocket a postcard and handed it to me. It bore an address at one side and on the other the words "43. K to Kt 8." "It is a correspondence game." he exclaimed. "That is my friend's last move, and I am considering my reply." "But you really must excuse me; the position seems utterly impossible. How on earth, for example--" "Ah!" he broke in smilingly. "I see; you are a beginner; you play to win." "Of course you wouldn't play to lose or draw!" He laughed aloud. "You have much to learn. My friend and myself do not play for results of that antiquated kind. We seek in chess the wonderful, the whimsical, the weird. Did you ever see a position like that?" I inwardly congratulated myself that I never had. "That position, sir, materializes the sinuous evolvements and syncretic, synthetic, and synchronous concatenations of two cerebral individualities. It is the product of an amphoteric and intercalatory interchange of--" "Have you seen the evening paper, sir?" interrupted the man opposite, holding out a newspaper. I noticed on the margin beside his thumb some pencilled writing. Thanking him, I took the paper and read--"Insane, but quite harmless. He is in my charge." After that I let the poor fellow run on in his wild way until both got out at the next station. But that queer position became fixed indelibly in my mind, with Black's last move 43. K to Kt 8; and a short time afterwards I found it actually possible to arrive at such a position in forty-three moves. Can the reader construct such a sequence? How did White get his rooks and king's bishop into their present positions, considering Black can never have moved his king's bishop? No odds were given, and every move was perfectly legitimate. MEASURING, WEIGHING, AND PACKING PUZZLES. "Measure still for measure." _Measure for Measure_, v. 1. Apparently the first printed puzzle involving the measuring of a given quantity of liquid by pouring from one vessel to others of known capacity was that propounded by Niccola Fontana, better known as "Tartaglia" (the stammerer), 1500-1559. It consists in dividing 24 oz. of valuable balsam into three equal parts, the only measures available being vessels holding 5, 11, and 13 ounces respectively. There are many different solutions to this puzzle in six manipulations, or pourings from one vessel to another. Bachet de Méziriac reprinted this and other of Tartaglia's puzzles in his _Problèmes plaisans et délectables_ (1612). It is the general opinion that puzzles of this class can only be solved by trial, but I think formulæ can be constructed for the solution generally of certain related cases. It is a practically unexplored field for investigation. The classic weighing problem is, of course, that proposed by Bachet. It entails the determination of the least number of weights that would serve to weigh any integral number of pounds from 1 lb. to 40 lbs. inclusive, when we are allowed to put a weight in either of the two pans. The answer is 1, 3, 9, and 27 lbs. Tartaglia had previously propounded the same puzzle with the condition that the weights may only be placed in one pan. The answer in that case is 1, 2, 4, 8, 16, 32 lbs. Major MacMahon has solved the problem quite generally. A full account will be found in Ball's _Mathematical Recreations_ (5th edition). Packing puzzles, in which we are required to pack a maximum number of articles of given dimensions into a box of known dimensions, are, I believe, of quite recent introduction. At least I cannot recall any example in the books of the old writers. One would rather expect to find in the toy shops the idea presented as a mechanical puzzle, but I do not think I have ever seen such a thing. The nearest approach to it would appear to be the puzzles of the jig-saw character, where there is only one depth of the pieces to be adjusted. 362.--THE WASSAIL BOWL. One Christmas Eve three Weary Willies came into possession of what was to them a veritable wassail bowl, in the form of a small barrel, containing exactly six quarts of fine ale. One of the men possessed a five-pint jug and another a three-pint jug, and the problem for them was to divide the liquor equally amongst them without waste. Of course, they are not to use any other vessels or measures. If you can show how it was to be done at all, then try to find the way that requires the fewest possible manipulations, every separate pouring from one vessel to another, or down a man's throat, counting as a manipulation. 363.--THE DOCTOR'S QUERY. "A curious little point occurred to me in my dispensary this morning," said a doctor. "I had a bottle containing ten ounces of spirits of wine, and another bottle containing ten ounces of water. I poured a quarter of an ounce of spirits into the water and shook them up together. The mixture was then clearly forty to one. Then I poured back a quarter-ounce of the mixture, so that the two bottles should again each contain the same quantity of fluid. What proportion of spirits to water did the spirits of wine bottle then contain?" 364.--THE BARREL PUZZLE. The men in the illustration are disputing over the liquid contents of a barrel. What the particular liquid is it is impossible to say, for we are unable to look into the barrel; so we will call it water. One man says that the barrel is more than half full, while the other insists that it is not half full. What is their easiest way of settling the point? It is not necessary to use stick, string, or implement of any kind for measuring. I give this merely as one of the simplest possible examples of the value of ordinary sagacity in the solving of puzzles. What are apparently very difficult problems may frequently be solved in a similarly easy manner if we only use a little common sense. [Illustration] 365.--NEW MEASURING PUZZLE. Here is a new poser in measuring liquids that will be found interesting. A man has two ten-quart vessels full of wine, and a five-quart and a four-quart measure. He wants to put exactly three quarts into each of the two measures. How is he to do it? And how many manipulations (pourings from one vessel to another) do you require? Of course, waste of wine, tilting, and other tricks are not allowed. 366.--THE HONEST DAIRYMAN. An honest dairyman in preparing his milk for public consumption employed a can marked B, containing milk, and a can marked A, containing water. From can A he poured enough to double the contents of can B. Then he poured from can B into can A enough to double its contents. Then he finally poured from can A into can B until their contents were exactly equal. After these operations he would send the can A to London, and the puzzle is to discover what are the relative proportions of milk and water that he provides for the Londoners' breakfast-tables. Do they get equal proportions of milk and water--or two parts of milk and one of water--or what? It is an interesting question, though, curiously enough, we are not told how much milk or water he puts into the cans at the start of his operations. 367.--WINE AND WATER. Mr. Goodfellow has adopted a capital idea of late. When he gives a little dinner party and the time arrives to smoke, after the departure of the ladies, he sometimes finds that the conversation is apt to become too political, too personal, too slow, or too scandalous. Then he always manages to introduce to the company some new poser that he has secreted up his sleeve for the occasion. This invariably results in no end of interesting discussion and debate, and puts everybody in a good humour. Here is a little puzzle that he propounded the other night, and it is extraordinary how the company differed in their answers. He filled a wine-glass half full of wine, and another glass twice the size one-third full of wine. Then he filled up each glass with water and emptied the contents of both into a tumbler. "Now," he said, "what part of the mixture is wine and what part water?" Can you give the correct answer? 368.--THE KEG OF WINE. Here is a curious little problem. A man had a ten-gallon keg full of wine and a jug. One day he drew off a jugful of wine and filled up the keg with water. Later on, when the wine and water had got thoroughly mixed, he drew off another jugful and again filled up the keg with water. It was then found that the keg contained equal proportions of wine and water. Can you find from these facts the capacity of the jug? 369.--MIXING THE TEA. "Mrs. Spooner called this morning," said the honest grocer to his assistant. "She wants twenty pounds of tea at 2s. 4½d. per lb. Of course we have a good 2s. 6d. tea, a slightly inferior at 2s. 3d., and a cheap Indian at 1s. 9d., but she is very particular always about her prices." "What do you propose to do?" asked the innocent assistant. "Do?" exclaimed the grocer. "Why, just mix up the three teas in different proportions so that the twenty pounds will work out fairly at the lady's price. Only don't put in more of the best tea than you can help, as we make less profit on that, and of course you will use only our complete pound packets. Don't do any weighing." How was the poor fellow to mix the three teas? Could you have shown him how to do it? 370.--A PACKING PUZZLE. As we all know by experience, considerable ingenuity is often required in packing articles into a box if space is not to be unduly wasted. A man once told me that he had a large number of iron balls, all exactly two inches in diameter, and he wished to pack as many of these as possible into a rectangular box 24+9/10 inches long, 22+4/5 inches wide, and 14 inches deep. Now, what is the greatest number of the balls that he could pack into that box? 371.--GOLD PACKING IN RUSSIA. The editor of the _Times_ newspaper was invited by a high Russian official to inspect the gold stored in reserve at St. Petersburg, in order that he might satisfy himself that it was not another "Humbert safe." He replied that it would be of no use whatever, for although the gold might appear to be there, he would be quite unable from a mere inspection to declare that what he saw was really gold. A correspondent of the _Daily Mail_ thereupon took up the challenge, but, although he was greatly impressed by what he saw, he was compelled to confess his incompetence (without emptying and counting the contents of every box and sack, and assaying every piece of gold) to give any assurance on the subject. In presenting the following little puzzle, I wish it to be also understood that I do not guarantee the real existence of the gold, and the point is not at all material to our purpose. Moreover, if the reader says that gold is not usually "put up" in slabs of the dimensions that I give, I can only claim problematic licence. Russian officials were engaged in packing 800 gold slabs, each measuring 12½ inches long, 11 inches wide, and 1 inch deep. What are the interior dimensions of a box of equal length and width, and necessary depth, that will exactly contain them without any space being left over? Not more than twelve slabs may be laid on edge, according to the rules of the government. It is an interesting little problem in packing, and not at all difficult. 372.--THE BARRELS OF HONEY. [Illustration] Once upon a time there was an aged merchant of Bagdad who was much respected by all who knew him. He had three sons, and it was a rule of his life to treat them all exactly alike. Whenever one received a present, the other two were each given one of equal value. One day this worthy man fell sick and died, bequeathing all his possessions to his three sons in equal shares. The only difficulty that arose was over the stock of honey. There were exactly twenty-one barrels. The old man had left instructions that not only should every son receive an equal quantity of honey, but should receive exactly the same number of barrels, and that no honey should be transferred from barrel to barrel on account of the waste involved. Now, as seven of these barrels were full of honey, seven were half-full, and seven were empty, this was found to be quite a puzzle, especially as each brother objected to taking more than four barrels of, the same description--full, half-full, or empty. Can you show how they succeeded in making a correct division of the property? CROSSING RIVER PROBLEMS "My boat is on the shore." BYRON. This is another mediæval class of puzzles. Probably the earliest example was by Abbot Alcuin, who was born in Yorkshire in 735 and died at Tours in 804. And everybody knows the story of the man with the wolf, goat, and basket of cabbages whose boat would only take one of the three at a time with the man himself. His difficulties arose from his being unable to leave the wolf alone with the goat, or the goat alone with the cabbages. These puzzles were considered by Tartaglia and Bachet, and have been later investigated by Lucas, De Fonteney, Delannoy, Tarry, and others. In the puzzles I give there will be found one or two new conditions which add to the complexity somewhat. I also include a pulley problem that practically involves the same principles. [Illustration] 373.--CROSSING THE STREAM. During a country ramble Mr. and Mrs. Softleigh found themselves in a pretty little dilemma. They had to cross a stream in a small boat which was capable of carrying only 150 lbs. weight. But Mr. Softleigh and his wife each weighed exactly 150 lbs., and each of their sons weighed 75 lbs. And then there was the dog, who could not be induced on any terms to swim. On the principle of "ladies first," they at once sent Mrs. Softleigh over; but this was a stupid oversight, because she had to come back again with the boat, so nothing was gained by that operation. How did they all succeed in getting across? The reader will find it much easier than the Softleigh family did, for their greatest enemy could not have truthfully called them a brilliant quartette--while the dog was a perfect fool. 374--CROSSING THE RIVER AXE. Many years ago, in the days of the smuggler known as "Rob Roy of the West," a piratical band buried on the coast of South Devon a quantity of treasure which was, of course, abandoned by them in the usual inexplicable way. Some time afterwards its whereabouts was discovered by three countrymen, who visited the spot one night and divided the spoil between them, Giles taking treasure to the value of £800, Jasper £500 worth, and Timothy £300 worth. In returning they had to cross the river Axe at a point where they had left a small boat in readiness. Here, however, was a difficulty they had not anticipated. The boat would only carry two men, or one man and a sack, and they had so little confidence in one another that no person could be left alone on the land or in the boat with more than his share of the spoil, though two persons (being a check on each other) might be left with more than their shares. The puzzle is to show how they got over the river in the fewest possible crossings, taking their treasure with them. No tricks, such as ropes, "flying bridges," currents, swimming, or similar dodges, may be employed. 375.--FIVE JEALOUS HUSBANDS. During certain local floods five married couples found themselves surrounded by water, and had to escape from their unpleasant position in a boat that would only hold three persons at a time. Every husband was so jealous that he would not allow his wife to be in the boat or on either bank with another man (or with other men) unless he was himself present. Show the quickest way of getting these five men and their wives across into safety. Call the men A, B, C, D, E, and their respective wives a, b, c, d, e. To go over and return counts as two crossings. No tricks such as ropes, swimming, currents, etc., are permitted. 376.--THE FOUR ELOPEMENTS. Colonel B---- was a widower of a very taciturn disposition. His treatment of his four daughters was unusually severe, almost cruel, and they not unnaturally felt disposed to resent it. Being charming girls with every virtue and many accomplishments, it is not surprising that each had a fond admirer. But the father forbade the young men to call at his house, intercepted all letters, and placed his daughters under stricter supervision than ever. But love, which scorns locks and keys and garden walls, was equal to the occasion, and the four youths conspired together and planned a general elopement. At the foot of the tennis lawn at the bottom of the garden ran the silver Thames, and one night, after the four girls had been safely conducted from a dormitory window to _terra firma_, they all crept softly down to the bank of the river, where a small boat belonging to the Colonel was moored. With this they proposed to cross to the opposite side and make their way to a lane where conveyances were waiting to carry them in their flight. Alas! here at the water's brink their difficulties already began. The young men were so extremely jealous that not one of them would allow his prospective bride to remain at any time in the company of another man, or men, unless he himself were present also. Now, the boat would only hold two persons, though it could, of course, be rowed by one, and it seemed impossible that the four couples would ever get across. But midway in the stream was a small island, and this seemed to present a way out of the difficulty, because a person or persons could be left there while the boat was rowed back or to the opposite shore. If they had been prepared for their difficulty they could have easily worked out a solution to the little poser at any other time. But they were now so hurried and excited in their flight that the confusion they soon got into was exceedingly amusing--or would have been to any one except themselves. As a consequence they took twice as long and crossed the river twice as often as was really necessary. Meanwhile, the Colonel, who was a very light sleeper, thought he heard a splash of oars. He quickly raised the alarm among his household, and the young ladies were found to be missing. Somebody was sent to the police-station, and a number of officers soon aided in the pursuit of the fugitives, who, in consequence of that delay in crossing the river, were quickly overtaken. The four girls returned sadly to their homes, and afterwards broke off their engagements in disgust. For a considerable time it was a mystery how the party of eight managed to cross the river in that little boat without any girl being ever left with a man, unless her betrothed was also present. The favourite method is to take eight counters or pieces of cardboard and mark them A, B, C, D, a, b, c, d, to represent the four men and their prospective brides, and carry them from one side of a table to the other in a matchbox (to represent the boat), a penny being placed in the middle of the table as the island. Readers are now asked to find the quickest method of getting the party across the river. How many passages are necessary from land to land? By "land" is understood either shore or island. Though the boat would not necessarily call at the island every time of crossing, the possibility of its doing so must be provided for. For example, it would not do for a man to be alone in the boat (though it were understood that he intended merely to cross from one bank to the opposite one) if there happened to be a girl alone on the island other than the one to whom he was engaged. 377.--STEALING THE CASTLE TREASURE. The ingenious manner in which a box of treasure, consisting principally of jewels and precious stones, was stolen from Gloomhurst Castle has been handed down as a tradition in the De Gourney family. The thieves consisted of a man, a youth, and a small boy, whose only mode of escape with the box of treasure was by means of a high window. Outside the window was fixed a pulley, over which ran a rope with a basket at each end. When one basket was on the ground the other was at the window. The rope was so disposed that the persons in the basket could neither help themselves by means of it nor receive help from others. In short, the only way the baskets could be used was by placing a heavier weight in one than in the other. Now, the man weighed 195 lbs., the youth 105 lbs., the boy 90 lbs., and the box of treasure 75 lbs. The weight in the descending basket could not exceed that in the other by more than 15 lbs. without causing a descent so rapid as to be most dangerous to a human being, though it would not injure the stolen property. Only two persons, or one person and the treasure, could be placed in the same basket at one time. How did they all manage to escape and take the box of treasure with them? The puzzle is to find the shortest way of performing the feat, which in itself is not difficult. Remember, a person cannot help himself by hanging on to the rope, the only way being to go down "with a bump," with the weight in the other basket as a counterpoise. PROBLEMS CONCERNING GAMES. "The little pleasure of the game." MATTHEW PRIOR. Every game lends itself to the propounding of a variety of puzzles. They can be made, as we have seen, out of the chessboard and the peculiar moves of the chess pieces. I will now give just a few examples of puzzles with playing cards and dominoes, and also go out of doors and consider one or two little posers in the cricket field, at the football match, and the horse race and motor-car race. 378.--DOMINOES IN PROGRESSION. [Illustration] It will be seen that I have played six dominoes, in the illustration, in accordance with the ordinary rules of the game, 4 against 4, 1 against 1, and so on, and yet the sum of the spots on the successive dominoes, 4, 5, 6, 7, 8, 9, are in arithmetical progression; that is, the numbers taken in order have a common difference of 1. In how many different ways may we play six dominoes, from an ordinary box of twenty-eight, so that the numbers on them may lie in arithmetical progression? We must always play from left to right, and numbers in decreasing arithmetical progression (such as 9, 8, 7, 6, 5, 4) are not admissible. 379.--THE FIVE DOMINOES. [Illustration] Here is a new little puzzle that is not difficult, but will probably be found entertaining by my readers. It will be seen that the five dominoes are so arranged in proper sequence (that is, with 1 against 1, 2 against 2, and so on), that the total number of pips on the two end dominoes is five, and the sum of the pips on the three dominoes in the middle is also five. There are just three other arrangements giving five for the additions. They are: -- (1--0) (0--0) (0--2) (2--1) (1--3) (4--0) (0--0) (0--2) (2--1) (1--0) (2--0) (0--0) (0--1) (1--3) (3--0) Now, how many similar arrangements are there of five dominoes that shall give six instead of five in the two additions? 380.--THE DOMINO FRAME PUZZLE. [Illustration] It will be seen in the illustration that the full set of twenty-eight dominoes is arranged in the form of a square frame, with 6 against 6, 2 against 2, blank against blank, and so on, as in the game. It will be found that the pips in the top row and left-hand column both add up 44. The pips in the other two sides sum to 59 and 32 respectively. The puzzle is to rearrange the dominoes in the same form so that all of the four sides shall sum to 44. Remember that the dominoes must be correctly placed one against another as in the game. 381.--THE CARD FRAME PUZZLE. In the illustration we have a frame constructed from the ten playing cards, ace to ten of diamonds. The children who made it wanted the pips on all four sides to add up alike, but they failed in their attempt and gave it up as impossible. It will be seen that the pips in the top row, the bottom row, and the left-hand side all add up 14, but the right-hand side sums to 23. Now, what they were trying to do is quite possible. Can you rearrange the ten cards in the same formation so that all four sides shall add up alike? Of course they need not add up 14, but any number you choose to select. [Illustration] 382.--THE CROSS OF CARDS. [Illustration] In this case we use only nine cards--the ace to nine of diamonds. The puzzle is to arrange them in the form of a cross, exactly in the way shown in the illustration, so that the pips in the vertical bar and in the horizontal bar add up alike. In the example given it will be found that both directions add up 23. What I want to know is, how many different ways are there of rearranging the cards in order to bring about this result? It will be seen that, without affecting the solution, we may exchange the 5 with the 6, the 5 with the 7, the 8 with the 3, and so on. Also we may make the horizontal and the vertical bars change places. But such obvious manipulations as these are not to be regarded as different solutions. They are all mere variations of one fundamental solution. Now, how many of these fundamentally different solutions are there? The pips need not, of course, always add up 23. 383.--THE "T" CARD PUZZLE. [Illustration] An entertaining little puzzle with cards is to take the nine cards of a suit, from ace to nine inclusive, and arrange them in the form of the letter "T," as shown in the illustration, so that the pips in the horizontal line shall count the same as those in the column. In the example given they add up twenty-three both ways. Now, it is quite easy to get a single correct arrangement. The puzzle is to discover in just how many different ways it may be done. Though the number is high, the solution is not really difficult if we attack the puzzle in the right manner. The reverse way obtained by reflecting the illustration in a mirror we will not count as different, but all other changes in the relative positions of the cards will here count. How many different ways are there? 384.--CARD TRIANGLES. Here you pick out the nine cards, ace to nine of diamonds, and arrange them in the form of a triangle, exactly as shown in the illustration, so that the pips add up the same on the three sides. In the example given it will be seen that they sum to 20 on each side, but the particular number is of no importance so long as it is the same on all three sides. The puzzle is to find out in just how many different ways this can be done. If you simply turn the cards round so that one of the other two sides is nearest to you this will not count as different, for the order will be the same. Also, if you make the 4, 9, 5 change places with the 7, 3, 8, and at the same time exchange the 1 and the 6, it will not be different. But if you only change the 1 and the 6 it will be different, because the order round the triangle is not the same. This explanation will prevent any doubt arising as to the conditions. [Illustration] 385.--"STRAND" PATIENCE. The idea for this came to me when considering the game of Patience that I gave in the _Strand Magazine_ for December, 1910, which has been reprinted in Ernest Bergholt's _Second Book of Patience Games_, under the new name of "King Albert." Make two piles of cards as follows: 9 D, 8 S, 7 D, 6 S, 5 D, 4 S, 3 D, 2 S, 1 D, and 9 H, 8 C, 7 H, 6 C, 5 H, 4 C, 3 H, 2 C, 1 H, with the 9 of diamonds at the bottom of one pile and the 9 of hearts at the bottom of the other. The point is to exchange the spades with the clubs, so that the diamonds and clubs are still in numerical order in one pile and the hearts and spades in the other. There are four vacant spaces in addition to the two spaces occupied by the piles, and any card may be laid on a space, but a card can only be laid on another of the next higher value--an ace on a two, a two on a three, and so on. Patience is required to discover the shortest way of doing this. When there are four vacant spaces you can pile four cards in seven moves, with only three spaces you can pile them in nine moves, and with two spaces you cannot pile more than two cards. When you have a grasp of these and similar facts you will be able to remove a number of cards bodily and write down 7, 9, or whatever the number of moves may be. The gradual shortening of play is fascinating, and first attempts are surprisingly lengthy. 386.--A TRICK WITH DICE. [Illustration] Here is a neat little trick with three dice. I ask you to throw the dice without my seeing them. Then I tell you to multiply the points of the first die by 2 and add 5; then multiply the result by 5 and add the points of the second die; then multiply the result by 10 and add the points of the third die. You then give me the total, and I can at once tell you the points thrown with the three dice. How do I do it? As an example, if you threw 1, 3, and 6, as in the illustration, the result you would give me would be 386, from which I could at once say what you had thrown. 387.--THE VILLAGE CRICKET MATCH. In a cricket match, Dingley Dell v. All Muggleton, the latter had the first innings. Mr. Dumkins and Mr. Podder were at the wickets, when the wary Dumkins made a splendid late cut, and Mr. Podder called on him to run. Four runs were apparently completed, but the vigilant umpires at each end called, "three short," making six short runs in all. What number did Mr. Dumkins score? When Dingley Dell took their turn at the wickets their champions were Mr. Luffey and Mr. Struggles. The latter made a magnificent off-drive, and invited his colleague to "come along," with the result that the observant spectators applauded them for what was supposed to have been three sharp runs. But the umpires declared that there had been two short runs at each end--four in all. To what extent, if any, did this manoeuvre increase Mr. Struggles's total? 388.--SLOW CRICKET. In the recent county match between Wessex and Nincomshire the former team were at the wickets all day, the last man being put out a few minutes before the time for drawing stumps. The play was so slow that most of the spectators were fast asleep, and, on being awakened by one of the officials clearing the ground, we learnt that two men had been put out leg-before-wicket for a combined score of 19 runs; four men were caught for a combined score or 17 runs; one man was run out for a duck's egg; and the others were all bowled for 3 runs each. There were no extras. We were not told which of the men was the captain, but he made exactly 15 more than the average of his team. What was the captain's score? 389.--THE FOOTBALL PLAYERS. "It is a glorious game!" an enthusiast was heard to exclaim. "At the close of last season, of the footballers of my acquaintance four had broken their left arm, five had broken their right arm, two had the right arm sound, and three had sound left arms." Can you discover from that statement what is the smallest number of players that the speaker could be acquainted with? It does not at all follow that there were as many as fourteen men, because, for example, two of the men who had broken the left arm might also be the two who had sound right arms. 390.--THE HORSE-RACE PUZZLE. There are no morals in puzzles. When we are solving the old puzzle of the captain who, having to throw half his crew overboard in a storm, arranged to draw lots, but so placed the men that only the Turks were sacrificed, and all the Christians left on board, we do not stop to discuss the questionable morality of the proceeding. And when we are dealing with a measuring problem, in which certain thirsty pilgrims are to make an equitable division of a barrel of beer, we do not object that, as total abstainers, it is against our conscience to have anything to do with intoxicating liquor. Therefore I make no apology for introducing a puzzle that deals with betting. Three horses--Acorn, Bluebottle, and Capsule--start in a race. The odds are 4 to 1, Acorn; 3 to 1, Bluebottle; 2 to 1, Capsule. Now, how much must I invest on each horse in order to win £13, no matter which horse comes in first? Supposing, as an example, that I betted £5 on each horse. Then, if Acorn won, I should receive £20 (four times £5), and have to pay £5 each for the other two horses; thereby winning £10. But it will be found that if Bluebottle was first I should only win £5, and if Capsule won I should gain nothing and lose nothing. This will make the question perfectly clear to the novice, who, like myself, is not interested in the calling of the fraternity who profess to be engaged in the noble task of "improving the breed of horses." 391.--THE MOTOR-CAR RACE. Sometimes a quite simple statement of fact, if worded in an unfamiliar manner, will cause considerable perplexity. Here is an example, and it will doubtless puzzle some of my more youthful readers just a little. I happened to be at a motor-car race at Brooklands, when one spectator said to another, while a number of cars were whirling round and round the circular track:-- "There's Gogglesmith--that man in the white car!" "Yes, I see," was the reply; "but how many cars are running in this race?" Then came this curious rejoinder:-- "One-third of the cars in front of Gogglesmith added to three-quarters of those behind him will give you the answer." Now, can you tell how many cars were running in the race? PUZZLE GAMES. "He that is beaten may be said To lie in honour's truckle bed." HUDIBRAS. It may be said generally that a game is a contest of skill for two or more persons, into which we enter either for amusement or to win a prize. A puzzle is something to be done or solved by the individual. For example, if it were possible for us so to master the complexities of the game of chess that we could be assured of always winning with the first or second move, as the case might be, or of always drawing, then it would cease to be a game and would become a puzzle. Of course among the young and uninformed, when the correct winning play is not understood, a puzzle may well make a very good game. Thus there is no doubt children will continue to play "Noughts and Crosses," though I have shown (No. 109, "_Canterbury Puzzles_") that between two players who both thoroughly understand the play, every game should be drawn. Neither player could ever win except through the blundering of his opponent. But I am writing from the point of view of the student of these things. The examples that I give in this class are apparently games, but, since I show in every case how one player may win if he only play correctly, they are in reality puzzles. Their interest, therefore, lies in attempting to discover the leading method of play. 392.--THE PEBBLE GAME. Here is an interesting little puzzle game that I used to play with an acquaintance on the beach at Slocomb-on-Sea. Two players place an odd number of pebbles, we will say fifteen, between them. Then each takes in turn one, two, or three pebbles (as he chooses), and the winner is the one who gets the odd number. Thus, if you get seven and your opponent eight, you win. If you get six and he gets nine, he wins. Ought the first or second player to win, and how? When you have settled the question with fifteen pebbles try again with, say, thirteen. 393.--THE TWO ROOKS. This is a puzzle game for two players. Each player has a single rook. The first player places his rook on any square of the board that he may choose to select, and then the second player does the same. They now play in turn, the point of each play being to capture the opponent's rook. But in this game you cannot play through a line of attack without being captured. That is to say, if in the diagram it is Black's turn to play, he cannot move his rook to his king's knight's square, or to his king's rook's square, because he would enter the "line of fire" when passing his king's bishop's square. For the same reason he cannot move to his queen's rook's seventh or eighth squares. Now, the game can never end in a draw. Sooner or later one of the rooks must fall, unless, of course, both players commit the absurdity of not trying to win. The trick of winning is ridiculously simple when you know it. Can you solve the puzzle? [Illustration] 394.--PUSS IN THE CORNER. [Illustration] This variation of the last puzzle is also played by two persons. One puts a counter on No. 6, and the other puts one on No. 55, and they play alternately by removing the counter to any other number in a line. If your opponent moves at any time on to one of the lines you occupy, or even crosses one of your lines, you immediately capture him and win. We will take an illustrative game. A moves from 55 to 52; B moves from 6 to 13; A advances to 23; B goes to 15; A retreats to 26; B retreats to 13; A advances to 21; B retreats to 2; A advances to 7; B goes to 3; A moves to 6; B must now go to 4; A establishes himself at 11, and B must be captured next move because he is compelled to cross a line on which A stands. Play this over and you will understand the game directly. Now, the puzzle part of the game is this: Which player should win, and how many moves are necessary? 395.--A WAR PUZZLE GAME. [Illustration] Here is another puzzle game. One player, representing the British general, places a counter at B, and the other player, representing the enemy, places his counter at E. The Britisher makes the first advance along one of the roads to the next town, then the enemy moves to one of his nearest towns, and so on in turns, until the British general gets into the same town as the enemy and captures him. Although each must always move along a road to the next town only, and the second player may do his utmost to avoid capture, the British general (as we should suppose, from the analogy of real life) must infallibly win. But how? That is the question. 396.--A MATCH MYSTERY. Here is a little game that is childishly simple in its conditions. But it is well worth investigation. Mr. Stubbs pulled a small table between himself and his friend, Mr. Wilson, and took a box of matches, from which he counted out thirty. "Here are thirty matches," he said. "I divide them into three unequal heaps. Let me see. We have 14, 11, and 5, as it happens. Now, the two players draw alternately any number from any one heap, and he who draws the last match loses the game. That's all! I will play with you, Wilson. I have formed the heaps, so you have the first draw." "As I can draw any number," Mr. Wilson said, "suppose I exhibit my usual moderation and take all the 14 heap." "That is the worst you could do, for it loses right away. I take 6 from the 11, leaving two equal heaps of 5, and to leave two equal heaps is a certain win (with the single exception of 1, 1), because whatever you do in one heap I can repeat in the other. If you leave 4 in one heap, I leave 4 in the other. If you then leave 2 in one heap, I leave 2 in the other. If you leave only 1 in one heap, then I take all the other heap. If you take all one heap, I take all but one in the other. No, you must never leave two heaps, unless they are equal heaps and more than 1, 1. Let's begin again." "Very well, then," said Mr. Wilson. "I will take 6 from the 14, and leave you 8, 11, 5." Mr. Stubbs then left 8, 11, 3; Mr. Wilson, 8, 5, 3; Mr. Stubbs, 6, 5, 3; Mr. Wilson,4, 5, 3; Mr. Stubbs, 4, 5, 1; Mr. Wilson, 4, 3, 1; Mr. Stubbs, 2, 3, 1; Mr. Wilson, 2, 1, 1; which Mr. Stubbs reduced to 1, 1, 1. "It is now quite clear that I must win," said Mr. Stubbs, because you must take 1, and then I take 1, leaving you the last match. You never had a chance. There are just thirteen different ways in which the matches may be grouped at the start for a certain win. In fact, the groups selected, 14, 11, 5, are a certain win, because for whatever your opponent may play there is another winning group you can secure, and so on and on down to the last match." 397.--THE MONTENEGRIN DICE GAME. It is said that the inhabitants of Montenegro have a little dice game that is both ingenious and well worth investigation. The two players first select two different pairs of odd numbers (always higher than 3) and then alternately toss three dice. Whichever first throws the dice so that they add up to one of his selected numbers wins. If they are both successful in two successive throws it is a draw and they try again. For example, one player may select 7 and 15 and the other 5 and 13. Then if the first player throws so that the three dice add up 7 or 15 he wins, unless the second man gets either 5 or 13 on his throw. The puzzle is to discover which two pairs of numbers should be selected in order to give both players an exactly even chance. 398.--THE CIGAR PUZZLE. I once propounded the following puzzle in a London club, and for a considerable period it absorbed the attention of the members. They could make nothing of it, and considered it quite impossible of solution. And yet, as I shall show, the answer is remarkably simple. Two men are seated at a square-topped table. One places an ordinary cigar (flat at one end, pointed at the other) on the table, then the other does the same, and so on alternately, a condition being that no cigar shall touch another. Which player should succeed in placing the last cigar, assuming that they each will play in the best possible manner? The size of the table top and the size of the cigar are not given, but in order to exclude the ridiculous answer that the table might be so diminutive as only to take one cigar, we will say that the table must not be less than 2 feet square and the cigar not more than 4½ inches long. With those restrictions you may take any dimensions you like. Of course we assume that all the cigars are exactly alike in every respect. Should the first player, or the second player, win? MAGIC SQUARE PROBLEMS. "By magic numbers." CONGREVE, _The Mourning Bride._ This is a very ancient branch of mathematical puzzledom, and it has an immense, though scattered, literature of its own. In their simple form of consecutive whole numbers arranged in a square so that every column, every row, and each of the two long diagonals shall add up alike, these magic squares offer three main lines of investigation: Construction, Enumeration, and Classification. Of recent years many ingenious methods have been devised for the construction of magics, and the law of their formation is so well understood that all the ancient mystery has evaporated and there is no longer any difficulty in making squares of any dimensions. Almost the last word has been said on this subject. The question of the enumeration of all the possible squares of a given order stands just where it did over two hundred years ago. Everybody knows that there is only one solution for the third order, three cells by three; and Frénicle published in 1693 diagrams of all the arrangements of the fourth order--880 in number--and his results have been verified over and over again. I may here refer to the general solution for this order, for numbers not necessarily consecutive, by E. Bergholt in _Nature_, May 26, 1910, as it is of the greatest importance to students of this subject. The enumeration of the examples of any higher order is a completely unsolved problem. As to classification, it is largely a matter of individual taste--perhaps an æsthetic question, for there is beauty in the law and order of numbers. A man once said that he divided the human race into two great classes: those who take snuff and those who do not. I am not sure that some of our classifications of magic squares are not almost as valueless. However, lovers of these things seem somewhat agreed that Nasik magic squares (so named by Mr. Frost, a student of them, after the town in India where he lived, and also called Diabolique and Pandiagonal) and Associated magic squares are of special interest, so I will just explain what these are for the benefit of the novice. [Illustration: SIMPLE] [Illustration: SEMI-NASIK] [Illustration: ASSOCIATED] [Illustration: NASIK] I published in _The Queen_ for January 15, 1910, an article that would enable the reader to write out, if he so desired, all the 880 magics of the fourth order, and the following is the complete classification that I gave. The first example is that of a Simple square that fulfils the simple conditions and no more. The second example is a Semi-Nasik, which has the additional property that the opposite short diagonals of two cells each together sum to 34. Thus, 14 + 4 + 11 + 5 = 34 and 12 + 6 + 13 + 3 = 34. The third example is not only Semi-Nasik but also Associated, because in it every number, if added to the number that is equidistant, in a straight line, from the centre gives 17. Thus, 1 + 16, 2 + 15, 3 + 14, etc. The fourth example, considered the most "perfect" of all, is a Nasik. Here all the broken diagonals sum to 34. Thus, for example, 15 + 14 + 2 + 3, and 10 + 4 + 7 + 13, and 15 + 5 + 2 + 12. As a consequence, its properties are such that if you repeat the square in all directions you may mark off a square, 4 × 4, wherever you please, and it will be magic. The following table not only gives a complete enumeration under the four forms described, but also a classification under the twelve graphic types indicated in the diagrams. The dots at the end of each line represent the relative positions of those complementary pairs, 1 + 16, 2 + 15, etc., which sum to 17. For example, it will be seen that the first and second magic squares given are of Type VI., that the third square is of Type III., and that the fourth is of Type I. Edouard Lucas indicated these types, but he dropped exactly half of them and did not attempt the classification. NASIK (Type I.) . . . . . 48 SEMI-NASIK (Type II., Transpositions of Nasik) . 48 " (Type III., Associated) 48 " (Type IV.) . . . 96 " (Type V.) . . . 96 192 ___ " (Type VI.) . . . 96 384 ___ SIMPLE. (Type VI.) . . . 208 " (Type VII.) . . . 56 " (Type VIII.). . . 56 " (Type IX.) . . . 56 " (Type X.) . . . 56 224 ___ " (Type XI.) . . . 8 " (Type XII.) . . . 8 16 448 ___ ___ ___ 880 ___ It is hardly necessary to say that every one of these squares will produce seven others by mere reversals and reflections, which we do not count as different. So that there are 7,040 squares of this order, 880 of which are fundamentally different. An infinite variety of puzzles may be made introducing new conditions into the magic square. In _The Canterbury Puzzles_ I have given examples of such squares with coins, with postage stamps, with cutting-out conditions, and other tricks. I will now give a few variants involving further novel conditions. 399.--THE TROUBLESOME EIGHT. Nearly everybody knows that a "magic square" is an arrangement of numbers in the form of a square so that every row, every column, and each of the two long diagonals adds up alike. For example, you would find little difficulty in merely placing a different number in each of the nine cells in the illustration so that the rows, columns, and diagonals shall all add up 15. And at your first attempt you will probably find that you have an 8 in one of the corners. The puzzle is to construct the magic square, under the same conditions, with the 8 in the position shown. [Illustration] 400.--THE MAGIC STRIPS. [Illustration] I happened to have lying on my table a number of strips of cardboard, with numbers printed on them from 1 upwards in numerical order. The idea suddenly came to me, as ideas have a way of unexpectedly coming, to make a little puzzle of this. I wonder whether many readers will arrive at the same solution that I did. Take seven strips of cardboard and lay them together as above. Then write on each of them the numbers 1, 2, 3, 4, 5, 6, 7, as shown, so that the numbers shall form seven rows and seven columns. Now, the puzzle is to cut these strips into the fewest possible pieces so that they may be placed together and form a magic square, the seven rows, seven columns, and two diagonals adding up the same number. No figures may be turned upside down or placed on their sides--that is, all the strips must lie in their original direction. Of course you could cut each strip into seven separate pieces, each piece containing a number, and the puzzle would then be very easy, but I need hardly say that forty-nine pieces is a long way from being the fewest possible. 401.--EIGHT JOLLY GAOL BIRDS. [Illustration] The illustration shows the plan of a prison of nine cells all communicating with one another by doorways. The eight prisoners have their numbers on their backs, and any one of them is allowed to exercise himself in whichever cell may happen to be vacant, subject to the rule that at no time shall two prisoners be in the same cell. The merry monarch in whose dominions the prison was situated offered them special comforts one Christmas Eve if, without breaking that rule, they could so place themselves that their numbers should form a magic square. Now, prisoner No. 7 happened to know a good deal about magic squares, so he worked out a scheme and naturally selected the method that was most expeditious--that is, one involving the fewest possible moves from cell to cell. But one man was a surly, obstinate fellow (quite unfit for the society of his jovial companions), and he refused to move out of his cell or take any part in the proceedings. But No. 7 was quite equal to the emergency, and found that he could still do what was required in the fewest possible moves without troubling the brute to leave his cell. The puzzle is to show how he did it and, incidentally, to discover which prisoner was so stupidly obstinate. Can you find the fellow? 402.--NINE JOLLY GAOL BIRDS. [Illustration] Shortly after the episode recorded in the last puzzle occurred, a ninth prisoner was placed in the vacant cell, and the merry monarch then offered them all complete liberty on the following strange conditions. They were required so to rearrange themselves in the cells that their numbers formed a magic square without their movements causing any two of them ever to be in the same cell together, except that at the start one man was allowed to be placed on the shoulders of another man, and thus add their numbers together, and move as one man. For example, No. 8 might be placed on the shoulders of No. 2, and then they would move about together as 10. The reader should seek first to solve the puzzle in the fewest possible moves, and then see that the man who is burdened has the least possible amount of work to do. 403.--THE SPANISH DUNGEON. Not fifty miles from Cadiz stood in the middle ages a castle, all traces of which have for centuries disappeared. Among other interesting features, this castle contained a particularly unpleasant dungeon divided into sixteen cells, all communicating with one another, as shown in the illustration. Now, the governor was a merry wight, and very fond of puzzles withal. One day he went to the dungeon and said to the prisoners, "By my halidame!" (or its equivalent in Spanish) "you shall all be set free if you can solve this puzzle. You must so arrange yourselves in the sixteen cells that the numbers on your backs shall form a magic square in which every column, every row, and each of the two diagonals shall add up the same. Only remember this: that in no case may two of you ever be together in the same cell." One of the prisoners, after working at the problem for two or three days, with a piece of chalk, undertook to obtain the liberty of himself and his fellow-prisoners if they would follow his directions and move through the doorway from cell to cell in the order in which he should call out their numbers. [Illustration] He succeeded in his attempt, and, what is more remarkable, it would seem from the account of his method recorded in the ancient manuscript lying before me, that he did so in the fewest possible moves. The reader is asked to show what these moves were. 404.--THE SIBERIAN DUNGEONS. [Illustration] The above is a trustworthy plan of a certain Russian prison in Siberia. All the cells are numbered, and the prisoners are numbered the same as the cells they occupy. The prison diet is so fattening that these political prisoners are in perpetual fear lest, should their pardon arrive, they might not be able to squeeze themselves through the narrow doorways and get out. And of course it would be an unreasonable thing to ask any government to pull down the walls of a prison just to liberate the prisoners, however innocent they might be. Therefore these men take all the healthy exercise they can in order to retard their increasing obesity, and one of their recreations will serve to furnish us with the following puzzle. Show, in the fewest possible moves, how the sixteen men may form themselves into a magic square, so that the numbers on their backs shall add up the same in each of the four columns, four rows, and two diagonals without two prisoners having been at any time in the same cell together. I had better say, for the information of those who have not yet been made acquainted with these places, that it is a peculiarity of prisons that you are not allowed to go outside their walls. Any prisoner may go any distance that is possible in a single move. 405.--CARD MAGIC SQUARES. [Illustration] Take an ordinary pack of cards and throw out the twelve court cards. Now, with nine of the remainder (different suits are of no consequence) form the above magic square. It will be seen that the pips add up fifteen in every row in every column, and in each of the two long diagonals. The puzzle is with the remaining cards (without disturbing this arrangement) to form three more such magic squares, so that each of the four shall add up to a different sum. There will, of course, be four cards in the reduced pack that will not be used. These four may be any that you choose. It is not a difficult puzzle, but requires just a little thought. 406.--THE EIGHTEEN DOMINOES. The illustration shows eighteen dominoes arranged in the form of a square so that the pips in every one of the six columns, six rows, and two long diagonals add up 13. This is the smallest summation possible with any selection of dominoes from an ordinary box of twenty-eight. The greatest possible summation is 23, and a solution for this number may be easily obtained by substituting for every number its complement to 6. Thus for every blank substitute a 6, for every 1 a 5, for every 2 a 4, for 3 a 3, for 4 a 2, for 5 a 1, and for 6 a blank. But the puzzle is to make a selection of eighteen dominoes and arrange them (in exactly the form shown) so that the summations shall be 18 in all the fourteen directions mentioned. [Illustration] SUBTRACTING, MULTIPLYING, AND DIVIDING MAGICS. Although the adding magic square is of such great antiquity, curiously enough the multiplying magic does not appear to have been mentioned until the end of the eighteenth century, when it was referred to slightly by one writer and then forgotten until I revived it in _Tit-Bits_ in 1897. The dividing magic was apparently first discussed by me in _The Weekly Dispatch_ in June 1898. The subtracting magic is here introduced for the first time. It will now be convenient to deal with all four kinds of magic squares together. [Illustration: ADDING SUBTRACTING MULTIPLYING DIVIDING] In these four diagrams we have examples in the third order of adding, subtracting, multiplying, and dividing squares. In the first the constant, 15, is obtained by the addition of the rows, columns, and two diagonals. In the second case you get the constant, 5, by subtracting the first number in a line from the second, and the result from the third. You can, of course, perform the operation in either direction; but, in order to avoid negative numbers, it is more convenient simply to deduct the middle number from the sum of the two extreme numbers. This is, in effect, the same thing. It will be seen that the constant of the adding square is n times that of the subtracting square derived from it, where n is the number of cells in the side of square. And the manner of derivation here is simply to reverse the two diagonals. Both squares are "associated"--a term I have explained in the introductory article to this department. The third square is a multiplying magic. The constant, 216, is obtained by multiplying together the three numbers in any line. It is "associated" by multiplication, instead of by addition. It is here necessary to remark that in an adding square it is not essential that the nine numbers should be consecutive. Write down any nine numbers in this way-- 1 3 5 4 6 8 7 9 11 so that the horizontal differences are all alike and the vertical differences also alike (here 2 and 3), and these numbers will form an adding magic square. By making the differences 1 and 3 we, of course, get consecutive numbers--a particular case, and nothing more. Now, in the case of the multiplying square we must take these numbers in geometrical instead of arithmetical progression, thus-- 1 3 9 2 6 18 4 12 36 Here each successive number in the rows is multiplied by 3, and in the columns by 2. Had we multiplied by 2 and 8 we should get the regular geometrical progression, 1, 2, 4, 8, 16, 32, 64, 128, and 256, but I wish to avoid high numbers. The numbers are arranged in the square in the same order as in the adding square. The fourth diagram is a dividing magic square. The constant 6 is here obtained by dividing the second number in a line by the first (in either direction) and the third number by the quotient. But, again, the process is simplified by dividing the product of the two extreme numbers by the middle number. This square is also "associated" by multiplication. It is derived from the multiplying square by merely reversing the diagonals, and the constant of the multiplying square is the cube of that of the dividing square derived from it. The next set of diagrams shows the solutions for the fifth order of square. They are all "associated" in the same way as before. The subtracting square is derived from the adding square by reversing the diagonals and exchanging opposite numbers in the centres of the borders, and the constant of one is again n times that of the other. The dividing square is derived from the multiplying square in the same way, and the constant of the latter is the 5th power (that is the nth) of that of the former. [Illustration] These squares are thus quite easy for odd orders. But the reader will probably find some difficulty over the even orders, concerning which I will leave him to make his own researches, merely propounding two little problems. 407.--TWO NEW MAGIC SQUARES. Construct a subtracting magic square with the first sixteen whole numbers that shall be "associated" by _subtraction_. The constant is, of course, obtained by subtracting the first number from the second in line, the result from the third, and the result again from the fourth. Also construct a dividing magic square of the same order that shall be "associated" by _division_. The constant is obtained by dividing the second number in a line by the first, the third by the quotient, and the fourth by the next quotient. 408.--MAGIC SQUARES OF TWO DEGREES. While reading a French mathematical work I happened to come across, the following statement: "A very remarkable magic square of 8, in two degrees, has been constructed by M. Pfeffermann. In other words, he has managed to dispose the sixty-four first numbers on the squares of a chessboard in such a way that the sum of the numbers in every line, every column, and in each of the two diagonals, shall be the same; and more, that if one substitutes for all the numbers their squares, the square still remains magic." I at once set to work to solve this problem, and, although it proved a very hard nut, one was rewarded by the discovery of some curious and beautiful laws that govern it. The reader may like to try his hand at the puzzle. MAGIC SQUARES OF PRIMES. The problem of constructing magic squares with prime numbers only was first discussed by myself in _The Weekly Dispatch_ for 22nd July and 5th August 1900; but during the last three or four years it has received great attention from American mathematicians. First, they have sought to form these squares with the lowest possible constants. Thus, the first nine prime numbers, 1 to 23 inclusive, sum to 99, which (being divisible by 3) is theoretically a suitable series; yet it has been demonstrated that the lowest possible constant is 111, and the required series as follows: 1, 7, 13, 31, 37, 43, 61, 67, and 73. Similarly, in the case of the fourth order, the lowest series of primes that are "theoretically suitable" will not serve. But in every other order, up to the 12th inclusive, magic squares have been constructed with the lowest series of primes theoretically possible. And the 12th is the lowest order in which a straight series of prime numbers, unbroken, from 1 upwards has been made to work. In other words, the first 144 odd prime numbers have actually been arranged in magic form. The following summary is taken from _The Monist_ (Chicago) for October 1913:-- Order of Totals of Lowest Squares Square. Series. Constants. made by-- (Henry E. 3rd 333 111 { Dudeney ( (1900). (Ernest Bergholt 4th 408 102 { and C. D. ( Shuldham. 5th 1065 213 H. A. Sayles. (C. D. Shuldham 6th 2448 408 { and J. ( N. Muncey. 7th 4893 699 do. 8th 8912 1114 do. 9th 15129 1681 do. 10th 24160 2416 J. N. Muncey. 11th 36095 3355 do. 12th 54168 4514 do. For further details the reader should consult the article itself, by W. S. Andrews and H. A. Sayles. These same investigators have also performed notable feats in constructing associated and bordered prime magics, and Mr. Shuldham has sent me a remarkable paper in which he gives examples of Nasik squares constructed with primes for all orders from the 4th to the 10th, with the exception of the 3rd (which is clearly impossible) and the 9th, which, up to the time of writing, has baffled all attempts. 409.--THE BASKETS OF PLUMS. [Illustration] This is the form in which I first introduced the question of magic squares with prime numbers. I will here warn the reader that there is a little trap. A fruit merchant had nine baskets. Every basket contained plums (all sound and ripe), and the number in every basket was different. When placed as shown in the illustration they formed a magic square, so that if he took any three baskets in a line in the eight possible directions there would always be the same number of plums. This part of the puzzle is easy enough to understand. But what follows seems at first sight a little queer. The merchant told one of his men to distribute the contents of any basket he chose among some children, giving plums to every child so that each should receive an equal number. But the man found it quite impossible, no matter which basket he selected and no matter how many children he included in the treat. Show, by giving contents of the nine baskets, how this could come about. 410.--THE MANDARIN'S "T" PUZZLE. [Illustration] Before Mr. Beauchamp Cholmondely Marjoribanks set out on his tour in the Far East, he prided himself on his knowledge of magic squares, a subject that he had made his special hobby; but he soon discovered that he had never really touched more than the fringe of the subject, and that the wily Chinee could beat him easily. I present a little problem that one learned mandarin propounded to our traveller, as depicted on the last page. The Chinaman, after remarking that the construction of the ordinary magic square of twenty-five cells is "too velly muchee easy," asked our countryman so to place the numbers 1 to 25 in the square that every column, every row, and each of the two diagonals should add up 65, with only prime numbers on the shaded "T." Of course the prime numbers available are 1, 2, 3, 5, 7, 11, 13, 17, 19, and 23, so you are at liberty to select any nine of these that will serve your purpose. Can you construct this curious little magic square? 411.--A MAGIC SQUARE OF COMPOSITES. As we have just discussed the construction of magic squares with prime numbers, the following forms an interesting companion problem. Make a magic square with nine consecutive composite numbers--the smallest possible. 412.--THE MAGIC KNIGHT'S TOUR. Here is a problem that has never yet been solved, nor has its impossibility been demonstrated. Play the knight once to every square of the chessboard in a complete tour, numbering the squares in the order visited, so that when completed the square shall be "magic," adding up to 260 in every column, every row, and each of the two long diagonals. I shall give the best answer that I have been able to obtain, in which there is a slight error in the diagonals alone. Can a perfect solution be found? I am convinced that it cannot, but it is only a "pious opinion." MAZES AND HOW TO THREAD THEM. "In wandering mazes lost." _Paradise Lost._ The Old English word "maze," signifying a labyrinth, probably comes from the Scandinavian, but its origin is somewhat uncertain. The late Professor Skeat thought that the substantive was derived from the verb, and as in old times to be mazed or amazed was to be "lost in thought," the transition to a maze in whose tortuous windings we are lost is natural and easy. The word "labyrinth" is derived from a Greek word signifying the passages of a mine. The ancient mines of Greece and elsewhere inspired fear and awe on account of their darkness and the danger of getting lost in their intricate passages. Legend was afterwards built round these mazes. The most familiar instance is the labyrinth made by Dædalus in Crete for King Minos. In the centre was placed the Minotaur, and no one who entered could find his way out again, but became the prey of the monster. Seven youths and seven maidens were sent regularly by the Athenians, and were duly devoured, until Theseus slew the monster and escaped from the maze by aid of the clue of thread provided by Ariadne; which accounts for our using to-day the expression "threading a maze." The various forms of construction of mazes include complicated ranges of caverns, architectural labyrinths, or sepulchral buildings, tortuous devices indicated by coloured marbles and tiled pavements, winding paths cut in the turf, and topiary mazes formed by clipped hedges. As a matter of fact, they may be said to have descended to us in precisely this order of variety. Mazes were used as ornaments on the state robes of Christian emperors before the ninth century, and were soon adopted in the decoration of cathedrals and other churches. The original idea was doubtless to employ them as symbols of the complicated folds of sin by which man is surrounded. They began to abound in the early part of the twelfth century, and I give an illustration of one of this period in the parish church at St. Quentin (Fig. 1). It formed a pavement of the nave, and its diameter is 34½ feet. The path here is the line itself. If you place your pencil at the point A and ignore the enclosing line, the line leads you to the centre by a long route over the entire area; but you never have any option as to direction during your course. As we shall find in similar cases, these early ecclesiastical mazes were generally not of a puzzle nature, but simply long, winding paths that took you over practically all the ground enclosed. [Illustration: FIG. 1.--Maze at St. Quentin.] [Illustration: FIG. 2.--Maze in Chartres Cathedral.] In the abbey church of St. Berlin, at St. Omer, is another of these curious floors, representing the Temple of Jerusalem, with stations for pilgrims. These mazes were actually visited and traversed by them as a compromise for not going to the Holy Land in fulfilment of a vow. They were also used as a means of penance, the penitent frequently being directed to go the whole course of the maze on hands and knees. [Illustration: FIG. 3.--Maze in Lucca Cathedral.] The maze in Chartres Cathedral, of which I give an illustration (Fig. 2), is 40 feet across, and was used by penitents following the procession of Calvary. A labyrinth in Amiens Cathedral was octagonal, similar to that at St. Quentin, measuring 42 feet across. It bore the date 1288, but was destroyed in 1708. In the chapter-house at Bayeux is a labyrinth formed of tiles, red, black, and encaustic, with a pattern of brown and yellow. Dr. Ducarel, in his "_Tour through Part of Normandy_" (printed in 1767), mentions the floor of the great guard-chamber in the abbey of St. Stephen, at Caen, "the middle whereof represents a maze or labyrinth about 10 feet diameter, and so artfully contrived that, were we to suppose a man following all the intricate meanders of its volutes, he could not travel less than a mile before he got from one end to the other." [Illustration: FIG. 4.--Maze at Saffron Walden, Essex.] Then these mazes were sometimes reduced in size and represented on a single tile (Fig. 3). I give an example from Lucca Cathedral. It is on one of the porch piers, and is 19½ inches in diameter. A writer in 1858 says that, "from the continual attrition it has received from thousands of tracing fingers, a central group of Theseus and the Minotaur has now been very nearly effaced." Other examples were, and perhaps still are, to be found in the Abbey of Toussarts, at Châlons-sur-Marne, in the very ancient church of St. Michele at Pavia, at Aix in Provence, in the cathedrals of Poitiers, Rheims, and Arras, in the church of Santa Maria in Aquiro in Rome, in San Vitale at Ravenna, in the Roman mosaic pavement found at Salzburg, and elsewhere. These mazes were sometimes called "Chemins de Jerusalem," as being emblematical of the difficulties attending a journey to the earthly Jerusalem and of those encountered by the Christian before he can reach the heavenly Jerusalem--where the centre was frequently called "Ciel." Common as these mazes were upon the Continent, it is probable that no example is to be found in any English church; at least I am not aware of the existence of any. But almost every county has, or has had, its specimens of mazes cut in the turf. Though these are frequently known as "miz-mazes" or "mize-mazes," it is not uncommon to find them locally called "Troy-towns," "shepherds' races," or "Julian's Bowers"--names that are misleading, as suggesting a false origin. From the facts alone that many of these English turf mazes are clearly copied from those in the Continental churches, and practically all are found close to some ecclesiastical building or near the site of an ancient one, we may regard it as certain that they were of church origin and not invented by the shepherds or other rustics. And curiously enough, these turf mazes are apparently unknown on the Continent. They are distinctly mentioned by Shakespeare:-- "The nine men's morris is filled up with mud, And the quaint mazes in the wanton green For lack of tread are undistinguishable." _A Midsummer Night's Dream_, ii. 1. "My old bones ache: here's a maze trod indeed, Through forth-rights and meanders!" _The Tempest_, iii. 3. [Illustration: FIG. 5.--Maze at Sneinton, Nottinghamshire.] There was such a maze at Comberton, in Cambridgeshire, and another, locally called the "miz-maze," at Leigh, in Dorset. The latter was on the highest part of a field on the top of a hill, a quarter of a mile from the village, and was slightly hollow in the middle and enclosed by a bank about 3 feet high. It was circular, and was thirty paces in diameter. In 1868 the turf had grown over the little trenches, and it was then impossible to trace the paths of the maze. The Comberton one was at the same date believed to be perfect, but whether either or both have now disappeared I cannot say. Nor have I been able to verify the existence or non-existence of the other examples of which I am able to give illustrations. I shall therefore write of them all in the past tense, retaining the hope that some are still preserved. [Illustration: FIG. 6.--Maze at Alkborough, Lincolnshire.] In the next two mazes given--that at Saffron Walden, Essex (110 feet in diameter, Fig. 4), and the one near St. Anne's Well, at Sneinton, Nottinghamshire (Fig. 5), which was ploughed up on February 27th, 1797 (51 feet in diameter, with a path 535 yards long)--the paths must in each case be understood to be on the lines, black or white, as the case may be. [Illustration: FIG. 7.--Maze at Boughton Green, Nottinghamshire.] I give in Fig. 6 a maze that was at Alkborough, Lincolnshire, overlooking the Humber. This was 44 feet in diameter, and the resemblance between it and the mazes at Chartres and Lucca (Figs. 2 and 3) will be at once perceived. A maze at Boughton Green, in Nottinghamshire, a place celebrated at one time for its fair (Fig. 7), was 37 feet in diameter. I also include the plan (Fig. 8) of one that used to be on the outskirts of the village of Wing, near Uppingham, Rutlandshire. This maze was 40 feet in diameter. [Illustration: FIG. 8.--Maze at Wing, Rutlandshire.] [Illustration: FIG. 9.--Maze on St. Catherine's Hill, Winchester.] The maze that was on St. Catherine's Hill, Winchester, in the parish of Chilcombe, was a poor specimen (Fig. 9), since, as will be seen, there was one short direct route to the centre, unless, as in Fig. 10 again, the path is the line itself from end to end. This maze was 86 feet square, cut in the turf, and was locally known as the "Mize-maze." It became very indistinct about 1858, and was then recut by the Warden of Winchester, with the aid of a plan possessed by a lady living in the neighbourhood. [Illustration: FIG. 10.--Maze on Ripon Common.] A maze formerly existed on Ripon Common, in Yorkshire (Fig. 10). It was ploughed up in 1827, but its plan was fortunately preserved. This example was 20 yards in diameter, and its path is said to have been 407 yards long. [Illustration: FIG. 11.--Maze at Theobalds, Hertfordshire.] In the case of the maze at Theobalds, Hertfordshire, after you have found the entrance within the four enclosing hedges, the path is forced (Fig. 11). As further illustrations of this class of maze, I give one taken from an Italian work on architecture by Serlio, published in 1537 (Fig. 12), and one by London and Wise, the designers of the Hampton Court maze, from their book, _The Retired Gard'ner_, published in 1706 (Fig. 13). Also, I add a Dutch maze (Fig. 14). [Illustration: FIG. 12.--Italian Maze of Sixteenth Century.] [Illustration: FIG. 13.--By the Designers of Hampton Court Maze.] [Illustration: FIG. 14.--A Dutch Maze.] So far our mazes have been of historical interest, but they have presented no difficulty in threading. After the Reformation period we find mazes converted into mediums for recreation, and they generally consisted of labyrinthine paths enclosed by thick and carefully trimmed hedges. These topiary hedges were known to the Romans, with whom the _topiarius_ was the ornamental gardener. This type of maze has of late years degenerated into the seaside "Puzzle Gardens. Teas, sixpence, including admission to the Maze." The Hampton Court Maze, sometimes called the "Wilderness," at the royal palace, was designed, as I have said, by London and Wise for William III., who had a liking for such things (Fig. 15). I have before me some three or four versions of it, all slightly different from one another; but the plan I select is taken from an old guide-book to the palace, and therefore ought to be trustworthy. The meaning of the dotted lines, etc., will be explained later on. [Illustration: FIG. 15.--Maze at Hampton Court Palace.] [Illustration: FIG. 16.--Maze at Hatfield House, Herts.] [Illustration: FIG. 17.--Maze formerly at South Kensington.] [Illustration: FIG. 18.--A German Maze.] The maze at Hatfield House (Fig. 16), the seat of the Marquis of Salisbury, like so many labyrinths, is not difficult on paper; but both this and the Hampton Court Maze may prove very puzzling to actually thread without knowing the plan. One reason is that one is so apt to go down the same blind alleys over and over again, if one proceeds without method. The maze planned by the desire of the Prince Consort for the Royal Horticultural Society's Gardens at South Kensington was allowed to go to ruin, and was then destroyed--no great loss, for it was a feeble thing. It will be seen that there were three entrances from the outside (Fig. 17), but the way to the centre is very easy to discover. I include a German maze that is curious, but not difficult to thread on paper (Fig. 18). The example of a labyrinth formerly existing at Pimperne, in Dorset, is in a class by itself (Fig. 19). It was formed of small ridges about a foot high, and covered nearly an acre of ground; but it was, unfortunately, ploughed up in 1730. [Illustration: FIG. 19.--Maze at Pimperne, Dorset.] We will now pass to the interesting subject of how to thread any maze. While being necessarily brief, I will try to make the matter clear to readers who have no knowledge of mathematics. And first of all we will assume that we are trying to enter a maze (that is, get to the "centre") of which we have no plan and about which we know nothing. The first rule is this: If a maze has no parts of its hedges detached from the rest, then if we always keep in touch with the hedge with the right hand (or always touch it with the left), going down to the stop in every blind alley and coming back on the other side, we shall pass through every part of the maze and make our exit where we went in. Therefore we must at one time or another enter the centre, and every alley will be traversed twice. [Illustration: FIG. 20.--M. Tremaux's Method of Solution.] [Illustration: FIG. 21.--How to thread the Hatfield Maze.] Now look at the Hampton Court plan. Follow, say to the right, the path indicated by the dotted line, and what I have said is clearly correct if we obliterate the two detached parts, or "islands," situated on each side of the star. But as these islands are there, you cannot by this method traverse every part of the maze; and if it had been so planned that the "centre" was, like the star, between the two islands, you would never pass through the "centre" at all. A glance at the Hatfield maze will show that there are three of these detached hedges or islands at the centre, so this method will never take you to the "centre" of that one. But the rule will at least always bring you safely out again unless you blunder in the following way. Suppose, when you were going in the direction of the arrow in the Hampton Court Maze, that you could not distinctly see the turning at the bottom, that you imagined you were in a blind alley and, to save time, crossed at once to the opposite hedge, then you would go round and round that U-shaped island with your right hand still always on the hedge--for ever after! [Illustration: FIG. 22. The Philadelphia Maze, and its Solution.] This blunder happened to me a few years ago in a little maze on the isle of Caldy, South Wales. I knew the maze was a small one, but after a very long walk I was amazed to find that I did not either reach the "centre" or get out again. So I threw a piece of paper on the ground, and soon came round to it; from which I knew that I had blundered over a supposed blind alley and was going round and round an island. Crossing to the opposite hedge and using more care, I was quickly at the centre and out again. Now, if I had made a similar mistake at Hampton Court, and discovered the error when at the star, I should merely have passed from one island to another! And if I had again discovered that I was on a detached part, I might with ill luck have recrossed to the first island again! We thus see that this "touching the hedge" method should always bring us safely out of a maze that we have entered; it may happen to take us through the "centre," and if we miss the centre we shall know there must be islands. But it has to be done with a little care, and in no case can we be sure that we have traversed every alley or that there are no detached parts. [Illustration: FIG. 23.--Simplified Diagram of Fig. 22.] If the maze has many islands, the traversing of the whole of it may be a matter of considerable difficulty. Here is a method for solving any maze, due to M. Trémaux, but it necessitates carefully marking in some way your entrances and exits where the galleries fork. I give a diagram of an imaginary maze of a very simple character that will serve our purpose just as well as something more complex (Fig. 20). The circles at the regions where we have a choice of turnings we may call nodes. A "new" path or node is one that has not been entered before on the route; an "old" path or node is one that has already been entered, 1. No path may be traversed more than twice. 2. When you come to a new node, take any path you like. 3. When by a new path you come to an old node or to the stop of a blind alley, return by the path you came. 4. When by an old path you come to an old node, take a new path if there is one; if not, an old path. The route indicated by the dotted line in the diagram is taken in accordance with these simple rules, and it will be seen that it leads us to the centre, although the maze consists of four islands. [Illustration: FIG. 24.--Can you find the Shortest Way to Centre?] Neither of the methods I have given will disclose to us the shortest way to the centre, nor the number of the different routes. But we can easily settle these points with a plan. Let us take the Hatfield maze (Fig. 21). It will be seen that I have suppressed all the blind alleys by the shading. I begin at the stop and work backwards until the path forks. These shaded parts, therefore, can never be entered without our having to retrace our steps. Then it is very clearly seen that if we enter at A we must come out at B; if we enter at C we must come out at D. Then we have merely to determine whether A, B, E, or C, D, E, is the shorter route. As a matter of fact, it will be found by rough measurement or calculation that the shortest route to the centre is by way of C, D, E, F. [Illustration: FIG. 25.--Rosamund's Bower.] I will now give three mazes that are simply puzzles on paper, for, so far as I know, they have never been constructed in any other way. The first I will call the Philadelphia maze (Fig. 22). Fourteen years ago a travelling salesman, living in Philadelphia, U.S.A., developed a curiously unrestrained passion for puzzles. He neglected his business, and soon his position was taken from him. His days and nights were now passed with the subject that fascinated him, and this little maze seems to have driven him into insanity. He had been puzzling over it for some time, and finally it sent him mad and caused him to fire a bullet through his brain. Goodness knows what his difficulties could have been! But there can be little doubt that he had a disordered mind, and that if this little puzzle had not caused him to lose his mental balance some other more or less trivial thing would in time have done so. There is no moral in the story, unless it be that of the Irish maxim, which applies to every occupation of life as much as to the solving of puzzles: "Take things aisy; if you can't take them aisy, take them as aisy as you can." And it is a bad and empirical way of solving any puzzle--by blowing your brains out. Now, how many different routes are there from A to B in this maze if we must never in any route go along the same passage twice? The four open spaces where four passages end are not reckoned as "passages." In the diagram (Fig. 22) it will be seen that I have again suppressed the blind alleys. It will be found that, in any case, we must go from A to C, and also from F to B. But when we have arrived at C there are three ways, marked 1, 2, 3, of getting to D. Similarly, when we get to E there are three ways, marked 4, 5, 6, of getting to F. We have also the dotted route from C to E, the other dotted route from D to F, and the passage from D to E, indicated by stars. We can, therefore, express the position of affairs by the little diagram annexed (Fig. 23). Here every condition of route exactly corresponds to that in the circular maze, only it is much less confusing to the eye. Now, the number of routes, under the conditions, from A to B on this simplified diagram is 640, and that is the required answer to the maze puzzle. Finally, I will leave two easy maze puzzles (Figs. 24, 25) for my readers to solve for themselves. The puzzle in each case is to find the shortest possible route to the centre. Everybody knows the story of Fair Rosamund and the Woodstock maze. What the maze was like or whether it ever existed except in imagination is not known, many writers believing that it was simply a badly-constructed house with a large number of confusing rooms and passages. At any rate, my sketch lacks the authority of the other mazes in this article. My "Rosamund's Bower" is simply designed to show that where you have the plan before you it often happens that the easiest way to find a route into a maze is by working backwards and first finding a way out. THE PARADOX PARTY. "Is not life itself a paradox?" C.L. DODGSON, _Pillow Problems_. "It is a wonderful age!" said Mr. Allgood, and everybody at the table turned towards him and assumed an attitude of expectancy. This was an ordinary Christmas dinner of the Allgood family, with a sprinkling of local friends. Nobody would have supposed that the above remark would lead, as it did, to a succession of curious puzzles and paradoxes, to which every member of the party contributed something of interest. The little symposium was quite unpremeditated, so we must not be too critical respecting a few of the posers that were forthcoming. The varied character of the contributions is just what we would expect on such an occasion, for it was a gathering not of expert mathematicians and logicians, but of quite ordinary folk. "It is a wonderful age!" repeated Mr. Allgood. "A man has just designed a square house in such a cunning manner that all the windows on the four sides have a south aspect." "That would appeal to me," said Mrs. Allgood, "for I cannot endure a room with a north aspect." "I cannot conceive how it is done," Uncle John confessed. "I suppose he puts bay windows on the east and west sides; but how on earth can be contrive to look south from the north side? Does he use mirrors, or something of that kind?" "No," replied Mr. Allgood, "nothing of the sort. All the windows are flush with the walls, and yet you get a southerly prospect from every one of them. You see, there is no real difficulty in designing the house if you select the proper spot for its erection. Now, this house is designed for a gentleman who proposes to build it exactly at the North Pole. If you think a moment you will realize that when you stand at the North Pole it is impossible, no matter which way you may turn, to look elsewhere than due south! There are no such directions as north, east, or west when you are exactly at the North Pole. Everything is due south!" "I am afraid, mother," said her son George, after the laughter had subsided, "that, however much you might like the aspect, the situation would be a little too bracing for you." "Ah, well!" she replied. "Your Uncle John fell also into the trap. I am no good at catches and puzzles. I suppose I haven't the right sort of brain. Perhaps some one will explain this to me. Only last week I remarked to my hairdresser that it had been said that there are more persons in the world than any one of them has hairs on his head. He replied, 'Then it follows, madam, that two persons, at least, must have exactly the same number of hairs on their heads.' If this is a fact, I confess I cannot see it." "How do the bald-headed affect the question?" asked Uncle John. "If there are such persons in existence," replied Mrs. Allgood, "who haven't a solitary hair on their heads discoverable under a magnifying-glass, we will leave them out of the question. Still, I don't see how you are to prove that at least two persons have exactly the same number to a hair." "I think I can make it clear," said Mr. Filkins, who had dropped in for the evening. "Assume the population of the world to be only one million. Any number will do as well as another. Then your statement was to the effect that no person has more than nine hundred and ninety-nine thousand nine hundred and ninety-nine hairs on his head. Is that so?" "Let me think," said Mrs. Allgood. "Yes--yes--that is correct." "Very well, then. As there are only nine hundred and ninety-nine thousand nine hundred and ninety-nine _different_ ways of bearing hair, it is clear that the millionth person must repeat one of those ways. Do you see?" "Yes; I see that--at least I think I see it." "Therefore two persons at least must have the same number of hairs on their heads; and as the number of people on the earth so greatly exceeds the number of hairs on any one person's head, there must, of course, be an immense number of these repetitions." "But, Mr. Filkins," said little Willie Allgood, "why could not the millionth man have, say, ten thousand hairs and a half?" "That is mere hair-splitting, Willie, and does not come into the question." "Here is a curious paradox," said George. "If a thousand soldiers are drawn up in battle array on a plane"--they understood him to mean "plain"--"only one man will stand upright." Nobody could see why. But George explained that, according to Euclid, a plane can touch a sphere only at one point, and that person only who stands at that point, with respect to the centre of the earth, will stand upright. "In the same way," he remarked, "if a billiard-table were quite level--that is, a perfect plane--the balls ought to roll to the centre." Though he tried to explain this by placing a visiting-card on an orange and expounding the law of gravitation, Mrs. Allgood declined to accept the statement. She could not see that the top of a true billiard-table must, theoretically, be spherical, just like a portion of the orange-peel that George cut out. Of course, the table is so small in proportion to the surface of the earth that the curvature is not appreciable, but it is nevertheless true in theory. A surface that we call level is not the same as our idea of a true geometrical plane. "Uncle John," broke in Willie Allgood, "there is a certain island situated between England and France, and yet that island is farther from France than England is. What is the island?" "That seems absurd, my boy; because if I place this tumbler, to represent the island, between these two plates, it seems impossible that the tumbler can be farther from either of the plates than they are from each other." "But isn't Guernsey between England and France?" asked Willie. "Yes, certainly." "Well, then, I think you will find, uncle, that Guernsey is about twenty-six miles from France, and England is only twenty-one miles from France, between Calais and Dover." "My mathematical master," said George, "has been trying to induce me to accept the axiom that 'if equals be multiplied by equals the products are equal.'" "It is self-evident," pointed out Mr. Filkins. "For example, if 3 feet equal 1 yard, then twice 3 feet will equal 2 yards. Do you see?" "But, Mr. Filkins," asked George, "is this tumbler half full of water equal to a similar glass half empty?" "Certainly, George." "Then it follows from the axiom that a glass full must equal a glass empty. Is that correct?" "No, clearly not. I never thought of it in that light." "Perhaps," suggested Mr. Allgood, "the rule does not apply to liquids." "Just what I was thinking, Allgood. It would seem that we must make an exception in the case of liquids." "But it would be awkward," said George, with a smile, "if we also had to except the case of solids. For instance, let us take the solid earth. One mile square equals one square mile. Therefore two miles square must equal two square miles. Is this so?" "Well, let me see! No, of course not," Mr. Filkins replied, "because two miles square is four square miles." "Then," said George, "if the axiom is not true in these cases, when is it true?" Mr. Filkins promised to look into the matter, and perhaps the reader will also like to give it consideration at leisure. "Look here, George," said his cousin Reginald Woolley: "by what fractional part does four-fourths exceed three-fourths?" "By one-fourth!" shouted everybody at once. "Try another one," George suggested. "With pleasure, when you have answered that one correctly," was Reginald's reply. "Do you mean to say that it isn't one-fourth?" "Certainly I do." Several members of the company failed to see that the correct answer is "one-third," although Reginald tried to explain that three of anything, if increased by one-third, becomes four. "Uncle John, how do you pronounce 't-o-o'?" asked Willie. "'Too," my boy." "And how do you pronounce 't-w-o'?" "That is also 'too.'" "Then how do you pronounce the second day of the week?" "Well, that I should pronounce 'Tuesday,' not 'Toosday.'" "Would you really? I should pronounce it 'Monday.'" "If you go on like this, Willie," said Uncle John, with mock severity, "you will soon be without a friend in the world." "Can any of you write down quickly in figures 'twelve thousand twelve hundred and twelve pounds'?" asked Mr. Allgood. His eldest daughter, Miss Mildred, was the only person who happened to have a pencil at hand. "It can't be done," she declared, after making an attempt on the white table-cloth; but Mr. Allgood showed her that it should be written, "£13,212." "Now it is my turn," said Mildred. "I have been waiting to ask you all a question. In the Massacre of the Innocents under Herod, a number of poor little children were buried in the sand with only their feet sticking out. How might you distinguish the boys from the girls?" "I suppose," said Mrs. Allgood, "it is a conundrum--something to do with their poor little 'souls.'" But after everybody had given it up, Mildred reminded the company that only boys were put to death. "Once upon a time," began George, "Achilles had a race with a tortoise--" "Stop, George!" interposed Mr. Allgood. "We won't have that one. I knew two men in my youth who were once the best of friends, but they quarrelled over that infernal thing of Zeno's, and they never spoke to one another again for the rest of their lives. I draw the line at that, and the other stupid thing by Zeno about the flying arrow. I don't believe anybody understands them, because I could never do so myself." "Oh, very well, then, father. Here is another. The Post-Office people were about to erect a line of telegraph-posts over a high hill from Turmitville to Wurzleton; but as it was found that a railway company was making a deep level cutting in the same direction, they arranged to put up the posts beside the line. Now, the posts were to be a hundred yards apart, the length of the road over the hill being five miles, and the length of the level cutting only four and a half miles. How many posts did they save by erecting them on the level?" "That is a very simple matter of calculation," said Mr. Filkins. "Find how many times one hundred yards will go in five miles, and how many times in four and a half miles. Then deduct one from the other, and you have the number of posts saved by the shorter route." "Quite right," confirmed Mr. Allgood. "Nothing could be easier." "That is just what the Post-Office people said," replied George, "but it is quite wrong. If you look at this sketch that I have just made, you will see that there is no difference whatever. If the posts are a hundred yards apart, just the same number will be required on the level as over the surface of the hill." [Illustration] "Surely you must be wrong, George," said Mrs. Allgood, "for if the posts are a hundred yards apart and it is half a mile farther over the hill, you have to put up posts on that extra half-mile." "Look at the diagram, mother. You will see that the distance from post to post is not the distance from base to base measured along the ground. I am just the same distance from you if I stand on this spot on the carpet or stand immediately above it on the chair." But Mrs. Allgood was not convinced. Mr. Smoothly, the curate, at the end of the table, said at this point that he had a little question to ask. "Suppose the earth were a perfect sphere with a smooth surface, and a girdle of steel were placed round the Equator so that it touched at every point." "'I'll put a girdle round about the earth in forty minutes,'" muttered George, quoting the words of Puck in _A Midsummer Night's Dream_. "Now, if six yards were added to the length of the girdle, what would then be the distance between the girdle and the earth, supposing that distance to be equal all round?" "In such a great length," said Mr. Allgood, "I do not suppose the distance would be worth mentioning." "What do you say, George?" asked Mr. Smoothly. "Well, without calculating I should imagine it would be a very minute fraction of an inch." Reginald and Mr. Filkins were of the same opinion. "I think it will surprise you all," said the curate, "to learn that those extra six yards would make the distance from the earth all round the girdle very nearly a yard!" "Very nearly a yard!" everybody exclaimed, with astonishment; but Mr. Smoothly was quite correct. The increase is independent of the original length of the girdle, which may be round the earth or round an orange; in any case the additional six yards will give a distance of nearly a yard all round. This is apt to surprise the non-mathematical mind. "Did you hear the story of the extraordinary precocity of Mrs. Perkins's baby that died last week?" asked Mrs. Allgood. "It was only three months old, and lying at the point of death, when the grief-stricken mother asked the doctor if nothing could save it. 'Absolutely nothing!' said the doctor. Then the infant looked up pitifully into its mother's face and said--absolutely nothing!" "Impossible!" insisted Mildred. "And only three months old!" "There have been extraordinary cases of infantile precocity," said Mr. Filkins, "the truth of which has often been carefully attested. But are you sure this really happened, Mrs. Allgood?" "Positive," replied the lady. "But do you really think it astonishing that a child of three months should say absolutely nothing? What would you expect it to say?" "Speaking of death," said Mr. Smoothly, solemnly, "I knew two men, father and son, who died in the same battle during the South African War. They were both named Andrew Johnson and buried side by side, but there was some difficulty in distinguishing them on the headstones. What would you have done?" "Quite simple," said Mr. Allgood. "They should have described one as 'Andrew Johnson, Senior,' and the other as 'Andrew Johnson, Junior.'" "But I forgot to tell you that the father died first." "What difference can that make?" "Well, you see, they wanted to be absolutely exact, and that was the difficulty." "But I don't see any difficulty," said Mr. Allgood, nor could anybody else. "Well," explained Mr. Smoothly, "it is like this. If the father died first, the son was then no longer 'Junior.' Is that so?" "To be strictly exact, yes." "That is just what they wanted--to be strictly exact. Now, if he was no longer 'Junior,' then he did not die 'Junior." Consequently it must be incorrect so to describe him on the headstone. Do you see the point?" "Here is a rather curious thing," said Mr. Filkins, "that I have just remembered. A man wrote to me the other day that he had recently discovered two old coins while digging in his garden. One was dated '51 B.C.,' and the other one marked 'George I.' How do I know that he was not writing the truth?" "Perhaps you know the man to be addicted to lying," said Reginald. "But that would be no proof that he was not telling the truth in this instance." "Perhaps," suggested Mildred, "you know that there were no coins made at those dates. "On the contrary, they were made at both periods." "Were they silver or copper coins?" asked Willie. "My friend did not state, and I really cannot see, Willie, that it makes any difference." "I see it!" shouted Reginald. "The letters 'B.C.' would never be used on a coin made before the birth of Christ. They never anticipated the event in that way. The letters were only adopted later to denote dates previous to those which we call 'A.D.' That is very good; but I cannot see why the other statement could not be correct." "Reginald is quite right," said Mr. Filkins, "about the first coin. The second one could not exist, because the first George would never be described in his lifetime as 'George I.'" "Why not?" asked Mrs. Allgood. "He _was_ George I." "Yes; but they would not know it until there was a George II." "Then there was no George II. until George III. came to the throne?" "That does not follow. The second George becomes 'George II.' on account of there having been a 'George I.'" "Then the first George was 'George I.' on account of there having been no king of that name before him." "Don't you see, mother," said George Allgood, "we did not call Queen Victoria 'Victoria I.;' but if there is ever a 'Victoria II.,' then she will be known that way." "But there _have_ been several Georges, and therefore he was 'George I.' There _haven't_ been several Victorias, so the two cases are not similar." They gave up the attempt to convince Mrs. Allgood, but the reader will, of course, see the point clearly. "Here is a question," said Mildred Allgood, "that I should like some of you to settle for me. I am accustomed to buy from our greengrocer bundles of asparagus, each 12 inches in circumference. I always put a tape measure round them to make sure I am getting the full quantity. The other day the man had no large bundles in stock, but handed me instead two small ones, each 6 inches in circumference. 'That is the same thing,' I said, 'and, of course, the price will be the same;' but he insisted that the two bundles together contained more than the large one, and charged me a few pence extra. Now, what I want to know is, which of us was correct? Would the two small bundles contain the same quantity as the large one? Or would they contain more?" "That is the ancient puzzle," said Reginald, laughing, "of the sack of corn that Sempronius borrowed from Caius, which your greengrocer, perhaps, had been reading about somewhere. He caught you beautifully." "Then they were equal?" "On the contrary, you were both wrong, and you were badly cheated. You only got half the quantity that would have been contained in a large bundle, and therefore ought to have been charged half the original price, instead of more." Yes, it was a bad swindle, undoubtedly. A circle with a circumference half that of another must have its area a quarter that of the other. Therefore the two small bundles contained together only half as much asparagus as a large one. "Mr. Filkins, can you answer this?" asked Willie. "There is a man in the next village who eats two eggs for breakfast every morning." "Nothing very extraordinary in that," George broke in. "If you told us that the two eggs ate the man it would be interesting." "Don't interrupt the boy, George," said his mother. "Well," Willie continued, "this man neither buys, borrows, barters, begs, steals, nor finds the eggs. He doesn't keep hens, and the eggs are not given to him. How does he get the eggs?" "Does he take them in exchange for something else?" asked Mildred. "That would be bartering them," Willie replied. "Perhaps some friend sends them to him," suggested Mrs. Allgood. "I said that they were not given to him." "I know," said George, with confidence. "A strange hen comes into his place and lays them." "But that would be finding them, wouldn't it?" "Does he hire them?" asked Reginald. "If so, he could not return them after they were eaten, so that would be stealing them." "Perhaps it is a pun on the word 'lay,'" Mr. Filkins said. "Does he lay them on the table?" "He would have to get them first, wouldn't he? The question was, How does he get them?" "Give it up!" said everybody. Then little Willie crept round to the protection of his mother, for George was apt to be rough on such occasions. "The man keeps ducks!" he cried, "and his servant collects the eggs every morning." "But you said he doesn't keep birds!" George protested. "I didn't, did I, Mr. Filkins? I said he doesn't keep hens." "But he finds them," said Reginald. "No; I said his servant finds them." "Well, then," Mildred interposed, "his servant gives them to him." "You cannot give a man his own property, can you?" All agreed that Willie's answer was quite satisfactory. Then Uncle John produced a little fallacy that "brought the proceedings to a close," as the newspapers say. 413.--A CHESSBOARD FALLACY. [Illustration] "Here is a diagram of a chessboard," he said. "You see there are sixty-four squares--eight by eight. Now I draw a straight line from the top left-hand corner, where the first and second squares meet, to the bottom right-hand corner. I cut along this line with the scissors, slide up the piece that I have marked B, and then clip off the little corner C by a cut along the first upright line. This little piece will exactly fit into its place at the top, and we now have an oblong with seven squares on one side and nine squares on the other. There are, therefore, now only sixty-three squares, because seven multiplied by nine makes sixty-three. Where on earth does that lost square go to? I have tried over and over again to catch the little beggar, but he always eludes me. For the life of me I cannot discover where he hides himself." "It seems to be like the other old chessboard fallacy, and perhaps the explanation is the same," said Reginald--"that the pieces do not exactly fit." "But they _do_ fit," said Uncle John. "Try it, and you will see." Later in the evening Reginald and George, were seen in a corner with their heads together, trying to catch that elusive little square, and it is only fair to record that before they retired for the night they succeeded in securing their prey, though some others of the company failed to see it when captured. Can the reader solve the little mystery? UNCLASSIFIED PROBLEMS. "A snapper up of unconsidered trifles." _Winter's Tale_, iv. 2. 414.--WHO WAS FIRST? Anderson, Biggs, and Carpenter were staying together at a place by the seaside. One day they went out in a boat and were a mile at sea when a rifle was fired on shore in their direction. Why or by whom the shot was fired fortunately does not concern us, as no information on these points is obtainable, but from the facts I picked up we can get material for a curious little puzzle for the novice. It seems that Anderson only heard the report of the gun, Biggs only saw the smoke, and Carpenter merely saw the bullet strike the water near them. Now, the question arises: Which of them first knew of the discharge of the rifle? 415.--A WONDERFUL VILLAGE. There is a certain village in Japan, situated in a very low valley, and yet the sun is nearer to the inhabitants every noon, by 3,000 miles and upwards, than when he either rises or sets to these people. In what part of the country is the village situated? 416.--A CALENDAR PUZZLE. If the end of the world should come on the first day of a new century, can you say what are the chances that it will happen on a Sunday? 417.--THE TIRING IRONS. [Illustration] The illustration represents one of the most ancient of all mechanical puzzles. Its origin is unknown. Cardan, the mathematician, wrote about it in 1550, and Wallis in 1693; while it is said still to be found in obscure English villages (sometimes deposited in strange places, such as a church belfry), made of iron, and appropriately called "tiring-irons," and to be used by the Norwegians to-day as a lock for boxes and bags. In the toyshops it is sometimes called the "Chinese rings," though there seems to be no authority for the description, and it more frequently goes by the unsatisfactory name of "the puzzling rings." The French call it "Baguenaudier." The puzzle will be seen to consist of a simple _loop_ of wire fixed in a handle to be held in the left hand, and a certain number of _rings_ secured by _wires_ which pass through holes in the _bar_ and are kept there by their blunted ends. The wires work freely in the bar, but cannot come apart from it, nor can the wires be removed from the rings. The general puzzle is to detach the loop completely from all the rings, and then to put them all on again. Now, it will be seen at a glance that the first ring (to the right) can be taken off at any time by sliding it over the end and dropping it through the loop; or it may be put on by reversing the operation. With this exception, the only ring that can ever be removed is the one that happens to be a contiguous second on the loop at the right-hand end. Thus, with all the rings on, the second can be dropped at once; with the first ring down, you cannot drop the second, but may remove the third; with the first three rings down, you cannot drop the fourth, but may remove the fifth; and so on. It will be found that the first and second rings can be dropped together or put on together; but to prevent confusion we will throughout disallow this exceptional double move, and say that only one ring may be put on or removed at a time. We can thus take off one ring in 1 move; two rings in 2 moves; three rings in 5 moves; four rings in 10 moves; five rings in 21 moves; and if we keep on doubling (and adding one where the number of rings is odd) we may easily ascertain the number of moves for completely removing any number of rings. To get off all the seven rings requires 85 moves. Let us look at the five moves made in removing the first three rings, the circles above the line standing for rings on the loop and those under for rings off the loop. Drop the first ring; drop the third; put up the first; drop the second; and drop the first--5 moves, as shown clearly in the diagrams. The dark circles show at each stage, from the starting position to the finish, which rings it is possible to drop. After move 2 it will be noticed that no ring can be dropped until one has been put on, because the first and second rings from the right now on the loop are not together. After the fifth move, if we wish to remove all seven rings we must now drop the fifth. But before we can then remove the fourth it is necessary to put on the first three and remove the first two. We shall then have 7, 6, 4, 3 on the loop, and may therefore drop the fourth. When we have put on 2 and 1 and removed 3, 2, 1, we may drop the seventh ring. The next operation then will be to get 6, 5, 4, 3, 2, 1 on the loop and remove 4, 3, 2, 1, when 6 will come off; then get 5, 4, 3, 2, 1 on the loop, and remove 3, 2, 1, when 5 will come off; then get 4, 3, 2, 1 on the loop and remove 2, 1, when 4 will come off; then get 3, 2, 1 on the loop and remove 1, when 3 will come off; then get 2, 1 on the loop, when 2 will come off; and 1 will fall through on the 85th move, leaving the loop quite free. The reader should now be able to understand the puzzle, whether or not he has it in his hand in a practical form. [Illustration] [Illustration: o o o o o * * {------------- o o o o * o 1{------------- o o o o o o 2{------------- o o o o o o * * 3{------------- o o o o o * 4{------------- o o o o * o 5{------------- o o o ] The particular problem I propose is simply this. Suppose there are altogether fourteen rings on the tiring-irons, and we proceed to take them all off in the correct way so as not to waste any moves. What will be the position of the rings after the 9,999th move has been made? 418.--SUCH A GETTING UPSTAIRS. In a suburban villa there is a small staircase with eight steps, not counting the landing. The little puzzle with which Tommy Smart perplexed his family is this. You are required to start from the bottom and land twice on the floor above (stopping there at the finish), having returned once to the ground floor. But you must be careful to use every tread the same number of times. In how few steps can you make the ascent? It seems a very simple matter, but it is more than likely that at your first attempt you will make a great many more steps than are necessary. Of course you must not go more than one riser at a time. Tommy knows the trick, and has shown it to his father, who professes to have a contempt for such things; but when the children are in bed the pater will often take friends out into the hall and enjoy a good laugh at their bewilderment. And yet it is all so very simple when you know how it is done. 419.--THE FIVE PENNIES. Here is a really hard puzzle, and yet its conditions are so absurdly simple. Every reader knows how to place four pennies so that they are equidistant from each other. All you have to do is to arrange three of them flat on the table so that they touch one another in the form of a triangle, and lay the fourth penny on top in the centre. Then, as every penny touches every other penny, they are all at equal distances from one another. Now try to do the same thing with five pennies--place them so that every penny shall touch every other penny--and you will find it a different matter altogether. 420.--THE INDUSTRIOUS BOOKWORM. [Illustration] Our friend Professor Rackbrane is seen in the illustration to be propounding another of his little posers. He is explaining that since he last had occasion to take down those three volumes of a learned book from their place on his shelves a bookworm has actually bored a hole straight through from the first page to the last. He says that the leaves are together three inches thick in each volume, and that every cover is exactly one-eighth of an inch thick, and he asks how long a tunnel had the industrious worm to bore in preparing his new tube railway. Can you tell him? 421.--A CHAIN PUZZLE. [Illustration] This is a puzzle based on a pretty little idea first dealt with by the late Mr. Sam Loyd. A man had nine pieces of chain, as shown in the illustration. He wanted to join these fifty links into one endless chain. It will cost a penny to open any link and twopence to weld a link together again, but he could buy a new endless chain of the same character and quality for 2s. 2d. What was the cheapest course for him to adopt? Unless the reader is cunning he may find himself a good way out in his answer. 422.--THE SABBATH PUZZLE. I have come across the following little poser in an old book. I wonder how many readers will see the author's intended solution to the riddle. Christians the week's _first_ day for Sabbath hold; The Jews the _seventh_, as they did of old; The Turks the _sixth_, as we have oft been told. How can these three, in the same place and day, Have each his own true Sabbath? tell, I pray. 423.--THE RUBY BROOCH. The annals of Scotland Yard contain some remarkable cases of jewel robberies, but one of the most perplexing was the theft of Lady Littlewood's rubies. There have, of course, been many greater robberies in point of value, but few so artfully conceived. Lady Littlewood, of Romley Manor, had a beautiful but rather eccentric heirloom in the form of a ruby brooch. While staying at her town house early in the eighties she took the jewel to a shop in Brompton for some slight repairs. "A fine collection of rubies, madam," said the shopkeeper, to whom her ladyship was a stranger. "Yes," she replied; "but curiously enough I have never actually counted them. My mother once pointed out to me that if you start from the centre and count up one line, along the outside and down the next line, there are always eight rubies. So I should always know if a stone were missing." [Illustration] Six months later a brother of Lady Littlewood's, who had returned from his regiment in India, noticed that his sister was wearing the ruby brooch one night at a county ball, and on their return home asked to look at it more closely. He immediately detected the fact that four of the stones were gone. "How can that possibly be?" said Lady Littlewood. "If you count up one line from the centre, along the edge, and down the next line, in any direction, there are always eight stones. This was always so and is so now. How, therefore, would it be possible to remove a stone without my detecting it?" "Nothing could be simpler," replied the brother. "I know the brooch well. It originally contained forty-five stones, and there are now only forty-one. Somebody has stolen four rubies, and then reset as small a number of the others as possible in such a way that there shall always be eight in any of the directions you have mentioned." There was not the slightest doubt that the Brompton jeweller was the thief, and the matter was placed in the hands of the police. But the man was wanted for other robberies, and had left the neighbourhood some time before. To this day he has never been found. The interesting little point that at first baffled the police, and which forms the subject of our puzzle, is this: How were the forty-five rubies originally arranged on the brooch? The illustration shows exactly how the forty-one were arranged after it came back from the jeweller; but although they count eight correctly in any of the directions mentioned, there are four stones missing. 424.--THE DOVETAILED BLOCK. [Illustration] Here is a curious mechanical puzzle that was given to me some years ago, but I cannot say who first invented it. It consists of two solid blocks of wood securely dovetailed together. On the other two vertical sides that are not visible the appearance is precisely the same as on those shown. How were the pieces put together? When I published this little puzzle in a London newspaper I received (though they were unsolicited) quite a stack of models, in oak, in teak, in mahogany, rosewood, satinwood, elm, and deal; some half a foot in length, and others varying in size right down to a delicate little model about half an inch square. It seemed to create considerable interest. 425.--JACK AND THE BEANSTALK. [Illustration] The illustration, by a British artist, is a sketch of Jack climbing the beanstalk. Now, the artist has made a serious blunder in this drawing. Can you find out what it is? 426.--THE HYMN-BOARD POSER. The worthy vicar of Chumpley St. Winifred is in great distress. A little church difficulty has arisen that all the combined intelligence of the parish seems unable to surmount. What this difficulty is I will state hereafter, but it may add to the interest of the problem if I first give a short account of the curious position that has been brought about. It all has to do with the church hymn-boards, the plates of which have become so damaged that they have ceased to fulfil the purpose for which they were devised. A generous parishioner has promised to pay for a new set of plates at a certain rate of cost; but strange as it may seem, no agreement can be come to as to what that cost should be. The proposed maker of the plates has named a price which the donor declares to be absurd. The good vicar thinks they are both wrong, so he asks the schoolmaster to work out the little sum. But this individual declares that he can find no rule bearing on the subject in any of his arithmetic books. An application having been made to the local medical practitioner, as a man of more than average intellect at Chumpley, he has assured the vicar that his practice is so heavy that he has not had time even to look at it, though his assistant whispers that the doctor has been sitting up unusually late for several nights past. Widow Wilson has a smart son, who is reputed to have once won a prize for puzzle-solving. He asserts that as he cannot find any solution to the problem it must have something to do with the squaring of the circle, the duplication of the cube, or the trisection of an angle; at any rate, he has never before seen a puzzle on the principle, and he gives it up. [Illustration] This was the state of affairs when the assistant curate (who, I should say, had frankly confessed from the first that a profound study of theology had knocked out of his head all the knowledge of mathematics he ever possessed) kindly sent me the puzzle. A church has three hymn-boards, each to indicate the numbers of five different hymns to be sung at a service. All the boards are in use at the same service. The hymn-book contains 700 hymns. A new set of numbers is required, and a kind parishioner offers to present a set painted on metal plates, but stipulates that only the smallest number of plates necessary shall be purchased. The cost of each plate is to be 6d., and for the painting of each plate the charges are to be: For one plate, 1s.; for two plates alike, 11¾d. each; for three plates alike, 11½d. each, and so on, the charge being one farthing less per plate for each similarly painted plate. Now, what should be the lowest cost? Readers will note that they are required to use every legitimate and practical method of economy. The illustration will make clear the nature of the three hymn-boards and plates. The five hymns are here indicated by means of twelve plates. These plates slide in separately at the back, and in the illustration there is room, of course, for three more plates. 427.--PHEASANT-SHOOTING. A Cockney friend, who is very apt to draw the long bow, and is evidently less of a sportsman than he pretends to be, relates to me the following not very credible yarn:-- "I've just been pheasant-shooting with my friend the duke. We had splendid sport, and I made some wonderful shots. What do you think of this, for instance? Perhaps you can twist it into a puzzle. The duke and I were crossing a field when suddenly twenty-four pheasants rose on the wing right in front of us. I fired, and two-thirds of them dropped dead at my feet. Then the duke had a shot at what were left, and brought down three-twenty-fourths of them, wounded in the wing. Now, out of those twenty-four birds, how many still remained?" It seems a simple enough question, but can the reader give a correct answer? 428.--THE GARDENER AND THE COOK. A correspondent, signing himself "Simple Simon," suggested that I should give a special catch puzzle in the issue of _The Weekly Dispatch_ for All Fools' Day, 1900. So I gave the following, and it caused considerable amusement; for out of a very large body of competitors, many quite expert, not a single person solved it, though it ran for nearly a month. [Illustration] "The illustration is a fancy sketch of my correspondent, 'Simple Simon,' in the act of trying to solve the following innocent little arithmetical puzzle. A race between a man and a woman that I happened to witness one All Fools' Day has fixed itself indelibly on my memory. It happened at a country-house, where the gardener and the cook decided to run a race to a point 100 feet straight away and return. I found that the gardener ran 3 feet at every bound and the cook only 2 feet, but then she made three bounds to his two. Now, what was the result of the race?" A fortnight after publication I added the following note: "It has been suggested that perhaps there is a catch in the 'return,' but there is not. The race is to a point 100 feet away and home again--that is, a distance of 200 feet. One correspondent asks whether they take exactly the same time in turning, to which I reply that they do. Another seems to suspect that it is really a conundrum, and that the answer is that 'the result of the race was a (matrimonial) tie.' But I had no such intention. The puzzle is an arithmetical one, as it purports to be." 429.--PLACING HALFPENNIES. [Illustration] Here is an interesting little puzzle suggested to me by Mr. W. T. Whyte. Mark off on a sheet of paper a rectangular space 5 inches by 3 inches, and then find the greatest number of halfpennies that can be placed within the enclosure under the following conditions. A halfpenny is exactly an inch in diameter. Place your first halfpenny where you like, then place your second coin at exactly the distance of an inch from the first, the third an inch distance from the second, and so on. No halfpenny may touch another halfpenny or cross the boundary. Our illustration will make the matter perfectly clear. No. 2 coin is an inch from No. 1; No. 3 an inch from No. 2; No. 4 an inch from No. 3; but after No. 10 is placed we can go no further in this attempt. Yet several more halfpennies might have been got in. How many can the reader place? 430.--FIND THE MAN'S WIFE. [Illustration] One summer day in 1903 I was loitering on the Brighton front, watching the people strolling about on the beach, when the friend who was with me suddenly drew my attention to an individual who was standing alone, and said, "Can you point out that man's wife? They are stopping at the same hotel as I am, and the lady is one of those in view." After a few minutes' observation, I was successful in indicating the lady correctly. My friend was curious to know by what method of reasoning I had arrived at the result. This was my answer:-- "We may at once exclude that Sister of Mercy and the girl in the short frock; also the woman selling oranges. It cannot be the lady in widows' weeds. It is not the lady in the bath chair, because she is not staying at your hotel, for I happened to see her come out of a private house this morning assisted by her maid. The two ladies in red breakfasted at my hotel this morning, and as they were not wearing outdoor dress I conclude they are staying there. It therefore rests between the lady in blue and the one with the green parasol. But the left hand that holds the parasol is, you see, ungloved and bears no wedding-ring. Consequently I am driven to the conclusion that the lady in blue is the man's wife--and you say this is correct." Now, as my friend was an artist, and as I thought an amusing puzzle might be devised on the lines of his question, I asked him to make me a drawing according to some directions that I gave him, and I have pleasure in presenting his production to my readers. It will be seen that the picture shows six men and six ladies: Nos. 1, 3, 5, 7, 9, and 11 are ladies, and Nos. 2, 4, 6, 8, 10, and 12 are men. These twelve individuals represent six married couples, all strangers to one another, who, in walking aimlessly about, have got mixed up. But we are only concerned with the man that is wearing a straw hat--Number 10. The puzzle is to find this man's wife. Examine the six ladies carefully, and see if you can determine which one of them it is. I showed the picture at the time to a few friends, and they expressed very different opinions on the matter. One said, "I don't believe he would marry a girl like Number 7." Another said, "I am sure a nice girl like Number 3 would not marry such a fellow!" Another said, "It must be Number 1, because she has got as far away as possible from the brute!" It was suggested, again, that it must be Number 11, because "he seems to be looking towards her;" but a cynic retorted, "For that very reason, if he is really looking at her, I should say that she is not his wife!" I now leave the question in the hands of my readers. Which is really Number 10's wife? The illustration is of necessity considerably reduced from the large scale on which it originally appeared in _The Weekly Dispatch_ (24th May 1903), but it is hoped that the details will be sufficiently clear to allow the reader to derive entertainment from its examination. In any case the solution given will enable him to follow the points with interest. SOLUTIONS. 1.--A POST-OFFICE PERPLEXITY. The young lady supplied 5 twopenny stamps, 30 penny stamps, and 8 twopence-halfpenny stamps, which delivery exactly fulfils the conditions and represents a cost of five shillings. 2.--YOUTHFUL PRECOCITY. The price of the banana must have been one penny farthing. Thus, 960 bananas would cost £5, and 480 sixpences would buy 2,304 bananas. 3.--AT A CATTLE MARKET. Jakes must have taken 7 animals to market, Hodge must have taken 11, and Durrant must have taken 21. There were thus 39 animals altogether. 4.--THE BEANFEAST PUZZLE. The cobblers spent 35s., the tailors spent also 35s., the hatters spent 42s., and the glovers spent 21s. Thus, they spent altogether £6,13s., while it will be found that the five cobblers spent as much as four tailors, twelve tailors as much as nine hatters, and six hatters as much as eight glovers. 5.--A QUEER COINCIDENCE. Puzzles of this class are generally solved in the old books by the tedious process of "working backwards." But a simple general solution is as follows: If there are n players, the amount held by every player at the end will be m(2^n), the last winner must have held m(n + 1) at the start, the next m(2n + 1), the next m(4n + 1), the next m(8n + 1), and so on to the first player, who must have held m(2^{n - 1}n + 1). Thus, in this case, n = 7, and the amount held by every player at the end was 2^7 farthings. Therefore m = 1, and G started with 8 farthings, F with 15, E with 29, D with 57, C with 113, B with 225, and A with 449 farthings. 6.--A CHARITABLE BEQUEST. There are seven different ways in which the money may be distributed: 5 women and 19 men, 10 women and 16 men, 15 women and 13 men, 20 women and 10 men, 25 women and 7 men, 30 women and 4 men, and 35 women and 1 man. But the last case must not be counted, because the condition was that there should be "men," and a single man is not men. Therefore the answer is six years. 7.--THE WIDOW'S LEGACY. The widow's share of the legacy must be £205, 2s. 6d. and 10/13 of a penny. 8.--INDISCRIMINATE CHARITY The gentleman must have had 3s. 6d. in his pocket when he set out for home. 9.--THE TWO AEROPLANES. The man must have paid £500 and £750 for the two machines, making together £1,250; but as he sold them for only £1,200, he lost £50 by the transaction. 10.--BUYING PRESENTS. Jorkins had originally £19, 18s. in his pocket, and spent £9, 19s. 11.--THE CYCLISTS' FEAST. There were ten cyclists at the feast. They should have paid 8s. each; but, owing to the departure of two persons, the remaining eight would pay 10s. each. 12.--A QUEER THING IN MONEY. The answer is as follows: £44,444, 4s. 4d. = 28, and, reduced to pence, 10,666,612=28. It is a curious little coincidence that in the answer 10,666,612 the four central figures indicate the only other answer, £66, 6s. 6d. 13.--A NEW MONEY PUZZLE. The smallest sum of money, in pounds, shillings, pence, and farthings, containing all the nine digits once, and once only, is £2,567, 18s. 9¾d. 14.--SQUARE MONEY. The answer is 1½d. and 3d. Added together they make 4½d., and 1½d. multiplied by 3 is also 4½d. 15.--POCKET MONEY. The largest possible sum is 15s. 9d., composed of a crown and a half-crown (or three half-crowns), four florins, and a threepenny piece. 16.--THE MILLIONAIRE'S PERPLEXITY. The answer to this quite easy puzzle may, of course, be readily obtained by trial, deducting the largest power of 7 that is contained in one million dollars, then the next largest power from the remainder, and so on. But the little problem is intended to illustrate a simple direct method. The answer is given at once by converting 1,000,000 to the septenary scale, and it is on this subject of scales of notation that I propose to write a few words for the benefit of those who have never sufficiently considered the matter. Our manner of figuring is a sort of perfected arithmetical shorthand, a system devised to enable us to manipulate numbers as rapidly and correctly as possible by means of symbols. If we write the number 2,341 to represent two thousand three hundred and forty-one dollars, we wish to imply 1 dollar, added to four times 10 dollars, added to three times 100 dollars, added to two times 1,000 dollars. From the number in the units place on the right, every figure to the left is understood to represent a multiple of the particular power of 10 that its position indicates, while a cipher (0) must be inserted where necessary in order to prevent confusion, for if instead of 207 we wrote 27 it would be obviously misleading. We thus only require ten figures, because directly a number exceeds 9 we put a second figure to the left, directly it exceeds 99 we put a third figure to the left, and so on. It will be seen that this is a purely arbitrary method. It is working in the denary (or ten) scale of notation, a system undoubtedly derived from the fact that our forefathers who devised it had ten fingers upon which they were accustomed to count, like our children of to-day. It is unnecessary for us ordinarily to state that we are using the denary scale, because this is always understood in the common affairs of life. But if a man said that he had 6,553 dollars in the septenary (or seven) scale of notation, you will find that this is precisely the same amount as 2,341 in our ordinary denary scale. Instead of using powers of ten, he uses powers of 7, so that he never needs any figure higher than 6, and 6,553 really stands for 3, added to five times 7, added to five times 49, added to six times 343 (in the ordinary notation), or 2,341. To reverse the operation, and convert 2,341 from the denary to the septenary scale, we divide it by 7, and get 334 and remainder 3; divide 334 by 7, and get 47 and remainder 5; and so keep on dividing by 7 as long as there is anything to divide. The remainders, read backwards, 6, 5, 5, 3, give us the answer, 6,553. Now, as I have said, our puzzle may be solved at once by merely converting 1,000,000 dollars to the septenary scale. Keep on dividing this number by 7 until there is nothing more left to divide, and the remainders will be found to be 11333311 which is 1,000,000 expressed in the septenary scale. Therefore, 1 gift of 1 dollar, 1 gift of 7 dollars, 3 gifts of 49 dollars, 3 gifts of 343 dollars, 3 gifts of 2,401 dollars, 3 gifts of 16,807 dollars, 1 gift of 117,649 dollars, and one substantial gift of 823,543 dollars, satisfactorily solves our problem. And it is the only possible solution. It is thus seen that no "trials" are necessary; by converting to the septenary scale of notation we go direct to the answer. 17.--THE PUZZLING MONEY BOXES. The correct answer to this puzzle is as follows: John put into his money-box two double florins (8s.), William a half-sovereign and a florin (12s.), Charles a crown (5s.), and Thomas a sovereign (20s.). There are six coins in all, of a total value of 45s. If John had 2s. more, William 2s. less, Charles twice as much, and Thomas half as much as they really possessed, they would each have had exactly 10s. 18.--THE MARKET WOMEN. The price received was in every case 105 farthings. Therefore the greatest number of women is eight, as the goods could only be sold at the following rates: 105 lbs. at 1 farthing, 35 at 3, 21 at 5, 15 at 7, 7 at 15, 5 at 21, 3 at 35, and 1 lb. at 105 farthings. 19.--THE NEW YEAR'S EVE SUPPERS. The company present on the occasion must have consisted of seven pairs, ten single men, and one single lady. Thus, there were twenty-five persons in all, and at the prices stated they would pay exactly £5 together. 20.--BEEF AND SAUSAGES. The lady bought 48 lbs. of beef at 2s., and the same quantity of sausages at 1s. 6d., thus spending £8, 8s. Had she bought 42 lbs. of beef and 56 lbs. of sausages she would have spent £4, 4s. on each, and have obtained 98 lbs. instead of 96 lbs.--a gain in weight of 2 lbs. 21.--A DEAL IN APPLES. I was first offered sixteen apples for my shilling, which would be at the rate of ninepence a dozen. The two extra apples gave me eighteen for a shilling, which is at the rate of eightpence a dozen, or one penny a dozen less than the first price asked. 22.--A DEAL IN EGGS. The man must have bought ten eggs at fivepence, ten eggs at one penny, and eighty eggs at a halfpenny. He would then have one hundred eggs at a cost of eight shillings and fourpence, and the same number of eggs of two of the qualities. 23.--THE CHRISTMAS-BOXES. The distribution took place "some years ago," when the fourpenny-piece was in circulation. Nineteen persons must each have received nineteen pence. There are five different ways in which this sum may have been paid in silver coins. We need only use two of these ways. Thus if fourteen men each received four four-penny-pieces and one threepenny-piece, and five men each received five threepenny-pieces and one fourpenny-piece, each man would receive nineteen pence, and there would be exactly one hundred coins of a total value of £1, 10s. 1d. 24.--A SHOPPING PERPLEXITY. The first purchase amounted to 1s. 5¾d., the second to 1s. 11½d., and together they make 3s. 5¼d. Not one of these three amounts can be paid in fewer than six current coins of the realm. 25.--CHINESE MONEY. As a ching-chang is worth twopence and four-fifteenths of a ching-chang, the remaining eleven-fifteenths of a ching-chang must be worth twopence. Therefore eleven ching-changs are worth exactly thirty pence, or half a crown. Now, the exchange must be made with seven round-holed coins and one square-holed coin. Thus it will be seen that 7 round-holed coins are worth seven-elevenths of 15 ching-changs, and 1 square-holed coin is worth one-eleventh of 16 ching-changs--that is, 77 rounds equal 105 ching-changs and 11 squares equal 16 ching-changs. Therefore 77 rounds added to 11 squares equal 121 ching-changs; or 7 rounds and 1 square equal 11 ching-changs, or its equivalent, half a crown. This is more simple in practice than it looks here. 26.--THE JUNIOR CLERKS' PUZZLE. Although Snoggs's _reason_ for wishing to take his rise at £2, 10s. half-yearly did not concern our puzzle, the _fact_ that he was duping his employer into paying him more than was intended did concern it. Many readers will be surprised to find that, although Moggs only received £350 in five years, the artful Snoggs actually obtained £362, 10s. in the same time. The rest is simplicity itself. It is evident that if Moggs saved £87, 10s. and Snoggs £181, 5s., the latter would be saving twice as great a proportion of his salary as the former (namely, one-half as against one-quarter), and the two sums added together make £268, 15s. 27.--GIVING CHANGE. The way to help the American tradesman out of his dilemma is this. Describing the coins by the number of cents that they represent, the tradesman puts on the counter 50 and 25; the buyer puts down 100, 3, and 2; the stranger adds his 10, 10, 5, 2, and 1. Now, considering that the cost of the purchase amounted to 34 cents, it is clear that out of this pooled money the tradesman has to receive 109, the buyer 71, and the stranger his 28 cents. Therefore it is obvious at a glance that the 100-piece must go to the tradesman, and it then follows that the 50-piece must go to the buyer, and then the 25-piece can only go to the stranger. Another glance will now make it clear that the two 10-cent pieces must go to the buyer, because the tradesman now only wants 9 and the stranger 3. Then it becomes obvious that the buyer must take the 1 cent, that the stranger must take the 3 cents, and the tradesman the 5, 2, and 2. To sum up, the tradesman takes 100, 5, 2, and 2; the buyer, 50, 10, 10, and 1; the stranger, 25 and 3. It will be seen that not one of the three persons retains any one of his own coins. 28.--DEFECTIVE OBSERVATION. Of course the date on a penny is on the same side as Britannia--the "tail" side. Six pennies may be laid around another penny, all flat on the table, so that every one of them touches the central one. The number of threepenny-pieces that may be laid on the surface of a half-crown, so that no piece lies on another or overlaps the edge of the half-crown, is one. A second threepenny-piece will overlap the edge of the larger coin. Few people guess fewer than three, and many persons give an absurdly high number. 29.--THE BROKEN COINS. If the three broken coins when perfect were worth 253 pence, and are now in their broken condition worth 240 pence, it should be obvious that 13/253 of the original value has been lost. And as the same fraction of each coin has been broken away, each coin has lost 13/253 of its original bulk. 30.--TWO QUESTIONS IN PROBABILITIES. In tossing with the five pennies all at the same time, it is obvious that there are 32 different ways in which the coins may fall, because the first coin may fall in either of two ways, then the second coin may also fall in either of two ways, and so on. Therefore five 2's multiplied together make 32. Now, how are these 32 ways made up? Here they are:-- (a) 5 heads 1 way (b) 5 tails 1 way (c) 4 heads and 1 tail 5 ways (d) 4 tails and 1 head 5 ways (e) 3 heads and 2 tails 10 ways (f) 3 tails and 2 heads 10 ways Now, it will be seen that the only favourable cases are a, b, c, and d--12 cases. The remaining 20 cases are unfavourable, because they do not give at least four heads or four tails. Therefore the chances are only 12 to 20 in your favour, or (which is the same thing) 3 to 5. Put another way, you have only 3 chances out of 8. The amount that should be paid for a draw from the bag that contains three sovereigns and one shilling is 15s. 3d. Many persons will say that, as one's chances of drawing a sovereign were 3 out of 4, one should pay three-fourths of a pound, or 15s., overlooking the fact that one must draw at least a shilling--there being no blanks. 31.--DOMESTIC ECONOMY. Without the hint that I gave, my readers would probably have been unanimous in deciding that Mr. Perkins's income must have been £1,710. But this is quite wrong. Mrs. Perkins says, "We have spent a third of his yearly income in rent," etc., etc.--that is, in two years they have spent an amount in rent, etc., equal to one-third of his yearly income. Note that she does _not_ say that they have spent _each year_ this sum, whatever it is, but that _during the two years_ that amount has been spent. The only possible answer, according to the exact reading of her words, is, therefore, that his income was £180 per annum. Thus the amount spent in two years, during which his income has amounted to £360, will be £60 in rent, etc., £90 in domestic expenses, £20 in other ways, leaving the balance of £190 in the bank as stated. 32.--THE EXCURSION TICKET PUZZLE. Nineteen shillings and ninepence may be paid in 458,908,622 different ways. I do not propose to give my method of solution. Any such explanation would occupy an amount of space out of proportion to its interest or value. If I could give within reasonable limits a general solution for all money payments, I would strain a point to find room; but such a solution would be extremely complex and cumbersome, and I do not consider it worth the labour of working out. Just to give an idea of what such a solution would involve, I will merely say that I find that, dealing only with those sums of money that are multiples of threepence, if we only use bronze coins any sum can be paid in (n + 1)² ways where n always represents the number of pence. If threepenny-pieces are admitted, there are 2n³ + 15n² + 33n --------------------- + 1 ways. 18 If sixpences are also used there are n^{4} + 22n³ + 159n² + 414n + 216 --------------------------------- 216 ways, when the sum is a multiple of sixpence, and the constant, 216, changes to 324 when the money is not such a multiple. And so the formulas increase in complexity in an accelerating ratio as we go on to the other coins. I will, however, add an interesting little table of the possible ways of changing our current coins which I believe has never been given in a book before. Change may be given for a Farthing in 0 way. Halfpenny in 1 way. Penny in 3 ways. Threepenny-piece in 16 ways. Sixpence in 66 ways. Shilling in 402 ways. Florin in 3,818 ways. Half-crown in 8,709 ways. Double florin in 60,239 ways. Crown in 166,651 ways. Half-sovereign in 6,261,622 ways. Sovereign in 500,291,833 ways. It is a little surprising to find that a sovereign may be changed in over five hundred million different ways. But I have no doubt as to the correctness of my figures. 33.--A PUZZLE IN REVERSALS. (i) £13. (2) £23, 19s. 11d. The words "the number of pounds exceeds that of the pence" exclude such sums of money as £2, 16s. 2d. and all sums under £1. 34.--THE GROCER AND DRAPER. The grocer was delayed half a minute and the draper eight minutes and a half (seventeen times as long as the grocer), making together nine minutes. Now, the grocer took twenty-four minutes to weigh out the sugar, and, with the half-minute delay, spent 24 min. 30 sec. over the task; but the draper had only to make _forty-seven_ cuts to divide the roll of cloth, containing forty-eight yards, into yard pieces! This took him 15 min. 40 sec., and when we add the eight minutes and a half delay we get 24 min. 10 sec., from which it is clear that the draper won the race by twenty seconds. The majority of solvers make forty-eight cuts to divide the roll into forty-eight pieces! 35.--JUDKINS'S CATTLE. As there were five droves with an equal number of animals in each drove, the number must be divisible by 5; and as every one of the eight dealers bought the same number of animals, the number must be divisible by 8. Therefore the number must be a multiple of 40. The highest possible multiple of 40 that will work will be found to be 120, and this number could be made up in one of two ways--1 ox, 23 pigs, and 96 sheep, or 3 oxen, 8 pigs, and 109 sheep. But the first is excluded by the statement that the animals consisted of "oxen, pigs, and sheep," because a single ox is not oxen. Therefore the second grouping is the correct answer. 36.--BUYING APPLES. As there were the same number of boys as girls, it is clear that the number of children must be even, and, apart from a careful and exact reading of the question, there would be three different answers. There might be two, six, or fourteen children. In the first of these cases there are ten different ways in which the apples could be bought. But we were told there was an equal number of "boys and girls," and one boy and one girl are not boys and girls, so this case has to be excluded. In the case of fourteen children, the only possible distribution is that each child receives one halfpenny apple. But we were told that each child was to receive an equal distribution of "apples," and one apple is not apples, so this case has also to be excluded. We are therefore driven back on our third case, which exactly fits in with all the conditions. Three boys and three girls each receive 1 halfpenny apple and 2 third-penny apples. The value of these 3 apples is one penny and one-sixth, which multiplied by six makes sevenpence. Consequently, the correct answer is that there were six children--three girls and three boys. 37.--BUYING CHESTNUTS. In solving this little puzzle we are concerned with the exact interpretation of the words used by the buyer and seller. I will give the question again, this time adding a few words to make the matter more clear. The added words are printed in italics. "A man went into a shop to buy chestnuts. He said he wanted a pennyworth, and was given five chestnuts. 'It is not enough; I ought to have a sixth _of a chestnut more_,' he remarked. 'But if I give you one chestnut more,' the shopman replied, 'you will have _five-sixths_ too many.' Now, strange to say, they were both right. How many chestnuts should the buyer receive for half a crown?" The answer is that the price was 155 chestnuts for half a crown. Divide this number by 30, and we find that the buyer was entitled to 5+1/6 chestnuts in exchange for his penny. He was, therefore, right when he said, after receiving five only, that he still wanted a sixth. And the salesman was also correct in saying that if he gave one chestnut more (that is, six chestnuts in all) he would be giving five-sixths of a chestnut in excess. 38.--THE BICYCLE THIEF. People give all sorts of absurd answers to this question, and yet it is perfectly simple if one just considers that the salesman cannot possibly have lost more than the cyclist actually stole. The latter rode away with a bicycle which cost the salesman eleven pounds, and the ten pounds "change;" he thus made off with twenty-one pounds, in exchange for a worthless bit of paper. This is the exact amount of the salesman's loss, and the other operations of changing the cheque and borrowing from a friend do not affect the question in the slightest. The loss of prospective profit on the sale of the bicycle is, of course, not direct loss of money out of pocket. 39.--THE COSTERMONGER'S PUZZLE. Bill must have paid 8s. per hundred for his oranges--that is, 125 for 10s. At 8s. 4d. per hundred, he would only have received 120 oranges for 10s. This exactly agrees with Bill's statement. 40.--MAMMA'S AGE. The age of Mamma must have been 29 years 2 months; that of Papa, 35 years; and that of the child, Tommy, 5 years 10 months. Added together, these make seventy years. The father is six times the age of the son, and, after 23 years 4 months have elapsed, their united ages will amount to 140 years, and Tommy will be just half the age of his father. 41.--THEIR AGES. The gentleman's age must have been 54 years and that of his wife 45 years. 42.--THE FAMILY AGES. The ages were as follows: Billie, 3½ years; Gertrude, 1¾ year; Henrietta, 5¼ years; Charlie, 10½; years; and Janet, 21 years. 43.--MRS. TIMPKINS'S AGE. The age of the younger at marriage is always the same as the number of years that expire before the elder becomes twice her age, if he was three times as old at marriage. In our case it was eighteen years afterwards; therefore Mrs. Timpkins was eighteen years of age on the wedding-day, and her husband fifty-four. 44.--A CENSUS PUZZLE. Miss Ada Jorkins must have been twenty-four and her little brother Johnnie three years of age, with thirteen brothers and sisters between. There was a trap for the solver in the words "seven times older than little Johnnie." Of course, "seven times older" is equal to eight times as old. It is surprising how many people hastily assume that it is the same as "seven times as old." Some of the best writers have committed this blunder. Probably many of my readers thought that the ages 24½ and 3½ were correct. 45.--MOTHER AND DAUGHTER. In four and a half years, when the daughter will be sixteen years and a half and the mother forty-nine and a half years of age. 46.--MARY AND MARMADUKE. Marmaduke's age must have been twenty-nine years and two-fifths, and Mary's nineteen years and three-fifths. When Marmaduke was aged nineteen and three-fifths, Mary was only nine and four-fifths; so Marmaduke was at that time twice her age. 47.--ROVER'S AGE. Rover's present age is ten years and Mildred's thirty years. Five years ago their respective ages were five and twenty-five. Remember that we said "four times older than the dog," which is the same as "five times as old." (See answer to No. 44.) 48.--CONCERNING TOMMY'S AGE. Tommy Smart's age must have been nine years and three-fifths. Ann's age was sixteen and four-fifths, the mother's thirty-eight and two-fifths, and the father's fifty and two-fifths. 49.--NEXT-DOOR NEIGHBOURS. Mr. Jupp 39, Mrs. Jupp 34, Julia 14, and Joe 13; Mr. Simkin 42; Mrs. Simkin 40; Sophy 10; and Sammy 8. 50.--THE BAG OF NUTS. It will be found that when Herbert takes twelve, Robert and Christopher will take nine and fourteen respectively, and that they will have together taken thirty-five nuts. As 35 is contained in 770 twenty-two times, we have merely to multiply 12, 9, and 14 by 22 to discover that Herbert's share was 264, Robert's 198, and Christopher's 308. Then, as the total of their ages is 17½ years or half the sum of 12, 9, and 14, their respective ages must be 6, 4½, and 7 years. 51.--HOW OLD WAS MARY? The age of Mary to that of Ann must be as 5 to 3. And as the sum of their ages was 44, Mary was 27½ and Ann 16½. One is exactly 11 years older than the other. I will now insert in brackets in the original statement the various ages specified: "Mary is (27½) twice as old as Ann was (13¾) when Mary was half as old (24¾) as Ann will be (49½) when Ann is three times as old (49½) as Mary was (16½) when Mary was (16½) three times as old as Ann (5½)." Now, check this backwards. When Mary was three times as old as Ann, Mary was 16½ and Ann 5½ (11 years younger). Then we get 49½ for the age Ann will be when she is three times as old as Mary was then. When Mary was half this she was 24¾. And at that time Ann must have been 13¾ (11 years younger). Therefore Mary is now twice as old--27½, and Ann 11 years younger--16½. 52.--QUEER RELATIONSHIPS. If a man marries a woman, who dies, and he then marries his deceased wife's sister and himself dies, it may be correctly said that he had (previously) married the sister of his widow. The youth was not the nephew of Jane Brown, because he happened to be her son. Her surname was the same as that of her brother, because she had married a man of the same name as herself. 53.--HEARD ON THE TUBE RAILWAY. The gentleman was the second lady's uncle. 54.--A FAMILY PARTY. The party consisted of two little girls and a boy, their father and mother, and their father's father and mother. 55.--A MIXED PEDIGREE. [Illustration: Thos. Bloggs m . . . . . | +------------------------+------------+ | | | | | | | W. Snoggs m Kate Bloggs. | | | | | | | . . m Henry Bloggs. | Joseph Bloggs m | | | | +--------+-------------+ | | | | | | | | | Jane John Alf. Mary Bloggs m Snoggs Snoggs m Bloggs ] The letter m stands for "married." It will be seen that John Snoggs can say to Joseph Bloggs, "You are my _father's brother-in-law_, because my father married your sister Kate; you are my _brother's father-in-law_, because my brother Alfred married your daughter Mary; and you are my _father-in-law's brother_, because my wife Jane was your brother Henry's daughter." 56.--WILSON'S POSER. If there are two men, each of whom marries the mother of the other, and there is a son of each marriage, then each of such sons will be at the same time uncle and nephew of the other. There are other ways in which the relationship may be brought about, but this is the simplest. 57.--WHAT WAS THE TIME? The time must have been 9.36 p.m. A quarter of the time since noon is 2 hr. 24 min., and a half of the time till noon next day is 7 hr. 12 min. These added together make 9 hr. 36 min. 58.--A TIME PUZZLE. Twenty-six minutes. 59.--A PUZZLING WATCH. If the 65 minutes be counted on the face of the same watch, then the problem would be impossible: for the hands must coincide every 65+5/11 minutes as shown by its face, and it matters not whether it runs fast or slow; but if it is measured by true time, it gains 5/11 of a minute in 65 minutes, or 60/143 of a minute per hour. 60.--THE WAPSHAW'S WHARF MYSTERY. There are eleven different times in twelve hours when the hour and minute hands of a clock are exactly one above the other. If we divide 12 hours by 11 we get 1 hr. 5 min. 27+3/11 sec., and this is the time after twelve o'clock when they are first together, and also the time that elapses between one occasion of the hands being together and the next. They are together for the second time at 2 hr. 10 min. 54+6/11 sec. (twice the above time); next at 3 hr. 16 min. 21+9/11 sec.; next at 4 hr. 21 min. 49+1/11 sec. This last is the only occasion on which the two hands are together with the second hand "just past the forty-ninth second." This, then, is the time at which the watch must have stopped. Guy Boothby, in the opening sentence of his _Across the World for a Wife_, says, "It was a cold, dreary winter's afternoon, and by the time the hands of the clock on my mantelpiece joined forces and stood at twenty minutes past four, my chambers were well-nigh as dark as midnight." It is evident that the author here made a slip, for, as we have seen above, he is 1 min. 49+1/11 sec. out in his reckoning. 61.--CHANGING PLACES. There are thirty-six pairs of times when the hands exactly change places between three p.m. and midnight. The number of pairs of times from any hour (n) to midnight is the sum of 12 - (n + 1) natural numbers. In the case of the puzzle n = 3; therefore 12 - (3 + 1) = 8 and 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, the required answer. The first pair of times is 3 hr. 21+57/143 min. and 4 hr. 16+112/143 min., and the last pair is 10 hr. 59+83/143 min. and 11 hr. 54+138/143 min. I will not give all the remainder of the thirty-six pairs of times, but supply a formula by which any of the sixty-six pairs that occur from midday to midnight may be at once found:-- 720b + 60a 720a + 60b min. a hr ---------- min. and b hr. --------------- 143 143 For the letter a may be substituted any hour from 0, 1, 2, 3 up to 10 (where nought stands for 12 o'clock midday); and b may represent any hour, later than a, up to 11. By the aid of this formula there is no difficulty in discovering the answer to the second question: a = 8 and b = 11 will give the pair 8 hr. 58+106/143 min. and 11 hr. 44+128/143 min., the latter being the time when the minute hand is nearest of all to the point IX--in fact, it is only 15/143 of a minute distant. Readers may find it instructive to make a table of all the sixty-six pairs of times when the hands of a clock change places. An easy way is as follows: Make a column for the first times and a second column for the second times of the pairs. By making a = 0 and b = 1 in the above expressions we find the first case, and enter hr. 5+5/143 min. at the head of the first column, and 1 hr. 0+60/143 min. at the head of the second column. Now, by successively adding 5+5/143 min. in the first, and 1 hr. 0+60/143 min. in the second column, we get all the _eleven_ pairs in which the first time is a certain number of minutes after nought, or mid-day. Then there is a "jump" in the times, but you can find the next pair by making a = 1 and b = 2, and then by successively adding these two times as before you will get all the _ten_ pairs after 1 o'clock. Then there is another "jump," and you will be able to get by addition all the _nine_ pairs after 2 o'clock. And so on to the end. I will leave readers to investigate for themselves the nature and cause of the "jumps." In this way we get under the successive hours, 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 66 pairs of times, which result agrees with the formula in the first paragraph of this article. Some time ago the principal of a Civil Service Training College, who conducts a "Civil Service Column" in one of the periodicals, had the query addressed to him, "How soon after XII o'clock will a clock with both hands of the same length be ambiguous?" His first answer was, "Some time past one o'clock," but he varied the answer from issue to issue. At length some of his readers convinced him that the answer is, "At 5+5/143 min. past XII;" and this he finally gave as correct, together with the reason for it that at that time _the time indicated is the same whichever hand you may assume as hour hand!_ 62.--THE CLUB CLOCK. The positions of the hands shown in the illustration could only indicate that the clock stopped at 44 min. 51+1143/1427 sec. after eleven o'clock. The second hand would next be "exactly midway between the other two hands" at 45 min. 52+496/1427 sec. after eleven o'clock. If we had been dealing with the points on the circle to which the three hands are directed, the answer would be 45 min. 22+106/1427 sec. after eleven; but the question applied to the hands, and the second hand would not be between the others at that time, but outside them. 63.--THE STOP-WATCH. The time indicated on the watch was 5+5/11 min. past 9, when the second hand would be at 27+3/11 sec. The next time the hands would be similar distances apart would be 54+6/11 min. past 2, when the second hand would be at 32+8/11 sec. But you need only hold the watch (or our previous illustration of it) in front of a mirror, when you will see the second time reflected in it! Of course, when reflected, you will read XI as I, X as II, and so on. 64.--THE THREE CLOCKS. As a mere arithmetical problem this question presents no difficulty. In order that the hands shall all point to twelve o'clock at the same time, it is necessary that B shall gain at least twelve hours and that C shall lose twelve hours. As B gains a minute in a day of twenty-four hours, and C loses a minute in precisely the same time, it is evident that one will have gained 720 minutes (just twelve hours) in 720 days, and the other will have lost 720 minutes in 720 days. Clock A keeping perfect time, all three clocks must indicate twelve o'clock simultaneously at noon on the 720th day from April 1, 1898. What day of the month will that be? I published this little puzzle in 1898 to see how many people were aware of the fact that 1900 would not be a leap year. It was surprising how many were then ignorant on the point. Every year that can be divided by four without a remainder is bissextile or leap year, with the exception that one leap year is cut off in the century. 1800 was not a leap year, nor was 1900. On the other hand, however, to make the calendar more nearly agree with the sun's course, every fourth hundred year is still considered bissextile. Consequently, 2000, 2400, 2800, 3200, etc., will all be leap years. May my readers live to see them. We therefore find that 720 days from noon of April 1, 1898, brings us to noon of March 22, 1900. 65.--THE RAILWAY STATION CLOCK. The time must have been 43+7/11 min. past two o'clock. 66.--THE VILLAGE SIMPLETON. The day of the week on which the conversation took place was Sunday. For when the day after to-morrow (Tuesday) is "yesterday," "to-day" will be Wednesday; and when the day before yesterday (Friday) was "to-morrow," "to-day" was Thursday. There are two days between Thursday and Sunday, and between Sunday and Wednesday. 67.--AVERAGE SPEED. The average speed is twelve miles an hour, not twelve and a half, as most people will hastily declare. Take any distance you like, say sixty miles. This would have taken six hours going and four hours returning. The double journey of 120 miles would thus take ten hours, and the average speed is clearly twelve miles an hour. 68.--THE TWO TRAINS. One train was running just twice as fast as the other. 69.--THE THREE VILLAGES. Calling the three villages by their initial letters, it is clear that the three roads form a triangle, A, B, C, with a perpendicular, measuring twelve miles, dropped from C to the base A, B. This divides our triangle into two right-angled triangles with a twelve-mile side in common. It is then found that the distance from A to C is 15 miles, from C to B 20 miles, and from A to B 25 (that is 9 and 16) miles. These figures are easily proved, for the square of 12 added to the square of 9 equals the square of 15, and the square of 12 added to the square of 16 equals the square of 20. 70.--DRAWING HER PENSION. The distance must be 6¾ miles. 71.--SIR EDWYN DE TUDOR. The distance must have been sixty miles. If Sir Edwyn left at noon and rode 15 miles an hour, he would arrive at four o'clock--an hour too soon. If he rode 10 miles an hour, he would arrive at six o'clock--an hour too late. But if he went at 12 miles an hour, he would reach the castle of the wicked baron exactly at five o'clock--the time appointed. 72.--THE HYDROPLANE QUESTION. The machine must have gone at the rate of seven-twenty-fourths of a mile per minute and the wind travelled five-twenty-fourths of a mile per minute. Thus, going, the wind would help, and the machine would do twelve-twenty-fourths, or half a mile a minute, and returning only two-twenty-fourths, or one-twelfth of a mile per minute, the wind being against it. The machine without any wind could therefore do the ten miles in thirty-four and two-sevenths minutes, since it could do seven miles in twenty-four minutes. 73.--DONKEY RIDING. The complete mile was run in nine minutes. From the facts stated we cannot determine the time taken over the first and second quarter-miles separately, but together they, of course, took four and a half minutes. The last two quarters were run in two and a quarter minutes each. 74.--THE BASKET OF POTATOES. Multiply together the number of potatoes, the number less one, and twice the number less one, then divide by 3. Thus 50, 49, and 99 multiplied together make 242,550, which, divided by 3, gives us 80,850 yards as the correct answer. The boy would thus have to travel 45 miles and fifteen-sixteenths--a nice little recreation after a day's work. 75.--THE PASSENGER'S FARE. Mr. Tompkins should have paid fifteen shillings as his correct share of the motor-car fare. He only shared half the distance travelled for £3, and therefore should pay half of thirty shillings, or fifteen shillings. 76.--THE BARREL OF BEER. Here the digital roots of the six numbers are 6, 4, 1, 2, 7, 9, which together sum to 29, whose digital root is 2. As the contents of the barrels sold must be a number divisible by 3, if one buyer purchased twice as much as the other, we must find a barrel with root 2, 5, or 8 to set on one side. There is only one barrel, that containing 20 gallons, that fulfils these conditions. So the man must have kept these 20 gallons of beer for his own use and sold one man 33 gallons (the 18-gallon and 15-gallon barrels) and sold the other man 66 gallons (the 16, 19, and 31 gallon barrels). 77.--DIGITS AND SQUARES. The top row must be one of the four following numbers: 192, 219, 273, 327. The first was the example given. 78.--ODD AND EVEN DIGITS. As we have to exclude complex and improper fractions and recurring decimals, the simplest solution is this: 79 + 5+1/3 and 84 + 2/6, both equal 84+1/3. Without any use of fractions it is obviously impossible. 79.--THE LOCKERS PUZZLE. The smallest possible total is 356 = 107 + 249, and the largest sum possible is 981 = 235 + 746, or 657+324. The middle sum may be either 720 = 134 + 586, or 702 = 134 + 568, or 407 = 138 + 269. The total in this case must be made up of three of the figures 0, 2, 4, 7, but no sum other than the three given can possibly be obtained. We have therefore no choice in the case of the first locker, an alternative in the case of the third, and any one of three arrangements in the case of the middle locker. Here is one solution:-- 107 134 235 249 586 746 --- --- --- 356 720 981 Of course, in each case figures in the first two lines may be exchanged vertically without altering the total, and as a result there are just 3,072 different ways in which the figures might be actually placed on the locker doors. I must content myself with showing one little principle involved in this puzzle. The sum of the digits in the total is always governed by the digit omitted. 9/9 - 7/10 - 5/11 -3/12 - 1/13 - 8/14 - 6/15 - 4/16 - 2/17 - 0/18. Whichever digit shown here in the upper line we omit, the sum of the digits in the total will be found beneath it. Thus in the case of locker A we omitted 8, and the figures in the total sum up to 14. If, therefore, we wanted to get 356, we may know at once to a certainty that it can only be obtained (if at all) by dropping the 8. 80.--THE THREE GROUPS. There are nine solutions to this puzzle, as follows, and no more:-- 12 × 483 = 5,796 27 × 198 = 5,346 42 × 138 = 5,796 39 × 186 = 7,254 18 × 297 = 5,346 48 × 159 = 7,632 28 × 157 = 4,396 4 × 1,738 = 6,952 4 × 1,963 = 7,852 The seventh answer is the one that is most likely to be overlooked by solvers of the puzzle. 81.--THE NINE COUNTERS. In this case a certain amount of mere "trial" is unavoidable. But there are two kinds of "trials"--those that are purely haphazard, and those that are methodical. The true puzzle lover is never satisfied with mere haphazard trials. The reader will find that by just reversing the figures in 23 and 46 (making the multipliers 32 and 64) both products will be 5,056. This is an improvement, but it is not the correct answer. We can get as large a product as 5,568 if we multiply 174 by 32 and 96 by 58, but this solution is not to be found without the exercise of some judgment and patience. 82.--THE TEN COUNTERS. As I pointed out, it is quite easy so to arrange the counters that they shall form a pair of simple multiplication sums, each of which will give the same product--in fact, this can be done by anybody in five minutes with a little patience. But it is quite another matter to find that pair which gives the largest product and that which gives the smallest product. Now, in order to get the smallest product, it is necessary to select as multipliers the two smallest possible numbers. If, therefore, we place 1 and 2 as multipliers, all we have to do is to arrange the remaining eight counters in such a way that they shall form two numbers, one of which is just double the other; and in doing this we must, of course, try to make the smaller number as low as possible. Of course the lowest number we could get would be 3,045; but this will not work, neither will 3,405, 3,45O, etc., and it may be ascertained that 3,485 is the lowest possible. One of the required answers is 3,485 × 2 = 6,970, and 6,970 × 1 = 6,970. The other part of the puzzle (finding the pair with the highest product) is, however, the real knotty point, for it is not at all easy to discover whether we should let the multiplier consist of one or of two figures, though it is clear that we must keep, so far as we can, the largest figures to the left in both multiplier and multiplicand. It will be seen that by the following arrangement so high a number as 58,560 may be obtained. Thus, 915 × 64 = 58,560, and 732 × 80 = 58,560. 83.--DIGITAL MULTIPLICATION. The solution that gives the smallest possible sum of digits in the common product is 23 × 174 = 58 × 69 = 4,002, and the solution that gives the largest possible sum of digits, 9×654 =18×327=5,886. In the first case the digits sum to 6 and in the second case to 27. There is no way of obtaining the solution but by actual trial. 84.--THE PIERROT'S PUZZLE. There are just six different solutions to this puzzle, as follows:-- 8 multiplied by 473 equals 3784 9 " 351 " 3159 15 " 93 " 1395 21 " 87 " 1287 27 " 81 " 2187 35 " 41 " 1435 It will be seen that in every case the two multipliers contain exactly the same figures as the product. 85.--THE CAB NUMBERS. The highest product is, I think, obtained by multiplying 8,745,231 by 96--namely, 839,542,176. Dealing here with the problem generally, I have shown in the last puzzle that with three digits there are only two possible solutions, and with four digits only six different solutions. These cases have all been given. With five digits there are just twenty-two solutions, as follows:-- 3 × 4128 = 12384 3 × 4281 = 12843 3 × 7125 = 21375 3 × 7251 = 21753 2541 × 6 = 15246 651 × 24 = 15624 678 × 42 = 28476 246 × 51 = 12546 57 × 834 = 47538 75 × 231 = 17325 624 × 78 = 48672 435 × 87 = 37845 ------ 9 × 7461 = 67149 72 × 936 = 67392 ------ 2 × 8714 = 17428 2 × 8741 = 17482 65 × 281 = 18265 65 × 983 = 63985 ------ 4973 × 8 = 39784 6521 × 8 = 52168 14 × 926 = 12964 86 × 251 = 21586 Now, if we took every possible combination and tested it by multiplication, we should need to make no fewer than 30,240 trials, or, if we at once rejected the number 1 as a multiplier, 28,560 trials--a task that I think most people would be inclined to shirk. But let us consider whether there be no shorter way of getting at the results required. I have already explained that if you add together the digits of any number and then, as often as necessary, add the digits of the result, you must ultimately get a number composed of one figure. This last number I call the "digital root." It is necessary in every solution of our problem that the root of the sum of the digital roots of our multipliers shall be the same as the root of their product. There are only four ways in which this can happen: when the digital roots of the multipliers are 3 and 6, or 9 and 9, or 2 and 2, or 5 and 8. I have divided the twenty-two answers above into these four classes. It is thus evident that the digital root of any product in the first two classes must be 9, and in the second two classes 4. Owing to the fact that no number of five figures can have a digital sum less than 15 or more than 35, we find that the figures of our product must sum to either 18 or 27 to produce the root 9, and to either 22 or 31 to produce the root 4. There are 3 ways of selecting five different figures that add up to 18, there are 11 ways of selecting five figures that add up to 27, there are 9 ways of selecting five figures that add up to 22, and 5 ways of selecting five figures that add up to 31. There are, therefore, 28 different groups, and no more, from any one of which a product may be formed. We next write out in a column these 28 sets of five figures, and proceed to tabulate the possible factors, or multipliers, into which they may be split. Roughly speaking, there would now appear to be about 2,000 possible cases to be tried, instead of the 30,240 mentioned above; but the process of elimination now begins, and if the reader has a quick eye and a clear head he can rapidly dispose of the large bulk of these cases, and there will be comparatively few test multiplications necessary. It would take far too much space to explain my own method in detail, but I will take the first set of figures in my table and show how easily it is done by the aid of little tricks and dodges that should occur to everybody as he goes along. My first product group of five figures is 84,321. Here, as we have seen, the root of each factor must be 3 or a multiple of 3. As there is no 6 or 9, the only single multiplier is 3. Now, the remaining four figures can be arranged in 24 different ways, but there is no need to make 24 multiplications. We see at a glance that, in order to get a five-figure product, either the 8 or the 4 must be the first figure to the left. But unless the 2 is preceded on the right by the 8, it will produce when multiplied either a 6 or a 7, which must not occur. We are, therefore, reduced at once to the two cases, 3 × 4,128 and 3 x 4,281, both of which give correct solutions. Suppose next that we are trying the two-figure factor, 21. Here we see that if the number to be multiplied is under 500 the product will either have only four figures or begin with 10. Therefore we have only to examine the cases 21 × 843 and 21 × 834. But we know that the first figure will be repeated, and that the second figure will be twice the first figure added to the second. Consequently, as twice 3 added to 4 produces a nought in our product, the first case is at once rejected. It only remains to try the remaining case by multiplication, when we find it does not give a correct answer. If we are next trying the factor 12, we see at the start that neither the 8 nor the 3 can be in the units place, because they would produce a 6, and so on. A sharp eye and an alert judgment will enable us thus to run through our table in a much shorter time than would be expected. The process took me a little more than three hours. I have not attempted to enumerate the solutions in the cases of six, seven, eight, and nine digits, but I have recorded nearly fifty examples with nine digits alone. 86.--QUEER MULTIPLICATION. If we multiply 32547891 by 6, we get the product, 195287346. In both cases all the nine digits are used once and once only. 87.--THE NUMBER CHECKS PUZZLE. Divide the ten checks into the following three groups: 7 1 5--4 6--3 2 8 9 0, and the first multiplied by the second produces the third. 88.--DIGITAL DIVISION. It is convenient to consider the digits as arranged to form fractions of the respective values, one-half, one-third, one-fourth, one-fifth, one-sixth, one-seventh, one-eighth, and one-ninth. I will first give the eight answers, as follows:-- 6729/13458 = 1/2 5823/17469 = 1/3 3942/15768 = 1/4 2697/13485 = 1/5 2943/17658 = 1/6 2394/16758 = 1/7 3187/25496 = 1/8 6381/57429 = 1/9 The sum of the numerator digits and the denominator digits will, of course, always be 45, and the "digital root" is 9. Now, if we separate the nine digits into any two groups, the sum of the two digital roots will always be 9. In fact, the two digital roots must be either 9--9, 8--1, 7--2, 6--3, or 5--4. In the first case the actual sum is 18, but then the digital root of this number is itself 9. The solutions in the cases of one-third, one-fourth, one-sixth, one-seventh, and one-ninth must be of the form 9--9; that is to say, the digital roots of both numerator and denominator will be 9. In the cases of one-half and one-fifth, however, the digital roots are 6--3, but of course the higher root may occur either in the numerator or in the denominator; thus 2697/13485, 2769/13845, 2973/14865, 3729/18645, where, in the first two arrangements, the roots of the numerator and denominator are respectively 6--3, and in the last two 3--6. The most curious case of all is, perhaps, one-eighth, for here the digital roots may be of any one of the five forms given above. The denominators of the fractions being regarded as the numerators multiplied by 2, 3, 4, 5, 6, 7, 8, and 9 respectively, we must pay attention to the "carryings over." In order to get five figures in the product there will, of course, always be a carry-over after multiplying the last figure to the left, and in every case higher than 4 we must carry over at least three times. Consequently in cases from one-fifth to one-ninth we cannot produce different solutions by a mere change of position of pairs of figures, as, for example, we may with 5832/17496 and 5823/17469, where the 2/6 and 3/9 change places. It is true that the same figures may often be differently arranged, as shown in the two pairs of values for one-fifth that I have given in the last paragraph, but here it will be found there is a general readjustment of figures and not a simple changing of the positions of pairs. There are other little points that would occur to every solver--such as that the figure 5 cannot ever appear to the extreme right of the numerator, as this would result in our getting either a nought or a second 5 in the denominator. Similarly 1 cannot ever appear in the same position, nor 6 in the fraction one-sixth, nor an even figure in the fraction one-fifth, and so on. The preliminary consideration of such points as I have touched upon will not only prevent our wasting a lot of time in trying to produce impossible forms, but will lead us more or less directly to the desired solutions. 89.--ADDING THE DIGITS. The smallest possible sum of money is £1, 8s. 9¾d., the digits of which add to 25. 90.--THE CENTURY PUZZLE. The problem of expressing the number 100 as a mixed number or fraction, using all the nine digits once, and once only, has, like all these digital puzzles, a fascinating side to it. The merest tyro can by patient trial obtain correct results, and there is a singular pleasure in discovering and recording each new arrangement akin to the delight of the botanist in finding some long-sought plant. It is simply a matter of arranging those nine figures correctly, and yet with the thousands of possible combinations that confront us the task is not so easy as might at first appear, if we are to get a considerable number of results. Here are eleven answers, including the one I gave as a specimen:-- 2148 1752 1428 1578 96 ----, 96 ----, 96 ----, 94 ----, 537 438 357 263 7524 5823 5742 3546 91 ----, 91 ----, 91 ----, 82 ----, 836 647 638 197 7524 5643 69258 81 ----, 81 ----, 3 -----. 396 297 714 Now, as all the fractions necessarily represent whole numbers, it will be convenient to deal with them in the following form: 96 + 4, 94 + 6, 91 + 9, 82 + 18, 81 + 19, and 3 + 97. With any whole number the digital roots of the fraction that brings it up to 100 will always be of one particular form. Thus, in the case of 96 + 4, one can say at once that if any answers are obtainable, then the roots of both the numerator and the denominator of the fraction will be 6. Examine the first three arrangements given above, and you will find that this is so. In the case of 94 + 6 the roots of the numerator and denominator will be respectively 3--2, in the case of 91 + 9 and of 82 + 18 they will be 9--8, in the case of 81 + 19 they will be 9--9, and in the case of 3 + 97 they will be 3--3. Every fraction that can be employed has, therefore, its particular digital root form, and you are only wasting your time in unconsciously attempting to break through this law. Every reader will have perceived that certain whole numbers are evidently impossible. Thus, if there is a 5 in the whole number, there will also be a nought or a second 5 in the fraction, which are barred by the conditions. Then multiples of 10, such as 90 and 80, cannot of course occur, nor can the whole number conclude with a 9, like 89 and 79, because the fraction, equal to 11 or 21, will have 1 in the last place, and will therefore repeat a figure. Whole numbers that repeat a figure, such as 88 and 77, are also clearly useless. These cases, as I have said, are all obvious to every reader. But when I declare that such combinations as 98 + 2, 92 + 8, 86 + 14, 83 + 17, 74 + 26, etc., etc., are to be at once dismissed as impossible, the reason is not so evident, and I unfortunately cannot spare space to explain it. But when all those combinations have been struck out that are known to be impossible, it does not follow that all the remaining "possible forms" will actually work. The elemental form may be right enough, but there are other and deeper considerations that creep in to defeat our attempts. For example, 98 + 2 is an impossible combination, because we are able to say at once that there is no possible form for the digital roots of the fraction equal to 2. But in the case of 97 + 3 there is a possible form for the digital roots of the fraction, namely, 6--5, and it is only on further investigation that we are able to determine that this form cannot in practice be obtained, owing to curious considerations. The working is greatly simplified by a process of elimination, based on such considerations as that certain multiplications produce a repetition of figures, and that the whole number cannot be from 12 to 23 inclusive, since in every such case sufficiently small denominators are not available for forming the fractional part. 91.--MORE MIXED FRACTIONS. The point of the present puzzle lies in the fact that the numbers 15 and 18 are not capable of solution. There is no way of determining this without trial. Here are answers for the ten possible numbers:-- 9+5472/1368 = 13; 9+6435/1287 = 14; 12+3576/894 = 16; 6+13258/947 = 20; 15+9432/786 = 27; 24+9756/813 = 36; 27+5148/396 = 40; 65+1892/473 = 69; 59+3614/278 = 72; 75+3648/192 = 94. I have only found the one arrangement for each of the numbers 16, 20, and 27; but the other numbers are all capable of being solved in more than one way. As for 15 and 18, though these may be easily solved as a simple fraction, yet a "mixed fraction" assumes the presence of a whole number; and though my own idea for dodging the conditions is the following, where the fraction is both complex and mixed, it will be fairer to keep exactly to the form indicated:-- 3952 ---- 746 = 15; 3 ---- 1 5742 ---- 638 = 18. 9 ---- 1 I have proved the possibility of solution for all numbers up to 100, except 1, 2, 3, 4, 15, and 18. The first three are easily shown to be impossible. I have also noticed that numbers whose digital root is 8--such as 26, 35, 44, 53, etc.--seem to lend themselves to the greatest number of answers. For the number 26 alone I have recorded no fewer than twenty-five different arrangements, and I have no doubt that there are many more. 92.--DIGITAL SQUARE NUMBERS. So far as I know, there are no published tables of square numbers that go sufficiently high to be available for the purposes of this puzzle. The lowest square number containing all the nine digits once, and once only, is 139,854,276, the square of 11,826. The highest square number under the same conditions is, 923,187,456, the square of 30,384. 93.--THE MYSTIC ELEVEN. Most people know that if the sum of the digits in the odd places of any number is the same as the sum of the digits in the even places, then the number is divisible by 11 without remainder. Thus in 896743012 the odd digits, 20468, add up 20, and the even digits, 1379, also add up 20. Therefore the number may be divided by 11. But few seem to know that if the difference between the sum of the odd and the even digits is 11, or a multiple of 11, the rule equally applies. This law enables us to find, with a very little trial, that the smallest number containing nine of the ten digits (calling nought a digit) that is divisible by 11 is 102,347,586, and the highest number possible, 987,652,413. 94.--THE DIGITAL CENTURY. There is a very large number of different ways in which arithmetical signs may be placed between the nine digits, arranged in numerical order, so as to give an expression equal to 100. In fact, unless the reader investigated the matter very closely, he might not suspect that so many ways are possible. It was for this reason that I added the condition that not only must the fewest possible signs be used, but also the fewest possible strokes. In this way we limit the problem to a single solution, and arrive at the simplest and therefore (in this case) the best result. Just as in the case of magic squares there are methods by which we may write down with the greatest ease a large number of solutions, but not all the solutions, so there are several ways in which we may quickly arrive at dozens of arrangements of the "Digital Century," without finding all the possible arrangements. There is, in fact, very little principle in the thing, and there is no certain way of demonstrating that we have got the best possible solution. All I can say is that the arrangement I shall give as the best is the best I have up to the present succeeded in discovering. I will give the reader a few interesting specimens, the first being the solution usually published, and the last the best solution that I know. Signs. Strokes. 1 + 2 + 3 + 4 + 5 + 6 + 7 + (8 × 9) = 100 ( 9 18) - (1 × 2) - 3 - 4 - 5 + (6 × 7) + (8 × 9) = 100 (12 20) 1 + (2 × 3) + (4 × 5) - 6 + 7 + (8 × 9) = 100 (11 21) (1 + 2 - 3 - 4)(5 - 6 - 7 - 8 - 9) = 100 ( 9 12) 1 + (2 × 3) + 4 + 5 + 67 + 8 + 9 =100 (8 16) (1 × 2) + 34 + 56 + 7 - 8 + 9 = 100 (7 13) 12 + 3 - 4 + 5 + 67 + 8 + 9 = 100 (6 11) 123 - 4 - 5 - 6 - 7 + 8 - 9 = 100 (6 7) 123 + 4 - 5 + 67 - 8 - 9 = 100 (4 6) 123 + 45 - 67 + 8 - 9 = 100 (4 6) 123 - 45 - 67 + 89 = 100 (3 4) It will be noticed that in the above I have counted the bracket as one sign and two strokes. The last solution is singularly simple, and I do not think it will ever be beaten. 95.--THE FOUR SEVENS. The way to write four sevens with simple arithmetical signs so that they represent 100 is as follows:-- 7 7 -- × -- = 100. .7 .7 Of course the fraction, 7 over decimal 7, equals 7 divided by 7/10, which is the same as 70 divided by 7, or 10. Then 10 multiplied by 10 is 100, and there you are! It will be seen that this solution applies equally to any number whatever that you may substitute for 7. 96.--THE DICE NUMBERS. The sum of all the numbers that can be formed with any given set of four different figures is always 6,666 multiplied by the sum of the four figures. Thus, 1, 2, 3, 4 add up 10, and ten times 6,666 is 66,660. Now, there are thirty-five different ways of selecting four figures from the seven on the dice--remembering the 6 and 9 trick. The figures of all these thirty-five groups add up to 600. Therefore 6,666 multiplied by 600 gives us 3,999,600 as the correct answer. Let us discard the dice and deal with the problem generally, using the nine digits, but excluding nought. Now, if you were given simply the sum of the digits--that is, if the condition were that you could use any four figures so long as they summed to a given amount--then we have to remember that several combinations of four digits will, in many cases, make the same sum. 10 11 12 13 14 15 16 17 18 19 20 1 1 2 3 5 6 8 9 11 11 12 21 22 23 24 25 26 27 28 29 30 11 11 9 8 6 5 3 2 1 1 Here the top row of numbers gives all the possible sums of four different figures, and the bottom row the number of different ways in which each sum may be made. For example 13 may be made in three ways: 1237, 1246, and 1345. It will be found that the numbers in the bottom row add up to 126, which is the number of combinations of nine figures taken four at a time. From this table we may at once calculate the answer to such a question as this: What is the sum of all the numbers composed of our different digits (nought excluded) that add up to 14? Multiply 14 by the number beneath t in the table, 5, and multiply the result by 6,666, and you will have the answer. It follows that, to know the sum of all the numbers composed of four different digits, if you multiply all the pairs in the two rows and then add the results together, you will get 2,520, which, multiplied by 6,666, gives the answer 16,798,320. The following general solution for any number of digits will doubtless interest readers. Let n represent number of digits, then 5 (10^n - 1) 8! divided by (9 - n)! equals the required sum. Note that 0! equals 1. This may be reduced to the following practical rule: Multiply together 4 × 7 × 6 × 5 ... to (n - 1) factors; now add (n + 1) ciphers to the right, and from this result subtract the same set of figures with a single cipher to the right. Thus for n = 4 (as in the case last mentioned), 4 × 7 × 6 = 168. Therefore 16,800,000 less 1,680 gives us 16,798,320 in another way. 97.--THE SPOT ON THE TABLE. The ordinary schoolboy would correctly treat this as a quadratic equation. Here is the actual arithmetic. Double the product of the two distances from the walls. This gives us 144, which is the square of 12. The sum of the two distances is 17. If we add these two numbers, 12 and 17, together, and also subtract one from the other, we get the two answers that 29 or 5 was the radius, or half-diameter, of the table. Consequently, the full diameter was 58 in. or 10 in. But a table of the latter dimensions would be absurd, and not at all in accordance with the illustration. Therefore the table must have been 58 in. in diameter. In this case the spot was on the edge nearest to the corner of the room--to which the boy was pointing. If the other answer were admissible, the spot would be on the edge farthest from the corner of the room. 98.--ACADEMIC COURTESIES. There must have been ten boys and twenty girls. The number of bows girl to girl was therefore 380, of boy to boy 90, of girl with boy 400, and of boys and girls to teacher 30, making together 900, as stated. It will be remembered that it was not said that the teacher himself returned the bows of any child. 99.--THE THIRTY-THREE PEARLS. The value of the large central pearl must have been £3,000. The pearl at one end (from which they increased in value by £100) was £1,400; the pearl at the other end, £600. 100.--THE LABOURER'S PUZZLE. The man said, "I am going twice as deep," not "as deep again." That is to say, he was still going twice as deep as he had gone already, so that when finished the hole would be three times its present depth. Then the answer is that at present the hole is 3 ft. 6 in. deep and the man 2 ft. 4 in. above ground. When completed the hole will be 10 ft. 6 in. deep, and therefore the man will then be 4 ft. 8 in. below the surface, or twice the distance that he is now above ground. 101.--THE TRUSSES OF HAY. Add together the ten weights and divide by 4, and we get 289 lbs. as the weight of the five trusses together. If we call the five trusses in the order of weight A, B, C, D, and E, the lightest being A and the heaviest E, then the lightest, no lbs., must be the weight of A and B; and the next lightest, 112 lbs., must be the weight of A and C. Then the two heaviest, D and E, must weigh 121 lbs., and C and E must weigh 120 lbs. We thus know that A, B, D, and E weigh together 231 lbs., which, deducted from 289 lbs. (the weight of the five trusses), gives us the weight of C as 58 lbs. Now, by mere subtraction, we find the weight of each of the five trusses--54 lbs., 56 lbs., 58 lbs., 59 lbs., and 62 lbs. respectively. 102.--MR. GUBBINS IN A FOG. The candles must have burnt for three hours and three-quarters. One candle had one-sixteenth of its total length left and the other four-sixteenths. 103.--PAINTING THE LAMP-POSTS. Pat must have painted six more posts than Tim, no matter how many lamp-posts there were. For example, suppose twelve on each side; then Pat painted fifteen and Tim nine. If a hundred on each side, Pat painted one hundred and three, and Tim only ninety-seven 104.--CATCHING THE THIEF. The constable took thirty steps. In the same time the thief would take forty-eight, which, added to his start of twenty-seven, carried him seventy-five steps. This distance would be exactly equal to thirty steps of the constable. 105.--THE PARISH COUNCIL ELECTION, The voter can vote for one candidate in 23 ways, for two in 253 ways, for three in 1,771, for four in 8,855, for five in 33,649, for six in 100,947, for seven in 245,157, for eight in 490,314, and for nine candidates in 817,190 different ways. Add these together, and we get the total of 1,698,159 ways of voting. 106.--THE MUDDLETOWN ELECTION. The numbers of votes polled respectively by the Liberal, the Conservative, the Independent, and the Socialist were 1,553, 1,535, 1,407, and 978 All that was necessary was to add the sum of the three majorities (739) to the total poll of 5,473 (making 6,212) and divide by 4, which gives us 1,553 as the poll of the Liberal. Then the polls of the other three candidates can, of course, be found by deducting the successive majorities from the last-mentioned number. 107.--THE SUFFRAGISTS' MEETING. Eighteen were present at the meeting and eleven left. If twelve had gone, two-thirds would have retired. If only nine had gone, the meeting would have lost half its members. 108.--THE LEAP-YEAR LADIES. The correct and only answer is that 11,616 ladies made proposals of marriage. Here are all the details, which the reader can check for himself with the original statements. Of 10,164 spinsters, 8,085 married bachelors, 627 married widowers, 1,221 were declined by bachelors, and 231 declined by widowers. Of the 1,452 widows, 1,155 married bachelors, and 297 married widowers. No widows were declined. The problem is not difficult, by algebra, when once we have succeeded in correctly stating it. 109.--THE GREAT SCRAMBLE. The smallest number of sugar plums that will fulfil the conditions is 26,880. The five boys obtained respectively: Andrew, 2,863; Bob, 6,335; Charlie, 2,438; David, 10,294; Edgar, 4,950. There is a little trap concealed in the words near the end, "one-fifth of the same," that seems at first sight to upset the whole account of the affair. But a little thought will show that the words could only mean "one-fifth of five-eighths", the fraction last mentioned--that is, one-eighth of the three-quarters that Bob and Andrew had last acquired. 110.--THE ABBOT'S PUZZLE. The only answer is that there were 5 men, 25 women, and 70 children. There were thus 100 persons in all, 5 times as many women as men, and as the men would together receive 15 bushels, the women 50 bushels, and the children 35 bushels, exactly 100 bushels would be distributed. 111.--REAPING THE CORN. The whole field must have contained 46.626 square rods. The side of the central square, left by the farmer, is 4.8284 rods, so it contains 23.313 square rods. The area of the field was thus something more than a quarter of an acre and less than one-third; to be more precise, .2914 of an acre. 112.--A PUZZLING LEGACY. As the share of Charles falls in through his death, we have merely to divide the whole hundred acres between Alfred and Benjamin in the proportion of one-third to one-fourth--that is in the proportion of four-twelfths to three-twelfths, which is the same as four to three. Therefore Alfred takes four-sevenths of the hundred acres and Benjamin three-sevenths. 113.--THE TORN NUMBER. The other number that answers all the requirements of the puzzle is 9,801. If we divide this in the middle into two numbers and add them together we get 99, which, multiplied by itself, produces 9,801. It is true that 2,025 may be treated in the same way, only this number is excluded by the condition which requires that no two figures should be alike. The general solution is curious. Call the number of figures in each half of the torn label n. Then, if we add 1 to each of the exponents of the prime factors (other than 3) of 10^n - 1 (1 being regarded as a factor with the constant exponent, 1), their product will be the number of solutions. Thus, for a label of six figures, n = 3. The factors of 10^n - 1 are 1¹ × 37¹ (not considering the 3³), and the product of 2 × 2 = 4, the number of solutions. This always includes the special cases 98 - 01, 00 - 01, 998 - 01, 000 - 001, etc. The solutions are obtained as follows:--Factorize 10³ - 1 in all possible ways, always keeping the powers of 3 together, thus, 37 × 27, 999 × 1. Then solve the equation 37x = 27y + 1. Here x = 19 and y = 26. Therefore, 19 × 37 = 703, the square of which gives one label, 494,209. A complementary solution (through 27x = 37x + 1) can at once be found by 10^n - 703 = 297, the square of which gives 088,209 for second label. (These non-significant noughts to the left must be included, though they lead to peculiar cases like 00238 - 04641 = 4879², where 0238 - 4641 would not work.) The special case 999 × 1 we can write at once 998,001, according to the law shown above, by adding nines on one half and noughts on the other, and its complementary will be 1 preceded by five noughts, or 000001. Thus we get the squares of 999 and 1. These are the four solutions. 114.--CURIOUS NUMBERS. The three smallest numbers, in addition to 48, are 1,680, 57,120, and 1,940,448. It will be found that 1,681 and 841, 57,121 and 28,561, 1,940,449 and 970,225, are respectively the squares of 41 and 29, 239 and 169, 1,393 and 985. 115.--A PRINTER'S ERROR. The answer is that 2^5 .9^2 is the same as 2592, and this is the only possible solution to the puzzle. 116.--THE CONVERTED MISER. As we are not told in what year Mr. Jasper Bullyon made the generous distribution of his accumulated wealth, but are required to find the lowest possible amount of money, it is clear that we must look for a year of the most favourable form. There are four cases to be considered--an ordinary year with fifty-two Sundays and with fifty-three Sundays, and a leap-year with fifty-two and fifty-three Sundays respectively. Here are the lowest possible amounts in each case:-- 313 weekdays, 52 Sundays £112,055 312 weekdays, 53 Sundays 19,345 314 weekdays, 52 Sundays No solution possible. 313 weekdays, 53 Sundays £69,174 The lowest possible amount, and therefore the correct answer, is £19,345, distributed in an ordinary year that began on a Sunday. The last year of this kind was 1911. He would have paid £53 on every day of the year, or £62 on every weekday, with £1 left over, as required, in the latter event. 117.--A FENCE PROBLEM. Though this puzzle presents no great difficulty to any one possessing a knowledge of algebra, it has perhaps rather interesting features. Seeing, as one does in the illustration, just one corner of the proposed square, one is scarcely prepared for the fact that the field, in order to comply with the conditions, must contain exactly 501,760 acres, the fence requiring the same number of rails. Yet this is the correct answer, and the only answer, and if that gentleman in Iowa carries out his intention, his field will be twenty-eight miles long on each side, and a little larger than the county of Westmorland. I am not aware that any limit has ever been fixed to the size of a "field," though they do not run so large as this in Great Britain. Still, out in Iowa, where my correspondent resides, they do these things on a very big scale. I have, however, reason to believe that when he finds the sort of task he has set himself, he will decide to abandon it; for if that cow decides to roam to fresh woods and pastures new, the milkmaid may have to start out a week in advance in order to obtain the morning's milk. Here is a little rule that will always apply where the length of the rail is half a pole. Multiply the number of rails in a hurdle by four, and the result is the exact number of miles in the side of a square field containing the same number of acres as there are rails in the complete fence. Thus, with a one-rail fence the field is four miles square; a two-rail fence gives eight miles square; a three-rail fence, twelve miles square; and so on, until we find that a seven-rail fence multiplied by four gives a field of twenty-eight miles square. In the case of our present problem, if the field be made smaller, then the number of rails will exceed the number of acres; while if the field be made larger, the number of rails will be less than the acres of the field. 118.--CIRCLING THE SQUARES. Though this problem might strike the novice as being rather difficult, it is, as a matter of fact, quite easy, and is made still easier by inserting four out of the ten numbers. First, it will be found that squares that are diametrically opposite have a common difference. For example, the difference between the square of 14 and the square of 2, in the diagram, is 192; and the difference between the square of 16 and the square of 8 is also 192. This must be so in every case. Then it should be remembered that the difference between squares of two consecutive numbers is always twice the smaller number plus 1, and that the difference between the squares of any two numbers can always be expressed as the difference of the numbers multiplied by their sum. Thus the square of 5 (25) less the square of 4 (16) equals (2 × 4) + 1, or 9; also, the square of 7 (49) less the square of 3 (9) equals (7 + 3) × (7 - 3), or 40. Now, the number 192, referred to above, may be divided into five different pairs of even factors: 2 × 96, 4 × 48, 6 × 32, 8 × 24, and 12 × 16, and these divided by 2 give us, 1 × 48, 2 × 24, 3 × 16, 4 × 12, and 6 × 8. The difference and sum respectively of each of these pairs in turn produce 47, 49; 22, 26; 13, 19; 8, 16; and 2, 14. These are the required numbers, four of which are already placed. The six numbers that have to be added may be placed in just six different ways, one of which is as follows, reading round the circle clockwise: 16, 2, 49, 22, 19, 8, 14, 47, 26, 13. I will just draw the reader's attention to one other little point. In all circles of this kind, the difference between diametrically opposite numbers increases by a certain ratio, the first numbers (with the exception of a circle of 6) being 4 and 6, and the others formed by doubling the next preceding but one. Thus, in the above case, the first difference is 2, and then the numbers increase by 4, 6, 8, and 12. Of course, an infinite number of solutions may be found if we admit fractions. The number of squares in a circle of this kind must, however, be of the form 4n + 6; that is, it must be a number composed of 6 plus a multiple of 4. 119.--RACKBRANE'S LITTLE LOSS. The professor must have started the game with thirteen shillings, Mr. Potts with four shillings, and Mrs. Potts with seven shillings. 120.--THE FARMER AND HIS SHEEP. The farmer had one sheep only! If he divided this sheep (which is best done by weight) into two parts, making one part two-thirds and the other part one-third, then the difference between these two numbers is the same as the difference between their squares--that is, one-third. Any two fractions will do if the denominator equals the sum of the two numerators. 121.--HEADS OR TAILS. Crooks must have lost, and the longer he went on the more he would lose. In two tosses he would be left with three-quarters of his money, in four tosses with nine-sixteenths of his money, in six tosses with twenty-seven sixty-fourths of his money, and so on. The order of the wins and losses makes no difference, so long as their number is in the end equal. 122.--THE SEE-SAW PUZZLE. The boy's weight must have been about 39.79 lbs. A brick weighed 3 lbs. Therefore 16 bricks weighed 48 lbs. and 11 bricks 33 lbs. Multiply 48 by 33 and take the square root. 123.--A LEGAL DIFFICULTY. It was clearly the intention of the deceased to give the son twice as much as the mother, or the daughter half as much as the mother. Therefore the most equitable division would be that the mother should take two-sevenths, the son four-sevenths, and the daughter one-seventh. 124.--A QUESTION OF DEFINITION. There is, of course, no difference in _area_ between a mile square and a square mile. But there may be considerable difference in _shape_. A mile square can be no other shape than square; the expression describes a surface of a certain specific size and shape. A square mile may be of any shape; the expression names a unit of area, but does not prescribe any particular shape. 125.--THE MINERS' HOLIDAY. Bill Harris must have spent thirteen shillings and sixpence, which would be three shillings more than the average for the seven men--half a guinea. 126.--SIMPLE MULTIPLICATION. The number required is 3,529,411,764,705,882, which may be multiplied by 3 and divided by 2, by the simple expedient of removing the 3 from one end of the row to the other. If you want a longer number, you can increase this one to any extent by repeating the sixteen figures in the same order. 127.--SIMPLE DIVISION. Subtract every number in turn from every other number, and we get 358 (twice), 716, 1,611, 1,253, and 895. Now, we see at a glance that, as 358 equals 2 × 179, the only number that can divide in every case without a remainder will be 179. On trial we find that this is such a divisor. Therefore, 179 is the divisor we want, which always leaves a remainder 164 in the case of the original numbers given. 128.--A PROBLEM IN SQUARES. The sides of the three boards measure 31 in., 41 in., and 49 in. The common difference of area is exactly five square feet. Three numbers whose squares are in A.P., with a common difference of 7, are 113/120, 337/120, 463/120; and with a common difference of 13 are 80929/19380, 106921/19380, and 127729/19380. In the case of whole square numbers the common difference will always be divisible by 24, so it is obvious that our squares must be fractional. Readers should now try to solve the case where the common difference is 23. It is rather a hard nut. 129.--THE BATTLE OF HASTINGS. Any number (not itself a square number) may be multiplied by a square that will give a product 1 less than another square. The given number must not itself be a square, because a square multiplied by a square produces a square, and no square plus 1 can be a square. My remarks throughout must be understood to apply to whole numbers, because fractional soldiers are not of much use in war. Now, of all the numbers from 2 to 99 inclusive, 61 happens to be the most awkward one to work, and the lowest possible answer to our puzzle is that Harold's army consisted of 3,119,882,982,860,264,400 men. That is, there would be 51,145,622,669,840,400 men (the square of 226,153,980) in each of the sixty-one squares. Add one man (Harold), and they could then form one large square with 1,766,319,049 men on every side. The general problem, of which this is a particular case, is known as the "Pellian Equation"--apparently because Pell neither first propounded the question nor first solved it! It was issued as a challenge by Fermat to the English mathematicians of his day. It is readily solved by the use of continued fractions. Next to 61, the most difficult number under 100 is 97, where 97 × 6,377,352² + 1 = a square. The reason why I assumed that there must be something wrong with the figures in the chronicle is that we can confidently say that Harold's army did not contain over three trillion men! If this army (not to mention the Normans) had had the whole surface of the earth (sea included) on which to encamp, each man would have had slightly more than a quarter of a square inch of space in which to move about! Put another way: Allowing one square foot of standing-room per man, each small square would have required all the space allowed by a globe three times the diameter of the earth. 130.--THE SCULPTOR'S PROBLEM. A little thought will make it clear that the answer must be fractional, and that in one case the numerator will be greater and in the other case less than the denominator. As a matter of fact, the height of the larger cube must be 8/7 ft., and of the smaller 3/7 ft., if we are to have the answer in the smallest possible figures. Here the lineal measurement is 11/7 ft.--that is, 1+4/7 ft. What are the cubic contents of the two cubes? First 8/7 × 3/7 × 8/7 = 512/343, and secondly 3/7 × 3/7 × 3/7 = 27/343. Add these together and the result is 539/343, which reduces to 11/7 or 1+4/7 ft. We thus see that the answers in cubic feet and lineal feet are precisely the same. The germ of the idea is to be found in the works of Diophantus of Alexandria, who wrote about the beginning of the fourth century. These fractional numbers appear in triads, and are obtained from three generators, a, b, c, where a is the largest and c the smallest. Then ab + c² = denominator, and a² - c², b² - c², and a² - b² will be the three numerators. Thus, using the generators 3, 2, 1, we get 8/7, 3/7, 5/7 and we can pair the first and second, as in the above solution, or the first and third for a second solution. The denominator must always be a prime number of the form 6n + 1, or composed of such primes. Thus you can have 13, 19, etc., as denominators, but not 25, 55, 187, etc. When the principle is understood there is no difficulty in writing down the dimensions of as many sets of cubes as the most exacting collector may require. If the reader would like one, for example, with plenty of nines, perhaps the following would satisfy him: 99999999/99990001 and 19999/99990001. 131.--THE SPANISH MISER. There must have been 386 doubloons in one box, 8,450 in another, and 16,514 in the third, because 386 is the smallest number that can occur. If I had asked for the smallest aggregate number of coins, the answer would have been 482, 3,362, and 6,242. It will be found in either case that if the contents of any two of the three boxes be combined, they form a square number of coins. It is a curious coincidence (nothing more, for it will not always happen) that in the first solution the digits of the three numbers add to 17 in every case, and in the second solution to 14. It should be noted that the middle one of the three numbers will always be half a square. 132.--THE NINE TREASURE BOXES. Here is the answer that fulfils the conditions:-- A = 4 B = 3,364 C = 6,724 D = 2,116 E = 5,476 F = 8,836 G = 9,409 H = 12,769 I = 16,129 Each of these is a square number, the roots, taken in alphabetical order, being 2, 58, 82, 46, 74, 94, 97, 113, and 127, while the required difference between A and B, B and C, D and E. etc., is in every case 3,360. 133.--THE FIVE BRIGANDS. The sum of 200 doubloons might have been held by the five brigands in any one of 6,627 different ways. Alfonso may have held any number from 1 to 11. If he held 1 doubloon, there are 1,005 different ways of distributing the remainder; if he held 2, there are 985 ways; if 3, there are 977 ways; if 4, there are 903 ways; if 5 doubloons, 832 ways; if 6 doubloons, 704 ways; if 7 doubloons, 570 ways; if 8 doubloons, 388 ways; if 9 doubloons, 200 ways; if 10 doubloons, 60 ways; and if Alfonso held 11 doubloons, the remainder could be distributed in 3 different ways. More than 11 doubloons he could not possibly have had. It will scarcely be expected that I shall give all these 6,627 ways at length. What I propose to do is to enable the reader, if he should feel so disposed, to write out all the answers where Alfonso has one and the same amount. Let us take the cases where Alfonso has 6 doubloons, and see how we may obtain all the 704 different ways indicated above. Here are two tables that will serve as keys to all these answers:-- Table I. Table II. A = 6. A = 6. B = n. B = n. C = (63 - 5n) + m. C = 1 + m. D = (128 + 4n) - 4m. D = (376 - 16n) - 4m. E = 3 + 3m. E = (15n - 183) + 3m. In the first table we may substitute for n any whole number from 1 to 12 inclusive, and m may be nought or any whole number from 1 to (31 + n) inclusive. In the second table n may have the value of any whole number from 13 to 23 inclusive, and m may be nought or any whole number from 1 to (93 - 4n) inclusive. The first table thus gives (32 + n) answers for every value of n; and the second table gives (94 - 4n) answers for every value of n. The former, therefore, produces 462 and the latter 242 answers, which together make 704, as already stated. Let us take Table I., and say n = 5 and m = 2; also in Table II. take n = 13 and m = 0. Then we at once get these two answers:-- A = 6 A = 6 B = 5 B = 13 C = 40 C = 1 D = 140 D = 168 E = 9 E = 12 --- --- 200 doubloons 200 doubloons. These will be found to work correctly. All the rest of the 704 answers, where Alfonso always holds six doubloons, may be obtained in this way from the two tables by substituting the different numbers for the letters m and n. Put in another way, for every holding of Alfonso the number of answers is the sum of two arithmetical progressions, the common difference in one case being 1 and in the other -4. Thus in the case where Alfonso holds 6 doubloons one progression is 33 + 34 + 35 + 36 + ... + 43 + 44, and the other 42 + 38 + 34 + 30 + ... + 6 + 2. The sum of the first series is 462, and of the second 242--results which again agree with the figures already given. The problem may be said to consist in finding the first and last terms of these progressions. I should remark that where Alfonso holds 9, 10, or 11 there is only one progression, of the second form. 134.--THE BANKER'S PUZZLE. In order that a number of sixpences may not be divisible into a number of equal piles, it is necessary that the number should be a prime. If the banker can bring about a prime number, he will win; and I will show how he can always do this, whatever the customer may put in the box, and that therefore the banker will win to a certainty. The banker must first deposit forty sixpences, and then, no matter how many the customer may add, he will desire the latter to transfer from the counter the square of the number next below what the customer put in. Thus, banker puts 40, customer, we will say, adds 6, then transfers from the counter 25 (the square of 5), which leaves 71 in all, a prime number. Try again. Banker puts 40, customer adds 12, then transfers 121 (the square of 11), as desired, which leaves 173, a prime number. The key to the puzzle is the curious fact that any number up to 39, if added to its square and the sum increased by 41, makes a prime number. This was first discovered by Euler, the great mathematician. It has been suggested that the banker might desire the customer to transfer sufficient to raise the contents of the box to a given number; but this would not only make the thing an absurdity, but breaks the rule that neither knows what the other puts in. 135.--THE STONEMASON'S PROBLEM. The puzzle amounts to this. Find the smallest square number that may be expressed as the sum of more than three consecutive cubes, the cube 1 being barred. As more than three heaps were to be supplied, this condition shuts out the otherwise smallest answer, 23³ + 24³ + 25³ = 204². But it admits the answer, 25³ + 26³ + 27³ + 28³ + 29³ = 315². The correct answer, however, requires more heaps, but a smaller aggregate number of blocks. Here it is: 14³ + 15³ + ... up to 25³ inclusive, or twelve heaps in all, which, added together, make 97,344 blocks of stone that may be laid out to form a square 312 × 312. I will just remark that one key to the solution lies in what are called triangular numbers. (See pp. 13, 25, and 166.) 136.--THE SULTAN'S ARMY. The smallest primes of the form 4n + 1 are 5, 13, 17, 29, and 37, and the smallest of the form 4n - 1 are 3, 7, 11, 19, and 23. Now, primes of the first form can always be expressed as the sum of two squares, and in only one way. Thus, 5 = 4 + 1; 13 = 9 + 4; 17 = 16 + 1; 29 = 25 + 4; 37 = 36 + 1. But primes of the second form can never be expressed as the sum of two squares in any way whatever. In order that a number may be expressed as the sum of two squares in several different ways, it is necessary that it shall be a composite number containing a certain number of primes of our first form. Thus, 5 or 13 alone can only be so expressed in one way; but 65, (5 × 13), can be expressed in two ways, 1,105, (5 × 13 × 17), in four ways, 32,045, (5 × 13 × 17 × 29), in eight ways. We thus get double as many ways for every new factor of this form that we introduce. Note, however, that I say _new_ factor, for the _repetition_ of factors is subject to another law. We cannot express 25, (5 × 5), in two ways, but only in one; yet 125, (5 × 5 × 5), can be given in two ways, and so can 625, (5 × 5 × 5 × 5); while if we take in yet another 5 we can express the number as the sum of two squares in three different ways. If a prime of the second form gets into your composite number, then that number cannot be the sum of two squares. Thus 15, (3 × 5), will not work, nor will 135, (3 × 3 × 3 × 5); but if we take in an even number of 3's it will work, because these 3's will themselves form a square number, but you will only get one solution. Thus, 45, (3 × 3 × 5, or 9 × 5) = 36 + 9. Similarly, the factor 2 may always occur, or any power of 2, such as 4, 8, 16, 32; but its introduction or omission will never affect the number of your solutions, except in such a case as 50, where it doubles a square and therefore gives you the two answers, 49 + 1 and 25 + 25. Now, directly a number is decomposed into its prime factors, it is possible to tell at a glance whether or not it can be split into two squares; and if it can be, the process of discovery in how many ways is so simple that it can be done in the head without any effort. The number I gave was 130. I at once saw that this was 2 × 5 × 13, and consequently that, as 65 can be expressed in two ways (64 + 1 and 49 + 16), 130 can also be expressed in two ways, the factor 2 not affecting the question. The smallest number that can be expressed as the sum of two squares in twelve different ways is 160,225, and this is therefore the smallest army that would answer the Sultan's purpose. The number is composed of the factors 5 × 5 × 13 × 17 × 29, each of which is of the required form. If they were all different factors, there would be sixteen ways; but as one of the factors is repeated, there are just twelve ways. Here are the sides of the twelve pairs of squares: (400 and 15), (399 and 32), (393 and 76), (392 and 81), (384 and 113), (375 and 140), (360 and 175), (356 and 183), (337 and 216), (329 and 228), (311 and 252), (265 and 300). Square the two numbers in each pair, add them together, and their sum will in every case be 160,225. 137.--A STUDY IN THRIFT. Mrs. Sandy McAllister will have to save a tremendous sum out of her housekeeping allowance if she is to win that sixth present that her canny husband promised her. And the allowance must be a very liberal one if it is to admit of such savings. The problem required that we should find five numbers higher than 36 the units of which may be displayed so as to form a square, a triangle, two triangles, and three triangles, using the complete number in every one of the four cases. Every triangular number is such that if we multiply it by 8 and add 1 the result is an odd square number. For example, multiply 1, 3, 6, 10, 15 respectively by 8 and add 1, and we get 9, 25, 49, 81, 121, which are the squares of the odd numbers 3, 5, 7, 9, 11. Therefore in every case where 8x² + 1 = a square number, x² is also a triangular. This point is dealt with in our puzzle, "The Battle of Hastings." I will now merely show again how, when the first solution is found, the others may be discovered without any difficulty. First of all, here are the figures:-- 8 × 1² + 1 = 3² 8 × 6² + 1 = 17² 8 × 35² + 1 = 99² 8 × 204² + 1 = 577² 8 × 1189² + 1 = 3363² 8 × 6930² + 1 = 19601² 8 × 40391² + 1 = 114243² The successive pairs of numbers are found in this way:-- (1 × 3) + (3 × 1) = 6 (8 × 1) + (3 × 3) = 17 (1 × 17) + (3 × 6) = 35 (8 × 6) + (3 × 17) = 99 (1 × 99) + (3 × 35) = 204 (8 × 35) + (3 × 99) = 577 and so on. Look for the numbers in the table above, and the method will explain itself. Thus we find that the numbers 36, 1225, 41616, 1413721, 48024900, and 1631432881 will form squares with sides of 6, 35, 204, 1189, 6930, and 40391; and they will also form single triangles with sides of 8, 49, 288, 1681, 9800, and 57121. These numbers may be obtained from the last column in the first table above in this way: simply divide the numbers by 2 and reject the remainder. Thus the integral halves of 17, 99, and 577 are 8, 49, and 288. All the numbers we have found will form either two or three triangles at will. The following little diagram will show you graphically at a glance that every square number must necessarily be the sum of two triangulars, and that the side of one triangle will be the same as the side of the corresponding square, while the other will be just 1 less. [Illustration +-----------+ +---------+ |. . . . ./.| |. . . ./.| |. . . ./. .| |. . ./. .| |. . ./. . .| |. ./. . .| |. ./. . . .| |./. . . .| |./. . . . .| /. . . . .| /. . . . . .| +---------+ +-----------+ ] Thus a square may always be divided easily into two triangles, and the sum of two consecutive triangulars will always make a square. In numbers it is equally clear, for if we examine the first triangulars--1, 3, 6, 10, 15, 21, 28--we find that by adding all the consecutive pairs in turn we get the series of square numbers--9, 16, 25, 36, 49, etc. The method of forming three triangles from our numbers is equally direct, and not at all a matter of trial. But I must content myself with giving actual figures, and just stating that every triangular higher than 6 will form three triangulars. I give the sides of the triangles, and readers will know from my remarks when stating the puzzle how to find from these sides the number of counters or coins in each, and so check the results if they so wish. +----------------------+-----------+---------------+-----------------------+ | Number | Side of | Side of | Sides of Two | Sides of Three | | | Square. | Triangle. | Triangles. | Triangles. | +------------+---------+-----------+---------------+-----------------------+ | 36 | 6 | 8 | 6 + 5 | 5 + 5 + 3 | | 1225 | 35 | 49 | 36 + 34 | 33 + 32 + 16 | | 41616 | 204 | 288 | 204 + 203 | 192 + 192 + 95 | | 1413721 | 1189 | 1681 | 1189 + 1188 | 1121 + 1120 + 560 | | 48024900 | 6930 | 9800 | 6930 + 6929 | 6533 + 6533 + 3267 | | 1631432881 | 40391 | 57121 | 40391 + 40390 | 38081 + 38080 + 19040 | +------------+---------+-----------+---------------+-----------------------+ I should perhaps explain that the arrangements given in the last two columns are not the only ways of forming two and three triangles. There are others, but one set of figures will fully serve our purpose. We thus see that before Mrs. McAllister can claim her sixth £5 present she must save the respectable sum of £1,631,432,881. 138.--THE ARTILLERYMEN'S DILEMMA. We were required to find the smallest number of cannon balls that we could lay on the ground to form a perfect square, and could pile into a square pyramid. I will try to make the matter clear to the merest novice. 1 2 3 4 5 6 7 1 3 6 10 15 21 28 1 4 10 20 35 56 84 1 5 14 30 55 91 140 Here in the first row we place in regular order the natural numbers. Each number in the second row represents the sum of the numbers in the row above, from the beginning to the number just over it. Thus 1, 2, 3, 4, added together, make 10. The third row is formed in exactly the same way as the second. In the fourth row every number is formed by adding together the number just above it and the preceding number. Thus 4 and 10 make 14, 20 and 35 make 55. Now, all the numbers in the second row are triangular numbers, which means that these numbers of cannon balls may be laid out on the ground so as to form equilateral triangles. The numbers in the third row will all form our triangular pyramids, while the numbers in the fourth row will all form square pyramids. Thus the very process of forming the above numbers shows us that every square pyramid is the sum of two triangular pyramids, one of which has the same number of balls in the side at the base, and the other one ball fewer. If we continue the above table to twenty-four places, we shall reach the number 4,900 in the fourth row. As this number is the square of 70, we can lay out the balls in a square, and can form a square pyramid with them. This manner of writing out the series until we come to a square number does not appeal to the mathematical mind, but it serves to show how the answer to the particular puzzle may be easily arrived at by anybody. As a matter of fact, I confess my failure to discover any number other than 4,900 that fulfils the conditions, nor have I found any rigid proof that this is the only answer. The problem is a difficult one, and the second answer, if it exists (which I do not believe), certainly runs into big figures. For the benefit of more advanced mathematicians I will add that the general expression for square pyramid numbers is (2n³ + 3n² + n)/6. For this expression to be also a square number (the special case of 1 excepted) it is necessary that n = p² - 1 = 6t², where 2p² - 1 = q² (the "Pellian Equation"). In the case of our solution above, n = 24, p = 5, t = 2, q = 7. 139.--THE DUTCHMEN'S WIVES. The money paid in every case was a square number of shillings, because they bought 1 at 1s., 2 at 2s., 3 at 3s., and so on. But every husband pays altogether 63s. more than his wife, so we have to find in how many ways 63 may be the difference between two square numbers. These are the three only possible ways: the square of 8 less the square of 1, the square of 12 less the square of 9, and the square of 32 less the square of 31. Here 1, 9, and 31 represent the number of pigs bought and the number of shillings per pig paid by each woman, and 8, 12, and 32 the same in the case of their respective husbands. From the further information given as to their purchases, we can now pair them off as follows: Cornelius and Gurtrün bought 8 and 1; Elas and Katrün bought 12 and 9; Hendrick and Anna bought 32 and 31. And these pairs represent correctly the three married couples. The reader may here desire to know how we may determine the maximum number of ways in which a number may be expressed as the difference between two squares, and how we are to find the actual squares. Any integer except 1, 4, and twice any odd number, may be expressed as the difference of two integral squares in as many ways as it can be split up into pairs of factors, counting 1 as a factor. Suppose the number to be 5,940. The factors are 2².3³.5.11. Here the exponents are 2, 3, 1, 1. Always deduct 1 from the exponents of 2 and add 1 to all the other exponents; then we get 1, 4, 2, 2, and half the product of these four numbers will be the required number of ways in which 5,940 may be the difference of two squares--that is, 8. To find these eight squares, as it is an _even_ number, we first divide by 4 and get 1485, the eight pairs of factors of which are 1 × 1485, 3 × 495, 5 × 297, 9 × 165, 11 × 135, 15 × 99, 27 × 55, and 33 × 45. The sum and difference of any one of these pairs will give the required numbers. Thus, the square of 1,486 less the square of 1,484 is 5,940, the square of 498 less the square of 492 is the same, and so on. In the case of 63 above, the number is _odd_; so we factorize at once, 1 × 63, 3 × 21, 7 × 9. Then we find that _half_ the sum and difference will give us the numbers 32 and 31, 12 and 9, and 8 and 1, as shown in the solution to the puzzle. The reverse problem, to find the factors of a number when you have expressed it as the difference of two squares, is obvious. For example, the sum and difference of any pair of numbers in the last sentence will give us the factors of 63. Every prime number (except 1 and 2) may be expressed as the difference of two squares in one way, and in one way only. If a number can be expressed as the difference of two squares in more than one way, it is composite; and having so expressed it, we may at once obtain the factors, as we have seen. Fermat showed in a letter to Mersenne or Frénicle, in 1643, how we may discover whether a number may be expressed as the difference of two squares in more than one way, or proved to be a prime. But the method, when dealing with large numbers, is necessarily tedious, though in practice it may be considerably shortened. In many cases it is the shortest method known for factorizing large numbers, and I have always held the opinion that Fermat used it in performing a certain feat in factorizing that is historical and wrapped in mystery. 140.--FIND ADA'S SURNAME. The girls' names were Ada Smith, Annie Brown, Emily Jones, Mary Robinson, and Bessie Evans. 141.--SATURDAY MARKETING. As every person's purchase was of the value of an exact number of shillings, and as the party possessed when they started out forty shilling coins altogether, there was no necessity for any lady to have any smaller change, or any evidence that they actually had such change. This being so, the only answer possible is that the women were named respectively Anne Jones, Mary Robinson, Jane Smith, and Kate Brown. It will now be found that there would be exactly eight shillings left, which may be divided equally among the eight persons in coin without any change being required. 142.--THE SILK PATCHWORK. [Illustration] Our illustration will show how to cut the stitches of the patchwork so as to get the square F entire, and four equal pieces, G, H, I, K, that will form a perfect Greek cross. The reader will know how to assemble these four pieces from Fig. 13 in the article. [Illustration: Fig. 1.] [Illustration: Fig. 2.] 143.--TWO CROSSES FROM ONE. It will be seen that one cross is cut out entire, as A in Fig. 1, while the four pieces marked B, C, D and E form the second cross, as in Fig. 2, which will be of exactly the same size as the other. I will leave the reader the pleasant task of discovering for himself the best way of finding the direction of the cuts. Note that the Swastika again appears. The difficult question now presents itself: How are we to cut three Greek crosses from one in the fewest possible pieces? As a matter of fact, this problem may be solved in as few as thirteen pieces; but as I know many of my readers, advanced geometricians, will be glad to have something to work on of which they are not shown the solution, I leave the mystery for the present undisclosed. 144.--THE CROSS AND THE TRIANGLE. The line A B in the following diagram represents the side of a square having the same area as the cross. I have shown elsewhere, as stated, how to make a square and equilateral triangle of equal area. I need not go, therefore, into the preliminary question of finding the dimensions of the triangle that is to equal our cross. We will assume that we have already found this, and the question then becomes, How are we to cut up one of these into pieces that will form the other? First draw the line A B where A and B are midway between the extremities of the two side arms. Next make the lines D C and E F equal in length to half the side of the triangle. Now from E and F describe with the same radius the intersecting arcs at G and draw F G. Finally make I K equal to H C and L B equal to A D. If we now draw I L, it should be parallel to F G, and all the six pieces are marked out. These fit together and form a perfect equilateral triangle, as shown in the second diagram. Or we might have first found the direction of the line M N in our triangle, then placed the point O over the point E in the cross and turned round the triangle over the cross until the line M N was parallel to A B. The piece 5 can then be marked off and the other pieces in succession. [Illustration] I have seen many attempts at a solution involving the assumption that the height of the triangle is exactly the same as the height of the cross. This is a fallacy: the cross will always be higher than the triangle of equal area. 145.--THE FOLDED CROSS. [Illustration: FIG. 1., FIG 2.] First fold the cross along the dotted line A B in Fig. 1. You then have it in the form shown in Fig. 2. Next fold it along the dotted line C D (where D is, of course, the centre of the cross), and you get the form shown in Fig. 3. Now take your scissors and cut from G to F, and the four pieces, all of the same size and shape, will fit together and form a square, as shown in Fig. 4. [Illustration: FIG. 3., FIG. 4.] 146.--AN EASY DISSECTION PUZZLE. [Illustration +===========+===========+- | · | · : \ | · | · : \ | · | · : \ | · | · : \ | · | · : \ +-----------+===========+===========+ | / : · | · : \ | / : · | · : \ | / : · | · : \ | / : · | · : \ | / : · | · : \ +===========+===========+===========+===========+ ] The solution to this puzzle is shown in the illustration. Divide the figure up into twelve equal triangles, and it is easy to discover the directions of the cuts, as indicated by the dark lines. 147.--AN EASY SQUARE PUZZLE. [Illustration +-----------------------------------------+ | . /| | . / | | . / | | / / | | / . / | | / . / | | / . / | | / ./ | | +--------------------+ | | / / | | / / | | / / | | / . / | | / . / | | / . / | | / . / | | / . | | / . | | / . | |/ . | +-----------------------------------------+ ] The diagram explains itself, one of the five pieces having been cut in two to form a square. 148.--THE BUN PUZZLE. [Illustration . . . . _ . . . |\ A . | \ . C | \ | | \ . | \ / . |______________________\/ | | . . . B . . . . . - _ . | . . | . . | . | | | D | E | | | . | . . | . . | . _ _ . | . . -+- . . . . . - - . | G| F | | - - . . . . . - _ _ - . . | . - -+- . . - - . | H | - - . . - _ _ - ] The secret of the bun puzzle lies in the fact that, with the relative dimensions of the circles as given, the three diameters will form a right-angled triangle, as shown by A, B, C. It follows that the two smaller buns are exactly equal to the large bun. Therefore, if we give David and Edgar the two halves marked D and E, they will have their fair shares--one quarter of the confectionery each. Then if we place the small bun, H, on the top of the remaining one and trace its circumference in the manner shown, Fred's piece, F, will exactly equal Harry's small bun, H, with the addition of the piece marked G--half the rim of the other. Thus each boy gets an exactly equal share, and there are only five pieces necessary. 149.--THE CHOCOLATE SQUARES. [Illustration] Square A is left entire; the two pieces marked B fit together and make a second square; the two pieces C make a third square; and the four pieces marked D will form the fourth square. 150.--DISSECTING A MITRE. The diagram on the next page shows how to cut into five pieces to form a square. The dotted lines are intended to show how to find the points C and F--the only difficulty. A B is half B D, and A E is parallel to B H. With the point of the compasses at B describe the arc H E, and A E will be the distance of C from B. Then F G equals B C less A B. This puzzle--with the added condition that it shall be cut into four parts of the same size and shape--I have not been able to trace to an earlier date than 1835. Strictly speaking, it is, in that form, impossible of solution; but I give the answer that is always presented, and that seems to satisfy most people. [Illustration] We are asked to assume that the two portions containing the same letter--AA, BB, CC, DD--are joined by "a mere hair," and are, therefore, only one piece. To the geometrician this is absurd, and the four shares are not equal in area unless they consist of two pieces each. If you make them equal in area, they will not be exactly alike in shape. [Illustration] 151.--THE JOINER'S PROBLEM. [Illustration] Nothing could be easier than the solution of this puzzle--when you know how to do it. And yet it is apt to perplex the novice a good deal if he wants to do it in the fewest possible pieces--three. All you have to do is to find the point A, midway between B and C, and then cut from A to D and from A to E. The three pieces then form a square in the manner shown. Of course, the proportions of the original figure must be correct; thus the triangle BEF is just a quarter of the square BCDF. Draw lines from B to D and from C to F and this will be clear. 152.--ANOTHER JOINER'S PROBLEM. [Illustration] THE point was to find a general rule for forming a perfect square out of another square combined with a "right-angled isosceles triangle." The triangle to which geometricians give this high-sounding name is, of course, nothing more or less than half a square that has been divided from corner to corner. The precise relative proportions of the square and triangle are of no consequence whatever. It is only necessary to cut the wood or material into five pieces. Suppose our original square to be ACLF in the above diagram and our triangle to be the shaded portion CED. Now, we first find half the length of the long side of the triangle (CD) and measure off this length at AB. Then we place the triangle in its present position against the square and make two cuts--one from B to F, and the other from B to E. Strange as it may seem, that is all that is necessary! If we now remove the pieces G, H, and M to their new places, as shown in the diagram, we get the perfect square BEKF. Take any two square pieces of paper, of different sizes but perfect squares, and cut the smaller one in half from corner to corner. Now proceed in the manner shown, and you will find that the two pieces may be combined to form a larger square by making these two simple cuts, and that no piece will be required to be turned over. The remark that the triangle might be "a little larger or a good deal smaller in proportion" was intended to bar cases where area of triangle is greater than area of square. In such cases six pieces are necessary, and if triangle and square are of equal area there is an obvious solution in three pieces, by simply cutting the square in half diagonally. 153.--A CUTTING-OUT PUZZLE. [Illustration] The illustration shows how to cut the four pieces and form with them a square. First find the side of the square (the mean proportional between the length and height of the rectangle), and the method is obvious. If our strip is exactly in the proportions 9 × 1, or 16 × 1, or 25 × 1, we can clearly cut it in 3, 4, or 5 rectangular pieces respectively to form a square. Excluding these special cases, the general law is that for a strip in length more than n² times the breadth, and not more than (n+1)² times the breadth, it may be cut in n+2 pieces to form a square, and there will be n-1 rectangular pieces like piece 4 in the diagram. Thus, for example, with a strip 24 × 1, the length is more than 16 and less than 25 times the breadth. Therefore it can be done in 6 pieces (n here being 4), 3 of which will be rectangular. In the case where n equals 1, the rectangle disappears and we get a solution in three pieces. Within these limits, of course, the sides need not be rational: the solution is purely geometrical. 154.--MRS. HOBSON'S HEARTHRUG. [Illustration] As I gave full measurements of the mutilated rug, it was quite an easy matter to find the precise dimensions for the square. The two pieces cut off would, if placed together, make an oblong piece 12 × 6, giving an area of 72 (inches or yards, as we please), and as the original complete rug measured 36 × 27, it had an area of 972. If, therefore, we deduct the pieces that have been cut away, we find that our new rug will contain 972 less 72, or 900; and as 900 is the square of 30, we know that the new rug must measure 30 × 30 to be a perfect square. This is a great help towards the solution, because we may safely conclude that the two horizontal sides measuring 30 each may be left intact. There is a very easy way of solving the puzzle in four pieces, and also a way in three pieces that can scarcely be called difficult, but the correct answer is in only two pieces. It will be seen that if, after the cuts are made, we insert the teeth of the piece B one tooth lower down, the two portions will fit together and form a square. 155.--THE PENTAGON AND SQUARE. A regular pentagon may be cut into as few as six pieces that will fit together without any turning over and form a square, as I shall show below. Hitherto the best answer has been in seven pieces--the solution produced some years ago by a foreign mathematician, Paul Busschop. We first form a parallelogram, and from that the square. The process will be seen in the diagram on the next page. The pentagon is ABCDE. By the cut AC and the cut FM (F being the middle point between A and C, and M being the same distance from A as F) we get two pieces that may be placed in position at GHEA and form the parallelogram GHDC. We then find the mean proportional between the length HD and the _height_ of the parallelogram. This distance we mark off from C at K, then draw CK, and from G drop the line GL, perpendicular to KC. The rest is easy and rather obvious. It will be seen that the six pieces will form either the pentagon or the square. I have received what purported to be a solution in five pieces, but the method was based on the rather subtle fallacy that half the diagonal plus half the side of a pentagon equals the side of a square of the same area. I say subtle, because it is an extremely close approximation that will deceive the eye, and is quite difficult to prove inexact. I am not aware that attention has before been drawn to this curious approximation. [Illustration] Another correspondent made the side of his square 1¼ of the side of the pentagon. As a matter of fact, the ratio is irrational. I calculate that if the side of the pentagon is 1--inch, foot, or anything else--the side of the square of equal area is 1.3117 nearly, or say roughly 1+3/10. So we can only hope to solve the puzzle by geometrical methods. 156.--THE DISSECTED TRIANGLE. Diagram A is our original triangle. We will say it measures 5 inches (or 5 feet) on each side. If we take off a slice at the bottom of any equilateral triangle by a cut parallel with the base, the portion that remains will always be an equilateral triangle; so we first cut off piece 1 and get a triangle 3 inches on every side. The manner of finding directions of the other cuts in A is obvious from the diagram. Now, if we want two triangles, 1 will be one of them, and 2, 3, 4, and 5 will fit together, as in B, to form the other. If we want three equilateral triangles, 1 will be one, 4 and 5 will form the second, as in C, and 2 and 3 will form the third, as in D. In B and C the piece 5 is turned over; but there can be no objection to this, as it is not forbidden, and is in no way opposed to the nature of the puzzle. [Illustration] 157.--THE TABLE-TOP AND STOOLS. [Illustration] One object that I had in view when presenting this little puzzle was to point out the uncertainty of the meaning conveyed by the word "oval." Though originally derived from the Latin word _ovum_, an egg, yet what we understand as the egg-shape (with one end smaller than the other) is only one of many forms of the oval; while some eggs are spherical in shape, and a sphere or circle is most certainly not an oval. If we speak of an ellipse--a conical ellipse--we are on safer ground, but here we must be careful of error. I recollect a Liverpool town councillor, many years ago, whose ignorance of the poultry-yard led him to substitute the word "hen" for "fowl," remarking, "We must remember, gentlemen, that although every cock is a hen, every hen is not a cock!" Similarly, we must always note that although every ellipse is an oval, every oval is not an ellipse. It is correct to say that an oval is an oblong curvilinear figure, having two unequal diameters, and bounded by a curve line returning into itself; and this includes the ellipse, but all other figures which in any way approach towards the form of an oval without necessarily having the properties above described are included in the term "oval." Thus the following solution that I give to our puzzle involves the pointed "oval," known among architects as the "vesica piscis." [Illustration: THE TWO STOOLS.] The dotted lines in the table are given for greater clearness, the cuts being made along the other lines. It will be seen that the eight pieces form two stools of exactly the same size and shape with similar hand-holes. These holes are a trifle longer than those in the schoolmaster's stools, but they are much narrower and of considerably smaller area. Of course 5 and 6 can be cut out in one piece--also 7 and 8--making only _six pieces_ in all. But I wished to keep the same number as in the original story. When I first gave the above puzzle in a London newspaper, in competition, no correct solution was received, but an ingenious and neatly executed attempt by a man lying in a London infirmary was accompanied by the following note: "Having no compasses here, I was compelled to improvise a pair with the aid of a small penknife, a bit of firewood from a bundle, a piece of tin from a toy engine, a tin tack, and two portions of a hairpin, for points. They are a fairly serviceable pair of compasses, and I shall keep them as a memento of your puzzle." 158.--THE GREAT MONAD. The areas of circles are to each other as the squares of their diameters. If you have a circle 2 in. in diameter and another 4 in. in diameter, then one circle will be four times as great in area as the other, because the square of 4 is four times as great as the square of 2. Now, if we refer to Diagram 1, we see how two equal squares may be cut into four pieces that will form one larger square; from which it is self-evident that any square has just half the area of the square of its diagonal. In Diagram 2 I have introduced a square as it often occurs in ancient drawings of the Monad; which was my reason for believing that the symbol had mathematical meanings, since it will be found to demonstrate the fact that the area of the outer ring or annulus is exactly equal to the area of the inner circle. Compare Diagram 2 with Diagram 1, and you will see that as the square of the diameter CD is double the square of the diameter of the inner circle, or CE, therefore the area of the larger circle is double the area of the smaller one, and consequently the area of the annulus is exactly equal to that of the inner circle. This answers our first question. [Illustration: 1 2 3 4] In Diagram 3 I show the simple solution to the second question. It is obviously correct, and may be proved by the cutting and superposition of parts. The dotted lines will also serve to make it evident. The third question is solved by the cut CD in Diagram 2, but it remains to be proved that the piece F is really one-half of the Yin or the Yan. This we will do in Diagram 4. The circle K has one-quarter the area of the circle containing Yin and Yan, because its diameter is just one-half the length. Also L in Diagram 3 is, we know, one-quarter the area. It is therefore evident that G is exactly equal to H, and therefore half G is equal to half H. So that what F loses from L it gains from K, and F must be half of Yin or Yan. 159.--THE SQUARE OF VENEER. [Illustration: +---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | :| | |: | | :| | |: | || | | | | :| | |: | | :| | |: | || | +---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | :| | |: | | :| | |: | || | | | | :| | |: | | :| | |: | || | +---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | :| | |: | | :| | |: | || | |_ _|___|__:|___|___|:__|___|__:|___|___|:__|__||___| +---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | :| | |: | | :| | |: | || | | | | :| | |: | | :| | |: | || | +---+---+---+---+---+---+---+---+---+---+===+===+---+ | | | :| | |: | | :| | ||: | | | | | | :| | |: | | :| | ||: | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+ |¯¯¯|¯¯¯|¯¯:|¯¯¯|¯¯¯|:¯¯|¯¯¯|¯¯:|¯¯¯|¯¯||:¯¯|¯¯¯|¯¯¯| | | | :| | |: | | :| | ||: | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | :| | |: | | :| | ||: | | | | | | :| | |: | B | :| | ||: | C | | +---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | :| | |: | | :| | ||: | | | |_ _|___|__:|___|___|:__|___|__:|___|__||:__|___|___| +---+---+---+---+---+---+---+---+===+===+===+===+===+ | | | :| | |: | | :|| | |: | | | | | | :| | |: | | :|| | |: | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | :| | |: | | :|| | |: | | | | | | :| | |: | | :|| | |: | A | | +---+---+---+---+---+---+---+---+---+---+---+---+---+ |¯¯¯|¯¯¯|¯¯:|¯¯¯|¯¯¯|:¯¯|¯¯¯|¯¯:||¯¯|¯¯¯|:¯¯|¯¯¯|¯¯¯| | | | :| | |: | | :|| | |: | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | :| | |: | | :|| | |: | | | | | | :| | |: | | :|| | |: | | | +===+===+===+===+===+===+===+===+---+---+---+---+---+ | | | :| | |: | | :|| | |: | | | | | | :| | D |: | | :|| | |: | | | +---+---+---+---+---+---+---+---+===+===+===+===+===+ ] Any square number may be expressed as the sum of two squares in an infinite number of different ways. The solution of the present puzzle forms a simple demonstration of this rule. It is a condition that we give actual dimensions. In this puzzle I ignore the known dimensions of our square and work on the assumption that it is 13n by 13n. The value of n we can afterwards determine. Divide the square as shown (where the dotted lines indicate the original markings) into 169 squares. As 169 is the sum of the two squares 144 and 25, we will proceed to divide the veneer into two squares, measuring respectively 12 × 12 and 5 × 5; and as we know that two squares may be formed from one square by dissection in four pieces, we seek a solution in this number. The dark lines in the diagram show where the cuts are to be made. The square 5 × 5 is cut out whole, and the larger square is formed from the remaining three pieces, B, C, and D, which the reader can easily fit together. Now, n is clearly 5/13 of an inch. Consequently our larger square must be 60/13 in. × 60/13 in., and our smaller square 25/13 in. × 25/13 in. The square of 60/13 added to the square of 25/13 is 25. The square is thus divided into as few as four pieces that form two squares of known dimensions, and all the sixteen nails are avoided. Here is a general formula for finding two squares whose sum shall equal a given square, say a². In the case of the solution of our puzzle p = 3, q = 2, and a = 5. ________________________ 2pqa \/ a²( p² + q²)² - (2pqa)² --------- = x; --------------------------- = y p² + q² p² + q² Here x² + y² = a². 160.--THE TWO HORSESHOES. The puzzle was to cut the two shoes (including the hoof contained within the outlines) into four pieces, two pieces each, that would fit together and form a perfect circle. It was also stipulated that all four pieces should be different in shape. As a matter of fact, it is a puzzle based on the principle contained in that curious Chinese symbol the Monad. (See No. 158.) [Illustration] The above diagrams give the correct solution to the problem. It will be noticed that 1 and 2 are cut into the required four pieces, all different in shape, that fit together and form the perfect circle shown in Diagram 3. It will further be observed that the two pieces A and B of one shoe and the two pieces C and D of the other form two exactly similar halves of the circle--the Yin and the Yan of the great Monad. It will be seen that the shape of the horseshoe is more easily determined from the circle than the dimensions of the circle from the horseshoe, though the latter presents no difficulty when you know that the curve of the long side of the shoe is part of the circumference of your circle. The difference between B and D is instructive, and the idea is useful in all such cases where it is a condition that the pieces must be different in shape. In forming D we simply add on a symmetrical piece, a curvilinear square, to the piece B. Therefore, in giving either B or D a quarter turn before placing in the new position, a precisely similar effect must be produced. 161.--THE BETSY ROSS PUZZLE. Fold the circular piece of paper in half along the dotted line shown in Fig. 1, and divide the upper half into five equal parts as indicated. Now fold the paper along the lines, and it will have the appearance shown in Fig. 2. If you want a star like Fig. 3, cut from A to B; if you wish one like Fig. 4, cut from A to C. Thus, the nearer you cut to the point at the bottom the longer will be the points of the star, and the farther off from the point that you cut the shorter will be the points of the star. [Illustration] 162.--THE CARDBOARD CHAIN. The reader will probably feel rewarded for any care and patience that he may bestow on cutting out the cardboard chain. We will suppose that he has a piece of cardboard measuring 8 in. by 2½ in., though the dimensions are of no importance. Yet if you want a long chain you must, of course, take a long strip of cardboard. First rule pencil lines B B and C C, half an inch from the edges, and also the short perpendicular lines half an inch apart. (See next page.) Rule lines on the other side in just the same way, and in order that they shall coincide it is well to prick through the card with a needle the points where the short lines end. Now take your penknife and split the card from A A down to B B, and from D D up to C C. Then cut right through the card along all the short perpendicular lines, and half through the card along the short portions of B B and C C that are not dotted. Next turn the card over and cut half through along the short lines on B B and C C at the places that are immediately beneath the dotted lines on the upper side. With a little careful separation of the parts with the penknife, the cardboard may now be divided into two interlacing ladder-like portions, as shown in Fig. 2; and if you cut away all the shaded parts you will get the chain, cut solidly out of the cardboard, without any join, as shown in the illustrations on page 40. It is an interesting variant of the puzzle to cut out two keys on a ring--in the same manner without join. [Illustration] 164.--THE POTATO PUZZLE. As many as twenty-two pieces may be obtained by the six cuts. The illustration shows a pretty symmetrical solution. The rule in such cases is that every cut shall intersect every other cut and no two intersections coincide; that is to say, every line passes through every other line, but more than two lines do not cross at the same point anywhere. There are other ways of making the cuts, but this rule must always be observed if we are to get the full number of pieces. The general formula is that with n cuts we can always produce ((n(n+1))/2)+1 . One of the problems proposed by the late Sam Loyd was to produce the maximum number of pieces by n straight cuts through a solid cheese. Of course, again, the pieces cut off may not be moved or piled. Here we have to deal with the intersection of planes (instead of lines), and the general formula is that with n cuts we may produce ((n - 1)n(n + 1))/6 + n + 1 pieces. It is extremely difficult to "see" the direction and effects of the successive cuts for more than a few of the lowest values of n. 165.--THE SEVEN PIGS. The illustration shows the direction for placing the three fences so as to enclose every pig in a separate sty. The greatest number of spaces that can be enclosed with three straight lines in a square is seven, as shown in the last puzzle. Bearing this fact in mind, the puzzle must be solved by trial. [Illustration: THE SEVEN PIGS.] 166.--THE LANDOWNER'S FENCES. Four fences only are necessary, as follows:-- [Illustration] 167.--THE WIZARD'S CATS. The illustration requires no explanation. It shows clearly how the three circles may be drawn so that every cat has a separate enclosure, and cannot approach another cat without crossing a line. [Illustration: THE WIZARDS' CATS.] 168.--THE CHRISTMAS PUDDING. The illustration shows how the pudding may be cut into two parts of exactly the same size and shape. The lines must necessarily pass through the points A, B, C, D, and E. But, subject to this condition, they may be varied in an infinite number of ways. For example, at a point midway between A and the edge, the line may be completed in an unlimited number of ways (straight or crooked), provided it be exactly reflected from E to the opposite edge. And similar variations may be introduced at other places. [Illustration] 169.--A TANGRAM PARADOX. The diagrams will show how the figures are constructed--each with the seven Tangrams. It will be noticed that in both cases the head, hat, and arm are precisely alike, and the width at the base of the body the same. But this body contains four pieces in the first case, and in the second design only three. The first is larger than the second by exactly that narrow strip indicated by the dotted line between A and B. This strip is therefore exactly equal in area to the piece forming the foot in the other design, though when thus distributed along the side of the body the increased dimension is not easily apparent to the eye. [Illustration] 170.--THE CUSHION COVERS. [Illustration] The two pieces of brocade marked A will fit together and form one perfect square cushion top, and the two pieces marked B will form the other. 171.--THE BANNER PUZZLE. The illustration explains itself. Divide the bunting into 25 squares (because this number is the sum of two other squares--16 and 9), and then cut along the thick lines. The two pieces marked A form one square, and the two pieces marked B form the other. [Illustration] 172.--MRS. SMILEY'S CHRISTMAS PRESENT. [Illustration] [Illustration] The first step is to find six different square numbers that sum to 196. For example, 1 + 4 + 25 + 36 + 49 + 81 = 196; 1 + 4 + 9 + 25 + 36 + 121 = 196; 1 + 9 + 16 + 25 + 64 + 81 = 196. The rest calls for individual judgment and ingenuity, and no definite rules can be given for procedure. The annexed diagrams will show solutions for the first two cases stated. Of course the three pieces marked A and those marked B will fit together and form a square in each case. The assembling of the parts may be slightly varied, and the reader may be interested in finding a solution for the third set of squares I have given. 173.--MRS. PERKINS'S QUILT. The following diagram shows how the quilt should be constructed. [Illustration] There is, I believe, practically only one solution to this puzzle. The fewest separate squares must be eleven. The portions must be of the sizes given, the three largest pieces must be arranged as shown, and the remaining group of eight squares may be "reflected," but cannot be differently arranged. 174.--THE SQUARES OF BROCADE. [Illustration: Diagram 1] So far as I have been able to discover, there is only one possible solution to fulfil the conditions. The pieces fit together as in Diagram 1, Diagrams 2 and 3 showing how the two original squares are to be cut. It will be seen that the pieces A and C have each twenty chequers, and are therefore of equal area. Diagram 4 (built up with the dissected square No. 5) solves the puzzle, except for the small condition contained in the words, "I cut the _two_ squares in the manner desired." In this case the smaller square is preserved intact. Still I give it as an illustration of a feature of the puzzle. It is impossible in a problem of this kind to give a _quarter-turn_ to any of the pieces if the pattern is to properly match, but (as in the case of F, in Diagram 4) we may give a symmetrical piece a _half-turn_--that is, turn it upside down. Whether or not a piece may be given a quarter-turn, a half-turn, or no turn at all in these chequered problems, depends on the character of the design, on the material employed, and also on the form of the piece itself. [Illustration: Diagram 2] [Illustration: Diagram 3] [Illustration: Diagram 4] [Illustration: Diagram 5] 175.--ANOTHER PATCHWORK PUZZLE. The lady need only unpick the stitches along the dark lines in the larger portion of patchwork, when the four pieces will fit together and form a square, as shown in our illustration. [Illustration] 176.--LINOLEUM CUTTING. There is only one solution that will enable us to retain the larger of the two pieces with as little as possible cut from it. Fig. 1 in the following diagram shows how the smaller piece is to be cut, and Fig. 2 how we should dissect the larger piece, while in Fig. 3 we have the new square 10 × 10 formed by the four pieces with all the chequers properly matched. It will be seen that the piece D contains fifty-two chequers, and this is the largest piece that it is possible to preserve under the conditions. [Illustration] 177.--ANOTHER LINOLEUM PUZZLE. Cut along the thick lines, and the four pieces will fit together and form a perfect square in the manner shown in the smaller diagram. [Illustration: ANOTHER LINOLEUM PUZZLE.] 178.--THE CARDBOARD BOX. The areas of the top and side multiplied together and divided by the area of the end give the square of the length. Similarly, the product of top and end divided by side gives the square of the breadth; and the product of side and end divided by the top gives the square of the depth. But we only need one of these operations. Let us take the first. Thus, 120 × 96 divided by 80 equals 144, the square of 12. Therefore the length is 12 inches, from which we can, of course, at once get the breadth and depth--10 in. and 8 in. respectively. 179.--STEALING THE BELL-ROPES. Whenever we have one side (a) of a right-angled triangle, and know the difference between the second side and the hypotenuse (which difference we will call b), then the length of the hypotenuse will be a² b --- + -. 2b 2 In the case of our puzzle this will be 48 × 48 ------- + 1½ in. = 32 ft. 1½ in., 6 which is the length of the rope. 180-- THE FOUR SONS. [Illustration] The diagram shows the most equitable division of the land possible, "so that each son shall receive land of exactly the same area and exactly similar in shape," and so that each shall have access to the well in the centre without trespass on another's land. The conditions do not require that each son's land shall be in one piece, but it is necessary that the two portions assigned to an individual should be kept apart, or two adjoining portions might be held to be one piece, in which case the condition as to shape would have to be broken. At present there is only one shape for each piece of land--half a square divided diagonally. And A, B, C, and D can each reach their land from the outside, and have each equal access to the well in the centre. 181.--THE THREE RAILWAY STATIONS. The three stations form a triangle, with sides 13, 14, and 15 miles. Make the 14 side the base; then the height of the triangle is 12 and the area 84. Multiply the three sides together and divide by four times the area. The result is eight miles and one-eighth, the distance required. 182.--THE GARDEN PUZZLE. Half the sum of the four sides is 144. From this deduct in turn the four sides, and we get 64, 99, 44, and 81. Multiply these together, and we have as the result the square of 4,752. Therefore the garden contained 4,752 square yards. Of course the tree being equidistant from the four corners shows that the garden is a quadrilateral that may be inscribed in a circle. 183.--DRAWING A SPIRAL. Make a fold in the paper, as shown by the dotted line in the illustration. Then, taking any two points, as A and B, describe semicircles on the line alternately from the centres B and A, being careful to make the ends join, and the thing is done. Of course this is not a _true_ spiral, but the puzzle was to produce the _particular_ spiral that was shown, and that was drawn in this simple manner. [Illustration] 184.--HOW TO DRAW AN OVAL. If you place your sheet of paper round the surface of a cylindrical bottle or canister, the oval can be drawn with one sweep of the compasses. 185.--ST. GEORGE'S BANNER. As the flag measures 4 ft. by 3 ft., the length of the diagonal (from corner to corner) is 5 ft. All you need do is to deduct half the length of this diagonal (2½ ft.) from a quarter of the distance all round the edge of the flag (3½ ft.)--a quarter of 14 ft. The difference (1 ft.) is the required width of the arm of the red cross. The area of the cross will then be the same as that of the white ground. 186.--THE CLOTHES LINE PUZZLE. Multiply together, and also add together, the heights of the two poles and divide one result by the other. That is, if the two heights are a and b respectively, then ab/(a + b) will give the height of the intersection. In the particular case of our puzzle, the intersection was therefore 2 ft. 11 in. from the ground. The distance that the poles are apart does not affect the answer. The reader who may have imagined that this was an accidental omission will perhaps be interested in discovering the reason why the distance between the poles may be ignored. 187.--THE MILKMAID PUZZLE. [Illustration: A |\ | \ | \ | \ B RIVER +----+-------------- | / \ | / \ | / \ |/ DOOR STOOL ] Draw a straight line, as shown in the diagram, from the milking-stool perpendicular to the near bank of the river, and continue it to the point A, which is the same distance from that bank as the stool. If you now draw the straight line from A to the door of the dairy, it will cut the river at B. Then the shortest route will be from the stool to B and thence to the door. Obviously the shortest distance from A to the door is the straight line, and as the distance from the stool to any point of the river is the same as from A to that point, the correctness of the solution will probably appeal to every reader without any acquaintance with geometry. 188.--THE BALL PROBLEM. If a round ball is placed on the level ground, six similar balls may be placed round it (all on the ground), so that they shall all touch the central ball. As for the second question, the ratio of the diameter of a circle to its circumference we call _pi_; and though we cannot express this ratio in exact numbers, we can get sufficiently near to it for all practical purposes. However, in this case it is not necessary to know the value of _pi_ at all. Because, to find the area of the surface of a sphere we multiply the square of the diameter by _pi_; to find the volume of a sphere we multiply the cube of the diameter by one-sixth of _pi_. Therefore we may ignore _pi_, and have merely to seek a number whose square shall equal one-sixth of its cube. This number is obviously 6. Therefore the ball was 6 ft. in diameter, for the area of its surface will be 36 times _pi_ in square feet, and its volume also 36 times _pi_ in cubic feet. 189.--THE YORKSHIRE ESTATES. The triangular piece of land that was not for sale contains exactly eleven acres. Of course it is not difficult to find the answer if we follow the eccentric and tricky tracks of intricate trigonometry; or I might say that the application of a well-known formula reduces the problem to finding one-quarter of the square root of (4 × 370 × 116) -(370 + 116 - 74)²--that is a quarter of the square root of 1936, which is one-quarter of 44, or 11 acres. But all that the reader really requires to know is the Pythagorean law on which many puzzles have been built, that in any right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. I shall dispense with all "surds" and similar absurdities, notwithstanding the fact that the sides of our triangle are clearly incommensurate, since we cannot exactly extract the square roots of the three square areas. [Illustration: A |\ | \. | \ . |5 \ . | 7 \ . E +--------- +C . | | ` . . | | `. . |4 |4 ` . . | 7 | ` .. D----------+----------------- B F ] In the above diagram ABC represents our triangle. ADB is a right-angled triangle, AD measuring 9 and BD measuring 17, because the square of 9 added to the square of 17 equals 370, the known area of the square on AB. Also AEC is a right-angled triangle, and the square of 5 added to the square of 7 equals 74, the square estate on A C. Similarly, CFB is a right-angled triangle, for the square of 4 added to the square of 10 equals 116, the square estate on BC. Now, although the sides of our triangular estate are incommensurate, we have in this diagram all the exact figures that we need to discover the area with precision. The area of our triangle ADB is clearly half of 9 × 17, or 76½ acres. The area of AEC is half of 5 × 7, or 17½ acres; the area of CFB is half of 4 × 10, or 20 acres; and the area of the oblong EDFC is obviously 4 × 7, or 28 acres. Now, if we add together 17½, 20, and 28 = 65½, and deduct this sum from the area of the large triangle ADB (which we have found to be 76½ acres), what remains must clearly be the area of ABC. That is to say, the area we want must be 76½ - 65½ = 11 acres exactly. 190.--FARMER WURZEL'S ESTATE. The area of the complete estate is exactly one hundred acres. To find this answer I use the following little formula, __________________ \/4ab - (a + b - c)² -------------------- 4 where a, b, c represent the three square areas, in any order. The expression gives the area of the triangle A. This will be found to be 9 acres. It can be easily proved that A, B, C, and D are all equal in area; so the answer is 26 + 20 + 18 + 9 + 9 + 9 + 9 = 100 acres. [Illustration] Here is the proof. If every little dotted square in the diagram represents an acre, this must be a correct plan of the estate, for the squares of 5 and 1 together equal 26; the squares of 4 and 2 equal 20; and the squares of 3 and 3 added together equal 18. Now we see at once that the area of the triangle E is 2½, F is 4½, and G is 4. These added together make 11 acres, which we deduct from the area of the rectangle, 20 acres, and we find that the field A contains exactly 9 acres. If you want to prove that B, C, and D are equal in size to A, divide them in two by a line from the middle of the longest side to the opposite angle, and you will find that the two pieces in every case, if cut out, will exactly fit together and form A. Or we can get our proof in a still easier way. The complete area of the squared diagram is 12 × 12 = 144 acres, and the portions 1, 2, 3, 4, not included in the estate, have the respective areas of 12½, 17½, 9½, and 4½. These added together make 44, which, deducted from 144, leaves 100 as the required area of the complete estate. 191.--THE CRESCENT PUZZLE. Referring to the original diagram, let AC be x, let CD be x - 9, and let EC be x - 5. Then x - 5 is a mean proportional between x - 9 and x, from which we find that x equals 25. Therefore the diameters are 50 in. and 41 in. respectively. 192.--THE PUZZLE WALL. [Illustration] The answer given in all the old books is that shown in Fig. 1, where the curved wall shuts out the cottages from access to the lake. But in seeking the direction for the "shortest possible" wall most readers to-day, remembering that the shortest distance between two points is a straight line, will adopt the method shown in Fig. 2. This is certainly an improvement, yet the correct answer is really that indicated in Fig. 3. A measurement of the lines will show that there is a considerable saving of length in this wall. 193.--THE SHEEP-FOLD. This is the answer that is always given and accepted as correct: Two more hurdles would be necessary, for the pen was twenty-four by one (as in Fig. A on next page), and by moving one of the sides and placing an extra hurdle at each end (as in Fig. B) the area would be doubled. The diagrams are not to scale. Now there is no condition in the puzzle that requires the sheep-fold to be of any particular form. But even if we accept the point that the pen was twenty-four by one, the answer utterly fails, for two extra hurdles are certainly not at all necessary. For example, I arrange the fifty hurdles as in Fig. C, and as the area is increased from twenty-four "square hurdles" to 156, there is now accommodation for 650 sheep. If it be held that the area must be exactly double that of the original pen, then I construct it (as in Fig. D) with twenty-eight hurdles only, and have twenty-two in hand for other purposes on the farm. Even if it were insisted that all the original hurdles must be used, then I should construct it as in Fig. E, where I can get the area as exact as any farmer could possibly require, even if we have to allow for the fact that the sheep might not be able to graze at the extreme ends. Thus we see that, from any point of view, the accepted answer to this ancient little puzzle breaks down. And yet attention has never before been drawn to the absurdity. [Illustration A 24 +--------------------------------+ | 24 |1 +--------------------------------+ B +--------------------------------+ | 48 |2 +--------------------------------+ 24 C +--------------------+ D | | +----------+ | | | | | |12 | 48 |6 | 156 | | | | | +----------+ | | 8 | | | | +--------------------+ 13 12 . E 13 . ' ' . . ' ' . ' . . ' 12 ' . ' 13 ] 194.--THE GARDEN WALLS. The puzzle was to divide the circular field into four equal parts by three walls, each wall being of exactly the same length. There are two essential difficulties in this problem. These are: (1) the thickness of the walls, and (2) the condition that these walls are three in number. As to the first point, since we are told that the walls are brick walls, we clearly cannot ignore their thickness, while we have to find a solution that will equally work, whether the walls be of a thickness of one, two, three, or more bricks. [Illustration] The second point requires a little more consideration. How are we to distinguish between a wall and walls? A straight wall without any bend in it, no matter how long, cannot ever become "walls," if it is neither broken nor intersected in any way. Also our circular field is clearly enclosed by one wall. But if it had happened to be a square or a triangular enclosure, would there be respectively four and three walls or only one enclosing wall in each case? It is true that we speak of "the four walls" of a square building or garden, but this is only a conventional way of saying "the four sides." If you were speaking of the actual brickwork, you would say, "I am going to enclose this square garden with a wall." Angles clearly do not affect the question, for we may have a zigzag wall just as well as a straight one, and the Great Wall of China is a good example of a wall with plenty of angles. Now, if you look at Diagrams 1, 2, and 3, you may be puzzled to declare whether there are in each case two or four new walls; but you cannot call them three, as required in our puzzle. The intersection either affects the question or it does not affect it. If you tie two pieces of string firmly together, or splice them in a nautical manner, they become "one piece of string." If you simply let them lie across one another or overlap, they remain "two pieces of string." It is all a question of joining and welding. It may similarly be held that if two walls be built into one another--I might almost say, if they be made homogeneous--they become one wall, in which case Diagrams 1, 2, and 3 might each be said to show one wall or two, if it be indicated that the four ends only touch, and are not really built into, the outer circular wall. The objection to Diagram 4 is that although it shows the three required walls (assuming the ends are not built into the outer circular wall), yet it is only absolutely correct when we assume the walls to have no thickness. A brick has thickness, and therefore the fact throws the whole method out and renders it only approximately correct. Diagram 5 shows, perhaps, the only correct and perfectly satisfactory solution. It will be noticed that, in addition to the circular wall, there are three new walls, which touch (and so enclose) but are not built into one another. This solution may be adapted to any desired thickness of wall, and its correctness as to area and length of wall space is so obvious that it is unnecessary to explain it. I will, however, just say that the semicircular piece of ground that each tenant gives to his neighbour is exactly equal to the semicircular piece that his neighbour gives to him, while any section of wall space found in one garden is precisely repeated in all the others. Of course there is an infinite number of ways in which this solution may be correctly varied. 195.--LADY BELINDA'S GARDEN. All that Lady Belinda need do was this: She should measure from A to B, fold her tape in four and mark off the point E, which is thus one quarter of the side. Then, in the same way, mark off the point F, one-fourth of the side AD Now, if she makes EG equal to AF, and GH equal to EF, then AH is the required width for the path in order that the bed shall be exactly half the area of the garden. An exact numerical measurement can only be obtained when the sum of the squares of the two sides is a square number. Thus, if the garden measured 12 poles by 5 poles (where the squares of 12 and 5, 144 and 25, sum to 169, the square of 13), then 12 added to 5, less 13, would equal four, and a quarter of this, 1 pole, would be the width of the path. 196.--THE TETHERED GOAT. [Illustration] This problem is quite simple if properly attacked. Let us suppose the triangle ABC to represent our half-acre field, and the shaded portion to be the quarter-acre over which the goat will graze when tethered to the corner C. Now, as six equal equilateral triangles placed together will form a regular hexagon, as shown, it is evident that the shaded pasture is just one-sixth of the complete area of a circle. Therefore all we require is the radius (CD) of a circle containing six quarter-acres or 1½ acres, which is equal to 9,408,960 square inches. As we only want our answer "to the nearest inch," it is sufficiently exact for our purpose if we assume that as 1 is to 3.1416, so is the diameter of a circle to its circumference. If, therefore, we divide the last number I gave by 3.1416, and extract the square root, we find that 1,731 inches, or 48 yards 3 inches, is the required length of the tether "to the nearest inch." 197.--THE COMPASSES PUZZLE. Let AB in the following diagram be the given straight line. With the centres A and B and radius AB describe the two circles. Mark off DE and EF equal to AD. With the centres A and F and radius DF describe arcs intersecting at G. With the centres A and B and distance BG describe arcs GHK and N. Make HK equal to AB and HL equal to HB. Then with centres K and L and radius AB describe arcs intersecting at I. Make BM equal to BI. Finally, with the centre M and radius MB cut the line in C, and the point C is the required middle of the line AB. For greater exactitude you can mark off R from A (as you did M from B), and from R describe another arc at C. This also solves the problem, to find a point midway between two given points without the straight line. [Illustration] I will put the young geometer in the way of a rigid proof. First prove that twice the square of the line AB equals the square of the distance BG, from which it follows that HABN are the four corners of a square. To prove that I is the centre of this square, draw a line from H to P through QIB and continue the arc HK to P. Then, conceiving the necessary lines to be drawn, the angle HKP, being in a semicircle, is a right angle. Let fall the perpendicular KQ, and by similar triangles, and from the fact that HKI is an isosceles triangle by the construction, it can be proved that HI is half of HB. We can similarly prove that C is the centre of the square of which AIB are three corners. I am aware that this is not the simplest possible solution. 198.--THE EIGHT STICKS. The first diagram is the answer that nearly every one will give to this puzzle, and at first sight it seems quite satisfactory. But consider the conditions. We have to lay "every one of the sticks on the table." Now, if a ladder be placed against a wall with only one end on the ground, it can hardly be said that it is "laid on the ground." And if we place the sticks in the above manner, it is only possible to make one end of two of them touch the table: to say that every one lies on the table would not be correct. To obtain a solution it is only necessary to have our sticks of proper dimensions. Say the long sticks are each 2 ft. in length and the short ones 1 ft. Then the sticks must be 3 in. thick, when the three equal squares may be enclosed, as shown in the second diagram. If I had said "matches" instead of "sticks," the puzzle would be impossible, because an ordinary match is about twenty-one times as long as it is broad, and the enclosed rectangles would not be squares. [Illustration] 199.--PAPA'S PUZZLE. I have found that a large number of people imagine that the following is a correct solution of the problem. Using the letters in the diagram below, they argue that if you make the distance BA one-third of BC, and therefore the area of the rectangle ABE equal to that of the triangular remainder, the card must hang with the long side horizontal. Readers will remember the jest of Charles II., who induced the Royal Society to meet and discuss the reason why the water in a vessel will not rise if you put a live fish in it; but in the middle of the proceedings one of the least distinguished among them quietly slipped out and made the experiment, when he found that the water _did_ rise! If my correspondents had similarly made the experiment with a piece of cardboard, they would have found at once their error. Area is one thing, but gravitation is quite another. The fact of that triangle sticking its leg out to D has to be compensated for by additional area in the rectangle. As a matter of fact, the ratio of BA to AC is as 1 is to the square root of 3, which latter cannot be given in an exact numerical measure, but is approximately 1.732. Now let us look at the correct general solution. There are many ways of arriving at the desired result, but the one I give is, I think, the simplest for beginners. [Illustration] Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC. Also mark off the point G so that DG shall equal DC. Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line. A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD. The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it. It will be seen at once that however much you may reduce the width of the card by bringing E nearer to B and D nearer to C, the line CG, being the diagonal of a square, will always lie in the same direction, and will cut BF in H. Finally, if you wish to get an approximate measure for the distance BA, all you have to do is to multiply the length of the card by the decimal .366. Thus, if the card were 7 inches long, we get 7 × .366 = 2.562, or a little more than 2½ inches, for the distance from B to A. But the real joke of the puzzle is this: We have seen that the position of the point A is independent of the width of the card, and depends entirely on the length. Now, in the illustration it will be found that both cards have the same length; consequently all the little maid had to do was to lay the clipped card on top of the other one and mark off the point A at precisely the same distance from the top left-hand corner! So, after all, Pappus' puzzle, as he presented it to his little maid, was quite an infantile problem, when he was able to show her how to perform the feat without first introducing her to the elements of statics and geometry. 200.--A KITE-FLYING PUZZLE. Solvers of this little puzzle, I have generally found, may be roughly divided into two classes: those who get within a mile of the correct answer by means of more or less complex calculations, involving "_pi_," and those whose arithmetical kites fly hundreds and thousands of miles away from the truth. The comparatively easy method that I shall show does not involve any consideration of the ratio that the diameter of a circle bears to its circumference. I call it the "hat-box method." [Illustration] Supposing we place our ball of wire, A, in a cylindrical hat-box, B, that exactly fits it, so that it touches the side all round and exactly touches the top and bottom, as shown in the illustration. Then, by an invariable law that should be known by everybody, that box contains exactly half as much again as the ball. Therefore, as the ball is 24 in. in diameter, a hat-box of the same circumference but two-thirds of the height (that is, 16 in. high) will have exactly the same contents as the ball. Now let us consider that this reduced hat-box is a cylinder of metal made up of an immense number of little wire cylinders close together like the hairs in a painter's brush. By the conditions of the puzzle we are allowed to consider that there are no spaces between the wires. How many of these cylinders one one-hundredth of an inch thick are equal to the large cylinder, which is 24 in. thick? Circles are to one another as the squares of their diameters. The square of 1/100 is 1/100000, and the square of 24 is 576; therefore the large cylinder contains 5,760,000 of the little wire cylinders. But we have seen that each of these wires is 16 in. long; hence 16 × 5,760,000 = 92,160,000 inches as the complete length of the wire. Reduce this to miles, and we get 1,454 miles 2,880 ft. as the length of the wire attached to the professor's kite. Whether a kite would fly at such a height, or support such a weight, are questions that do not enter into the problem. 201.--HOW TO MAKE CISTERNS. Here is a general formula for solving this problem. Call the two sides of the rectangle a and b. Then a + b - (a² + b² - ab)^½ --------------------------- 6 equals the side of the little square pieces to cut away. The measurements given were 8 ft. by 3 ft., and the above rule gives 8 in. as the side of the square pieces that have to be cut away. Of course it will not always come out exact, as in this case (on account of that square root), but you can get as near as you like with decimals. 202.--THE CONE PUZZLE. The simple rule is that the cone must be cut at one-third of its altitude. 203.--CONCERNING WHEELS. If you mark a point A on the circumference of a wheel that runs on the surface of a level road, like an ordinary cart-wheel, the curve described by that point will be a common cycloid, as in Fig. 1. But if you mark a point B on the circumference of the flange of a locomotive-wheel, the curve will be a curtate cycloid, as in Fig. 2, terminating in nodes. Now, if we consider one of these nodes or loops, we shall see that "at any given moment" certain points at the bottom of the loop must be moving in the opposite direction to the train. As there is an infinite number of such points on the flange's circumference, there must be an infinite number of these loops being described while the train is in motion. In fact, at any given moment certain points on the flanges are always moving in a direction opposite to that in which the train is going. [Illustration: 1] [Illustration: 2] In the case of the two wheels, the wheel that runs round the stationary one makes two revolutions round its own centre. As both wheels are of the same size, it is obvious that if at the start we mark a point on the circumference of the upper wheel, at the very top, this point will be in contact with the lower wheel at its lowest part when half the journey has been made. Therefore this point is again at the top of the moving wheel, and one revolution has been made. Consequently there are two such revolutions in the complete journey. 204.--A NEW MATCH PUZZLE. 1. The easiest way is to arrange the eighteen matches as in Diagrams 1 and 2, making the length of the perpendicular AB equal to a match and a half. Then, if the matches are an inch in length, Fig. 1 contains two square inches and Fig. 2 contains six square inches--4 × 1½. The second case (2) is a little more difficult to solve. The solution is given in Figs. 3 and 4. For the purpose of construction, place matches temporarily on the dotted lines. Then it will be seen that as 3 contains five equal equilateral triangles and 4 contains fifteen similar triangles, one figure is three times as large as the other, and exactly eighteen matches are used. [Illustration: Figures 1, 2, 3, 4.] 205.--THE SIX SHEEP-PENS. [Illustration] Place the twelve matches in the manner shown in the illustration, and you will have six pens of equal size. 206.--THE KING AND THE CASTLES. There are various ways of building the ten castles so that they shall form five rows with four castles in every row, but the arrangement in the next column is the only one that also provides that two castles (the greatest number possible) shall not be approachable from the outside. It will be seen that you must cross the walls to reach these two. [Illustration: The King and the Castles] 207.--CHERRIES AND PLUMS. There are several ways in which this problem might be solved were it not for the condition that as few cherries and plums as possible shall be planted on the north and east sides of the orchard. The best possible arrangement is that shown in the diagram, where the cherries, plums, and apples are indicated respectively by the letters C, P, and A. The dotted lines connect the cherries, and the other lines the plums. It will be seen that the ten cherry trees and the ten plum trees are so planted that each fruit forms five lines with four trees of its kind in line. This is the only arrangement that allows of so few as two cherries or plums being planted on the north and east outside rows. [Illustration] 208.--A PLANTATION PUZZLE. The illustration shows the ten trees that must be left to form five rows with four trees in every row. The dots represent the positions of the trees that have been cut down. [Illustration] 209.--THE TWENTY-ONE TREES. I give two pleasing arrangements of the trees. In each case there are twelve straight rows with five trees in every row. [Illustration: Figure 1, Figure 2.] 210.--THE TEN COINS. The answer is that there are just 2,400 different ways. Any three coins may be taken from one side to combine with one coin taken from the other side. I give four examples on this and the next page. We may thus select three from the top in ten ways and one from the bottom in five ways, making fifty. But we may also select three from the bottom and one from the top in fifty ways. We may thus select the four coins in one hundred ways, and the four removed may be arranged by permutation in twenty-four ways. Thus there are 24 × 100 = 2,400 different solutions. [Illustration] As all the points and lines puzzles that I have given so far, excepting the last, are variations of the case of ten points arranged to form five lines of four, it will be well to consider this particular case generally. There are six fundamental solutions, and no more, as shown in the six diagrams. These, for the sake of convenience, I named some years ago the Star, the Dart, the Compasses, the Funnel, the Scissors, and the Nail. (See next page.) Readers will understand that any one of these forms may be distorted in an infinite number of different ways without destroying its real character. In "The King and the Castles" we have the Star, and its solution gives the Compasses. In the "Cherries and Plums" solution we find that the Cherries represent the Funnel and the Plums the Dart. The solution of the "Plantation Puzzle" is an example of the Dart distorted. Any solution to the "Ten Coins" will represent the Scissors. Thus examples of all have been given except the Nail. On a reduced chessboard, 7 by 7, we may place the ten pawns in just three different ways, but they must all represent the Dart. The "Plantation" shows one way, the Plums show a second way, and the reader may like to find the third way for himself. On an ordinary chessboard, 8 by 8, we can also get in a beautiful example of the Funnel--symmetrical in relation to the diagonal of the board. The smallest board that will take a Star is one 9 by 7. The Nail requires a board 11 by 7, the Scissors [Illustration] 11 by 9, and the Compasses 17 by 12. At least these are the best results recorded in my note-book. They may be beaten, but I do not think so. If you divide a chessboard into two parts by a diagonal zigzag line, so that the larger part contains 36 squares and the smaller part 28 squares, you can place three separate schemes on the larger part and one on the smaller part (all Darts) without their conflicting--that is, they occupy forty different squares. They can be placed in other ways without a division of the board. The smallest square board that will contain six different schemes (not fundamentally different), without any line of one scheme crossing the line of another, is 14 by 14; and the smallest board that will contain one scheme entirely enclosed within the lines of a second scheme, without any of the lines of the one, when drawn from point to point, crossing a line of the other, is 14 by 12. [Illustration: STAR DART COMPASSES FUNNEL SCISSORS NAIL] 211.--THE TWELVE MINCE-PIES. If you ignore the four black pies in our illustration, the remaining twelve are in their original positions. Now remove the four detached pies to the places occupied by the black ones, and you will have your seven straight rows of four, as shown by the dotted lines. [Illustration: The Twelve Mince Pies.] 212.--THE BURMESE PLANTATION. The arrangement on the next page is the most symmetrical answer that can probably be found for twenty-one rows, which is, I believe, the greatest number of rows possible. There are several ways of doing it. 213.--TURKS AND RUSSIANS. The main point is to discover the smallest possible number of Russians that there could have been. As the enemy opened fire from all directions, it is clearly necessary to find what is the smallest number of heads that could form sixteen lines with three heads in every line. Note that I say sixteen, and not thirty-two, because every line taken by a bullet may be also taken by another bullet fired in exactly the opposite direction. Now, as few as eleven points, or heads, may be arranged to form the required sixteen lines of three, but the discovery of this arrangement is a hard nut. The diagram at the foot of this page will show exactly how the thing is to be done. [Illustration] If, therefore, eleven Russians were in the positions shown by the stars, and the thirty-two Turks in the positions indicated by the black dots, it will be seen, by the lines shown, that each Turk may fire exactly over the heads of three Russians. But as each bullet kills a man, it is essential that every Turk shall shoot one of his comrades and be shot by him in turn; otherwise we should have to provide extra Russians to be shot, which would be destructive of the correct solution of our problem. As the firing was simultaneous, this point presents no difficulties. The answer we thus see is that there were at least eleven Russians amongst whom there was no casualty, and that all the thirty-two Turks were shot by one another. It was not stated whether the Russians fired any shots, but it will be evident that even if they did their firing could not have been effective: for if one of their bullets killed a Turk, then we have immediately to provide another man for one of the Turkish bullets to kill; and as the Turks were known to be thirty-two in number, this would necessitate our introducing another Russian soldier and, of course, destroying the solution. I repeat that the difficulty of the puzzle consists in finding how to arrange eleven points so that they shall form sixteen lines of three. I am told that the possibility of doing this was first discovered by the Rev. Mr. Wilkinson some twenty years ago. 214.--THE SIX FROGS. Move the frogs in the following order: 2, 4, 6, 5, 3, 1 (repeat these moves in the same order twice more), 2, 4, 6. This is a solution in twenty-one moves--the fewest possible. If n, the number of frogs, be even, we require (n² + n)/2 moves, of which (n² - n)/2 will be leaps and n simple moves. If n be odd, we shall need ((n² + 3n)/2) - 4 moves, of which (n² - n)/2 will be leaps and 2n - 4 simple moves. In the even cases write, for the moves, all the even numbers in ascending order and the odd numbers in descending order. This series must be repeated ½n times and followed by the even numbers in ascending order once only. Thus the solution for 14 frogs will be (2, 4, 6, 8, 10, 12, 14, 13, 11, 9, 7, 5, 3, 1) repeated 7 times and followed by 2, 4, 6, 8, 10, 12, 14 = 105 moves. In the odd cases, write the even numbers in ascending order and the odd numbers in descending order, repeat this series ½(n - 1) times, follow with the even numbers in ascending order (omitting n - 1), the odd numbers in descending order (omitting 1), and conclude with all the numbers (odd and even) in their natural order (omitting 1 and n). Thus for 11 frogs: (2, 4, 6, 8, 10, 11, 9, 7, 5, 3, 1) repeated 5 times, 2, 4, 6, 8, 11, 9, 7, 5, 3, and 2, 3, 4, 5, 6, 7, 8, 9, 10 = 73 moves. This complete general solution is published here for the first time. 215.--THE GRASSHOPPER PUZZLE. Move the counters in the following order. The moves in brackets are to be made four times in succession. 12, 1, 3, 2, 12, 11, 1, 3, 2 (5, 7, 9, 10, 8, 6, 4), 3, 2, 12, 11, 2, 1, 2. The grasshoppers will then be reversed in forty-four moves. The general solution of this problem is very difficult. Of course it can always be solved by the method given in the solution of the last puzzle, if we have no desire to use the fewest possible moves. But to employ a full economy of moves we have two main points to consider. There are always what I call a lower movement (L) and an upper movement (U). L consists in exchanging certain of the highest numbers, such as 12, 11, 10 in our "Grasshopper Puzzle," with certain of the lower numbers, 1, 2, 3; the former moving in a clockwise direction, the latter in a non-clockwise direction. U consists in reversing the intermediate counters. In the above solution for 12, it will be seen that 12, 11, and 1, 2, 3 are engaged in the L movement, and 4, 5, 6, 7, 8, 9, 10 in the U movement. The L movement needs 16 moves and U 28, making together 44. We might also involve 10 in the L movement, which would result in L 23, U 21, making also together 44 moves. These I call the first and second methods. But any other scheme will entail an increase of moves. You always get these two methods (of equal economy) for odd or even counters, but the point is to determine just how many to involve in L and how many in U. Here is the solution in table form. But first note, in giving values to n, that 2, 3, and 4 counters are special cases, requiring respectively 3, 3, and 6 moves, and that 5 and 6 counters do not give a minimum solution by the second method--only by the first. FIRST METHOD. +----------+---------------------------+-----------------------+-----------+ | Total No.| L MOVEMENT. | U MOVEMENT. | | | of +-------------+-------------+----------+------------+ Total No. | | Counters.| No. of | No. of | No. of | No. of | of Moves. | | | Counters. | Moves. |Counters. | Moves. | | +----------+-------------+-------------+----------+------------+-----------+ | 4n | n-1 and n |2(n-1)²+5n-7 | 2n+1 |2n²+3n+1 |4(n²+n-1) | | 4n-2 | n-1 " n |2(n-1)²+5n-7 | 2n-1 |2(n-1)²+3n-2|4n²-5 | | 4n+1 | n " n+1 |2n²+5n-2 | 2n |2n²+3n-4 |2(2n²+4n-3)| | 4n-1 | n-1 " n |2(n-1)²+5n-7 | 2n |2n²+3n-4 |4n²+4n-9 | +----------+-------------+-------------+----------+------------+-----------+ SECOND METHOD. +---------+--------------------------+-------------------------+-----------+ |Total No.| L MOVEMENT. | U MOVEMENT. | | | of +-------------+------------+----------+--------------+ Total No. | |Counters.| No. of | No. of | No. of | No. of | of Moves. | | | Counters. | Moves. | Counters.| Moves. | | +---------+-------------+------------+----------+--------------+-----------+ | 4n | n and n |2n²+3n-4 | 2n | 2(n-1)²+5n-2 |4(n²+n-1) | | 4n-2 | n-1 " n-1 |2(n-1)²+3n-7| 2n | 2(n-1)²+5n-2 |4n²-5 | | 4n+1 | n " n |2n²+3n-4 | 2n+1 | 2n²+5n-2 |2(2n²+4n-3)| | 4n-1 | n " n |2n²+3n-4 | 2n-1 | 2(n-1)²+5n-7 |4n²+4n-9 | +---------+-------------+------------+----------+--------------+-----------+ More generally we may say that with m counters, where m is even and greater than 4, we require (m² + 4m - 16)/4 moves; and where m is odd and greater than 3, (m² + 6m - 31)/4 moves. I have thus shown the reader how to find the minimum number of moves for any case, and the character and direction of the moves. I will leave him to discover for himself how the actual order of moves is to be determined. This is a hard nut, and requires careful adjustment of the L and the U movements, so that they may be mutually accommodating. 216.--THE EDUCATED FROGS. The following leaps solve the puzzle in ten moves: 2 to 1, 5 to 2, 3 to 5, 6 to 3, 7 to 6, 4 to 7, 1 to 4, 3 to 1, 6 to 3, 7 to 6. 217.--THE TWICKENHAM PUZZLE. Play the counters in the following order: K C E K W T C E H M K W T A N C E H M I K C E H M T, and there you are, at Twickenham. The position itself will always determine whether you are to make a leap or a simple move. 218.--THE VICTORIA CROSS PUZZLE. In solving this puzzle there were two things to be achieved: first, so to manipulate the counters that the word VICTORIA should read round the cross in the same direction, only with the V on one of the dark arms; and secondly, to perform the feat in the fewest possible moves. Now, as a matter of fact, it would be impossible to perform the first part in any way whatever if all the letters of the word were different; but as there are two I's, it can be done by making these letters change places--that is, the first I changes from the 2nd place to the 7th, and the second I from the 7th place to the 2nd. But the point I referred to, when introducing the puzzle, as a little remarkable is this: that a solution in twenty-two moves is obtainable by moving the letters in the order of the following words: "A VICTOR! A VICTOR! A VICTOR I!" There are, however, just six solutions in eighteen moves, and the following is one of them: I (1), V, A, I (2), R, O, T, I (1), I (2), A, V, I (2), I (1), C, I (2), V, A, I (1). The first and second I in the word are distinguished by the numbers 1 and 2. It will be noticed that in the first solution given above one of the I's never moves, though the movements of the other letters cause it to change its relative position. There is another peculiarity I may point out--that there is a solution in twenty-eight moves requiring no letter to move to the central division except the I's. I may also mention that, in each of the solutions in eighteen moves, the letters C, T, O, R move once only, while the second I always moves four times, the V always being transferred to the right arm of the cross. 219.--THE LETTER BLOCK PUZZLE. This puzzle can be solved in 23 moves--the fewest possible. Move the blocks in the following order: A, B, F, E, C, A, B, F, E, C, A, B, D, H, G, A, B, D, H, G, D, E, F. 220.--A LODGING-HOUSE DIFFICULTY. The shortest possible way is to move the articles in the following order: Piano, bookcase, wardrobe, piano, cabinet, chest of drawers, piano, wardrobe, bookcase, cabinet, wardrobe, piano, chest of drawers, wardrobe, cabinet, bookcase, piano. Thus seventeen removals are necessary. The landlady could then move chest of drawers, wardrobe, and cabinet. Mr. Dobson did not mind the wardrobe and chest of drawers changing rooms so long as he secured the piano. 221.--THE EIGHT ENGINES. The solution to the Eight Engines Puzzle is as follows: The engine that has had its fire drawn and therefore cannot move is No. 5. Move the other engines in the following order: 7, 6, 3, 7, 6, 1, 2, 4, 1, 3, 8, 1, 3, 2, 4, 3, 2, seventeen moves in all, leaving the eight engines in the required order. There are two other slightly different solutions. 222.--A RAILWAY PUZZLE. This little puzzle may be solved in as few as nine moves. Play the engines as follows: From 9 to 10, from 6 to 9, from 5 to 6, from 2 to 5, from 1 to 2, from 7 to 1, from 8 to 7, from 9 to 8, and from 10 to 9. You will then have engines A, B, and C on each of the three circles and on each of the three straight lines. This is the shortest solution that is possible. 223.--A RAILWAY MUDDLE. [Illustration: 1] [Illustration: 2] [Illustration: 3] [Illustration: 4] [Illustration: 5] [Illustration: 6] Only six reversals are necessary. The white train (from A to D) is divided into three sections, engine and 7 wagons, 8 wagons, and 1 wagon. The black train (D to A) never uncouples anything throughout. Fig. 1 is original position with 8 and 1 uncoupled. The black train proceeds to position in Fig. 2 (no reversal). The engine and 7 proceed towards D, and black train backs, leaves 8 on loop, and takes up position in Fig. 3 (first reversal). Black train goes to position in Fig. 4 to fetch single wagon (second reversal). Black train pushes 8 off loop and leaves single wagon there, proceeding on its journey, as in Fig. 5 (third and fourth reversals). White train now backs on to loop to pick up single car and goes right away to D (fifth and sixth reversals). 224.--THE MOTOR-GARAGE PUZZLE. The exchange of cars can be made in forty-three moves, as follows: 6-G, 2-B, 1-E, 3-H, 4-I, 3-L, 6-K, 4-G, 1-I, 2-J, 5-H, 4-A, 7-F, 8-E, 4-D, 8-C, 7-A, 8-G, 5-C, 2-B, 1-E, 8-I, 1-G, 2-J, 7-H, 1-A, 7-G, 2-B, 6-E, 3-H, 8-L, 3-I, 7-K, 3-G, 6-I, 2-J, 5-H, 3-C, 5-G, 2-B, 6-E, 5-I, 6-J. Of course, "6-G" means that the car numbered "6" moves to the point "G." There are other ways in forty-three moves. 225.--THE TEN PRISONERS. [Illustration] It will be seen in the illustration how the prisoners may be arranged so as to produce as many as sixteen even rows. There are 4 such vertical rows, 4 horizontal rows, 5 diagonal rows in one direction, and 3 diagonal rows in the other direction. The arrows here show the movements of the four prisoners, and it will be seen that the infirm man in the bottom corner has not been moved. 226.--ROUND THE COAST. In order to place words round the circle under the conditions, it is necessary to select words in which letters are repeated in certain relative positions. Thus, the word that solves our puzzle is "Swansea," in which the first and fifth letters are the same, and the third and seventh the same. We make out jumps as follows, taking the letters of the word in their proper order: 2-5, 7-2, 4-7, 1-4, 6-1, 3-6, 8-3. Or we could place a word like "Tarapur" (in which the second and fourth letters, and the third and seventh, are alike) with these moves: 6-1, 7-4, 2-7, 5--2, 8-5, 3-6, 8-3. But "Swansea" is the only word, apparently, that will fulfil the conditions of the puzzle. This puzzle should be compared with Sharp's Puzzle, referred to in my solution to No. 341, "The Four Frogs." The condition "touch and jump over two" is identical with "touch and move along a line." 227.--CENTRAL SOLITAIRE. Here is a solution in nineteen moves; the moves enclosed in brackets count as one move only: 19-17, 16-18, (29-17, 17-19), 30-18, 27-25, (22-24, 24-26), 31-23, (4-16, 16-28), 7-9, 10-8, 12-10, 3-11, 18-6, (1-3, 3-11), (13-27, 27-25), (21-7, 7-9), (33-31, 31-23), (10-8, 8-22, 22-24, 24-26, 26-12, 12-10), 5-17. All the counters are now removed except one, which is left in the central hole. The solution needs judgment, as one is tempted to make several jumps in one move, where it would be the reverse of good play. For example, after playing the first 3-11 above, one is inclined to increase the length of the move by continuing with 11-25, 25-27, or with 11-9, 9-7. I do not think the number of moves can be reduced. 228.--THE TEN APPLES. Number the plates (1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16) in successive rows from the top to the bottom. Then transfer the apple from 8 to 10 and play as follows, always removing the apple jumped over: 9-11, 1-9, 13-5, 16-8, 4-12, 12-10, 3-1, 1-9, 9-11. 229.--THE NINE ALMONDS. This puzzle may be solved in as few as four moves, in the following manner: Move 5 over 8, 9, 3, 1. Move 7 over 4. Move 6 over 2 and 7. Move 5 over 6, and all the counters are removed except 5, which is left in the central square that it originally occupied. 230.--THE TWELVE PENNIES. Here is one of several solutions. Move 12 to 3, 7 to 4, 10 to 6, 8 to 1, 9 to 5, 11 to 2. 231.--PLATES AND COINS. Number the plates from 1 to 12 in the order that the boy is seen to be going in the illustration. Starting from 1, proceed as follows, where "1 to 4" means that you take the coin from plate No. 1 and transfer it to plate No. 4: 1 to 4, 5 to 8, 9 to 12, 3 to 6, 7 to 10, 11 to 2, and complete the last revolution to 1, making three revolutions in all. Or you can proceed this way: 4 to 7, 8 to 11, 12 to 3, 2 to 5, 6 to 9, 10 to 1. It is easy to solve in four revolutions, but the solutions in three are more difficult to discover. This is "The Riddle of the Fishpond" (No. 41, _Canterbury Puzzles_) in a different dress. 232.--CATCHING THE MICE. In order that the cat should eat every thirteenth mouse, and the white mouse last of all, it is necessary that the count should begin at the seventh mouse (calling the white one the first)--that is, at the one nearest the tip of the cat's tail. In this case it is not at all necessary to try starting at all the mice in turn until you come to the right one, for you can just start anywhere and note how far distant the last one eaten is from the starting point. You will find it to be the eighth, and therefore must start at the eighth, counting backwards from the white mouse. This is the one I have indicated. In the case of the second puzzle, where you have to find the smallest number with which the cat may start at the white mouse and eat this one last of all, unless you have mastered the general solution of the problem, which is very difficult, there is no better course open to you than to try every number in succession until you come to one that works correctly. The smallest number is twenty-one. If you have to proceed by trial, you will shorten your labour a great deal by only counting out the remainders when the number is divided successively by 13, 12, 11, 10, etc. Thus, in the case of 21, we have the remainders 8, 9, 10, 1, 3, 5, 7, 3, 1, 1, 3, 1, 1. Note that I do not give the remainders of 7, 3, and 1 as nought, but as 7, 3, and 1. Now, count round each of these numbers in turn, and you will find that the white mouse is killed last of all. Of course, if we wanted simply any number, not the smallest, the solution is very easy, for we merely take the least common multiple of 13, 12, 11, 10, etc. down to 2. This is 360360, and you will find that the first count kills the thirteenth mouse, the next the twelfth, the next the eleventh, and so on down to the first. But the most arithmetically inclined cat could not be expected to take such a big number when a small one like twenty-one would equally serve its purpose. In the third case, the smallest number is 100. The number 1,000 would also do, and there are just seventy-two other numbers between these that the cat might employ with equal success. 233.--THE ECCENTRIC CHEESEMONGER. To leave the three piles at the extreme ends of the rows, the cheeses may be moved as follows--the numbers refer to the cheeses and not to their positions in the row: 7-2, 8-7, 9-8, 10-15, 6-10, 5-6, 14-16, 13-14, 12-13, 3-1, 4-3, 11-4. This is probably the easiest solution of all to find. To get three of the piles on cheeses 13, 14, and 15, play thus: 9-4, 10-9, 11-10, 6-14, 5-6, 12-15, 8-12, 7-8, 16-5, 3-13, 2-3, 1-2. To leave the piles on cheeses 3, 5, 12, and 14, play thus: 8-3, 9-14, 16-12, 1-5, 10-9, 7-10, 11-8, 2-1, 4-16, 13-2, 6-11, 15-4. 234.--THE EXCHANGE PUZZLE. Make the following exchanges of pairs: H-K, H-E, H-C, H-A, I-L, I-F, I-D, K-L, G-J, J-A, F-K, L-E, D-K, E-F, E-D, E-B, B-K. It will be found that, although the white counters can be moved to their proper places in 11 moves, if we omit all consideration of exchanges, yet the black cannot be so moved in fewer than 17 moves. So we have to introduce waste moves with the white counters to equal the minimum required by the black. Thus fewer than 17 moves must be impossible. Some of the moves are, of course, interchangeable. 235.--TORPEDO PRACTICE. [Illustration: 10 6 7 \ |/ 4 u u 2 \ u / 3-u u u u u u u u u u -----9--- / u 8 u u / \ 1 5 ] If the enemy's fleet be anchored in the formation shown in the illustration, it will be seen that as many as ten out of the sixteen ships may be blown up by discharging the torpedoes in the order indicated by the numbers and in the directions indicated by the arrows. As each torpedo in succession passes under three ships and sinks the fourth, strike out each vessel with the pencil as it is sunk. 236.--THE HAT PUZZLE. [Illustration: 1 2 3 4 5 6 7 8 9 10 11 12 +--+--+--+--+--+--+--+--+--+--+--+--+ | *| o| *| O| *| O| *| O| *| O| | | +--+--+--+--+--+--+--+--+--+--+--+--+ | *| | | O| *| O| *| O| *| O| O| *| +--+--+--+--+--+--+--+--+--+--+--+--+ | *| *| O| O| *| O| | | *| O| O| *| +--+--+--+--+--+--+--+--+--+--+--+--+ | *| *| O| | | O| O| *| *| O| O| *| +--+--+--+--+--+--+--+--+--+--+--+--+ | *| *| O| O| O| O| O| *| *| | | *| +--+--+--+--+--+--+--+--+--+--+--+--+ | | | O| O| O| O| O| *| *| *| *| *| +--+--+--+--+--+--+--+--+--+--+--+--+ ] I suggested that the reader should try this puzzle with counters, so I give my solution in that form. The silk hats are represented by black counters and the felt hats by white counters. The first row shows the hats in their original positions, and then each successive row shows how they appear after one of the five manipulations. It will thus be seen that we first move hats 2 and 3, then 7 and 8, then 4 and 5, then 10 and 11, and, finally, 1 and 2, leaving the four silk hats together, the four felt hats together, and the two vacant pegs at one end of the row. The first three pairs moved are dissimilar hats, the last two pairs being similar. There are other ways of solving the puzzle. 237.--BOYS AND GIRLS. There are a good many different solutions to this puzzle. Any contiguous pair, except 7-8, may be moved first, and after the first move there are variations. The following solution shows the position from the start right through each successive move to the end:-- . . 1 2 3 4 5 6 7 8 4 3 1 2 . . 5 6 7 8 4 3 1 2 7 6 5 . . 8 4 3 1 2 7 . . 5 6 8 4 . . 2 7 1 3 5 6 8 4 8 6 2 7 1 3 5 . . 238.--ARRANGING THE JAM POTS. Two of the pots, 13 and 19, were in their proper places. As every interchange may result in a pot being put in its place, it is clear that twenty-two interchanges will get them all in order. But this number of moves is not the fewest possible, the correct answer being seventeen. Exchange the following pairs: (3-1, 2-3), (15-4, 16-15), (17-7, 20-17), (24-10, 11-24, 12-11), (8-5, 6-8, 21-6, 23-21, 22-23, 14-22, 9-14, 18-9). When you have made the interchanges within any pair of brackets, all numbers within those brackets are in their places. There are five pairs of brackets, and 5 from 22 gives the number of changes required--17. 239.--A JUVENILE PUZZLE. [Illustration: +-----------------+ | C E | | | | | | D F | +---------------B | G | A | | | H | +-----------------+ ] As the conditions are generally understood, this puzzle is incapable of solution. This can be demonstrated quite easily. So we have to look for some catch or quibble in the statement of what we are asked to do. Now if you fold the paper and then push the point of your pencil down between the fold, you can with one stroke make the two lines CD and EF in our diagram. Then start at A, and describe the line ending at B. Finally put in the last line GH, and the thing is done strictly within the conditions, since folding the paper is not actually forbidden. Of course the lines are here left unjoined for the purpose of clearness. In the rubbing out form of the puzzle, first rub out A to B with a single finger in one stroke. Then rub out the line GH with one finger. Finally, rub out the remaining two vertical lines with two fingers at once! That is the old trick. 240.--THE UNION JACK. [Illustration: +-------+ +----- A B | | / \ | | / |\ \ | | / /| | \ \ | | / / | | \ \| |/ / | | \ | / / | | \ |\ /| / | +-----\-|-\/-|-/-----+ \| /\ |/ |/ \/ |\ /\ /| \/ |\ +-----/-|-/\-|-\-----+ | / / \| \ | | / | \ \ | | / /| |\ \ | | / / | | \ \ | |/ / | | \ \| / | | \ / | | \ -----+ +----- ] There are just sixteen points (all on the outside) where three roads may be said to join. These are called by mathematicians "odd nodes." There is a rule that tells us that in the case of a drawing like the present one, where there are sixteen odd nodes, it requires eight separate strokes or routes (that is, half as many as there are odd nodes) to complete it. As we have to produce as much as possible with only one of these eight strokes, it is clearly necessary to contrive that the seven strokes from odd node to odd node shall be as short as possible. Start at A and end at B, or go the reverse way. 241.--THE DISSECTED CIRCLE. [Illustration: /---------------\ / \ / /------B \ / / | /^\ \ / / |\ | / \ \ / / | \ | / \ \ / / | \ | / A \ \ / / | \ | / | \ \ | / | \|/ | \ | | | -----+-----*-----+----- | | | | \ | /|\ | / | | | | \ | / | \ | / | | | | \ | / | \ | / | | | | \ | / | \ | / | | | | \|/ | \|/ | | D-+------*-----+-----*----E | | | /|\ | /|\ | | | / | \ | / | \ | | | / | \ | / | \ | | | / | \ | / | \ | | | / | \|/ | \ | | | -----+-----*-----+----- | | \ | /|\ | / | \ | / | \ | / / \ | / | \ | / / \ | / | \ | / / \ |/ | \| / / \ | / / \------+------/ / | / C-------/ ] It can be done in twelve continuous strokes, thus: Start at A in the illustration, and eight strokes, forming the star, will bring you back to A; then one stroke round the circle to B, one stroke to C, one round the circle to D, and one final stroke to E--twelve in all. Of course, in practice the second circular stroke will be over the first one; it is separated in the diagram, and the points of the star not joined to the circle, to make the solution clear to the eye. 242.--THE TUBE INSPECTOR'S PUZZLE. The inspector need only travel nineteen miles if he starts at B and takes the following route: BADGDEFIFCBEHKLIHGJK. Thus the only portions of line travelled over twice are the two sections D to G and F to I. Of course, the route may be varied, but it cannot be shortened. 243.--VISITING THE TOWNS. Note that there are six towns, from which only two roads issue. Thus 1 must lie between 9 and 12 in the circular route. Mark these two roads as settled. Similarly mark 9, 5, 14, and 4, 8, 14, and 10, 6, 15, and 10, 2, 13, and 3, 7, 13. All these roads must be taken. Then you will find that he must go from 4 to 15, as 13 is closed, and that he is compelled to take 3, 11, 16, and also 16, 12. Thus, there is only one route, as follows: 1, 9, 5, 14, 8, 4, 15, 6, 10, 2, 13, 7, 3, 11, 16, 12, 1, or its reverse--reading the line the other way. Seven roads are not used. 244.--THE FIFTEEN TURNINGS. [Illustration] It will be seen from the illustration (where the roads not used are omitted) that the traveller can go as far as seventy miles in fifteen turnings. The turnings are all numbered in the order in which they are taken. It will be seen that he never visits nineteen of the towns. He might visit them all in fifteen turnings, never entering any town twice, and end at the black town from which he starts (see "The Rook's Tour," No. 320), but such a tour would only take him sixty-four miles. 245.--THE FLY ON THE OCTAHEDRON. [Illustration] Though we cannot really see all the sides of the octahedron at once, we can make a projection of it that suits our purpose just as well. In the diagram the six points represent the six angles of the octahedron, and four lines proceed from every point under exactly the same conditions as the twelve edges of the solid. Therefore if we start at the point A and go over all the lines once, we must always end our route at A. And the number of different routes is just 1,488, counting the reverse way of any route as different. It would take too much space to show how I make the count. It can be done in about five minutes, but an explanation of the method is difficult. The reader is therefore asked to accept my answer as correct. 246.--THE ICOSAHEDRON PUZZLE. [Illustration] There are thirty edges, of which eighteen were visible in the original illustration, represented in the following diagram by the hexagon NAESGD. By this projection of the solid we get an imaginary view of the remaining twelve edges, and are able to see at once their direction and the twelve points at which all the edges meet. The difference in the length of the lines is of no importance; all we want is to present their direction in a graphic manner. But in case the novice should be puzzled at only finding nineteen triangles instead of the required twenty, I will point out that the apparently missing triangle is the outline HIK. In this case there are twelve odd nodes; therefore six distinct and disconnected routes will be needful if we are not to go over any lines twice. Let us therefore find the greatest distance that we may so travel in one route. It will be noticed that I have struck out with little cross strokes five lines or edges in the diagram. These five lines may be struck out anywhere so long as they do not join one another, and so long as one of them does not connect with N, the North Pole, from which we are to start. It will be seen that the result of striking out these five lines is that all the nodes are now even except N and S. Consequently if we begin at N and stop at S we may go over all the lines, except the five crossed out, without traversing any line twice. There are many ways of doing this. Here is one route: N to H, I, K, S, I, E, S, G, K, D, H, A, N, B, A, E, F, B, C, G, D, N, C, F, S. By thus making five of the routes as short as is possible--simply from one node to the next--we are able to get the greatest possible length for our sixth line. A greater distance in one route, without going over the same ground twice, it is not possible to get. It is now readily seen that those five erased lines must be gone over twice, and they may be "picked up," so to speak, at any points of our route. Thus, whenever the traveller happens to be at I he can run up to A and back before proceeding on his route, or he may wait until he is at A and then run down to I and back to A. And so with the other lines that have to be traced twice. It is, therefore, clear that he can go over 25 of the lines once only (25 × 10,000 miles = 250,000 miles) and 5 of the lines twice (5 × 20,000 miles = 100,000 miles), the total, 350,000 miles, being the length of his travels and the shortest distance that is possible in visiting the whole body. It will be noticed that I have made him end his travels at S, the South Pole, but this is not imperative. I might have made him finish at any of the other nodes, except the one from which he started. Suppose it had been required to bring him home again to N at the end of his travels. Then instead of suppressing the line AI we might leave that open and close IS. This would enable him to complete his 350,000 miles tour at A, and another 10,000 miles would take him to his own fireside. There are a great many different routes, but as the lengths of the edges are all alike, one course is as good as another. To make the complete 350,000 miles tour from N to S absolutely clear to everybody, I will give it entire: N to H, I, A, I, K, H, K, S, I, E, S, G, F, G, K, D, C, D, H, A, N, B, E, B, A, E, F, B, C, G, D, N, C, F, S--that is, thirty-five lines of 10,000 miles each. 247.--INSPECTING A MINE. Starting from A, the inspector need only travel 36 furlongs if he takes the following route: A to B, G, H, C, D, I, H, M, N, I, J, O, N, S, R, M, L, G, F, K, L, Q, R, S, T, O, J, E, D, C, B, A, F, K, P, Q. He thus passes between A and B twice, between C and D twice, between F and K twice, between J and O twice, and between R and S twice--five repetitions. Therefore 31 passages plus 5 repeated equal 36 furlongs. The little pitfall in this puzzle lies in the fact that we start from an even node. Otherwise we need only travel 35 furlongs. 248.--THE CYCLIST'S TOUR. When Mr. Maggs replied, "No way, I'm sure," he was not saying that the thing was impossible, but was really giving the actual route by which the problem can be solved. Starting from the star, if you visit the towns in the order, NO WAY, I'M SURE, you will visit every town once, and only once, and end at E. So both men were correct. This was the little joke of the puzzle, which is not by any means difficult. 249.--THE SAILOR'S PUZZLE. [Illustration] There are only four different routes (or eight, if we count the reverse ways) by which the sailor can start at the island marked A, visit all the islands once, and once only, and return again to A. Here they are:-- A I P T L O E H R Q D C F U G N S K M B A A I P T S N G L O E U F C D K M B Q R H A A B M K S N G L T P I O E U F C D Q R H A A I P T L O E U G N S K M B Q D C F R H A Now, if the sailor takes the first route he will make C his 12th island (counting A as 1); by the second route he will make C his 13th island; by the third route, his 16th island; and by the fourth route, his 17th island. If he goes the reverse way, C will be respectively his 10th, 9th, 6th, and 5th island. As these are the only possible routes, it is evident that if the sailor puts off his visit to C as long as possible, he must take the last route reading from left to right. This route I show by the dark lines in the diagram, and it is the correct answer to the puzzle. The map may be greatly simplified by the "buttons and string" method, explained in the solution to No. 341, "The Four Frogs." 250.--THE GRAND TOUR. The first thing to do in trying to solve a puzzle like this is to attempt to simplify it. If you look at Fig. 1, you will see that it is a simplified version of the map. Imagine the circular towns to be buttons and the railways to be connecting strings. (See solution to No. 341.) Then, it will be seen, we have simply "straightened out" the previous diagram without affecting the conditions. Now we can further simplify by converting Fig. 1 into Fig. 2, which is a portion of a chessboard. Here the directions of the railways will resemble the moves of a rook in chess--that is, we may move in any direction parallel to the sides of the diagram, but not diagonally. Therefore the first town (or square) visited must be a black one; the second must be a white; the third must be a black; and so on. Every odd square visited will thus be black and every even one white. Now, we have 23 squares to visit (an odd number), so the last square visited must be black. But Z happens to be white, so the puzzle would seem to be impossible of solution. [Illustration: Fig. 1.] [Illustration: Fig. 2.] As we were told that the man "succeeded" in carrying put his plan, we must try to find some loophole in the conditions. He was to "enter every town once and only once," and we find no prohibition against his entering once the town A after leaving it, especially as he has never left it since he was born, and would thus be "entering" it for the first time in his life. But he must return at once from the first town he visits, and then he will have only 22 towns to visit, and as 22 is an even number, there is no reason why he should not end on the white square Z. A possible route for him is indicated by the dotted line from A to Z. This route is repeated by the dark lines in Fig. 1, and the reader will now have no difficulty in applying; it to the original map. We have thus proved that the puzzle can only be solved by a return to A immediately after leaving it. 251.--WATER, GAS, AND ELECTRICITY. [Illustration] According to the conditions, in the strict sense in which one at first understands them, there is no possible solution to this puzzle. In such a dilemma one always has to look for some verbal quibble or trick. If the owner of house A will allow the water company to run their pipe for house C through his property (and we are not bound to assume that he would object), then the difficulty is got over, as shown in our illustration. It will be seen that the dotted line from W to C passes through house A, but no pipe ever crosses another pipe. 252.--A PUZZLE FOR MOTORISTS. [Illustration] The routes taken by the eight drivers are shown in the illustration, where the dotted line roads are omitted to make the paths clearer to the eye. 253.--A BANK HOLIDAY PUZZLE. The simplest way is to write in the number of routes to all the towns in this manner. Put a 1 on all the towns in the top row and in the first column. Then the number of routes to any town will be the sum of the routes to the town immediately above and to the town immediately to the left. Thus the routes in the second row will be 1, 2, 3, 4, 5, 6, etc., in the third row, 1, 3, 6, 10, 15, 21, etc.; and so on with the other rows. It will then be seen that the only town to which there are exactly 1,365 different routes is the twelfth town in the fifth row--the one immediately over the letter E. This town was therefore the cyclist's destination. The general formula for the number of routes from one corner to the corner diagonally opposite on any such rectangular reticulated arrangement, under the conditions as to direction, is (m+n)!/m!n!, where m is the number of towns on one side, less one, and n the number on the other side, less one. Our solution involves the case where there are 12 towns by 5. Therefore m = 11 and n = 4. Then the formula gives us the answer 1,365 as above. 254.-- THE MOTOR-CAR TOUR. First of all I will ask the reader to compare the original square diagram with the circular one shown in Figs. 1, 2, and 3 below. If for the moment we ignore the shading (the purpose of which I shall proceed to explain), we find that the circular diagram in each case is merely a simplification of the original square one--that is, the roads from A lead to B, E, and M in both cases, the roads from L (London) lead to I, K, and S, and so on. The form below, being circular and symmetrical, answers my purpose better in applying a mechanical solution, and I therefore adopt it without altering in any way the conditions of the puzzle. If such a question as distances from town to town came into the problem, the new diagrams might require the addition of numbers to indicate these distances, or they might conceivably not be at all practicable. [Illustration: Figs. 1, 2, and 3] Now, I draw the three circular diagrams, as shown, on a sheet of paper and then cut out three pieces of cardboard of the forms indicated by the shaded parts of these diagrams. It can be shown that every route, if marked out with a red pencil, will form one or other of the designs indicated by the edges of the cards, or a reflection thereof. Let us direct our attention to Fig. 1. Here the card is so placed that the star is at the town T; it therefore gives us (by following the edge of the card) one of the circular routes from London: L, S, R, T, M, A, E, P, O, J, D, C, B, G, N, Q, K, H, F, I, L. If we went the other way, we should get L, I, F, H, K, Q, etc., but these reverse routes were not to be counted. When we have written out this first route we revolve the card until the star is at M, when we get another different route, at A a third route, at E a fourth route, and at P a fifth route. We have thus obtained five different routes by revolving the card as it lies. But it is evident that if we now take up the card and replace it with the other side uppermost, we shall in the same manner get five other routes by revolution. We therefore see how, by using the revolving card in Fig. 1, we may, without any difficulty, at once write out ten routes. And if we employ the cards in Figs. 2 and 3, we similarly obtain in each case ten other routes. These thirty routes are all that are possible. I do not give the actual proof that the three cards exhaust all the possible cases, but leave the reader to reason that out for himself. If he works out any route at haphazard, he will certainly find that it falls into one or other of the three categories. 255.--THE LEVEL PUZZLE. Let us confine our attention to the L in the top left-hand corner. Suppose we go by way of the E on the right: we must then go straight on to the V, from which letter the word may be completed in four ways, for there are four E's available through which we may reach an L. There are therefore four ways of reading through the right-hand E. It is also clear that there must be the same number of ways through the E that is immediately below our starting point. That makes eight. If, however, we take the third route through the E on the diagonal, we then have the option of any one of the three V's, by means of each of which we may complete the word in four ways. We can therefore spell LEVEL in twelve ways through the diagonal E. Twelve added to eight gives twenty readings, all emanating from the L in the top left-hand corner; and as the four corners are equal, the answer must be four times twenty, or eighty different ways. 256.--THE DIAMOND PUZZLE. There are 252 different ways. The general formula is that, for words of n letters (not palindromes, as in the case of the next puzzle), when grouped in this manner, there are always 2^(n+1) - 4 different readings. This does not allow diagonal readings, such as you would get if you used instead such a word as DIGGING, where it would be possible to pass from one G to another G by a diagonal step. 257.--THE DEIFIED PUZZLE. The correct answer is 1,992 different ways. Every F is either a corner F or a side F--standing next to a corner in its own square of F's. Now, FIED may be read _from_ a corner F in 16 ways; therefore DEIF may be read _into_ a corner F also in 16 ways; hence DEIFIED may be read _through_ a corner F in 16 × 16 = 256 ways. Consequently, the four corner F's give 4 × 256 = 1,024 ways. Then FIED may be read from a side F in 11 ways, and DEIFIED therefore in 121 ways. But there are eight side F's; consequently these give together 8 × 121 = 968 ways. Add 968 to 1,024 and we get the answer, 1,992. In this form the solution will depend on whether the number of letters in the palindrome be odd or even. For example, if you apply the word NUN in precisely the same manner, you will get 64 different readings; but if you use the word NOON, you will only get 56, because you cannot use the same letter twice in immediate succession (since you must "always pass from one letter to another") or diagonal readings, and every reading must involve the use of the central N. The reader may like to find for himself the general formula in this case, which is complex and difficult. I will merely add that for such a case as MADAM, dealt with in the same way as DEIFIED, the number of readings is 400. 258.-- THE VOTERS' PUZZLE. THE number of readings here is 63,504, as in the case of "WAS IT A RAT I SAW" (No. 30, _Canterbury Puzzles_). The general formula is that for palindromic sentences containing 2n + 1 letters there are (4(2^n -1))² readings. 259.-- HANNAH'S PUZZLE. Starting from any one of the N's, there are 17 different readings of NAH, or 68 (4 times 17) for the 4 N's. Therefore there are also 68 ways of spelling HAN. If we were allowed to use the same N twice in a spelling, the answer would be 68 times 68, or 4,624 ways. But the conditions were, "always passing from one letter to another." Therefore, for every one of the 17 ways of spelling HAN with a particular N, there would be 51 ways (3 times 17) of completing the NAH, or 867 (17 times 51) ways for the complete word. Hence, as there are four N's to use in HAN, the correct solution of the puzzle is 3,468 (4 times 867) different ways. 260.--THE HONEYCOMB PUZZLE. The required proverb is, "There is many a slip 'twixt the cup and the lip." Start at the T on the outside at the bottom right-hand corner, pass to the H above it, and the rest is easy. 261.-- THE MONK AND THE BRIDGES. [Illustration] The problem of the Bridges may be reduced to the simple diagram shown in illustration. The point M represents the Monk, the point I the Island, and the point Y the Monastery. Now the only direct ways from M to I are by the bridges a and b; the only direct ways from I to Y are by the bridges c and d; and there is a direct way from M to Y by the bridge e. Now, what we have to do is to count all the routes that will lead from M to Y, passing over all the bridges, a, b, c, d, and e once and once only. With the simple diagram under the eye it is quite easy, without any elaborate rule, to count these routes methodically. Thus, starting from a, b, we find there are only two ways of completing the route; with _a, c_, there are only two routes; with a, d, only two routes; and so on. It will be found that there are sixteen such routes in all, as in the following list:-- a b e c d b c d a e a b e d c b c e a d a c d b e b d c a e a c e b d b d e a c a d e b c e c a b d a d c b e e c b a d b a e c d e d a b c b a e d c e d b a c If the reader will transfer the letters indicating the bridges from the diagram to the corresponding bridges in the original illustration, everything will be quite obvious. 262.--THOSE FIFTEEN SHEEP. If we read the exact words of the writer in the cyclopædia, we find that we are not told that the pens were all necessarily empty! In fact, if the reader will refer back to the illustration, he will see that one sheep is already in one of the pens. It was just at this point that the wily farmer said to me, "_Now_ I'm going to start placing the fifteen sheep." He thereupon proceeded to drive three from his flock into the already occupied pen, and then placed four sheep in each of the other three pens. "There," says he, "you have seen me place fifteen sheep in four pens so that there shall be the same number of sheep in every pen." I was, of course, forced to admit that he was perfectly correct, according to the exact wording of the question. 263.--KING ARTHUR'S KNIGHTS. On the second evening King Arthur arranged the knights and himself in the following order round the table: A, F, B, D, G, E, C. On the third evening they sat thus, A, E, B, G, C, F, D. He thus had B next but one to him on both occasions (the nearest possible), and G was the third from him at both sittings (the furthest position possible). No other way of sitting the knights would have been so satisfactory. 264.--THE CITY LUNCHEONS. The men may be grouped as follows, where each line represents a day and each column a table:-- AB CD EF GH IJ KL AE DL GK FI CB HJ AG LJ FH KC DE IB AF JB KI HD LG CE AK BE HC IL JF DG AH EG ID CJ BK LF AI GF CL DB EH JK AC FK DJ LE GI BH AD KH LB JG FC EI AL HI JE BF KD GC AJ IC BG EK HL FD Note that in every column (except in the case of the A's) all the letters descend cyclically in the same order, B, E, G, F, up to J, which is followed by B. 265.--A PUZZLE FOR CARD-PLAYERS. In the following solution each of the eleven lines represents a sitting, each column a table, and each pair of letters a pair of partners. A B -- I L | E J -- G K | F H -- C D A C -- J B | F K -- H L | G I -- D E A D -- K C | G L -- I B | H J -- E F A E -- L D | H B -- J C | I K -- F G A F -- B E | I C -- K D | J L -- G H A G -- C F | J D -- L E | K B -- H I A H -- D G | K E -- B F | L C -- I J A I -- E H | L F -- C G | B D -- J K A J -- F I | B G -- D H | C E -- K L A K -- G J | C H -- E I | D F -- L B A L -- H K | D I -- F J | E G -- B C It will be seen that the letters B, C, D ...L descend cyclically. The solution given above is absolutely perfect in all respects. It will be found that every player has every other player once as his partner and twice as his opponent. 266.--A TENNIS TOURNAMENT. Call the men A, B, D, E, and their wives a, b, d, e. Then they may play as follows without any person ever playing twice with or against any other person:-- First Court. Second Court. 1st Day | A d against B e | D a against E b 2nd Day | A e " D b | E a " B d 3rd Day | A b " E d | B a " D e It will be seen that no man ever plays with or against his own wife--an ideal arrangement. If the reader wants a hard puzzle, let him try to arrange eight married couples (in four courts on seven days) under exactly similar conditions. It can be done, but I leave the reader in this case the pleasure of seeking the answer and the general solution. 267.--THE WRONG HATS. The number of different ways in which eight persons, with eight hats, can each take the wrong hat, is 14,833. Here are the successive solutions for any number of persons from one to eight:-- 1 = 0 2 = 1 3 = 2 4 = 9 5 = 44 6 = 265 7 = 1,854 8 = 14,833 To get these numbers, multiply successively by 2, 3, 4, 5, etc. When the multiplier is even, add 1; when odd, deduct 1. Thus, 3 × 1 - 1 = 2; 4 × 2 + 1 = 9; 5 × 9 - 1 = 44; and so on. Or you can multiply the sum of the number of ways for n - 1 and n - 2 persons by n - 1, and so get the solution for n persons. Thus, 4(2 + 9) = 44; 5(9 + 44) = 265; and so on. 268.--THE PEAL OF BELLS. The bells should be rung as follows:-- 1 2 3 4 2 1 4 3 2 4 1 3 4 2 3 1 4 3 2 1 3 4 1 2 3 1 4 2 1 3 2 4 3 1 2 4 1 3 4 2 1 4 3 2 4 1 2 3 4 2 1 3 2 4 3 1 2 3 4 1 3 2 1 4 2 3 1 4 3 2 4 1 3 4 2 1 4 3 1 2 4 1 3 2 1 4 2 3 1 2 4 3 2 1 3 4 I have constructed peals for five and six bells respectively, and a solution is possible for any number of bells under the conditions previously stated. 269.--THREE MEN IN A BOAT. If there were no conditions whatever, except that the men were all to go out together, in threes, they could row in an immense number of different ways. If the reader wishes to know how many, the number is 455^7. And with the condition that no two may ever be together more than once, there are no fewer than 15,567,552,000 different solutions--that is, different ways of arranging the men. With one solution before him, the reader will realize why this must be, for although, as an example, A must go out once with B and once with C, it does not necessarily follow that he must go out with C on the same occasion that he goes with B. He might take any other letter with him on that occasion, though the fact of his taking other than B would have its effect on the arrangement of the other triplets. Of course only a certain number of all these arrangements are available when we have that other condition of using the smallest possible number of boats. As a matter of fact we need employ only ten different boats. Here is one the arrangements:-- 1 2 3 4 5 1st Day (ABC) (DBF) (GHI) (JKL) (MNO) 8 6 7 9 10 2nd Day (ADG) (BKN) (COL) (JEI) (MHF) 3 5 4 1 2 3rd Day (AJM) (BEH) (CFI) (DKO) (GNL) 7 6 8 9 1 4th Day (AEK) (CGM) (BOI) (DHL) (JNF) 4 5 3 10 2 5th Day (AHN) (CDJ) (BFL) (GEO) (MKI) 6 7 8 10 1 6th Day (AFO) (BGJ) (CKH) (DNI) (MEL) 5 4 3 9 2 7th Day (AIL) (BDM) (CEN) (GKF) (JHO) It will be found that no two men ever go out twice together, and that no man ever goes out twice in the same boat. This is an extension of the well-known problem of the "Fifteen Schoolgirls," by Kirkman. The original conditions were simply that fifteen girls walked out on seven days in triplets without any girl ever walking twice in a triplet with another girl. Attempts at a general solution of this puzzle had exercised the ingenuity of mathematicians since 1850, when the question was first propounded, until recently. In 1908 and the two following years I indicated (see _Educational Times Reprints_, Vols. XIV., XV., and XVII.) that all our trouble had arisen from a failure to discover that 15 is a special case (too small to enter into the general law for all higher numbers of girls of the form 6n+3), and showed what that general law is and how the groups should be posed for any number of girls. I gave actual arrangements for numbers that had previously baffled all attempts to manipulate, and the problem may now be considered generally solved. Readers will find an excellent full account of the puzzle in W.W. Rouse Ball's _Mathematical Recreations_, 5th edition. 270.--THE GLASS BALLS. There are, in all, sixteen balls to be broken, or sixteen places in the order of breaking. Call the four strings A, B, C, and D--order is here of no importance. The breaking of the balls on A may occupy any 4 out of these 16 places--that is, the combinations of 16 things, taken 4 together, will be 13 × 14 × 15 × 16 ----------------- = 1,820 1 × 2 × 3 × 4 ways for A. In every one of these cases B may occupy any 4 out of the remaining 12 places, making 9 × 10 × 11 × 12 ----------------- = 495 1 × 2 × 3 × 4 ways. Thus 1,820 × 495 = 900,900 different placings are open to A and B. But for every one of these cases C may occupy 5 × 6 × 7 × 8 ------------- = 70 1 × 2 × 3 × 4 different places; so that 900,900 × 70 = 63,063,000 different placings are open to A, B, and C. In every one of these cases, D has no choice but to take the four places that remain. Therefore the correct answer is that the balls may be broken in 63,063,000 different ways under the conditions. Readers should compare this problem with No. 345, "The Two Pawns," which they will then know how to solve for cases where there are three, four, or more pawns on the board. 271.--FIFTEEN LETTER PUZZLE. The following will be found to comply with the conditions of grouping:-- ALE MET MOP BLM BAG CAP YOU CLT IRE OIL LUG LNR NAY BIT BUN BPR AIM BEY RUM GMY OAR GIN PLY CGR PEG ICY TRY CMN CUE COB TAU PNT ONE GOT PIU The fifteen letters used are A, E, I, O, U, Y, and B, C, G, L, M, N, P, R, T. The number of words is 27, and these are all shown in the first three columns. The last word, PIU, is a musical term in common use; but although it has crept into some of our dictionaries, it is Italian, meaning "a little; slightly." The remaining twenty-six are good words. Of course a TAU-cross is a T-shaped cross, also called the cross of St. Anthony, and borne on a badge in the Bishop's Palace at Exeter. It is also a name for the toad-fish. We thus have twenty-six good words and one doubtful, obtained under the required conditions, and I do not think it will be easy to improve on this answer. Of course we are not bound by dictionaries but by common usage. If we went by the dictionary only in a case of this kind, we should find ourselves involved in prefixes, contractions, and such absurdities as I.O.U., which Nuttall actually gives as a word. 272.--THE NINE SCHOOLBOYS. The boys can walk out as follows:-- 1st Day. 2nd Day. 3rd Day. A B C B F H F A G D E F E I A I D B G H I C G D H C E 4th Day. 5th Day. 6th Day. A D H G B I D C A B E G C F D E H B F I C H A E I G F Every boy will then have walked by the side of every other boy once and once only. Dealing with the problem generally, 12n+9 boys may walk out in triplets under the conditions on 9n+6 days, where n may be nought or any integer. Every possible pair will occur once. Call the number of boys m. Then every boy will pair m-1 times, of which (m-1)/4 times he will be in the middle of a triplet and (m-1)/2 times on the outside. Thus, if we refer to the solution above, we find that every boy is in the middle twice (making 4 pairs) and four times on the outside (making the remaining 4 pairs of his 8). The reader may now like to try his hand at solving the two next cases of 21 boys on 15 days, and 33 boys on 24 days. It is, perhaps, interesting to note that a school of 489 boys could thus walk out daily in one leap year, but it would take 731 girls (referred to in the solution to No. 269) to perform their particular feat by a daily walk in a year of 365 days. 273.--THE ROUND TABLE. The history of this problem will be found in _The Canterbury Puzzles_ (No. 90). Since the publication of that book in 1907, so far as I know, nobody has succeeded in solving the case for that unlucky number of persons, 13, seated at a table on 66 occasions. A solution is possible for any number of persons, and I have recorded schedules for every number up to 25 persons inclusive and for 33. But as I know a good many mathematicians are still considering the case of 13, I will not at this stage rob them of the pleasure of solving it by showing the answer. But I will now display the solutions for all the cases up to 12 persons inclusive. Some of these solutions are now published for the first time, and they may afford useful clues to investigators. The solution for the case of 3 persons seated on 1 occasion needs no remark. A solution for the case of 4 persons on 3 occasions is as follows:-- 1 2 3 4 1 3 4 2 1 4 2 3 Each line represents the order for a sitting, and the person represented by the last number in a line must, of course, be regarded as sitting next to the first person in the same line, when placed at the round table. The case of 5 persons on 6 occasions may be solved as follows:-- 1 2 3 4 5 1 2 4 5 3 1 2 5 3 4 --------- 1 3 2 5 4 1 4 2 3 5 1 5 2 4 3 The case for 6 persons on 10 occasions is solved thus:-- 1 2 3 6 4 5 1 3 4 2 5 6 1 4 5 3 6 2 1 5 6 4 2 3 1 6 2 5 3 4 ----------- 1 2 4 5 6 3 1 3 5 6 2 4 1 4 6 2 3 5 1 5 2 3 4 6 1 6 3 4 5 2 It will now no longer be necessary to give the solutions in full, for reasons that I will explain. It will be seen in the examples above that the 1 (and, in the case of 5 persons, also the 2) is repeated down the column. Such a number I call a "repeater." The other numbers descend in cyclical order. Thus, for 6 persons we get the cycle, 2, 3, 4, 5, 6, 2, and so on, in every column. So it is only necessary to give the two lines 1 2 3 6 4 5 and 1 2 4 5 6 3, and denote the cycle and repeaters, to enable any one to write out the full solution straight away. The reader may wonder why I do not start the last solution with the numbers in their natural order, 1 2 3 4 5 6. If I did so the numbers in the descending cycle would not be in their natural order, and it is more convenient to have a regular cycle than to consider the order in the first line. The difficult case of 7 persons on 15 occasions is solved as follows, and was given by me in _The Canterbury Puzzles_:-- 1 2 3 4 5 7 6 1 6 2 7 5 3 4 1 3 5 2 6 7 4 1 5 7 4 3 6 2 1 5 2 7 3 4 6 In this case the 1 is a repeater, and there are _two_ separate cycles, 2, 3, 4, 2, and 5, 6, 7, 5. We thus get five groups of three lines each, for a fourth line in any group will merely repeat the first line. A solution for 8 persons on 21 occasions is as follows:-- 1 8 6 3 4 5 2 7 1 8 4 5 7 2 3 6 1 8 2 7 3 6 4 5 The 1 is here a repeater, and the cycle 2, 3, 4, 5, 6, 7, 8. Every one of the 3 groups will give 7 lines. Here is my solution for 9 persons on 28 occasions:-- 2 1 9 7 4 5 6 3 8 2 9 5 1 6 8 3 4 7 2 9 3 1 8 4 7 5 6 2 9 1 5 6 4 7 8 3 There are here two repeaters, 1 and 2, and the cycle is 3, 4, 5, 6, 7, 8, 9. We thus get 4 groups of 7 lines each. The case of 10 persons on 36 occasions is solved as follows:-- 1 10 8 3 6 5 4 7 2 9 1 10 6 5 2 9 7 4 3 8 1 10 2 9 3 8 6 5 7 4 1 10 7 4 8 3 2 9 5 6 The repeater is 1, and the cycle, 2, 3, 4, 5, 6, 7, 8, 9, 10. We here have 4 groups of 9 lines each. My solution for 11 persons on 45 occasions is as follows:-- 2 11 9 4 7 6 5 1 8 3 10 2 1 11 7 6 3 10 8 5 4 9 2 11 10 3 9 4 8 5 1 7 6 2 11 5 8 1 3 10 6 7 9 4 2 11 1 10 3 4 9 6 7 5 8 There are two repeaters, 1 and 2, and the cycle is, 3, 4, 5,... 11. We thus get 5 groups of 9 lines each. The case of 12 persons on 55 occasions is solved thus:-- 1 2 3 12 4 11 5 10 6 9 7 8 1 2 4 11 6 9 8 7 10 5 12 3 1 2 5 10 8 7 11 4 3 12 6 9 1 2 6 9 10 5 3 12 7 8 11 4 1 2 7 8 12 3 6 9 11 4 5 10 Here 1 is a repeater, and the cycle is 2, 3, 4, 5,... 12. We thus get 5 groups of 11 lines each. 274.--THE MOUSE-TRAP PUZZLE. If we interchange cards 6 and 13 and begin our count at 14, we may take up all the twenty-one cards--that is, make twenty-one "catches"--in the following order: 6, 8, 13, 2, 10, 1, 11, 4, 14, 3, 5, 7, 21, 12, 15, 20, 9, 16, 18, 17, 19. We may also exchange 10 and 14 and start at 16, or exchange 6 and 8 and start at 19. 275.--THE SIXTEEN SHEEP. The six diagrams on next page show solutions for the cases where we replace 2, 3, 4, 5, 6, and 7 hurdles. The dark lines indicate the hurdles that have been replaced. There are, of course, other ways of making the removals. 276.--THE EIGHT VILLAS. There are several ways of solving the puzzle, but there is very little difference between them. The solver should, however, first of all bear in mind that in making his calculations he need only consider the four villas that stand at the corners, because the intermediate villas can never vary when the corners are known. One way is to place the numbers nought to 9 one at a time in the top left-hand corner, and then consider each case in turn. Now, if we place 9 in the corner as shown in the Diagram A, two of the corners cannot be occupied, while the corner that is diagonally opposite may be filled by 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 persons. We thus see that there are 10 [Illustration: +---+---+ +-+-----+ +---+---+ |O OHO O| |OHO O O| |O OHO O| | H | | + | | +=+ | |O OHO O| |OHO O O| |O OHOHO| +-+ +-+-+ +-+-----+ +---+ + | |O|O O|O| |O|O O O| |O O O|O| | +---+ | | +-+-+ | | +-+ | |O O O O| |O O OHO| |O O|O O| +-------+ +-------+ +-------+ 2 3 4 +-----+-+ +-+-----+ +-------+ |O O OHO| |OHO O O| |O O O O| | +=+ | | +=+ | | +=+=+=+ |O OHO O| |OHOHO O| |OHOHO O| | +-+-+ + | + +-+ | + + + | |O|O O|O| |O|O O|O| |O|OHO O| +=+ +=+ | + +=+ +=+ + | |O O O O| |OHO O O| |O O|O O| +-------+ +-+-----+ +---+---+ 5 6 7 THE SIXTEEN SHEEP ] solutions with a 9 in the corner. If, however, we substitute 8, the two corners in the same row and column may contain 0, 0, or 1, 1, or 0, 1, or 1, 0. In the case of B, ten different selections may be made for the fourth corner; but in each of the cases C, D, and E, only nine selections are possible, because we cannot use the 9. Therefore with 8 in the top left-hand corner there are 10 + (3 × 9) = 37 different solutions. If we then try 7 in the corner, the result will be 10 + 27 + 40, or 77 solutions. With 6 we get 10 + 27 + 40 + 49 = 126; with 5, 10 + 27 + 40 + 49 + 54 = 180; with 4, the same as with 5, + 55 = 235 ; with 3, the same as with 4, + 52 = 287; with 2, the same as with 3, + 45 = 332; with 1, the same as with 2, + 34 = 366, and with nought in the top left-hand corner the number of solutions will be found to be 10 + 27 + 40 + 49 + 54 + 55 + 52 + 45 + 34 + 19 = 385. As there is no other number to be placed in the top left-hand corner, we have now only to add these totals together thus, 10 + 37 + 77 + 126 + 180 + 235 + 287 + 332 + 366 + 385 = 2,035. We therefore find that the total number of ways in which tenants may occupy some or all of the eight villas so that there shall be always nine persons living along each side of the square is 2,035. Of course, this method must obviously cover all the reversals and reflections, since each corner in turn is occupied by every number in all possible combinations with the other two corners that are in line with it. [Illustration: A B C D E +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ |9| |0| |8| |0| |8| |1| |8| |0| |8| |1| +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ | |*| | | |*| | | |*| | | |*| | | |*| | +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ |0| | | |0| | | |1| | | |1| | | |0| | | +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ ] Here is a general formula for solving the puzzle: (n² + 3n + 2)(n² + 3n + 3)/6. Whatever may be the stipulated number of residents along each of the sides (which number is represented by n), the total number of different arrangements may be thus ascertained. In our particular case the number of residents was nine. Therefore (81 + 27 + 2) × (81 + 27 + 3) and the product, divided by 6, gives 2,035. If the number of residents had been 0, 1, 2, 3, 4, 5, 6, 7, or 8, the total arrangements would be 1, 7, 26, 70, 155, 301, 532, 876, or 1,365 respectively. 277.--COUNTER CROSSES. Let us first deal with the Greek Cross. There are just eighteen forms in which the numbers may be paired for the two arms. Here they are:-- 12978 13968 14958 34956 24957 23967 23958 13769 14759 14967 24758 23768 12589 23759 13579 34567 14768 24568 14569 23569 14379 23578 14578 25368 15369 24369 23189 24378 15378 45167 24179 25169 34169 35168 34178 25178 Of course, the number in the middle is common to both arms. The first pair is the one I gave as an example. I will suppose that we have written out all these crosses, always placing the first row of a pair in the upright and the second row in the horizontal arm. Now, if we leave the central figure fixed, there are 24 ways in which the numbers in the upright may be varied, for the four counters may be changed in 1 × 2 × 3 × 4 = 24 ways. And as the four in the horizontal may also be changed in 24 ways for every arrangement on the other arm, we find that there are 24 × 24 = 576 variations for every form; therefore, as there are 18 forms, we get 18 × 576 = 10,368 ways. But this will include half the four reversals and half the four reflections that we barred, so we must divide this by 4 to obtain the correct answer to the Greek Cross, which is thus 2,592 different ways. The division is by 4 and not by 8, because we provided against half the reversals and reflections by always reserving one number for the upright and the other for the horizontal. In the case of the Latin Cross, it is obvious that we have to deal with the same 18 forms of pairing. The total number of different ways in this case is the full number, 18 × 576. Owing to the fact that the upper and lower arms are unequal in length, permutations will repeat by reflection, but not by reversal, for we cannot reverse. Therefore this fact only entails division by 2. But in every pair we may exchange the figures in the upright with those in the horizontal (which we could not do in the case of the Greek Cross, as the arms are there all alike); consequently we must multiply by 2. This multiplication by 2 and division by 2 cancel one another. Hence 10,368 is here the correct answer. 278.--A DORMITORY PUZZLE. [Illustration: MON. TUES. WED. +---+---+---+ +---+---+---+ +---+---+---+ | 1 | 2 | 1 | | 1 | 3 | 1 | | 1 | 4 | 1 | +---+---+---+ +---+---+---+ +---+---+---+ | 2 | | 2 | | 1 | | 1 | | 1 | | 1 | +---+---+---+ +---+---+---+ +---+---+---+ | 1 | 22| 1 | | 3 | 19| 3 | | 4 | 16| 4 | +---+---+---+ +---+---+---+ +---+---+---+ THURS. FRI. SAT. +---+---+---+ +---+---+---+ +---+---+---+ | 1 | 5 | 1 | | 2 | 6 | 2 | | 4 | 4 | 4 | +---+---+---+ +---+---+---+ +---+---+---+ | 2 | | 2 | | 1 | | 1 | | 4 | | 4 | +---+---+---+ +---+---+---+ +---+---+---+ | 4 | 13| 4 | | 7 | 6 | 7 | | 4 | 4 | 4 | +---+---+---+ +---+---+---+ +---+---+---+ ] Arrange the nuns from day to day as shown in the six diagrams. The smallest possible number of nuns would be thirty-two, and the arrangements on the last three days admit of variation. 279.--THE BARRELS OF BALSAM. This is quite easy to solve for any number of barrels--if you know how. This is the way to do it. There are five barrels in each row Multiply the numbers 1, 2, 3, 4, 5 together; and also multiply 6, 7, 8, 9, 10 together. Divide one result by the other, and we get the number of different combinations or selections of ten things taken five at a time. This is here 252. Now, if we divide this by 6 (1 more than the number in the row) we get 42, which is the correct answer to the puzzle, for there are 42 different ways of arranging the barrels. Try this method of solution in the case of six barrels, three in each row, and you will find the answer is 5 ways. If you check this by trial, you will discover the five arrangements with 123, 124, 125, 134, 135 respectively in the top row, and you will find no others. The general solution to the problem is, in fact, this: n C 2n ----- n + 1 where 2n equals the number of barrels. The symbol C, of course, implies that we have to find how many combinations, or selections, we can make of 2n things, taken n at a time. 280.--BUILDING THE TETRAHEDRON. Take your constructed pyramid and hold it so that one stick only lies on the table. Now, four sticks must branch off from it in different directions--two at each end. Any one of the five sticks may be left out of this connection; therefore the four may be selected in 5 different ways. But these four matches may be placed in 24 different orders. And as any match may be joined at either of its ends, they may further be varied (after their situations are settled for any particular arrangement) in 16 different ways. In every arrangement the sixth stick may be added in 2 different ways. Now multiply these results together, and we get 5 × 24 × 16 × 2 = 3,840 as the exact number of ways in which the pyramid may be constructed. This method excludes all possibility of error. A common cause of error is this. If you calculate your combinations by working upwards from a basic triangle lying on the table, you will get half the correct number of ways, because you overlook the fact that an equal number of pyramids may be built on that triangle downwards, so to speak, through the table. They are, in fact, reflections of the others, and examples from the two sets of pyramids cannot be set up to resemble one another--except under fourth dimensional conditions! 281.--PAINTING A PYRAMID. It will be convenient to imagine that we are painting our pyramids on the flat cardboard, as in the diagrams, before folding up. Now, if we take any _four_ colours (say red, blue, green, and yellow), they may be applied in only 2 distinctive ways, as shown in Figs, 1 and 2. Any other way will only result in one of these when the pyramids are folded up. If we take any _three_ colours, they may be applied in the 3 ways shown in Figs. 3, 4, and 5. If we take any _two_ colours, they may be applied in the 3 ways shown in Figs. 6, 7, and 8. If we take any _single_ colour, it may obviously be applied in only 1 way. But four colours may be selected in 35 ways out of seven; three in 35 ways; two in 21 ways; and one colour in 7 ways. Therefore 35 applied in 2 ways = 70; 35 in 3 ways = 105; 21 in 3 ways = 63; and 7 in 1 way = 7. Consequently the pyramid may be painted in 245 different ways (70 + 105 + 63 + 7), using the seven colours of the solar spectrum in accordance with the conditions of the puzzle. [Illustration: 1 2 +---------------+ +---------------+ \ R / \ B / \ B / \ R / \ / \ / \ / \ / \ / G \ / \ / G \ / \-------/ \-------/ \ / \ / \ Y / \ Y / \ / \ / ' ' 3 4 5 +---------------+ +---------------+ +---------------+ \ R / \ R / \ R / \ G / \ Y / \ R / \ / \ / \ / \ / \ / \ / \ / G \ / \ / G \ / \ / G \ / \-------/ \-------/ \-------/ \ / \ / \ / \ Y / \ Y / \ Y / \ / \ / \ / ' ' ' 6 7 8 +---------------+ +---------------+ +---------------+ \ G / \ Y / \ Y / \ Y / \ G / \ G / \ / \ / \ / \ / \ / \ / \ / G \ / \ / G \ / \ / G \ / \-------/ \-------/ \-------/ \ / \ / \ / \ Y / \ Y / \ Y / \ / \ / \ / ' ' ' ] 282.--THE ANTIQUARY'S CHAIN. [Illustration] THE number of ways in which nine things may be arranged in a row without any restrictions is 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 = 362,880. But we are told that the two circular rings must never be together; therefore we must deduct the number of times that this would occur. The number is 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 = 40,320 × 2 = 80,640, because if we consider the two circular links to be inseparably joined together they become as one link, and eight links are capable of 40,320 arrangements; but as these two links may always be put on in the orders AB or BA, we have to double this number, it being a question of arrangement and not of design. The deduction required reduces our total to 282,240. Then one of our links is of a peculiar form, like an 8. We have therefore the option of joining on either one end or the other on every occasion, so we must double the last result. This brings up our total to 564,480. We now come to the point to which I directed the reader's attention--that every link may be put on in one of two ways. If we join the first finger and thumb of our left hand horizontally, and then link the first finger and thumb of the right hand, we see that the right thumb may be either above or below. But in the case of our chain we must remember that although that 8-shaped link has two independent _ends_ it is like every other link in having only two _sides_--that is, you cannot turn over one end without turning the other at the same time. We will, for convenience, assume that each link has a black side and a side painted white. Now, if it were stipulated that (with the chain lying on the table, and every successive link falling over its predecessor in the same way, as in the diagram) only the white sides should be uppermost as in A, then the answer would be 564,480, as above--ignoring for the present all reversals of the completed chain. If, however, the first link were allowed to be placed either side up, then we could have either A or B, and the answer would be 2 × 564,480 = 1,128,960; if two links might be placed either way up, the answer would be 4 × 564,480; if three links, then 8 × 564,480, and so on. Since, therefore, every link may be placed either side up, the number will be 564,480 multiplied by 2^9, or by 512. This raises our total to 289,013,760. But there is still one more point to be considered. We have not yet allowed for the fact that with any given arrangement three of the other arrangements may be obtained by simply turning the chain over through its entire length and by reversing the ends. Thus C is really the same as A, and if we turn this page upside down, then A and C give two other arrangements that are still really identical. Thus to get the correct answer to the puzzle we must divide our last total by 4, when we find that there are just 72,253,440 different ways in which the smith might have put those links together. In other words, if the nine links had originally formed a piece of chain, and it was known that the two circular links were separated, then it would be 72,253,439 chances to 1 that the smith would not have put the links together again precisely as they were arranged before! 283.--THE FIFTEEN DOMINOES. The reader may have noticed that at each end of the line I give is a four, so that, if we like, we can form a ring instead of a line. It can easily be proved that this must always be so. Every line arrangement will make a circular arrangement if we like to join the ends. Now, curious as it may at first appear, the following diagram exactly represents the conditions when we leave the doubles out of the question and devote our attention to forming circular arrangements. Each number, or half domino, is in line with every other number, so that if we start at any one of the five numbers and go over all the lines of the pentagon once and once only we shall come back to the starting place, and the order of our route will give us one of the circular arrangements for the ten dominoes. Take your pencil and follow out the following route, starting at the 4: 41304210234. You have been over all the lines once only, and by repeating all these figures in this way, 41--13--30--04--42--21--10--02--23--34, you get an arrangement of the dominoes (without the doubles) which will be perfectly clear. Take other routes and you will get other arrangements. If, therefore, we can ascertain just how many of these circular routes are obtainable from the pentagon, then the rest is very easy. Well, the number of different circular routes over the pentagon is 264. How I arrive at these figures I will not at present explain, because it would take a lot of space. The dominoes may, therefore, be arranged in a circle in just 264 different ways, leaving out the doubles. Now, in any one of these circles the five doubles may be inserted in 2^5 = 32 different ways. Therefore when we include the doubles there are 264 × 32 = 8,448 different circular arrangements. But each of those circles may be broken (so as to form our straight line) in any one of 15 different places. Consequently, 8,448 × 15 gives 126,720 different ways as the correct answer to the puzzle. [Illustration: ----- | | / | | \ / ----- \ / . . \ ----- . . ----- | | . . | o o | | o | -.--------.--- | | | | . . . | o o | ----- . . .. ----- \ . . . . / ----- .. ----- | o | . . |o | | | --------- | o | | o |. .| o| ----- ----- ] I purposely refrained from asking the reader to discover in just how many different ways the full set of twenty-eight dominoes may be arranged in a straight line in accordance with the ordinary rules of the game, left to right and right to left of any arrangement counting as different ways. It is an exceedingly difficult problem, but the correct answer is 7,959,229,931,520 ways. The method of solving is very complex. 284.--THE CROSS TARGET. [Illustration: -- -- (CD)( ) -- -- (AE)(A ) -- -- -- -- -- -- (CE)(E )(A )(AB)(C )(D ) -- -- -- -- -- -- (D )( )(B )(E )(EB)( ) -- -- -- -- -- -- (C )(B ) -- -- ( )(ED) -- -- ] Twenty-one different squares may be selected. Of these nine will be of the size shown by the four A's in the diagram, four of the size shown by the B's, four of the size shown by the C's, two of the size shown by the D's, and two of the size indicated by the upper single A, the upper single E, the lower single C, and the EB. It is an interesting fact that you cannot form any one of these twenty-one squares without using at least one of the six circles marked E. 285.--THE FOUR POSTAGE STAMPS. Referring to the original diagram, the four stamps may be given in the shape 1, 2, 3, 4, in three ways; in the shape 1, 2, 5, 6, in six ways; in the shape 1, 2, 3, 5, or 1, 2, 3, 7, or 1, 5, 6, 7, or 3, 5, 6, 7, in twenty-eight ways; in shape 1, 2, 3, 6, or 2, 5, 6, 7, in fourteen ways; in shape 1, 2, 6, 7, or 2, 3, 5, 6, or 1, 5, 6, 10, or 2, 5, 6, 9, in fourteen ways. Thus there are sixty-five ways in all. 286.--PAINTING THE DIE. The 1 can be marked on any one of six different sides. For every side occupied by 1 we have a selection of four sides for the 2. For every situation of the 2 we have two places for the 3. (The 6, 5, and 4 need not be considered, as their positions are determined by the 1, 2, and 3.) Therefore 6, 4, and 2 multiplied together make 48 different ways--the correct answer. 287.--AN ACROSTIC PUZZLE. There are twenty-six letters in the alphabet, giving 325 different pairs. Every one of these pairs may be reversed, making 650 ways. But every initial letter may be repeated as the final, producing 26 other ways. The total is therefore 676 different pairs. In other words, the answer is the square of the number of letters in the alphabet. 288.--CHEQUERED BOARD DIVISIONS. There are 255 different ways of cutting the board into two pieces of exactly the same size and shape. Every way must involve one of the five cuts shown in Diagrams A, B, C, D, and E. To avoid repetitions by reversal and reflection, we need only consider cuts that enter at the points a, b, and c. But the exit must always be at a point in a straight line from the entry through the centre. This is the most important condition to remember. In case B you cannot enter at a, or you will get the cut provided for in E. Similarly in C or D, you must not enter the key-line in the same direction as itself, or you will get A or B. If you are working on A or C and entering at a, you must consider joins at one end only of the key-line, or you will get repetitions. In other cases you must consider joins at both ends of the key; but after leaving a in case D, turn always either to right or left--use one direction only. Figs. 1 and 2 are examples under A; 3 and 4 are examples under B; 5 and 6 come under C; [Illustration] and 7 is a pretty example of D. Of course, E is a peculiar type, and obviously admits of only one way of cutting, for you clearly cannot enter at b or c. Here is a table of the results:-- a b c Ways. A = 8 + 17 + 21 = 46 B = 0 + 17 + 21 = 38 C = 15 + 31 + 39 = 85 D = 17 + 29 + 39 = 85 E = 1 + 0 + 0 = 1 -- -- -- --- 41 94 120 255 I have not attempted the task of enumerating the ways of dividing a board 8 × 8--that is, an ordinary chessboard. Whatever the method adopted, the solution would entail considerable labour. 289.--LIONS AND CROWNS. [Illustration] Here is the solution. It will be seen that each of the four pieces (after making the cuts along the thick lines) is of exactly the same size and shape, and that each piece contains a lion and a crown. Two of the pieces are shaded so as to make the solution quite clear to the eye. 290.--BOARDS WITH AN ODD NUMBER OF SQUARES. There are fifteen different ways of cutting the 5 × 5 board (with the central square removed) into two pieces of the same size and shape. Limitations of space will not allow me to give diagrams of all these, but I will enable the reader to draw them all out for himself without the slightest difficulty. At whatever point on the edge your cut enters, it must always end at a point on the edge, exactly opposite in a line through the centre of the square. Thus, if you enter at point 1 (see Fig. 1) at the top, you must leave at point 1 at the bottom. Now, 1 and 2 are the only two really different points of entry; if we use any others they will simply produce similar solutions. The directions of the cuts in the following fifteen [Illustration: Fig. 1. Fig. 2.] solutions are indicated by the numbers on the diagram. The duplication of the numbers can lead to no confusion, since every successive number is contiguous to the previous one. But whichever direction you take from the top downwards you must repeat from the bottom upwards, one direction being an exact reflection of the other. 1, 4, 8. 1, 4, 3, 7, 8. 1, 4, 3, 7, 10, 9. 1, 4, 3, 7, 10, 6, 5, 9. 1, 4, 5, 9. 1, 4, 5, 6, 10, 9. 1, 4, 5, 6, 10, 7, 8. 2, 3, 4, 8. 2, 3, 4, 5, 9. 2, 3, 4, 5, 6, 10, 9. 2, 3, 4, 5, 6, 10, 7, 8. 2, 3, 7, 8. 2, 3, 7, 10, 9. 2, 3, 7, 10, 6, 5, 9. 2, 3, 7, 10, 6, 5, 4, 8. It will be seen that the fourth direction (1, 4, 3, 7, 10, 6, 5, 9) produces the solution shown in Fig. 2. The thirteenth produces the solution given in propounding the puzzle, where the cut entered at the side instead of at the top. The pieces, however, will be of the same shape if turned over, which, as it was stated in the conditions, would not constitute a different solution. 291.--THE GRAND LAMA'S PROBLEM. The method of dividing the chessboard so that each of the four parts shall be of exactly the same size and shape, and contain one of the gems, is shown in the diagram. The method of shading the squares is adopted to make the shape of the pieces clear to the eye. Two of the pieces are shaded and two left white. The reader may find it interesting to compare this puzzle with that of the "Weaver" (No. 14, _Canterbury Puzzles_). [Illustration: THE GRAND LAMA'S PROBLEM. +===+===+===+===+===+===+===+===+ |:o:| : : : : : : : I...I...+===+===+===+===+===+===+ |:::| o |:::::::::::::::::::::::| I...I...I...+===+===+===+===+...I |:::| |:o:| : : : |:::| I...I...I...I...I===+===+...I...I |:::| |:::| o |:::::::| |:::| I...I...I...+===I===+...I...I...I |:::| |:::::::| |:::| |:::| I...I...+===+===+...+...I...I...I |:::| : : : |:::| |:::| I...+===+===+===+===I...I...I...I |:::::::::::::::::::::::| |:::| +===+===+===+===+===+===+...I...I | : : : : : : |:::| +===+===+===+===+===+===+===+===+ ] 292.--THE ABBOT'S WINDOW. THE man who was "learned in strange mysteries" pointed out to Father John that the orders of the Lord Abbot of St. Edmondsbury might be easily carried out by blocking up twelve of the lights in the window as shown by the dark squares in the following sketch:-- [Illustration: +===+===+===+===+===+===+===+===+ | : : : : : : : | I...+===+...+...+...+...+===+...I | IIIII : : : IIIII | I...+===+===+...+...+===+===+...I | : IIIII : IIIII : | I...+...+===+===+===+===+...+...I | : : IIIIIIIII : : | I...+...+...+===+===+...+...+...I | : : IIIIIIIII : : | I...+...+===+===+===+===+...+...I | : IIIII : IIIII : | I...+===+===+...+...+===+===+...I | IIIII : : : IIIII | I...+===+...+...+...+...+===+...I | : : : : : : : | +===+===+===+===+===+===+===+===+ ] Father John held that the four corners should also be darkened, but the sage explained that it was desired to obstruct no more light than was absolutely necessary, and he said, anticipating Lord Dundreary, "A single pane can no more be in a _line_ with itself than one bird can go into a corner and flock in solitude. The Abbot's condition was that no diagonal _lines_ should contain an odd number of lights." Now, when the holy man saw what had been done he was well pleased, and said, "Truly, Father John, thou art a man of deep wisdom, in that thou hast done that which seemed impossible, and yet withal adorned our window with a device of the cross of St. Andrew, whose name I received from my godfathers and godmothers." Thereafter he slept well and arose refreshed. The window might be seen intact to-day in the monastery of St. Edmondsbury, if it existed, which, alas! the window does not. 293.--THE CHINESE CHESSBOARD. +===I===+===+===+===I===+===+===+ | |:::: 2 ::::| 3 |:::| 5 |:6:| I...+===+...+===+...I...I...+===I |:::: 1 |:::| ::::| 4 |:::| 7 | I...+===+===+...I===I...I===+===I | |:::: |:::| ::::| 9 |:::| I===I...I===============I...I...I |:::: 11|:::: ::::: 10|:::| 8 | I=======I===I===========I...I...I | ::::: 12|:::: 13::::| |:::| I=======+...I...+===+===|===+===I |:::: 14|:::| |:::| 16::::| 17| I...+...I===I===+...+...+===+...I | ::::| ::::: 15|:::| ::::| I=======+===========+===+=======I |:::: ::::: 18::::: ::::: | +===+===+===+===+===+===+===+===+ +===+===I===I===+===I===+===+===+ | ::::| |:::: |:::| ::::| I...+===I...I=======I...I===+...I |:::| |:::: |:::: |:::| | I...I===I===============I===I...I | |:::: ::::| ::::: |:::| I===I=======I=======I=======I===I |:::| ::::| ::::| ::::| | I...I===+...I...+...I...+===+...I | ::::| |:::: |:::| ::::| I...+===I...+===I===+...I===+...I |:::| |:::: |:::: |:::| | I===I...+=======I=======+...I===I | |:::: ::::| ::::: |:::| I...+=======+...I...+=======+...I |:::: ::::| |:::| ::::: | +===+===+===+===+===+===+===+===+ Eighteen is the maximum number of pieces. I give two solutions. The numbered diagram is so cut that the eighteenth piece has the largest area--eight squares--that is possible under the conditions. The second diagram was prepared under the added condition that no piece should contain more than five squares. No. 74 in _The Canterbury Puzzles_ shows how to cut the board into twelve pieces, all different, each containing five squares, with one square piece of four squares. 294.--THE CHESSBOARD SENTENCE. +===I===I===I===I=======I=======+ | |:::| |:::| ::::| ::::| I===I...I===I...I...+===I...+===I |:::| ::::: |:::| ::::: | |...|...+===I...I...+===+...+===I | |:::| |:::| ::::| ::::| |...+===+...+===I===I===I=======I |:::: ::::: |:::| ::::: | I===========I===I...I===I===+...| | ::::: |:::| |:::| |:::| |...+===+...|...|...|...I===+...| |:::| |:::| |:::| |:::: | |...|...|...|...I===+...+===+...| | |:::| |:::| ::::: |:::| I===+...+===I...+=======I===+...| |:::: ::::| ::::: |:::: | +===========I===================+ The pieces may be fitted together, as shown in the illustration, to form a perfect chessboard. 295.--THE EIGHT ROOKS. Obviously there must be a rook in every row and every column. Starting with the top row, it is clear that we may put our first rook on any one of eight different squares. Wherever it is placed, we have the option of seven squares for the second rook in the second row. Then we have six squares from which to select the third row, five in the fourth, and so on. Therefore the number of our different ways must be 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320 (that is 8!), which is the correct answer. How many ways there are if mere reversals and reflections are not counted as different has not yet been determined; it is a difficult problem. But this point, on a smaller square, is considered in the next puzzle. 296.--THE FOUR LIONS. There are only seven different ways under the conditions. They are as follows: 1 2 3 4, 1 2 4 3, 1 3 2 4, 1 3 4 2, 1 4 3 2, 2 1 4 3, 2 4 1 3. Taking the last example, this notation means that we place a lion in the second square of first row, fourth square of second row, first square of third row, and third square of fourth row. The first example is, of course, the one we gave when setting the puzzle. 297.--BISHOPS--UNGUARDED. +...+...+...+...+...+...+...+...+ : ::::: ::::: ::::: ::::: +...+...+...+...+...+...+...+...+ ::::: ::::: ::::: ::::: : +...+...+...+...+...+...+...+...+ : ::::: ::::: ::::: ::::: +...+...+...+...+...+...+...+...+ ::B:: B ::B:: B ::B:: B ::B:: B : +...+...+...+...+...+...+...+...+ : ::::: ::::: ::::: ::::: +...+...+...+...+...+...+...+...+ ::::: ::::: ::::: ::::: : +...+...+...+...+...+...+...+...+ ::::: ::::: ::::: ::::: : +...+...+...+...+...+...+...+...+ : ::::: ::::: ::::: ::::: +...+...+...+...+...+...+...+...+ This cannot be done with fewer bishops than eight, and the simplest solution is to place the bishops in line along the fourth or fifth row of the board (see diagram). But it will be noticed that no bishop is here guarded by another, so we consider that point in the next puzzle. 298.--BISHOPS--GUARDED. +...+...+...+...+.......+.......+ : ::::: ::::: ::::: ::::: +...+...+...+...+...+...+...+...+ ::::: ::::: ::::: ::::: : +...+...+...+...+...+...+...+...+ : ::::: ::::: ::::: ::::: +...+...+...+...+...+...+.......+ ::::: B ::B:: B ::::: B ::B:: : +...........+...+...+...+...+...+ : ::B:: B ::B:: ::B:: B ::::: +...+...+...+...+...+...+...+...+ ::::: ::::: ::::: ::::: : +...+...+...+...+...+...+...+...+ : ::::: ::::: ::::: ::::: +...+...+...+...+.......+...+...+ ::::: ::::: ::::: ::::: : +...+...+...+...+.......+...+...+ This puzzle is quite easy if you first of all give it a little thought. You need only consider squares of one colour, for whatever can be done in the case of the white squares can always be repeated on the black, and they are here quite independent of one another. This equality, of course, is in consequence of the fact that the number of squares on an ordinary chessboard, sixty-four, is an even number. If a square chequered board has an odd number of squares, then there will always be one more square of one colour than of the other. Ten bishops are necessary in order that every square shall be attacked and every bishop guarded by another bishop. I give one way of arranging them in the diagram. It will be noticed that the two central bishops in the group of six on the left-hand side of the board serve no purpose, except to protect those bishops that are on adjoining squares. Another solution would therefore be obtained by simply raising the upper one of these one square and placing the other a square lower down. 299.--BISHOPS IN CONVOCATION. The fourteen bishops may be placed in 256 different ways. But every bishop must always be placed on one of the sides of the board--that is, somewhere on a row or file on the extreme edge. The puzzle, therefore, consists in counting the number of different ways that we can arrange the fourteen round the edge of the board without attack. This is not a difficult matter. On a chessboard of n² squares 2n - 2 bishops (the maximum number) may always be placed in 2^n ways without attacking. On an ordinary chessboard n would be 8; therefore 14 bishops may be placed in 256 different ways. It is rather curious that the general result should come out in so simple a form. [Illustration] 300.--THE EIGHT QUEENS. [Illustration] The solution to this puzzle is shown in the diagram. It will be found that no queen attacks another, and also that no three queens are in a straight line in any oblique direction. This is the only arrangement out of the twelve fundamentally different ways of placing eight queens without attack that fulfils the last condition. 301.--THE EIGHT STARS. The solution of this puzzle is shown in the first diagram. It is the only possible solution within the conditions stated. But if one of the eight stars had not already been placed as shown, there would then have been eight ways of arranging the stars according to this scheme, if we count reversals and reflections as different. If you turn this page round so that each side is in turn at the bottom, you will get the four reversals; and if you reflect each of these in a mirror, you will get the four reflections. These are, therefore, merely eight aspects of one "fundamental solution." But without that first star being so placed, there is another fundamental solution, as shown in the second diagram. But this arrangement being in a way symmetrical, only produces four different aspects by reversal and reflection. [Illustration] 302.--A PROBLEM IN MOSAICS. [Illustration] The diagram shows how the tiles may be rearranged. As before, one yellow and one purple tile are dispensed with. I will here point out that in the previous arrangement the yellow and purple tiles in the seventh row might have changed places, but no other arrangement was possible. 303.--UNDER THE VEIL. Some schemes give more diagonal readings of four letters than others, and we are at first tempted to favour these; but this is a false scent, because what you appear to gain in this direction you lose in others. Of course it immediately occurs to the solver that every LIVE or EVIL is worth twice as much as any other word, since it reads both ways and always counts as 2. This is an important consideration, though sometimes those arrangements that contain most readings of these two words are fruitless in other words, and we lose in the general count. [Illustration: _ _ I V E L _ _ E V L _ _ I _ _ L _ _ I _ _ V E I _ V E _ _ _ L _ E _ _ L V _ I _ L I _ _ I _ E V /V _ E L _ _ I _ _ I _ _ V E L _\ ] The above diagram is in accordance with the conditions requiring no letter to be in line with another similar letter, and it gives twenty readings of the five words--six horizontally, six vertically, four in the diagonals indicated by the arrows on the left, and four in the diagonals indicated by the arrows on the right. This is the maximum. Four sets of eight letters may be placed on the board of sixty-four squares in as many as 604 different ways, without any letter ever being in line with a similar one. This does not count reversals and reflections as different, and it does not take into consideration the actual permutations of the letters among themselves; that is, for example, making the L's change places with the E's. Now it is a singular fact that not only do the twenty word-readings that I have given prove to be the real maximum, but there is actually only that one arrangement from which this maximum may be obtained. But if you make the V's change places with the I's, and the L's with the E's, in the solution given, you still get twenty readings--the same number as before in every direction. Therefore there are two ways of getting the maximum from the same arrangement. The minimum number of readings is zero--that is, the letters can be so arranged that no word can be read in any of the directions. 304.--BACHET'S SQUARE. [Illustration: 1] [Illustration: 2] [Illustration: 3] [Illustration: 4] Let us use the letters A, K, Q, J, to denote ace, king, queen, jack; and D, S, H, C, to denote diamonds, spades, hearts, clubs. In Diagrams 1 and 2 we have the two available ways of arranging either group of letters so that no two similar letters shall be in line--though a quarter-turn of 1 will give us the arrangement in 2. If we superimpose or combine these two squares, we get the arrangement of Diagram 3, which is one solution. But in each square we may put the letters in the top line in twenty-four different ways without altering the scheme of arrangement. Thus, in Diagram 4 the S's are similarly placed to the D's in 2, the H's to the S's, the C's to the H's, and the D's to the C's. It clearly follows that there must be 24×24 = 576 ways of combining the two primitive arrangements. But the error that Labosne fell into was that of assuming that the A, K, Q, J must be arranged in the form 1, and the D, S, H, C in the form 2. He thus included reflections and half-turns, but not quarter-turns. They may obviously be interchanged. So that the correct answer is 2 × 576 = 1,152, counting reflections and reversals as different. Put in another manner, the pairs in the top row may be written in 16 × 9 × 4 × 1 = 576 different ways, and the square then completed in 2 ways, making 1,152 ways in all. 305.--THE THIRTY-SIX LETTER BLOCKS. I pointed out that it was impossible to get all the letters into the box under the conditions, but the puzzle was to place as many as possible. This requires a little judgment and careful investigation, or we are liable to jump to the hasty conclusion that the proper way to solve the puzzle must be first to place all six of one letter, then all six of another letter, and so on. As there is only one scheme (with its reversals) for placing six similar letters so that no two shall be in a line in any direction, the reader will find that after he has placed four different kinds of letters, six times each, every place is occupied except those twelve that form the two long diagonals. He is, therefore, unable to place more than two each of his last two letters, and there are eight blanks left. I give such an arrangement in Diagram 1. [Illustration: 1] [Illustration: 2] The secret, however, consists in not trying thus to place all six of each letter. It will be found that if we content ourselves with placing only five of each letter, this number (thirty in all) may be got into the box, and there will be only six blanks. But the correct solution is to place six of each of two letters and five of each of the remaining four. An examination of Diagram 2 will show that there are six each of C and D, and five each of A, B, E, and F. There are, therefore, only four blanks left, and no letter is in line with a similar letter in any direction. 306.--THE CROWDED CHESSBOARD. [Illustration] Here is the solution. Only 8 queens or 8 rooks can be placed on the board without attack, while the greatest number of bishops is 14, and of knights 32. But as all these knights must be placed on squares of the same colour, while the queens occupy four of each colour and the bishops 7 of each colour, it follows that only 21 knights can be placed on the same colour in this puzzle. More than 21 knights can be placed alone on the board if we use both colours, but I have not succeeded in placing more than 21 on the "crowded chessboard." I believe the above solution contains the maximum number of pieces, but possibly some ingenious reader may succeed in getting in another knight. 307.--THE COLOURED COUNTERS. The counters may be arranged in this order:-- R1, B2, Y3, O4, GS. Y4, O5, G1, R2, B3. G2, R3, B4, Y5, O1. B5, Y1, O2, G3, R4. O3, G4, R5, B1, Y2. 308.--THE GENTLE ART OF STAMP-LICKING. The following arrangement shows how sixteen stamps may be stuck on the card, under the conditions, of a total value of fifty pence, or 4s. 2d.:-- [Illustration] If, after placing the four 5d. stamps, the reader is tempted to place four 4d. stamps also, he can afterwards only place two of each of the three other denominations, thus losing two spaces and counting no more than forty-eight pence, or 4s. This is the pitfall that was hinted at. (Compare with No. 43, _Canterbury Puzzles_.) 309.--THE FORTY-NINE COUNTERS. The counters may be arranged in this order:-- A1, B2, C3, D4, E5, F6, G7. F4, G5, A6, B7, C1, D2, E3. D7, E1, F2, G3, A4, B5, C6. B3, C4, D5, E6, F7, G1, A2. G6, A7, B1, C2, D3, E4, F5. E2, F3, G4, A5, B6, C7, D1. C5, D6, E7, F1, G2, A3, B4. 310.--THE THREE SHEEP. The number of different ways in which the three sheep may be placed so that every pen shall always be either occupied or in line with at least one sheep is forty-seven. The following table, if used with the key in Diagram 1, will enable the reader to place them in all these ways:-- +------------+---------------------------+----------+ | | | No. of | | Two Sheep. | Third Sheep. | Ways. | +------------+---------------------------+----------+ | A and B | C, E, G, K, L, N, or P | 7 | | A and C | I, J, K, or O | 4 | | A and D | M, N, or J | 3 | | A and F | J, K, L, or P | 4 | | A and G | H, J, K, N, O, or P | 6 | | A and H | K, L, N, or O | 4 | | A and O | K or L | 2 | | B and C | N | 1 | | B and E | F, H, K, or L | 4 | | B and F | G, J, N, or O | 4 | | B and G | K, L, or N | 3 | | B and H | J or N | 2 | | B and J | K or L | 2 | | F and G | J | 1 | | | | ---- | | | | 47 | +------------+---------------------------+----------+ This, of course, means that if you place sheep in the pens marked A and B, then there are seven different pens in which you may place the third sheep, giving seven different solutions. It was understood that reversals and reflections do not count as different. If one pen at least is to be _not_ in line with a sheep, there would be thirty solutions to that problem. If we counted all the reversals and reflections of these 47 and 30 cases respectively as different, their total would be 560, which is the number of different ways in which the sheep may be placed in three pens without any conditions. I will remark that there are three ways in which two sheep may be placed so that every pen is occupied or in line, as in Diagrams 2, 3, and 4, but in every case each sheep is in line with its companion. There are only two ways in which three sheep may be so placed that every pen shall be occupied or in line, but no sheep in line with another. These I show in Diagrams 5 and 6. Finally, there is only one way in which three sheep may be placed so that at least one pen shall not be in line with a sheep and yet no sheep in line with another. Place the sheep in C, E, L. This is practically all there is to be said on this pleasant pastoral subject. [Illustration] 311.--THE FIVE DOGS PUZZLE. The diagrams show four fundamentally different solutions. In the case of A we can reverse the order, so that the single dog is in the bottom row and the other four shifted up two squares. Also we may use the next column to the right and both of the two central horizontal rows. Thus A gives 8 solutions. Then B may be reversed and placed in either diagonal, giving 4 solutions. Similarly C will give 4 solutions. The line in D being symmetrical, its reversal will not be different, but it may be disposed in 4 different directions. We thus have in all 20 different solutions. [Illustration] 312.--THE FIVE CRESCENTS OF BYZANTIUM. [Illustration] If that ancient architect had arranged his five crescent tiles in the manner shown in the following diagram, every tile would have been watched over by, or in a line with, at least one crescent, and space would have been reserved for a perfectly square carpet equal in area to exactly half of the pavement. It is a very curious fact that, although there are two or three solutions allowing a carpet to be laid down within the conditions so as to cover an area of nearly twenty-nine of the tiles, this is the only possible solution giving exactly half the area of the pavement, which is the largest space obtainable. 313.--QUEENS AND BISHOP PUZZLE. [Illustration: FIG. 1.] [Illustration: FIG. 2.] The bishop is on the square originally occupied by the rook, and the four queens are so placed that every square is either occupied or attacked by a piece. (Fig. 1.) I pointed out in 1899 that if four queens are placed as shown in the diagram (Fig. 2), then the fifth queen may be placed on any one of the twelve squares marked a, b, c, d, and e; or a rook on the two squares, c; or a bishop on the eight squares, a, b, and e; or a pawn on the square b; or a king on the four squares, b, c, and e. The only known arrangement for four queens and a knight is that given by Mr. J. Wallis in _The Strand Magazine_ for August 1908, here reproduced. (Fig. 3.) [Illustration: FIG. 3.] I have recorded a large number of solutions with four queens and a rook, or bishop, but the only arrangement, I believe, with three queens and two rooks in which all the pieces are guarded is that of which I give an illustration (Fig. 4), first published by Dr. C. Planck. But I have since found the accompanying solution with three queens, a rook, and a bishop, though the pieces do not protect one another. (Fig. 5.) [Illustration: FIG. 4.] [Illustration: FIG. 5.] 314.--THE SOUTHERN CROSS. My readers have been so familiarized with the fact that it requires at least five planets to attack every one of a square arrangement of sixty-four stars that many of them have, perhaps, got to believe that a larger square arrangement of stars must need an increase of planets. It was to correct this possible error of reasoning, and so warn readers against another of those numerous little pitfalls in the world of puzzledom, that I devised this new stellar problem. Let me then state at once that, in the case of a square arrangement of eighty one stars, there are several ways of placing five planets so that every star shall be in line with at least one planet vertically, horizontally, or diagonally. Here is the solution to the "Southern Cross": -- It will be remembered that I said that the five planets in their new positions "will, of course, obscure five other stars in place of those at present covered." This was to exclude an easier solution in which only four planets need be moved. 315.--THE HAT-PEG PUZZLE. The moves will be made quite clear by a reference to the diagrams, which show the position on the board after each of the four moves. The darts indicate the successive removals that have been made. It will be seen that at every stage all the squares are either attacked or occupied, and that after the fourth move no queen attacks any other. In the case of the last move the queen in the top row might also have been moved one square farther to the left. This is, I believe, the only solution to the puzzle. [Illustration: 1] [Illustration: 2] [Illustration: 3] [Illustration: 4] 316.--THE AMAZONS. It will be seen that only three queens have been removed from their positions on the edge of the board, and that, as a consequence, eleven squares (indicated by the black dots) are left unattacked by any queen. I will hazard the statement that eight queens cannot be placed on the chessboard so as to leave more than eleven squares unattacked. It is true that we have no rigid proof of this yet, but I have entirely convinced myself of the truth of the statement. There are at least five different ways of arranging the queens so as to leave eleven squares unattacked. [Illustration] 317.--A PUZZLE WITH PAWNS. Sixteen pawns may be placed so that no three shall be in a straight line in any possible direction, as in the diagram. We regard, as the conditions required, the pawns as mere points on a plane. [Illustration] 318.--LION-HUNTING. There are 6,480 ways of placing the man and the lion, if there are no restrictions whatever except that they must be on different spots. This is obvious, because the man may be placed on any one of the 81 spots, and in every case there are 80 spots remaining for the lion; therefore 81 × 80 = 6,480. Now, if we deduct the number of ways in which the lion and the man may be placed on the same path, the result must be the number of ways in which they will not be on the same path. The number of ways in which they may be in line is found without much difficulty to be 816. Consequently, 6,480 - 816 = 5,664, the required answer. The general solution is this: 1/3n(n - 1)(3n² - n + 2). This is, of course, equivalent to saying that if we call the number of squares on the side of a "chessboard" n, then the formula shows the number of ways in which two bishops may be placed without attacking one another. Only in this case we must divide by two, because the two bishops have no distinct individuality, and cannot produce a different solution by mere exchange of places. 319.--THE KNIGHT-GUARDS. [Illustration: DIAGRAM 1.] [Illustration: DIAGRAM 2.] The smallest possible number of knights with which this puzzle can be solved is fourteen. It has sometimes been assumed that there are a great many different solutions. As a matter of fact, there are only three arrangements--not counting mere reversals and reflections as different. Curiously enough, nobody seems ever to have hit on the following simple proof, or to have thought of dealing with the black and the white squares separately. [Illustration: DIAGRAM 3.] [Illustration: DIAGRAM 4.] [Illustration: DIAGRAM 5.] Seven knights can be placed on the board on white squares so as to attack every black square in two ways only. These are shown in Diagrams 1 and 2. Note that three knights occupy the same position in both arrangements. It is therefore clear that if we turn the board so that a black square shall be in the top left-hand corner instead of a white, and place the knights in exactly the same positions, we shall have two similar ways of attacking all the white squares. I will assume the reader has made the two last described diagrams on transparent paper, and marked them _1a_ and _2a_. Now, by placing the transparent Diagram _1a_ over 1 you will be able to obtain the solution in Diagram 3, by placing _2a_ over 2 you will get Diagram 4, and by placing _2a_ over 1 you will get Diagram 5. You may now try all possible combinations of those two pairs of diagrams, but you will only get the three arrangements I have given, or their reversals and reflections. Therefore these three solutions are all that exist. 320.--THE ROOK'S TOUR. [Illustration] The only possible minimum solutions are shown in the two diagrams, where it will be seen that only sixteen moves are required to perform the feat. Most people find it difficult to reduce the number of moves below seventeen*. [Illustration: THE ROOK'S TOUR.] 321.--THE ROOK'S JOURNEY. [Illustration] I show the route in the diagram. It will be seen that the tenth move lands us at the square marked "10," and that the last move, the twenty-first, brings us to a halt on square "21." 322.--THE LANGUISHING MAIDEN. The dotted line shows the route in twenty-two straight paths by which the knight may rescue the maiden. It is necessary, after entering the first cell, immediately to return before entering another. Otherwise a solution would not be possible. (See "The Grand Tour," p. 200.) 323.--A DUNGEON PUZZLE. If the prisoner takes the route shown in the diagram--where for clearness the doorways are omitted--he will succeed in visiting every cell once, and only once, in as many as fifty-seven straight lines. No rook's path over the chessboard can exceed this number of moves. [Illustration: THE LANGUISHING MAIDEN] [Illustration: A DUNGEON PUZZLE.] 324.--THE LION AND THE MAN. First of all, the fewest possible straight lines in each case are twenty-two, and in order that no cell may be visited twice it is absolutely necessary that each should pass into one cell and then immediately "visit" the one from which he started, afterwards proceeding by way of the second available cell. In the following diagram the man's route is indicated by the unbroken lines, and the lion's by the dotted lines. It will be found, if the two routes are followed cell by cell with two pencil points, that the lion and the man never meet. But there was one little point that ought not to be overlooked--"they occasionally got glimpses of one another." Now, if we take one route for the man and merely reverse it for the lion, we invariably find that, going at the same speed, they never get a glimpse of one another. But in our diagram it will be found that the man and the lion are in the cells marked A at the same moment, and may see one another through the open doorways; while the same happens when they are in the two cells marked B, the upper letters indicating the man and the lower the lion. In the first case the lion goes straight for the man, while the man appears to attempt to get in the rear of the lion; in the second case it looks suspiciously like running away from one another! [Illustration] 325.--AN EPISCOPAL VISITATION. [Illustration] In the diagram I show how the bishop may be made to visit every one of his white parishes in seventeen moves. It is obvious that we must start from one corner square and end at the one that is diagonally opposite to it. The puzzle cannot be solved in fewer than seventeen moves. 326.--A NEW COUNTER PUZZLE. Play as follows: 2--3, 9--4, 10--7, 3--8, 4--2, 7--5, 8--6, 5--10, 6--9, 2--5, 1--6, 6--4, 5--3, 10--8, 4--7, 3--2, 8--1, 7--10. The white counters have now changed places with the red ones, in eighteen moves, without breaking the conditions. 327.--A NEW BISHOP'S PUZZLE. [Illustration: A] [Illustration: B] Play as follows, using the notation indicated by the numbered squares in Diagram A:-- White. | Black. | White. | Black. 1. 18--15 | 1. 3--6 | 10. 20--10 | 10. 1--11 2. 17--8 | 2. 4--13 | 11. 3--9 | 11. 18--12 3. 19--14 | 3. 2--7 | 12. 10--13 | 12. 11--8 4. 15--5 | 4. 6--16 | 13. 19--16 | 13. 2--5 5. 8--3 | 5. 13-18 | 14. 16--1 | 14. 5--20 6. 14--9 | 6. 7--12 | 15. 9--6 | 15. 12--15 7. 5--10 | 7. 16-11 | 16. 13-7 | 16. 8--14 8. 9--19 | 8. 12--2 | 17. 6--3 | 17. 15-18 9. 10--4 | 9. 11-17 | 18. 7--2 | 18. 14--19 Diagram B shows the position after the ninth move. Bishops at 1 and 20 have not yet moved, but 2 and 19 have sallied forth and returned. In the end, 1 and 19, 2 and 20, 3 and 17, and 4 and 18 will have exchanged places. Note the position after the thirteenth move. 328.--THE QUEEN'S TOUR. [Illustration] The annexed diagram shows a second way of performing the Queen's Tour. If you break the line at the point J and erase the shorter portion of that line, you will have the required path solution for any J square. If you break the line at I, you will have a non-re-entrant solution starting from any I square. And if you break the line at G, you will have a solution for any G square. The Queen's Tour previously given may be similarly broken at three different places, but I seized the opportunity of exhibiting a second tour. 329.--THE STAR PUZZLE. The illustration explains itself. The stars are all struck out in fourteen straight strokes, starting and ending at a white star. [Illustration] 330.--THE YACHT RACE. The diagram explains itself. The numbers will show the direction of the lines in their proper order, and it will be seen that the seventh course ends at the flag-buoy, as stipulated. [Illustration] 331.--THE SCIENTIFIC SKATER. In this case we go beyond the boundary of the square. Apart from that, the moves are all queen moves. There are three or four ways in which it can be done. Here is one way of performing the feat:-- [Illustration] It will be seen that the skater strikes out all the stars in one continuous journey of fourteen straight lines, returning to the point from which he started. To follow the skater's course in the diagram it is necessary always to go as far as we can in a straight line before turning. 332.--THE FORTY-NINE STARS. The illustration shows how all the stars may be struck out in twelve straight strokes, beginning and ending at a black star. [Illustration] 333.--THE QUEEN'S JOURNEY. The correct solution to this puzzle is shown in the diagram by the dark line. The five moves indicated will take the queen the greatest distance that it is possible for her to go in five moves, within the conditions. The dotted line shows the route that most people suggest, but it is not quite so long as the other. Let us assume that the distance from the centre of any square to the centre of the next in the same horizontal or vertical line is 2 inches, and that the queen travels from the centre of her original square to the centre of the one at which she rests. Then the first route will be found to exceed 67.9 inches, while the dotted route is less than 67.8 inches. The difference is small, but it is sufficient to settle the point as to the longer route. All other routes are shorter still than these two. [Illustration] 334.--ST. GEORGE AND THE DRAGON. We select for the solution of this puzzle one of the prettiest designs that can be formed by representing the moves of the knight by lines from square to square. The chequering of the squares is omitted to give greater clearness. St. George thus slays the Dragon in strict accordance with the conditions and in the elegant manner we should expect of him. [Illustration: St. George and the Dragon.] 335.--FARMER LAWRENCE'S CORNFIELDS. There are numerous solutions to this little agricultural problem. The version I give in the next column is rather curious on account of the long parallel straight lines formed by some of the moves. [Illustration: Farmer Lawrence's Cornfields.] 336.--THE GREYHOUND PUZZLE. There are several interesting points involved in this question. In the first place, if we had made no stipulation as to the positions of the two ends of the string, it is quite impossible to form any such string unless we begin and end in the top and bottom row of kennels. We may begin in the top row and end in the bottom (or, of course, the reverse), or we may begin in one of these rows and end in the same. But we can never begin or end in one of the two central rows. Our places of starting and ending, however, were fixed for us. Yet the first half of our route must be confined entirely to those squares that are distinguished in the following diagram by circles, and the second half will therefore be confined to the squares that are not circled. The squares reserved for the two half-strings will be seen to be symmetrical and similar. The next point is that the first half-string must end in one of the central rows, and the second half-string must begin in one of these rows. This is now obvious, because they have to link together to form the complete string, and every square on an outside row is connected by a knight's move with similar squares only--that is, circled or non-circled as the case may be. The half-strings can, therefore, only be linked in the two central rows. [Illustration] Now, there are just eight different first half-strings, and consequently also eight second half-strings. We shall see that these combine to form twelve complete strings, which is the total number that exist and the correct solution of our puzzle. I do not propose to give all the routes at length, but I will so far indicate them that if the reader has dropped any he will be able to discover which they are and work them out for himself without any difficulty. The following numbers apply to those in the above diagram. The eight first half-strings are: 1 to 6 (2 routes); 1 to 8 (1 route); 1 to 10 (3 routes); 1 to 12 (1 route); and 1 to 14 (1 route). The eight second half-strings are: 7 to 20 (1 route); 9 to 20 (1 route); 11 to 20 (3 routes); 13 to 20 (1 route); and 15 to 20 (2 routes). Every different way in which you can link one half-string to another gives a different solution. These linkings will be found to be as follows: 6 to 13 (2 cases); 10 to 13 (3 cases); 8 to 11 (3 cases); 8 to 15 (2 cases); 12 to 9 (1 case); and 14 to 7 (1 case). There are, therefore, twelve different linkings and twelve different answers to the puzzle. The route given in the illustration with the greyhound will be found to consist of one of the three half-strings 1 to 10, linked to the half-string 13 to 20. It should be noted that ten of the solutions are produced by five distinctive routes and their reversals--that is, if you indicate these five routes by lines and then turn the diagrams upside down you will get the five other routes. The remaining two solutions are symmetrical (these are the cases where 12 to 9 and 14 to 7 are the links), and consequently they do not produce new solutions by reversal. 337.--THE FOUR KANGAROOS. [Illustration] A pretty symmetrical solution to this puzzle is shown in the diagram. Each of the four kangaroos makes his little excursion and returns to his corner, without ever entering a square that has been visited by another kangaroo and without crossing the central line. It will at once occur to the reader, as a possible improvement of the puzzle, to divide the board by a central vertical line and make the condition that this also shall not be crossed. This would mean that each kangaroo had to confine himself to a square 4 by 4, but it would be quite impossible, as I shall explain in the next two puzzles. 338.--THE BOARD IN COMPARTMENTS. [Illustration] In attempting to solve this problem it is first necessary to take the two distinctive compartments of twenty and twelve squares respectively and analyse them with a view to determining where the necessary points of entry and exit lie. In the case of the larger compartment it will be found that to complete a tour of it we must begin and end on two of the outside squares on the long sides. But though you may start at any one of these ten squares, you are restricted as to those at which you can end, or (which is the same thing) you may end at whichever of these you like, provided you begin your tour at certain particular squares. In the case of the smaller compartment you are compelled to begin and end at one of the six squares lying at the two narrow ends of the compartments, but similar restrictions apply as in the other instance. A very little thought will show that in the case of the two small compartments you must begin and finish at the ends that lie together, and it then follows that the tours in the larger compartments must also start and end on the contiguous sides. In the diagram given of one of the possible solutions it will be seen that there are eight places at which we may start this particular tour; but there is only one route in each case, because we must complete the compartment in which we find ourself before passing into another. In any solution we shall find that the squares distinguished by stars must be entering or exit points, but the law of reversals leaves us the option of making the other connections either at the diamonds or at the circles. In the solution worked out the diamonds are used, but other variations occur in which the circle squares are employed instead. I think these remarks explain all the essential points in the puzzle, which is distinctly instructive and interesting. 339.--THE FOUR KNIGHTS' TOURS. [Illustration] It will be seen in the illustration how a chessboard may be divided into four parts, each of the same size and shape, so that a complete re-entrant knight's tour may be made on each portion. There is only one possible route for each knight and its reversal. 340.--THE CUBIC KNIGHT'S TOUR. [Illustration] If the reader should cut out the above diagram, fold it in the form of a cube, and stick it together by the strips left for that purpose at the edges, he would have an interesting little curiosity. Or he can make one on a larger scale for himself. It will be found that if we imagine the cube to have a complete chessboard on each of its sides, we may start with the knight on any one of the 384 squares, and make a complete tour of the cube, always returning to the starting-point. The method of passing from one side of the cube to another is easily understood, but, of course, the difficulty consisted in finding the proper points of entry and exit on each board, the order in which the different boards should be taken, and in getting arrangements that would comply with the required conditions. 341.--THE FOUR FROGS. The fewest possible moves, counting every move separately, are sixteen. But the puzzle may be solved in seven plays, as follows, if any number of successive moves by one frog count as a single play. All the moves contained within a bracket are a single play; the numbers refer to the toadstools: (1--5), (3--7, 7--1), (8--4, 4--3, 3--7), (6--2, 2--8, 8--4, 4--3), (5--6, 6--2, 2--8), (1--5, 5--6), (7--1). This is the familiar old puzzle by Guarini, propounded in 1512, and I give it here in order to explain my "buttons and string" method of solving this class of moving-counter problem. Diagram A shows the old way of presenting Guarini's puzzle, the point being to make the white knights change places with the black ones. In "The Four Frogs" presentation of the idea the possible directions of the moves are indicated by lines, to obviate the necessity of the reader's understanding the nature of the knight's move in chess. But it will at once be seen that the two problems are identical. The central square can, of course, be ignored, since no knight can ever enter it. Now, regard the toadstools as buttons and the connecting lines as strings, as in Diagram B. Then by disentangling these strings we can clearly present the diagram in the form shown in Diagram C, where the relationship between the buttons is precisely the same as in B. Any solution on C will be applicable to B, and to A. Place your white knights on 1 and 3 and your black knights on 6 and 8 in the C diagram, and the simplicity of the solution will be very evident. You have simply to move the knights round the circle in one direction or the other. Play over the moves given above, and you will find that every little difficulty has disappeared. [Illustrations: A B C D E] In Diagram D I give another familiar puzzle that first appeared in a book published in Brussels in 1789, _Les Petites Aventures de Jerome Sharp_. Place seven counters on seven of the eight points in the following manner. You must always touch a point that is vacant with a counter, and then move it along a straight line leading from that point to the next vacant point (in either direction), where you deposit the counter. You proceed in the same way until all the counters are placed. Remember you always touch a vacant place and slide the counter from it to the next place, which must be also vacant. Now, by the "buttons and string" method of simplification we can transform the diagram into E. Then the solution becomes obvious. "Always move _to_ the point that you last moved _from_." This is not, of course, the only way of placing the counters, but it is the simplest solution to carry in the mind. There are several puzzles in this book that the reader will find lend themselves readily to this method. 342.--THE MANDARIN'S PUZZLE. The rather perplexing point that the solver has to decide for himself in attacking this puzzle is whether the shaded numbers (those that are shown in their right places) are mere dummies or not. Ninety-nine persons out of a hundred might form the opinion that there can be no advantage in moving any of them, but if so they would be wrong. The shortest solution without moving any shaded number is in thirty-two moves. But the puzzle can be solved in thirty moves. The trick lies in moving the 6, or the 15, on the second move and replacing it on the nineteenth move. Here is the solution: 2, 6, 13, 4, 1, 21, 4, 1, 10, 2, 21, 10, 2, 5, 22, 16, 1, 13, 6, 19, 11, 2, 5, 22, 16, 5, 13, 4, 10, 21. Thirty moves. 343.--EXERCISE FOR PRISONERS. There are eighty different arrangements of the numbers in the form of a perfect knight's path, but only forty of these can be reached without two men ever being in a cell at the same time. Two is the greatest number of men that can be given a complete rest, and though the knight's path can be arranged so as to leave either 7 and 13, 8 and 13, 5 and 7, or 5 and 13 in their original positions, the following four arrangements, in which 7 and 13 are unmoved, are the only ones that can be reached under the moving conditions. It therefore resolves itself into finding the fewest possible moves that will lead up to one of these positions. This is certainly no easy matter, and no rigid rules can be laid down for arriving at the correct answer. It is largely a matter for individual judgment, patient experiment, and a sharp eye for revolutions and position. A +--+--+--+--+ | 6| 1|10|15| +--+--+--+--+ | 9|12| 7| 4| +--+--+--+--+ | 2| 5|14|11| +--+--+--+--+ |13| 8| 3|**| +--+--+--+--+ B +--+--+--+--+ | 6| 1|10|15| +--+--+--+--+ |11|14| 7| 4| +--+--+--+--+ | 2| 5|12| 9| +--+--+--+--+ |13| 8| 3|**| +--+--+--+--+ C +--+--+--+--+ | 6| 9| 4|15| +--+--+--+--+ | 1|12| 7|10| +--+--+--+--+ | 8| 5|14| 3| +--+--+--+--+ |13| 2|11|**| +--+--+--+--+ D +--+--+--+--+ | 6|11| 4|15| +--+--+--+--+ | 1|14| 7|10| +--+--+--+--+ | 8| 5|12| 3| +--+--+--+--+ |13| 2| 9|**| +--+--+--+--+ [Illustration: A, B, C, D] As a matter of fact, the position C can be reached in as few as sixty-six moves in the following manner: 12, 11, 15, 12, 11, 8, 4, 3, 2, 6, 5, 1, 6, 5, 10, 15, 8, 4, 3, 2, 5, 10, 15, 8, 4, 3, 2, 5, 10, 15, 8, 4, 12, 11, 3, 2, 5, 10, 15, 6, 1, 8, 4, 9, 8, 1, 6, 4, 9, 12, 2, 5, 10, 15, 4, 9, 12, 2, 5, 3, 11, 14, 2, 5, 14, 11 = 66 moves. Though this is the shortest that I know of, and I do not think it can be beaten, I cannot state positively that there is not a shorter way yet to be discovered. The most tempting arrangement is certainly A; but things are not what they seem, and C is really the easiest to reach. If the bottom left-hand corner cell might be left vacant, the following is a solution in forty-five moves by Mr. R. Elrick: 15, 11, 10, 9, 13, 14, 11, 10, 7, 8, 4, 3, 8, 6, 9, 7, 12, 4, 6, 9, 5, 13, 7, 5, 13, 1, 2, 13, 5, 7, 1, 2, 13, 8, 3, 6, 9, 12, 7, 11, 14, 1, 11, 14, 1. But every man has moved. 344.--THE KENNEL PUZZLE. The first point is to make a choice of the most promising knight's string and then consider the question of reaching the arrangement in the fewest moves. I am strongly of opinion that the best string is the one represented in the following diagram, in which it will be seen that each successive number is a knight's move from the preceding one, and that five of the dogs (1, 5, 10, 15, and 20) never leave their original kennels. +-----+------+------+------+------+ |1 |2 |3 |4 |5 | | | | | | | | 1 | 18 | 9 | 14 | 5 | | | | | | | +-----+------+------+------+------+ |6 |7 |8 |9 |10 | | | | | | | | 8 | 13 | 4 | 19 | 10 | | | | | | | +-----+------+------+------+------+ |11 |12 |13 |14 |15 | | | | | | | | 17 | 2 | 11 | 6 | 15 | | | | | | | +-----+------+------+------+------+ |16 |17 |18 |19 |20 | | | | | | | | 12 | 7 | 16 | 3 | 20 | | | | | | | +-----+------+------+------+------+ |21 |22 |23 |24 |25 | | | | | | | | | | | | | | | | | | | +-----+------+------+------+------+ [Illustration] This position may be arrived at in as few as forty-six moves, as follows: 16--21, 16--22, 16--23, 17--16, 12--17, 12--22, 12--21,7--12, 7--17, 7--22, 11--12, 11--17, 2--7, 2--12, 6--11, 8--7, 8--6, 13--8, 18--13, 11--18, 2--17, 18--12, 18--7, 18--2, 13--7, 3--8, 3--13, 4--3, 4--8, 9--4, 9--3, 14--9, 14--4, 19--14, 19--9, 3--14, 3--19, 6--12, 6--13, 6--14, 17--11, 12--16, 2--12, 7--17, 11--13, 16--18 = 46 moves. I am, of course, not able to say positively that a solution cannot be discovered in fewer moves, but I believe it will be found a very hard task to reduce the number. 345.--THE TWO PAWNS. Call one pawn A and the other B. Now, owing to that optional first move, either pawn may make either 5 or 6 moves in reaching the eighth square. There are, therefore, four cases to be considered: (1) A 6 moves and B 6 moves; (2) A 6 moves and B 5 moves; (3) A 5 moves and B 6 moves; (4) A 5 moves and B 5 moves. In case (1) there are 12 moves, and we may select any 6 of these for A. Therefore 7×8×9×10×11×12 divided by 1×2×3×4×5×6 gives us the number of variations for this case--that is, 924. Similarly for case (2), 6 selections out of 11 will be 462; in case (3), 5 selections out of 11 will also be 462; and in case (4), 5 selections out of 10 will be 252. Add these four numbers together and we get 2,100, which is the correct number of different ways in which the pawns may advance under the conditions. (See No. 270, on p. 204.) 346.--SETTING THE BOARD. The White pawns may be arranged in 40,320 ways, the White rooks in 2 ways, the bishops in 2 ways, and the knights in 2 ways. Multiply these numbers together, and we find that the White pieces may be placed in 322,560 different ways. The Black pieces may, of course, be placed in the same number of ways. Therefore the men may be set up in 322,560 × 322,560 = 104,044,953,600 ways. But the point that nearly everybody overlooks is that the board may be placed in two different ways for every arrangement. Therefore the answer is doubled, and is 208,089,907,200 different ways. 347.--COUNTING THE RECTANGLES. There are 1,296 different rectangles in all, 204 of which are squares, counting the square board itself as one, and 1,092 rectangles that are not squares. The general formula is that a board of n² squares contains ((n² + n)²)/4 rectangles, of which (2n³ + 3n² + n)/6 are squares and (3n^4 + 2n³ - 3n² - 2n)/12 are rectangles that are not squares. It is curious and interesting that the total number of rectangles is always the square of the triangular number whose side is n. 348.--THE ROOKERY. The answer involves the little point that in the final position the numbered rooks must be in numerical order in the direction contrary to that in which they appear in the original diagram, otherwise it cannot be solved. Play the rooks in the following order of their numbers. As there is never more than one square to which a rook can move (except on the final move), the notation is obvious--5, 6, 7, 5, 6, 4, 3, 6, 4, 7, 5, 4, 7, 3, 6, 7, 3, 5, 4, 3, 1, 8, 3, 4, 5, 6, 7, 1, 8, 2, 1, and rook takes bishop, checkmate. These are the fewest possible moves--thirty-two. The Black king's moves are all forced, and need not be given. 349.--STALEMATE. Working independently, the same position was arrived at by Messrs. S. Loyd, E.N. Frankenstein, W.H. Thompson, and myself. So the following may be accepted as the best solution possible to this curious problem :-- White. Black. 1. P--Q4 1. P--K4 2. Q--Q3 2. Q--R5 3. Q--KKt3 3. B--Kt5 ch 4. Kt--Q2 4. P--QR4 5. P--R4 5. P--Q3 6. P--R3 6. B--K3 7. R--R3 7. P--KB4 8. Q--R2 8. P--B4 9. R--KKt3 9. B--Kt6 10. P--QB4 10. P--B5 11. P--B3 11. P--K5 12. P--Q5 12. P--K6 And White is stalemated. We give a diagram of the curious position arrived at. It will be seen that not one of White's pieces may be moved. [Illustration] +-+-+-+-+-+-+-+-+ |r|n| | |k| |n|r| +-+-+-+-+-+-+-+-+ | |p| | | | |p|p| +-+-+-+-+-+-+-+-+ | | | |p| | | | | +-+-+-+-+-+-+-+-+ |p| |p|P| | | | | +-+-+-+-+-+-+-+-+ |P|b|P| | |p| |q| +-+-+-+-+-+-+-+-+ | |b| | |p|P|R|P| +-+-+-+-+-+-+-+-+ | |P| |N|P| |P|Q| +-+-+-+-+-+-+-+-+ | | |B| |K|B|N|R| +-+-+-+-+-+-+-+-+ 350.--THE FORSAKEN KING. Play as follows:-- White. Black. 1. P to K 4th 1. Any move 2. Q to Kt 4th 2. Any move except on KB file (a) 3. Q to Kt 7th 3. K moves to royal row 4. B to Kt 5th 4. Any move 5. Mate in two moves If 3. K other than to royal row 4. P to Q 4th 4. Any move 5. Mate in two moves (a) If 2. Any move on KB file 3. Q to Q 7th 3. K moves to royal row 4. P to Q Kt 3rd 4. Any move 5. Mate in two moves If 3. K other than to royal row 4. P to Q 4th 4. Any move 5. Mate in two moves Of course, by "royal row" is meant the row on which the king originally stands at the beginning of a game. Though, if Black plays badly, he may, in certain positions, be mated in fewer moves, the above provides for every variation he can possibly bring about. 351.--THE CRUSADER. White. Black. 1. Kt to QB 3rd 1. P to Q 4th 2. Kt takes QP 2. Kt to QB 3rd 3. Kt takes KP 3. P to KKt 4th 4. Kt takes B 4. Kt to KB 3rd 5. Kt takes P 5. Kt to K 5th 6. Kt takes Kt 6. Kt to B 6th 7. Kt takes Q 7. R to KKt sq 8. Kt takes BP 8. R to KKt 3rd 9. Kt takes P 9. R to K 3rd 10. Kt takes P 10. Kt to Kt 8th 11. Kt takes B 11. R to R 6th 12. Kt takes R 12. P to Kt 4th 13. Kt takes P (ch) 13. K to B 2nd 14. Kt takes P 14. K to Kt 3rd 15. Kt takes R 15. K to R 4th 16. Kt takes Kt 16. K to R 5th White now mates in three moves. 17. P to Q 4th 17. K to R 4th 18. Q to Q 3rd 18. K moves 19. Q to KR 3rd (mate) If 17. K to Kt 5th 18. P to K 4th (dis. ch) 18. K moves 19. P to KKt 3rd (mate) The position after the sixteenth move, with the mate in three moves, was first given by S. Loyd in _Chess Nuts_. 352.--IMMOVABLE PAWNS. 1. Kt to KB 3 2. Kt to KR 4 3. Kt to Kt 6 4. Kt takes R 5. Kt to Kt 6 6. Kt takes B 7. K takes Kt 8. Kt to QB 3 9. Kt to R 4 10. Kt to Kt 6 11. Kt takes R 12. Kt to Kt 6 13. Kt takes B 14. Kt to Q 6 15. Q to K sq 16. Kt takes Q 17. K takes Kt, and the position is reached. Black plays precisely the same moves as White, and therefore we give one set of moves only. The above seventeen moves are the fewest possible. 353.--THIRTY-SIX MATES. Place the remaining eight White pieces thus: K at KB 4th, Q at QKt 6th, R at Q 6th, R at KKt 7th, B at Q 5th, B at KR 8th, Kt at QR 5th, and Kt at QB 5th. The following mates can then be given:-- By discovery from Q 8 By discovery from R at Q 6th 13 By discovery from B at R 8th 11 Given by Kt at R 5th 2 Given by pawns 2 -- Total 36 Is it possible to construct a position in which more than thirty-six different mates on the move can be given? So far as I know, nobody has yet beaten my arrangement. 354.--AN AMAZING DILEMMA. Mr Black left his king on his queen's knight's 7th, and no matter what piece White chooses for his pawn, Black cannot be checkmated. As we said, the Black king takes no notice of checks and never moves. White may queen his pawn, capture the Black rook, and bring his three pieces up to the attack, but mate is quite impossible. The Black king cannot be left on any other square without a checkmate being possible. The late Sam Loyd first pointed out the peculiarity on which this puzzle is based. 355.--CHECKMATE! Remove the White pawn from B 6th to K 4th and place a Black pawn on Black's KB 2nd. Now, White plays P to K 5th, check, and Black must play P to B 4th. Then White plays P takes P _en passant_, checkmate. This was therefore White's last move, and leaves the position given. It is the only possible solution. 356.--QUEER CHESS. +-+-+-+-+-+-+-+-+ | | | | | | | | | +-+-+-+-+-+-+-+-+ | | |R|k|R|N| | | +-+-+-+-+-+-+-+-+ | | | | | | | | | +-+-+-+-+-+-+-+-+ If you place the pieces as follows (where only a portion of the board is given, to save space), the Black king is in check, with no possible move open to him. The reader will now see why I avoided the term "checkmate," apart from the fact that there is no White king. The position is impossible in the game of chess, because Black could not be given check by both rooks at the same time, nor could he have moved into check on his last move. I believe the position was first published by the late S. Loyd. 357.--ANCIENT CHINESE PUZZLE. Play as follows:-- 1. R--Q 6 2. K--R 7 3. R (R 6)--B 6 (mate). Black's moves are forced, so need not be given. 358.--THE SIX PAWNS. The general formula for six pawns on all squares greater than 2² is this: Six times the square of the number of combinations of n things taken three at a time, where n represents the number of squares on the side of the board. Of course, where n is even the unoccupied squares in the rows and columns will be even, and where n is odd the number of squares will be odd. Here n is 8, so the answer is 18,816 different ways. This is "The Dyer's Puzzle" (_Canterbury Puzzles_, No. 27) in another form. I repeat it here in order to explain a method of solving that will be readily grasped by the novice. First of all, it is evident that if we put a pawn on any line, we must put a second one in that line in order that the remainder may be even in number. We cannot put four or six in any row without making it impossible to get an even number in all the columns interfered with. We have, therefore, to put two pawns in each of three rows and in each of three columns. Now, there are just six schemes or arrangements that fulfil these conditions, and these are shown in Diagrams A to F, inclusive, on next page. [Illustration] I will just remark in passing that A and B are the only distinctive arrangements, because, if you give A a quarter-turn, you get F; and if you give B three quarter-turns in the direction that a clock hand moves, you will get successively C, D, and E. No matter how you may place your six pawns, if you have complied with the conditions of the puzzle they will fall under one of these arrangements. Of course it will be understood that mere expansions do not destroy the essential character of the arrangements. Thus G is only an expansion of form A. The solution therefore consists in finding the number of these expansions. Supposing we confine our operations to the first three rows, as in G, then with the pairs a and b placed in the first and second columns the pair c may be disposed in any one of the remaining six columns, and so give six solutions. Now slide pair b into the third column, and there are five possible positions for c. Slide b into the fourth column, and c may produce four new solutions. And so on, until (still leaving a in the first column) you have b in the seventh column, and there is only one place for c--in the eighth column. Then you may put a in the second column, b in the third, and c in the fourth, and start sliding c and b as before for another series of solutions. We find thus that, by using form A alone and confining our operations to the three top rows, we get as many answers as there are combinations of 8 things taken 3 at a time. This is (8 × 7 × 6)/(1 × 2 × 3) = 56. And it will at once strike the reader that if there are 56 different ways of electing the columns, there must be for each of these ways just 56 ways of selecting the rows, for we may simultaneously work that "sliding" process downwards to the very bottom in exactly the same way as we have worked from left to right. Therefore the total number of ways in which form A may be applied is 56 × 6 = 3,136. But there are, as we have seen, six arrangements, and we have only dealt with one of these, A. We must, therefore, multiply this result by 6, which gives us 3,136 × 6 = 18,816, which is the total number of ways, as we have already stated. 359.--COUNTER SOLITAIRE. Play as follows: 3--11, 9--10, 1--2, 7--15, 8--16, 8--7, 5--13, 1--4, 8--5, 6--14, 3--8, 6--3, 6--12, 1--6, 1--9, and all the counters will have been removed, with the exception of No. 1, as required by the conditions. 360.--CHESSBOARD SOLITAIRE. Play as follows: 7--15, 8--16, 8--7, 2--10, 1--9, 1--2, 5--13, 3--4, 6--3, 11--1, 14--8, 6--12, 5--6, 5--11, 31--23, 32--24, 32--31, 26--18, 25--17, 25--26, 22--32, 14--22, 29--21, 14--29, 27--28, 30--27, 25--14, 30--20, 25--30, 25--5. The two counters left on the board are 25 and 19--both belonging to the same group, as stipulated--and 19 has never been moved from its original place. I do not think any solution is possible in which only one counter is left on the board. 361.--THE MONSTROSITY. White Black, 1. P to KB 4 P to QB 3 2. K to B 2 Q to R 4 3. K to K 3 K to Q sq 4. P to B 5 K to B 2 5. Q to K sq K to Kt 3 6. Q to Kt 3 Kt to QR 3 7. Q to Kt 8 P to KR 4 8. Kt to KB 3 R to R 3 9. Kt to K 5 R to Kt 3 10. Q takes B R to Kt 6, ch 11. P takes R K to Kt 4 12. R to R 4 P to B 3 13. R to Q 4 P takes Kt 14. P to QKt 4 P takes R, ch 15. K to B 4 P to R 5 16. Q to K 8 P to R 6 17. Kt to B 3, ch P takes Kt 18. B to R 3 P to R 7 19. R to Kt sq P to R 8 (Q) 20. R to Kt 2 P takes R 21. K to Kt 5 Q to KKt 8 22. Q to R 5 K to R 5 23. P to Kt 5 R to B sq 24. P to Kt 6 R to B 2 25. P takes R P to Kt 8 (B) 26. P to B 8 (R) Q to B 2 27. B to Q 6 Kt to Kt 5 28. K to Kt 6 K to R 6 29. R to R 8 K to Kt 7 30. P to R 4 Q (Kt 8) to Kt 3 31. P to R 5 K to B 8 32. P takes Q K to Q 8 33. P takes Q K to K 8 34. K to B 7 Kt to KR 3, ch 35. K to K 8 B to R 7 36. P to B 6 B to Kt sq 37. P to B 7 K takes B 38. P to B 8 (B) Kt to Q 4 39. B to Kt 8 Kt to B 3, ch 40. K to Q 8 Kt to K sq 41. P takes Kt (R) Kt to B 2, ch 42. K to B 7 Kt to Q sq 43. Q to B 7, ch K to Kt 8 And the position is reached. The order of the moves is immaterial, and this order may be greatly varied. But, although many attempts have been made, nobody has succeeded in reducing the number of my moves. 362.--THE WASSAIL BOWL. The division of the twelve pints of ale can be made in eleven manipulations, as below. The six columns show at a glance the quantity of ale in the barrel, the five-pint jug, the three-pint jug, and the tramps X, Y, and Z respectively after each manipulation. Barrel. 5-pint. 3-pint. X. Y. Z. 7 .. 5 .. 0 .. 0 .. 0 .. 0 7 .. 2 .. 3 .. 0 .. 0 .. 0 7 .. 0 .. 3 .. 2 .. 0 .. 0 7 .. 3 .. 0 .. 2 .. 0 .. 0 4 .. 3 .. 3 .. 2 .. 0 .. 0 0 .. 3 .. 3 .. 2 .. 4 .. 0 0 .. 5 .. 1 .. 2 .. 4 .. 0 0 .. 5 .. 0 .. 2 .. 4 .. 1 0 .. 2 .. 3 .. 2 .. 4 .. 1 0 .. 0 .. 3 .. 4 .. 4 .. 1 0 .. 0 .. 0 .. 4 .. 4 .. 4 And each man has received his four pints of ale. 363.--THE DOCTOR'S QUERY. The mixture of spirits of wine and water is in the proportion of 40 to 1, just as in the other bottle it was in the proportion of 1 to 40. 364.--THE BARREL PUZZLE. [Illustration: Figs. 1, 2, and 3] All that is necessary is to tilt the barrel as in Fig. 1, and if the edge of the surface of the water exactly touches the lip a at the same time that it touches the edge of the bottom b, it will be just half full. To be more exact, if the bottom is an inch or so from the ground, then we can allow for that, and the thickness of the bottom, at the top. If when the surface of the water reached the lip a it had risen to the point c in Fig. 2, then it would be more than half full. If, as in Fig. 3, some portion of the bottom were visible and the level of the water fell to the point d, then it would be less than half full. This method applies to all symmetrically constructed vessels. 365.--NEW MEASURING PUZZLE. The following solution in eleven manipulations shows the contents of every vessel at the start and after every manipulation:-- 10-quart. 10-quart. 5-quart. 4-quart. 10 .. 10 .. 0 .. 0 5 .. 10 .. 5 .. 0 5 .. 10 .. 1 .. 4 9 .. 10 .. 1 .. 0 9 .. 6 .. 1 .. 4 9 .. 7 .. 0 .. 4 9 .. 7 .. 4 .. 0 9 .. 3 .. 4 .. 4 9 .. 3 .. 5 .. 3 9 .. 8 .. 0 .. 3 4 .. 8 .. 5 .. 3 4 .. 10 .. 3 .. 3 366.--THE HONEST DAIRYMAN. Whatever the respective quantities of milk and water, the relative proportion sent to London would always be three parts of water to one of milk. But there are one or two points to be observed. There must originally be more water than milk, or there will be no water in A to double in the second transaction. And the water must not be more than three times the quantity of milk, or there will not be enough liquid in B to effect the second transaction. The third transaction has no effect on A, as the relative proportions in it must be the same as after the second transaction. It was introduced to prevent a quibble if the quantity of milk and water were originally the same; for though double "nothing" would be "nothing," yet the third transaction in such a case could not take place. 367.--WINE AND WATER. The wine in small glass was one-sixth of the total liquid, and the wine in large glass two-ninths of total. Add these together, and we find that the wine was seven-eighteenths of total fluid, and therefore the water eleven-eighteenths. 368.--THE KEG OF WINE. The capacity of the jug must have been a little less than three gallons. To be more exact, it was 2.93 gallons. 369.--MIXING THE TEA. There are three ways of mixing the teas. Taking them in the order of quality, 2s. 6d., 2s. 3d., 1s. 9p., mix 16 lbs., 1 lb., 3 lbs.; or 14 lbs., 4 lbs., 2 lbs.; or 12 lbs., 7 lbs., 1 lb. In every case the twenty pounds mixture should be worth 2s. 4½d. per pound; but the last case requires the smallest quantity of the best tea, therefore it is the correct answer. 370.--A PACKING PUZZLE. On the side of the box, 14 by 22+4/5, we can arrange 13 rows containing alternately 7 and 6 balls, or 85 in all. Above this we can place another layer consisting of 12 rows of 7 and 6 alternately, or a total of 78. In the length of 24+9/10 inches 15 such layers may be packed, the alternate layers containing 85 and 78 balls. Thus 8 times 85 added to 7 times 78 gives us 1,226 for the full contents of the box. 371.--GOLD PACKING IN RUSSIA. The box should be 100 inches by 100 inches by 11 inches deep, internal dimensions. We can lay flat at the bottom a row of eight slabs, lengthways, end to end, which will just fill one side, and nine of these rows will dispose of seventy-two slabs (all on the bottom), with a space left over on the bottom measuring 100 inches by 1 inch by 1 inch. Now make eleven depths of such seventy-two slabs, and we have packed 792, and have a space 100 inches by 1 inch by 11 inches deep. In this we may exactly pack the remaining eight slabs on edge, end to end. 372.--THE BARRELS OF HONEY. The only way in which the barrels could be equally divided among the three brothers, so that each should receive his 3½ barrels of honey and his 7 barrels, is as follows:-- Full. Half-full. Empty. A 3 1 3 B 2 3 2 C 2 3 2 There is one other way in which the division could be made, were it not for the objection that all the brothers made to taking more than four barrels of the same description. Except for this difficulty, they might have given B his quantity in exactly the same way as A above, and then have left C one full barrel, five half-full barrels, and one empty barrel. It will thus be seen that in any case two brothers would have to receive their allowance in the same way. 373.--CROSSING THE STREAM. First, the two sons cross, and one returns Then the man crosses and the other son returns. Then both sons cross and one returns. Then the lady crosses and the other son returns Then the two sons cross and one of them returns for the dog. Eleven crossings in all. It would appear that no general rule can be given for solving these river-crossing puzzles. A formula can be found for a particular case (say on No. 375 or 376) that would apply to any number of individuals under the restricted conditions; but it is not of much use, for some little added stipulation will entirely upset it. As in the case of the measuring puzzles, we generally have to rely on individual ingenuity. 374.--CROSSING THE RIVER AXE. Here is the solution:-- | {J 5) | G T8 3 5 | ( J } | G T8 3 5 | {G 3) | JT8 53 | ( G } | JT8 53 | {J T) | G 8 J 5 | (T 3} | G 8 J 5 | {G 8) | T 3 G 8 | (J 5} | T G 8 | {J T) | 53 JT8 | ( G } | 53 JT8 | {G 3) | 5 G T8 3 | ( J } | 5 G T8 3 | {J 5) | G, J, and T stand for Giles, Jasper, and Timothy; and 8, 5, 3, for £800, £500, and £300 respectively. The two side columns represent the left bank and the right bank, and the middle column the river. Thirteen crossings are necessary, and each line shows the position when the boat is in mid-stream during a crossing, the point of the bracket indicating the direction. It will be found that not only is no person left alone on the land or in the boat with more than his share of the spoil, but that also no two persons are left with more than their joint shares, though this last point was not insisted upon in the conditions. 375.--FIVE JEALOUS HUSBANDS. It is obvious that there must be an odd number of crossings, and that if the five husbands had not been jealous of one another the party might have all got over in nine crossings. But no wife was to be in the company of a man or men unless her husband was present. This entails two more crossings, eleven in all. The following shows how it might have been done. The capital letters stand for the husbands, and the small letters for their respective wives. The position of affairs is shown at the start, and after each crossing between the left bank and the right, and the boat is represented by the asterisk. So you can see at a glance that a, b, and c went over at the first crossing, that b and c returned at the second crossing, and so on. ABCDE abcde *|..| | | 1. ABCDE de |..|* abc 2. ABCDE bcde *|..| a 3. ABCDE e |..|* abcd 4. ABCDE de *|..| abc 5. DE de |,,|* ABC abc 6. CDE cde *|..| AB ab 7. cde |..|* ABCDE ab 8. bcde *|..| ABCDE a 9. e |..|* ABCDE abcd 10. bc e *|..| ABCDE a d 11. |..|* ABCDE abcde There is a little subtlety concealed in the words "show the _quickest_ way." Everybody correctly assumes that, as we are told nothing of the rowing capabilities of the party, we must take it that they all row equally well. But it is obvious that two such persons should row more quickly than one. Therefore in the second and third crossings two of the ladies should take back the boat to fetch d, not one of them only. This does not affect the number of landings, so no time is lost on that account. A similar opportunity occurs in crossings 10 and 11, where the party again had the option of sending over two ladies or one only. To those who think they have solved the puzzle in nine crossings I would say that in every case they will find that they are wrong. No such jealous husband would, in the circumstances, send his wife over to the other bank to a man or men, even if she assured him that she was coming back next time in the boat. If readers will have this fact in mind, they will at once discover their errors. 376.--THE FOUR ELOPEMENTS. If there had been only three couples, the island might have been dispensed with, but with four or more couples it is absolutely necessary in order to cross under the conditions laid down. It can be done in seventeen passages from land to land (though French mathematicians have declared in their books that in such circumstances twenty-four are needed), and it cannot be done in fewer. I will give one way. A, B, C, and D are the young men, and a, b, c, and d are the girls to whom they are respectively engaged. The three columns show the positions of the different individuals on the lawn, the island, and the opposite shore before starting and after each passage, while the asterisk indicates the position of the boat on every occasion. Lawn. | Island. | Shore. | | ABCDabcd * | | ABCD cd | | ab * ABCD bcd * | | a ABCD d | bc * | a ABCD cd * | b | a CD cd | b | AB a * BCD cd * | b | A a BCD | bcd * | A a BCD d * | bc | A a D d | bc | ABC a * D d | abc * | ABC D d | b | ABC a c * B D d * | b | A C a c d | b | ABCD a c * d | bc * | ABCD a d | | ABCD abc * cd * | | ABCD ab | | ABCD abcd * Having found the fewest possible passages, we should consider two other points in deciding on the "quickest method": Which persons were the most expert in handling the oars, and which method entails the fewest possible delays in getting in and out of the boat? We have no data upon which to decide the first point, though it is probable that, as the boat belonged to the girls' household, they would be capable oarswomen. The other point, however, is important, and in the solution I have given (where the girls do 8-13ths of the rowing and A and D need not row at all) there are only sixteen gettings-in and sixteen gettings-out. A man and a girl are never in the boat together, and no man ever lands on the island. There are other methods that require several more exchanges of places. 377.--STEALING THE CASTLE TREASURE. Here is the best answer, in eleven manipulations:-- Treasure down. Boy down--treasure up. Youth down--boy up. Treasure down. Man down--youth and treasure up. Treasure down. Boy down--treasure up. Treasure down. Youth down--boy up. Boy down--treasure up. Treasure down. 378.--DOMINOES IN PROGRESSION. There are twenty-three different ways. You may start with any domino, except the 4--4 and those that bear a 5 or 6, though only certain initial dominoes may be played either way round. If you are given the common difference and the first domino is played, you have no option as to the other dominoes. Therefore all I need do is to give the initial domino for all the twenty-three ways, and state the common difference. This I will do as follows:-- With a common difference of 1, the first domino may be either of these: 0--0, 0--1, 1--0, 0--2, 1--1, 2--0, 0--3, 1--2, 2--1, 3--0, 0--4, 1--3, 2--2, 3--1, 1--4, 2--3, 3--2, 2--4, 3--3, 3--4. With a difference of 2, the first domino may be 0--0, 0--2, or 0--1. Take the last case of all as an example. Having played the 0--1, and the difference being 2, we are compelled to continue with 1--2, 2--3, 3--4. 4--5, 5--6. There are three dominoes that can never be used at all. These are 0--5, 0--6, and 1--6. If we used a box of dominoes extending to 9--9, there would be forty different ways. 379.--THE FIVE DOMINOES. There are just ten different ways of arranging the dominoes. Here is one of them:-- (2--0) (0--0) (0--1) (1--4) (4--0). I will leave my readers to find the remaining nine for themselves. 380.--THE DOMINO FRAME PUZZLE. [Illustration: +---+-------+-------+-------+-------+-------+-------+-------+ | 2 | 2 | 5 | 5 | 6 | 6 | 6 | 6 | 1 | 1 | | | | | 4 | | - +-------+-------+-------+-------+-------+-------+---+---+ | 2 | | 4 | +---+ | - | | 2 | | 3 | | - | +---+ | 6 | | 3 | +---+ T H E | - | | 6 | | 3 | | - | +---+ | 3 | | 3 | +---+ | - | | 3 | | 1 | | - | D O M I N O F R A M E +---+ | | | 1 | +---+ | - | | | | 1 | | - | +---+ | 5 | | 1 | +---+ -S-O-L-U-T-I-O-N- | - | | 5 | | 4 | | - | +---+ | 3 | | 4 | +---+ | - | | 3 | | 6 | | - | +---+ | 2 | | 6 | +---+---+-------+-------+-------+-------+-------+-------+ - | | 2 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 4 | 4 | 4 | 2 | 2 | | | +-------+-------+-------+-------+-------+-------+-------+---+ ] The illustration is a solution. It will be found that all four sides of the frame add up 44. The sum of the pips on all the dominoes is 168, and if we wish to make the sides sum to 44, we must take care that the four corners sum to 8, because these corners are counted twice, and 168 added to 8 will equal 4 times 44, which is necessary. There are many different solutions. Even in the example given certain interchanges are possible to produce different arrangements. For example, on the left-hand side the string of dominoes from 2--2 down to 3--2 may be reversed, or from 2--6 to 3--2, or from 3--0 to 5--3. Also, on the right-hand side we may reverse from 4--3 to 1--4. These changes will not affect the correctness of the solution. 381.--THE CARD FRAME PUZZLE. The sum of all the pips on the ten cards is 55. Suppose we are trying to get 14 pips on every side. Then 4 times 14 is 56. But each of the four corner cards is added in twice, so that 55 deducted from 56, or 1, must represent the sum of the four corner cards. This is clearly impossible; therefore 14 is also impossible. But suppose we came to trying 18. Then 4 times 18 is 72, and if we deduct 55 we get 17 as the sum of the corners. We need then only try different arrangements with the four corners always summing to 17, and we soon discover the following solution:-- [Illustration: +-------+-------+-------+ | 2 | 10 | 6 | +---+---+------ +---+---+ | | | | | 3 | | 7 | | | | | +---+ +---+ | | | | | 8 | | 1 | | | | | +---+---+-------+--+----+ | 5 | 9 | 4 | +-------+-------+-------+ ] The final trials are very limited in number, and must with a little judgment either bring us to a correct solution or satisfy us that a solution is impossible under the conditions we are attempting. The two centre cards on the upright sides can, of course, always be interchanged, but I do not call these different solutions. If you reflect in a mirror you get another arrangement, which also is not considered different. In the answer given, however, we may exchange the 5 with the 8 and the 4 with the 1. This is a different solution. There are two solutions with 18, four with 19, two with 20, and two with 22--ten arrangements in all. Readers may like to find all these for themselves. 382.--THE CROSS OF CARDS. There are eighteen fundamental arrangements, as follows, where I only give the numbers in the horizontal bar, since the remainder must naturally fall into their places. 5 6 1 7 4 2 4 5 6 8 3 5 1 6 8 3 4 5 6 7 3 4 1 7 8 1 4 7 6 8 2 5 1 7 8 2 3 7 6 8 2 5 3 6 8 2 4 7 5 8 1 5 3 7 8 3 4 9 5 6 2 4 3 7 8 2 4 9 5 7 1 4 5 7 8 1 4 9 6 7 2 3 5 7 8 2 3 9 6 7 It will be noticed that there must always be an odd number in the centre, that there are four ways each of adding up 23, 25, and 27, but only three ways each of summing to 24 and 26. 383.--THE "T" CARD PUZZLE. If we remove the ace, the remaining cards may he divided into two groups (each adding up alike) in four ways; if we remove 3, there are three ways; if 5, there are four ways; if 7, there are three ways; and if we remove 9, there are four ways of making two equal groups. There are thus eighteen different ways of grouping, and if we take any one of these and keep the odd card (that I have called "removed") at the head of the column, then one set of numbers can be varied in order in twenty-four ways in the column and the other four twenty-four ways in the horizontal, or together they may be varied in 24 × 24 = 576 ways. And as there are eighteen such cases, we multiply this number by 18 and get 10,368, the correct number of ways of placing the cards. As this number includes the reflections, we must divide by 2, but we have also to remember that every horizontal row can change places with a vertical row, necessitating our multiplying by 2; so one operation cancels the other. 384.--CARD TRIANGLES. The following arrangements of the cards show (1) the smallest possible sum, 17; and (2) the largest possible, 23. 1 7 9 6 4 2 4 8 3 6 3 7 5 2 9 5 1 8 It will be seen that the two cards in the middle of any side may always be interchanged without affecting the conditions. Thus there are eight ways of presenting every fundamental arrangement. The number of fundamentals is eighteen, as follows: two summing to 17, four summing to 19, six summing to 20, four summing to 21, and two summing to 23. These eighteen fundamentals, multiplied by eight (for the reason stated above), give 144 as the total number of different ways of placing the cards. 385.--"STRAND" PATIENCE. The reader may find a solution quite easy in a little over 200 moves, but, surprising as it may at first appear, not more than 62 moves are required. Here is the play: By "4 C up" I mean a transfer of the 4 of clubs with all the cards that rest on it. 1 D on space, 2 S on space, 3 D on space, 2 S on 3 D, 1 H on 2 S, 2 C on space, 1 D on 2 C, 4 S on space, 3 H on 4 S (9 moves so far), 2 S up on 3 H (3 moves), 5 H and 5 D exchanged, and 4 C on 5 D (6 moves), 3 D on 4 C (1), 6 S (with 5 H) on space (3), 4 C up on 5 H (3), 2 C up on 3 D (3), 7 D on space (1), 6 C up on 7 D (3), 8 S on space (1), 7 H on 8 S (1), 8 C on 9 D (1), 7 H on 8 C (1), 8 S on 9 H (1), 7 H on 8 S (1), 7 D up on 8 C (5), 4 C up on 5 D (9), 6 S up on 7 H (3), 4 S up on 5 H (7) = 62 moves in all. This is my record; perhaps the reader can beat it. 386.--A TRICK WITH DICE. All you have to do is to deduct 250 from the result given, and the three figures in the answer will be the three points thrown with the dice. Thus, in the throw we gave, the number given would be 386; and when we deduct 250 we get 136, from which we know that the throws were 1, 3, and 6. The process merely consists in giving 100a + 10b + c + 250, where a, b, and c represent the three throws. The result is obvious. 387.--THE VILLAGE CRICKET MATCH. [Illustration: | Mr. Dumkins >>--> |------------------------> | | <------------------- | | -------------------> | 1 |<----------------------- | | | | <------------------------| | -------------------> | | <------------------- | | ----------------------->| | <--<< Mr. Podder | | Mr. Luffey >>--> |------------------------> | | <------------------- | | ----------------------->| 2 | | |<----------------------- | | -------------------> | | <------------------------| <--<< Mr. Struggles | ] The diagram No. 1 will show that as neither Mr. Podder nor Mr. Dumkins can ever have been within the crease opposite to that from which he started, Mr. Dumkins would score nothing by his performance. Diagram No. 2 will, however, make it clear that since Mr. Luffey and Mr. Struggles have, notwithstanding their energetic but careless movements, contrived to change places, the manoeuvre must increase Mr. Struggles's total by one run. 388.--SLOW CRICKET. The captain must have been "not out" and scored 21. Thus:-- 2 men (each lbw) 19 4 men (each caught) 17 1 man (run out) 0 3 men (each bowled) 9 1 man (captain--not out) 21 -- -- 11 66 The captain thus scored exactly 15 more than the average of the team. The "others" who were bowled could only refer to three men, as the eleventh man would be "not out." The reader can discover for himself why the captain must have been that eleventh man. It would not necessarily follow with any figures. 389.--THE FOOTBALL PLAYERS. The smallest possible number of men is seven. They could be accounted for in three different ways: 1. Two with both arms sound, one with broken right arm, and four with both arms broken. 2. One with both arms sound, one with broken left arm, two with broken right arm, and three with both arms broken. 3. Two with left arm broken, three with right arm broken, and two with both arms broken. But if every man was injured, the last case is the only one that would apply. 390.--THE HORSE-RACE PUZZLE. The answer is: £12 on Acorn, £15 on Bluebottle, £20 on Capsule. 391.--THE MOTOR-CAR RACE. The first point is to appreciate the fact that, in a race round a circular track, there are the same number of cars behind one as there are before. All the others are both behind and before. There were thirteen cars in the race, including Gogglesmith's car. Then one-third of twelve added to three-quarters of twelve will give us thirteen--the correct answer. 392.--THE PEBBLE GAME. In the case of fifteen pebbles, the first player wins if he first takes two. Then when he holds an odd number and leaves 1, 8, or 9 he wins, and when he holds an even number and leaves 4, 5, or 12 he also wins. He can always do one or other of these things until the end of the game, and so defeat his opponent. In the case of thirteen pebbles the first player must lose if his opponent plays correctly. In fact, the only numbers with which the first player ought to lose are 5 and multiples of 8 added to 5, such as 13, 21, 29, etc. 393.--THE TWO ROOKS. The second player can always win, but to ensure his doing so he must always place his rook, at the start and on every subsequent move, on the same diagonal as his opponent's rook. He can then force his opponent into a corner and win. Supposing the diagram to represent the positions of the rooks at the start, then, if Black played first, White might have placed his rook at A and won next move. Any square on that diagonal from A to H will win, but the best play is always to restrict the moves of the opposing rook as much as possible. If White played first, then Black should have placed his rook at B (F would not be so good, as it gives White more scope); then if White goes to C, Black moves to D; White to E, Black to F; White to G, Black to C; White to H, Black to I; and Black must win next move. If at any time Black had failed to move on to the same diagonal as White, then White could take Black's diagonal and win. r: black rook R: white rook +-+-+-+-+-+-+-+-+ |r| | | | | | | | +-+-+-+-+-+-+-+-+ | |A| | | | | | | +-+-+-+-+-+-+-+-+ | | | | | | | | | +-+-+-+-+-+-+-+-+ | | | | | | | | | +-+-+-+-+-+-+-+-+ | | | | |B|D|F| | +-+-+-+-+-+-+-+-+ | | | | | |R|C|E| +-+-+-+-+-+-+-+-+ | | | | | | |I|G| +-+-+-+-+-+-+-+-+ | | | | | | | |H| +-+-+-+-+-+-+-+-+ THE TWO ROOKS. 394.--PUSS IN THE CORNER. No matter whether he plays first or second, the player A, who starts the game at 55, must win. Assuming that B adopts the very best lines of play in order to prolong as much as possible his existence, A, if he has first move, can always on his 12th move capture B; and if he has the second move, A can always on his 14th move make the capture. His point is always to get diagonally in line with his opponent, and by going to 33, if he has first move, he prevents B getting diagonally in line with himself. Here are two good games. The number in front of the hyphen is always A's move; that after the hyphen is B's:-- 33-8, 32-15, 31-22, 30-21, 29-14, 22-7, 15-6, 14-2, 7-3, 6-4, 11-, and A must capture on his next (12th) move, -13, 54-20, 53-27, 52-34, 51-41, 50-34, 42-27, 35-20, 28-13, 21-6, 14-2, 7-3, 6-4, 11-, and A must capture on his next (14th) move. 395.--A WAR PUZZLE GAME. The Britisher can always catch the enemy, no matter how clever and elusive that astute individual may be; but curious though it may seem, the British general can only do so after he has paid a somewhat mysterious visit to the particular town marked "1" in the map, going in by 3 and leaving by 2, or entering by 2 and leaving by 3. The three towns that are shaded and have no numbers do not really come into the question, as some may suppose, for the simple reason that the Britisher never needs to enter any one of them, while the enemy cannot be forced to go into them, and would be clearly ill-advised to do so voluntarily. We may therefore leave these out of consideration altogether. No matter what the enemy may do, the Britisher should make the following first nine moves: He should visit towns 24, 20, 19, 15, 11, 7, 3, 1, 2. If the enemy takes it into his head also to go to town 1, it will be found that he will have to beat a precipitate retreat _the same way that he went in_, or the Britisher will infallibly catch him in towns 2 or 3, as the case may be. So the enemy will be wise to avoid that north-west corner of the map altogether. [Illustration] Now, when the British general has made the nine moves that I have given, the enemy will be, after his own ninth move, in one of the towns marked 5, 8, 11, 13, 14, 16, 19, 21, 24, or 27. Of course, if he imprudently goes to 3 or 6 at this point he will be caught at once. Wherever he may happen to be, the Britisher "goes for him," and has no longer any difficulty in catching him in eight more moves at most (seventeen in all) in one of the following ways. The Britisher will get to 8 when the enemy is at 5, and win next move; or he will get to 19 when the enemy is at 22, and win next move; or he will get to 24 when the enemy is at 27, and so win next move. It will be found that he can be forced into one or other of these fatal positions. In short, the strategy really amounts to this: the Britisher plays the first nine moves that I have given, and although the enemy does his very best to escape, our general goes after his antagonist and always driving him away from that north-west corner ultimately closes in with him, and wins. As I have said, the Britisher never need make more than seventeen moves in all, and may win in fewer moves if the enemy plays badly. But after playing those first nine moves it does not matter even if the Britisher makes a few bad ones. He may lose time, but cannot lose his advantage so long as he now keeps the enemy from town 1, and must eventually catch him. This is a complete explanation of the puzzle. It may seem a little complex in print, but in practice the winning play will now be quite easy to the reader. Make those nine moves, and there ought to be no difficulty whatever in finding the concluding line of play. Indeed, it might almost be said that then it is difficult for the British general _not_ to catch the enemy. It is a question of what in chess we call the "opposition," and the visit by the Britisher to town 1 "gives him the jump" on the enemy, as the man in the street would say. Here is an illustrative example in which the enemy avoids capture as long as it is possible for him to do so. The Britisher's moves are above the line and the enemy's below it. Play them alternately. 24 20 19 15 11 7 3 1 2 6 10 14 18 19 20 24 ----------------------------------------------- 13 9 13 17 21 20 24 23 19 15 19 23 24 25 27 The enemy must now go to 25 or B, in either of which towns he is immediately captured. 396.--A MATCH MYSTERY. If you form the three heaps (and are therefore the second to draw), any one of the following thirteen groupings will give you a win if you play correctly: 15, 14, 1; 15, 13, 2; 15, 12, 3; 15, 11, 4; 15, 10, 5; 15, 9, 6; 15, 8, 7; 14, 13, 3; 14, 11, 5; 14, 9, 7; 13, 11, 6; 13, 10, 7; 12, 11, 7. The beautiful general solution of this problem is as follows. Express the number in every heap in powers of 2, avoiding repetitions and remembering that 2^0 = 1. Then if you so leave the matches to your opponent that there is an even number of every power, you can win. And if at the start you leave the powers even, you can always continue to do so throughout the game. Take, as example, the last grouping given above--12, 11, 7. Expressed in powers of 2 we have-- 12 = 8 4 - - 11 = 8 - 2 1 7 = - 4 2 1 ------- 2 2 2 2 ------- As there are thus two of every power, you must win. Say your opponent takes 7 from the 12 heap. He then leaves-- 5 = - 4 - 1 11 = 8 - 2 1 7 = - 4 2 1 ------- 1 2 2 3 ------- Here the powers are not all even in number, but by taking 9 from the 11 heap you immediately restore your winning position, thus-- 5 = - 4 - 1 2 = - - 2 - 7 = - 4 2 1 ------- - 2 2 2 ------- And so on to the end. This solution is quite general, and applies to any number of matches and any number of heaps. A correspondent informs me that this puzzle game was first propounded by Mr. W.M.F. Mellor, but when or where it was published I have not been able to ascertain. 397.--THE MONTENEGRIN DICE GAME. The players should select the pairs 5 and 9, and 13 and 15, if the chances of winning are to be quite equal. There are 216 different ways in which the three dice may fall. They may add up 5 in 6 different ways and 9 in 25 different ways, making 31 chances out of 216 for the player who selects these numbers. Also the dice may add up 13 in 21 different ways, and 15 in 10 different ways, thus giving the other player also 31 chances in 216. 398.--THE CIGAR PUZZLE. Not a single member of the club mastered this puzzle, and yet I shall show that it is so simple that the merest child can understand its solution--when it is pointed out to him! The large majority of my friends expressed their entire bewilderment. Many considered that "the theoretical result, in any case, is determined by the relationship between the table and the cigars;" others, regarding it as a problem in the theory of Probabilities, arrived at the conclusion that the chances are slightly in favour of the first or second player, as the case may be. One man took a table and a cigar of particular dimensions, divided the table into equal sections, and proceeded to make the two players fill up these sections so that the second player should win. But why should the first player be so accommodating? At any stage he has only to throw down a cigar obliquely across several of these sections entirely to upset Mr. 2's calculations! We have to assume that each player plays the best possible; not that one accommodates the other. The theories of some other friends would be quite sound if the shape of the cigar were that of a torpedo--perfectly symmetrical and pointed at both ends. I will show that the first player should infallibly win, if he always plays in the best possible manner. Examine carefully the following diagram, No. 1, and all will be clear. [Illustration: 1] [Illustration: 2] The first player must place his first cigar _on end_ in the exact centre of the table, as indicated by the little circle. Now, whatever the second player may do throughout, the first player must always repeat it in an exactly diametrically opposite position. Thus, if the second player places a cigar at A, I put one at AA; he places one at B, I put one at BB; he places one at C, I put one at CC; he places one at D, I put one at DD; he places one at E, I put one at EE; and so on until no more cigars can be placed without touching. As the cigars are supposed to be exactly alike in every respect, it is perfectly clear that for every move that the second player may choose to make, it is possible exactly to repeat it on a line drawn through the centre of the table. The second player can always duplicate the first player's move, no matter where he may place a cigar, or whether he places it on end or on its side. As the cigars are all alike in every respect, one will obviously balance over the edge of the table at precisely the same point as another. Of course, as each player is supposed to play in the best possible manner, it becomes a matter of theory. It is no valid objection to say that in actual practice one would not be sufficiently exact to be sure of winning. If as the first player you did not win, it would be in consequence of your _not_ having played the best possible. The second diagram will serve to show why the first cigar must be placed on end. (And here I will say that the first cigar that I selected from a box I was able so to stand on end, and I am allowed to assume that all the other cigars would do the same.) If the first cigar were placed on its side, as at F, then the second player could place a cigar as at G--as near as possible, but not actually touching F. Now, in this position you cannot repeat his play on the opposite side, because the two ends of the cigar are not alike. It will be seen that GG, when placed on the opposite side in the same relation to the centre, intersects, or lies on top of, F, whereas the cigars are not allowed to touch. You must therefore put the cigar farther away from the centre, which would result in your having insufficient room between the centre and the bottom left-hand corner to repeat everything that the other player would do between G and the top right-hand corner. Therefore the result would not be a certain win for the first player. 399.--THE TROUBLESOME EIGHT. [Illustration: +---+---+---+ | 4½| 8 | 2½| +---+---+---+ | 3 | 5 | 7 | +---+---+---+ | 7½| 2 | 5½| +---+---+---+ ] The conditions were to place a different number in each of the nine cells so that the three rows, three columns, and two diagonals should each add up 15. Probably the reader at first set himself an impossible task through reading into these conditions something which is not there--a common error in puzzle-solving. If I had said "a different figure," instead of "a different number," it would have been quite impossible with the 8 placed anywhere but in a corner. And it would have been equally impossible if I had said "a different whole number." But a number may, of course, be fractional, and therein lies the secret of the puzzle. The arrangement shown in the figure will be found to comply exactly with the conditions: all the numbers are different, and the square adds up 15 in all the required eight ways. 400.--THE MAGIC STRIPS. There are of course six different places between the seven figures in which a cut may be made, and the secret lies in keeping one strip intact and cutting each of the other six in a different place. After the cuts have been made there are a large number of ways in which the thirteen pieces may be placed together so as to form a magic square. Here is one of them:-- [Illustration: +-------------+ |1 2 3 4 5 6 7| +---------+---+ |3 4 5 6 7|1 2| +-----+---+---+ |5 6 7|1 2 3 4| +-+---+-------+ |7|1 2 3 4 5 6| +-+---------+-+ |2 3 4 5 6 7|1| +-------+---+-+ |4 5 6 7|1 2 3| +---+---+-----+ |6 7|1 2 3 4 5| +---+---------+ ] The arrangement has some rather interesting features. It will be seen that the uncut strip is at the top, but it will be found that if the bottom row of figures be placed at the top the numbers will still form a magic square, and that every successive removal from the bottom to the top (carrying the uncut strip stage by stage to the bottom) will produce the same result. If we imagine the numbers to be on seven complete _perpendicular_ strips, it will be found that these columns could also be moved in succession from left to right or from right to left, each time producing a magic square. 401.--EIGHT JOLLY GAOL-BIRDS. There are eight ways of forming the magic square--all merely different aspects of one fundamental arrangement. Thus, if you give our first square a quarter turn you will get the second square; and as the four sides may be in turn brought to the top, there are four aspects. These four in turn reflected in a mirror produce the remaining four aspects. Now, of these eight arrangements only four can possibly be reached under the conditions, and only two of these four can be reached in the fewest possible moves, which is nineteen. These two arrangements are shown. Move the men in the following order: 5, 3, 2, 5, 7, 6, 4, 1, 5, 7, 6, 4, 1, 6, 4, 8, 3, 2, 7, and you get the first square. Move them thus: 4, 1, 2, 4, 1, 6, 7, 1, 5, 8, 1, 5, 6, 7, 5, 6, 4, 2, 7, and you have the arrangement in the second square. In the first case every man has moved, but in the second case the man numbered 3 has never left his cell. Therefore No. 3 must be the obstinate prisoner, and the second square must be the required arrangement. [Illustration: +---+---+---+ +---+---+---+ | | | | | | | | | 5 7 | | 7 4 3 | | | | | | | | | +- -+- -+- -+ +- -+- -+- -+ | | | | | | | | | 6 4 2 | | 4 8 | | | | | | | | | +- -+- -+- -+ +- -+- -+- -+ | | | | | | | | | 1 8 3 | | 5 6 1 | | | | | | | | | +---+---+---+ +---+---+---+ ] 402.--NINE JOLLY GAOL BIRDS. There is a pitfall set for the unwary in this little puzzle. At the start one man is allowed to be placed on the shoulders of another, so as to give always one empty cell to enable the prisoners to move about without any two ever being in a cell together. The two united prisoners are allowed to add their numbers together, and are, of course, permitted to remain together at the completion of the magic square. But they are obviously not compelled so to remain together, provided that one of the pair on his final move does not break the condition of entering a cell already occupied. After the acute solver has noticed this point, it is for him to determine which method is the better one--for the two to be together at the count or to separate. As a matter of fact, the puzzle can be solved in seventeen moves if the men are to remain together; but if they separate at the end, they may actually save a move and perform the feat in sixteen! The trick consists in placing the man in the centre on the back of one of the corner men, and then working the pair into the centre before their final separation. [Illustration: A B +---+---+---+ +---+---+---+ | | | | | | | | | 2 9 4 | | 6 7 2 | | | | | | | | | +- -+- -+- -+ +- -+- -+- -+ | | | | | | | | | 7 5 3 | | 1 5 9 | | | | | | | | | +- -+- -+- -+ +- -+- -+- -+ | | | | | | | | | 6 1 8 | | 8 3 4 | | | | | | | | | +---+---+---+ +---+---+---+ ] Here are the moves for getting the men into one or other of the above two positions. The numbers are those of the men in the order in which they move into the cell that is for the time being vacant. The pair is shown in brackets:-- Place 5 on 1. Then, 6, 9, 8, 6, 4, (6), 2, 4, 9, 3, 4, 9, (6), 7, 6, 1. Place 5 on 9. Then, 4, 1, 2, 4, 6, (14), 8, 6, 1, 7, 6, 1, (14), 3, 4, 9. Place 5 on 3. Then, 6, (8), 2, 6, 4, 7, 8, 4, 7, 1, 6, 7, (8), 9, 4, 3. Place 5 on 7. Then, 4, (12), 8, 4, 6, 3, 2, 6, 3, 9, 4, 3, (12), 1, 6, 7. The first and second solutions produce Diagram A; the second and third produce Diagram B. There are only sixteen moves in every case. Having found the fewest moves, we had to consider how we were to make the burdened man do as little work as possible. It will at once be seen that as the pair have to go into the centre before separating they must take at fewest two moves. The labour of the burdened man can only be reduced by adopting the other method of solution, which, however, forces us to take another move. 403.--THE SPANISH DUNGEON. [Illustration] +-----+-----+-----+-----+ +-----+-----+-----+-----+ | | | | | | | | | | | 1 | 2 | 3 | 4 | | 10 | 9 | 7 | 4 | |_____|_____|_____|_____| |_____|_____|_____|_____| | | | | | | | | | | | 5 | 6 | 7 | 8 | | 6 | 5 | 11 | 8 | |_____|_____|_____|_____| |_____|_____|_____|_____| | | | | | | | | | | | 9 | 10 | 11 | 12 | | 1 | 2 | 12 | 15 | |_____|_____|_____|_____| |_____|_____|_____|_____| | | | | | | | | | | | 13 | 14 | 15 | | | 13 | 14 | | 3 | | | | | | | | | | | +-----+-----+-----+-----+ +-----+-----+-----+-----+ This can best be solved by working backwards--that is to say, you must first catch your square, and then work back to the original position. We must first construct those squares which are found to require the least amount of readjustment of the numbers. Many of these we know cannot possibly be reached. When we have before us the most favourable possible arrangements, it then becomes a question of careful analysis to discover which position can be reached in the fewest moves. I am afraid, however, it is only after considerable study and experience that the solver is able to get such a grasp of the various "areas of disturbance" and methods of circulation that his judgment is of much value to him. The second diagram is a most favourable magic square position. It will be seen that prisoners 4, 8, 13, and 14 are left in their original cells. This position may be reached in as few as thirty-seven moves. Here are the moves: 15, 14, 10, 6, 7, 3, 2, 7, 6, 11, 3, 2, 7, 6, 11, 10, 14, 3, 2, 11, 10, 9, 5, 1, 6, 10, 9, 5, 1, 6, 10, 9, 5, 2, 12, 15, 3. This short solution will probably surprise many readers who may not find a way under from sixty to a hundred moves. The clever prisoner was No. 6, who in the original illustration will be seen with his arms extended calling out the moves. He and No. 10 did most of the work, each changing his cell five times. No. 12, the man with the crooked leg, was lame, and therefore fortunately had only to pass from his cell into the next one when his time came round. 404.--THE SIBERIAN DUNGEONS. [Illustration] +-----+-----+-----+-----+ | | | | | | 8 | 5 | 10 | 11 | |_____|_____|_____|_____| | | | | | | 16 | 13 | 2 | 3 | |_____|_____|_____|_____| | | | | | | 1 | 12 | 7 | 14 | |_____|_____|_____|_____| | | | | | | 9 | 4 | 15 | 6 | | | | | | +-----+-----+-----+-----+ In attempting to solve this puzzle it is clearly necessary to seek such magic squares as seem the most favourable for our purpose, and then carefully examine and try them for "fewest moves." Of course it at once occurs to us that if we can adopt a square in which a certain number of men need not leave their original cells, we may save moves on the one hand, but we may obstruct our movements on the other. For example, a magic square may be formed with the 6, 7, 13, and 16 unmoved; but in such case it is obvious that a solution is impossible, since cells 14 and 15 can neither be left nor entered without breaking the condition of no two men ever being in the same cell together. The following solution in fourteen moves was found by Mr. G. Wotherspoon: 8-17, 16-21, 6-16, 14-8, 5-18, 4-14, 3-24, 11-20, 10-19, 2-23, 13-22, 12-6, 1-5, 9-13. As this solution is in what I consider the theoretical minimum number of moves, I am confident that it cannot be improved upon, and on this point Mr. Wotherspoon is of the same opinion. 405.--CARD MAGIC SQUARES. Arrange the cards as follows for the three new squares:-- 3 2 4 6 5 7 9 8 10 4 3 2 7 6 5 10 9 8 2 4 3 5 7 6 8 10 9 Three aces and one ten are not used. The summations of the four squares are thus: 9, 15, 18, and 27--all different, as required. 406.--THE EIGHTEEN DOMINOES. [Illustration] The illustration explains itself. It will be found that the pips in every column, row, and long diagonal add up 18, as required. 407.--TWO NEW MAGIC SQUARES. Here are two solutions that fulfil the conditions:-- [Illustration: SUBTRACTING DIVIDING 11 4 14 13 36 8 54 27 16 7 1 2 216 12 1 2 6 5 3 12 6 3 4 72 9 19 8 15 9 18 24 108 ] The first, by subtracting, has a constant 8, and the associated pairs all have a difference of 4. The second square, by dividing, has a constant 9, and all the associated pairs produce 3 by division. These are two remarkable and instructive squares. 408.--MAGIC SQUARES OF TWO DEGREES. The following is the square that I constructed. As it stands the constant is 260. If for every number you substitute, in its allotted place, its square, then the constant will be 11,180. Readers can write out for themselves the second degree square. [Illustration: 7 53 | 41 27 | 2 52 | 48 30 12 58 | 38 24 | 13 63 | 35 17 ------+-------+-------+------ 51 1 | 29 47 | 54 8 | 28 42 64 14 | 18 36 | 57 11 | 23 37 ------+-------+-------+------ 25 43 | 55 5 | 32 46 | 50 4 22 40 | 60 10 | 19 33 | 61 15 ------+-------+-------+------ 45 31 | 3 49 | 44 26 | 6 56 34 20 | 16 62 | 39 21 | 9 59 ] The main key to the solution is the pretty law that if eight numbers sum to 260 and their squares to 11,180, then the same will happen in the case of the eight numbers that are complementary to 65. Thus 1 + 18 + 23 + 26 + 31 + 48 + 56 + 57 = 260, and the sum of their squares is 11,180. Therefore 64 + 47 + 42 + 39 + 34 + 17 + 9 + 8 (obtained by subtracting each of the above numbers from 65) will sum to 260 and their squares to 11,180. Note that in every one of the sixteen smaller squares the two diagonals sum to 65. There are four columns and four rows with their complementary columns and rows. Let us pick out the numbers found in the 2nd, 1st, 4th, and 3rd rows and arrange them thus :-- [Illustration: 1 8 28 29 42 47 51 54 2 7 27 30 41 48 52 53 3 6 26 31 44 45 49 56 4 5 25 32 43 46 50 55 ] Here each column contains four consecutive numbers cyclically arranged, four running in one direction and four in the other. The numbers in the 2nd, 5th, 3rd, and 8th columns of the square may be similarly grouped. The great difficulty lies in discovering the conditions governing these groups of numbers, the pairing of the complementaries in the squares of four and the formation of the diagonals. But when a correct solution is shown, as above, it discloses all the more important keys to the mystery. I am inclined to think this square of two degrees the most elegant thing that exists in magics. I believe such a magic square cannot be constructed in the case of any order lower than 8. 409.--THE BASKETS OF PLUMS. As the merchant told his man to distribute the contents of one of the baskets of plums "among some children," it would not be permissible to give the complete basketful to one child; and as it was also directed that the man was to give "plums to every child, so that each should receive an equal number," it would also not be allowed to select just as many children as there were plums in a basket and give each child a single plum. Consequently, if the number of plums in every basket was a prime number, then the man would be correct in saying that the proposed distribution was quite impossible. Our puzzle, therefore, resolves itself into forming a magic square with nine different prime numbers. [Illustration] A B +-----+-----+-----+ +-----+-----+-----+ | | | | | | | | | 7 | 61 | 43 | | 83 | 29 | 101 | |_____|_____|_____| |_____|_____|_____| | | | | | | | | | 73 | 37 | 1 | | 89 | 71 | 53 | |_____|_____|_____| |_____|_____|_____| | | | | | | | | | 31 | 13 | 67 | | 41 | 113 | 59 | | | | | | | | | +-----+-----+-----+ +-----+-----+-----+ C D +-----+-----+-----+ +-----+-----+-----+ | | | | | | | | | 103 | 79 | 37 | |1669 | 199 |1249 | |_____|_____|_____| |_____|_____|_____| | | | | | | | | | 7 | 73 | 139 | | 619 |1039 |1459 | |_____|_____|_____| |_____|_____|_____| | | | | | | | | | 109 | 67 | 43 | | 829 |1879 | 409 | | | | | | | | | +-----+-----+-----+ +-----+-----+-----+ In Diagram A we have a magic square in prime numbers, and it is the one giving the smallest constant sum that is possible. As to the little trap I mentioned, it is clear that Diagram A is barred out by the words "every basket contained plums," for one plum is not plums. And as we were referred to the baskets, "as shown in the illustration," it is perfectly evident, without actually attempting to count the plums, that there are at any rate more than 7 plums in every basket. Therefore C is also, strictly speaking, barred. Numbers over 20 and under, say, 250 would certainly come well within the range of possibility, and a large number of arrangements would come within these limits. Diagram B is one of them. Of course we can allow for the false bottoms that are so frequently used in the baskets of fruitsellers to make the basket appear to contain more fruit than it really does. Several correspondents assumed (on what grounds I cannot think) that in the case of this problem the numbers cannot be in consecutive arithmetical progression, so I give Diagram D to show that they were mistaken. The numbers are 199, 409, 619, 829, 1,039, 1,249, 1,459, 1,669, and 1,879--all primes with a common difference of 210. 410.--THE MANDARIN'S "T" PUZZLE. There are many different ways of arranging the numbers, and either the 2 or the 3 may be omitted from the "T" enclosure. The arrangement that I give is a "nasik" square. Out of the total of 28,800 nasik squares of the fifth order this is the only one (with its one reflection) that fulfils the "T" condition. This puzzle was suggested to me by Dr. C. Planck. [Illustration: THE MANDARIN'S "T" PUZZLE. +-----+-----+-----+-----+-----+ | | | | | | | 19 | 23 | 11 | 5 | 7 | |_____|_____|_____|_____|_____| | | | | | | | 1 | 10 | 17 | 24 | 13 | |_____|_____|_____|_____|_____| | | | | | | | 22 | 14 | 3 | 6 | 20 | |_____|_____|_____|_____|_____| | | | | | | | 8 | 16 | 25 | 12 | 4 | |_____|_____|_____|_____|_____| | | | | | | | 15 | 2 | 9 | 18 | 21 | | | | | | | +-----+-----+-----+-----+-----+ 411.--A MAGIC SQUARE OF COMPOSITES. The problem really amounts to finding the smallest prime such that the next higher prime shall exceed it by 10 at least. If we write out a little list of primes, we shall not need to exceed 150 to discover what we require, for after 113 the next prime is 127. We can then form the square in the diagram, where every number is composite. This is the solution in the smallest numbers. We thus see that the answer is arrived at quite easily, in a square of the third order, by trial. But I propose to show how we may get an answer (not, it is true, the one in smallest numbers) without any tables or trials, but in a very direct and rapid manner. [Illustration] +-----+-----+-----+ | | | | | 121 | 114 | 119 | |_____|_____|_____| | | | | | 116 | 118 | 120 | |_____|_____|_____| | | | | | 117 | 122 | 115 | | | | | +-----+-----+-----+ First write down any consecutive numbers, the smallest being greater than 1--say, 2, 3, 4, 5, 6, 7, 8, 9, 10. The only factors in these numbers are 2, 3, 5, and 7. We therefore multiply these four numbers together and add the product, 210, to each of the nine numbers. The result is the nine consecutive composite numbers, 212 to 220 inclusive, with which we can form the required square. Every number will necessarily be divisible by its difference from 210. It will be very obvious that by this method we may find as many consecutive composites as ever we please. Suppose, for example, we wish to form a magic square of sixteen such numbers; then the numbers 2 to 17 contain the factors 2, 3, 5, 7, 11, 13, and 17, which, multiplied together, make 510510 to be added to produce the sixteen numbers 510512 to 510527 inclusive, all of which are composite as before. But, as I have said, these are not the answers in the smallest numbers: for if we add 523 to the numbers 1 to 16, we get sixteen consecutive composites; and if we add 1,327 to the numbers 1 to 25, we get twenty-five consecutive composites, in each case the smallest numbers possible. Yet if we required to form a magic square of a hundred such numbers, we should find it a big task by means of tables, though by the process I have shown it is quite a simple matter. Even to find thirty-six such numbers you will search the tables up to 10,000 without success, and the difficulty increases in an accelerating ratio with each square of a larger order. 412.--THE MAGIC KNIGHT'S TOUR. +----+----+----+----+----+----+----+----+ | 46 | 55 | 44 | 19 | 58 | 9 | 22 | 7 | +----+----+----+----+----+----+----+----+ | 43 | 18 | 47 | 56 | 21 | 6 | 59 | 10 | +----+----+----+----+----+----+----+----+ | 54 | 45 | 20 | 41 | 12 | 57 | 8 | 23 | +----+----+----+----+----+----+----+----+ | 17 | 42 | 53 | 48 | 5 | 24 | 11 | 60 | +----+----+----+----+----+----+----+----+ | 52 | 3 | 32 | 13 | 40 | 61 | 34 | 25 | +----+----+----+----+----+----+----+----+ | 31 | 16 | 49 | 4 | 33 | 28 | 37 | 62 | +----+----+----+----+----+----+----+----+ | 2 | 51 | 14 | 29 | 64 | 39 | 26 | 35 | +----+----+----+----+----+----+----+----+ | 15 | 30 | 1 | 50 | 27 | 36 | 63 | 38 | +----+----+----+----+----+----+----+----+ Here each successive number (in numerical order) is a knight's move from the preceding number, and as 64 is a knight's move from 1, the tour is "re-entrant." All the columns and rows add up 260. Unfortunately, it is not a perfect magic square, because the diagonals are incorrect, one adding up 264 and the other 256--requiring only the transfer of 4 from one diagonal to the other. I think this is the best result that has ever been obtained (either re-entrant or not), and nobody can yet say whether a perfect solution is possible or impossible. 413.--A CHESSBOARD FALLACY. [Illustration] The explanation of this little fallacy is as follows. The error lies in assuming that the little triangular piece, marked C, is exactly the same height as one of the little squares of the board. As a matter of fact, its height (if we make the sixty-four squares each a square inch) will be 1+1/7 in. Consequently the rectangle is really 9+1/7 in. by 7 in., so that the area is sixty-four square inches in either case. Now, although the pieces do fit together exactly to form the perfect rectangle, yet the directions of the horizontal lines in the pieces will not coincide. The new diagram above will make everything quite clear to the reader. 414.--WHO WAS FIRST? Biggs, who saw the smoke, would be first; Carpenter, who saw the bullet strike the water, would be second; and Anderson, who heard the report, would be last of all. 415.--A WONDERFUL VILLAGE. When the sun is in the horizon of any place (whether in Japan or elsewhere), he is the length of half the earth's diameter more distant from that place than in his meridian at noon. As the earth's semi-diameter is nearly 4,000 miles, the sun must be considerably more than 3,000 miles nearer at noon than at his rising, there being no valley even the hundredth part of 1,000 miles deep. 416.--A CALENDAR PUZZLE. The first day of a century can never fall on a Sunday; nor on a Wednesday or a Friday. 417.--THE TIRING-IRONS. I will give my complete working of the solution, so that readers may see how easy it is when you know how to proceed. And first of all, as there is an even number of rings, I will say that they may all be taken off in one-third of (2^(n + 1) - 2) moves; and since n in our case is 14, all the rings may be taken off in 10,922 moves. Then I say 10,922 - 9,999 = 923, and proceed to find the position when only 923 out of the 10,922 moves remain to be made. Here is the curious method of doing this. It is based on the binary scale method used by Monsieur L. Gros, for an account of which see W.W. Rouse Ball's _Mathematical Recreations_. Divide 923 by 2, and we get 461 and the remainder 1; divide 461 by 2, and we get 230 and the remainder 1; divide 230 by 2, and we get 115 and the remainder nought. Keep on dividing by 2 in this way as long as possible, and all the remainders will be found to be 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, the last remainder being to the left and the first remainder to the right. As there are fourteen rings and only ten figures, we place the difference, in the form of four noughts, in brackets to the left, and bracket all those figures that repeat a figure on their left. Then we get the following arrangement: (0 0 0 0) 1 (1 1) 0 (0) 1 (1) 0 1 (1). This is the correct answer to the puzzle, for if we now place rings below the line to represent the figures in brackets and rings on the line for the other figures, we get the solution in the required form, as below:-- O O O OO ------------------------- OOOO OO O O O This is the exact position of the rings after the 9,999th move has been made, and the reader will find that the method shown will solve any similar question, no matter how many rings are on the tiring-irons. But in working the inverse process, where you are required to ascertain the number of moves necessary in order to reach a given position of the rings, the rule will require a little modification, because it does not necessarily follow that the position is one that is actually reached in course of taking off all the rings on the irons, as the reader will presently see. I will here state that where the total number of rings is odd the number of moves required to take them all off is one-third of (2^(n + 1) - 1). With n rings (where n is _odd_) there are 2^n positions counting all on and all off. In (1/3)(2^(n + 1) + 2) positions they are all removed. The number of positions not used is (1/3)(2^n - 2). With n rings (where n is _even_) there are 2^n positions counting all on and all off. In (2^(n + 1) + 1) positions they are all removed. The number of positions not used is here (1/3)(2^n - 1). It will be convenient to tabulate a few cases. +--------+------------+-----------+-----------+ | No. of | Total | Positions | Positions | | Rings. | Positions. | used. | not used. | +--------+------------+-----------+-----------+ | 1 | 2 | 2 | 0 | | 3 | 8 | 6 | 2 | | 5 | 32 | 22 | 10 | | 7 | 128 | 86 | 42 | | 9 | 512 | 342 | 170 | | | | | | | 2 | 4 | 3 | 1 | | 4 | 16 | 11 | 5 | | 6 | 64 | 43 | 21 | | 8 | 256 | 171 | 85 | | 10 | 1024 | 683 | 341 | +--------+------------+-----------+-----------+ Note first that the number of _positions used_ is one more than the number of _moves_ required to take all the rings off, because we are including "all on" which is a position but not a move. Then note that the number of _positions not used_ is the same as the number of _moves used_ to take off a set that has one ring fewer. For example, it takes 85 moves to remove 7 rings, and the 42 positions not used are exactly the number of moves required to take off a set of 6 rings. The fact is that if there are 7 rings and you take off the first 6, and then wish to remove the 7th ring, there is no course open to you but to reverse all those 42 moves that never ought to have been made. In other words, you must replace all the 7 rings on the loop and start afresh! You ought first to have taken off 5 rings, to do which you should have taken off 3 rings, and previously to that 1 ring. To take off 6 you first remove 2 and then 4 rings. 418.--SUCH A GETTING UPSTAIRS. Number the treads in regular order upwards, 1 to 8. Then proceed as follows: 1 (step back to floor), 1, 2, 3 (2), 3, 4, 5 (4), 5, 6, 7 (6), 7, 8, landing (8), landing. The steps in brackets are taken in a backward direction. It will thus be seen that by returning to the floor after the first step, and then always going three steps forward for one step backward, we perform the required feat in nineteen steps. 419.--THE FIVE PENNIES. [Illustration] First lay three of the pennies in the way shown in Fig. 1. Now hold the remaining two pennies in the position shown in Fig. 2, so that they touch one another at the top, and at the base are in contact with the three horizontally placed coins. Then the five pennies will be equidistant, for every penny will touch every other penny. 420.--THE INDUSTRIOUS BOOKWORM. The hasty reader will assume that the bookworm, in boring from the first to the last page of a book in three volumes, standing in their proper order on the shelves, has to go through all three volumes and four covers. This, in our case, would mean a distance of 9½ in., which is a long way from the correct answer. You will find, on examining any three consecutive volumes on your shelves, that the first page of Vol. I. and the last page of Vol. III. are actually the pages that are nearest to Vol. II., so that the worm would only have to penetrate four covers (together, ½ in.) and the leaves in the second volume (3 in.), or a distance of 3½ inches, in order to tunnel from the first page to the last. 421.--A CHAIN PUZZLE. To open and rejoin a link costs threepence. Therefore to join the nine pieces into an endless chain would cost 2s. 3d., whereas a new chain would cost 2s. 2d. But if we break up the piece of eight links, these eight will join together the remaining eight pieces at a cost of 2s. But there is a subtle way of even improving on this. Break up the two pieces containing three and four links respectively, and these seven will join together the remaining seven pieces at a cost of only 1s. 9d. 422.--THE SABBATH PUZZLE. The way the author of the old poser proposed to solve the difficulty was as follows: From the Jew's abode let the Christian and the Turk set out on a tour round the globe, the Christian going due east and the Turk due west. Readers of Edgar Allan Poe's story, _Three Sundays in a Week_, or of Jules Verne's _Round the World in Eighty Days_, will know that such a proceeding will result in the Christian's gaining a day and in the Turk's losing a day, so that when they meet again at the house of the Jew their reckoning will agree with his, and all three may keep their Sabbath on the same day. The correctness of this answer, of course, depends on the popular notion as to the definition of a day--the average duration between successive sun-rises. It is an old quibble, and quite sound enough for puzzle purposes. Strictly speaking, the two travellers ought to change their reckonings on passing the 180th meridian; otherwise we have to admit that at the North or South Pole there would only be one Sabbath in seven years. 423.--THE RUBY BROOCH. In this case we were shown a sketch of the brooch exactly as it appeared after the four rubies had been stolen from it. The reader was asked to show the positions from which the stones "may have been taken;" for it is not possible to show precisely how the gems were originally placed, because there are many such ways. But an important point was the statement by Lady Littlewood's brother: "I know the brooch well. It originally contained forty-five stones, and there are now only forty-one. Somebody has stolen four rubies, and then reset as small a number as possible in such a way that there shall always be eight stones in any of the directions you have mentioned." [Illustration] The diagram shows the arrangement before the robbery. It will be seen that it was only necessary to reset one ruby--the one in the centre. Any solution involving the resetting of more than one stone is not in accordance with the brother's statement, and must therefore be wrong. The original arrangement was, of course, a little unsymmetrical, and for this reason the brooch was described as "rather eccentric." 424.--THE DOVETAILED BLOCK. [Illustration] The mystery is made clear by the illustration. It will be seen at once how the two pieces slide together in a diagonal direction. 425.--JACK AND THE BEANSTALK. The serious blunder that the artist made in this drawing was in depicting the tendrils of [Illustration] the bean climbing spirally as at A above, whereas the French bean, or scarlet runner, the variety clearly selected by the artist in the absence of any authoritative information on the point, always climbs as shown at B. Very few seem to be aware of this curious little fact. Though the bean always insists on a sinistrorsal growth, as B, the hop prefers to climb in a dextrorsal manner, as A. Why, is one of the mysteries that Nature has not yet unfolded. 426.--THE HYMN-BOARD POSER. This puzzle is not nearly so easy as it looks at first sight. It was required to find the smallest possible number of plates that would be necessary to form a set for three hymn-boards, each of which would show the five hymns sung at any particular service, and then to discover the lowest possible cost for the same. The hymn-book contains 700 hymns, and therefore no higher number than 700 could possibly be needed. Now, as we are required to use every legitimate and practical method of economy, it should at once occur to us that the plates must be painted on both sides; indeed, this is such a common practice in cases of this kind that it would readily occur to most solvers. We should also remember that some of the figures may possibly be reversed to form other figures; but as we were given a sketch of the actual shapes of these figures when painted on the plates, it would be seen that though the 6's may be turned upside down to make 9's, none of the other figures can be so treated. It will be found that in the case of the figures 1, 2, 3, 4, and 5, thirty-three of each will be required in order to provide for every possible emergency; in the case of 7, 8, and 0, we can only need thirty of each; while in the case of the figure 6 (which may be reversed for the figure 9) it is necessary to provide exactly forty-two. It is therefore clear that the total number of figures necessary is 297; but as the figures are painted on both sides of the plates, only 149 such plates are required. At first it would appear as if one of the plates need only have a number on one side, the other side being left blank. But here we come to a rather subtle point in the problem. Readers may have remarked that in real life it is sometimes cheaper when making a purchase to buy more articles than we require, on the principle of a reduction on taking a quantity: we get more articles and we pay less. Thus, if we want to buy ten apples, and the price asked is a penny each if bought singly, or ninepence a dozen, we should both save a penny and get two apples more than we wanted by buying the full twelve. In the same way, since there is a regular scale of reduction for plates painted alike, we actually save by having two figures painted on that odd plate. Supposing, for example, that we have thirty plates painted alike with 5 on one side and 6 on the other. The rate would be 4¾d., and the cost 11s. 10½d. But if the odd plate with, say, only a 5 on one side of it have a 6 painted on the other side, we get thirty-one plates at the reduced rate of 4½d., thus saving a farthing on each of the previous thirty, and reducing the cost of the last one from 1s. to 4½d. But even after these points are all seen there comes in a new difficulty: for although it will be found that all the 8's may be on the backs of the 7's, we cannot have all the 2's on the backs of the 1's, nor all the 4 on the backs of the 3's, etc. There is a great danger, in our attempts to get as many as possible painted alike, of our so adjusting the figures that some particular combination of hymns cannot be represented. Here is the solution of the difficulty that was sent to the vicar of Chumpley St. Winifred. Where the sign X is placed between two figures, it implies that one of these figures is on one side of the plate and the other on the other side. d. £ s. d. 31 plates painted 5 X 9 @ 4½ = 0 11 7½ 30 " 7 X 8 @ 4¾ = 0 11 10½ 21 " 1 X 2 @ 7 = 0 12 3 21 " 3 X 0 @ 7 = 0 12 3 12 " 1 X 3 @ 9¼ = 0 9 3 12 " 2 X 4 @ 9¼ = 0 9 3 12 " 9 X 4 @ 9¼ = 0 9 3 8 " 4 X 0 @ 10¼ = 0 6 10 1 " 5 X 4 @ 12 = 0 1 0 1 " 5 X 0 @ 12 = 0 1 0 149 plates @ 6d. each = 3 14 6 ---------- £7 19 1 Of course, if we could increase the number of plates, we might get the painting done for nothing, but such a contingency is prevented by the condition that the fewest possible plates must be provided. This puzzle appeared in _Tit-Bits_, and the following remarks, made by me in the issue for 11th December 1897, may be of interest. The "Hymn-Board Poser" seems to have created extraordinary interest. The immense number of attempts at its solution sent to me from all parts of the United Kingdom and from several Continental countries show a very kind disposition amongst our readers to help the worthy vicar of Chumpley St. Winifred over his parochial difficulty. Every conceivable estimate, from a few shillings up to as high a sum as £1,347, 10s., seems to have come to hand. But the astonishing part of it is that, after going carefully through the tremendous pile of correspondence, I find that only one competitor has succeeded in maintaining the reputation of the _Tit-Bits_ solvers for their capacity to solve anything, and his solution is substantially the same as the one given above, the cost being identical. Some of his figures are differently combined, but his grouping of the plates, as shown in the first column, is exactly the same. Though a large majority of competitors clearly hit upon all the essential points of the puzzle, they completely collapsed in the actual arrangement of the figures. According to their methods, some possible selection of hymns, such as 111, 112, 121, 122,211, cannot be set up. A few correspondents suggested that it might be possible so to paint the 7's that upside down they would appear as 2's or 4's; but this would, of course, be barred out by the fact that a representation of the actual figures to be used was given. 427.--PHEASANT-SHOOTING. The arithmetic of this puzzle is very easy indeed. There were clearly 24 pheasants at the start. Of these 16 were shot dead, 1 was wounded in the wing, and 7 got away. The reader may have concluded that the answer is, therefore, that "seven remained." But as they flew away it is clearly absurd to say that they "remained." Had they done so they would certainly have been killed. Must we then conclude that the 17 that were shot remained, because the others flew away? No; because the question was not "how many remained?" but "how many still remained?" Now the poor bird that was wounded in the wing, though unable to fly, was very active in its painful struggles to run away. The answer is, therefore, that the 16 birds that were shot dead "still remained," or "remained still." 428.--THE GARDENER AND THE COOK. Nobody succeeded in solving the puzzle, so I had to let the cat out of the bag--an operation that was dimly foreshadowed by the puss in the original illustration. But I first reminded the reader that this puzzle appeared on April 1, a day on which none of us ever resents being made an "April Fool;" though, as I practically "gave the thing away" by specially drawing attention to the fact that it was All Fools' Day, it was quite remarkable that my correspondents, without a single exception, fell into the trap. One large body of correspondents held that what the cook loses in stride is exactly made up in greater speed; consequently both advance at the same rate, and the result must be a tie. But another considerable section saw that, though this might be so in a race 200 ft. straight away, it could not really be, because they each go a stated distance at "every bound," and as 100 is not an exact multiple of 3, the gardener at his thirty-fourth bound will go 2 ft. beyond the mark. The gardener will, therefore, run to a point 102 ft. straight away and return (204 ft. in all), and so lose by 4 ft. This point certainly comes into the puzzle. But the most important fact of all is this, that it so happens that the gardener was a pupil from the Horticultural College for Lady Gardeners at, if I remember aright, Swanley; while the cook was a very accomplished French chef of the hemale persuasion! Therefore "she (the gardener) made three bounds to his (the cook's) two." It will now be found that while the gardener is running her 204 ft. in 68 bounds of 3 ft., the somewhat infirm old cook can only make 45+1/3 of his 2 ft. bounds, which equals 90 ft. 8 in. The result is that the lady gardener wins the race by 109 ft. 4 in. at a moment when the cook is in the air, one-third through his 46th bound. The moral of this puzzle is twofold: (1) Never take things for granted in attempting to solve puzzles; (2) always remember All Fools' Day when it comes round. I was not writing of _any_ gardener and cook, but of a _particular_ couple, in "a race that I witnessed." The statement of the eye-witness must therefore be accepted: as the reader was not there, he cannot contradict it. Of course the information supplied was insufficient, but the correct reply was: "Assuming the gardener to be the 'he,' the cook wins by 4 ft.; but if the gardener is the 'she,' then the gardener wins by 109 ft. 4 in." This would have won the prize. Curiously enough, one solitary competitor got on to the right track, but failed to follow it up. He said: "Is this a regular April 1 catch, meaning that they only ran 6 ft. each, and consequently the race was unfinished? If not, I think the following must be the solution, supposing the gardener to be the 'he' and the cook the 'she.'" Though his solution was wrong even in the case he supposed, yet he was the only person who suspected the question of sex. 429.--PLACING HALFPENNIES. Thirteen coins may be placed as shown on page 252. 430.--FIND THE MAN'S WIFE. There is no guessing required in this puzzle. It is all a question of elimination. If we can pair off any five of the ladies with their respective husbands, other than husband No. 10, then the remaining lady must be No. 10's wife. [Illustration: PLACING HALFPENNIES.] I will show how this may be done. No. 8 is seen carrying a lady's parasol in the same hand with his walking-stick. But every lady is provided with a parasol, except No. 3; therefore No. 3 may be safely said to be the wife of No. 8. Then No. 12 is holding a bicycle, and the dress-guard and make disclose the fact that it is a lady's bicycle. The only lady in a cycling skirt is No. 5; therefore we conclude that No. 5 is No. 12's wife. Next, the man No. 6 has a dog, and lady No. 11 is seen carrying a dog chain. So we may safely pair No. 6 with No. 11. Then we see that man No. 2 is paying a newsboy for a paper. But we do not pay for newspapers in this way before receiving them, and the gentleman has apparently not taken one from the boy. But lady No. 9 is seen reading a paper. The inference is obvious--that she has sent the boy to her husband for a penny. We therefore pair No. 2 with No. 9. We have now disposed of all the ladies except Nos. 1 and 7, and of all the men except Nos. 4 and 10. On looking at No. 4 we find that he is carrying a coat over his arm, and that the buttons are on the left side;--not on the right, as a man wears them. So it is a lady's coat. But the coat clearly does not belong to No. 1, as she is seen to be wearing a coat already, while No. 7 lady is very lightly clad. We therefore pair No. 7 lady with man No. 4. Now the only lady left is No. 1, and we are consequently forced to the conclusion that she is the wife of No. 10. This is therefore the correct answer. INDEX. Abbot's Puzzle, The, 20, 161. ---- Window, The, 87, 213. Academic Courtesies, 18, 160. Acrostic Puzzle, An, 84, 210. Adam and Eve and the Apples, 18. Aeroplanes, The Two, 2, 148. Age and Kinship Puzzles, 6. ---- Concerning Tommy's, 7, 153. ---- Mamma's, 7, 152. ---- Mrs. Timpkins's, 7, 152. ---- Rover's, 7, 152. Ages, The Family, 7, 152. ---- Their, 7, 152. Alcuin, Abbot, 20, 112. Almonds, The Nine, 64, 195. Amazons, The, 94, 221. Andrews, W.S., 125. Apples, A Deal in, 3, 149. ---- Buying, 6, 151. ---- The Ten, 64, 195. Approximations in Dissection, 28. Arithmetical and Algebraical Problems, 1. ---- Various, 17. Arthur's Knights, King, 77, 203. Artillerymen's Dilemma, 26, 167. Asparagus, Bundles of, 140. Aspects all due South, 137. Associated Magic Squares, 120. Axiom, A Puzzling, 138. Bachet de Méziriac, 90, 109, 112. Bachet's Square, 90, 216. Ball Problem, The, 51, 183. Ball, W.W. Rouse, 109, 204, 248. Balls, The Glass, 78, 204. Banker's Puzzle, The, 25, 165. Bank Holiday Puzzle, A, 73, 201. Banner Puzzle, The, 46, 179. ---- St. George's, 50, 182. Barrel Puzzle, The, 109, 235. Barrels of Balsam, The, 82, 208. Beanfeast Puzzle, A, 2, 148. Beef and Sausages, 3, 149. Beer, The Barrel of, 13, 155. Bell-ropes, Stealing the, 49, 181. Bells, The Peal of, 78, 204. Bergholt, E., 116, 119, 125. Betsy Ross Puzzle, The, 40, 176. Bicycle Thief, The, 6, 152. Bishops--Guarded, 88, 214. ---- in Convocation, 89, 215. ---- Puzzle, A New, 98, 225. ---- Unguarded, 88, 214. Board, The Chess-, 85. ---- in Compartments, The, 102, 228. ---- Setting the, 105, 231. Boards with Odd Number of Squares, 86, 212. Boat, Three Men in a, 78, 204. Bookworm, The Industrious, 143, 248. Boothby, Guy, 154. Box, The Cardboard, 49, 181. ---- The Paper, 40. Boys and Girls, 67, 197. Bridges, The Monk and the, 75, 202. Brigands, The Five, 25, 164. Brocade, The Squares of, 47, 180. Bun Puzzle, The, 35, 170. Busschop, Paul, 172. Buttons and String Method, 230. Cab Numbers, The, 15, 157. Calendar Puzzle, A, 142, 247. _Canterbury Puzzles, The_, 14, 28, 58, 117, 121, 195, 202, 205, 206, 212, 213, 217, 233. Card Frame Puzzle, The, 114, 238. ---- Magic Squares, 123, 244. ---- Players, A Puzzle for, 78, 203. ---- Puzzle, The "T," 115, 239. ---- Triangles, 115, 239. Cards, The Cross of, 115, 238. Cardan, 142. Carroll, Lewis, 43. Castle Treasure, Stealing the, 113, 237. Cats, the Wizard's, 42, 178. Cattle, Judkins's, 6, 151. ---- Market, At a, 1, 148. Census Puzzle, A, 7, 152. Century Puzzle, The, 16, 158. ---- The Digital, 16, 159. Chain Puzzle, A, 144, 249. ---- The Antiquary's, 83, 209. ---- The Cardboard, 40, 176. Change, Giving, 4, 150. ---- Ways of giving, 151. Changing Places, 10, 154. Channel Island, 138. Charitable Bequest, A, 2, 148 Charity, Indiscriminate, 2, 148. Checkmate, 107, 233. Cheesemonger, The Eccentric, 66, 196. Chequered Board Divisions, 85, 210. Cherries and Plums, 56, 189. Chess Puzzles, Dynamical, 96. ---- Statical, 88. ---- Various, 105. ---- Queer, 107, 233. Chessboard, The, 85. ---- Fallacy, A, 141, 247. ---- Guarded, 95. ---- Non-attacking Arrangements, 96. ---- Problems, 84. ---- Sentence, The, 87, 214. ---- Solitaire, 108, 234. ---- The Chinese, 87, 213. ---- The Crowded, 91, 217. Chestnuts, Buying, 6, 152. Chinese Money, 4, 150. ---- Puzzle, Ancient, 107, 233. ---- ---- _The Fashionable_, 43. Christmas Boxes, The, 4, 150. ---- Present, Mrs. Smiley's, 46, 179. ---- Pudding, The, 43, 178. Cigar Puzzle, The, 119, 242. Circle, The Dissected, 69, 197. Cisterns, How to Make, 54, 188. Civil Service "Howler," 154. Clare, John, 58. Clock Formulæ, 154. ---- Puzzles, 9. ---- The Club, 10, 154. ---- The Railway Station, 11, 155. Clocks, The Three, 11, 154. Clothes Line Puzzle, The, 50, 182. Coast, Round the, 63, 195. Coincidence, A Queer, 2, 148. Coins, The Broken, 5, 150. ---- The Ten, 57, 190. ---- Two Ancient, 140. Combination and Group Problems, 76. Compasses Puzzle, The, 53, 186. Composite Magic Squares, 127, 246. Cone Puzzle, The, 55, 188. Corn, Reaping the, 20, 161. Cornfields, Farmer Lawrence's, 101, 227. Costermonger's Puzzle, The, 6, 152. Counter Problems, Moving, 58. ---- Puzzle, A New, 98, 225. ---- Solitaire, 107, 234. Counters, The Coloured, 91, 217. ---- The Forty-nine, 92, 217. ---- The Nine, 14, 156. ---- The Ten, 15, 156. Crescent Puzzle, The, 52, 184. Crescents of Byzantium, The Five, 92, 219. Cricket Match, The Village, 116, 239. ---- Slow, 116, 239. Cross and Triangle, 35, 169. ---- of Cards, 115, 238. ---- The Folded, 35, 169. ---- The Southern, 93, 220. Crosses, Counter, 81, 207. ---- from One, Two, 35, 168. ---- ---- Three, 169. Crossing River Problems, 112. Crusader, The, 106, 232. Cubes, Sums of, 165. Cushion Covers, The, 46, 179. Cutting-out Puzzle, A, 37, 172. Cyclists' Feast, The, 2, 148. Dairyman, The Honest, 110, 235. Definition, A Question of, 23, 163. De Fonteney, 112. Deified Puzzle, The, 74, 202. Delannoy, 112. De Morgan, A., 27. De Tudor, Sir Edwyn, 12, 155. Diabolique Magic Squares, 120. Diamond Puzzle, The, 74, 202. Dice, A Trick with, 116, 239. ---- Game, The Montenegrin, 119, 242. ---- Numbers, The, 17, 160. Die, Painting the, 84, 210, Digital Analysis, 157, 158. ---- Division, 16, 158. ---- Multiplication, 15, 156. ---- Puzzles, 13. Digits, Adding the, 16, 158. ---- and Squares, 14, 155. ---- Odd and Even, 14, 156. Dilemma, An Amazing, 106, 233. Diophantine Problem, 164. Dissection Puzzle, An Easy, 35, 170. ---- Puzzles, 27. ---- ---- Various, 35. Dividing Magic Squares, 124. Division, Digital, 16, 158. ---- Simple, 23, 163. Doctor's Query, The, 109, 235. Dogs Puzzle, The Five, 92, 218. Domestic Economy, 5, 151. Domino Frame Puzzle, The, 114, 238. Dominoes in Progression, 114, 237. ---- The Eighteen, 123, 245. ---- The Fifteen, 83, 209. ---- The Five, 114, 238. Donkey Riding, 13, 155. Dormitory Puzzle, A, 81, 208. Dovetailed Block, The, 145, 249. Drayton's _Polyolbion_, 58. Dungeon Puzzle, A, 97, 224. Dungeons, The Siberian, 123, 244. ---- The Spanish, 122, 244. Dutchmen's Wives, The, 26, 167. Dynamical Chess Puzzles, 96. Earth's Girdle, The, 139. _Educational Times Reprints_, 204. Eggs, A Deal in, 3, 149. ---- Obtaining the, 140. Election, The Muddletown, 19, 161. ---- The Parish Council, 19, 161. Eleven, The Mystic, 16, 159. Elopements, The Four, 113, 237. Elrick, E., 231. Engines, The Eight, 61, 194. Episcopal Visitation, An, 98, 225. Estate, Farmer Wurzel's, 51, 184. Estates, The Yorkshire, 51, 183. Euclid, 31, 138. Euler, L., 165. Exchange Puzzle, The, 66, 196. Fallacy, A Chessboard, 141, 247. Family Party, A, 8, 153. Fare, The Passenger's, 13, 155. Farmer and his Sheep, The, 22, 163. Fence Problem, A, 21, 162. Fences, The Landowner's, 42, 178. Fermat, 164, 168. Find the Man's Wife, 147, 251. Fly on the Octahedron, The, 70, 198. Fog, Mr. Gubbins in a, 18, 161. Football Players, The, 116, 240. Fraction, A Puzzling, 138. Fractions, More Mixed, 16, 159. Frame Puzzle, The Card, 114, 238. ---- ---- The Domino, 114, 238. Frankenstein, E.N., 232. Frénicle, B., 119, 168. Frogs, The Educated, 59, 194- ---- The Four, 103, 229. ---- The Six, 59, 193. Frost, A.H., 120. Games, Puzzle, 117. ---- Problems concerning, 114. Garden, Lady Belinda's, 52, 186. ---- Puzzle, The, 49, 182. Gardener and the Cook, The, 146, 251. Geometrical Problems, 27. ---- Puzzles, Various, 49. George and the Dragon, St., 101, 227. Getting Upstairs, Such a, 143, 248. Girdle, the Earth's, 139. Goat, The Tethered, 53, 186. Grand Lama's Problem, The, 86, 212. Grasshopper Puzzle, The, 59, 193. Greek Cross Puzzles, 28. ---- ---- Three from One, 169. Greyhound Puzzle, The, 101, 227. Grocer and Draper, The, 5, 151. Gros, L., 248. Group Problems, Combination and, 76. Groups, The Three, 14, 156. Guarini, 229. Hairdresser's Puzzle, The, 137. Halfpennies, Placing, 147, 251. Hampton Court Maze solved, 133. Hannah's Puzzle, 75, 202. Hastings, The Battle of, 23, 164. Hatfield Maze solved, 136. Hat Puzzle, The, 67, 196. Hat-peg Puzzle, The, 93, 221. Hats, The Wrong, 78, 203. Hay, The Trusses of, 18, 161. Heads or Tails, 22, 163. Hearthrug, Mrs. Hobson's, 37, 172. Helmholtz, Von, 41. Honey, The Barrels of, 111, 236. Honeycomb Puzzle, The, 75, 202. Horse Race Puzzle, The, 117, 240. Horseshoes, The Two, 40, 175. Houdin, 68. Hydroplane Question, The, 12, 155. Hymn-board Poser, The, 145, 250. Icosahedron Puzzle, The, 70, 198. Jack and the Beanstalk, 145, 249. Jackson, John, 56. Jaenisch, C.F. de, 92. Jampots, Arranging the, 68, 197. Jealous Husbands, Five, 113, 236. Joiner's Problem, The, 36, 171. ---- ---- Another, 37, 171. Jolly Gaol-Birds, Eight, 122, 243. ---- ---- Nine, 122, 243. Journey, The Queen's, 100, 227. ---- The Rook's, 96, 224. Junior Clerks' Puzzle, The, 4, 150. Juvenile Puzzle, A, 68, 197. Kangaroos, The Four, 102, 228. Kelvin, Lord, 41. Kennel Puzzle, The, 105, 231. King and the Castles, The, 56, 189. ---- The Forsaken, 106, 232. Kite-flying Puzzle, A, 54, 187. Knight-guards, The, 95, 222. Knights, King Arthur's, 77, 203. ---- Tour, Magic, 127, 247. ---- ---- The Cubic, 103, 229. ---- ---- The Four, 103, 229. Labosne, A., 25, 90, 216. Labourer's Puzzle, The, 18, 160. _Ladies' Diary_, 26. Lagrange, J.L., 9. Laisant, C.A., 76. Lamp-posts, Painting the, 19, 161. Leap Year, 155. ---- ---- Ladies, The, 19, 161. Legacy, A Puzzling, 20, 161. Legal Difficulty, A, 23, 163. Le Plongeon, Dr., 29. Letter Block Puzzle, The, 60, 194. ---- Blocks, The Thirty-six, 91, 216. ---- Puzzle, The Fifteen, 79, 205. Level Puzzle, The, 74, 202. Linoleum Cutting, 48, 181. ---- Puzzle, Another, 49, 181. Lion and the Man, The, 97, 224. ---- Hunting, 94, 222. Lions and Crowns, 85, 212. ---- The Four, 88, 214. Lockers Puzzle, The, 14, 156. Locomotion and Speed Puzzles, 11. Lodging-house Difficulty, A, 61, 194. London and Wise, 131. Loyd, Sam, 8, 43, 44, 98, 144, 232, 233. Lucas, Edouard, 16, 76, 112, 121. Luncheons, The City, 77, 203. MacMahon, Major, 109. Magic Knight's Tour, 127, 247. ---- Square Problems, 119. ---- ---- Card, 123, 244. ---- ---- of Composites, 127, 246. ---- ---- of Primes, 125. ---- ---- of Two Degrees, 125, 245. ---- ---- Two New, 125, 245. ---- Strips, 121, 243. Magics, Subtracting, Multiplying, and Dividing, 124. Maiden, The Languishing, 97, 224. Mandarin's Puzzle, The, 103, 230. ---- "T" Puzzle, The, 126, 246. Marketing, Saturday, 27, 168. Market Women, The, 3, 149. Mary and Marmaduke, 7, 152. Mary, How Old was, 8, 153. Massacre of Innocents, 139. Match Mystery, A, 118, 241. ---- Puzzle, A New, 55, 188. Mates, Thirty-six, 106, 233. Mazes and how to thread Them, 127. Measuring, Weighing, and Packing Puzzles, 109. ---- Puzzle, New, 110, 235. Meeting, The Suffragists', 19, 161. Mellor, W.M.F., 242. Ménages, Problême de, 76. Mersenne, M., 168. Mice, Catching the, 65, 196. Milkmaid Puzzle, The, 50, 183. Millionaire's Perplexity, The, 3, 149. Mince Pies, The Twelve, 57, 191. Mine, Inspecting a, 71, 199. Miners' Holiday, The, 23, 163. Miser, The Converted, 21, 162. Mitre, Dissecting a, 35, 170. Monad, The Great, 39, 174. Money, A Queer Thing in, 2, 148. ---- Boxes, The Puzzling, 3, 149. ----, Pocket, 3, 149. ---- Puzzles, 1. ---- Puzzle, A New, 2, 148. ----, Square, 3, 149. _Monist, The_, 125. Monk and the Bridges, The, 75, 202. Monstrosity, The, 108, 234. Montenegrin Dice Game, The, 119, 242. Moreau, 76. Morris, Nine Men's, 58. Mosaics, A Problem in, 90, 215. Mother and Daughter, 7, 152. Motor-car Race, The, 117, 240. ---- Tour, The, 74, 201. ---- Garage Puzzle, The, 62, 195. Motorists, A Puzzle for, 73, 201. Mouse-trap Puzzle, The, 80, 206. Moving Counter Problems, 58. Multiplication, Digital, 15, 156. ---- Queer, 15, 157. ---- Simple, 23, 163. Multiplying Magic Squares, 124. Muncey, J.N., 125. Murray, Sir James, 44. Napoleon, 43, 44. Nasik Magic Squares, 120. Neighbours, Next-Door, 8, 153. Newton, Sir Isaac, 56. Nine Men's Morris, 58. Notation, Scales of, 149. Noughts and Crosses, 58, 117. _Nouvelles Annales de Mathématiques_, 14. Number Checks Puzzle, The, 16, 158. Numbers, Curious, 20, 162. Nuts, The Bag of, 8, 153. Observation, Defective, 4, 150. Octahedron, The Fly on the, 70, 198. Oval, How to draw an, 50, 182. Ovid's Game, 58. Packing in Russia, Gold, 111, 236. ---- Puzzles, Measuring, Weighing, and, 109. ---- Puzzle, A, 111, 236. Pandiagonal Magic Squares, 120. Papa's Puzzle, 53, 187. Pappus, 53. Paradox Party, The, 137. Party, A Family, 8, 153. Patchwork Puzzles, 46. ---- Puzzle, Another, 48, 180. ---- The Silk, 34, 168. Patience, _Strand_, 116, 239. Pawns, A Puzzle with, 94, 222. ---- Immovable, 106, 233. ---- The Six, 107, 233. ---- The Two, 105, 231. Pearls, The Thirty-three, 18, 160. Pebble Game, The, 117, 240. Pedigree, A Mixed, 8, 153. Pellian Equation, 164, 167. Pennies, The Five, 143, 248. ---- The Twelve, 65, 195. Pension, Drawing her, 12, 155. Pentagon and Square, The, 37, 172. ---- Drawing a, 37. Pfeffermann, M., 125. Pheasant-Shooting, 146, 251. Philadelphia Maze solved, 137. Pierrot's Puzzle, The, 15, 156. Pigs, The Seven, 41, 177. Planck, C., 220, 246. Plane Paradox, 138. Plantation Puzzle, A, 57, 189. ---- The Burmese, 58, 191. Plates and Coins, 65, 195. Plums, The Baskets of, 126, 245. Poe, E.A., 249. Points and Lines Problems, 56. Postage Stamps, The Four, 84, 210. Post-Office Perplexity, A, 1, 148. Potato Puzzle, The, 41, 177. Potatoes, The Basket of, 13, 155. Precocious Baby, The, 139. Presents, Buying, 2, 148. Prime Magic Squares, 125. Printer's Error, A, 20, 162. Prisoners, Exercise for, 104, 230. ---- The Ten, 62, 195. Probabilities, Two Questions in, 5, 150. Problems concerning Games, 114. Puss in the Corner, 118, 240. Puzzle Games, 117. Pyramid, Painting a, 83, 208. Pyramids, Square and Triangular, 167. Pythagoras, 31. "Queen, The," 120. Queens and Bishop Puzzle, 93, 219. ---- The Eight, 89, 215. Queen's Journey, The, 100, 227. ---- Tour, The, 98, 225. Quilt, Mrs. Perkins's, 47, 180. Race Puzzle, The Horse-, 117, 240. ---- The Motor-car, 117, 240. Rackbrane's Little Loss, 21, 163. Railway Muddle, A, 62, 194. ---- Puzzle, A, 61, 194. ---- Stations, The Three, 49, 182. _Rational Amusement for Winter Evenings_, 56. Rectangles, Counting the, 105, 232. Reiss, M., 58. Relationships, Queer, 8, 153. Reversals, A Puzzle in, 5, 151. River Axe, Crossing the, 112, 236. River Problems, Crossing, 112. Rookery, The, 105, 232. Rook's Journey, The, 96, 224. ---- Tour, The, 96, 223. Rooks, The Eight, 88, 214. ---- The Two, 117, 240. Round Table, The, 80, 205. Route Problems, Unicursal and, 68. Ruby Brooch, The, 144, 249. Sabbath Puzzle, The, 144, 249. Sailor's Puzzle, The, 71, 199. Sayles, H.A., 125. Schoolboys, The Nine, 80, 205. Schoolgirls, The Fifteen, 80, 204. Scramble, The Great, 19, 161. Sculptor's Problem, The, 23, 164. Second Day of Week, 139. See-Saw Puzzle, The, 22, 163. Semi-Nasik Magic Squares, 120. Senior and Junior, 140. Sevens, The Four, 17, 160. Sharp's Puzzle, 230. Sheepfold, The, 52, 184. Sheep Pens, The Six, 55, 189. ---- The Sixteen, 80, 206. ---- The Three, 92, 217. ---- Those Fifteen, 77, 203. Shopping Perplexity, A, 4, 150. Shuldham, C.D., 125, 126. Siberian Dungeons, The, 123, 244. Simpleton, The Village, 11, 155. Skater, The Scientific, 100, 226. Skeat, Professor, 127. Solitaire, Central, 63, 195. ---- Chessboard, 108, 234. ---- Counter, 107, 234. Sons, The Four, 49, 181. Spanish Dungeons, The, 122, 244. ---- Miser, The, 24, 164. Speed and Locomotion Puzzles, 11. ---- Average, 11, 155. Spiral, Drawing a, 50, 182. Spot on the Table, The, 17, 160. Square Numbers, Check for, 13. ---- ---- Digital, 16, 159. ---- of Veneer, The, 39, 175. ---- Puzzle, An Easy, 35, 170. Squares, A Problem in, 23, 163. ---- Circling the, 21, 162. ---- Difference of Two, 167. ---- Magic, 119. ---- Sum of Two, 165, 175. ---- The Chocolate, 35, 170. Stalemate, 106, 232. Stamp-licking, The Gentle Art of, 91, 217. Star Puzzle, The, 99, 226. Stars, The Eight, 89, 215. ---- The Forty-nine, 100, 226. Statical Chess Puzzles, 88. Sticks, The Eight, 53, 186. Stonemason's Problem, The, 25, 165. Stop-watch, The, 11, 154. _Strand Magazine, The_, 44, 116, 220. _Strand_ Patience, 116, 239. Stream, Crossing the, 112, 236. Strutt, Joseph, 59. Subtracting Magic Squares, 124. Sultan's Army, The, 25, 165. Suppers, The New Year's Eve, 3, 149. Surname, Find Ada's, 27, 168. Swastika, The, 29, 31, 169. "T" Card Puzzle, The, 115, 239. Table, The Round, 80, 205. Table-top and Stools, The, 38, 173. Tangram Paradox, A, 43, 178. Target, The Cross, 84, 210. Tarry, 112. Tartaglia, 25, 109, 112. Tea, Mixing the, 111, 235. Telegraph Posts, The, 139. Tennis Tournament, A, 78, 203. Tetrahedron, Building the, 82, 208. Thief, Catching the, 19, 161. Thrift, A Study in, 25, 166. Thompson, W.H., 232. Ticket Puzzle, The Excursion, 5, 151. Time Puzzle, A, 10, 153. ---- What was the, 10, 153. Tiring Irons, The, 142, 247. _Tit-Bits_, 58, 79, 124, 251. Torn Number, The, 20, 162. Torpedo Practice, 67, 196. Tour, The Cyclists', 71, 199. ---- The Grand, 72, 200. ---- The Queen's, 98, 225. ---- The Rook's, 96, 223. Towns, Visiting the, 70, 198. Trains, The Two, 11, 155. Treasure Boxes, The Nine, 24, 164. Trees, The Twenty-one, 57, 190. Trémaux, M., 133, 135. Triangle, The Dissected, 38, 173. Triangular Numbers, 13, 25, 166. ---- ---- Check for, 13. Troublesome Eight, The, 121, 242. Tube Inspector's Puzzle, The, 69, 198. ---- Railway, Heard on the, 8, 153. Turks and Russians, 58, 191. Turnings, The Fifteen, 70, 198. Twickenham Puzzle, The, 60, 194. Two Pieces Problem, The, 96. Unclassified Puzzles, 142. Unicursal and Route Problems, 68. Union Jack, The, 50, 69, 197. Vandermonde, A., 58, 103. Veil, Under the, 90, 216. Verne, Jules, 249. Victoria Cross Puzzle, The, 60, 194. Village, A Wonderful, 142, 247. Villages, The Three, 12, 155. Villas, The Eight, 80, 206. Vortex Rings, 40. Voter's Puzzle, The, 75, 202. Wall, The Puzzle, 52, 184. Wallis, J., 142. ---- (Another), 220. Walls, The Garden, 52, 185. Wapshaw's Wharf Mystery, The, 10, 153. War Puzzle Game, The, 118, 240. Wassail Bowl, The, 109, 235. Watch, A Puzzling, 10, 153. Water, Gas, and Electricity, 73, 200. _Weekly Dispatch, The_, 28, 124, 125, 146, 148. Weighing Puzzles, Measuring, Packing, and, 109. Wheels, Concerning, 55, 188. Who was First? 142, 247. Whyte, W.T., 147. Widow's Legacy, The, 2, 148. Wife, Find the Man's, 147, 251. Wilkinson, Rev. Mr., 193. Wilson, Professor, 29. Wilson's Poser, 9, 153. Wine and Water, 110, 235. ---- The Keg of, 110, 235. Wotherspoon, G., 244. Yacht race, The, 99, 226. Youthful Precocity, 1, 148. Zeno, 139. THE END. 
5768	----	Copyright (C) 2002 by Sam Williams. Free As in Freedom: Richard Stallman's Crusade for Free Software. By Sam Williams Available on the web at: http://www.faifzilla.org/ Produced under the Free Documentation License Table of Contents Chapter 1 For Want of a Printer Chapter 2 2001: A Hacker's Odyssey Chapter 3 A Portrait of the Hacker as a Young Man Chapter 4 Impeach God Chapter 5 Small Puddle of Freedom Chapter 6 The Emacs Commune Chapter 7 A Stark Moral Choice Chapter 8 St. Ignucius Chapter 9 The GNU General Public License Chapter 10 GNU/Linux Chapter 11 Open Source Chapter 12 A Brief Journey Through Hacker Hell Chapter 13 Continuing the Fight Chapter 14 Epilogue: Chapter 15 Appendix A : Terminology Chapter 16 Appendix B Hack, Hackers, and Hacking Chapter 17 Appendix C GNU Free Documentation License (GFDL) Preface The work of Richard M. Stallman literally speaks for itself. From the documented source code to the published papers to the recorded speeches, few people have expressed as much willingness to lay their thoughts and their work on the line. Such openness-if one can pardon a momentary un-Stallman adjective-is refreshing. After all, we live in a society that treats information, especially personal information, as a valuable commodity. The question quickly arises. Why would anybody want to part with so much information and yet appear to demand nothing in return? As we shall see in later chapters, Stallman does not part with his words or his work altruistically. Every program, speech, and on-the-record bon mot comes with a price, albeit not the kind of price most people are used to paying. I bring this up not as a warning, but as an admission. As a person who has spent the last year digging up facts on Stallman's personal history, it's more than a little intimidating going up against the Stallman oeuvre. "Never pick a fight with a man who buys his ink by the barrel," goes the old Mark Twain adage. In the case of Stallman, never attempt the definitive biography of a man who trusts his every thought to the public record. For the readers who have decided to trust a few hours of their time to exploring this book, I can confidently state that there are facts and quotes in here that one won't find in any Slashdot story or Google search. Gaining access to these facts involves paying a price, however. In the case of the book version, you can pay for these facts the traditional manner, i.e., by purchasing the book. In the case of the electronic versions, you can pay for these facts in the free software manner. Thanks to the folks at O'Reilly & Associates, this book is being distributed under the GNU Free Documentation License, meaning you can help to improve the work or create a personalized version and release that version under the same license. If you are reading an electronic version and prefer to accept the latter payment option, that is, if you want to improve or expand this book for future readers, I welcome your input. Starting in June, 2002, I will be publishing a bare bones HTML version of the book on the web site, http://www.faifzilla.org. My aim is to update it regularly and expand the Free as in Freedom story as events warrant. If you choose to take the latter course, please review Appendix C of this book. It provides a copy of your rights under the GNU Free Documentation License. For those who just plan to sit back and read, online or elsewhere, I consider your attention an equally valuable form of payment. Don't be surprised, though, if you, too, find yourself looking for other ways to reward the good will that made this work possible. One final note: this is a work of journalism, but it is also a work of technical documentation. In the process of writing and editing this book, the editors and I have weighed the comments and factual input of various participants in the story, including Richard Stallman himself. We realize there are many technical details in this story that may benefit from additional or refined information. As this book is released under the GFDL, we are accepting patches just like we would with any free software program. Accepted changes will be posted electronically and will eventually be incorporated into future printed versions of this work. If you would like to contribute to the further improvement of this book, you can reach me at sam@inow.com. Comments and Questions Please address comments and questions concerning this book to the publisher: O'Reilly & Associates, Inc. 1005 Gravenstein Highway North Sebastopol, CA 95472 (800) 998-9938 (in the United States or Canada) (707) 829-0515 (international/local) (707) 829-0104 (fax) There is a web page for this book, which lists errata, examples, or any additional information. The site also includes a link to a forum where you can discuss the book with the author and other readers. You can access this site at: http://www.oreilly.com/catalog/freedom/ To comment or ask technical questions about this book, send email to: bookquestions@oreilly.com For more information about books, conferences, Resource Centers, and the O'Reilly Network, see the O'Reilly web site at: http://www.oreilly.com Acknowledgments Special thanks to Henning Gutmann for sticking by this book. Special thanks to Aaron Oas for suggesting the idea to Tracy in the first place. Thanks to Laurie Petrycki, Jeffrey Holcomb, and all the others at O'Reilly & Associates. Thanks to Tim O'Reilly for backing this book. Thanks to all the first-draft reviewers: Bruce Perens, Eric Raymond, Eric Allman, Jon Orwant, Julie and Gerald Jay Sussman, Hal Abelson, and Guy Steele. I hope you enjoy this typo-free version. Thanks to Alice Lippman for the interviews, cookies, and photographs. Thanks to my family, Steve, Jane, Tish, and Dave. And finally, last but not least: thanks to Richard Stallman for having the guts and endurance to "show us the code." Sam Williams For Want of a Printer I fear the Greeks. Even when they bring gifts. ---Virgil The Aeneid The new printer was jammed, again. Richard M. Stallman, a staff software programmer at the Massachusetts Institute of Technology's Artificial Intelligence Laboratory (AI Lab), discovered the malfunction the hard way. An hour after sending off a 50-page file to the office laser printer, Stallman, 27, broke off a productive work session to retrieve his documents. Upon arrival, he found only four pages in the printer's tray. To make matters even more frustrating, the four pages belonged to another user, meaning that Stallman's print job and the unfinished portion of somebody else's print job were still trapped somewhere within the electrical plumbing of the lab's computer network. Waiting for machines is an occupational hazard when you're a software programmer, so Stallman took his frustration with a grain of salt. Still, the difference between waiting for a machine and waiting on a machine is a sizable one. It wasn't the first time he'd been forced to stand over the printer, watching pages print out one by one. As a person who spent the bulk of his days and nights improving the efficiency of machines and the software programs that controlled them, Stallman felt a natural urge to open up the machine, look at the guts, and seek out the root of the problem. Unfortunately, Stallman's skills as a computer programmer did not extend to the mechanical-engineering realm. As freshly printed documents poured out of the machine, Stallman had a chance to reflect on other ways to circumvent the printing jam problem. How long ago had it been that the staff members at the AI Lab had welcomed the new printer with open arms? Stallman wondered. The machine had been a donation from the Xerox Corporation. A cutting edge prototype, it was a modified version of the popular Xerox photocopier. Only instead of making copies, it relied on software data piped in over a computer network to turn that data into professional-looking documents. Created by engineers at the world-famous Xerox Palo Alto Research Facility, it was, quite simply, an early taste of the desktop-printing revolution that would seize the rest of the computing industry by the end of the decade. Driven by an instinctual urge to play with the best new equipment, programmers at the AI Lab promptly integrated the new machine into the lab's sophisticated computing infrastructure. The results had been immediately pleasing. Unlike the lab's old laser printer, the new Xerox machine was fast. Pages came flying out at a rate of one per second, turning a 20-minute print job into a 2-minute print job. The new machine was also more precise. Circles came out looking like circles, not ovals. Straight lines came out looking like straight lines, not low-amplitude sine waves. It was, for all intents and purposes, a gift too good to refuse. It wasn't until a few weeks after its arrival that the machine's flaws began to surface. Chief among the drawbacks was the machine's inherent susceptibility to paper jams. Engineering-minded programmers quickly understood the reason behind the flaw. As a photocopier, the machine generally required the direct oversight of a human operator. Figuring that these human operators would always be on hand to fix a paper jam, if it occurred, Xerox engineers had devoted their time and energies to eliminating other pesky problems. In engineering terms, user diligence was built into the system. In modifying the machine for printer use, Xerox engineers had changed the user-machine relationship in a subtle but profound way. Instead of making the machine subservient to an individual human operator, they made it subservient to an entire networked population of human operators. Instead of standing directly over the machine, a human user on one end of the network sent his print command through an extended bucket-brigade of machines, expecting the desired content to arrive at the targeted destination and in proper form. It wasn't until he finally went to check up on the final output that he realized how little of the desired content had made it through. Stallman himself had been of the first to identify the problem and the first to suggest a remedy. Years before, when the lab was still using its old printer, Stallman had solved a similar problem by opening up the software program that regulated the printer on the lab's PDP-11 machine. Stallman couldn't eliminate paper jams, but he could insert a software command that ordered the PDP-11 to check the printer periodically and report back to the PDP-10, the lab's central computer. To ensure that one user's negligence didn't bog down an entire line of print jobs, Stallman also inserted a software command that instructed the PDP-10 to notify every user with a waiting print job that the printer was jammed. The notice was simple, something along the lines of "The printer is jammed, please fix it," and because it went out to the people with the most pressing need to fix the problem, chances were higher that the problem got fixed in due time. As fixes go, Stallman's was oblique but elegant. It didn't fix the mechanical side of the problem, but it did the next best thing by closing the information loop between user and machine. Thanks to a few additional lines of software code, AI Lab employees could eliminate the 10 or 15 minutes wasted each week in running back and forth to check on the printer. In programming terms, Stallman's fix took advantage of the amplified intelligence of the overall network. "If you got that message, you couldn't assume somebody else would fix it," says Stallman, recalling the logic. "You had to go to the printer. A minute or two after the printer got in trouble, the two or three people who got messages arrive to fix the machine. Of those two or three people, one of them, at least, would usually know how to fix the problem." Such clever fixes were a trademark of the AI Lab and its indigenous population of programmers. Indeed, the best programmers at the AI Lab disdained the term programmer, preferring the more slangy occupational title of hacker instead. The job title covered a host of activities-everything from creative mirth making to the improvement of existing software and computer systems. Implicit within the title, however, was the old-fashioned notion of Yankee ingenuity. To be a hacker, one had to accept the philosophy that writing a software program was only the beginning. Improving a program was the true test of a hacker's skills.For more on the term "hacker," see Appendix B. Such a philosophy was a major reason why companies like Xerox made it a policy to donate their machines and software programs to places where hackers typically congregated. If hackers improved the software, companies could borrow back the improvements, incorporating them into update versions for the commercial marketplace. In corporate terms, hackers were a leveragable community asset, an auxiliary research-and-development division available at minimal cost. It was because of this give-and-take philosophy that when Stallman spotted the print-jam defect in the Xerox laser printer, he didn't panic. He simply looked for a way to update the old fix or " hack" for the new system. In the course of looking up the Xerox laser-printer software, however, Stallman made a troubling discovery. The printer didn't have any software, at least nothing Stallman or a fellow programmer could read. Until then, most companies had made it a form of courtesy to publish source-code files-readable text files that documented the individual software commands that told a machine what to do. Xerox, in this instance, had provided software files in precompiled, or binary, form. Programmers were free to open the files up if they wanted to, but unless they were an expert in deciphering an endless stream of ones and zeroes, the resulting text was pure gibberish. Although Stallman knew plenty about computers, he was not an expert in translating binary files. As a hacker, however, he had other resources at his disposal. The notion of information sharing was so central to the hacker culture that Stallman knew it was only a matter of time before some hacker in some university lab or corporate computer room proffered a version of the laser-printer source code with the desired source-code files. After the first few printer jams, Stallman comforted himself with the memory of a similar situation years before. The lab had needed a cross-network program to help the PDP-11 work more efficiently with the PDP-10. The lab's hackers were more than up to the task, but Stallman, a Harvard alumnus, recalled a similar program written by programmers at the Harvard computer-science department. The Harvard computer lab used the same model computer, the PDP-10, albeit with a different operating system. The Harvard computer lab also had a policy requiring that all programs installed on the PDP-10 had to come with published source-code files. Taking advantage of his access to the Harvard computer lab, Stallman dropped in, made a copy of the cross-network source code, and brought it back to the AI Lab. He then rewrote the source code to make it more suitable for the AI Lab's operating system. With no muss and little fuss, the AI Lab shored up a major gap in its software infrastructure. Stallman even added a few features not found in the original Harvard program, making the program even more useful. "We wound up using it for several years," Stallman says. From the perspective of a 1970s-era programmer, the transaction was the software equivalent of a neighbor stopping by to borrow a power tool or a cup of sugar from a neighbor. The only difference was that in borrowing a copy of the software for the AI Lab, Stallman had done nothing to deprive Harvard hackers the use of their original program. If anything, Harvard hackers gained in the process, because Stallman had introduced his own additional features to the program, features that hackers at Harvard were perfectly free to borrow in return. Although nobody at Harvard ever came over to borrow the program back, Stallman does recall a programmer at the private engineering firm, Bolt, Beranek & Newman, borrowing the program and adding a few additional features, which Stallman eventually reintegrated into the AI Lab's own source-code archive. "A program would develop the way a city develops," says Stallman, recalling the software infrastructure of the AI Lab. "Parts would get replaced and rebuilt. New things would get added on. But you could always look at a certain part and say, `Hmm, by the style, I see this part was written back in the early 60s and this part was written in the mid-1970s.'" Through this simple system of intellectual accretion, hackers at the AI Lab and other places built up robust creations. On the west coast, computer scientists at UC Berkeley, working in cooperation with a few low-level engineers at AT&T, had built up an entire operating system using this system. Dubbed Unix, a play on an older, more academically respectable operating system called Multics, the software system was available to any programmer willing to pay for the cost of copying the program onto a new magnetic tape and shipping it. Not every programmer participating in this culture described himself as a hacker, but most shared the sentiments of Richard M. Stallman. If a program or software fix was good enough to solve your problems, it was good enough to solve somebody else's problems. Why not share it out of a simple desire for good karma? The fact that Xerox had been unwilling to share its source-code files seemed a minor annoyance at first. In tracking down a copy of the source-code files, Stallman says he didn't even bother contacting Xerox. "They had already given us the laser printer," Stallman says. "Why should I bug them for more?" When the desired files failed to surface, however, Stallman began to grow suspicious. The year before, Stallman had experienced a blow up with a doctoral student at Carnegie Mellon University. The student, Brian Reid, was the author of a useful text-formatting program dubbed Scribe. One of the first programs that gave a user the power to define fonts and type styles when sending a document over a computer network, the program was an early harbinger of HTML, the lingua franca of the World Wide Web. In 1979, Reid made the decision to sell Scribe to a Pittsburgh-area software company called Unilogic. His graduate-student career ending, Reid says he simply was looking for a way to unload the program on a set of developers that would take pains to keep it from slipping into the public domain. To sweeten the deal, Reid also agreed to insert a set of time-dependent functions- "time bombs" in software-programmer parlance-that deactivated freely copied versions of the program after a 90-day expiration date. To avoid deactivation, users paid the software company, which then issued a code that defused the internal time-bomb feature. For Reid, the deal was a win-win. Scribe didn't fall into the public domain, and Unilogic recouped on its investment. For Stallman, it was a betrayal of the programmer ethos, pure and simple. Instead of honoring the notion of share-and-share alike, Reid had inserted a way for companies to compel programmers to pay for information access. As the weeks passed and his attempts to track down Xerox laser-printer source code hit a brick wall, Stallman began to sense a similar money-for-code scenario at work. Before Stallman could do or say anything about it, however, good news finally trickled in via the programmer grapevine. Word had it that a scientist at the computer-science department at Carnegie Mellon University had just departed a job at the Xerox Palo Alto Research Center. Not only had the scientist worked on the laser printer in question, but according to rumor, he was still working on it as part of his research duties at Carnegie Mellon. Casting aside his initial suspicion, Stallman made a firm resolution to seek out the person in question during his next visit to the Carnegie Mellon campus. He didn't have to wait long. Carnegie Mellon also had a lab specializing in artificial-intelligence research, and within a few months, Stallman had a business-related reason to visit the Carnegie Mellon campus. During that visit, he made sure to stop by the computer-science department. Department employees directed him to the office of the faculty member leading the Xerox project. When Stallman reached the office, he found the professor working there. In true engineer-to-engineer fashion, the conversation was cordial but blunt. After briefly introducing himself as a visitor from MIT, Stallman requested a copy of the laser-printer source code so that he could port it to the PDP-11. To his surprise, the professor refused to grant his request. "He told me that he had promised not to give me a copy," Stallman says. Memory is a funny thing. Twenty years after the fact, Stallman's mental history tape is notoriously blank in places. Not only does he not remember the motivating reason for the trip or even the time of year during which he took it, he also has no recollection of the professor or doctoral student on the other end of the conversation. According to Reid, the person most likely to have fielded Stallman's request is Robert Sproull, a former Xerox PARC researcher and current director of Sun Laboratories, a research division of the computer-technology conglomerate Sun Microsystems. During the 1970s, Sproull had been the primary developer of the laser-printer software in question while at Xerox PARC. Around 1980, Sproull took a faculty research position at Carnegie Mellon where he continued his laser-printer work amid other projects. "The code that Stallman was asking for was leading-edge state-of-the-art code that Sproull had written in the year or so before going to Carnegie Mellon," recalls Reid. "I suspect that Sproull had been at Carnegie Mellon less than a month before this request came in." When asked directly about the request, however, Sproull draws a blank. "I can't make a factual comment," writes Sproull via email. "I have absolutely no recollection of the incident." With both participants in the brief conversation struggling to recall key details-including whether the conversation even took place-it's hard to gauge the bluntness of Sproull's refusal, at least as recalled by Stallman. In talking to audiences, Stallman has made repeated reference to the incident, noting that Sproull's unwillingness to hand over the source code stemmed from a nondisclosure agreement, a contractual agreement between Sproull and the Xerox Corporation giving Sproull, or any other signatory, access the software source code in exchange for a promise of secrecy. Now a standard item of business in the software industry, the nondisclosure agreement, or NDA, was a novel development at the time, a reflection of both the commercial value of the laser printer to Xerox and the information needed to run it. "Xerox was at the time trying to make a commercial product out of the laser printer," recalls Reid. "They would have been insane to give away the source code." For Stallman, however, the NDA was something else entirely. It was a refusal on the part of Xerox and Sproull, or whomever the person was that turned down his source-code request that day, to participate in a system that, until then, had encouraged software programmers to regard programs as communal resources. Like a peasant whose centuries-old irrigation ditch had grown suddenly dry, Stallman had followed the ditch to its source only to find a brand-spanking-new hydroelectric dam bearing the Xerox logo. For Stallman, the realization that Xerox had compelled a fellow programmer to participate in this newfangled system of compelled secrecy took a while to sink in. At first, all he could focus on was the personal nature of the refusal. As a person who felt awkward and out of sync in most face-to-face encounters, Stallman's attempt to drop in on a fellow programmer unannounced had been intended as a demonstration of neighborliness. Now that the request had been refused, it felt like a major blunder. "I was so angry I couldn't think of a way to express it. So I just turned away and walked out without another word," Stallman recalls. "I might have slammed the door. Who knows? All I remember is wanting to get out of there." Twenty years after the fact, the anger still lingers, so much so that Stallman has elevated the event into a major turning point. Within the next few months, a series of events would befall both Stallman and the AI Lab hacker community that would make 30 seconds worth of tension in a remote Carnegie Mellon office seem trivial by comparison. Nevertheless, when it comes time to sort out the events that would transform Stallman from a lone hacker, instinctively suspicious of centralized authority, to a crusading activist applying traditional notions of liberty, equality, and fraternity to the world of software development, Stallman singles out the Carnegie Mellon encounter for special attention. "It encouraged me to think about something that I'd already been thinking about," says Stallman. "I already had an idea that software should be shared, but I wasn't sure how to think about that. My thoughts weren't clear and organized to the point where I could express them in a concise fashion to the rest of the world." Although previous events had raised Stallman's ire, he says it wasn't until his Carnegie Mellon encounter that he realized the events were beginning to intrude on a culture he had long considered sacrosanct. As an elite programmer at one of the world's elite institutions, Stallman had been perfectly willing to ignore the compromises and bargains of his fellow programmers just so long as they didn't interfere with his own work. Until the arrival of the Xerox laser printer, Stallman had been content to look down on the machines and programs other computer users grimly tolerated. On the rare occasion that such a program breached the AI Lab's walls-when the lab replaced its venerable Incompatible Time Sharing operating system with a commercial variant, the TOPS 20, for example-Stallman and his hacker colleagues had been free to rewrite, reshape, and rename the software according to personal taste. Now that the laser printer had insinuated itself within the AI Lab's network, however, something had changed. The machine worked fine, barring the occasional paper jam, but the ability to modify according to personal taste had disappeared. From the viewpoint of the entire software industry, the printer was a wake-up call. Software had become such a valuable asset that companies no longer felt the need to publicize source code, especially when publication meant giving potential competitors a chance to duplicate something cheaply. From Stallman's viewpoint, the printer was a Trojan Horse. After a decade of failure, privately owned software-future hackers would use the term " proprietary" software-had gained a foothold inside the AI Lab through the sneakiest of methods. It had come disguised as a gift. That Xerox had offered some programmers access to additional gifts in exchange for secrecy was also galling, but Stallman takes pains to note that, if presented with such a quid pro quo bargain at a younger age, he just might have taken the Xerox Corporation up on its offer. The awkwardness of the Carnegie Mellon encounter, however, had a firming effect on Stallman's own moral lassitude. Not only did it give him the necessary anger to view all future entreaties with suspicion, it also forced him to ask the uncomfortable question: what if a fellow hacker dropped into Stallman's office someday and it suddenly became Stallman's job to refuse the hacker's request for source code? "It was my first encounter with a nondisclosure agreement, and it immediately taught me that nondisclosure agreements have victims," says Stallman, firmly. "In this case I was the victim. [My lab and I] were victims." It was a lesson Stallman would carry with him through the tumultuous years of the 1980s, a decade during which many of his MIT colleagues would depart the AI Lab and sign nondisclosure agreements of their own. Because most nondisclosure aggreements (NDAs) had expiration dates, few hackers who did sign them saw little need for personal introspection. Sooner or later, they reasoned, the software would become public knowledge. In the meantime, promising to keep the software secret during its earliest development stages was all a part of the compromise deal that allowed hackers to work on the best projects. For Stallman, however, it was the first step down a slippery slope. "When somebody invited me to betray all my colleagues in that way, I remembered how angry I was when somebody else had done that to me and my whole lab," Stallman says. "So I said, `Thank you very much for offering me this nice software package, but I can't accept it on the conditions that you're asking for, so I'm going to do without it.'" As Stallman would quickly learn, refusing such requests involved more than personal sacrifice. It involved segregating himself from fellow hackers who, though sharing a similar distaste for secrecy, tended to express that distaste in a more morally flexible fashion. It wasn't long before Stallman, increasingly an outcast even within the AI Lab, began billing himself as "the last true hacker," isolating himself further and further from a marketplace dominated by proprietary software. Refusing another's request for source code, Stallman decided, was not only a betrayal of the scientific mission that had nurtured software development since the end of World War II, it was a violation of the Golden Rule, the baseline moral dictate to do unto others as you would have them do unto you. Hence the importance of the laser printer and the encounter that resulted from it. Without it, Stallman says, his life might have followed a more ordinary path, one balancing the riches of a commercial programmer with the ultimate frustration of a life spent writing invisible software code. There would have been no sense of clarity, no urgency to address a problem others weren't addressing. Most importantly, there would have been no righteous anger, an emotion that, as we soon shall see, has propelled Stallman's career as surely as any political ideology or ethical belief. "From that day forward, I decided this was something I could never participate in," says Stallman, alluding to the practice of trading personal liberty for the sake of convenience-Stallman's description of the NDA bargain-as well as the overall culture that encouraged such ethically suspect deal-making in the first place. "I decided never to make other people victims just like I had been a victim." 2001: A Hacker's Odyssey The New York University computer-science department sits inside Warren Weaver Hall, a fortress-like building located two blocks east of Washington Square Park. Industrial-strength air-conditioning vents create a surrounding moat of hot air, discouraging loiterers and solicitors alike. Visitors who breach the moat encounter another formidable barrier, a security check-in counter immediately inside the building's single entryway. Beyond the security checkpoint, the atmosphere relaxes somewhat. Still, numerous signs scattered throughout the first floor preach the dangers of unsecured doors and propped-open fire exits. Taken as a whole, the signs offer a reminder: even in the relatively tranquil confines of pre-September 11, 2001, New York, one can never be too careful or too suspicious. The signs offer an interesting thematic counterpoint to the growing number of visitors gathering in the hall's interior atrium. A few look like NYU students. Most look like shaggy-aired concert-goers milling outside a music hall in anticipation of the main act. For one brief morning, the masses have taken over Warren Weaver Hall, leaving the nearby security attendant with nothing better to do but watch Ricki Lake on TV and shrug her shoulders toward the nearby auditorium whenever visitors ask about "the speech." Once inside the auditorium, a visitor finds the person who has forced this temporary shutdown of building security procedures. The person is Richard M. Stallman, founder of the GNU Project, original president of the Free Software Foundation, winner of the 1990 MacArthur Fellowship, winner of the Association of Computing Machinery's Grace Murray Hopper Award (also in 1990), corecipient of the Takeda Foundation's 2001 Takeda Award, and former AI Lab hacker. As announced over a host of hacker-related web sites, including the GNU Project's own http://www.gnu.org site, Stallman is in Manhattan, his former hometown, to deliver a much anticipated speech in rebuttal to the Microsoft Corporation's recent campaign against the GNU General Public License. The subject of Stallman's speech is the history and future of the free software movement. The location is significant. Less than a month before, Microsoft senior vice president Craig Mundie appeared at the nearby NYU Stern School of Business, delivering a speech blasting the General Public License, or GPL, a legal device originally conceived by Stallman 16 years before. Built to counteract the growing wave of software secrecy overtaking the computer industry-a wave first noticed by Stallman during his 1980 troubles with the Xerox laser printer-the GPL has evolved into a central tool of the free software community. In simplest terms, the GPL locks software programs into a form of communal ownership-what today's legal scholars now call the "digital commons"-through the legal weight of copyright. Once locked, programs remain unremovable. Derivative versions must carry the same copyright protection-even derivative versions that bear only a small snippet of the original source code. For this reason, some within the software industry have taken to calling the GPL a "viral" license, because it spreads itself to every software program it touches. Actually, the GPL's powers are not quite that potent. According to section 10 of the GNU General Public License, Version 2 (1991), the viral nature of the license depends heavily on the Free Software Foundation's willingness to view a program as a derivative work, not to mention the existing license the GPL would replace. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software that is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally. "To compare something to a virus is very harsh," says Stallman. "A spider plant is a more accurate comparison; it goes to another place if you actively take a cutting." For more information on the GNU General Public License, visit [http://www.gnu.org/copyleft/gpl.html.] In an information economy increasingly dependent on software and increasingly beholden to software standards, the GPL has become the proverbial "big stick." Even companies that once laughed it off as software socialism have come around to recognize the benefits. Linux, the Unix-like kernel developed by Finnish college student Linus Torvalds in 1991, is licensed under the GPL, as are many of the world's most popular programming tools: GNU Emacs, the GNU Debugger, the GNU C Compiler, etc. Together, these tools form the components of a free software operating system developed, nurtured, and owned by the worldwide hacker community. Instead of viewing this community as a threat, high-tech companies like IBM, Hewlett Packard, and Sun Microsystems have come to rely upon it, selling software applications and services built to ride atop the ever-growing free software infrastructure. They've also come to rely upon it as a strategic weapon in the hacker community's perennial war against Microsoft, the Redmond, Washington-based company that, for better or worse, has dominated the PC-software marketplace since the late 1980s. As owner of the popular Windows operating system, Microsoft stands to lose the most in an industry-wide shift to the GPL license. Almost every line of source code in the Windows colossus is protected by copyrights reaffirming the private nature of the underlying source code or, at the very least, reaffirming Microsoft's legal ability to treat it as such. From the Microsoft viewpoint, incorporating programs protected by the "viral" GPL into the Windows colossus would be the software equivalent of Superman downing a bottle of Kryptonite pills. Rival companies could suddenly copy, modify, and sell improved versions of Windows, rendering the company's indomitable position as the No. 1 provider of consumer-oriented software instantly vulnerable. Hence the company's growing concern over the GPL's rate of adoption. Hence the recent Mundie speech blasting the GPL and the " open source" approach to software development and sales. And hence Stallman's decision to deliver a public rebuttal to that speech on the same campus here today. 20 years is a long time in the software industry. Consider this: in 1980, when Richard Stallman was cursing the AI Lab's Xerox laser printer, Microsoft, the company modern hackers view as the most powerful force in the worldwide software industry, was still a privately held startup. IBM, the company hackers used to regard as the most powerful force in the worldwide software industry, had yet to to introduce its first personal computer, thereby igniting the current low-cost PC market. Many of the technologies we now take for granted-the World Wide Web, satellite television, 32-bit video-game consoles-didn't even exist. The same goes for many of the companies that now fill the upper echelons of the corporate establishment, companies like AOL, Sun Microsystems, Amazon.com, Compaq, and Dell. The list goes on and on. The fact that the high-technology marketplace has come so far in such little time is fuel for both sides of the GPL debate. GPL-proponents point to the short lifespan of most computer hardware platforms. Facing the risk of buying an obsolete product, consumers tend to flock to companies with the best long-term survival. As a result, the software marketplace has become a winner-take-all arena.See Shubha Ghosh, "Revealing the Microsoft Windows Source Code," Gigalaw.com (January, 2000). http://www.gigalaw.com/articles/ghosh-2000-01-p1.html The current, privately owned software environment, GPL-proponents say, leads to monopoly abuse and stagnation. Strong companies suck all the oxygen out of the marketplace for rival competitors and innovative startups. GPL-opponents argue just the opposite. Selling software is just as risky, if not more risky, than buying software, they say. Without the legal guarantees provided by private software licenses, not to mention the economic prospects of a privately owned "killer app" (i.e., a breakthrough technology that launches an entirely new market),Killer apps don't have to be proprietary. Witness, of course, the legendary Mosaic browser, a program whose copyright permits noncommercial derivatives with certain restrictions. Still, I think the reader gets the point: the software marketplace is like the lottery. The bigger the potential payoff, the more people want to participate. For a good summary of the killer-app phenomenon, see Philip Ben-David, "Whatever Happened to the `Killer App'?" e-Commerce News (December 7, 2000). companies lose the incentive to participate. Once again, the market stagnates and innovation declines. As Mundie himself noted in his May 3 address on the same campus, the GPL's "viral" nature "poses a threat" to any company that relies on the uniqueness of its software as a competitive asset. Added Mundie: It also fundamentally undermines the independent commercial software sector because it effectively makes it impossible to distribute software on a basis where recipients pay for the product rather than just the cost of distributionSee Craig Mundie, "The Commercial Software Model," senior vice president, Microsoft Corp. Excerpted from an online transcript of Mundie's May 3,speech to the New York University Stern School of Business. http://www.ecommercetimes.com/perl/story/5893.html 001, http://www.microsoft.com/presspass/exec/craig/05-03sharedsource.asp The mutual success of GNU/ LinuxThe acronym GNU stands for "GNU's not Unix." In another portion of the May 29, 2001, NYU speech, Stallman summed up the acronym's origin: We hackers always look for a funny or naughty name for a program, because naming a program is half the fun of writing the program. We also had a tradition of recursive acronyms, to say that the program that you're writing is similar to some existing program . . . I looked for a recursive acronym for Something Is Not UNIX. And I tried all 26 letters and discovered that none of them was a word. I decided to make it a contraction. That way I could have a three-letter acronym, for Something's Not UNIX. And I tried letters, and I came across the word "GNU." That was it. Although a fan of puns, Stallman recommends that software users pronounce the "g" at the beginning of the acronym (i.e., "gah-new"). Not only does this avoid confusion with the word "gnu," the name of the African antelope, Connochaetes gnou , it also avoids confusion with the adjective "new." "We've been working on it for 17 years now, so it is not exactly new any more," Stallman says. Source: author notes and online transcript of "Free Software: Freedom and Cooperation," Richard Stallman's May 29, 2001, speech at New York University. http://www.gnu.org/events/rms-nyu-2001-transcript.txt , the amalgamated operating system built around the GPL-protected Linux kernel, and Windows over the last 10 years reveals the wisdom of both perspectives. Nevertheless, the battle for momentum is an important one in the software industry. Even powerful vendors such as Microsoft rely on the support of third-party software developers whose tools, programs, and computer games make an underlying software platform such as Windows more attractive to the mainstream consumer. Citing the rapid evolution of the technology marketplace over the last 20 years, not to mention his own company's admirable track record during that period, Mundie advised listeners to not get too carried away by the free software movement's recent momentum: Two decades of experience have shown that an economic model that protects intellectual property and a business model that recoups research and development costs can create impressive economic benefits and distribute them very broadly. Such admonitions serve as the backdrop for Stallman's speech today. Less than a month after their utterance, Stallman stands with his back to one of the chalk boards at the front of the room, edgy to begin. If the last two decades have brought dramatic changes to the software marketplace, they have brought even more dramatic changes to Stallman himself. Gone is the skinny, clean-shaven hacker who once spent his entire days communing with his beloved PDP-10. In his place stands a heavy-set middle-aged man with long hair and rabbinical beard, a man who now spends the bulk of his time writing and answering email, haranguing fellow programmers, and giving speeches like the one today. Dressed in an aqua-colored T-shirt and brown polyester pants, Stallman looks like a desert hermit who just stepped out of a Salvation Army dressing room. The crowd is filled with visitors who share Stallman's fashion and grooming tastes. Many come bearing laptop computers and cellular modems, all the better to record and transmit Stallman's words to a waiting Internet audience. The gender ratio is roughly 15 males to 1 female, and 1 of the 7 or 8 females in the room comes in bearing a stuffed penguin, the official Linux mascot, while another carries a stuffed teddy bear. <Graphic file:/home/craigm/books/free_0201.png> Richard Stallman, circa 2000. "I decided I would develop a free software operating system or die trying . . of old age of course." Photo courtesy of http://www.stallman.org. Agitated, Stallman leaves his post at the front of the room and takes a seat in a front-row chair, tapping a few commands into an already-opened laptop. For the next 10 minutes Stallman is oblivious to the growing number of students, professors, and fans circulating in front of him at the foot of the auditorium stage. Before the speech can begin, the baroque rituals of academic formality must be observed. Stallman's appearance merits not one but two introductions. Mike Uretsky, codirector of the Stern School's Center for Advanced Technology, provides the first. "The role of a university is to foster debate and to have interesting discussions," Uretsky says. "This particular presentation, this seminar falls right into that mold. I find the discussion of open source particularly interesting." Before Uretsky can get another sentence out, Stallman is on his feet waving him down like a stranded motorist. "I do free software," Stallman says to rising laughter. "Open source is a different movement." The laughter gives way to applause. The room is stocked with Stallman partisans, people who know of his reputation for verbal exactitude, not to mention his much publicized 1998 falling out with the open source software proponents. Most have come to anticipate such outbursts the same way radio fans once waited for Jack Benny's trademark, "Now cut that out!" phrase during each radio program. Uretsky hastily finishes his introduction and cedes the stage to Edmond Schonberg, a professor in the NYU computer-science department. As a computer programmer and GNU Project contributor, Schonberg knows which linguistic land mines to avoid. He deftly summarizes Stallman's career from the perspective of a modern-day programmer. "Richard is the perfect example of somebody who, by acting locally, started thinking globally [about] problems concerning the unavailability of source code," says Schonberg. "He has developed a coherent philosophy that has forced all of us to reexamine our ideas of how software is produced, of what intellectual property means, and of what the software community actually represents." Schonberg welcomes Stallman to more applause. Stallman takes a moment to shut off his laptop, rises out of his chair, and takes the stage. At first, Stallman's address seems more Catskills comedy routine than political speech. "I'd like to thank Microsoft for providing me the opportunity to be on this platform," Stallman wisecracks. "For the past few weeks, I have felt like an author whose book was fortuitously banned somewhere." For the uninitiated, Stallman dives into a quick free software warm-up analogy. He likens a software program to a cooking recipe. Both provide useful step-by-step instructions on how to complete a desired task and can be easily modified if a user has special desires or circumstances. "You don't have to follow a recipe exactly," Stallman notes. "You can leave out some ingredients. Add some mushrooms, 'cause you like mushrooms. Put in less salt because your doctor said you should cut down on salt-whatever." Most importantly, Stallman says, software programs and recipes are both easy to share. In giving a recipe to a dinner guest, a cook loses little more than time and the cost of the paper the recipe was written on. Software programs require even less, usually a few mouse-clicks and a modicum of electricity. In both instances, however, the person giving the information gains two things: increased friendship and the ability to borrow interesting recipes in return. "Imagine what it would be like if recipes were packaged inside black boxes," Stallman says, shifting gears. "You couldn't see what ingredients they're using, let alone change them, and imagine if you made a copy for a friend. They would call you a pirate and try to put you in prison for years. That world would create tremendous outrage from all the people who are used to sharing recipes. But that is exactly what the world of proprietary software is like. A world in which common decency towards other people is prohibited or prevented." With this introductory analogy out of the way, Stallman launches into a retelling of the Xerox laser-printer episode. Like the recipe analogy, the laser-printer story is a useful rhetorical device. With its parable-like structure, it dramatizes just how quickly things can change in the software world. Drawing listeners back to an era before Amazon.com one-click shopping, Microsoft Windows, and Oracle databases, it asks the listener to examine the notion of software ownership free of its current corporate logos. Stallman delivers the story with all the polish and practice of a local district attorney conducting a closing argument. When he gets to the part about the Carnegie Mellon professor refusing to lend him a copy of the printer source code, Stallman pauses. "He had betrayed us," Stallman says. "But he didn't just do it to us. Chances are he did it to you." On the word "you," Stallman points his index finger accusingly at an unsuspecting member of the audience. The targeted audience member's eyebrows flinch slightly, but Stallman's own eyes have moved on. Slowly and deliberately, Stallman picks out a second listener to nervous titters from the crowd. "And I think, mostly likely, he did it to you, too," he says, pointing at an audience member three rows behind the first. By the time Stallman has a third audience member picked out, the titters have given away to general laughter. The gesture seems a bit staged, because it is. Still, when it comes time to wrap up the Xerox laser-printer story, Stallman does so with a showman's flourish. "He probably did it to most of the people here in this room-except a few, maybe, who weren't born yet in 1980," Stallman says, drawing more laughs. "[That's] because he had promised to refuse to cooperate with just about the entire population of the planet Earth." Stallman lets the comment sink in for a half-beat. "He had signed a nondisclosure agreement," Stallman adds. Richard Matthew Stallman's rise from frustrated academic to political leader over the last 20 years speaks to many things. It speaks to Stallman's stubborn nature and prodigious will. It speaks to the clearly articulated vision and values of the free software movement Stallman helped build. It speaks to the high-quality software programs Stallman has built, programs that have cemented Stallman's reputation as a programming legend. It speaks to the growing momentum of the GPL, a legal innovation that many Stallman observers see as his most momentous accomplishment. Most importantly, it speaks to the changing nature of political power in a world increasingly beholden to computer technology and the software programs that power that technology. Maybe that's why, even at a time when most high-technology stars are on the wane, Stallman's star has grown. Since launching the GNU Project in 1984,5 Stallman has been at turns ignored, satirized, vilified, and attacked-both from within and without the free software movement. Through it all, the GNU Project has managed to meet its milestones, albeit with a few notorious delays, and stay relevant in a software marketplace several orders of magnitude more complex than the one it entered 18 years ago. So too has the free software ideology, an ideology meticulously groomed by Stallman himself. To understand the reasons behind this currency, it helps to examine Richard Stallman both in his own words and in the words of the people who have collaborated and battled with him along the way. The Richard Stallman character sketch is not a complicated one. If any person exemplifies the old adage "what you see is what you get," it's Stallman. "I think if you want to understand Richard Stallman the human being, you really need to see all of the parts as a consistent whole," advises Eben Moglen, legal counsel to the Free Software Foundation and professor of law at Columbia University Law School. "All those personal eccentricities that lots of people see as obstacles to getting to know Stallman really are Stallman: Richard's strong sense of personal frustration, his enormous sense of principled ethical commitment, his inability to compromise, especially on issues he considers fundamental. These are all the very reasons Richard did what he did when he did." Explaining how a journey that started with a laser printer would eventually lead to a sparring match with the world's richest corporation is no easy task. It requires a thoughtful examination of the forces that have made software ownership so important in today's society. It also requires a thoughtful examination of a man who, like many political leaders before him, understands the malleability of human memory. It requires an ability to interpret the myths and politically laden code words that have built up around Stallman over time. Finally, it requires an understanding of Stallman's genius as a programmer and his failures and successes in translating that genius to other pursuits. When it comes to offering his own summary of the journey, Stallman acknowledges the fusion of personality and principle observed by Moglen. "Stubbornness is my strong suit," he says. "Most people who attempt to do anything of any great difficulty eventually get discouraged and give up. I never gave up." He also credits blind chance. Had it not been for that run-in over the Xerox laser printer, had it not been for the personal and political conflicts that closed out his career as an MIT employee, had it not been for a half dozen other timely factors, Stallman finds it very easy to picture his life following a different career path. That being said, Stallman gives thanks to the forces and circumstances that put him in the position to make a difference. "I had just the right skills," says Stallman, summing up his decision for launching the GNU Project to the audience. "Nobody was there but me, so I felt like, `I'm elected. I have to work on this. If not me , who?'" Endnotes 1. Actually, the GPL's powers are not quite that potent. According to section 10 of the GNU General Public License, Version 2 (1991), the viral nature of the license depends heavily on the Free Software Foundation's willingness to view a program as a derivative work, not to mention the existing license the GPL would replace. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software that is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally. "To compare something to a virus is very harsh," says Stallman. "A spider plant is a more accurate comparison; it goes to another place if you actively take a cutting." For more information on the GNU General Public License, visit [http://www.gnu.org/copyleft/gpl.html.] A Portrait of the Hacker as a Young Man Richard Stallman's mother, Alice Lippman, still remembers the moment she realized her son had a special gift. "I think it was when he was eight," Lippman recalls. The year was 1961, and Lippman, a recently divorced single mother, was wiling away a weekend afternoon within the family's tiny one-bedroom apartment on Manhattan's Upper West Side. Leafing through a copy of Scientific American, Lippman came upon her favorite section, the Martin Gardner-authored column titled "Mathematical Games." A substitute art teacher, Lippman always enjoyed Gardner's column for the brain-teasers it provided. With her son already ensconced in a book on the nearby sofa, Lippman decided to take a crack at solving the week's feature puzzle. "I wasn't the best person when it came to solving the puzzles," she admits. "But as an artist, I found they really helped me work through conceptual barriers." Lippman says her attempt to solve the puzzle met an immediate brick wall. About to throw the magazine down in disgust, Lippman was surprised by a gentle tug on her shirt sleeve. "It was Richard," she recalls, "He wanted to know if I needed any help." Looking back and forth, between the puzzle and her son, Lippman says she initially regarded the offer with skepticism. "I asked Richard if he'd read the magazine," she says. "He told me that, yes, he had and what's more he'd already solved the puzzle. The next thing I know, he starts explaining to me how to solve it." Hearing the logic of her son's approach, Lippman's skepticism quickly gave way to incredulity. "I mean, I always knew he was a bright boy," she says, "but this was the first time I'd seen anything that suggested how advanced he really was." Thirty years after the fact, Lippman punctuates the memory with a laugh. "To tell you the truth, I don't think I ever figured out how to solve that puzzle," she says. "All I remember is being amazed he knew the answer." Seated at the dining-room table of her second Manhattan apartment-the same spacious three-bedroom complex she and her son moved to following her 1967 marriage to Maurice Lippman, now deceased-Alice Lippman exudes a Jewish mother's mixture of pride and bemusement when recalling her son's early years. The nearby dining-room credenza offers an eight-by-ten photo of Stallman glowering in full beard and doctoral robes. The image dwarfs accompanying photos of Lippman's nieces and nephews, but before a visitor can make too much of it, Lippman makes sure to balance its prominent placement with an ironic wisecrack. "Richard insisted I have it after he received his honorary doctorate at the University of Glasgow," says Lippman. "He said to me, `Guess what, mom? It's the first graduation I ever attended.'"1 Such comments reflect the sense of humor that comes with raising a child prodigy. Make no mistake, for every story Lippman hears and reads about her son's stubbornness and unusual behavior, she can deliver at least a dozen in return. "He used to be so conservative," she says, throwing up her hands in mock exasperation. "We used to have the worst arguments right here at this table. I was part of the first group of public city school teachers that struck to form a union, and Richard was very angry with me. He saw unions as corrupt. He was also very opposed to social security. He thought people could make much more money investing it on their own. Who knew that within 10 years he would become so idealistic? All I remember is his stepsister coming to me and saying, `What is he going to be when he grows up? A fascist?'" As a single parent for nearly a decade-she and Richard's father, Daniel Stallman, were married in 1948, divorced in 1958, and split custody of their son afterwards-Lippman can attest to her son's aversion to authority. She can also attest to her son's lust for knowledge. It was during the times when the two forces intertwined, Lippman says, that she and her son experienced their biggest battles. "It was like he never wanted to eat," says Lippman, recalling the behavior pattern that set in around age eight and didn't let up until her son's high-school graduation in 1970. "I'd call him for dinner, and he'd never hear me. I'd have to call him 9 or 10 times just to get his attention. He was totally immersed." Stallman, for his part, remembers things in a similar fashion, albeit with a political twist. "I enjoyed reading," he says. "If I wanted to read, and my mother told me to go to the kitchen and eat or go to sleep, I wasn't going to listen. I saw no reason why I couldn't read. No reason why she should be able to tell me what to do, period. Essentially, what I had read about, ideas such as democracy and individual freedom, I applied to myself. I didn't see any reason to exclude children from these principles." The belief in individual freedom over arbitrary authority extended to school as well. Two years ahead of his classmates by age 11, Stallman endured all the usual frustrations of a gifted public-school student. It wasn't long after the puzzle incident that his mother attended the first in what would become a long string of parent-teacher conferences. "He absolutely refused to write papers," says Lippman, recalling an early controversy. "I think the last paper he wrote before his senior year in high school was an essay on the history of the number system in the west for a fourth-grade teacher." Gifted in anything that required analytical thinking, Stallman gravitated toward math and science at the expense of his other studies. What some teachers saw as single-mindedness, however, Lippman saw as impatience. Math and science offered simply too much opportunity to learn, especially in comparison to subjects and pursuits for which her son seemed less naturally inclined. Around age 10 or 11, when the boys in Stallman's class began playing a regular game of touch football, she remembers her son coming home in a rage. "He wanted to play so badly, but he just didn't have the coordination skills," Lippman recalls. "It made him so angry." The anger eventually drove her son to focus on math and science all the more. Even in the realm of science, however, her son's impatience could be problematic. Poring through calculus textbooks by age seven, Stallman saw little need to dumb down his discourse for adults. Sometime, during his middle-school years, Lippman hired a student from nearby Columbia University to play big brother to her son. The student left the family's apartment after the first session and never came back. "I think what Richard was talking about went over his head," Lippman speculates. Another favorite maternal anecdote dates back to the early 1960s, shortly after the puzzle incident. Around age seven, two years after the divorce and relocation from Queens, Richard took up the hobby of launching model rockets in nearby Riverside Drive Park. What started as aimless fun soon took on an earnest edge as her son began recording the data from each launch. Like the interest in mathematical games, the pursuit drew little attention until one day, just before a major NASA launch, Lippman checked in on her son to see if he wanted to watch. "He was fuming," Lippman says. "All he could say to me was, `But I'm not published yet.' Apparently he had something that he really wanted to show NASA." Such anecdotes offer early evidence of the intensity that would become Stallman's chief trademark throughout life. When other kids came to the table, Stallman stayed in his room and read. When other kids played Johnny Unitas, Stallman played Werner von Braun. "I was weird," Stallman says, summing up his early years succinctly in a 1999 interview. "After a certain age, the only friends I had were teachers."See Michael Gross, "Richard Stallman: High School Misfit, Symbol of Free Software, MacArthur-certified Genius" (1999). This interview is one of the most candid Stallman interviews on the record. I recommend it highly. http://www.mgross.com/interviews/stallman1.html Although it meant courting more run-ins at school, Lippman decided to indulge her son's passion. By age 12, Richard was attending science camps during the summer and private school during the school year. When a teacher recommended her son enroll in the Columbia Science Honors Program, a post-Sputnik program designed for gifted middle- and high-school students in New York City, Stallman added to his extracurriculars and was soon commuting uptown to the Columbia University campus on Saturdays. Dan Chess, a fellow classmate in the Columbia Science Honors Program, recalls Richard Stallman seeming a bit weird even among the students who shared a similar lust for math and science. "We were all geeks and nerds, but he was unusually poorly adjusted," recalls Chess, now a mathematics professor at Hunter College. "He was also smart as shit. I've known a lot of smart people, but I think he was the smartest person I've ever known." Seth Breidbart, a fellow Columbia Science Honors Program alumnus, offers bolstering testimony. A computer programmer who has kept in touch with Stallman thanks to a shared passion for science fiction and science-fiction conventions, he recalls the 15-year-old, buzz-cut-wearing Stallman as "scary," especially to a fellow 15-year-old. "It's hard to describe," Breidbart says. "It wasn't like he was unapproachable. He was just very intense. [He was] very knowledgeable but also very hardheaded in some ways." Such descriptions give rise to speculation: are judgment-laden adjectives like "intense" and "hardheaded" simply a way to describe traits that today might be categorized under juvenile behavioral disorder? A December, 2001, Wired magazine article titled "The Geek Syndrome" paints the portrait of several scientifically gifted children diagnosed with high-functioning autism or Asperger Syndrome. In many ways, the parental recollections recorded in the Wired article are eerily similar to the ones offered by Lippman. Even Stallman has indulged in psychiatric revisionism from time to time. During a 2000 profile for the Toronto Star, Stallman described himself to an interviewer as "borderline autistic,"See Judy Steed, Toronto Star, BUSINESS, (October 9, 2000): C03. His vision of free software and social cooperation stands in stark contrast to the isolated nature of his private life. A Glenn Gould-like eccentric, the Canadian pianist was similarly brilliant, articulate, and lonely. Stallman considers himself afflicted, to some degree, by autism: a condition that, he says, makes it difficult for him to interact with people. a description that goes a long way toward explaining a lifelong tendency toward social and emotional isolation and the equally lifelong effort to overcome it. Such speculation benefits from the fast and loose nature of most so-called " behavioral disorders" nowadays, of course. As Steve Silberman, author of " The Geek Syndrome," notes, American psychiatrists have only recently come to accept Asperger Syndrome as a valid umbrella term covering a wide set of behavioral traits. The traits range from poor motor skills and poor socialization to high intelligence and an almost obsessive affinity for numbers, computers, and ordered systems.See Steve Silberman, "The Geek Syndrome," Wired (December, 2001). Reflecting on the broad nature of this umbrella, Stallman says its possible that, if born 40 years later, he might have merited just such a diagnosis. Then again, so would many of his computer-world colleagues. "It's possible I could have had something like that," he says. "On the other hand, one of the aspects of that syndrome is difficulty following rhythms. I can dance. In fact, I love following the most complicated rhythms. It's not clear cut enough to know." Chess, for one, rejects such attempts at back-diagnosis. "I never thought of him [as] having that sort of thing," he says. "He was just very unsocialized, but then, we all were." Lippman, on the other hand, entertains the possibility. She recalls a few stories from her son's infancy, however, that provide fodder for speculation. A prominent symptom of autism is an oversensitivity to noises and colors, and Lippman recalls two anecdotes that stand out in this regard. "When Richard was an infant, we'd take him to the beach," she says. "He would start screaming two or three blocks before we reached the surf. It wasn't until the third time that we figured out what was going on: the sound of the surf was hurting his ears." She also recalls a similar screaming reaction in relation to color: "My mother had bright red hair, and every time she'd stoop down to pick him up, he'd let out a wail." In recent years, Lippman says she has taken to reading books about autism and believes that such episodes were more than coincidental. "I do feel that Richard had some of the qualities of an autistic child," she says. "I regret that so little was known about autism back then." Over time, however, Lippman says her son learned to adjust. By age seven, she says, her son had become fond of standing at the front window of subway trains, mapping out and memorizing the labyrinthian system of railroad tracks underneath the city. It was a hobby that relied on an ability to accommodate the loud noises that accompanied each train ride. "Only the initial noise seemed to bother him," says Lippman. "It was as if he got shocked by the sound but his nerves learned how to make the adjustment." For the most part, Lippman recalls her son exhibiting the excitement, energy, and social skills of any normal boy. It wasn't until after a series of traumatic events battered the Stallman household, she says, that her son became introverted and emotionally distant. The first traumatic event was the divorce of Alice and Daniel Stallman, Richard's father. Although Lippman says both she and her ex-husband tried to prepare their son for the blow, she says the blow was devastating nonetheless. "He sort of didn't pay attention when we first told him what was happening," Lippman recalls. "But the reality smacked him in the face when he and I moved into a new apartment. The first thing he said was, `Where's Dad's furniture?'" For the next decade, Stallman would spend his weekdays at his mother's apartment in Manhattan and his weekends at his father's home in Queens. The shuttling back and forth gave him a chance to study a pair of contrasting parenting styles that, to this day, leaves Stallman firmly opposed to the idea of raising children himself. Speaking about his father, a World War II vet who passed away in early 2001, Stallman balances respect with anger. On one hand, there is the man whose moral commitment led him to learn French just so he could be more helpful to Allies when they'd finally come. On the other hand, there was the parent who always knew how to craft a put-down for cruel effect.Regrettably, I did not get a chance to interview Daniel Stallman for this book. During the early research for this book, Stallman informed me that his father suffered from Alzheimer's. When I resumed research in late 2001, I learned, sadly, that Daniel Stallman had died earlier in the year. "My father had a horrible temper," Stallman says. "He never screamed, but he always found a way to criticize you in a cold, designed-to-crush way." As for life in his mother's apartment, Stallman is less equivocal. "That was war," he says. "I used to say in my misery, `I want to go home,' meaning to the nonexistent place that I'll never have." For the first few years after the divorce, Stallman found the tranquility that eluded him in the home of his paternal grandparents. Then, around age 10 his grandparents passed away in short succession. For Stallman, the loss was devastating. "I used to go and visit and feel I was in a loving, gentle environment," Stallman recalls. "It was the only place I ever found one, until I went away to college." Lippman lists the death of Richard's paternal grandparents as the second traumatic event. "It really upset him," she says. He was very close to both his grandparents. Before they died, he was very outgoing, almost a leader-of-the-pack type with the other kids. After they died, he became much more emotionally withdrawn." From Stallman's perspective, the emotional withdrawal was merely an attempt to deal with the agony of adolescence. Labeling his teenage years a "pure horror," Stallman says he often felt like a deaf person amid a crowd of chattering music listeners. "I often had the feeling that I couldn't understand what other people were saying," says Stallman, recalling the emotional bubble that insulated him from the rest of the adolescent and adult world. "I could understand the words, but something was going on underneath the conversations that I didn't understand. I couldn't understand why people were interested in the things other people said." For all the agony it produced, adolescence would have a encouraging effect on Stallman's sense of individuality. At a time when most of his classmates were growing their hair out, Stallman preferred to keep his short. At a time when the whole teenage world was listening to rock and roll, Stallman preferred classical music. A devoted fan of science fiction, Mad magazine, and late-night TV, Stallman cultivated a distinctly off-the-wall personality that fed off the incomprehension of parents and peers alike. "Oh, the puns," says Lippman, still exasperated by the memory of her son's teenage personality. "There wasn't a thing you could say at the dinner table that he couldn't throw back at you as a pun." Outside the home, Stallman saved the jokes for the adults who tended to indulge his gifted nature. One of the first was a summer-camp counselor who handed Stallman a print-out manual for the IBM 7094 computer during his 12th year. To a preteenager fascinated with numbers and science, the gift was a godsend.Stallman, an atheist, would probably quibble with this description. Suffice it to say, it was something Stallman welcomed. See previous note 1: "As soon as I heard about computers, I wanted to see one and play with one." By the end of summer, Stallman was writing out paper programs according to the 7094's internal specifications, anxiously anticipating getting a chance to try them out on a real machine. With the first personal computer still a decade away, Stallman would be forced to wait a few years before getting access to his first computer. His first chance finally came during his junior year of high school. Hired on at the IBM New York Scientific Center, a now-defunct research facility in downtown Manhattan, Stallman spent the summer after high-school graduation writing his first program, a pre-processor for the 7094 written in the programming language PL/I. "I first wrote it in PL/I, then started over in assembler language when the PL/I program was too big to fit in the computer," he recalls. After that job at the IBM Scientific Center, Stallman had held a laboratory-assistant position in the biology department at Rockefeller University. Although he was already moving toward a career in math or physics, Stallman's analytical mind impressed the lab director enough that a few years after Stallman departed for college, Lippman received an unexpected phone call. "It was the professor at Rockefeller," Lippman says. "He wanted to know how Richard was doing. He was surprised to learn that he was working in computers. He'd always thought Richard had a great future ahead of him as a biologist." Stallman's analytical skills impressed faculty members at Columbia as well, even when Stallman himself became a target of their ire. "Typically once or twice an hour [Stallman] would catch some mistake in the lecture," says Breidbart. "And he was not shy about letting the professors know it immediately. It got him a lot of respect but not much popularity." Hearing Breidbart's anecdote retold elicits a wry smile from Stallman. "I may have been a bit of a jerk sometimes," he admits. "But I found kindred spirits among the teachers, because they, too, liked to learn. Kids, for the most part, didn't. At least not in the same way." Hanging out with the advanced kids on Saturday nevertheless encouraged Stallman to think more about the merits of increased socialization. With college fast approaching, Stallman, like many in his Columbia Science Honors Program, had narrowed his list of desired schools down to two choices: Harvard and MIT. Hearing of her son's desire to move on to the Ivy League, Lippman became concerned. As a 15-year-old high-school junior, Stallman was still having run-ins with teachers and administrators. Only the year before, he had pulled straight A's in American History, Chemistry, French, and Algebra, but a glaring F in English reflected the ongoing boycott of writing assignments. Such miscues might draw a knowing chuckle at MIT, but at Harvard, they were a red flag. During her son's junior year, Lippman says she scheduled an appointment with a therapist. The therapist expressed instant concern over Stallman's unwillingness to write papers and his run-ins with teachers. Her son certainly had the intellectual wherewithal to succeed at Harvard, but did he have the patience to sit through college classes that required a term paper? The therapist suggested a trial run. If Stallman could make it through a full year in New York City public schools, including an English class that required term papers, he could probably make it at Harvard. Following the completion of his junior year, Stallman promptly enrolled in summer school at Louis D. Brandeis High School, a public school located on 84th Street, and began making up the mandatory art classes he had shunned earlier in his high-school career. By fall, Stallman was back within the mainstream population of New York City high-school students. It wasn't easy sitting through classes that seemed remedial in comparison with his Saturday studies at Columbia, but Lippman recalls proudly her son's ability to toe the line. "He was forced to kowtow to a certain degree, but he did it," Lippman says. "I only got called in once, which was a bit of a miracle. It was the calculus teacher complaining that Richard was interrupting his lesson. I asked how he was interrupting. He said Richard was always accusing the teacher of using a false proof. I said, `Well, is he right?' The teacher said, `Yeah, but I can't tell that to the class. They wouldn't understand.'" By the end of his first semester at Brandeis, things were falling into place. A 96 in English wiped away much of the stigma of the 60 earned 2 years before. For good measure, Stallman backed it up with top marks in American History, Advanced Placement Calculus, and Microbiology. The crowning touch was a perfect 100 in Physics. Though still a social outcast, Stallman finished his 11 months at Brandeis as the fourth-ranked student in a class of 789. <Graphic file:/home/craigm/books/free_0306.png> Stallman's senior-year transcript at Louis D. Brandeis H.S., November, 1969. Note turnaround in English class performance. "He was forced to kowtow to a certain degree," says his mother, "but he did it." Outside the classroom, Stallman pursued his studies with even more diligence, rushing off to fulfill his laboratory-assistant duties at Rockefeller University during the week and dodging the Vietnam protesters on his way to Saturday school at Columbia. It was there, while the rest of the Science Honors Program students sat around discussing their college choices, that Stallman finally took a moment to participate in the preclass bull session. Recalls Breidbart, "Most of the students were going to Harvard and MIT, of course, but you had a few going to other Ivy League schools. As the conversation circled the room, it became apparent that Richard hadn't said anything yet. I don't know who it was, but somebody got up the courage to ask him what he planned to do." Thirty years later, Breidbart remembers the moment clearly. As soon as Stallman broke the news that he, too, would be attending Harvard University in the fall, an awkward silence filled the room. Almost as if on cue, the corners of Stallman's mouth slowly turned upward into a self-satisfied smile. Says Breidbart, "It was his silent way of saying, `That's right. You haven't got rid of me yet.'" Impeach God Although their relationship was fraught with tension, Richard Stallman would inherit one noteworthy trait from his mother: a passion for progressive politics. It was an inherited trait that would take several decades to emerge, however. For the first few years of his life, Stallman lived in what he now admits was a "political vacuum."See Michael Gross, "Richard Stallman: High School Misfit, Symbol of Free Software, MacArthur-certified Genius" (1999). Like most Americans during the Eisenhower age, the Stallman family spent the 50s trying to recapture the normalcy lost during the wartime years of the 1940s. "Richard's father and I were Democrats but happy enough to leave it at that," says Lippman, recalling the family's years in Queens. "We didn't get involved much in local or national politics." That all began to change, however, in the late 1950s when Alice divorced Daniel Stallman. The move back to Manhattan represented more than a change of address; it represented a new, independent identity and a jarring loss of tranquility. "I think my first taste of political activism came when I went to the Queens public library and discovered there was only a single book on divorce in the whole library," recalls Lippman. "It was very controlled by the Catholic church, at least in Elmhurst, where we lived. I think that was the first inkling I had of the forces that quietly control our lives." Returning to her childhood neighborhood, Manhattan's Upper West Side, Lippman was shocked by the changes that had taken place since her departure to Hunter College a decade and a half before. The skyrocketing demand for postwar housing had turned the neighborhood into a political battleground. On one side stood the pro-development city-hall politicians and businessmen hoping to rebuild many of the neighborhood's blocks to accommodate the growing number of white-collar workers moving into the city. On the other side stood the poor Irish and Puerto Rican tenants who had found an affordable haven in the neighborhood. At first, Lippman didn't know which side to choose. As a new resident, she felt the need for new housing. As a single mother with minimal income, however, she shared the poorer tenants' concern over the growing number of development projects catering mainly to wealthy residents. Indignant, Lippman began looking for ways to combat the political machine that was attempting to turn her neighborhood into a clone of the Upper East Side. Lippman says her first visit to the local Democratic party headquarters came in 1958. Looking for a day-care center to take care of her son while she worked, she had been appalled by the conditions encountered at one of the city-owned centers that catered to low-income residents. "All I remember is the stench of rotten milk, the dark hallways, the paucity of supplies. I had been a teacher in private nursery schools. The contrast was so great. We took one look at that room and left. That stirred me up." The visit to the party headquarters proved disappointing, however. Describing it as "the proverbial smoke-filled room," Lippman says she became aware for the first time that corruption within the party might actually be the reason behind the city's thinly disguised hostility toward poor residents. Instead of going back to the headquarters, Lippman decided to join up with one of the many clubs aimed at reforming the Democratic party and ousting the last vestiges of the Tammany Hall machine. Dubbed the Woodrow Wilson/FDR Reform Democratic Club, Lippman and her club began showing up at planning and city-council meetings, demanding a greater say. "Our primary goal was to fight Tammany Hall, Carmine DeSapio and his henchman,"Carmine DeSapio holds the dubious distinction of being the first Italian-American boss of Tammany Hall, the New York City political machine. For more information on DeSapio and the politics of post-war New York, see John Davenport, "Skinning the Tiger: Carmine DeSapio and the End of the Tammany Era," New York Affairs (1975): 3:1. says Lippman. "I was the representative to the city council and was very much involved in creating a viable urban-renewal plan that went beyond simply adding more luxury housing to the neighborhood." Such involvement would blossom into greater political activity during the 1960s. By 1965, Lippman had become an "outspoken" supporter for political candidates like William Fitts Ryan, a Democratic elected to Congress with the help of reform clubs and one of the first U.S. representatives to speak out against the Vietnam War. It wasn't long before Lippman, too, was an outspoken opponent of U.S. involvement in Indochina. "I was against the Vietnam war from the time Kennedy sent troops," she says. "I had read the stories by reporters and journalists sent to cover the early stages of the conflict. I really believed their forecast that it would become a quagmire." Such opposition permeated the Stallman-Lippman household. In 1967, Lippman remarried. Her new husband, Maurice Lippman, a major in the Air National Guard, resigned his commission to demonstrate his opposition to the war. Lippman's stepson, Andrew Lippman, was at MIT and temporarily eligible for a student deferment. Still, the threat of induction should that deferment disappear, as it eventually did, made the risk of U.S. escalation all the more immediate. Finally, there was Richard who, though younger, faced the prospect of choosing between Vietnam or Canada when the war lasted into the 1970s. "Vietnam was a major issue in our household," says Lippman. "We talked about it constantly: what would we do if the war continued, what steps Richard or his stepbrother would take if they got drafted. We were all opposed to the war and the draft. We really thought it was immoral." For Stallman, the Vietnam War elicited a complex mixture of emotions: confusion, horror, and, ultimately, a profound sense of political impotence. As a kid who could barely cope in the mild authoritarian universe of private school, Stallman experienced a shiver whenever the thought of Army boot camp presented itself. "I was devastated by the fear, but I couldn't imagine what to do and didn't have the guts to go demonstrate," recalls Stallman, whose March 18th birthday earned him a dreaded low number in the draft lottery when the federal government finally eliminated college deferments in 1971. "I couldn't envision moving to Canada or Sweden. The idea of getting up by myself and moving somewhere. How could I do that? I didn't know how to live by myself. I wasn't the kind of person who felt confident in approaching things like that." Stallman says he was both impressed and shamed by the family members who did speak out. Recalling a bumper sticker on his father's car likening the My Lai massacre to similar Nazi atrocities in World War II, he says he was "excited" by his father's gesture of outrage. "I admired him for doing it," Stallman says. "But I didn't imagine that I could do anything. I was afraid that the juggernaut of the draft was going to destroy me." Although descriptions of his own unwillingness to speak out carry a tinge of nostalgic regret, Stallman says he was ultimately turned off by the tone and direction of the anti-war movement. Like other members of the Science Honors Program, he saw the weekend demonstrations at Columbia as little more than a distracting spectacle.Chess, another Columbia Science Honors Program alum, describes the protests as "background noise." "We were all political," he says, "but the SHP was imporant. We would never have skipped it for a demonstration." Ultimately, Stallman says, the irrational forces driving the anti-war movement became indistinguishable from the irrational forces driving the rest of youth culture. Instead of worshiping the Beatles, girls in Stallman's age group were suddenly worshiping firebrands like Abbie Hoffman and Jerry Rubin. To a kid already struggling to comprehend his teenage peers, escapist slogans like "make love not war" had a taunting quality. Not only was it a reminder that Stallman, the short-haired outsider who hated rock 'n' roll, detested drugs, and didn't participate in campus demonstrations, wasn't getting it politically; he wasn't "getting it" sexually either. "I didn't like the counter culture much," Stallman admits. "I didn't like the music. I didn't like the drugs. I was scared of the drugs. I especially didn't like the anti-intellectualism, and I didn't like the prejudice against technology. After all, I loved a computer. And I didn't like the mindless anti-Americanism that I often encountered. There were people whose thinking was so simplistic that if they disapproved of the conduct of the U.S. in the Vietnam War, they had to support the North Vietnamese. They couldn't imagine a more complicated position, I guess." Such comments alleviate feelings of timidity. They also underline a trait that would become the key to Stallman's own political maturation. For Stallman, political confidence was directly proportionate to personal confidence. By 1970, Stallman had become confident in few things outside the realm of math and science. Nevertheless, confidence in math gave him enough of a foundation to examine the anti-war movement in purely logical terms. In the process of doing so, Stallman had found the logic wanting. Although opposed to the war in Vietnam, Stallman saw no reason to disavow war as a means for defending liberty or correcting injustice. Rather than widen the breach between himself and his peers, however, Stallman elected to keep the analysis to himself. In 1970, Stallman left behind the nightly dinnertime conversations about politics and the Vietnam War as he departed for Harvard. Looking back, Stallman describes the transition from his mother's Manhattan apartment to life in a Cambridge dorm as an "escape." Peers who watched Stallman make the transition, however, saw little to suggest a liberating experience. "He seemed pretty miserable for the first while at Harvard," recalls Dan Chess, a classmate in the Science Honors Program who also matriculated at Harvard. "You could tell that human interaction was really difficult for him, and there was no way of avoiding it at Harvard. Harvard was an intensely social kind of place." To ease the transition, Stallman fell back on his strengths: math and science. Like most members of the Science Honors Program, Stallman breezed through the qualifying exam for Math 55, the legendary "boot camp" class for freshman mathematics "concentrators" at Harvard. Within the class, members of the Science Honors Program formed a durable unit. "We were the math mafia," says Chess with a laugh. "Harvard was nothing, at least compared with the SHP." To earn the right to boast, however, Stallman, Chess, and the other SHP alumni had to get through Math 55. Promising four years worth of math in two semesters, the course favored only the truly devout. "It was an amazing class," says David Harbater, a former "math mafia" member and now a professor of mathematics at the University of Pennsylvania. "It's probably safe to say there has never been a class for beginning college students that was that intense and that advanced. The phrase I say to people just to get it across is that, among other things, by the second semester we were discussing the differential geometry of Banach manifolds. That's usually when their eyes bug out, because most people don't start talking about Banach manifolds until their second year of graduate school." Starting with 75 students, the class quickly melted down to 20 by the end of the second semester. Of that 20, says Harbater, "only 10 really knew what they were doing." Of that 10, 8 would go on to become future mathematics professors, 1 would go on to teach physics. "The other one," emphasizes Harbater, "was Richard Stallman." Seth Breidbart, a fellow Math 55 classmate, remembers Stallman distinguishing himself from his peers even then. "He was a stickler in some very strange ways," says Breidbart. There is a standard technique in math which everybody does wrong. It's an abuse of notation where you have to define a function for something and what you do is you define a function and then you prove that it's well defined. Except the first time he did and presented it, he defined a relation and proved that it's a function. It's the exact same proof, but he used the correct terminology, which no one else did. That's just the way he was." It was in Math 55 that Richard Stallman began to cultivate a reputation for brilliance. Breidbart agrees, but Chess, whose competitive streak refused to yield, says the realization that Stallman might be the best mathematician in the class didn't set in until the next year. "It was during a class on Real Analysis, which I took with Richard the next year," says Chess, now a math professor at Hunter College. "I actually remember in a proof about complex valued measures that Richard came up with an idea that was basically a metaphor from the calculus of variations. It was the first time I ever saw somebody solve a problem in a brilliantly original way." Chess makes no bones about it: watching Stallman's solution unfold on the chalkboard was a devastating blow. As a kid who'd always taken pride in being the smartest mathematician the room, it was like catching a glimpse of his own mortality. Years later, as Chess slowly came to accept the professional rank of a good-but-not-great mathematician, he had Stallman's sophomore-year proof to look back on as a taunting early indicator. "That's the thing about mathematics," says Chess. "You don't have to be a first-rank mathematician to recognize first-rate mathematical talent. I could tell I was up there, but I could also tell I wasn't at the first rank. If Richard had chosen to be a mathematician, he would have been a first-rank mathematician." For Stallman, success in the classroom was balanced by the same lack of success in the social arena. Even as other members of the math mafia gathered to take on the Math 55 problem sets, Stallman preferred to work alone. The same went for living arrangements. On the housing application for Harvard, Stallman clearly spelled out his preferences. "I said I preferred an invisible, inaudible, intangible roommate," he says. In a rare stroke of bureaucratic foresight, Harvard's housing office accepted the request, giving Stallman a one-room single for his freshman year. Breidbart, the only math-mafia member to share a dorm with Stallman that freshman year, says Stallman slowly but surely learned how to interact with other students. He recalls how other dorm mates, impressed by Stallman's logical acumen, began welcoming his input whenever an intellectual debate broke out in the dining club or dorm commons. "We had the usual bull sessions about solving the world's problems or what would be the result of something," recalls Breidbart. "Say somebody discovers an immortality serum. What do you do? What are the political results? If you give it to everybody, the world gets overcrowded and everybody dies. If you limit it, if you say everyone who's alive now can have it but their children can't, then you end up with an underclass of people without it. Richard was just better able than most to see the unforeseen circumstances of any decision." Stallman remembers the discussions vividly. "I was always in favor of immortality," he says. "I was shocked that most people regarded immortality as a bad thing. How else would we be able to see what the world is like 200 years from now?" Although a first-rank mathematician and first-rate debater, Stallman shied away from clear-cut competitive events that might have sealed his brilliant reputation. Near the end of freshman year at Harvard, Breidbart recalls how Stallman conspicuously ducked the Putnam exam, a prestigious test open to math students throughout the U.S. and Canada. In addition to giving students a chance to measure their knowledge in relation to their peers, the Putnam served as a chief recruiting tool for academic math departments. According to campus legend, the top scorer automatically qualified for a graduate fellowship at any school of his choice, including Harvard. Like Math 55, the Putnam was a brutal test of merit. A six-hour exam in two parts, it seemed explicitly designed to separate the wheat from the chaff. Breidbart, a veteran of both the Science Honors Program and Math 55, describes it as easily the most difficult test he ever took. "Just to give you an idea of how difficult it was," says Breidbart, "the top score was a 120, and my score the first year was in the 30s. That score was still good enough to place me 101st in the country." Surprised that Stallman, the best student in the class, had passed on the test, Breidbart says he and a fellow classmate cornered him in the dining common and demanded an explanation. "He said he was afraid of not doing well," Breidbart recalls. Breidbart and the friend quickly wrote down a few problems from memory and gave them to Stallman. "He solved all of them," Breidbart says, "leading me to conclude that by not doing well, he either meant coming in second or getting something wrong." Stallman remembers the episode a bit differently. "I remember that they did bring me the questions and it's possible that I solved one of them, but I'm pretty sure I didn't solve them all," he says. Nevertheless, Stallman agrees with Breidbart's recollection that fear was the primary reason for not taking the test. Despite a demonstrated willingness to point out the intellectual weaknesses of his peers and professors in the classroom, Stallman hated the notion of head-to-head competition. "It's the same reason I never liked chess," says Stallman. "Whenever I'd play, I would become so consumed by the fear of making a single mistake that I would start making stupid mistakes very early in the game. The fear became a self-fulfilling prophecy." Whether such fears ultimately prompted Stallman to shy away from a mathematical career is a moot issue. By the end of his freshman year at Harvard, Stallman had other interests pulling him away from the field. Computer programming, a latent fascination throughout Stallman's high-school years, was becoming a full-fledged passion. Where other math students sought occasional refuge in art and history classes, Stallman sought it in the computer-science laboratory. For Stallman, the first taste of real computer programming at the IBM New York Scientific Center had triggered a desire to learn more. "Toward the end of my first year at Harvard school, I started to have enough courage to go visit computer labs and see what they had. I'd ask them if they had extra copies of any manuals that I could read." Taking the manuals home, Stallman would examine machine specifications, compare them with other machines he already knew, and concoct a trial program, which he would then bring back to the lab along with the borrowed manual. Although some labs balked at the notion of a strange kid coming off the street and working on the lab machinery, most recognized competence when they saw it and let Stallman run the programs he had created. One day, near the end of freshman year, Stallman heard about a special laboratory near MIT. The laboratory was located on the ninth floor an off-campus building in Tech Square, the newly built facility dedicated to advanced research. According to the rumors, the lab itself was dedicated to the cutting-edge science of artificial intelligence and boasted the cutting-edge machines and software programs to match. Intrigued, Stallman decided to pay a visit. The trip was short, about 2 miles on foot, 10 minutes by train, but as Stallman would soon find out, MIT and Harvard can feel like opposite poles of the same planet. With its maze-like tangle of interconnected office buildings, the Institute's campus offered an aesthetic yin to Harvard's spacious colonial-village yang. The same could be said for the student body, a geeky collection of ex-high school misfits known more for its predilection for pranks than its politically powerful alumni. The yin-yang relationship extended to the AI Lab as well. Unlike Harvard computer labs, there was no grad-student gatekeeper, no clipboard waiting list for terminal access, no explicit atmosphere of "look but don't touch." Instead, Stallman found only a collection of open terminals and robotic arms, presumably the artifacts of some A.I. experiment. Although the rumors said anybody could sit down at the terminals, Stallman decided to stick with the original plan. When he encountered a lab employee, he asked if the lab had any spare manuals it could loan to an inquisitive student. "They had some, but a lot of things weren't documented," Stallman recalls. "They were hackers after all." Stallman left with something even better than a manual: a job. Although he doesn't remember what the first project was, he does remember coming back to the AI Lab the next week, grabbing an open terminal and writing software code. Looking back, Stallman sees nothing unusual in the AI Lab's willingness to accept an unproven outsider at first glance. "That's the way it was back then," he says. "That's the way it still is now. I'll hire somebody when I meet him if I see he's good. Why wait? Stuffy people who insist on putting bureaucracy into everything really miss the point. If a person is good, he shouldn't have to go through a long, detailed hiring process; he should be sitting at a computer writing code." To get a taste of "bureaucratic and stuffy," Stallman need only visit the computer labs at Harvard. There, access to the terminals was doled out according to academic rank. As an undergrad, Stallman usually had to sign up or wait until midnight, about the time most professors and grad students finished their daily work assignments. The waiting wasn't difficult, but it was frustrating. Waiting for a public terminal, knowing all the while that a half dozen equally usable machines were sitting idle inside professors' locked offices, seemed the height of illogic. Although Stallman paid the occasional visit to the Harvard computer labs, he preferred the more egalitarian policies of the AI Lab. "It was a breath of fresh air," he says. "At the AI Lab, people seemed more concerned about work than status." Stallman quickly learned that the AI Lab's first-come, first-served policy owed much to the efforts of a vigilant few. Many were holdovers from the days of Project MAC, the Department of Defense-funded research program that had given birth to the first time-share operating systems. A few were already legends in the computing world. There was Richard Greenblatt, the lab's in-house Lisp expert and author of MacHack, the computer chess program that had once humbled A.I. critic Hubert Dreyfus. There was Gerald Sussman, original author of the robotic block-stacking program HACKER. And there was Bill Gosper, the in-house math whiz already in the midst of an 18-month hacking bender triggered by the philosophical implications of the computer game LIFE.See Steven Levy, Hackers (Penguin USA [paperback], 1984): 144. Levy devotes about five pages to describing Gosper's fascination with LIFE, a math-based software game first created by British mathematician John Conway. I heartily recommend this book as a supplement, perhaps even a prerequisite, to this one. Members of the tight-knit group called themselves " hackers." Over time, they extended the "hacker" description to Stallman as well. In the process of doing so, they inculcated Stallman in the ethical traditions of the "hacker ethic ." To be a hacker meant more than just writing programs, Stallman learned. It meant writing the best possible programs. It meant sitting at a terminal for 36 hours straight if that's what it took to write the best possible programs. Most importantly, it meant having access to the best possible machines and the most useful information at all times. Hackers spoke openly about changing the world through software, and Stallman learned the instinctual hacker disdain for any obstacle that prevented a hacker from fulfilling this noble cause. Chief among these obstacles were poor software, academic bureaucracy, and selfish behavior. Stallman also learned the lore, stories of how hackers, when presented with an obstacle, had circumvented it in creative ways. Stallman learned about " lock hacking," the art of breaking into professors' offices to "liberate" sequestered terminals. Unlike their pampered Harvard counterparts, MIT faculty members knew better than to treat the AI Lab's terminal as private property. If a faculty member made the mistake of locking away a terminal for the night, hackers were quick to correct the error. Hackers were equally quick to send a message if the mistake repeated itself. "I was actually shown a cart with a heavy cylinder of metal on it that had been used to break down the door of one professor's office,"Gerald Sussman, an MIT faculty member and hacker whose work at the AI Lab predates Stallman's, disputes this memory. According to Sussman, the hackers never broke any doors to retrieve terminals. Stallman says. Such methods, while lacking in subtlety, served a purpose. Although professors and administrators outnumbered hackers two-to-one inside the AI Lab, the hacker ethic prevailed. Indeed, by the time of Stallman's arrival at the AI Lab, hackers and the AI Lab administration had coevolved into something of a symbiotic relationship. In exchange for fixing the machines and keeping the software up and running, hackers earned the right to work on favorite pet projects. Often, the pet projects revolved around improving the machines and software programs even further. Like teenage hot-rodders, most hackers viewed tinkering with machines as its own form of entertainment. Nowhere was this tinkering impulse better reflected than in the operating system that powered the lab's central PDP-6 mini-computer. Dubbed ITS, short for the Incompatible Time Sharing system, the operating system incorporated the hacking ethic into its very design. Hackers had built it as a protest to Project MAC's original operating system, the Compatible Time Sharing System, CTSS, and named it accordingly. At the time, hackers felt the CTSS design too restrictive, limiting programmers' power to modify and improve the program's own internal architecture if needed. According to one legend passed down by hackers, the decision to build ITS had political overtones as well. Unlike CTSS, which had been designed for the IBM 7094, ITS was built specifically for the PDP-6. In letting hackers write the systems themselves, AI Lab administrators guaranteed that only hackers would feel comfortable using the PDP-6. In the feudal world of academic research, the gambit worked. Although the PDP-6 was co-owned in conjunction with other departments, A.I. researchers soon had it to themselves. ITS boasted features most commercial operating systems wouldn't offer for years, features such as multitasking, debugging, and full-screen editing capability. Using it and the PDP-6 as a foundation, the Lab had been able to declare independence from Project MAC shortly before Stallman's arrival.I apologize for the whirlwind summary of ITS' genesis, an operating system many hackers still regard as the epitome of the hacker ethos. For more information on the program's political significance, see Simson Garfinkel, Architects of the Information Society: Thirty-Five Years of the Laboratory for Computer Science at MIT (MIT Press, 1999). As an apprentice hacker, Stallman quickly became enamored with ITS. Although forbidding to most newcomers, the program contained many built-in features that provided a lesson in software development to hacker apprentices such as himself. "ITS had a very elegant internal mechanism for one program to examine another," says Stallman, recalling the program. "You could examine all sorts of status about another program in a very clean, well-specified way." Using this feature, Stallman was able to watch how programs written by hackers processed instructions as they ran. Another favorite feature would allow the monitoring program to freeze the monitored program's job between instructions. In other operating systems, such a command would have resulted in half-computed gibberish or an automatic systems crash. In ITS, it provided yet another way to monitor the step-by-step performance. "If you said, `Stop the job,' it would always be stopped in user mode. It would be stopped between two user-mode instructions, and everything about the job would be consistent for that point," Stallman says. "If you said, `Resume the job,' it would continue properly. Not only that, but if you were to change the status of the job and then change it back, everything would be consistent. There was no hidden status anywhere." By the end of 1970, hacking at the AI Lab had become a regular part of Stallman's weekly schedule. From Monday to Thursday, Stallman devoted his waking hours to his Harvard classes. As soon as Friday afternoon arrived, however, he was on the T, heading down to MIT for the weekend. Stallman usually timed his arrival to coincide with the ritual food run. Joining five or six other hackers in their nightly quest for Chinese food, he would jump inside a beat-up car and head across the Harvard Bridge into nearby Boston. For the next two hours, he and his hacker colleagues would discuss everything from ITS to the internal logic of the Chinese language and pictograph system. Following dinner, the group would return to MIT and hack code until dawn. For the geeky outcast who rarely associated with his high-school peers, it was a heady experience, suddenly hanging out with people who shared the same predilection for computers, science fiction, and Chinese food. "I remember many sunrises seen from a car coming back from Chinatown," Stallman would recall nostalgically, 15 years after the fact in a speech at the Swedish Royal Technical Institute. "It was actually a very beautiful thing to see a sunrise, 'cause that's such a calm time of day. It's a wonderful time of day to get ready to go to bed. It's so nice to walk home with the light just brightening and the birds starting to chirp; you can get a real feeling of gentle satisfaction, of tranquility about the work that you have done that night."See Richard Stallman, "RMS lecture at KTH (Sweden)," (October 30, 1986). http://www.gnu.org/philosophy/stallman-kth.html The more Stallman hung out with the hackers, the more he adopted the hacker worldview. Already committed to the notion of personal liberty, Stallman began to infuse his actions with a sense of communal responsibility. When others violated the communal code, Stallman was quick to speak out. Within a year of his first visit, Stallman was the one breaking into locked offices, trying to recover the sequestered terminals that belonged to the lab community as a whole. In true hacker fashion, Stallman also sought to make his own personal contribution to the art of lock hacking. One of the most artful door-opening tricks, commonly attributed to Greenblatt, involved bending a stiff wire into a cane and attaching a loop of tape to the long end. Sliding the wire under the door, a hacker could twist and rotate the wire so that the long end touched the door knob. Provided the adhesive on the tape held, a hacker could open the doorknob with a few sharp twists. When Stallman tried the trick, he found it good but wanting in a few places. Getting the tape to stick wasn't always easy, and twisting the wire in a way that turned the doorknob was similarly difficult. Stallman remembered that the hallway ceiling possessed tiles that could be slid away. Some hackers, in fact, had used the false ceiling as a way to get around locked doors, an approach that generally covered the perpetrator in fiberglass but got the job done. Stallman considered an alternative approach. What if, instead of slipping a wire under the door, a hacker slid away one of the panels and stood over the door jamb? Stallman took it upon himself to try it out. Instead of using a wire, Stallman draped out a long U-shaped loop of magnetic tape, fastening a loop of adhesive tape at the base of the U. Standing over the door jamb, he dangled the tape until it looped under the doorknob. Lifting the tape until the adhesive fastened, he then pulled on the left end of the tape, twisting the doorknob counter-clockwise. Sure enough, the door opened. Stallman had added a new twist to the art of lock hacking. "Sometimes you had to kick the door after you turned the door knob," says Stallman, recalling the lingering bugginess of the new method. "It took a little bit of balance to pull it off." Such activities reflected a growing willingness on Stallman's part to speak and act out in defense of political beliefs. The AI Lab's spirit of direct action had proved inspirational enough for Stallman to break out of the timid impotence of his teenage years. Breaking into an office to free a terminal wasn't the same as taking part in a protest march, but it was effective in ways that most protests weren't. It solved the problem at hand. By the time of his last years at Harvard, Stallman was beginning to apply the whimsical and irreverent lessons of the AI Lab back at school. "Did he tell you about the snake?" his mother asks at one point during an interview. "He and his dorm mates put a snake up for student election. Apparently it got a considerable number of votes." Stallman verifies the snake candidacy with a few caveats. The snake was a candidate for election within Currier House, Stallman's dorm, not the campus-wide student council. Stallman does remember the snake attracting a fairly significant number of votes, thanks in large part to the fact that both the snake and its owner both shared the same last name. "People may have voted for it, because they thought they were voting for the owner," Stallman says. "Campaign posters said that the snake was `slithering for' the office. We also said it was an `at large' candidate, since it had climbed into the wall through the ventilating unit a few weeks before and nobody knew where it was." Running a snake for dorm council was just one of several election-related pranks. In a later election, Stallman and his dorm mates nominated the house master's son. "His platform was mandatory retirement at age seven," Stallman recalls. Such pranks paled in comparison to the fake-candidate pranks on the MIT campus, however. One of the most successful fake-candidate pranks was a cat named Woodstock, which actually managed to outdraw most of the human candidates in a campus-wide election. "They never announced how many votes Woodstock got, and they treated those votes as spoiled ballots," Stallman recalls. "But the large number of spoiled ballots in that election suggested that Woodstock had actually won. A couple of years later, Woodstock was suspiciously run over by a car. Nobody knows if the driver was working for the MIT administration." Stallman says he had nothing to do with Woodstock's candidacy, "but I admired it."In an email shortly after this book went into its final edit cycle, Stallman says he drew political inspiration from the Harvard campus as well. "In my first year of Harvard, in a Chinese History class, I read the story of the first revolt against the Chin dynasty," he says. "The story is not reliable history, but it was very moving." At the AI Lab, Stallman's political activities had a sharper-edged tone. During the 1970s, hackers faced the constant challenge of faculty members and administrators pulling an end-run around ITS and its hacker-friendly design. One of the first attempts came in the mid-1970s, as more and more faculty members began calling for a file security system to protect research data. Most other computer labs had installed such systems during late 1960s, but the AI Lab, through the insistence of Stallman and other hackers, remained a security-free zone. For Stallman, the opposition to security was both ethical and practical. On the ethical side, Stallman pointed out that the entire art of hacking relied on intellectual openness and trust. On the practical side, he pointed to the internal structure of ITS being built to foster this spirit of openness, and any attempt to reverse that design required a major overhaul. "The hackers who wrote the Incompatible Timesharing System decided that file protection was usually used by a self-styled system manager to get power over everyone else," Stallman would later explain. "They didn't want anyone to be able to get power over them that way, so they didn't implement that kind of a feature. The result was, that whenever something in the system was broken, you could always fix it."See Richard Stallman (1986). Through such vigilance, hackers managed to keep the AI Lab's machines security-free. Over at the nearby MIT Laboratory for Computer Sciences, however, security-minded faculty members won the day. The LCS installed its first password-based system in 1977. Once again, Stallman took it upon himself to correct what he saw as ethical laxity. Gaining access to the software code that controlled the password system, Stallman implanted a software command that sent out a message to any LCS user who attempted to choose a unique password. If a user entered "starfish," for example, the message came back something like: I see you chose the password "starfish." I suggest that you switch to the password "carriage return." It's much easier to type, and also it stands up to the principle that there should be no passwords.See Steven Levy, Hackers (Penguin USA [paperback], 1984): 417. I have modified this quote, which Levy also uses as an excerpt, to illustrate more directly how the program might reveal the false security of the system. Levy uses the placeholder "[such and such]." Users who did enter "carriage return"-that is, users who simply pressed the Enter or Return button, entering a blank string instead of a unique password-left their accounts accessible to the world at large. As scary as that might have been for some users, it reinforced the hacker notion that Institute computers, and even Institute computer files, belonged to the public, not private individuals. Stallman, speaking in an interview for the 1984 book Hackers, proudly noted that one-fifth of the LCS staff accepted this argument and employed the blank-string password.See Steven Levy, Hackers (Penguin USA [paperback], 1984): 417. Stallman's null-string crusade would prove ultimately futile. By the early 1980s, even the AI Lab's machines were sporting password-based security systems. Even so, it represents a major milestone in terms of Stallman's personal and political maturation. To the objective observer familiar with Stallman's later career, it offers a convenient inflection point between the timid teenager afraid to speak out even on issues of life-threatening importance and the adult activist who would soon turn needling and cajoling into a full-time occupation. In voicing his opposition to computer security, Stallman drew on many of the forces that had shaped his early life: hunger for knowledge, distaste for authority, and frustration over hidden procedures and rules that rendered some people clueless outcasts. He would also draw on the ethical concepts that would shape his adult life: communal responsibility, trust, and the hacker spirit of direct action. Expressed in software-computing terms, the null string represents the 1.0 version of the Richard Stallman political worldview-incomplete in a few places but, for the most part, fully mature. Looking back, Stallman hesitates to impart too much significance to an event so early in his hacking career. "In that early stage there were a lot of people who shared my feelings," he says. "The large number of people who adopted the null string as their password was a sign that many people agreed that it was the proper thing to do. I was simply inclined to be an activist about it." Stallman does credit the AI Lab for awakening that activist spirit, however. As a teenager, Stallman had observed political events with little idea as to how a single individual could do or say anything of importance. As a young adult, Stallman was speaking out on matters in which he felt supremely confident, matters such as software design, communal responsibility, and individual freedom. "I joined this community which had a way of life which involved respecting each other's freedom," he says. "It didn't take me long to figure out that that was a good thing. It took me longer to come to the conclusion that this was a moral issue." Hacking at the AI Lab wasn't the only activity helping to boost Stallman's esteem. During the middle of his sophomore year at Harvard, Stallman had joined up with a dance troupe that specialized in folk dances . What began as a simple attempt to meet women and expand his social horizons soon expanded into yet another passion alongside hacking. Dancing in front of audiences dressed in the native garb of a Balkan peasant, Stallman no longer felt like the awkward, uncoordinated 10-year-old whose attempts to play football had ended in frustration. He felt confident, agile, and alive. For a brief moment, he even felt a hint of emotional connection. He soon found being in front of an audience fun, and it wasn't long thereafter that he began craving the performance side of dancing almost as much as the social side. Although the dancing and hacking did little to improve Stallman's social standing, they helped him overcome the feelings of weirdness that had clouded his pre-Harvard life. Instead of lamenting his weird nature, Stallman found ways to celebrate it. In 1977, while attending a science-fiction convention, he came across a woman selling custom-made buttons. Excited, Stallman ordered a button with the words "Impeach God" emblazoned on it. For Stallman, the "Impeach God" message worked on many levels. An atheist since early childhood, Stallman first saw it as an attempt to set a "second front" in the ongoing debate on religion. "Back then everybody was arguing about God being dead or alive," Stallman recalls. "`Impeach God' approached the subject of God from a completely different viewpoint. If God was so powerful as to create the world and yet do nothing to correct the problems in it, why would we ever want to worship such a God? Wouldn't it be better to put him on trial?" At the same time, "Impeach God" was a satirical take on America and the American political system. The Watergate scandal of the 1970s affected Stallman deeply. As a child, Stallman had grown up mistrusting authority. Now, as an adult, his mistrust had been solidified by the culture of the AI Lab hacker community. To the hackers, Watergate was merely a Shakespearean rendition of the daily power struggles that made life such a hassle for those without privilege. It was an outsized parable for what happened when people traded liberty and openness for security and convenience. Buoyed by growing confidence, Stallman wore the button proudly. People curious enough to ask him about it received the same well-prepared spiel. "My name is Jehovah," Stallman would say. "I have a special plan to save the universe, but because of heavenly security reasons I can't tell you what that plan is. You're just going to have to put your faith in me, because I see the picture and you don't. You know I'm good because I told you so. If you don't believe me, I'll throw you on my enemies list and throw you in a pit where Infernal Revenue Service will audit your taxes for eternity." Those who interpreted the spiel as a word-for-word parody of the Watergate hearings only got half the message. For Stallman, the other half of the message was something only his fellow hackers seemed to be hearing. One hundred years after Lord Acton warned about absolute power corrupting absolutely, Americans seemed to have forgotten the first part of Acton's truism: power, itself, corrupts. Rather than point out the numerous examples of petty corruption, Stallman felt content voicing his outrage toward an entire system that trusted power in the first place. "I figured why stop with the small fry," says Stallman, recalling the button and its message. "If we went after Nixon, why not going after Mr. Big. The way I see it, any being that has power and abuses it deserves to have that power taken away." Small Puddle of Freedom Ask anyone who's spent more than a minute in Richard Stallman's presence, and you'll get the same recollection: forget the long hair. Forget the quirky demeanor. The first thing you notice is the gaze. One look into Stallman's green eyes, and you know you're in the presence of a true believer. To call the Stallman gaze intense is an understatement. Stallman's eyes don't just look at you; they look through you. Even when your own eyes momentarily shift away out of simple primate politeness, Stallman's eyes remain locked-in, sizzling away at the side of your head like twin photon beams. Maybe that's why most writers, when describing Stallman, tend to go for the religious angle. In a 1998 Salon.com article titled "The Saint of Free Software," Andrew Leonard describes Stallman's green eyes as "radiating the power of an Old Testament prophet."See Andrew Leonard, "The Saint of Free Software," Salon.com (August 1998). http://www.salon.com/21st/feature/1998/08/cov_31feature.html A 1999 Wired magazine article describes the Stallman beard as "Rasputin-like,"See Leander Kahney, "Linux's Forgotten Man," Wired News (March 5, 1999). http://www.wired.com/news/print/0,1294,18291,00.html while a London Guardian profile describes the Stallman smile as the smile of "a disciple seeing Jesus."See "Programmer on moral high ground; Free software is a moral issue for Richard Stallman believes in freedom and free software." London Guardian (November 6, 1999). These are just a small sampling of the religious comparisons. To date, the most extreme comparison has to go to Linus Torvalds, who, in his autobiography-see Linus Torvalds and David Diamond, Just For Fun: The Story of an Accidentaly Revolutionary (HarperCollins Publishers, Inc., 2001): 58-writes "Richard Stallman is the God of Free Software." Honorable mention goes to Larry Lessig, who, in a footnote description of Stallman in his book-see Larry Lessig, The Future of Ideas (Random House, 2001): 270-likens Stallman to Moses: . . . as with Moses, it was another leader, Linus Torvalds, who finally carried the movement into the promised land by facilitating the development of the final part of the OS puzzle. Like Moses, too, Stallman is both respected and reviled by allies within the movement. He is [an] unforgiving, and hence for many inspiring, leader of a critically important aspect of modern culture. I have deep respect for the principle and commitment of this extraordinary individual, though I also have great respect for those who are courageous enough to question his thinking and then sustain his wrath. In a final interview with Stallman, I asked him his thoughts about the religious comparisons. "Some people do compare me with an Old Testament prophent, and the reason is Old Testament prophets said certain social practices were wrong. They wouldn't compromise on moral issues. They couldn't be bought off, and they were usually treated with contempt." Such analogies serve a purpose, but they ultimately fall short. That's because they fail to take into account the vulnerable side of the Stallman persona. Watch the Stallman gaze for an extended period of time, and you will begin to notice a subtle change. What appears at first to be an attempt to intimidate or hypnotize reveals itself upon second and third viewing as a frustrated attempt to build and maintain contact. If, as Stallman himself has suspected from time to time, his personality is the product of autism or Asperger Syndrome, his eyes certainly confirm the diagnosis. Even at their most high-beam level of intensity, they have a tendency to grow cloudy and distant, like the eyes of a wounded animal preparing to give up the ghost. My own first encounter with the legendary Stallman gaze dates back to the March, 1999, LinuxWorld Convention and Expo in San Jose, California. Billed as a "coming out party" for the Linux software community, the convention also stands out as the event that reintroduced Stallman to the technology media. Determined to push for his proper share of credit, Stallman used the event to instruct spectators and reporters alike on the history of the GNU Project and the project's overt political objectives. As a reporter sent to cover the event, I received my own Stallman tutorial during a press conference announcing the release of GNOME 1.0, a free software graphic user interface. Unwittingly, I push an entire bank of hot buttons when I throw out my very first question to Stallman himself: do you think GNOME's maturity will affect the commercial popularity of the Linux operating system? "I ask that you please stop calling the operating system Linux," Stallman responds, eyes immediately zeroing in on mine. "The Linux kernel is just a small part of the operating system. Many of the software programs that make up the operating system you call Linux were not developed by Linus Torvalds at all. They were created by GNU Project volunteers, putting in their own personal time so that users might have a free operating system like the one we have today. To not acknowledge the contribution of those programmers is both impolite and a misrepresentation of history. That's why I ask that when you refer to the operating system, please call it by its proper name, GNU/Linux." Taking the words down in my reporter's notebook, I notice an eerie silence in the crowded room. When I finally look up, I find Stallman's unblinking eyes waiting for me. Timidly, a second reporter throws out a question, making sure to use the term " GNU/Linux" instead of Linux. Miguel de Icaza, leader of the GNOME project, fields the question. It isn't until halfway through de Icaza's answer, however, that Stallman's eyes finally unlock from mine. As soon as they do, a mild shiver rolls down my back. When Stallman starts lecturing another reporter over a perceived error in diction, I feel a guilty tinge of relief. At least he isn't looking at me, I tell myself. For Stallman, such face-to-face moments would serve their purpose. By the end of the first LinuxWorld show, most reporters know better than to use the term "Linux" in his presence, and wired.com is running a story comparing Stallman to a pre-Stalinist revolutionary erased from the history books by hackers and entrepreneurs eager to downplay the GNU Project's overly political objectives.2 Other articles follow, and while few reporters call the operating system GNU/Linux in print, most are quick to credit Stallman for launching the drive to build a free software operating system 15 years before. I won't meet Stallman again for another 17 months. During the interim, Stallman will revisit Silicon Valley once more for the August, 1999 LinuxWorld show. Although not invited to speak, Stallman does managed to deliver the event's best line. Accepting the show's Linus Torvalds Award for Community Service-an award named after Linux creator Linus Torvalds-on behalf of the Free Software Foundation, Stallman wisecracks, "Giving the Linus Torvalds Award to the Free Software Foundation is a bit like giving the Han Solo Award to the Rebel Alliance." This time around, however, the comments fail to make much of a media dent. Midway through the week, Red Hat, Inc., a prominent GNU/Linux vendor, goes public. The news merely confirms what many reporters such as myself already suspect: "Linux" has become a Wall Street buzzword, much like "e-commerce" and "dot-com" before it. With the stock market approaching the Y2K rollover like a hyperbola approaching its vertical asymptote, all talk of free software or open source as a political phenomenon falls by the wayside. Maybe that's why, when LinuxWorld follows up its first two shows with a third LinuxWorld show in August, 2000, Stallman is conspicuously absent. My second encounter with Stallman and his trademark gaze comes shortly after that third LinuxWorld show. Hearing that Stallman is going to be in Silicon Valley, I set up a lunch interview in Palo Alto, California. The meeting place seems ironic, not only because of the recent no-show but also because of the overall backdrop. Outside of Redmond, Washington, few cities offer a more direct testament to the economic value of proprietary software. Curious to see how Stallman, a man who has spent the better part of his life railing against our culture's predilection toward greed and selfishness, is coping in a city where even garage-sized bungalows run in the half-million-dollar price range, I make the drive down from Oakland. I follow the directions Stallman has given me, until I reach the headquarters of Art.net, a nonprofit "virtual artists collective." Located in a hedge-shrouded house in the northern corner of the city, the Art.net headquarters are refreshingly run-down. Suddenly, the idea of Stallman lurking in the heart of Silicon Valley doesn't seem so strange after all. I find Stallman sitting in a darkened room, tapping away on his gray laptop computer. He looks up as soon as I enter the room, giving me a full blast of his 200-watt gaze. When he offers a soothing "Hello," I offer a return greeting. Before the words come out, however, his eyes have already shifted back to the laptop screen. "I'm just finishing an article on the spirit of hacking," Stallman says, fingers still tapping. "Take a look." I take a look. The room is dimly lit, and the text appears as greenish-white letters on a black background, a reversal of the color scheme used by most desktop word-processing programs, so it takes my eyes a moment to adjust. When they do, I find myself reading Stallman's account of a recent meal at a Korean restaurant. Before the meal, Stallman makes an interesting discovery: the person setting the table has left six chopsticks instead of the usual two in front of Stallman's place setting. Where most restaurant goers would have ignored the redundant pairs, Stallman takes it as challenge: find a way to use all six chopsticks at once. Like many software hacks, the successful solution is both clever and silly at the same time. Hence Stallman's decision to use it as an illustration. As I read the story, I feel Stallman watching me intently. I look over to notice a proud but child-like half smile on his face. When I praise the essay, my comment barely merits a raised eyebrow. "I'll be ready to go in a moment," he says. Stallman goes back to tapping away at his laptop. The laptop is gray and boxy, not like the sleek, modern laptops that seemed to be a programmer favorite at the recent LinuxWorld show. Above the keyboard rides a smaller, lighter keyboard, a testament to Stallman's aging hands. During the late 1980s, when Stallman was putting in 70- and 80-hour work weeks writing the first free software tools and programs for the GNU Project, the pain in Stallman's hands became so unbearable that he had to hire a typist. Today, Stallman relies on a keyboard whose keys require less pressure than a typical computer keyboard. Stallman has a tendency to block out all external stimuli while working. Watching his eyes lock onto the screen and his fingers dance, one quickly gets the sense of two old friends locked in deep conversation. The session ends with a few loud keystrokes and the slow disassembly of the laptop. "Ready for lunch?" Stallman asks. We walk to my car. Pleading a sore ankle, Stallman limps along slowly. Stallman blames the injury on a tendon in his left foot. The injury is three years old and has gotten so bad that Stallman, a huge fan of folk dancing, has been forced to give up all dancing activities. "I love folk dancing inherently," Stallman laments. "Not being able to dance has been a tragedy for me." Stallman's body bears witness to the tragedy. Lack of exercise has left Stallman with swollen cheeks and a pot belly that was much less visible the year before. You can tell the weight gain has been dramatic, because when Stallman walks, he arches his back like a pregnant woman trying to accommodate an unfamiliar load. The walk is further slowed by Stallman's willingness to stop and smell the roses, literally. Spotting a particularly beautiful blossom, he tickles the innermost petals with his prodigious nose, takes a deep sniff and steps back with a contented sigh. "Mmm, rhinophytophilia,"At the time, I thought Stallman was referring to the flower's scientific name. Months later, I would learn that rhinophytophilia was in fact a humorous reference to the activity, i.e., Stallman sticking his nose into a flower and enjoying the moment. For another humorous Stallman flower incident, visit: http://www.stallman.org/texas.html he says, rubbing his back. The drive to the restaurant takes less than three minutes. Upon recommendation from Tim Ney, former executive director of the Free Software Foundation, I have let Stallman choose the restaurant. While some reporters zero in on Stallman's monk-like lifestyle, the truth is, Stallman is a committed epicure when it comes to food. One of the fringe benefits of being a traveling missionary for the free software cause is the ability to sample delicious food from around the world. "Visit almost any major city in the world, and chances are Richard knows the best restaurant in town," says Ney. "Richard also takes great pride in knowing what's on the menu and ordering for the entire table." For today's meal, Stallman has chosen a Cantonese-style dim sum restaurant two blocks off University Avenue, Palo Alto's main drag. The choice is partially inspired by Stallman's recent visit to China, including a lecture stop in Guangdong province, in addition to Stallman's personal aversion to spicier Hunanese and Szechuan cuisine. "I'm not a big fan of spicy," Stallman admits. We arrive a few minutes after 11 a.m. and find ourselves already subject to a 20-minute wait. Given the hacker aversion to lost time, I hold my breath momentarily, fearing an outburst. Stallman, contrary to expectations, takes the news in stride. "It's too bad we couldn't have found somebody else to join us," he tells me. "It's always more fun to eat with a group of people." During the wait, Stallman practices a few dance steps. His moves are tentative but skilled. We discuss current events. Stallman says his only regret about not attending LinuxWorld was missing out on a press conference announcing the launch of the GNOME Foundation. Backed by Sun Microsystems and IBM, the foundation is in many ways a vindication for Stallman, who has long championed that free software and free-market economics need not be mutually exclusive. Nevertheless, Stallman remains dissatisfied by the message that came out. "The way it was presented, the companies were talking about Linux with no mention of the GNU Project at all," Stallman says. Such disappointments merely contrast the warm response coming from overseas, especially Asia, Stallman notes. A quick glance at the Stallman 2000 travel itinerary bespeaks the growing popularity of the free software message. Between recent visits to India, China, and Brazil, Stallman has spent 12 of the last 115 days on United States soil. His travels have given him an opportunity to see how the free software concept translates into different languages of cultures. "In India many people are interested in free software, because they see it as a way to build their computing infrastructure without spending a lot of money," Stallman says. "In China, the concept has been much slower to catch on. Comparing free software to free speech is harder to do when you don't have any free speech. Still, the level of interest in free software during my last visit was profound." The conversation shifts to Napster, the San Mateo, California software company, which has become something of a media cause c�l�bre in recent months. The company markets a controversial software tool that lets music fans browse and copy the music files of other music fans. Thanks to the magnifying powers of the Internet, this so-called "peer-to-peer" program has evolved into a de facto online juke box, giving ordinary music fans a way to listen to MP3 music files over the computer without paying a royalty or fee, much to record companies' chagrin. Although based on proprietary software, the Napster system draws inspiration from the long-held Stallman contention that once a work enters the digital realm-in other words, once making a copy is less a matter of duplicating sounds or duplicating atoms and more a matter of duplicating information-the natural human impulse to share a work becomes harder to restrict. Rather than impose additional restrictions, Napster execs have decided to take advantage of the impulse. Giving music listeners a central place to trade music files, the company has gambled on its ability to steer the resulting user traffic toward other commercial opportunities. The sudden success of the Napster model has put the fear in traditional record companies, with good reason. Just days before my Palo Alto meeting with Stallman, U.S. District Court Judge Marilyn Patel granted a request filed by the Recording Industry Association of America for an injunction against the file-sharing service. The injunction was subsequently suspended by the U.S. Ninth District Court of Appeals, but by early 2001, the Court of Appeals, too, would find the San Mateo-based company in breach of copyright law,5 a decision RIAA spokesperson Hillary Rosen would later proclaim proclaim a "clear victory for the creative content community and the legitimate online marketplace."See "A Clear Victory for Recording Industry in Napster Case," RIAA press release (February 12, 2001). http://www.riaa.com/PR_story.cfm?id=372 For hackers such as Stallman, the Napster business model is scary in different ways. The company's eagerness to appropriate time-worn hacker principles such as file sharing and communal information ownership, while at the same time selling a service based on proprietary software, sends a distressing mixed message. As a person who already has a hard enough time getting his own carefully articulated message into the media stream, Stallman is understandably reticent when it comes to speaking out about the company. Still, Stallman does admit to learning a thing or two from the social side of the Napster phenomenon. "Before Napster, I thought it might be OK for people to privately redistribute works of entertainment," Stallman says. "The number of people who find Napster useful, however, tells me that the right to redistribute copies not only on a neighbor-to-neighbor basis, but to the public at large, is essential and therefore may not be taken away." No sooner does Stallman say this than the door to the restaurant swings open and we are invited back inside by the host. Within a few seconds, we are seated in a side corner of the restaurant next to a large mirrored wall. The restaurant's menu doubles as an order form, and Stallman is quickly checking off boxes before the host has even brought water to the table. "Deep-fried shrimp roll wrapped in bean-curd skin," Stallman reads. "Bean-curd skin. It offers such an interesting texture. I think we should get it." This comment leads to an impromptu discussion of Chinese food and Stallman's recent visit to China. "The food in China is utterly exquisite," Stallman says, his voice gaining an edge of emotion for the first time this morning. "So many different things that I've never seen in the U.S., local things made from local mushrooms and local vegetables. It got to the point where I started keeping a journal just to keep track of every wonderful meal." The conversation segues into a discussion of Korean cuisine. During the same June, 2000, Asian tour, Stallman paid a visit to South Korea. His arrival ignited a mini-firestorm in the local media thanks to a Korean software conference attended by Microsoft founder and chairman Bill Gates that same week. Next to getting his photo above Gates's photo on the front page of the top Seoul newspaper, Stallman says the best thing about the trip was the food. "I had a bowl of naeng myun, which is cold noodles," says Stallman. "These were a very interesting feeling noodle. Most places don't use quite the same kind of noodles for your naeng myun, so I can say with complete certainty that this was the most exquisite naeng myun I ever had." The term "exquisite" is high praise coming from Stallman. I know this, because a few moments after listening to Stallman rhapsodize about naeng myun, I feel his laser-beam eyes singeing the top of my right shoulder. "There is the most exquisite woman sitting just behind you," Stallman says. I turn to look, catching a glimpse of a woman's back. The woman is young, somewhere in her mid-20s, and is wearing a white sequinned dress. She and her male lunch companion are in the final stages of paying the check. When both get up from the table to leave the restaurant, I can tell without looking, because Stallman's eyes suddenly dim in intensity. "Oh, no," he says. "They're gone. And to think, I'll probably never even get to see her again." After a brief sigh, Stallman recovers. The moment gives me a chance to discuss Stallman's reputation vis-ý-vis the fairer sex. The reputation is a bit contradictory at times. A number of hackers report Stallman's predilection for greeting females with a kiss on the back of the hand.See Mae Ling Mak, "Mae Ling's Story" (December 17, 1998). http://www.crackmonkey.org/pipermail/crackmonkey/1998q4/003006.htm So far, Mak is the only person I've found willing to speak on the record in regard to this practice, although I've heard this from a few other female sources. Mak, despite expressing initial revulsion at it, later managed to put aside her misgivings and dance with Stallman at a 1999 LinuxWorld show. http://www.linux.com/interact/potd.phtml?potd_id=44 A May 26, 2000 Salon.com article, meanwhile, portrays Stallman as a bit of a hacker lothario. Documenting the free software-free love connection, reporter Annalee Newitz presents Stallman as rejecting traditional family values, telling her, "I believe in love, but not monogamy."See Annalee Newitz, "If Code is Free Why Not Me?" Salon.com (May 26, 2000). Stallman lets his menu drop a little when I bring this up. "Well, most men seem to want sex and seem to have a rather contemptuous attitude towards women," he says. "Even women they're involved with. I can't understand it at all." I mention a passage from the 1999 book Open Sources in which Stallman confesses to wanting to name the ill-fated GNU kernel after a girlfriend at the time. The girlfriend's name was Alix, a name that fit perfectly with the Unix developer convention of putting an "x" at the end of any new kernel name-e.g., "Linux." Because the woman was a Unix system administrator, Stallman says it would have been an even more touching tribute. Unfortunately, Stallman notes, the kernel project's eventual main developer renamed the kernel HURD.See Richard Stallman, "The GNU Operating System and the Free Software Movement," Open Sources (O'Reilly & Associates, Inc., 1999): 65. Although Stallman and the girlfriend later broke up, the story triggers an automatic question: for all the media imagery depicting him as a wild-eyed fanatic, is Richard Stallman really just a hopeless romantic, a wandering Quixote tilting at corporate windmills in an effort to impress some as-yet-unidentified Dulcinea? "I wasn't really trying to be romantic," Stallman says, recalling the Alix story. "It was more of a teasing thing. I mean, it was romantic, but it was also teasing, you know? It would have been a delightful surprise." For the first time all morning, Stallman smiles. I bring up the hand kissing. "Yes, I do do that," Stallman says. "I've found it's a way of offering some affection that a lot of women will enjoy. It's a chance to give some affection and to be appreciated for it." Affection is a thread that runs clear through Richard Stallman's life, and he is painfully candid about it when questions arise. "There really hasn't been much affection in my life, except in my mind," he says. Still, the discussion quickly grows awkward. After a few one-word replies, Stallman finally lifts up his menu, cutting off the inquiry. "Would you like some shimai?" he asks. When the food comes out, the conversation slaloms between the arriving courses. We discuss the oft-noted hacker affection for Chinese food, the weekly dinner runs into Boston's Chinatown district during Stallman's days as a staff programmer at the AI Lab, and the underlying logic of the Chinese language and its associated writing system. Each thrust on my part elicits a well-informed parry on Stallman's part. "I heard some people speaking Shanghainese the last time I was in China," Stallman says. "It was interesting to hear. It sounded quite different [from Mandarin]. I had them tell me some cognate words in Mandarin and Shanghainese. In some cases you can see the resemblance, but one question I was wondering about was whether tones would be similar. They're not. That's interesting to me, because there's a theory that the tones evolved from additional syllables that got lost and replaced. Their effect survives in the tone. If that's true, and I've seen claims that that happened within historic times, the dialects must have diverged before the loss of these final syllables." The first dish, a plate of pan-fried turnip cakes, has arrived. Both Stallman and I take a moment to carve up the large rectangular cakes, which smell like boiled cabbage but taste like potato latkes fried in bacon. I decide to bring up the outcast issue again, wondering if Stallman's teenage years conditioned him to take unpopular stands, most notably his uphill battle since 1994 to get computer users and the media to replace the popular term "Linux" with "GNU/Linux." "I believe it did help me," Stallman says, chewing on a dumpling. "I have never understood what peer pressure does to other people. I think the reason is that I was so hopelessly rejected that for me, there wasn't anything to gain by trying to follow any of the fads. It wouldn't have made any difference. I'd still be just as rejected, so I didn't try." Stallman points to his taste in music as a key example of his contrarian tendencies. As a teenager, when most of his high school classmates were listening to Motown and acid rock, Stallman preferred classical music. The memory leads to a rare humorous episode from Stallman's middle-school years. Following the Beatles' 1964 appearance on the Ed Sullivan Show, most of Stallman's classmates rushed out to purchase the latest Beatles albums and singles. Right then and there, Stallman says, he made a decision to boycott the Fab Four. "I liked some of the pre-Beatles popular music," Stallman says. "But I didn't like the Beatles. I especially disliked the wild way people reacted to them. It was like: who was going to have a Beatles assembly to adulate the Beatles the most?" When his Beatles boycott failed to take hold, Stallman looked for other ways to point out the herd-mentality of his peers. Stallman says he briefly considered putting together a rock band himself dedicated to satirizing the Liverpool group. "I wanted to call it Tokyo Rose and the Japanese Beetles." Given his current love for international folk music, I ask Stallman if he had a similar affinity for Bob Dylan and the other folk musicians of the early 1960s. Stallman shakes his head. "I did like Peter, Paul and Mary," he says. "That reminds me of a great filk." When I ask for a definition of "filk," Stallman explains the concept. A filk, he says, is a popular song whose lyrics have been replaced with parody lyrics. The process of writing a filk is called filking, and it is a popular activity among hackers and science-fiction aficionados. Classic filks include "On Top of Spaghetti," a rewrite of "On Top of Old Smokey," and "Yoda," filk-master "Weird" Al Yankovic's Star Wars-oriented rendition of the Kinks tune, "Lola." Stallman asks me if I would be interested in hearing the folk filk. As soon as I say yes, Stallman's voice begins singing in an unexpectedly clear tone: How much wood could a woodchuck chuck,If a woodchuck could chuck wood?How many poles could a polak lock,If a polak could lock poles?How many knees could a negro grow,If a negro could grow knees?The answer, my dear, is stick it in your ear.The answer is to stick it in your ear. The singing ends, and Stallman's lips curl into another child-like half smile. I glance around at the nearby tables. The Asian families enjoying their Sunday lunch pay little attention to the bearded alto in their midst.For more Stallman filks, visit http://www.stallman.org/doggerel.html. To hear Stallman singing "The Free Software Song," visit http://www.gnu.org/music/free-software-song.html. After a few moments of hesitation, I finally smile too. "Do you want that last cornball?" Stallman asks, eyes twinkling. Before I can screw up the punch line, Stallman grabs the corn-encrusted dumpling with his two chopsticks and lifts it proudly. "Maybe I'm the one who should get the cornball," he says. The food gone, our conversation assumes the dynamics of a normal interview. Stallman reclines in his chair and cradles a cup of tea in his hands. We resume talking about Napster and its relation to the free software movement. Should the principles of free software be extended to similar arenas such as music publishing? I ask. "It's a mistake to transfer answers from one thing to another," says Stallman, contrasting songs with software programs. "The right approach is to look at each type of work and see what conclusion you get." When it comes to copyrighted works, Stallman says he divides the world into three categories. The first category involves "functional" works-e.g., software programs, dictionaries, and textbooks. The second category involves works that might best be described as "testimonial"-e.g., scientific papers and historical documents. Such works serve a purpose that would be undermined if subsequent readers or authors were free to modify the work at will. The final category involves works of personal expression-e.g., diaries, journals, and autobiographies. To modify such documents would be to alter a person's recollections or point of view-action Stallman considers ethically unjustifiable. Of the three categories, the first should give users the unlimited right to make modified versions, while the second and third should regulate that right according to the will of the original author. Regardless of category, however, the freedom to copy and redistribute noncommercially should remain unabridged at all times, Stallman insists. If that means giving Internet users the right to generate a hundred copies of an article, image, song, or book and then email the copies to a hundred strangers, so be it. "It's clear that private occasional redistribution must be permitted, because only a police state can stop that," Stallman says. "It's antisocial to come between people and their friends. Napster has convinced me that we also need to permit, must permit, even noncommercial redistribution to the public for the fun of it. Because so many people want to do that and find it so useful." When I ask whether the courts would accept such a permissive outlook, Stallman cuts me off. "That's the wrong question," he says. "I mean now you've changed the subject entirely from one of ethics to one of interpreting laws. And those are two totally different questions in the same field. It's useless to jump from one to the other. How the courts would interpret the existing laws is mainly in a harsh way, because that's the way these laws have been bought by publishers." The comment provides an insight into Stallman's political philosophy: just because the legal system currently backs up businesses' ability to treat copyright as the software equivalent of land title doesn't mean computer users have to play the game according to those rules. Freedom is an ethical issue, not a legal issue. "I'm looking beyond what the existing laws are to what they should be," Stallman says. "I'm not trying to draft legislation. I'm thinking about what should the law do? I consider the law prohibiting the sharing of copies with your friend the moral equivalent of Jim Crow. It does not deserve respect." The invocation of Jim Crow prompts another question. How much influence or inspiration does Stallman draw from past political leaders? Like the civil-rights movement of the 1950s and 1960s, his attempt to drive social change is based on an appeal to timeless values: freedom, justice, and fair play. Stallman divides his attention between my analogy and a particularly tangled strand of hair. When I stretch the analogy to the point where I'm comparing Stallman with Dr. Martin Luther King, Jr., Stallman, after breaking off a split end and popping it into his mouth, cuts me off. "I'm not in his league, but I do play the same game," he says, chewing. I suggest Malcolm X as another point of comparison. Like the former Nation of Islam spokesperson, Stallman has built up a reputation for courting controversy, alienating potential allies, and preaching a message favoring self-sufficiency over cultural integration. Chewing on another split end, Stallman rejects the comparison. "My message is closer to King's message," he says. "It's a universal message. It's a message of firm condemnation of certain practices that mistreat others. It's not a message of hatred for anyone. And it's not aimed at a narrow group of people. I invite anyone to value freedom and to have freedom." Even so, a suspicious attitude toward political alliances remains a fundamental Stallman character trait. In the case of his well-publicized distaste for the term "open source," the unwillingness to participate in recent coalition-building projects seems understandable. As a man who has spent the last two decades stumping on the behalf of free software, Stallman's political capital is deeply invested in the term. Still, comments such as the "Han Solo" wisecrack at the 1999 LinuxWorld have only reinforced the Stallman's reputation in the software industry as a disgrunted mossback unwilling to roll with political or marketing trends. "I admire and respect Richard for all the work he's done," says Red Hat president Robert Young, summing up Stallman's paradoxical political nature. "My only critique is that sometimes Richard treats his friends worse than his enemies." Stallman's unwillingness to seek alliances seems equally perplexing when you consider his political interests outside of the free software movement. Visit Stallman's offices at MIT, and you instantly find a clearinghouse of left-leaning news articles covering civil-rights abuses around the globe. Visit his web site, and you'll find diatribes on the Digital Millennium Copyright Act, the War on Drugs, and the World Trade Organization. Given his activist tendencies, I ask, why hasn't Stallman sought a larger voice? Why hasn't he used his visibility in the hacker world as a platform to boost rather than reduce his political voice. Stallman lets his tangled hair drop and contemplates the question for a moment. "I hesitate to exaggerate the importance of this little puddle of freedom," he says. "Because the more well-known and conventional areas of working for freedom and a better society are tremendously important. I wouldn't say that free software is as important as they are. It's the responsibility I undertook, because it dropped in my lap and I saw a way I could do something about it. But, for example, to end police brutality, to end the war on drugs, to end the kinds of racism we still have, to help everyone have a comfortable life, to protect the rights of people who do abortions, to protect us from theocracy, these are tremendously important issues, far more important than what I do. I just wish I knew how to do something about them." Once again, Stallman presents his political activity as a function of personal confidence. Given the amount of time it has taken him to develop and hone the free software movement's core tenets, Stallman is hesitant to jump aboard any issues or trends that might transport him into uncharted territory. "I wish I knew I how to make a major difference on those bigger issues, because I would be tremendously proud if I could, but they're very hard and lots of people who are probably better than I am have been working on them and have gotten only so far," he says. "But as I see it, while other people were defending against these big visible threats, I saw another threat that was unguarded. And so I went to defend against that threat. It may not be as big a threat, but I was the only one there." Chewing a final split end, Stallman suggests paying the check. Before the waiter can take it away, however, Stallman pulls out a white-colored dollar bill and throws it on the pile. The bill looks so clearly counterfeit, I can't help but pick it up and read it. Sure enough, it is counterfeit. Instead of bearing the image of a George Washington or Abe Lincoln, the bill's front side bears the image of a cartoon pig. Instead of the United States of America, the banner above the pig reads "United Swines of Avarice." The bill is for zero dollars, and when the waiter picks up the money, Stallman makes sure to tug on his sleeve. "I added an extra zero to your tip," Stallman says, yet another half smile creeping across his lips. The waiter, uncomprehending or fooled by the look of the bill, smiles and scurries away. "I think that means we're free to go," Stallman says. The Emacs Commune The AI Lab of the 1970s was by all accounts a special place. Cutting-edge projects and top-flight researchers gave it an esteemed position in the world of computer science. The internal hacker culture and its anarchic policies lent a rebellious mystique as well. Only later, when many of the lab's scientists and software superstars had departed, would hackers fully realize the unique and ephemeral world they had once inhabited. "It was a bit like the Garden of Eden," says Stallman, summing up the lab and its software-sharing ethos in a 1998 Forbes article. "It hadn't occurred to us not to cooperate."See Josh McHugh, "For the Love of Hacking," Forbes (August 10, 1998). http://www.forbes.com/forbes/1998/0810/6203094a.html Such mythological descriptions, while extreme, underline an important fact. The ninth floor of 545 Tech Square was more than a workplace for many. For hackers such as Stallman, it was home. The word "home" is a weighted term in the Stallman lexicon. In a pointed swipe at his parents, Stallman, to this day, refuses to acknowledge any home before Currier House, the dorm he lived in during his days at Harvard. He has also been known to describe leaving that home in tragicomic terms. Once, while describing his years at Harvard, Stallman said his only regret was getting kicked out. It wasn't until I asked Stallman what precipitated his ouster, that I realized I had walked into a classic Stallman setup line. "At Harvard they have this policy where if you pass too many classes they ask you to leave," Stallman says. With no dorm and no desire to return to New York, Stallman followed a path blazed by Greenblatt, Gosper, Sussman, and the many other hackers before him. Enrolling at MIT as a grad student, Stallman rented an apartment in nearby Cambridge but soon viewed the AI Lab itself as his de facto home. In a 1986 speech, Stallman recalled his memories of the AI Lab during this period: I may have done a little bit more living at the lab than most people, because every year or two for some reason or other I'd have no apartment and I would spend a few months living at the lab. And I've always found it very comfortable, as well as nice and cool in the summer. But it was not at all uncommon to find people falling asleep at the lab, again because of their enthusiasm; you stay up as long as you possibly can hacking, because you just don't want to stop. And then when you're completely exhausted, you climb over to the nearest soft horizontal surface. A very informal atmosphere.See Stallman (1986). The lab's home-like atmosphere could be a problem at times. What some saw as a dorm, others viewed as an electronic opium den. In the 1976 book Computer Power and Human Reason, MIT researcher Joseph Weizenbaum offered a withering critique of the " computer bum," Weizenbaum's term for the hackers who populated computer rooms such as the AI Lab. "Their rumpled clothes, their unwashed hair and unshaved faces, and their uncombed hair all testify that they are oblivious to their bodies and to the world in which they move," Weizenbaum wrote. "[Computer bums] exist, at least when so engaged, only through and for the computers."See Joseph Weizenbaum, Computer Power and Human Reason: From Judgment to Calculation (W. H. Freeman, 1976): 116. Almost a quarter century after its publication, Stallman still bristles when hearing Weizenbaum's "computer bum" description, discussing it in the present tense as if Weizenbaum himself was still in the room. "He wants people to be just professionals, doing it for the money and wanting to get away from it and forget about it as soon as possible," Stallman says. "What he sees as a normal state of affairs, I see as a tragedy." Hacker life, however, was not without tragedy. Stallman characterizes his transition from weekend hacker to full-time AI Lab denizen as a series of painful misfortunes that could only be eased through the euphoria of hacking. As Stallman himself has said, the first misfortune was his graduation from Harvard. Eager to continue his studies in physics, Stallman enrolled as a graduate student at MIT. The choice of schools was a natural one. Not only did it give Stallman the chance to follow the footsteps of great MIT alumni: William Shockley ('36), Richard P. Feynman ('39), and Murray Gell-Mann ('51), it also put him two miles closer to the AI Lab and its new PDP-10 computer. "My attention was going toward programming, but I still thought, well, maybe I can do both," Stallman says. Toiling in the fields of graduate-level science by day and programming in the monastic confines of the AI Lab by night, Stallman tried to achieve a perfect balance. The fulcrum of this geek teeter-totter was his weekly outing with the folk-dance troupe, his one social outlet that guaranteed at least a modicum of interaction with the opposite sex. Near the end of that first year at MIT, however, disaster struck. A knee injury forced Stallman to drop out of the troupe. At first, Stallman viewed the injury as a temporary problem, devoting the spare time he would have spent dancing to working at the AI Lab even more. By the end of the summer, when the knee still ached and classes reconvened, Stallman began to worry. "My knee wasn't getting any better," Stallman recalls, "which meant I had to stop dancing completely. I was heartbroken." With no dorm and no dancing, Stallman's social universe imploded. Like an astronaut experiencing the aftereffects of zero-gravity, Stallman found that his ability to interact with nonhackers, especially female nonhackers, had atrophied significantly. After 16 weeks in the AI Lab, the self confidence he'd been quietly accumulating during his 4 years at Harvard was virtually gone. "I felt basically that I'd lost all my energy," Stallman recalls. "I'd lost my energy to do anything but what was most immediately tempting. The energy to do something else was gone. I was in total despair." Stallman retreated from the world even further, focusing entirely on his work at the AI Lab. By October, 1975, he dropped out of MIT, never to go back. Software hacking, once a hobby, had become his calling. Looking back on that period, Stallman sees the transition from full-time student to full-time hacker as inevitable. Sooner or later, he believes, the siren's call of computer hacking would have overpowered his interest in other professional pursuits. "With physics and math, I could never figure out a way to contribute," says Stallman, recalling his struggles prior to the knee injury. "I would have been proud to advance either one of those fields, but I could never see a way to do that. I didn't know where to start. With software, I saw right away how to write things that would run and be useful. The pleasure of that knowledge led me to want to do it more." Stallman wasn't the first to equate hacking with pleasure. Many of the hackers who staffed the AI Lab boasted similar, incomplete academic r�sum�s. Most had come in pursuing degrees in math or electrical engineering only to surrender their academic careers and professional ambitions to the sheer exhilaration that came with solving problems never before addressed. Like St. Thomas Aquinas, the scholastic known for working so long on his theological summae that he sometimes achieved spiritual visions, hackers reached transcendent internal states through sheer mental focus and physical exhaustion. Although Stallman shunned drugs, like most hackers, he enjoyed the "high" that came near the end of a 20-hour coding bender. Perhaps the most enjoyable emotion, however, was the sense of personal fulfillment. When it came to hacking, Stallman was a natural. A childhood's worth of late-night study sessions gave him the ability to work long hours with little sleep. As a social outcast since age 10, he had little difficulty working alone. And as a mathematician with built-in gift for logic and foresight, Stallman possessed the ability to circumvent design barriers that left most hackers spinning their wheels. "He was special," recalls Gerald Sussman, an MIT faculty member and former AI Lab researcher. Describing Stallman as a "clear thinker and a clear designer," Sussman employed Stallman as a research-project assistant beginning in 1975. The project was complex, involving the creation of an AI program that could analyze circuit diagrams. Not only did it involve an expert's command of Lisp, a programming language built specifically for AI applications, but it also required an understanding of how a human might approach the same task. When he wasn't working on official projects such as Sussman's automated circuit-analysis program, Stallman devoted his time to pet projects. It was in a hacker's best interest to improve the lab's software infrastructure, and one of Stallman's biggest pet projects during this period was the lab's editor program TECO. The story of Stallman's work on TECO during the 1970s is inextricably linked with Stallman's later leadership of the free software movement. It is also a significant stage in the history of computer evolution, so much so that a brief recapitulation of that evolution is necessary. During the 1950s and 1960s, when computers were first appearing at universities, computer programming was an incredibly abstract pursuit. To communicate with the machine, programmers created a series of punch cards, with each card representing an individual software command. Programmers would then hand the cards over to a central system administrator who would then insert them, one by one, into the machine, waiting for the machine to spit out a new set of punch cards, which the programmer would then decipher as output. This process, known as " batch processing," was cumbersome and time consuming. It was also prone to abuses of authority. One of the motivating factors behind hackers' inbred aversion to centralization was the power held by early system operators in dictating which jobs held top priority. In 1962, computer scientists and hackers involved in MIT's Project MAC, an early forerunner of the AI Lab, took steps to alleviate this frustration. Time-sharing, originally known as "time stealing," made it possible for multiple programs to take advantage of a machine's operational capabilities. Teletype interfaces also made it possible to communicate with a machine not through a series of punched holes but through actual text. A programmer typed in commands and read the line-by-line output generated by the machine. During the late 1960s, interface design made additional leaps. In a famous 1968 lecture, Doug Engelbart, a scientist then working at the Stanford Research Institute, unveiled a prototype of the modern graphical interface. Rigging up a television set to the computer and adding a pointer device which Engelbart dubbed a " mouse," the scientist created a system even more interactive than the time-sharing system developed a MIT. Treating the video display like a high-speed printer, Engelbart's system gave a user the ability to move the cursor around the screen and see the cursor position updated by the computer in real time. The user suddenly had the ability to position text anywhere on the screen. Such innovations would take another two decades to make their way into the commercial marketplace. Still, by the 1970s, video screens had started to replace teletypes as display terminals, creating the potential for full-screen-as opposed to line-by-line-editing capabilities. One of the first programs to take advantage of this full-screen capability was the MIT AI Lab's TECO. Short for Text Editor and COrrector, the program had been upgraded by hackers from an old teletype line editor for the lab's PDP-6 machine.ccording to the Jargon File, TECO's name originally stood for Tape Editor and Corrector. TECO was a substantial improvement over old editors, but it still had its drawbacks. To create and edit a document, a programmer had to enter a series of software commands specifying each edit. It was an abstract process. Unlike modern word processors, which update text with each keystroke, TECO demanded that the user enter an extended series of editing instructions followed by an "end of command" sequence just to change the text.Over time, a hacker grew proficient enough to write entire documents in edit mode, but as Stallman himself would later point out, the process required "a mental skill like that of blindfold chess."See Richard Stallman, "EMACS: The Extensible, Customizable, Display Editor," AI Lab Memo (1979). An updated HTML version of this memo, from which I am quoting, is available at http://www.gnu.org/software/emacs/emacs-paper.html. To facilitate the process, AI Lab hackers had built a system that displayed both the "source" and "display" modes on a split screen. Despite this innovative hack, switching from mode to mode was still a nuisance. TECO wasn't the only full-screen editor floating around the computer world at this time. During a visit to the Stanford Artificial Intelligence Lab in 1976, Stallman encountered an edit program named E. The program contained an internal feature, which allowed a user to update display text after each command keystroke. In the language of 1970s programming, E was one of the first rudimentary WYSIWYG editors. Short for "what you see is what you get," WYSIWYG meant that a user could manipulate the file by moving through the displayed text, as opposed to working through a back-end editor program."See Richard Stallman, "Emacs the Full Screen Editor" (1987). http://www.lysator.liu.se/history/garb/txt/87-1-emacs.txt Impressed by the hack, Stallman looked for ways to expand TECO's functionality in similar fashion upon his return to MIT. He found a TECO feature called Control-R, written by Carl Mikkelson and named after the two-key combination that triggered it. Mikkelson's hack switched TECO from its usual abstract command-execution mode to a more intuitive keystroke-by-keystroke mode. Stallman revised the feature in a subtle but significant way. He made it possible to trigger other TECO command strings, or " macros," using other, two-key combinations. Where users had once entered command strings and discarded them after entering then, Stallman's hack made it possible to save macro tricks on file and call them up at will. Mikkelson's hack had raised TECO to the level of a WYSIWYG editor. Stallman's hack had raised it to the level of a user-programmable WYSIWYG editor. "That was the real breakthrough," says Guy Steele, a fellow AI Lab hacker at the time. By Stallman's own recollection, the macro hack touched off an explosion of further innovation. "Everybody and his brother was writing his own collection of redefined screen-editor commands, a command for everything he typically liked to do," Stallman would later recall. "People would pass them around and improve them, making them more powerful and more general. The collections of redefinitions gradually became system programs in their own right." So many people found the macro innovations useful and had incorporated it into their own TECO programs that the TECO editor had become secondary to the macro mania it inspired. "We started to categorize it mentally as a programming language rather than as an editor," Stallman says. Users were experiencing their own pleasure tweaking the software and trading new ideas. Two years after the explosion, the rate of innovation began to exhibit dangerous side effects. The explosive growth had provided an exciting validation of the collaborative hacker approach, but it had also led to over-complexity. "We had a Tower of Babel effect," says Guy Steele. The effect threatened to kill the spirit that had created it, Steele says. Hackers had designed ITS to facilitate programmers' ability to share knowledge and improve each other's work. That meant being able to sit down at another programmer's desk, open up a programmer's work and make comments and modifications directly within the software. "Sometimes the easiest way to show somebody how to program or debug something was simply to sit down at the terminal and do it for them," explains Steele. The macro feature, after its second year, began to foil this capability. In their eagerness to embrace the new full-screen capabilities, hackers had customized their versions of TECO to the point where a hacker sitting down at another hacker's terminal usually had to spend the first hour just figuring out what macro commands did what. Frustrated, Steele took it upon himself to the solve the problem. He gathered together the four different macro packages and began assembling a chart documenting the most useful macro commands. In the course of implementing the design specified by the chart, Steele says he attracted Stallman's attention. "He started looking over my shoulder, asking me what I was doing," recalls Steele. For Steele, a soft-spoken hacker who interacted with Stallman infrequently, the memory still sticks out. Looking over another hacker's shoulder while he worked was a common activity at the AI Lab. Stallman, the TECO maintainer at the lab, deemed Steele's work "interesting" and quickly set off to complete it. "As I like to say, I did the first 0.001 percent of the implementation, and Stallman did the rest," says Steele with a laugh. The project's new name, Emacs, came courtesy of Stallman. Short for "editing macros," it signified the evolutionary transcendence that had taken place during the macros explosion two years before. It also took advantage of a gap in the software programming lexicon. Noting a lack of programs on ITS starting with the letter "E," Stallman chose Emacs, making it possible to reference the program with a single letter. Once again, the hacker lust for efficiency had left its mark. In the course of developing a standard system of macro commands, Stallman and Steele had to traverse a political tightrope. In creating a standard program, Stallman was in clear violation of the fundamental hacker tenet-"promote decentralization." He was also threatening to hobble the very flexibility that had fueled TECO's explosive innovation in the first place. "On the one hand, we were trying to make a uniform command set again; on the other hand, we wanted to keep it open ended, because the programmability was important," recalls Steele. To solve the problem, Stallman, Steele, and fellow hackers David Moon and Dan Weinreib limited their standardization effort to the WYSIWYG commands that controlled how text appeared on-screen. The rest of the Emacs effort would be devoted to retaining the program's Tinker Toy-style extensibility. Stallman now faced another conundrum: if users made changes but didn't communicate those changes back to the rest of the community, the Tower of Babel effect would simply emerge in other places. Falling back on the hacker doctrine of sharing innovation, Stallman embedded a statement within the source code that set the terms of use. Users were free to modify and redistribute the code on the condition that they gave back all the extensions they made. Stallman dubbed it the " Emacs Commune." Just as TECO had become more than a simple editor, Emacs had become more than a simple software program. To Stallman, it was a social contract. In an early memo documenting the project, Stallman spelled out the contract terms. "EMACS," he wrote, "was distributed on a basis of communal sharing, which means that all improvements must be given back to me to be incorporated and distributed."See Stallman (1979): #SEC34. Not everybody accepted the contract. The explosive innovation continued throughout the decade, resulting in a host of Emacs-like programs with varying degrees of cross-compatibility. A few cited their relation to Stallman's original Emacs with humorously recursive names: Sine (Sine is not Emacs), Eine (Eine is not Emacs), and Zwei (Zwei was Eine initially). As a devoted exponent of the hacker ethic, Stallman saw no reason to halt this innovation through legal harassment. Still, the fact that some people would so eagerly take software from the community chest, alter it, and slap a new name on the resulting software displayed a stunning lack of courtesy. Such rude behavior was reflected against other, unsettling developments in the hacker community. Brian Reid's 1979 decision to embed "time bombs" in Scribe, making it possible for Unilogic to limit unpaid user access to the software, was a dark omen to Stallman. "He considered it the most Nazi thing he ever saw in his life," recalls Reid. Despite going on to later Internet fame as the cocreator of the Usenet alt heirarchy, Reid says he still has yet to live down that 1979 decision, at least in Stallman's eyes. "He said that all software should be free and the prospect of charging money for software was a crime against humanity."In a 1996 interview with online magazine MEME , Stallman cited Scribe's sale as irksome, but hesitated to mention Reid by name. "The problem was nobody censured or punished this student for what he did," Stallman said. "The result was other people got tempted to follow his example." See MEME 2.04. http://memex.org/meme2-04.html Although Stallman had been powerless to head off Reid's sale, he did possess the ability to curtail other forms of behavior deemed contrary to the hacker ethos. As central source-code maintainer for the Emacs "commune," Stallman began to wield his power for political effect. During his final stages of conflict with the administrators at the Laboratory for Computer Science over password systems, Stallman initiated a software " strike,"See Steven Levy, Hackers (Penguin USA [paperback], 1984): 419. refusing to send lab members the latest version of Emacs until they rejected the security system on the lab's computers. The move did little to improve Stallman's growing reputation as an extremist, but it got the point across: commune members were expected to speak up for basic hacker values. "A lot of people were angry with me, saying I was trying to hold them hostage or blackmail them, which in a sense I was," Stallman would later tell author Steven Levy. "I was engaging in violence against them because I thought they were engaging in violence to everyone at large." Over time, Emacs became a sales tool for the hacker ethic. The flexibility Stallman and built into the software not only encouraged collaboration, it demanded it. Users who didn't keep abreast of the latest developments in Emacs evolution or didn't contribute their contributions back to Stallman ran the risk of missing out on the latest breakthroughs. And the breakthroughs were many. Twenty years later, users had modified Emacs for so many different uses-using it as a spreadsheet, calculator, database, and web browser-that later Emacs developers adopted an overflowing sink to represent its versatile functionality. "That's the idea that we wanted to convey," says Stallman. "The amount of stuff it has contained within it is both wonderful and awful at the same time." Stallman's AI Lab contemporaries are more charitable. Hal Abelson, an MIT grad student who worked with Stallman during the 1970s and would later assist Stallman as a charter boardmember of the Free Software Foundation, describes Emacs as "an absolutely brilliant creation." In giving programmers a way to add new software libraries and features without messing up the system, Abelson says, Stallman paved the way for future large-scale collaborative software projects. "Its structure was robust enough that you'd have people all over the world who were loosely collaborating [and] contributing to it," Abelson says. "I don't know if that had been done before."In writing this chapter, I've elected to focus more on the social significance of Emacs than the software significance. To read more about the software side, I recommend Stallman's 1979 memo. I particularly recommend the section titled "Research Through Development of Installed Tools" (#SEC27). Not only is it accessible to the nontechnical reader, it also sheds light on how closely intertwined Stallman's political philosophies are with his software-design philosophies. A sample excerpt follows: EMACS could not have been reached by a process of careful design, because such processes arrive only at goals which are visible at the outset, and whose desirability is established on the bottom line at the outset. Neither I nor anyone else visualized an extensible editor until I had made one, nor appreciated its value until he had experienced it. EMACS exists because I felt free to make individually useful small improvements on a path whose end was not in sight. Guy Steele expresses similar admiration. Currently a research scientist for Sun Microsystems, he remembers Stallman primarily as a "brilliant programmer with the ability to generate large quantities of relatively bug-free code." Although their personalities didn't exactly mesh, Steele and Stallman collaborated long enough for Steele to get a glimpse of Stallman's intense coding style. He recalls a notable episode in the late 1970s when the two programmers banded together to write the editor's "pretty print" feature. Originally conceived by Steele, pretty print was another keystroke-triggerd feature that reformatted Emacs' source code so that it was both more readable and took up less space, further bolstering the program's WYSIWIG qualities. The feature was strategic enough to attract Stallman's active interest, and it wasn't long before Steele wrote that he and Stallman were planning an improved version. "We sat down one morning," recalls Steele. "I was at the keyboard, and he was at my elbow," says Steele. "He was perfectly willing to let me type, but he was also telling me what to type. The programming session lasted 10 hours. Throughout that entire time, Steele says, neither he nor Stallman took a break or made any small talk. By the end of the session, they had managed to hack the pretty print source code to just under 100 lines. "My fingers were on the keyboard the whole time," Steele recalls, "but it felt like both of our ideas were flowing onto the screen. He told me what to type, and I typed it." The length of the session revealed itself when Steele finally left the AI Lab. Standing outside the building at 545 Tech Square, he was surprised to find himself surrounded by nighttime darkness. As a programmer, Steele was used to marathon coding sessions. Still, something about this session was different. Working with Stallman had forced Steele to block out all external stimuli and focus his entire mental energies on the task at hand. Looking back, Steele says he found the Stallman mind-meld both exhilarating and scary at the same time. "My first thought afterward was: it was a great experience, very intense, and that I never wanted to do it again in my life." A Stark Moral Choice On September 27, 1983, computer programmers logging on to the Usenet newsgroup net.unix-wizards encountered an unusual message. Posted in the small hours of the morning, 12:30 a.m. to be exact, and signed by rms@mit-oz , the message's subject line was terse but attention-grabbing. "New UNIX implementation," it read. Instead of introducing a newly released version of Unix, however, the message's opening paragraph issued a call to arms: Starting this Thanksgiving I am going to write a complete Unix-compatible software system called GNU (for Gnu's Not Unix), and give it away free to everyone who can use it. Contributions of time, money, programs and equipment are greatly needed.1 To an experienced Unix developer, the message was a mixture of idealism and hubris. Not only did the author pledge to rebuild the already mature Unix operating system from the ground up, he also proposed to improve it in places. The new GNU system, the author predicted, would carry all the usual components-a text editor, a shell program to run Unix-compatible applications, a compiler, "and a few other things."See Richard Stallman, "Initial GNU Announcement" (September 1983). http://www.gnu.ai.mit.edu/gnu/initial-announcement.html It would also contain many enticing features that other Unix systems didn't yet offer: a graphic user interface based on the Lisp programming language, a crash-proof file system, and networking protocols built according to MIT's internal networking system. "GNU will be able to run Unix programs, but will not be identical to Unix," the author wrote. "We will make all improvements that are convenient, based on our experience with other operating systems." Anticipating a skeptical response on some readers' part, the author made sure to follow up his operating-system outline with a brief biographical sketch titled, "Who am I?": I am Richard Stallman, inventor of the original much-imitated EMACS editor, now at the Artificial Intelligence Lab at MIT. I have worked extensively on compilers, editors, debuggers, command interpreters, the Incompatible Timesharing System and the Lisp Machine operating system. I pioneered terminal-independent display support in ITS. In addition I have implemented one crashproof file system and two window systems for Lisp machines. As fate would have it, Stallman's fanciful GNU Project missed its Thanksgiving launch date. By January, 1984, however, Stallman made good on his promise and fully immersed himself in the world of Unix software development. For a software architect raised on ITS, it was like designing suburban shopping malls instead of Moorish palaces. Even so, building a Unix-like operating system had its hidden advantages. ITS had been powerful, but it also possessed an Achilles' heel: MIT hackers had designed it to take specific advantage of the DEC-built PDP line. When AI Lab administrators elected to phase out the lab's powerful PDP-10 machine in the early 1980s, the operating system that hackers once likened to a vibrant city became an instant ghost town. Unix, on the other hand, was designed for mobility and long-term survival. Originally developed by junior scientists at AT&T, the program had slipped out under corporate-management radar, finding a happy home in the cash-strapped world of academic computer systems. With fewer resources than their MIT brethren, Unix developers had customized the software to ride atop a motley assortment of hardware systems: everything from the 16-bit PDP-11-a machine considered fit for only small tasks by most AI Lab hackers-to 32-bit mainframes such as the VAX 11/780. By 1983, a few companies, most notably Sun Microsystems, were even going so far as to develop a new generation of microcomputers, dubbed "workstations," to take advantage of the increasingly ubiquitous operating system. To facilitate this process, the developers in charge of designing the dominant Unix strains made sure to keep an extra layer of abstraction between the software and the machine. Instead of tailoring the operating system to take advantage of a specific machine's resources-as the AI Lab hackers had done with ITS and the PDP-10-Unix developers favored a more generic, off-the-rack approach. Focusing more on the interlocking standards and specifications that held the operating system's many subcomponents together, rather than the actual components themselves, they created a system that could be quickly modified to suit the tastes of any machine. If a user quibbled with a certain portion, the standards made it possible to pull out an individual subcomponent and either fix it or replace it with something better. Simply put, what the Unix approach lacked in terms of style or aesthetics, it more than made up for in terms of flexibility and economy, hence its rapid adoption.See Marshall Kirk McKusick, "Twenty Years of Berkeley Unix," Open Sources (O'Reilly & Associates, Inc., 1999): 38. Stallman's decision to start developing the GNU system was triggered by the end of the ITS system that the AI Lab hackers had nurtured for so long. The demise of ITS had been a traumatic blow to Stallman. Coming on the heels of the Xerox laser printer episode, it offered further evidence that the AI Lab hacker culture was losing its immunity to business practices in the outside world. Like the software code that composed it, the roots of ITS' demise stretched way back. Defense spending, long a major font for computer-science research, had dried up during the post-Vietnam years. In a desperate quest for new funds, laboratories and universities turned to the private sector. In the case of the AI Lab, winning over private investors was an easy sell. Home to some of the most ambitious computer-science projects of the post-war era, the lab became a quick incubator of technology. Indeed, by 1980, most of the lab's staff, including many hackers, were dividing its time between Institute and commercial projects. What at first seemed like a win-win deal-hackers got to work on the best projects, giving the lab first look at many of the newest computer technologies coming down the pike-soon revealed itself as a Faustian bargain. The more time hackers devoted to cutting-edge commercial projects, the less time they had to devote to general maintenance on the lab's baroque software infrastructure. Soon, companies began hiring away hackers outright in an attempt to monopolize their time and attention. With fewer hackers to mind the shop, programs and machines took longer to fix. Even worse, Stallman says, the lab began to undergo a "demographic change." The hackers who had once formed a vocal minority within the AI Lab were losing membership while "the professors and the students who didn't really love the [PDP-10] were just as numerous as before."See Richard Stallman (1986). The breaking point came in 1982. That was the year the lab's administration decided to upgrade its main computer, the PDP-10. Digital, the corporation that manufactured the PDP-10, had discontinued the line. Although the company still offered a high-powered mainframe, dubbed the KL-10, the new machine required a drastic rewrite or "port" of ITS if hackers wanted to continue running the same operating system. Fearful that the lab had lost its critical mass of in-house programming talent, AI Lab faculty members pressed for Twenex, a commercial operating system developed by Digital. Outnumbered, the hackers had no choice but to comply. "Without hackers to maintain the system, [faculty members] said, `We're going to have a disaster; we must have commercial software,'" Stallman would recall a few years later. "They said, `We can expect the company to maintain it.' It proved that they were utterly wrong, but that's what they did." At first, hackers viewed the Twenex system as yet another authoritarian symbol begging to be subverted. The system's name itself was a protest. Officially dubbed TOPS-20 by DEC, it was a successor to TOPS-10, a commercial operating system DEC marketed for the PDP-10. Bolt Beranek Newman had deveoped an improved version, dubbed Tenex, which TOPS-20 drew upon.Multiple sources: see Richard Stallman interview, Gerald Sussman email, and Jargon File 3.0.0. http://www.clueless.com/jargon3.0.0/TWENEX.html Stallman, the hacker who coined the Twenex term, says he came up with the name as a way to avoid using the TOPS-20 name. "The system was far from tops, so there was no way I was going to call it that," Stallman recalls. "So I decided to insert a `w' in the Tenex name and call it Twenex." The machine that ran the Twenex/TOPS-20 system had its own derisive nickname: Oz. According to one hacker legend, the machine got its nickname because it required a smaller PDP-11 machine to power its terminal. One hacker, upon viewing the KL-10-PDP-11 setup for the first time, likened it to the wizard's bombastic onscreen introduction in the Wizard of Oz. "I am the great and powerful Oz," the hacker intoned. "Pay no attention to the PDP-11 behind that console."See http://www.as.cmu.edu/~geek/humor/See_Figure_1.txt If hackers laughed when they first encountered the KL-10, their laughter quickly died when they encountered Twenex. Not only did Twenex boast built-in security, but the system's software engineers had designed the tools and applications with the security system in mind. What once had been a cat-and-mouse game over passwords in the case of the Laboratory for Computer Science's security system, now became an out-and-out battle over system management. System administrators argued that without security, the Oz system was more prone to accidental crashes. Hackers argued that crashes could be better prevented by overhauling the source code. Unfortunately, the number of hackers with the time and inclination to perform this sort of overhaul had dwindled to the point that the system-administrator argument prevailed. Cadging passwords and deliberately crashing the system in order to glean evidence from the resulting wreckage, Stallman successfully foiled the system administrators' attempt to assert control. After one foiled "coup d'etat," Stallman issued an alert to the entire AI staff. "There has been another attempt to seize power," Stallman wrote. "So far, the aristocratic forces have been defeated." To protect his identity, Stallman signed the message "Radio Free OZ." The disguise was a thin one at best. By 1982, Stallman's aversion to passwords and secrecy had become so well known that users outside the AI Laboratory were using his account as a stepping stone to the ARPAnet, the research-funded computer network that would serve as a foundation for today's Internet. One such "tourist" during the early 1980s was Don Hopkins, a California programmer who learned through the hacking grapevine that all an outsider needed to do to gain access to MIT's vaunted ITS system was to log in under the initials RMS and enter the same three-letter monogram when the system requested a password. "I'm eternally grateful that MIT let me and many other people use their computers for free," says Hopkins. "It meant a lot to many people." This so-called "tourist" policy, which had been openly tolerated by MIT management during the ITS years,See "MIT AI Lab Tourist Policy." http://catalog.com/hopkins/text/tourist-policy.html fell by the wayside when Oz became the lab's primary link to the ARPAnet. At first, Stallman continued his policy of repeating his login ID as a password so outside users could follow in his footsteps. Over time, however, the Oz's fragility prompted administrators to bar outsiders who, through sheer accident or malicious intent, might bring down the system. When those same administrators eventually demanded that Stallman stop publishing his password, Stallman, citing personal ethics, refused to do so and ceased using the Oz system altogether.3 "[When] passwords first appeared at the MIT AI Lab I [decided] to follow my belief that there should be no passwords," Stallman would later say. "Because I don't believe that it's really desirable to have security on a computer, I shouldn't be willing to help uphold the security regime." Stallman's refusal to bow before the great and powerful Oz symbolized the growing tension between hackers and AI Lab management during the early 1980s. This tension paled in comparison to the conflict that raged within the hacker community itself. By the time the KL-10 arrived, the hacker community had already divided into two camps. The first centered around a software company called Symbolics, Inc. The second centered around Symbolics chief rival, Lisp Machines, Inc. (LMI). Both companies were in a race to market the Lisp Machine, a device built to take full advantage of the Lisp programming language. Created by artificial-intelligence research pioneer John McCarthy, a MIT artificial-intelligence researcher during the late 1950s, Lisp is an elegant language well-suited for programs charged with heavy-duty sorting and processing. The language's name is a shortened version of LISt Processing. Following McCarthy's departure to the Stanford Artificial Intelligence Laboratory, MIT hackers refined the language into a local dialect dubbed MACLISP. The "MAC" stood for Project MAC, the DARPA-funded research project that gave birth to the AI Lab and the Laboratory for Computer Science. Led by AI Lab arch-hacker Richard Greenblatt, AI Lab programmers during the 1970s built up an entire Lisp-based operating system, dubbed the Lisp Machine operating system. By 1980, the Lisp Machine project had generated two commercial spin-offs. Symbolics was headed by Russell Noftsker, a former AI Lab administrator, and Lisp Machines, Inc., was headed by Greenblatt. The Lisp Machine software was hacker-built, meaning it was owned by MIT but available for anyone to copy as per hacker custom. Such a system limited the marketing advantage of any company hoping to license the software from MIT and market it as unique. To secure an advantage, and to bolster the aspects of the operating system that customers might consider attractive, the companies recruited various AI Lab hackers and set them working on various components of the Lisp Machine operating system outside the auspices of the AI Lab. The most aggressive in this strategy was Symbolics. By the end of 1980, the company had hired 14 AI Lab staffers as part-time consultants to develop its version of the Lisp Machine. Apart from Stallman, the rest signed on to help LMI.See H. P. Newquist, The Brain Makers: Genius, Ego, and Greed in the Quest for Machines that Think (Sams Publishing, 1994): 172. At first, Stallman accepted both companies' attempt to commercialize the Lisp machine, even though it meant more work for him. Both licensed the Lisp Machine OS source code from MIT, and it was Stallman's job to update the lab's own Lisp Machine to keep pace with the latest innovations. Although Symbolics' license with MIT gave Stallman the right to review, but not copy, Symbolics' source code, Stallman says a "gentleman's agreement" between Symbolics management and the AI Lab made it possible to borrow attractive snippets in traditional hacker fashion. On March 16, 1982, a date Stallman remembers well because it was his birthday, Symbolics executives decided to end this gentlemen's agreement. The move was largely strategic. LMI, the primary competition in the Lisp Machine marketplace, was essentially using a copy of the AI Lab Lisp Machine. Rather than subsidize the development of a market rival, Symbolics executives elected to enforce the letter of the license. If the AI Lab wanted its operating system to stay current with the Symbolics operating system, the lab would have to switch over to a Symbolics machine and sever its connection to LMI. As the person responsible for keeping up the lab's Lisp Machine, Stallman was incensed. Viewing this announcement as an "ultimatum," he retaliated by disconnecting Symbolics' microwave communications link to the laboratory. He then vowed never to work on a Symbolics machine and pledged his immediate allegiance to LMI. "The way I saw it, the AI Lab was a neutral country, like Belgium in World War I," Stallman says. "If Germany invades Belgium, Belgium declares war on Germany and sides with Britain and France." The circumstances of the so-called "Symbolics War" of 1982-1983 depend heavily on the source doing the telling. When Symbolics executives noticed that their latest features were still appearing in the AI Lab Lisp Machine and, by extension, the LMI Lisp machine, they installed a "spy" program on Stallman's computer terminal. Stallman says he was rewriting the features from scratch, taking advantage of the license's review clause but also taking pains to make the source code as different as possible. Symbolics executives argued otherwise and took their case to MIT administration. According to 1994 book, The Brain Makers: Genius, Ego, and Greed, and the Quest for Machines That Think, written by Harvey Newquist, the administration responded with a warning to Stallman to "stay away" from the Lisp Machine project.Ibid.: 196. According to Stallman, MIT administrators backed Stallman up. "I was never threatened," he says. "I did make changes in my practices, though. Just to be ultra safe, I no longer read their source code. I used only the documentation and wrote the code from that." Whatever the outcome, the bickering solidified Stallman's resolve. With no source code to review, Stallman filled in the software gaps according to his own tastes and enlisted members of the AI Lab to provide a continuous stream of bug reports. He also made sure LMI programmers had direct access to the changes. "I was going to punish Symbolics if it was the last thing I did," Stallman says. Such statements are revealing. Not only do they shed light on Stallman's nonpacifist nature, they also reflect the intense level of emotion triggered by the conflict. According to another Newquist-related story, Stallman became so irate at one point that he issued an email threatening to "wrap myself in dynamite and walk into Symbolics' offices."Ibid. Newquist, who says this anecdote was confirmed by several Symbolics executives, writes, "The message caused a brief flurry of excitement and speculation on the part of Symbolics' employees, but ultimately, no one took Stallman's outburst that seriously." Although Stallman would deny any memory of the email and still describes its existence as a "vicious rumor," he acknowledges that such thoughts did enter his head. "I definitely did have fantasies of killing myself and destroying their building in the process," Stallman says. "I thought my life was over." The level of despair owed much to what Stallman viewed as the "destruction" of his "home"-i.e., the demise of the AI Lab's close-knit hacker subculture. In a later email interview with Levy, Stallman would liken himself to the historical figure Ishi, the last surviving member of the Yahi, a Pacific Northwest tribe wiped out during the Indian wars of the 1860s and 1870s. The analogy casts Stallman's survival in epic, almost mythical, terms. In reality, however, it glosses over the tension between Stallman and his fellow AI Lab hackers prior to the Symbolics-LMI schism. Instead of seeing Symbolics as an exterminating force, many of Stallman's colleagues saw it as a belated bid for relevance. In commercializing the Lisp Machine, the company pushed hacker principles of engineer-driven software design out of the ivory-tower confines of the AI Lab and into the corporate marketplace where manager-driven design principles held sway. Rather than viewing Stallman as a holdout, many hackers saw him as a troubling anachronism. Stallman does not dispute this alternate view of historical events. In fact, he says it was yet another reason for the hostility triggered by the Symbolics "ultimatum." Even before Symbolics hired away most of the AI Lab's hacker staff, Stallman says many of the hackers who later joined Symbolics were shunning him. "I was no longer getting invited to go to Chinatown," Stallman recalls. "The custom started by Greenblatt was that if you went out to dinner, you went around or sent a message asking anybody at the lab if they also wanted to go. Sometime around 1980-1981, I stopped getting asked. They were not only not inviting me, but one person later confessed that he had been pressured to lie to me to keep their going away to dinner without me a secret." Although Stallman felt anger toward the hackers who orchestrated this petty form of ostracism, the Symbolics controversy dredged up a new kind of anger, the anger of a person about to lose his home. When Symbolics stopped sending over its source-code changes, Stallman responded by holing up in his MIT offices and rewriting each new software feature and tool from scratch. Frustrating as it may have been, it guaranteed that future Lisp Machine users had unfettered access to the same features as Symbolics users. It also guaranteed Stallman's legendary status within the hacker community. Already renowned for his work with Emacs, Stallman's ability to match the output of an entire team of Symbolics programmers-a team that included more than a few legendary hackers itself-still stands has one of the major human accomplishments of the Information Age, or of any age for that matter. Dubbing it a "master hack" and Stallman himself a "virtual John Henry of computer code," author Steven Levy notes that many of his Symbolics-employed rivals had no choice but to pay their idealistic former comrade grudging respect. Levy quotes Bill Gosper, a hacker who eventually went to work for Symbolics in the company's Palo Alto office, expressing amazement over Stallman's output during this period: I can see something Stallman wrote, and I might decide it was bad (probably not, but somebody could convince me it was bad), and I would still say, "But wait a minute-Stallman doesn't have anybody to argue with all night over there. He's working alone! It's incredible anyone could do this alone!"See Steven Levy, Hackers (Penguin USA [paperback], 1984): 426. For Stallman, the months spent playing catch up with Symbolics evoke a mixture of pride and profound sadness. As a dyed-in-the-wool liberal whose father had served in World War II, Stallman is no pacifist. In many ways, the Symbolics war offered the rite of passage toward which Stallman had been careening ever since joining the AI Lab staff a decade before. At the same time, however, it coincided with the traumatic destruction of the AI Lab hacker culture that had nurtured Stallman since his teenage years. One day, while taking a break from writing code, Stallman experienced a traumatic moment passing through the lab's equipment room. There, Stallman encountered the hulking, unused frame of the PDP-10 machine. Startled by the dormant lights, lights that once actively blinked out a silent code indicating the status of the internal program, Stallman says the emotional impact was not unlike coming across a beloved family member's well-preserved corpse. "I started crying right there in the equipment room," he says. "Seeing the machine there, dead, with nobody left to fix it, it all drove home how completely my community had been destroyed." Stallman would have little opportunity to mourn. The Lisp Machine, despite all the furor it invoked and all the labor that had gone into making it, was merely a sideshow to the large battles in the technology marketplace. The relentless pace of computer miniaturization was bringing in newer, more powerful microprocessors that would soon incorporate the machine's hardware and software capabilities like a modern metropolis swallowing up an ancient desert village. Riding atop this microprocessor wave were hundreds-thousands-of commercial software programs, each protected by a patchwork of user licenses and nondisclosure agreements that made it impossible for hackers to review or share source code. The licenses were crude and ill-fitting, but by 1983 they had become strong enough to satisfy the courts and scare away would-be interlopers. Software, once a form of garnish most hardware companies gave away to make their expensive computer systems more flavorful, was quickly becoming the main dish. In their increasing hunger for new games and features, users were putting aside the traditional demand to review the recipe after every meal. Nowhere was this state of affairs more evident than in the realm of personal computer systems. Companies such as Apple Computer and Commodore were minting fresh millionaires selling machines with built-in operating systems. Unaware of the hacker culture and its distaste for binary-only software, many of these users saw little need to protest when these companies failed to attach the accompanying source-code files. A few anarchic adherents of the hacker ethic helped propel that ethic into this new marketplace, but for the most part, the marketplace rewarded the programmers speedy enough to write new programs and savvy enough to copyright them as legally protected works. One of the most notorious of these programmers was Bill Gates, a Harvard dropout two years Stallman's junior. Although Stallman didn't know it at the time, seven years before sending out his message to the n et.unix-wizards newsgroup, Gates, a budding entrepreneur and general partner with the Albuquerque-based software firm Micro-Soft, later spelled as Microsoft, had sent out his own open letter to the software-developer community. Written in response to the PC users copying Micro-Soft's software programs, Gates' " Open Letter to Hobbyists" had excoriated the notion of communal software development. "Who can afford to do professional work for nothing?" asked Gates. "What hobbyist can put three man-years into programming, finding all bugs, documenting his product, and distributing it for free?"See Bill Gates, "An Open Letter to Hobbyists" (February 3, 1976). To view an online copy of this letter, go to http://www.blinkenlights.com/classiccmp/gateswhine.html. Although few hackers at the AI Lab saw the missive, Gates' 1976 letter nevertheless represented the changing attitude toward software both among commercial software companies and commercial software developers. Why treat software as a zero-cost commodity when the market said otherwise? As the 1970s gave way to the 1980s, selling software became more than a way to recoup costs; it became a political statement. At a time when the Reagan Administration was rushing to dismantle many of the federal regulations and spending programs that had been built up during the half century following the Great Depression, more than a few software programmers saw the hacker ethic as anticompetitive and, by extension, un-American. At best, it was a throwback to the anticorporate attitudes of the late 1960s and early 1970s. Like a Wall Street banker discovering an old tie-dyed shirt hiding between French-cuffed shirts and double-breasted suits, many computer programmers treated the hacker ethic as an embarrassing reminder of an idealistic age. For a man who had spent the entire 1960s as an embarrassing throwback to the 1950s, Stallman didn't mind living out of step with his peers. As a programmer used to working with the best machines and the best software, however, Stallman faced what he could only describe as a "stark moral choice": either get over his ethical objection for " proprietary" software-the term Stallman and his fellow hackers used to describe any program that carried private copyright or end-user license that restricted copying and modification-or dedicate his life to building an alternate, nonproprietary system of software programs. Given his recent months-long ordeal with Symbolics, Stallman felt more comfortable with the latter option. "I suppose I could have stopped working on computers altogether," Stallman says. "I had no special skills, but I'm sure I could have become a waiter. Not at a fancy restaurant, probably, but I could've been a waiter somewhere." Being a waiter-i.e., dropping out of programming altogether-would have meant completely giving up an activity, computer programming, that had given him so much pleasure. Looking back on his life since moving to Cambridge, Stallman finds it easy to identify lengthy periods when software programming provided the only pleasure. Rather than drop out, Stallman decided to stick it out. An atheist, Stallman rejects notions such as fate, dharma, or a divine calling in life. Nevertheless, he does feel that the decision to shun proprietary software and build an operating system to help others do the same was a natural one. After all, it was Stallman's own personal combination of stubbornness, foresight, and coding virtuosity that led him to consider a fork in the road most others didn't know existed. In describing the decision in a chapter for the 1999 book, Open Sources, Stallman cites the spirit encapsulated in the words of the Jewish sage Hillel: If I am not for myself, who will be for me?If I am only for myself, what am I?If not now, when?See Richard Stallman, Open Sources (O'Reilly & Associates, Inc., 1999): 56. Stallman adds his own footnote to this statement, writing, "As an atheist, I don't follow any religious leaders, but I sometimes find I admire something one of them has said." Speaking to audiences, Stallman avoids the religious route and expresses the decision in pragmatic terms. "I asked myself: what could I, an operating-system developer, do to improve the situation? It wasn't until I examined the question for a while that I realized an operating-system developer was exactly what was needed to solve the problem." Once he reached that decision, Stallman says, everything else "fell into place." He would abstain from using software programs that forced him to compromise his ethical beliefs, while at the same time devoting his life to the creation of software that would make it easier for others to follow the same path. Pledging to build a free software operating system "or die trying-of old age, of course," Stallman quips, he resigned from the MIT staff in January, 1984, to build GNU. The resignation distanced Stallman's work from the legal auspices of MIT. Still, Stallman had enough friends and allies within the AI Lab to retain rent-free access to his MIT office. He also had the ability to secure outside consulting gigs to underwrite the early stages of the GNU Project. In resigning from MIT, however, Stallman negated any debate about conflict of interest or Institute ownership of the software. The man whose early adulthood fear of social isolation had driven him deeper and deeper into the AI Lab's embrace was now building a legal firewall between himself and that environment. For the first few months, Stallman operated in isolation from the Unix community as well. Although his announcement to the net.unix-wizards group had attracted sympathetic responses, few volunteers signed on to join the crusade in its early stages. "The community reaction was pretty much uniform," recalls Rich Morin, leader of a Unix user group at the time. "People said, `Oh, that's a great idea. Show us your code. Show us it can be done.'" In true hacker fashion, Stallman began looking for existing programs and tools that could be converted into GNU programs and tools. One of the first was a compiler named VUCK, which converted programs written in the popular C programming language into machine-readable code. Translated from the Dutch, the program's acronym stood for the Free University Compiler Kit. Optimistic, Stallman asked the program's author if the program was free. When the author informed him that the words "Free University" were a reference to the Vrije Universiteit in Amsterdam, Stallman was chagrined. "He responded derisively, stating that the university was free but the compiler was not," recalls Stallman. "I therefore decided that my first program for the GNU Project would be a multi-language, multi-platform compiler." Eventually Stallman found a Pastel language compiler written by programmers at Lawrence Livermore National Lab. According to Stallman's knowledge at the time, the compiler was free to copy and modify. Unfortunately, the program possessed a sizable design flaw: it saved each program into core memory, tying up precious space for other software activities. On mainframe systems this design flaw had been forgivable. On Unix systems it was a crippling barrier, since the machines that ran Unix were too small to handle the large files generated. Stallman made substantial progress at first, building a C-compatible frontend to the compiler. By summer, however, he had come to the conclusion that he would have to build a totally new compiler from scratch. In September of 1984, Stallman shelved compiler development for the near term and began searching for lower-lying fruit. He began development of a GNU version of Emacs, the program he himself had been supervising for a decade. The decision was strategic. Within the Unix community, the two native editor programs were vi, written by Sun Microsystems cofounder Bill Joy, and ed, written by Bell Labs scientist (and Unix cocreator) Ken Thompson. Both were useful and popular, but neither offered the endlessly expandable nature of Emacs. In rewriting Emacs for the Unix audience, Stallman stood a better chance of showing off his skills. It also stood to reason that Emacs users might be more attuned to the Stallman mentality. Looking back, Stallman says he didn't view the decision in strategic terms. "I wanted an Emacs, and I had a good opportunity to develop one." Once again, the notion of reinventing the wheel grated on Stallman's efficient hacker sensibilities. In writing a Unix version of Emacs, Stallman was soon following the footsteps of Carnegie Mellon graduate student James Gosling, author of a C-based version dubbed Gosling Emacs or GOSMACS. Gosling's version of Emacs included an interpreter that exploited a simplified offshoot of the Lisp language called MOCKLISP. Determined to build GNU Emacs on a similar Lisp foundation, Stallman borrowed copiously from Gosling's innovations. Although Gosling had put GOSMACS under copyright and had sold the rights to UniPress, a privately held software company, Stallman cited the assurances of a fellow developer who had participated in the early MOCKLISP interpreter. According to the developer, Gosling, while a Ph.D. student at Carnegie Mellon, had assured early collaborators that their work would remain accessible. When UniPress caught wind of Stallman's project, however, the company threatened to enforce the copyright. Once again, Stallman faced the prospect of building from the ground up. In the course of reverse-engineering Gosling's interpreter, Stallman would create a fully functional Lisp interpreter, rendering the need for Gosling's original interpreter moot. Nevertheless, the notion of developers selling off software rights-indeed, the very notion of developers having software rights to sell in the first place-rankled Stallman. In a 1986 speech at the Swedish Royal Technical Institute, Stallman cited the UniPress incident as yet another example of the dangers associated with proprietary software. "Sometimes I think that perhaps one of the best things I could do with my life is find a gigantic pile of proprietary software that was a trade secret, and start handing out copies on a street corner so it wouldn't be a trade secret any more," said Stallman. "Perhaps that would be a much more efficient way for me to give people new free software than actually writing it myself; but everyone is too cowardly to even take it." Despite the stress it generated, the dispute over Gosling's innovations would assist both Stallman and the free software movement in the long term. It would force Stallman to address the weaknesses of the Emacs Commune and the informal trust system that had allowed problematic offshoots to emerge. It would also force Stallman to sharpen the free software movement's political objectives. Following the release of GNU Emacs in 1985, Stallman issued " The GNU Manifesto," an expansion of the original announcement posted in September, 1983. Stallman included within the document a lengthy section devoted to the many arguments used by commercial and academic programmers to justify the proliferation of proprietary software programs. One argument, "Don't programmers deserve a reward for their creativity," earned a response encapsulating Stallman's anger over the recent Gosling Emacs episode: "If anything deserves a reward, it is social contribution," Stallman wrote. "Creativity can be a social contribution, but only in so far [sic] as society is free to use the results. If programmers deserve to be rewarded for creating innovative programs, by the same token they deserve to be punished if they restrict the use of these programs."See Richard Stallman, "The GNU Manifesto" (1985). http://www.gnu.org/manifesto.html With the release of GNU Emacs, the GNU Project finally had code to show. It also had the burdens of any software-based enterprise. As more and more Unix developers began playing with the software, money, gifts, and requests for tapes began to pour in. To address the business side of the GNU Project, Stallman drafted a few of his colleagues and formed the Free Software Foundation (FSF), a nonprofit organization dedicated to speeding the GNU Project towards its goal. With Stallman as president and various hacker allies as board members, the FSF helped provide a corporate face for the GNU Project. Robert Chassell, a programmer then working at Lisp Machines, Inc., became one of five charter board members at the Free Software Foundation following a dinner conversation with Stallman. Chassell also served as the organization's treasurer, a role that started small but quickly grew. "I think in '85 our total expenses and revenue were something in the order of $23,000, give or take," Chassell recalls. "Richard had his office, and we borrowed space. I put all the stuff, especially the tapes, under my desk. It wasn't until sometime later LMI loaned us some space where we could store tapes and things of that sort." In addition to providing a face, the Free Software Foundation provided a center of gravity for other disenchanted programmers. The Unix market that had seemed so collegial even at the time of Stallman's initial GNU announcement was becoming increasingly competitive. In an attempt to tighten their hold on customers, companies were starting to close off access to Unix source code, a trend that only speeded the number of inquiries into ongoing GNU software projects. The Unix wizards who once regarded Stallman as a noisy kook were now beginning to see him as a software Cassandra. "A lot of people don't realize, until they've had it happen to them, how frustrating it can be to spend a few years working on a software program only to have it taken away," says Chassell, summarizing the feelings and opinions of the correspondents writing in to the FSF during the early years. "After that happens a couple of times, you start to say to yourself, `Hey, wait a minute.'" For Chassell, the decision to participate in the Free Software Foundation came down to his own personal feelings of loss. Prior to LMI, Chassell had been working for hire, writing an introductory book on Unix for Cadmus, Inc., a Cambridge-area software company. When Cadmus folded, taking the rights to the book down with it, Chassell says he attempted to buy the rights back with no success. "As far as I know, that book is still sitting on shelf somewhere, unusable, uncopyable, just taken out of the system," Chassell says. "It was quite a good introduction if I may say so myself. It would have taken maybe three or four months to convert [the book] into a perfectly usable introduction to GNU/Linux today. The whole experience, aside from what I have in my memory, was lost." Forced to watch his work sink into the mire while his erstwhile employer struggled through bankruptcy, Chassell says he felt a hint of the anger that drove Stallman to fits of apoplexy. "The main clarity, for me, was the sense that if you want to have a decent life, you don't want to have bits of it closed off," Chassell says. "This whole idea of having the freedom to go in and to fix something and modify it, whatever it may be, it really makes a difference. It makes one think happily that after you've lived a few years that what you've done is worthwhile. Because otherwise it just gets taken away and thrown out or abandoned or, at the very least, you no longer have any relation to it. It's like losing a bit of your life." St. Ignucius The Maui High Performance Computing Center is located in a single-story building in the dusty red hills just above the town of Kihei. Framed by million-dollar views and the multimillion dollar real estate of the Silversword Golf Course, the center seems like the ultimate scientific boondoggle. Far from the boxy, sterile confines of Tech Square or even the sprawling research metropolises of Argonne, Illinois and Los Alamos, New Mexico, the MHPCC seems like the kind of place where scientists spend more time on their tans than their post-doctoral research projects. The image is only half true. Although researchers at the MHPCC do take advantage of the local recreational opportunities, they also take their work seriously. According to Top500.org, a web site that tracks the most powerful supercomputers in the world, the IBM SP Power3 supercomputer housed within the MHPCC clocks in at 837 billion floating-point operations per second, making it one of 25 most powerful computers in the world. Co-owned and operated by the University of Hawaii and the U.S. Air Force, the machine divides its computer cycles between the number crunching tasks associated with military logistics and high-temperature physics research. Simply put, the MHPCC is a unique place, a place where the brainy culture of science and engineering and the laid-back culture of the Hawaiian islands coexist in peaceful equilibrium. A slogan on the lab's 2000 web site sums it up: "Computing in paradise." It's not exactly the kind of place you'd expect to find Richard Stallman, a man who, when taking in the beautiful view of the nearby Maui Channel through the picture windows of a staffer's office, mutters a terse critique: "Too much sun." Still, as an emissary from one computing paradise to another, Stallman has a message to deliver, even if it means subjecting his pale hacker skin to the hazards of tropical exposure. The conference room is already full by the time I arrive to catch Stallman's speech. The gender breakdown is a little better than at the New York speech, 85% male, 15% female, but not by much. About half of the audience members wear khaki pants and logo-encrusted golf shirts. The other half seems to have gone native. Dressed in the gaudy flower-print shirts so popular in this corner of the world, their faces are a deep shade of ochre. The only residual indication of geek status are the gadgets: Nokia cell phones, Palm Pilots, and Sony VAIO laptops. Needless to say, Stallman, who stands in front of the room dressed in plain blue T-shirt, brown polyester slacks, and white socks, sticks out like a sore thumb. The fluorescent lights of the conference room help bring out the unhealthy color of his sun-starved skin. His beard and hair are enough to trigger beads of sweat on even the coolest Hawaiian neck. Short of having the words "mainlander" tattooed on his forehead, Stallman couldn't look more alien if he tried. As Stallman putters around the front of the room, a few audience members wearing T-shirts with the logo of the Maui FreeBSD Users Group (MFUG) race to set up camera and audio equipment. FreeBSD, a free software offshoot of the Berkeley Software Distribution, the venerable 1970s academic version of Unix, is technically a competitor to the GNU/Linux operating system. Still, in the hacking world, Stallman speeches are documented with a fervor reminiscent of the Grateful Dead and its legendary army of amateur archivists. As the local free software heads, it's up to the MFUG members to make sure fellow programmers in Hamburg, Mumbai, and Novosibirsk don't miss out on the latest pearls of RMS wisdom. The analogy to the Grateful Dead is apt. Often, when describing the business opportunities inherent within the free software model, Stallman has held up the Grateful Dead as an example. In refusing to restrict fans' ability to record live concerts, the Grateful Dead became more than a rock group. They became the center of a tribal community dedicated to Grateful Dead music. Over time, that tribal community became so large and so devoted that the band shunned record contracts and supported itself solely through musical tours and live appearances. In 1994, the band's last year as a touring act, the Grateful Dead drew $52 million in gate receipts alone.See "Grateful Dead Time Capsule: 1985-1995 North American Tour Grosses." http://www.accessplace.com/gdtc/1197.htm While few software companies have been able to match that success, the tribal aspect of the free software community is one reason many in the latter half of the 1990s started to accept the notion that publishing software source code might be a good thing. Hoping to build their own loyal followings, companies such as IBM, Sun Microsystems, and Hewlett Packard have come to accept the letter, if not the spirit, of the Stallman free software message. Describing the GPL as the information-technology industry's "Magna Carta," ZDNet software columnist Evan Leibovitch sees the growing affection for all things GNU as more than just a trend. "This societal shift is letting users take back control of their futures," Leibovitch writes. "Just as the Magna Carta gave rights to British subjects, the GPL enforces consumer rights and freedoms on behalf of the users of computer software."See Evan Leibovitch, "Who's Afraid of Big Bad Wolves," ZDNet Tech Update (December 15, 2000). http://techupdate.zdnet.com/techupdate/stories/main/0Y/A The tribal aspect of the free software community also helps explain why 40-odd programmers, who might otherwise be working on physics projects or surfing the Web for windsurfing buoy reports, have packed into a conference room to hear Stallman speak. Unlike the New York speech, Stallman gets no introduction. He also offers no self-introduction. When the FreeBSD people finally get their equipment up and running, Stallman simply steps forward, starts speaking, and steamrolls over every other voice in the room. "Most of the time when people consider the question of what rules society should have for using software, the people considering it are from software companies, and they consider the question from a self-serving perspective," says Stallman, opening his speech. "What rules can we impose on everybody else so they have to pay us lots of money? I had the good fortune in the 1970s to be part of a community of programmers who shared software. And because of this I always like to look at the same issue from a different direction to ask: what kind of rules make possible a good society that is good for the people who are in it? And therefore I reach completely different answers." Once again, Stallman quickly segues into the parable of the Xerox laser printer, taking a moment to deliver the same dramatic finger-pointing gestures to the crowd. He also devotes a minute or two to the GNU/Linux name. "Some people say to me, `Why make such a fuss about getting credit for this system? After all, the important thing is the job is done, not whether you get recognition for it.' Well, this would be wise advice if it were true. But the job wasn't to build an operating system; the job is to spread freedom to the users of computers. And to do that we have to make it possible to do everything with computers in freedom."For narrative purposes, I have hesitated to go in-depth when describing Stallman's full definition of software "freedom." The GNU Project web site lists four fundamental components: The freedom to run a program, for any purpose (freedom 0). The freedom to study how a program works, and adapt it to your needs (freedom 1). The freedom to redistribute copies of a program so you can help your neighbor (freedom 2). The freedom to improve the program, and release your improvements to the public, so that the whole community benefits (freedom 3). For more information, please visit "The Free Software Definition" at http://www.gnu.org/philosophy/free-sw.html. Adds Stallman, "There's a lot more work to do." For some in the audience, this is old material. For others, it's a little arcane. When a member of the golf-shirt contingent starts dozing off, Stallman stops the speech and asks somebody to wake the person up. "Somebody once said my voice was so soothing, he asked if I was some kind of healer," says Stallman, drawing a quick laugh from the crowd. "I guess that probably means I can help you drift gently into a blissful, relaxing sleep. And some of you might need that. I guess I shouldn't object if you do. If you need to sleep, by all means do." The speech ends with a brief discussion of software patents, a growing issue of concern both within the software industry and within the free software community. Like Napster, software patents reflect the awkward nature of applying laws and concepts written for the physical world to the frictionless universe of information technology. The difference between protecting a program under copyright and protecting a program under software patents is subtle but significant. In the case of copyright, a software creator can restrict duplication of the source code but not duplication of the idea or functionality that the source code addresses. In other words, if a developer chooses not to use a software program under the original developer's terms, that second developer is still free to reverse-engineer the program-i.e., duplicate the software program's functionality by rewriting the source code from scratch. Such duplication of ideas is common within the commercial software industry, where companies often isolate reverse-engineering teams to head off accusations of corporate espionage or developer hanky-panky. In the jargon of modern software development, companies refer to this technique as "clean room" engineering. Software patents work differently. According to the U.S. Patent Office, companies and individuals may secure patents for innovative algorithms provided they submit their claims to a public review. In theory, this allows the patent-holder to trade off disclosure of their invention for a limited monopoly of a minimum of 20 years after the patent filing. In practice, the disclosure is of limited value, since the operation of the program is often self-evident. Unlike copyright, a patent gives its holder the ability to head off the independent development of software programs with the same or similar functionality. In the software industry, where 20 years can cover the entire life cycle of a marketplace, patents take on a strategic weight. Where companies such as Microsoft and Apple once battled over copyright and the "look and feel" of various technologies, today's Internet companies use patents as a way to stake out individual applications and business models, the most notorious example being Amazon.com's 2000 attempt to patent the company's "one-click" online shopping process. For most companies, however, software patents have become a defensive tool, with cross-licensing deals balancing one set of corporate patents against another in a tense form of corporate detente. Still, in a few notable cases of computer encryption and graphic imaging algorithms, software vendors have successfully stifled rival technologies. For Stallman, the software-patent issue dramatizes the need for eternal hacker vigilance. It also underlines the importance of stressing the political benefits of free software programs over the competitive benefits. Pointing to software patents' ability to create sheltered regions in the marketplace, Stallman says competitive performance and price, two areas where free software operating systems such as GNU/Linux and FreeBSD already hold a distinct advantage over their proprietary counterparts, are red herrings compared to the large issues of user and developer freedom. "It's not because we don't have the talent to make better software," says Stallman. "It's because we don't have the right. Somebody has prohibited us from serving the public. So what's going to happen when users encounter these gaps in free software? Well, if they have been persuaded by the open source movement that these freedoms are good because they lead to more-powerful reliable software, they're likely to say, `You didn't deliver what you promised. This software's not more powerful. It's missing this feature. You lied to me.' But if they have come to agree with the free software movement, that the freedom is important in itself, then they will say, `How dare those people stop me from having this feature and my freedom too.' And with that kind of response, we may survive the hits that we're going to take as these patents explode." Such comments involve a hefty dose of spin, of course. Most open source advocates are equally, if not more, vociferous as Stallman when it comes to opposing software patents. Still, the underlying logic of Stallman's argument-that open source advocates emphasize the utilitarian advantages of free software over the political advantages-remains uncontested. Rather than stress the political significance of free software programs, open source advocates have chosen to stress the engineering integrity of the hacker development model. Citing the power of peer review, the open source argument paints programs such as GNU/Linux or FreeBSD as better built, better inspected and, by extension, more trushworthy to the average user. That's not to say the term "open source" doesn't have its political implications. For open source advocates, the term open source serves two purposes. First, it eliminates the confusion associated with the word "free," a word many businesses interpret as meaning "zero cost." Second, it allows companies to examine the free software phenomenon on a technological, rather than ethical, basis. Eric Raymond, cofounder of the Open Source Initiative and one of the leading hackers to endorse the term, effectively summed up the frustration of following Stallman down the political path in a 1999 essay, titled " Shut Up and Show Them the Code": RMS's rhetoric is very seductive to the kind of people we are. We hackers are thinkers and idealists who readily resonate with appeals to "principle" and "freedom" and "rights." Even when we disagree with bits of his program, we want RMS's rhetorical style to work; we think it ought to work; we tend to be puzzled and disbelieving when it fails on the 95% of people who aren't wired like we are.4 Included among that 95%, Raymond writes, are the bulk of business managers, investors, and nonhacker computer users who, through sheer weight of numbers, tend to decide the overall direction of the commercial software marketplace. Without a way to win these people over, Raymond argues, programmers are doomed to pursue their ideology on the periphery of society: When RMS insists that we talk about "computer users' rights," he's issuing a dangerously attractive invitation to us to repeat old failures. It's one we should reject-not because his principles are wrong, but because that kind of language, applied to software, simply does not persuade anybody but us. In fact, it confuses and repels most people outside our culture.4 Watching Stallman deliver his political message in person, it is hard to see anything confusing or repellent. Stallman's appearance may seem off-putting, but his message is logical. When an audience member asks if, in shunning proprietary software, free software proponents lose the ability to keep up with the latest technological advancements, Stallman answers the question in terms of his own personal beliefs. "I think that freedom is more important than mere technical advance," he says. "I would always choose a less advanced free program rather than a more advanced nonfree program, because I won't give up my freedom for something like that. My rule is, if I can't share it with you, I won't take it." Such answers, however, reinforce the quasi-religious nature of the Stallman message. Like a Jew keeping kosher or a Mormon refusing to drink alcohol, Stallman paints his decision to use free software in the place of proprietary in the color of tradition and personal belief. As software evangelists go, Stallman avoids forcing those beliefs down listeners' throats. Then again, a listener rarely leaves a Stallman speech not knowing where the true path to software righteousness lies. As if to drive home this message, Stallman punctuates his speech with an unusual ritual. Pulling a black robe out of a plastic grocery bag, Stallman puts it on. Out of a second bag, he pulls a reflective yellow computer disk and places it on his head. The crowd lets out a startled laugh. "I am St. Ignucius of the Church of Emacs," says Stallman, raising his right hand in mock-blessing. "I bless your computer, my child." <Graphic file:books/free_0801.png> Stallman dressed as St. Ignucius. Photo by Wouter van Oortmerssen. The laughter turns into full-blown applause after a few seconds. As audience members clap, the computer disk on Stallman's head catches the glare of an overhead light, eliciting a perfect halo effect. In the blink of an eye, Stallman goes from awkward haole to Russian religious icon. " Emacs was initially a text editor," says Stallman, explaining the getup. "Eventually it became a way of life for many and a religion for some. We call this religion the Church of Emacs." The skit is a lighthearted moment of self-pardoy, a humorous return-jab at the many people who might see Stallman's form of software asceticism as religious fanaticism in disguise. It is also the sound of the other shoe dropping-loudly. It's as if, in donning his robe and halo, Stallman is finally letting listeners of the hook, saying, "It's OK to laugh. I know I'm weird." Discussing the St. Ignucius persona afterward, Stallman says he first came up with it in 1996, long after the creation of Emacs but well before the emergence of the "open source" term and the struggle for hacker-community leadership that precipitated it. At the time, Stallman says, he wanted a way to "poke fun at himself," to remind listeners that, though stubborn, Stallman was not the fanatic some made him out to be. It was only later, Stallman adds, that others seized the persona as a convenient way to play up his reputation as software ideologue, as Eric Raymond did in an 1999 interview with the linux.com web site: When I say RMS calibrates what he does, I'm not belittling or accusing him of insincerity. I'm saying that like all good communicators he's got a theatrical streak. Sometimes it's conscious-have you ever seen him in his St. Ignucius drag, blessing software with a disk platter on his head? Mostly it's unconscious; he's just learned the degree of irritating stimulus that works, that holds attention without (usually) freaking people out.See "Guest Interview: Eric S. Raymond," Linux.com (May 18, 1999). http://www.linux.com/interviews/19990518/8/ Stallman takes issue with the Raymond analysis. "It's simply my way of making fun of myself," he says. "The fact that others see it as anything more than that is a reflection of their agenda, not mine." That said, Stallman does admit to being a ham. "Are you kidding?" he says at one point. "I love being the center of attention." To facilitate that process, Stallman says he once enrolled in Toastmasters, an organization that helps members bolster their public-speaking skills and one Stallman recommends highly to others. He possesses a stage presence that would be the envy of most theatrical performers and feels a link to vaudevillians of years past. A few days after the Maui High Performance Computing Center speech, I allude to the 1999 LinuxWorld performace and ask Stallman if he has a Groucho Marx complex-i.e., the unwillingness to belong to any club that would have him as a member. Stallman's response is immediate: "No, but I admire Groucho Marx in a lot of ways and certainly have been in some things I say inspired by him. But then I've also been inspired in some ways by Harpo." The Groucho Marx influence is certainly evident in Stallman's lifelong fondness for punning. Then again, punning and wordplay are common hacker traits. Perhaps the most Groucho-like aspect of Stallman's personality, however, is the deadpan manner in which the puns are delivered. Most come so stealthily-without even the hint of a raised eyebrow or upturned smile-you almost have to wonder if Stallman's laughing at his audience more than the audience is laughing at him. Watching members of the Maui High Performance Computer Center laugh at the St. Ignucius parody, such concerns evaporate. While not exactly a standup act, Stallman certainly possesses the chops to keep a roomful of engineers in stitches. "To be a saint in the Church of Emacs does not require celibacy, but it does require making a commitment to living a life of moral purity," he tells the Maui audience. "You must exorcise the evil proprietary operating system from all your computer and then install a wholly [holy] free operating system. And then you must install only free software on top of that. If you make this commitment and live by it, then you too will be a saint in the Church of Emacs, and you too may have a halo." The St. Ignucius skit ends with a brief inside joke. On most Unix systems and Unix-related offshoots, the primary competitor program to Emacs is vi, a text-editing program developed by former UC Berkeley student and current Sun Microsystems chief scientist, Bill Joy. Before doffing his "halo," Stallman pokes fun at the rival program. "People sometimes ask me if it is a sin in the Church of Emacs to use vi," he says. "Using a free version of vi is not a sin; it is a penance. So happy hacking." After a brief question-and-answer session, audience members gather around Stallman. A few ask for autographs. "I'll sign this," says Stallman, holding up one woman's print out of the GNU General Public License, "but only if you promise me to use the term GNU/Linux instead of Linux and tell all your friends to do likewise." The comment merely confirms a private observation. Unlike other stage performers and political figures, Stallman has no "off" mode. Aside from the St. Ignucius character, the ideologue you see onstage is the ideologue you meet backstage. Later that evening, during a dinner conversation in which a programmer mentions his affinity for "open source" programs, Stallman, between bites, upbraids his tablemate: "You mean free software. That's the proper way to refer to it." During the question-and-answer session, Stallman admits to playing the pedagogue at times. "There are many people who say, `Well, first let's invite people to join the community, and then let's teach them about freedom.' And that could be a reasonable strategy, but what we have is almost everybody's inviting people to join the community, and hardly anybody's teaching them about freedom once they come in." The result, Stallman says, is something akin to a third-world city. People move in, hoping to strike it rich or at the very least to take part in a vibrant, open culture, and yet those who hold the true power keep evolving new tricks and strategies-i.e., software patents-to keep the masses out. "You have millions of people moving in and building shantytowns, but nobody's working on step two: getting them out of those shantytowns. If you think talking about software freedom is a good strategy, please join in doing step two. There are plenty working on step one. We need more people working on step two." Working on "step two" means driving home the issue that freedom, not acceptance, is the root issue of the free software movement. Those who hope to reform the proprietary software industry from the inside are on a fool's errand. "Change from the inside is risky," Stallman stays. "Unless you're working at the level of a Gorbachev, you're going to be neutralized." Hands pop up. Stallman points to a member of the golf shirt-wearing contingent. "Without patents, how would you suggest dealing with commercial espionage?" "Well, those two questions have nothing to do with each other, really," says Stallman. "But I mean if someone wants to steal another company's piece of software." Stallman's recoils as if hit by a poisonous spray. "Wait a second," Stallman says. "Steal? I'm sorry, there's so much prejudice in that statement that the only thing I can say is that I reject that prejudice. Companies that develop nonfree software and other things keep lots and lots of trade secrets, and so that's not really likely to change. In the old days-even in the 1980s-for the most part programmers were not aware that there were even software patents and were paying no attention to them. What happened was that people published the interesting ideas, and if they were not in the free software movement, they kept secret the little details. And now they patent those broad ideas and keep secret the little details. So as far as what you're describing, patents really make no difference to it one way or another." "But if it doesn't affect their publication," a new audience member jumps in, his voice trailing off almost as soon as he starts speaking. "But it does," Stallman says. "Their publication is telling you that this is an idea that's off limits to the rest of the community for 20 years. And what the hell good is that? Besides, they've written it in such a hard way to read, both to obfuscate the idea and to make the patent as broad as possible, that it's basically useless looking at the published information to learn anything anyway. The only reason to look at patents is to see the bad news of what you can't do." The audience falls silent. The speech, which began at 3:15, is now nearing the 5:00 whistle, and most listeners are already squirming in their seats, antsy to get a jump start on the weekend. Sensing the fatigue, Stallman glances around the room and hastily shuts things down. "So it looks like we're done," he says, following the observation with an auctioneer's "going, going, gone" to flush out any last-minute questioners. When nobody throws their hand up, Stallman signs off with a traditional exit line. "Happy hacking," he says. Endnotes 1. See "Grateful Dead Time Capsule: 1985-1995 North American Tour Grosses." http://www.accessplace.com/gdtc/1197.htm 2. See Evan Leibovitch, "Who's Afraid of Big Bad Wolves," ZDNet Tech Update (December 15, 2000). http://techupdate.zdnet.com/techupdate/stories/main/0Y/A> 3. For narrative purposes, I have hesitated to go in-depth when describing Stallman's full definition of software "freedom." The GNU Project web site lists four fundamental components: The freedom to run a program, for any purpose (freedom 0). The freedom to study how a program works, and adapt it to your needs (freedom 1). The freedom to redistribute copies of a program so you can help your neighbor (freedom 2). The freedom to improve the program, and release your improvements to the public, so that the whole community benefits (freedom 3). For more information, please visit "The Free Software Definition" at http://www.gnu.org/philosophy/free-sw.html. 4. See Eric Raymond, "Shut Up and Show Them the Code," online essay, (June 28, 1999). 5. See "Guest Interview: Eric S. Raymond," Linux.com (May 18, 1999). http://www.linux.com/interviews/19990518/8/ The GNU General Public License By the spring of 1985, Richard Stallman had settled on the GNU Project's first milestone-a Lisp-based free software version of Emacs. To meet this goal, however, he faced two challenges. First, he had to rebuild Emacs in a way that made it platform independent. Second, he had to rebuild the Emacs Commune in a similar fashion. The dispute with UniPress had highlighted a flaw in the Emacs Commune social contract. Where users relied on Stallman's expert insight, the Commune's rules held. In areas where Stallman no longer held the position of alpha hacker-pre-1984 Unix systems, for example-individuals and companies were free to make their own rules. The tension between the freedom to modify and the freedom to exert authorial privilege had been building before GOSMACS. The Copyright Act of 1976 had overhauled U.S. copyright law, extending the legal protection of copyright to software programs. According to Section 102(b) of the Act, individuals and companies now possessed the ability to copyright the "expression" of a software program but not the "actual processes or methods embodied in the program."See Hal Abelson, Mike Fischer, and Joanne Costello, "Software and Copyright Law," updated version (1998). Translated, programmers and companies had the ability to treat software programs like a story or song. Other programmers could take inspiration from the work, but to make a direct copy or nonsatirical derivative, they first had to secure permission from the original creator. Although the new law guaranteed that even programs without copyright notices carried copyright protection, programmers quickly asserted their rights, attaching coypright notices to their software programs. At first, Stallman viewed these notices with alarm. Rare was the software program that didn't borrow source code from past programs, and yet, with a single stroke of the president's pen, Congress had given programmers and companies the power to assert individual authorship over communally built programs. It also injected a dose of formality into what had otherwise been an informal system. Even if hackers could demonstrate how a given program's source-code bloodlines stretched back years, if not decades, the resources and money that went into battling each copyright notice were beyond most hackers' means. Simply put, disputes that had once been settled hacker-to-hacker were now settled lawyer-to-lawyer. In such a system, companies, not hackers, held the automatic advantage. Proponents of software copyright had their counter-arguments: without copyright, works might otherwise slip into the public domain. Putting a copyright notice on a work also served as a statement of quality. Programmers or companies who attached their name to the copyright attached their reputations as well. Finally, it was a contract, as well as a statement of ownership. Using copyright as a flexible form of license, an author could give away certain rights in exchange for certain forms of behavior on the part of the user. For example, an author could give away the right to suppress unauthorized copies just so long as the end user agreed not to create a commercial offshoot. It was this last argument that eventually softened Stallman's resistance to software copyright notices. Looking back on the years leading up to the GNU Project, Stallman says he began to sense the beneficial nature of copyright sometime around the release of Emacs 15.0, the last significant pre-GNU Project upgrade of Emacs. "I had seen email messages with copyright notices plus simple `verbatim copying permitted' licenses," Stallman recalls. "Those definitely were [an] inspiration." For Emacs 15, Stallman drafted a copyright that gave users the right to make and distribute copies. It also gave users the right to make modified versions, but not the right to claim sole ownership of those modified versions, as in the case of GOSMACS. Although helpful in codifying the social contract of the Emacs Commune, the Emacs 15 license remained too "informal" for the purposes of the GNU Project, Stallman says. Soon after starting work on a GNU version of Emacs, Stallman began consulting with the other members of the Free Software Foundation on how to shore up the license's language. He also consulted with the attorneys who had helped him set up the Free Software Foundation. Mark Fischer, a Boston attorney specializing in intellectual-property law, recalls discussing the license with Stallman during this period. "Richard had very strong views about how it should work," Fischer says, "He had two principles. The first was to make the software absolutely as open as possible. The second was to encourage others to adopt the same licensing practices." Encouraging others to adopt the same licensing practices meant closing off the escape hatch that had allowed privately owned versions of Emacs to emerge. To close that escape hatch, Stallman and his free software colleagues came up with a solution: users would be free to modify GNU Emacs just so long as they published their modifications. In addition, the resulting "derivative" works would also have carry the same GNU Emacs License. The revolutionary nature of this final condition would take a while to sink in. At the time, Fischer says, he simply viewed the GNU Emacs License as a simple contract. It put a price tag on GNU Emacs' use. Instead of money, Stallman was charging users access to their own later modifications. That said, Fischer does remember the contract terms as unique. "I think asking other people to accept the price was, if not unique, highly unusual at that time," he says. The GNU Emacs License made its debut when Stallman finally released GNU Emacs in 1985. Following the release, Stallman welcomed input from the general hacker community on how to improve the license's language. One hacker to take up the offer was future software activist John Gilmore, then working as a consultant to Sun Microsystems. As part of his consulting work, Gilmore had ported Emacs over to SunOS, the company's in-house version of Unix. In the process of doing so, Gilmore had published the changes as per the demands of the GNU Emacs License. Instead of viewing the license as a liability, Gilmore saw it as clear and concise expression of the hacker ethos. "Up until then, most licenses were very informal," Gilmore recalls. As an example of this informality, Gilmore cites a copyright notice for trn, a Unix utility. Written by Larry Wall, future creator of the Perl programming language, patch made it simple for Unix programmers to insert source-code fixes-" patches" in hacker jargon-into any large program. Recognizing the utility of this feature, Wall put the following copyright notice in the program's accompanying README file: Copyright (c) 1985, Larry Wall You may copy the trn kit in whole or in part as long as you don't try to make money off it, or pretend that you wrote it.See Trn Kit README. http://www.za.debian.org/doc/trn/trn-readme Such statements, while reflective of the hacker ethic, also reflected the difficulty of translating the loose, informal nature of that ethic into the rigid, legal language of copyright. In writing the GNU Emacs License, Stallman had done more than close up the escape hatch that permitted proprietary offshoots. He had expressed the hacker ethic in a manner understandable to both lawyer and hacker alike. It wasn't long, Gilmore says, before other hackers began discussing ways to "port" the GNU Emacs License over to their own programs. Prompted by a conversation on Usenet, Gilmore sent an email to Stallman in November, 1986, suggesting modification: You should probably remove "EMACS" from the license and replace it with "SOFTWARE" or something. Soon, we hope, Emacs will not be the biggest part of the GNU system, and the license applies to all of it.See John Gilmore, quoted from email to author. Gilmore wasn't the only person suggesting a more general approach. By the end of 1986, Stallman himself was at work with GNU Project's next major milestone, a source-code debugger, and was looking for ways to revamp the Emacs license so that it might apply to both programs. Stallman's solution: remove all specific references to Emacs and convert the license into a generic copyright umbrella for GNU Project software. The GNU General Public License, GPL for short, was born. In fashioning the GPL, Stallman followed the software convention of using decimal numbers to indicate prototype versions and whole numbers to indicate mature versions. Stallman published Version 1.0 of the GPL in 1989 (a project Stallman was developing in 1985), almost a full year after the release of the GNU Debugger, Stallman's second major foray into the realm of Unix programming. The license contained a preamble spelling out its political intentions: The General Public License is designed to make sure that you have the freedom to give away or sell copies of free software, that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things. To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it.See Richard Stallman, et al., "GNU General Public License: Version 1," (February, 1989). http://www.gnu.org/copyleft/copying-1.0.html In fashioning the GPL, Stallman had been forced to make an additional adjustment to the informal tenets of the old Emacs Commune. Where he had once demanded that Commune members publish any and all changes, Stallman now demanded publication only in instances when programmers circulated their derivative versions in the same public manner as Stallman. In other words, programmers who simply modified Emacs for private use no longer needed to send the source-code changes back to Stallman. In what would become a rare compromise of free software doctrine, Stallman slashed the price tag for free software. Users could innovate without Stallman looking over their shoulders just so long as they didn't bar Stallman and the rest of the hacker community from future exchanges of the same program. Looking back, Stallman says the GPL compromise was fueled by his own dissatisfaction with the Big Brother aspect of the original Emacs Commune social contract. As much as he liked peering into other hackers' systems, the knowledge that some future source-code maintainer might use that power to ill effect forced him to temper the GPL. "It was wrong to require people to publish all changes," says Stallman. "It was wrong to require them to be sent to one privileged developer. That kind of centralization and privilege for one was not consistent with a society in which all had equal rights." As hacks go, the GPL stands as one of Stallman's best. It created a system of communal ownership within the normally proprietary confines of copyright law. More importantly, it demonstrated the intellectual similarity between legal code and software code. Implicit within the GPL's preamble was a profound message: instead of viewing copyright law with suspicion, hackers should view it as yet another system begging to be hacked. "The GPL developed much like any piece of free software with a large community discussing its structure, its respect or the opposite in their observation, needs for tweaking and even to compromise it mildly for greater acceptance," says Jerry Cohen, another attorney who helped Stallman with the creation of the license. "The process worked very well and GPL in its several versions has gone from widespread skeptical and at times hostile response to widespread acceptance." In a 1986 interview with Byte magazine, Stallman summed up the GPL in colorful terms. In addition to proclaiming hacker values, Stallman said, readers should also "see it as a form of intellectual jujitsu, using the legal system that software hoarders have set up against them."See David Betz and Jon Edwards, "Richard Stallman discusses his public-domain [sic] Unix-compatible software system with BYTE editors," BYTE (July, 1996). (Reprinted on the GNU Project web site: http://www.gnu.org/gnu/byte-interview.html.) This interview offers an interesting, not to mention candid, glimpse at Stallman's political attitudes during the earliest days of the GNU Project. It is also helpful in tracing the evolution of Stallman's rhetoric. Describing the purpose of the GPL, Stallman says, "I'm trying to change the way people approach knowledge and information in general. I think that to try to own knowledge, to try to control whether people are allowed to use it, or to try to stop other people from sharing it, is sabotage." Contrast this with a statement to the author in August 2000: "I urge you not to use the term `intellectual property' in your thinking. It will lead you to misunderstand things, because that term generalizes about copyrights, patents, and trademarks. And those things are so different in their effects that it is entirely foolish to try to talk about them at once. If you hear somebody saying something about intellectual property, without quotes, then he's not thinking very clearly and you shouldn't join." Years later, Stallman would describe the GPL's creation in less hostile terms. "I was thinking about issues that were in a sense ethical and in a sense political and in a sense legal," he says. "I had to try to do what could be sustained by the legal system that we're in. In spirit the job was that of legislating the basis for a new society, but since I wasn't a government, I couldn't actually change any laws. I had to try to do this by building on top of the existing legal system, which had not been designed for anything like this." About the time Stallman was pondering the ethical, political, and legal issues associated with free software, a California hacker named Don Hopkins mailed him a manual for the 68000 microprocessor. Hopkins, a Unix hacker and fellow science-fiction buff, had borrowed the manual from Stallman a while earlier. As a display of gratitude, Hopkins decorated the return envelope with a number of stickers obtained at a local science-fiction convention. One sticker in particular caught Stallman's eye. It read, "Copyleft (L), All Rights Reversed." Following the release of the first version of GPL, Stallman paid tribute to the sticker, nicknaming the free software license "Copyleft." Over time, the nickname and its shorthand symbol, a backwards "C," would become an official Free Software Foundation synonym for the GPL. The German sociologist Max Weber once proposed that all great religions are built upon the "routinization" or "institutionalization" of charisma. Every successful religion, Weber argued, converts the charisma or message of the original religious leader into a social, political, and ethical apparatus more easily translatable across cultures and time. While not religious per se, the GNU GPL certainly qualifies as an interesting example of this "routinization" process at work in the modern, decentralized world of software development. Since its unveiling, programmers and companies who have otherwise expressed little loyalty or allegiance to Stallman have willingly accepted the GPL bargain at face value. A few have even accepted the GPL as a preemptive protective mechanism for their own software programs. Even those who reject the GPL contract as too compulsory, still credit it as influential. One hacker falling into this latter group was Keith Bostic, a University of California employee at the time of the GPL 1.0 release. Bostic's department, the Computer Systems Research Group (SRG), had been involved in Unix development since the late 1970s and was responsible for many key parts of Unix, including the TCP/IP networking protocol, the cornerstone of modern Internet communications. By the late 1980s, AT&T, the original owner of the Unix brand name, began to focus on commercializing Unix and began looking to the Berkeley Software Distribution, or BSD, the academic version of Unix developed by Bostic and his Berkeley peers, as a key source of commercial technology. Although the Berkeley BSD source code was shared among researchers and commercial programmers with a source-code license, this commercialization presented a problem. The Berkeley code was intermixed with proprietary AT&T code. As a result, Berkeley distributions were available only to institutions that already had a Unix source license from AT&T. As AT&T raised its license fees, this arrangement, which had at first seemed innocuous, became increasingly burdensome. Hired in 1986, Bostic had taken on the personal project of porting BSD over to the Digital Equipment Corporation's PDP-11 computer. It was during this period, Bostic says, that he came into close interaction with Stallman during Stallman's occasional forays out to the west coast. "I remember vividly arguing copyright with Stallman while he sat at borrowed workstations at CSRG," says Bostic. "We'd go to dinner afterward and continue arguing about copyright over dinner." The arguments eventually took hold, although not in the way Stallman would have liked. In June, 1989, Berkeley separated its networking code from the rest of the AT&T-owned operating system and distributed it under a University of California license. The contract terms were liberal. All a licensee had to do was give credit to the university in advertisements touting derivative programs.The University of California's "obnoxious advertising clause" would later prove to be a problem. Looking for a less restrictive alternative to the GPL, some hackers used the University of California, replacing "University of California" with the name of their own instution. The result: free software programs that borrowed from dozens of other programs would have to cite dozens of institutions in advertisements. In 1999, after a decade of lobbying on Stallman's part, the University of California agreed to drop this clause. In contrast to the GPL, proprietary offshoots were permissible. Only one problem hampered the license's rapid adoption: the BSD Networking release wasn't a complete operating system. People could study the code, but it could only be run in conjunction with other proprietary-licensed code. Over the next few years, Bostic and other University of California employees worked to replace the missing components and turn BSD into a complete, freely redistributable operating system. Although delayed by a legal challenge from Unix Systems Laboratories-the AT&T spin-off that retained ownership of the Unix brand name-the effort would finally bear fruit in the early 1990s. Even before then, however, many of the Berkeley utilities would make their way into Stallman's GNU Project. "I think it's highly unlikely that we ever would have gone as strongly as we did without the GNU influence," says Bostic, looking back. "It was clearly something where they were pushing hard and we liked the idea." By the end of the 1980s, the GPL was beginning to exert a gravitational effect on the free software community. A program didn't have to carry the GPL to qualify as free software-witness the case of the BSD utilities-but putting a program under the GPL sent a definite message. "I think the very existence of the GPL inspired people to think through whether they were making free software, and how they would license it," says Bruce Perens, creator of Electric Fence, a popular Unix utility, and future leader of the Debian GNU/Linux development team. A few years after the release of the GPL, Perens says he decided to discard Electric Fence's homegrown license in favor of Stallman's lawyer-vetted copyright. "It was actually pretty easy to do," Perens recalls. Rich Morin, the programmer who had viewed Stallman's initial GNU announcement with a degree of skepticism, recalls being impressed by the software that began to gather under the GPL umbrella. As the leader of a SunOS user group, one of Morin's primary duties during the 1980s had been to send out distribution tapes containing the best freeware or free software utilities. The job often mandated calling up original program authors to verify whether their programs were copyright protected or whether they had been consigned to the public domain. Around 1989, Morin says, he began to notice that the best software programs typically fell under the GPL license. "As a software distributor, as soon as I saw the word GPL, I knew I was home free," recalls Morin. To compensate for the prior hassles that went into compiling distribution tapes to the Sun User Group, Morin had charged recipients a convenience fee. Now, with programs moving over to the GPL, Morin was suddenly getting his tapes put together in half the time, turning a tidy profit in the process. Sensing a commercial opportunity, Morin rechristened his hobby as a business: Prime Time Freeware. Such commercial exploitation was completely within the confines of the free software agenda. "When we speak of free software, we are referring to freedom, not price," advised Stallman in the GPL's preamble. By the late 1980s, Stallman had refined it to a more simple mnemonic: "Don't think free as in free beer; think free as in free speech." For the most part, businesses ignored Stallman's entreaties. Still, for a few entrepreneurs, the freedom associated with free software was the same freedom associated with free markets. Take software ownership out of the commercial equation, and you had a situation where even the smallest software company was free to compete against the IBMs and DECs of the world. One of the first entrepreneurs to grasp this concept was Michael Tiemann, a software programmer and graduate student at Stanford University. During the 1980s, Tiemann had followed the GNU Project like an aspiring jazz musician following a favorite artist. It wasn't until the release of the GNU C Compiler in 1987, however, that he began to grasp the full potential of free software. Dubbing GCC a "bombshell," Tiemann says the program's own existence underlined Stallman's determination as a programmer. "Just as every writer dreams of writing the great American novel, every programmer back in the 1980s talked about writing the great American compiler," Tiemman recalls. "Suddenly Stallman had done it. It was very humbling." "You talk about single points of failure, GCC was it," echoes Bostic. "Nobody had a compiler back then, until GCC came along." Rather than compete with Stallman, Tiemann decided to build on top of his work. The original version of GCC weighed in at 110,000 lines of code, but Tiemann recalls the program as surprisingly easy to understand. So easy in fact that Tiemann says it took less than five days to master and another week to port the software to a new hardware platform, National Semiconductor's 32032 microchip. Over the next year, Tiemann began playing around with the source code, creating a native compiler for the C+ programming language. One day, while delivering a lecture on the program at Bell Labs, Tiemann ran into some AT&T developers struggling to pull off the same thing. "There were about 40 or 50 people in the room, and I asked how many people were working on the native code compiler," Tiemann recalls. "My host said the information was confidential but added that if I took a look around the room I might get a good general idea." It wasn't long after, Tiemann says, that the light bulb went off in his head. "I had been working on that project for six months," Tiemann says. I just thought to myself, whether it's me or the code this is a level of efficiency that the free market should be ready to reward." Tiemann found added inspiration in the GNU Manifesto, which, while excoriating the greed of some software vendors, encourages other vendors to consider the advantages of free software from a consumer point of view. By removing the power of monopoly from the commerical software question, the GPL makes it possible for the smartest vendors to compete on the basis of service and consulting, the two most profit-rich corners of the software marketplace. In a 1999 essay, Tiemann recalls the impact of Stallman's Manifesto. "It read like a socialist polemic, but I saw something different. I saw a business plan in disguise."7. See Michael Tiemann, "Future of Cygnus Solutions: An Entrepreneur's Account," Open Sources (O'Reilly & Associates, Inc., 1999): 139. Teaming up with John Gilmore, another GNU Project fan, Tiemann launched a software consulting service dedicated to customizing GNU programs. Dubbed Cygnus Support, the company signed its first development contract in February, 1990. By the end of the year, the company had $725,000 worth of support and development contracts. GNU Emacs, GDB, and GCC were the "big three" of developer-oriented tools, but they weren't the only ones developed by Stallman during the GNU Project's first half decade. By 1990, Stallman had also generated GNU versions of the Bourne Shell (rechristened the Bourne Again Shell, or BASH), YACC (rechristened Bison), and awk (rechristened gawk). Like GCC , every GNU program had to be designed to run on multiple systems, not just a single vendor's platform. In the process of making programs more flexible, Stallman and his collaborators often made them more useful as well. Recalling the GNU universalist approach, Prime Time Freeware's Morin points to a critical, albeit mundane, software package called hello. "It's the hello world program which is five lines of C, packaged up as if it were a GNU distribution," Morin says. "And so it's got the Texinfo stuff and the configure stuff. It's got all the other software engineering goo that the GNU Project has come up with to allow packages to port to all these different environments smoothly. That's tremendously important work, and it affects not only all of [Stallman's] software, but also all of the other GNU Project software." According to Stallman, improving software programs was secondary to building them in the first place. "With each piece I may or may not find a way to improve it," said Stallman to Byte. "To some extent I am getting the benefit of reimplementation, which makes many systems much better. To some extent it's because I have been in the field a long time and worked on many other systems. I therefore have many ideas to bring to bear."See Richard Stallman, BYTE (1986). Nevertheless, as GNU tools made their mark in the late 1980s, Stallman's AI Lab-honed reputation for design fastidiousness soon became legendary throughout the entire software-development community. Jeremy Allison, a Sun user during the late 1980s and programmer destined to run his own free software project, Samba, in the 1990s, recalls that reputation with a laugh. During the late 1980s, Allison began using Emacs. Inspired by the program's community-development model, Allison says he sent in a snippet of source code only to have it rejected by Stallman. "It was like the Onion headline," Allison says. "`Child's prayers to God answered: No.'" Stallman's growing stature as a software programmer, however, was balanced by his struggles as a project manager. Although the GNU Project moved from success to success in creation of developer-oriented tools, its inability to generate a working kernel-the central "traffic cop" program in all Unix systems that determines which devices and applications get access to the microprocessor and when-was starting to elicit grumbles as the 1980s came to a close. As with most GNU Project efforts, Stallman had started kernel development by looking for an existing program to modify. According to a January 1987 "Gnusletter," Stallman was already working to overhaul TRIX, a Unix kernel developed at MIT. A review of GNU Project "GNUsletters" of the late 1980s reflects the management tension. In January, 1987, Stallman announced to the world that the GNU Project was working to overhaul TRIX, a Unix kernel developed at MIT. A year later, in February of 1988, the GNU Project announced that it had shifted its attentions to Mach, a lightweight "micro-kernel" developed at Carnegie Mellon. All told, however, official GNU Project kernel development wouldn't commence until 1990.See "HURD History." http://www.gnu.org/software/hurd/history.html The delays in kernel development were just one of many concerns weighing on Stallman during this period. In 1989, Lotus Development Corporation filed suit against rival software company, Paperback Software International, for copying menu commands in Lotus' popular 1-2-3 Spreadsheet program. Lotus' suit, coupled with the Apple -Microsoft "look and feel" battle, provided a troublesome backdrop for the GNU Project. Although both suits fell outside the scope of the GNU Project, both revolved around operating systems and software applications developed for the personal computer, not Unix-compatible hardware systems-they threatened to impose a chilling effect on the entire culture of software development. Determined to do something, Stallman recruited a few programmer friends and composed a magazine ad blasting the lawsuits. He then followed up the ad by helping to organize a group to protest the corporations filing the suit. Calling itself the League of Programming Freedom, the group held protests outside the offices of Lotus, Inc. and the Boston courtroom hosting the Lotus trial. The protests were notable.According to a League of Programming Freedom Press, the protests were notable for featuring the first hexadecimal protest chant: 1-2-3-4, toss the lawyers out the door; 5-6-7-8, innovate don't litigate; 9-A-B-C, 1-2-3 is not for me; D-E-F-O, look and feel have got to go http://lpf.ai.mit.edu/Links/prep.ai.mit.edu/demo.final.release They document the evolving nature of software industry. Applications had quietly replaced operating systems as the primary corporate battleground. In its unfulfilled quest to build a free software operating system, the GNU Project seemed hopelessly behind the times. Indeed, the very fact that Stallman had felt it necessary to put together an entirely new group dedicated to battling the "look and feel" lawsuits reinforced that obsolescence in the eyes of some observers. In 1990, the John D. and Catherine T. MacArthur Foundation cerified Stallman's genius status when it granted Stallman a MacArthur fellowship, therefore making him a recipient for the organization's so-called "genius grant." The grant, a $240,000 reward for launching the GNU Project and giving voice to the free software philosophy, relieved a number of short-term concerns. First and foremost, it gave Stallman, a nonsalaried employee of the FSF who had been supporting himself through consulting contracts, the ability to devote more time to writing GNU code.I use the term "writing" here loosely. About the time of the MacArthur award, Stallman began suffering chronic pain in his hands and was dictating his work to FSF-employed typists. Although some have speculated that the hand pain was the result of repetitive stress injury, or RSI, an injury common among software programmers, Stallman is not 100% sure. "It was NOT carpal tunnel syndrome," he writes. "My hand problem was in the hands themselves, not in the wrists." Stallman has since learned to work without typists after switching to a keyboard with a lighter touch. Ironically, the award also made it possible for Stallman to vote. Months before the award, a fire in Stallman's apartment house had consumed his few earthly possessions. By the time of the award, Stallman was listing himself as a "squatter"See Reuven Lerner, "Stallman wins $240,000 MacArthur award," MIT, The Tech (July 18, 1990). http://the-tech.mit.edu/V110/N30/rms.30n.html at 545 Technology Square. "[The registrar of voters] didn't want to accept that as my address," Stallman would later recall. "A newspaper article about the MacArthur grant said that and then they let me register."See Michael Gross, "Richard Stallman: High School Misfit, Symbol of Free Software, MacArthur-certified Genius" (1999). Most importantly, the MacArthur money gave Stallman more freedom. Already dedicated to the issue of software freedom, Stallman chose to use the additional freedom to increase his travels in support of the GNU Project mission. Interestingly, the ultimate success of the GNU Project and the free software movement in general would stem from one of these trips. In 1990, Stallman paid a visit to the Polytechnic University in Helsinki, Finland. Among the audience members was 21-year-old Linus Torvalds, future developer of the Linux kernel-the free software kernel destined to fill the GNU Project's most sizable gap. A student at the nearby University of Helsinki at the time, Torvalds regarded Stallman with bemusement. "I saw, for the first time in my life, the stereotypical long-haired, bearded hacker type," recalls Torvalds in his 2001 autobiography Just for Fun. "We don't have much of them in Helsinki."See Linus Torvalds and David Diamond, Just For Fun: The Story of an Accidentaly Revolutionary (HarperCollins Publishers, Inc., 2001): 58-59. While not exactly attuned to the "sociopolitical" side of the Stallman agenda, Torvalds nevertheless appreciated the agenda's underlying logic: no programmer writes error-free code. By sharing software, hackers put a program's improvement ahead of individual motivations such as greed or ego protection. Like many programmers of his generation, Torvalds had cut his teeth not on mainframe computers like the IBM 7094, but on a motley assortment of home-built computer systems. As university student, Torvalds had made the step up from C programming to Unix, using the university's MicroVAX. This ladder-like progression had given Torvalds a different perspective on the barriers to machine access. For Stallman, the chief barriers were bureaucracy and privilege. For Torvalds, the chief barriers were geography and the harsh Helsinki winter. Forced to trek across the University of Helsinki just to log in to his Unix account, Torvalds quickly began looking for a way to log in from the warm confines of his off-campus apartment. The search led Torvalds to the operating system Minix, a lightweight version of Unix developed for instructional purposes by Dutch university professor Andrew Tanenbaum. The program fit within the memory confines of a 386 PC, the most powerful machine Torvalds could afford, but still lacked a few necessary features. It most notably lacked terminal emulation, the feature that allowed Torvalds' machine to mimic a university terminal, making it possible to log in to the MicroVAX from home. During the summer of 1991, Torvalds rewrote Minix from the ground up, adding other features as he did so. By the end of the summer, Torvalds was referring to his evolving work as the "GNU/Emacs of terminal emulation programs."See Linus Torvalds and David Diamond, Just For Fun: The Story of an Accidentaly Revolutionary (HarperCollins Publishers, Inc., 2001): 78. Feeling confident, he solicited a Minix newsgroup for copies of the POSIX standards, the software blue prints that determined whether a program was Unix compatible. A few weeks later, Torvalds was posting a message eerily reminiscent of Stallman's original 1983 GNU posting: Hello everybody out there using minix- I'm doing a (free) operating system (just a hobby, won't be big and professional like gnu for 386 (486) AT clones). This has been brewing since April, and is starting to get ready. I'd like any feedback on things people like/dislike in minix, as my OS resembles it somewhat (same physical layout of the file-system (due to practical reasons) among other things).See "Linux 10th Anniversary." http://www.linux10.org/history/ The posting drew a smattering of responses and within a month, Torvalds had posted a 0.01 version of the operating system-i.e., the earliest possible version fit for outside review-on an Internet FTP site. In the course of doing so, Torvalds had to come up with a name for the new system. On his own PC hard drive, Torvalds had saved the program as Linux, a name that paid its respects to the software convention of giving each Unix variant a name that ended with the letter X. Deeming the name too "egotistical," Torvalds changed it to Freax, only to have the FTP site manager change it back. Although Torvalds had set out build a full operating system, both he and other developers knew at the time that most of the functional tools needed to do so were already available, thanks to the work of GNU, BSD, and other free software developers. One of the first tools the Linux development team took advantage of was the GNU C Compiler, a tool that made it possible to process programs written in the C programming language. Integrating GCC improved the performance of Linux. It also raised issues. Although the GPL's "viral" powers didn't apply to the Linux kernel, Torvald's willingness to borrow GCC for the purposes of his own free software operating system indicated a certain obligation to let other users borrow back. As Torvalds would later put it: "I had hoisted myself up on the shoulders of giants."See Linus Torvalds and David Diamond, Just For Fun: The Story of an Accidentaly Revolutionary (HarperCollins Publishers, Inc., 2001): 96-97. Not surprisingly, he began to think about what would happen when other people looked to him for similar support. A decade after the decision, Torvalds echoes the Free Software Foundation's Robert Chassel when he sums up his thoughts at the time: You put six months of your life into this thing and you want to make it available and you want to get something out of it, but you don't want people to take advantage of it. I wanted people to be able to see [Linux], and to make changes and improvements to their hearts' content. But I also wanted to make sure that what I got out of it was to see what they were doing. I wanted to always have access to the sources so that if they made improvements, I could make those improvements myself.See Linus Torvalds and David Diamond, Just For Fun: The Story of an Accidentaly Revolutionary (HarperCollins Publishers, Inc., 2001): 94-95. When it was time to release the 0.12 version of Linux, the first to include a fully integrated version of GCC, Torvalds decided to voice his allegiance with the free software movement. He discarded the old kernel license and replaced it with the GPL. The decision triggered a porting spree, as Torvalds and his collaborators looked to other GNU programs to fold into the growing Linux stew. Within three years, Linux developers were offering their first production release, Linux 1.0, including fully modified versions of GCC, GDB, and a host of BSD tools. By 1994, the amalgamated operating system had earned enough respect in the hacker world to make some observers wonder if Torvalds hadn't given away the farm by switching to the GPL in the project's initial months. In the first issue of Linux Journal, publisher Robert Young sat down with Torvalds for an interview. When Young asked the Finnish programmer if he felt regret at giving up private ownership of the Linux source code, Torvalds said no. "Even with 20/20 hindsight," Torvalds said, he considered the GPL "one of the very best design decisions" made during the early stages of the Linux project.See Robert Young, "Interview with Linus, the Author of Linux," Linux Journal (March 1, 1994). http://www.linuxjournal.com/article.php?sid=2736 That the decision had been made with zero appeal or deference to Stallman and the Free Software Foundation speaks to the GPL's growing portability. Although it would take a few years to be recognized by Stallman, the explosiveness of Linux development conjured flashbacks of Emacs. This time around, however, the innovation triggering the explosion wasn't a software hack like Control-R but the novelty of running a Unix-like system on the PC architecture. The motives may have been different, but the end result certainly fit the ethical specifications: a fully functional operating system composed entirely of free software. As his initial email message to the comp.os.minix newsgroup indicates, it would take a few months before Torvalds saw Linux as anything less than a holdover until the GNU developers delivered on the HURD kernel. This initial unwillingness to see Linux in political terms would represent a major blow to the Free Software Foundation. As far as Torvalds was concerned, he was simply the latest in a long line of kids taking apart and reassembling things just for fun. Nevertheless, when summing up the runaway success of a project that could have just as easily spent the rest of its days on an abandoned computer hard drive, Torvalds credits his younger self for having the wisdom to give up control and accept the GPL bargain. "I may not have seen the light," writes Torvalds, reflecting on Stallman's 1991 Polytechnic University speech and his subsequent decision to switch to the GPL. "But I guess something from his speech sunk in ."See Linus Torvalds and David Diamond, Just For Fun: The Story of an Accidentaly Revolutionary (HarperCollins Publishers, Inc., 2001): 59. interview offers an interesting, not to mention candid, glimpse at Stallman's political attitudes during the earliest days of the GNU Project. It is also helpful in tracing the evolution of Stallman's rhetoric. Describing the purpose of the GPL, Stallman says, "I'm trying to change the way people approach knowledge and information in general. I think that to try to own knowledge, to try to control whether people are allowed to use it, or to try to stop other people from sharing it, is sabotage." Contrast this with a statement to the author in August 2000: "I urge you not to use the term `intellectual property' in your thinking. It will lead you to misunderstand things, because that term generalizes about copyrights, patents, and trademarks. And those things are so different in their effects that it is entirely foolish to try to talk about them at once. If you hear somebody saying something about intellectual property, without quotes, then he's not thinking very clearly and you shouldn't join." GNU/Linux By 1993, the free software movement was at a crossroads. To the optimistically inclined, all signs pointed toward success for the hacker cultur. Wired magazine, a funky, new publication offering stories on data encryption, Usenet, and software freedom, was flying off magazine racks. The Internet, once a slang term used only by hackers and research scientists, had found its way into mainstream lexicon. Even President Clinton was using it. The personal computer, once a hobbyist's toy, had grown to full-scale respectability, giving a whole new generation of computer users access to hacker-built software. And while the GNU Project had not yet reached its goal of a fully intact, free software operating system, curious users could still try Linux in the interim. Any way you sliced it, the news was good, or so it seemed. After a decade of struggle, hackers and hacker values were finally gaining acceptance in mainstream society. People were getting it. Or were they? To the pessimistically inclined, each sign of acceptance carried its own troubling countersign. Sure, being a hacker was suddenly cool, but was cool good for a community that thrived on alienation? Sure, the White House was saying all the right things about the Internet, even going so far as to register its own domain name, whitehouse.gov, but it was also meeting with the companies, censorship advocates, and law-enforcement officials looking to tame the Internet's Wild West culture. Sure, PCs were more powerful, but in commoditizing the PC marketplace with its chips, Intel had created a situation in which proprietary software vendors now held the power. For every new user won over to the free software cause via Linux, hundreds, perhaps thousands, were booting up Microsoft Windows for the first time. Finally, there was the curious nature of Linux itself. Unrestricted by design bugs (like GNU) and legal disputes (like BSD), Linux' high-speed evolution had been so unplanned, its success so accidental, that programmers closest to the software code itself didn't know what to make of it. More compilation album than operating system, it was comprised of a hacker medley of greatest hits: everything from GCC, GDB, and glibc (the GNU Project's newly developed C Library) to X (a Unix-based graphic user interface developed by MIT's Laboratory for Computer Science) to BSD-developed tools such as BIND (the Berkeley Internet Naming Daemon, which lets users substitute easy-to-remember Internet domain names for numeric IP addresses) and TCP/IP. The arch's capstone, of course, was the Linux kernel-itself a bored-out, super-charged version of Minix. Rather than building their operating system from scratch, Torvalds and his rapidly expanding Linux development team had followed the old Picasso adage, "good artists borrow; great artists steal." Or as Torvalds himself would later translate it when describing the secret of his success: "I'm basically a very lazy person who likes to take credit for things other people actually do."Torvalds has offered this quote in many different settings. To date, however, the quote's most notable appearance is in the Eric Raymond essay, "The Cathedral and the Bazaar" (May, 1997). http://www.tuxedo.org/~esr/writings/cathedral-bazaar/cathedral-bazaar/index.html Such laziness, while admirable from an efficiency perspective, was troubling from a political perspective. For one thing, it underlined the lack of an ideological agenda on Torvalds' part. Unlike the GNU developers, Torvalds hadn't built an operating system out of a desire to give his fellow hackers something to work with; he'd built it to have something he himself could play with. Like Tom Sawyer whitewashing a fence, Torvalds' genius lay less in the overall vision and more in his ability to recruit other hackers to speed the process. That Torvalds and his recruits had succeeded where others had not raised its own troubling question: what, exactly, was Linux? Was it a manifestation of the free software philosophy first articulated by Stallman in the GNU Manifesto? Or was it simply an amalgamation of nifty software tools that any user, similarly motivated, could assemble on his own home system? By late 1993, a growing number of Linux users had begun to lean toward the latter definition and began brewing private variations on the Linux theme. They even became bold enough to bottle and sell their variations-or "distributions"-to fellow Unix aficionados. The results were spotty at best. "This was back before Red Hat and the other commercial distributions," remembers Ian Murdock, then a computer science student at Purdue University. "You'd flip through Unix magazines and find all these business card-sized ads proclaiming `Linux.' Most of the companies were fly-by-night operations that saw nothing wrong with slipping a little of their own source code into the mix." Murdock, a Unix programmer, remembers being "swept away" by Linux when he first downloaded and installed it on his home PC system. "It was just a lot of fun," he says. "It made me want to get involved." The explosion of poorly built distributions began to dampen his early enthusiasm, however. Deciding that the best way to get involved was to build a version of Linux free of additives, Murdock set about putting a list of the best free software tools available with the intention of folding them into his own distribution. "I wanted something that would live up to the Linux name," Murdock says. In a bid to "stir up some interest," Murdock posted his intentions on the Internet, including Usenet's comp.os.linux newsgroup. One of the first responding email messages was from rms@ai.mit.edu . As a hacker, Murdock instantly recognized the address. It was Richard M. Stallman, founder of the GNU Project and a man Murdock knew even back then as "the hacker of hackers." Seeing the address in his mail queue, Murdock was puzzled. Why on Earth would Stallman, a person leading his own operating-system project, care about Murdock's gripes over Linux? Murdock opened the message. "He said the Free Software Foundation was starting to look closely at Linux and that the FSF was interested in possibly doing a Linux system, too. Basically, it looked to Stallman like our goals were in line with their philosophy." The message represented a dramatic about-face on Stallman's part. Until 1993, Stallman had been content to keep his nose out of the Linux community's affairs. In fact, he had all but shunned the renegade operating system when it first appeared on the Unix programming landscape in 1991. After receiving the first notification of a Unix-like operating system that ran on PCs, Stallman says he delegated the task of examining the new operating system to a friend. Recalls Stallman, "He reported back that the software was modeled after System V, which was the inferior version of Unix. He also told me it wasn't portable." The friend's report was correct. Built to run on 386-based machines, Linux was firmly rooted to its low-cost hardware platform. What the friend failed to report, however, was the sizable advantage Linux enjoyed as the only freely modifiable operating system in the marketplace. In other words, while Stallman spent the next three years listening to bug reports from his HURD team, Torvalds was winning over the programmers who would later uproot and replant the operating system onto new platforms. By 1993, the GNU Project's inability to deliver a working kernel was leading to problems both within the GNU Project and within the free software movement at large. A March, 1993, a Wired magazine article by Simson Garfinkel described the GNU Project as "bogged down" despite the success of the project's many tools.See Simson Garfinkel, "Is Stallman Stalled?" Wired (March, 1993). Those within the project and its nonprofit adjunct, the Free Software Foundation, remember the mood as being even worse than Garfinkel's article let on. "It was very clear, at least to me at the time, that there was a window of opportunity to introduce a new operating system," says Chassell. "And once that window was closed, people would become less interested. Which is in fact exactly what happened."Chassel's concern about there being a 36-month "window" for a new operating system is not unique to the GNU Project. During the early 1990s, free software versions of the Berkeley Software Distribution were held up by Unix System Laboratories' lawsuit restricting the release of BSD-derived software. While many users consider BSD offshoots such as FreeBSD and OpenBSD to be demonstrably superior to GNU/Linux both in terms of performance and security, the number of FreeBSD and OpenBSD users remains a fraction of the total GNU/Linux user population. To view a sample analysis of the relative success of GNU/Linux in relation to other free software operating systems, see the essay by New Zealand hacker, Liam Greenwood, "Why is Linux Successful" (1999). Much has been made about the GNU Project's struggles during the 1990-1993 period. While some place the blame on Stallman for those struggles, Eric Raymond, an early member of the GNU Emacs team and later Stallman critic, says the problem was largely institutional. "The FSF got arrogant," Raymond says. "They moved away from the goal of doing a production-ready operating system to doing operating-system research." Even worse, "They thought nothing outside the FSF could affect them." Murdock, a person less privy to the inner dealings of the GNU Project, adopts a more charitable view. "I think part of the problem is they were a little too ambitious and they threw good money after bad," he says. "Micro-kernels in the late 80s and early 90s were a hot topic. Unfortunately, that was about the time that the GNU Project started to design their kernel. They ended up with alot of baggage and it would have taken a lot of backpedaling to lose it." Stallman cites a number of issues when explaining the delay. The Lotus and Apple lawsuits had provided political distractions, which, coupled with Stallman's inability to type, made it difficult for Stallman to lend a helping hand to the HURD team. Stallman also cites poor communication between various portions of the GNU Project. "We had to do a lot of work to get the debugging environment to work," he recalls. "And the people maintaining GDB at the time were not that cooperative." Mostly, however, Stallman says he and the other members of the GNU Project team underestimated the difficulty of expanding the Mach microkernal into a full-fledged Unix kernel. "I figured, OK, the [Mach] part that has to talk to the machine has already been debugged," Stallman says, recalling the HURD team's troubles in a 2000 speech. "With that head start, we should be able to get it done faster. But instead, it turned out that debugging these asynchronous multithreaded programs was really hard. There were timing books that would clobber the files, and that's no fun. The end result was that it took many, many years to produce a test version."See Maui High Performance Computing Center Speech. Whatever the excuse, or excuses, the concurrent success of the Linux-kernel team created a tense situation. Sure, the Linux kernel had been licensed under the GPL, but as Murdock himself had noted, the desire to treat Linux as a purely free software operating system was far from uniform. By late 1993, the total Linux user population had grown from a dozen or so Minix enthusiasts to somewhere between 20,000 and 100,000.GNU/Linux user-population numbers are sketchy at best, which is why I've provided such a broad range. The 100,000 total comes from the Red Hat "Milestones" site, http://www.redhat.com/about/corporate/milestones.html. What had once been a hobby was now a marketplace ripe for exploitation. Like Winston Churchill watching Soviet troops sweep into Berlin, Stallman felt an understandable set of mixed emotions when it came time to celebrate the Linux "victory."I wrote this Winston Churchill analogy before Stallman himself sent me his own unsolicited comment on Churchill: World War II and the determination needed to win it was a very strong memory as I was growing up. Statements such as Churchill's, "We will fight them in the landing zones, we will fight them on the beaches . . . we will never surrender," have always resonated for me. Although late to the party, Stallman still had clout. As soon as the FSF announced that it would lend its money and moral support to Murdock's software project, other offers of support began rolling in. Murdock dubbed the new project Debian-a compression of his and his wife, Deborah's, names-and within a few weeks was rolling out the first distribution. "[Richard's support] catapulted Debian almost overnight from this interesting little project to something people within the community had to pay attention to," Murdock says. In January of 1994, Murdock issued the " Debian Manifesto." Written in the spirit of Stallman's "GNU Manifesto" from a decade before, it explained the importance of working closely with the Free Software Foundation. Murdock wrote: The Free Software Foundation plays an extremely important role in the future of Debian. By the simple fact that they will be distributing it, a message is sent to the world that Linux is not a commercial product and that it never should be, but that this does not mean that Linux will never be able to compete commercially. For those of you who disagree, I challenge you to rationalize the success of GNU Emacs and GCC, which are not commercial software but which have had quite an impact on the commercial market regardless of that fact. The time has come to concentrate on the future of Linux rather than on the destructive goal of enriching oneself at the expense of the entire Linux community and its future. The development and distribution of Debian may not be the answer to the problems that I have outlined in the Manifesto, but I hope that it will at least attract enough attention to these problems to allow them to be solved. Shortly after the Manifesto's release, the Free Software Foundation made its first major request. Stallman wanted Murdock to call its distribution "GNU/Linux." At first, Murdock says, Stallman had wanted to use the term " Lignux"-"as in Linux with GNU at the heart of it"-but a sample testing of the term on Usenet and in various impromptu hacker focus groups had merited enough catcalls to convince Stallman to go with the less awkward GNU/Linux. Although some would dismiss Stallman's attempt to add the "GNU" prefix as a belated quest for credit, Murdock saw it differently. Looking back, Murdock saw it as an attempt to counteract the growing tension between GNU Project and Linux-kernel developers. "There was a split emerging," Murdock recalls. "Richard was concerned." The deepest split, Murdock says, was over glibc. Short for GNU C Library, glibc is the package that lets programmers make "system calls" directed at the kernel. Over the course of 1993-1994, glibc emerged as a troublesome bottleneck in Linux development. Because so many new users were adding new functions to the Linux kernel, the GNU Project's glibc maintainers were soon overwhelmed with suggested changes. Frustrated by delays and the GNU Project's growing reputation for foot-dragging, some Linux developers suggested creating a " fork"-i.e., a Linux-specific C Library parallel to glibc. In the hacker world, forks are an interesting phenomenon. Although the hacker ethic permits a programmer to do anything he wants with a given program's source code, most hackers prefer to pour their innovations into a central source-code file or " tree" to ensure compatibility with other people's programs. To fork glibc this early in the development of Linux would have meant losing the potential input of hundreds, even thousands, of Linux developers. It would also mean growing incompatibility between Linux and the GNU system that Stallman and the GNU team still hoped to develop. As leader of the GNU Project, Stallman had already experienced the negative effects of a software fork in 1991. A group of Emacs developers working for a software company named Lucid had a falling out over Stallman's unwillingness to fold changes back into the GNU Emacs code base. The fork had given birth to a parallel version, Lucid Emacs, and hard feelings all around.Jamie Zawinski, a former Lucid programmer who would go on to head the Mozilla development team, has a web site that documents the Lucid/GNU Emacs fork, titled, "The Lemacs/FSFmacs Schism." http://www.jwz.org/doc/lemacs.html Murdock says Debian was mounting work on a similar fork in glibc source code that motivated Stallman to insist on adding the GNU prefix when Debian rolled out its software distribution. "The fork has since converged. Still, at the time, there was a concern that if the Linux community saw itself as a different thing as the GNU community, it might be a force for disunity." Stallman seconds Murdock's recollection. In fact, he says there were nascent forks appearing in relation to every major GNU component. At first, Stallman says he considered the forks to be a product of sour grapes. In contrast to the fast and informal dynamics of the Linux-kernel team, GNU source-code maintainers tended to be slower and more circumspect in making changes that might affect a program's long-term viability. They also were unafraid of harshly critiquing other people's code. Over time, however, Stallman began to sense that there was an underlying lack of awareness of the GNU Project and its objectives when reading Linux developers' emails. "We discovered that the people who considered themselves Linux users didn't care about the GNU Project," Stallman says. "They said, `Why should I bother doing these things? I don't care about the GNU Project. It's working for me. It's working for us Linux users, and nothing else matters to us.' And that was quite surprising given that people were essentially using a variant of the GNU system, and they cared so little. They cared less than anybody else about GNU." While some viewed descriptions of Linux as a "variant" of the GNU Project as politically grasping, Murdock, already sympathetic to the free software cause, saw Stallman's request to call Debian's version GNU/Linux as reasonable. "It was more for unity than for credit," he says. Requests of a more technical nature quickly followed. Although Murdock had been accommodating on political issues, he struck a firmer pose when it came to the design and development model of the actual software. What had begun as a show of solidarity soon became of model of other GNU projects. "I can tell you that I've had my share of disagreements with him," says Murdock with a laugh. "In all honesty Richard can be a fairly difficult person to work with." In 1996, Murdock, following his graduation from Purdue, decided to hand over the reins of the growing Debian project. He had already been ceding management duties to Bruce Perens, the hacker best known for his work on Electric Fence, a Unix utility released under the GPL. Perens, like Murdock, was a Unix programmer who had become enamored of GNU/Linux as soon as the program's Unix-like abilities became manifest. Like Murdock, Perens sympathized with the political agenda of Stallman and the Free Software Foundation, albeit from afar. "I remember after Stallman had already come out with the GNU Manifesto, GNU Emacs, and GCC, I read an article that said he was working as a consultant for Intel," says Perens, recalling his first brush with Stallman in the late 1980s. "I wrote him asking how he could be advocating free software on the one hand and working for Intel on the other. He wrote back saying, `I work as a consultant to produce free software.' He was perfectly polite about it, and I thought his answer made perfect sense." As a prominent Debian developer, however, Perens regarded Murdock's design battles with Stallman with dismay. Upon assuming leadership of the development team, Perens says he made the command decision to distance Debian from the Free Software Foundation. "I decided we did not want Richard's style of micro-management," he says. According to Perens, Stallman was taken aback by the decision but had the wisdom to roll with it. "He gave it some time to cool off and sent a message that we really needed a relationship. He requested that we call it GNU/Linux and left it at that. I decided that was fine. I made the decision unilaterally. Everybody breathed a sigh of relief." Over time, Debian would develop a reputation as the hacker's version of Linux, alongside Slackware, another popular distribution founded during the same 1993-1994 period. Outside the realm of hacker-oriented systems, however, Linux was picking up steam in the commercial Unix marketplace. In North Carolina, a Unix company billing itself as Red Hat was revamping its business to focus on Linux. The chief executive officer was Robert Young, the former Linux Journal editor who in 1994 had put the question to Linus Torvalds, asking whether he had any regrets about putting the kernel under the GPL. To Young, Torvalds' response had a "profound" impact on his own view toward Linux. Instead of looking for a way to corner the GNU/Linux market via traditional software tactics, Young began to consider what might happen if a company adopted the same approach as Debian-i.e., building an operating system completely out of free software parts. Cygnus Solutions, the company founded by Michael Tiemann and John Gilmore in 1990, was already demonstrating the ability to sell free software based on quality and customizability. What if Red Hat took the same approach with GNU/Linux? "In the western scientific tradition we stand on the shoulders of giants," says Young, echoing both Torvalds and Sir Isaac Newton before him. "In business, this translates to not having to reinvent wheels as we go along. The beauty of [the GPL] model is you put your code into the public domain.Young uses the term "public domain" incorrectly here. Public domain means not protected by copyright. GPL-protected programs are by definition protected by copyright. If you're an independent software vendor and you're trying to build some application and you need a modem-dialer, well, why reinvent modem dialers? You can just steal PPP off of Red Hat Linux and use that as the core of your modem-dialing tool. If you need a graphic tool set, you don't have to write your own graphic library. Just download GTK. Suddenly you have the ability to reuse the best of what went before. And suddenly your focus as an application vendor is less on software management and more on writing the applications specific to your customer's needs." Young wasn't the only software executive intrigued by the business efficiencies of free software. By late 1996, most Unix companies were starting to wake up and smell the brewing source code. The Linux sector was still a good year or two away from full commercial breakout mode, but those close enough to the hacker community could feel it: something big was happening. The Intel 386 chip, the Internet, and the World Wide Web had hit the marketplace like a set of monster waves, and Linux-and the host of software programs that echoed it in terms of source-code accessibility and permissive licensing-seemed like the largest wave yet. For Ian Murdock, the programmer courted by Stallman and then later turned off by Stallman's micromanagement style, the wave seemed both a fitting tribute and a fitting punishment for the man who had spent so much time giving the free software movement an identity. Like many Linux aficionados, Murdock had seen the original postings. He'd seen Torvalds's original admonition that Linux was "just a hobby." He'd also seen Torvalds's admission to Minix creator Andrew Tanenbaum: "If the GNU kernel had been ready last spring, I'd not have bothered to even start my project."This quote is taken from the much-publicized Torvalds-Tanenbaum "flame war" following the initial release of Linux. In the process of defending his choice of a nonportable monolithic kernel design, Torvalds says he started working on Linux as a way to learn more about his new 386 PC. "If the GNU kernel had been ready last spring, I'd not have bothered to even start my project." See Chris DiBona et al., Open Sources (O'Reilly & Associates, Inc., 1999): 224. Like many, Murdock knew the opportunities that had been squandered. He also knew the excitement of watching new opportunities come seeping out of the very fabric of the Internet. "Being involved with Linux in those early days was fun," recalls Murdock. "At the same time, it was something to do, something to pass the time. If you go back and read those old [comp.os.minix] exchanges, you'll see the sentiment: this is something we can play with until the HURD is ready. People were anxious. It's funny, but in a lot of ways, I suspect that Linux would never have happened if the HURD had come along more quickly." By the end of 1996, however, such "what if" questions were already moot. Call it Linux, call it GNU/Linux; the users had spoken. The 36-month window had closed, meaning that even if the GNU Project had rolled out its HURD kernel, chances were slim anybody outside the hard-core hacker community would have noticed. The first Unix-like free software operating system was here, and it had momentum. All hackers had left to do was sit back and wait for the next major wave to come crashing down on their heads. Even the shaggy-haired head of one Richard M. Stallman. Ready or not. Open Source In November , 1995, Peter Salus, a member of the Free Software Foundation and author of the 1994 book, A Quarter Century of Unix , issued a call for papers to members of the GNU Project's "system-discuss" mailing list. Salus, the conference's scheduled chairman, wanted to tip off fellow hackers about the upcoming Conference on Freely Redistributable Software in Cambridge, Massachusetts. Slated for February, 1996 and sponsored by the Free Software Foundation, the event promised to be the first engineering conference solely dedicated to free software and, in a show of unity with other free software programmers, welcomed papers on "any aspect of GNU, Linux, NetBSD, 386BSD, FreeBSD, Perl, Tcl/tk, and other tools for which the code is accessible and redistributable." Salus wrote: Over the past 15 years, free and low-cost software has become ubiquitous. This conference will bring together implementers of several different types of freely redistributable software and publishers of such software (on various media). There will be tutorials and refereed papers, as well as keynotes by Linus Torvalds and Richard Stallman.See Peter Salus, "FYI-Conference on Freely Redistributable Software, 2/2, Cambridge" (1995) (archived by Terry Winograd). http://hci.stanford.edu/pcd-archives/pcd-fyi/1995/0078.html One of the first people to receive Salus' email was conference committee member Eric S. Raymond. Although not the leader of a project or company like the various other members of the list, Raymond had built a tidy reputation within the hacker community as a major contributor to GNU Emacs and as editor of The New Hacker Dictionary, a book version of the hacking community's decade-old Jargon File. For Raymond, the 1996 conference was a welcome event. Active in the GNU Project during the 1980s, Raymond had distanced himself from the project in 1992, citing, like many others before him, Stallman's "micro-management" style. "Richard kicked up a fuss about my making unauthorized modifications when I was cleaning up the Emacs LISP libraries," Raymond recalls. "It frustrated me so much that I decided I didn't want to work with him anymore." Despite the falling out, Raymond remained active in the free software community. So much so that when Salus suggested a conference pairing Stallman and Torvalds as keynote speakers, Raymond eagerly seconded the idea. With Stallman representing the older, wiser contingent of ITS/Unix hackers and Torvalds representing the younger, more energetic crop of Linux hackers, the pairing indicated a symbolic show of unity that could only be beneficial, especially to ambitious younger (i.e., below 40) hackers such as Raymond. "I sort of had a foot in both camps," Raymond says. By the time of the conference, the tension between those two camps had become palpable. Both groups had one thing in common, though: the conference was their first chance to meet the Finnish wunderkind in the flesh. Surprisingly, Torvalds proved himself to be a charming, affable speaker. Possessing only a slight Swedish accent, Torvalds surprised audience members with his quick, self-effacing wit.Although Linus Torvalds is Finnish, his mother tongue is Swedish. "The Rampantly Unofficial Linus FAQ" offers a brief explanation: Finland has a significant (about 6%) Swedish-speaking minority population. They call themselves "finlandssvensk" or "finlandssvenskar" and consider themselves Finns; many of their families have lived in Finland for centuries. Swedish is one of Finland's two official languages. http://tuxedo.org/~esr/faqs/linus/ Even more surprising, says Raymond, was Torvalds' equal willingness to take potshots at other prominent hackers, including the most prominent hacker of all, Richard Stallman. By the end of the conference, Torvalds' half-hacker, half-slacker manner was winning over older and younger conference-goers alike. "It was a pivotal moment," recalls Raymond. "Before 1996, Richard was the only credible claimant to being the ideological leader of the entire culture. People who dissented didn't do so in public. The person who broke that taboo was Torvalds." The ultimate breach of taboo would come near the end of the show. During a discussion on the growing market dominance of Microsoft Windows or some similar topic, Torvalds admitted to being a fan of Microsoft's PowerPoint slideshow software program. From the perspective of old-line software purists, it was like a Mormon bragging in church about his fondness of whiskey. From the perspective of Torvalds and his growing band of followers, it was simply common sense. Why shun worthy proprietary software programs just to make a point? Being a hacker wasn't about suffering, it was about getting the job done. "That was a pretty shocking thing to say," Raymond remembers. "Then again, he was able to do that, because by 1995 and 1996, he was rapidly acquiring clout." Stallman, for his part, doesn't remember any tension at the 1996 conference, but he does remember later feeling the sting of Torvalds' celebrated cheekiness. "There was a thing in the Linux documentation which says print out the GNU coding standards and then tear them up," says Stallman, recalling one example. "OK, so he disagrees with some of our conventions. That's fine, but he picked a singularly nasty way of saying so. He could have just said `Here's the way I think you should indent your code.' Fine. There should be no hostility there." For Raymond, the warm reception other hackers gave to Torvalds' comments merely confirmed his suspicions. The dividing line separating Linux developers from GNU/Linux developers was largely generational. Many Linux hackers, like Torvalds, had grown up in a world of proprietary software. Unless a program was clearly inferior, most saw little reason to rail against a program on licensing issues alone. Somewhere in the universe of free software systems lurked a program that hackers might someday turn into a free software alternative to PowerPoint. Until then, why begrudge Microsoft the initiative of developing the program and reserving the rights to it? As a former GNU Project member, Raymond sensed an added dynamic to the tension between Stallman and Torvalds. In the decade since launching the GNU Project, Stallman had built up a fearsome reputation as a programmer. He had also built up a reputation for intransigence both in terms of software design and people management. Shortly before the 1996 conference, the Free Software Foundation would experience a full-scale staff defection, blamed in large part on Stallman. Brian Youmans, a current FSF staffer hired by Salus in the wake of the resignations, recalls the scene: "At one point, Peter [Salus] was the only staff member working in the office." For Raymond, the defection merely confirmed a growing suspicion: recent delays such as the HURD and recent troubles such as the Lucid-Emacs schism reflected problems normally associated with software project management, not software code development. Shortly after the Freely Redistributable Software Conference, Raymond began working on his own pet software project, a popmail utility called " fetchmail." Taking a cue from Torvalds, Raymond issued his program with a tacked-on promise to update the source code as early and as often as possible. When users began sending in bug reports and feature suggestions, Raymond, at first anticipating a tangled mess, found the resulting software surprisingly sturdy. Analyzing the success of the Torvalds approach, Raymond issued a quick analysis: using the Internet as his "petri dish" and the harsh scrutiny of the hacker community as a form of natural selection, Torvalds had created an evolutionary model free of central planning. What's more, Raymond decided, Torvalds had found a way around Brooks' Law. First articulated by Fred P. Brooks, manager of IBM's OS/360 project and author of the 1975 book, The Mythical Man-Month , Brooks' Law held that adding developers to a project only resulted in further project delays. Believing as most hackers that software, like soup, benefits from a limited number of cooks, Raymond sensed something revolutionary at work. In inviting more and more cooks into the kitchen, Torvalds had actually found away to make the resulting software better.Brooks' Law is the shorthand summary of the following quote taken from Brooks' book: Since software construction is inherently a systems effort-an exercise in complex interrelationships-communication effort is great, and it quickly dominates the decrease in individual task time brought about by partitioning. Adding more men then lengthens, not shortens, the schedule. See Fred P. Brooks, The Mythical Man-Month (Addison Wesley Publishing, 1995) Raymond put his observations on paper. He crafted them into a speech, which he promptly delivered before a group of friends and neighbors in Chester County, Pennsylvania. Dubbed " The Cathedral and the Bazaar," the speech contrasted the management styles of the GNU Project with the management style of Torvalds and the kernel hackers. Raymond says the response was enthusiastic, but not nearly as enthusiastic as the one he received during the 1997 Linux Kongress, a gathering of Linux users in Germany the next spring. "At the Kongress, they gave me a standing ovation at the end of the speech," Raymond recalls. "I took that as significant for two reasons. For one thing, it meant they were excited by what they were hearing. For another thing, it meant they were excited even after hearing the speech delivered through a language barrier." Eventually, Raymond would convert the speech into a paper, also titled "The Cathedral and the Bazaar." The paper drew its name from Raymond's central analogy. GNU programs were "cathedrals," impressive, centrally planned monuments to the hacker ethic, built to stand the test of time. Linux, on the other hand, was more like "a great babbling bazaar," a software program developed through the loose decentralizing dynamics of the Internet. Implicit within each analogy was a comparison of Stallman and Torvalds. Where Stallman served as the classic model of the cathedral architect-i.e., a programming "wizard" who could disappear for 18 months and return with something like the GNU C Compiler-Torvalds was more like a genial dinner-party host. In letting others lead the Linux design discussion and stepping in only when the entire table needed a referee, Torvalds had created a development model very much reflective of his own laid-back personality. From the Torvalds' perspective, the most important managerial task was not imposing control but keeping the ideas flowing. Summarized Raymond, "I think Linus's cleverest and most consequential hack was not the construction of the Linux kernel itself, but rather his invention of the Linux development model."See Eric Raymond, "The Cathredral and the Bazaar" (1997). In summarizing the secrets of Torvalds' managerial success, Raymond himself had pulled off a coup. One of the audience members at the Linux Kongress was Tim O'Reilly, publisher of O'Reilly & Associates, a company specializing in software manuals and software-related books (and the publisher of this book). After hearing Raymond's Kongress speech, O'Reilly promptly invited Raymond to deliver it again at the company's inaugural Perl Conference later that year in Monterey, California. Although the conference was supposed to focus on Perl, a scripting language created by Unix hacker Larry Wall, O'Reilly assured Raymond that the conference would address other free software technologies. Given the growing commercial interest in Linux and Apache, a popular free software web server, O'Reilly hoped to use the event to publicize the role of free software in creating the entire infrastructure of the Internet. From web-friendly languages such as Perl and Python to back-room programs such as BIND (the Berkeley Internet Naming Daemon), a software tool that lets users replace arcane IP numbers with the easy-to-remember domain-name addresses (e.g., amazon.com), and sendmail, the most popular mail program on the Internet, free software had become an emergent phenomenon. Like a colony of ants creating a beautiful nest one grain of sand at a time, the only thing missing was the communal self-awareness. O'Reilly saw Raymond's speech as a good way to inspire that self-awareness, to drive home the point that free software development didn't start and end with the GNU Project. Programming languages, such as Perl and Python, and Internet software, such as BIND, sendmail, and Apache, demonstrated that free software was already ubiquitous and influential. He also assured Raymond an even warmer reception than the one at Linux Kongress. O'Reilly was right. "This time, I got the standing ovation before the speech," says Raymond, laughing. As predicted, the audience was stocked not only with hackers, but with other people interested in the growing power of the free software movement. One contingent included a group from Netscape, the Mountain View, California startup then nearing the end game of its three-year battle with Microsoft for control of the web-browser market. Intrigued by Raymond's speech and anxious to win back lost market share, Netscape executives took the message back to corporate headquarters. A few months later, in January, 1998, the company announced its plan to publish the source code of its flagship Navigator web browser in the hopes of enlisting hacker support in future development. When Netscape CEO Jim Barksdale cited Raymond's "Cathedral and the Bazaar" essay as a major influence upon the company's decision, the company instantly elevated Raymond to the level of hacker celebrity. Determined not to squander the opportunity, Raymond traveled west to deliver interviews, advise Netscape executives, and take part in the eventual party celebrating the publication of Netscape Navigator's source code. The code name for Navigator's source code was "Mozilla": a reference both to the program's gargantuan size-30 million lines of code-and to its heritage. Developed as a proprietary offshoot of Mosaic, the web browser created by Marc Andreessen at the University of Illinois, Mozilla was proof, yet again, that when it came to building new programs, most programmers preferred to borrow on older, modifiable programs. While in California, Raymond also managed to squeeze in a visit to VA Research, a Santa Clara-based company selling workstations with the GNU/Linux operating system preinstalled. Convened by Raymond, the meeting was small. The invite list included VA founder Larry Augustin, a few VA employees, and Christine Peterson, president of the Foresight Institute, a Silicon Valley think tank specializing in nanotechnology. "The meeting's agenda boiled down to one item: how to take advantage of Netscape's decision so that other companies might follow suit?" Raymond doesn't recall the conversation that took place, but he does remember the first complaint addressed. Despite the best efforts of Stallman and other hackers to remind people that the word "free" in free software stood for freedom and not price, the message still wasn't getting through. Most business executives, upon hearing the term for the first time, interpreted the word as synonymous with "zero cost," tuning out any follow up messages in short order. Until hackers found a way to get past this cognitive dissonance, the free software movement faced an uphill climb, even after Netscape. Peterson, whose organization had taken an active interest in advancing the free software cause, offered an alternative: open source. Looking back, Peterson says she came up with the open source term while discussing Netscape's decision with a friend in the public relations industry. She doesn't remember where she came upon the term or if she borrowed it from another field, but she does remember her friend disliking the term.5 At the meeting, Peterson says, the response was dramatically different. "I was hesitant about suggesting it," Peterson recalls. "I had no standing with the group, so started using it casually, not highlighting it as a new term." To Peterson's surprise, the term caught on. By the end of the meeting, most of the attendees, including Raymond, seemed pleased by it. Raymond says he didn't publicly use the term "open source" as a substitute for free software until a day or two after the Mozilla launch party, when O'Reilly had scheduled a meeting to talk about free software. Calling his meeting "the Freeware Summit," O'Reilly says he wanted to direct media and community attention to the other deserving projects that had also encouraged Netscape to release Mozilla. "All these guys had so much in common, and I was surprised they didn't all know each other," says O'Reilly. "I also wanted to let the world know just how great an impact the free software culture had already made. People were missing out on a large part of the free software tradition." In putting together the invite list, however, O'Reilly made a decision that would have long-term political consequences. He decided to limit the list to west-coast developers such as Wall, Eric Allman, creator of sendmail, and Paul Vixie, creator of BIND. There were exceptions, of course: Pennsylvania-resident Raymond, who was already in town thanks to the Mozilla launch, earned a quick invite. So did Virginia-resident Guido van Rossum, creator of Python. "Frank Willison, my editor in chief and champion of Python within the company, invited him without first checking in with me," O'Reilly recalls. "I was happy to have him there, but when I started, it really was just a local gathering." For some observers, the unwillingness to include Stallman's name on the list qualified as a snub. "I decided not to go to the event because of it," says Perens, remembering the summit. Raymond, who did go, says he argued for Stallman's inclusion to no avail. The snub rumor gained additional strength from the fact that O'Reilly, the event's host, had feuded publicly with Stallman over the issue of software-manual copyrights. Prior to the meeting, Stallman had argued that free software manuals should be as freely copyable and modifiable as free software programs. O'Reilly, meanwhile, argued that a value-added market for nonfree books increased the utility of free software by making it more accessible to a wider community. The two had also disputed the title of the event, with Stallman insisting on "Free Software" over the less politically laden "Freeware." Looking back, O'Reilly doesn't see the decision to leave Stallman's name off the invite list as a snub. "At that time, I had never met Richard in person, but in our email interactions, he'd been inflexible and unwilling to engage in dialogue. I wanted to make sure the GNU tradition was represented at the meeting, so I invited John Gilmore and Michael Tiemann, whom I knew personally, and whom I knew were passionate about the value of the GPL but seemed more willing to engage in a frank back-and-forth about the strengths and weaknesses of the various free software projects and traditions. Given all the later brouhaha, I do wish I'd invited Richard as well, but I certainly don't think that my failure to do so should be interpreted as a lack of respect for the GNU Project or for Richard personally." Snub or no snub, both O'Reilly and Raymond say the term "open source" won over just enough summit-goers to qualify as a success. The attendees shared ideas and experiences and brainstormed on how to improve free software's image. Of key concern was how to point out the successes of free software, particularly in the realm of Internet infrastructure, as opposed to playing up the GNU/Linux challenge to Microsoft Windows. But like the earlier meeting at VA, the discussion soon turned to the problems associated with the term "free software." O'Reilly, the summit host, remembers a particularly insightful comment from Torvalds, a summit attendee. "Linus had just moved to Silicon Valley at that point, and he explained how only recently that he had learned that the word `free' had two meanings-free as in `libre' and free as in `gratis'-in English." Michael Tiemann, founder of Cygnus, proposed an alternative to the troublesome "free software" term: sourceware. "Nobody got too excited about it," O'Reilly recalls. "That's when Eric threw out the term `open source.'" Although the term appealed to some, support for a change in official terminology was far from unanimous. At the end of the one-day conference, attendees put the three terms-free software, open source, or sourceware-to a vote. According to O'Reilly, 9 out of the 15 attendees voted for "open source." Although some still quibbled with the term, all attendees agreed to use it in future discussions with the press. "We wanted to go out with a solidarity message," O'Reilly says. The term didn't take long to enter the national lexicon. Shortly after the summit, O'Reilly shepherded summit attendees to a press conference attended by reporters from the New York Times, the Wall Street Journal, and other prominent publications. Within a few months, Torvalds' face was appearing on the cover of Forbes magazine, with the faces of Stallman, Perl creator Larry Wall, and Apache team leader Brian Behlendorf featured in the interior spread. Open source was open for business. For summit attendees such as Tiemann, the solidarity message was the most important thing. Although his company had achieved a fair amount of success selling free software tools and services, he sensed the difficulty other programmers and entrepreneurs faced. "There's no question that the use of the word free was confusing in a lot of situations," Tiemann says. "Open source positioned itself as being business friendly and business sensible. Free software positioned itself as morally righteous. For better or worse we figured it was more advantageous to align with the open source crowd. For Stallman, the response to the new "open source" term was slow in coming. Raymond says Stallman briefly considered adopting the term, only to discard it. "I know because I had direct personal conversations about it," Raymond says. By the end of 1998, Stallman had formulated a position: open source, while helpful in communicating the technical advantages of free software, also encouraged speakers to soft-pedal the issue of software freedom. Given this drawback, Stallman would stick with the term free software. Summing up his position at the 1999 LinuxWorld Convention and Expo, an event billed by Torvalds himself as a "coming out party" for the Linux community, Stallman implored his fellow hackers to resist the lure of easy compromise. "Because we've shown how much we can do, we don't have to be desperate to work with companies or compromise our goals," Stallman said during a panel discussion. "Let them offer and we'll accept. We don't have to change what we're doing to get them to help us. You can take a single step towards a goal, then another and then more and more and you'll actually reach your goal. Or, you can take a half measure that means you don't ever take another step and you'll never get there." Even before the LinuxWorld show, however, Stallman was showing an increased willingness to alienate his more conciliatory peers. A few months after the Freeware Summit, O'Reilly hosted its second annual Perl Conference. This time around, Stallman was in attendance. During a panel discussion lauding IBM's decision to employ the free software Apache web server in its commercial offerings, Stallman, taking advantage of an audience microphone, disrupted the proceedings with a tirade against panelist John Ousterhout, creator of the Tcl scripting language. Stallman branded Ousterhout a "parasite" on the free software community for marketing a proprietary version of Tcl via Ousterhout's startup company, Scriptics. "I don't think Scriptics is necessary for the continued existence of Tcl," Stallman said to hisses from the fellow audience members.See Malcolm Maclachlan, "Profit Motive Splits Open Source Movement," TechWeb News (August 26, 1998). http://content.techweb.com/wire/story/TWB19980824S0012 "It was a pretty ugly scene," recalls Prime Time Freeware's Rich Morin. "John's done some pretty respectable things: Tcl, Tk, Sprite. He's a real contributor." Despite his sympathies for Stallman and Stallman's position, Morin felt empathy for those troubled by Stallman's discordant behavior. Stallman's Perl Conference outburst would momentarily chase off another potential sympathizer, Bruce Perens. In 1998, Eric Raymond proposed launching the Open Source Initiative, or OSI, an organization that would police the use of the term "open source" and provide a definition for companies interested in making their own programs. Raymond recruited Perens to draft the definition.See Bruce Perens et al., "The Open Source Definition," The Open Source Initiative (1998). http://www.opensource.org/docs/definition.html Perens would later resign from the OSI, expressing regret that the organization had set itself up in opposition to Stallman and the FSF. Still, looking back on the need for a free software definition outside the Free Software Foundation's auspices, Perens understands why other hackers might still feel the need for distance. "I really like and admire Richard," says Perens. "I do think Richard would do his job better if Richard had more balance. That includes going away from free software for a couple of months." Stallman's monomaniacal energies would do little to counteract the public-relations momentum of open source proponents. In August of 1998, when chip-maker Intel purchased a stake in GNU/Linux vendor Red Hat, an accompanying New York Times article described the company as the product of a movement "known alternatively as free software and open source."See Amy Harmon, "For Sale: Free Operating System," New York Times (September 28, 1998). http://www.nytimes.com/library/tech/98/09/biztech/articles/28linux.html Six months later, a John Markoff article on Apple Computer was proclaiming the company's adoption of the "open source" Apache server in the article headline.See John Markoff, "Apple Adopts `Open Source' for its Server Computers," New York Times (March 17, 1999). http://www.nytimes.com/library/tech/99/03/biztech/articles/17apple.html Such momentum would coincide with the growing momentum of companies that actively embraced the "open source" term. By August of 1999, Red Hat, a company that now eagerly billed itself as "open source," was selling shares on Nasdaq. In December, VA Linux-formerly VA Research-was floating its own IPO to historical effect. Opening at $30 per share, the company's stock price exploded past the $300 mark in initial trading only to settle back down to the $239 level. Shareholders lucky enough to get in at the bottom and stay until the end experienced a 698% increase in paper wealth, a Nasdaq record. Among those lucky shareholders was Eric Raymond, who, as a company board member since the Mozilla launch, had received 150,000 shares of VA Linux stock. Stunned by the realization that his essay contrasting the Stallman-Torvalds managerial styles had netted him $36 million in potential wealth, Raymond penned a follow-up essay. In it, Raymond mused on the relationship between the hacker ethic and monetary wealth: Reporters often ask me these days if I think the open-source community will be corrupted by the influx of big money. I tell them what I believe, which is this: commercial demand for programmers has been so intense for so long that anyone who can be seriously distracted by money is already gone. Our community has been self-selected for caring about other things-accomplishment, pride, artistic passion, and each other.See Eric Raymond, "Surprised by Wealth," Linux Today (December 10, 1999). http://linuxtoday.com/news_story.php3?ltsn=1999-12-10-001-05-NW-LF Whether or not such comments allayed suspicions that Raymond and other open source proponents had simply been in it for the money, they drove home the open source community's ultimate message: all you needed to sell the free software concept is a friendly face and a sensible message. Instead of fighting the marketplace head-on as Stallman had done, Raymond, Torvalds, and other new leaders of the hacker community had adopted a more relaxed approach-ignoring the marketplace in some areas, leveraging it in others. Instead of playing the role of high-school outcasts, they had played the game of celebrity, magnifying their power in the process. "On his worst days Richard believes that Linus Torvalds and I conspired to hijack his revolution," Raymond says. "Richard's rejection of the term open source and his deliberate creation of an ideological fissure in my view comes from an odd mix of idealism and territoriality. There are people out there who think it's all Richard's personal ego. I don't believe that. It's more that he so personally associates himself with the free software idea that he sees any threat to that as a threat to himself." Ironically, the success of open source and open source advocates such as Raymond would not diminish Stallman's role as a leader. If anything, it gave Stallman new followers to convert. Still, the Raymond territoriality charge is a damning one. There are numerous instances of Stallman sticking to his guns more out of habit than out of principle: his initial dismissal of the Linux kernel, for example, and his current unwillingness as a political figure to venture outside the realm of software issues. Then again, as the recent debate over open source also shows, in instances when Stallman has stuck to his guns, he's usually found a way to gain ground because of it. "One of Stallman's primary character traits is the fact he doesn't budge," says Ian Murdock. "He'll wait up to a decade for people to come around to his point of view if that's what it takes." Murdock, for one, finds that unbudgeable nature both refreshing and valuable. Stallman may no longer be the solitary leader of the free software movement, but he is still the polestar of the free software community. "You always know that he's going to be consistent in his views," Murdock says. "Most people aren't like that. Whether you agree with him or not, you really have to respect that." A Brief Journey Through Hacker Hell Richard Stallman stares, unblinking, through the windshield of a rental car, waiting for the light to change as we make our way through downtown Kihei. The two of us are headed to the nearby town of Pa'ia, where we are scheduled to meet up with some software programmers and their wives for dinner in about an hour or so. It's about two hours after Stallman's speech at the Maui High Performance Center, and Kihei, a town that seemed so inviting before the speech, now seems profoundly uncooperative. Like most beach cities, Kihei is a one-dimensional exercise in suburban sprawl. Driving down its main drag, with its endless succession of burger stands, realty agencies, and bikini shops, it's hard not to feel like a steel-coated morsel passing through the alimentary canal of a giant commercial tapeworm. The feeling is exacerbated by the lack of side roads. With nowhere to go but forward, traffic moves in spring-like lurches. 200 yards ahead, a light turns green. By the time we are moving, the light is yellow again. For Stallman, a lifetime resident of the east coast, the prospect of spending the better part of a sunny Hawaiian afternoon trapped in slow traffic is enough to trigger an embolism. Even worse is the knowledge that, with just a few quick right turns a quarter mile back, this whole situation easily could have been avoided. Unfortunately, we are at the mercy of the driver ahead of us, a programmer from the lab who knows the way and who has decided to take us to Pa'ia via the scenic route instead of via the nearby Pilani Highway. "This is terrible," says Stallman between frustrated sighs. "Why didn't we take the other route?" Again, the light a quarter mile ahead of us turns green. Again, we creep forward a few more car lengths. This process continues for another 10 minutes, until we finally reach a major crossroad promising access to the adjacent highway. The driver ahead of us ignores it and continues through the intersection. "Why isn't he turning?" moans Stallman, throwing up his hands in frustration. "Can you believe this?" I decide not to answer either. I find the fact that I am sitting in a car with Stallman in the driver seat, in Maui no less, unbelievable enough. Until two hours ago, I didn't even know Stallman knew how to drive. Now, listening to Yo-Yo Ma's cello playing the mournful bass notes of "Appalachian Journey" on the car stereo and watching the sunset pass by on our left, I do my best to fade into the upholstery. When the next opportunity to turn finally comes up, Stallman hits his right turn signal in an attempt to cue the driver ahead of us. No such luck. Once again, we creep slowly through the intersection, coming to a stop a good 200 yards before the next light. By now, Stallman is livid. "It's like he's deliberately ignoring us," he says, gesturing and pantomiming like an air craft carrier landing-signals officer in a futile attempt to catch our guide's eye. The guide appears unfazed, and for the next five minutes all we see is a small portion of his head in the rearview mirror. I look out Stallman's window. Nearby Kahoolawe and Lanai Islands provide an ideal frame for the setting sun. It's a breathtaking view, the kind that makes moments like this a bit more bearable if you're a Hawaiian native, I suppose. I try to direct Stallman's attention to it, but Stallman, by now obsessed by the inattentiveness of the driver ahead of us, blows me off. When the driver passes through another green light, completely ignoring a "Pilani Highway Next Right," I grit my teeth. I remember an early warning relayed to me by BSD programmer Keith Bostic. "Stallman does not suffer fools gladly," Bostic warned me. "If somebody says or does something stupid, he'll look them in the eye and say, `That's stupid.'" Looking at the oblivious driver ahead of us, I realize that it's the stupidity, not the inconvenience, that's killing Stallman right now. "It's as if he picked this route with absolutely no thought on how to get there efficiently," Stallman says. The word "efficiently" hangs in the air like a bad odor. Few things irritate the hacker mind more than inefficiency. It was the inefficiency of checking the Xerox laser printer two or three times a day that triggered Stallman's initial inquiry into the printer source code. It was the inefficiency of rewriting software tools hijacked by commercial software vendors that led Stallman to battle Symbolics and to launch the GNU Project. If, as Jean Paul Sartre once opined, hell is other people, hacker hell is duplicating other people's stupid mistakes, and it's no exaggeration to say that Stallman's entire life has been an attempt to save mankind from these fiery depths. This hell metaphor becomes all the more apparent as we take in the slowly passing scenery. With its multitude of shops, parking lots, and poorly timed street lights, Kihei seems less like a city and more like a poorly designed software program writ large. Instead of rerouting traffic and distributing vehicles through side streets and expressways, city planners have elected to run everything through a single main drag. From a hacker perspective, sitting in a car amidst all this mess is like listening to a CD rendition of nails on a chalkboard at full volume. "Imperfect systems infuriate hackers," observes Steven Levy, another warning I should have listened to before climbing into the car with Stallman. "This is one reason why hackers generally hate driving cars-the system of randomly programmed red lights and oddly laid out one-way streets causes delays which are so goddamn unnecessary [Levy's emphasis] that the impulse is to rearrange signs, open up traffic-light control boxes . . . redesign the entire system."See Steven Levy, Hackers (Penguin USA [paperback], 1984): 40. More frustrating, however, is the duplicity of our trusted guide. Instead of searching out a clever shortcut-as any true hacker would do on instinct-the driver ahead of us has instead chosen to play along with the city planners' game. Like Virgil in Dante's Inferno, our guide is determined to give us the full guided tour of this hacker hell whether we want it or not. Before I can make this observation to Stallman, the driver finally hits his right turn signal. Stallman's hunched shoulders relax slightly, and for a moment the air of tension within the car dissipates. The tension comes back, however, as the driver in front of us slows down. "Construction Ahead" signs line both sides of the street, and even though the Pilani Highway lies less than a quarter mile off in the distance, the two-lane road between us and the highway is blocked by a dormant bulldozer and two large mounds of dirt. It takes Stallman a few seconds to register what's going on as our guide begins executing a clumsy five-point U-turn in front of us. When he catches a glimpse of the bulldozer and the "No Through Access" signs just beyond, Stallman finally boils over. "Why, why, why?" he whines, throwing his head back. "You should have known the road was blocked. You should have known this way wouldn't work. You did this deliberately." The driver finishes the turn and passes us on the way back toward the main drag. As he does so, he shakes his head and gives us an apologetic shrug. Coupled with a toothy grin, the driver's gesture reveals a touch of mainlander frustration but is tempered with a protective dose of islander fatalism. Coming through the sealed windows of our rental car, it spells out a succinct message: "Hey, it's Maui; what are you gonna do?" Stallman can take it no longer. "Don't you fucking smile!" he shouts, fogging up the glass as he does so. "It's your fucking fault. This all could have been so much easier if we had just done it my way." Stallman accents the words "my way" by gripping the steering wheel and pulling himself towards it twice. The image of Stallman's lurching frame is like that of a child throwing a temper tantrum in a car seat, an image further underlined by the tone of Stallman's voice. Halfway between anger and anguish, Stallman seems to be on the verge of tears. Fortunately, the tears do not arrive. Like a summer cloudburst, the tantrum ends almost as soon as it begins. After a few whiny gasps, Stallman shifts the car into reverse and begins executing his own U-turn. By the time we are back on the main drag, his face is as impassive as it was when we left the hotel 30 minutes earlier. It takes less than five minutes to reach the next cross-street. This one offers easy highway access, and within seconds, we are soon speeding off toward Pa'ia at a relaxing rate of speed. The sun that once loomed bright and yellow over Stallman's left shoulder is now burning a cool orange-red in our rearview mirror. It lends its color to the gauntlet wili wili trees flying past us on both sides of the highway. For the next 20 minutes, the only sound in our vehicle, aside from the ambient hum of the car's engine and tires, is the sound of a cello and a violin trio playing the mournful strains of an Appalachian folk tune. Endnote Continuing the Fight For Richard Stallman, time may not heal all wounds, but it does provide a convenient ally. Four years after " The Cathedral and the Bazaar," Stallman still chafes over the Raymond critique. He also grumbles over Linus Torvalds' elevation to the role of world's most famous hacker. He recalls a popular T-shirt that began showing at Linux tradeshows around 1999. Designed to mimic the original promotional poster for Star Wars, the shirt depicted Torvalds brandishing a lightsaber like Luke Skywalker, while Stallman's face rides atop R2D2. The shirt still grates on Stallmans nerves not only because it depicts him as a Torvalds' sidekick, but also because it elevates Torvalds to the leadership role in the free software/open source community, a role even Torvalds himself is loath to accept. "It's ironic," says Stallman mournfully. "Picking up that sword is exactly what Linus refuses to do. He gets everybody focusing on him as the symbol of the movement, and then he won't fight. What good is it?" Then again, it is that same unwillingness to "pick up the sword," on Torvalds part, that has left the door open for Stallman to bolster his reputation as the hacker community's ethical arbiter. Despite his grievances, Stallman has to admit that the last few years have been quite good, both to himself and to his organization. Relegated to the periphery by the unforeseen success of GNU/Linux, Stallman has nonetheless successfully recaptured the initiative. His speaking schedule between January 2000 and December 2001 included stops on six continents and visits to countries where the notion of software freedom carries heavy overtones-China and India, for example. Outside the bully pulpit, Stallman has also learned how to leverage his power as costeward of the GNU General Public License (GPL). During the summer of 2000, while the air was rapidly leaking out of the 1999 Linux IPO bubble, Stallman and the Free Software Foundation scored two major victories. In July, 2000, Troll Tech, a Norwegian software company and developer of Qt, a valuable suite of graphics tools for the GNU/Linux operating system, announced it was licensing its software under the GPL. A few weeks later, Sun Microsystems, a company that, until then, had been warily trying to ride the open source bandwagon without giving up total control of its software properties, finally relented and announced that it, too, was dual licensing its new OpenOffice application suite under the Lesser GNU Public License (LGPL) and the Sun Industry Standards Source License (SISSL). Underlining each victory was the fact that Stallman had done little to fight for them. In the case of Troll Tech, Stallman had simply played the role of free software pontiff. In 1999, the company had come up with a license that met the conditions laid out by the Free Software Foundation, but in examining the license further, Stallman detected legal incompatibles that would make it impossible to bundle Qt with GPL-protected software programs. Tired of battling Stallman, Troll Tech management finally decided to split the Qt into two versions, one GPL-protected and one QPL-protected, giving developers a way around the compatibility issues cited by Stallman. In the case of Sun, they desired to play according to the Free Software Foundation's conditions. At the 1999 O'Reilly Open Source Conference, Sun Microsystems cofounder and chief scientist Bill Joy defended his company's "community source" license, essentially a watered-down compromise letting users copy and modify Sun-owned software but not charge a fee for said software without negotiating a royalty agreement with Sun. A year after Joy's speech, Sun Microsystems vice president Marco Boerries was appearing on the same stage spelling out the company's new licensing compromise in the case of OpenOffice, an office-application suite designed specifically for the GNU/Linux operating system. "I can spell it out in three letters," said Boerries. "GPL." At the time, Boerries said his company's decision had little to do with Stallman and more to do with the momentum of GPL-protected programs. "What basically happened was the recognition that different products attracted different communities, and the license you use depends on what type of community you want to attract," said Boerries. "With [OpenOffice], it was clear we had the highest correlation with the GPL community."See Marco Boerries, interview with author (July, 2000). Such comments point out the under-recognized strength of the GPL and, indirectly, the political genius of man who played the largest role in creating it. "There isn't a lawyer on earth who would have drafted the GPL the way it is," says Eben Moglen, Columbia University law professor and Free Software Foundation general counsel. "But it works. And it works because of Richard's philosophy of design." A former professional programmer, Moglen traces his pro bono work with Stallman back to 1990 when Stallman requested Moglen's legal assistance on a private affair. Moglen, then working with encryption expert Phillip Zimmerman during Zimmerman's legal battles with the National Security Administration, says he was honored by the request. "I told him I used Emacs every day of my life, and it would take an awful lot of lawyering on my part to pay off the debt." Since then, Moglen, perhaps more than any other individual, has had the best chance to observe the crossover of Stallman's hacker philosophies into the legal realm. Moglen says the difference between Stallman's approach to legal code and software code are largely the same. "I have to say, as a lawyer, the idea that what you should do with a legal document is to take out all the bugs doesn't make much sense," Moglen says. "There is uncertainty in every legal process, and what most lawyers want to do is to capture the benefits of uncertainty for their client. Richard's goal is the complete opposite. His goal is to remove uncertainty, which is inherently impossible. It is inherently impossible to draft one license to control all circumstances in all legal systems all over the world. But if you were to go at it, you would have to go at it his way. And the resulting elegance, the resulting simplicity in design almost achieves what it has to achieve. And from there a little lawyering will carry you quite far." As the person charged with pushing the Stallman agenda, Moglen understands the frustration of would-be allies. "Richard is a man who does not want to compromise over matters that he thinks of as fundamental," Moglen says, "and he does not take easily the twisting of words or even just the seeking of artful ambiguity, which human society often requires from a lot of people." Because of the Free Software Foundation's unwillingness to weigh in on issues outside the purview of GNU development and GPL enforcement, Moglen has taken to devoting his excess energies to assisting the Electronic Frontier Foundation, the organization providing legal aid to recent copyright defendants such as Dmitri Skylarov. In 2000, Moglen also served as direct counsel to a collection of hackers that were joined together from circulating the DVD decryption program deCSS. Despite the silence of his main client in both cases, Moglen has learned to appreciate the value of Stallman's stubbornness. "There have been times over the years where I've gone to Richard and said, `We have to do this. We have to do that. Here's the strategic situation. Here's the next move. Here's what he have to do.' And Richard's response has always been, `We don't have to do anything.' Just wait. What needs doing will get done." "And you know what?" Moglen adds. "Generally, he's been right." Such comments disavow Stallman's own self-assessment: "I'm not good at playing games," Stallman says, addressing the many unseen critics who see him as a shrewd strategist. "I'm not good at looking ahead and anticipating what somebody else might do. My approach has always been to focus on the foundation, to say `Let's make the foundation as strong as we can make it.'" The GPL's expanding popularity and continuing gravitational strength are the best tributes to the foundation laid by Stallman and his GNU colleagues. While no longer capable of billing himself as the "last true hacker," Stallman nevertheless can take sole credit for building the free software movement's ethical framework. Whether or not other modern programmers feel comfortable working inside that framework is immaterial. The fact that they even have a choice at all is Stallman's greatest legacy. Discussing Stallman's legacy at this point seems a bit premature. Stallman, 48 at the time of this writing, still has a few years left to add to or subtract from that legacy. Still, the autopilot nature of the free software movement makes it tempting to examine Stallman's life outside the day-to-day battles of the software industry and within a more august, historical setting. To his credit, Stallman refuses all opportunities to speculate. "I've never been able to work out detailed plans of what the future was going to be like," says Stallman, offering his own premature epitaph. "I just said `I'm going to fight. Who knows where I'll get?'" There's no question that in picking his fights, Stallman has alienated the very people who might otherwise have been his greatest champions. It is also a testament to his forthright, ethical nature that many of Stallman's erstwhile political opponents still manage to put in a few good words for him when pressed. The tension between Stallman the ideologue and Stallman the hacker genius, however, leads a biographer to wonder: how will people view Stallman when Stallman's own personality is no longer there to get in the way? In early drafts of this book, I dubbed this question the "100 year" question. Hoping to stimulate an objective view of Stallman and his work, I asked various software-industry luminaries to take themselves out of the current timeframe and put themselves in a position of a historian looking back on the free software movement 100 years in the future. From the current vantage point, it is easy to see similarities between Stallman and past Americans who, while somewhat marginal during their lifetime, have attained heightened historical importance in relation to their age. Easy comparisons include Henry David Thoreau, transcendentalist philosopher and author of On Civil Disobedience, and John Muir, founder of the Sierra Club and progenitor of the modern environmental movement. It is also easy to see similarities in men like William Jennings Bryan, a.k.a. "The Great Commoner," leader of the populist movement, enemy of monopolies, and a man who, though powerful, seems to have faded into historical insignificance. Although not the first person to view software as public property, Stallman is guaranteed a footnote in future history books thanks to the GPL. Given that fact, it seems worthwhile to step back and examine Richard Stallman's legacy outside the current time frame. Will the GPL still be something software programmers use in the year 2102, or will it have long since fallen by the wayside? Will the term "free software" seem as politically quaint as "free silver" does today, or will it seem eerily prescient in light of later political events? Predicting the future is risky sport, but most people, when presented with the question, seemed eager to bite. "One hundred years from now, Richard and a couple of other people are going to deserve more than a footnote," says Moglen. "They're going to be viewed as the main line of the story." The "couple other people" Moglen nominates for future textbook chapters include John Gilmore, Stallman's GPL advisor and future founder of the Electronic Frontier Foundation, and Theodor Holm Nelson, a.k.a. Ted Nelson, author of the 1982 book, Literary Machines . Moglen says Stallman, Nelson, and Gilmore each stand out in historically significant, nonoverlapping ways. He credits Nelson, commonly considered to have coined the term "hypertext," for identifying the predicament of information ownership in the digital age. Gilmore and Stallman, meanwhile, earn notable credit for identifying the negative political effects of information control and building organizations-the Electronic Frontier Foundation in the case of Gilmore and the Free Software Foundation in the case of Stallman-to counteract those effects. Of the two, however, Moglen sees Stallman's activities as more personal and less political in nature. "Richard was unique in that the ethical implications of unfree software were particularly clear to him at an early moment," says Moglen. "This has a lot to do with Richard's personality, which lots of people will, when writing about him, try to depict as epiphenomenal or even a drawback in Richard Stallman's own life work." Gilmore, who describes his inclusion between the erratic Nelson and the irascible Stallman as something of a "mixed honor," nevertheless seconds the Moglen argument. Writes Gilmore: My guess is that Stallman's writings will stand up as well as Thomas Jefferson's have; he's a pretty clear writer and also clear on his principles . . . Whether Richard will be as influential as Jefferson will depend on whether the abstractions we call "civil rights" end up more important a hundred years from now than the abstractions that we call "software" or "technically imposed restrictions." Another element of the Stallman legacy not to be overlooked, Gilmore writes, is the collaborative software-development model pioneered by the GNU Project. Although flawed at times, the model has nevertheless evolved into a standard within the software-development industry. All told, Gilmore says, this collaborative software-development model may end up being even more influential than the GNU Project, the GPL License, or any particular software program developed by Stallman: Before the Internet, it was quite hard to collaborate over distance on software, even among teams that know and trust each other. Richard pioneered collaborative development of software, particularly by disorganized volunteers who seldom meet each other. Richard didn't build any of the basic tools for doing this (the TCP protocol, email lists, diff and patch, tar files, RCS or CVS or remote-CVS), but he used the ones that were available to form social groups of programmers who could effectively collaborate. Lawrence Lessig, Stanford law professor and author of the 2001 book, The Future of Ideas , is similarly bullish. Like many legal scholars, Lessig sees the GPL as a major bulwark of the current so-called "digital commons," the vast agglomeration of community-owned software programs, network and telecommunication standards that have triggered the Internet's exponential growth over the last three decades. Rather than connect Stallman with other Internet pioneers, men such as Vannevar Bush, Vinton Cerf, and J. C. R. Licklider who convinced others to see computer technology on a wider scale, Lessig sees Stallman's impact as more personal, introspective, and, ultimately, unique: [Stallman] changed the debate from is to ought. He made people see how much was at stake, and he built a device to carry these ideals forward . . . That said, I don't quite know how to place him in the context of Cerf or Licklider. The innovation is different. It is not just about a certain kind of code, or enabling the Internet. [It's] much more about getting people to see the value in a certain kind of Internet. I don't think there is anyone else in that class, before or after. Not everybody sees the Stallman legacy as set in stone, of course. Eric Raymond, the open source proponent who feels that Stallman's leadership role has diminished significantly since 1996, sees mixed signals when looking into the 2102 crystal ball: I think Stallman's artifacts (GPL, Emacs, GCC) will be seen as revolutionary works, as foundation-stones of the information world. I think history will be less kind to some of the theories from which RMS operated, and not kind at all to his personal tendency towards territorial, cult-leader behavior. As for Stallman himself, he, too, sees mixed signals: What history says about the GNU Project, twenty years from now, will depend on who wins the battle of freedom to use public knowledge. If we lose, we will be just a footnote. If we win, it is uncertain whether people will know the role of the GNU operating system-if they think the system is "Linux," they will build a false picture of what happened and why. But even if we win, what history people learn a hundred years from now is likely to depend on who dominates politically. Searching for his own 19th-century historical analogy, Stallman summons the figure of John Brown, the militant abolitionist regarded as a hero on one side of the Mason Dixon line and a madman on the other. John Brown's slave revolt never got going, but during his subsequent trial he effectively roused national demand for abolition. During the Civil War, John Brown was a hero; 100 years after, and for much of the 1900s, history textbooks taught that he was crazy. During the era of legal segregation, while bigotry was shameless, the US partly accepted the story that the South wanted to tell about itself, and history textbooks said many untrue things about the Civil War and related events. Such comparisons document both the self-perceived peripheral nature of Stallman's current work and the binary nature of his current reputation. Although it's hard to see Stallman's reputation falling to the level of infamy as Brown's did during the post-Reconstruction period-Stallman, despite his occasional war-like analogies, has done little to inspire violence-it's easy to envision a future in which Stallman's ideas wind up on the ash-heap. In fashioning the free software cause not as a mass movement but as a collection of private battles against the forces of proprietary temptation, Stallman seems to have created a unwinnable situation, especially for the many acolytes with the same stubborn will. Then again, it is that very will that may someday prove to be Stallman's greatest lasting legacy. Moglen, a close observer over the last decade, warns those who mistake the Stallman personality as counter-productive or epiphenomenal to the "artifacts" of Stalllman's life. Without that personality, Moglen says, there would be precious few artifiacts to discuss. Says Moglen, a former Supreme Court clerk: Look, the greatest man I ever worked for was Thurgood Marshall. I knew what made him a great man. I knew why he had been able to change the world in his possible way. I would be going out on a limb a little bit if I were to make a comparison, because they could not be more different. Thurgood Marshall was a man in society, representing an outcast society to the society that enclosed it, but still a man in society. His skill was social skills. But he was all of a piece, too. Different as they were in every other respect, that the person I most now compare him to in that sense, all of a piece, compact, made of the substance that makes stars, all the way through, is Stallman. In an effort to drive that image home, Moglen reflects on a shared moment in the spring of 2000. The success of the VA Linux IPO was still resonating in the business media, and a half dozen free software-related issues were swimming through the news. Surrounded by a swirling hurricane of issues and stories each begging for comment, Moglen recalls sitting down for lunch with Stallman and feeling like a castaway dropped into the eye of the storm. For the next hour, he says, the conversation calmly revolved around a single topic: strengthening the GPL. "We were sitting there talking about what we were going to do about some problems in Eastern Europe and what we were going to do when the problem of the ownership of content began to threaten free software," Moglen recalls. "As we were talking, I briefly thought about how we must have looked to people passing by. Here we are, these two little bearded anarchists, plotting and planning the next steps. And, of course, Richard is plucking the knots from his hair and dropping them in the soup and behaving in his usual way. Anybody listening in on our conversation would have thought we were crazy, but I knew: I knew the revolution's right here at this table. This is what's making it happen. And this man is the person making it happen." Moglen says that moment, more than any other, drove home the elemental simplicity of the Stallman style. "It was funny," recalls Moglen. "I said to him, `Richard, you know, you and I are the two guys who didn't make any money out of this revolution.' And then I paid for the lunch, because I knew he didn't have the money to pay for it .'" Endnote Epilogue: Crushing Loneliness Writing the biography of a living person is a bit like producing a play. The drama in front of the curtain often pales in comparison to the drama backstage. In The Autobiography of Malcolm X, Alex Haley gives readers a rare glimpse of that backstage drama. Stepping out of the ghostwriter role, Haley delivers the book's epilogue in his own voice. The epilogue explains how a freelance reporter originally dismissed as a "tool" and "spy" by the Nation of Islam spokesperson managed to work through personal and political barriers to get Malcolm X's life story on paper. While I hesitate to compare this book with The Autobiography of Malcolm X, I do owe a debt of gratitude to Haley for his candid epilogue. Over the last 12 months, it has served as a sort of instruction manual on how to deal with a biographical subject who has built an entire career on being disagreeable. From the outset, I envisioned closing this biography with a similar epilogue, both as an homage to Haley and as a way to let readers know how this book came to be. The story behind this story starts in an Oakland apartment, winding its way through the various locales mentioned in the book-Silicon Valley, Maui, Boston, and Cambridge. Ultimately, however, it is a tale of two cities: New York, New York, the book-publishing capital of the world, and Sebastopol, California, the book-publishing capital of Sonoma County. The story starts in April, 2000. At the time, I was writing stories for the ill-fated BeOpen web site (http://www.beopen.com/). One of my first assignments was a phone interview with Richard M. Stallman. The interview went well, so well that Slashdot (http://www.slashdot.org/), the popular "news for nerds" site owned by VA Software, Inc. (formerly VA Linux Systems and before that, VA Research), gave it a link in its daily list of feature stories. Within hours, the web servers at BeOpen were heating up as readers clicked over to the site. For all intents and purposes, the story should have ended there. Three months after the interview, while attending the O'Reilly Open Source Conference in Monterey, California, I received the following email message from Tracy Pattison, foreign-rights manager at a large New York publishing house: To: sam@BeOpen.com Subject: RMS InterviewDate: Mon, 10 Jul 2000 15:56:37 -0400Dear Mr. Williams, I read your interview with Richard Stallman on BeOpen with great interest. I've been intrigued by RMS and his work for some time now and was delighted to find your piece which I really think you did a great job of capturing some of the spirit of what Stallman is trying to do with GNU-Linux and the Free Software Foundation. What I'd love to do, however, is read more - and I don't think I'm alone. Do you think there is more information and/or sources out there to expand and update your interview and adapt it into more of a profile of Stallman? Perhaps including some more anecdotal information about his personality and background that might really interest and enlighten readers outside the more hardcore programming scene? The email asked that I give Tracy a call to discuss the idea further. I did just that. Tracy told me her company was launching a new electronic book line, and it wanted stories that appealed to an early-adopter audience. The e-book format was 30,000 words, about 100 pages, and she had pitched her bosses on the idea of profiling a major figure in the hacker community. Her bosses liked the idea, and in the process of searching for interesting people to profile, she had come across my BeOpen interview with Stallman. Hence her email to me. That's when Tracy asked me: would I be willing to expand the interview into a full-length feature profile? My answer was instant: yes. Before accepting it, Tracy suggested I put together a story proposal she could show her superiors. Two days later, I sent her a polished proposal. A week later, Tracy sent me a follow up email. Her bosses had given it the green light. I have to admit, getting Stallman to participate in an e-book project was an afterthought on my part. As a reporter who covered the open source beat, I knew Stallman was a stickler. I'd already received a half dozen emails at that point upbraiding me for the use of "Linux" instead of "GNU/Linux." Then again, I also knew Stallman was looking for ways to get his message out to the general public. Perhaps if I presented the project to him that way, he would be more receptive. If not, I could always rely upon the copious amounts of documents, interviews, and recorded online conversations Stallman had left lying around the Internet and do an unauthorized biography. During my research, I came across an essay titled "Freedom-Or Copyright?" Written by Stallman and published in the June, 2000, edition of the MIT Technology Review, the essay blasted e-books for an assortment of software sins. Not only did readers have to use proprietary software programs to read them, Stallman lamented, but the methods used to prevent unauthorized copying were overly harsh. Instead of downloading a transferable HTML or PDF file, readers downloaded an encrypted file. In essence, purchasing an e-book meant purchasing a nontransferable key to unscramble the encrypted content. Any attempt to open a book's content without an authorized key constituted a criminal violation of the Digital Millennium Copyright Act, the 1998 law designed to bolster copyright enforcement on the Internet. Similar penalties held for readers who converted a book's content into an open file format, even if their only intention was to read the book on a different computer in their home. Unlike a normal book, the reader no longer held the right to lend, copy, or resell an e-book. They only had the right to read it on an authorized machine, warned Stallman: We still have the same old freedoms in using paper books. But if e-books replace printed books, that exception will do little good. With "electronic ink," which makes it possible to download new text onto an apparently printed piece of paper, even newspapers could become ephemeral. Imagine: no more used book stores; no more lending a book to your friend; no more borrowing one from the public library-no more "leaks" that might give someone a chance to read without paying. (And judging from the ads for Microsoft Reader, no more anonymous purchasing of books either.) This is the world publishers have in mind for us.See "Safari Tech Books Online; Subscriber Agreement: Terms of Service." http://safari.oreilly.com/mainhlp.asp?help=service Needless to say, the essay caused some concern. Neither Tracy nor I had discussed the software her company would use nor had we discussed the type of copyright that would govern the e-book's usage. I mentioned the Technology Review article and asked if she could give me information on her company's e-book policies. Tracy promised to get back to me. Eager to get started, I decided to call Stallman anyway and mention the book idea to him. When I did, he expressed immediate interest and immediate concern. "Did you read my essay on e-books?" he asked. When I told him, yes, I had read the essay and was waiting to hear back from the publisher, Stallman laid out two conditions: he didn't want to lend support to an e-book licensing mechanism he fundamentally opposed, and he didn't want to come off as lending support. "I don't want to participate in anything that makes me look like a hypocrite," he said. For Stallman, the software issue was secondary to the copyright issue. He said he was willing to ignore whatever software the publisher or its third-party vendors employed just so long as the company specified within the copyright that readers were free to make and distribute verbatim copies of the e-book's content. Stallman pointed to Stephen King's The Plant as a possible model. In June, 2000, King announced on his official web site that he was self-publishing The Plant in serial form. According to the announcement, the book's total cost would be $13, spread out over a series of $1 installments. As long as at least 75% of the readers paid for each chapter, King promised to continue releasing new installments. By August, the plan seemed to be working, as King had published the first two chapters with a third on the way. "I'd be willing to accept something like that," Stallman said. "As long as it also permitted verbatim copying." I forwarded the information to Tracy. Feeling confident that she and I might be able to work out an equitable arrangement, I called up Stallman and set up the first interview for the book. Stallman agreed to the interview without making a second inquiry into the status issue. Shortly after the first interview, I raced to set up a second interview (this one in Kihei), squeezing it in before Stallman headed off on a 14-day vacation to Tahiti. It was during Stallman's vacation that the bad news came from Tracy. Her company's legal department didn't want to adjust its copyright notice on the e-books. Readers who wanted to make their books transferable would either have to crack the encryption code or convert the book to an open format such as HTML. Either way, the would be breaking the law and facing criminal penalties. With two fresh interviews under my belt, I didn't see any way to write the book without resorting to the new material. I quickly set up a trip to New York to meet with my agent and with Tracy to see if there was a compromise solution. When I flew to New York, I met my agent, Henning Guttman. It was our first face-to-face meeting, and Henning seemed pessimistic about our chances of forcing a compromise, at least on the publisher's end. The large, established publishing houses already viewed the e-book format with enough suspicion and weren't in the mood to experiment with copyright language that made it easier for readers to avoid payment. As an agent who specialized in technology books, however, Henning was intrigued by the novel nature of my predicament. I told him about the two interviews I'd already gathered and the promise not to publish the book in a way that made Stallman "look like a hypocrite." Agreeing that I was in an ethical bind, Henning suggested we make that our negotiating point. Barring that, Henning said, we could always take the carrot-and-stick approach. The carrot would be the publicity that came with publishing an e-book that honored the hacker community's internal ethics. The stick would be the risks associated with publishing an e-book that didn't. Nine months before Dmitri Skylarov became an Internet cause celebre, we knew it was only a matter of time before an enterprising programmer revealed how to hack e-books. We also knew that a major publishing house releasing an encryption-protected e-book on Richard M. Stallman was the software equivalent of putting "Steal This E-Book" on the cover. After my meeting with Henning, I put a call into Stallman. Hoping to make the carrot more enticing, I discussed a number of potential compromises. What if the publisher released the book's content under a split license, something similar to what Sun Microsystems had done with Open Office, the free software desktop applications suite? The publisher could then release commercial versions of the e-book under a normal format, taking advantage of all the bells and whistles that went with the e-book software, while releasing the copyable version under a less aesthetically pleasing HTML format. Stallman told me he didn't mind the split-license idea, but he did dislike the idea of making the freely copyable version inferior to the restricted version. Besides, he said, the idea was too cumbersome. Split licenses worked in the case of Sun's Open Office only because he had no control over the decision making. In this case, Stallman said, he did have a way to control the outcome. He could refuse to cooperate. I made a few more suggestions with little effect. About the only thing I could get out of Stallman was a concession that the e-book's copyright restrict all forms of file sharing to "noncommercial redistribution." Before I signed off, Stallman suggested I tell the publisher that I'd promised Stallman that the work would be free. I told Stallman I couldn't agree to that statement but that I did view the book as unfinishable without his cooperation. Seemingly satisfied, Stallman hung up with his usual sign-off line: "Happy hacking." Henning and I met with Tracy the next day. Tracy said her company was willing to publish copyable excerpts in a unencrypted format but would limit the excerpts to 500 words. Henning informed her that this wouldn't be enough for me to get around my ethical obligation to Stallman. Tracy mentioned her own company's contractual obligation to online vendors such as Amazon.com. Even if the company decided to open up its e-book content this one time, it faced the risk of its partners calling it a breach of contract. Barring a change of heart in the executive suite or on the part of Stallman, the decision was up to me. I could use the interviews and go against my earlier agreement with Stallman, or I could plead journalistic ethics and back out of the verbal agreement to do the book. Following the meeting, my agent and I relocated to a pub on Third Ave. I used his cell phone to call Stallman, leaving a message when nobody answered. Henning left for a moment, giving me time to collect my thoughts. When he returned, he was holding up the cell phone. "It's Stallman," Henning said. The conversation got off badly from the start. I relayed Tracy's comment about the publisher's contractual obligations. "So," Stallman said bluntly. "Why should I give a damn about their contractual obligations?" Because asking a major publishing house to risk a legal battle with its vendors over a 30,000 word e-book is a tall order, I suggested. "Don't you see?" Stallman said. "That's exactly why I'm doing this. I want a signal victory. I want them to make a choice between freedom and business as usual." As the words "signal victory" echoed in my head, I felt my attention wander momentarily to the passing foot traffic on the sidewalk. Coming into the bar, I had been pleased to notice that the location was less than half a block away from the street corner memorialized in the 1976 Ramones song, "53rd and 3rd," a song I always enjoyed playing in my days as a musician. Like the perpetually frustrated street hustler depicted in that song, I could feel things falling apart as quickly as they had come together. The irony was palpable. After weeks of gleefully recording other people's laments, I found myself in the position of trying to pull off the rarest of feats: a Richard Stallman compromise. When I continued hemming and hawing, pleading the publisher's position and revealing my growing sympathy for it, Stallman, like an animal smelling blood, attacked. "So that's it? You're just going to screw me? You're just going to bend to their will?" I brought up the issue of a dual-copyright again. "You mean license," Stallman said curtly. "Yeah, license. Copyright. Whatever," I said, feeling suddenly like a wounded tuna trailing a rich plume of plasma in the water. "Aw, why didn't you just fucking do what I told you to do!" he shouted. I must have been arguing on behalf of the publisher to the very end, because in my notes I managed to save a final Stallman chestnut: "I don't care. What they're doing is evil. I can't support evil. Good-bye." As soon as I put the phone down, my agent slid a freshly poured Guinness to me. "I figured you might need this," he said with a laugh. "I could see you shaking there towards the end." I was indeed shaking. The shaking wouldn't stop until the Guinness was more than halfway gone. It felt weird, hearing myself characterized as an emissary of "evil." It felt weirder still, knowing that three months before, I was sitting in an Oakland apartment trying to come up with my next story idea. Now, I was sitting in a part of the world I'd only known through rock songs, taking meetings with publishing executives and drinking beer with an agent I'd never even laid eyes on until the day before. It was all too surreal, like watching my life reflected back as a movie montage. About that time, my internal absurdity meter kicked in. The initial shaking gave way to convulsions of laughter. To my agent, I must have looked like a another fragile author undergoing an untimely emotional breakdown. To me, I was just starting to appreciate the cynical beauty of my situation. Deal or no deal, I already had the makings of a pretty good story. It was only a matter of finding a place to tell it. When my laughing convulsions finally subsided, I held up my drink in a toast. "Welcome to the front lines, my friend," I said, clinking pints with my agent. "Might as well enjoy it." If this story really were a play, here's where it would take a momentary, romantic interlude. Disheartened by the tense nature of our meeting, Tracy invited Henning and I to go out for drinks with her and some of her coworkers. We left the bar on Third Ave., headed down to the East Village, and caught up with Tracy and her friends. Once there, I spoke with Tracy, careful to avoid shop talk. Our conversation was pleasant, relaxed. Before parting, we agreed to meet the next night. Once again, the conversation was pleasant, so pleasant that the Stallman e-book became almost a distant memory. When I got back to Oakland, I called around to various journalist friends and acquaintances. I recounted my predicament. Most upbraided me for giving up too much ground to Stallman in the preinterview negotiation. A former j-school professor suggested I ignore Stallman's "hypocrite" comment and just write the story. Reporters who knew of Stallman's media-savviness expressed sympathy but uniformly offered the same response: it's your call. I decided to put the book on the back burner. Even with the interviews, I wasn't making much progress. Besides, it gave me a chance to speak with Tracy without running things past Henning first. By Christmas we had traded visits: she flying out to the west coast once, me flying out to New York a second time. The day before New Year's Eve, I proposed. Deciding which coast to live on, I picked New York. By February, I packed up my laptop computer and all my research notes related to the Stallman biography, and we winged our way to JFK Airport. Tracy and I were married on May 11. So much for failed book deals. During the summer, I began to contemplate turning my interview notes into a magazine article. Ethically, I felt in the clear doing so, since the original interview terms said nothing about traditional print media. To be honest, I also felt a bit more comfortable writing about Stallman after eight months of radio silence. Since our telephone conversation in September, I'd only received two emails from Stallman. Both chastised me for using "Linux" instead of "GNU/Linux" in a pair of articles for the web magazine Upside Today. Aside from that, I had enjoyed the silence. In June, about a week after the New York University speech, I took a crack at writing a 5,000-word magazine-length story about Stallman. This time, the words flowed. The distance had helped restore my lost sense of emotional perspective, I suppose. In July, a full year after the original email from Tracy, I got a call from Henning. He told me that O'Reilly & Associates, a publishing house out of Sebastopol, California, was interested in the running the Stallman story as a biography. The news pleased me. Of all the publishing houses in the world, O'Reilly, the same company that had published Eric Raymond's The Cathedral and the Bazaar, seemed the most sensitive to the issues that had killed the earlier e-book. As a reporter, I had relied heavily on the O'Reilly book Open Sources as a historical reference. I also knew that various chapters of the book, including a chapter written by Stallman, had been published with copyright notices that permitted redistribution. Such knowledge would come in handy if the issue of electronic publication ever came up again. Sure enough, the issue did come up. I learned through Henning that O'Reilly intended to publish the biography both as a book and as part of its new Safari Tech Books Online subscription service. The Safari user license would involve special restrictions,1 Henning warned, but O'Reilly was willing to allow for a copyright that permitted users to copy and share and the book's text regardless of medium. Basically, as author, I had the choice between two licenses: the Open Publication License or the GNU Free Documentation License. I checked out the contents and background of each license. The Open Publication License (OPL)See "The Open Publication License: Draft v1.0" (June 8, 1999). http://opencontent.org/openpub/ gives readers the right to reproduce and distribute a work, in whole or in part, in any medium "physical or electronic," provided the copied work retains the Open Publication License. It also permits modification of a work, provided certain conditions are met. Finally, the Open Publication License includes a number of options, which, if selected by the author, can limit the creation of "substantively modified" versions or book-form derivatives without prior author approval. The GNU Free Documentation License (GFDL),See "The GNU Free Documentation License: Version 1.1" (March, 2000). http://www.gnu.org/copyleft/fdl.html meanwhile, permits the copying and distribution of a document in any medium, provided the resulting work carries the same license. It also permits the modification of a document provided certain conditions. Unlike the OPL, however, it does not give authors the option to restrict certain modifications. It also does not give authors the right to reject modifications that might result in a competitive book product. It does require certain forms of front- and back-cover information if a party other than the copyright holder wishes to publish more than 100 copies of a protected work, however. In the course of researching the licenses, I also made sure to visit the GNU Project web page titled "Various Licenses and Comments About Them."See http://www.gnu.org/philosophy/license-list.html On that page, I found a Stallman critique of the Open Publication License. Stallman's critique related to the creation of modified works and the ability of an author to select either one of the OPL's options to restrict modification. If an author didn't want to select either option, it was better to use the GFDL instead, Stallman noted, since it minimized the risk of the nonselected options popping up in modified versions of a document. The importance of modification in both licenses was a reflection of their original purpose-namely, to give software-manual owners a chance to improve their manuals and publicize those improvements to the rest of the community. Since my book wasn't a manual, I had little concern about the modification clause in either license. My only concern was giving users the freedom to exchange copies of the book or make copies of the content, the same freedom they would have enjoyed if they purchased a hardcover book. Deeming either license suitable for this purpose, I signed the O'Reilly contract when it came to me. Still, the notion of unrestricted modification intrigued me. In my early negotiations with Tracy, I had pitched the merits of a GPL-style license for the e-book's content. At worst, I said, the license would guarantee a lot of positive publicity for the e-book. At best, it would encourage readers to participate in the book-writing process. As an author, I was willing to let other people amend my work just so long as my name always got top billing. Besides, it might even be interesting to watch the book evolve. I pictured later editions looking much like online versions of the Talmud, my original text in a central column surrounded by illuminating, third-party commentary in the margins. My idea drew inspiration from Project Xanadu (http://www.xanadu.com/), the legendary software concept originally conceived by Ted Nelson in 1960. During the O'Reilly Open Source Conference in 1999, I had seen the first demonstration of the project's open source offshoot Udanax and had been wowed by the result. In one demonstration sequence, Udanax displayed a parent document and a derivative work in a similar two-column, plain-text format. With a click of the button, the program introduced lines linking each sentence in the parent to its conceptual offshoot in the derivative. An e-book biography of Richard M. Stallman didn't have to be Udanax-enabled, but given such technological possibilities, why not give users a chance to play around?Anybody willing to "port" this book over to Udanax, the free software version of Xanadu, will receive enthusiastic support from me. To find out more about this intriguing technology, visit http://www.udanax.com/. When Laurie Petrycki, my editor at O'Reilly, gave me a choice between the OPL or the GFDL, I indulged the fantasy once again. By September of 2001, the month I signed the contract, e-books had become almost a dead topic. Many publishing houses, Tracy's included, were shutting down their e-book imprints for lack of interest. I had to wonder. If these companies had treated e-books not as a form of publication but as a form of community building, would those imprints have survived? After I signed the contract, I notified Stallman that the book project was back on. I mentioned the choice O'Reilly was giving me between the Open Publication License and the GNU Free Documentation License. I told him I was leaning toward the OPL, if only for the fact I saw no reason to give O'Reilly's competitors a chance to print the same book under a different cover. Stallman wrote back, arguing in favor of the GFDL, noting that O'Reilly had already used it several times in the past. Despite the events of the past year, I suggested a deal. I would choose the GFDL if it gave me the possibility to do more interviews and if Stallman agreed to help O'Reilly publicize the book. Stallman agreed to participate in more interviews but said that his participation in publicity-related events would depend on the content of the book. Viewing this as only fair, I set up an interview for December 17, 2001 in Cambridge. I set up the interview to coincide with a business trip my wife Tracy was taking to Boston. Two days before leaving, Tracy suggested I invite Stallman out to dinner. "After all," she said, "he is the one who brought us together." I sent an email to Stallman, who promptly sent a return email accepting the offer. When I drove up to Boston the next day, I met Tracy at her hotel and hopped the T to head over to MIT. When we got to Tech Square, I found Stallman in the middle of a conversation just as we knocked on the door. "I hope you don't mind," he said, pulling the door open far enough so that Tracy and I could just barely hear Stallman's conversational counterpart. It was a youngish woman, mid-20s I'd say, named Sarah. "I took the liberty of inviting somebody else to have dinner with us," Stallman said, matter-of-factly, giving me the same cat-like smile he gave me back in that Palo Alto restaurant. To be honest, I wasn't too surprised. The news that Stallman had a new female friend had reached me a few weeks before, courtesy of Stallman's mother. "In fact, they both went to Japan last month when Richard went over to accept the Takeda Award," Lippman told me at the time.Alas, I didn't find out about the Takeda Foundation's decision to award Stallman, along with Linus Torvalds and Ken Sakamura, with its first-ever award for "Techno-Entrepreneurial Achievement for Social/Economic Well-Being" until after Stallman had made the trip to Japan to accept the award. For more information about the award and its accompanying $1 million prize, visit the Takeda site, http://www.takeda-foundation.jp/. On the way over to the restaurant, I learned the circumstances of Sarah and Richard's first meeting. Interestingly, the circumstances were very familiar. Working on her own fictional book, Sarah said she heard about Stallman and what an interesting character he was. She promptly decided to create a character in her book on Stallman and, in the interests of researching the character, set up an interview with Stallman. Things quickly went from there. The two had been dating since the beginning of 2001, she said. "I really admired the way Richard built up an entire political movement to address an issue of profound personal concern," Sarah said, explaining her attraction to Stallman. My wife immediately threw back the question: "What was the issue?" "Crushing loneliness." During dinner, I let the women do the talking and spent most of the time trying to detect clues as to whether the last 12 months had softened Stallman in any significant way. I didn't see anything to suggest they had. Although more flirtatious than I remembered-a flirtatiousness spoiled somewhat by the number of times Stallman's eyes seemed to fixate on my wife's chest-Stallman retained the same general level of prickliness. At one point, my wife uttered an emphatic "God forbid" only to receive a typical Stallman rebuke. "I hate to break it to you, but there is no God," Stallman said. Afterwards, when the dinner was complete and Sarah had departed, Stallman seemed to let his guard down a little. As we walked to a nearby bookstore, he admitted that the last 12 months had dramatically changed his outlook on life. "I thought I was going to be alone forever," he said. "I'm glad I was wrong." Before parting, Stallman handed me his "pleasure card," a business card listing Stallman's address, phone number, and favorite pastimes ("sharing good books, good food and exotic music and dance") so that I might set up a final interview. <Graphic file:/home/craigm/books/free_ep10.png> Stallman's "pleasure" card, handed to me the night of our dinner. The next day, over another meal of dim sum, Stallman seemed even more lovestruck than the night before. Recalling his debates with Currier House dorm maters over the benefits and drawbacks of an immortality serum, Stallman expressed hope that scientists might some day come up with the key to immortality. "Now that I'm finally starting to have happiness in my life, I want to have more," he said. When I mentioned Sarah's "crushing loneliness" comment, Stallman failed to see a connection between loneliness on a physical or spiritual level and loneliness on a hacker level. "The impulse to share code is about friendship but friendship at a much lower level," he said. Later, however, when the subject came up again, Stallman did admit that loneliness, or the fear of perpetual loneliness, had played a major role in fueling his determination during the earliest days of the GNU Project. "My fascination with computers was not a consequence of anything else," he said. "I wouldn't have been less fascinated with computers if I had been popular and all the women flocked to me. However, it's certainly true the experience of feeling I didn't have a home, finding one and losing it, finding another and having it destroyed, affected me deeply. The one I lost was the dorm. The one that was destroyed was the AI Lab. The precariousness of not having any kind of home or community was very powerful. It made me want to fight to get it back." After the interview, I couldn't help but feel a certain sense of emotional symmetry. Hearing Sarah describe what attracted her to Stallman and hearing Stallman himself describe the emotions that prompted him to take up the free software cause, I was reminded of my own reasons for writing this book. Since July, 2000, I have learned to appreciate both the seductive and the repellent sides of the Richard Stallman persona. Like Eben Moglen before me, I feel that dismissing that persona as epiphenomenal or distracting in relation to the overall free software movement would be a grievous mistake. In many ways the two are so mutually defining as to be indistinguishable. While I'm sure not every reader feels the same level of affinity for Stallman-indeed, after reading this book, some might feel zero affinity-I'm sure most will agree. Few individuals offer as singular a human portrait as Richard M. Stallman. It is my sincere hope that, with this initial portrait complete and with the help of the GFDL, others will feel a similar urge to add their own perspective to that portrait. Appendix A : Terminology For the most part, I have chosen to use the term GNU/Linux in reference to the free software operating system and Linux when referring specifically to the kernel that drives the operating system. The most notable exception to this rule comes in Chapter 9 . In the final part of that chapter, I describe the early evolution of Linux as an offshoot of Minix. It is safe to say that during the first two years of the project's development, the operating system Torvalds and his colleagues were working on bore little similarity to the GNU system envisioned by Stallman, even though it gradually began to share key components, such as the GNU C Compiler and the GNU Debugger. This decision further benefits from the fact that, prior to 1993, Stallman saw little need to insist on credit. Some might view the decision to use GNU/Linux for later versions of the same operating system as arbitrary. I would like to point out that it was in no way a prerequisite for gaining Stallman's cooperation in the making of this book. I came to it of my own accord, partly because of the operating system's modular nature and the community surrounding it, and partly because of the apolitical nature of the Linux name. Given that this is a biography of Richard Stallman, it seemed inappropriate to define the operating system in apolitical terms. In the final phases of the book, when it became clear that O'Reilly & Associates would be the book's publisher, Stallman did make it a condition that I use "GNU/Linux" instead of Linux if O'Reilly expected him to provide promotional support for the book after publication. When informed of this, I relayed my earlier decision and left it up to Stallman to judge whether the resulting book met this condition or not. At the time of this writing, I have no idea what Stallman's judgment will be. A similar situation surrounds the terms "free software" and "open source." Again, I have opted for the more politically laden "free software" term when describing software programs that come with freely copyable and freely modifiable source code. Although more popular, I have chosen to use the term "open source" only when referring to groups and businesses that have championed its usage. But for a few instances, the terms are completely interchangeable, and in making this decision I have followed the advice of Christine Peterson, the person generally credited with coining the term. "The `free software' term should still be used in circumstances where it works better," Peterson writes. "[`Open source'] caught on mainly because a new term was greatly needed, not because it's ideal." Appendix B Hack, Hackers, and Hacking To understand the full meaning of the word " hacker," it helps to examine the word's etymology over the years. The New Hacker Dictionary , an online compendium of software-programmer jargon, officially lists nine different connotations of the word "hack" and a similar number for "hacker." Then again, the same publication also includes an accompanying essay that quotes Phil Agre, an MIT hacker who warns readers not to be fooled by the word's perceived flexibility. "Hack has only one meaning," argues Agre. "An extremely subtle and profound one which defies articulation." Regardless of the width or narrowness of the definition, most modern hackers trace the word back to MIT, where the term bubbled up as popular item of student jargon in the early 1950s. In 1990 the MIT Museum put together a journal documenting the hacking phenomenon. According to the journal, students who attended the institute during the fifties used the word "hack" the way a modern student might use the word "goof." Hanging a jalopy out a dormitory window was a "hack," but anything harsh or malicious-e.g., egging a rival dorm's windows or defacing a campus statue-fell outside the bounds. Implicit within the definition of "hack" was a spirit of harmless, creative fun. This spirit would inspire the word's gerund form: "hacking." A 1950s student who spent the better part of the afternoon talking on the phone or dismantling a radio might describe the activity as "hacking." Again, a modern speaker would substitute the verb form of "goof"-"goofing" or "goofing off"-to describe the same activity. As the 1950s progressed, the word "hack" acquired a sharper, more rebellious edge. The MIT of the 1950s was overly competitive, and hacking emerged as both a reaction to and extension of that competitive culture. Goofs and pranks suddenly became a way to blow off steam, thumb one's nose at campus administration, and indulge creative thinking and behavior stifled by the Institute's rigorous undergraduate curriculum. With its myriad hallways and underground steam tunnels, the Institute offered plenty of exploration opportunities for the student undaunted by locked doors and "No Trespassing" signs. Students began to refer to their off-limits explorations as "tunnel hacking." Above ground, the campus phone system offered similar opportunities. Through casual experimentation and due diligence, students learned how to perform humorous tricks. Drawing inspiration from the more traditional pursuit of tunnel hacking, students quickly dubbed this new activity "phone hacking." The combined emphasis on creative play and restriction-free exploration would serve as the basis for the future mutations of the hacking term. The first self-described computer hackers of the 1960s MIT campus originated from a late 1950s student group called the Tech Model Railroad Club. A tight clique within the club was the Signals and Power (S&P) Committee-the group behind the railroad club's electrical circuitry system. The system was a sophisticated assortment of relays and switches similar to the kind that controlled the local campus phone system. To control it, a member of the group simply dialed in commands via a connected phone and watched the trains do his bidding. The nascent electrical engineers responsible for building and maintaining this system saw their activity as similar in spirit to phone hacking. Adopting the hacking term, they began refining it even further. From the S&P hacker point of view, using one less relay to operate a particular stretch of track meant having one more relay for future play. Hacking subtly shifted from a synonym for idle play to a synonym for idle play that improved the overall performance or efficiency of the club's railroad system at the same time. Soon S&P committee members proudly referred to the entire activity of improving and reshaping the track's underlying circuitry as "hacking" and to the people who did it as "hackers." Given their affinity for sophisticated electronics-not to mention the traditional MIT-student disregard for closed doors and "No Trespassing" signs-it didn't take long before the hackers caught wind of a new machine on campus. Dubbed the TX-0, the machine was one of the first commercially marketed computers. By the end of the 1950s, the entire S&P clique had migrated en masse over to the TX-0 control room, bringing the spirit of creative play with them. The wide-open realm of computer programming would encourage yet another mutation in etymology. "To hack" no longer meant soldering unusual looking circuits, but cobbling together software programs with little regard to "official" methods or software-writing procedures. It also meant improving the efficiency and speed of already-existing programs that tended to hog up machine resources. True to the word's roots, it also meant writing programs that served no other purpose than to amuse or entertain. A classic example of this expanded hacking definition is the game Spacewar, the first interactive video game. Developed by MIT hackers in the early 1960s, Spacewar had all the traditional hacking definitions: it was goofy and random, serving little useful purpose other than providing a nightly distraction for the dozen or so hackers who delighted in playing it. From a software perspective, however, it was a monumental testament to innovation of programming skill. It was also completely free. Because hackers had built it for fun, they saw no reason to guard their creation, sharing it extensively with other programmers. By the end of the 1960s, Spacewar had become a favorite diversion for mainframe programmers around the world. This notion of collective innovation and communal software ownership distanced the act of computer hacking in the 1960s from the tunnel hacking and phone hacking of the 1950s. The latter pursuits tended to be solo or small-group activities. Tunnel and phone hackers relied heavily on campus lore, but the off-limits nature of their activity discouraged the open circulation of new discoveries. Computer hackers, on the other hand, did their work amid a scientific field biased toward collaboration and the rewarding of innovation. Hackers and "official" computer scientists weren't always the best of allies, but in the rapid evolution of the field, the two species of computer programmer evolved a cooperative-some might say symbiotic-relationship. It is a testament to the original computer hackers' prodigious skill that later programmers, including Richard M. Stallman, aspired to wear the same hacker mantle. By the mid to late 1970s, the term "hacker" had acquired elite connotations. In a general sense, a computer hacker was any person who wrote software code for the sake of writing software code. In the particular sense, however, it was a testament to programming skill. Like the term "artist," the meaning carried tribal overtones. To describe a fellow programmer as hacker was a sign of respect. To describe oneself as a hacker was a sign of immense personal confidence. Either way, the original looseness of the computer-hacker appellation diminished as computers became more common. As the definition tightened, "computer" hacking acquired additional semantic overtones. To be a hacker, a person had to do more than write interesting software; a person had to belong to the hacker "culture" and honor its traditions the same way a medieval wine maker might pledge membership to a vintners' guild. The social structure wasn't as rigidly outlined as that of a guild, but hackers at elite institutions such as MIT, Stanford, and Carnegie Mellon began to speak openly of a "hacker ethic": the yet-unwritten rules that governed a hacker's day-to-day behavior. In the 1984 book Hackers, author Steven Levy, after much research and consultation, codified the hacker ethic as five core hacker tenets. In many ways, the core tenets listed by Levy continue to define the culture of computer hacking. Still, the guild-like image of the hacker community was undermined by the overwhelmingly populist bias of the software industry. By the early 1980s, computers were popping up everywhere, and programmers who once would have had to travel to top-rank institutions or businesses just to gain access to a machine suddenly had the ability to rub elbows with major-league hackers via the ARPAnet. The more these programmers rubbed elbows, the more they began to appropriate the anarchic philosophies of the hacker culture in places like MIT. Lost within the cultural transfer, however, was the native MIT cultural taboo against malicious behavior. As younger programmers began employing their computer skills to harmful ends-creating and disseminating computer viruses, breaking into military computer systems, deliberately causing machines such as MIT Oz, a popular ARPAnet gateway, to crash-the term "hacker" acquired a punk, nihilistic edge. When police and businesses began tracing computer-related crimes back to a few renegade programmers who cited convenient portions of the hacking ethic in defense of their activities, the word "hacker" began appearing in newspapers and magazine stories in a negative light. Although books like Hackers did much to document the original spirit of exploration that gave rise to the hacking culture, for most news reporters, "computer hacker" became a synonym for "electronic burglar." Although hackers have railed against this perceived misusage for nearly two decades, the term's rebellious connotations dating back to the 1950s make it hard to discern the 15-year-old writing software programs that circumvent modern encryption programs from the 1960s college student, picking locks and battering down doors to gain access to the lone, office computer terminal. One person's creative subversion of authority is another person's security headache, after all. Even so, the central taboo against malicious or deliberately harmful behavior remains strong enough that most hackers prefer to use the term " cracker"-i.e., a person who deliberately cracks a computer security system to steal or vandalize data-to describe the subset of hackers who apply their computing skills maliciously. This central taboo against maliciousness remains the primary cultural link between the notion of hacking in the early 21st century and hacking in the 1950s. It is important to note that, as the idea of computer hacking has evolved over the last four decades, the original notion of hacking-i.e., performing pranks or exploring underground tunnels-remains intact. In the fall of 2000, the MIT Museum paid tradition to the Institute's age-old hacking tradition with a dedicated exhibit, the Hall of Hacks. The exhibit includes a number of photographs dating back to the 1920s, including one involving a mock police cruiser. In 1993, students paid homage to the original MIT notion of hacking by placing the same police cruiser, lights flashing, atop the Institute's main dome. The cruiser's vanity license plate read IHTFP, a popular MIT acronym with many meanings. The most noteworthy version, itself dating back to the pressure-filled world of MIT student life in the 1950s, is "I hate this fucking place." In 1990, however, the Museum used the acronym as a basis for a journal on the history of hacks. Titled, The Institute for Hacks Tomfoolery and Pranks, the journal offers an adept summary of the hacking. "In the culture of hacking, an elegant, simple creation is as highly valued as it is in pure science," writes Boston Globe reporter Randolph Ryan in a 1993 article attached to the police car exhibit. "A Hack differs from the ordinary college prank in that the event usually requires careful planning, engineering and finesse, and has an underlying wit and inventiveness," Ryan writes. "The unwritten rule holds that a hack should be good-natured, non-destructive and safe. In fact, hackers sometimes assist in dismantling their own handiwork." The urge to confine the culture of computer hacking within the same ethical boundaries is well-meaning but impossible. Although most software hacks aspire to the same spirit of elegance and simplicity, the software medium offers less chance for reversibility. Dismantling a police cruiser is easy compared with dismantling an idea, especially an idea whose time has come. Hence the growing distinction between "black hat" and "white hat"-i.e., hackers who turn new ideas toward destructive, malicious ends versus hackers who turn new ideas toward positive or, at the very least, informative ends. Once a vague item of obscure student jargon, the word "hacker" has become a linguistic billiard ball, subject to political spin and ethical nuances. Perhaps this is why so many hackers and journalists enjoy using it. Where that ball bounces next, however, is anybody's guess. Appendix C GNU Free Documentation License (GFDL) GNU Free Documentation License Version 1.1, March 2000 Copyright (C) 2000 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 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45691	----	THE MYSTERY OF SPACE _The domain of the senses, in nature, is almost infinitely small in comparison with the vast region accessible to thought which lies beyond them._--TYNDALL, "On Radiant Heat." THE MYSTERY OF SPACE _A Study of the Hyperspace Movement in the Light of the Evolution of New Psychic Faculties and An Inquiry into the Genesis and Essential Nature of Space_ BY ROBERT T. BROWNE NEW YORK E. P. DUTTON & COMPANY 681 FIFTH AVENUE COPYRIGHT 1919, BY E. P. DUTTON & COMPANY _All Rights Reserved_ _Printed in the United States of America_ TO THE CHERISHED MEMORY OF Mylie De Prè WHOSE WIFELY DEVOTION, SYMPATHETIC ASSISTANCE AND ENCOURAGEMENT DURING THE EARLY LABORS ON THE TEXT WERE A CONSTANT SOURCE OF INSPIRATION AND FORTITUDE TO THE AUTHOR, THIS BOOK IS DEDICATED AFFECTIONATELY PREFACE Mathematics is the biometer of intellectual evolution. Hence, the determination of the _status quo_ of the intellect at any time can be accomplished most satisfactorily by applying to it the rigorous measure of the mathematical method. The intellect has but one true divining rod and that is mathematics. By day and by night it points the way unerringly, so long as it leads through materiality; but, falteringly, blindly, fatally, when that way veers into the territory of vitality and spirituality. Wherefore, when we have wished to ascertain the real status of the intellect, as well as its limitations, tendencies, possibilities, we have turned to its conduct in the field of mathesis where it is least trammeled in its ingressive and egressive motivations because of the natural and easy accommodation which is offered for intellectual movement. Whether there are signs of moribundity or symptomatic evidences of marked growth or of a termination of intellectual regnancy, or whatever may be the occasion for the examination, no surer index than the mathematical may be found for the purpose. Full logical justification is, therefore, claimed for the choice of mathematical evidences to test the assumption that a new era of conscious mental conquest is opening for the vanguard of humanity and sequently for the human family. The treatment of these evidences has fallen logically into two divisions, namely, the first, a brief and elementary review of the principles of the non-Euclidean geometry and their bearings upon the question of space as the subject of mathematical study; and second, the consideration of space as a psychological, vital and dynamic or creative phenomenon. In Part I an effort has been made to trace the growth of the notion of hyperspace and to show that it is a symbol of a new epoch of intellectual expansion, an actual seizure of a new domain of awareness by the mind. And for this purpose a critical examination of the fundamental question of dimensionality is entered upon from which it develops that the status of this primary norm of mathematical thought exhibits a relatively inchoate character because of its insufficiency as a definitive quantity, and further, because of its rather superfoetated aspect when utilized as a panacea for mathetic symptoms. Also, it has been found necessary to survey the field of the four-space which has been accorded such a prominent place in the mathematical thought of the day. The reader should find in the chapter devoted to it adequate material for thought and sufficient comprehension of its meanings as a mathematical contrivance. In Part II an endeavor is made to interpret the evidences offered by high mathematic considerations in the light of the larger psychogenetic movement. For this undertaking the quality of awareness has been studied with the view to establishing its priority as a determining agency in the consideration of space in aspects broader and less restricted than those embraced in the mathematical premises. Wherefore, it appears that there are massive implications arising out of the hyperspace propaganda which have hitherto been neglected in the preliminaries incident to the fabrication of its structure. A very brief, and more or less symbolic, sketch of the genesis of space has served to demonstrate its essential nature as a consubstantive of materiality, vitality and intellectuality, the three major objective processes. Consequently it becomes imperatively necessary that any view of space which neglects its kosmic intent and purpose should be regarded as gravely fragmentary and insufficient. It is only by linking up the two aspects of space, the mathematical and the psychological, in such a manner that the one shall supplement the other, that we shall be able to arrive at a truly satisfactory understanding of its nature. In Chapter IX attention is invited to some of the extremities of mathematical laws wherein it is shown that, because mathematical goods are strictly of intellectual texture and fabric, vain is the hope of reaching any reliable certainty with respect to many vital questions, even regarding space itself, by means of the mathematical method. The intellect, and, therefore, mathematics encounter the most formidable stricture when effort is made to maneuver in the field of vitality or realism. In addition, it is shown that, when pushed to the utmost logical limits, metageometry proves not only futile, but emphasizes the need for a sharp turning of the path of search from the intellectual or material to the spiritual or intuitional. Indeed, it becomes painfully certain that the Golden Fleece of profounder knowledge will be discovered never by an expedition whose bark has its sails set for the winds of mathematical seas. But, contrarily, a new bark, moored at the furthermost shore of the sea of intellectuality with sails set for the winds which come from the realm of intuitional perception, must be seized. Whereupon, by the straightest line, we shall, at the last, land upon the shore of realism, of truth all inclusive. Mathematical evidences have been used in these discussions because they, of all lines of knowledge, afford a more just exemplification of intellectual evolution. The science of mathematics is the measure of the quality of intellectual growth and, therefore, its data, its postulates, hypotheses and advances clearly mark the stages of the intellectual movement. Chapter X is the natural and logical sequence of the inquiry into the question of spatiality. The conclusions reached therein and the obvious inferences which should be drawn from the arguments presented flow inevitably not only from the evidences of mathematical data but of the common observations of life. And while we disclaim any intention of demanding acceptance of them as final, authoritative declarations, we shall be satisfied if the readers of this volume be incited to solve for themselves the problems which these queries naturally suggest. Happy indeed shall be the outcome if there be any who, following the path sketched herein, shall find the solution of the _Mystery of Space_ and apply its meanings to the enhancement of the values of the intuitive life. In conclusion, the author esteems it a special obligation of gratitude that he should here acknowledge the debt which he owes to all of his friends who have in any way assisted or encouraged him in the completion of this work. Among those whom he is permitted to thank in this way is Mr. JAMES RINDFLEISCH who, having very kindly prepared the illustrations for the photo-engravers, is deserving of special mention. ROBERT T. BROWNE. New York City, 1919. CONTENTS PAGE PREFACE vii INTRODUCTION: Explanatory Notes x _PART ONE_ CHAPTER I THE PROLOGUE On the Variability of Psychic Powers--The Discovery of the Fourth Dimension Marks a Distinct Stage in Psychogenesis--The non-Methodical Character of Discoveries--The Three Periods of Psychogenetic Development--The Scope and Permissibility of Mathetic License--Kosmic Unitariness Underlying Diversity 23 CHAPTER II HISTORICAL SKETCH OF THE HYPERSPACE MOVEMENT Egypt the Birthplace of Geometry--Precursors: Nasir-Eddin, Christoph Clavius, Saccheri, Lambert, La Grange, Kant--Influence of the _Mecanique Analytique_--The Parallel Postulate the Root and Substance of the Non-Euclidean Geometry--The Three Great Periods: The Formative, Determinative and Elaborative--Riemann and the Properties of Analytic Spaces 44 CHAPTER III ESSENTIALS OF THE NON-EUCLIDEAN GEOMETRY The Non-Euclidean Geometry Concerned with Conceptual Space Entirely--Outcome of Failures at Solving the Parallel-Postulate--The Basis of the Non-Euclidean Geometry--Space Curvature and Manifoldness--Some Elements of the Non-Euclidean Geometry--Certainty, Necessity and Universality as Bulwarks of Geometry--Some Consequences of Efforts at Solving the Parallel-Postulate--The Final Issue of the Non-Euclidean Geometry--Extended Consciousness 69 CHAPTER IV DIMENSIONALITY Arbitrary Character of Dimensionality--Various Definitions of Dimension--Real Space and Geometric Space Differentiated--The Finity of Space--Difference Between the Purely Formal and the Actual--Space as Dynamic Appearance--The _A Priori_ and the _A Posteriori_ as Defined by Paul Carus 92 CHAPTER V THE FOURTH DIMENSION The Ideal and the Representative Nature of Objects in the Sensible World--The Fluxional, the Basis of Mental Differences--Natural Symbols and Artificial Symbols--Use of Analogies to Prove the Existence of a Fourth Dimension--The Generation of a Hypercube or Tesseract--Possibilities in the World of the Fourth Dimension--Some Logical Difficulties Inhering in the Four-Space Conception--The Fallacy of the Plane-Rotation Hypothesis--C. H. Hinton and Major Ellis on the Fourth Dimension 118 _PART TWO_ SPATIALITY: AN INQUIRY INTO THE ESSENTIAL NATURE OF SPACE AS DISTINGUISHED FROM THE MATHEMATICAL INTERPRETATION CHAPTER VI CONSCIOUSNESS THE NORM OF SPACE DETERMINATIONS Realism Is Determined by Awareness--Succession of Degrees of Realism--Sufficiency of Tridimensionality--The Insufficiency of Self-consistency as a Norm of Truth--General Forward Movement in the Evolution of Consciousness Implied in the Hyperspace Concept--The Hypothetical Nature of Our Knowledge--Hyperspace the Symbol of a More Extensive Realm of Awareness--Variations in the Methods of Interpreting Intellectual Notions--The Tuitional and the Intuitional Faculties--The Illusionary Character of the Phenomenal--Consciousness and the Degrees of Realism 161 CHAPTER VII THE GENESIS AND NATURE OF SPACE Symbology of Mathematical Knowledge--Manifestation and Non-manifestation Defined--The Pyknon and Pyknosis--The Kosmic Engenderment of Space--On the Consubstantiality of Spatiality, Intellectuality, Materiality, Vitality and Kosmic Geometrism--Chaos-Theos-Kosmos--Chaogeny and Chaomorphogeny--N. Malebranche on God and the World--The Space-Mind--Space and Mind Are One--The Kosmic Pentoglyph 203 CHAPTER VIII THE MYSTERY OF SPACE The Thinker and the Ego--Increscent Automatism of the Intellect--The Egopsyche and the Omnipsyche--Kosmic Order or Geometrism--Life as Engendering Element--The Mystery of Space Stated--Kathekos and Kathekotic Consciousness--Function of the Ideal--The Path of Search for an Understanding of the Nature and Extent of Space Must Proceed in an Inverse Direction 242 CHAPTER IX METAGEOMETRICAL NEAR-TRUTHS Realism Is Psychological and Vital--The Impermanence of Facts--On the Tendency of the Intellect to Fragmentate--The Intellect and Logic--The Passage of Space--Kosmometer and Zoometer, Instruments for the Measurement of the Passage of Space and the Flow of Life--The Disposal of Life and the Power to Create--Space, a Dynamic, Creative Process--Numbers and Kosmogony--Kosmic Significance of the Circle and the Pi-proportion--Mechanical Tendence of the Intellect and Its Inaptitude for the Understanding of Life--The Criterion of Truth 284 CHAPTER X MEDIA OF NEW PERCEPTIVE FACULTIES The Spiritualization of Matter Is the End of Evolution--Sequence and Design in the Evolution of Human Faculties--The Upspringing Intuition--Evidences of Supernormal Powers of Perception and the Possibility of Attainment--The Influence and Place of the Pituitary Body and the Pineal Gland in the Evolution of Additional Faculties--The Skeptical Attitude of Empirical Science and the Need for a More Liberal Posture--The General Results of Pituitarial Awakening Upon Man and the Theory of Knowledge 327 BIBLIOGRAPHY 359 INDEX 367 THE MYSTERY OF SPACE INTRODUCTION EXPLANATORY NOTES The following interpretation of words, phrases and notions occurring in the text, and also biographical sketches which the scope and purpose of the book itself make it impracticable to elaborate, are appended with the view to facilitating its perusal. AT-ONE-MENT (state of unity, unitariness); denotes the ultimate state of oneness towards which all evolutionary movement tends; applied to consciousness, indicates the final expansion of consciousness wherein it coincides with the universal consciousness in extent and quality of comprehension. As applied to things, denotes the unification of all movements, tendencies, and evolutions as a singularity; the end of all evolutionary activity (vide p. 270). BELTRAMI, EUGENIO, was born at Cremona, November 16, 1835; there he attended the elementary schools, the gymnasium and the lyceum, excepting the scholastic year 1848-49 when he was at the Gymnasium of Venice, now known as Marco Polo. He finished his lyceal studies in the summer of 1853, and in the following autumn (November) became a student in the _Mathematical Faculty of the University of Pavia_, after having obtained a scholarship there on the Castiglioni Foundation in the _Collegio Ghisleri_. In 1854, the succeeding year, he was expelled from this college in company with five of his colleagues who were accused of promoting "disorders" against the Abbot LEONARDI, rector of the college. The expulsion brought him many hardships and disappointments, and for two years he drifted along merely existing as his family was too poor to have him matriculated at another university. But in 1856, he went to Verona where he succeeded in securing employment as secretary to the engineer, DIDAY, in the Government service of Lombardy-Venice. On January 10, 1857, he was dismissed from this position "for political reasons"; but as the annexation of Lombardy to Piedmont occurred soon thereafter, he became again attached to the office of DIDAY, his former employer, when it was transferred to Milan as a consequence of political changes. At Milan BELTRAMI took up his mathematical education in real earnest as he now had access to Professor BRIOSCHI, his former tutor, and also LUIGI CREMONA. Through the influence of these two men he was designated (October 18, 1862) "Professore straordinario" in the University of Bologna. His work on _Surfaces of Constant Negative Curvature_, as the pseudosphere, and his application of the expression given by LOBACHEVSKI (q.v.) for the angle of parallelism, very definitely secure for him a place among the foremost workers in the field of the non-Euclidean geometry. He postulated a theorem, known as _Beltrami's Theorem_, which he stated as follows: "The center of a circle circumscribing a triangle is the center of gravity of the centers of its inscribed and escribed circles." He died in the year 1900. (Vide _Amer. Math. Mo._, Vol. IX, p. 59.) BOLYAI, JANOS (1802-1860), was born at Kleansenburg, Hungary. He is said to have inherited his mathematical genius from his father, BOLYAI FARKAS (1775-1856), who was born at Bolya, Hungary. Being a very spirited youth, his progress in his studies was most remarkable. He completed the curriculum at the Latin school when only twelve years of age. Was graduated from the Philosophical Curriculum as a result of two years of study and then entered the Viennese Academy of Engineers. Was appointed lieutenant at Temesvárlin, 1823, whence on November 3, 1823, he wrote his father: "I have discovered such magnificent things that I am myself astonished at them. It would be damage eternal if they were lost. When you see them, my father, you yourself will acknowledge it. Now, I cannot say more, only so much: that from nothing I have created another wholly new world." This letter was written in the Magyar language and has been preserved at the Marcos Vásárhely, Hungary. The mathematical conceptions formulated by him became the appendix of the _Tentamen_, a book which his father had written on the Theory of Parallels. His _Science Absolute of Space_ was translated into the French in 1868 by the French mathematician, J. HOÜEL, to whom belongs the credit of popularizing the works both of BOLYAI and LOBACHEVSKI. (Vide _Science_, n. s., Vol. 35, No. 906, 1912.) CAYLEY, ARTHUR, born at Richmond, Surrey, England, August 16, 1821; studied at King's College school; entered Trinity College, Cambridge, already a well equipped mathematician at the age of seventeen. When but twenty-one years of age he took two of the highest honors in the University of Cambridge. He was Senior Wrangler and First Smith's Prizeman. He published his first paper in 1841 and this was followed by eight hundred memoirs. For fourteen years he practiced as Conveyancer. In 1863 Lady SADLER'S various trusts were consolidated, and a new Sadlerian professorship of Pure Mathematics was created for the express purpose of affording a place for Cayley. Meanwhile, as early as 1852 he was a fellow of the Royal Society; in 1858 he joined SYLVESTER and STOKES in publishing the _Quarterly Journal of Pure and Applied Mathematics_. He was for a considerable time principal adviser as to the merits of all mathematical papers which were presented for publication to the Royal Society, the Astronomical Society, the Mathematical Society and the Cambridge Philosophical Society. He is said to have been the "most learned and erudite of mathematicians," and much of the material, therefore, which now constitutes the basis of the non-Euclidean geometry is due to his laborious efforts and comprehensive knowledge of mathematics. (Vide _Review of Reviews_, Vol. II, 1895, Sketch, reprinted from _Monist_.) CHAOGENY (Gr. Chaos, disorder, geny--generating, evolution); the evolution of chaos into order. A kosmic process involving the elaboration of the original, formless world-plasm into the first faint signs of orderliness; the beginnings of the movement of life or the Creative Logos in preparation of the field of evolution. CHAOMORPHOGENY (Gr. Chaos, disorder--Morphe, form--geny, becoming, generating); evolution of the space-form, the universe; the establishment of the metes and bounds of the universe; also, the origination and characterization of all forms as to tendence, purpose and limitations. CONCEPTUALIZATION--The act of conceptualizing, the formulation of concepts; the process by which the Thinker arrives at concepts; the logical procedure by which the consistency of a scheme of thought is established. CONSTRUCTION, IDEAL--A purely formal conception; a theory, hypothesis; a logical determination not necessarily based upon facts, but possessing virtue because of consistency; a self-consistent scheme of thought. COSMOS--Whenever the term "cosmos" appears in the text spelled as here shown, it refers to phenomena pertaining to the earth or the solar system; when spelled "kosmos" reference is made to the universe as a whole. CRITERION OF TRUTH--Defined in the text as a four-fold standard of reference, embracing the following elements, namely, the causal, the sustentative, relational and developmental. Lacking any one of these, no view of truth is more than fragmentary. Applied to space, it contemplates an inquiry into the genesis or causal aspect, an accounting for the duration aspect, a recognition of its relation to the totality of objects, and lastly, a prophecy of its telestic or perfective culmination. This test has been applied to the study of space as sketched in the text and the conclusions reached are an outcome of the inquiry directed along these lines. CURVATURE OF SPACE--A doctrine formulated by RIEMANN and which maintains that space is curved, and consequently, all lines drawn therein are curved lines. Professor PICKERING aptly describes the results of movements in a curved space by pointing out that if we go far enough east we arrive at the west; north, we arrive at the south; towards the zenith, we arrive at the nadir, and _vice versa_. DEIFORM--The basic idea indicated is that the universe is the form or body of the supreme deity, since He is not only immanent in the kosmos, but sustains it by His life; that in order to create a manifested universe, it was necessary to limit, or sacrifice, in a measure, His own illimitability. Viewed in this light, the kosmos assumes an added significance. DIMENSION--(L. Dimensio, to measure), measurement; a system of space measurement. The Euclidean geometry recognizes three dimensions or coördinates as being necessary to establish a point position; witness, the corners of a cube to form which three of the edges come together at a point. These edges represent coördinates. For the purposes of metageometry, the term dimension has been variously defined, as, direction, extent, a system of space measurement, or a system of coördinates. Regarded as a series of coördinates, it became possible to postulate a system which required four coördinates to establish the position of a point, as in the hypercube. There may be five, six, seven, eight, or any number of such coördinate systems according to the kind of space involved in the calculations. Determinations based upon the logical necessities of the various coördinate systems have been found to be self-consistent throughout and, therefore, valid for metageometrical purposes. Much depends upon the definition; for, after the definition has been once determined it remains then merely to make inferences and conclusions conform to the intent of the definition. DIVERSITY--Philosophically, the idea indicated has reference to all dissimilarities, differences, inequalities, divergent tendencies, movements and characteristics to be noted in the universum of life; the antithesis of kosmic unity; the natural outcome of life in seeking expression; the result of the fragmentative tendency of life. DUODIM (duo, two; dim, abbreviation of dimension)--A hypothetical being supposed to be possessed with a consciousness adapted only to two dimensions; a dweller in "Flatland" or two-space whose scope of motility is limited to two directions, as on the face of a plane; a term invented by hyperspace advocates for the purpose of establishing by analogy some of the characteristics of the four space and also its rationality. DUOPYKNON (duo, two--pyknon, primary unit in the process of kosmic involution, a condensation)--Secondary phase in the elaboration of chaos into kosmic order. DUOPYKNOSIS (duo, two, secondary--pyknosis, process of condensation and origination)--The second period of the involutionary movement of life during which the duadic plane of the kosmos is being established; the second, in the series of seven distinct phases, of space-genesis; dual differentiation of kosmic plasm. Duopyknosis contemplates that, in the passage of the kosmos-to-be from the plane of non-manifestation to the plane of manifestation, there are seven distinct, though interdependent and interrelational, stages through which life passes, and that, of these, it is the second. It relates to the plane of non-manifestation and is, therefore, beyond the ken of the intellectuality, being a symbol. EGOPSYCHE (Ego, the self-conscious I--psyche, soul)--The mental, emotional and physical mechanisms of man, the Thinker. These include the purely mental system, the emotional or affective mechanism, the nerve-systems (cerebro-spinal and sympathetic) and the brain; the objective or sense-derived consciousness of the Thinker which is elaborated from the total mass of perceptions transmitted through the senses; the medium of self-consciousness; the intellectual consciousness as distinguished from the intuitional or omnipsychic consciousness (q.v.). The notion of the egopsychic consciousness is based upon data already empirically determined from the mass of evidences everywhere observable. It seems to be apparent that there is a consciousness, a seat of knowledge, in man the content of which is unknown to the sense-consciousness. Dreams, premonitions, intuitions, impressions and the totality of all such phenomena substantiate this view. Furthermore, it is agreed that the source of the intuition is not identical with that of the intellect. The egopsychic consciousness, accordingly, is purely intellectual. FLUXION, PSYCHIC--The difference between a mental image and an object; an image is the representation of certain salient or cardinal characteristics of an object, sufficient for identification; but an image is not congruent, in every respect, with the object. Thus when we perceive an object, although as BERGSON contends we perceive it in the place where it is and not in the brain, it is the image of the object which takes its place in memory and not the object itself. There is, of course, a marked disparity between this memory-image and the object. Even if the image possessed one of the properties of the object, as, size, it could not take its place in memory, and neither could it do so if it possessed any of the properties of the real object. Consciousness is such that all due allowances are made for these conditions and the mind is able to retain more or less exact knowledge of these properties in the image; but there is a difference, small though it may be. This difference is the psychic fluxional. FOHAT (Skt)--A term applied to the Creative Logos who is said to be the generating element in the differentiation of chaos into kosmic orderliness; the supreme deity in the rôle of Creator. FORM, PURE--An abstraction arrived at by subtracting the last vestige of materiality or substantiality from an idea and viewing the remains as a pure unsubstantial form or idealization; the shell or frame-work of a material object or condition; existing in idea or thought only; a mental conception regarded as a type or norm; a purely hypothetical construction. FOUR SPACE--Often referred to as the fourth dimension (vide Chapter V); a space in which four coördinates (four lines drawn perpendicular to each other) are necessary and sufficient to establish the position of a point, as, a hypercube. GAUSS, CHARLES FREDERICK, born at Brunswick, April 30, 1777. His father, being a bricklayer, had intended that he should follow the same occupation. So, in 1784, Charles was sent to the Bütner Public School in Brunswick, in order that he might be taught the ordinary elements of education. But during his attendance at this school, his unusual intelligence and aptitude attracted the attention and friendship of Professor BARTELS who later became the Professor of Mathematics at Dorpat. In 1792, through the kindly representations of Professor BARTELS to the Duke of Brunswick, young Gauss was sent to the Collegium Carolinum. This greatly displeased his father as he saw in this move the frustration of his plans for Charles. In 1794, however, GAUSS entered the University of Göttingen still undecided whether he should make mathematics or philology his life work. While residing at Göttingen, he made his celebrated discoveries in analysis and these turned his attention definitely to the field of mathematics. He completed his studies at Göttingen and returned to Brunswick in 1798, residing at Helmstadt where he had access to the Library in the preparation of his _Disquisitiones Arithmeticae_ which was published in 1801. He received his doctorate degree (Ph.D.) on July 16, 1799. His next notable work was the invention of a method by which he calculated the elements of the orbit of the planet _Ceres_ which had been discovered by PIAZZI, January 1, 1800, and who had left no record of his calculations by which other astronomers could locate the planet. GAUSS also calculated an ephemeris of _Ceres'_ motion by means of which DE ZACH rediscovered the planet December 31, 1800. His _Theoria Motus Corporum Coelestium in Conicis Sectionibus Solem Ambientium_, in which the author gives a "complete system of formulæ and processes for computing the movements of a heavenly body revolving in a conic section" is an outgrowth of his early researches and brought him lasting fame. Through the influence of his friend OLBERS, he was appointed, July 9, 1807, first director of the new Göttingen Observatory, and Professor of Astronomy in the University, a position which he held until the end of his life. He died February 23, 1855. (Vide _Astronomical Society Notices_, Vol. 16, p. 80, 1856; also _Nature_, Vol. XV, pp. 533-537, 1877.) GEOMETRISM--Of geometrical quality; a notion derived from PLATO'S declaration: "God geometrizes." It was his belief that the creative acts of the deity are executed in accordance with geometric design and laws; that in the totality of such acts there necessarily inheres a latent geometric quality. KANT closely adhered to this notion in his discussions of space as an aspect of divine intelligence. He believed that the intellect merely rediscovers this latent geometrism when it turns to the study of materiality, and this belief is shared by BERGSON, the foremost metaphysician of the present time. HYPERSPACE (hyper, above, beyond, transcending--space)--That species of space constructed by the intellect for convenience of measurement; an idealized construction; a purely arbitrary, conventional mathematical determination; the fourth dimension; any space that requires more than three coördinates to fix a point position in it, as, a five space, an _n_-space. INTUITOGRAPH--The means by which the omnipsychic consciousness transmits intuitional impressions to the egopsychic or intellectual consciousness. An intuitogram is a direct cognition, an intuition; a primary truth projected into the egopsychic consciousness by the Thinker. It is recognized that, under the necessities of the present schematism of things, it is exceedingly difficult to propagate an intuition, especially with the same degree of ease as concepts are propagable; yet, this is believed to be a condition which will be overcome as the evolution of the higher faculties proceeds. INVOLUTION--Process of enfolding, involving; antithesis of evolution; philosophically, the doctrine of involution maintains that, during the process of kosmic pyknosis (space-genesis), _all_ that is to be expressed, developed and perfected as a result of the evolutionary movement was first involved, enfolded or deposited as latent archetypal tendencies and radicles in the original world-plasm; that, as the involutionary movement proceeded through the various phases of space-genesis, these became more and more phenomenal until at last they terminated in the elaboration of a manifest universe: each stage, accordingly, of the involutionary procedure became the basic substructure of a plane of specialized substance or materiality and consciousness. Thus it appears that evolution really begins where involution ends (vide Fig. 18), and the two opposing processes constitute the dualism of life as generating element. This notion has been symbolized in the _Lingam yoni_ of Hellenistic philosophies, also in _Yang_ and _Yin_ of Chinese philosophy, which represent the original pair of opposites. KATHEKOS--A purely arbitrary term devised for the express purpose of providing a convenient symbol to convey the idea embodied in the triglyph, _Chaos-Theos-Kosmos_, and is composed of the first three letters in each one of the terms of the triglyph; hence, symbolizes the triunity and interaction involved in the resolution of chaos into an orderly kosmos by the will of the Creative Logos. Thus, "kathekos" embodies a quadruplicate notion, namely, chaos, Creative Logos, manifested kosmos, and the creational activity of the Logos in the transmutation of disorder into order. The justification for this term, therefore, resides in its convenience, brevity and comprehensiveness. By referring to figure 17, it will be seen that Kathekos divides into two kinds--involutionary, or that which pertains to involution, and evolutionary, or that that pertains to evolution. It thus comprises the beginning and the end of the world age or cycle and pertains to non-manifestation. The _raison d'être_ of this differentiation is embodied in the notion that, on the involutionary arc of the cycle, the chaogenic period represents a phase of the world age when space-genesis is in an archetypal state wherein are involved all possibilities that are to become manifest in the kosmos, and on the evolutionary arc, the kathekotic period which is parallel to the chaogenic and represents a phase of the world age when the kosmos has reached ultimate perfection, embodying the perfected results of the possibilities which inhered in the chaogenic period or in involutionary kathekos. Thus, kathekos is dual in nature, on the one hand representing kosmic potency, and on the other, kosmic perfection of these potencies. It is Alpha, as related to involution, and Omega, as related to evolution. KATHEKOSITY--A derivative, signifying creative activity and all that it implies; the state of consciousness or cognition corresponding thereto. KLEIN, FELIX (1849--), born at Dusseldorf; studied at Bonn, and when only seventeen years of age was made assistant to the noted PLÜCKER in the Physical Institute. He took his doctorate degree in 1868, then went to Berlin, and later to Göttingen where he assisted in editing PLÜCKER'S works. He entered the Göttingen faculty in 1871; became Professor of Mathematics at Erlangen in 1872; and subsequently held professorships at Munich, 1875; Leipzig, 1880, and Göttingen, 1886. No one else in Germany has exerted so great influence upon American mathematics as he. KOSMOS--_See Cosmos_. LA GRANGE, JOSEPH LOUIS, born at Turin, January 25, 1736; died at Paris, April 10, 1813; regarded as the greatest mathematician since the time of Newton. It may be interesting to note that LA GRANGE remarked that mechanics is really a branch of pure mathematics analogous to a geometry of four dimensions, namely, time, and the three coördinates of the point in space. (Vide _Ball's Account of the History of Mathematics_.) LIE, SOPHUS, a noted mathematician, referred to as the "great comparative anatomist of geometric theories, creator of the doctrines of Contact Transformations and Infinite Continuous Groups, and revolutionizer of the Theory of Differential Equations." LOGOS--The supreme deity of the phenomenal universe; Creator; Fohat; a planetary god; the deity of a solar system. MANVANTARA (Skt.)--A world age; the periods of involution and evolution combined; the stage during which the universe is in manifestation; a Day of Brahma. MATHESIS (Gr. mathein, to learn)--Erudition; profound learning; the realm of metaphysical conceptions; the field of higher mathematics; the sphere of conceivability; the theoretical. MENTOGRAPH--A cognitive factor consisting of a complete perception fused or in coalescence with a memory-image. Pure memory, of itself, is without utility as an aid to cognition; but, when nourished or supplemented by the substance of perception it becomes the basis of intellectual consciousness. METAGEOMETRY (Gr. Meta, beyond, transcending--geometry)--Commonly, any kind of geometry that differs from the Euclidean, as the non-Euclidean; a geometry based upon the assumption that the angular sum of a triangle is greater or less than two right angles; the highest form of geometry; a system of idealized mathematical constructions. Sometimes called "pangeometry"; designated by GAUSS as "Astral Geometry"; the geometry of hyperspace. It consists of results arrived at by geometers in seeking a proof of the parallel-postulate. META-SELF--The higher self in man; the universal self; the one self of which all individual selves are but fragments or parts. In man, it is coördinate with the omnipsyche (q.v.) and as such is the medium of kosmic consciousness. MORPHOGENY (Gr. Morphe, form, vehicle, body--geny, evolution)--The evolution of forms, the production of individual bodies or vehicles for life, including organs and faculties. Morphogenic--a derivative; pertaining to morphogeny; a kosmic process (vide figs. 17 and 18). _N_-DIMENSIONALITY--Quality of conceptual space by virtue of which it may be regarded as possessing an indefinite number of dimensions. NEAR-TRUTH--Any statement or view which is based upon partial knowledge; predicates concerning a class or genus derived from limited acquaintance with particulars of the class or genus; statements based upon logical determinations inhering in idealized constructions and applied to concrete or objective conditions; an abstraction viewed as a reality; the application of the qualities of abstractions to realities. NEUROGRAM--Psychologically, a movement received by the afferent nerves in the form of a stimulation and transmitted through the brain and efferent nerves as either a reflex or voluntary action; a nerve impulse; a perception; a primary unit of intellectual consciousness; cf. _Intuitogram_. NEWCOMB, SIMON (1835-1909), born at Wallace, Nova Scotia; educated in his father's school and came to the United States in 1853. Began, in 1854, teaching in Maryland; was appointed computer on _Nautical Almanac_ at Cambridge in 1857; was graduated at Lawrence Scientific School in 1858; appointed Professor of Mathematics in the U. S. Navy in 1861. He supervised the construction of the 26-inch equatorial telescope at the Naval Observatory, and was secretary of the Transit of Venus Commission; was a member of nearly all of the Imperial and Royal societies of Europe and of the various societies in the United States, receiving the Copley Medal in 1874; the Huygens, 1878; the Royal Society, 1890, and the Bruce Medal in 1898; held the presidency of the following learned societies, viz: American Association for the Advancement of Science, 1877; Society for Psychical Research, 1885-1886; American Mathematical Society, 1897-1898; the Astronomical and Astrophysical Society of America from its foundation in 1899. He rendered notable service in popularizing the doctrine of hyperspace. NORM--An authoritative standard; model or type; standard of reference. The choice of a norm for spatial determinations cannot abide in any premise except that which naturally, and not artificially and conventionally, conforms to what is actually perceived; if so, there should be justification for challenging the wisdom and utility of the present schematism of things. There is an inherent conformity of space with intellect and intellect with space, and because of this natural complementarity of part with part and whole with whole, space cannot be otherwise than the intellectuality normally conceives it to be, provided, of course, that the cognitive movement is free and untrammeled by arbitrary hindrances. Consciousness, therefore, is the norm or standard of reference for all questions arising out of a consideration of spatiality. OMNIPSYCHE--A term used to denote the Thinker's cognitive apparatus; the universal soul manifesting in individuals; the consciousness of the Thinker in virtue of which he is at-one with the universal consciousness; the medium of kosmic consciousness; the source of the intuition, cf. _Egopsyche_. The divinity in man (which is taken for granted), or his highest self can in no way be said justly to take its rise from sense-experience or from any bodily process. If divine, then eternal, and therefore, persistent. Broadly, the doctrine of evolution recognizes the passage of life from form to form, adding a little to each successive form and inevitably pushing each to a higher degree of perfection. Now, what is it that passes from form to form? Is it undifferentiated life or is it a specialized form of life? From every evidence, it would be judged that the life that ensouls an individual form is a specialized principle, i.e., limited to the execution of a given purpose. If life as a specialized principle, limited to the execution of a given purpose in each form, passes on, it must preserve, at least, the sublimated results obtained during its residence in each individual form. It would thus become a sort of reservoir containing all these transmuted results. The omnipsyche, within the meaning of the text, is precisely this specialized life principle. PARALLEL-POSTULATE--Variously referred to as the XIth, XIIth and XIIIth axiom of the _Elements of Euclid_; stated by MANNING, in _Non-Euclidean Geometry_, p. 91, in the following form: "If two lines are cut by a third, and the sum of the interior angles on the same side of the cutting line is less than two right angles, the lines will meet on that side when sufficiently produced." This celebrated postulate has proven to be the most fruitful ever devised; for it embodies in itself the possibility of three geometries based respectively upon the following assumptions, namely: I. That there exists a triangle, the sum of whose angles is congruent to a straight angle, the Euclidean; II. That there exists a triangle the sum of whose angles is less than a straight angle, the LOBACHEVSKIAN; III. That there exists a triangle the sum of whose angles is greater than a straight angle, the CAYLEY-KLEIN. Speaking of the content of the last two named, EDWARD MOFFAT WEYER[1] says: "Hypothetical realms, wherein the dimensions of space are assumed to be greater in number than three, yield strange geometries, which are only card castles, products of a sort of intellectual play in the construction of which the laws of logic supply the rules of the game. The character of each is determined by whatsoever assumption its builder lays down at the start." [1] Vide _Popular Science Monthly_, vol. 78, p. 554, 1911. PASSAGE OF SPACE--A phrase connoting the movement of space from chaos to perfect order, a process believed to be infinite. The genesis of space necessarily implies an elaboration, a procedure, by which the metamorphosis from disorder to kosmic order is made, and this movement is referred to as the "passage of space," a phenomenon thought to be measurable by means of a suitable instrumentality. PERISOPHISM--See _Near-Truth_. PSEUDOSPHERE--A surface of constant negative curvature; basis of BELTRAMI'S metageometrical calculations; surface resembling a champagne glass or common spool. The assumption that space is pseudospherical has given rise to the notion of space-curvature and various other conceptions. PSYCHOGENY (Gr. Psyche, Soul--geny)--History of the evolution of the soul or the development of the senso-mechanism in organisms. ERNST HAECKEL has traced the psychogeny of man through twenty-two different stages from the moneron to the anthropoid apes, and man. PRALAYA (Skt.)--Kosmic quietude; the period during which the universe is not in manifestation; gestatory period; kosmic inactivity; opposed to manvantara (q.v.); figuratively, the kosmic womb; world egg. PYKNON (Gr. pyknon, hard)--The principle of kosmic condensation; the primary basis of space-genesis; the initiation of the process by virtue of which chaos is elaborated into kosmic order. PYKNOSIS--The process of spatial engenderment. There are seven of these processes, each indicating a phase of duration, namely: MONOPYKNOSIS, the primary phase; DUOPYKNOSIS, secondary; TRIPYKNOSIS, tertiary. These three pertain to the plane of non-manifestation, the pralayic or gestatory duration-phase. The results arrived at during these duration-phases are concentrated in the Quartopyknotic which corresponds to the causal plane of manifestation or pure kosmic spirituality. QUINTOPYKNOSIS, a process concerned in the genesis of mentality; SEXTOPYKNOSIS, kosmic sensibility; SEPTOPYKNOSIS, kosmic materiality. These seven phases of duration constitute the scope of space genesis or kosmogenesis, and incidentally depose the substructure of kosmic materiality, sensibility, intellectuality and spirituality, as well as the higher trinity of kosmic modes. The ramifications of these principles are innumerable and omnipresent. (See Chapter VII.) QUARTODIM--A hypothetical being assumed to have a consciousness adapted to hyperspace or the fourth dimension, and whose scope of action is encompassed within a space which requires four coördinates, as the four-space. REALITY (Realism)--Life; the harmony existing among the parts to maintain their equilibrium in the whole; the principle of integrity subsisting among parts; kosmic vitality. RIEMANN, GEORGE FREDERICH BERNHARD, was born September 17, 1826, in the village of Breselenz, near Dannenburg, in Hanover. Until he was eight years of age his father was his sole tutor, but even at this age he exhibited great powers of arithmetical calculation. In the Spring of 1840 young RIEMANN was sent to the Hanover Lyceum where he remained for two years, leaving in 1842 for the Gymnasium at Luneburg. Here, under the direction of Professor SCHMALFUSS, he learned very rapidly, and is said to have required only one week thoroughly to familiarize himself with LEGENDRE'S _Theory of Numbers_. On April 12, 1846 (Easter), he entered the University of Göttingen as a student of Theology in accordance with his father's wishes. His passion for mathematics, however, was so aroused by the lectures of GAUSS that He begged his father to be allowed to devote himself entirely to the studies of his choice. For two years he studied under JACOBI at Berlin. He then returned to Göttingen, and was graduated, his thesis being a dissertation on the foundations of a general theory of functions of a variable complex magnitude. In 1854 he qualified as a teacher by giving a lecture on the "Hypothesis on which Geometry is Founded." In 1857 he became "Professor Extraordinarius," and in 1859 was elected Corresponding Member of the Academy of Sciences of Berlin and in 1860 a member of the Academy of Sciences of Göttingen. After four years of failing health, during which he visited Messina, Palermo, Naples, Rome, Florence, Pisa and Milan, he died at Lago Maggiore, July 20, 1866, in full possession of his faculties and conscious of his approaching end. SCHWEIKART, FERDINAND KARL (1780-1857), studied from 1796 to 1798 in Marburg, attending the mathematical lectures of J. K. F. HAUFF. In 1812 he became professor in Charkov, a position which he held for four years. In 1816 he became a tutor in the City of Marburg where he remained until 1820 when he transferred his labors to Königsberg. It was during his tutorship at Charkov, Marburg and Königsberg that he, entirely alone and without the slightest suggestion from any man, developed and taught a non-Euclidean geometry to the students under his care. For copy of his treatise on non-Euclidean geometry, see _Historical Sketch of the Hyperspace Movement_, Chapter II. SCOPOGRAPHIC IMPRESSIONS--Sight perceptions fused with an associated memory-image, and forming the basis of action on external phenomena. SENSOGRAPHIC IMPRESSIONS--Perceptions or impulses transmitted through the nerves of a sense-organ; any impression acting through the media of the senses. SENSIBLE WORLD--The world of the senses; that which responds to the senses; the domain of perception; the phenomenal world; world of perceptual space. SPACE-CURVATURE (see _Curvature of Space_). SPACE-GENESIS--The process of spatial engenderment; the movement of life as engendering agent in bringing into manifestation the kosmos; the story of the appearance of the organized kosmos. The genesis of space can only be symbolized, as has been done in the text, for the limitations of human consciousness do not otherwise admit of the empirical establishment of the notion of its detailed procedure. SPATIALITY--Space as a dynamic, creative movement; kosmic order, as opposed to disorder; the path of the engendering movement of life; the place of life. Spatiality, materiality, intellectuality and geometricity or the latent geometrism of the kosmos are thought of as being consubstantial and interdependent; but, of these, spatiality is regarded as the substance out of which the latter three are elaborated. SUPERCONCEPTUAL--The purely intuitional; an act of cognition performed without the detailed work of conception derived from sense-data; conception of intuitions and their inter-relations; the Thinker's consciousness freed from intellectual characterization. SUPERPERCEPTION--Perception of conceptual relations; a state of cognition wherein, instead of receiving percepts or images from the external world, then elaborating them into concepts, the Thinker apprehends composite images or concepts at first hand. It is a power which the liberated mind of the future will possess owing to the growing automatism of the intellect and the more facile expression of the intuitional consciousness. TESSERACT (Gr. Tessera, four, cube, tessella)--A hypercube (see Chapter V.) THINKER (Skt. Manu, thinker)--The real, spiritual man, as differentiated from his perceptive vehicles--mind, emotions and physical body; the omnipsychic intelligence who receives, classifies, interprets and preserves percepts; the manipulator of concepts; in fine, the higher, spiritual man. The Thinker uses the various perceptive instrumentalities as so many tentacles or antennae by which he contacts the sensible world and makes the necessary adaptations to environment. He is the pure intelligence which is the source of all cognitive motivation; opposed to ego, because the egopsychic instrumentality is essentially an individualizing, separative agency; while the Thinker's omnipsychic intelligence is the basis of his unity with the universal intelligence. This conception of the Thinker implies that, as a spiritual intelligence, he is within and without the body, filling it as the ocean fills the sponge, encompassing, enveloping it and, at the same time, originating the totality of activities which manifest in and through the body. He is limited, therefore, in his manifestations in the sensible world only by the pliability of his vehicles. TRANSFINITY--A state or condition that is incomprehensible to finite intelligence; that which transcends the finite, yet is not infinite; less than infinity and greater than finity. Space is referred to as being transfinite rather than infinite in extent. But space transfinite should be distinguished from space "finite though unbounded." For, there would seem to be little worthy of choice between a "finite, unbounded space" and an infinite one. The absence of boundary would naturally suggest an infinite extent. And although RIEMANN who is the author of the "unbounded" space arbitrarily determined that such a space should be a manifold possessing a measure of curvature which could be determined either by counting or actual measurement, he undoubtedly knew, nevertheless, that while each manifold might be an "unbounded" space the totality of such manifolds, infinite in number, must also be infinite in extent. It would seem to do violence to common sense, if not to logical necessity, to view space both as "unbounded" and finite in extent, yet there would be no such difficulty in the recognition of space as being both transfinite and finite; because it is conceivable that the extent and character of space finite should transcend a finite intellectuality, and yet not be infinite. TRIDIM--A being whose scope of consciousness is limited to a space of three dimensions, as ordinary human beings. TRIDIMENSIONALITY--That quality possessed by perceptual space by virtue of which it is necessary and sufficient to have three coördinates, and only three, to establish the position of a point. UNODIM--A hypothetical being assumed to have a consciousness limited to linear or one-space. ZONES OF AFFINITY--Regions in the domain of intellectuality wherein minds, possessing a common differential, rate of vibration or quality, adhere to certain tenets from choice. Schools of philosophy, religions, and all those major divisions of intellectual effort which divide and subdivide intellectual allegiance are believed to take their rise in this property of intellectuality in virtue of which all minds having a similar coefficient gravitate towards a common agreement, especially where the movement is voluntary and untrammeled. _PART ONE_ THE ESSENTIALS OF THE GEOMETRY OF HYPERSPACE AND THEIR SIGNIFICATIONS CHAPTER I THE PROLOGUE On the Variability of Psychic Powers--The Discovery of the Fourth Dimension Marks a Distinct Stage in Psychogenesis--The Non-Methodical Character of Discoveries--The Three Periods of Psychogenetic Development--The Scope and Permissibility of Mathetic License--Kosmic Unitariness Underlying Diversity. In presenting this volume to the public profound apologies are made to the professional mathematician for the temerity which is shown thereby. All technical discussion of the problems pertinent to the geometry of hyperspace, however, has been carefully avoided. The reader is, therefore, referred to the bibliography published at the end of this volume for matter relating to this aspect of the subject. The aim rather has been to outline briefly the progress of mathematical thought which has led up to the idea of the multiple dimensionality of space; to state the cardinal principles of the Non-Euclidean geometry and to offer an interpretation of the metageometrical concept in the light of the evolutionary nature of human faculties and material characteristics and properties. The onus of this treatise is, therefore, to distinguish between what is commonly known as sensible space and that other species of space known as geometric spaces. Also to show that the notion which has been styled _hyperspace_ is nothing more nor less than an evidence of the faint, early outcroppings in the human mind of a faculty which, in the course of time, will become the normal possession of the entire human race. Thus the weight of all presentations will be to give currency to the belief, very strongly held, that humanity, now in its infancy, is yet to evolve faculties and capabilities, both mental and spiritual, to a degree hitherto viewed as inconceivable. On this view it must appear that the faculty of thought including the powers of imagination and conceptualization are not psychological invariants, but, on the other hand, are true variants. They are, consequently, answerable to the principle of evolution just as all vital phenomena are. Some have thought that no matter what idea may come into the mind of the human race or at what time the idea may be born the mind always has been able to conceive it. That is, many believe that the nature of mind is such that no matter how complex an idea may be there has always been in the mind the power of conceiving it. But this view cannot be said to have the support of any trustworthy testimony. If so, then the mind must at once be recognized as fully matured and capable during every epoch of human evolution, no less in the first than in the latest, which, of course, is absurd. It is undoubtedly more reasonable and correct to believe that the powers of conceptualization are matters of evolutionary concern. For instance, the assertion that the mind was incapable of conceiving, in the realm of theology, a non-anthropomorphic god, or, in the field of biology, the doctrine of evolution, or, in the domain of invention, the wireless telegraph, or, in mathematics, the concept of hyperspace before the actual time of these conceptions, cannot be successfully controverted. In fact, it may be laid down as one of the first principles of psychogenesis that the mind rarely, if ever, conceives an idea until it has previously developed the power of conceptualizing it and giving it expression in the terms of prior experience. As in the growth of the body there are certain processes which require the full development of the organ of expression before they can be safely executed so in the phyletic development of faculties there are certain ideas, conceptions and scopes of mental vision which cannot be visualized or conceptualized until the basis for such mentation has been laid by the appearance of previously developed faculties of expression. And especially is this true of the intellect. Inasmuch as the entire content of the intellect is constituted of sense-derived knowledge, with the exception of intuitions which are not of intellectual origin though dependent upon the intellect for interpretation, there can be no doubt as to the necessity of there being first deposed in the intellect a sense-derived basis for intellection before it can become manifest. The Sensationalists, led by LEIBNITZ, propounded as their fundamental premise this dictum: "_There is nothing in the intellect which has not first been in the senses except the intellect itself_," and this has never been gainsaid by any school that could disprove it. The intuitionalist does not deny it: he merely claims that we are the recipients of another form of knowledge, the intuitional, which, instead of being derived from sense-experience, is projected into the intellectual consciousness from another source which we designate the Thinker. Thus, from the two forms of consciousness, come into the area of awareness truths that spring from entirely different sources. From the one source a steady stream of impressions flow constituting the substance of intellectual consciousness; from the other only a drop, every now and then, falls into the great inrushing mass so as to add a dim phosphorescence to an otherwise unilluminated pool. Obviously, when there is a lack of sensuous data from which a certain concept may be elaborated there can be no conception based upon them, and as the variety and quality of concepts are in exact proportion to the variety and quality of sense-experience there can be no demand for a particular species of notions such as might be elaborated out of the absent or non-existent perception. Hence, the power of conceiving springs forth from sense-experience. Sense-experience is essentially a mass of perceptions: these, creating a demand for additional adaptations, conspire, as if, to evoke the power or faculty to meet the demand, and consequently, an added conceptualization is made. Progress in human thought is made in a manner similar to that which prevails in the development of other natural processes, such as, the power of speech in the child. In the development of this faculty there are certain definite stages which appear in due sequence. The child is not gifted with the power of speech at once. It comes, by gradual and sometimes painful growth, into a full use of this faculty. Now, much the same principle holds true in the evolution of the mind in the human species. It is an established biologic principle that the ontogenetic processes manifested in the individual are but a recapitulation of the phylogenetic processes which are observable in the progress of the entire species. The view becomes even more cogent when note is taken of the fact that the foetus, during embryogenesis, passes successively through stages of growth which have been shown to be analogous, if not identical, with those stages through which the human species has developed, namely, the mineral, vegetal and animal. Wherefore it may be said that the fourth dimensional concept marks a distinct stage in psychogenesis or evolution of mind. It required, as will be shown in Chapter II, nearly two thousand years for it to germinate, take root and come to full fruition. For it was not until the early years of the nineteenth century that mathematicians, taking inspiration from RIEMANN (1826-1866) fully recognized the concept as a metaphysical possibility, or even the idea was conceived at all. Serious doubt is entertained as to the possibility of its conception by any human mind before this date, that is, the time when it was actually born. Prior to that time, mathematical thought was taking upon itself that shape and tendence which would eventually lead to the discovery of hyperspace; but it could not have reached the zenith of its upward strivings at one bound. That would have been unnatural. Such is the constitution of the mind that although it is the quantity which bridges the chasm between the two stages of man's evolution when he merely thinks and when he really knows it is entirely under the domain of law and must observe the times and seasons, as it were, in the performance of its functions. The scope of psychogenesis is very broad, perhaps unlimited; but its various stages are very clearly defined notwithstanding the breadth of its scope of motility. And while the distance from _moneron_ to man, or from feeling to thinking is vast, the gulf which separates man, the Thinker, from man, the knower, is vaster still. Who, therefore, can say what are the delights yet in store for the mind as it approaches, by slow paces, the goal whereat it will not need to struggle through the devious paths of perceiving, conceiving, analyzing, comparing, generalizing, inferring and judging; but will be able to know definitely, absolutely and instantaneously? That some such consummation as this shall crown the labors of mental evolution seems only natural and logical. It may be thought by some that the character and content of revelational impressions constitute a variation from the requirements of the law above referred to, but a little thought will expose the fallacy of this view. The nature of a revealed message is such as to make it thoroughly amenable to the restrictions imposed by the evolutionary aspects of mind in general. That this is true becomes apparent upon an examination of the four cardinal characteristics of such impressions. First, we have to consider the indefinite character of an apocalyptic ideograph which is due to its symbolic nature. This is a feature which relieves the impression of any pragmatic value whatsoever, especially for the period embracing its promulgation. Then, such cryptic messages may or may not be understood by the recipient in which latter case it is nonpropagable. Second, the necessity of previous experience in the mind of the recipient in order that he may be able to interpret to his own mind the psychic impingement. The basis which such experience affords must necessarily be present in order that there may be an adequate medium of mental qualities and powers in which the ideogram may be preserved. A third characteristic is that revelations quite invariably presuppose a contemplative attitude of mind which, in the very nature of the case, superinduces a state of preparedness in the mind for the proper entertainment of the concept involved. This fact proves quite conclusively that revelational impressions are not exceptions to the general rule. Lastly, a dissatisfaction with the conditions with which the symbolism deals or to which it pertains is also a prerequisite. This condition is really that which calls forth the cryptic annunciation, and yet, preceding it is a long series of causes which have produced both the conditions and the revolt which the revelator feels at their presence. In view of the foregoing, it would appear that objections based upon the alleged nonconformity of the revealed or inspired cannot be entertained as it must be manifest that it, too, falls within the scope of the laws of mental growth. Discoveries, whether of philosophical or mechanical nature, or whether of ethical or purely mathematical tendence, are never the results of a deliberate, methodical or purposive reflection. For instance, let us take LIE'S "transformation groups," mathematic contrivances used in the solution of certain theorems. Now, it ought to be obvious that these mathetic machinations were not discovered by SOPHUS LIE as a consequence of any methodic or purposeful intention on his part. That is, he did not set out deliberately to discover "transformation groups." For back of the "groups" lay the entire range of analytic investigations; the mathematical thought of more than a thousand years furnished the substructure upon which Lie built the conception of his "groups." Similarly, it may be said with equal assurance that no matter how great the intensity of thought, nor how purposeful, nor of how long duration the series of concentrated abstractions which led up to the invention of the printing press, the linotype or multiplex printing press of our day could not have been produced abruptly, nor by use of the mental dynamics of the human mind of remoter days. Its production had to follow the path outlaid by the laws of psychogenesis and await the development of those powers which alone could give it birth. The whole question resolves itself, therefore, into the idea of the complete subserviency of the mind, in all matters of special moment, to the laws aforementioned. The supersession of the law of its own life by the mind is well-nigh unthinkable, if not quite so. If we now view the history of the mind as manifested in the human species, three great epochs which divide the scope of mental evolution into more or less well-defined stages present themselves. These are: first, the _formative stage_; second, _the determinative stage_; third, the stage of _freedom_, or the _elaborative_ stage. In all of the early races of men, through every step which even preceded the _genus homo_, the generic mind was being formulated. It was being given shape, outline and direction. All of the first stage, the _formative_, was devoted to organization and direction. Those elementary sensations which constituted the basis of mind in the primitive man were accordingly strongly determinative of what the mind should be in these latter days. To this general result were contributed the effects of the activity of cells, nerves, bones, fibers, muscles and the blood. The _formative_ period naturally covered a very extensive area in the history of mind or psychogenetic development. It was followed closely, but almost insensibly, by the _determinative_ period during which all the latent powers, capacities and faculties which were the direct products of the _formative_ period were being utilized in meeting the demands of the law of necessity. The making of provisions against domestic want, against the attacks of external foes; the combating of diseases, physical inefficiency, the weather, wild beasts, the asperities of tribal enmities; as well as furthering the production of art, music, sculpture, the various branches of handiwork, literature, philosophies, religions and the effectuation of all those things which now appear as the result of the mental activity of the present-day man make up the essence and purpose of the determinative period. Signs of the dawn of the _elaborative_ stage, also called the stage of _freedom_, have been manifest now for upwards of three centuries and it is, therefore, in its beginnings. It is not fully upon us. Not yet can we fully realize what it may mean, nor can we unerringly forecast its ultimate outcome; but we feel that it is even now here in all the glories of its matutinal freshness. And the mind is beginning to be free from the grinding necessities of the constructive period having already freed itself from the restrictive handicaps of the primeval formulation period. Already the upgrowing rejuvenescences so common at the beginning of a new period are commencing to show themselves in every department of human activity in the almost universal desire for greater freedom. And this is particularly noticeable in the many political upheavals which, from time to time, are coming to the surface as well as in the countless other aspects of the wide-spread renaissance. Perhaps the time may come, never quite fully, when there will be no longer any necessity to provide against the external exigencies of life; perhaps, the time will never be when the mind shall no more be bound by the law of self-preservation, not even when it has attained unto the immortality of absolute knowledge; yet, it is intuitively felt that it must come to pass that the mind shall be vastly freer than it is to-day. And with this new freedom must come liberation from the necessities of the elementary problems of mere physical existence. The inference is, therefore, drawn that the fourth dimensional concept, and all that it connotes of hyperspace or spaces of _n_-dimensionality are some of the evidences that this stage of freedom is dawning. And the mind, joyous at the prospect of unbounded liberty which these concepts offer, cannot restrain itself but has already begun to revel in the sunlit glories of a newer day. What the end shall be; what effect this new liberty will have on man's spiritual and economic life; and what it may mean in the upward strivings of the Thinker for that sublime perpetuity which is always the property of immediate knowledge no one can hope, at the present time, to fathom. It is, however, believed with KEYSER that "it is by the creation of hyperspaces that the rational spirit secures release from limitation"; for, as he says, "in them it lives ever joyously, sustained by an unfailing sense of infinite freedom." The elevating influence of abstract thinking, such as excogitation upon problems dealing with entities inhabiting the domain of _mathesis_ is, without doubt, incalculable in view of the fact that it is only through this kind of thought that the spirit is enabled to reach its highest possibilities. This is undoubtedly the philosophy of those religious and occult exercises known as "meditations," and this perhaps was the main idea in the mind of the Hebrew poet when he exclaimed: "Let the words of my mouth and the meditation of my heart be acceptable in thy sight, O Lord, my strength and my Redeemer." The principal, if not the only, value possessed by the "summitless hierarchies of hyperspaces" which the mathematician constructs in the world of pure thought is the enrichening and ennobling influence which they exert upon the mind. But admittedly this unbounded domain of mathetic territory which he explores and which he finds "peopled with ideas, ensembles, propositions, relations and implications in endless variety and multiplicity" is quite real to him and subsists under a reign of law the penalties of which, while not as austere and unreasonable as some which we find in our tridimensional world, are nevertheless quite as palpable and as much to be feared. For the orthodoxy of mathematics is as cold and intolerant as ever the religious fanatic can be. But the reality and even the actuality which may be imputed to the domain of mathesis is of an entirely different quality from that which we experience in our world of triune dimensionality and it is a regrettable error of judgment to identify them. It ought, therefore, never be expected, nor is it logically reasonable to assume that the entities which inhabit the mathetic realm of the analyst should be submissive to the laws of sensible space; nor that the conditions which may be found therein can ever be made conformable to the conditions which exist in perceptual space. It was PLATO'S belief that ideas alone possessed reality and what we regard as actual and real is on account of its ephemerality and evanescence not real but illusionary. This view has been shared by a number of eminent thinkers who followed, with some ostentation, the lead established by PLATO. For a considerable period of time this school of thinkers had many adherents; but the principles at length fell into disrepute owing to the absurdities indulged in by some of the less careful followers. The realism, or for that matter, the actuality of ideas cannot be denied; yet it is a realism which is neither to be compared with the physical reality of sense-impressions nor its phenomena. The character and peculiarity of ideas are in a class apart from similar notions of perceptual space content. It is as if we were considering the potentialities of the spirit world and the entities therein in connection with incarnate entities which in the very nature of the case is not allowable. Furthermore, it is unreasonable to suppose that the conditions on a higher plane than the physical can be made responsible to a similar set of conditions on the physical plane. There are certain astronomers who base their speculations as to the habitability of other planets upon the absurd hypothesis that the conditions of life upon all planets must be the same as those on the earth, forgetting that the extent of the universe and the scope of motility of life itself are of such a nature as to admit of endless variations and adaptations. There is a realism of ideas and a realism of perceptual space. Yet this is no reason why the two should be identified. On the other hand, owing to the diversity in the universe, every consideration would naturally lead to the assumption that they are dissimilar. To invest ideas, notions, implications and inferences with a reality need not logically or otherwise affect the reality of a stone, a fig, or even of a sense-impression. To a being on the spirit levels our grossest realities must appear as non-existent. They are neither palpable nor contactable in any manner within the ordinary range of physical possibilities. For us his gravest experiences can have no reality whatsoever; for no matter how real an experience may be to him it is altogether beyond our powers of perception, and therefore, to us non-existent also. It should, however, be stated that the state of our knowledge about a given condition can in no way affect its existence. It merely establishes the fact that two or more realities may exist independent of one another and further that the gamut of realism in the universe is infinite and approaches a final state when its occlusion into absolute being follows as a logical sequence. Recurring to the consideration of the reality of spirit-realms as compared with that of sensible space, it comes to view that our idealism, that is, the idealism which is a quality of conceptualization, may be regarded as identical with their realism, at least as being on the same plane as it. Stated differently, the things that are ideal to us and which constitute the data of our consciousness may be as real to them as the commonest object of sense-knowledge is to us. What, therefore, appears to us as the most ethereal and idealistic may have quite a realistic character for them. Ultimately, however, and in the final deeps of analysis it will be found undoubtedly that both our realism and our idealism as well as similar qualities of the spirit world are in all essential considerations quite illusionary. All knowledge gained in a condition short of divinity itself is sadly relative. Even mathematical knowledge falls far short of the absolute, the fondest claims of the orthodox mathematician to the contrary notwithstanding. It has been said frequently that a mathematical fact is an absolute fact and that its verity, necessity and certainty cannot be questioned anywhere in the universe whether on Jupiter, Neptune, Fomalhaut, Canopus or Spica. But having so declared, the fact of the sheer relativity of our knowledge is not disturbed thereby nor controverted. Happily, neither distance nor a lack of distance can in any way affect the quality of human knowledge, mathematical knowledge not excepted. That can only be affected by conditions which cause it to approach perfection and nothing but evolution can do that. In the light of results obtained in analytic investigations the question of the flexibility of mathematical applications becomes evident and one instead of being convinced of the vaunted invariability of the laws obtaining in the world of mathesis is, on the other hand, made aware of the remarkable and seemingly unrestrained facility with which these laws may be made to apply to any conditions or set of assumptions within the range of the mind's powers of conception. Mathematicians have deified the _definition_ and endowed it with omnific powers imputing unto it all the attributes of divinity--immutability, invariance, and sempiternity. In this they have erred grievously although, perhaps, necessarily. Mathetic conclusions are entirely conditional and depend for their certainty upon the imputed certitude of other propositions which in turn are dependent, in ever increasing and endlessly complex relations, upon previously assumed postulates. These facts make it exceedingly difficult to understand the attitude of mind which has obscured the utter mutability and consequent ultimate unreliability of the fine-spun theories of analytic machinations. The apriority of all mathematical knowledge is open to serious questioning. And although there is no hesitancy in admitting the basic agreement of the most primary facts of mathematical knowledge with the essential character of the intellect the existence of well-defined limits for such congruence cannot be gainsaid. The subjunctive quality of geometric and analytical propositions is made apparent by an examination of the possibilities falling within the scope of permissibility offered by mathetic license. For instance, privileged to proceed according to the analytic method it is allowable to reconstruct the sequence of values in our ordinary system of enumeration so as to admit of the specification of a new value for say, the entire series of odd numbers. This value might be assumed to be a plus-or-minus one, dependent upon its posture in the series. That is, all odd numbers in the series beginning with the digit 3, and continuing, 5, 7, 9, 11, 13, 15, 17, 19, ... _n_, could be assumed to have only a place value which might be regarded as a constant-variable. The series of even numbers, 2, 4, 6, 8, 10, 12, 14, 16, ... _n_, may be assumed to retain their present sequence values. Under this system the digit 1 would have an absolute value; all other odd numbers would have a constant-variable value; constant, because always no more nor less than 1 dependent upon their place in the operations and whether their values were to be applied by addition or subtraction to or from one of the values in the even number series; variable, because their values would be determinable by their application and algebraic use. There would, of course, be utilitarian objection to a system of this kind; but under the conditions of a suppositionary hypothesis, it would be self-consistent throughout, and if given universal assent would suit our purposes equally as well as our present system. But the fact that this can be done under the mathematic method verily proves the violability of mathematical laws and completely negatives the assumption that the sum of any two digits, as say 2 plus 2 equals 4, is necessarily and unavoidably immutable. For it can be seen that the sum-value of all numbers may be made dependent upon the assumed value which may be assigned to them or to any collection thereof. Furthermore, it is a matter of historical knowledge that it was the custom of ancient races of men to account for values by an entirely different method from what we use to-day. The latter is a result of evolution and while experience teaches that it is by far the most convenient, it is nevertheless true that earlier men managed at least fairly well on a different basis. Then, too, the fact of the utility and universal applicability of our present system, based upon universal assent, does not obviate the conclusion that any other system, consistent in itself, might be made to serve our purposes as well. It ought to be said, however, in justice to the rather utilitarian results obtained by LA GRANGE, HELMHOLTZ, FECHNER, and others who strove to make use of their discoveries in analysis in solving mechanical, physiological and other problems of more or less pragmatic import that, in so far as this is true, mathematical knowledge must be recognized as being consistent with the necessities of _a priori_ requirements. But even these results may not be regarded as transcending the scope of the most fundamental principles of sense-experience. It will be discovered finally, perhaps, that the energy spent in elaborating complicate series of analytic curiosities has been misappropriated. It will then be necessary to turn the attention definitely to the study of that which lies not at the terminus of the intellect's _modus vivendi_, but which is both the origin of the intellect and its eternal sustainer--the intuition, or life itself. This can result in nothing less than the complete spiritualization of man's mental outlook and the consequent inevitable recognition of the underlying and ever-sustaining _one-ness_ of all vital manifestations. One of the curiosities of the tendency in man's mind to specialize in analytics, whether in the field of pure mathematics or metaphysics, is the fact that it almost invariably leads to an attempt to account for cosmic origins on the basis of paralogic theories. This in times past has given rise to the theory of the purely mechanical origin of the universe as well as many other fantastic fallacies the chief error of which lay in the failure to distinguish between the realism of mental concepts and that of the sensible world. In spite of this, however, one is bound to appreciate the beneficial effects of analytic operations because they serve as invigorants to mental growth. It could not, therefore, be wished that there were no such thing as analytics; for the equilibria-restoring property of the mind may at all times be relied upon to minimize the danger of excesses in either direction. Just as the tide flowing in flows out again, thereby restoring the ocean's equilibrium, so the mind ascending in one generation beyond the safety mark has its equilibrium restored in the next by a relinquishment of the follies of the former. The four-space is one of the curiosities of analytics; yet it need not be a menace to the sane contemplation of the variegated products of analysis. Safety here abides in the restraint which should characterize all discussion and application of the concept. If enthusiasts would be content not to transport the so-called fourth dimensional space out of the sphere of hyperspace and cease trying to speculate upon the results of its interposal into three space conditions, which is in every way a constructual impossibility, there could not be any possible objection to its due consideration. This would obviate the danger of calling into question either the sincerity or perspicacity of those whose enthusiasm tempts them to transgress the limits of propriety in their behavior towards the inquiry. There is but one life, one mind, one extension, one quantity, one quality, one being, one state, one condition, one mood, one affection, one desire, one feeling, one consciousness. There is also but one number and that is unity. All so-called integers are but fractional parts of this kosmic unity. The idea represented by the word _two_ really connotates two parts of unity and the same is true of a decillion, or any number of parts. These are merely the infinitesimals of unity and they grow less in size and consequence as the divisions increase in number. The analysis of unity into an infinity of parts is purely an _a posteriori_ procedure. That it is an inherent mind-process is a fallacy. All our common quantities, as the mile, kilometer, yard, foot, inch, gallon, quart, are conventional and arbitrary and susceptible of wide variations. As the basis of all physical phenomena is unity; it is only in the ephemeral manifestations of sensuous objects that they appear as separate and distinct quantities. We see on a tree many leaves, many apples or cherries; on a cob many grains of corn. We have learned to assign to each of these quantities in their summation a sequence value. But this is an empirical notion and cannot be said to inhere in the mind itself. Let us take, for instance, the mustard seed. If it were true that in one of these seeds there existed all the subsequent seeds which appear in the mustard plant as separate and identifiable quantities, and not in essence, then there would perhaps be warrant for the notion that diversity, as the calculable element, is an _a priori_ conception. But, as this is not the case and since diversity is purely empirical and pertains only to the efflorescence of the one life it is manifestly absurd to take that view. Under the most charitable allowances, therefore, there can be but two quantities--unity and diversity; yet not two, for these are one. Unity is the _one_ quantity and diversity is the division of unity into a transfinity of parts. Unity is infinite, absolute and all-inclusive. Diversity is finite although it may be admitted to be transfinite, or greater than any assignable value. Unity alone is incomprehensible. In order to understand something of its nature we divide it into a diversity of parts; and because we fail to understand the transfinity of the multitude of parts we mistakenly call them infinite. When analysis shall have proceeded far enough into the abysmal mysteries of diversity; when the mathematical mind shall have been overcome by the overwhelming perplexity of the maze of diverse parts, it shall then fall asleep and upon awaking shall find that wonderfully simple thing--_unity_. It is the one quantity that is endowed with a magnitude which is both inconceivable and irresolvable. The one ineluctable fact in the universe is the incomprehensibility and all-inclusivity of _one-ness_. It is incomprehensible, inconceivable and infinite at the present stage of mind development. But the goal of mind is to understand the essential character of unity, of life. Its evolution will then stop, for it will have reached the prize of divinity itself whereupon the intellect exalted by and united with the intuition shall also become one with the divine consciousness. CHAPTER II HISTORICAL SKETCH OF THE HYPERSPACE MOVEMENT Egypt the Birthplace of Geometry--Precursors: NASIR-EDDIN, CHRISTOPH CLAVIUS, SACCHERI, LAMBERT, LA GRANGE, KANT--Influence of the _Mecanique Analytique_--The Parallel-Postulate the Root and Substance of the Non-Euclidean Geometry--The Three Great Periods: The Formative, Determinative and Elaborative--RIEMANN and the Properties of Analytic Spaces. The evolution of the idea of a fourth dimension of space covers a long period of years. The earliest known record of the beginnings of the study of space is found in a hieratic papyrus which forms a part of the Rhind Collection in the British Museum and which has been deciphered by EISENLOHR. It is believed to be a copy of an older manuscript of date 3400 B. C., and is entitled "_Directions for Knowing All Dark Things_" The copy is said to have been made by AHMES, an Egyptian priest between 1700 and 1100 B. C. It begins by giving the dimensions of barns; then follows the consideration of various rectilineal figures, circles, pyramids, and the value of pi ([Greek: p]). Although many of the solutions given in the manuscript have been found to be incorrect in minor particulars, the fact remains that Egypt is really the birth-place of geometry. And this fact is buttressed by the knowledge that THALES, long before he founded the Ionian School which was the beginning of Greek influence in the study of mathematics, is found studying geometry and astronomy in Egypt. The concept of hyperspace began to germinate in the latter part of the first century, B. C. For it was at this date that GEMINOS of RHODES (B. C. 70) began to think seriously of the mathematical labyrinth into which EUCLID'S parallel-postulate most certainly would lead if an attempt at demonstrating its certitude were made. He recognized the difficulties which would engage the attention of those who might venture to delve into the mysterious possibilities of the problem. There is no doubt, too, but that EUCLID himself was aware, in some measure at least, of these difficulties; for his own attitude towards this postulate seems to have been one of noncommittance. It is, therefore, not strange that the astronomer, PTOLEMY (A. D. 87-165), should be found seeking to prove the postulate by a consideration of the possibilities of interstellar triangles. His researches, however, brought him no relief from the general dissatisfaction which he felt with respect to the validity of the problem itself. For nearly one thousand years after the attempts at solving the postulate by GEMINOS and PTOLEMY, the field of mathematics lay undisturbed. For it was at this time that there arose a strange phenomenon, more commonly known as the "Dark Ages," which put an effectual check to further research or independent investigations. Mathematicians throughout this long lapse of time were content to accept EUCLID as the one incontrovertible, unimpeachable authority, and even such investigations as were made did not have a rebellious tendence, but were mainly endeavors to substantiate his claims. Accordingly, it was not until about the first half of the thirteenth century that any real advance was made. At this time there appeared an Arab, NASIR-EDDIN (1201-1274) who attempted to make an improvement on the problem of parallelism. His work on EUCLID was printed in Rome in 1594 A. D., about three hundred and twenty years after his demise and was communicated in 1651 by JOHN WALLIS (1616-1703) to the mathematicians of Oxford University. Although his calculations and conclusions were respectfully received by the Oxford authorities no definite results were regarded as accomplished by what he had done. It is believed, however, that his work reopened speculation upon the problem and served as a basis, however slight, for the greater work that was to be done by those who followed him during the next succeeding eight hundred years. About twenty years before the printing of the work of NASIR-EDDIN, CHRISTOPH CLAVIUS (1574) deduced the axiom of parallels from the assumption that a line whose points are all equidistant from a straight line is itself straight. In his consideration of the parallel-postulate he is said to have regarded it as EUCLID'S XIIIth axiom. Later BOLYAI spoke of it as the XIth and later still, TODHUNTER treated it as the XIIth. Hence, there does not seem to have been any general unanimity of opinion as to the exact status of the parallel-postulate, and especially is this true in view of the uncertainty now known to have existed in EUCLID'S mind concerning it. GIROLAMO SACCHERI (1667-1733), a learned Jesuit, born at San Remo, came next upon the stage. And so important was his work that it will perpetuate the memory of his name in the history of mathematics. He was a teacher of grammar in the Jesuit _Collegio di Brera_ where TOMMASO CEVA, a brother of GIOVANNI, the well-known mathematician, was teacher of mathematics. His association with the CEVA brothers was especially beneficial to him. He made use of CEVA'S very ingenious methods in his first published book, 1693, entitled _Solutions of Six Geometrical Problems Proposed by Count Roger Ventimiglia_. A +--------------------------------+ B | | | | | | | | | | | | C +--------------------------------+ D FIG. 1. SACCHERI attacked the problem of parallels in quite a new way. Examining a quadrilateral, _ABCD_, in which the angles _A_ and _B_ are right angles and the sides _AC_ and _BD_ are equal, he determined to show that the angles _C_ and _D_ are equal. He also sought to prove that they are either right angles, obtus acute. He undertook to prove the falsity of the latter two propositions (that they are either obtuse or acute), leaving as the only possibility that they must be right angles. In doing so, he found that his assumptions led him into contradictions which he experienced difficulty in explaining. His labors in connection with the solution of the problems proposed by COUNT VENTIMIGLIA, including his work on the question of parallels, led directly into the field of metageometrical researches, and perhaps to him as to no other who had preceded him, or at least to him in a larger degree, belongs the credit for a continued renewal of interest in that series of investigations which resulted in the formulation of the non-Euclidean geometry. The last published work of SACCHERI was a recital of his endeavors at demonstrating the parallel-postulate. This received the "Imprimatur" of the Inquisition, July 13, 1733; the Provincial Company of Jesus took possession of the book for perusal on August 16, 1733; but unfortunately within two months after it had been reviewed by these authorities, SACCHERI passed away. All efforts which had been made prior to the work of SACCHERI were based upon the assumption that there must be an equivalent postulate which, if it could be demonstrated, would lead to a direct, positive proof of EUCLID'S proposition. Although these and all other attempts at reaching such a proof have signally failed and although it may correctly be said that the entire history of demonstrations aiming at the solution of the famous postulate has been one long series of utter failures, it can be asserted with equal certitude that it has proven to be one of the most fruitful problems in the history of mathematical thought. For out of these failures has been built a superstructure of analytical investigations which surpasses the most sanguine expectations of those who had labored and failed. In 1766 JOHN LAMBERT (1728-1777) wrote a paper upon the _Theory of Parallels_ dated Sept. 5, 1766, first published in 1786, from the papers left by F. BERNOULLI, which contained the following assertions:[2] 1. The parallel-axiom needs proof, since it does not hold for geometry on the surface of the sphere. 2. In order to make intuitive a geometry in which the triangle's sum is less than two right angles, we need an "imaginary" sphere (the pseudosphere). 3. In a space in which the triangle's sum is different from two right angles there is an absolute measure (a natural unit for length). At this time IMMANUEL KANT (1724-1804), the noted German metaphysician, was in the midst of his philosophical labors. And it is believed that it was he who first suggested the idea of different _spaces_. Below is given a statement taken from his _Prolegomena_[3] which corroborates this view. "That complete space (which is itself no longer the boundary of another space) has three dimensions, and that space in general cannot have more, is based on the proposition that not more than three lines can intersect at right angles in one point.... That we can require a line to be drawn to infinity, a series of changes to be continued (for example, _spaces_ passed through by motion) in indefinitum, presupposes a representation of space and time which can only attach to intuition." [2] Vide _New York Mathematical Society Bulletin_, Vol. III, 1893-4, p. 79, G. B. HALSTEAD on _Lambert's Non-Euclidean Geometry_. [3] _Prolegomena_, KANT, p. 37, Trans. by J. P. MAHAFFY and J. H. BERNARD. His differentiation between space in general and space which may be considered as the "boundary of another space" shows, in the light of the subsequent developments of the mathematical idea of space that he very fully appreciated the marvelous scope of analytic spaces. His conception of space, therefore, must have had a profound influence upon the mathematic thought of the day causing it to undergo a rapid reconstruction at the hands of geometers who came after him. Under the masterly influence of LA GRANGE (1736-1813) the idea of different spaces began to take definite shape and direction; the geometry of hyperspace began to crystallize; and the field of mathesis prepared for the growth of a conception the comprehension of which was destined to be the profoundest undertaking ever attempted by the human mind. Unlike most great men whom the world learns tardily to admire, LA GRANGE lived to see his talents and genius fully recognized by his compeers; for he was the recipient of many honors both from his countrymen and his admirers in foreign lands. He spent twenty years in Prussia where he went upon the invitation of FREDERICK the Great who in the Royal summons referred to himself as the "greatest king in Europe" and to LA GRANGE as the "greatest mathematician" in Europe. In Prussia the _Mecanique Analytique_ and a long series of memoirs which were published in the Berlin and Turin Transactions were produced. LA GRANGE did not exhibit any marked taste for mathematics until he was 17 years of age. Soon thereafter he came into possession of a memoir by HALLEY quite by accident and this so aroused his latent genius that within one year after he had reviewed HALLEY'S memoir he became an accomplished mathematician. He created the calculus of variations, solved most of the problems proposed by Fermat, adding a number of theorems of his own contrivance; raised the theory of differential equations to the position of a science rather than a series of ingenious methods for the solution of special problems and furnished a solution for the famous isoperimetrical problem which had baffled the skill of the foremost mathematicians for nearly half a century. All these stupendous tasks he performed by the time he reached the age of nineteen. The _Mecanique Analytique_ is his greatest and most comprehensive work. In this he established the law of virtual work from which, by the aid of his calculus of variations, he deduced the whole of mechanics, including both solids and liquids. It was his object in the _Analytique_ to show that the whole subject of mechanics is implicitly embraced in a single principle, and to lay down certain formulae from which any particular result can be obtained. He frequently made the assertion that he had, in the _Mecanique Analytique_, transformed mechanics which he persistently defined as a "geometry of four dimensions"[4] into a branch of analytics and had shown the so-called mechanical principles to be the simple results of the calculus. Hence, there can be no doubt but that LA GRANGE not only completed the foundation, but provided most of the material in his analyses and other "abstract results of great generality" which he obtained in his numerous calculations, for the superstructure subsequently known as the geometry of hyperspace, and in which the fourth dimensional concept occupies a very fundamental place. [4] In 1754 D'ALEMBERT (1717-1783) published an article in the famous old _Encyclopedia_ edited by DIDEROT and himself on _Dimension_. In this article the idea of the fourth dimension is dwelt upon at length. The view which he expressed in this article, of course, served greatly to popularize the conception among the learned men of the day, and owing to the close relationship existing between D'ALEMBERT and LA GRANGE, it is not surprising that the latter should have been very much enamored of the idea. It is as if for nearly seventeen hundred years workmen, such as GEMINOS, of RHODES, PTOLEMY, SACCHERI, NASIR-EDDIN, LAMBERT, CLAVIUS, and hundred of others who struggled with the problem of parallels, had made more or less sporadic attempts at the excavation of the land whereon a marvelously intricate building was to be constructed. There is no historical evidence to show that any of them ever dreamed that the results of their labors would be utilized in the manner in which they have been used. Then came KANT with the wonderfully penetrating searchlight of his masterful intellect who from the elevation which he occupied saw that the site had great possibilities, but he had not the mathematical talent to undertake the work of actual, methodical construction. Indeed his task was of a different sort. However, he succeeded in opening the way for LA GRANGE and others who followed him. LA GRANGE immediately seized upon the idea which for more than a thousand years had been impinging upon the minds of mathematicians vainly seeking lodgment and began the elaboration of a plan in accordance with which minds better skilled in the pragmatic application of abstract principles than his could complete the work begun. Unfortunately, on account of his intense devotion and loyalty to the study of pure mathematics, and when he had reached the summit of his greatness where he stood "without a rival as the foremost living mathematician," his health became seriously affected, causing him to suffer constant attacks of profound melancholia from which he died on April 10, 1813. We come now to one of the most remarkable periods in the history of mental development. During the six hundred years between the birth of NASIR-EDDIN and the death of LA GRANGE the entire world of mathesis was being reconstituted. Since there had been gradually going on an internal process which, when completed, forever would liberate the mind from the narrow confines of consciousness limited to the three-space, it is not surprising that we should find, in the mathematical thought of the time, an absolutely epoch-making departure. The innumerable attempts at the solution of the parallel-postulate, all failures in the sense that they did not prove, have intensified greatly the esteem in which the never-dying elements of EUCLID are held to-day. And despite the fact that there may come a time when his axioms and conclusions may be found to be incongruent with the facts of sensuous reality; and though all of his fundamental conceptions of space in general, his theorems, propositions and postulates may have to give way before the searching glare of a deeper knowledge because of some revealed fault, the perfection of his work in the realm of pure mathematics will remain forever a master piece demanding the undiminished admiration of mankind. The parallel-postulate, as stated by EUCLID in his _Elements of Geometry_, reads as follows: "If a straight line meet two straight lines so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles." On this postulate hang all the "law and the prophets" of the non-Euclidean Geometry. In it are the virtual elements of three possible geometries. Furthermore, it is both the warp and the woof of the loom of present-day metageometrical researches. It is the golden egg laid by the god SEB at the beginning of a new life cycle in psychogenesis. Its progeny are numerous--hyperspaces, sects, straights, digons, equidistantials, polars, planars, coplanars, invariants, quaternions, complex variables, groups and many others. A wonderfully interesting breed, full of meaning and pregnant with the power of final emancipations for the human intellect! When the conclusions which were systematically formulated as a result of the investigations along the lines of hypotheses which controverted the parallel-postulate were examined it was found that they fell into three main divisions, namely: the synthetic or hyperbolic; the analytic or RIEMANNIAN and the elliptic or CAYLEY-KLEIN. These divisions or groups are based upon the three possibilities which inhere in the conception taken of the sum of the angles referred to in the above postulate as to whether it is equal to, greater or less than two right angles. The assumption that the angular sum is congruent to a straight angle is called the Euclidean or parabolic hypothesis and is to be distinguished from the synthetic or hyperbolic hypothesis established by GAUSS, LOBACHEVSKI and BOLYAI and which assumes that the angular sum is less than a straight angle. The elliptic or CAYLEY-KLEIN hypothesis assumes that the angular sum is greater than a straight angle. LOBACHEVSKI, however, not satisfied with the statement of the parallel-postulate as given by EUCLID and which had caused the age-long controversy, substituted for it the following: "All straight lines which, in a plane, radiate from a given point, can, with respect to any other straight line, in the same plane, be divided into two classes--the intersecting and the non-intersecting. The boundary line of the one and the other class is called parallel to the given line." This is but another way of saying about the same thing that EUCLID had declared before, and yet, curiously enough it afforded just the liberty that LOBACHEVSKI needed to enable him to elaborate his theory. For the purposes of this sketch the field of the development of non-Euclidean geometry is divided into three periods to be known as: (1) the _formative_ period in which mathematical thought was being formulated for the new departure; (2) the _determinative_ period during which the mathematical ideas were given direction, purpose and a general tendence; (3) the _elaborative_ period during which the results of the former periods were elaborated into definite kinds of geometries and attempts made at popularizing the hypotheses. The Formative Period CHARLES FREDERICH GAUSS (1777-1855) by some has been regarded as the most influential mathematician that figured in the formulation of the non-Euclidean geometry; but closer examination into his efforts at investigating the properties of a triangle shows that while his researches led to the establishment of the theorem that a regular polygon of seventeen sides (or of any number which is prime, and also one more than a power of two) can be inscribed, under the Euclidean restrictions as to means, in a circle, and also that the common spherical angle on the surface of a sphere is closely connected with the constitution of the area inclosed thereby, he cannot justly be designated as the leader of those who formulated the synthetic school. And this, too, for the simple reason that, as he himself admits in one of his letters to TAURINUS, he had not "published anything on the subject." In this same letter he informs TAURINUS that he had pondered the subject for more than thirty years and expressed the belief that there could not be any one who had "concerned himself more exhaustively with this second part (that the sum of the angles of a triangle cannot be more than 180 degrees)" than he had. Writing from Göttingen to TAURINUS, November 8, 1824, and commenting upon the geometric value of the sum of the angles of a triangle, he says: "Your presentation of the demonstration that the sum of the angles of a plane triangle cannot be greater than 180 degrees does, indeed, leave something to be desired in point of geometrical precision. But this could be supplied, and there is no doubt that the impossibility in question admits of the most rigorous demonstration. But the case is quite different with the second part, namely, that the sum of the angles cannot be smaller than 180 degrees; this is the real difficulty, the rock upon which all endeavors are wrecked.... The assumption that the sum of the three angles is smaller than 180 degrees leads to a new geometry entirely different from our Euclidean--a geometry which is throughout consistent with itself, and which I have elaborated in a manner entirely satisfactory to myself, so that I can solve every problem in it with the exception of the determining of a constant which is not _a priori_ obtainable." It appears from this correspondence that GAUSS had in the privacy of his own study elaborated a complete non-Euclidean geometry, and had so thoroughly familiarized himself with its characteristics and possibilities that the solution of every problem embraced within it was very clear to him except that of the determination of a constant. He concluded the above letter by saying: "All my endeavors to discover contradiction or inconsistencies in this non-Euclidean geometry have been in vain, and the only thing in it that conflicts with our reason is the fact that if it were true there would necessarily exist in space a linear magnitude quite determinate in itself; yet unknown to us." Judging from the correspondence between GAUSS and GERLING (1788-1857), BESSEL (1784-1846), SCHUMACHER and TAURINUS, the nephew of SCHWEIKART, and that between SCHWEIKART and GERLING, there had grown up a general dissatisfaction in the minds of mathematicians of this period with Euclidean geometry and especially the parallel-postulate and its connotations. BESSEL expresses this general discontent in one of his letters to GAUSS, dated February 10, 1829, in which he says: "Through that which LAMBERT said and what SCHWEIKART disclosed orally, it has become clear to me that our geometry is incomplete, and should receive a correction, which is hypothetical, and if the sum of the angles of the plane triangle is equal to 180 degrees, vanishes." The opinion of leading mathematicians at this time seems to have been crystallizing very rapidly. Unconsciously the men of this formative period were adducing evidence which would give form and tendence to the developments in the field of mathesis at a later date. They appear to have been reaching out for that which, ignis fatuus-like, was always within easy reach, but not quite apprehensible. A bolder student than GAUSS was FERDINAND CARL SCHWEIKART (1780-1857) who also has been credited with the founding of the non-Euclidean geometry. In fact, if judged by the same standards as GAUSS, he would be called the "father of the geometry of hyperspace"; for he really published the first treatise on the subject. This was in the nature of an inclosure which he inserted between the leaves of a book he loaned to GERLING. He also asked that it be shown to GAUSS that he might give his judgment as to its merits. SCHWEIKART'S treatise, dated Marburg, December, 1818, is here quoted in full: "There is a two-fold geometry--a geometry in the narrower sense, the Euclidean, and an astral science of magnitude. "The triangles of the latter have the peculiarity that the sum of the three angles is not equal to two right angles. "This presumed, it can be most rigorously proven: (_a_) That the sum of the three angles in the triangle is less than two right angles. "(_b_) That this sum becomes ever smaller, the more content the angle incloses. (_c_) That the altitude of an isosceles right-angled triangle indeed ever increases, the more one lengthens the side; that it, however, cannot surpass a certain line which I call the constant." Squares have consequently the following form: [Illustration: FIG. 2.] "If this constant were for us the radius of the earth (so that every line drawn in the universe, from one fixed star to another, distant 90° from the first, would be a tangent to the surface of the earth) it would be infinitely great in comparison with the spaces which occur in daily life." The above, being the first published, not printed, treatise on the new geometry occupies a unique place in the history of higher mathematics. It gave additional strength to the formative tendencies which characterized this period and marked SCHWEIKART as a constructive and original thinker. The nascent aspects of this stage received a fruitful contribution when NICOLAI LOBACHEVSKI (1793-1847) created his _Imaginary Geometry_ and JANOS BOLYAI (1802-1860) published as an appendix to his father's _Tentamen_, his _Science Absolute of Space_. LOBACHEVSKI and BOLYAI have been called the "Creators of the Non-Euclidean Geometry." And this appellation seems richly to be deserved by these pioneers. Their work gave just the impetus most needed to fix the status of the new line of researches which led to such remarkable discoveries in the more recent years. The _Imaginary Geometry_ and the _Science Absolute of Space_ were translated by the French mathematician, J. HOÜEL in 1868 and by him elevated out of their forty-five years of obscurity and non-effectiveness to a position where they became available for the mathematical public. To BOLYAI and LOBACHEVSKI, consequently, belong the honor of starting the movement which resulted in the development of metageometry and hence that which has proved to be the gateway of a new mathematical freedom. GAUSS, SCHWEIKART, LOBACHEVSKI, WOLFGANG and JANOS BOLYAI were the principal figures of the formative period and the value of their work with respect to the formulation of principles upon which was constructed the Temple of Metageometry cannot be overestimated. The Determinative Period This period is characterized chiefly by its close relationship to the theory of surfaces. RIEMANN'S Habilitation Lecture on _The Hypotheses Which Constitute the Bases of Geometry_ marks the beginning of this epoch. In this dissertation, RIEMANN not only promulgated the system upon which GAUSS had spent more than thirty years of his life in elaborating, for he was a disciple of GAUSS; but he disclosed his own views with respect to space which he regarded as a particular case of manifold. His work contains two fundamental concepts, namely, the _manifold_ and the _measure of curvature_ of a continuous manifold, possessed of what he called _flatness_ in the smallest parts. The conception of the measure of curvature is extended by RIEMANN from surfaces to spaces and a new kind of space, finite, but unbounded, is shown to be possible. He showed that the dimensions of any space are determined by the number of measurements necessary to establish the position of a point in that space. Conceiving, therefore, that space is a manifold of finite, but unbounded, extension, he established the fact that the passage from one element of a manifold to another may be either discrete or continuous and that the manifold is discrete or continuous according to the manner of passage. Where the manifold is regarded as discrete two portions of it can be compared, as to magnitude, by counting; where continuous, by measurement. If the whole manifold be caused to pass over into another manifold each of its elements passing through a one-dimensional manifold, a two-dimensional manifold is thus generated. In this way, a manifold of _n_-dimensions can be generated. On the other hand, a manifold of _n_-dimensions can be analyzed into one of one dimension and one of (_n_-1) dimensions. To RIEMANN, then, is due the credit for first promulgating the idea that space being a special case of manifold is generable, and therefore, _finite_. He laid the foundation for the establishment of a special kind of geometry known as the "elliptic." Space, as viewed by him, possessed the following properties, viz.: generability, divisibility, measurability, ponderability, finity and flexity. These are the six pillars upon which rests the structure of hyperspace analyses.[5] [5] Vide _Nature_, Vol. VIII, pp. 14-17; 36, 37 (1873); also _Mathematical Papers_, pp. 65-71. _Generability_ is that property of geometric space by virtue of which it may be generated, or constructed, by the movement of a line, plane, surface or solid in a direction without itself. _Divisibility_ is that property of geometric space by virtue of which it may be segmented or divided into separate parts and superposed, or inserted, upon or between each other. _Measurability_ is that property by virtue of which geometric space is determined to be a manifold of either a positive or negative curvature, also by which its extent may be measured. _Ponderability_ is that property of geometric space by virtue of which it may be regarded as a quantity which can be manipulated, assorted, shelved or otherwise disposed of. _Finity_ is that property by virtue of which geometric space is limited to the scope of the individual consciousness of a unodim, a duodim or a tridim and by virtue of which it is finite in extent. _Flexity_ is that property by virtue of which geometric space is regarded as possessing curvature, and in consequence of which progress through it is made in a curved, rather than a geodetic line, also by virtue of which it may be flexed without disruption or dilatation. RIEMANN who thus prepared the way for entrance into a veritable labyrinth of hyperspaces is, therefore, correctly styled "The father of metageometry," and the fourth dimension is his eldest born. He died while but forty years of age and never lived long enough fully to elaborate his theory with respect to its application to the measure of curvature of space. This was left for his very energetic disciple, EUGENIO BELTRAMI (1835-1900) who was born nine years after RIEMANN and lived thirty-four years longer than he. His labors mark the characteristic standpoint of the determinative period. BELTRAMI'S mathematical investigations were devoted mainly to the non-Euclidean geometry. These led him to the rather remarkable conclusion that the propositions embodied therein relate to figures lying upon surfaces of constant negative curvature. BELTRAMI sought to show that such surfaces partake of the nature of the pseudosphere, and in doing so, made use of the following illustration: [Illustration: FIG. 3.] [Illustration: FIG. 4.] If the plane figure _aabb_ is made to revolve upon its axis of symmetry _AB_ the two arcs, _ab_ and _ab_ will describe a pseudospherical concave-convex surface like that of a solid anchor ring. Above and below, toward _aa_ and _bb_, the surface will turn outward with ever-increasing flexure till it becomes perpendicular to the axis and ends at the edge with one curvature infinite. Or, the half of a pseudospherical surface may be rolled up into the shape of a champagne glass, as in Fig. 4. In this way, the two straightest lines of the pseudospherical surface may be indefinitely produced, giving a kind of space (pseudospherical) in which the axiom of parallels does not hold true. The determinative period marks the most important stage in the development of non-Euclidean geometry and certainly the most significant in the evolution of the idea of hyperspaces and multiple dimensionality. RIEMANN and BELTRAMI are chief among those whose labors characterize the scope of this period. Their work gave direction and general outline for later developments and all subsequent researches along these lines have been conducted in strict conformity with the principles laid down by these pioneer constructionists. They laid out the field and designated its confines beyond which no adventurer has since dared to pass. The great importance of the work of RIEMANN at this time may be seen further in the fact that it not only marked the beginning of a new epoch in geometry; but his pronouncement of the hypothesis that space is unbounded, though finite, is really the first time in the history of human thought that expression was ever given to the idea that space may yet be only of limited extent. Before that time the minds of all men seemed to have been unanimous in the consideration of space as an illimitable and infinite quantity. The Elaborative Period The elaborative stage includes the work of all those who, working upon the bases laid down by LOBACHEVSKI, BOLYAI, SCHWEIKART and RIEMANN, have sought to amplify the conclusions reached by them. Among those whose investigations have greatly multiplied the applications of hyperspace conceptions are HOÜEL (1866) and FLYE ST. MARIE (1871) of France; HELMHOLTZ (1868), FRISCHAUF (1872), KLEIN (1849), and BALTZER (1877) of Germany; BELTRAMI (1872) of Italy; DE TILLY (1879) of Belgium; CLIFFORD and CAYLEY (1821) of England; NEWCOMB (1835) and HALSTEAD of America. These have been most active in popularizing the subject of non-Euclidean geometry and incidentally the idea of the fourth dimension. The great mass of non-professional mathematical readers, therefore, owe these men an immeasurable debt of gratitude for the work that they have done in the matter of rendering the conceptions which constitute the fabric of metageometry understandable and thinkable. A glance at the bibliography appended at the end of this volume will give some idea of the enormous amount of labor that has been expended in an effort to translate the most abstract mathematical principles into a language that could easily be comprehended by the average intelligent person. The characteristic standpoint of this period is the popular comprehension of the hyperspace concept and the consequent mental liberation which follows. For there is no doubt but that unheard of possibilities of thought have been revealed by investigations into the nature of space. An entirely new world has been opened to view and only a beginning has been made at the exploration of its extent and resources. One of the notable incidents of the early years of this period is the position taken by FELIX KLEIN who stands in about the same relation to CAYLEY as BELTRAMI does to RIEMANN, in that he assumed the task of completing the work of his predecessor. KLEIN held that there are only two kinds of RIEMANNIAN _space_--the elliptical and the spherical. Or in other words, that there are only two possible kinds of space in which the propositions announced by RIEMANN could apply. SOPHUS LIE, called the "great comparative anatomist of geometric theories," carried his classifications to a final conclusion in connection with spaces of all kinds and decided that there are possible only four kinds of three dimensional spaces. But whether men with peering, microscopic, histological vision shall establish the existence of one or many spaces, and regardless of the mathematic rigor with which they shall demonstrate the self-consistency of the doctrines which they hold, the fact remains that the hypotheses thus maintained, while they may be regarded as true descriptions of the spaces concerned, are, nevertheless, incompatible. All of them cannot be valid. It will perhaps be found that none of them are valid, especially objectively so. The only true view, therefore, of these systems of hyperspaces is that which assigns them to their rightful place in the infinitely vast world of pure mathesis where their validity may go unchallenged and their existence unquestioned; for in that domain of unconfined mentation, in that realm of divine intuitability, the marvelous wonderland of ideas and notions, one is not only disinclined to doubt their logical actuality, but is quite willing to accede their claims. CHAPTER III ESSENTIALS OF THE NON-EUCLIDEAN GEOMETRY The Non-Euclidean Geometry Concerned with Conceptual Space Entirely--Outcome of Failures at Solving the Parallel-Postulate--The Basis of the Non-Euclidean Geometry--Space Curvature and Manifoldness--Some Elements of the Non-Euclidean Geometry--Certainty, Necessity and Universality as Bulwarks of Geometry--Some Consequences of Efforts at Solving the Parallel-Postulate--The Final Issue of the Non-Euclidean Geometry--Extended Consciousness. The term "non-Euclidean" is used to designate any system of geometry which is not strictly Euclidean in content. It is interesting to note how the term came to be used. It appears to have been employed first by GAUSS. He did not strike upon it suddenly, however, as in the correspondence between him and WACHTER in 1816 he used the designation "anti-Euclidean" and then, later, following SCHWEIKART, he adopted the latter's terminology and called it "Astral Geometry." This he found in SCHWEIKART'S first published _treatise_ known by that name and which made its appearance at Marburg in December, 1818. Finally, in his correspondence with TAURINUS in 1824, GAUSS first used the expression "non-Euclidean" to designate the system which he had elaborated and continued to use it in his correspondence with SCHUMACHER in 1831. "Non-Legendrean," "semi-Euclidean" and "non-Archimedean" are titles used by M. DEHN to denote all kinds of geometries which represented variations from the hypotheses laid down by LEGENDRE, EUCLID and ARCHIMEDES. The semi-Euclidean is a system of geometry in which the sum of the angles of a triangle is said to be equal to two right angles, but in which one may draw an infinity of parallels to a straight line through a given point. The non-Euclidean geometry embraces all the results obtained as a consequence of efforts made at finding a satisfactory proof of the parallel-postulate and is, therefore, based upon a conception of space which is at variance with that held by EUCLID. According to the Ionian school space is an infinite continuum possessing uniformity throughout its entire extent. The non-Euclideans maintain that space is not an infinite extension; but a finite though unbounded manifold capable of being generated by the movement of a point, line or plane in a direction without itself. It is also held that space is curved and exists in the shape of a sphere or pseudosphere and is consequently elliptical. The inapplicability of EUCLID's parallel-postulate to lines drawn upon the surface of a sphere suggested the possibility of a space in which the postulate could apply to all possible surfaces or that space itself may be spherical in which case the postulate would be invalidated altogether. Hence, it is quite natural that mathematicians finding themselves unable to prove the postulate with due mathetic precision should turn their attention to the conceptually possible. In this virtual abandonment of the perceptual for the conceptual lies the fundamental difference between the Euclidean and the non-Euclidean geometries. It may be said to the credit of the Euclideans that they have sought to make their geometric conceptions conform as closely as possible to the actual nature of things in the sensuous world while at the same time they must have perceived that at best their spatial notions were only approximations to the sensuous actuality of objects in space. On the other hand, non-Euclideans make no pretense at discovering any congruency between their notions and things as they actually are. The attitude of the metageometricians in this respect is very aptly described by CASSIUS JACKSON KEYSER who says: "He constructs in thought a summitless hierarchy of hyperspaces, an endless series of orderly worlds, worlds that are possible and logically actual, and he is content not to know if any of them be otherwise actual or actualized."[6] [6] _Mathematics_, by C. J. KEYSER, Adrian Professor of Mathematics, Columbia University. The non-Euclidean is, therefore, not concerned about the applicability of ensembles, notions and propositions to real, perceptual space conditions. It is sufficient for him to know that his creations are thinkable. As soon as he can resolve the nebulosity of his consciousness into the conceptual "star-forms" of definite ideas and notions, he sits down to the feast which he finds provided by superfoetated hypotheses fabricated in the deeps of mind and logical actualities imperturbed and unmindful of the weal of perceptual space in its homogeneity of form and dimensionality. Fundamentally, the non-Euclidean geometry is constructed upon the basis of conceptual space almost entirely. Knowledge of its content is accordingly derived from a superperceptual representation of relations and interrelations subsisting between and among notions, ideas, propositions and magnitudes arising out of a conceptual consideration thereof. In other words, representations of the non-Euclidean magnitudes, cannot be said to be strictly perceptual in the same sense that three-space magnitudes are perceived; for three-space magnitudes are really sense objects while hyperspace magnitudes are not sense objects. They are far removed from the sensuous world and in order to conceive them one must raise his consciousness from the sensuous plane to the conceptual plane and become aware of a class of perceptions which are not perceptions in the strict sense of the word, but superperceptions; because they are representations of concepts rather than precepts. Notions of perceptual space are constituted of the triple presentations arising out of the visual, tactual and motor sensations which are fused together in their final delivery to the consciousness. The synthesis of these three sense-deliveries is accomplished by equilibrating their respective differences and by correcting the perceptions of one sense by those of another in such a way as to obtain a completely reliable perception of the object. This is the manner in which the characteristics of Euclidean space are established. The characteristics of non-Euclidean space are not arrived at exactly in this way. Being beyond the scope of the visual, tactile and motor sense apprehensions, it cannot be said to represent judgments derived from any consideration or elaboration of the deliveries presented through these media. Yet, the substance of metageometry, or the science of the measurement of hyperspaces, may not be regarded as an _a priori_ substructure upon which the system is founded. That is, the conceptual space of non-Euclidean geometry is not presented to the consciousness as an _a priori_ notion. On the other hand, the _a posterioristic_ quality of metageometric spaces marks the entire scope of motility of the notions appertaining thereto. The notions, therefore, of conceptual space are derivable only from the perception of concepts, or, otherwise consist of judgments concerning interconceptual relations. The process of apperception involved in the recognition of relations which may be methodically determined is much removed from the primary procedure of perceiving sense-impressions and fusing them into final deliveries to the consciousness for conceptualization or the elaboration into concepts or general notions. It is a procedure which is in every way superconceptual and extra-sensuous. The metageometrician or analyst in no way relies upon sense-deliveries for the data of his constructions; for, if he did, he should, then, be reduced to the necessity of confining his conclusions to the sphere of motility imposed by the sensible world with the result that we should be able to verify empirically all his postulations. But, contrarily, he goes to the extra-sensuous, and there in the realm of pure conceptuality, he finds the requisite freedom for his theories; thus, environed by a sort of intellectual anarchism, he pursues analytical pleasures quite unrestrainedly. The difference between the two mental processes--that which leads from the sensible world to conception and that which veers into the fields beyond--is so great that it is hardly permissible to view the results arrived at in the outcome of the separate processes as being identical. To illustrate this difference, let us draw an analogy. The miner digs the iron ore out of the ground. The iron is separated from the extraneous material and delivered to the furnaces where the metal is melted and turned out as pig iron. It is further treated, and steel, of various grades, cast iron and other kinds of iron are produced. The treatment of the iron ore up to this stage is similar to the treatment of sense-impressions by the Thinker. Steel, cast iron, et cetera, are similar to mental concepts. Later, the steel and other products are converted into instruments and numerous articles. This represents the superperceptual process. Trafficking in iron ore products, such as instruments of precision, watch springs, and the like, represents a stage still farther removed from the primary treatment of the ore and is similar to that to which concepts are treated when the metageometrician manipulates them in the construction of conceptual space-forms. Perception is the dealing with raw iron ore while conception is analogous to the production of the finished product. Superperception would be analogous to the trafficking in the finished product as such and without any reference to the source or the preceding processes. Thus the notions and judgments of the non-Euclidean geometry are arrived at as a result of a triple process of perception, conception and superperception the latter being merely superconceived as formal space-notions. But it is obvious that the more complex the processes by which judgments purporting to relate to perceptual things are derived the more likely are those judgments to be at variance with the nature of the things themselves. In view of the foregoing, the dangers resulting from identifying the products of the two processes are very obvious indeed. But the difference between the two procedures is the difference between Euclidean and non-Euclidean geometries or the difference between perceptual space notions and conceptual space notions. Hence, it is not understood just how or why it has occurred to anyone that the two notions could be made congruent. Magnitudes in perceptual, sensible space are things apart from those that may be said to exist in mathematical space or that space whose qualities and properties have no existence outside of the mind which has conceived them. It is believed to be quite impossible to approach the study of metageometrical propositions with a clear, open mind without previously understanding the fundamental distinctions which exist between them. It follows, therefore, as a logical conclusion that geometric space of whatsoever nature is a purely formal construction of the intellect, and for this reason is completely under the sovereignty of the intellect however whimsical its demands may be. Being thus the creature of the intellect, its possibilities are limited only by the limitations of the intellect itself. Perceptual space, being neither the creature of the intellect nor necessarily an _a priori_ notion resident in the mental substructure, but existing entirely independent of the intellect or its apprehension thereof, cannot be expected to conform to the purely formal restrictions imposed by the mind except in so far as those restrictions may be determined by the nature of perceptual space. And for that matter, it should not be forgotten that, as yet, we have no means of determining whether or not the testimony of the intellect is thoroughly credible simply because there is no other standard by which we may prove its testimony. It is possible to justify the deliveries of the eye by the sense of touch, or vice versa. It is also possible to prove all our sense-deliveries by one or the other of the senses. But we have no such good fortune with the deliveries of the intellect. We have simply to accept its testimony as final; because we cannot do any better. But if it were possible to correct the testimony of the intellect by some other faculty or power which by nature might be more accurate than the intellect it should be found that the intellect itself is sadly limited. The possible curvature of space is a notion which also characterizes the content of the non-Euclidean geometry. It is upon this notion that the question of the finity and unboundedness of space, in the mathematical sense, rests. In the curved space, the straightest line is a curved line which returns upon itself. Progression eastward brings one to the west; progression northward brings one to the south, et cetera. On this view space is finite, but may not be regarded as possessing boundaries. Space-curvature, reinforced by the idea that space is also a manifold is the enabling clause of metageometry and without them the analyst dares not proceed. Here again, we are led to the confession that however fantastic these two notions may seem and evidently are, there is nevertheless to be recognized in them a "dim glimpse" of a veritable reality--a slight foreshadowing of the revelation of some great kosmic mystery. The manifoldness of space is the fiat of analysis. It is the inevitable outcome of the analyst's method of procedure. His education, training and view of things in general inhibit his arriving at any other result and he may be pardoned with good grace for his manufacture of the space-manifold. For by it perhaps a better appreciation of that wonderful extension of consciousness in the nature of which is involved the explanation of the perplexing problems which the manifold and other metageometrical expedients faintly adumbrate may be gained. It is pertinent, in the light of the above, to examine into some of the relative merits of the three formal bulwarks of geometrical knowledge. These are _certainty_, _necessity_ and _universality_. Geometric certainty is derived solely from the nature of the premises upon which it is based. If the premises be contradictory, it is, of course, defective. But if the premises are non-contradictory or self-evident, then the certainty of geometric notions and conclusions is valid. Another consideration of prime importance in this connection is the _definition_. From it all premises proceed. Hence, the definition is even more important than the premise; for it is the persisting determinant of all geometric conclusions while the premise is dependent upon the limitations of the definition. The determinative character of the definition has led to its apotheosis; but this, admittedly, has been necessary in order to give stability and permanency to the conclusions which followed. But in spite of this it would appear that the certainty of geometric conclusions is not a quality to be reckoned as absolute or final. With the same certainty that it can be said the sum of the angles of the triangle is equal to two right angles it may be asserted that that sum is also greater or less than two right angles. Certainty which is based upon the inherent congruity of definitions, premises and propositions is an entirely different matter from that certainty which arises out of the real, abiding validity of a scheme of thought. But this difference is not lessened by the fact that the latter is dependent, in a measure, upon the correct systematization of our spatial experiences by means of methodical processes. Euclidean geometry, accordingly, is not so certain in its applications as it is utilitarian; but non-Euclidean geometry is even less certain than the former and consequently more lacking in its utilitarian possibilities. The necessity of geometrical determinations is merely the necessity which inheres in logical inferences or deductions. These may or may not be valid. Inasmuch as the necessariness of deductions is primarily based upon the conditional certainty of premises and definitions it appears that this quality is in no way peculiar to geometry whether Euclidean or non-Euclidean. In like manner, the universality of geometric judgments may not properly be regarded as a peculiarity of geometry; but is explicable upon the basis of the formal character of the assumptions which underlie it. The chief value, then, of non-Euclidean geometry seems to abide in the fact that it clarifies our understanding as to the complex processes by which it is possible to organize and systematize our spatial experiences for assimilation and use in other branches of knowledge. With the above statement of the case of the non-Euclidean geometry it is now thought permissible to state briefly some of the elements thereof.[7] [7] The science of pure mathematics is perhaps indebted to no one in so great a degree as to GEORGE BRUCE HALSTEAD, formerly of the University of Texas, whose labors in connection with the popular exposition of the non-Euclidean geometry have been most untiring and effectual. Vide _Popular Astronomy_, Vol. VII and VIII, 1900, Dr. G. B. HALSTEAD. Below will be found some of the elements obtained as a consequence of efforts made both at proving and disproving the parallel-postulate of Euclid: "If two points determine a line it is called a straight." "If two straights make with a transversal equal alternate angles they have a common perpendicular." "A piece of a straight is called a sect." "If two equal coplanar sects are erected perpendicular to a straight, if they do not meet, then the sect joining their extremities makes equal angles with them and is bisected by a perpendicular erected midway between their feet." "The sum of the angles of a rectilineal triangle is a straight angle, in the hypothesis of the right (angle); is greater than a straight angle in the hypothesis of the obtuse (angle); is less than a straight angle in the hypothesis of the acute (angle)." "The hypothesis of right is Euclidean; the hypothesis of the acute is BOLYAI-LOBACHEVSKIAN; the hypothesis of obtuse is RIEMANNIAN." "If one straight is parallel to a second the second is parallel to the first." "Parallels continually approach each other." "The perpendiculars erected at the middle point of the sides of a triangle are all parallel, if two are parallel." "If the foot of a perpendicular slides on a straight its extremity describes a curve called an equidistant curve, or an equidistantial." "An equidistantial will slide on its trace." "In the hypothesis of the obtuse a straight is of finite size and returns into itself." "Two straights always intersect." "Two straights perpendicular to a third straight intersect at a point half a straight from the third either way." "A pole is half a straight from its polar." "A polar is the locus of coplanar points half a straight from its pole. Therefore, if the pole of one straight lies on another straight the pole of this second straight is on the first straight." "The cross of two straights is the pole of the join of their poles." "Any two straights inclose a plane figure, a digon." "Two digons are congruent if their angles are equal." "The equidistantial is a circle with center at the poles of its basal straight." A typical postulate based upon the BOLYAI hypothesis of the acute angle is the following: "From any point _P_ drop _PC_, a perpendicular to any given straight line _AB_. If _D_ move off indefinitely on the ray _CB_, the sect will approach as limit _PF_ copunctal with _AB_ at infinity. [Illustration: FIG. 5.] _PD_ is said to be at _P_ the parallel to _AB_ toward _B_. _PF_ makes with _PC_ an angle _CPF_ which is called the angle of parallelism for the perpendicular _PC_. It is less than a right angle by an amount which is the limit of the deficiency of the triangle _PCD_. On the other side of _PC_, an equal angle of parallelism gives the parallel _P_ to _BA_ towards _AM_.[8] Thus at any point there are two parallels to a straight. A straight has, therefore, two separate points at infinity." "Straights through _P_ which make with _PC_ an angle greater than the angle of parallelism and less than its supplement do not meet the straight _AB_ at all not even at infinity." [8] NOTE.--_M_ may be any point on the line _BA_ indefinitely produced. The parallel-postulate is stated in the non-Euclidean geometry as follows: "If a straight line meeting two straight lines make those angles which are inward and upon the same side of it less than two right angles the two straight lines being produced indefinitely will meet each other on this side where the angles are less than two right angles." It is stated by MANNING[9] in the following language: "If two lines are cut by a third and the sum of the interior angles on the same side of the cutting line is less than two right angles the line will meet on that side when sufficiently produced." It is rather significant that in this postulate which is really a definition of space should be found grounds for such diverse interpretations as to its nature. Of course, the moment the mind seeks to understand the infinite by interpreting it in the unmodified terms of the apparently unchangeable finite it entangles itself into insurmountable difficulties. As a drowning man grasps after straws so the mind, immersed in endless abysses of infinity, fails to conduct itself in a seemly manner; but gasps, struggles and flounders and is happy if it can, in the depths of its perplexity, discover a way of logical escape. The pure mathematician has a hankering after the logically consistent in all his pursuits; to him it is the "Holy Grail" of his highest aspirations. He seeks it as the devotee seeks immortality. It is to him a philosopher's stone, the elixir of perpetual youth, the eternal criterion of all knowledge. [9] Vide _Non-Euclidean Geometry_, p. 91. Failures to demonstrate the celebrated postulate of EUCLID led, as a matter of course, to the substitution of various other postulates more or less equivalent to it in that each of them may be deduced from the other without the aid of any new hypothesis. Among those who sought proof by a restatement of the problem are the following: 1. PTOLEMY: The internal angles which two parallels make with a transversal on the same side are supplementary. 2. CLAVIUS: Two parallel straight lines are equidistant. 3. PROCLUS: If a straight line intersects one of two parallels it also intersects the other. 4. WALLIS: A triangle being given another triangle can be constructed similar to the given one and of any size whatever. 5. BOLYAI (W.): Through three points not lying on a straight line a sphere can always be drawn. 6. LORENZ: Through a point between the lines bounding an angle a straight line can always be drawn which will intersect these two lines. 7. SACCHERI: The sum of the angles of a triangle is equal to two right angles. There were, of course, many other statements and substitutions used by mathematicians in their endeavors satisfactorily to establish the truth of the parallel-postulate. That their labors should have terminated, first, by doubting it, then by denying, and finally, by building up a system of geometries which altogether ignores the postulate is just what might naturally be expected of these men who have given to the world the non-Euclidean geometry. In doing what they did many, if not all of them, were not aware in any measure of the proportions of the imposing superstructure that would be built upon their apparent failures. All of them undoubtedly must have sensed the vague adumbrations forecast by the unfolding mysteries which they sought to lay bare; all of them must have felt as they executed the early tasks of those crepuscular days of pure mathematics that the way which they were traveling would lead to the inner shrine of a higher knowledge and a wider freedom; they may have been led by divine intuition to strike out on this new path and yet they could not have known how fully their dreams would be realized by the mathematicians of the twentieth century. If so, they were truly gods and mathesis is their kingdom. The analyst proceeds upon a basis entirely at variance with that which guides the ordinary investigator in the formulation of his conclusions. The empirical scientist in arriving at his theories or hypotheses is governed at all times by the degree of conformity which his postulates exhibit to the actual phenomena of nature. He endeavors to ascertain just how far or in what degree his hypothesis is congruent with things found in nature. If the dissidence is found to predominate he abandons his theory and makes another statement and again sets out to determine the degree of conformity. If he then finds that the natural phenomena agree with his theory he accepts it as for the time being finally settling the question. In all things he is limited by the answer which nature gives to his queries. Not so with the exponent of pure mathematics. For him the truth of hypotheses and postulates is not dependent upon the fact that physical nature contains phenomena which answer to them. The sole determining factor for him is whether or not he is able to state with _rational consistency_ the assumed first principles and then logically develop their consequences. If he can do this, that is, if he can state his hypotheses with consistency and develop their consequences into a logical system of thought, he is quite satisfied and well pleased with his performances. But the fact that this is true is of vital significance for all who seek clearly to understand the essential character of hyperspatiality. It appears, therefore, that the science of consequences is the radical essence of pure geometry. The metageometrician enjoys unlimited freedom in the choice of his postulates and suffers curtailment only when it comes to the question of consistency. He is at liberty to formulate as many systems of geometry as the barriers of consistency will permit and these are practically innumerable. So long then as the laws of compatibility remain inviolate his multiplication of postulate-systems may proceed indefinitely. Is it strange then that under conditions where an investigator has such unbridled liberty he should be found indulging in mathetic excesses? KANT held that the axioms of geometry are synthetic judgments _a priori_; but it appears that in the strictest sense this is not the case. It depends upon the type of mind which is taken as a standard of reference. If it be the uncultivated mind, it is certain that to it the relations expressed by an axiom would never appear spontaneously. If on the other hand, the standard be that of a cultivated mind it is also equally certain that to it these relations would be discovered only after methodical operations. All judgments arrived at as a result of logical processes should, it seems, be regarded as judgments _a posteriori_, i.e., the results of empirical operations. Confessedly, the facts adduced in course of experimentation serve as guides in choosing among all of the many possible logical conventions; but our choice remains untrammeled except by the compulsion arising out of a fear of inconsistency. The real criterion then of all geometries is neither truth, conformability nor necessity, but consistency and convenience. The difficulty with the non-Euclideans resolved itself into the question as to whether it is more consistent, as well as convenient, to establish a proof of the postulate by taking advantage of the support to be found in other postulates or whether, by seeking a demonstration based upon the deliveries of sense-experience as to the nature of space and its properties, a still more consistent conclusion might be reached. They had further perplexity, however, when it came to a decision as to whether the organic world is produced and maintained in Euclidean space or in a purely conceptual space which alone can be apprehended by the mind's powers of representation. Unwilling to admit the existence of the world in Euclidean space, they turned their attention to the examination of the properties of another kind of space so-called which unlike the space of the Ionian school could be made to answer not only all the purposes of plane and solid figures, but of spherics as well. And so, the manifold space was invented by RIEMANN and later underwent some remarkable improvements at the hands of his disciple, BELTRAMI. But it may be said here, parenthetically, that the truth of the whole matter is that our world is neither in Euclidean nor non-Euclidean space, both of which, in the last analysis, are conceptual abstractions. Although it may not be denied that the Euclidean space is the more compatible. The problem of devising a space, if only a very limited portion, in which could be demonstrated the assumed alternative hypothesis and its consequences logically developed, occasioned no inconsiderable concern for the non-Euclidean investigators; but neither LOBACHEVSKI, BOLYAI nor RIEMANN were to be baffled by the difficulties which they met. These only cited them to more laborious toil. Having succeeded in mentally constructing the particular kind of space which was adaptable to their rigorous mathetic requirements it immediately occurred to them that all the qualities of the limited space thus devised might logically be amplified and extended to the entire world of space and that what is true of figures constructed in the segmented portion of space which they used for experimental purposes is also true of figures drawn anywhere in the universe of this space as all lines drawn in the finite, bounded portion could be extended indefinitely and all magnitudes similarly treated. From these results, it was but a single step to the conclusion which followed--that either an entirely new world of space had been discovered or that our notion of the space in which the organic world was produced is wholly wrong and needs revision. But notwithstanding the insurmountable obstacles which stood in the way of the investigators who made the attempt to discover the homology which might exist between the characteristics of the newly fabricated space and the phenomenal world, investigations were carried forward with almost amazing recklessness and loyalty to the mathetic spirit until it was discovered that all efforts to trace out any definite lines of correspondence were futile. Then the policy of ignoring the question of conformability was adopted and has since been pursued with unchangeable regularity by the analytical investigator. Among the results obtained by the non-Euclideans in their profound researches into the nature of hyperspace are these: 1. It was found that the angular sum of a triangle, being ordinarily assumed to be a variable quantity, is either less or greater than two right angles so that a strictly Euclidean rectangle could not be constructed. 2. The angle sums of two triangles of equal area are equal. 3. No two triangles not equal can have the same angles so that similar triangles are impossible unless they are of the same size. 4. If two equal perpendiculars are erected to the same line, their distance apart increases with their length. 5. A line every point of which is equally distant from a given straight line is a curved line. 6. Any two lines which do not meet, even at infinity, have one common perpendicular which measures their minimum distance. 7. _Lines which meet at infinity are parallel._ But it is apparent that these results have not followed upon any mathematical consequence of other supporting postulates or axioms such as would place them on a coördinate basis with those used as a support for the parallel-postulate; for they are based upon the envisagement of an entirely new principle of space-perception and belong to a wholly different set of space qualities. The final issue then of the non-Euclidean geometry is neither in the utility of its processes and conclusions nor in the increscent inclination towards a new outlook upon the world of mathesis; but resides solely in the possibilities yet to be developed in that vast domain of analytical thought which it has discovered and opened to view. To say that it sheds any light upon the nature of the universe is perhaps to take the radical view; yet it cannot be doubted that the researches incident to the formulation of the non-Euclidean geometry have greatly extended the scope of consciousness. Whether the extension is valid and normal or simply a hypertrophic excrescence of mental feverishness; whether by virtue of it we shall more closely approach an understanding of the true nature of the mind of the Infinite, or shall all fall into insanity, are certainly debatable questions. It nevertheless appears evident that humanity has gained something of real, abiding permanence by this new departure. If that something be merely an extended consciousness or an awakening to the fact that there are stages of awareness beyond the strictly sensuous, and every observable evidence points to this, then there has only begun the process by which the faculty of conscious functioning in this new world shall become the normal possession of the human species. But this new world cannot be said to be of mathematical import; for it is doubtful if mathematical laws such as have been devised up to the present time, would obtain therein. So that if anything, it must be psychological and vital. On this view the worlds of hyperspace inlaid with analytic manifoldnesses and constant curvatures are but the primal excitants which will finally awaken in the mind the faculty of awareness in the new domain of psychological content. Then will come the blooming of the diurnal flower of the mind's immortality and the outputting of the organ of consciousness wherewith the infinite stretches of hyperspaces, the low-lying valleys of reals and imaginaries and the uplifting hills of finites and infinites shall be divested of their mysteries and stand out in their unitariness no longer draped in the veil of the inscrutable and the incomprehensible. The fourth dimension, regarded by some as a new scope of motion for objects in space, by others as a new and strange direction of spatial extent and by others still as the doorway of the temple of exegesis wherein an explanation may be found for the entire congeries of mysteries and supermysteries which now perplex the human mind, may also be said to be the key to the non-Euclidean geometry. But it really complicates the situation; for one has to be capable of prolonged abstract thought even to envisage is as a conceptual possibility. POINCARÉ[10] says: "Any one who should dedicate his life to it could, perhaps, eventually imagine the fourth dimension," implying thereby that a lifetime of prolonged abstract thought is necessary to bring the mind to that point of ecstasy where it could even so much as imagine this additional dimension. Nevertheless by it (the fourth dimension) was the non-Euclidean geometry made and without it was not any of the hyperspaces made that were made. It is the view which geometers have taken of space in general that has made the fourth dimension possible, and not only the fourth, but dimensions of all degrees. The basis of the non-Euclidean geometry may be found then in the notion of space which has been predominant in the minds of the investigators. [10] Vide _Nature_, Vol. XLV, 1892. Finally, it should be pointed out that the non-Euclidean geometry, though a consistent system of postulates, has been constructed upon a misconception based upon the identification of real, perceptual space with systems of space-measurements. Hyperspaces which are not spaces at all should not be confounded with _real space_. But they constitute the substance of non-Euclidean geometry; they are its blood and sinews. Their study is interesting, because of the possibilities of speculation which it offers. No mind that has thought deeply upon the intricacies of the fourth dimension, or hyperspace, remains the same after the process. It is bound to experience a certain sense of humility, and yet some pride born of a knowledge that it has been in the presence of a great mystery and has delved into the fearful deeps of kosmic mind. To the mind that has thus been anointed by the sacred chrism of the inner mysteries of creative mentality there always come that stillness and calm such as characterize the aftermath of reflection upon the incomprehensible and the transfinite. CHAPTER IV DIMENSIONALITY Arbitrary Character of Dimensionality--Various Definitions of Dimension--Real Space and Geometric Space Differentiated--The Finity of Space--Difference Between the Purely Formal and the Actual--Space as Dynamic Appearance--The _A Priori_ and the _A Posteriori_ as Defined by PAUL CARUS. In previous chapters we have traced the growth and development of the non-Euclidean geometry showing that the so-called fourth dimension is an aspect thereof. It is now deemed fitting that we should enter into a more detailed study of the question of dimensionality with a view to examining some of the difficulties which encompass it. The question of dimension is as old as geometry itself. Without it geometric conclusions are void and meaningless. Yet the conception of dimensionality itself is purely conventional. In its application to space there is involved a great deal of confusion because of the inferential character of its definition. For instance, commonly we measure a body in space and arbitrarily assign three elements to determine its position. The simplest standard for this purpose is the cube having three of its edges terminating at one of its corners. D + A | \ | \ | \ | \ | \| +------------------------- B C FIG. 6. Thus because it is found that the entire volume of a cube is actually comprehended within the directions indicated by the lines _ab_, _bc_ and _db_ it is determined that the three coördinates of the point _b_ are necessary and sufficient to establish the dimensions of the cube and consequently of the space in which it rests. The conception may be stated in this way: If a collection of elements, say points or lines, be of such a nature or order that it is sufficient to know a certain definite number of facts about it in order to be able to distinguish every one of the elements from all the others, then the assemblage or collection of elements is said to be of the same number of dimensions as there are elements necessary to its determination. In the above figure there are three elements, namely, the lines _ab_, _bc_, and _db_, which are necessary and sufficient for the determination of the position of the point _b_. In this way geometers have determined that our space is tridimensional; but it is obvious that this conclusion is based not upon any examination of space itself but upon the measurement of bodies in space. Upon this view it is seen that conclusions based upon such a procedure render our notion of the extension of bodies in space identical with the notion of spatial extensity. In other words, we take bodies in space and by examining their characteristics and properties arrive at an alleged apodeictic judgment of space. It is by means of this conventional norm of geometric knowledge that various other spaces, notably the one-, two-, four-and _n_-space, have been devised. It would appear that if some more absolute standard of measurement or definition of space were adopted the confusion which now clings to the conception of dimension could be obviated. For if it be true that three and only three elements are necessary to determine a point-position in our space and that in this determination we also find the number of dimensions of space, then it may also be true that _n_-coördinates would just as truly determine the dimensionality of an _n_-space, which is granted. But then the _n_-space would be just as legitimate as the three-space; for it is determined by exactly the same standards. It is both quantitatively and qualitatively the same. If, however, on account of the exigencies that might arise, we are forced to seek solace in the notion of an _n_-space whither shall we turn for it? It cannot be found; for it is imperceptible, uninhabitable, non-existent, and therefore, absolutely and purely an abstraction. Consequently, there must be something radically wrong with the definition of space or with its determinants. The purely arbitrary character of dimensionality is very aptly described by CASSIUS JACKSON KEYSER, who says: "... The dimensionality of a given space is not unique, but depends upon the choice of the geometric entity for primary or generating element. A space being given, its dimensionality is not therewith determined, but depends upon the will of the investigator who by a proper choice of generating element endows the space with any dimensionality he pleases. That fact is of cardinal significance for science and philosophy."[11] [11] Vide _Monist_, Vol. XVI, 1896, Mathematical Emancipations. It is a fact of "cardinal significance" for science; because it emphasizes the necessity for some more rational procedure than that of the geometrician in arriving at an absolutely unique method of determining the dimension and essential nature of real space. Its significance for philosophy lies in the need of a logical, rigidly exclusive and absolutely peculiar standard of space definition. The definition of perceptual space should be such as rigorously inhibits its inclusion as a particular in any general class. The necessity for this is warranted by its universality and uniqueness. The lines of demarkation between what is recognized as perceptual space and what has been called geometric or conceptual space should be very sharply drawn. So that when reference is made to either there will be no doubt as to which is meant. And then, too, conceptual space is no space at all, properly speaking. It is merely a system of space-measurement. And as such has no logical right to be put in the same category as perceptual space. Real space is unique. Geometric space belongs to a class whose members are capable of indefinite multiplication. It is certainly most illogical to identify them. Perceptual space, figuratively speaking, is a quantity; analytic space is the foot-rule, the yard-stick, the kilometer, by which it is measured and apportioned. It is logically impossible to predicate the same conclusion for both of them. That is, to do so causes a profound fracture of the fundamental norms of logic. Such conclusions being thus illegitimate it is rather surprising that an error of this nature should have been made. It is perhaps accountable for on the grounds of the geometer's complete _insouciance_ as to how his postulates shall stand in their relation to things in the phenomenal world. It is agreed that as convenient as is Euclid's system of space-measurement it is not by any means congruent with the extension of real space objects. It does, however, approximate congruity with these objects as nearly as possible. How then could it be expected that a system of space-measurement so far removed from this primary congruence as the non-Euclidean system is should exhibit more obvious signs of correspondence? But the advocates of the _n_-dimensionality of space have illatively asserted the identity of space and its dimensions. Accordingly, there is not recognized any distinction between their conception of space itself and its qualitative peculiarities. They use the terms interchangeably. So that dimension means space and _vice versa_. In this lack of discrimination may be found the source of much of the confusion which attaches to the conception of space. If it were arguable that the relation between space and its dimensions is the same as that between matter and its properties then the restriction of this relation to three and only three directions of extent would be disallowed; for the reason that if, as is commonly done, dimension be made to mean direction of extent, there would be an unlimited number of directions of extent and they would all be perceptible. But this is really another fundamental fault. Non-Euclideans have stretched the meaning of the term dimension so that it not only includes the idea of direction but an entirely new class of qualities--the fourth dimension. And despite this reformation of the original conception, they demand that it shall be called space. We have just shown that the generic concept of dimensionality is that three and only three coördinates are necessary and sufficient for its determination. Granting that this is true, are we not compelled consequently to see that we have, by adding a fourth or _n_-dimensions, involved ourselves into a more complex situation than before? For by postulating a fourth dimension either we have created a new world whose dimensions are four in number or we have explicitly admitted that the three dimensions have a fourth. Aside from the logical difficulties which beset these conclusions there is also set up a condition which is at variance with the most elementary requirements of common sense. Thus far mathematical thought has not served to clarify our notions of space nor to shed any new light upon the vital processes which are alleged to have their explanation in the new discovery. Simply stated, metageometricians have brought us to the place where we must either recognize that the fourth dimension is another sphere lying dangerously near the earth in which space extends in four primary directions and in which four coördinates are necessary for its determination or we are driven to the other horn of the dilemma where we are brought face to face with the conclusion that the three perceptual space dimensions have in common a hitherto unknown property or extension in virtue of which it may be viewed as having an unlimited number of dimensions. To accept the latter view is equivalent to saying that, in the above figure, the three lines _ab_, _bc_ and _db_ have formed a triple _entente_ by which they have mutually and severally acquired a new domain, hyperspace, and in which, because of the vast resources of the region, they are able to perform wondrous things. Let us examine briefly the various current definitions of dimension. It is assumed by not a few that dimension is the same as _direction_. But can we grant this wholly to be true? If so, then a mere child may see that there are and must necessarily be as many dimensions as there are directions. Primarily, there are six directions of space and an unlimited number of subsidiary directions. On this view it is not necessary to invent a new domain of space if the object be merely to discover and utilize a greater number of dimensions than has heretofore been allowed. For the identification of the term dimension with direction already makes available an almost infinite number of dimensions. But this view is objected to by the advocates, for it is contrary to the hypothesis of _n_-dimensionality. Dimension also means _extent_. This is partially true. It cannot be wholly true. For, if it were, then, space would have only one dimension which is also not allowable under the hypothesis. Then the definition leaves out of account the idea that space is at the same time a direction or collection of directions. The term extension is generic and when applied to space means extension in all possible directions and not in any one direction. So that it is not permissible to say that space extends in this direction or that because it extends in all directions simultaneously and equally. Geometers claim that space is a system of coördinates necessary for the establishment of a point-position in it. This view, however, identifies space with a system of space-measurement and is therefore faulty. According to this view there may be as many spaces as there are systems of space-measurement and the latter may be limitless. But if the totality of spaces are to be viewed as one space then we shall have one space with an indefinite number of dimensions; also an indefinite number of space measurements which would be confusing. Much, if not all, of such a system's utility and convenience would be unavailable or useless. That, too, would be in violation of the avowed purpose of these investigations which is to enhance the utility and convenience of mathematic operations. Now it is evident that space is neither direction, extension, a system of space-measurement nor a system of manifolds whose dimensions are generable. And this is so for the same reason that a piece of cloth is not the elements of measurement--inches, feet, yards--by which it is apportioned. And because we find that the fabric of space lends itself accommodatingly to our conventional norms of measurement is not sufficient reason for identifying it with these norms. Here we have the source of all error in mathematical conclusions about the nature of space; because all such conclusions are based not upon the intrinsic nature of space, but upon artificial forms which we choose to impose upon it for our own convenience. But it should be remembered that the irregularities which we note are not in space itself but inhere in the forms which we use. For these purposes space is extremely elastic and accommodates itself to the shape and scope of any construction we may decide to try upon it. In this respect it is like water which has no regard for the shape, size or kind of vessel into which it may be posited. There is one thing certain that judging from the above considerations there has been not yet any absolute, all-satisfying definition devised for space by mathematicians. The best definitions hitherto constructed are purely artificial and arbitrary determinations. It is rather anomalous that there should be so little unanimity about what is the most fundamental consideration of mathematical conclusions which are supposed to be so certain, so necessary and universal as to be incontrovertible. Confessedly, it is a condition which raises again the question as to just what are the limits of mathematical certainty and necessity and just how far we shall depend upon the validity of mathematics to determine for us absolutely certain conclusions about the nature of space. In view of the uncertainty noted, are we justified in following too closely the mathematic lead even in matters of logic, to say nothing of our conception of space? It seems that we shall have necessarily, on account of the recognized limitations of mathematics in this matter, to turn to some more tenable source for the norms of our knowledge concerning space. For in the light of the rather indefensible position which metageometricians have involved themselves there appears to be no hope in this direction. It is undoubtedly safer not to rely altogether upon the purely abstract, even in the world of mathesis, for any absolute criterion of knowledge. It is perhaps well that we should expunge the word _absolute_ from our vocabularies. It is really a misnomer and has no meaning in the lexicon of nature. There is in reality no _absolute_ in the sense of final absolution from all conditions or restrictions. In the ultimate analysis there is unquestionably no hue, tone, quality, condition nor any imaginable posture of life, being or manifestation that is absolved from every other one of its class or from the totality. All these are relational and interdependent. There is no room for the absolute. In fact, it is a quality which cannot in any way be ascribed to any aspect of kosmic manifestation. It has existence only in the mind and has been devised for the purpose of marking the limits of its scope. All being is relative; all life is relative and is destined to change its qualities as it evolves. All knowledge is also relative and what is true of one state may not be true of another; what is true of one life may not be true of another life; the limitations of one degree of knowledge may not have any bearings upon another degree. The norms of one will not satisfy the conditions of another stage of manifestation. It is always within limits that the criterion of knowledge will be found to satisfy a given set of conditions. Hence within certain limits mathematical conclusions will maintain their validity. Error is committed by pushing the validity of these limits to a position without the sphere of limitations. This seems to be the crux of the whole matter. Mathematicians, notably non-Euclideans, have sought to extend the comparatively small sphere of limits of congruence between mathematic and perceptual space to such an extent as to cause it to encroach upon forbidden territory. In doing this they have erred grievously, causing serious offense to the more sensitive spirit of the high-caste mathematicians among whom are none more truly conservative than PAUL CARUS,[12] who says: "Metageometricians are a hot-headed race and display sometimes all the characteristics of sectarian fanatics. To them it is quite clear there may be two straight lines through one and the same point which do not coincide and yet are both parallel to a third line." [12] Vide _Monist_, Vol. XIX, p. 402 (1909). To the student who has carefully followed the development of the non-Euclidean geometry and the notion of hyperspace the above characterization is none too severe nor ill-deserved. Nothing could more vividly yet correctly portray the impious tactics of the metageometrician and establish his perceptual obliquity more surely than the mere fact, mentioned by CARUS, that he can with evident lack of mental perturbation proclaim that two straight lines, noncoincident with each other, may pass through a point and yet be parallel to a third line. But this is a mere trifle, a bagatelle, to the many other infractions of which he is guilty. The wonder is that he is able to secure such obsequious acceptance of his offerings as many of the most serious minded mathematicians are inclined to give. Is it to be wondered at that, despite the profuse protestations of the advocates, many who take up the study of the question of hyperspace should experience a deep revulsion from the posture assumed by metageometricians with respect to these queries? Linked with the idea of dimensionality is the notion that space is infinite. This is a conception which has its roots imbedded in the depths of antiquity. Primitive man, looking up into the heavens at what appeared to him as a never ending extension, was awed by its vastness; but the minds of the most learned of the present-day men are not free from this innate dread of infinity. It permeates the thought life of all alike and none seems to be able to rise above it. Mathematicians, philosophers, scientists all share in the general belief that space is without limit, unending in extent and eternally existent. RIEMANN, whose thought life found its most convenient mode of expression by means of pure mathematics, was the first in the history of human thought to surmise that space is not infinite but limited even though unbounded. But his conception has been much vitiated on account of its entanglement with an _idealized_ construction by which space is regarded as a thing to be manipulated and generated by act of thought. Were it not for this his conception would indeed mark the beginning of a new era in psychogenesis. As it is, when all the nonsensical effusions have been cleared away from our space conceptions and men come really to understand something of the essential nature of space this new era will find its true beginnings in the mind of RIEMANN. Although it must be said, as is the case with all progressive movements, the later development of a rationale for this conclusion will vary greatly from his original conception. For he had in mind a space that is generable and therefore a logical construction while ultimately the mind will swing back to a consideration of real space. Already men are beginning to see a new light. Already they are beginning to take a new view of space in general. The departure is especially noticeable in the attitude assumed by HIRAM M. STANLEY.[13] He says: "If we seek the most satisfactory understanding of space we shall look neither to mathematics nor Psychology but to Physics. The trend of Physics, say with such a representative as OSTWALD, is to make things the expression of force; the constitution and appearance of things are determined by dynamism; and we may best interpret space as a mode of this dynamic appearance." Space, as a mode of dynamic appearance is a slight improvement upon the old idea of a pure vacuity; for in the light of what we now know about space content much of the dignity of that view has been lost. Men now know that space is not an empty void. They know that the atmosphere fills a great deal of space. They also have extended their conception in this direction to include the ether and occultism goes further and postulates four kinds of ether--the chemical, life, light and psychographic ethers. But it does not stop here. It postulates a series of grades of finer matter than the physical which fills space and permeates its entire extent even to identification with its essential nature. [13] _Philosophical Review_, Vol. VII (1898). STANLEY continues: "Everything does not, as commonly conceived, fall into some pre-existent space convenient for it; but everything makes its own spaciousness by its own defensive and offensive force, and the totality of all appearance is space in general." According to STANLEY, not only do physical, perceptual objects, by their "offensive and defensive force" make their own space but the appearance of that in which no physical object is makes room for itself by its own dynamic force. In other words, that which we call "pure extensity" is by virtue of its dynamism the cause of its own existence. At first hand there appears to be little worthy of serious consideration in this view of STANLEY; yet, if carried to its logical conclusion, the merit of the hypothesis becomes apparent. Accordingly, interstellar distances which are commonly said to be even without air or life of any kind are really an appearance possessed of a dynamism peculiar to itself. And this very force-appearance, constituting space, is that which makes it perceivable. For instance, let us say the space that exists between the earth and the moon, is not really empty nor does it have an existence prior to itself, but is a mode of dynamic appearance which is the cause of its own existence. Its dynamic character makes it to appear perceptible to our senses. Logically, if the dynamism were removed there would remain neither space nor the appearance of space. If this were true, and it is worthy of serious thought, then space is certainly finite, as in its totality, according to STANLEY'S view, it would have to be regarded as a "phenomenon of the inner and finite life of the infinite." It is believed that we may go a step further and unqualifiedly assert that _space is finite_, even denying its infinity as a "general mode of the activity of the whole." Yet it is transfinite in the sense that it transcends the comprehension of finite minds or processes. It is _finite_ because it is in _manifestation_. Everything that is in manifestation is finite. The infinite is not in manifestation. Infinity has to be limited always to become manifest. The Deity has limited His being in order that there may be a manifested universe. All things, all appearances are finite; because they are phenomena connected with manifestation. This question may be viewed from another standpoint. All things in manifestation or existence are polar in their constitution. For instance: there cannot be a "here" without a "there." There cannot be an "upper" without a "lower." Right is copolar with wrong; good is copolar with evil; night with day; manifestation with non-manifestation; truth with falsity; infinity with finity and so on, throughout the whole gamut of the pairs of opposites. What is the logical inference? Space is paired with a lack of space. There cannot be what we call _space_ without there being at the same time the possibility, at least, of the _lack of space or spacelessness_. This is a conclusion that is rigorously logical and incontrovertible. But it has been urged that it is impossible for the mind to imagine a condition where there is no space. It even has been asserted that it is contrary to the constitution of the mind itself to imagine "no space." But whether imaginable or not has no effect whatever upon the validity of the conception. Neither, it is said, can we imagine a fourth dimension but the mind has come dangerously near to imagining it. The distance from excogitating upon, discussing and describing the properties of four-space to imagining it is not so great after all. Truly it is difficult indeed, it seems, to be able to describe a thing yet not be able to imagine or make a mental image of it. There is an evident fallacy here. Either the description of four-space is no description at all or it is a true delineation of an idealized construction which is well within the mind's powers of imagination. Indeed the question of imaginability is not determinative in itself; for what the mind may now be unable to imagine, because of its more or less nebulous character, and owing to its infancy may in the course of time be easily accomplished. The universe is a compacted _plenum_. It is chock-full of mind, of life, of energy and matter. These four are basically one. They exist, of course, in varying degrees of tenuity and intensity and answer to a wide range of vibrations. Together, in their manifestation of action and interaction, in their _dynamic appearance_, if you please, they constitute space. If these were removed with all that their existence implies there would result a condition of spacelessness in which no one of the appearances which we now perceive would be possible. Even sheer extensity would be non-existent. All scope of motility would be lacking. Dimension, coördinates, direction, space-relations--all would be impossible. A straight line is an ideal construction of the mind. It does not exist in nature. It can never be actualized in the phenomenal universe. Between the ideal and the real, or actual, there is a kosmic chasm. It broadens or narrows according as the phenomenal appearance approaches or recedes from the ideal. What, therefore, can be postulated of the one will not apply with equal force to the other. They are not congruent and can never be in the actualized universe. The moment the actual becomes identified with the ideal it ceases to be the actual. The universe does not exist as _pure form_, neither does space. As purely formal constructions of the intellect these can have no perceptible existence. The phenomenal or sensible may not be judged by exactly the same standard as the formal. The phenomenal or sensible represents things as they appear to the senses, or, so far as the actualized universe is concerned, _as they really are_. The _formal_ represents things as they are made to appear by the mind. It cannot be actualized. It may be said that the purely formal is the limit of evolution. The phenomenal may approach the ideal as a limit, but can never become fully congruent with it. _The difference between the ideal and the actual is a dynamic one_; it is by virtue of this difference that the universe is held in manifestation. Evolution is the decrement of this difference between the purely formal and the actual. So long then as a kosmic differential is maintained the phenomenal continues to be manifest: when it is finally reduced to nothing it goes out of manifestation. The phenomenal is finite; the ideal infinite. Wherefore, it is undoubtedly improper to refer to space as being infinite. The term really is inapplicable. Transfinity is much better and more accurate. Space is transfinite because its scope is greater than any finite scope of motility can encompass, because it exceeds finite comprehensibility. RIEMANN'S notion that space is limited gains weight in the light of the foregoing considerations. But he could not conceive of the limitability and unboundedness of space as such in its pure essence; but was compelled, by his own limitations, to make an idealized construction in which he could actualize his conception. And for real, dynamic space, he substituted his ideal construction and proceeded upon that basis. And of course, his view while it had no reference to perceptual space nevertheless possessed an illative relation thereto and should be recognized as construable in that light. The process of squaring the circle recognized as a geometric impossibility is significant of the fluxional nature of the universal residuum perpetually maintained between the archetypal and the manifested kosmos. It seems that there is a profound truth embodied in this problem. There is a lesson that may be learned by mathematicians, philosophers, scientists and thinkers in general. There is an element of eternal necessity and universality about it which is truly symbolic of the finity of the universe and the infinity of the archetypal. Just as a square or a series of polygonal figures inscribed in a circle cannot be made to coincide exactly with the circle so cannot the actual be made to coincide with the ideal. The circumference of the circle is the unapproachable limit of inscribed squares. If it were possible so to multiply squares thus inscribed that a figure coincident with the circumference of a circle might be constructed, such a figure would not be a square but a circle. The manifested universe is like that--the process of inscribing squares within a circle. It is ever _becoming_, _evolving_, _developing_, but never quite attains. Infinity is a process. But no single stage in that process is infinite. Each is finite and their totality makes the infinity of the process. The universe manifested to the senses or the intellect is finite. "Space," says PAUL CARUS, "is the possibility of motion in all directions."[14] To be sure, it is admitted that space offers opportunity for motion in all directions. But is space this opportunity of motility? Or is possibility of motion space? The possibility of motion must rest in the thing that moves. It implies a potency in the moving entity, not in space. If it is meant that space is the potency that resides in the moving element it is still more difficult to understand the connotation. But even granting this view, are we not compelled to recognize the dynamism of space as a necessary inference? Another definition which CARUS gives is that space is a "_pure form of extension_." If it be granted that space is a pure form of extension we should have to conclude that it has no actual existence; for _pure form_ does not exist except as an idealized construction. It cannot be found in nature. Pure form is _ideal_. Impure or natural form is actual. Therefore the space in which we live and in which the universe exists cannot be a "pure form" because life cannot exist in the purely formal. It is useless to talk about space as mere form so long as it maintains life. The difficulty which this phase of the question presents is another evidence of the inadequacy of our definitions. [14] Vide _Foundations of Mathematics_, p. 107. It is also found to be impossible to concur in CARUS' conception of knowledge _a priori_. His notion of the _a priori_ varies somewhat from the Kantian view. He defines it as an "idealized construction," the "mind made," "abstract thought," and places it in the same category as a concept. This is undoubtedly born of his desire to get rid of KANT'S "innate ideas" which seem to be distasteful to him. But in doing so it appears that the real _a priori_ has been overlooked. Let us examine for a moment this important question. The _a posteriori_ connotates all knowledge gained through the senses, or sense experience. All knowledge therefore whose origin can be traced to the senses is knowledge _a posteriori_. Now, knowledge _a priori_ should be just the opposite of this. It should indicate such knowledge as that which does not have its origin in the senses, or which is not dependent upon the ordinary avenues of sense-experience. Abstract thought is as truly experience as smelling, seeing or hearing. It is by traversing its scope of motility that the mind finds out what the norms of logic are. It could not remain quiescent and discover them. It has to be active, examining, comparing and judging. Almost the entire range of thought, its entire scope, is characterized by the _a posterioristic method_. In fact, all thought is _a posterioristic_. Despite the fact that, in thinking in the abstract, it is necessary mentally to remove all elements of concreteness, all materiality and all actuality, the conclusions reached have to be referred to the standards maintained by the actual, the concrete and the material. Then we do not start with the abstract in our thinking. We begin with the concrete and by mentally removing all physical qualities arrive at the abstract. The mind has a constitution. It acts in a given way because it is its nature so to act. Not because it has learned to act in that manner. It performs certain functions intuitively without previous instruction or experience for the same reason that water dampens or heat warms. It is natural for it to do so. This naturalness, this performance of function without being taught or without experience constitute the principle of _apriority_ in the mind. _Aprioriness_ is a principle of mind partaking of the very nature and essence of mind. It is the very mainspring of mentality. Perception and conception are processes which the mind performs intuitively. The mind perceives and conceives because it is impossible for the normal mind to do otherwise. We take a view upon a given question; we assume certain mental attitudes of affirmation, negation or indifference because we have learned to do so by virtue of the tuitional capability of mind. These describe the _a posteriori_. That is, all knowledge obtained as a result of voluntary mental processes constitutes the mass of knowledge _a posteriori_. The _a priori_ is what the mind is by nature: the _a posteriori_ is what the mind becomes. It is the mind-content. The _a priori_ is not a mental construction; it is an essential principle of mind. It should not be identified with the "purely formal," as is done by PAUL CARUS:[15] [15] Vide _Foundations of Mathematics_, p. 42. He says: "The _a priori_ is identical with the purely formal which originates in our mind by abstraction. When we limit our attention to the purely relational, dropping all other features out of sight, we produce a field of abstraction in which we can construct purely formal combination, such as numbers, or the ideas of types and species. Thus we create a world of pure thought which has the advantage of being applicable to any purely formal consideration and we work out systems of numbers which, when counting, we can use as standards of reference for our experience in practical life." Thus CARUS definitely links up the _a priori_ to a factor which is nothing more nor less than a mental by-product. For such is the category in which would be placed both the process of abstraction and its results. It is therefore exceedingly difficult to understand why so cursory a consideration should have been given to the principle of _apriority_ than which no other element of mind is more essentially a part of the mind itself. The formal is symbolic. It signifies an informing quantity. Pure form itself is but a negation of that which formerly filled it. Then, too, the formal is purely artificial because it is a mental construction. Essentially there is as much difference between the purely formal and the _a priori_ as between creator and creature, as between potter and clay. The one is the builder, the other is the material; the one the knower and the other the known. Thus, the only reason that the formal is found to be answerable to the _a priori_ at all is due to the fact that it is construable only upon the basis of the _a priori_. But being so is not sufficient warrant for its identification with the _a priori_. The formal merely represents the totality of possibilities in the universe as viewed by the mind; but as the number of possibilities open to the mind is, on account of its nature and purpose limited, it is not to be supposed that it (the mind) shall measure up to all the possibilities offered by the formal. Moreover, it is certain that no sane mind cherishes the hope that there shall ever be found in the universe of life and form a congruence for all of the possibilities held out by the purely formal. As an eternal principle of mind, the _a priori_ is in agreement with the divine mind of the kosmos. In its _aposteriority_ the mind is of diverse tendences, qualities and characteristics. Apriorily, it acts in unison with the eternal purpose of life and the universal mind. In its aposteriority, it often goes awry. In its _apriority_ it can never be insane; insanity is a symptom of the morbid _a posteriori_. The mind in man acts the same as mind in the vegetal and lower animal kingdoms. Metabolism and katabolism, indeed all cell-activity, are _a priori_ performances of the mind. Growth and all its phenomena, the cyclicism of natural processes, and every activity connected therewith belong to the category of the _a priori_. Cells multiply, divide, build up and tear down tissues and they do it intuitively. Most certainly these functions are performed without any assistance from the intellect. All the myriad activities in nature with which the intellect in man has not the slightest concern, truly acting in accord with some primordial impetus, are activities _a priori_. Now what is the attitude of the intellect, in the light of the _a priori_, towards space and the question of dimensionality? It is evident that no matter what this attitude may be it is in agreement with the constitution of things and of the universe. And if so, it is right and without illusion. It is also evident that whatever notion _a posteriori_ the intellect may entertain with respect to these questions is unavoidably liable to the illusionary drawbacks common to conclusions based upon limited experience. The geometric view of space belongs to the category of the _a posteriori_. Hence it is subject to the usual imposition of error. Tersely stated, KANT's view of space is that it is a form of intuition, a form _a priori_, a transcendental form. As such he considered it to be a native form of perception not belonging to the category of sense-deliveries. Accordingly, space is a form of intuition arising out of and inhering in the constitution of mind. It is a notion which constitutes the universal and eternal prerequisite of mind and is, therefore, intrinsically necessary to all phases of mentation. Now, this being true just what may be said to be the relation of dimensionality to this _a priori_ form of space which is found to exist in the mind as an eternal aspect of its nature? Does the mind intuitively measure its contents or its operations by the empirical standard of space-measurement known as dimension? Is the attitude of the mind towards the objectively real one of discrimination _a priori_ as to the direction or dimension in which a percept may originate? In other words, does the mind habitually and intuitively refer its data to a system of coördinates for final determination? There is no other answer but that the mind makes no such reference and is dependent upon no kind of coördinate system in any of its operations _a priori_. As a form of intuition, the space notion is present in the mind as a scope of existence, of motility, of being and of sheer roominess. The notion of direction or dimension, being an artificial construction, does not enter into this form of intuition at all. It is only when the mind comes to elaborate upon its perceptive performances and possibilities that the questions of relations, positions and directions arise. But this latter is a matter separate and distinct from the state of awareness which embodies the notion of space. Dimension is an arbitrary norm constructed by the mind for the determination of various positions in space. It is an accident or by-product of the process of elaborative cognition, a convenient and appropriate means of measurement for objects in space and their space-relations. But it is no more _a priori_ than a foot rule or a square. But being purely an empirical product it may be said to be an aspect of psychogenesis because it relates to the evolutionary aspect of mind. The assumption may therefore be allowed that the mind may, in the course of its evolution, find it convenient and appropriate to devise an additional ordinate or dimension to satisfy the necessities of its more complex ramifications into the nature of things and to determine their greatly increased space-relations. It may be even possible for the mind to function normally in a space of four dimensions. But this would simply be a new adjustment, not a change in the essential nature of mind. It would be like the series of adjustments to environments which man has made in the onward movement of civilization. There has been no serious change in the manhood _per se_ of man. That has remained the same; there has been merely a complication of environmental influences. Similarly, in the acquisition of four-dimensional powers, granting that such an acquisition is possible, there is nothing to be added to the _aprioriness_ of mind _itself_. Is it not, therefore, logical to assume that the discovery of a fourth coördinate and the consequent conceptualization of the same, point to the development in the mind of a greatly extended faculty, more keenly penetrative powers of cognition and a further diversification of its environments than it has hitherto enjoyed? Indeed, it seems so. CHAPTER V THE FOURTH DIMENSION The Ideal and the Representative Nature of Objects in the Sensible World--The Psychic Fluxional the Basis of Mental Differences--Natural and Artificial Symbols--Use of Analogies to Prove the Existence of a Fourth Dimension--The Generation of a Hypercube or Tesseract--Possibilities in the World of the Fourth Dimension--Some Logical Difficulties Inhering in the Four-Space Conception--The Fallacy of the Plane-Rotation Hypothesis--C. H. HINTON and Major ELLIS on the Fourth Dimension. The world of mathesis is truly a marvelous domain. Vast are its possibilities and vaster still its sweep of conceivability. It is the kingdom of the mind where, in regal freedom, it may perform feats which it is impossible to actualize in the phenomenal universe. In fact, there is no necessity to consider the limitations imposed by the actualities of the sensuous world. Logic is the architect of this region, and for it there is no limit to the admissibility of hypotheses. These may be multiplied at will, and legitimately so. The chief error lies in the attempt to make them appear as actual facts of the physical world. Mathematicians, speculating upon the possibilities of mathetic constructions and forgetting the necessary distinctions which should be recognized as differentiating the two worlds, in their enthusiasm have been led into the error of postulating as qualities of the phenomenal world the characteristics of the conceptual. Accordingly, a great deal of confusion as to the proper limits and restrictions of these conceptions has arisen and there still may be found those who are enthusiastically endeavoring to push the actualities of the physical over into the conceptual. But in assuming any attitude towards mathetic propositions, especially with a view to demonstrating their actuality, very careful discrimination as to the essential qualities and their connotations should be made. Hence, before taking up a brief study of the fourth dimension proper, it is deemed fitting to indicate some of the fundamental distinctions which every student of these questions should be able to make with reference to the data which he meets. All objects of the sensible world have both an essential or ideal nature and a representative or sensuous nature. That is, they may be studied from the standpoint of the ideal as well as the sensuous. The representative nature is that which we recognize as the mode of appearance to our senses which, as KANT held, is not the essential or ideal character of the thing itself. For there is quite as much difference between the sensuous percept and the real thing itself as between an object and its shadow. In fact, a concept viewed in this light, may be seen to have all the characteristics of an ordinary shadow; for instance, the shadow of a tree. View it as the sun is rising; it will then be seen to appear very much elongated, becoming less in length and more distinct in outline as the sun rises to a position directly overhead. The elongation may again be seen when the sun is setting. Throughout the day as the sun assumes different angles with reference to the tree the proportions and definiteness of the shadow vary accordingly. Thus the angularity of the sun, the intensity and fullness of the light and the shape and size of the tree operate to determine the character of the shadow. Much the same thing is true of a sensuous representation. If we examine carefully our ideas of geometric quantities and magnitudes, it will be found that the concepts themselves are not identical with the objects of the physical world, but mere mental shadows of them. The angularity of consciousness, or the distinctness of one's state of awareness, being analogous to similar attitudes in the solar influence are the main determinants of the character of the mental shadow or concept. Wherefore mathematical "spaces" or magnitudes are not sensuous things and have therefore no more real existence than a shadow, and strictly speaking not as much; for a shadow may be seen, while such magnitudes can only be conceived. It may be urged that since we can conceive of such things they must have existence of some kind. And so they have, but it is an existence of a different kind from that which we recognize as belonging to things in the sensible world. They have a conceptual existence, but not a sensuous one. Therein lies the great difference. To be sure, a shadow is a more or less true representation of the thing to which it pertains. That this is true can be established empirically. Similarly, the degree of congruity between objects and concepts likewise may be determined. If this were not true we should be very much disappointed with what we find in the phenomenal world and could never be quite sure that the mentograph existing in our minds was a faithful representation of the thing which we might be examining. But really the foundation for such a disappointment is present in every concept, every percept with which the mind deals. This disappointment, although in actual experience is reduced to an almost negligible quantity, is due to the failure of sensuous objects to conform wholly to the specific details of the mental shadow or mentograph. This lack of congruence between the mental picture and the object itself is necessary for obvious reasons. It is markedly observable in the early efforts of a child in learning distances, weights, resistances, temperatures and the like. No inconsiderable time is required for the child to be able correctly to harmonize his sense-deliveries with actual conditions. Otherwise, the child would never make any of the ludicrous mistakes of judgment of which it is guilty when trying to get its bearings in the world of the senses. In the course of time the child gradually learns by experience that certain things are true of objects, distances, temperatures, resistances, etc., and that certain things are not true of them. He learns these things by actually contacting various objects. He is then competent to render correct judgments, within certain limits, as to the conditions which he finds in the sensible world. And the allowances, equations and corrections which his motor, sensory and psychic mechanisms learn to make in childhood serve for all subsequent time. And this is important to remember; for the mature mind is apt to forget or overlook the adaptations which the child-mind has made in its growth. If there were no such differences between the concept and the thing itself, actual physical contact would not be necessary. For one could rely wholly upon the sense-deliveries and each sense might operate entirely independently of all the others as there would be no necessity to correct the delivery of one by those of the others. This, of course, raises the question as to the necessity of sense-experience at all under conditions where there would be no disparity between the thing itself and the ideal representation of it in the mind. The absence of this variable quantity would open to the mind the possibility of really knowing the essential nature of objects in the phenomenal world, a condition of affairs which is admittedly now without the range of the powers of the mind. At any rate, the essential "thingness" of objects can never be comprehended by the mind until the diminution of this disparity between the object of sense and the mental picture of it which exists in the consciousness has proceeded to such a limit as either completely to have obliterated it or to such an extent that the psychic fluxion is so slight as not to matter. It is believed that the results of mental evolution, as the mind approaches the transfinite as a limit, will operate to minimize the fluxional quantity which subsists between all objects of sense and their ideal representation as data of consciousness. The conclusion that the mind of early men who lived hundreds of thousands and perhaps millions of years ago on this planet consumed a much longer time in learning the adjustments between the objects which it contacted in the sensuous world and the elementary representations which were registered in its youthful consciousness than is to-day required for similar processes seems to be demanded, and substantiated as well, by what is known of the phyletic development of the mind in the human race. In view of the above, it is thought that the duration of such simple mental processes served not only to prolong the physical life of the man of those early days, but may also account for the puerility and incapacity of the mind at that stage. Not that the slow mental processes were active causative agencies in lengthening the life of man, but that they together with the crass physicality of man necessitated a longer physical life. This, perhaps in a larger sense than any other consideration, accounts for the fundamental discrepancies in the mind of the primitive man in comparison with the efficiency of the mind of the present-day man. In view of the potential character of mind and in the light of the well graduated scale of its accomplishments, it is undoubtedly safe to conclude that the quality of mental capacities is proportional to the psychic fluxional which may exist at any time between the ideal and the essential or real. Mental differences and potentialities in general may be due to the magnitude of the psychic fluxional or differential that exists between the conceptual and the perceptual universe. In some minds it may be greater than in others. The chasm between things-in-themselves and the mental notion pertaining thereto may vary in a direct ratio to the individual mind's place in psychogenesis, and therefore, be the key to all mental differences in this respect. Most certain it is that there may be marked fluctuations in the judicial approach of minds towards any psychic end. In other words, there is not only a fluxional or differential between the object and its representation, but also a differential between the approach of one mind and another in the judicial determination of notions concerning ideas. In this way, differences of opinions as to the right and wrong of judgments arise. Indeed, there seem to be zones of affinity for minds of similar characteristics, or minds that have the same degree of differential; so that, in choosing among the many possible judgments predicable upon a species of data, all those minds having the same degree of psychic differential discover a special affinity or agreement among themselves. Hence, we have cults, schools of thought, and various other sectional bodies that find a basis of agreement for their operations in this way. The outcome of this remarkable intellectual phenomenon is that there are as many different kinds of judgments as there are zones of affinity among minds. Various systems of philosophy owe their existence to these considerations, and the considerations themselves flow from the fact that all intellectual operations are essentially superficial; because there is no means by which they may penetrate to the steady flowing stream of reality which pervades and sustains objects in the sensible world. In view, therefore, of the foregoing and with special reference to geometric constructions, it is necessary in approaching a study of the four-space that it be understood at the outset that the fourth dimension can neither be actualized nor made objectively possible even in the slightest degree in the perceptual world; because it belongs to the world of pure thought and exists there as an "extra personal affair," separate and distinct from the world of the senses. As says SIMON NEWCOMB:[16] "The experience of the race and all the refinements of modern science may be regarded as showing quite conclusively that, within the limits of our experience, there is no motion of material masses, in the direction of a fourth dimension, no physical agency which we can assume to have its origin in regions to which matter cannot move, when it has three degrees of freedom." There is, however, no logical objection to the study of the fourth dimension as a purely hypothetical question, if by pursuit of the same an improvement of methods of research and of the outlook upon the field of the actual may be gained. Hence, it is with this attitude of mind that we approach the consideration of the fourth dimension. [16] Vide _Science_, Vol. VII, p. 2, No. 158, 1898. Various efforts have been made to render the conception of a fourth dimension of space thinkable. The student of space has reasoned: "We say that there are three dimensions of space. Why should we stop here? May there not be spaces of four dimensions and more?" Or he has said: "If 'A' may represent the side of a square, A^2 its area, and A^3 the volume of a cube with edge equal to A; what may A^4, A^5 or A^{_n_th} represent in our space? The conclusion, with respect to the quantity A^4, has been that it should represent a space of four dimensions." Algebraic quantities, however, represent neither objects in space nor space qualities except in a purely conventional manner. All efforts to justify the objective existence of a fourth dimension based upon such reasoning will, therefore, fail; because the basis of such arguments is itself faulty. In the sentence: "The man loves his bottle," the thing meant is not the bottle, but what the bottle contains. For the purpose of the figure the bottle signifies its contents. There is no more real connection between the bottle and what it contains than between any word and the object for which it stands. Words are said to be symbols of ideas. But they are not natural symbols; they are conventional symbols, made for the purpose of cataloguing, indexing and systematizing our knowledge. Words can be divorced from ideas and objects, or rather have never had any real connection with them. There are two classes of natural symbols, namely; _objects_ and _ideas_. These, objects and ideas, symbolize realities. Realities are imperceptible and incomprehensible to the intellect which has aptitude only for a slight comprehension of the symbols of realities. For instance, a tree is a natural symbol. It represents an actuality which is imperceptible to the intellect. The intellect can deal only with the sensible symbol. It is a natural symbol; because it is possible directly to trace a living connection between the tree and the _tree-reality_. That is, it would be possible so to trace out the vital connection between the tree and its reality if the intellect had aptitude for such tracery. But, in reality, since it has no such aptitude, it remains for the work of that higher faculty than the intellect which recognizes both the connection and the intellect's inability to trace it. Further, an object is called a natural symbol because it is the bridge between sensuous representation and reality. It is as if one could begin at the surface of an object and by a subtle process of elimination and excortication arrive at the heart of the universum of reality. No such consummation may be reached by dealing with words which have merely an artificial relationship with the objects which they signify. Again, ideas, that is, ideas that are universal in application and have their roots in the great ocean of reality, are natural symbols; because if it were possible to handle an idea with the physical hands it would be possible to arrive at the heart of that which it symbolized without ever losing our connection with the idea itself. In other words, ideas and objects, unlike words, can never be divorced from that which they symbolize. Both, being of the same class, are the opposite poles of realities. This then is the difference between natural symbols and artificial symbols--that a natural symbol, such as objects and ideas, is copolar with reality whereas an artificial symbol, such as words, geometric constructions and the like not only lacks this copolarity but is itself a symbol of natural symbols. It is, therefore, inconceivable that because the algebraic quantity A^3 has been arbitrarily decreed to be a representation of the volume of a cube, every such quantity in the algebraic series shall actually represent some object or set of objects in the physical world. Even if it be granted that such may be the case, is it not certain that there is a limit to things in the objective universe? Yet there may not be any limit to algebraic or mathematical determinations. The material universe is limited and conditioned; the world of mathesis is unlimited and unconditioned save by its own limitations and conditions. It is irrational to expect that physical phenomena shall justify all mathematical predicates. _There is perhaps no single mathematical desideratum or consideration which may be said to be the natural symbolism of realities; for the whole of mathematical conclusions is a mass of artificial and arbitrary but concordant symbols of the crasser or nether pole of the antipodes of realism._ It is exceedingly dangerous, therefore, to predicate upon such a far-fetched symbolism as mathematics furnishes anything purporting to deal with ultimate realities. And those who insist upon doing so are either blind themselves to these limitations or are madly endeavoring to befog the minds of others who are dependent upon them for leadership in questions of mathematical import. Analogies have been unsparingly used in efforts to popularize the four-space conception and much of the violence which has been done to the notion is due to this vagary. The mathematical publicist, in trying to give a mental picture of the fourth dimension, examines the appearances of three dimensional beings as they might appear to a two dimensional being or _duodim_. He imagines a race of beings endowed with all the human faculties except that they live in a land of but two dimensions--length and breadth. He thinks of them as shadows of three dimensional beings to whom there are no such conceptions as "up" and "down." They can see nothing nor sense anything in any way that is without their plane. They can move in any direction within the plane in which they live, but can have no idea of any movement that might carry them without that plane. A house for such beings might be simply a series of rectangles. One of them might be as safe behind a line as a _tridim_ or three dimensional being would be behind a stone wall. A bank safe for the _unodim_ would be a mere circle. A _duodim_ in any two dimensional prison might be rescued by a tridim without the opening of doors or the breaking of walls. An action of a _tridim_ performed so as to contact their plane would be to them a miracle, absolutely unaccountable upon the basis of any known fact to the _unodim_ or _duodim_. A _tridim_ might go into a house where lived a family of _duodims_, appear and disappear without being detected or its ever being discovered how he accomplished such a marvelous feat. Our miracles, after the same fashion, are said to be the antics of some four dimensional being who has similar access to our three dimensional world and whose actions are similarly inexplicable to us. So the analogies have been multiplied. But the temptation to apply the consequences of such reasoning to actual three-space conditions has been so great that many have yielded to it and have consequently sought actually to explain physical phenomena upon the basis of the fourth dimension. The utilitarian side of the question of hyperspace has not been neglected either. And so, early in the development of the hypothesis and its various connotations, the attention of investigators was turned to this aspect of the inquiry. Strange possibilities were revealed as a result. For instance, it was found that an expert fourth dimensional operator is possessed of extraordinary advantages over ordinary tridimensional beings. Operating from his mysterious hiding place in hyperspace, he could easily appear and disappear in so mysterious a manner that even the most strongly sealed chests of treasures would be easily and entirely at his disposal. No city police, Scotland Yard detective nor gendarme could have any terrors for him. DRS. JEKYLL and MESSRS. HYDE might abound everywhere without fear of detection. Objects as well as persons might be made to pass into or out of closed rooms "without penetrating the walls," thus making escape easy for the imprisoned. No tridimensional state, condition or system of arrangements, accordingly, would be safe from the ravages of evilly inclined four dimensional entities. Objects that now are limited to a point or line rotation could in the fourth dimension rotate about a plane and thus further increase the perplexities of our engineering and mechanical problems; four lines could be erected perpendicular to each other whereas in three space only three such lines can be erected; the right hand could be maneuvered into the fourth dimension and be recovered as a left hand; the mysteries of growth, decay and death would find a satisfactory explanation on the basis of the fourth dimensional hypothesis and many, if not all, of the perplexing problems of physiology, chemistry, physics, astronomy, anthropology and psychology would yield up their mysteries to the skill of the fourth dimensional operator. Marvelous possibilities these and much to be desired! But the most remarkable thing about these so-called possibilities is their impossibility. It is this kind of erratic reasoning that has brought the conception of a fourth dimension into general disrepute with the popular mind. It is to be regretted, too, for the notion is a perfectly legitimate one in the domain of mathesis where it originated and rightly belongs. It is not to be wondered at that metageometricians and others should at first surmise that, in the four-space, they had found the key to the deep mysteries of nature in all branches of inquiry. For so vast was the domain and so marvelous were the possibilities which the new movement revealed that it was to be expected that those who were privileged to get the first glimpses thereof would not be able to realize fully their significance. But the stound of their minds and the attendant magnification of the elements which they discovered were but incidents in the larger and more comprehensive process of adjustment to the great outstanding facts of psychogenesis which is only faintly foreshadowed in the so-called hyperdimensional. The whole scope of inquiry connected with hyperspace is not an end in itself. It is merely a means to an end. And that is the preparation of the human mind for the inborning of a new faculty and consequently more largely extended powers of cognition. Metageometrical discoveries are therefore the excrescences of a deeper, more significant world process of mental unfoldment. They belong to the matutinal phenomena incident to this new stage of mental evolution. All such investigations are but the preliminary exercises which give birth to new tendencies which are destined to flower forth into additional faculties and capacities. So that it is well that the evolutionary aspect of the question be not overlooked; for there is danger of this on account of the magnitude and kosmic importance of its scope of motility. A geometric line is said to be a space of one dimension. A plane is a space of two dimensions and a cube, a space of three dimensions. In figure 7 below, the line _ab_ is said to be one dimensional; because only one coördinate is necessary to locate a point-position in it. The plane, _abcd_, figure 8, is said to be two dimensional because two coördinates, _ab_ and _db_ are required to locate a point, as the point _b_. The cube _abcdefgh_, figure 9, is said to be tridimensional, because, in order to locate the point _b_, for instance, it is necessary to have three coördinates, _ab_, _bc_ and _gb_. The tesseract is said to be four dimensional, because, in order to locate the point _b_, in the tesseract, it is necessary to have four coördinates, _ab_, _bc_, _bb'_ and _h'b_, figure 10. A -------------------------------- B FIG. 7. A +------------------------------+ B | | | | | | | | | | | | C +------------------------------+ D FIG. 8. It will be noted that in figures 8, 9 and 10, the element of perpendicularity enters as a necessary determination. In figure 8, the lines _ab_ and _bd_ are perpendicular to each other. Similarly, in Fig. 10, lines _ab_, _bc_, _bb'_ and _h'b_ are perpendicular to one another. That is, at their intersections, they make right angles. Similarly, figures representing any number of dimensions may be constructed. [Illustration: FIG. 9.] [Illustration: FIG. 10.--The Tesseract.] The line _ab_ represents a one-space. An entity living in a one space is called a "unodim." The plane, _abcd_, represents a two-space, and entities living in such a space are called _duodims_. The cube, _abcdefgh_, represents a three-space and entities inhabiting such a space are called _tridims_. Figure 10 represents a four-space, and its inhabitants are called _quartodims_. Each of the above-mentioned spaces is said to have certain limitations peculiar to itself. The fourth dimension is said to lie in a direction at right angles to each of our three-space directions. This, of course, gives rise to the possibility of generating a new kind of volume, the hypervolume. The hypercube or tesseract is described by moving the generating cube in the direction in which the fourth dimension extends. For instance, if the cube, Fig. 9, were moved in a direction at right angles to each of its sides a distance equal to one of its sides, a figure of four dimensions, the tesseract, would result. The initial cube, _abcc'e'fhh'_, when moved in a direction at right angles to each of its faces, generates the hypercube, Fig. 10. The lines, _aa'_, _bb'_, _cc'_, _dd'_, _ee'_, _ff'_, _gg'_, _hh'_, are assumed to be perpendicular to the lines meeting at the points, _a_, _b_, _c_, _d_, _e_, _f_, _g_, _h_. Hence _a'b'_, _b'd_, _dd'_, _d'a'_, _ef_, _fg_, _gg'_, _g'e_, represent the final cube resulting from the hyperspace movement. Counting the number of cubes that compose the hypercube we find that there are eight. The generating cube, _abcc'e'f'hh'_, and the final cube, _a'b'_, _b'd_, _dd'_, _d'a'_, _ef_, _fg_, _gg'_, _g'e_, make two cubes; and each face generates a cube making eight in all. A tesseract, therefore, is a figure bounded by eight cubes. To find the different elements of a tesseract, the following rules will apply: 1. _To find the number of lines_: Multiply the number of lines in the generating cube by two, and add a line for each point or corner in it. E.g., 2 × 12 = 24 + 8 = 32. 2. _To find the number of planes, faces or squares_: Multiply the number of planes in the generating cube by 2 and add a plane for each line in it. E.g., 2 × 6 + 12 = 24. 3. _To find the number of cubes in a hypercube_: Multiply the number of cubes in the generating cube, one, by two and add a cube for each plane in it. E.g., 2 × 1 + 6 = 8. 4. _To find the number of points or corners_: Multiply the number of corners in the generating cube by 2. E.g., 2 × 8 = 16. * * * * * In a plane there may be three points each equally distant from one another. These may be joined, forming an equilateral triangle in which there are three vertices or points, three lines or sides and one surface. In three-space there may be four points each equidistant from the others. At the vertices of a regular tetrahedron may be found such points. The tetrahedron has four points, one at each vertex, 6 lines and 4 equilateral triangles, as in Fig. 11. In four-space, we have 5 points each equidistant from all the rest, giving the hypertetrahedron. This four dimensional figure may be generated by moving the tetrahedron in the direction of the fourth dimension, as in Fig. 12. If a plane be passed through each of the six edges of the tetrahedron and the new vertex there will be six new planes or faces, making 10 in all, counting the original four. From the new vertex there is also a tetrahedron resting upon each base of the original tetrahedron so that there are five tetrahedra in all. _A hypertetrahedron is a four-dimensional figure consisting of five tetrahedra, ten faces, 10 lines and 5 points._ [Illustration: FIG. 11.--Tetrahedron.] [Illustration: FIG. 12.--Hypertetrahedron.] PAUL CARUS[17] suggests the use of mirrors so arranged that they give eight representations of a cube when placed at their point of intersection. He says: "If we build up three mirrors at right angles and place any object in the intersecting corner we shall see the object not once, but eight times. The body is reflected below and the object thus doubled is mirrored not only on both upright sides but in addition in the corner beyond, appearing in either of the upright mirrors coincidingly in the same place. Thus the total multiplication of our tridimensional boundaries of a four dimensional complex is rendered eight-fold. "We must now bear in mind that this representation of a fourth dimension suffers from all the faults of the analogous figure of a cube in two dimensional space. The several figures are not eight independent bodies but are mere boundaries and the four dimensional space is conditioned by their interrelation. It is that unrepresentable something which they inclose, or in other words, of which they are assumed to be boundaries. If we were four dimensional beings we could naturally and easily enter into the mirrored space and transfer tridimensional bodies or parts of them into those other objects reflected here in the mirrors representing the boundaries of the four dimensional object. While thus on the one hand the mirrored pictures would be as real as the original object, they would not take up the space of our three dimensions, and in this respect, our method of representing the fourth dimension by mirrors would be quite analogous to the cube pictured on a plane surface, for the space to which we (being limited to our tridimensional space-conception), would naturally relegate the seven additional mirrored images is unoccupied and if we should make the trial, we would find it empty." [17] Vide _Foundations of Mathematics_, pp. 93-94. The utility of such a representation as that which CARUS outlines in the above is granted, i.e., so far as the purpose which it serves in giving a general idea of what a four-space object might be imagined to be like, but the illustration does not demonstrate the existence of a fourth dimension. It only shows what might be if there were a four-space in which objects could exist and be examined. We, of course, have no right to assume that because it can be shown by analogous reasoning that certain characteristics of the fourth dimensional object can be represented in three-space the possible existence of such an object is thereby established. Not at all. For there is no imaginable condition of tridimensional mechanics in which an object may be said to have an objective existence similar to that represented by the mirrored cube. But there are discrepancies in this representation which well might be considered. They have virtually the force of invalidating somewhat the conception which the analogy is designed to illustrate. For instance, in the case of the mirrored object placed at the point of intersection of the three mirrors built up at right angles to each other. Upon examination of such a construction it is found that the reflection of the object in the mirrors has not any perceptible connection with the object itself. And this, too, despite the fact that they are regarded as boundaries of the hypercube; especially is this true when it is noted that these reflections are called upon to play the part of real, palpable boundaries. If a fourth dimensional object were really like the mirror-representation it would be open to serious objections from all viewpoints. The replacement of any of the boundaries required in the analogy would necessarily mean the replacement of the hypercube itself. In other words, if the real cube be removed from its position at the intersection of the mirrors no reflection will be seen, and hence no boundaries and no hypercube. The analogy while admittedly possessing some slight value in the direction meant, is nevertheless valueless so far as a detailed representation is concerned. So the analogy falls down; but once again is the question raised as to whether the so-called fourth dimension can be established or proven at all upon purely mathematical grounds. It also emphasizes the necessity for a clearer conception of the meaning of dimension and space. The logical difficulties which beset the hyperspace conception are dwelt upon at length by JAMES H. HYSLOP. He says:[18] "The supposition that there are three dimensions instead of one, or that there are only three dimensions is purely arbitrary, though convenient for certain practical purposes. Here the supposition expresses only differences of directions from an assumed point. Thus what would be said to lie in a plane in one relation would lie in the third dimension in another. There is nothing to determine absolutely what is the first, second, or third dimension. If the plane horizontal to the sensorium be called plane dimension, the plane vertical to it will be called solid, or the third dimension, but a change of position will change the names of these dimensions without involving the slightest qualitative change or difference in meaning. "Moreover, we usually select three lines or planes terminating vertically at the same point, the lines connecting the three surfaces of a cube with the same point, as the representative of what is meant by three dimensions, and reduce all other lines and planes to these. But interesting facts are observable here. 1. If the vertical relation between two lines be necessary for defining a dimension, then all other lines than the specified ones are either not in any dimension at all, or they are outside the three given dimensions. This is denied by all parties, which only shows that a vertical relation to other lines is not necessary to the determination of a dimension. 2. If lines outside the three vertically intersecting lines still lie in dimension or are reducible to the other dimensions they may lie in more than one dimension at the same time which after all is a fact. This only shows that qualitatively all three dimensions are the same and that any line outside of another can only represent a dimension in the sense of _direction_ from a given point or line, and we are entitled to assume as many dimensions as we please, all within three dimensions. "This mode of treatment shows the source of the illusion about the 'fourth dimension.' The term in its generic import denotes commensurable quality and denotes only one such quality, so that the property supposed to determine non-Euclidean geometry must be qualitatively different from this, if its figures involve the necessary qualitative differentiation from Euclidean mathematics. But this would shut out the idea of 'dimension' as its basis which is contrary to the supposition. On the other hand, the term has a specific meaning which as different qualitatively from the generic includes a right to use the generic term to describe them differentially, but if used only quantitatively, that is, to express direction as it, in fact, does in these cases, involves the admission of the actual, not a supposititious, existence of a fourth dimension which again is contrary to the supposition of the non-Euclidean geometry. Stated briefly, dimension as commensurable quality makes the existence of the fourth dimension a transcendental problem, but as mere direction, an empirical problem. And the last conception satisfies all the requirements of the case because it conforms to the purely quantitative differences which exist between Euclidean and non-Euclidean geometry as the very language about 'surfaces,' 'triangles,' etc., in spite of the prefix 'pseudo,' necessarily implies." [18] Vide _Philosophical Review_, Vol. V, 1896, p. 352, et. seq. Thus it would seem that those who have been most diligent in constructing the hyperspace conception have been the least careful of the logical difficulties which beset the elaboration of their assumptions. Yet it sometimes requires the illogical, the absurd and the aberrant to bring us to a right conception of the truth, and when we come to a comparison of the two, truth and absurdity, we are the more surprised that error could have gained so great foothold in face of so overwhelming evidences to the contrary. The entire situation is, accordingly, aptly set forth by HYSLOP when he says, continuing: "There are either a confusion of the abstract with the concrete or of quantitative with qualitative logic, ... so that all discussion about a fourth dimension is simply an extended mass of equivocations turning upon the various meanings of the term 'dimension.' This when once discovered, either makes the controversy ridiculous or the claim for non-Euclidean properties a mere truism, but effectually explodes the logical claims for a new dimensional quality of space as a piece of mere jugglery in which the juggler is as badly deceived as his spectators. It simply forces mathematics to transcend its own functions as defined by its own advocates and to assume the prerogatives of metaphysics." Shall we, therefore, assent to the imperialistic policy of mathematicians who would fain usurp the preserves of the metaphysician in order that they may exploit a superfoetated hypothesis? It is not believed that the harshness of HYSLOP'S judgment in this respect is undeserved. It is, however, regretted that the notions of mathematicians have been so inchoate as to justify this rather caustic, though appropriate criticism. For it does appear that the moment the mathematician deserts the province of his restricted sphere of motility and enters the realm of the transcendental, that moment he loses his way and becomes an inexperienced mariner on an uncharted sea. It is interesting to note that CASSIUS JACKSON KEYSER,[19] while recognizing the purely arbitrary character of the so-called dimensionality of space, nevertheless lends himself to the view that "if we think of the line as generating element we shall find that our space has four dimensions. That fact may be seen in various ways, as follows: "A line is determined by any two of its points. Every line pierces every plane. By joining the points of one plane to all the points of another, all the lines of space are obtained. To determine a line, it is, then, enough to determine two of its points, one in the one plane and one in the other. For each of these determinations two data, as before explained, are necessary and sufficient. The position of the line is thus seen to depend upon four independent variables, and the four dimensionality of our space _in lines_ is obvious." [19: Vide _Monist_, Vol. XVI, 1896, Mathematical Emancipations.] Similarly he argues for the four dimensionality of space in spheres: "We may view our space as an assemblage of its spheres. To distinguish a sphere from all other spheres, we need to know four and but four independent facts about it, as say, three that shall determine its center and one its size. Hence our space is four dimensional also in spheres. In circles, its dimensionality is six; in surfaces of second order (those that are pierced by a straight line in two points), nine; and so on ad infinitum." The view taken by KEYSER is a typical one. It is the mathematical view and is characterized by a certain lack of restraint which is found to be peculiar to the whole scheme of thought relating to hyperspace. It is clear that the kind of space that will permit of such radical changes in its nature as to be at one time three dimensional, at another time four dimensional, then six, nine and even _n_-dimensional is not the kind of space in which the objective world is known to exist. Indeed, it is not the kind of space that really exists at all. In the first place, a line cannot generate perceptual space. Neither can a circle, nor a sphere nor any other geometrical construction. It is, therefore, not permissible, except mathematically, to view our space either as "an assemblage of its spheres," its circles or its surfaces; for obviously perceptual space is not a geometrical construction even though the intellect naturally finds inhering in it a sort of latent geometrism which is kosmical. For there is a wide difference between that kosmic order which is space and the finely elaborated abstraction which the geometer deceives himself into identifying with space. There is absolutely neither perceptible nor imperceptible means by which perceptual space in anywise can be affected by an act of will, ideation or movement. Just why mathematicians persist in vagarizing upon the generability of space by movement of lines, circles, planes, etc., is confessedly not easily understood especially when the natural outcome of such procedure is self-stultification. It is far better to recognize, as a guiding principle in all mathematical disquisitions respecting the nature of space that the possibilities found to inhere in an idealized construction cannot be objectified in kosmic, sensible space. The line of demarkation should be drawn once for all, and all metageometrical calculations and theories should be prefaced by the remark that: "if objective space were amenable to the peculiarities of an idealized construction such and such a result would be possible," or words to that effect. This mode of procedure would serve to clarify many if not all of the hyperspace conceptions for the non-mathematician as well as for the metageometricians themselves, especially those who are unwilling to recognize the utter impossibility of their constructions as applied to perceptual space. We should then cease to have the spectacle of otherwise well-demeanored men committing the error of trying to realize abstractions or abstractionizing realities. Herein is the crux of the whole matter, that mathematicians, rather than be content with realities as they find them in the kosmos, should seek to reduce them to abstractions, or, on the other hand, make their abstractions appear to be realities. KEYSER proceeds to show how the concept of the generability of hyperspace may be conceived by beginning with the point, moving it in a direction without itself and generating a line; beginning with the line, treating it similarly, and generating a plane; taking the plane, moving it in a direction at right angles to itself and generating a cube; finally, using the cube as generating element and constructing a four-space figure, the tesseract. Now, as a matter of fact, a point being intangible cannot be moved in any direction neither can a point-portion of sensible space be removed. Nevertheless, we quite agree with him when he asserts: "Certainly there is naught of absurdity in supposing that _under suitable stimulation the human mind may, in the course of time, speedily develop a spatial intuition of four or more dimensions_." (The italics in the above quotation are ours.) Here we have a tacit implication that the notion which geometers have heretofore designated as "dimension" really is a matter of consciousness, of intuition, and therefore, determinable only by the limitations of consciousness and the deliveries of our intuitive cognitions. As a more detailed discussion of this phase of the subject shall be entered into when we come to a consideration of Chapter VI on "Consciousness as the Norm of Space Determinations" further comment is deferred until then. Now, as it appears certain that what geometers are accustomed to call "dimension" is both relative and interchangeable in meaning--the one becoming the other according as it is viewed--the conclusion very naturally follows that neither constructive nor symbolic geometry is based upon dimension as commensurable quality. The real basis of the non-Euclidean geometry is dimension as direction. For whatever else may be said of the fourth dimension so-called it is certainly unthinkable, even to the metageometricians, when it is absolved from direction although no specific direction can be assigned to it. It is agreed perhaps among all non-Euclidean publicists that the fourth dimension must lie in a "direction which is at right angles to all the three dimensions." But if they are asked how this direction may be ascertained or even imagined they are nonplused because they simply do not know. The difficulty in this connection seems to hinge about the question of identifying the conditions of the world of phantasy with those of the world of sense. There are distortions, ramifications, submersibles, duplex convolutions and other mathetic acrobatics which can be performed in the realm of the conceptual the execution of which could never be actualized in the objective world. Because these antics are possible in the premises of the mathematical imagination is scarce justification for the attempts at reproduction in an actualized and phenomenal universe. One of the proudest boasts of the fourth dimensionist is that hyperspace offers the possibility of a new species of rotation, namely, _rotation about a plane_. He refers to the fact that in the so-called one-space, rotation can take place only about a point. For instance in Figure 7, the line _ab_ represents a one-space in which rotation can take place only about one of the two points _a_ and _b_. In Figure 8 which represents a two-space, rotation may take place about the line _ab_ or the line _cd_, etc., or, in other words, the plane _abcd_ can be rotated on the axial line _ab_ in the direction of the third dimension. In tridimensional space only two kinds of rotation are possible, namely, rotation about a point and about a line. In the fourth dimension it is claimed that rotation can take place about a plane. For example, the cube in Figure 9, by manipulation in the direction of the fourth dimension, can be made to rotate about the side _abgf_. A very ingenious argument is used to show how rotation about a plane is thinkable and possible in hyperspace. But with this, as with the entire fabric of hyperspace speculations, dependence is placed almost entirely upon analogous and symbolic conceptions for evidence as to the consistency and rationality of the conclusions arrived at. D C D1 +-------------+............. | | : | | : | | : | | : | | : +-------------+............. A B A1 FIG. 13. It is urged that inasmuch as the rotation about the line _bc_ in Figure 13 would be incomprehensible or unimaginable to a plane being for the reason that such a rotation involves a movement of the plane into the third dimension, a dimension of which the plane being has no knowledge, in like manner rotation about a plane is also unimaginable or incomprehensible to a tridim or a three dimensional being. It is shown, however, that the plane being, by making use of the possibilities of an "assumed" tridimension, could arrive at a rational explanation of line rotation. [Illustration: FIG. 14.] Figure 14 offers an illustration by means of which a two dimensional mathematician could demonstrate the possibility of line rotation. He is already acquainted with rotation about a point; for it is the only possible rotation that is observable in his two dimensional world. By conceiving of a line as an infinity or succession of points extending in the same direction; by imagining the movement of his plane in the direction of the third dimension thereby generating a cube and at the same time assuming that the lines thus generated were merely successions of points extending in the same direction, he could demonstrate that the entire cube Figure 14, could be rotated about the line _BHX_ used as an axis. For upon this hypothesis it would be arguable that a cube is a succession of planes piled one upon the other and limited only by the length of the cube which would be extending in the, to him, unknown direction of the third dimension. He could very logically conclude that as a plane can rotate about a point, a succession of planes constituting a tridimensional cube, could also be conceived as rotating about a line which would be a succession of points under the condition of the hypothesis. His demonstration, therefore, that the cube, Figure 14, can be made to rotate around the line _BHX_ would be thoroughly rational. He could thus prove line-rotation without even being able to actualize in his experience such a rotation. Analogously, it is sought by metageometricians to prove in like manner the possibility of rotation about a plane. Thus in Figure 16 is shown a cube which has been rotated about one of its faces and changed from its initial position to the position it would occupy when the rotation had been completed or its final position attained. [Illustration: FIG. 15.] [Illustration: FIG. 16.--Plane Rotation] The gist of the arguments put forward as a basis for plane-rotation is briefly stated thus: The face _cefg_ is conceived as consisting of an infinity of lines. A cube, as in Figure 15, is imagined or assumed to be sected into an infinity of such lines, each line being the terminus of one of the planes which make up the cube. Each one of the constituting planes is thought of as rotating about its line-boundary which intersects the side of the cube. The process is continued indefinitely until the entire series of planes is rotated, one by one, around the series of lines which constitute the axial plane. Hence, in order that the cube, Figure 16, may change from its initial position to its final position each one of the infinitesimal planes of which the cube is assumed to be composed must be made to rotate about each one of the infinitesimal lines of which the plane used as an axis is composed. In this way, it is shown that the entire cube has been made to rotate about its face, _cefg_. This concisely, is the "QUOD ERAT DEMONSTRANDUM" of the metageometrician who sets out to prove rotation about a plane. Thus it is made to appear that in order that tridimensional beings may be enabled to conceive of four-space rotation, as in Figures 15 and 16, in which the rotation must also be thought of as taking place in the direction of the fourth dimension, they must adopt the same tactics that a two dimensional being would use to understand some of the possibilities of the tridimensional world. It is, of course, unwise to assume that because a thing can be shown to be possible by analogical reasoning its actuality is thereby established. This consideration cannot be too emphatically insisted upon; for many have been led into the error by relying too confidentially upon results based upon this line of argumentation. There is a vast difference between mentally doing what may be assumed to be possible, the hypothetical, and the doing of what is actually possible, the practical. In the first place, plane-rotation in the actual universe is a structural impossibility. The very nature and constitution of material bodies will not admit of such contortion as that required by the rotation of a body, say a cube, about one of its faces. Let us examine some of the results of plane rotation. 1. The rotation must take place in the direction of the fourth dimension. Now, as it is utterly impossible for any one, whether layman or metageometrician, even to imagine or conceive, in any way that is practical, the direction of the fourth dimension it is also impossible for one to move or rotate a plane, surface, line or any other body in that direction. We are in the very beginning of the process of plane-rotation so-called confronted with a physical impossibility. 2. Plane rotation necessarily involves the orbital diversion of every particle in the cube. This alone is sufficient to prohibit such a rotation; for it is obvious that the moment a particle or any series of particles is diverted from its established orbital path disruption of that portion of the cube must necessarily follow. This upon the assumption that the particles of matter are in motion and revolving in their corpuscular orbits. 3. Plane-rotation necessitates a radical change in the absolute motion of each individual particle, electron, atom or molecule of matter in the cube and a consequent retardation or acceleration of this motion. This upon the hypothesis that the particles of matter are vibrating at the rate of absolute motion. 4. It presupposes a reconstitution of each atom, molecule or particle in the cube, changing the path of intra-corpuscular rotation either from a right to left direction or from a left to right direction, as the case may be. The particles of matter in the cube will be acted upon in much the same manner as the particles in a glove when it is maneuvered in the fourth dimension. In describing this phenomenon, MANNING says:[20] "Every part by itself, in its own place is turned over with only a slight possible stretching and slight changing of positions of the different particles of matter which go to make up the glove." [20] Vide _Fourth Dimension, Simply Explained_, edited by H. P. MANNING, p. 28. The slight stretching and slight changing of the positions of the particles referred to would be of small consequence if applied to ponderable bodies. But when used in connection with particles of matter which are themselves of very infinitesimal size means far more--enough, as we have said, to militate severely against the integrity of the cube. It is not deemed necessary to go further into the physical aspects of plane-rotation as it is believed sufficient has been said to negative the assumption from a purely structural viewpoint. Among the vagaries of hyperspace publicists none is perhaps more notable than the view taken by C. H. HINTON:[21] "If it could be shown that the electric current in the negative direction were exactly alike the electric current in the positive direction, except for a reversal of the components of the motion in three dimensional space, then the dissimilarity of the discharge from the positive and negative poles would be an indication of the one-sidedness of our space. The only cause of difference in the two discharges would be due to a component in the fourth dimension, which directed in one direction transverse to our space, met with a different resistance to that which it met when directed in the opposite direction." [21] Vide _Fourth Dimension_, p. 75, C. H. HINTON. To be sure. And with equal certainty it might be said that if the moon were made of green cheese it might well be the ambition of the world's chefs to be able at some time to flavor macaroni with it, thus serving a rare dish. Even so, if there were an actual, objective fourth dimension to our space we might be able to shove into it all the perplexing problems of life and let it solve them for us. But the fact that the fourth dimensional hypothesis is itself a mere supposition seems to have been overlooked or rather completely ignored by HINTON. Or else, ought it not be an obvious folly to hope to construct a rational explanation of perplexing physical conditions upon the basis of a purely suppositionary, and therefore unproven, hypothesis? The recognized domain of the four-space, mathematically considered, is according to the most generous allowance very small, so small, in fact, that the disposition of some to crowd into it the essential content of the manifested universe is a matter of profound amazement. Then, too, it cannot be denied that there is no appreciable urgency or necessity for having recourse to a purely hypothetical construction for explicatory data regarding a phenomenon which has not been shown to be without the scope of ordinary scientific methods of procedure to unravel. The claim of certain spiritualists, notably ZOLLNER of Leipsig, that the phenomena of spiritism is accountable for on the grounds that the fourth dimension affords a residential area for discarnate beings whence spiritistic forayers may impose their presence upon unprotected three dimensional beings is no less fatuous than the original supposition itself. For upon this latter is built the entire fabric of meaningless speculations so gleefully indulged in by those who glibly proclaim the reality of the four-space. Indeed, clearer second thought will reveal that, when the pendulum of erratic thinking and trafficking in mental constructions swings back, hyperspaces, after all, are but the _ignes fatuii_ of mathetic obscurantism. Then, why should it be deemed necessary to discover some more mysterious realm of four dimensional proportions in which the spirits of the dead may find a habitation? Are the spiritualists, too, reduced to the necessity of further mystifying their already adequately mysterious phenomena? If there were not quite enough of physicality upon the basis of which all the antics of these entities can be explained, and that satisfactorily, one would, as a matter of course, be inclined to lend some credence to these claims; but as it is clear that all organized beings have some power, if no more than that which maintains their organization, and as it ought also be an acceptable fact that such a being is directed by mind; and further, that owing to the nature of a spirit body it can penetrate solid matter or matter of any other degree of density below the coefficient of spirit matter, it ought likewise be unnecessary to go without the province of strictly tridimensional mechanics for an explanation of spiritistic phenomena. Equally unnecessary and uncalled for is the attempt of certain others who lean toward the view of speculative chemists to account for the none too securely established hypothesis that eight different alcohols, each having the formula C_{5}H_{12}O may be produced without variation. This is said to be due to the fact that certain of the component atoms, notably the carbon atoms, take a fourth dimensional position in the compound and thus produce the unusual spectacle of eight alcohols from one formula. Have chemists actually exhausted all purely physical means of reaching an understanding of the carbon compounds and are therefore compelled to resort to questionable means in order to make additional progress in their field? It is incredible. Hence the more facetious appears the mathematical extravaganza in which originates the tendence among the more sanguine advocates to make of the fourth dimension a sort of "jack of all trades," a veritable "Aladdin's lamp" wherewith all kosmic profundities may be illuminated and made plain. Not until the perfection of instruments of precision has been reached, and not until human ingenuity has been exhausted in its efforts to produce more refined methods of research should it be permissible even to venture into untried and more or less debatable fields in search of a relief which after all is unobtainable. Notwithstanding the fact that all attempts at accounting for physical phenomena on the basis of _n_-dimensionality (which is itself by all the standards of objective reference a non-existent quantity and therefore irreconcilable with perceptual space requirements) are to be characterized simply as a senseless dalliance with otherwise deeply profound questions, many have fallen into a complete forgetfulness of the logical barriers inhering in and hedging about the query and have committed other and less excusable errors in the premises. Take, for instance, the suggestion that the action of a tartrate upon a beam of polarized light is due to the assumption of a fourth dimensional direction by some component in the acid. This for the reason that experimentation has shown that tartaric acid, in one form, will turn the plane of polarized light to the right while in another form will turn it to the left. It is not believed, however, that there is any warrant for such an assumption. There is also another kind of tartrate which seems to be neutral in that it has no effect whatever upon the beam of light, turning it neither to the right nor to the left nor having other visible or determinable effect upon it. Indeed, it is not clear how it is hoped to prove such a case by constituting as a norm a hypothesis which is essentially indemonstrable. A more logical procedure would be first to establish the objective, discoverable posture of four-space; show the actual movement of matter and entities therein; locate it by empirical methods of research, and then, basing our assertions upon apodeictic evidences, assume a new attitude toward these phenomena because of the support found in established and verifiable facts. Some hope of gaining a respectful hearing might then be entertained; but at least to do so now appears to be quite untimely. MAJOR WILMOT E. ELLIS, Coast Artillery Corps, United States Army, in _The Fourth Dimension Simply Explained_,[22] remarks: "... in the ether, if anywhere, we should expect to find some fourth dimensional characteristics. Gravitation, electricity, magnetism and light are known to be due to stresses in, or motions of, the infinitesimal particles of the ether. The real nature of these phenomena has never been fully explained by three dimensional mathematical analysis. Indeed, the unexplained residuum would seem to indicate that so far we have merely been considering the three dimensional aspects of four dimensional processes. As one illustration of many, it has been shown both mathematically and experimentally that no more than five corpuscles may have an independent grouping in an atom." [22] Q. v., p. 242, edited by H. P. Manning. The weakness of this view may be due to the fact that at that time MAJOR ELLIS was emphasizing in his own mind the necessity of simplifying the conception so as to make it of easy comprehension rather than the establishment of any fealty to truth or the spirit of mathesis in his examination of the problem. What therefore of reality the student fails to find in his view may be attributed to the sacrifice which the writer (MAJOR ELLIS) felt himself called upon to make for the sake of simplicity. Hence a certain expressed connivance at his position is allowable. But, on the other hand, if such were not the conscious intent of MAJOR ELLIS it is not understood how it should appear that "the unexplained residuum would seem to indicate that so far we have merely been considering the three dimensional aspects of four dimensional processes." Contrarily, it has yet to be proved that three dimensional space does not afford ample scope of motility for all observable or recognizable physical processes and that there is no necessity for reference to hyperspace phenomena for an explanation of the "unexplained residuum." It is, of course, understood that many of the possibilities predicated for hyperspace are purely nonsensical so far as their actual realization is concerned. Our concern is, therefore, not with that class of predicates, but with those wherein reside some slight show of probability of their response to the conditions of n-dimensionality either as a system of space-measurement or a so-called space or series of spaces. MAJOR ELLIS concludes his simple study of four-space by proposing the following query: "May not birth be an unfolding through the ether into the symmetrical life-cell, and death, the reverse process of a folding-up into four dimensional unity?" It is confessed that there seems to be nothing to warrant the giving of an affirmative reply to this query. It is, perhaps, sentimentally speaking a very beautiful thing to contemplate death as a painless, unconscious involvement into a glorious _one-ness_ with all life, and birth, as the reverse of all this. But where is the utility of such a dream if it be merely a dream and impossible of realization? SIMON NEWCOMB,[23] at one time one of the outstanding figures in the early development of the fourth dimensional hypothesis, openly declared that "there is no proof that the molecule may not vibrate in a fourth dimension. There are facts which seem to indicate at least the possibility of molecular motion or change of some sort not expressible in terms of time and the three coördinates in space." Of course, there is no proof that a molecule may not at times be ensconced in a four-space neither is there proof nor probability that it is so hidden. Indeed, there is no proof that there is such a thing as a molecule for that matter. [23] Vide _Science_, Vol. VII, 158, 1898, p. 4. In all of the foregoing proposals it is assumed that the fourth dimension really exists and that it lies just beneath the surface of the visible, palpable limits of the material universe; that lying in close juxtaposition to all that we are able to see, to hear or sense in any way is this mysterious, eternally prolific, all-powerful something, hyperspace, ever-ready to nourish and sustain the forms which have the nether parts firmly encysted in one or the other of her _n_-dimensional berths. Thus it would seem that while yet functioning in a strictly tridimensional atmosphere, some one, more reckless than the rest, should at last stumble upon some up-lying portion of it and be instantly transformed into a mathetic fay of etherealized four-dimensional stuff. _PART TWO_ SPATIALITY AN INQUIRY INTO THE ESSENTIAL NATURE OF SPACE AS DISTINGUISHED FROM THE MATHEMATICAL INTERPRETATION CHAPTER VI CONSCIOUSNESS THE NORM OF SPACE DETERMINATIONS Realism Is Determined by Awareness--Succession of Degrees of Realism--Sufficiency of Tridimensionality--The Insufficiency of Self-Consistency as a Norm of Truth--General Forward Movement in the Evolution of Consciousness Implied in the Hyperspace Concept--The Hypothetical Nature of Our Knowledge--Hyperspace the Symbol of a More Extensive Realm of Awareness--Variations in the Method of Interpreting Intellectual Notions--The Tuitional and the Intuitional Faculties--The Illusionary Character of the Phenomenal--Consciousness and the Degrees of Realism. Things have value for us only to the extent to which we can become aware of their being. The appraisement of all objects, conditions, states or qualities is determined directly by the degree or quality of awareness with which we apprehend them. Those elements which are without the intellect's scope of awareness have no interest and hence no value so far as the individual intellect is concerned. And this is true of all degrees and states of consciousness from the lowest to the highest, from the human to the divine. There enter into all conscious determinations three factors, namely: (_a_) the scope, or totality, of adaptations which an organism can make in the sensible world, (_b_) the power of consciousness to make adaptations and (_c_) environment. These three are interdependent. The totality of adaptations depends primarily, of course, upon the quality of conscious powers or faculties, and also, in a lesser degree, upon opportunities afforded by environment. Faculties of consciousness are derived directly from the influences exerted upon the organism by his environment and the results of the struggle to overcome them. Environment is of two kinds, artificial and natural. The artificial environment is such as has been modified by our conscious action upon external phenomena. The residue is natural. And thus the scope of adaptability becomes an unvarying witness to the quality of consciousness manifesting through a given organism. The universe is so constructed that the essential character of its various states and qualities is a fixed quantity for a given scope of consciousness and varies only as the sphere of consciousness varies. States of existence or scopes of adaptation which are registering upon a higher plane or in a more subtle sphere of existence than that in which we may at any time be functioning can only appear evidential to us when the mechanism of our consciousness becomes congruently adjusted therewith. So that the focus of consciousness must always be a variable quantity adaptable, under proper conditions, to any plane in the kosmos. Consciousness, then, becomes the sphere of limits both of knowledge and adaptability. But lest we seem to admit implicitly part of the contentions which mathematical publicists have made in postulating the unodim and duodim consciousness, it is necessary carefully to differentiate between the results arrived at as a result of the two procedures. In the first place, analysts _assume_ the existence of a unodim and duodim plane of consciousness and proceed to construct thereon an analogy designed to show the feasibility of another assumption, the fourth dimension. While, in laying the foundation of consciousness upon a tridimensional plane we do not start with an _assumption_, but with a fact. Therein lies the difference. Enormous advantages inhere in a procedure based upon facts, but in a series of planes built upon assumptions no such advantages are discovered. For however much the series of hypothetical planes may be extended or elaborated there must inhere necessarily throughout the series an assumptional value which vitiates the conclusions no less than the premises. The sanity and integrity of intellectual operations depend almost entirely upon the differentiation which we make between the necessities arising out of assumptions and those which spring up empirically from established facts. No procedure is necessary to establish the value of such a differentiation, nevertheless it may be suggested that it is allowable, under the rules of logic, to make any assumption whatsoever so long as care is taken to see that the conclusions embody in themselves the characteristics of the original premise. For instance, it is permissible to assume that space is curved. Under such an assumption, it is only necessary that the constructions which follow shall be self-consistent. But the case is different when we come to deal with spatiality and vitality. These are quantities which cannot, in the last analysis, be made to conform to the rules of the game of logic. Thus, when it is intimated that realism lends itself to an apparent division into degrees, and that each degree has a corresponding state of consciousness, it is by no means to be inferred that such apparent divisions are of mathematical import. For, in reality, i.e., when the consciousness has expanded so as to become congruent with the limits of even the space mind (vide Fig. 20), there appear to be no divisions in realism. It is only because of the fragmentariness of our outlook upon the kosmos that realism appears to be divided into various planes; for all of these planes are one. The divisions exist for relative knowledge, but not for complete knowledge; they exist for a finite intelligence, but not for a transfinite intelligence. That is why we view realism as a series of planes. It is because we discover that, as we proceed, as our consciousness expands and we take in more and more of the vital activities of the kosmos and understand better the causes underlying that which we contact, we have passed from a state of lesser knowledge to one of greater knowledge. And so we say we have passed from one degree of realism to another, whereas, really we have not passed from one degree of realism to another degree. Instead, it is our consciousness that has expanded. If now, we conceive reality to be a scale extending from one extremity to another (that is, from supreme consciousness to entire unconsciousness, from final knowledge to total ignorance), and the intellectual consciousness as the indicator which traverses the scale denoting at all times the precise degree of our comprehension of reality, and hence the degree of expansion of consciousness, we shall constitute a similitude closely approximating the real _status quo_ of humanity with respect to the sensible and supersensible worlds. The quantity or force which causes the indicator to move along the scale is the quality of awareness. And this varies directly as the scope of adaptability varies. Realism is homogeneous throughout its extent; but the scale marked upon it registers from _naught_ to _unity_. And between these every conceivable degree of awareness may be registered. The indicator moves only as the scope widens, and thus is shown a change in the quality of awareness. For, however paradoxical it may seem, the wider the scope of knowledge the better its quality: the more one knows, the more complete and of higher quality becomes that which he knows. The intellect is of scientific tendence, studiously rejecting all phenomena which do not yield to its senso-mechanisms. Even intuitions suffer the humility of rejection and do not escape the limitations which the intellect imposes upon them. This is so, because, as yet, there is no adequate perceptive and conceptive apparatus for the propagation and classification of intuitions, as apart from concepts. The outcome of these proscriptions is that intuitions--free, mobile, and more or less formless in themselves, must first be rehabilitated and vestured in garments _a la intellect_ to conform to the prevailing mode. But intuitions thus treated are no longer intuitions, but empirical concepts. True intuitions are like aqueous vapor--amorphous, permeating, diffusive: axioms or empirical concepts are like cakes of ice--formal, inflexible and conforming to the shape of the mold into which they are poured. Because of this--the scientific tendence of the intellect and the consequent necessity of reforming so much of the data which constitute its substructure, of pressing, condensing and reshaping it to suit its own ready-made patterns--it can be perceived how profound is the influence of the intellectual consciousness in determining the character of the totality of data which the sensible world, and for that matter, the supersensible, offer us. The intellect is the only means at hand for the interpretation of the meaning and significance of the world of phenomena. Consequently, whatever meaning or significance we are led to attach to that part of the universe which we contact, in any way, is dictated by the intellectual consciousness. There is no escape from the decisions of the intellect so long as the present scheme of things endures. Thus, by whatever standard of reference the matter may be determined, it remains indisputably established that the intellectual consciousness is the sole determinant of the phenomenal value of everything within our scope of awareness or adaptability. And whatever the fault, incongruity or discrepancy that may be revealed by a more intimate knowledge of the genesis and character of the appearance of the sensible world, it will be found to be due to the peculiar cut and mode of the intellect and not to things themselves. The value, qualitative or existential, which the intellect irrevocably assigns to objects and conditions in the world of the senses is the exclusive _norm_ not only by which these are judged, but also, by which our action upon them and their action upon us are determined. Images or objects which do not act upon us and upon which we cannot act have no interest for us. But as an integral part of the totality of images or objects in the sensible world, we must inevitably act upon all that is outside of ourselves, and these, in turn, must react upon us. On the other hand, there must be objects and images in the universe of life and form upon which, because of their inherent nature and on account of the lack of our interest in them and their interest in us, we can neither act nor become the object of their action. But herein is a mystery. For, either we act upon and are recipients of the action of the totality of images or objects in both the sensible and supersensible worlds, or we are so placed in the grand scheme of things that both ourselves and the sphere of our interests and possible actions are closed circuits, hermetically sealed and non-communicative with the other like spheres, which do not and cannot act upon us. There is yet a third possibility--that we are so fashioned, in the entirety of our being, that some part of us is exactly congruent with some part of every sphere of possible actions and interests in the kosmos, and therefore, each of us has being or consciousness of a kind that is keyed to and registering in the totality of such spheres; and that, at present, because our interests and possible actions are limited to the plane of sensibility, we are conscious only there. And further, that although those spheres of our consciousness which are fixed to register in other planes do not answer to the lowest on which we now operate, having a character of which we are unaware, they nevertheless cannot be said not to exist, because of the lack of communication between them. Among these three possible choices, we have no hesitancy in expressing a decided preference for the last mentioned--that the range of our being is co-extensive with the range of reality, and like a pendulum, we oscillate, at long intervals, between two kosmic extremities--nescience and omniscience. The intellectual consciousness is the touch-stone of realism. It is like a spreading light which, as it expands, reveals previously darkened corners and conditions, only it has power both to reveal and to bring into manifestation. In its present state, man's consciousness is like a searchlight. It illumines and takes cognizance of everything that falls within its scope of motility and is consequently able to study in detail that which it reveals. But that which is beyond its scope is as if it never existed so far as the individual consciousness is concerned. It is not reasonable to predict that the same characteristics that are observable in any given state shall persist throughout all the various scopes through which the consciousness must proceed in its evolutionary expansion. For the scale of kosmic realism is one grand panorama extending from the grossest to the most subtle and refined. While in general the thread of realism may pervade the entire scale it is nevertheless marked by many and diverse changes in its characteristics as it is followed from one stage to another. So that the realistic character of one stage may vary greatly from that which next preceded it or from that which will succeed it. It would appear, therefore, that in passing from one stage of realism to another there need not remain anything but the mere fact of reality in its connection with ultimate reality; for it is obvious that in every condition of realism which may be encountered in the kosmos there must be a basic thread of ultimate reality running through the whole. The entire gamut of realism may accordingly be traversed without the danger of being diverted from the golden thread of realism which thus permeates all. It is always the phenomena of realism with which we are concerned and which we are trying to understand rather than realism itself. It is this that confounds us. If it were not for the phenomena, which is the way realism or life presents itself to our consciousness, we should experience no trouble in discovering the reality, all other things being equal. For the former ever obscures the latter. It is the supreme task of mental evolution to break through the clouds of phenomena in the search for the eternal substratum of reality which runs through the sensible universe of things. The first view of conditions that the mind takes upon awakening to consciousness in any new sphere of cognition is necessarily hazy and inchoate. There is more or less of astonishment, wonder and bewilderment upon first becoming aware of a new scope of realism. In this state it is natural that the mind should overlook or ignore much that is essential and perhaps all that is so even escaping the true import of the phenomena which it senses. It is reasonable, too, that in such a state the main outlines of what is really seen may be greatly distorted and exaggerated so that it is well-nigh impossible to secure a correct comprehension of the character of a new scope of realism from any early survey. It is not until later years, after much study and circumspection that the mind, becoming used to the new conditions, begins to get correct impressions and to make valid judgments as to that which it discerns. And even then, it not infrequently happens that the resultant view of things in general is found to be in need of revision and correction. Hence, after everything is sifted down to the ultimate allowance for the illusion incident to too much enthusiasm and wonder we have only a very small residuum of truth upon which to build and this latter we often find to be the single thread of reality which runs through all the phenomena and which is, therefore, the only quantity that remains worthy of much consideration. Thus it is with religion. The path of progress over which our religious conceptions have come need not be outlined here, but to any one at all acquainted with the history of religious thought and ideals it at once must be patent that it has been one continuous surrender of the old for the new, of one degree of realism for another newer and higher degree; that always it has been the phenomena, the flora of the ideals which have had to give way, while nothing was left but the roots of realism from which they have sprung. It has been the same with scientific knowledge. Facts have been collected and hypotheses proposed to synthesize them and yet these have had to give way for others, and others still, until the data of scientific knowledge to-day are quite different from what they were in earlier days. And yet permeating the scientific knowledge of all times has been the golden thread of reality, and of all facts and systems of facts which man has successively assumed and surrendered nothing has remained but the reality; indeed, nothing could so remain, but reality. So it is with air phenomena with which consciousness has to deal. This perhaps is due to the fact that the mind interprets phenomena in accordance with the quality of its awareness, and as consciousness is a variable quantity, its standards of interpretation will likewise vary. Each new scope of awareness, after this manner, yields higher and more exact standards of interpretation. And then, progressing in awareness from the segment to the whole a fuller view of the phenomena as well as of reality itself is gained and also a more comprehensive judgment of the relations which exist between the segment and the whole. In other words, as the scope of consciousness widens it becomes more and more apparent that what was first thought to be a separate segment is in reality identified with the whole in an indissoluble manner. For the Thinker is then not only aware of the segment as such, but he is also conscious of the fact that it has definite relations with the entirety and that what he needs is merely a more extended consciousness. In denying the existence of the four-space or spaces of _n_-dimensionality as described and defined by geometricians, we do not thereby deny the existence of a plane of consciousness which is as much unlike the conditions of the tridimensional world as it is said to be unlike the four-dimensional world; but what we do deny is that such a higher plane of existence has necessarily to be conditioned by such characteristics as the metageometricians have proposed. It is maintained that there is no basis in consciousness for a world of four dimensions; that the consciousness has no tendency for action in four-space. Neither has matter nor life any inclination or potency to behave in a four-dimensional manner. It is indeed more rational to suppose that there is a higher plane, in fact, a series of higher planes, in which the thread of realism is continuous, not broken as it necessarily would have to be in extending to hyperspace, nor curved as in a manifold; that this series of subtler and finer planes of consciousness are merely an elongation of our three dimensional scope of realism. It, therefore, remains only to master the phenomena of each in just the same manner as we have, in a measure, mastered the phenomena of tridimensionality. For it is easily conceivable that the quality of consciousness is such that it may adapt itself to a far wider range of possibilities than may be discovered in hyperspace and still be a tri-space quantity. It is believed, however, that in all the new and higher planes of consciousness tridimensionality is the norm both of the phenomena and of the reality peculiar to them. And that, being such, does not change or vary, but is a fixed quantity regardless of the plane of consciousness. Furthermore, it is believed that the highest state of consciousness in the entire kosmos could easily exist, and does so exist, upon the basis of three-space as the norm of its extent. A sharp line of demarkation should be drawn between the reality which is life and consciousness and that which belongs to the realm of phantasy. For it is the prerogative of the intellect to create, out of the remains and deposits which it finds in the pathway of life, whatsoever it wills. This it does continuously; but it scarcely can be expected that such creations shall be endowed with the same dynamic character as that which life bestows upon its creations. The creations of the one are merely dead carcasses while those of the other are vital and real. Between them the same marked difference exists as between the growing tree and the lumber which the builder converts into a house. The organization which we witness when we look upon a building made of the dead body of a tree is not the same kind of organization as that which we see when we view the living, growing, vital tree. The dead tree is a deposit of life cast off by it when it passed on. Whatever the intellect can do in disposing of the remains of the tree-life is conventional and artificial. If it convert it into an edifice it will then bestow upon it a sort of consistency which is quite sufficient for all purposes. But the consistency which holds the organization of an edifice together is not the kind of consistency which holds a living tree together. In fact, there is a consistency that is not consistent. Such is the consistency of metageometry. It is self-consistent and yet inconsistent with the consistency of the kosmos and its norm of being which is consciousness. Self-consistency is one thing and kosmic consistency is quite another. It does not necessarily follow that because a given scheme of thought is consistent in all its parts that it is also consistent with universal truth or with life. This very vital fact was overlooked by GAUSS and all those who followed in his wake when he discovered that his _Astral Geometry_ was consistent throughout in all its parts. There is only one norm of truth and that is kosmic consistency. It matters little that a thing shall be self-consistent; it matters much whether it is consistent with the universal standard. It has been shown to be logically possible to elaborate at least two different systems of geometry, namely, the geometry of the acute angle and that of the obtuse, which, while each of them is self-consistent throughout, are nevertheless inconsistent with each other and with the geometry of the right angle (Euclidean). This, it would seem, appears to be sufficient for the invalidation of either one or both of the non-Euclidean systems of geometric thought. Indeed, if it can be shown that the Euclidean geometry is more representative of the true approach to the norm of space-genesis and of creation so far as its mode of manifestation is concerned, and consequently true of the norm set up by consciousness, the rejection of both systems of non-Euclidean geometry seems to be thoroughly warranted. But this is obvious and requires no demonstration nor comment to make it clear. We have only to ask ourselves whether it has ever occurred to us that consciousness has either a tendency to or adaptability for action in a curvilinear manner; or, if when we contemplate ideas or idea-relations we have the impression of perceiving a curvilinear or manifold tendence in them either of a positive or negative nature, and also whether it has been observed that our thought processes naturally assume four-dimensional attitudes. If we find that such a query must be answered negatively, and indeed we must so find, then, we have no basis for the assumption that any one of the systems of non-Euclidean geometry is valid either for the present status of consciousness or for a future existence, since it is true that the future is but an elongation of the present. Evolution is to bring no radical changes in the norms of reality; it has merely to deepen and widen and make more intense, efficient and comprehensive the present scope of our consciousness and thereby, while the Thinker is passing from one degree of realism to another, to bring him into a clearer conception of what his own limited scope of motility means to the whole. The four-space is a mathetic divertisement. That is, it cannot be said to lie in the direction of a straight line which proceeds either in a forward or lateral direction. Neither does it lie in a plane which is vertical or horizontal to the sensorium. It is, therefore, a fractural departure from any conceivable tridimensional direction, a geometric anomaly. Evolution, despite the minor aspects of its movement, undoubtedly proceeds in a straight line and not by a zigzag nor discontinuous line and hence it is irrational to assume that it will, when it passes on to the next advanced stage, emerge into the realm of the four-space. For the so-called hyperspace of geometry cannot by any standards of reference be said to lie in the plane of any straight line which can be described in three-space. If life is to evolve more efficient forms and if the forms are to evolve into more perfect organizations and mind and consciousness to become more intense and comprehensive expressions of the divine mind of the kosmos it is certainly not in the domain of hyperspace that these shall find the substructure of their higher development; but, if at all, it shall be found, as in all times past, in the realm of perceptual space where bodies are said to have three and only three possibilities of motion. What then is the significance of the more than a thousand years of mathematical labors; of all that has been said and done in an endeavor to bring into the popular consciousness a conception of hyperspace? Is it a question of _"Love's Labour's Lost?"_ Or is it a mere prostitution of mathematical talent? To answer these queries is the burden of this treatise and it is hoped that as the text continues the reader may be able to arrive at his own conclusions as to the relative value of the work of the mathematicians in this respect and be able to judge for himself the true significance of it all. The specific value of consciousness as a determinative factor in space-measurement has been recognized by all who have sought to arrive at a logical justification for the conception of four-dimensionality by analogous reasoning. The existence of the _unodim_ with consciousness limited to a line or point has been assumed and it has been shown how greatly such a being would be handicapped by his limited area of consciousness, it having been proposed to confine his consciousness to one dimension. An _unodim_ would, of course, be entirely unaware of any other dimension than that in which he could consciously function. So that with respect to his own consciousness no other dimension would be necessary for the continuance of his life processes. He might live his life without any knowledge even of any limitations or barriers to other things higher than those of his plane. He would be content to exist in the one-space and enjoy the benefits which it offered. He could have no notion of the two-space, but it has been allowed that a _super-unodim_, an _unodim_ metageometrician, if you please, could reason out a mental conception of what the two-space might be. Passing on to a space of two dimensions, the domain of the _duodim_, a greater freedom of movement is allowed and instead of being able to function in only one dimension the inhabitants of this plane would find themselves able to move about in at least two directions. Consciousness would accordingly enjoy a more comprehensive scope. But in a manner similar to that used by the _unodim_ metageometrician it is held that the _duodim_ could get a conception of the three-space by analogous reasoning and so gradually become conscious of a higher degree of spatiality than his own. In the conscious reasoning of both, however, is the condition of perpendicularity. That is, it must be assumed by both the _unodim_ and the _duodim_ that the new dimension must lie in a plane perpendicular to their space. So, the _unodim_ would postulate that the two-space must lie in a direction at right angles to his space, and yet he would not be able to indicate the direction owing to his ignorance of any experience that would acquaint him with the new space as well as the want of possibility of motion therein. Similarly, the _duodim_ would arrive at a conception of three-space. Thus, it has been argued that _tridims_, or people living in our tridimensional world, could, by using a like line of argument or reasoning, arrive at a conception or understanding of the four-space, which, of course, must also lie in a direction at right angles to three-space. The implications of this mode of thought show how thoroughly the mathematician recognizes the limitations which consciousness imposes upon our knowledge of the world and the subtler conditions about us. But, moreover, it is even obvious to all who stop to think about it; for it can readily be seen how little those things which do not enter our scope of awareness affect us either physically, mentally or spiritually. But no one can be so bold as to deny utterly that anything exists but what is found in our consciousnesses. It is even true that in the great centers of population where people are compelled to live many families in the same house, it is the usual thing for these individual families to live in complete forgetfulness of all the others in the house and live their lives so completely that it would be exceedingly difficult to measure the effect the one has upon the other. The mathematician, as is shown by the hyperspace movement, recognizes that there are planes of superconsciousness the nature and character of which persons confined to limited areas of consciousness can have no knowledge and may only arrive at that knowledge by serious thought and contemplation. In other words, they tacitly admit the existence of higher planes of consciousness as well as the necessity of elevating the personal consciousness in order to comprehend them. Although it was not expressly allowable in the analogy of the _unodim_, it is nevertheless one of the strongest implications of the process of reasoning that the _unodim_ could have easily raised the plane of his consciousness by continuing his researches until he, too, became conscious of the three-space, mathematically, as well as the two-space. For it was not necessary for him to raise the plane of consciousness in order to contact the two-space. He had need only to widen it. But in order to comprehend the mathematical three-space it would be necessary for him to elevate his consciousness. The fundamental error in the foregoing line of thought rests in the fact that awareness in the human family has not developed in the manner outlined. The human species has not come into conscious relations with the three-space by outgrowing the one-space and the two-space in succession. The fact of the matter is that when consciousness first dawned it must have encompassed all three dimensions simultaneously and equally and there is nothing to indicate that its rise was otherwise. Then, specifically there is no evidence that the evolution of consciousness has proceeded in a rectangular manner. Indeed, there is undoubtedly no warrant for the assumption that it has progressed in ways that are mathematically determinable at all. The question very naturally rises in view of the above as to the relative value of mathematical knowledge in the scheme of psychogenesis. Can mathematical knowledge or laws be said actually and finally to settle once for all time any question in which consciousness or life enters as a factor? Upon the response to this question hinges unanswerably the decision as to the category which mathematical knowledge should by right occupy in the entire schematism of life. If it can be successfully maintained that final judicative power abides in mathematics in the determination of these questions, then it would be useless to struggle against the fiat of mathematics and mathematicians; verily, we should be compelled to accept _nolens volens_ all that mathematicians have devised about hyperspace and its connotations. If, on the other hand, it can be shown that no such judicative power inheres in mathematical knowledge we shall then be able to establish for mathematics a true category and to dispose of the hyperspace movement in a manner that shall at once be logical and necessary. That the discovery of hyperspace by the mathematician is merely an aspect of a general forward movement in the evolution of consciousness can be shown by a brief correlative study. Hyperspace is the artificial symbol of a higher and more extensive realm of awareness. For it cannot be doubted that to be able to think in the terms of hyperspace, to study the various relations and interrelations upspringing from the original premises, actually to become conscious in the hyperspatial realm thus constructed, requires a different species or quality of consciousness than that required for ordinary thinking. The period covering the rise of artificial spatiality is contemporaneous with the rise of the phenomena identified with the spiritual life of SWEDENBORG; for during the same time he began a series of visions which revealed to him great knowledge of the unseen and supersensuous realities of life and existence. His consciousness was being flooded with the light from so-called celestial spheres and he was gradually becoming conscious of a "new dimension," a new space, a higher world that is altogether unlike the world of the senses. During this period, too, DANTE, the great kosmic seer began to jot down the results of his "hyperspace" experiences, after which he wrote his _Divina Commedia_ in which he describes more or less minutely some of the characteristics of the hyperspace domain which was revealed to his consciousness as he saw and interpreted it. Both SWEDENBORG and DANTE being deeply religious and pious-minded had their reports of the new world colored by their own mental experiences and proclivities. PLATO had at an earlier day set down what he conceived to be the ethical and civic characteristics of the new age, the _Utopia_ of mankind living on a higher plane. It was during these days that the doctrine of evolution was born, although it remained for DARWIN to formulate and buttress it with a stupendous congeries of facts. MARTIN LUTHER, the great religious reformer, likewise contacted the radiating light-glow of a higher consciousness into which the race was coming but of which only the foremost were able to get advance glimpses. KANT, one of the peerless leaders of the vanguard of humanity, at this time also, conceived and wrote his _Critique of Pure Reason_ which is likewise an evidence of the upliftment of his consciousness on the side of pure intellectuality and the commencement of a general period of illumination. And then, later, but embraced within the same period, artists began to get glimpses of this higher consciousness which showed itself in a new and strange departure in art. In rapid succession new schools sprang up and came to be known as the "cubist," "post-impressionist," "futurist," "orphimist," the "synchromist" and the "vorticist." Art really began the search for the "plastic essence of the world" trying to portray its conception of the "real image of the spirit" of the world. Color acquired a new kind of splendor and painting gave birth to a new intrinsic beauty of material and sheer loveliness of texture. All of which were evidences of an intellectual up-reaching in response to the sharp appulsions from above. DARWIN's mind, being of scientific bent, saw and interpreted everything in the terms of materialistic science; but there is no doubt but that the expansion of the area of awareness which his mind experienced in his great conception of evolution as a continuous process and all that it implies thereby was a result of the universal appulsion of the human intellect against the new domain of consciousness. And KANT's conception of space in general perhaps may be said to have been the seed-thought for the metageometrician. But thus it will be noted that in all the cases mentioned in the foregoing there is always present the personal element of the investigator, and that the reports of each of these have been colored and characterized by their individual consciousness and experiences. That all reports would agree with respect to details connected with the new domain of consciousness could scarcely be expected owing to the wonder and bewilderment which seize the intellect under such circumstances. No implication that the mathematicians have been unduly excited by what they have discovered after years of patient research in this direction is indicated by the foregoing observations; but it cannot be denied that the enthusiasm of the moment and the consequent minimization of all other phenomena but the special line being investigated serve very effectively to obscure the mental vision of the more partisan. It perhaps is sufficient that the investigator should set down in as orderly manner as possible the things which he conceives, and that he should interpret them according to the standards of his own intellect. More than this cannot be expected. Moreover, it usually suffices that the future investigator, far removed from the beclouding influences of partisanship, who successfully raises his consciousness to that higher plane shall be able to synthesize the findings of all and thereby with the aid which comes to him from a more advantageous position arrive at sounder views and a more reliable judgment. It will thus be seen that the metageometrician's method of interpretation is no more entitled to final credence and general acceptance than that of the spiritualist, SWEDENBORG, or the occult seer, DANTE. For in their best moods and at their highest points of mental efficiency these have only succeeded in vaguely symbolizing what they have conceived of the realities of the supersensuous realm in terms of their own experiences. Is there any more cogent reason, then, for accepting the analyst's conception of a world of hyperspace peopled with ensembles, propositions, spaces of _n_-dimensionality and other mathetic contrivances than the _Inferno_ of DANTE, inhabited by hideous shapes and repellent denizens, the remains of ill-spent earth-lives or SWEDENBORG'S _Celestial Realm_, wherein dwell numerous beings of celestial character performing various tasks in the work of the world? These observations should not lead the reader to come to the conclusion that the visions of DANTE and SWEDENBORG are deemed to be more worthy of credence than mathematical knowledge when that knowledge is limited to the sphere where it rightfully belongs; but the proper view is that which would make it appear that it is the way these widely differing workers interpret what they have seen; that it is the adaptation of the unseen realms to the peculiarities of the mentalities which observe them. The mathematicians have simply portrayed as well as they could their conception of the new stage of consciousness and its contents, and following the _modus vivendi_ of all intellects have interpreted these things in the terms of mathematics, merely because mathematics constituted the best available symbology at hand for the purpose. Similarly, the painter sees a new world of color; the politician, a new era of political freedom; the religious enthusiast, a new religious conception; the scientist, a new condition of matter and energy, and so on, to the most ordinary mind, each sees something new while at the same time is necessarily limited to the confines of his own mentality when he comes to interpret what he sees and conceives. Hence, there would appear to be only one way to regard all these advances and that is by synthesizing them, by correlating, and by tracing them to a common source, and finally by seeing them as one general forward movement of intellectual evolution. Man, the Thinker, who in essence is a pure intelligence, has two mental mechanisms or organs of consciousness. One of these is the brain-consciousness or the egoic. It is so called because the brain is its organ of expression and impression. It manifests through the brain and uses it to further the various objective cognitive processes. The brain-consciousness is a child of the physical body and its life is intimately identified with the life of the body. This consciousness may be called the _a posterioristic_ mechanism or organ of the Thinker and is therefore his means of interpreting the phenomena of the objective world. Cell-consciousness is a phase of the ergonic functions of the _a posterioristic_ mechanism. The other organ of consciousness is an aspect of the intelligence of the Thinker himself and perhaps may be said to be the active, organized portion of that intelligence. It is separate and distinct from the _a posterioristic_ consciousness yet sustaining a substructural relationship with it, being the source of the egoic or brain-consciousness. It may be called the _a priori_ consciousness. Its roots are buried deep within the heart of the space-mind and it is therefrom nourished and developed by what it receives in the way of intelligence. It is the intuitional faculty; knows without being taught; conceives without reason; interprets according to the norms of the space-mind or the divine mind of the kosmos. It always resides on a higher plane than that of the brain-mind or consciousness, only at rare intervals being able to contact it with flashes of its own intelligence as intuitions. The _a priori_ consciousness being the intuitive faculty of the Thinker is, therefore, a phase of his mental life on a higher plane than the sensuous. All its conceptions constitute the _a priori_ knowledge of the brain mind so-called. The _a priori_ faculty of man's higher consciousness gives the character possessed by that form of knowledge known to philosophy as the _a priori_. So that the _a priori_ has a more substantial basis than has hitherto been surmised. It is not only that which may be said to transcend experience but that which is the organ of contact with the supersensuous realities as well as of expression through the brain-consciousness. The mind's method of apprehending objective phenomena is not a direct process but an indirect process by virtue of which neurograms or nerve-impacts registered in the brain are interpreted. External sense-impressions are, of course, conveyed to the cortical area by means of appropriate vibrations which traverse the lines of the neural mechanism. These are recorded in the brain areas just as a telegraphic communication is registered in the apparatus of the receiving end, and in being so, they make terminal registrations which man, the Thinker, interprets after a psychic code which has been built up empirically. That is, he comes to know that certain rates of vibration and certain peculiarities therein mean certain things when referred to the sensorium. He then interprets according to this experience the symbolism of all neurographical impressions. But it is obvious that under such circumstances, where the interpreter is far removed from the thing itself and finds it necessary to interpret rates of vibration or symbols in order to arrive at a knowledge of the intelligence which is conveyed to him, the chances of inadequate conception are very great indeed. In fact, it is not possible through the use of neurographical symbols alone to get any complete notion of the phenomena considered. And thus there stands between the Thinker and absolute knowledge a barrier which prevents his arriving at a state of certitude in his knowledge of the world of sensible objects. It is, however, a barrier which will always remain, checking ever his approach to finality in his understanding of the universum of appearances. A markedly different condition obtains in the realm of the _a priori_ or intuitional for the reason that the barriers which inhere in the neurographical or _a posterioristic_ method are absent and the Thinker has a more direct approach to the objects of cognition. Hence the chance for error is very small indeed. This will account, therefore, for the superiority of the intuitional over the rational or the perceptual. Indeed, it is doubtful whether the purely rational possesses any value whatsoever when its _modus vivendi_ is unsanctioned by the intuitional. Else why can we not be certain that the results of our rational processes are correct at all times? Is it not because we lack the power to perceive whether our premises are correct in the first place? Quite truly. For if the Thinker can intuit the necessity and certitude of any given premise it follows that the consequences of that premise are true. It would, therefore, appear that the more the intuitional faculty is developed the clearer will be our perceptions not alone of abstract values but of objective things themselves. Further, it is doubtlessly true that the more the space-mind is developed in the human race the deeper will become our perceptions of the essential _be-ness_ of things so that whatever may be the presentations of the space-mind to the brain-mind they will be by far more accurate than the impressions we receive through the latter as a medium of apprehension. It is but natural, however, that in the present more or less chaotic condition in which the faculty of the intuitional is found it should be difficult even to interpret its presentations accurately. It is perhaps due to the fact that we are unused to its deliveries and mode of registration as well as to the fact that it has been overshadowed by the intellectual or rational faculty. But the mere fact that it is present and functioning, even if but rudimentarily, is evidence of its potentiality and the possibility of its future development to a still higher degree of efficiency. There is no doubt but that the original impulse which resulted in the creation of the faculty of perceptibility in the Thinker also marked out the metes and bounds of our entire range of perceivability which includes not only the intuitional but something higher still. There is no doubting either the obvious fact that these metes and bounds cannot have been other than rudimentary or general lines of denotation, and that the work of their further elaboration and refinement is a matter of evolutionary detail. For if we assume that the general principles of evolution are true we immediately recognize the cogency of this view. That which we now call the hand has not always been the perfect instrument that it is nor has the ear always been so keenly adjusted as at present. It has required undoubtedly many million of years for the eye to reach its present degree of complexity and adaptability. Yet in all these cases the different organs existed in potentiality from the beginning; the metes and bounds of the hand, the ear and the eye were laid out primordially. Evolution has specialized and adjusted them to environments and needs. Thus it will be seen that while the intuitional faculty was designed for manifestation from the beginning it has nevertheless required ages for its appearance even in the most rudimentary fashion. Almost the entire content of human knowledge is based upon assumptions or hypotheses; in fact, is but a mass of these, and especially is this true of mathematics, science and philosophy. Of course, there are certain minor observable facts which by reason of the seeming permanence of their existence have been eliminated from the category of assumptions. But even this elimination when it is traced to its depths may be found to be erroneous, and perhaps after all, when we have really begun to know something of the reality of things, may have again to be placed in this category. And then, too, the hypothetical nature of our knowledge is due largely to the Thinker's method of contacting the objective world which is the subject of his knowledge. It is because it is necessary for him to interpret the neurographical symbols which sense-impressions make in the brain matter according to a psychic code that renders his knowledge of things in general hypothetical. His interpretations are based upon an assumed value which experience has taught him to give to each neurogram. But even when his interpretations leave nothing to be desired in respect to their accuracy of apprehension of what the neurographical vibration implies there is that further barrier to his cognition of reality which is due to his remote removal from the object itself and the consequent extreme difficulty, if not present impossibility, of identifying his consciousness with the essence of the objects which he contemplates. When the Thinker's consciousness is presented with a neurograph of say, a cube, it is not the cube itself which he contemplates or observes; it is the neurograph or psychic symbol which the sense-impressions make in the brain. His consciousness deals not with the object but with the symbols. It is true that when he verifies one neurograph by another, as the _scopographic_ or sight impressions by the _tactographic_ or touch impressions it is found that the delivery thus determined is a true enough representation. It is also true that the Thinker, as a rule, does not accept a neurograph as valid until it has been verified by at least one or more presentations through his outer sense organs. It occurs, therefore, that all such deliveries are verified and corrected by one or more sense witnesses before final acceptance by the Thinker; but even then it cannot be said that his notions thus gained are in all respects correct and true to the standards set up by the brain-consciousness not to mention higher forms of consciousness. And then, when we consider that in addition to the numerous chances for error which naturally inhere in this method of cognition it must also be apparent that the Thinker's approach to the reality of things is much impeded by his separation therefrom, the unreliability of our ordinary methods of cognition is much emphasized. But aside from the egoic or brain-consciousness there is the higher consciousness of the Thinker himself. For the brain-consciousness is merely his method of regarding and comprehending the neurographical deliveries, the psychic code by which he systematizes and organizes his cognitions or impressions of the sensible world. This higher consciousness constitutes the faculty _a priori_ for the Thinker on a higher plane of existence, and because it deals with elements altogether unlike those which make up the data of brain-consciousness is, accordingly, less liable to error in its judgments of the supersensuous presentations than is the objective or brain-mind. In fact, it is difficult to conceive of a state or conditions wherein, supported as this view contemplates, the intuition should err in judgment. Viewed from the standpoint of external impedimenta, this condition may be said to be due to the absence of sensuous barriers which would otherwise prevent the near approach of the Thinker's consciousness to the essence of things which is the object of his consciousness on this higher plane. Directly, however, it is undoubtedly due to the fact that, following the lead of life itself, yea, as the veritable handmaid of life, it cannot err where life is concerned. When dealing with notions _a priori_ or intuitograms the Thinker is relieved of the onerous necessities and limitations incident to the examination and determination of neurographic symbols registered in the brain cortex and so is free to study, to examine and judge at first hand the impressions which are received from his own plane of intuition. The difference is about the same as that which should exist between the methods of communication between two telegraphic operators when in one instance they would have to depend upon the deliveries conveyed over the wires, while in the other, when they stood face to face with each other, they could communicate by direct conversation. In the one case the method of communication is direct and simple, while in the other it is indirect, circuitous and complex. It can, therefore, be readily seen that in all cases where the approach is made in a direct, simple manner the probability of error is much less than in cases where the intellectual approach is less direct and more complicated. Hence in drawing conclusions as to the relative importance of the two mechanisms of consciousness, the _a posterioristic_ and the _a priori_, it is necessary to bear in mind the comparative superiority of the one over the other as a means of cognition. It matters little that the intuitional faculty is not so well developed as the tuitional because it is but natural that inasmuch as the Thinker's needs are subserved in the sensuous realm by the tuitional consciousness it should, from more active use, gain somewhat over the intuitional in facility of expression and general utility. And the more so, because the two faculties serve different purposes; one is attuned to receive impressions from a subtler plane while the other is fitted for contact with the phenomenal universe; one is related to the conceptual while the other is related to the perceptual. They differ not only in function, but in nature as well. There is, of course, a natural barrier consisting of the inherent limitations of each faculty which prevents the full mergence or unification of the two states of consciousness so that there exists a state of consciousness the data of which the brain-mind is unaware, it being able only at rare intervals even to receive so much as slight impressions from it in the nature of intuitional flashes or inspirations and the like. Viewed in this light it would appear that the cognitions which are most truthworthy are those which are presented by the intuitional faculty because they are nearer to the essential reality of things; they have to do more specifically with the nature of that which _appears_ while the tuitional mind can only regard that which is the appearance. Herein lies the whole difference. The natural outcome of this division of labor between the tuitional and the intuitional is the establishment of the fact of man's relationship both to the phenomenal and the real; that in his psychic nature must reside the faculty of apprehending the real and that he shall one day awaken into activity this now latent faculty whereby he may make a direct and naked contact with reality. If it be true that, as PLATO said, God does geometrize, and that the divine geometry, as will appear, is based upon a system, an alphabet which taken together are the point . , the line ----, the triangle [Illustration], the square [Illustration] and the circle [Illustration], then, we have in this geometric alphabet the very secret of the divine geometry. With these, and in the kosmic laboratory of _chaogeny_, the Creative Logos has measured off the limits and confines of space; with them He has traced out its dimensions the archeological evidences of which we may view in the space-mind itself; and with them he has established the manner of its appearance to the Thinker. In dimensions, three, and yet not three, but one, Space, the eternal progenitor of all forms and energies, having received the divine fiat in the beginning that thus far it should extend and no further, persists in faithful obedience to the law of its being--tridimensionality. It must be so because it is thus sanctioned by the highest faculty in man that can render judgment thereon. If tridimensionality inhere in the space-mind, as the law of its being and in the intuitional consciousness as the norm of its essential nature and as the easiest and simplest expression of the tuitional mind, how can it be gainsaid that these considerations obviate the necessity of the mathetical hyperspace? If the reality of things is hidden from us and if we are, therefore, unable to perceive their real essences it is because our mode of thought and our consciousness have obscured our vision and limited us to this state of paucity of perception. It is not because reality is itself a hidden, inscrutable quantity nor that its _modus vivendi_ is "unknowable"; but because we being multiformly limited, "cabined, cribbed and confined" are resultantly lacking in the power to discern that which otherwise would be most obvious to us. It may well be set down as axiomatic that when, in the process of our thinking, we arrive at the inscrutable, the unknowable and the infinite, it is evident that our thought processes are dealing with a form of realism which is higher and beyond the possibilities of our loftiest thought-reaches. And in order to symbolize to itself this condition the intellect poses such terms as "inscrutable," "unknowable" and the "infinite" simply because that is the best it can do. Hence when it is said that space is infinite it is apparent that the mind recognizes that when it contemplates space it is dealing with something whose degree of realism transcends its powers of comprehension. Infinity is a relative term, and in fact, decreases in extensity in the proportion that the consciousness expands and comprehends. It is not unlikely that should the intellect one day discover that it had awakened into union with the space-mind it would immediately reject its preconceived notion of the infinity of space. But we need not wait until the coming of this far off event in the path of psychogenesis; for we can here and now perceive with what must be a higher faculty than the intellect the verity of this conclusion. But certain it is that the intellect, in the pride and arrogance of its traditional heritage, will not without a great struggle yield the ground and prestige it has held for an aeon of time; and in vain does the intuition serve notice of dispossessal in these premises; but however stubbornly fought the battle, however tenaciously held the position time will discover the weakening of the intellect's hand. Death for the intellect may ensue as a result of the conflict but it will be a death wherefrom it will arise, quickened, revivified and uplifted by its disposer, the intuition, upon the remains of its dead self to a higher and grander state than it has ever enjoyed before. Space is not static. It is dynamic, potential and kinetic. It is a process, a becoming. Its duration as a process is never ending. Its extensity is limited and finite. The so-called infinity of space is one of the capital illusions of the intellect which can only be removed by an expansion of the consciousness, by a mergence of the individual consciousness with the space-consciousness. In the ever-widening circle of the individual consciousness lesser realities give way to greater as the darkness recedes from the light--the lesser appearing in comparison with the greater, as the consciousness broadens, as matter to spirit, as night to day or as limitation to non-limitation. Thus the most solid facts and conditions of our limited life are but the shadows of the deeper realities which shall be revealed to the Thinker in the days of his larger and more glorious life of freedom from limitations. And now it will appear that the whole fabric of our knowledge shall have to be reduced to the bare warp and woof; for nothing is real but these. It is as if the Thinker, using the tuitional mind, had been in all times past studying the design woven in the surface of a very thick plush carpet. There are the warp and woof, the long vertical threads which make the plush and the intricate design appearing on the surface. Our knowledge may be likened to the design. It represents the contents of our knowledge. We have not even so much as begun the study of the nature of the vertical threads as they appear beneath the surface to say nothing of beginning the study of the warp and woof. The warp and the woof are the realism of the kosmos; the vertical threads are the roots and stem of the phenomenal world; the design is our sensible world as it appears to the intellect. The life of the intellect has been spent in contemplating this design; while of the hands which wove the carpet, of the mind which directed the hands and of the spirit which vitalized all, it knows nothing nor indeed can it know anything. Where shall we say are those hands, that mind and that spirit which made the carpet possible and an actuality? In vain do we search among the remains, among the soft, glistening threads of the carpet or among the intricacies of the design. For they are not there. They have passed on. The intellect looks at the design or at the vertical threads and because it is unable to follow them to their source, it decides that they are infinite, inscrutable and unknowable. But not so. All that is required are eyes to see and a mind (or shall we say a mind vitalized by the intuition) trained to discern the threads as they point upward with their termini firmly rooted in the warp and woof of the fabric. But we must first master the design, and then turning to the threads, master them. Then shall the doors of kosmic reality swing wide and the Thinker shall be ushered into the eternal palace of kosmic realism wherein he shall find the great secret, the heart, the purpose, the beginning and the end, the very nature of things-in-themselves. The nature of every degree or condition of realism is so constituted that its qualities, characteristics and limitations are exactly adequate for the satisfaction and fulfillment of all the requirements and needs of every possible state of normal consciousness. So that each degree of reality and each state of normal consciousness is sufficient and complete in itself and mutually satisfies the necessities of each other. The substratum of reality or life which extends from the heart of the kosmos to the extreme limits of the phenomenal universe exists in degrees, not discrete, but continuous. And these merge into one another by insensible stages. Such is the imperceptible continuity of the whole as each degree is gradually immerged into the other that only the limitations of consciousness itself make it to appear as if it were discontinuous. For every stage of realism there is a state of consciousness which answers to it completely and sufficiently. So both the state of consciousness and that of reality, manifesting at any given stage, seem to be complete and final for that stage. Realism or life and consciousness possess only a relative finality fashioned upon the necessities and requirements for any given state of being. Consciousness alone fixes the apparent limits of life; it also determines the state of our knowledge of life. And thus when the Thinker is confined to any stage of reality and congruent degree of consciousness it appears that what he there finds is ample for all his purposes. Accordingly he is convinced that that stage is the final consideration of his scope of motility. It is only when he is able to raise his consciousness to a point where he can contact higher realities that he becomes aware that there are higher stages in which his consciousness may manifest. This peculiarity of the Thinker's consciousness is accentuated when he allows himself to become wholly engrossed with a study of the phenomena of that stage in which he can consciously function. Hence it constantly occurs that men exhausting the study of the phenomenal find themselves floundering upon the beach of the outlying shores of consciousness where in sheer desperation they fall into the illusion that they have indeed reached the limits of manifested life and that beyond those limits there is no organized being. Unconscious are they that in ever widening circles the fertile lands extend and await the awakening of their consciousness when they may till the fallow ground of this new domain and begin again the search for the ultimately real. With respect to the present powers of consciousness, it cannot be successfully controverted that the concept of tridimensionality of space is sufficient for all purposes. It must be so for it is not only an aspect of the phenomena of space but of reality as well. This fact is attested by the nature of mind that answers to the nature of space. Tridimensionality characterizes the entire extent of consciousness and life, and therefore, of space itself. This characterization may be traced to the very doors of the heart of space where the three become one. Nor would this conception be in the least vitiated if it were allowed that the mass of the phenomena of the supersensuous world, lying in close proximity to the sensuous world, does present itself to the consciousness in a four-dimensional manner and that the phenomena of a still higher plane present themselves in a five or _n_-dimensional manner to that state of consciousness which may be congruent with them; because then we should be making allowances for the changes in phenomena and their mode of presentation to the consciousness which by no means implies a corresponding change in reality or life. All phenomena are fashioned by the intellect. The phenomenal world is just what the intellect interprets it to be. It is that and nothing more. Its qualities, attributes and characteristics are such as the consciousness gives to it. It exists only for the purposes of the evolving consciousness. And, as an instrument of consciousness, its existence is strictly subject to the evolutionary needs thereof. In that moment that the immediate needs of the consciousness shall no longer be able to find satisfaction in the phenomena of any plane of nature, in that moment the phenomena of that plane disappear, recede and are swallowed up in the maelstrom of eternal reality. In the gradual expansion of consciousness as it passes through the infinite series of grades of awareness meantime becoming deeper, broader and more comprehensive as it proceeds, there may be observed running through all these planes and orders that which is neither the phenomena of the various planes nor the consciousness; but which must be the substructural basis of both, remaining the same, unchanged and unchangeable. That is the thread of reality, the passage of life itself which is the eternal basis of all. Now it is to this reality, life, that the space-mind is related and in which its roots, its heart and the very center of its being are at one with the divine mind of the kosmos. The question of dimensionality is solely a concern of the objective or brain-mind which is the intellect. It is one of the ways in which the intellect endeavors to understand phenomena. It is an arbitrary contrivance devised by the intellect for its convenience in studying the world of things. Without it, as obviously appears, the intellect would not be able to go very far in its consideration of the minor problems which inhere in material things. The fourth dimension is but another attitude, another contrivance, which the intellect has devised in order that it may study from another angle the evanescent phenomena of the world of appearances. Having apparently exhausted the possibilities of motion in three dimensions, and being driven on to the acquirement of more picturesque views by the very necessity of its continued growth, it has betaken itself to another higher mountain peak, called "hyperspace" where with larger lenses and higher powered instruments it is beginning to scan the landscapes of a new intellectual realm of consciousness. Yet the celestial wonders of this new-found realm of consciousness remain in undisturbed forgetfulness or neglect. But it is not by a scrutiny of mathetic landscapes nor by a study of the celestial wonders that the Thinker shall one day realize the object of his eagerly pushed quest after the real; for he shall find it, if at all, in the temple of the kosmic mind which is not made by the intellect nor meted and bounded by geometric systems of space-measurement. In all the learned pother incident to the mastery of the phenomenal, the furniture of the world of the senses, it is as if the self in man, the Thinker, sat secluded in a six-walled tenement, and hence six times removed from the subject of his study, and endeavored to interpret that which appeared to his vision. And thus, thinking that what he sees is the only reality, he remains in inglorious nescience of the reality of that upon which he himself stands, unconscious that the tunnel-shaped aperture through which he peers leads not outward, but _backward_ and within to the habitation of the real of which he himself is a part. Men are deeply and well-nigh hopelessly concerned with appearances, with static views of life, with instantaneous exposures. Life, reality and all the eternal verities pass on and assume countless postures, attitudes, moods, tenses and nuances. The intellect is content to occupy itself with a single tense or mood. Indeed, it has no aptitude or power to consider more than one at a single time. It thus misses the continuity, the ceaselessness, of life. What is more, every singularity, every attitude, mood or tense which the intellect grasps for consideration is immediately remade so as to fit its own moods and tenses. And upon each and every nuance the intellect immediately imposes its own form--actually and literally rehabilitates them with its own habiliments. Unfortunately, this peculiarity occludes the intellect from any approach to the true nature of that phase which it can grasp. Hyperspace is one of the illusions of the phenomenal; it is the dress which the intellect has superimposed upon a single nuance; it is a mask which is an exact replica of the mood of the intellect. Yet through this mask the intellect grandly hopes to approach reality. The period through which the mind is now passing is a repetition of the evil days of scholasticism when men set out to determine the exact number of celestial beings that could be perched upon the extremity of a needle point. It is a time when men's minds easily assume grotesque and hideous shapes and their thoughts become the embodiment of fantastic entities. The exclusive occupation of such minds becomes the fabrication of mathetic monstrosities which rapidly deliquesce upon the first approach of the real or the appearance of the first ray of intuition which may escape through the dim and misty condition of the intellectual over-hangings. It will not be ever thus; for the Thinker will one day pass from a study of the arrangement of phenomena in space and by well-ordered steps come once again to himself. And then through the maze of it all set out upon the true path----the tridimensionality of space following which he will inevitably approach the citadel of the real, the kosmic space-mind. CHAPTER VII THE GENESIS AND NATURE OF SPACE Symbology of Mathematical Knowledge--Manifestation and Non-manifestation Defined--The Pyknon and Pyknosis--The Kosmic Engenderment of Space--On the Consubstantiality of Spatiality, Intellectuality, Materiality, Vitality and Kosmic Geometrism--Chaos-Theos-Kosmos--Chaogeny and Chaomorphogeny--N. MALEBRANCHE On God and the World--The Space-Mind--Space and Mind Are One--The Kosmic Pentoglyph. Geometry is concerned primarily with a study of the measurement of magnitudes in space. Three coördinates are necessary and sufficient for all of its determinations. Metageometry comprehends the study of the measurement of magnitudes in conceptual space. For its purposes four or _n_-coördinates are necessary and sufficient. Perceptual space is that form of extension in which the physical universe is recognized to have been created and in which it now exists. Conceptual space is an idealized conception belonging to the domain of mathesis and has no actual, physical existence outside of the mind. Mathematical space represents the idealism of perceptual space. Geometrical magnitudes may be defined as symbols of physical objects and geometry as a treatise on the symbology of forms in space. In fact, all cognitive processes are simply efforts at interpreting the symbolism of sense-deliveries; and the difference between mere knowledge and wisdom, which is the essence of all knowledge, is the difference between the understanding of a symbol and the comprehension of the essential nature of the thing symbolized. So long as knowledge of space is limited to the understanding of a symbol or symbols by which it is presented to the consciousness so long will it fall short of the comprehension of the essential nature of space. In vain have we sought in times past to understand space by studying relations, positions and the characteristics of forms in space; in vain have we based our conclusions as to its real nature upon the fragmentary evidences which our senses present to our consciousnesses. It is as if one had busied himself with one of the meshes in a great net and confined his entire attention to what he found there, meanwhile remaining in complete ignorance of the nature of the net, how it came to be there, of what it is made and how great its extent may be. There is ever a marked difference between a symbol and the thing which it symbolizes. Words are the symbols of ideas; ideas, as they exist in the mind, are the symbols of eternal verities as they exist in the consciousness of the Logos of the universe. There may be a wide diversity of symbolic forms which represent one single idea; as, for instance, the variety of word forms which represent the idea of deity in the various languages. Likewise there may be a multiplicity of ideas which represent a single verity. But neither is the idea nor the word the real thing in itself. That quality of a life-aspect which we call its _thingness_ has an essential nature which cannot become the object of consciousness except by virtue of its representation through ideas and their symbolisms, and even then, the thing which we conceive is not the nature of a quality of the life-aspect but an idea of it--a symbol which stands for that idea. In order, therefore, for the mind to arrive at an understanding of an eternal verity, such as space, it must first be able to synthesize all of the representative ideas and then abstract from their compositeness a notion of its essential nature. But this can be done only by identifying the consciousness with the essential being of the object considered. In other words, the consciousness and the intrinsic being of forms, principles, forces and processes must embrace each other in the intimacies of direct cognition; the life which is consciousness and that life which is essential being, being coeval, coördinate and mutually responsive, must in so close a contact as here intimated reach an understanding of the realism shared by both. That is, the human consciousness, following in the wake of life and consisting of a specialized aspect of life itself, will, by such an intimate approach to the life-principle of forms, readily understand; for it has only to recognize a replica of itself in rendering its judgment. But it is not claimed that such a state of recognition by the consciousness of life itself can be attained at all by ordinary means, neither is it believed that it is the next stage in conscious evolution. However, it is not doubted but that such an exaltation of the consciousness is possible, yea practical; but the difficulties which beset the path of attainment in this direction are so great that it may as well be considered unattainable. The mere fact of these difficulties, however, only re-emphasizes the insufficiency of the intellectual method. The identification of consciousness with essential being is a procedure which cannot be accomplished by an act of will directly and immediately. Because it is a process, a series of unfoldments, an adjustment of the focus of consciousness to the kosmic essentialities which constitute the substructure of the manifested universe. In the very nature of things, a kosmic essentiality cannot be viewed as being in manifestation especially in the same degree as ordinary physical objects are manifest. The former is a state, a potentiality, a dynamic force, an existence which should be thought of as an extra-kosmic affair dwelling on the plane of unity or kosmic origins; while the latter are the exact opposite of this. The one can be seen, felt and sensed while the other is the roots which are not seen but lie buried deeply in the heart of the universal plasm of being and beyond the ken of sensuous apperception. The term _manifestation_ is both relative and flexible in its use. It is relative because it will apply equally to all stages of cognition. A thing is in manifestation when it is presentable to the ordinary means of cognition belonging to any stage of conscious functioning; it is not in manifestation when it is beyond the scope of the Thinker's schematism of cognitive powers. Its flexibility is seen in its ready yieldance to the entire range of implications inhering in the process of cognition, fitting the simplest as well as the highest and most complex. Great is the gulf which is interposed between manifestation and non-manifestation; and yet the two, in essence, are one. They are linked together as the stem of a flower is joined to its roots. Likewise one is visible, palpable while the other is invisible, impalpable though no less real and abiding. As the thin crust of earth separates the stem, leaves and flowers from the roots so the limitations of man's consciousness separate the manifest from the unmanifest. Similarly, as when the surrounding earth is removed from the roots and they are laid bare revealing their continuity and unity with the outputting stem and flowers, so, when the limitations of consciousness are removed by the subtle process of expansion to which the consciousness is amenable so that it can encompass the erstwhile unmanifest, it, too, will reveal the eternal unity of the kosmic polars--manifestation and unmanifestation. There is but one barrier to ultimate knowledge and that is the human consciousness however paradoxical this may seem. The unutterable darkness which shuts out the so-called "unknowable" from our cognition is the limitation of man's upreaching consciousness. These limitations constitute the difference between the human intellect and the mind of the Logos. Nevertheless the outlying frontiers of man's consciousness gradually are being pushed farther and farther without. Every new idea gained, each new emotion indulged, each new conception conquered, and every mental foray which the Thinker makes into the realm of the conceptual, every exploration into the abysmal labyrinth of man's inner nature are the self's expeditionary forces which are gradually annihilating the frontier barriers of consciousness and thus approaching more closely upon the _Ultima Thule_ of man's spiritual possibilities. Space is in manifestation. It exists and has being whether it is viewed as an object, an entity or the mere possibility of motion. That it offers an opportunity of motion and renders it possible for objects to move freely from point to point cannot be denied and yet this fact has no bearing whatever upon the essential nature of space. The very fact of its appearance, its manifestation, makes it obvious that it is the nether pole of that eternal pair of opposites--manifestation and non-manifestation, being and non-being, which are essentially one. It will be seen from figure 17 that the period of involution embraces seven separate stages, the _monopyknon_, _duopyknon_, _tripyknon_, etc., being the unit principle or engendering elements of the respective stages. Involution comprises all creative activity from the first faint stirrings of the void and formless chaos until the universe has actually become manifest and dense physical matter has appeared. It is divided into two cardinal periods, namely, the chaogenic period during which primordial chaos is given its character and directive tendencies for the world age. It is a phase of duration wherein the fiat of kosmic order is promulgated, and consist of three stages, monopyknosis, duopyknosis, tripyknosis[24] gradually, insensibly, gradating into the fourth or _quartopyknotic_; the chaomorphogenic period is likewise divided into three stages--_quintopkynosis_, _sextopyknosis_ and _septopyknosis_, developing out of the fourth gradually. The quartopyknotic stage is the stage of metamorphosis or transmutation wherein the transition from non-manifestation to manifestation is completed; it is also the stage of kosmic causation, because from it spring the matured causes or "vital impetus" which engender all that follow. [24] See Fig. 18. [Illustration: FIG. 17.--Involution and Evolution] The close of the involutionary phase of the world age is marked by the final deposition of dense physical matter and this is closely followed by the beginnings of the evolutionary movement which, like the involutionary movement, is divided into two cardinal periods, namely, the morphogenic (during which are produced, in turn, insensible forms, sensible forms and spiritual forms) and the kathekotic period which marks the perfection, the consummation of the evolutionary movement. These two cardinal periods of the evolutionary phase of duration and the two cardinal periods of the involutionary phase complete the kosmic age, the "Great Day of Brahma." The concentric circles, beginning with the dot and ending with the seven concentric circles, and designated as a, b, c, d, e, f, g, are representations of the constitution of the respective units corresponding to each of the seven subdivisions. They symbolize the seven degrees of condensation or pyknosis which comprise the genesis of space, on the one hand, and on the other, the stages of unfoldment. Because, during involution all potencies, powers and characters were being infolded, involved; but during evolution, these are being unfolded, expressed, evolved. The figure 18 is another view of these two major movements, involution and evolution. The genesis of space is here shown symbolized by the Kosmic Egg. The seven stages of involution are referred to as, the monopyknotic, duopyknotic, tripyknotic, quartopyknotic, quintopyknotic, sextopyknotic and the septopyknotic; while the corresponding stages of evolution are referred to as, the physical, the sentient, mental, causative or spiritual, the triadic, duadic and monadic, indicating that the principle of physicality is succeeded by the principles of sentience, mentality, spirituality, and the three forms of kathekotic being. This symbolism, it should be stated, is designed with respect to the universe and man and has no reference to other possible evolutions than the human and contemporaneous animal, plant and mineral evolutions. [Illustration: FIG. 18.--The Genesis of Space] To follow the ramifications of the symbolism above would involve a survey of all branches of knowledge, and indeed, would be out of place in this book. Only the widest general outlines can be suggested here and it is believed that this is sufficient to enable the reader to grasp the magnitude of the symbol and to understand its purpose and intent. It would appear, therefore, that if it is possible for the intellect to traverse by means of a study of kosmic symbolism, used as a standard of reference, the entire length of the bridge which engages the antipodes into an eternal unity, something may be gained in the way of a more definite and clearer understanding of the essential nature of space in its relation to kosmogenesis. In the diagram, figure 17, is shown a table which represents the stages of space-genesis. It will be noted that the whole scheme is divided into seven stages. It is not an arbitrary division simply but a symbolic one and represents fullness, completeness, entirety. The names given, namely, "monopyknotic, duopyknotic," etc., represent the symbolic characteristics of each stage in its relation to the universe in the process of becoming. The terms "monadic," "duadic," "triadic," etc., are representative of the seven planes of matter in the universe. A _pyknon_ is a kosmic principle and represents the typal aspect of kosmogenesis. It is a generic term and may be identified in its relation to the various stages by the prefix. The monopyknon belongs to the ulterior pole of the antipodes on the side of non-manifestation. So do the _duopyknon_ and the _tripyknon_. Pyknosis is a process of kosmic condensation, or limitation for purposes of manifestation. It is a stage in the descent of the kosmic Spirit-Life, a degradation of non-manifestation into manifestation, and is, therefore, the cardinal causative principle of creation. The term pyknon being generic is applicable alike to a particle of matter, a state of being, a condition of existence, a process or a principle. The _monopyknon_ is, accordingly, the primary aspect of the process of kosmic pyknosis. It is the archetype and therefore all inclusive and omnipotential. But whether regarded as a singularity or as a whole it should never be divorced in thought from the primal act of creation. It represents the first act of material differentiation in the being of the creative Logos on the plane of non-manifestation. It is the beginning of every great planetary or kosmic _manvantara_ or period of manifestation. During either a planetary, solar or kosmic _pralaya_, or gestatory period, the kosmic plasm is in a quiescent, undifferentiated condition. This undifferentiated plasm when acted upon by the will of the Creative Logos, _Fohat_, as He is sometimes called in Eastern Philosophies, begins to become conditioned, begins to differentiate. The primal act or stage in such a process is the formation or appearance of a monopyknon. It then becomes the characteristic aspect of that stage. Monopyknons are the quiescent, unawakened, though potential and archetypal principles peculiar to the monopyknotic period of space-genesis which are ultimately to become, on the physical plane, singularities of life of whatsoever kind. _Thus the lineage of every single life-form or principle in the universe runs unbroken back from the present of the Now to the present of monopyknosis. So great is the design of the kosmos that the entity which is now man or the atom was started on its journey to this culmination at the break of the Great Kosmic Day when the omni-pregnant wheels of monopyknosis first began to turn._ Duopyknons and tripyknons constitute the two remaining stages on the plane of non-manifestation. And their correspondences in every stage of involution or the descent of spirit into matter are eternal and kosmic. Likewise the lineage of the dual and triple aspects of all life forms on the path of evolution may be traced rearward to the duopyknotic and tripyknotic stages of kosmogenesis. The metamorphosis by which the monopyknon becomes a duopyknon contrives the differentiation of the pristine plasm of kosmic being so that the first becomes the ensouling or vitalizing principle of the second; and, in turn, the second becomes the vitalizing or inner principle of the third; the third of the fourth and the fourth of the fifth and so on throughout the series until the last is reached which is the _septopyknon_. The septopyknon is, therefore, a seven-principled form. It is both unitary and septenary--unitary in the sense that the seven are really one and septenary for the reason that each of the seven principles, in the course of evolution, becomes a separate process specially adapted to functioning upon its peculiar plane of matter. Thus it is seen that the utmost significance attaches to this septenary pyknosis of the kosmic plasm of life. The implications of this conception are, of course, too vast and multifarious to be set down here. We shall have to dismiss it with one observation only, and that is: _every single appearance of life and form in the totality of such appearances is rooted in kosmic pyknosis where it has received its inner vitalizing force, its form and the law of its mode and manner of appearance together with the metes and bounds of its existence_. These processes, monopyknosis and duopyknosis, are to be regarded as taking place, each in its own period, everywhere throughout the Body of Being of the Logos but on the plane of non-manifestation. They are states of preparation for manifestation analogous to the germinative period of the seed or the egg. They represent the first stirrings of the kosmic plasm and contain the promise and potency of all that is to succeed them. There is one other stage, coördinate with these two, and that is the tripyknotic which completes the unmanifest trinity and constitutes the archetypal vehicle whence proceeds the manifested universe. The ensouling principle of the tripyknon is the duopyknotic principle. But when the descent has reached the tripyknotic stage the three have been merged into one and the characteristics which were peculiar to each as a separate pyknon are then fused into a single quality having three aspects which are mutually interdependent and coördinate. The unmanifest trinity, now complete as a result of the triple pyknotic process, is the imperishable and ever sustaining radix, the all-mother of the manifested universe. It is the Golden Egg laid by the god _Seb_ at the beginning of a great life-cycle. It is also _Chaos-Theos-Kosmos_, i.e., kosmic disorder, divine will or generating element, and kosmic order or space. In it, as in an egg, resides in kosmic potency all that the universe is to become in any "Great Day of Brahma" or any Great Kosmic Life Cycle. Its eldest born is SPACE, physiological and perceptual, and the latter is the eternal father-mother of the universe. _Space is, therefore, the male-female principle of manifestation; in its kosmic womb all forms are created, developed, evolved and sustained._ Into it again, at the close of the Great Day, all existences, forms and all principles matured and ripened by the vicissitudes of kosmic evolution will be inhaled with the return of the Great Breath of Life. The unmanifest trinity is the archetype, and therefore, the pattern or model for the manifest, embodying in potency all that the manifest may ever become. Forth from the unmanifest proceeds space as a dynamic process endued with the potentiality for generating all that the manifested universe contains. The terms "unmanifest," "unconditioned" and "unlimited" have a special meaning here and are used in the same sense as the mathematical term "transfinite," and therefore, imply a transcending of any finite or assignable degree or quality of manifestation. Hence they should be distinguished from the term "absolute" which has a different implication. So that although the triple process outlined above may not be viewed except as a characterization of the plane of non-manifestation, and hence of the primordial activity of the Creative Logos, there is nothing in the symbolism to warrant the identification of this process in any way with the Logos in absolution. For on this view Absolute Being is, in a large measure, sacrificed when the monopyknotic process is begun and the monopyknon (kosmic principle) begins to appear. Absolute Being, while it may not be defined, delineated or described may be symbolized by the ideograph: "action in inaction"; "being in non-being"; "manifestation in non-manifestation"; for these are symbols merely and do not describe or delineate. We have observed the subtle connection which exists between manifestation and non-manifestation and have seen how that, as the roots of a plant sustain the outer growth of stem and flowers, the former being the matrix out of which proceeds the latter, and that in like manner does the unmanifest sustain the manifest; it should, therefore, be clear that the Body of Being of the Unmanifest Logos, in a similar manner, is the basis and cause of all that is manifest or in existence; _and yet it is more, it is essentially all that is and all that the all is to be in manifestation at any time throughout the immeasurable process of kosmogenesis_. _It is the Self of the Universe, the meta-self of the great diversity of selves._ The Self, however, although it cannot be said to exist except as a simple, homogeneous quantity is nevertheless, in the very nature of things, a triunity, and in essence, not only the basal element of the All-Space and the concrete forms which exist therein, but is also identical in essence and substance with space. A stage has now been reached in the description of our symbolism when it may be assumed, upon the basis of the foregoing, that the meta-self has become manifest, i.e., in potential and dynamic appearance, as a result of the triple process of pyknosis hereinbefore outlined. The meta-self may then be identified with the Supreme Manifest Deity; for there is ever a subtle identification of the manifest with the unmanifest. Out of their action and interaction the universe is made manifest and phenomenal and is thereby sustained. The reciprocity of action between these two kosmic polars, is the metamorphotic key to creation; it is the symbol of the procedure of Creative Will in the act of creating. It is the transitional process whereby the passage is made from non-being to being; from the unconditioned to the conditioned; from undifferentiation to differentiation and is reflected and symbolized in every natural process wherein matter is transformed from one state to another, or life and mind and spirit diffused, centralized and organized into ever new and higher forms and expressions. Another important notion to be gained in this connection is the fact that it appears as a logical sequence to the foregoing that the being of the manifested Logos must necessarily fill all space, yea more, is that space in every conceivable essentiality. His limitations are the limitations of space. His qualities, properties and attributes are the qualities, properties and attributes of space and are only different from the original spatial character when manifested through a diversity of forms by whose very inner constitution and exterior form the modifications are accomplished. _All matter in the universe, all energy, and indeed, all manifestations or emanations of whatsoever character, hue, tone or quality are, in reality, His being and nothing but His being. There is, therefore, no form nor ensouling principle whether of life, mind or sheer dynamism which can exist outside of His being and be, even in the slightest degree, absolved from an eternal identity therewith._ Once this idea is grasped and its varied implications noted it then is no longer conceivable that any other order or schematism can be possible in our universe, and that, too, despite the multiform conceptions peculiar to the varied systems of philosophy. The matutinal dawn of creation came at the close of the tripyknotic movement which resulted in the elaboration of materials, the initiation of principles, processes and types, and the final preparation of the field of evolution. The three processes or aspects of non-manifestation projected in preparation for manifestation, namely, monopyknosis, duopyknosis and tripyknosis represent the earliest stages of germinal development. When these had closed, the Great Kosmic Egg began to germinate; the first faint, indescribable signs of manifested life began to appear. Involution set in. The fourth or quartopyknotic stage, though only slightly differentiated, or rather representing that period of kosmic involution when that which is to become the manifested universe first begins to fall under the sway of kosmic order, is nevertheless the basis of all great world processes. It is just midway between the poles--manifestation and non-manifestation. During this stage the life elements are receiving the imprints of character, being endowed with directive tendencies and stored with such dynamism as will persist throughout the Great Life Cycle in which they are to manifest. It is here that begins the movement of involution, the storing away of those elements and factors, no more and no less, that are to show forth on the upward path of evolution; it is here that matter begins to assume form; electrons, ions and atoms created, or, that those minute processes which on the evolutionary side are to culminate in these are originated. This is the metamorphotic stage. It is the laboratory of the universe wherein _Fohat_, the Creative Logos, prepares the materials out of which and in which the vast diversity of _morphons_ or forms is created. Quartopyknosis, accordingly, is the first active step, on the plane of manifestation, which results in the appearance of perceptual space and consequently of physical matter itself as well as all the other grades of matter in the kosmos. Space, brought into existence by the act of the Creative Logos in imposing limitation upon His being, is in its primordial form composed of quartopyknons or quadruplicate principles and tendencies which act in unison and to the accomplishment of a single end or purpose. On this plane or during the continuance of this period of space-creation, the roots of universal law and order are produced. In it are planted the principles of good and evil and a sharp line of demarkation established between all the conceivable pairs of opposites which exist. It accounts for the duality of life and form. Male-female; father-mother; positive-negative; Rajah-Tamas (action-inaction)--all these find in this process their eternal origination. It is the stage of harmony, bliss, ideality, perfection, perfect equilibrium and balance. Here, innumerable ages before they actually appear, the glow-worm and the daisy, the amoeba and the dynosaur, man and the planetary gods alike abode their time awaiting the toppling of the scales of kosmic potency when all would be plunged headlong into the endless labyrinth of becoming. The quartopyknotic process is similar in all details to the three preceding processes, these latter being prototypes of all succeeding stages of involution. The quintopyknon, accordingly, symbolizes the quintuplicative action of life in its descending movement toward the creation of matter in its densest form. That is, it is a five-fold principle acting in unison and kosmic consistency, infolding in the universal wherewithal that which is to become mental matter on the side of evolution. Just, as may be seen in the diagram, Figure 17, the quartopyknotic process symbolizes, on the involutionary side of the life current that which is to become on the evolutionary side, spiritual essence, so the quintopyknotic deposits the seeds of that which is to become mental matter. During both the quartopyknotic and the quintopyknotic processes all the potencies and promises, residing on the plane of non-manifestation and destined to show forth as spirit and mind are brought into a fuller and more marked degree of manifestation and become the seeds of spirituality and mentality which are to ripen and be ladened with the fruitage thereof many ages hence. The reader should bear in mind that the processes here described are thought of as taking place at the beginning and as having their roots planted in eternal duration; that they refer to a period long before the universe even resembled anything like its present aspect; at a time even before there were individual minds to perceive it, before even the gods--solar, planetary, super-solar and super-planetary were in existence to take part in the matutinal ceremonials of creation's vast hour of stillness. The mind must accustom itself to go back of appearances, back of time, back of space itself and discern the foundations of time, space and appearances being laid and to perceive that which is no less than the action of the Supreme Deity Himself in brooding over the primordial formlessness which is Himself, and from which will gradually evolve the universe of qualities, conditions and appearances. The quintopyknon is, therefore, the base of the mind-principle in the kosmos. All the qualities of mind whether in man or in the planetary gods, whether in the moneron or the tyrannosaur, in the mountain or the oak, reside in kosmic potency in the quintopyknon. NICHOLAS MALEBRANCHE,[25] in one of his very lucid moments, beheld the essential character of the symbology of space and was led to the conclusion that God is space itself. To him it was equally certain that all our ideas of space, geometrical or purely physiological, as well as our notions of the great suprasensual domain of ideas, exist in the kosmic deiform, or body of the Logos of Being. He saw "all things in God." God did not create ideas; they are a part of God Himself; God did not create mind; it is a part of Himself; no kind of matter did He create; it is a veritable part of Himself and indissoluble from Himself. The great outstanding implication of this philosophy is that our consciousness of God is but a part of God's consciousness of Himself; our consciousness of self and the not-self is but God's consciousness of these things. There is no existence of anything, either of the self or the not-self, except in this consciousness. It is refreshing, therefore, to note that although the approach is made from another and entirely different direction, almost the same conclusion as to the ultimate resolution of all chaogenetic elements into what is the very systasis or consistence of the great kosmic deiform, is reached. [25] Vide _Recherche_, Chap. VII, also Philosophical Review, V. 15, p. 401.--MALEBRANCHE. But a marvelous vision comes with the dawn of this truth upon the lower mind. It establishes clearly the truth of KANT'S notion when he said: "Since everything we conceive is conceived as being in space, there is nothing which comes before our mind from which the idea of space can be derived; it is equally present in the most rudimentary perception and the most complete." The mind cannot get away from the conception of space, because, out of the very essence of space, as a result of the quintopyknotic process, it was produced, created and organized. The idea of space is, therefore, not derived from things in space nor from their relations in space. _It sprang up with self-consciousness._ As soon as the Thinker became conscious of himself he became aware of space. The very state of self-consciousness implies space. The self in man is a specialized aspect of space. Indeed, it is a projection of space. The moment the self can say: "I am," it also can complete the declaration by saying: "_I am Space_." When the self looks out from his six-walled cabin of imprisonment into the immensity of what we call space he looks out into that which is himself and his immensity; he perceives the source and the ever-present sustenance of his being and recognizes his identity therewith, provided he does not allow himself to become entangled in the philosophical difficulties which the intellect is prone to throw around the simple, yet marvelously complex, notion of self-consciousness. This should settle, once for all, the question of _apriority_. The _a priori_ inheres in quintopyknosis or kosmic psychogenesis. It is the essential nature of mind; it is the mind's lines of organization; it is the law of the mind's being and action. All mental perception originates from things in space. No thought of any detail, of any state or condition, whether limited or unlimited, related or isolated can be conceived except it be of things in space. And this is so, because _mind and space are one_. It is not so with our conception of time. Time is merely an aspect of consciousness in its limitation and does not inhere in the mind in the same manner that does space. In fact, it is not a part of the mind's nature as it has been shown that space is. It would, therefore, seem to be a grave mistake so to coördinate the two notions. Space is the progenitor of mind and is continually identified with mind. Time is the child of consciousness. That is, it is one of the illusions of consciousness which the ego will shed as his consciousness expands. Duration alone is a coördinate of space. The mind now recognizes space as something apart and separate from itself only because of its unconsciousness of the identity existing between it and space. Just so, it is not by mind alone that the _at-one_ state of consciousness shall be attained; for although in one form or another it is able to gain some knowledge of the apparent oneness of all life it cannot directly realize this oneness. In order to do this fully it must be able consciously to identify itself with the life, feel what it feels and experience what it experiences and otherwise come into a conscious relationship with the root and source of life. Space-consciousness is a simple, direct cognitive process; while time-consciousness is a complex, and therefore, indirect process. The former cannot be analyzed. That is, no analysis is necessary to its sufficient comprehension; the latter must always be analyzed and categorized for its sufficient apprehension. Every moment of time whether past, present or future, when presented to the consciousness, is determined by its relationship to some other moment of time. Space is indivisible; time is divisible. Space is an intuitional concept; time is an intellectual concept. Time belongs to phenomena. Space is the root and source of phenomena. Time is the leaves of a tree while space is the life of the tree. Space-consciousness, in its relation to the present status of mind-development, is itself an illusion; for despite the fact that the Thinker's apperception of it as a state is simple, direct and fundamental, it is only so because of the inability to realize to itself the unity of the seeming two. The attainment of space-consciousness or the space mind, which contrives the understanding of the identity of mind and space also annihilates the consciousness of space as a separate notion from the mind. Once the Thinker's consciousness has arisen to that state where it perceives its unity with space all sense of separateness is lost. Just as when two molecules of hydrogen uniting with one molecule of oxygen to form a new compound lose their identity in the new realization of unity, so does the consciousness when by the alchemy of psychogenesis it becomes identified with space, not only lose its identity as such, but also any consciousness whatever that space exists as something separate and distinct from itself. Imagine the whole of the duration aspect of kosmogenesis crowded into an infinitesimal instant and the bulk of all matter, suns, stars, worlds and planets, condensed into a space less than the magnitude of an hydrogen ion and in this way a symbol of what it may mean to attain unto absolute knowledge or unto the space-mind, may be obtained. Recurring to the process of quintopyknosis, it may be noted that the quintopyknon or five-fold kosmic principle of life which we have seen to be identical to the seeds of mental matter brought into existence by the reaction of Fohatic energy or the Will of the Creative Logos upon the substance of the quartopyknotic stage, is, symbolically speaking, more dense and compact than the pyknons of the preceding stages. It is ensouled by the quartopyknon. It is a rather complex state of ensoulment consisting of four condensations or pyknoses. The next stage in the process of kosmic involution which is also concerned with the preparation of the evolutionary field is that of sextopyknosis and implies the senary condensation of the original world-stuff with the view to the formation of emotional or "astral" matter. Identically the same process of ensoulment or involution obtains upon the plane of sextopyknosis as have been observed to obtain upon the preceding planes of involution. Involution must necessarily precede evolution. That which has not been involved, enfolded or ensouled cannot be evolved or unfolded. Whatever potencies, powers and capabilities or qualities and characteristics that may appear at any time in the universe of life and form must have first been involved or enfolded or else they could not have been evolved. Space itself is an evolution. It is a process of becoming, of unfolding, of flowering forth. As it evolves more and more there will appear new and added characteristics and qualities of life and form. New possibilities will arise and in the end a supernal vision of a glorified universe will burst into view. The scheme of space-genesis is completed during the septopyknotic process wherein the basal elements of dense physical matter and its various gradations are produced and given character, form and direction. But this completion means merely a temporary estopment in the process of kosmogenesis which actually results in the formation of physical matter in its crassest state. It does not mean a final arrest of the entire process which is conceived of as continuing only in a regressive manner back to a kathekotic[26] condition wherein it embodies the fruitage of the entire scheme. The septopyknon, accordingly, is a seven-ply pyknon in which are embodied, in varying degrees of manifestation and phanerobiogenic (life-exhibiting) quality, the essentialities of all that has preceded on all planes and during all stages of space-genesis. That is to say--in the physical life of the universe is confined the essence of all the series of grades of life in the kosmos. In man's physical body are wrapped up all the glories attainable in his long, almost unending pilgrimage of evolution; in it are stored all the possibilities of the spirit; all powers, all qualities, all characteristics, ever intended for man's attainment are in the physical. But they must be evolved, they must be unfolded and expressed. The physical must be _glorified_, _spiritualized_, _deified_. For by the way of the glorification and spiritualization of the flesh man may attain unto oneness with the divinity in himself and consequently with the divine life of the world. [26] _Kathekosis_ (from Chaos-Theos-Kosmos) is to evolution what "chaogeny" is to involution. It is the end of evolution, but also the beginning of involution, and in the latter function is known as "chaogeny." See diagram No. 17. To summarize: The genesis of space embraces seven stages, namely, the monopyknotic, the duopyknotic and the tripyknotic which belong to the plane of non-manifestation and are the primordial world-stuff and together make up the unmanifest body of the Logos of Being. These become the seed-germ of the universe of spatiality. The quartopyknotic is the fourth stage in the process of space-genesis and is the _metamorphotic_ or crucial stage during which non-manifestation is metamorphosed into manifestation. In it the unmanifest becomes the manifest. It corresponds to the plane of pure spirit, and indeed, embodies within itself all the qualities which spirituality is to show forth during the life of the kosmos. The quintopyknotic is the fifth stage and corresponds to the mental plane, embodying in itself all qualities of mentality in the universe and furnishing the basis and essence of that which is to become the kosmic mind in manifestation. The sextopyknotic is the sixth process and symbolizes the sixth stage which embodies all the characteristics and properties of emotional matter in the universe and is the basal element of the plastic essence of sentient existence in the kosmos. The septopyknotic is the seventh and final stage corresponding to the physical plane of the kosmos and contains in its seven-fold constitution the seeds and potencies of all the preceding stages, as well as all the characteristics and properties which physical matter is destined to show forth during the _manvantara_ or world age. These seven processes result in the dynamic appearance of space, the mother and container of all things, and complete the involutionary aspects of kosmogenesis. Evolution began where involution ceased and will end for this _manvantara_ when the last vestige of those powers, capabilities and potencies which were involved shall have been evolved unto kosmic perfection. The measure of the Great Kosmic Space-form was sealed at the close of the involutionary movement of the Great Life Wave. Then its metes and bounds were fixed by the fringe of kathekosity which circumscribed it. If it be true that the reader found it extremely difficult to grant the connotations of the symbolism when the mental or quintopyknotic stage was reached when illative cognizance was given to the fact that space is also composed of mental matter, it may be still more difficult to grant the claim that physical matter is also essentially a part of space. But this is the implication. _And, therefore, it follows that all matter, all energy, all life and all mind wherever it may be found in the Great Space-Form is space itself and nothing but space._ Hence, it appears that space is indeed the dynamism of the universe. In its kosmic womb the great world egg was formed of its own substance solely and in it still the universe of form persists and evolves withal. If it be suitable for the physicist to talk of gravitation, electricity, magnetism and force let him do so, for these terms serve the present category of human knowledge; but the human mind will not lament the day when it comes to recognize that these things, these forces, these aspects are nothing more than space-activities and space-phenomena. If a planet's place be preserved in space it is because space, vital, dynamic, creative space, sustains it and from its gentle, yet eternally firm grasp there is no escape. All that the planets, suns and worlds are and all that they may ever become in this _manvantara_ or world age have been derived from space, yea, are of the very essence of spatiality. If the chemist choose to talk of chemism, negative and positive, of combining properties and dissociative phenomena let him also become aware that these phenomena are but the external aspects of the inner and ephemeral life-processes of space-forms and that ultimately these, too, may be traced back into an eternal originality within the bosom of the all-mother, spatiality. Dense physical matter, such as constitutes the physicality of celestial bodies in its ultimate dissociation would, accordingly, be resolved into the original chaogenetic formlessness which marked the chaogeny of non-manifestation although it would naturally be orderly and progressive passing through the seven stages, septopyknosis, sextopyknosis, quintopyknosis, etc., until the end had been reached, meanwhile exhibiting in each plane the phenomena peculiar to the dissociative processes thereof. On this view space is a _plenum_ of matter of varying degrees of intensity, ranging from the densest physical to the most tenuous and formless matter of the highest levels of the manifested universe. But as neither the dense material forms nor the other grades of matter have an eternally enduring quality, being alike subject to mutation, space likewise falls under the law of becoming whereby it, too, must yield to the edict of kosmic disorder. Some may be inclined to argue that since space and mind are one and the same thing it must necessarily follow that whatever possibilities of measurement may be found to exist in the mind would logically be found to exist in space; and that since all the necessary conditions of hyperspatial operations are proved to be existent in the mind the case of the hyperspatiality of perceptual space is proved thereby. In other words, if the fourth dimension can be proved to be mentally construable it is also possible in perceptual space. But these hypotheses are not granted, and neither will they be acceptable to those minds who choose to take that view when it is known that there is a marked difference between the mind that is purely intellectual and mind that is purely intuitional or mind _a priori_. The intellect is fashioned for matter only; it is so constructed as to fit squarely into every nook and cranny, every groove and interstice in matter; yet for the generating element, life, it has no aptitude nor suitable congruence. The attainment of the space-mind or kosmic consciousness would then imply a mastery of all fundamental possibilities pertaining to all degrees of matter. Thus by becoming conscious in the matter of all the planes one makes a certain definite approach to this ultimate state of consciousness until all the barriers between ordinary self-consciousness and the consciousness of the space-mind have been entirely obliterated. Pyknosis, in all of its septenary aspects, is concerned primarily with involution or the preparation of the chaogenetic elements for the work of kosmic evolution. It may be thought of as being divided into two great divisions, namely: _chaogeny_ and _chaomorphogeny_. It is concerned with the organization of chaos, the establishment of kosmic geometrism in the formless, void, arupic substance and preparation for evolution. Chaogeny, of course, is that kosmic process by virtue of which space itself becomes manifest and in which there is no established order. Chaomorphogeny (from _Chaos + Morphe + Geny_) signifies the activities of the creative Logos in laying the foundations in primordial space-matter of the various star-forms, including nebulæ, worlds, planets, suns, etc., of which Canopus, Jupiter, Fomalhaut and Sirius and our own sun are examples, giving direction and general tendence to their varied life-processes. Both these processes are concerned with the preparation of the field and its consequent fertilization in anticipation of its cultivation and harvest. These two constitute kosmic involution or the great life wave's passage on the downward arc of the Great World Egg or Circle. It is during the chaomorphogenic cycle that the constitution of the universe of manifestation is promulgated; when laws for its government during that _manvantara_ are sketched out in the world of nascent spatiality; when the archetype of every imaginable or possible form is projected upon the impregnated screen of creation, then folded in, pushed toward the center, involved, awaiting that time when the life wave begins its passage upon the upward arc and evolution ensues, calling forth all that has been enfolded in the bosom of the pyknotic centers of manifestation. It is easily conceivable that here during the troublous times of the chaomorphogenic enfoldment the now known six directions of space were among the eternal edicts of space-genesis and that that law which now makes it appear that three coördinates, and only three, are sufficient for the determination of a point position in space was imprinted in the very nature of that which was to become space. The kosmic field having been prepared as a result of the chaomorphogenic activities, lowly and scarcely organized forms begin to appear and the ascent upon the upward arc of the Great Cycle commences. Evolution begins. Its scope is likewise divided into two great stages, namely: (_a_) Morphogeny, the purpose of which is the development of life-forms or pyknons which are to appear on the various planets, stars, worlds and suns of the universe. It embraces the whole span of the life-aspect on the evolutionary side of manifestation. In this aspect is included also every conceivable adaptation of the universal principle of life from the beginning of its movement to the end. The universe is now functioning in and progressing through the vicissitudes of this stage. That is, all the present observable adaptations which the life-pyknon or principle is making, has made or will make, are embraced within the scope of what is here designated as the morphogenetic aspect of evolution. (_b_) The second and last division of the arc of evolution is called _Kathekos_, thus symbolizing the syncretism of the trinitarian aspects of kosmogenesis, _Chaos-Theos-Kosmos_, perfected and united as a result of the labors of manifestation. In this final summation of the labors of the life-wave as it has progressed from involution through all the devious manifestations of evolution are embodied the perfection and ultimate elaboration and expression of all the pyknotic tendencies which were established during the entire scope of space-genesis. Thus it will be seen that the first three stages of space-genesis, called chaogeny, encompass the first three pyknotic processes or are analogous thereto while the latter called chaomorphogeny, the organization and ensoulment of space-forms, embraces the latter four, quartopyknosis, quintopyknosis, sextopyknosis and septopyknosis. This division obtains on the involutionary side of the great life-cycle. The upward arc of the Great Kosmic Egg or Cycle is also divided into two great stages, namely, Morphogeny (manifestation of life through the various forms which it assumes) and _Kathekos_, or the kathekotic plane of perfected triunity which is represented by the evolutionary union of _Chaos-Theos-Kosmos_. _Kathekos_ would, therefore, symbolize the ultimate elaboration of Chaos into a well-ordered kosmos wherein are expressed all the possibilities which inhered in the archetypal plan of the Creative Logos or _Theos_ and in which all had reached the ultimate perfection in the body of being of the Logos Himself. But the kathekotic plane is to be distinguished from the original _Chaos-Theos-Kosmos_ represented as functionating upon the plane of non-manifestation during chaogeny. _Kathekos_ symbolizes the perfected manifestations of the triune aspects of the Creative Logos through the perfected forms resulting from the labors of kosmic evolution, while _Chaos-Theos-Kosmos_ symbolized, as a triune _glyph_, the Unmanifest Trinity in the primordial beginnings of space-genesis. One is the seed; the other the fully matured plant; one the egg; while the other is the full grown bird; one the root; the other the fruitage; one Alpha, the other Omega; one the beginning, the other the end. The end, however, is reached only that, in due time, the entire scheme may be commenced again, once more utilizing the results of the preceding scheme of evolution as the basis of the ensuing one. Thus after every Kosmic Day, commences the Kosmic Night. The succession of kosmic days and nights is infinite. This infinity of _becomings_ in the life of the kosmos is a necessary outcome of eternal duration. The above, thus briefly set down, is the symbolism of space-genesis. It is commended to the reader as a basis for the conception that the real, essential, perceptual space is something far more wonderful, more fundamental than either the geometrician or the metageometrician has ever dreamed of, and yet the latter's consciousness is undoubtedly being appulsed by the fingers of a new species of conceptualizations which, one day in the not too distant future, will arouse in it the faint hungerings after the realization of the real space-nature. These mathetic appetites thus brought into being will finally lead the human mind into the Elysian fields of kosmic consciousness where for another million years, perhaps, it may feed upon the mysteries and hypermysteries to be found in the granaries of the Space-Mind. The study of space in its wider and deeper meanings is necessary in order that a clearer understanding of its true significance, as the subject of geometric researches, may be gained. It is confessed, however, that there is neither direct evidence nor implicative authority for any assumption that the view herein outlined affords any justification for the notion of the n-dimensionality of space. For, although the line of reasoning indulged in must lead inevitably to the conclusion that the worlds of spatiality, materiality, intellectuality and spirituality, essentially and fundamentally one so far as origins and qualities are concerned, were alike engendered by the same generating element, life; and that spatiality being the primal basis of the others is, nevertheless, under the exigencies of this aspect of the kosmos, highly susceptible to the mensurative requirements of the grossest, there appears to be no necessity for calling upon extraneous considerations for assistance in our efforts to comprehend the various connotations of the symbolism. Then, too, it is easily conceivable that under conditions where these elements, spatiality, intellectuality and materiality, are not only co-extensive but interpenetrative, there is no justification for the assumption that they must exist in layers or manifoldnesses or in discrete degrees, separated from one another as if they were constituted of different substances and occupied different spheres. For every single point in perceptual space is a focus for lines drawn through every conceivable grade of materiality, spatiality or intellectuality in the kosmos. And the same system of coördinates which is necessary and sufficient for the localization of a point in our space is also sufficient for the location of a point anywhere in the entire world of spatiality, intellectuality or spirituality. In fact, the external, visible worlds of materiality and spatiality are nothing more than the _mass-termini_ of lines extending from divinity to physicality; from primordial originality to kosmic modernity and it is intellectually conceivable that progression back over the grooves made by these mass-termini of lines would lead directly and unerringly to originality itself. In spite of the manifold pyknoses which we have shown to characterize the symbolism of space-genesis it is a very simple matter; for the entire scheme could and must have proceeded along strictly tridimensional lines. Tridimensionality must have inhered in the primeval archetype of space or else it could not appear as an outstanding fact of perceptual space now; for all that we can now observe in space as characteristics must have first been included, enfolded, involved, before it could have been evolved. Hence, it is to be remembered that we are to-day dealing with the expressions of tendencies and principles which inhered in the manifested universe as potentialities in the very beginning. The alphabet of space-genesis consists of five characters, namely, the point, the line, the triangle, the square and the circle. These are the pentagrammaton of space, of intellectuality, materiality and of spirituality. They constitute the basis of kosmic geometrism. With these all geometrical figures may be constructed; with them all magnitudes may be delineated and projected. They describe every conceivable activity of the Creative Logos and designate the bounds of the entire scope of motility of kosmogenesis. In figure 19, are shown the dot, the line, triangle, square and the circle which together form the kosmic pentoglyph. The point symbolizes kosmic inertia, inactivity or the beginning of motion; the line is the first aspect of motion, the beginning of creation; the triple aspect of kosmogenesis is symbolized by the triangle, _Chaos-Theos-Kosmos_, the Unmanifest Trinity; the square emblematizes kosmic being in evolution; while the circle is the syncretism of all these and stands for the perfected kosmos, or the kosmos in process of perfection. Very truly did PLATO remark: "God geometrizes"; for the _pentagrammaton_--the point-line-triangle-square-circle--is the deity's way of manifesting Himself. But there is here no need for space-curvature nor for triangles whose value is greater or less than 180 degrees; there is no need even for the mathematical fourth dimension. [Illustration: FIG. 19.--Kosmic Pentoglyph] It cannot be believed, however, that metageometricians are really in earnest in what they suggest of hyperspace and _n_-dimensionality; it cannot be believed that they are entirely satisfied with what they have found of the so-called hyperspatial, and yet, some of them are fanatically patriotic over the new-found domain; some are even intolerant. But there are others who look upon the fabric of metageometry as a stepping stone to space-realities, a mile-post on the path to the realization of a higher consciousness, the consciousness of the space-mind or kosmic consciousness. And may this not, after all, be the goal of the human intellect, now slightly distraught by the exuberances of youth and the joys of a new mental freedom? The work of the future mathematicians will be the destruction of the tumorous inconsistences to be found in the various non-Euclidean systems of geometrical thought, the elimination of the novelties and the nonsensicals, the synthesizing of those elements which are sanctioned by the space-mind and the building thereon a sane interpretation of space-phenomena in the light of illuminations received from greatly extended faculties and a participation in that larger consciousness into which the human race seems slowly to be immerging. The domain of hyperspace is but the fairy-land of mathesis, peopled with goblins, gnomes, kobolds, elves and fays which are the spaces, dimensions, propositions, ensembles and theorems of the metageometrician. But like the fairies and nature spirits of the unseen about us, they have their bases in the real, objective world of facts however difficult it may be to establish their direct connection with it. As the gay, invisible sprites of phantom-land represent intelligent natural forces at work in the furtherance of the evolution of forms, so the impalpable things of mathesis are emblems of kosmic forces at work in the upbuilding of structures of higher consciousness which shall be towers of vision for the human soul whence it may view the hill-crests of infinite knowledge and the low-lying plains of kosmic mysteries. Finally, it has been noted that space is the very consistence of the kosmos; it is the life, the form and both the outer and the inner manifestation of the combined life and form; it is reality, also illusion; it is concrete, also ideal. We have noted also that mind is consubstantial with space and that space gives it its inner life and nature as well as nourishes its outer growth and development. In fact, we have seen that space and mind are _one_ essentially and that they exist as aspects of the same thing, life. In whatever way, then, that the mind normally views space that is the natural way. All attempts to deviate from the natural way are, therefore, unsanctioned by the nature of things. So long as geometry remains true to the nature of mind and space so long will it be valid universally and possessed of kosmic necessity and invariance. It behaves most unseemly when it departs from its fealty to the nature of things _per se_. Both the outlook of the mind upon the objective world as well as its inlook upon its own states of consciousness or the subjective world are tridimensional. Its growth is tridimensional, its nature is likewise tridimensional, and there is not even the slightest tendence either to perceive, conceive or perform in a four-dimensional manner, mathematically speaking. Trace out the biologic development of each mental faculty, from the mind of the moneron to the mind of the most highly developed man and it will be found that everywhere and always, without variation or exception, the nature of each of these has been to express itself tridimensionally and naturally. There is not even the slightest sign of so much as a germinal appetence for the four-space; it would, therefore, seem almost a prostitution of mental faculty to divert mental energy into the seemingly useless channel of present-day metageometrical researches; yet, it must be admitted that even though the end sought cannot be attained, the final results of the intellectual delvings into the dread homogeneity of kosmic origins and the consequent realization of the awesome coevalism of mind and space whence shall arise the recognition of the wondrous unitariness of all existences, will be that we shall come upon that thrice mysterious contrivance--the Heart of Divinity, the Kosmic Space-Center in which abide the roots of the _Great All_ in a marvelously indescribable unity and infinite originality. We may conclude, then, that hyperspatiality and all its appurtenances are but the toys of the childhood of humanity. But, as the years pass and the days of maturity come on apace, it, toy-like, will also be discarded. And the mind will seize then upon the seriousness of reality just as the matured youth responds to the stern realities of life and manhood responsibilities. But no one can say that the toys of childhood are wholly useless; no one can say that the joys which they bring are entirely fatuous and unreal nor shall we attempt to intimate that mathetic contrivances are without utility, without purpose and significance in the life of the growing mind of humanity. But they, too, will pass away. CHAPTER VIII THE MYSTERY OF SPACE The Thinker and the Ego--Increscent Automatism of the Intellect--The Egopsyche and the Omnipsyche--Kosmic Order or Geometrism--Life as Engendering Element--The Mystery of Space Stated--Kathekos and Kathekotic Consciousness--Function of the Ideal--The Path of Search for an Understanding of the Nature and Extent of Space Must Proceed in an Inverse Direction. The fragmentariness of the Thinker's outlook upon the universe of spatiality is due to the inhibitive action set up by the constrictive bonds which his complicate mechanism of intellectuality interposes between himself and reality. The Thinker, who stands back of and uses the various media of objective consciousness, such as the neural mechanisms, brain, emotions, his individualized life-force and the mind which together make up the instruments with which he contacts the sensuous domain, by adapting his consciousness to these means, as the artisan utilizes his tools, constitutes his own intellectuality. The intellectuality, then, is the totality of media by which consciousness effects its entrance into the sensuous world and by which it receives impressions therefrom. In other words, it is the sum of all those qualities, operations, processes and mechanisms which are recognized as constituting the _modus vivendi_ of man's intellectuality, and these are, in reality, nothing more than the ego himself. Many have been inclined to regard that which has been called the ego as the highest sovereign power in the state of manhood. He has been looked upon as the final consideration in the constitution of the human being. But the ego is an evolutionary product and the concomitant of self-consciousness which is the I-making faculty in man's psychic life. It is that quality of consciousness which makes man conceive of himself as a separate, detached and independent being. It is a purely intellectual or tuitional product, and, as such, is to be differentiated from the intuitional or life-quality which is the essence of man's real selfhood. With respect to the Thinker, the ego occupies precisely the same status as the agent to his principal. As the agent is the representative of the principal in all matters which come within the scope of his prescribed jurisdiction so is the ego the agent of the Thinker who is a spiritual intelligence. Accordingly, from an ethical viewpoint the Thinker is responsible for the acts of the agent and can in no wise escape the penalties accruing as a result of the agent's violations. Just as a commercial firm sends out a representative for the collection of data concerning certain phases of its business or it may be of any business or the entire world market so the Thinker projects his own consciousness into the mechanisms which are in their totality the egoic life. That is, he sends out his agent, the ego, into life and into the objective world of facts and demands that he shall convey to him, from all points of the territory which he is expected to cover, reports of his findings. Of course, these reports which are transmitted by the ego (the intellectual mechanism of the Thinker) are more or less well prepared summations of his individual observations and deductions. These are the percepts which the ego presents to the Thinker's consciousness. Concepts are formed by the Thinker in his treatment of these sense-presentations. It very frequently happens that the ego transmits reports which, for one reason or another, give very imperfect knowledge of the matter which his reports are designed to cover. Often it is necessary that additional and supplemental reports be made about the same thing, and even then, it is well-nigh impossible, if not quite so, for him fully to cover every detail of the matter under consideration and in no case is it possible for him to do more than report on the superficialities of the question under scrutiny. If the ego, in his operations, be imagined to be hampered by similar circumstances and difficulties as those which would ordinarily beset a commercial attaché it will then be clear that his reports must ever be fragmentary because of the inaccessibility of much of the data which would be necessary for a full report, and further, because of the inadequacy of his methods and means of gathering data due to the inherent limitations of his capabilities, endurance and perspicacity and innumerable other limitations and difficulties which must be faced in all search for the real. So that, while the sufficiency of the means which the ego enjoys at this stage for all practical purposes is granted no hesitancy is entertained when it comes to a discovery of the reals of knowledge in declaring their insufficiency. Then, too, when it is remembered that these egoic reports are in the nature of neurographical communications which are similar to telegraphic despatches and must pass through several stations, as ganglia, etc., often being relayed from one to another, it will be quite apparent that much, even of the original quality of the missives forwarded, will have been lost or radically changed in some way before it is finally delivered for the inspection of the Thinker himself. It not infrequently happens, even in perfectly normal beings, that the ego in filing, recording, transcribing, interpreting, translating and otherwise preparing these data for the Thinker's use, lets a cog slip, misplaces some of the data, loses or destroys fragments of it and so is unable to maintain a complete portfolio of his materials. As the Thinker is entirely dependent upon his agent, the ego, for the trustworthiness of his information covering the matter of the sensuous world it is obvious that at best his information is very fragmentary indeed, and necessarily so when it is considered that the _modus operandi_ of his agent and the difficulty of his operations are so complicate as to magnify the obstructions in the way to complete freedom in this regard. To continue the similitude of principal and agent it may be asserted that it is also true that the commercial house that sends out its attaches frequently will send letters containing directions as to procedure, sometimes censuring for past delinquencies and sometimes commending for praiseworthy deeds; and this, too, in addition to the original instructions which were given at the outset. It even comes to pass that the home office, because of some meretricious accomplishment, as the marked increase of efficiency shown by the agent's close application to his duties and the consequent success of his operations, confers certain favors upon the agent or removes some of the restrictions which were originally imposed, gives an increase in salary or promotes the agent to a higher and more lucrative office with larger powers and greater authority. This is analogous to what the Thinker does for the ego. For he not only receives reports from the ego, but often, in the shape of intuitions, gives additional information as to the proper manner of doing things, sheds more light upon some obscure operation, commends for duty well performed, condemns for failures or for wrong-doing, rewards arduous toil with greater powers of vision, keener insight, greater capabilities; in fact, promotes the ego in the sight of other egos by marking him out as an exceptional ego. But the curious aspect of this procedure is that, in time and after the ego has been repeatedly commended and promoted and otherwise favored by the Thinker, he begins to think that he owns the firm, that he is the life and main support of the whole corporation. He becomes arrogant, self-willed and finally falls into the illusion that he alone is responsible for the phenomenal success of the firm. This is the source of that illusion of the intellect which makes itself think that it, the ego, is all there is to man, that his instruments of operation in the objective world are the only kind of instruments that may be used; that his method of gathering data about things is the only safe and sure method; and so it develops that the intellectuality is the source of man's separateness, his individuality and his apparent aloofness from other men and things. It is, of course, needless to point out that in this way the intellect comes to be the tyrant of man, ruling with a rigid monopoly and as an all-exclusive autocracy. From the above implications it would appear that the intellect and the intuitive faculty are two separate and distinct processes, and so they are. One is the inverse of the other. The tendence of the egoic life or the intellect is for the external while the intuition is an internal process. The intellect acts from without towards the interior while the intuition acts from within outward. The intellect is the product of the intuition which is another term for the consciousness of the Thinker on his own plane. Just as the child lives a separate and distinct, though dependent, life from the parents so the intellect has a _modus vivendi_ which is distinct and separate from that of the Thinker, and yet it is in all points dependent upon the life of the Thinker. Here again, we find an analogy in the relation of the child to the parent. As some children are more amenable to the will of the parent than others, so, in some persons, the intellect is more amenable to the action of the intuition than in others. Yet it is a certain fact that the more the outward life is governed by the intuition, i.e., the more the intellect responds to the intuitive faculty of the Thinker, the higher the order of the life of the ego and the more accurate his decisions and judgments. In fact, it assuredly may be asserted that the place of every individual in the scale of evolution is determined in a very large measure by the degree of agreement between the intuition and the intellect or by the ease with which the intuition may operate through the intellect as a medium. At least, the quality of one's life may be determined directly by these considerations. The Thinker being himself a pure spiritual intelligence, living upon the plane of spirit and therefore unhampered by the difficulties which the ego meets in his operations in the objective sensorium, and possessed of far greater knowledge, is correspondingly free from the limitations of the ego and very naturally closer to kosmic realities. Hence, he is better situated for the procurement of correct notions of relations, essentialities and the like. It is believed, therefore, that in the proportion that these two processes, the intellectual and the intuitional, are brought, in the course of evolution, to a closer and more rigid agreement, in the proportion that the Thinker is able to transmit the intuitograms in the shape of concepts or that the intuition is made more and more conceptual, in just that proportion is humanity becoming perfect and its evolution complete. The difficulty found to inhere in the conceptualization of intuitions so that they may be propagated from man to man seems not to lie in the Thinker himself, but more essentially in the ego, in the intellectuality and its complicate schematism or plan of action. It would appear, therefore, that the only way of escaping or transcending this difficulty is for the ego so to refine his vehicles or so facilitate his plan of action by eliminating the numerous relays or sub-stations intervening between the consciousness of the Thinker and that which may be said to be his own that the transmission of intuitograms may be accomplished with the greatest ease and clearness. While no attempt will be made to indicate the probable line of action which the ego or objective man will adopt for this purpose, it is believed that it may be said without pedanticism that the only true method of attaining unto this much desired state of things is, first of all, by assuming a sympathetic attitude not only towards the question of the intuition itself but to all phenomena which are an outgrowth of, or incident to, the manifestations of the intuitive faculty through the intellectuality, and second, by the practice of prolonged abstract thought, this latter procedure effecting a suspension of the intellectuality temporarily at the same time allowing it to experience an undisturbed contact with the intuitional consciousness, thereby laying the basis for future recognition of its nature and quality. It would seem that these two conditions are absolutely necessary in order that a more congruent relationship may be promoted between these two cognitive faculties. Ordinarily, it would appear that the philosopher who is undoubtedly inured to the necessities of continuous abstraction or the mathematician whose most common tasks naturally fall in this category would be among all men most apt to develop to the point of conceptualizing intuitograms readily, yet it seems that this is not the case. And there is good reason for it. The mind of the philosopher and the mathematician is intellectual rather than intuitional and is, therefore, wedded to matter, to the action and reaction of matter against matter and hence operating in a direction at variance with the trend of an intuitional mind. And this condition is undoubtedly due to a lack of a sympathetic attitude towards this species of consciousness. At any rate, it is thought that a too great anxiety in this respect need not be entertained by humanity at all, for the reason that in the case of a faculty, the rudimentary outcroppings of which are so marked and universally observable and existing in greater or lesser degrees in various human beings, there is ample evidence for the belief that it is being carefully and duly promoted by a well-directed evolution of psychic faculties and powers, so that at the proper time, determinable by the state of perfection reached by the intellectuality or the ego in the operation of his cognitive processes, the much desired agreement of these two faculties will have been realized and the conceptualization of intuitograms into propagable conceptions an accomplished fact. Until this goal shall have been reached and the intuition shall have overshadowed the intellect as the intellect now overshadows the intuition; or the consciousness of the ego, derived from the interplay of the Thinker's consciousness among the various elements which constitute the ego himself, shall have been merged with that of the Thinker, the outlook must remain fragmentary, only becoming a well-ordered whole as the barriers of dissidence are broken down in succession. The evolution of consciousness, from the simple, undifferentiated _moneron_ to the differentiated cell and from that to the cell-colony and from the cell-colony to the organism, traversing in successive paces through all the stages of lower life--mineral, vegetable and animal--to the stages of the simple, communal consciousness of the higher animals, to the self or individual consciousness of the human being, each requiring millions of years for its perfection before a more advanced stage is entered, has been one continuous relinquishment of the lower and less complicate for the higher and more complex expression of itself through the given media. When a newer and higher stage of consciousness is being entered by humanity its appearance or manifestation is first made in the most advanced of the race and that only in a dim, vague way. This rudimentary condition persists for some time, perhaps many thousands of years, then the faculty becomes more general in appearance, the number of advanced individuals increases, and consequently, as in the case of the intuitive faculty, it becomes universally prevalent in all humanity; becomes transmissible as so-called "acquired characters," and then appears as the normal faculty of the entire human family cropping out in each individual. Thus, in passing from the few advanced ones in the beginning to that stage where it becomes the common possession of all, a faculty requires many thousands of years for its perfection, and especially has this been true in the past history of the development of human faculties. But it is believed that the sweep of the life current as it proceeds from form to form, from faculty to faculty, gains in momentum as it proceeds, so that in these latter years due to the already highly developed vehicular mechanisms at its disposal not so great a period of time as formerly is required for the _out-bringing_ of a new faculty. It might well be that while in the past hundreds of thousands of years were necessary in the perfection of organs and faculties, in these latter days only a few thousand, perhaps hundreds, may be necessary and that in the days of the future not even so many years may be required to universalize a faculty. And especially does this appear to be true in a state of affairs where so large a number of persons are beginning consciously to take their evolution in hand and by volitional activities are supplying greatly increased impetus to their psychic processes which under ordinary, natural methods would be considerably slower in their development. It is quite obvious that all cultural efforts when applied to the betterment of a given plant, animal or faculty result in a corresponding hastening of the process of growth far in excess of what that growth would be under normal, natural conditions. All the present faculties possessed by man are remarkably susceptible to cultural influences; in fact, the standing edict of ethical and social law is that the human faculties must be cultivated as highly as possible, thereby giving the spirit a more perfect medium of expression. These observations, therefore, lead irresistibly and unavoidably to the conclusion that the time for the upspringing of the intuitional faculty in the human organism is even now upon us, that undoubtedly in certain very advanced ones it has already reached a notable degree of perfection and is rather more general than would appear in the absence of careful investigation. Now, just as the intellect has made for individuality, has emphasized the separateness of the Thinker's existence from that of other thinkers, has developed self-consciousness to a very high degree, even pushing it far over into the domain of the higher consciousness to the temporary obscuration of the latter, so the intuitional will make for union, for the brotherhood of man, for co-operation and for the common weal. Through it man will come gradually into the consciousness that fundamentally, in his inner nature, in every respect of vital concern, he is at-one with his fellowmen and not only with the apparent units of life but with all life as expressed in whatsoever form throughout the universe. Then, too, he will be closer to the reality of things, of actions and natural processes; in fine, he will have begun the development of the space-mind which will bring him to the knowledge that he is one with space also and, therefore, with the divine life of the world. One of the peculiarities of the vital force which shows itself in the consciousness as man's intellect, is its growing _automatism_, or that tendency which enables the consciousness to perform its functions automatically and thus allow opportunity for the development of newer and higher faculties. Actions, oft repeated, tend to become automatic. This is also true of thought and consciousness. It is one of the beneficent results of abstract thought that it develops, or tends to develop, a kind of automatism whereby a marked saving in time and energy is effected. This affords opportunity for other things. It is undoubtedly true that in the days of the truly primitive man his consciousness was more completely engaged in the execution of the ergonic functions of cells, organs and tissues; that all those processes which are now said to be involuntary and reflexive were at one time, in the distant past of man's evolution, the results of conscious volitions. This is a condition which must have preceded even the development of the intellect itself. Indeed, there could be no intellect in a state where the entire modicum of consciousness was being utilized in the performance of cellular and histologic functions. The rise of the intellect must have been in direct ratio to the development of automatism among the cells, tissues and organs, so that as these came gradually to perform their special labors reflexively the intellect began to be formulated and to grow, at first only incipiently, then more and more completely until it reached its present state. At the present stage of its evolution, a great deal of the labor of the intellect is beginning to fall into a kind of increscent automatism, although only rudimentarily, in many instances. Yet, as a result of this tendency, quite the whole of the phenomena of perception is characterized by a sort of automatic action. And the mind perceives without conscious volition. Many of the steps of conceptualization are automatic, in part, if not wholly. Certain it is that impulses once set in operation whether consciously or unconsciously continue to act along the same line until exhausted or until the end has been attained. Consequently, it is a proven fact that often serious mathematical and philosophic problems have been solved by the mind long after any conscious effort to solve them had ceased. Often solutions have been arrived at during sleep. Many such cases might be cited, but the phenomenon is now so common that almost every one can cite some experience in his own life that will substantiate the claim. There is no doubt but that these phenomena are evidences of a reflexive development in the intellect. The time will come undoubtedly, and necessarily so if the intellect is to give way to a higher faculty, which shall be as much above the intellect in its grasp of things as the intellect is now above the simple consciousness of the lower animal, when quite the entirety of our intellectual processes will become automatic or self-performing. What then remains of the egoic schematism, after its transmutation or elevation as the organ of the intuitional consciousness will be utilized as the organ of the Thinker's involuntary cognitive processes. This will mean that all of that laborious ideation which is now the abstract thought of the Thinker will be performed automatically, leaving the higher aspect of the egoic consciousness free to conceptualize or intuitograph the intuitions. Perceptualization then will be replaced by conceptualization. This latter will occupy about the same status as the former does now. And necessarily, perception will become more complex. In other words, while we now perceive simple percepts which are again arranged into concepts making a composite picture of the object, we shall then be taking in the composite picture of the object at first hand, thereby dispensing with the rather slow process of perception as it now operates. We shall still be perceiving, but what we perceive will be concepts rather than percepts, as at present. The increased powers of intellection gained as a result of the increscent _automatism_ in the intellect, the flowering forth of the intuitive faculty and the general enhancement of the intellect throughout all its processes will enable it to entertain concepts or composite picture of things just as readily and as perfectly as it can at present deal with a single percept. Concepts will be replaced by super-concepts or intuitographs. Increased perspicacity will enable the Thinker to manipulate the concepts and intuitographs with the same ease and readiness and withal the mind will have attained unto an almost unrealizable freedom in its search after truth. The outcome of this new adjustment which, of course, will not spring up at once, but by insensible degrees, will be the clarification and unification of our knowledge. It will mean also the simplification of it; the obviation of diversities of opinions, the springing up of a new and winnowed system of philosophy which shall be the true one; further, it will imply the lessening of the probability of error in our judgments and conclusions; the removal of illusion to a much larger degree than to-day is possible and the realization by every one of something of the essence of things, of causes and effects, of actions, operations, natural forces and laws; in fact, a condition of mind which will present to the consciousness the simple truth above every conceivable phase of kosmic life which may come within the scope of the Thinker's observation. The further implications of this view are that there is a difference between the Thinker and the intellectuality. The Thinker is eternal and partakes, therefore, of the very essence of primordial originality while the mentality is an artificial process, the resultant of the adaptation of the Thinker's consciousness to his vehicular contrivances of objective cognition and the interplay of his life among them. If the appearance of a choppy sea disturbed by the passage of a brisk breeze over its surface be imagined, a similitude of the great ocean of life may be envisaged. The wavelet crests symbolize the egos; the base of the wavelet which is one with the great sea of water represents the Thinker which is one with the divine life and consciousness of the kosmos. Just as wavelet crests are continually springing up and falling back into the sea, so are egos continually being cast forth and reabsorbed into the universality of life only to be recast, as a wavelet crest or ego, upon the surface of the moving ocean of life. _And so, in this respect, the universum of life and consciousness which are essentially one is in a constant state of ever-becoming, un-becoming and re-becoming._ Another implication is that, on account of the diversity and complexity of the means of contact with the external world, it is not possible for the ego to arrive at more than a fragmentary understanding of even the latent geometrism of life, mind and materiality. In our examination of the sensuous world, we are very much like the three blind men set to examining an elephant. One set to scrutinizing his trunk by means of his sense of feeling. When asked for his judgment as to what the elephant was he declared it was a snake; a second who began with the legs found it to be like huge pillars; and a third who caught hold of the elephant's tail and declared the elephant to be like a rope. Each one of the blind men described what he was able to perceive. To each what he felt was all there was upon which he could render judgment. And so, artists, philosophers, mathematicians, musicians, mechanicians, religious seers, metaphysicians and all other types of mind, are just so many blind men set to the examination of an elephant, or the sensuous world. Each one confidently believes his view to be correct; each one is satisfied with the deliveries of his senses. Yet no one of them is wholly correct, no one of them has seen every phase and aspect of the problem. Does it not, therefore, appear to be the more reasonable and urgent that the view which synthesizes the judgments of all the possible examiners thereby constructing a composite idea of the entire mass of judgments is the more reliable and the more correct? Referring again to the dual intelligence, the ego and the Thinker, which together constitute man, it is deemed necessary, in order to present the concept of this duality to the mind of the reader in the way that shall enable him easily to recall it, to designate the egoic intelligence as the _egopsyche_, and the Thinker's intelligence as the _omnipsyche_. The egopsyche is the I-making faculty, the faculty of self-consciousness and the synthesis of all those psychic states and functions known as the intellect or mind and includes the ethical aspect of man's nature. The omnipsyche is the organism of kosmic consciousness, the space-mind, or man's higher self and that which connects with or allies him to all life; it is the basis of human unity and of unity with divinity, just as the egopsyche is the basis of separation and individuality; it is the organ of direct and instantaneous cognition and the permanent essence which has persisted through every form which the being, man, has ever assumed and through every stage of human evolution. In it are stored up the memories of the Thinker's past, the secrets of life, mind, being, reality, and the history of life from the beginning; in it also the plan of action for the future of the life-wave as it passes from plane to plane, from stage to stage, and from form to form. It is the spark from the flame that is never quite free from its source; it is the continuous spark, the prolonged ray which does not go out and cannot be extinguished. It is that in man which when full union therewith has been attained makes him a god in full consciousness. The omnipsyche is really a neglected and overlooked factor in the doctrine of evolution. Evolutionists, while they claim life to be continuous and that man has come through all the kingdoms of nature in succession and has spent millions of years in the perfection of his various organs, faculties and stages of consciousness, make no ample allowance for what is in reality the basal element in evolution--a continuous, persisting, permanent life-force which does not lose its identity from the beginning to the end of the process. This fact--that that spark of life which set out upon the evolutionary journey as a moneron has glowed steadily from that stage to manhood, maintaining meantime its original purposiveness and intent--seems to be the most obvious consideration of the whole doctrine, yet it has been more or less completely ignored. The elementary requirements of evolution would seem to establish clearly the necessity for some such eternally persisting principle as the omnipsyche which is capable of such subtle adaptations to every conceivable form of life and in which should be gathered up the evolutionary results of every life-cycle. For this purpose the omnipsyche or unifying principle in man was designed from the beginning and it is that which constitutes the basis of his intellectual nature while in a far larger sense it is the divinity in man himself. It is indeed strange that so important a factor as the omnipsyche should have been omitted by evolutionists. Yet it can be accounted for upon the grounds of the purely mechanistic character of all intellectual attempts at solving the problems of vital manifestations. But so long as men rely upon mechanical explanations of such phenomena so long will they be prone to overlook the very essentialities of the problems which they devoutly wish to solve. The continuity of the physical germ-plasm of the human species,[27] now quite generally admitted, would suggest, it seems, an analogous condition to the continuity of the psychic plasm called the omnipsyche, the only difference being that the omnipsyche is an intelligent factor while the physical plasm is a medium of transmission though non-intelligent. The omnipsyche is, therefore, the psychic reservoir of evolution into which are stored the transmuted psychics of moneron, amoeba, jellyfish and every other form which it has ensouled and acts as the storeroom of man's psychic operations as well as the source of his intellectuality. We turn now from the study of a sketch of the mechanism of man's consciousness which gives at its best only a fragmentary view of the universe of spatiality to a consideration of space itself in the light of its interrelational bearings upon the question of intellectuality. [27] See _The Germ Plasm; A Theory of Heredity_, by A. WEISSMAN. In the chapter on the "Genesis and Nature of Space" we have, in tracing out the engenderment of space, proved it to be basically one with matter (and indeed the progenitor of matter), also with life and consciousness. Further, it has been shown that all the characteristics of materiality are due to the adaptation of consciousness to it and that out of this adaptation grew the intellectuality. A close approximation to this view was maintained by KANT when he discovered that our faculty of thinking or the intellect only finds again in matter the mathematical order or properties which our faculty of perceiving or consciousness has deposed there. It appears, therefore, that when the intellect approaches matter or spatiality it finds always a ready yieldance to its demands simply because intellectuality has previously established therein the delineation or map of the path over which it necessarily must traverse in its examination of the object of its pursuit. In other words, the kosmic mind in engendering materiality and spatiality has set up therein a kosmic order or geometrism. Both motor and intellectual progress, therefore, can be made through the world of spatiality because of the immanence of this kosmic geometrism which lies latent in the very fabric of the world of substance fashioning both the character and the nature of the intellect as well as of space itself. So that there is a perfect congruity subsisting between spatiality and intellectuality. Accordingly it is impossible for either one or the other to transcend the grim grasp of the mathematical order which binds them in such lasting and fundamental agreement. Extra-spatiality may degrade itself into spatiality, and indeed in the very nature of the case, does so degrade itself, yet spatiality can never raise itself beyond the limits set by its engendering parent. Materiality may become more and more spatialized and consciousness more and more intellectualized, but they must proceed hand-in-hand one not superseding the other. Being the essence of the natural geometry which is everywhere immanent in the universum of matter, space becomes an organized and ordered extension, in fact is the totality of such organized and ordered extension, which conforms to the latent geometrism the engenderment of which it is the sole cause in the last analysis. Does it not appear then that all that mass of artificial geometry which has sprung up as a result of departures made from this natural geometry is utterly baseless and most certainly lacking in the kosmic agreement which spatiality lends to our primary conceptions? Of course, it is admittedly possible to devise certain conventional forms of logic and endow them with all the evidences of a rigid consistency but which, because of their purely artificial character, will fall far short of any real conformity to the potential geometrism which has been established in spatiality. And this fact is of utmost significance for all those who seek to find justification either logically or naturally for the existence of a multi-dimensional quality in space; for, if a clear, discriminative conception as to the categorical relationship, each to each, of the two kinds of geometry be carried in mind, it will not be easy to confound them neither will it be difficult to discern where the one ends and the other begins. Now, the fourth dimension and the entirety of those mathematical speculations touching upon the question of hyperspace, dimensionality, space-curvature and the manifoldness of space are purely conventional and arbitrary contrivances and do not meet with any agreement or authority in the native geometrism which we find inhering in space and which the intellect recognizes there. This conclusion seems to be obvious for the reason that, in the first place, the non-Euclidean geometries have been constructed upon the basis of a negation of the latent geometrism of space and intellectuality; and if so, is it reasonable to expect that either they or any of their conclusions should accord with the nature of that form of geometry so admirably delineated by EUCLID? Obviously not. It is a matter of historical knowledge that the whole of the artificial non-Euclidean geometries consists of those purely conventional results which investigators arrived at when they denied or controverted the norms supplied by the natural geometry. When metageometricians found that they could neither prove nor disprove the Euclidean parallel-postulate they then set upon the examination of idealized constructions which negatived the postulate. The results, thus obtained, although self-consistent enough, were compiled into systems of geometry which naturally were at variance with each other and with this inherent geometrism which is found in spatiality and answered to by the intellect both normally and logically. Furthermore, there is another consideration which to us seems to be equally if not more forbidding, in its objections to the coördination of the two systems of geometry, and that is the fact that the geometry of hyperspace is denied the corroborative testimony of experience and this is true of practically the whole of its data. Indeed, there is perhaps no single element in its entire constitution which claims the authority of experience. This is undoubtedly the weakest point in the structure of the hyperspatial geometries. Contrarily, such is not the case with the natural geometry; for, in this, the intellect in retracing its steps over the path laid out by that movement which has at the same time created both the intellect and spatiality, finds an orderly and commodious arrangement into which it naturally and easily falls. So exact is this agreement of the intellect with the kosmic order that if it were possible to remove the whole of spatiality and materiality there would still be left the frame work which is this latent geometrism of kosmogenesis. But the fact that the intellect naturally fills all the interstices of materiality and spatiality, fitting snugly into all of them as if molded for just that purpose, by no means warrants the assumption that it would or does also fit the engendering factor which has created these interstices. The frame work, the order or the geometrism of the kosmos has been established by life acting consciously upon the universum of materiality. And in order to establish this geometrism life had to be mobile, active, creative. It could not remain static, immobile, and accomplish it. Being mobile, dynamic, creative, it passes on. It is like a fashioning tool which the cabinet makers use in cutting out designs upon a piece of wood. It moves, and keeps moving until the design is finished, and then it is ready for more designing. Life is like that. It cuts out the designs in materiality, fashions the form, molds the material, and passes on to other forms. The intellect fits into these designs gracefully. But what it finds is not life itself, only the design which life has made. Hence, as there is neither an empirical spatiality nor materiality in conformity with which the artificial geometry of the analyst may be said to exist, and as it may not be said to conform to the path which life has made in passing through either of these, it is absurd to predicate it upon the same basis as the natural geometry. And so, we are forced, in the light of these considerations to deny the validity and hence the acceptability of the non-Euclidean geometries as either reasonable or warrantable substitutes for the Euclidean, and denying which we also formally ignore the claims of the fourth dimension, as mathematically designed, to any legitimate anchorage in either our vital or intellectual movements. It has been shown that the flow of life, as it describes that movement which we call evolution, engenders simultaneously and consubstantially spatiality, materiality and intellectuality, and these, in turn, the natural order or geometrism everywhere immanent in the universe; and that automatically, one out of the other and each out of the all, these constitute the totality of kosmic fundamentals. Also we have sketched the mechanism of man's consciousness and discovered how, in its evolutionary development it has divided into two aspects, the egopsychic and the omnipsychic, and these two factors ally him definitely and adequately to the world of the senses and to the world of supersensuous cognitions. And thus we have cleared up some of the misconceptions which had to be confronted and made more easy the approach to the central idea, thereby conserving the substantiating influence which a general and more comprehensive view of the whole would naturally give. _The totality of kosmic order is space. It is circumscribed by an orderless envelope of chaos just as the germ of an egg is surrounded by the egg-plasm. The organized kosmos is the germ, kernel or central, nucleated mass, enduring in a state of becoming. Involutionary kathekos or primordial chaos is the egg-plasm which nourishes the germ or the kosmos and is that out of which the germ evolves. Kathekos or chaos is the unmanifest, unorganized, unconditioned, unlimited and undifferentiated plasm. Space is the manifest, limited, finite, organized germ that, feeding upon the enveloping chaos, exists in a perpetual state of alternate manifestation and non-manifestation--appearing, disappearing and reappearing indefinitely._ The appearance of the kosmos as an orderly elaboration of the involutionary phase of kosmogenesis, in so far as kosmic order may be said to be an accomplished fact, marked the turning point in that procedure whose function it was to make manifest a universe possessing certain definite characteristics of orderliness; but the kosmos, as it now stands, may not be thought of as having attained unto a state of ultimate orderliness. The idea meant to be conveyed is that between the point of becoming and the actually pyknosed, or solidified stage in the process of creation there is a more or less well defined line of demarkation cutting off that which is spatiality from that which is non-spatiality. Beyond the limits of spatiality is an absence of geometric order. Here geometry breaks down, becomes impotent, because it is an intellectual construction; at least, it is not so apparent as in the manifested kosmos. It is a state about which it is utterly futile to predicate anything; because no words can describe it. The most that may be said is that it is absence of geometric order as it inheres in space. And if so, all those movements comprehended under the general notions of spatiality, materiality, intellectuality and geometricity have both their extensive and detensive or inverse movements nullified in their approach to it. Involutionary _Kathekos_, therefore, may be said to be the primordial wilderness of disorder which outskirts the well laid-out and carefully planned garden of the spatial universe. We may excogitate upon some of the obvious functions of this kathekotic world-plasm; but in doing so we must leave off all attempts at a description of its appearance, its magnitude, extent or other qualities, and think only of its kosmic function. We cannot say that there is back of it a spatiality nor can we say that it is a spatiality; for whatever may be its extent or volume, it suffices that it may not be said to be space. It is chaos. Space is order, organization, geometricity. It cannot be said that there is a latent geometrism in chaos; because geometric order is found only in spatiality and is that which distinguishes spatiality from kathekosity or non-spatiality. Chaos is the lack of spatiality. This, of course, implies that it is impenetrable to the intellectuality or to vitality. All inverse movement such as is discovered as taking place in spatiality and which results in the phenomenalization of space runs aground when it strikes against the rock-bound coast of kathekosity. We can only say that it is both the point of origin for the evolving universe of life and form and its terminus. It is the nebulosity out of which the whole came and into which all is ultimately occluded. A great and far-reaching error is made in all our thinking with respect to the kosmogonic processes when we postulate the complete absorption of chaos as an early act of kosmogony. Customarily, we think of kosmic chaos as a primordial condition whose existence was done away as soon as the universe came into active manifestation. This because it has been exceedingly difficult, if not quite impossible, for those whose privilege it was to determine the trend of philosophic thought to free themselves from the bondage of a dogma which owed its existence to a traditional or legendary interpretation of facts that ought never have been so interpreted. Chaos IS and EVER SHALL BE, so long as the universe itself lacks completion, fullness or perfection in purpose, extent and possibility. It is undoubtedly being diminished, however, in proportion as the kosmos is approaching absolute perfection. And when the last vestige of chaos disappears from the outerskirts of the maturing kosmos there shall appear a _glorified universe_ of indescribable qualities. Space being a perception _a priori_ cannot be determined wholly by purely objective methods. The yard-stick, the telescope and the light-year are objects which belong exclusively to the phenomenal and with them alone never can we arrive at a true conception of the nature of space. We can no more demonstrate the nature of space by the use of objective instruments and movements than we can measure the spirit in a balance. Certainly, then, it cannot be hoped that by taking the measurement of space-distances in light-years, or studying the nature of material bodies, we shall be able to fathom this most objectively incomprehensible and ineluctable thing which we call space. It is such that every Thinker must, in his own inner consciousness, come into the realization of that awfully mysterious something which is the nature of space both as to existence and extent by his own subjective efforts unaided, uncharted and alone. When we measure, weigh, apportion and otherwise try to determine a thing we are dealing with the phenomenal which is no more the thing itself than a shadow is the object which casts it. What does it matter that metageometricians shall be able to demonstrate that space exhibits itself to the senses in a four-or _n_-dimensional manner? Granting that they may be able to do this, if merely for the sake of the discussion, when they have finished, it will not be space that they have determined, but the phenomena of space, its arborescence, while space itself remain indeterminate and unapproachable by phenomenal methods. If there are curvature, manifoldness and _n_-dimensionality these are not properties of space, but of intellectuality in its cultured state and when it is, therefore, removed from the native state of conception. Scientists may be able to weigh the human body, count every cell, name and describe every nerve, muscle and fiber; they may even be able to know it in every conceivable part and from every physical angle and relationship, and yet know nothing of the life which vitalizes that body and makes it appear the phenomenal thing that it is. So it is not by instruments which man may devise that we shall be able to determine the true nature and purpose of space. We must adopt other methods and means and assume other angles of approach than the purely objective in order to comprehend space which, being the sole inherent aspect of consciousness, can be understood best by applying the measures which the latter provides for its understanding. It would appear, therefore, that the best study of space is the consciousness itself, knowing which we shall know space. The universum of space, including the phenomenal universe, and its relation to consciousness may be likened to a conical funnel whose base represents the phenomenal world of the senses and whose apex or smallest point represents ultimate reality. In Figure 20, we have endeavored to symbolize graphically this conception of space. The base marked "Sensorium" represents the sensible world. That marked "Realism" symbolizes the ultimate plane of reality, the inner essence of the world, the plane of "things-in-themselves." The cone arising from the base "sensorium" symbolizes the objective world as compared with consciousness; the subverted cone, with apex in the sensorium, represents the evolving human consciousness. The successive bases have the following symbology: Self-consciousness, Communal Consciousness, Mikrocosmic Consciousness, Makrokosmic or Universal Consciousness, the Plane of the Space-Mind Consciousness, Divine Consciousness, Kathekotic Consciousness, or the Plane of Final Union with the Manifest Logos. Self-consciousness is that form of consciousness which enables the ego to become aware of himself as distinguished from other selves or the Not-self; the Omnipsychic or Communal Consciousness is that form of consciousness from which arises the realization by the Thinker of his oneness with all other thinkers and with other forms of life. Mikrocosmic consciousness denotes a still higher form of consciousness, as that which enables the Thinker to become conscious of his living identity with the life of the world or the planet on which he lives. It represents a stage in the expansion of consciousness when it becomes one with the consciousness of the planet upon which it may be functioning. Makrokosmic consciousness accomplishes the awareness of the Thinker's unity with the life of the kosmos or universe. The space-mind and the consciousness which constitutes it enable the Thinker to comprehend the originality and the terminality of kosmic processes. It is archetypal so far as the life-cycle of the universe is concerned because the beginning, the intermediate portion and the ending of the kosmos are encompassed within it. Divine consciousness is that form of consciousness which arises upon the unification of the Thinker's consciousness with that of the manifest deity; it is, in fact, omniscience. The kathekotic consciousness belongs to the ultimate plane of reality; to kosmic origins and chaogeny, and therefore, pertains to the plane of non-manifestation. [Illustration: FIG. 20.--Kosmos and Consciousness] The implications are that in comparison with the sensorium, the Thinker's consciousness is a mere point in space. It is, in reality, so small and insignificant that the extensity of the physical world or universe seems unlimited, unfathomable in meaning and infinite in extent. But as his consciousness expands, as it passes, in evolutionary succession from one plane of reality to another and higher one, the illimitability, the incomprehensibility and infinity of the universe grow ever smaller and smaller, until the plane of divine consciousness is reached. Then the previously incomprehensible dwindles into insignificance, lost in the real illimitability, infinity and unfathomability of consciousness itself. Kosmic psychogenesis, as exhibited and specialized for the purposes of the evolution of the Thinker, can have no other destiny than the flowering forth as the _ne plus ultra_ of manifestation which is nothing short of unification with the highest form of consciousness existent in the kosmos. It is not to be considered really that the scope of space is diminished but that the growing, expanding consciousness of the Thinker will so reduce the relative extension of it that illimitability will be swallowed up in its extensity. Consciousness, in becoming infinite in comprehension, annihilates the imaginary infinity of space. Accordingly, that which now appears to be beyond mental encompassment undergoes a corresponding diminution in every respect as the consciousness expands and becomes more comprehensive. _The mystery of space decreases as the scope of consciousness increases._ As the Thinker's consciousness expands the extensity of the manifested universe decreases. Thus the mystery of every aspect of kosmic life lessens, and fades away, as the intimacy of our knowledge concerning it becomes more and more complete. There is no mystery where knowledge is. Mysteriousness is a symbol of ignorance or unconsciousness, and that which we do not understand acts as a Flaming Sword keeping the way of the Temple of Reality lest ignorance break in and despoil the treasures thereof. Figure 21 is a graph showing a sectional view of consciousness on all planes represented as seven concentric circles. This describes the analogous enveilment of the consciousness when it ensouls a physical body or when bound to the purely objective world of the senses. The overcoming of the barriers of reality, represented by the circumscribing circles is the work of the Thinker who is forever seeking to expand and to know. For only at its center, as symbolized here, is the consciousness at one with the highest aspect of kosmic consciousness and there alone is the mystery of space despoiled of its habiliments. [Illustration: FIG. 21.--Septenary Enveilment of Consciousness] Accordingly, as consciousness or the Thinker is more and more divested of carnal barriers and illusions there develops a gradual recognition of the unitariness of spatial extent and magnitude; there arises the certain knowledge that space has but one dimension and that dimension is sheer _extension_. The Thinker's sphere of awareness is represented as if it begins as a point in space and develops into a line which divides into two lines, the boundaries of the space cones. Thus it may be perceived that the ancients had a similar conception in mind when they symbolized kosmogenesis with the dot (.), the line, and the circle with diameter inscribed, which together represent the universe in manifestation. We realize the impossibility of adequately depicting the full significance of the inverse ratio existing between the extensity of space and the increscent inclusivity of consciousness by means of graphs; for neither words nor diagrams can portray the scope and meaning of the conception in its entirety. Yet they aid the intellect to grasp a ray of light, an intimation of what the Thinker sees and understands interiorly. In this connection it is interesting to note the function of the ideal in the evolution and expansion of consciousness. The ideal has no perceptual value; it has no status in the world of the senses. It is unapproachable either in thought or action, and therefore, lies beyond the grasp of both the intellectuality and the vitality. It is indescribable, inconceptible and searchless; for the moment that we describe, define, or approach the ideal, either intellectually or vitally, in that moment it ceases to be ideal, but actual. It flees from even the slightest approach; it never remains the same; it cannot be attained, at least its attainment causes it to lose its idealty. It is then no longer the ideal. It is like an _ignis fatuus_; the closer we come to it the farther away it recedes. It hangs suspended before the mind like the luscious grapes which hung before the mouth of the hungry Tantalus. As the grapes and the water receded from his reach at every effort he made to seize them so the ideal remains eternally unseizable and unattainable. Whatever, therefore, is in our thought processes, or in our knowledge, that may be said to be _ideal_, does not really exist. The ideal is a phantom growing out of the nature and essence of the intellectuality. Its purpose is to lead merely the mind on; to allure it, to tantalize, and compel it to grow by exertion, by the struggle to attain, by the desire to overcome. In this respect, it serves well its function in the economy of intellectual evolution. It is a mysterious aspect of the original and eternal desire to live which is the kosmic _urge_ present in all organized being and has its roots hidden in the divine purpose of creation. Idealized constructions, then, are like Arabian feasts conjured up by a famishing mentality. They are like the dreams of a starving man in which he actualizes in phantom-stuff the choicest viands in abundant supply for his imaginary delectation. The mind that is satisfied never idealizes, never makes an idealized construction. It is only when an "aching void" is felt, when a longing for the realization of that which it has not arises within itself, when a feeling of distinct lack, a want, a hankering after something not in its reach, takes possession of the mind that it begins to idealize. That is why some minds are without ideals. It is because they are satisfied with what they have and can understand. They feel no hungering for better and grander things; they have no desire to understand the unknown and the mysterious; hence they do not idealize; they make no attempt to represent unto themselves a picture of that which is beyond them. Such minds are dormant, hibernant, asleep, unfeeling and unresponsive to the divine urge. But the ideal is neither obtainable objectively nor subjectively, neither phenomenally nor really, so that when we come upon the ideal in our mode of thinking we have arrived not at a finality or end, but at that which is designed to lead us on to something higher, to nobler accomplishments and more extensive conquests. When we have devised idealized constructions, therefore, we should not therewith be content but should scrutinize them, examine and study them for their implications; for thereby we may discover the path and the guide-posts to a new domain, a new ideal, following which we shall, in time, come to a point in our search for the real where the fluxional is at a minimum; we shall reach that something which will admit of no further struggle--the last chasm between the phenomenal and the real--and standing on the bridge, consciousness, which engages the twain, shall have a complete view of that Sacred and Imperishable Land of Kosmic Realism where like a fleeting cloud of sheerest vapor shall be seen the phantom-ideal deliquescing and disappearing in the cold, thin air of the real and the eternal. Since space is judged to be infinite by the intellect occluded in such clouds of illusion and hampered by such constrictive bonds of limitations, as it now endures, we have no right to conclude that the concept of infinity would still linger before the mind's eyes when the illusionary veil is removed; in fact, there is ample reason to believe, nay for the assertion, that the recession of the veil will reveal just the opposite of this illusion, namely that space is finite, and even _bounded_ by the fringe of chaogenetic disorderliness. Either we perceive the real or we do not; either the pure _thingness_ of all objects can be perceived or it cannot be perceived. If not, granting that there is such a thing as the real, it must be within the ultimate range of conceivability. It also seems reasonable that realism exists somewhere, and if so, must be sought in a direction inverse to that in which we find the phenomenal and the approach thereto must necessarily be gradual, continuous and direct and not by abrupt breaks, by twists and turns. The phenomenal must lie at the terminus of the real, and _vice versa_. So that by retracing the path blazed out by the real in coming to phenomenalization we shall perhaps find that which casts our shadowy world, just as by tracing a shadow in a direction inverse to that in which it extends we may find the object which projects it. _It is not out and beyond that we shall find the end of space; it is not by counting tens of thousands of light-years that the supposed limits of space shall be attained. The path of search must project in an opposite direction--not star-ward but Thinker-ward, toward the subtle habitation of the consciousness itself._ We err greatly when we think that by measuring distances we shall encompass space; for that which we measure and determine is but the clouds caused by the vapor of reality. It is, therefore, not without, but within, in an inverse direction that the search must proceed. Going back over the life-stream, beginning where it strikes against the shores of solid objectivity, deeper and deeper still, past the innermost mile-stone of the self-consciousness, back into the very heart of the imperturbable interior of being where the Thinker's castle opens its doors to the Great Kosmic Self, from that open door-way we may step out into that great mystery of space--limited, yet not limited, multi-dimensioned, and yet having only one dimension, veritably real and fundamental, the Father-Mother of all phenomena. Here the great mystery of mysteries is revealed as the citadel of the universal and the ultimate real. In this citadel, the plane of kosmic consciousness, space loses its spaciousness and time its timeliness, diversity its multiplicity and oneness alone reigns supreme. But the movement towards the center and circumference of space, after this manner, requires aid neither from the notion of space-curvature nor that of the space-manifold, except, indeed, only in so far as a state of consciousness or a degree of realism may be said to be a tridimensional manifold. The feeling that space is single-pointed, and yet ubiquitously centered, has been indulged by mathematicians and others in a more or less modified form; but they have imagined it in the terms of an indefinite proceeding outward until in some manner unaccountable alike to all we come back to the point of origin. It has been expressed by PICKERING when he says that if we go far enough east we shall arrive at the west; far enough north we shall come to the south; far enough into the zenith we shall come to the nadir. But this conception is based upon a notion of space which is the exclusive result of mathematical determinations and subject to all the restrictions of mathetic rigorousness. It requires that we shall allow space to be curved. This we decline to do for the reason that it is both unnecessary and contrary to the most fundamental affirmations of the _a priori_ faculty of the Thinker's cognitive apparatus. It would seem to be necessary only that we should extend our consciousness backward, revert it into the direction whence life came to find that which we seek. By extension of consciousness is meant the ability to function consciously upon the various superkosmic spheres or planes just as we do on the physical. Yet it should be quite as easy to devise an idealized construction which would imagify the results of this ingressive movement of the consciousness as to represent the results of a progressive outward movement star-ward. Having done so the examination of them could be conducted along lines similar to those followed in the scrutiny of objective results. What would it mean to the Thinker if he were able to identify his consciousness with the ether in all its varying degrees; what would it mean if he were able to identify his consciousness with life and with the pure mind-matter of the kosmos; and lastly, with the spiritual essence of the universe? What if his various vehicles of awareness were available for his purposes of cognition? What, indeed, if he could traverse consciously the entire gamut of realism and consciousness from man to the divine consciousness? Does it not appear reasonable that as he assumed each of these various vestures of consciousness, in succession, he would gradually and finally, come to a full understanding of reality itself? It seems so. This view is even more cogent when it is considered that the limitations, and consequent obscuration of consciousness are proportional to the number of vehicles or barriers through which the Thinker is required to act in contacting the phenomenal universe. Common sense suggests that freedom of motility is determined by the presence or absence (more particularly the latter) of bonds and barriers; that the less the number of such barriers the greater the scope of motility and consequently greater the knowledge. PLATO evidently had this in mind when he imagined the life of men spent in cave-walled prisons in which their bodies were so fixed that they were compelled to sit in one prescribed position, and therefore, be unable to see anything except the shadows of persons or objects as they passed by. He conceived that men thus conditioned would, in time, suffer the diminution of their scope of consciousness to such an extent as to reduce it to identification with the shadows on the walls. Their consciousness would be mere shadow-consciousnesses the entire data of which would be shadowgraphs. So that for them the only reality would be the shadows which they constantly saw. A similar thing really happens to man's consciousness limited to the plane of the objective world. Things which are not objective do not appear as real to him, if they do appear at all. It is not that there are no other realities than those which appear to the egopsychic consciousness or that fall within its scope; but that this form of consciousness is incompetent to judge of the nature and appearance of those realities which do not answer to the limitations under which it exists. And so, with men whose data of consciousness or whose outlook upon the world of facts, or rather life, are confined to the narrow bounds of mathematic rigor and exclusiveness, there may appear to exist no realities which may not be defined in the terms of mathematics. Similarly, to the empiricist, used to measurements of magnitudes, weights, and rates of motion, there may also appear to be no realities which are not amenable to the mold of his empirical contrivances--the balance, the chromatic and the scalpel. All of these are shadow men constricted to the metes and bounds of shadows which they observe only because they are ignorant of the realities which lie without their plane. Life has so many ways of exhibiting its remains to the intellect; and these remains have so many facets or viewpoints from which they may be studied, that nothing short of a panoramic view of all the modes of exhibition and of all the facets and angles of appearances will suffice to present a trustworthy and comprehensive view of the whole. Then, life itself is so illusive, so unseizable by the intellect that the testimony of all investigators are required to summarize its modes of appearance. And, therefore, eventual contentment shall be secured only when the mass of diverse testimonies is reduced to the lowest common divisor, and for this purpose the operations of every class of investigators must be viewed as the work of specialists upon separate phases, facets and angles of life's remains. And so it is manifestly absurd for the empiricists, by taking note of the dimension, extent, quality and character of the shadows, or one single class of angles, to hope to predicate any trustworthy judgments about either the realities which cast the shadows or underlie the angles; because whatever notion or conception they may be able to gain must of necessity be merely fragmentary and entirely inadequate. Despite this fact, however, we still have the spectacle of men who, studying the sensible universum of space-content, endeavor either to make it appear as a finality in itself, or that the world of the real must necessarily be conformable to the precise standards which they arbitrarily set up in their examination of the objective world. It can be said with assurance that we shall never be able even so much as to approach a true understanding of the unseen, real world until we shall have changed our mental attitude towards it and ceased to expect that it shall necessarily be fashioned and ordered in exactly the same way as the world of our senses, or that it shall be understood by applying the same methods of procedure as those which we use in our examination of the phenomenal, sensuous world. It is a matter of logical necessity that, as there are no senses which can respond to the real, as there are no organs which vibrate in accord with the rates of vibration of the real, there can be no reasonable hope of understanding it by means of sensuous contrivances and standards. Let the consciousness, therefore, be turned not outward, but _inward_ where is situated the temple of divine life; let there be taken away the outward sheaths which enshrine the pure intelligence of the Thinker; let him grow and expand his sphere of awareness; let there be an exploration of the abysmal deeps of mind, of life and consciousness; for buried deeply in man's own inner nature is the answer to all queries which may vex his impuissant intellectuality. CHAPTER IX METAGEOMETRICAL NEAR-TRUTHS Realism is Psychological and Vital--The Impermanence of Facts--On the Tendency of the Intellect to Fragmentate--The Intellect and Logic--The Passage of Space, the Kosmometer and Zoometer, Instruments for the Measurement of the Passage of Space and the Flow of Life--The Disposal of Life and the Power to Create--Space a Dynamic, Creative Process--Numbers and Kosmogony--The Kosmic Significance of the Circle and the Pi-Proportion--Mechanical Tendence of the Intellect and its Inaptitude for the Understanding of Life--The Criterion of Truth. Kosmic truth has many facets. The rays of light which we see darting from its surface do not always come from the core. Often they are reflections of rays whose light stops short at the superfice; and these, in turn, are reflections of deeper realities. Thus the reflected light may be traced to its source by following the lead of external reflections. It is now known that moonlight, and perhaps, in many cases, starlight, are reflections of sunlight, if not of our sun, some other in the universe. But it is only at certain times and under certain conditions that we can see the sun which is the source of the other kinds of light. The stars which owe their light to suns are so many facets of sunlight. The moon is a facet of sunlight also. Facts are facets of truth. They are so many faces of eternal truth. They represent the many ways reality exhibits itself, or rather its effects, to the consciousness. When we, therefore, become aware of facts we have not in virtue thereof become aware of the reality which produces the facts. We have come to know only something of the termini of realism while the complexities and internal ramifications which lie between realism itself and these termini altogether elude our cognition. Let us examine briefly an icosahedron, for instance. An icosahedron is a figure comprehended under twenty equal sides. These various sides are so many faces by means of which the figure presents itself to the consciousness. These faces, however, are not the real object. The figure may be examined by viewing it from any one of its sides; yet, by simply examining a single face, or any number of faces, less the total number, we arrive at no satisfactory knowledge of the magnitude or its substance. We must first become conscious of all the faces, holding them in mind as a composite picture, before we can even begin to have anything like a complete notion of the icosahedron. Then by continuing the examination we may find that the magnitude is composed of wood fiber or stone or metal, as the case may be. In this way we might carry the examination to indefinite limits and finally arrive at a very comprehensive knowledge of the icosahedron and yet be unaware altogether of the forces which have been at work in the production of the magnitude or of the reality which lies back of it. Realism is psychological and vital. In essence it is mind, spirit, life. Yet these three are one. Mind is the outward vehicle of life; spirit is the form or the interior vehicle which life assumes in order to express itself. Realism, then, is life. Is the logician dealing with reality when he collects and coördinates the various modes of interpretation by which we learn to understand the symbolism of life? Obviously not. The data of logic are simply a collection of rules for interpreting concepts. It is a compendium of indices for the Book of Life. It is no more the book itself than a table of contents is a book. But logic occupies about the same category as does an index to a volume. A book, however, is more than its conventional contents. It is the thought that is symbolized therein. The book of life, accordingly, is the sum total of life's expressions; but it is not life itself. That is the subtle, evasive something which the contents of the book of life symbolize. Nature, both in her palpable and impalpable aspects, may be said to be the book of life wherein are recorded the movements, the expressions, and the diacritics of life. The whole is a magnitude of many facets (little faces). We shall have to know all the faces before we can say that we have a comprehensive knowledge of nature. For so long as we have only a fragmentary knowledge of the whole, so long even as we have merely a superficial knowledge of any aspect of nature, just so long will our knowledge be in vain. Just as it frequently happens that, on account of the partial view of things, we are led to make incorrect judgments concerning them, so when we come to make assertions about life or nature in general, we are apt to fall into the error of rendering judgments upon insufficient data. And it is not at all likely that judgments thus arrived at can possess true validity because it may happen, and undoubtedly does always so happen under the present limitations of human knowledge, that the very elements which are ignored or neglected in forming a judgment possess enough of virtue to alter the intrinsic value of determinations based upon otherwise insufficient data. Hence it develops not infrequently that our judgments repeatedly have to be changed in proportion as our data are made more and more comprehensive. Men searching eagerly for the truth sometimes allow themselves to be carried away by the enthusiasm of the moment which arises upon the discovery of a new facet of truth; but if all searchers were to bear in mind the fact that reality presents itself to consciousness in myriad ways and that there are innumerable facets all leading eventually back to the source of all they not so easily would be induced to jump to the conclusion that they had covered the entire ground. For when we have discovered a million facts, or many millions of them, about nature we may say that we have only merely begun and that what we have found is not to be compared with the totality even of the directly observable phases of nature. Logic, therefore, deals with the symbolism existing between and among facets of truth, and not directly with truth itself, although the conclusions reached by the logicians may be true enough from an intrinsic standpoint. Logic is not truth, however; it is merely the consistence of relations and inter-relations between facts and among groups of facts. Truth is not established by logic; it stands in no need of the light of logic for its revelation; indeed, more apt than not is logic to obscure truth. Truth is its own proof; it is self-evident. Logic is a mere modeler of facts; it is static, immobile, fixed. All truly logical processes need a starting point, a foundation, a premise, a base. Truth, being eternal, mobile, dynamic, vital, needs no starting point; needs no foundation because it is itself fundamental; it requires no premise because no premise is comprehensive enough to encompass it. There is only one way of arriving at truth and that is not to arrive at all--just to recognize it without procedure. The fact that facts are, and the fact of their relations and inter-relations, their sequence and implications, can be arrived at only by logical processes. Life, in its passage through the universum of spatiality, carefully diacriticizes between the realm of facts and the domain of truth, marking each off from the other by unmistakable signs and barriers. Truth is perceived as an axiomatic, self-evident principle and no amount of logic could prove or establish its verity. Facts are intellectual creatures; truth is intuitional, vital. The intellect conceives the consistence of facts while the intuition recognizes truth--is truth, and therefore, follows in the wake of life as consciousness. There is no permanence in facts and the intellectual recognition of their consistence. The discovery of a single new fact may destroy the consistence of a whole mass of previously correlated facts. Thus is revealed the miracle power of logic over facts. It can take a mass of facts, related or unrelated, mold them into hypotheses, endow them with a sort of interior consistency, and make these hypotheses take the posture of truth. Hence logic is often an effective mask which the intellect commonly imposes upon its material; but it does so instinctively and can no more escape the rigorous compulsion of this instinctive functioning than water can escape its liquidity. Wherefore, we conclude that true permanence abides alone in truth because truth is duration itself. For the foundations of the whole structure of facts in religion, science, art and philosophy which man has toilfully built up in the last million years might easily be destroyed or overturned by the discovery of some great fact or by appreciating the true value of truth. Let us suppose it should suddenly be realized by men that they are really and truly gods capable of creating and possessing all the other virtues, powers and capabilities which we are accustomed to impute to supreme divinity; and suppose that the fact of their omnipotence and divine omniscience always had been obvious but that men were so engaged upon details and the non-essentials of life and matter that they had not noticed nor realized it before, would not this realization make a vast difference in the character of our knowledge and the attitude which we would necessarily assume thereafter towards matter, life and the problems which they present? Would not it completely revolutionize our arts, our sciences and our philosophies? How much, then, of the facts of these would be left when the light of omniscience had been turned on--when truth itself could be perceived and interiorly realized? Not much, to be sure. We should undoubtedly have to dispense with the entirety of our fact-mass, for it should then be entirely useless and meaningless in the light of the resplendent omniscience of truth. As at present constituted consciousness is focused upon the material plane for the purposes of superficial observances. But if the focus of consciousness should be changed so as to reveal conditions upon what must be a higher and more interior level, the aspect of things would be entirely changed and the whole of our theory of knowledge would have to be reconstituted. It is conceivable, yea obvious, that the stern reality of being is far removed into the Great Interior of that which is; and there is a point in the path to the interiority of being where there is no illusion, no appearance, indeed, nothing but the cold, illuminating body of reality itself. It must appear also that along the journey interior-ward there are many apparent levels or planes, each of which requires a new focus. It is unreasonable, then, to suppose that the conclusions arrived at as a result of purely logical processes, confined to the lowest levels of reality, are pertinent and valid for the entirety of realism which is neither of mathematical nor logical import. For instance, if we take the purely axiomatic assertion: _x_ equals _x_, the intellect is at once certain that this is so, and cannot be otherwise, and yet a proposition of this kind is purely conceptual, conventional and arbitrary. _x_ may also equal 1, 2, 3, 4, or any other quantity. Then, if each _x_ in the above equation be replaced, by say, a horse, there immediately arises a difficulty. For it is not possible to find two horses which are in all respects mutually equal. So that as soon as we pass from the conceptual into the actual, whether on the side of objective reality or that of absolute reality, the validity of the axiom is immediately exposed to serious questioning. The truth of the matter is that on both sides of the conceptual it is always found that there is a variance from the standards set up by the conceptual, this variance being more marked on the side nearest to absolute reality than on the side of objectivism. Objectively, the conformity of the sensible with the conceptual is of such approximation as to lend trustworthy utility to the conceptual in its application to the sensuous. Thus by simply eliminating the vital factors from our equations we are enabled to proceed in a reasonably safe manner with our judgments. Really, however, no such approximate congruence can be found; for on the side of reality we are dealing with an indivisible something--something that is eternally and absolutely unitary in its constitution while when we transfer the scene of our observations to the objective world we discover a contrary situation. Here we are everywhere beset by diversities, multiplicities and dissimilarities. This is so because the intellect naturally tends toward the objective where it finds a most comfortable atmosphere for its operations. The conceptual is related to the objective as a train of cars is related to the railway. That is to say, the constitution of the intellect is such that it finds its most facile expression in the objective world and is about as comfortable in the domain of realism as the same train of cars would be on the ocean. The intellectuality is designed to deal with facets of truth; it is made to manipulate segments, parts, fractions, and cannot chart its way through a continuum such as reality. Being constitutionally a conventionality of the Thinker's own contrivance, and arising out of the subtle adaptation of his vehicles to the environment afforded by the sensuous world, it can only find congruence in that conventionality which is the instrumentality of a higher intellectuality expressed in a diversity of forms, into which reality divides itself for manifestation. The human intellect is, therefore, the bridge over which is made the passage from the individual consciousness to the All-Consciousness; simultaneously, the medium whereby the physics of the brain are converted into the psychics of unconsciousness. It may be likened to a pair of specially constructed tongs which are so formed as to fit exactly the objects which a higher intellectuality has made. It is without the province of the intellect to take note of what intervenes between physics and psychics; it is always oblivious of interstices while taking cognizance of objects or things. In this respect, the intellect is much like a steerage passenger on board an ocean liner who sees only his port of departure and port of arrival, knowing nothing in the meantime of what happens during the voyage, nothing of what the other passengers on the upper decks may experience and taking no part in any of the passing show until he lands. So that the passage of the intellect from fact to fact is an altogether uninteresting voyage; it may as well be made unconsciously, and to all intents and purposes, is so made. Accordingly, the advocates of _n_-dimensionality find it quite impossible to predicate anything whatsoever of the passage, say, from tridimensionality to quartodimensionality. They find themselves at ease in tridimensionality and have even contrived to find pleasant environs in the four-space having made therein such idealized constructions as will afford ample hospitality to the intellect. But the questions as to how the passage from the three-space to the four-space is to be made and how the intellect shall demean itself during the passage have been completely ignored and, therefore, left unanswered. What, then, shall be said of an explorer who says he has found a new land and yet can give no intimation as to how one may proceed to arrive at the new land, what changes are to be made en route, nor the slightest suggestion as to the direction one should take in setting out for it? It is not likely that the report of such an explorer, in practical life, would be taken seriously; and yet, there are those who, relying utterly upon similar reports made by certain enthusiastic analysts, dare to place credence in their asseverations. Not only have they given wide credence to these reports, but have, indeed, sought to rehabilitate their own territory in accordance with the strange descriptions given by unhappy analytical explorers. Now the question of greatest concern, granting for the nonce that there is such a domain as hyperspace, is the _passage_. How shall we make the passage? Or, is the passage possible? In vain do we interrogate the analyst; for he does not know, nor does he confess to know. Evidently it is impossible for him to know by means of the intellect alone; for the intellect not being fitted to take cognizance of the "passage," but only the starts and stops, has no aptitude for such questions. Hence, what seems to be the most important phase of the entire question will have to remain utterly inscrutable until the intellect nourished by the intuition shall be aroused from its lethargy and brought to a certain high point of illumination where it, too, may take note of the passage. Space is the path which life makes in its downward sweep through all the stages of pyknosis or kosmic condensation by virtue of which it accomplishes the engenderment of materiality as also the path marked out by it in its upward swing whereby it accomplishes the spiritualization of matter. It is the kosmic order which life establishes by means of its outgoings and incomings. When we look out into space we perceive that which is a dynamic appearance of life itself, and not a pure form. Nothing that is a pure form can exist in nature and in as much as space is not only indissoluble from nature but partakes of its very essence it cannot be said to be a pure form. The intellect, however, prone to follow the grooves laid out by pure logic, never fails to seek to make everything that it contacts conform to these logical necessities. But, if the analyst were to make careful discrimination as to the respective categories--that into which life falls and that in which the intellect is forced by its nature to proceed--he not so easily would be led into the fault of attempting to shape realities upon models which being strictly conventional were not meant for such uses. But neither the logician nor the mathematician can be condemned for such generosity if such condemnation were justifiable. For they everywhere and at all times insist upon _realizing_ abstractions and _abstractionizing_ realities, and they do this with an _insouciance_ that is at times surprising. Yet it is in this very vagary that is discovered the true nature of the intellect. There is a sort of dual tendence observed in the method of the intellect's operation. A polarity is maintained throughout: the abstractive and the concretional. It vacillates continually between the abstract and the concrete and no sooner has it found a concrete than it begins to set up an abstract for it; and _vice versa_--as soon as it is has constructed an abstract it immediately seeks either its concrete or sets out to hew some other concrete into such shape as will fit it. And between these two extremes numerous excuses are found for exercising this peculiar characteristic, and that too, without regard to consequences. It would seem that the intellect, in thus functioning, was really engaged at a sort of sensuous play out of which it derived an intense and not altogether unselfish pleasure. Of course, it must be granted that diversity has its specific and withal necessary uses in that it affords a field for the operation of human intellectuality and represents the adaptation of the kosmic intellect to the human for the purposes of evolution. This adaptation while necessary for the intellectual development is, however, not an end in itself. It is merely a means to a higher purpose. In fact, if we regard materiality as a deposit of life, carried by it as a kind of impedimentum, and consciousness, which _is_ life, as being identical with the intellectuality which makes these adaptations, there should be no grounds for the statement that the one is adaptable to the other at all. And as this is really the view which we assume it would perhaps be more strict to regard the adaptation as subsisting between the human intellect and materiality both of which having been constructed by kosmic intellectuality. Pursuant to the diversity of uses to which materiality lends itself there arises in the intellect a supreme tendency to segment, to break up into separate parts, to multiply and diversify. It is not content unless it is at this favorite and natural pastime. It delights in taking a whole and dividing it into innumerable parts. This it will do again and again; because all its muscles, sinews and nerves are molded in that mold and can no more cease in their tendency to fragmentation than can the muscles of a dancing mouse cease in their circular twirling of the mouse's body. Yet, in this it is but creating a well-nigh endless task for itself--which task must be performed to the uttermost. But in its performance, that is, in the intellect's complete understanding of the diversity of parts, in the knowledge of their relations and inter-relations and in their synthesis, it may arrive at that one ineluctable something which is called _unity_. And so doing, become ultimately free. In view of the foregoing, it is not surprising that the intellect should have, finally, fallen upon the notion of _n_-dimensionality. It has come to that as naturally as it has performed its most common task. Left alone and unhampered in its movements, it has simply followed the lead of the Great Highway through the domain of materiality. And now it has arrived at a stage where it thinks it has succeeded in fractionalizing space. Time has long ago yielded to fragmentation, been divided into minute parts and each part carefully measured. Space, not having a visible indicator like time to denote its passage or parts, suffered a long and tedious delay before it could boast of a measurer. As the sun-dial measured time in the past and became the forerunner of the modern clock so _n_-dimensionality measures space for the mathematician. What more practical instrument for this purpose may yet be devised is not ours to prophesy; yet it is not to be despaired of that some one shall find a suitable means for this purpose. Seriously, however, it is not without possibility that should some subtle mind devise an instrument for marking the passage of space as we have for denoting the passage of time a great stride forward would be accomplished in the evolution of the human intellect. For the general outcome of the intellect's attention being turned to the _passage of space_ would undoubtedly be to recognize not only its dynamism but its _becoming-ness_, as a process of kosmogenesis. Because such an instrument would have to be so constructed as to take note of the movement of life, and for this reason, it would have to be extremely sensitive necessarily and keyed to the subtleties of vitality and not to materiality. Mathematics shall have failed utterly in the utilitarian aspects of this phase of its latest diversion if it do not justify its claims by crowning its work in the field of hyperspace with a "Kosmometer," an instrument devised for the measurement of the movement of space or a "Zoometer," an instrument devised for the measurement of the passage of life. We should like to encourage inventive minds to turn their attention life-ward and space-ward with the end in view of constructing such an instrument. When once we have learned accurately to measure life we shall then be able to dispose of it--to _create_. It is not doubted that if ever humanity is to arrive at that point in its evolution where it can understand life; if ever it is to attain unto the supreme mastery both of vitality and materiality and to come to the ultimate attainment of divine consciousness (all of which we confidently believe to be in store for humanity) it must be accomplished after this manner: first, by syncretizing materiality with vitality, and then, by intuitionally recognizing the truth of the implications of the syncretism. The history of consciousness in the human family is identical with the history of man's conquest over matter and physical forces. And this is clearly indicated in the incidentals contingent upon the toilsome rise of the _genus homo_ from the earliest caveman whose status denoted a comparatively negligible transcendence of material forces, to the present-day man who has gained a markedly notable conquest over these forces. Always consciousness seeks the means of adequately expressing itself in the sensible world. And to this end it engenders faculties, organs and processes in the bodily mechanism, and, in matter, devises instruments of application whereupon and wherewith it may test, analyze, combine and recombine the forces and materials it finds. The unlimited range of expressions lying open to the consciousness makes it necessary continually to devise higher and higher grades of appliances to meet its needs as it expands. It will not be gainsaid that the telescope has served actually to lay bare to the consciousness an immeasurable realm of knowledge nor that the microscope, turning its attention in an opposite direction, marvelously has enlarged and enrichened our knowledge of the world about us. And similar declaration may be made anent almost every invention, discovery and conquest which man has made over natural phenomena. Thus, by externally applying mechanical implements to the subject of his consciousness, man has extended actually his consciousness, his sphere of knowledge; has greatly enhanced its quality, and, in the process, has urged the intellect to endeavors that have wrought its present unequaled mastery of things. Nor have the spiritual aspects of our advance along these lines been the least notable. For these have enjoyed the essence of all that has been gained in the process and have, therefore, kept pace with the onward movement of the intellectual consciousness. But heretofore no advance has been made as a result of methodic or reflexive determinations. That is, men did not set out from the beginning, equipped with foreknowledge of what their efforts would bring, to develop the present quality of human consciousness. They simply worked on, their attention being absorbed by the problems that lay nearest and demanded earliest consideration. So the advance has come as a resultant of man's close application to his ever-present needs--shelter, clothing, food, protection and other preservative measures--and it has come naturally and inevitably and without prepense. Nevertheless, if man, knowing what to expect from the syncretization of matter and mind, after this fashion, should set out deliberately to accelerate the intensification, expansion and growth of his consciousness, there is no doubt but that the consequence would be most far-reaching and satisfactory. But the path that leads to this grand consummation does not lie in the direction of diversity; it lies in the opposite direction. In vain, then, does the intellect fractionalize in the hope that by doing so it shall come to the solid substructure of life; in vain does the analyst segment space into any number of parts or orders; in vain does he ask how many and how much; for by answering none of these queries will he find the satisfaction which he vaguely seeks. If it be true that it is not by analysis but by synthesis that the true norm of life, and therefore, of reality shall be found it is futile to entertain serious hope of finding it in any other way. As a perisophism or near-truth, then, _n_-dimensionality takes foremost rank. And this is so for the reason that when we proceed in the direction of multiple dimensions, that is, one dimension piled upon another dimension or inserted between two others we are traveling in a direction which, the more we multiply our dimensions, leads us farther and farther away from the truth. This is a simple truism. If we take, for instance, a wooden ball and cut it up into four quarters, and divide each one of these quarters into eighths, into sixteenths, thirty-seconds, sixty-fourths, etc., indefinitely, we shall have a multiplicity of parts, each one unlike the original ball. But from no examination of the multipartite segments can we derive anything like an adequate conception of the original ball. Something, of course, can be learned, but not enough to enable the rendering of a correct judgment as to the nature, size, shape and general appearance of the ball. But this is precisely what happens when the analyst divides space into many dimensions. He cuts it up into _n_-dimensional parts and the more minutely he divides it into parts the more remote will each part be in its similarity to the original shape and form of space, and the farther away from the true conception of the nature of space he is led thereby. Now, _n_-dimensionality or that phase of metageometry which regards space as being divisible into any number of dimensions or systems of coördinates is a direct and inevitable product of that tendency of the intellect to individuate and to singularize phenomena. Biologically speaking, it is a peculiarity which harks back to the time when life was manifested through the cell-colony and when the individual cells began, because of increasing consciousness, to detach themselves from the colony and set out for themselves, and thus each intellect recapitulates in its _modus vivendi_ the salient tendencies of phylogenesis. Let it suffice, then, to point out that this universal tendency to segment and fragmentate which rigorously characterizes intellectual operations upon every phenomenon with which it deals is a culmination of the primordial tendency among cells to divide, inasmuch as this phase of cell life must be the work of the kosmic intellect. The natural inference is that from the extreme of individualization there shall be a gradual turning, whether of the intellect _per se_ or of the intellect joined to the intuition does not matter, towards that other extreme of _communalization_. And from this latter shall grow up, as one of the inevitable and ineluctable tendencies of the Thinker's consciousness a torrentious movement in human society towards coöperation, brotherhood, mutuality and union in everything. So that whereas in the past and at the present time the intellect has been developing under the dominant note of individuality it will then be coming gradually under another dominant note--_communality_. The result of this development will be the unification of all things, and instead of many dimensions of space, many measures of time, and a general diversification of all phenomena, we shall come to the only true notion of these things and realize pragmatically the true value and extent of unity in the universe. It is admitted that the intellect, in treating objects singly and dealing only with the starts and stops of a movement, is withal loyal to the kosmic order, design and purpose which have priorly characterized manifested phenomena by segmentation. And in this loyalty it has been following merely a natural lead which, while admitting of the widest development and experience, nevertheless at the same time beneficently obscures the underlying reality in order that in its adaptation to the sensuous world the intellect might have the greatest freedom for the development suited to the given stage of its evolution. But in thus admitting the natural congruence between the intellectuality and the phenomenal or sensuous we do not thereby unite with those who already believe that this kosmic agreement is the _ne plus ultra_ of psychogenesis. On the other hand, it is maintained that this is merely a phase of psychogenesis which shall be outgrown in just the same measure as other phases have been outgrown. And notwithstanding the fact that judgments of the intellect with respect to inter-factual relations or the ens of facts themselves are as valid as its judicial determination of self-consciousness, no more and no less, we are, by the very rigor and exclusiveness of this logical necessity and inherent limitation, led to view the intellect's interpretation of phenomena as partial and fragmentary; for the reason that the necessitous confinement of its understanding and interpretative powers to fact-relations quite effectively inhibits the use of these powers for the contemplation of the deeper causative agencies which have operated to produce the phenomena. But it is apparent that just as the transmuted results of other phases of psychogenesis are now being utilized as a basis for the efficient operation of the intellect in the sensuous world, thereby enabling the attainment of a very high mastery over matter, so will the functional dynamism acquired by it in the pursuit and comprehension of diversity serve well when, in later days, it has acquired the power to deal directly with reality, to _create_ and dispose of life just as the kosmic intellect has and is now using it in the execution of the infinite process of _becoming_ through which creation is proceeding. It would seem that the necessary prerequisite to the development of any higher functional capability is that the intellect should be capable of disposing of innumerable details, indeed the totality of kosmic detail, before it can come wholly into the power and capacity to understand and manipulate life. Furthermore, it appears that the acquirement of this power quite necessarily has been delayed awaiting that time when, dominated by the intuition, the intellect shall have attained the requisite managerial ability for marshaling an exceedingly large number of details. The supreme tendency of life is expression. And this expression, singularly enough, reaches its most perfect phenomenalization by means of that movement which results in the multiplication of forms. Despite the fact, therefore, that the comprehension of reality involves a gradual turning away from the exclusive occupation of organizing a multitude of separate and apparently unrelated facts to a monistic view which at once recognizes the unitariness and co-originality of all things, of life, mind and form, the intellect will need the training and development which come from the mastery of diversity. It is, then, not difficult to perceive the wise utilitarianism of the present schematism of things as shown in the universal tendency in the intellect to devote itself exclusively to parts or segments of truth. Whenever an individual intellectuality, on account of prolonged thought and the consequent inurement of the mind to higher and higher vibrations of the kosmic intellect, brings itself to such a high point of sensitiveness that it can receive so much as an intimation of some great truth, it begins to sense, in a more or less vague way, something of the substance and general tendence of the underlying reality of that which foreshadows its appearance. Then, confounded by the multiformal characteristics of kosmic truth because of the fact that it presents itself in such numerous ways and forms, men often are induced to attempt the reformation of all facts, or a great mass of kindred facts, in accordance with the newly-found fact or principle. They forget evidently that no fact in the universe can be at variance with any other fact and still be a fact. So that in the totality of facts every separate and distinct fact must be congruent with every other fact forming a beautiful, harmonious and symmetrical whole; but often the whole is made to suffer in the attempt at making it conform to the substance of a mere intimation. Moreover, it is conceivable that even the totality of facts may lack a rigid conformity with reality in all its parts and that having compassed the entire mass of facts one may fall short of the understanding of realism. This is practically what has happened in the mind of the metageometrician who having received an intimation as to the real nature of space as that whose center is everywhere and yet nowhere and whose nature is psychological and vital rather than mathematical and logical, misses the great outstanding facts and clings to the intimations which he experiences as to the nature of space. He, therefore, concludes that the form of space is that of a flexure or curve. There is a valid element in the notion of the curvature of space but not enough of truth wholly to validate the notion. Since the very reality of space is a matter which can be determined only by the conformance of the consciousness with it in such a manner as to render the conception of it entirely unintelligible to the intellect except in so far as it may be able to identify itself with the space-process, there is much room for the serious questioning of the mathematic conclusion upon the grounds of its fragmentariness if not entirely upon the basis of its invalidity. Wherefore it may be seen that any search for either the center or the extreme outer limits which proceeds in a manner conformable to the external indications of the intellectual order is vain, indeed. Although it is undoubtedly true that the attainment of a central or frontier position in space does not involve any lineal progression whatsoever, the same being attainable, not by progression nor by overcoming distances, but by a subtle adjustment, yea, a sort of attachment of the consciousness to the order of becoming which binds the appearance of space, wherever one may be, it is nevertheless difficult and painful for the intellect to grasp the totality of this truth at one sweep. Indeed, it is not possible for it, alone and unaided by the intuition, to grasp it at all. Hence, the mathematician who depends entirely upon the deliveries of the intellect which conform, in their passage from the conceptual to the written or spoken word, to all the rigors of mathetic requirements, fails utterly in perceiving the magnitude of this conception and all its connotations; he fails because his prejudices and the woof and warp of his intellectual habits prevent his assuming a sympathetic attitude toward it and thereby precluding at the start any calm consideration of it. And not only is this true of the mathematician but of all those whose endeavors are confined to the plane of purely sensuous and logical data. It would, therefore, appear that our entire attitude towards things spatial must be changed before we can even begin to perceive the reality which is really the object of all researches in this domain. But, on the surface, there is after all little difference between the ultimate facts involved in these two totally different conceptions. Mathematically speaking, all progression eastward would terminate at the west, and _vice versa_; and the same would be true regardless of the point from which progression might originate. Always the terminus would be the opposite of the starting point. Then, too, it might be said that if we sought the space-center we should arrive at the circumference. The difficulty with this view is that there is a very remote, though important, connection between it and the truth of the matter. But the partiality of this view, and the absence of either experience or intuition to intimate a more reasonable view, serve effectively to buttress it as a hypothesis acceptable to many. Thus it is ever more difficult to supplant a near-truth than it is to gain credence for the whole truth. On the other hand, according to the view which we maintain here, it is quite true that the seeking of the kosmic space-center will reveal the circumference; that the search for the nadir will uncover the zenith; the east effloresces as the west, and a northward journey will wind up at the south, etc., but in quite a different manner from that which the mathematician has in mind when he postulates the curvature of space. Our view involves no space curvature nor any other spatial distortion. _It deals with space as reality, as a dynamic process, a flux which, like the sea, is continually casting itself upon the shores of chaos and falling back upon itself only to be recast against the rock-bound coast of its chaotic limits._ Now, that which falls back upon itself and rolls in a recurrent movement upon its own surface is _life_ which, in its recession is the natural and kosmic limitations of itself, generates matter in all its varied expressions. Space, in its extensity, cannot transcend life; for it is the path which life makes in its _out-coming_, its manifestation. Of the chaotic fringe which circumscribes the manifested universe it is absurd to say that it is vital or psychological in any sense of these terms. For notwithstanding the fact that out of its very substance are engendered life, intellectuality, spatiality and materiality, it is nevertheless none of these in its primary essence. It is Chaos-Kosmos; because from its content the kosmos is evolved, and it still remains; it is chaos-spatiality; chaos-materiality; chaos-intellectuality; chaos-geometricity; because these are engendered by the movement of life in chaos while at the same time there remains a residuum of the chaogenetic substance which constitutes the limitations of all these subsequent processes. In this sense, the chaogenetic fringe becomes the limits of the manifested universe so that it would appear that all those major processes outlined above are finite manifestations of the eternal chaos. But none of those possibilities of motion which are found in these major movements of the kosmos can be logically said to exist in chaos. It is the embodiment of everything that is the opposite of those qualities which may be found in them, that is, in materiality, vitality, spatiality, intellectuality and geometricity. Apropos to this phase of the discussion let us examine briefly one of its most significant implications, both mathematical and kosmic, which arises out of the fact that space is an engendered product of life that is bound by the fringe of chaos which sustains and limits it. The chaotic fringe plus manifesting kosmos constitute the absolute magnitude of the kosmos. The manifestation factor is complemented by the chaos factor and together the two define the _full_ universe. Kosmogony is the universal movement of all kosmic elements or factors in diminishing the chaotic complement and reducing it to kosmic order or geometrism. It is undoubtedly impossible to determine mathematically the exact volume of either complement or the ratio of the one to the other; yet it is conceivable that the chaotic fringe is greater in extent than the ordered portion of the kosmic uni-circle or universe. It is even conceivable that the difference, upon the basis of the meaning of the Pythagorean Tetragrammaton and the view outlined in the Chapter on the "Mystery of Space," is as seven to three wherefrom the conclusion might be drawn that the universe has yet seven complete stages more or less of evolution before the close of the Great Cycle of Manifestation when the fringe of chaos shall have been totally used up in the work of creation. But for those who may experience impatience at the infinitude of the process when viewed in this light the terms may be reversed and the difference may be conceived as the ratio of three to seven wherefrom the conclusion would follow that the kosmogonic process is seven-tenths complete, as it will not vary the seeming infinitude either way it may be determined. The notion, despite its speculative character, offers an explanation of otherwise inexplicable conditions, and, on account of its profound connotations, may even be found to be productive of the highest good in its equilibrating influence upon our mode of thinking. In any event, there does appear to be a subtle relation subsisting between true numbers and kosmogony. Number is a phase of the kosmogonic movement, a measurer of the intellect and the establisher of the geometrism of space, answering tentatively to the numericity of pure being. In fact, being actually expresses number and number itself is an evolution and not a thing posited once for all as a pure, invariable form in the universe. It is, like the kosmos, in a state of becoming and there may yet appear to our cognitive powers a whole series of new numbers pure in itself and altogether conformable to the conditions reigning at the time. The symbology of the circle, in all times recognized as the true symbol of the kosmos in eternity, of eternity itself, of the archetypal, of space, duration and Ultimate Perfection, is replete with profound significations. But it should be understood that the circle is a symbol of the _perfected universe_ and not the universe in a state of evolution. It symbolizes perfection, completion and the ultimate union of the manifesting with the archetypal which results in the crowning deed of Perfection. The circle is, therefore, not a symbol of the universe as it now stands; it does not represent a snapshot view of the kosmos but the universe as a _full_. It cannot be a _full_ until it has attained the _ne plus ultra_ of completion; for a kosmic full is that state to be attained by the manifested kosmos upon the termination of all the fundamental processes now in operation. But it is this state that the circle really represents, and by virtue of which it possesses its intrinsic qualities and also in virtue of which the intellect recognizes these qualities. The properties of understanding and recognizance in the intellect are veritably fixed by the _status quo_ of the universe during every stage. That is, the focus of the intellect, like the focus of a chromatic lens, is adjusted by the fiat of the nature and eternal fitness of things to correspond exactly with every state through which the kosmos itself passes. This is one of the obvious implications of the phanerobiogenic behavior of the kosmos and is necessarily resident in the notion of the genesis of space and intellectuality as consubstantial and coördinate factors. _Wherefore the more cogent is the reason for the belief that the inherent qualities of the kosmogonic fundamentals; as, vitality, materiality, spatiality, intellectuality and geometricity, are true variants, and that their variability is proportional to the progress of these major movements toward the ultimate satisfaction of the original creative impulse._ May it not be, therefore, that the indeterminate character of the ratio of the diameter to the circumference (3.1415926...), is due to causes far more profound than the crudity of our micrometers or the mere supposed fact of the circle's peculiarity? May it not also be true that the _pi proportion_ shall become a whole number, and in its integration, keep apace with the perfecting process of the kosmos, diminishing, by retrogression to one or increasing, by progression, to ten which, after all, is essentially unity, being the perfect numeral? It is not without the utmost assurance that these queries will be categorically questioned by the orthodox, creed-loyal, strictly intellectual type that we sketch these implications, but it is felt to be an urgent duty to remind all such that the most effective barrier to realization in the field of philosophy is an intolerant attitude towards all lines of thought which suggest the impermanence of conditions as we find them in the kosmos at the present time. The fact is that our lives are so distressingly short that we have neither time nor opportunity to watch the changing moods of the kosmos nor discern the gradual reduction of mere appearance to the firm basis of reality, and accordingly, the intellect tenaciously clings to those notions which it derives from the instant-exposure which the lens of intellectual conceivability allots to it. Once the view is taken it is immediately invested with everlastingness. This everlastingness is then imputed to the kosmos in that particular pose, attitude or state. Always the intellect beholds in that passing view, snatched from the fleeting panorama of eternal duration, a picture of itself which it mistakes for the reality of the not-self. The inclination of the axis of the earth toward the plane of its orbit is approximately twenty-three and one-half degrees. No well-informed astronomer, however, doubts now the fact that this ecliptic angle is being gradually lessened; because, as a result of centuries of observation, it has been found to be decreasing at the rate of about 46.3 seconds per century. Yet no intellect is able to perceive in any given lifetime the actual decrement of this angle. It is only by careful measurements after centuries of waiting that a difference can be discovered at all. Thus it may even be so with the ratio of the diameter to the circumference of a circle, the only difference being that it has not yet been determined whether there is a _decrement_ or an _increase_ in the size of the ratio. The _pi_ proportion is, then, a register or measurer of the slow, measured approach of the manifesting kosmos to the standard of ultimate perfection. Therefore, and in view of these considerations, we may not hesitate to confirm our belief in the validity of the notion that it actually and literally expresses the key to the evolutionary status of kosmogony. The mathematical determination which limits it as an unchangeable, inelastic quantity is, consequently, only partially true and leads to the inclusion of this quantity under the category of mathematical near-truths, for such it appears to be in spite of its rigorous establishment. The formal topography into which the intellect spreads when seeking the ideal and the abstract is not a condition which is derivable from the real essence of life or matter, but, on the other hand, is a product of the intellect itself partaking of its nature rather than of the nature of reality. There is, therefore, a very important distinction to be made between all deliveries of the intellect and the realism both of the objects and conditions to which the intellectual deliveries pertain. One of the most marked peculiarities of the human intellect is the fact that it always unavoidably stamps its own nature and features upon every datum which passes through it to the consciousness. The utmost importance attaches to this phenomenon, for the reason that it points to the necessity of carefully scrutinizing intellectual deliveries and the making of allowances for those ever-present characteristics which the intellect superimposes upon its data. Perhaps the inherent colorific quality which it imposes upon our knowledge would be better understood if a similitude were indulged at this juncture. The intellect may be likened to a color-bearing instrument which, when it has once handled an object, leaves forever its own color transfused into every cell and fiber of the object so that when the same object is presented to the consciousness for purposes of cognition it bears always the same peculiar marks and colorations which the intellect, in its manipulation of it, places thereon. In this respect the intellect may also be said to be like a potter who has but one mold and that of a peculiar formation. Hence, whatever wares it presents to the consciousness will invariably be found to be molded in conformity with that particular mold. If it were possible to view reality or the essential nature of things the difficulty which now the intellect lays in the path of direct and uncolored cognition would be obviated; for then there no longer would be any necessity of viewing things as they are colored or molded by the intellect. The intuition, being a process of pure consciousness, will, when it has arisen to a position where it may dominate the intellect as the intellect now dominates it, so modify this tendency which we see so ineradicably bound up in the very nature of the intellect that the apparently insurmountable difficulties which it has interposed between mere perception and a direct cognitive operation will be quite completely overcome. Thus, in the above, is discovered another obstacle which posits itself between the notion of space as reality and the intellectual determination of it which the mathematician examines and to which his consciousness is necessarily limited. Furthermore, it may be perceived also how easily the mind may be deluded into thinking that the intellectual notion which it entertains of space is necessarily correct, when obversely, it is simply examining a concept which has been remade by the intellect into a form which is not at all unlike its own peculiar nature, and therefore, as much short of reality as the intellect itself is. Similarly, if the mathematical mind succeed in catching a glimpse of the reality of space in the form of an intimation, which, in itself though fragmentary, is nevertheless true, its consciousness is finally deprived of the true validity thereof simply because of the behavior of the intellect in its manipulation of it. The importance of these intellectual difficulties cannot be over-estimated for they furnish the grounds for the ineptitude of intellectual determinations made in a sphere of motility to which the intellect is a stranger. And this fact will appear more evident when it is perceived that quite the entire content of human knowledge has been thoroughly vitiated by them. So that only in those very rare moments which (in a highly sensitive mentality) enable the intuition to gain a momentary ascendancy over the intellect is it possible for the Thinker to catch hold of realism itself, and project the truth of what he sees into the lower, intellectual consciousness. But so small is that portion of our knowledge which owes its origin to the intuition that when compared with the totality of that which we seem to understand it is well-nigh negligible. And then, when it is considered that at present there is no way of conceptualizing adequately the intuitograph so as to make it propagable the insignificance of this form of knowledge is even more notable. It can now be seen in how large a measure the notion of the curvature of space is merely an intellectual translation of a true intuition into the terms of the intellect which, in the very nature of the case, can only approximate the truth because of its colorific habits. A similar declaration may be made of that other datum of metageometrical knowledge which postulates the ultimate convergence of parallel lines. In fact, what has been said as to the perisophical nature of the notion of space-curvature will apply with equal force to the idea of parallel convergence since the latter is a derivative of the former. But there is yet another consideration, apart from the colorific influence of the intellect, which, although it partakes of the nature of this quality, is nevertheless a near-truth of quite a different order. This may be better understood by referring to the _graph_ showing the inverse ratio of objective space to the consciousness.[28] Let us suppose that the _graph_ may also represent the Thinker's outlook into the world of spatiality. It then appears that, because of that movement of consciousness in its pursuit of life which, as it expands, makes the objective world to appear to be diminished in proportion to the extent of its expansion, it is quite natural, under such circumstances, that parallel lines drawn anywhere in the limits of the objective world should seem to come to a point in the ultimate extension of themselves. While this _graph_ is not meant to depict such a view, it may be found nevertheless, to be a true delineation of the topography of that state of mind into which the metageometrician brings himself when he visualizes space as _curved_; for there is no doubt but that a state of intellectual ecstasy, such as that in which the mind of the metageometrician must be functioning in order to perceive space in that form, is quite different from the normal and, therefore, in need of a different topographical survey. But, if we grant that in the creational aspects of space there is conceivable an ever-present tendency to convolution, or a rolling back upon itself, it is imaginable that parallel lines inscribed either upon its surface or in its texture need not necessarily meet but maintain their parallelism regardless of the complexity of the convolutions. The convergence of parallel lines is much like a tangent in the outgrowth of the idea from the notion of space-curvature. The more a tangential line is extended the farther away from the circumference it becomes and consequently less in agreement therewith. The more subsidiary propositions or corollaries are multiplied the more remote from the truth the determinations become and especially is this true of the hypothesis of space curvature. [28] Figure 20. In the notion of the manifoldness of space, by virtue of which it is conceived as existing in a series of superimposable and generable manifolds of varying degrees of complexity, are discernible traces of that intuitional intimation which underlies the assertion that because of the necessary phenomenalization of reality for the purpose of manifestation to the intellect it appears to exist in a series of separate degrees, each one more refined and subtle than the preceding one and requiring a more highly developed species of consciousness for its comprehension. In other words, that intuitional glimpse of the essential character of reality, as viewed by the human consciousness, which impinged upon the minds of RIEMANN and BELTRAMI leading them to postulate as a corollary proposition to space-curvature, its manifoldness, is nothing more nor less than the intuition that the universum of spatiality cannot otherwise present itself to the intellect, owing to its peculiar adaptation to the sensuous, except by a series of continuous degrees which are perceptible only in proportion as the understanding is magnified to conform with it. After all, however, it is not improbable that the very objectivism of the universe in manifestation subsists in just the manner in which this intuitive glimpse implies and that the wisdom and utilitariness of the kosmogonic process which engendered spatiality are clearly demonstrated in that arrangement of the contents of the kosmos which presents the grossest elements of phenomena first to the intellect in its most impotent state while reserving the less crass for that time when the Thinker shall have evolved a cognitive organ adaptable to its presentations. Those metageometricians who cling to the idea of the manifoldness of space, based as we have shown upon the pseudo-interpretation of a rather vague hint arising out of an unquestionably true intuition, have allowed themselves to fall into the unconscious error of magnifying the importance of the mere insinuation as to the space-nature to such an extent as wholly to obscure in their own minds and in the minds of those who think after them whatever of the true vision that may have been grasped by them. Furthermore, it is indubitably true that that same peculiarity of arrangement by which impalpable and invisible forces really subtend gross matter producing that subtle schematism in virtue of which the visible is subjoined to the invisible, the sensuous to the non-sensuous, spirit to matter, etc., also characterizes the appearance of spatiality to the human understanding. While there is a superficial semblance of separate and discrete manifolds into which space may be divided there are, in reality, no such sharp lines of demarkation between the subtle and the gross, between the visible and the invisible or between spirit and matter, each of these being capable of reduction, by insensible degrees, into the other regardless as to whether the reductional process originates on the side of the most refined or on that of the grossest. Accordingly, there are no reasonable grounds upon which the notion of a space-manifold may be justified except as a metageometrical near-truth. In addition to the foregoing, there are yet other very fundamental considerations which would seem to debar the totality of analytical conclusions as to the nature of space from any claim to ultimate reliability and trustworthiness. These are _first_: the fact that analyses are absolutely incapable of dealing with life; that being the direct product of a sort of mechanical consistency which marks the intellectual operations it has adaptability only for dealing with fragments or disconnected parts, and that without any reference whatsoever to the current of life or the flow of reality which has produced the parts. This fact is clearly shown in that attitude of the understanding which inevitably leads it to the declaration that a line is an infinite series of points, a plane an infinite series of lines, and a cube, an infinite series of planes, and so on, indefinitely. To do this, to look upon all phenomena as a series of parts similar to each other and piled, one upon the other, or juxtaposed in the manner which they are discovered in the sensible world, is the natural tendency of the intellect and this tendency finds its most facile expression in analytics. Inadaptability of this sort is especially observable in all problems of arithmetical analysis in which the vital element is a factor. When these analyses are carried to their logical conclusion, as has been shown in the chapter on "The Fourth Dimension," invariably they end in an evident absurdity. But it is at their very conclusion where the life-element is encountered, where reality is approached, that they break down. The failure of analysis, then, to encompass life, to fit into its requirements and to satisfy its natural outcome seems clearly to establish the basis of the perisophical nature of the entirety of analytical claims, especially that species of analysis which seeks the remoter fields of the conceptual for its determinations. _Second_: the close connection which has been seen to subsist between space and life as joint products of the same movement makes it obvious that the same ultimate rule of interpretation must be applied to both in order to insure correct and dependable judgments regarding them. How different would be the intellectual attitude towards space if it were considered in the same light as vitality, provided one really understood anything about vitality! Moreover, as it appears certain that the path of the intellect does not run in the same direction as the path which life makes, but in an inverse direction, it is clear that the judgments of the former, as to the action and essence of the latter, must necessarily be ultimately unreliable. It can readily be seen, however, that should the intellect be focused so as to follow the path of life, to attach itself to the very stream of life, it would have necessarily to neglect materiality. And such an adjustment would, of course, obviate the need of a material life at all for humanity. In fact, a physical life with an intellect would be impossible under such conditions. It is well to recognize the suitability of the present schematism and not to become unwisely restive because of it; but it is also fitting that we should discriminate between that which is possible for the intellect chained to materiality and that which is impossible for it, in such a state, when foraying in a territory foreign to its nature, and beyond its powers to master. The predominating tendency in the intellect to account for the universe of life, mind and matter upon a strictly mechanical basis is undoubtedly due to the constitution of the intellect which does not admit of its direct consideration of the vital essence of things. We are bound ineluctably to the surface of things. All our knowledge is therefore superficial. We are even bound to the surface of ideas, and cannot penetrate to the interior of these realities. Our art is the reproductions of superfices; our philosophies are the husks of eternalities; our religions, the habiliments of relations, and while it cannot be doubted that this arrangement is pre-eminently the best possible one for the present stage of man's evolution, it is nevertheless worth while to note that it is this very restricted activity of the intellect which shuts out from man's consciousness those very elements about which he is most concerned when he goes into the field of philosophy in search of a solution to his unanswerable queries. But some progress most surely is made when the mind is enabled to see its plight and recognize what are the difficulties and limitations that lie in its path of ultimate attainment. It is believed that the mechanistic, or true, character of the intellect reached its zenith in the mind of LAGRANGE when he succeeded in reducing the entirety of physics to certain mechanical laws and formulæ which he embodied in his "_Mecanique Analytique_" This work is undoubtedly the capstone of intellectual endeavors and stands as a monument which marks the culmination of the present stage of intellectual development. In thus placing the _Mecanique_ at the apex of intellectual endeavors it is not thereby meant to be implied that the intellect shall not make more progress nor that other formulæ, equally as marvelous as those which LAGRANGE discovered, may be devised, nor that other laws, heretofore undreamed of, may be found; but what is maintained is the fact that while there will be growth and development these will run along other channels, perhaps in the realm of the intuitable, and not any longer, especially so notably as now, in an opposite direction against the current of life and reality; and further, that there will be a gradual turning away from mechanics to biogenetics, from diversity to unity, from the purely intellectual to the intuitional, and withal a final getting rid of the bonds of illusion, of that thralldom of mechanics, whereupon will slowly arise the obsolescence of all those disparities which may now be recognized in our knowledge and in the applications of the intellect to the data of the objective world. Because the intellect is unsuited to deal with reality, and because of its peculiar adaptation for diversity, for multiplicity, due to its mechanistic _modus vivendi_, there has grown up a voluminous catalogue of systems of philosophy. These embody such a multitudinous array of beliefs, ideas, conceptions, theories and conjectures and constitute a movement in human thought which oscillates between the empiricist on the one hand and the transcendentalist on the other; between the idealist and the realist, leaning sometimes towards the Platonic, the Cartesian and the Kantian and at other times towards SPINOZA, ARISTOTLE, SPENCER and SOCRATES, always terminating by multiplying the number of diverse beliefs rather than unifying them that the conclusion is unavoidable that so marked a lack of unanimity is indicative of a profound mental prestriction. It was, therefore, inevitable that mathematics should fall under the same spell and brook no let nor hindrance until it had succeeded in devising several diverse systems of geometry which it has done for the mere joy of doing something, of following its instinctive aptitudes. There is no other basis for the heterogeneity of our philosophies, our mathematics, indeed our beliefs than this mechanical, and hence, radically illusionary character of the intellect in consequence of which we have had to be satisfied with mere glimpses, hints, intimations and faint glimmerings of reality, of life, and of those kosmic movements which, if we had the ability to trace them from their source outward, would lead us unerringly to a truer and deeper knowledge of those things that under the present schematism must remain for us a closed book. The criterion of truth for us, constituted as we are and wedged in between the stream of life and its shore of materiality, must be that which relates our knowledge both to the stream and to the shore. It must be so that all predicates which purport to approach it shall exhibit a dual reference--one that relates to materiality and another that relates to vitality, and yet a third that shall combine these two relations into one. All assertions, therefore, which pertain exclusively to either of these elements--to materiality or to life--are necessarily partial, fragmentary and perisophical in nature. Mathematics, because it relates to matter and the mechanical forces set up by matter acting against matter cannot be said to agree with such a criterion; art, because it relates to snapshots or static views of matter is even more remote in its agreement; philosophy, as it has been known in the past and is known to-day, because it seeks to deal with a vitality fashioned after the image of materiality has failed when posited alongside of this criterion; and thus, the intellectual toil of millions of years has been in vain in so far as it has not succeeded even in raising a corner of the cover which hides reality from our view. A near-truth is any variation from this standard, this norm or criterion. It may be either logical, cognitive, scientific or even metaphysical. To define: a logical truth is a predicate based upon and involving the coherency and consistency of thoughts themselves; a cognitive truth is the conformity of knowledge with so much of reality as is known; scientific truth is the conformity of thoughts to things and conditions. All of these are obviously near-truths. Then, too, a near-truth may be defined as an assertion based upon the criterion of truth but falling within the category of cognitive truths owing to insufficiency of data or vision. Such indeed are those metageometrical predicates--_n_-dimensionality, space-flexure, space-manifoldness and all other assertions based upon these in general and specifically. Any recognition of truth must clearly embrace both the vital and material aspects of its subject in order to be adequately inclusive, that is, it should include the causative, the sustentative, the relational and the developmental factors. These four factors are considered necessary and sufficient to determine the conformity of any view to the criterion of truth for when we are cognizant of the cause of a subject, understand the sustentative factors which keep it in existence, are conversant with its relations to other subjects and can follow its developmental variations until we come to its final status, why then, our knowledge is both sufficient and ultimate so far as that subject is concerned. Is it asking too much of mathematics or of philosophy or any system of thought that it conform to these standards or to this criterion before we shall accept it as final? Or shall we be satisfied with less than this? Let us hope not. In the foregoing presentation stress has been placed upon the fragmentary, and therefore, illusionary character of the intellect in order to arrive at an understanding of the difficulties under which real knowledge has to be acquired and to indicate the inanity of all attempts to resolve the riddle of space by means of mathematics though regarded as the most typical exemplification of the mechanistic nature of the intellect. And further, to show that, on account of the radical incongruity which estranges life, the producer of spatiality, from the intellect which returns again to scrutinize the passage of life in its outward expressions, no hope of ever gaining the true viewpoint by means of the intellect need be entertained. But in doing so, it is deemed fitting that a note of warning should be sounded against any abortive attempts that may be made to obscure or distort the results of such a close discrimination lest the true import of the examination be lost for, if we emphasize the vanity of the intellect in the pursuit of that which it is by nature unsuited to attain we also equally stress the wise utilitarianism which limits it to the performance of the tasks assigned while at the same time reserving for the function of more highly evolved powers, and indeed, for the intuition itself, the solution of the riddle of spatiality. And if we declare the futility of the mathematical method in all endeavors aimed at unveiling the mysteries of life and mind, and of that movement which has its roots set in eternal duration from which it proceeds in an endless continuity of purpose and promise, we do also recognize that in the science of mathematics the intellect shall, as in no other method of cognition, most fully fulfill the kosmic intent of its existence; and moreover, in the pursuit thereof it shall push the frontiers of its possibilities outward until it can be said almost to be able to make disposition of life itself--at least to that point where, when the intuition shall have come into its own, the passage from the mechanics of matter to the dynamics of life, shall be comparatively easy and natural. CHAPTER X THE MEDIA OF NEW PERCEPTIVE FACULTIES The Spiritualization of Matter the End of Evolution--Sequence and Design in the Evolution of Human Faculties--The Upspringing Intuition--Evidences of Supernormal Powers of Perception and the Possibility of Attainment--The Influence and Place of the Pituitary Body and the Pineal Gland in the Evolution of Additional Faculties--The Skeptical Attitude of Empirical Science and the Need for a More Liberal Posture--The General Results of Pituitarial Awakening Upon Man and the Theory of Knowledge. Evolution is a continuous process and the primal impetus back of the great on-flowing ocean of life acts infinitively. It is not terminated when life has succeeded in perfecting a form for the perfection of forms in themselves is not the end of vital activity. _The end of evolution is the complete spiritualization of matter._ So that it does not matter how perfect a form may be either subjectively or in its adaptation to environments; it does not matter how faultless a medium for the ensouling life it may be, there is ever the eternal necessity that life must drive it back over the path of its genesis until it shall be transmuted into pure spirit. Adaptation succeeds adaptation and with each there is a change in the form and this process continues until there is a more or less perfect congruence between form and juxtaposed environmental conditions. But no sooner than agreement has been attained under one set of conditions new conditions arise and require a new setting, new adaptative movements. Thus there is a continuous proceeding from stage to stage, going from the grossest to the subtlest and most refined, always the form is being pushed onward and upward by life. But adaptation is not undergone for the benefit of the form, but more truly for the informing principle. It is the progression of the life-element which constitutes the adaptation of form to form and to their peculiar environs. The form is a tool or instrument of life which it discards the moment it fails to respond to its requirements. Thus forms are constantly being assumed and as constantly being relinquished. But no effort of life is lost regardless as to whether the action is performed in one or another form. The totality of matter is perpetually being acted upon by the totality of life. Every appulse of life against matter means an added push in the direction of spiritualization. The totality of such appulses of life against matter may seem infinitely small in the visible results which they produce in the process of spiritualization; but with each there is an eternal gain in that movement that shall end in the complete transmutation of materiality into spirituality. This action of life in metamorphosing matter, the nether pole of the great pair of opposites, into spirituality, its copolar factor, in its outward, visible effects, is what we vaguely call evolution. And such it is; for life is merely unfolding that which it has enfolded. Matter, having been involved as a phase of kosmic involution, is now being evolved. In the genesis of the kosmos there appear to be three great undulations in the universal current of life. The first of these prepares the field by depositing that elemental essence which is to become the world-plasm; the second precipitates the universum of materiality, spatiality and intellectuality, not as we now know them, of course, but as potencies; the third great undulation in the current of life effects the endowment of the world-plasm with those tendencies that are to build around themselves forms appropriate to their fulfillment. This ensoulment of the world-plasm with tendencies and the consequent segmentation of it into separate forms by these tendencies constitute the primary stages of that procedure of life which results ultimately in the up-raisement of matter and its final exaltation into pure spirit. Hence, the entire mass of materiality is besieged on all sides by the sum-total of life and the former is being raised slowly and irresistibly to heights that are immeasurably more sublime than its present degree of grossness. It appears paradoxical, therefore, that life, although in all respects vastly superior to matter, should become the apparent vassal of materiality and give itself up to all the strict rules of imprisonment which are imposed upon it by the properties and qualities which we observe in matter. It seems so subject to every whim and fancy of matter that one is inclined to think that matter and not life is the chief designer of universal destiny. This is not a condition to be wondered at so much, for the reason that this apparent vassalage, this seeming enslavement of life by matter, is due to that superior and most marvelous adaptability of life which it enjoys in contradistinction to the relative unpliability of matter, and due also to the fact that life is kinetic and matter, being a mere deposit of life, is static. Life is mobility while matter is immobility and thus in possessing a greater range of freedom is, of course, correspondingly superior; but in this adaptation of itself to the labyrinthine cavities and multiformed interstices in matter it exhibits but a seeming serfdom which is really not a serfdom but a mastery. It is as if a man had taken lumber, hardware and stone and built a house wherein he might dwell--life has merely used matter, molded and fashioned it so as to make for itself a medium, a dwelling-place wherein it operates, not as a slave but as a master possessing unlimited freedom of motility. In the production of a form life stamps upon it, once for all time, the path of its engendering action. It leaves its finger-prints upon the mold which it makes for itself. So that if we would know where life _has been_ or where it _is_ we should look for its finger-prints (organization); we should observe the sinuosities which mark its pathway, remembering always that it is life that has formed the intricacies and complexities of the form into which it pours itself so accommodatingly in order that it may raise that form, develop and transmute it into something higher and better. When we speak of _form_ it must not be understood thereby that reference is made only to the gross physical form, but to the entire range of vital assumptions or vehicles which life ensouls for purposes of manifestation. This range we believe to cover the whole path of kosmogenesis seriating from the densest to the most subtle. Our chief concern, however, is the immediate effect which the totality of life's operations will have upon humanity or the form which it ensouls as the human organism. For it is impossible that humanity shall escape either the general or the specific results of the exalting power which life exerts over materiality and its appurtenances. It is, of course, impossible here to go into the various implications of this general forward movement of the universum of materiality or even to outline briefly the divergent lines of operation into which a satisfactory exposition of this view would naturally lead. And then to do so would be inappropriate in a volume of this kind. So we shall have to be content at this juncture to limit our study to a consideration of what we believe to be some of the immediate indications of this vast and most far-reaching phenomenon. In the chapter on the "Genesis and Nature of Space" it is shown that the material universe is engendered at the same time and by the same movement or process as the universum of spatiality and intellectuality and that as the passage from chaos to kosmos proceeds the function of this movement is changed gradually from engenderment to exaltation wherein materiality is transmuted into spirituality. It is, of course, obvious that as materiality is exalted so are spatiality and intellectuality; and that as the one becomes more and more refined, capable of answering to higher and yet higher requirements so do all the others. For, at work in all and through all of these, is the current of life which pervades them, engendering, sustaining and elevating as it proceeds. So that as matter has evolved added characteristics and properties, each answering to a given need and arising out of the necessities inhering in the stage at which it appeared, so has the intellect evolved faculties to correspond therewith. In other words, the evolution of faculties for the expression of the human intellect has proceeded synchronously with the evolution of material qualities. And whenever a new faculty or an additional scope of motility is achieved by humanity there is always found a set of kosmic conditions which answers thereto. The cardinal principles of the doctrine of evolution are not, therefore, adverse to the conclusion that the organs of sense-perception--hearing, touch, sight, taste and smell--have not been endowed upon the human race or attained by it at one time; but rather that each answering to a newly acquired need and opening a wider scope of motility for the intellect has been evolved separately and in due order. It would also seem that the quality of consciousness, as it has been manifested in the various stages of life through which it has passed, and especially the mineral, vegetal and lower animal, has not always been of the same degree of efficiency. Nor has it enjoyed the same kind of freedom which it now enjoys in the highly evolved _genus homo_. It is equally apparent that matter itself has not always been in possession of the same qualities and characteristics which it now exhibits; but that it, too, has gone through various stages of evolution bringing forward into each new stage the transmuted results of each preceding one as a basis for further evolution and expansion. The innumerable archæological evidences which support this view make it unnecessary to do more than state the facts, as they appear to be substantiated by indubitable testimonies. Furthermore, it is believed that the outstanding implications of these phenomena will not be successfully controverted by those who are disinclined to see such implications in the evolutionary process. In a previous chapter we have briefly sketched the characteristics which mark the upspringing of a new faculty showing how, at first, it appears as an abnormality which exhibits itself in a very few individuals only, and that in a more or less indefinite manner; and how later the number of individuals in which it appears gradually increases, the definiteness of the faculty, at the same time, appearing more marked; then, like a tidal wave, it recurs in a still larger number of persons until, at last after a long period of time usually several thousands of years, it becomes universal exhibiting itself in every individual and appearing as a hereditary characteristic of the entire human race. It is, therefore, not without assurance as to the ultimate soundness of this view that we make the assertions which follow this brief introduction. It has already been stated that for a very obvious reason, namely, the satisfaction of the needs of our present humanity, the intuition is for the time dominated by the intellect and held in subjugation by it so that all of man's external operations are governed and dictated almost entirely by the intellectuality, allowing the intuition only rare moments when it can come to the fore at all. This is the rule in the evolution of faculties and characteristics. The higher faculty, although potentially present in every way, is ever held in abeyance while the lower is brought, under the rigors of its own evolution, to a point where its joint operation with the higher may be executed with the least possible friction and retardation as also with the greatest possible coördination and coöperation. Accordingly, notwithstanding the fact that materiality must possess in potentiality all the qualities which it will at any time reveal, it is nevertheless necessary that these qualities shall come forth gradually and in due order. Similarly, humanity has come into possession of its various faculties of mind, and powers of physiological functions, by insensible degrees, the higher always being held in abeyance until the lower is fully developed. Those faculties which are to bestow added powers, additional freedom and a greater scope of motility are the ones which appear later than those which are truly primitive in character. These facts have been amply demonstrated by the science of embryology wherein it is shown that _ontogeny_ is a recapitulation of _phylogeny_. That is, the history of the development of the individual is a recapitulation of the development of the species. Thus the various stages of development through which the human embryo passes while _in utero_ are but a repetition of similar stages through which the entire human species has passed in its phylogenetic development. Wherefore, it is certain that humanity has not attained, at one and the same time, all the powers of mind and body which it now possesses; that the childhood of the human race represented a time when it had but few faculties or organs of sense-perception--indeed a time when the higher sense-organs of smell, taste and sight were entirely lacking although residing in potentiality therein. It is undoubtedly true that the earth has passed through a similar evolution with respect to its own material characteristics, that its childhood was, in all points, analogous to the childhood of humanity; that the air, earth and water were wholly absent, except in potentiality, during the nebulous youth of its genesis. It is even probable that there are at work to-day processes which in the future shall culminate in the evolution of newer, higher and more complicately organized species of plants, animals and minerals. Every year brings fresh evidences that crystallize the conviction that the earth has been the scene for the appearance of many strange orders of animal life. Fossiliferous strata are continually yielding incontestable testimonies of changing flora and fauna. We count the animal and vegetal life of to-day as being more highly developed than that of any other previous age, and it is well that this is so, for simplicity of organization and primality of manifestation are always succeeded by complexity and a greater scope of adaptability. We have said that the whole of that movement of the intellect which has brought forth the metageometrical creations of hyperspaces, the curvature of space and its manifoldness together with the entire assemblage of mathetic contrivances are merely the early evidences of the appearance in the human race of a new faculty, a new medium of perception whereby the Thinker shall acquire a still greater range of motility than that now offered by the intellect. Attention has been called also to the fact that this phenomenon has been manifested not alone in the field of mathematics, but in art, religion, politics and also in science in which we have only to witness the marvelous strides already made in the discovery of radio-active substances, the ROENTGEN, BECQUEREL, LEONARD and other kinds of rays. It Is quite confidently believed that these forward movements in every branch of intellectual pursuit, these combined efforts of the intellect, in peering into the occult side of material things, are in response to the evolutionary needs of the Thinker, and in addition, are the evidences, and shall in time be the cause, of the development of an additional set of faculties. Function, or the performance of acts, determines faculty or the power of action and ultimately the organ itself. Thus the mere wish to perform aroused by desire and vitalized by the will actually terminates, in the course of time, in the genesis of a faculty, or the power to perform. The constant upreaching yearnings of the Thinker through his intellect for greater freedom and a larger scope of action, the desire to peer into the mysteries of life and mind, the infantile out-feelings of the mentality after some safer and surer basis for its theory of knowledge cannot fail in producing not only the faculty or power to satisfy these cravings but the very organ or medium by virtue of which the satisfaction may be attained. It is not strange that in mathematics the intellect should have found first the clue to the existence of a higher sphere of intellectual research wherein it might become the creator of the various entities which peopled the new found domain; it is not strange that the mathematician should, in this instance, have assumed the role of the prophet proclaiming by various mathetic contrivances (although unconsciously) that the human race is nearing that time when it shall actually be able to function consciously in some higher sphere; neither is it to be wondered at that the voice of the prophet is heard and respected throughout the earth; for, indeed the mathematician is a spokesman who, as a rule, is unmoved by sudden outbursts of passion and ecstatic frenzies of emotions but calmly and dispassionately verifies his conclusions, tests them for consistency and having found them to satisfy the most rigorous mathetic requirements hesitates not to propound them. For this cause humanity respects the mathematician, and when he speaks listens to his voice. It is well, too, that this is so; for the history of mathematics is clearly the history of the development of the intellect. So exact a determinator of the quality of intellectual efficiency is it that its reign may be said to be an absolute monarchism whose lines of dominance extend to the minutest desire or appetency. It has always been the guide of the intellect, going before, as it were, blazing the trail, pushing back the frontiers of the intellect's domain and clearing away the _debris_ so that the intellect with its retinue of servitors might have an easy path of progress. Mathematics, however, has not the aptitude to serve the intuition as it serves the intellect. So the path into which the intuition would lead humanity the mathematician, because of his training and peculiar functions, is unprepared to enter. It is for this reason that when mathematics leads the intellect up to that point where it encounters life it fails, it becomes confused and its dictatorship becomes a mockery, its decrees remain unexecuted and futile. In taking this view we have certainly no desire to offend the mathematician or to detract from the glory of his monarchistic rulership over the intellectual progress of the race; for, in truth, mathematics is the diadem of gold wherewith man has crowned his intellect. Yet it is well, yea imperative in the light of recent developments in the realm of hyperspace, that a careful discrimination should be made as between the sphere of the intellect and that to which the intuition shall attain. The intuition, long held in abeyance until the intellect should be fully crowned and reach the zenith of its powers, is now coming to the front. It will be many centuries perhaps before it shall have grown to such proportions as those already attained by the intellect; perhaps a few thousand years may pass before the intuition shall have evolved to that point where it may labor as coadjutor to the intellect; but undoubtedly the time will come when it, too, shall reward the Thinker's labors with that which shall be more precious than the crown of gold which the intellect has won. Then, the intellect, grown old and decrepit with years of reigning shall become dim and crystal-shaped and finally pass into automatism or reflexive movements where without the urge of volitional impulses it will perform with exactness, precision and utter loyalty the tasks which it has learned so well to execute in the days of its forgotten glory. Mankind will then be free. A new freedom, wherein the erstwhile lightning flashes of intuition will become fused into one glorious sheen of all-revealing light, shall come to men and thus the race resplendent will walk the earth enshrined in the majesty of divine powers attained as a result of millions of years of aspiration. That there are supersensuous realms so far above the range of our senses as to be entirely beyond their ken needs now no proof or argument; for the scientist has demonstrated, by the invention of instruments of extreme delicacy and precision, that such a world does really exist. Already we know of stars so distant that, though light traverses in the brief space of an hour six hundred million miles, they might have ceased to shine before the pyramids were built and yet be visible to us in the skies. If the human eye were as sensitive as the spectroscope many thousand tints and shades might be added to the world of color; if they possessed the magnifying powers of the microscope we should live in constant terror and awe of the monstrous entities that teem in the water which we drink and in the air which we breathe; and if our ears could detect the microphonic vibrations which register in the delicate apparatus of some microphones the dead, vacuum-stillness of nature's great silences would appear as a babel of voices by the seaside. The sense of touch, responding to the same range of vibrations as the micrometer, would reveal actually the interstices between particles of the densest elements; and gold, silver, platinum and mercury would seem but honeycombs of matter. But, to the forward-looking there is no element of absurdity in the expectation that all these senses shall, one day, be able to dispense with the artificial aid of physical apparatus and perform, with even greater precision and faithfulness, the task which they now perform so crudely and ineffectively. There are without doubt vibrations of taste and smell which are so far above the range of these senses that they have no effect upon them whatsoever. Notwithstanding the fact, however, that the galvanometer, microscope, the microphone, the spectroscope and the telescope have extended thus the sphere of sense-knowledge there are yet subtler vibrations to which these delicate instruments do not and ought not be expected to respond. But to say, as do many empiricists, that since these phenomena cannot be detected by scientific instruments they do not, therefore, exist seems to be expecting too much of material means as well as exposing oneself unnecessarily to criticism on the grounds of extreme materialistic appetences. There is indeed need of a more liberal attitude among men of science towards the world of the unseen. Intolerance of the data which it offers will for a time perhaps preserve the aloofness of scientific dogmatism inviolate but there will most surely come a reaction against the dogmatism of science and men will seek freedom and attain it despite their fetters. Sir OLIVER LODGE, in his book, the _Survival of Man_,[29] says: "Man's outlook upon the universe is entering upon a new phase. Simultaneously with the beginning of a revolutionary increase in his powers of physical locomotion--which will soon be extended to a third dimension and no longer limited to a solid or liquid surface--his power of reciprocal mental intercourse is also in process of being enlarged; for there are signs that it will some day be no longer limited to contemporary denizens of earth, but will permit a utilization of knowledge and powers superior to his own, even to the extent of ultimately attaining trustworthy information concerning other conditions of existence." [29] See pp. 338, 341. It is the author's good fortune that he has for a period extending over several years been able to verify the conclusions which Sir OLIVER LODGE expresses in the above, and thus to satisfy his own mind that the process by which man's mental powers are "being enlarged" is indeed demonstrable by actual observation and experimental methods. LODGE continues: "The boundary between the two states--the known and the unknown, is still substantial, but it is wearing thin in places, and like excavators engaged in boring a tunnel from opposite ends amid the roar of water and other noises, we are beginning to hear now and again the strokes of the pick-axes of our comrades on the other side." CAMILLE FLAMMARION[30] cites 436 cases of psychic manifestations examined by himself and which establish beyond any reasonable doubt that there are certain perceptive faculties, namely, clairvoyance and clairaudience, that crop out in certain individuals, sometimes in abnormal conditions and sometimes normally, the very unusual character of which proves their rudimentary nature and the potency of their maturescence in the humanity of the future. Among the cases cited by FLAMMARION are 186 instances of manifestations from the dying received by persons who were awake; 70 cases were manifestations received by persons asleep; 57 were observations of direct transmission of thought without the aid of sight, hearing or touch or other physical means; 49 were cases of sight at a distance or clairvoyance by persons awake, in dreams or in somnambulism and 74 cases of premonitory dreams or predictions of the future. Indeed, there are few persons now living who have not had similar experiences, if not exactly like these, of the same nature. These examples, of course, may be greatly multiplied in every country in the world, and it is unnecessary to enumerate them further; for, when once the existence of such faculties has been demonstrated in persons, either in a normal or an abnormal condition, their presence can no longer be questioned by the fair-minded. It is, then, only a question of evolution before they will appear in the normal way and their universalization, as transmissible characters, be an accomplished fact. When we are brought face to face with this sort of phenomenon which seems to be increasing rapidly the conclusion is inevitably forced upon us that since evolution must be a continuous process and matter destined to yield higher and more refined powers and humanity to come into a far more extensive scope of motility because of the opening avenues of knowledge, it is not impossible that these acuter senses, these new faculties are now existing in the human race in a rudimentary stage and are designed to become the universal possession of all. That this is to be the almost immediate outcome of the perpetual exalting power which life exercises not alone over materiality but over human organs and faculties as well, seems to be the one big, outstanding implication of the evolutionary process. The presence of such functions as the ability to sense the invisible and the inaudible, to answer to vibrations far subtler than anything in the scope of our external sense-organs, certainly indicates the existence of rudimentary faculties which make these functions possible. _Back of these vague, indefinite functions, back of every supernormal or abnormal manifestation of man's mentality and back of all that class of phenomena which take their rise out of supersensuous areas must lie, in ever increasing potency, faculties and organs, however rudimentary, which are the source of these manifestations._ Life, that ineluctable agent of creation, which is incessantly pushing outward the confines of the intellect's scope of motility, never wearying, never tiring nor sleeping, has long ago, in the dim and distant past of man's evolution, laid the foundations; and in fact, with one stroke of its creative hand, has molded the organs which are to become the active media of these new faculties. And now, these incipient demonstrations, these infantile struggles which we see now and again outputting from them, are but the specializing processes through which, in their later development, these organs are proceeding. These are the outward signs which should tell us that life is breaking up these organs into special parts, assigning to each a certain division of labor and making of each a perfect coördinate of all the others. It is, by these very dispread exhibitions, cutting up, specializing and by slow degrees determining the function, character and general tendence of the organs of expression wherewith these manifestations shall be centralized and put into effective operation. In doing this, it is but following its accustomed procedure, the procedure which it adopted when it produced the eye, the ear, the heart and the spleen. We shall, therefore, gauge our understanding of the purport and end of evolution; in fact, we shall determine our exact intellectual comprehension of life itself, by the attitude which we adopt towards it and the mode of its appearance. Much depends, accordingly, upon the posture which we assume towards life--whether we shall say the totality of life's creative powers has been dissipated in the bringing of the human body to its present degree of perfection; whether we shall say that it is neither necessary nor possible for life to produce other organs and faculties which shall respond to the unseen world about us revealing its glories in a way far more perfect than do our external sense-organs reveal the wonders of the world of sensation; or, whether we shall conclude from these most palpable evidences that life has yet other powers and faculties which it designs to bestow upon the human mind and other organs and capabilities with which it shall endow the human body so that man, in his evolution, shall be enabled to rise to still higher spheres while yet incarnate. There may be, and undoubtedly are, those who, for various reasons prefer to take the former positions and there are certainly those who like LODGE, FLAMMARION, HUDSON, CROOKES and a host of others, preferring the latter view, would rather believe in the strength of the great mass of corroborative testimonies that we are even to-day in the midst of the matutinal hours of a newer, a better and a far more efficient era of human evolution than any through which we have hitherto come. [30] See _Unknown_, p. 485, et. seq. Already, recent scientific investigations and the results obtained therefrom have begun to turn the attention of medical authorities to the activities of two very small organs situated in the mid-brain and known as the _pineal gland_ and the _pituitary body_. These organs, and especially the _pineal gland_ hitherto supposed to be a vestige of the past, are now beginning to be recognized as rudimentary organs belonging to the future evolution of humanity. Dr. CHARLES DE M. SAJOUS, who is an authority on the _pituitary body_, believes that it has no active internal secretions but is an "epithelio-nervous organ" which controls, through nerves leading to the adrenals and thyroid bone, the processes of general oxygenation, metabolism and nutrition. Little is known of the functions of the pineal gland except that it is an ovoid, reddish organ attached to the posterior cerebral commissure projecting downward and backward between the anterior pair of the _corpora quadrigemina_. It is otherwise known as the "_conarium_" the "_pinus_" or "_epiphysis_." Situated at the base of the brain, it is held in position by a fold of the _pia mater_ while its base is connected with the cerebrum by two pedicles. It contains amylaceous and gritty, calcareous particles constituting the brain sand. There are, however, marked structural resemblances between the _pineal gland_ and the _pituitary body_ and their formation is perhaps the most interesting feature of the development of the _thalamencephalon_ or mid-brain. The _hypophysis cerebri_ or _pituitary body_ is a small, ovoid, pale, reddish mass varying in weight from five to ten grains and situated at the basal extremity of the brain in a depression of the cranium known as the _sella turcica_, a configuration very much like a Turkish saddle in shape. It is a composite, ductless gland and consists of two divisions, an anterior and a posterior, connected by an intermedial portion--all of which are attached to the base of the cerebrum by the _infundibulum_. The anterior lobe is larger than the posterior and very vascular, springing in its development from the buccal cavity of the embryo; the posterior lobe is situated in a depression of the anterior and is a brain-process. The _pituitary body_ itself is lodged in a cavity of the _sphenoid_ bone called the _pituitary fossa_. This is a most remarkable position, for the reason that the _sphenoid, or wedge-shaped_, bone which lies at the base of the skull articulates from behind with the occipital and in front with the _frontal_ and _ethmoid_ bones and by lateral processes with the _frontal_, _parietal_ and _temporal_ bones. From this position it binds together all the bones of the cranium, and moreover, articulates with many bones of the face. It is upon the upper surface of the _sphenoid_ bone which occupies such a prominent and commanding position in the cranium, in a deep depression, that the _pituitary gland_ is located. Each nasal chamber is lined by a mucous membrane called the _pituitary_ or _Schneiderian_. This membrane is prolonged into the meatuses and air sinuses which open into the nasal chambers. The _pituitary_ membrane is thick and soft and diminishes the size of the meatuses and air sinuses. It is covered by a ciliated columnar epithelium and contains numerous racemose glands for the secretion of mucous or _pituita_. It is also vascular and the veins which ramify it have a plexiform or net-work like arrangement. It divides into two membranes--a respiratory, which is concerned in breathing, and an olfactory region. The respiratory region corresponds to the floor of the nose, to the inferior turbinated bone and to the lower third of the nasal septum. The olfactory region is the seat and distribution of the olfactory nerve and corresponds to the base of the nose, to the superior and middle turbinals and the upper two-thirds of the nasal septum. Recent developments prove that this gland has a profound influence over the animal economy. It is believed by some that the _pituitary body_ actually destroys certain substances which have a toxic influence on the nervous system; others believe that it secretes material media for the proper action of the trophic or nutritive apparatus; still others believe that it influences blood-pressure. It is known, however, from experimentation, that its removal in dogs, cats, mice and guinea pigs causes a fall of temperature, lassitude, muscular twitchings, dyspnoea or difficult breathing, and even speedy death. Hypertrophy of the gland is directly associated with certain diseases, such as _giantism_ and _acromegaly_. The latter is a disease which causes a general enlargement of the bones of the head, feet and hands, usually occurring between the ages of twenty and forty years, and most frequently in females. The fact that these diseases are so closely associated with a hypertrophic condition of the pituitary gland has led to the conclusion that perhaps the giants or Cyclops of ancient times were cases of _giantism_ or _acromegaly_. This view, while interesting from the standpoint of the functions of the _pituitary_ gland, is not necessarily a correct one; for the age of giants, when men attained to a much larger stature than at present, can be accounted for on other grounds, namely; that the early mesozoic man, on account of his having to live among animals, trees and other vegetation of such huge size, had naturally to be fitted with a frame proportional to other animals in order that he might successfully cope with his environing conditions. Nature thus wisely fitted him for the conditions which she had prepared for the scenes of his life. The facts adduced in the foregoing description are purely empirical and may be verified by any who seek to establish their correctness or incorrectness. But we are about to introduce a species of testimony which while it may also be verified will not be found so easy of verification as the above-mentioned physiological facts, and not by the same means; yet they are nevertheless deserving of a place here. It is the liberal attitude that we must adopt towards all phenomena, excluding none that give promise of the widening and deepening of our knowledge and an explanation of much that has seemed heretofore unaccountable. We have noted how subtle is the physical connection between these two bodies, the _pineal gland_ and the _pituitary body_; we have seen how profound is the effect which the latter has been demonstrated, in a measure, to have over the entire bodily economy; but there is even other testimony to the effect that those gifted with the inner vision can observe the "pulsating aura" in each body, a movement which is not unlike the pulsations of the heart and which never ceases throughout life. In the development of clairvoyance it is known that this motion becomes intensified, the auric vibrations becoming stronger and more pronounced. The _pituitary body_ is the _energizer_ of the _pineal gland_ and, as its pulsating arc rises more and more until it contacts the _pineal gland_, it awakens and arouses it into a renewed activity in much the same manner as current electricity excites nervous tissue. When the _pineal gland_ is thus aroused clairvoyant perception is said to become possible. These are facts which cannot be proved by the materialistic man of science nor can they be demonstrated to the layman who has to depend alone upon sense-deliveries for his knowledge. This is true for the reason that, in the first place, it is necessary that he shall either feel in his own mid-brain the energizing activity of these two organs and have his entire nerve-body shaken from crown to toe by the down rushing currents of that subtle energy with which the _pituitary body_ floods it or be himself the perceiver of its activities. Nevertheless attention is here called to these phenomena and the conclusions drawn therefrom are offered as a means of denoting the probable line of investigations which will establish the directions which we should pursue and the source whence we shall find outcropping the new faculties and their organs of expression. We confess to a knowledge of the fact that men of empirical science have long maintained a rather skeptical, if not contemptuous, attitude towards all these phenomena but it is also felt that there is far more of discredit in their attitude than of credit; for, in so doing, they have voluntarily adopted measures by means of which the knowledge that they so eagerly seek is shut out from their attainment. In vain, then, is appeal made to the intellect to remove the barriers which it unconsciously interposes between itself and the goal of its pursuit; in vain do we appeal to the materialist to give ear to testimony the data of which cannot be made amenable to his knife and scalpel neither to the microscope nor microphone; in sheer vanity is he adjured to look _within_--into the interior of life, of mind and the things which he handles with his instruments--for the answers to his queries, for the path which leads into the wake of life and consciousness. Because his utter loyalty and devotion to the _modus vivendi_ of the intellect will not permit this; but, after all, it is not wholly wise to allure him away unbetimes from his search after truth through superficialities nor to inveigle him into giving up his tenacious prosecution of the physically determinable. We would not have it so; for, perchance, he, too, one fine day, in the quiet of his laboratory shall come upon the data which may substantiate in his own mind the long settled conclusions of the occultist who, frequently and not without cause, grows impatient at the scientist's obstinate delay. These two workers, the empiricist and the occultist, must ultimately come together as collaborators--the one working upon the form, the vehicle, physical matter and the other seeking to understand the life, the interior forces which produce, the creative element. They cannot remain always aloof from one another; for they, too, are as men digging a tunnel from opposite ends. Finally, the partition will break and thus will dawn a new day for the knowledge of humanity and men will see the rationale, the truth and good sense of coöperation in this respect. It can be said with confidence that whatever in the future may be learned as to the physiologic functions of the _pituitary body_ and the _pineal gland_, it suffices to know that it is life which they express and that, too, in a far superior manner than any of the other sense organs. The _modus_ of these two glands differs in a very marked way from that of the organs of sight, hearing, taste, smell and feeling. For these latter are designed for contact with the external, objective world of sensations, their growth and evolution being dependent upon stimuli received from without while with the former the case is far different, in fact, just the opposite. The mode of life of the _pituitary body_ and the _pineal gland_, instead of receiving sustenance and impetus from external stimuli, is rather dependent upon impacts received from the Thinker's own consciousness and made to impinge upon them by an exclusively interior process. Situated in the mid-brain, safely secluded from all external interference, they are naturally limited to stimuli which come from within, or it may be said, they are responsive to excitations that are more spiritual than those which come through the external sense-organs. If, as has been said they control the internal processes of metabolism (anabolism and katabolism), oxygenation, nutrition, and other important internal movements, none of which can be said to be under the control of the intellect, is it not, therefore, justly assumed that their response is directed towards stimuli which arise interiorly or upon a plane higher than the intellectual? It is a matter of scientific knowledge that those persons gifted with clairvoyance, and commonly known as "sensitives" are far more responsive to nervous excitation than those who are not so gifted. This would seem to imply that, on account of the superactivity of these two organs, the entire nerve-body has, in consequence, become more delicately and subtly organized. They seem to act as a switchboard for the regulation of the flow of the current of life through the body. Not only do they come more nearly to an adequate expression of the physiologic function of life, but, as their energization means an enlargement of the scope of perception by giving the Thinker's active consciousness access to hitherto unapproachable realities and by penetrating the outer mask which life ensouls and also laying bare a domain of unlimited knowledge the manifestation of which is far more real than anything the senses can disclose, it is evident that they constitute, in their collaborative functions, a more highly adaptable medium for the expression of the Thinker's consciousness. And if so, for the kosmic consciousness which is the source of all forms of consciousness, they furnish a specializing and _adaptizing_ agency. Now, in all those cases of inspirations, revelations, telepathic communications, clairaudience, clairvoyance, dreams, visions, etc., wherein the Thinker is enabled to perceive facts and verities which are then presented to his consciousness in a manner clearly without the province of the common sense-organs, it must be apparent that these manifestations are apprehended by a perceptual mechanism which is entirely independent of external sense presentations but which is an interior and subtler form of psychic activity. Sounds which are heard by so-called "sensitives" and objects which are perceived by eyes that are keener than those organs said to have been evolved from the "medusa" cannot be heard by other persons nor perceived by them in any way. Thus it would seem that there are inner organs of perception which respond to these finer vibrations and which enable the person so gifted to apprehend them. There are those who, presumably basing their assertions upon actual observation and knowledge, unqualifiedly assert that in order "to gain contact with the inner worlds all that remains to be done is the awakening of the _pituitary body_ and the _pineal gland_. When this is accomplished man will again possess the faculty of perception in the higher worlds, but on a grander scale than formerly (when humanity was in its infancy and exercised a lower form of psychic power only); because it will be in connection with the voluntary nervous system, and therefore, under the control of the will. Through this inner perceptive faculty all avenues of knowledge will be opened to him and he will have at his service a means of acquiring information compared with which all other methods of investigation are but child's play."[31] It is the lack of this ability to see, with our physical eyes, as it were, by the "Roentgen rays," to penetrate the inwardness of things that has baffled and confounded men for so long a time and which has eventually led certain mathematicians and others to conjecture such strange, and in many cases, illogical possibilities for the denizens of four-space. This inability together with the desire to fathom the innermost complexities of solids and to handle, albeit with unholy hands, the supersensuous, the mysterious and the unapproachable identity of "things-in-themselves" have induced the more zealous among them to contrive some kind of hypothesis which would, at least, offer an explanation of these phenomena. It has driven them to wrestle with metaphysical possibilities in a vain endeavor to grasp that which, _ignis-fatuus_ like, ever evades their slightest intellectual approach. But why this prolonged struggle, why this intellectual maneuvering and sophistry? "We can calculate, compute, excogitate," says PAUL CARUS,[32] "and describe all the characteristics of four-dimensional space, so long as we remain in the realm of abstract thought and do not venture to make use of our motility and execute our plans in an actualized construction of motion; but as soon as we make an _a priori_ construction of the scope of our motility, we find out the incompatibility of the whole scheme." Thus mathematicians are forced to relinquish all hopes of transforming the world of life into a sort of four-space dwelling place where everything is done according to the laws of mathematics. But whether they shall accept it or not there is a wider, truer and more rational view which recognizes all metageometrical investigations, as well as all kindred phenomena, as universal evidences indeed, as the very causes which, in the future humanity, will actually awaken and cause to be accelerated in their development these little inner sense-organs, the _pineal gland_ and the _pituitary body_, whose perfect development promises to provide for the Thinker's consciousness an avenue of expression such as humanity has possessed never before. And too, it is not without full knowledge of the fact that it has been customary, among certain scientists or perhaps all of them, to regard these bodies, at least the _pineal gland_, as vestigal organs belonging to the past of human evolution, that we make these assertions. Yet, as man proceeds in the perfection of mechanical science, in the development of instruments of precision that aid his external senses, responds more and more to the subtle vibrations teeming everywhere in the atmosphere about him, and comes, in the course of time, naturally to possess a more sensitively keyed nervous mechanism, a finer body and higher spiritual aspirations, there will be a corresponding widening of his scope of vision and the attainment of larger powers of perception which must inevitably, in the very nature of things, tend towards a deeper and truer knowledge. [31] _Rosicrucian Cosmo-Conception_, p. 477, MAX HEINDEL. [32] FOUNDATIONS OF MATHEMATICS, p. 90. In view of the foregoing, it is believed that the general results of this pituitarial awakening which may be expected as humanity continues to evolve should be seen in the marked effects which will be wrought in the entire metabolistic area of the human body whereby a gradual intensification and sensitization of the whole neural mechanism will raise the peculiar efficiency of all the senses whether purely physiologic or psychic. For there are undoubtedly notes so delicate in their intensity that they transcend the grasp of the audital nerves; scents and fragrances so subtle in their excelling purity that it is beyond the powers of our present olfactorial contrivances to detect them; colors and other external stimuli so sublimely supersensuous that a nervous mechanism perhaps ten-fold more delicate and responsive than ours is required to apprehend them. All these, and more than at present is conceivable, will come, with the aid of pituitarial stimulation, within the purview of a more highly developed humanity of the future. And because mathematics have led a movement into the very camp of the intellectuals--logic-bound and tethered by the severest rigors of mathesis--whereby the intolerant intellect has been compelled, by rules of its own making, to recognize the existence of the supersensuous, and by looking into the glaring light of the sun of the intuitable to gain strength of vision and boldness to press forward, a great and far-reaching service has been wrought for humanity. And in the tower of hyperspace mathematics have erected a monument to the intellect which, as long as the human race remains, will mark the great turning point in man's path to the highest life. What if it were possible that the scientist, when he had carried instruments to their utmost precision and penetration, should suddenly, or otherwise, be endowed with a clear-perceptivity of sight, hearing and smell, so that he could with his own powers of vision, feeling and hearing take up the task where the microscope, the microphone and the micrometer left off and delve into depths far too unfathomable for his appliances, perceiving the innermost realities of things and processes? What if it were possible for him, with these added powers, to see and examine without the aid of the magnifying lens the electron, the atom and the molecule? What if the cell, the bacterium, and other invisible forms of life would then deliver up their secrets to his knowing mind? What if he could sense with his own inner vision, the ultra-violet and the infra-red rays; what indeed, if spirit itself, the innermost sheath of life, should be visible and palpable to him and he could note the internal processes, the action and movements of the infinitesimals of life? Think you not that such direct contact, such immediate and incontrovertible knowledge would be far superior to any advantage which his manufactured devices now bestow? It is even so. Thus will react upon man's perceptive apparatus the flood of light which the awakened intuition will shed upon them and thus will man rise higher, driven on by the current of life with the mass of materiality, to a point of complete spiritualization and take additional steps in that direction which leads to Raja Yoga or the Royal Union with the divine life of the universe. Before this step is taken, however, and before the passage from mechanics to biogenetics is made, as made it must be, man must win a complete mastery over matter. But this he will do; for more and more he is learning to put all those forms of labor which are so exacting as to leave him no time for the development of his higher powers into the hands of machinery. He will not be free until he has done this well-nigh completely. This is the task of the intellect and with it man must win his way to these higher faculties which are destined to succeed the intellect whereupon he will be ushered out of a life bound and restricted by mechanics to a life of unimaginable freedom, the _intuitive life_. The outcome of these new faculties of perception and the development of the intuition will be the springing up of a new species of art that, turning away from appearances and sinking beneath or rising above, superficialities, will seek to portray in newly found colors, the plastic essence of things so that we shall have an art which pertains to the real, superseding that which pertains to the phenomenal. Language and the need of it will pass away; for man will have outgrown the use of signs and symbols in his communion with his fellows and will use the language of the intuition--direct and instantaneous cognition. Philosophy will be regenerated, re-created. Speculation will give way to truth and there shall be but one philosophy and that shall be the _knowledge of the real_. Mathematics, the royal insignia of the intellectual life, because it can deal only with immobilities, with segments and parts and has no aptitude for the continuous flow, will yield its kingdom to a higher form of kinetics which will serve the intuitive faculty as mathematics now serve the intellect. Science will then be no longer empirical in its method; but a system of direct and incontrovertible truths. Religion will rise to meet these changes which will come in the path of human evolution; and faith will surrender its place to knowledge. Ethics, recast in a new mold, will deal with the new aspect of man's relation to his fellowmen. Man, for whose highest good these ultimate changes will come, will be a new creature, a higher and better man; and humanity shall evolve a new race. There shall, indeed, be "a new heaven and a new earth." THE END. BIBLIOGRAPHY AMERICAN MATHEMATICAL MONTHLY: Vol. VIII, p. 161, P. BARBARIN, for _Le Mathematiche_, Trans, by GEO. B. HALSTED. 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INDEX A Abeyance, higher faculties held in, until lower are fully developed, 334 -- intuition kept in, 338, _ff_ Ability, to sense the inaudible and the invisible, 342 Absolute, no room for the, 101 Abstract thought, invigorating power of, 33, 40 Abstractions, realizing, 144, 294 Absurdity, analysis inevitably ends in an, 319-20 Acrobatics, mathetic, 146 Acromegaly, 347 Action, automatic, of the intellect, 253-4 -- engendering, of life on the form, 330 Activities, space, as physical and chemical phenomena, 229-30 Actualities of the physical world pushed over into the conceptual, 119 Actuality imperceptible to the intellect, 126 Adaptability of life, 330 -- -- mathematical laws, 37-39 Adaptation, purpose of, 327-8 Adaptations made by an organism, 162 Additional freedom bestowed by higher faculties, 334 Affinity, zones of, 22, 124 Agency, interpretative, intellect as an, 166-7 Agent, ego as an, of the Thinker, 243, _ff_ -- engendering, life as an, 331 -- of creation, life as an, 343 Ahmes, an Egyptian priest, 44 Alchemy of psychogenesis, 225 Alcohols, eight different, from one formula, 155 Algebraic quantities and space, 125-7 Allowability of the rules of logic, 163 Aloofness, the, of scientific dogmatism, 340 Alphabet of space-genesis, 237 -- the geometric, 193 Amenability of mind to the laws of evolution, 28-31 Analogies, use of, to popularize the four-space, 128-9 Analogy, difference between two processes illustrated by, 74 Analysis incapable of dealing with life, 319 -- the manifold, the fiat of, 77 Analyst and the manufacture of the space-manifold, 77 -- disregards the conformity of the nature of things, 84 Analytics and the mechanical origin of the universe, 40-1 -- perisophical nature of, 320, _ff_ -- the four-space, a curiosity of, 40 Anchorage, fourth dimension denied, 265 Angularity of consciousness, 120 Answer of the senses to new needs, 332 -- the, of consciousness to realism, 97 Anti-Euclidean geometry, 69 Apotheosis of the definition, 78 Apparatus, man's perceptive, 357 Apparent vassalage of life, 329-30 Appearance, dynamic, space as, 104-7 Appetences, materialistic, of empiricists, 340 Appulses of life against matter, 328 Appurtenances of materiality, 331 _A priori_, 185, _ff_ -- -- knowledge, 110-5 Aprioriness, 112, 116 Apriority, the principle of, 113, 223 Arbitrariness of the common mensurative quantities, 41 Archeological evidences, 332 Archimedes, 70 Aristotle, 322 Arrangement of the contents of the kosmos, 317-8 Art and the criterion of truth, 323 -- -- -- higher consciousness, 181 -- evidences of new faculties in the field of, 335 Assemblage of mathetic contrivances, 335 Assumptions, category of, 189 -- procedure based upon, disadvantages of, 163 At-one-ment, 224, 252, 270 -- defined as the end of evolutionary activity, 1 Attachment of the consciousness to the order of becoming, 305 Attainment, the difficulty of, by the intellectual method, 206 -- of space-consciousness, 225 Attitude, the need of a more liberal, 340 -- towards things spatial, 306 Audital nerves, notes that transcend the grasp of, 355 Aura, pulsating, 348 Automatism of the intellect, 253, _ff_; 338 Avenues of knowledge opened to inner perceptive faculties, 353 Awakening of the faculty of awareness in a new domain, 90 -- -- -- pituitary body and the pineal gland, 353 -- pituitarial, general results of, 355-8 Awareness, a determinant of conception, 120 -- as a gauge of the existence of things, 161-202 -- degrees of, 165 -- hyperspace, a symbol of a more extensive, 180 -- progress in, 171 -- Thinker's sphere of, 274, 283 B Baltzer, 66 Barrier, the, to the Thinker's certitude, 186-8 Barriers, freedom determined by absence of, 280 Becoming, endless labyrinth of, 220 -- the kosmos, in a state of, 265 -- the order of, 305 Becomings, infinity of, 234 Becquerel rays, 336 Being, identification of consciousness with, 205 -- kathekotic, 210 Being, the interiority of, 290 -- the numericity of, 309 -- the range of, co-extensive with reality, 168 Beings, discarnate, 154 Beltrami, Eugenio, 1-2, 63-5, 66-7 -- -- and the manifold, 317 -- -- and the pseudosphere, 64-5 Besiegement of matter by life, 329-30 Bessel, on geometry as incomplete, 58 Bewilderment, new realities cause, 169-70 -- of the mind, 182 Biogenetics, passage from mechanics to, 357 -- turning to, from mechanics, 322 Birth, as a fourth dimensional process, 159 Blind men and the elephant, 257 Bolyai, Janos, 2-3, 46, 60, 66, 83, 87 Boundaries of a hypercube, 136-8 Brain, the physics of, 292 Brain-consciousness, 184, 186 -- standards of, 190 Brotherhood, and the tendencies of the Thinker's consciousness, 301 -- the keynote of the intuition, 252 Bulwarks, the formal, of geometry, 77-9 C Carpet, the sensible world as a, 196 Carus, Paul, on four-dimensional space, 254 -- -- on metageometricians, 102 -- -- on space, 110 -- -- on the representation of the tesseract, 136-7 Cayley, 3-4, 54-5, 66-7 Cell-activity as a performance _a priori_, 114 Cell-colony, 250, 301 Cell-consciousness as an aposterioristic phenomenon, 185 Centers, pyknotic, 232 Certainty, geometric, the basis of, 77-8 -- of mathetic conclusions, provisional, 37 Chaogenic period of involution, 208, _ff_ Chaogeny, 231, 233 -- the evolution of chaos into order, 4 -- the laboratory of, 193 Chaomorphogeny, defined, 4, 231-2 Chaos, as egg-plasm, 265 -- duration of, 268 -- movement of life in, 307 -- shores of, 307 Chaos-geometricity, 307 -- -intellectuality, 307 -- -materiality, 307 -- -spatiality, 307 -- -Theos-Kosmos, 215, 233-4, 237 -- -- -- a triglyph symbolizing kathekos, 11-2 Character of the universe, fixed by consciousness, 162 Characteristics, changes in the, of reality, 169 -- fourth dimensional, of the ether, 157 -- minds of similar, fall into zones of affinity, 124 -- of non-Euclidean space, 72-3 Chasm, kosmic, between the real and the ideal, 107 Chemists, speculative, and the ourth dimension, 155 Chrism, sacred, of creative mentality, 91 Circle, significance of process of squaring, 109 Circle, the symbology of, 309 Circuits, closed, our interests as, 167 Clairaudience, 352 Clairvoyance, 352 Clairvoyant perception, 349 Clause, enabling, of metageometry, 76-7 Clavius, Christoph, 46, 52, 83 Clifford, 66 Code, psychic, for systematizing cognitions, 190-1 Cognition, instantaneous, organ of, 258 -- intimacies of direct, 205 -- the method of, 186-8 Cognitions, intuitive, 145, 192-3 -- psychic code for systematizing, 190-1 Commensurable quality, dimension as, 140 Communal consciousness, 270, _ff_ Communalization and the intuition, 301 Conarium, the, 345 Conceivability and evolution of mind, 24 -- ultimate range of, 278 Conceiving, power of, derived from sense-experience, 26 Concept, as a shadow, 119 -- the hyperspace, gradual rise of, 27-8 Conception and awareness, 120 -- every, based upon prior experience, 25 -- freedom, the, of the mind from, 28 Conceptions, symbolic, dependence placed upon, 147 Concepts, and the Thinker, 244 -- intuitograms as, 248 -- quality of, dependent upon sense-experience, 26 Concepts, the perception of, 255 Conceptual, related to the objective, 291 -- the, and mathematicians, 71 Conceptualization, the act of, 4 -- the power of, dependent upon evolution of mind, 25 Conclusions, mathetic, complex dependence of, 37 Concrete hewn into shape by the intellect, 295 Congruency, the, between intellect and the universe, 319, _ff_ Congruity, the, between concepts and objects, 120 Connection, the, traced out between reality and object, 126 Co-originality of things, 303 Consciousness and kosmos, graph representing, 271 -- -- the character of the sensible world, 166 -- -- -- wake of life, 205 -- -- time, 224 -- angularity of, 120 -- _a priori_, 185 -- a replica of, and judgments, 205 -- as a scale, 165 -- -- barrier to ultimate knowledge, 207 -- -- determinant of dimension, 145 -- -- life, 295 -- -- variable quantity, 171 -- attachment of, to the order of becoming, 305 -- dawn of, and dimension, 179 -- degrees of, 162 -- deprived of the validity of notions, 314 -- evolution of, 250 -- expansion of, 164, 272 -- extended, 89-90 -- focus of, 290 Consciousness and the planes of the kosmos, 163 -- fragmentary view of, 260 -- freedom of, in the _genus homo_, 332 -- habitation of, 278 -- higher, and art, 181 -- -- planes of, 178 -- history of, 298-9 -- identification of, with objects, 189-90 -- intuitional, 255 -- keyed to the entire range of reality, 167-8 -- kinds of, 270, _ff_ -- kosmic, Elysian fields of, 235 -- of the primitive man, 253 -- organ of, 184, _ff_; 90 -- psychics of, 292 -- raised from the sensuous to the conceptual, 72 -- recipient of truths from dual sources, 26 -- sensible world as instrument of, 199 -- triple presentation of notions of space to, 72 -- turned inward, 283 -- unification of the states of, 192 -- youthful, of mankind, 122 Consequences, the science of, 85 Considerations, mathematical, and reality, 128 Consistency, kosmic, 174 -- self, of hyperspatial hypotheses, 67 -- the criterion of geometry, 86 -- -- inconsistence of, 173 -- -- metageometrician curtailed by, 85 Construction, idealized, cannot be objectified, 144, 276 -- ideal, the meaning of, 4 -- mental, trafficking in, 154 Consubstantiality of intellectuality and spatiality, 331, _ff_ Continuity of the psychic plasm, 260 Contrivances, mathetic, the passing of, 241 Convergence of parallel lines, 315 Convolutions, duplex, 146 Coördinates, as spatial determinants, 203 -- of hyperspace, 94, 97, 99, 115-6 -- systems of, 301, _ff_ -- three, 132, 232 Copolarity of ideas and objects, 127 _Corpora quadrigemina_, 345 Corpuscular orbits of particles, 152 Cosmos, significance of, 4 Creation, impregnated screen of, 232 Creations, the, of the intellect, 173 Criterion of geometry as consistency and convenience, 86 -- -- truth, 323-4 Crookes, 344 Cube, as succession of planes, 148 -- generation of the, 144 -- illustrated, 133 -- the generating, 134 Curiosities, analytic, energy spent in elaborating misappropriated, 39 Current, electric, as component in the fourth dimension, 153 Curvature of space, the doctrine of, formulated by Riemann, 5 -- -- -- valid element in the notion of, 305 -- the measure of, 61 Curved space and metageometricians, 316 D D'Alembert, _note_, 51 Dante, 181 Darwin, 181 Day, the Great Kosmic, 213 De Tilly, 66 Death, as fourth dimensional unity, 159 -- of the intellect, 195 Decrement of the diametrical ratio to the circumference of circle, 312 Deeps, fearful, of kosmic mind, 71, 91 Definition, apotheosis of, 77-8 -- as an arbitrary determination, 100 -- the deification of, 37 Degrees of realism, 164-5 Deiform, basic idea of the, 5 Demarkation between reality and phantasy, the line of, 173 Deposits of life, 173 Descartes, 322 Designs, cut out in materiality by life, 264 Details, the power to dispose of innumerable, 303 Determinations, geometrical, the necessity of, inheres in logical deductions, 78 -- the factors of conscious, 162 Determinative period of mental development, 31 Development of the intuitive faculty, future, 188 Diacritics of life, 286 Diameter, the ratio of, to the circumference, 311-2 Difference between concept and thing, 121-2 -- between mathematical and perceptual space, 120 -- between the ideal and the actual is dynamic, 108 Differential, among minds, 124 Difficulty of imagining the fourth dimension, 90 -- of propagating an intuition, 315 Difficulties in the acquisition of real knowledge, 325 -- of hyperspace, the logical, 139-41 Dilemma of metageometricians, 97-8 Dimension, a distinct stage in psychogenesis, 27 -- -- system of space measurement, 5 -- and analysis, 163 -- -- D'Alembert, _note_, 51 -- -- the action of a tartrate in, 156 -- as an arbitrary contrivance, 262 -- -- -- assemblage of elements, 93 -- -- direction, 140 -- -- extent, 98 -- -- lying near the surface of things, 159-60 -- -- space, 96 -- current definitions of, 98-100 -- denied legitimate anchorage, 265 -- determined by consciousness, 145 -- does not explain spiritualism, 155 -- evolution of the fourth, 44 -- fourth, direction of, 134 -- as "jack of all trades," 156 -- impossible to actualize, 124 -- key, to non-Euclidean geometry, 90 -- movement of a plane into, 147-8 -- no motion of material masses into, 125 -- not _a priori_, 116 -- proof of the existence of the fourth, 136 Dimensionality, and the intellect, 200 -- as dependent upon the will of the investigator, 95 Dimensionality, conception of, purely conventional, 92 -- logical difficulties which beset, 97 -- of perceptual space, 72 -- -- space, the four, 142-3 Dimensions, four, no basis in consciousness for, 172 -- piled upon one another, 300 -- vanity, the, of segmenting space into many, 299 Direction as dimension, 99 Disappointment with the sensible world, 120-1 Discoveries, metageometrical, as excrescences of mental unfoldment, 131 -- never the result of methodic reflection, 29-30 Disorder, the edict of, and space, 230 Disorderliness, the fringe of, 277 Dispossessal of the intellect by the intuition, 195 Dissimilarities, besetment of, 291 Distinctions, between conceptual and perceptual space, fundamental, 73-5 Diversity, a result of the fragmentative tendency of life, 6 -- as transfinite quantity, 42 Divinity, prize of, won by the intellect, 43 Duadic phase of evolution, 210-1 Duodim, defined, 6; 128, 129, 134, 177 -- consciousness, 163 Duopyknon, 212, 213 -- the meaning of, 6 Duopyknosis, 208-10, 218 -- as a stage in the evolution of space, 6 Duration and space, 224 -- eternal, 234 Dynamic appearance, space as, 104, 107 Dynamism, of the intellect, functional, 303 E Earth, the nebulous youth of, 335 -- a new, 358 East effloresces as West, 306 Ego, the, 270 -- and percepts, 244 -- as sovereign, 244 -- Thinker's dependence upon, 245 -- -- treatment of, 246 Egopsyche, 265 -- as an agency of self-consciousness, 258 -- defined, 7 Egos, compared to choppy sea, 256-7 Egypt, birth-place of geometry, 44 Eisenlohr, 44 Elaborative period of the non-Euclidean geometry, 66-70 -- -- of mental development, 31-32 Element of evolution, basal, 259 Elements of the non-Euclidean geometry, 79-80 Elephant and blind men, 257 Ellis, Wilmot E., on ether as four-dimensional, 157-9 Embrace of direct cognition, 205 Engenderment of space, 260 Ensoulment of the world-plasm, 329 Entities, hyperspace, and the phenomenal world, 128-130 Enumeration, reformation of the system of, 38 Enveilment of consciousness, 273-4 Environment, artificial and natural, 162 Epiphysis, the, 345 Equidistantial, described by hyperspatial perpendicular, 80 Essence, elemental, as world-plasm, 329 Ether as possessing fourth dimensional characteristics, 157 Ethics recast in a new mold, 358 Euclid, 46, 70, 83, 263 -- and the parallel-postulate, 45 -- never-dying elements of, 53 Evidences, mathematical, exemplifies intellectual evolution, x -- the vanity of fragmentary, 204 Evolution, a continuous process, 327 -- and the norms of reality, 175 -- basal element of, 259 -- commencement of, 232-3 -- intellectual, forward movement of, 184 -- kosmic, vicissitudes of, 215 -- laws of, govern mind, 28 -- mental, results of, 122 -- of faculties, synchronous with evolution of matter, 332 -- -- material characteristics of the earth, 335 -- -- organs, time required for, 188 -- preparation of the field of, 218 Exaltation of matter into spirit, 329 Examples of new perceptive evidences, 341-3 Existence on a higher plane, states of, 162 Experience, corroborative testimony of, denied hyperspace, 263 -- prior, and conception, 25 Experiences, spatial, systematization of, 78 Extension, space as an unbounded, 61 Extra-spatiality, degraded into spatiality, 261 Extravaganza, mathematical, and the fourth dimension, 156 Eyes, as Roentgen rays, 353 F Fact-mass, 289 Factors of conscious determinations, 162 -- four, of the criterion of truth, 324-5 Facts as facets of truth, 284 -- attempts to reform, 304 -- logic as symbolism of, 287 -- -- the modeler of, 288 Faculties, dual derivation of, 162 -- evolution of, synchronous with evolution of material qualities, 332, _ff_ -- extended, 239 -- foreshadowed by the hyperdimensional, 131 -- higher, man must win his way to, 357 -- new faculties evidenced by four-space, 24-29 -- rudimentary nature of, 341, _ff_ -- the source of, 349 Faculty and the intellect, 247, _ff_ -- as transmissible character, 251 -- determined by function, 336 -- greatly extended, 117 -- higher than the intellect, 126 -- I-making, self-consciousness as, 243 -- intuitive, 185 -- of perception in higher worlds, 353 -- -- awareness, the awakening of, 89-91 -- overshadowed by the intellect, 188 -- of perception in higher worlds, 353 -- rudimentary condition of, 192, _ff_ Faculty, outcropping of, 249-51 Failure of efforts to justify the objective existence of four-space, 125-6 Failures at solving the parallel-postulate, outcome of, 48, 83 Faith, dispossessed by knowledge, 358 Fay, mathetic, 160 Fechner, 39 Finity and unboundedness of space, 76 Flammarion, Camille, 341-2, 344 Flexity, as property of hyperspace, 63 Flexure, space as a, 305 Fluxional, between sense objects and ideal representations, 122-3 Fluxion, psychic, as difference between memory image and object, 7-8; 122 Focus of consciousness, 163, 290 -- -- the intellect, 310 Fohat, and the creation of morphons, 219 -- as creative energy, 213 -- Creator, 8 Fohatic energy, 226 Form, as vehicle of life, 330 -- definition of, 8 -- driven back over the path of its genesis, 327 -- pure, cannot exist in nature, 294 -- the idealty of, 110 -- the universe not a pure, 108 Formative period of mental development, 31 -- -- of non-Euclidean geometry, 55 Formula, eight different alcohols from one, 155 Four-dimensionality, justification of, 176 Four-space, a curiosity of analytics, 40 -- and Riemann, 27 -- as a divertisement, 175 -- consciousness does not act in the, 172 -- existence of, denied, 171-2 -- movement of matter in, 157 -- reality of, glibly proclaimed, 154 -- the, 8, 240 -- -- denizens of, 353 -- -- domain of, 154 -- -- study of, 124 -- use of analogies to popularize, 128 Fourth dimension, analogical reasoning of, 177-9 -- as a transcendental problem, 140 -- -- an attitude of the intellect, 200 -- electric current as a component in, 153 -- imaginability of, 106-7 -- Simon Newcomb on, 125 Fragmentariness of the intellectual method, 164 Fragmentary view of the universe, 260 Fragmentation, tendency to, 296 -- harks back to cell-division, 301 Freedom, a new, 338 -- determined by absence of bonds, 280 -- mathematical, 60 -- mental, 66-7 -- now dawning for the mind, 32 -- of consciousness in _genus homo_, 332 -- regal freedom of the mind, 118 -- three degrees of, 125 -- unrealizable, for the Thinker, 255 Fringe, chaogenetic, 308, 309 -- of disorderliness, 257 -- -- kathekosity, 229 Frischauf, 66 Full, the universe as a, 310 Function determines faculty, 336 Functions back of latent faculties, indefinite, 343 -- cellular and histologic, 253 Functioning, instinctive, of the intellect, 289 Fundamentals, totality of, kosmic, 265 G Gamut of realism, 169 Gauge, awareness as a, 161-202 Gauss, Charles Frederick, 8-9, 56-8, 59, 174 -- -- -- as formulator of the non-Euclidean geometry, 57 Geminos of Rhodes, 45, 52 Generability, as property of space, 62 -- of hyperspace, Keyser on, 144 -- -- space by lines, 143 Generation, the, of the hypercube, 134 -- of the hypertetrahedron, 135-6 Genesis, of space, 211, 227, _ff_ -- -- the earth, its nebulous youth, 335 -- -- -- form, 327 -- -- -- sensible world, 167 _Genus homo_, freedom of consciousness in, 332 Geometricity, 266 Geometries, non-Euclidean, based upon a negation of the latent geometrism, 262-3 -- three possible, 54 Geometrism, engenderment of, 262 -- established by life, 264 -- kosmic geometrism, establishment of, 231 -- latent, 257 -- native geometrism of space, 262 -- rediscovered by the intellect, 9-10 Geometrism, the basis of, 237, 261 Geometry, anti-Euclidean, 69 -- and the study of magnitudes in space, 203 -- artificial, 262-4 -- a two-fold, 59 -- breakdown of, 266 -- determinative period of the non-Euclidean, 61-4 -- diverse systems of, 323 -- Euclid's Elements of, 53-4 -- formal bulwarks of, 77-9 -- _Imaginary Geometry, The_, 60 -- natural geometry, 261 -- Plato and the divine geometry, 193 -- possible systems of, 174 -- radical essence of pure, 85 Geometry, non-Euclidean, at variance with the parallel-postulate, 83-5 -- -- based upon a misconception, 91 -- -- determined by qualitative differences, 140 -- -- elaborative period of, 66-70 -- -- first published treatise on, 60 -- -- formative period of, 55 -- -- the final issue of, 88-9 -- -- growth and development of, 92 -- -- invalidation of, 174 -- -- key to, 90 -- Lambert's Non-Euclidean geometry, _note_, 51 -- -- popularization of, 66-7 -- -- Schweikart's treatise on, 59 -- -- self-consistence of, 57 -- -- some elements of, 79-80 -- -- superperceptual knowledge of, 72 Geometry, non-Legendrean, 70 -- symbolic, and commensurable quality, 145 Gerling, as correspondent of Gauss, 58 Germ-plasm, continuity of, 260 Giantism, 347 Glimpse, a, of the reality of space, 314 Glorification of the flesh, 227 Groups, transformation, discovery of, 30 Guide-posts to a new domain, 273, 277 Gulf, interposed between manifestation and non-manifestation, 206 H Halley, memoir of, mastered by La Grange, 50 Halstead, G. B., _note_, 49, 66; _note_, 79 Heindel, Max, _note_, 353 Helmholtz, 39, 66 Hewer, the, of the concrete, the intellect as, 295 Hinton, C. H., 154 -- on the fourth dimension, 153 History, the, of mind, three great epochs of, 31-32 Homogeneity, the, of realism, 165 Hoüel, J., 60, 66 Hudson, 344 Humanity and the exalting power of life, 331, _ff_ Hypercube, boundaries of, and mirrors, 136-9 -- the generation of, 134, 145 Hyperdimensional, as a prophecy of new faculties, 131 Hyperspace, 175 -- a monument to the intellect, 356 -- and the involved procedure of arriving at a recognition of its relations, 73 -- and the passage thither, 293, _ff_ Hyperspace as a figurative mountain-peak, 200 -- -- -- movement, 44-68 -- -- an all-powerful something, 160 -- -- -- idealized construction, 10 -- -- evidence of new faculties, 24 -- -- _ignes fatuii_, 154 -- -- illusion, 202 -- a symbol of higher consciousness, 180-4 -- concept of, as an evolutionary quantity, 44-5 -- confounded with real space, 91 -- creation of, 335 -- denied the corroborative testimony of experience, 263 -- direction of, baffling to mathematicians, 146 -- discovery of, a sign of mental evolution, 180 -- domain of, a fairy-land, 239 -- magnitudes of, the non-sensuous, 72 -- mysterious hiding place in, 129 -- the logical difficulties of, 139-141 -- the six pillars of, 62-3 Hyperspatiality as the toys of childhood, 241 Hypertetrahedron, the, 135 Hypertrophy of the pituitary body, 347 Hypervolume, the, 134 Hypothesis, a superfoetated, 141 -- four-space, utility of, 129; 154 Hypotheses, admissibility of, 118 -- incompatibility of the non-Euclidean, 67; 71 -- Riemannian, 80 -- solution of, Bolyai, 31 Hyslop, James H., on the logical difficulties of hyperspace, 139-141 I Icosahedron, examination of the, 285 Ideal, and the real, kosmic chasm between, 107 -- perceptual value of, 275, _ff_ Ideas and words as symbols, 126-7 -- Malebranche on, 222 -- realism of, 24-35 -- the symbolism of, 205 Identification of consciousness with being, 205-6 -- with the objects of study, 189-90 Identity of things-in-themselves, 353 _Ignes fatuii_ and hyperspaces, 154 Illustration of plane-rotation, 148-150 -- -- the tesseract, 133 Images, totality of, recoils upon us, 167 Imaginability of the fourth dimension, 90, 106-7 Imagination, premises of, the mathematical, 146 Impossibility of plane-rotation, the structural, 151-3 Impressions, the symbolism of, neurographic, 186 Impulse, the satisfaction of the original creative, 310-1 Incomprehensibility of reality to the intellect, 126 Incongruity, life estranged by a radical, 325 Individual as space, 223 Ineptitude of intellectual determinations for vitality, 314 Infinite, interpreted in the terms of the finite, 82 Infinitesimals of unity, numbers as, 41 Infinity as a process, 109 -- of becomings, 234 Infinity of parallels through a given point, 70 -- -- space, a capital illusion, 195 -- the concept of, 277 -- -- innate dread of, 103 -- -- relativity of, 194 Influence of abstract thought, 33 -- -- Kant on the non-Euclidean geometry, 49-50 -- -- La Grange, 51-2 -- -- the intellect, 315 Infundibulum, the, 345 Inner organs of perception, 352 Innermost, the, realities of things, 356 _Insouciance_ of the geometer, 96, 294 Instant-exposure and intellect, 311 Instrument, intellect likened to a color-bearing, 313 -- for the measurement of the passage of space, 297 -- of consciousness, the sensible world as, 199 -- -- life, form as an, 328 Integers, as fractional parts of unity, 41 Intellect and its domination of the intuition, 333 -- -- -- final union with the space-mind, 194 -- -- -- topography, 312 -- -- spatiality, 263 -- -- the deposits of life, 173 -- -- -- designs cut by life in materiality, 264 -- -- -- dictum of Sensationalists, 26 -- -- -- instant-exposure, 311 -- -- -- intuitive faculty, 247, _ff_ -- -- -- prize of divinity, 43 -- as a color-bearing instrument, 313-4 -- -- -- searchlight, 168 Intellect as a fashioner of phenomena, 199, _ff_ -- -- hewer of the concrete, 295 -- -- sole interpretative agency, 166-7 -- automatism of, 253, 338 -- cannot seize life, 282 -- crowned by a diadem of gold, 338 -- dominated by the intuition, 250 -- fashioned for matter only, 231 -- follows in the grooves of logic, 294 -- hyperspace as a monument to, 356 -- in the field of vitality, ix -- its aptitude for starts and stops, 292, 302 -- -- instinctive tendency to fragmentate, 296 -- quality, determined by mathematics, 337 -- makes for individuality, 252 -- misses the ceaselessness of life, 201 -- _modus vivendi_ of, its influence upon knowledge, 184 -- the constitution of, 291 -- -- cut and mode of, 167 -- -- focus of, 310 -- -- illusion of, 246 -- -- illusionary character of, 323 -- -- incomprehensibility of reality to, 126 -- -- instinctive functioning of, 289 -- -- judgments of, 302 -- -- moods of, 202 -- -- predominating tendency in, 320-1 -- -- scientific tendence of, 165-6 -- -- struggle of, against dispossessal, 195 -- unsuited to deal with reality, 322, _ff_ Intellectuality and reality, 304 -- -- spatiality, consubstantial, 331 Intellectuality as co-extensive with spatiality and materiality, 236 -- the source of, 260 -- -- Thinker makes his own, 242 Intelligence and automatism, 253, _ff_ -- the Thinker as a pure, 243 -- transfinite intelligence and the degrees of realism, 164 Intent, the kosmic, of the intellect, 326 Interests, the sphere of our, as closed circuits, 167 Interior, the great, 290 Interiority of being, 290 Interpretation, the standards of, vary as consciousness varies, 171 Interstices of materiality, 264 Intuition and brotherhood, 252 -- -- communalization, 301 -- -- the riddle of spatiality, 325 -- as dispossessor of the intellect, 195 -- cannot be served by mathematics, 337 -- dominated by the intellect, 333 -- held in abeyance, 338, _ff_ -- its domination of the intellect, 250 -- the development of a spatial, not absurd, 145 -- -- need of a sympathetic attitude towards, 249 -- -- results of the development of, 357 Intuitional consciousness, 255 -- the superiority of, over the rational, 187 Intuitions and the lead of life, 191 -- -- -- Thinker, 27 -- free, mobile and formless, 166 -- the conceptualization of, 248 -- -- humility of, 165 -- -- nature of, 185 Intuitograms, as concepts, 248 Intuitograph as means of contacting the egopsychic consciousness by the Thinker, 10 -- as super-concepts, 255 -- the difficulty of transmitting, 315 Invariability, the vaunted, of the laws of mathesis, 37 Invariants, psychological, 24 Investigations, metageometrical, and the new sense-organs, 354 Involution, as antithesis of evolution, 10-11 -- kosmic involution, 226 -- of matter, 328 -- the movement of, 219 -- -- seven stages of, 208-12 Ions, creation of, 219 -- magnitude of a hydrogen ion, 225 J Judgments _a priori_, Kant on, 85-6 -- and the faculty _a priori_, 190-1 -- -- zones of affinity, 124 -- based upon a replica of consciousness, 205 -- no trustworthy, can be predicated upon fragmentary knowledge, 282 -- of the intellect, 302 -- the lessening of error in, 256 -- -- more complex the more at variance with the nature of things, 75 -- -- synthesis of, 257-8 -- valid judgments long delayed, 170-1 Judicative power of mathematics, 180 Justification for a multi-dimensional quality in space, 262 -- of four-dimensionality, 176 Justification of sense-deliveries by one another, 76 -- -- the existence of the fourth dimension, 125 K Kant, 85, 181, 182, 322 -- and the faculty of thinking, 261 -- -- -- idea of space, 322-3 -- influence of, on the hyperspace movement, 49-50 -- on the nature of things, 119 -- -- space as an intuition, 115 Kathekos, 233-4 -- as chaos, 266 -- -- symbol of Chaos-Theos-Kosmos, 11-12 -- symbology of, 234 Kathekosis, _note_, 227 Kathekosity, fringe of, 229 -- rock-bound coast of, 267 -- significance of, 12 Kathekotic consciousness, 272 -- period, 209 Key to the mysteries of nature, the fourth dimension as, 131 Keyser, Cassius Jackson, and freedom of the mind, 33 -- -- -- on attitude of metageometricians, 71 -- -- -- -- dimensionality, 94-5 -- -- -- -- four-dimensionality of space, 142-3 -- -- -- -- generability of hyperspace, 144 Klein, Felix, 12, 54-5, 66-7 Knowledge, all, relative, 101 -- barrier to the certitude of the Thinker's, 186-7 -- fabric of, 196 -- hypothetical nature of, 189, _ff_ -- immeasurable realm of, laid bare by the telescope, 298 Knowledge, mathematical, apriority of, questioned, 37 -- nature of the non-Euclidean, superperceptual, 72 -- real, difficulties of acquiring, 325 -- related to the stream of life and to the shore of materiality, 323 -- relative, degrees exist for, 164 -- sphere of, 184 -- systematization of, 126 -- ultimate, consciousness as a barrier to, 207 -- unification of, 256 Kosmogenesis, the latent geometrism of, 264 -- -- scope of, 237 Kosmometer, the, 297 Kosmos, and consciousness, graph representing, 271 -- arrangements of the contents of, 317-8 -- in a state of becoming, 265 -- magnitude of, 308 -- moods of, 311 -- space as the consistence of, 239 -- _see_ Cosmos, 12 L Labor, division of, between the tuitional and the intuitional faculties, 193-4 -- mathematical labors, significance of, 176 La Grange, Joseph Louis, 12, 39, 50-3, 321, 322 -- -- -- -- and the parallel-postulate, 51-2 Lambert, John, 58 -- -- and the theory of parallels, 48-9 Language, the passing of, 357 Legendre, 70 Leibnitz' dictum, 26 Leonard rays, 336 License, mathematic, permissibility of, 38 Lie, Sophus, 12, 67 -- -- and transformation groups, 30 Life, analysis incapable of dealing with, 319 -- and consciousness as one, 224 -- -- form rooted in pyknosis, 214, 328 -- -- the fourth dimension, 172 -- -- -- inaptitude of mathematics for, 179 -- -- -- power to create, 297 -- as agent of creation, 343 -- -- creative agent, 264 -- -- expression, 303 -- -- vassal of materiality, 329-30 -- causative agencies in prolonging, 123 -- current of, as engendering element, 331 -- deposits of, and the intellect, 173 -- estranged by a radical incongruity, 325 -- exalting power of, and humanity, 331, _ff_; 342 -- exhibition of its remains, 282 -- flow of, 265, 297 -- form as an instrument of, 328 -- indescribable signs of, 219 -- infinitive action of, 327 -- intellect has no aptitude for, 231 -- intuitive, values of, x -- larger life of the Thinker, 196 -- lead of, followed by intuition, 191 -- limits of, fixed by consciousness, 198, _ff_ -- most solid facts of, as shadows, 195 -- motility of, admits of endless variations, 35 Life, movement of chaos in, 307 -- passage of, 201 -- passage through spatiality, 288 -- power to manipulate, 303 -- recurrent movement of, 307 -- totality of egoic, 243 -- undulations in the current of, 329 -- uniqueness of, 41 -- wake of, and consciousness, 205, 288 Life-cycle, evolutionary results of the, 259 Life-stream, the, 278 Light, polarized, and the fourth dimension, 156 -- consciousness as a spreading, 168 Light-years and space, 278 Limits, the sphere of, consciousness as, 163 Line, as generating element, 144 -- the straight, a curved, 76 Lineage of every principle runs back to monopyknosis, 213 Lines, perpendicularity of, in four space, 130 Lobachevski, 55, 60, 66, 87 Lodge, Sir Oliver, 340-1; 344 Logic as architect, 118 -- conventional forms of, 262 -- data of, 286 -- intellect follows in the grooves laid out by, 294 -- miracle power of, over facts, 288-9 -- rules of, allowability under, 163 -- -- -- the game of, 164 -- as symbolism of facts, 287 Logos and the limits of space, 193 -- being of, 218 -- body of being of, 214 -- consciousness of, 204 -- creative, 12, 213 -- the, 216-7 Lorenz, 83 Luther, Martin, 181 M Magnitudes, geometry, a study of, 203 -- non-sensuous, of hyperspace, 72 Makrokosmic consciousness, 270 Malebranche, N., 222; see _note_ Manifestation and non-manifestation, 206 Manifold, finite, though unbounded, 70 -- manufactured by the analyst, 77 -- the, 61-2, 172 Manifoldness, analytical manifoldnesses, as mental excitants, 89-90 -- as a conventional construction, 262 -- -- an intuition, 318 -- of space, as a near-truth, 317 Manning, 82, 157 -- and maneuvers in the fourth dimension, 152 Manvantara, 213, 228-9, 232 -- as evolution and involution combined, 13 Mask, logic as a, 288-9 Mass-termini, of lines, 236 Mastery of life over matter, 330 -- of the sensible world, 196-7 Materiality as a deposit of life, 295 -- -- consubstantial with spatiality, 331, _ff_ -- becoming spatialized, 261 -- characteristics of, 260 -- engendered by kosmic mind, 261 -- engenderment of, 294 -- interpenetrative with spirituality, 236 -- interstices of, 264 -- neglect of, 320 Materiality, shore of, 323 -- transmuted into spirituality, 328 Mathematicians and the definition, 37 -- -- -- limitations of consciousness, 178 -- -- -- phenomenal world, 118-9 -- as prophets, 336 -- respected by humanity, 337 -- the gods of mathesis, 84 Mathematics, a determinant of the quality of the intellect, vii -- and the criterion of truth, 323 -- -- -- kosmic intent of the intellect, 326 -- as symbology, 184 -- Euclidean, 140 -- fails when it encounters life, 337 -- its inaptitude for life, 179 -- -- kingdom yielded to kinetics, 358 -- possesses no judicative power over life, 180 -- the orthodoxy of, 34 Mathesis, and conceptional space, 203 -- conduct of the intellect in the field of, vii -- definition of, 13 -- domain of, as origin of fourth dimension, 130 -- gods of, 84 -- marvelous domain of, 118 -- realm of, not submissive to laws of sensible space, 34 -- things of, as emblems of kosmic forces, 239 -- world of, 67, 127 Matter and mind, syncretization of, 299 -- appulses of life against, 328 -- as a deposit of life, 330 -- honeycombs of, 339 Matter metamorphosed by life, 328 -- mind as wedded to, 249 -- movement of, in four-space, 157 -- qualities of, and faculties, 332 -- seven planes of, 212 -- spiritualization of, as end of evolution, 327 -- totality of, acted upon by totality of life, 328 -- unity of, with space, 260 -- universum of, 261 Matutinal ceremonials of creation, 221 Measurability, as property of space, 62-3 Measurement of hyperspaces, the science of, 73 Measurements, space determined by the number of, 61 -- systems of space, 91 Measurer of space, _n_-dimensionality as a, 296-7 _Mecanique Analytique_, 50-1, 321 Mechanics of matter, the passage from, to the dynamics of life, 326 -- to biogenetics, the passage from 357 -- the turning from, to biogenetics, 322, _ff_ Mechanism, the doctrine of, due to analytics, 40 Memories, stored in the omnipsyche, 258 Men as gods, 289 Mental evolution, determinative period of, 31 -- -- elaborative period of, 31-32 -- -- formative period of, 31 Mentality, inner mysteries of creative, 91 -- infantile out-feelings of the, 336 -- the principle of, 210, _ff_ Mentalities, the adaptation of phenomena to, 184 Mentograph, the, 121 -- the basis of intellectual consciousness, 13 Mesozoic man and his environments, 347-8 Metageometrical investigations and the new sense-organs, 354 Metageometricians and hyperspace, 238 -- -- proof of rotation about a plane, 149-151 -- -- the curved space, 316 -- -- -- fourth dimension, 145 -- -- -- key to the mysteries of nature, 131 -- baffled by the direction of hyperspace, 146 -- dilemma of, 97-8 -- eschew sense-data, 73-4 -- Keyser on the attitude of, 71 -- perceptual obliquity of, 102 -- -- seed thought of, 182 -- turned to idealized constructions, 263 Metageometry, as a stepping stone, 238 -- defined, 13 -- Riemann, the father of, 63 -- the fabric of, 66 Metamorphosis of matter by life, 328 -- of the monopyknon, 214 Metamorphotic stage of space-genesis, 228 Metaphysician, the preserves of, usurped by mathematicians, 141 Meta-self as medium of kosmic consciousness, 13, 217 Methods of the ego, 344 -- scientific, and the ample explication of phenomena, 154 Mikrokosmic consciousness, 270 Mind, amenability of, to the evolutionary movement, 28-32 -- and freedom from conception, 28 -- -- -- -- problems of physical existence, 32 -- -- the new freedom, 33 -- as vehicle of life, 285-6 -- -- wedded to matter, 249 -- coevalism of, and space, 240 -- consubstantial with space, 239-40 -- divine, of the kosmos, 176 -- evolution of, and conceivability, 24 -- of the duodim, 128 -- powers of, not attained simultaneously, 334 -- profound influence of hyperspace on the, 91 -- space as progenitor of, 224 -- three great epochs of the history of, 30-31 -- tuitional, the limitations of, 192-3 -- the unity of, with space, 224 Mind-principle, the quintopyknon as the basis of, 221, _ff_ Mirrors, three, and the hypercube, 136-8 Mobility, life as, 264 Molecule, the four-dimensional, Newcomb on, 159 Monadic phase of evolution, 210-1 Moneron and man, the gulf between, 28 Monopyknon, the, 213, 216 Monopyknosis, defined, 17, 208-10, 218 Monstrosities, mathetic, 202 Moods of the kosmos, 311 Morphogeny, 13, 233-4 Morphons, the creation of, 219 Motility, the scope of, 131 Movement, forward, in knowledge, 336 -- hyperspace as an evolutionary, 44-68 -- -- influence of Kant on, 49-50 -- of life in chaos, 307 Multi-dimensional quality of space, justification for, 262 Multiplication of hypotheses, 118 Mutuality, intuition as promoter of, 301, _ff_ Mysteries, inner, of creative mentality, 91 -- kosmic, low-lying plains of, 239 --- of life and mind, 326 -- -- the sciences and the four-space operator, 131 Mystery, kosmic, the revelation of, foreshadowed in manifold, 77 -- of reaction, 167-8 -- -- space decreases as copes of consciousness increases, 273, _ff_ N _n_-Coördinates, 203 _n_-Dimensionality, as a phase of geometry, 300 -- -- property of intellectuality, 269 -- -- quality of conceptual space, 14 -- irreconcilable with perceptual space, 156 -- no justification for, 235 -- not surprising that the intellect fell upon, 296, _ff_ -- predicates concerning, 158 _n_-Space, imperceptibility of, 94 Nasir-Eddin, 46, 53 Natural symbols, ideas as, 126 Nature, representative, of things, 119 -- the vacuum-stillness of, 339 _Naught_ to _unity_, consciousness registers from, 165 Near-truth, 300, _ff_ -- as variation from criterion of truth, 324 -- based upon partial knowledge, 14 -- space-curvature as a, 306 -- space-manifold as a, 318 Nebulosity, 267 Necessity, as a bulwark of geometry, 77-8 Nerve-body, the, 349 Neurogram, 14 Neurograph, the Thinker's scrutiny of, 190 Neurographical communications, 244 Newcomb, Simon, 14-5, 66 -- -- and the four-dimensional molecule, 159 -- -- on the fourth dimension, 125 Non-Euclideans and the meaning of dimension, 97 -- -- things as they actually are, 71 -- dilemma of, 86 -- meaning of the term, 69 -- results obtained by, 88 Non-manifestation, as antithesis of manifestation, 106 -- and manifestation, 206 Non-methodic advancement of human consciousness, 299 Non-spatiality, cut off from spatiality, 266 Norm, as a value assigned by the intellect, 167 -- consciousness as a, 15; 161-202 -- of life, found by synthesis, 300 -- -- the natural geometry, 263 Note, the dominant, of communality, 301 Notions of space, triple presentations to the consciousness, 72 Number, as a phase of kosmogenesis, 309 -- numbers as infinitesimals of unity, 41 Numericity of being, 309 O Objects and ideas, 126-7 -- passage of, into the fourth dimension, 130 -- of study, identification of consciousness with, 189 Obliquity, perceptual, of metageometricians, 102 Obscurantism, mathetic, 154 Occultist and the scientist, 350 Omnipsyche, 265 -- as agency of kosmic consciousness, 258 -- -- neglected factor of evolution, 259 -- defined, 15 One-space represented by a line, 134 Ontogeny recapitulates phylogeny, 27, 334 Order, kosmic, the difference between, and hyperspace, 143 -- mathematical, discovered by the intellect, 261 -- the fiat of, 208 -- the totality of, 265 Orderliness, 266 Organ, determined by function to be performed, 336 Organs broken up by life into special parts, 343 -- the evolution of, 188 Orthodoxy of mathematics, 34 Outcome of new adjustments, 256 Out-feelings of the Thinker, infantile, 336 Outlook, the spiritualization of our mental, 40 P Papyrus, a hieratic, 44 Parallel-postulate and the surface of a sphere, 70 -- as basis of non-Euclidean geometry, 54 -- -- stated by Euclid, 55 -- -- -- -- Lobachevski, 55 -- -- -- -- Manning, 82 -- failures to demonstrate, 48, 82-3 -- in the Elements of Euclid, 16 -- Saccheri's proof of, 47 Parallels, convergence of, 80 -- infinity of, through a given point, 70 -- meet at infinity, 88 -- theory of, 47-9 Passage from mechanics to biogenetics, 357 -- from mechanics to dynamics, 326 -- from three-space to four-space, 292-3 -- of mathetic contrivances, 241 -- of space, 16, 297 Pathway of life, sinuosities in, 330 Patterns of the intellect, ready-made, 166 Pentagrammaton of space, 237 Perceptibility in the Thinker, the faculty of, 188 Perception, domain of, and the sensible world, 19 -- inner organs of, 352 -- replaced by conception, 255 Percepts, and the ego, 244 Period, chaomorphogenic, 208, _ff_ -- determinative, 61-5 -- kathekotic, 210 Perisophical nature of analytics, 320, _ff_ Perisophism, a, 300 Permissibility of mathetic license, unlimited, 38 Perpendicularity, 132, 177 -- of lines in the four-space, 130 Phantasy and reality, line of demarkation between, 173 -- the world of, 146 Phantom-ideal, the, 277 Phase of the world age, evolutionary, 209 Phenomena, efforts to explain, on the basis of the fourth dimension, 129 -- fashioned by the intellect, 199 -- fragmentary interpretation of, 302 -- mind's method of apprehending, 186 -- physical phenomena amply explained by scientific methods, 154 Phenomenal, the inverse of realism, 278 Philosophy and the criterion of truth, 323-4 -- regeneration of, 358 -- systems of, and zones of affinity, 124 Phosphorescence, dim, added to the unilluminated pool of sense-consciousness by the intuition, 26 Phylogeny and ontogeny, 334 -- represented in ontogeny, 27 Physical as embodiment of all possibilities, 227 -- phenomena as space-activities, 229 Physicality, the principle of, 210, _ff_ Physics of the brain, 292 _Pi_ proportion, significance of, 311-2 Pickering on space-curvature, 279 Pineal gland, energized by the pituitary body, 348-9 -- -- the, 344, _ff_ -- -- not a vestigal organ, 354 Pituitarial awakening, general results of, 355-8 Pituitary body and the Thinker, 351 -- -- as an organ, 344-352 -- -- C. De M. Sajous on, 345 -- -- hypertrophy of, 347 -- -- location of, 246 -- -- the, 344, _ff_ Plane, as generating element, 144 -- movement of, into the fourth dimension, 147 -- realism appears to be divided into, 164 -- rotation about a, 130, 146-151 -- the seven planes of matter, 212 Planes, a cube as a succession of, 148-9 Plasm, kosmic, strivings of, 215 -- psychic, 260 -- the differentiation of, 214 Plato, 322 -- and the divine geometry, 193, 237 -- -- -- shadow consciousness, 281 -- on God as geometrizer, 181 -- -- ideas, 34 Play, sensuous, of the intellect, 295 Plenum, space as a, 230 -- the universe as a, 107 Poincairé, 90 Point, an infinity of parallels through a given, 70 -- as generating element, 144 -- position of, 132 -- succession of points, a line as, 148 Polarity between concrete and abstract, 294-5 Ponderability, as property of hyperspace, 63 Popularization of the fourth dimension, 128 -- of the non-Euclidean geometry, 66-7 Possibilities in the world of hyperspace, 129 -- of four-space, the marvelous, 130 Postulate-systems, the multiplication of, 85 Powers of mind, not attained at one and the same time, 334 Pralaya, 213 -- as kosmic quietude, 17 Predicates, all mathematical, not justified by the phenomenal, 128 Principle, ensouling, 214-5 Printing press, the invention of, and the laws of psychogenesis, 30 Problem, the fourth dimension as a transcendental, 140 Procedure, the involved, of arriving at hyperspace notions, 73 Process, evolution a continuous, 327 -- space as a dynamic, 307 Proclus, 83 Profundities, kosmic, and hyperspace, 156 Progression eastward terminates at the west, 306 Proof of rotation about a plane, 146-151 Propagation of intuitographs, difficulty of, 315 Propositions, geometric, subjunctive quality of, 38 Pseudosphere and the shape of space, 70 -- as basis of Beltrami's calculations, 17 -- the nature of, 64-5 Psychics of consciousness, 292 -- transmuted, 260 Psychogenesis, alchemy of, 225 -- kosmic, 272 -- laws of, and the printing press, 30, 116 Psychogenesis, mind's place in, 123 -- outgrown phase of, 302 -- outstanding facts of, 131 Psychogeny, defined, 17 Ptolemy, 45, 52, 83 Publicist, the mathematical, and the duodim, 128 -- the non-Euclidean, 146 Pyknon as a kosmic principle, 212 -- -- basis of space-genesis, 17 Pyknosis, 212 -- and involution, 210, 231 -- as a metamorphotic process, 214 -- the seven processes of, 17, 294 Q Quality, dimension as commensurable, 140 Quantities, algebraic, cannot represent space, 125-7 Quantity, consciousness as a tri-space, 172 Quartodim, 18, 134 Quartodimensionality, 292 Quartopyknosis, 17, 208-10 Quartopyknotic, principle as basis of spirituality, 228 Quintopyknon, 220 Quintopyknosis and the _a priori_, 223 -- meaning of, 17, 208-10 Quintopyknotic principle as basis of mentality, 228 R Race, humanity to evolve a new, 358 -- resplendent, 338 Rajah-Tamas, 220 Rajah Yoga, 357 Rational, valueless when unsanctioned by the intuition, 187 Reaction, as a mystery, 167-8 Real, Thinker as a part of the, 201 Realism and its degrees, 164-5, 279 -- -- the Thinker, 315 -- as life, 286 -- consciousness as touch-stone of, 168 -- found in the direction inverse to the phenomenal world, 278 -- homogeneity of, 165 -- illusionary nature of, 36 -- infinite gamut of, 36 -- its degrees and the states of consciousness, 197 -- new scope of, bewilders the mind, 170-1 -- not of mathematical import, 290 -- of concepts and the sensible world, 40 -- -- the domain of mathesis, 33-4 -- psychological quality of, 285 -- the thread of, 169 -- three dimensional scope of, 172 Realities, abstractionizing, 144, 294 -- incomprehensible to the intellect, 126 -- innermost, of things, 356 -- lesser give way to greater, 195 -- natural symbolism of, 128 -- our, non-existent to beings on spirit-levels, 35 -- shaped upon conventional models, 294 -- supersensuous, 180 Reality, as life and consciousness, 173 -- barrier to the cognition of, 189, _ff_, 273 -- comprehension of, 165, 303 -- current of, 322 -- defined as life, 18 -- flow of, 319 -- flowing stream of, 124 -- myriad ways of presenting itself, 287 -- naked contact with, 193 -- not an inscrutable quantity, 194 -- obscured, 302 -- of the four-space glibly asserted, 154 -- thread of, 200 -- universum of, 127 Realms, supersensuous, 338 Re-becoming, life and consciousness as a, 257 Reciprocity of consciousness and realism, 197-8 -- of the manifest and the unmanifest, 217 Recoil of images upon us, 167-8 Reflexive development of the intellect, 254 Relativity of all knowledge, 101 Religion and realism, 170 -- the changes in, 358 Remains of life exhibited to the intellect, 283 Remaking of moods by the intellect, 201 Replica of consciousness as basis of judgments, 205 Representation, sensuous, compared with a shadow, 119-120 Research, improvement in the methods of, 125 Reservoir, psychic, of evolution, 260 Residuum, the unexplained, 157-8 Revelational impressions, 29 Riddle of spatiality and the intuition, 325 Riemann, G. F. B., 18, 19, 54, 66-7, 87 -- and a limited space, 108-9 -- -- the curvature of space, 5 -- -- -- determinative period, 65 -- -- finite space, 103 -- -- -- four-space, 27 -- -- -- manifold, 317 -- -- -- as inventor of the manifold, 86 -- on the bases of geometry, 61-4 Roentgen rays, 336 Rotation about a line, 147-8 -- about a plane, 146-151 -- -- -- -- illustrated, 148-50 -- intra-corpuscular, 152 Rudimentary organs, vague functions lying back of, 343 Rule in the evolution of faculties, 333 S Saccheri, Girolamo, 47-8, 52, 83 Sajous, C. De M., on the pituitary body, 345 Schematism, the suitability of the present, 320-1 Schweikart, Ferdinand Karl, 19, 66, 69 -- on the non-Euclidean geometry, 58-60 _Science Absolute of Space_, 60 Science and reality, 170-1 -- no longer empirical, 358 -- skeptical attitude of, 349 Scientist, method of the, opposed to the analyst, 84-5 Scopographic impressions, 19, 190 Screen, the impregnated, of creation, 232 Sea, choppy, and egos, 256-7 Search, the path of, for spatial understanding, 278 Seb, the god, 215 Sects, coplanar, 79 Self and the immensity of space, 223 -- of the universe, 217 -- the kosmic, 278 Self-consciousness, 270 -- and space, 223 -- as I-making faculty, 243 -- determination of, 302 Self-consistency, 173 _Sella turcica_, the, 345 Semi-Euclidean geometry, the, 70 Sensationalists on the intellect, 26 Sense-data, spurned by the metageometrician, 73 Sense-delivery, one, justified by the other, 76 -- symbolism of, 203-4 Sense-knowledge, the sphere of, extended, 339-40 Senses, the answer of, to new needs, 332 Sensible world, as domain of perception, 19 Sensographic impressions, as perceptions, 19 Senso-mechanisms of the intellect, 165 Sentience, the principle of, 210, _ff_ Septopyknon, the, 227 Septopyknosis, as kosmic materiality, 18, 208-10 Septopyknotic, the, principle, 228 Seven planes of matter, 212 Sextopyknon, the, 209 Sextopyknosis, defined as kosmic sensibility, 18, 208-10 Sextopyknotic, as emotional principle, 228 Shadow likened to a sense presentation, 119 Shadowgraphs, 281 Silences, nature's great, 339 Similitude of agent and principle, 243-6 Sinuosities in the pathway of life, 330 Socrates, 322 Solitariness of intellectual testimony, 76 Space and edict of disorder, 230 -- -- four-dimensional entities, 128-130 Space and quartopyknons, 219-20 -- -- self-consciousness, 223 -- -- the idealized construction, 144 -- as a dynamic process, 216 -- -- all-mother, 229-30 -- -- an assemblage of spheres, 143 -- -- becoming, 195 -- -- boundary of another space, 49-50 -- -- consistence of the kosmos, 239 -- -- curved, 164 -- -- dynamic appearance, 228, 307 -- -- eldest born of kathekosis, 215 -- -- engendered product of life, 308, _ff_ -- -- evolution, 226, _ff_ -- -- finite extension, 62, 106 -- -- generable quantity, 103 -- -- infinite continuum, 70 -- -- intuitional concept, 225 -- -- kosmic order, 265, 267 -- -- nether pole of non-manifestation, 207-8 -- -- path of life, 293-4 -- -- pseudosphere, 70 -- -- sheer roominess, 115 -- -- system of coördinates, 99 -- -- unbounded extension, 61 -- best study of, the consciousness, 269 -- conceptual, as basis of non-Euclidean geometry, 72 -- -- fundamentally distinct from perceptual, 72-75 -- elliptical, 67 -- engenderment of, 260 -- essential nature of, 212 -- fabric of, the, lends itself to measurement, 99-100 -- four-, 8 -- fractionalized space, 296 -- foundations of, 221 -- four-dimensional, 41 -- Paul Carus on, 354 -- fourth dimension of, efforts at making it thinkable, 125 -- generability of, by lines, 143 -- genesis of, 227, _ff_ -- geometric, purely formal construction, 75-6 -- limits of and the Creative Logos, 193 -- manifoldness of, 77 -- mind and, the coevalism of, 240 -- mystery of, 278, _ff_ -- nature of, answered to by mind, 198-9 -- nether pole of non-manifestation, 207-8 -- non-Euclidean, the characteristics of, 72-3 -- not a pure form, 110 -- passage of, 16 -- path of search for, must be Thinker-ward, 278 -- perceptual, irreconcilable with the fourth dimension, 156 -- possible curvature of, and the non-Euclidean geometry, 76-7 -- primeval, and tridimensionality, 236 -- problem of devising a, 87-8 -- psychological nature of, 305 -- real, confounded with hyperspace, 91 -- Riemann on the curvature of, 5 -- the unity of, 224 -- -- -- -- with matter, 260 Space-activities, as chemism and physicism, 229-30 Space-center, the, 306-7 Space-consciousness, 225 -- as a direct process, 224 -- mergence of, with the individual consciousness, 195 Space-curvature, as an arbitrary construction, 262 -- as near-truth, 306 -- nature of, 279 -- no need for, 237 -- translation of, in terms of the intellect, 315 Space-forms, the construction of, 74 Space-genesis, 20, 232, 233 -- alphabet of, 237 -- completion of, 226 -- norm of, 174 -- pyknon as a basis of, 17 -- symbolism of, 208, _ff_, 235 Space-manifold, the, 279 Space-measurement, and reality, Euclid's system of, 96 -- systems of, 201 Space-mind and archeological evidences, 193 -- -- knowledge, 187 -- -- the Thinker, 253, 272 -- attainment of, 225, 231 -- granaries of, 235 -- the realization of, 238 Space-process, the, 305 Space-realities, hyperspace as a stepping stone to, 238 Spark, the omnipsyche as a, 258 Spatiality and the intellect, 263-4 -- consubstantial with materiality, 20, 331, _ff_ -- cut-off from non-spatiality, 266 -- interpenetrative with materiality, 236 -- nonconformity of, with logic, 164 -- riddle of, and the intuition, 325 --, rise of, 180 -- Thinker's outlook in, 316 -- world of nascent, 232 Spencer, Herbert, 322 Spinoza, 322 Spiritism, the phenomena of, 154 Spiritualists, the claim of, regarding the fourth dimension, 154-5 Spirituality, materiality transmuted into, 328 -- principle of, 210, _ff_ -- seeds of, 221 Spiritualization of the flesh, 227 -- -- man's mental outlook, 40 -- -- matter, the end of evolution, 327 Squares, the form of, in the non-Euclidean geometry, 59 St. Marie, Flye, 66 Stage of pyknosis, 208, _ff_ --, quartopyknotic, 219, _ff_ Stanley, Hiram M., on space as dynamism, 104-7 States, the two, of consciousness; one unaware of the other, 192 Straight, a, determined by two points, 79 Stuff, etherealized four-dimensional, 160 Sum, angular, 79 -- triangular, less than two right angles, 59 Superconcepts, as intuitographs, 255 Superconceptual, as the intuitional, 20 Superperception and automatism of the intellect, 20 Superperceptual, the, 74 Superunodim, the, 177 Swedenborg, 180, 184 Symbol, psychic, and the brain, 190 -- words as symbols of ideas, 126-7 Symbolism of life, 286 -- -- neurographical impressions, 186 -- -- realities, 128 -- -- sense-deliveries, 204 -- -- space-genesis, 208, _ff_ Symbology, depicting planes of consciousness, 270 -- of mathematical knowledge, 203, _ff_ Synchronous evolution of faculties and properties of matter, 332 Syncretization of mind and matter, 299 Synthesis and the norm of reality, 300 Synthetic school of geometry, 56 T Tactographic impressions, 190 Tartrate and a component in the fourth dimension, 156-7 Task of mental evolution, the supreme, 169 Taurinus, 56-7, 69 Telescope, its effect upon consciousness, 298 Tendence, intellect as scientific, 165-6 Tendencies, destined to flower as faculties, 89-91, 332 Tendency to fragmentate, the intellect and, 296 Tesseract as represented by Carus, 136-8 -- illustrated, 133 -- the, 20, 145 -- the elements of, 135 -- the four coördinates of, 132 Testimony of the intellect, incapable of comparison, 76 Tetragrammaton, the, 308 Tetrahedron, the, 135 Thalamencephalon, the, 345 Thales, 44 Theory of knowledge, 290 Thingness of objects and the impossibility of comprehending, 122 Things, the dual nature of, 119 -- -- value of, 161 Things-in-themselves, the identity of, 353 Thinkability of the fourth dimension, 125 Thinker as a pure intelligence, 184, 248 -- -- engenderer of the intellect, 291 -- -- principal to the ego, 243, _ff_ -- -- six times removed from the sensible world, 201 -- -- the source of the intuition, 26 -- and concepts, 244 -- -- his approach to the kosmic mind, 202, 223 -- -- -- larger life, 196 -- -- -- treatment of the ego, 246 -- -- -- of sense-impressions, 74-5 -- -- realism, 315 -- -- space, 193 -- -- a segment of reality, 171 -- the, and the amplitude of the degrees of realism, 198, 270 -- and the barriers of consciousness, 207 -- -- -- brotherhood of man, 301 -- -- -- degrees of realism, 175 -- -- -- manipulation of concepts, 255 -- -- -- nature of space, 268 -- -- -- neurograph, 190 -- -- -- new faculty, 335 -- -- -- perceptive faculty, 188 -- -- -- pituitary body, 351 -- defined as the spiritual man, 21 -- his consciousness and the ether, 280 -- -- dependence upon the ego, 245 -- -- method of contacting the external world, 189-191 -- -- outlook upon the universe of spatiality, 242 Thinker, his schematism of cognitive powers, 206 -- -- sphere of awareness, 283 -- the difference between, and the intellectuality, 256 -- -- evolutionary needs of, 336 -- -- upreaching yearnings of, 336 Thinker-ward, space must be sought in a direction, 278 Thinking, abstract, elevating influence of, 33 Thought, prolonged abstract, benefits of, 249 Time, as an aspect of consciousness, 224 -- divested of timeliness, 279 Todhunter, 46 Tool, fashioning, life as a, 264 Topography of the intellect, 312 Touchstone, consciousness as a, 168 Tracery of connection between ideas and objects, 126 Trafficking in mental constructions, 154 Transcendental, the realm of, and mathematicians, 142 Transfinite as a limit, 122, 216 Transfinity, 21-2 Tree-reality, 126 Triadic phase of evolution, 210-1 Triangles, angular sum of, 78, 88 Tridim, the, 22, 129, 134, 147, 177 Tridimension, the, 147 Tridimensionality and primeval space, 236 -- and the space-mind, 193 -- a quality of perceptual space, 22 -- mastery of the phenomena of, 172 -- the sufficiency of, 198 Tripyknon, the, 212, 213 Tripyknosis, 17, 208-10, 219 Truth compared with facts, 288 -- criterion of, 5, 323-4 -- facets of, 284 -- kinds of, 324 -- logic does not illuminate, 287-8 Tuitive, the, and the intuitive faculties, 192-3 U Ultima Thule, the, 207 Unbounded, the, as a finite extension, 70 Undulations, three, in the current of life, 329 Unfoldment, mental, and metageometrical discoveries, 131 Uni-circle, the, 308 Unification of all knowledge, 256 Uniqueness of real space, 95 Unitariness of all existences, 241 Unity, as end of analysis, 42 -- death as a fourth dimensional, 159 -- kosmic, integers as fractional parts of, 41 -- of mind and space, 224, 230 -- the new realization of, 225 Universality, geometric, based upon the formal character of assumptions, 77-8 Universe, a glorified, 226, 268 -- as a full, 308 -- -- -- plenum, 107 -- and the seven planes of matter, 212 -- character of, fixed by consciousness, 162 -- limited and conditioned, 127 -- not a pure form, 108 -- the perfected, and the circle, 310 -- -- theory of the mechanical origin of, due to analytics, 40 -- unity in the, 301-2 Universum of appearance, 187 -- of life and consciousness, 257 Universum of reality, 127 -- -- space, 269 Unknowable as a symbol, 194 -- the darkness which shuts out the, 207 Unodim, the, 129, 134, 176-7 -- -- consciousness, 163 -- defined, 22 Up-raisement of matter, 329 Upspringing of a new faculty, 333 V Vacuum-stillness of nature, 339 Validity of mathematical conclusions, 101 Value assigned by the intellect to the sensible world, 167 -- assumptional, vitiating influence of, 163 -- -- of the ideal, 275 Vanity of fragmentary evidences, 204 -- of intellectual method, 325, _ff_ -- -- segmenting space into many dimensions, 299 Vassal, life as the, of materiality, 329 Vehicle of life, form as, 330 -- -- -- mind as, 286 Vicissitudes of kosmic evolution, 215 Vision, the inner, 356 W Wachter, 69 Wallis, John, 46, 83 Weissman, _note_, 260 Words, as symbols of ideas, 126-7, 204 World and the child mind, 121 -- as instrument of consciousness, 199, 298 -- fabric of, and geometrism, 261 -- impossibility of objectifying the fourth dimension in the perceptual, 124 -- of phantasy, conditions of, identified with sensible realm, 146 -- phenomenal, and hyperspace entities, 128-130 -- sensible, as a carpet, 196 -- the domain of a perception, 19 -- -- genesis of, 167 World-plasm, kathekotic, 267 -- as elemental essence, 329 Y Yoga, Rajah, 357 Youth of the earth, the nebulous, 335 Z Zollner, and the claims of the spiritualists, 154 Zones of Affinity, 22, 124 Zoometer, the, 297 Transcriber's Notes Variations in spelling, punctuation and hyphenation have been retained except in obvious cases of typographical errors. Inconsistencies between the text and index have been resolved in favour of the text. Duplication of the sub title (The Mystery of Space) on consecutive pages has been removed. Italics are shown thus _italic_. In the five cases where illustrations could be realistically rendered using only text symbols this has been done. 
39300	----	THE MACMILLAN COMPANY NEW YORK · BOSTON · CHICAGO · DALLAS ATLANTA · SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON · BOMBAY · CALCUTTA MELBOURNE THE MACMILLAN COMPANY OF CANADA, LIMITED TORONTO THE PSYCHOLOGY OF ARITHMETIC BY EDWARD L. THORNDIKE TEACHERS COLLEGE, COLUMBIA UNIVERSITY New York THE MACMILLAN COMPANY 1929 _All rights reserved_ COPYRIGHT, 1922, BY THE MACMILLAN COMPANY. Set up and electrotyped. Published January, 1922. Reprinted October, 1924; May, 1926; August, 1927; October, 1929. · PRINTED IN THE UNITED STATES OF AMERICA · PREFACE Within recent years there have been three lines of advance in psychology which are of notable significance for teaching. The first is the new point of view concerning the general process of learning. We now understand that learning is essentially the formation of connections or bonds between situations and responses, that the satisfyingness of the result is the chief force that forms them, and that habit rules in the realm of thought as truly and as fully as in the realm of action. The second is the great increase in knowledge of the amount, rate, and conditions of improvement in those organized groups or hierarchies of habits which we call abilities, such as ability to add or ability to read. Practice and improvement are no longer vague generalities, but concern changes which are definable and measurable by standard tests and scales. The third is the better understanding of the so-called "higher processes" of analysis, abstraction, the formation of general notions, and reasoning. The older view of a mental chemistry whereby sensations were compounded into percepts, percepts were duplicated by images, percepts and images were amalgamated into abstractions and concepts, and these were manipulated by reasoning, has given way to the understanding of the laws of response to elements or aspects of situations and to many situations or elements thereof in combination. James' view of reasoning as "selection of essentials" and "thinking things together" in a revised and clarified form has important applications in the teaching of all the school subjects. This book presents the applications of this newer dynamic psychology to the teaching of arithmetic. Its contents are substantially what have been included in a course of lectures on the psychology of the elementary school subjects given by the author for some years to students of elementary education at Teachers College. Many of these former students, now in supervisory charge of elementary schools, have urged that these lectures be made available to teachers in general. So they are now published in spite of the author's desire to clarify and reinforce certain matters by further researches. A word of explanation is necessary concerning the exercises and problems cited to illustrate various matters, especially erroneous pedagogy. These are all genuine, having their source in actual textbooks, courses of study, state examinations, and the like. To avoid any possibility of invidious comparisons they are not quotations, but equivalent problems such as represent accurately the spirit and intent of the originals. I take pleasure in acknowledging the courtesy of Mr. S. A. Courtis, Ginn and Company, D. C. Heath and Company, The Macmillan Company, The Oxford University Press, Rand, McNally and Company, Dr. C. W. Stone, The Teachers College Bureau of Publications, and The World Book Company, in permitting various quotations. EDWARD L. THORNDIKE. TEACHERS COLLEGE COLUMBIA UNIVERSITY April 1, 1920 CONTENTS CHAPTER PAGE INTRODUCTION: THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS xi I. THE NATURE OF ARITHMETICAL ABILITIES 1 Knowledge of the Meanings of Numbers Arithmetical Language Problem Solving Arithmetical Reasoning Summary The Sociology of Arithmetic II. THE MEASUREMENT OF ARITHMETICAL ABILITIES 27 A Sample Measurement of an Arithmetical Ability Ability to Add Integers Measurements of Ability in Computation Measurements of Ability in Applied Arithmetic: the Solution of Problems III. THE CONSTITUTION OF ARITHMETICAL ABILITIES 51 The Elementary Functions of Arithmetical Learning Knowledge of the Meaning of a Fraction Learning the Processes of Computation IV. THE CONSTITUTION OF ARITHMETICAL ABILITIES (_continued_) 70 The Selection of the Bonds to Be Formed The Importance of Habit Formation Desirable Bonds Now Often Neglected Wasteful and Harmful Bonds Guiding Principles V. THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS 102 The Need of Stronger Elementary Bonds Early Mastery The Strength of Bonds for Temporary Service The Strength of Bonds with Technical Facts and Terms The Strength of Bonds Concerning the Reasons for Arithmetical Processes Propædeutic Bonds VI. THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF PRACTICE AND THE ORGANIZATION OF ABILITIES 122 The Amount of Practice Under-learning and Over-learning The Organization of Abilities VII. THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION OF BONDS 141 Conventional _versus_ Effective Orders Decreasing Interference and Increasing Facilitation Interest General Principles VIII. THE DISTRIBUTION OF PRACTICE 156 The Problem Sample Distributions Possible Improvements IX. THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND GENERAL NOTIONS IN ARITHMETIC 169 Responses to Elements and Classes Facilitating the Analysis of Elements Systematic and Opportunistic Stimuli to Analysis Adaptations to Elementary-school Pupils X. THE PSYCHOLOGY OF THINKING: REASONING IN ARITHMETIC 185 The Essentials of Arithmetical Reasoning Reasoning as the Coöperation of Organized Habits XI. ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE SCHOOL 195 The Utilization of Instinctive Interests The Order of Development of Original Tendencies Inventories of Arithmetical Knowledge and Skill The Perception of Number and Quantity The Early Awareness of Number XII. INTEREST IN ARITHMETIC 209 Censuses of Pupils' Interests Relieving Eye Strain Significance for Related Activities Intrinsic Interest in Arithmetical Learning XIII. THE CONDITIONS OF LEARNING 227 External Conditions The Hygiene of the Eyes in Arithmetic The Use of Concrete Objects in Arithmetic Oral, Mental, and Written Arithmetic XIV. THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE 266 Illustrative Cases General Principles Difficulty and Success as Stimuli False Inferences XV. INDIVIDUAL DIFFERENCES 285 Nature and Amount Differences within One Class The Causes of Individual Differences The Interrelations of Individual Differences BIBLIOGRAPHY OF REFERENCES 302 INDEX 311 GENERAL INTRODUCTION THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS The psychology of the elementary school subjects is concerned with the connections whereby a child is able to respond to the sight of printed words by thoughts of their meanings, to the thought of "six and eight" by thinking "fourteen," to certain sorts of stories, poems, songs, and pictures by appreciation thereof, to certain situations by acts of skill, to certain others by acts of courtesy and justice, and so on and on through the series of situations and responses which are provided by the systematic training of the school subjects and the less systematic training of school life during their study. The aims of elementary education, when fully defined, will be found to be the production of changes in human nature represented by an almost countless list of connections or bonds whereby the pupil thinks or feels or acts in certain ways in response to the situations the school has organized and is influenced to think and feel and act similarly to similar situations when life outside of school confronts him with them. We are not at present able to define the work of the elementary school in detail as the formation of such and such bonds between certain detached situations and certain specified responses. As elsewhere in human learning, we are at present forced to think somewhat vaguely in terms of mental functions, like "ability to read the vernacular," "ability to spell common words," "ability to add, subtract, multiply, and divide with integers," "knowledge of the history of the United States," "honesty in examinations," and "appreciation of good music," defined by some general results obtained rather than by the elementary bonds which constitute them. The psychology of the school subjects begins where our common sense knowledge of these functions leaves off and tries to define the knowledge, interest, power, skill, or ideal in question more adequately, to measure improvement in it, to analyze it into its constituent bonds, to decide what bonds need to be formed and in what order as means to the most economical attainment of the desired improvement, to survey the original tendencies and the tendencies already acquired before entrance to school which help or hinder progress in the elementary school subjects, to examine the motives that are or may be used to make the desired connections satisfying, to examine any other special conditions of improvement, and to note any facts concerning individual differences that are of special importance to the conduct of elementary school work. Put in terms of problems, the task of the psychology of the elementary school subjects is, in each case:-- (1) _What is the function?_ For example, just what is "ability to read"? Just what does "the understanding of decimal notation" mean? Just what are "the moral effects to be sought from the teaching of literature"? (2) _How are degrees of ability or attainment, and degrees of progress or improvement in the function or a part of the function measured?_ For example, how can we determine how well a pupil should write, or how hard words we expect him to spell, or what good taste we expect him to show? How can we define to ourselves what knowledge of the meaning of a fraction we shall try to secure in grade 4? (3) _What can be done toward reducing the function to terms of particular situation-response connections, whose formation can be more surely and easily controlled?_ For example, how far does ability to spell involve the formation one by one of bonds between the thought of almost every word in the language and the thought of that word's letters in their correct order; and how far does, say, the bond leading from the situation of the sound of _ceive_ in _receive_ and _deceive_ to their correct spelling insure the correct spelling of that part of _perceive_? Does "ability to add" involve special bonds leading from "27 and 4" to "31," from "27 and 5" to "32," and "27 and 6" to "33"; or will the bonds leading from "7 and 4" to "11," "7 and 5" to "12" and "7 and 6" to "13" (each plus a simple inference) serve as well? What are the situations and responses that represent in actual behavior the quality that we call school patriotism? (4) _In almost every case a certain desired change of knowledge or skill or power can be attained by any one of several sets of bonds. Which of them is the best? What are the advantages of each?_ For example, learning to add may include the bonds "0 and 0 are 0," "0 and 1 are 1," "0 and 2 are 2," "1 and 0 are 1," "2 and 0 are 2," etc.; or these may be all left unformed, the pupil being taught the habits of entering 0 as the sum of a column that is composed of zeros and otherwise neglecting 0 in addition. Are the rules of usage worth teaching as a means toward correct speech, or is the time better spent in detailed practice in correct speech itself? (5) _A bond to be formed may be formed in any one of many degrees of strength. Which of these is, at any given stage of learning the subject, the most desirable, all things considered?_ For example, shall the dates of all the early settlements of North America be learned so that the exact year will be remembered for ten years, or so that the exact date will be remembered for ten minutes and the date with an error plus or minus of ten years will be remembered for a year or two? Shall the tables of inches, feet, and yards, and pints, quarts, and gallons be learned at their first appearance so as to be remembered for a year, or shall they be learned only well enough to be usable in the work of that week, which in turn fixes them to last for a month or so? Should a pupil in the first year of study of French have such perfect connections between the sounds of French words and their meanings that he can understand simple sentences containing them spoken at an ordinary rate of speaking? Or is slow speech permissible, and even imperative, on the part of the teacher, with gradual increase of rate? (6) _In almost every case, any set of bonds may produce the desired change when presented in any one of several orders. Which is the best order? What are the advantages of each?_ Certain systems for teaching handwriting perfect the elementary movements one at a time and then teach their combination in words and sentences. Others begin and continue with the complex movement-series that actual words require. What do the latter lose and gain? The bonds constituting knowledge of the metric system are now formed late in the pupil's course. Would it be better if they were formed early as a means of facilitating knowledge of decimal fractions? (7) _What are the original tendencies and pre-school acquisitions upon which the connection-forming of the elementary school may be based or which it has to counteract?_ For example, if a pupil knows the meaning of a heard word, he may read it understandingly from getting its sound, as by phonic reconstruction. What words does the average beginner so know? What are the individual differences in this respect? What do the instincts of gregariousness, attention-getting, approval, and helpfulness recommend concerning group-work _versus_ individual-work, and concerning the size of a group that is most desirable? The original tendency of the eyes is certainly not to move along a line from left to right of a page, then back in one sweep and along the next line. What is their original tendency when confronted with the printed page, and what must we do with it in teaching reading? (8) _What armament of satisfiers and annoyers, of positive and negative interests and motives, stands ready for use in the formation of the intrinsically uninteresting connections between black marks and meanings, numerical exercises and their answers, words and their spelling, and the like?_ School practice has tried, more or less at random, incentives and deterrents from quasi-physical pain to the most sentimental fondling, from sheer cajolery to philosophical argument, from appeals to assumed savage and primitive traits to appeals to the interest in automobiles, flying-machines, and wireless telegraphy. Can not psychology give some rules for guidance, or at least limit experimentation to its more hopeful fields? (9) _The general conditions of efficient learning are described in manuals of educational psychology. How do these apply in the case of each task of the elementary school?_ For example, the arrangement of school drills in addition and in short division in the form of practice experiments has been found very effective in producing interest in the work and in improvement at it. In what other arithmetical functions may we expect the same? (10) _Beside the general principles concerning the nature and causation of individual differences, there must obviously be, in existence or obtainable as a possible result of proper investigation, a great fund of knowledge of special differences relevant to the learning of reading, spelling, geography, arithmetic, and the like. What are the facts as far as known? What are the means of learning more of them?_ Courtis finds that a child may be specially strong in addition and yet be specially weak in subtraction in comparison with others of his age and grade. It even seems that such subtle and intricate tendencies are inherited. How far is such specialization the rule? Is it, for example, the case that a child may have a special gift for spelling certain sorts of words, for drawing faces rather than flowers, for learning ancient history rather than modern? * * * * * Such are our problems: this volume discusses them in the case of arithmetic. The student who wishes to relate the discussion to the general pedagogy of arithmetic may profitably read, in connection with this volume: The Teaching of Elementary Mathematics, by D. E. Smith ['01], The Teaching of Primary Arithmetic, by H. Suzzallo ['11], How to Teach Arithmetic, by J. C. Brown and L. D. Coffman ['14], The Teaching of Arithmetic, by Paul Klapper ['16], and The New Methods in Arithmetic, by the author ['21]. THE PSYCHOLOGY OF ARITHMETIC CHAPTER I THE NATURE OF ARITHMETICAL ABILITIES According to common sense, the task of the elementary school is to teach:--(1) the meanings of numbers, (2) the nature of our system of decimal notation, (3) the meanings of addition, subtraction, multiplication, and division, and (4) the nature and relations of certain common measures; to secure (5) the ability to add, subtract, multiply, and divide with integers, common and decimal fractions, and denominate numbers, (6) the ability to apply the knowledge and power represented by (1) to (5) in solving problems, and (7) certain specific abilities to solve problems concerning percentage, interest, and other common occurrences in business life. This statement of the functions to be developed and improved is sound and useful so far as it goes, but it does not go far enough to make the task entirely clear. If teachers had nothing but the statement above as a guide to what changes they were to make in their pupils, they would often leave out important features of arithmetical training, and put in forms of training that a wise educational plan would not tolerate. It is also the case that different leaders in arithmetical teaching, though they might all subscribe to the general statement of the previous paragraph, certainly do not in practice have identical notions of what arithmetic should be for the elementary school pupil. The ordinary view of the nature of arithmetical learning is obscure or inadequate in four respects. It does not define what 'knowledge of the meanings of numbers' is; it does not take account of the very large amount of teaching of _language_ which is done and should be done as a part of the teaching of arithmetic; it does not distinguish between the ability to meet certain quantitative problems as life offers them and the ability to meet the problems provided by textbooks and courses of study; it leaves 'the ability to apply arithmetical knowledge and power' as a rather mystical general faculty to be improved by some educational magic. The four necessary amendments may be discussed briefly. KNOWLEDGE OF THE MEANINGS OF NUMBERS Knowledge of the meanings of the numbers from one to ten may mean knowledge that 'one' means a single thing of the sort named, that two means one more than one, that three means one more than two, and so on. This we may call the _series_ meaning. To know the meaning of 'six' in this sense is to know that it is one more than five and one less than seven--that it is between five and seven in the number series. Or we may mean by knowledge of the meanings of numbers, knowledge that two fits a collection of two units, that three fits a collection of three units, and so on, each number being a name for a certain sized collection of discrete things, such as apples, pennies, boys, balls, fingers, and the other customary objects of enumeration in the primary school. This we may call the _collection_ meaning. To know the meaning of six in this sense is to be able to name correctly any collection of six separate, easily distinguishable individual objects. In the third place, knowledge of the numbers from one to ten may mean knowledge that two is twice whatever is called one, that three is three times whatever is one, and so on. This is, of course, the _ratio_ meaning. To know the meaning of six in this sense is to know that if ___________ is one, a line half a foot long is six, that if [___] is one, [____________] is about six, while if [__] is one, [______] is about six, and the like. In the fourth place, the meaning of a number may be a smaller or larger fraction of its _implications_--its numerical relations, facts about it. To know six in this sense is to know that it is more than five or four, less than seven or eight, twice three, three times two, the sum of five and one, or of four and two, or of three and three, two less than eight--that with four it makes ten, that it is half of twelve, and the like. This we may call the '_nucleus of facts_' or _relational_ meaning of a number. Ordinary school practice has commonly accepted the second meaning as that which it is the task of the school to teach beginners, but each of the other meanings has been alleged to be the essential one--the series idea by Phillips ['97], the ratio idea by McLellan and Dewey ['95] and Speer ['97], and the relational idea by Grube and his followers. This diversity of views concerning what the function is that is to be improved in the case of learning the meanings of the numbers one to ten is not a trifling matter of definition, but produces very great differences in school practice. Consider, for example, the predominant value assigned to counting by Phillips in the passage quoted below, and the samples of the sort of work at which children were kept employed for months by too ardent followers of Speer and Grube. THE SERIES IDEA OVEREMPHASIZED "This is essentially the counting period, and any words that can be arranged into a series furnish all that is necessary. Counting is fundamental, and counting that is spontaneous, free from sensible observation, and from the strain of reason. A study of these original methods shows that multiplication was developed out of counting, and not from addition as nearly all textbooks treat it. Multiplication is counting. When children count by 4's, etc., they accent the same as counting gymnastics or music. When a child now counts on its fingers it simply reproduces a stage in the growth of the civilization of all nations. I would emphasize again that during the counting period there is a somewhat spontaneous development of the number series-idea which Preyer has discussed in his Arithmogenesis; that an immense momentum is given by a systematic series of names; and that these names are generally first learned and applied to objects later. A lady teacher told me that the Superintendent did not wish the teachers to allow the children to count on their fingers, but she failed to see why counting with horse-chestnuts was any better. Her children could hardly avoid using their fingers in counting other objects yet they followed the series to 100 without hesitation or reference to their fingers. This spontaneous counting period, or naming and following the series, should precede its application to objects." [D.E. Phillips, '97, p. 238.] THE RATIO IDEA OVEREMPHASIZED [Illustration: FIG. 1.] "Ratios.--1. Select solids having the relation, or ratio, of _a_, _b_, _c_, _d_, _o_, _e_. 2. Name the solids, _a_, _b_, _c_, _d_, _o_, _e_. The means of expressing must be as freely supplied as the means of discovery. The pupil is not expected to invent terms. 3. Tell all you can about the relation of these units. 4. Unite units and tell what the sum equals. 5. Make statements like this: _o_ less _e_ equals _b_. 6. _c_ can be separated into how many _d_'s? into how many _b_'s? 7. _c_ can be separated into how many _b_'s? What is the name of the largest unit that can be found in both _c_ and _d_ an exact number of times? 8. Each of the other units equals what part of _c_? 9. If _b_ is 1, what is each of the other units? 10. If _a_ is 1, what is each of the other units? 11. If _b_ is 1, how many 1's are there in each of the other units? 12. If _d_ is 1, how many 1's and parts of 1 in each of the other units? 13. 2 is the relation of what units? 14. 3 is the relation of what units? 15. 1/2 is the relation of what units? 16. 2/3 is the relation of what units? 17. Which units have the relation 3/2? 18. Which unit is 3 times as large as 1/2 of _b_? 19. _c_ equals 6 times 1/3 of what unit? 20. 1/3 of what unit equals 1/6 of _c_? 21. What equals 1/2 of _c_? _d_ equals how many sixths of _c_? 22. _o_ equals 5 times 1/3 of what unit? 23. 1/3 of what unit equals 1/5 of _o_? 24. 2/3 of _d_ equals what unit? _b_ equals how many thirds of _d_? 25. 2 is the ratio of _d_ to 1/3 of what unit? 3 is the ratio of _d_ to 1/2 of what unit? 26. _d_ equals 3/4 of what unit? 3/4 is the ratio of what units?" [Speer, '97, p. 9f.] THE RELATIONAL IDEA OVEREMPHASIZED An inspection of books of the eighties which followed the "Grube method" (for example, the _New Elementary Arithmetic_ by E.E. White ['83]) will show undue emphasis on the relational ideas. There will be over a hundred and fifty successive tasks all, or nearly all, on +7 and -7. There will be much written work of the sort shown below: _Add:_ 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1 2 -- -- -- which must have sorely tried the eyes of all concerned. Pupils are taught to "give the analysis and synthesis of each of the nine digits." Yet the author states that he does not carry the principle of the Grube method "to the extreme of useless repetition and mechanism." It should be obvious that all four meanings have claims upon the attention of the elementary school. Four is the thing between three and five in the number series; it is the name for a certain sized collection of discrete objects; it is also the name for a continuous magnitude equal to four units--for four quarts of milk in a gallon pail as truly as for four separate quart-pails of milk; it is also, if we know it well, the thing got by adding one to three or subtracting six from ten or taking two two's or half of eight. To know the meaning of a number means to know somewhat about it in all of these respects. The difficulty has been the narrow vision of the extremists. A child must not be left interminably counting; in fact the one-more-ness of the number series can almost be had as a by-product. A child must not be restricted to exercises with collections objectified as in Fig. 2 or stated in words as so many apples, oranges, hats, pens, etc., when work with measurement of continuous quantities with varying units--inches, feet, yards, glassfuls, pints, quarts, seconds, minutes, hours, and the like--is so easy and so significant. On the other hand, the elaboration of artificial problems with fictitious units of measure just to have relative magnitudes as in the exercises on page 5 is a wasteful sacrifice. Similarly, special drills emphasizing the fact that eighteen is eleven and seven, twelve and six, three less than twenty-one, and the like, are simply idolatrous; these facts about eighteen, so far as they are needed, are better learned in the course of actual column-addition and -subtraction. [Illustration: FIG. 2.] ARITHMETICAL LANGUAGE The second improvement to be made in the ordinary notion of what the functions to be improved are in the case of arithmetic is to include among these functions the knowledge of certain words. The understanding of such words as _both_, _all_, _in all_, _together_, _less_, _difference_, _sum_, _whole_, _part_, _equal_, _buy_, _sell_, _have left_, _measure_, _is contained in_, and the like, is necessary in arithmetic as truly as is the understanding of numbers themselves. It must be provided for by the school; for pre-school and extra-school training does not furnish it, or furnishes it too late. It can be provided for much better in connection with the teaching of arithmetic than in connection with the teaching of English. It has not been provided for. An examination of the first fifty pages of eight recent textbooks for beginners in arithmetic reveals very slight attention to this matter at the best and no attention at all in some cases. Three of the books do not even use the word _sum_, and one uses it only once in the fifty pages. In all the four hundred pages the word _difference_ occurs only twenty times. When the words are used, no great ingenuity or care appears in the means of making sure that their meanings are understood. The chief reason why it has not been provided for is precisely that the common notion of what the functions are that arithmetic is to develop has left out of account this function of intelligent response to quantitative terms, other than the names of the numbers and processes. Knowledge of language over a much wider range is a necessary element in arithmetical ability in so far as the latter includes ability to solve verbally stated problems. As arithmetic is now taught, it does include that ability, and a large part of the time of wise teaching is given to improving the function 'knowing what a problem states and what it asks for.' Since, however, this understanding of verbally stated problems may not be an absolutely necessary element of arithmetic, it is best to defer its consideration until we have seen what the general function of problem-solving is. PROBLEM-SOLVING The third respect in which the function, 'ability in arithmetic,' needs clearer definition, is this 'problem-solving.' The aim of the elementary school is to provide for correct and economical response to genuine problems, such as knowing the total due for certain real quantities at certain real prices, knowing the correct change to give or get, keeping household accounts, calculating wages due, computing areas, percentages, and discounts, estimating quantities needed of certain materials to make certain household or shop products, and the like. Life brings these problems usually either with a real situation (as when one buys and counts the cost and his change), or with a situation that one imagines or describes to himself (as when one figures out how much money he must save per week to be able to buy a forty-dollar bicycle before a certain date). Sometimes, however, the problem is described in words to the person who must solve it by another person (as when a life insurance agent says, 'You pay only 25 cents a week from now till--and you get $250 then'; or when an employer says, 'Your wages would be 9 dollars a week, with luncheon furnished and bonuses of such and such amounts'). Sometimes also the problem is described in printed or written words to the person who must solve it (as in an advertisement or in the letter of a customer asking for an estimate on this or that). The problem may be in part real, in part imagined or described to oneself, and in part described to one orally or in printed or written words (as when the proposed articles for purchase lie before one, the amount of money one has in the bank is imagined, the shopkeeper offers 10 percent discount, and the printed price list is there to be read). To fit pupils to solve these real, personally imagined, or self-described problems, and 'described-by-another' problems, schools have relied almost exclusively on training with problems of the last sort only. The following page taken almost at random from one of the best recent textbooks could be paralleled by thousands of others; and the oral problems put by teachers have, as a rule, no real situation supporting them. 1. At 70 cents per 100 pounds, what will be the amount of duty on an invoice of 3622 steel rails, each rail being 27 feet long and weighing 60 pounds to the yard? 2. A man had property valued at $6500. What will be his taxes at the rate of $10.80 per $1000? 3. Multiply seventy thousand fourteen hundred-thousandths by one hundred nine millionths, and divide the product by five hundred forty-five. 4. What number multiplied by 43-3/4 will produce 265-5/8? 5. What decimal of a bushel is 3 quarts? 6. A man sells 5/8 of an acre of land for $93.75. What would be the value of his farm of 150-3/4 acres at the same rate? 7. A coal dealer buys 375 tons coal at $4.25 per ton of 2240 pounds. He sells it at $4.50 per ton of 2000 pounds. What is his profit? 8. Bought 60 yards of cloth at the rate of 2 yards for $5, and 80 yards more at the rate of 4 yards for $9. I immediately sold the whole of it at the rate of 5 yards for $12. How much did I gain? 9. A man purchased 40 bushels of apples at $1.50 per bushel. Twenty-five hundredths of them were damaged, and he sold them at 20 cents per peck. He sold the remainder at 50 cents per peck. How much did he gain or lose? 10. If oranges are 37-1/2 cents per dozen, how many boxes, each containing 480, can be bought for $60? 11. A man can do a piece of work in 18-3/4 days. What part of it can he do in 6-2/3 days? 12. How old to-day is a boy that was born Oct. 29, 1896? [Walsh, '06, Part I, p. 165.] As a result, teachers and textbook writers have come to think of the functions of solving arithmetical problems as identical with the function of solving the described problems which they give in school in books, examination papers, and the like. If they do not think explicitly that this is so, they still act in training and in testing pupils as if it were so. It is not. Problems should be solved in school to the end that pupils may solve the problems which life offers. To know what change one should receive after a given real purchase, to keep one's accounts accurately, to adapt a recipe for six so as to make enough of the article for four persons, to estimate the amount of seed required for a plot of a given size from the statement of the amount required per acre, to make with surety the applications that the household, small stores, and ordinary trades require--such is the ability that the elementary school should develop. Other things being equal, the school should set problems in arithmetic which life then and later will set, should favor the situations which life itself offers and the responses which life itself demands. Other things are not always equal. The same amount of time and effort will often be more productive toward the final end if directed during school to 'made-up' problems. The keeping of personal financial accounts as a school exercise is usually impracticable, partly because some of the children have no earnings or allowance--no accounts to keep, and partly because the task of supervising work when each child has a different problem is too great for the teacher. The use of real household and shop problems will be easy only when the school program includes the household arts and industrial education, and when these subjects themselves are taught so as to improve the functions used by real life. Very often the most efficient course is to make sure that arithmetical procedures are applied to the real and personally initiated problems which they fit, by having a certain number of such problems arise and be solved; then to make sure that the similarity between these real problems and certain described problems of the textbook or teacher's giving is appreciated; and then to give the needed drill work with described problems. In many cases the school practice is fairly well justified in assuming that solving described problems will prepare the pupil to solve the corresponding real problems actually much better than the same amount of time spent on the real problems themselves. All this is true, yet the general principle remains that, other things being equal, the school should favor real situations, should present issues as life will present them. Where other things make the use of verbally described problems of the ordinary type desirable, these should be chosen so as to give a maximum of preparation for the real applications of arithmetic in life. We should not, for example, carelessly use any problem that comes to mind in applying a certain principle, but should stop to consider just what the situations of life really require and show clearly the application of that principle. For example, contrast these two problems applying cancellation:-- A. A man sold 24 lambs at $18 apiece on each of six days, and bought 8 pounds of metal with the proceeds. How much did he pay per ounce for the metal? B. How tall must a rectangular tank 16" long by 8" wide be to hold as much as a rectangular tank 24" by 18" by 6"? The first problem not only presents a situation that would rarely or never occur, but also takes a way to find the answer that would not, in that situation, be taken since the price set by another would determine the amount. Much thought and ingenuity should in the future be expended in eliminating problems whose solution does not improve the real function to be improved by applied arithmetic, or improves it at too great cost, and in devising problems which prepare directly for life's demands and still can fit into a curriculum that can be administered by one teacher in charge of thirty or forty pupils, under the limitations of school life. The following illustrations will to some extent show concretely what the ability to apply the knowledge and power represented by abstract or pure arithmetic--the so-called fundamentals--in solving problems should mean and what it should not mean. _Samples of Desirable Applications of Arithmetic in Problems where the Situation is Actually Present to Sense in Whole or in Part_ Keeping the scores and deciding which side beat and by how much in appropriate classroom games, spelling matches, and the like. Computing costs, making and inspecting change, taking inventories, and the like with a real or play store. Mapping the school garden, dividing it into allotments, planning for the purchase of seeds, and the like. Measuring one's own achievement and progress in tests of word-knowledge, spelling, addition, subtraction, speed of writing, and the like. Measuring the rate of improvement per hour of practice or per week of school life, and the like. Estimating costs of food cooked in the school kitchen, articles made in the school shops, and the like. Computing the cost of telegrams, postage, expressage, for a real message or package, from the published tariffs. Computing costs from mail order catalogues and the like. _Samples of Desirable Applications of Arithmetic where the Situation is Not Present to Sense_ The samples given here all concern the subtraction of fractions. Samples concerning any other arithmetical principle may be found in the appropriate pages of any text which contains problem-material selected with consideration of life's needs. A 1. Dora is making jelly. The recipe calls for 24 cups of sugar and she has only 21-1/2. She has no time to go to the store so she has to borrow the sugar from a neighbor. How much must she get? _Subtract_ 24 _Think "1/2 and 1/2 = 1." Write 1/2._ 21-1/2 _Think "2 and 2 = 4." Write the 2._ -------- 2-1/2 2. A box full of soap weighs 29-1/2 lb. The empty box weighs 3-1/2 lb. How much does the soap alone weigh? 3. On July 1, Mr. Lewis bought a 50-lb. bag of ice-cream salt. On July 15 there were just 11-1/2 lb. left. How much had he used in the two weeks? 4. Grace promised to pick 30 qt. blueberries for her mother. So far she has picked 18-1/2 qt. How many more quarts must she pick? B This table of numbers tells Weight of Mary Adams what Nell's baby sister Mary When born 7-3/8 lb. weighed every two months from 2 months old 11-1/4 lb. the time she was born till she 4 months old 14-1/8 lb. was a year old. 6 months old 15-3/4 lb. 8 months old 17-5/8 lb. 10 months old 19-1/2 lb. 12 months old 21-3/8 lb. 1. How much did the Adams baby gain in the first two months? 2. How much did the Adams baby gain in the second two months? 3. In the third two months? 4. In the fourth two months? 5. From the time it was 8 months old till it was 10 months old? 6. In the last two months? 7. From the time it was born till it was 6 months old? C 1. Helen's exact average for December was 87-1/3. Kate's was 84-1/2. How much higher was Helen's than Kate's? 87-1/3 How do you think of 1/2 and 1/3? 84-1/2 How do you think of 1-2/6? ------ How do you change the 4? 2. Find the exact average for each girl in the following list. Write the answers clearly so that you can see them easily. You will use them in solving problems 3, 4, 5, 6, 7, and 8. Alice Dora Emma Grace Louise Mary Nell Rebecca Reading 91 87 83 81 79 77 76 73 Language 88 78 82 79 73 78 73 75 Arithmetic 89 85 79 75 84 87 89 80 Spelling 90 79 75 80 82 91 68 81 Geography 91 87 83 75 78 85 73 79 Writing 90 88 75 72 93 92 95 78 3. Which girl had the highest average? 4. How much higher was her average than the next highest? 5. How much difference was there between the highest and the lowest girl? 6. Was Emma's average higher or lower than Louise's? How much? 7. How much difference was there between Alice's average and Dora's? 8. How much difference was there between Mary's average and Nell's? 9. Write five other problems about these averages, and solve each of them. _Samples of Undesirable Applications of Arithmetic_[1] Will has XXI marbles, XII jackstones, XXXVI pieces of string. How many things had he? George's kite rose CDXXXV feet and Tom's went LXIII feet higher. How high did Tom's kite rise? If from DCIV we take CCIV the result will be a number IV times as large as the number of dollars Mr. Dane paid for his horse. How much did he pay for his horse? Hannah has 5/8 of a dollar, Susie 7/25, Nellie 3/4, Norah 13/16. How much money have they all together? A man saves 3-17/80 dollars a week. How much does he save in a year? A tree fell and was broken into 4 pieces, 13-1/6 feet, 10-3/7 feet, 8-1/2 feet, and 7-16/21 feet long. How tall was the tree? Annie's father gave her 20 apples to divide among her friends. She gave each one 2-2/9 apples apiece. How many playmates had she? John had 17-2/5 apples. He divided his whole apples into fifths. How many pieces had he in all? A landlady has 3-3/7 pies to be divided among her 8 boarders. How much will each boarder receive? There are twenty columns of spelling words in Mary's lesson and 16 words in each column. How many words are in her lesson? There are 9 nuts in a pint. How many pints in a pile of 5,888,673 nuts? The Adams school contains eight rooms; each room contains 48 pupils; if each pupil has eight cents, how much have they together? A pile of wood in the form of a cube contains 15-1/2 cords. What are the dimensions to the nearest inch? A man 6 ft. high weighs 175 lb. How tall is his wife who is of similar build, and weighs 125 lb.? A stick of timber is in the shape of the frustum of a square pyramid, the lower base being 22 in. square and the upper 14 in. square. How many cubic feet in the log, if it is 22 ft. long? Mr. Ames, being asked his age, replied: "If you cube one half of my age and add 41,472 to the result, the sum will be one half the cube of my age. How old am I?" [1] The following and later problems are taken from actual textbooks or courses of study or state examinations; to avoid invidious comparisons, they are not exact quotations, but are equivalents in principle and form, as stated in the preface. These samples, just given, of the kind of problem-solving that should not be emphasized in school training refer in some cases to books of forty years back, but the following represent the results of a collection made in 1910 from books then in excellent repute. It required only about an hour to collect them; and I am confident that a thousand such problems describing situations that the pupil will never encounter in real life, or putting questions that he will never be asked in real life, could easily be found in any ten textbooks of the decade from 1900 to 1910. If there are 250 kernels of corn on one ear, how many are there on 24 ears of corn the same size? Maud is four times as old as her sister, who is 4 years old. What is the sum of their ages? If the first century began with the year 1, with what year does it end? Every spider has 8 compound eyes. How many eyes have 21 spiders? A nail 4 inches long is driven through a board so that it projects 1.695 inches on one side and 1.428 on the other. How thick is the board? Find the perimeter of an envelope 5 in. by 3-1/4 in. How many minutes in 5/9 of 9/4 of an hour? Mrs. Knox is 3/4 as old as Mr. Knox, who is 48 years old. Their son Edward is 4/9 as old as his mother. How old is Edward? Suppose a pie to be exactly round and 10-1/2 miles in diameter. If it were cut into 6 equal pieces, how long would the curved edge of each piece be? 8-1/3% of a class of 36 boys were absent on a rainy day. 33-1/3% of those present went out of the room to the school yard. How many were left in the room? Just after a ton of hay was weighed in market, a horse ate one pound of it. What was the ratio of what he ate to what was left? If a fan having 15 rays opens out so that the outer rays form a straight line, how many degrees are there between any two adjacent rays? One half of the distance between St. Louis and New Orleans is 280 miles more than 1/10 of the distance; what is the distance between these places? If the pressure of the atmosphere is 14.7 lb. per square inch what is the pressure on the top of a table 1-1/4 yd. long and 2/3 yd. wide? 13/28 of the total acreage of barley in 1900 was 100,000 acres; what was the total acreage? What is the least number of bananas that a mother can exactly divide between her 2 sons, or among her 4 daughters, or among all her children? If Alice were two years older than four times her actual age she would be as old as her aunt, who is 38 years old. How old is Alice? Three men walk around a circular island, the circumference of which is 360 miles. A walks 15 miles a day, B 18 miles a day, and C 24 miles a day. If they start together and walk in the same direction, how many days will elapse before they will be together again? With only thirty or forty dollars a year to spend on a pupil's education, of which perhaps eight dollars are spent on improving his arithmetical abilities, the immediate guidance of his responses to real situations and personally initiated problems has to be supplemented largely by guidance of his responses to problems described in words, diagrams, pictures, and the like. Of these latter, words will be used most often. As a consequence the understanding of the words used in these descriptions becomes a part of the ability required in arithmetic. Such word knowledge is also required in so far as the problems to be solved in real life are at times described, as in advertisements, business letters, and the like. This is recognized by everybody in the case of words like _remainder_, _profit_, _loss_, _gain_, _interest_, _cubic capacity_, _gross_, _net_, and _discount_, but holds equally of _let_, _suppose_, _balance_, _average_, _total_, _borrowed_, _retained_, and many such semi-technical words, and may hold also of hundreds of other words unless the textbook and teacher are careful to use only words and sentence structures which daily life and the class work in English have made well known to the pupils. To apply arithmetic to a problem a pupil must understand what the problem is; problem-solving depends on problem-reading. In actual school practice training in problem-reading will be less and less necessary as we get rid of problems to be solved simply for the sake of solving them, unnecessarily unreal problems, and clumsy descriptions, but it will remain to some extent as an important joint task for the 'arithmetic' and 'reading' of the elementary school. ARITHMETICAL REASONING The last respect in which the nature of arithmetical abilities requires definition concerns arithmetical reasoning. An adequate treatment of the reasoning that may be expected of pupils in the elementary school and of the most efficient ways to encourage and improve it cannot be given until we have studied the formation of habits. For reasoning is essentially the organization and control of habits of thought. Certain matters may, however, be decided here. The first concerns the use of computation and problems merely for discipline,--that is, the emphasis on training in reasoning regardless of whether the problem is otherwise worth reasoning about. It used to be thought that the mind was a set of faculties or abilities or powers which grew strong and competent by being exercised in a certain way, no matter on what they were exercised. Problems that could not occur in life, and were entirely devoid of any worthy interest, save the intellectual interest in solving them, were supposed to be nearly or quite as useful in training the mind to reason as the genuine problems of the home, shop, or trade. Anything that gave the mind a chance to reason would do; and pupils labored to find when the minute hand and hour hand would be together, or how many sheep a shepherd had if half of what he had plus ten was one third of twice what he had! We now know that the training depends largely on the particular data used, so that efficient discipline in reasoning requires that the pupil reason about matters of real importance. There is no magic essence or faculty of reasoning that works in general and irrespective of the particular facts and relations reasoned about. So we should try to find problems which not only stimulate the pupil to reason, but also direct his reasoning in useful channels and reward it by results that are of real significance. We should replace the purely disciplinary problems by problems that are also valuable as special training for important particular situations of life. Reasoning sought for reasoning's sake alone is too wasteful an expenditure of time and is also likely to be inferior as reasoning. The second matter concerns the relative merits of 'catch' problems, where the pupil has to go against some customary habit of thinking, and what we may call 'routine' problems, where the regular ways of thinking that have served him in the past will, except for some blunder, guide him rightly. Consider, for example, these four problems: 1. "A man bought ten dozen eggs for $2.50 and sold them for 30 cents a dozen. How many cents did he lose?" 2. "I went into Smith's store at 9 A.M. and remained until 10 A.M. I bought six yards of gingham at 40 cents a yard and three yards of muslin at 20 cents a yard and gave a $5.00 bill. How long was I in the store?" 3. "What must you divide 48 by to get half of twice 6?" 4. "What must you add to 19 to get 30?" The 'catch' problem is now in disrepute, the wise teacher feeling by a sort of intuition that to willfully require a pupil to reason to a result sharply contrary to that to which previous habits lead him is risky. The four illustrations just given show, however, that mere 'catchiness' or 'contra-previous-habit-ness' in a problem is not enough to condemn it. The fourth problem is a catch problem, but so useful a one that it has been adopted in many modern books as a routine drill! The first problem, on the contrary, all, save those who demand no higher criterion for a problem than that it make the pupil 'think,' would reject. It demands the reversal of fixed habits _to no valid purpose_; for in life the question in such case would never (or almost never) be 'How many cents did he lose?' but 'What was the result?' or simply 'What of it?' This problem weakens without excuse the child's confidence in the training he has had. Problems like (2) are given by teachers of excellent reputation, but probably do more harm than good. If a pupil should interrupt his teacher during the recitation in arithmetic by saying, "I got up at 7 o'clock to multiply 9 by 2-3/4 and got 24-3/4 for my answer; was that the right time to get up?" the teacher would not thank fortune for the stimulus to thought but would think the child a fool. Such catch questions may be fairly useful as an object lesson on the value of search for the essential element in a situation if a great variety of them are given one after another with routine problems intermixed and with warning of the general nature of the exercise at the beginning. Even so, it should be remembered that reasoning should be chiefly a force organizing habits, not opposing them; and also that there are enough bad habits to be opposed to give all necessary training. Fabricated puzzle situations wherein a peculiar hidden element of the situation makes the good habits called up by the situation misleading are useful therefore rather as a relief and amusing variation in arithmetical work than as stimuli to thought. Problems like the third quoted above we might call puzzling rather than 'catch' problems. They have value as drills in analysis of a situation into its elements that will amuse the gifted children, and as tests of certain abilities. They also require that of many confusing habits, the right one be chosen, rather than that ordinary habits be set aside by some hidden element in the situation. Not enough is known about their effect to enable us to decide whether or not the elementary school should include special facility with them as one of the arithmetical functions that it specially trains. The fourth 'catch' quoted above, which all would admit is a good problem, is good because it opposes a good habit for the sake of another good habit, forces the analysis of an element whose analysis life very much requires, and does it with no obvious waste. It is not safe to leave a child with the one habit of responding to 'add, 19, 30' by 49, for in life the 'have 19, must get .... to have 30' situation is very frequent and important. On the whole, the ordinary problems which ordinary life proffers seem to be the sort that should be reasoned out, though the elementary school may include the less noxious forms of pure mental gymnastics for those pupils who like them. SUMMARY These discussions of the meanings of numbers, the linguistic demands of arithmetic, the distinction between scholastic and real applications of arithmetic, and the possible restrictions of training in reasoning,--may serve as illustrations of the significance of the question, "What are the functions that the elementary school tries to improve in its teaching of arithmetic?" Other matters might well be considered in this connection, but the main outline of the work of the elementary school is now fairly clear. The arithmetical functions or abilities which it seeks to improve are, we may say:-- (1) Working knowledge of the meanings of numbers as names for certain sized collections, for certain relative magnitudes, the magnitude of unity being known, and for certain centers or nuclei of relations to other numbers. (2) Working knowledge of the system of decimal notation. (3) Working knowledge of the meanings of addition, subtraction, multiplication, and division. (4) Working knowledge of the nature and relations of certain common measures. (5) Working ability to add, subtract, multiply, and divide with integers, common and decimal fractions, and denominate numbers, all being real positive numbers. (6) Working knowledge of words, symbols, diagrams, and the like as required by life's simpler arithmetical demands or by economical preparation therefor. (7) The ability to apply all the above as required by life's simpler arithmetical demands or by economical preparation therefor, including (7 _a_) certain specific abilities to solve problems concerning areas of rectangles, volumes of rectangular solids, percents, interest, and certain other common occurrences in household, factory, and business life. THE SOCIOLOGY OF ARITHMETIC The phrase 'life's simpler arithmetical demands' is necessarily left vague. Just what use is being made of arithmetic in this country in 1920 by each person therein, we know only very roughly. What may be called a 'sociology' of arithmetic is very much needed to investigate this matter. For rare or difficult demands the elementary school should not prepare; there are too many other desirable abilities that it should improve. A most interesting beginning at such an inventory of the actual uses of arithmetic has been made by Wilson ['19] and Mitchell.[2] Although their studies need to be much extended and checked by other methods of inquiry, two main facts seem fairly certain. First, the great majority of people in the great majority of their doings use only very elementary arithmetical processes. In 1737 cases of addition reported by Wilson, seven eighths were of five numbers or less. Over half of the multipliers reported were one-figure numbers. Over 95 per cent of the fractions operated with were included in this list: 1/2 1/4 3/4 1/3 2/3 1/8 3/8 1/5 2/5 4/5. Three fourths of all the cases reported were simple one-step computations with integers or United States money. Second, they often use these very elementary processes, not because such are the quickest and most convenient, but because they have lost, or maybe never had, mastery of the more advanced processes which would do the work better. The 5 and 10 cent stores, the counter with "Anything on this counter for 25¢," and the arrangements for payments on the installment plan are familiar instances of human avoidance of arithmetic. Wilson found very slight use of decimals; and Mitchell found men computing with 49ths as common fractions when the use of decimals would have been more efficient. If given 120 seconds to do a test like that shown below, leading lawyers, physicians, manufacturers, and business men and their wives will, according to my experience, get only about half the work right. Many women, finding on their meat bill "7-3/8 lb. roast beef $2.36," will spend time and money to telephone the butcher asking how much roast beef was per pound, because they have no sure power in dividing by a mixed number. [2] The work of Mitchell has not been published, but the author has had the privilege of examining it. Test Perform the operations indicated. Express all fractions in answers in lowest terms. _Add:_ 3/4 + 1/6 + .25 4 yr. 6 mo. 1 yr. 2 mo. 6 yr. 9 mo. 3 yr. 6 mo. 4 yr. 5 mo. ----------- _Subtract:_ 8.6 - 6.05007 7/8 - 2/3 = 5-7/16 - 2-3/16 = _Multiply:_ 29 ft. 6 in. 7 × 8 × 4-1/2 = 8 ------------ _Divide:_ 4-1/2 ÷ 7 = It seems probable that the school training in arithmetic of the past has not given enough attention to perfecting the more elementary abilities. And we shall later find further evidence of this. On the other hand, the fact that people in general do not at present use a process may not mean that they ought not to use it. Life's simpler arithmetical demands certainly do not include matters like the rules for finding cube root or true discount, which no sensible person uses. They should not include matters like computing the lateral surface or volume of pyramids and cones, or knowing the customs of plasterers and paper hangers, which are used only by highly specialized trades. They should not include matters like interest on call loans, usury, exact interest, and the rediscounting of notes, which concern only brokers, bank clerks, and rich men. They should not include the technique of customs which are vanishing from efficient practice, such as simple interest on amount for times longer than a year, days of grace, or extremes and means in proportions. They should not include any elaborate practice with very large numbers, or decimals beyond thousandths, or the addition and subtraction of fractions which not one person in a hundred has to add or subtract oftener than once a year. When we have an adequate sociology of arithmetic, stating accurately just who should use each arithmetical ability and how often, we shall be able to define the task of the elementary school in this respect. For the present, we may proceed by common sense, guided by two limiting rules. The first is,--"It is no more desirable for the elementary school to teach all the facts of arithmetic than to teach all the words in the English language, or all the topography of the globe, or all the details of human physiology." The second is,--"It is not desirable to eliminate any element of arithmetical training until you have something better to put in its place." CHAPTER II THE MEASUREMENT OF ARITHMETICAL ABILITIES One of the best ways to clear up notions of what the functions are which schools should develop and improve is to get measures of them. If any given knowledge or skill or power or ideal exists, it exists in some amount. A series of amounts of it, varying from less to more, defines the ability itself in a way that no general verbal description can do. Thus, a series of weights, 1 lb., 2 lb., 3 lb., 4 lb., etc., helps to tell us what we mean by weight. By finding a series of words like _only_, _smoke_, _another_, _pretty_, _answer_, _tailor_, _circus_, _telephone_, _saucy_, and _beginning_, which are spelled correctly by known and decreasing percentages of children of the same age, or of the same school grade, we know better what we mean by 'spelling-difficulty.' Indeed, until we can measure the efficiency and improvement of a function, we are likely to be vague and loose in our ideas of what the function is. A SAMPLE MEASUREMENT OF AN ARITHMETICAL ABILITY: THE ABILITY TO ADD INTEGERS Consider first, as a sample, the measurement of ability to add integers. The following were the examples used in the measurements made by Stone ['08]: 596 4695 428 872 2375 94 7948 4052 75 6786 6354 304 567 260 645 858 5041 984 9447 1543 897 7499 ---- --- ---- The scoring was as follows: Credit of 1 for each column added correctly. Stone combined measures of other abilities with this in a total score for amount done correctly in 12 minutes. Stone also scored the correctness of the additions in certain work in multiplication. Courtis uses a sheet of twenty-four tasks or 'examples,' each consisting of the addition of nine three-place numbers as shown below. Eight minutes is allowed. He scores the amount done by the number of examples, and also scores the number of examples done correctly, but does not suggest any combination of these two into a general-efficiency score. 927 379 756 837 924 110 854 965 344 --- The author long ago proposed that pupils be measured also with series like _a_ to _g_ shown below, in which the difficulty increases step by step. _a._ 3 2 2 3 2 2 1 2 2 3 1 2 4 5 5 1 4 2 3 3 3 2 2 2 - - - - - - - - _b._ 21 32 12 24 34 34 22 12 23 12 52 31 33 12 23 13 24 25 15 14 32 23 43 61 -- -- -- -- -- -- -- -- _c._ 22 3 4 35 32 83 22 3 3 31 3 2 33 11 3 21 38 45 52 52 2 4 33 64 -- -- -- -- -- -- -- -- _d._ 30 20 10 22 10 20 52 12 20 50 40 43 30 4 6 22 40 17 24 13 40 23 30 44 -- -- -- -- -- -- -- -- _e._ 4 5 20 12 12 20 10 20 30 3 40 4 11 20 20 10 30 20 4 1 23 7 2 20 2 40 23 40 11 10 30 20 20 10 11 20 22 30 25 -- -- -- -- -- -- -- -- _f._ 19 9 9 14 2 19 24 9 4 13 9 14 13 12 13 13 9 14 17 23 13 15 15 34 12 25 26 29 18 19 25 28 18 39 -- -- -- -- -- -- -- -- _g._ 13 13 9 14 12 9 9 13 12 9 14 24 23 19 19 29 9 9 13 21 28 26 26 14 8 8 29 23 29 16 15 19 17 19 19 22 -- -- -- -- -- -- -- -- Woody ['16] has constructed his well-known tests on this principle, though he uses only one example at each step of difficulty instead of eight or ten as suggested above. His test, so far as addition of integers goes, is:-- SERIES A. ADDITION SCALE (in part) By Clifford Woody (1) (2) (3) (4) (5) (6) (7) (8) (9) 2 2 17 53 72 60 3 + 1 = 2 + 5 + 1 = 20 3 4 2 45 26 37 10 -- 3 -- -- -- -- 2 -- 30 25 -- (10) (11) (12) (13) (14) (15) (16) (17) (18) 21 32 43 23 25 + 42 = 100 9 199 2563 33 59 1 25 33 24 194 1387 35 17 2 16 45 12 295 4954 -- -- 13 -- 201 15 156 2065 -- 46 19 --- ---- --- -- (19) (20) (21) (22) $ .75 $12.50 $8.00 547 1.25 16.75 5.75 197 .49 15.75 2.33 685 ----- ------ 4.16 678 .94 456 6.32 393 ----- 525 240 152 --- In his original report, Woody gives no scheme for scoring an individual, wisely assuming that, with so few samples at each degree of difficulty, a pupil's score would be too unreliable for individual diagnosis. The test is reliable for a class; and for a class Woody used the degree of difficulty such that a stated fraction of the class can do the work correctly, if twenty minutes is allowed for the thirty-eight examples of the entire test. The measurement of even so simple a matter as the efficiency of a pupil's responses to these tests in adding integers is really rather complex. There is first of all the problem of combining speed and accuracy into some single estimate. Stone gives no credit for a column unless it is correctly added. Courtis evades the difficulty by reporting both number done and number correct. The author's scheme, which gives specified weights to speed and accuracy at each step of the series, involves a rather intricate computation. This difficulty of equating speed and accuracy in adding means precisely that we have inadequate notions of what the ability is that the elementary school should improve. Until, for example, we have decided whether, for a given group of pupils, fifteen Courtis attempts with ten right, is or is not a better achievement than ten Courtis attempts with nine right, we have not decided just what the business of the teacher of addition is, in the case of that group of pupils. There is also the difficulty of comparing results when short and long columns are used. Correctness with a short column, say of five figures, testifies to knowledge of the process and to the power to do four successive single additions without error. Correctness with a long column, say of ten digits, testifies to knowledge of the process and to the power to do nine successive single additions without error. Now if a pupil's precision was such that on the average he made one mistake in eight single additions, he would get about half of his five-digit columns right and almost none of his ten-digit columns right. (He would do this, that is, if he added in the customary way. If he were taught to check results by repeated addition, by adding in half-columns and the like, his percentages of accurate answers might be greatly increased in both cases and be made approximately equal.) Length of column in a test of addition under ordinary conditions thus automatically overweights precision in the single additions as compared with knowledge of the process, and ability at carrying. Further, in the case of a column of whatever size, the result as ordinarily scored does not distinguish between one, two, three, or more (up to the limit) errors in the single additions. Yet, obviously, a pupil who, adding with ten-digit columns, has half of his answer-figures wrong, probably often makes two or more errors within a column, whereas a pupil who has only one column-answer in ten wrong, probably almost never makes more than one error within a column. A short-column test is then advisable as a means of interpreting the results of a long-column test. Finally, the choice of a short-column or of a long-column test is indicative of the measurer's notion of the kind of efficiency the world properly demands of the school. Twenty years ago the author would have been readier to accept a long-column test than he now is. In the world at large, long-column addition is being more and more done by machine, though it persists still in great frequency in the bookkeeping of weekly and monthly accounts in local groceries, butcher shops, and the like. The search for a measure of ability to add thus puts the problem of speed _versus_ precision, and of short-column _versus_ long-column additions clearly before us. The latter problem has hardly been realized at all by the ordinary definitions of ability to add. It may be said further that the measurement of ability to add gives the scientific student a shock by the lack of precision found everywhere in schools. Of what value is it to a graduate of the elementary school to be able to add with examples like those of the Courtis test, getting only eight out of ten right? Nobody would pay a computer for that ability. The pupil could not keep his own accounts with it. The supposed disciplinary value of habits of precision runs the risk of turning negative in such a case. It appears, at least to the author, imperative that checking should be taught and required until a pupil can add single columns of ten digits with not over one wrong answer in twenty columns. Speed is useful, especially indirectly as an indication of control of the separate higher-decade additions, but the social demand for addition below a certain standard of precision is _nil_, and its disciplinary value is _nil_ or negative. This will be made a matter of further study later. MEASUREMENTS OF ABILITIES IN COMPUTATION Measurements of these abilities may be of two sorts--(1) of the speed and accuracy shown in doing one same sort of task, as illustrated by the Courtis test for addition shown on page 28; and (2) of how hard a task can be done perfectly (or with some specified precision) within a certain assigned time or less, as illustrated by the author's rough test for addition shown on pages 28 and 29, and by the Woody tests, when extended to include alternative forms. The Courtis tests, originated as an improvement on the Stone tests and since elaborated by the persistent devotion of their author, are a standard instrument of the first sort for measuring the so-called 'fundamental' arithmetical abilities with integers. They are shown on this and the following page. Tests of the second sort are the Woody tests, which include operations with integers, common and decimal fractions, and denominate numbers, the Ballou test for common fractions ['16], and the "Ladder" exercises of the Thorndike arithmetics. Some of these are shown on pages 36 to 41. Courtis Test Arithmetic. Test No. 1. Addition Series B You will be given eight minutes to find the answers to as many of these addition examples as possible. Write the answers on this paper directly underneath the examples. You are not expected to be able to do them all. You will be marked for both speed and accuracy, but it is more important to have your answers right than to try a great many examples. 927 297 136 486 384 176 277 837 379 925 340 765 477 783 445 882 756 473 988 524 881 697 682 959 837 983 386 140 266 200 594 603 924 315 353 812 679 366 481 118 110 661 904 466 241 851 778 781 854 794 547 355 796 535 849 756 965 177 192 834 850 323 157 222 344 124 439 567 733 229 953 525 --- --- --- --- --- --- --- --- and sixteen more addition examples of nine three-place numbers. Courtis Test Arithmetic. Test No. 2. Subtraction Series B You will be given four minutes to find the answers to as many of these subtraction examples as possible. Write the answers on this paper directly underneath the examples. You are not expected to be able to do them all. You will be marked for both speed and accuracy, but it is more important to have your answers right than to try a great many examples. 107795491 75088824 91500053 87939983 77197029 57406394 19901563 72207316 --------- -------- -------- -------- and twenty more tasks of the same sort. Courtis Test Arithmetic. Test No. 3. Multiplication Series B You will be given six minutes to work as many of these multiplication examples as possible. You are not expected to be able to do them all. Do your work directly on this paper; use no other. You will be marked for both speed and accuracy, but it is more important to get correct answers than to try a large number of examples. 8246 7843 4837 3478 6482 29 702 83 15 46 ---- ---- ---- ---- ---- and twenty more multiplication examples of the same sort. Courtis Test Arithmetic. Test No. 4. Division Series B You will be given eight minutes to work as many of these division examples as possible. You are not expected to be able to do them all. Do your work directly on this paper; use no other. You will be marked for both speed and accuracy, but it is more important to get correct answers than to try a large number of examples. _____ ______ _____ ______ 25)6775 94)85352 37)9990 86)80066 and twenty more division examples of the same sort. SERIES B. MULTIPLICATION SCALE By Clifford Woody (1) (3) (4) (5) 3 × 7 = 2 × 3 = 4 × 8 = 23 3 -- (8) (9) (11) (12) 50 254 1036 5096 3 6 8 6 -- --- ---- ---- (13) (16) (18) (20) 8754 7898 24 287 8 9 234 .05 ---- ---- --- --- (24) (26) (27) (29) 16 9742 6.25 1/8 × 2 = 2-5/8 59 3.2 --- ---- ---- (33) (35) (37) (38) 2-1/2 × 3-1/2 = 987-3/4 2-1/4 × 4-1/2 × 1-1/2 = .0963-1/8 25 .084 ---- ----- SERIES B. DIVISION SCALE By Clifford Woody (1) (2) (7) (8) __ ___ ___ 3)6 9)27 4 ÷ 2 = 9)0 (11) (14) (15) (17) ___ _____ 2)13 8)5856 1/4 of 128 = 50 ÷ 7 = (19) (23) (27) (28) ____ ______ 248 ÷ 7 = 23)469 7/8 of 624 = .003).0936 (30) (34) (36) ______________ 3/4 ÷ 5 = 62.50 ÷ 1-1/4 = 9)69 lbs. 9 oz. Ballou Test Addition of Fractions _Test 1_ _Test 2_ (1) 1/4 (2) 3/14 (1) 1/3 (2) 2/7 1/4 1/14 1/6 3/14 --- ---- --- ---- _Test 3_ _Test 4_ (1) 3/5 (2) 5/6 (1) 1/7 (2) 7/9 11/15 1/2 9/10 1/4 ----- --- ---- --- _Test 5_ _Test 6_ (1) 1/10 (2) 4/9 (1) 1/6 (2) 5/6 1/6 5/12 9/10 3/8 ---- ---- ---- --- An Addition Ladder [Thorndike, '17, III, 5] Begin at the bottom of the ladder. See if you can climb to the top without making a mistake. Be sure to copy the numbers correctly. #Step 6.# _a._ Add 1-1/3 yd., 7/8 yd., 1-1/4 yd., 3/4 yd., 7/8 yd., and 1-1/2 yd. _b._ Add 62-1/2¢, 66-2/3¢, 56-1/4¢, 60¢, and 62-1/2¢. _c._ Add 1-5/16, 1-9/32, 1-3/8, 1-11/32, and 1-7/16. _d._ Add 1-1/3 yd., 1-1/4 yd., 1-1/2 yd., 2 yd., 3/4 yd., and 2/3 yd. #Step 5.# _a._ Add 4 ft. 6-1/2 in., 53-1/4 in., 5 ft. 1/2 in., 56-3/4 in., and 5 ft. _b._ Add 7 lb., 6 lb. 11 oz., 7-1/2 lb., 6 lb. 4-1/2 oz., and 8-1/2 lb. _c._ Add 1 hr. 6 min. 20 sec., 58 min. 15 sec., 1 hr. 4 min., and 55 min. _d._ Add 7 dollars, 13 half dollars, 21 quarters, 17 dimes, and 19 nickels. #Step 4.# _a._ Add .05-1/2, .06, .04-3/4, .02-3/4, and .05-1/4. _b._ Add .33-1/3, .12-1/2, .18, .16-2/3, .08-1/3 and .15. _c._ Add .08-1/3, .06-1/4, .21, .03-3/4, and .16-2/3. _d._ Add .62, .64-1/2, .66-2/3, .10-1/4, and .68. #Step 3.# _a._ Add 7-1/4, 6-1/2, 8-3/8, 5-3/4, 9-5/8 and 3-7/8. _b._ Add 4-5/8, 12, 7-1/2, 8-3/4, 6 and 5-1/4. _c._ Add 9-3/4, 5-7/8, 4-1/8, 6-1/2, 7, 3-5/8. _d._ Add 12, 8-1/2, 7-1/3, 5, 6-2/3, and 9-1/2. #Step 2.# _a._ Add 12.04, .96, 4.7, 9.625, 3.25, and 20. _b._ Add .58, 6.03, .079, 4.206, 2.75, and 10.4. _c._ Add 52, 29.8, 41.07, 1.913, 2.6, and 110. _d._ Add 29.7, 315, 26.75, 19.004, 8.793, and 20.05. #Step 1.# _a._ Add 10-3/5, 11-1/5, 10-4/5, 11, 11-2/5, 10-3/5, and 11. _b._ Add 7-3/8, 6-5/8, 8, 9-1/8, 7-7/8, 5-3/8, and 8-1/8. _c._ Add 21-1/2, 18-3/4, 31-1/2, 19-1/4, 17-1/4, 22, and 16-1/2. _d._ Add 14-5/12, 12-7/12, 9-11/12, 6-1/12, and 5. A Subtraction Ladder [Thorndike, '17, III, 11] #Step 9.# _a._ 2.16 mi. - 1-3/4 mi. _b._ 5.72 ft. - 5 ft. 3 in. _c._ 2 min. 10-1/2 sec. - 93.4 sec. _d._ 30.28 A. - 10-1/5 A. _e._ 10 gal. 2-1/2 qt. - 4.623 gal. #Step 8.# _a_ _b_ _c_ _d_ _e_ 25-7/12 10-1/4 9-5/16 5-7/16 4-2/3 12-3/4 7-1/3 6-3/8 2-3/4 1-3/4 ------- ------ ------ ------ ----- #Step 7.# _a_ _b_ _c_ _d_ _e_ 28-3/4 40-1/2 10-1/4 24-1/3 37-1/2 16-1/8 14-3/8 6-1/2 11-1/2 14-3/4 ------ ------ ------ ------ ------ #Step 6.# _a_ _b_ _c_ _d_ _e_ 10-1/3 7-1/4 15-1/8 12-1/5 4-1/16 4-2/3 2-3/4 6-3/8 11-4/5 2-7/16 ------ ----- ------ ------ ------ #Step 5.# _a_ _b_ _c_ _d_ _e_ 58-4/5 66-2/3 28-7/8 62-1/2 9-7/12 52-1/5 33-1/3 7-5/8 37-1/2 4-5/12 ------ ------ ------ ------ ------ #Step 4.# _a._ 4 hr. - 2 hr. 17 min. _b._ 4 lb. 7 oz. - 2 lb. 11 oz. _c._ 1 lb. 5 oz. - 13 oz. _d._ 7 ft. - 2 ft. 8 in. _e._ 1 bu. - 1 pk. #Step 3.# _a_ _b_ _c_ _d_ _e_ 92 mi. 6735 mi. $3 - 89¢ 28.4 mi. $508.40 84.15 mi. 6689 mi. 18.04 mi. 208.62 --------- -------- -------- --------- ------- #Step 2.# _a_ _b_ _c_ _d_ _e_ $25.00 $100.00 $750.00 6124 sq. mi. 7846 sq. mi. 9.36 71.28 736.50 2494 sq. mi. 2789 sq. mi. ------ ------- ------- ------------ ------------ #Step 1.# _a_ _b_ _c_ _d_ _e_ $18.64 $25.39 $56.70 819.4 mi. 67.55 mi. 7.40 13.37 45.60 209.2 mi. 36.14 mi. ------ ------ ------ --------- --------- An Average Ladder [Thorndike, '17, III, 132] Find the average of the quantities on each line. Begin with #Step 1#. Climb to the top without making a mistake. Be sure to copy the numbers correctly. Extend the division to two decimal places if necessary. #Step 6.# _a_. 2-2/3, 1-7/8, 2-3/4, 4-1/4, 3-5/8, 3-1/2 _b_. 62-1/2¢, 66-2/3¢, 40¢, 83-1/3¢, $1.75, $2.25 _c_. 3-11/16, 3-9/32, 3-3/8, 3-17/32, 3-7/16 _d_. .17, 19, .16-2/3, .15-1/2, .23-1/4, .18 #Step 5.# _a_. 5 ft. 3-1/2 in., 61-1/4 in., 58-3/4 in., 4 ft. 11 in. _b_. 6 lb. 9 oz., 6 lb. 11 oz., 7-1/4 lb., 7-3/8 lb. _c_. 1 hr. 4 min. 40 sec., 58 min. 35 sec., 1-1/4 hr. _d_. 2.8 miles, 3-1/2 miles, 2.72 miles #Step 4.# _a._ .03-1/2, .06, .04-3/4, .05-1/2, .05-1/4 _b._ .043, .045, .049, .047, .046, .045 _c._ 2.20, .87-1/2, 1.18, .93-3/4, 1.2925, .80 _d._ .14-1/2, .12-1/2, .33-1/3, .16-2/3, .15, .17 #Step 3.# _a._ 5-1/4, 4-1/2, 8-3/8, 7-3/4, 6-5/8, 9-3/8 _b._ 9-5/8, 12, 8-1/2, 8-3/4, 6, 5-1/4, 9 _c._ 9-3/8, 5-3/4, 4-1/8, 7-1/2, 6 _d._ 11, 9-1/2, 10-1/3, 13, 16-2/3, 9-1/2 #Step 2.# _a._ 13.05, .97, 4.8, 10.625, 3.37 _b._ 1.48, 7.02, .93, 5.307, 4.1, 7, 10.4 _c._ 68, 71.4, 59.8, 112, 96.1, 79.8 _d._ 2.079, 3.908, 4.165, 2.74 #Step 1.# _a._ 4, 9-1/2, 6, 5, 7-1/2, 8, 10, 9 _b._ 6, 5, 3.9, 7.1, 8 _c._ 1086, 1141, 1059, 1302, 1284 _d._ $100.82, $206.49, $317.25, $244.73 As such tests are widened to cover the whole task of the elementary school in respect to arithmetic, and accepted by competent authorities as adequate measures of achievement in computing, they will give, as has been said, a working definition of the task. The reader will observe, for example, that work such as the following, though still found in many textbooks and classrooms, does not, in general, appear in the modern tests and scales. Reduce the following improper fractions to mixed numbers:-- 19/13 43/21 176/25 198/14 Reduce to integral or mixed numbers:-- 61381/37 2134/67 413/413 697/225 Simplify:-- 3/4 of 8/9 of 3/5 of 15/22 Reduce to lowest terms:-- 357/527 264/312 492/779 418/874 854/1769 30/735 44/242 77/847 18/243 96/224 Find differences:-- 6-2/7 8-5/11 8-4/13 5-1/4 7-1/8 3-1/14 5-1/7 3-7/13 2-11/14 2-1/7 ------ ------ ------ ------- ------ Square:-- 2/3 4/5 5/7 6/9 10/11 12/13 2/7 15/16 19/20 17/18 25/30 41/53 Multiply:-- 2/11 × 33 32 × 3/14 39 × 2/13 60 × 11/28 77 × 4/11 63 × 2/27 54 × 8/45 65 × 3/13 344-16/21 432-2/7 MEASUREMENTS OF ABILITY IN APPLIED ARITHMETIC: THE SOLUTION OF PROBLEMS Stone ['08] measured achievement with the following problems, fifteen minutes being the time allowed. "Solve as many of the following problems as you have time for; work them in order as numbered: 1. If you buy 2 tablets at 7 cents each and a book for 65 cents, how much change should you receive from a two-dollar bill? 2. John sold 4 Saturday Evening Posts at 5 cents each. He kept 1/2 the money and with the other 1/2 he bought Sunday papers at 2 cents each. How many did he buy? 3. If James had 4 times as much money as George, he would have $16. How much money has George? 4. How many pencils can you buy for 50 cents at the rate of 2 for 5 cents? 5. The uniforms for a baseball nine cost $2.50 each. The shoes cost $2 a pair. What was the total cost of uniforms and shoes for the nine? 6. In the schools of a certain city there are 2200 pupils; 1/2 are in the primary grades, 1/4 in the grammar grades, 1/8 in the high school, and the rest in the night school. How many pupils are there in the night school? 7. If 3-1/2 tons of coal cost $21, what will 5-1/2 tons cost? 8. A news dealer bought some magazines for $1. He sold them for $1.20, gaining 5 cents on each magazine. How many magazines were there? 9. A girl spent 1/8 of her money for car fare, and three times as much for clothes. Half of what she had left was 80 cents. How much money did she have at first? 10. Two girls receive $2.10 for making buttonholes. One makes 42, the other 28. How shall they divide the money? 11. Mr. Brown paid one third of the cost of a building; Mr. Johnson paid 1/2 the cost. Mr. Johnson received $500 more annual rent than Mr. Brown. How much did each receive? 12. A freight train left Albany for New York at 6 o'clock. An express left on the same track at 8 o'clock. It went at the rate of 40 miles an hour. At what time of day will it overtake the freight train if the freight train stops after it has gone 56 miles?" The criteria he had in mind in selecting the problems were as follows:-- "The main purpose of the reasoning test is the determination of the ability of VI A children to reason in arithmetic. To this end, the problems, as selected and arranged, are meant to embody the following conditions:-- 1. Situations equally concrete to all VI A children. 2. Graduated difficulties. _a._ As to arithmetical thinking. _b._ As to familiarity with the situation presented. 3. The omission of _a._ Large numbers. _b._ Particular memory requirements. _c._ Catch problems. _d._ All subject matter except whole numbers, fractions, and United States money. The test is purposely so long that only very rarely did any pupil fully complete it in the fifteen minute limit." Credits were given of 1, for each of the first five problems, 1.4, 1.2, and 1.6 respectively for problems 6, 7, and 8, and of 2 for each of the others. Courtis sought to improve the Stone test of problem-solving, replacing it by the two tests reproduced below. ARITHMETIC--Test No. 6. Speed Test--Reasoning #Do not work# the following examples. Read each example through, make up your mind what operation you would use if you were going to work it, then write the name of the operation selected in the blank space after the example. Use the following abbreviations:--"Add." for addition, "Sub." for subtraction, "Mul." for multiplication, and "Div." for division. +-----------+----+ | OPERATION | | |-----------+----| 1. A girl brought a collection of 37 colored postal | | | cards to school one day, and gave away 19 cards to | | | her friends. How many cards did she have left to | | | take home? | | | |-----------+----| 2. Five boys played marbles. When the game was | | | over, each boy had the same number of marbles. If | | | there were 45 marbles altogether, how many did each | | | boy have? | | | |-----------+----| 3. A girl, watching from a window, saw 27 | | | automobiles pass the school the first hour, and | | | 33 the second. How many autos passed by the | | | school in the two hours? | | | |-----------+----| 4. In a certain school there were eight rooms and | | | each room had seats for 50 children. When all the | | | places were taken, how many children were there in | | | the school? | | | |-----------+----| 5. A club of boys sent their treasurer to buy | | | baseballs. They gave him $3.15 to spend. How many | | | balls did they expect him to buy, if the balls cost | | | 45¢. apiece? | | | |-----------+----| 6. A teacher weighed all the girls in a certain | | | grade. If one girl weighed 79 pounds and another | | | 110 pounds, how many pounds heavier was one girl | | | than the other? | | | |-----------+----| 7. A girl wanted to buy a 5-pound box of candy to | | | give as a present to a friend. She decided to get | | | the kind worth 35¢. a pound. What did she pay for | | | the present? | | | |-----------+----| 8. One day in vacation a boy went on a fishing trip | | | and caught 12 fish in the morning, and 7 in the | | | afternoon. How many fish did he catch altogether? | | | |-----------+----| 9. A boy lived 15 blocks east of a school; his chum | | | lived on the same street, but 11 blocks west of the | | | school. How many blocks apart were the two boys' | | | houses? | | | |-----------+----| 10. A girl was 5 times as strong as her small | | | sister. If the little girl could lift a weight of | | | 20 pounds, how large a weight could the older girl | | | lift? | | | |-----------+----| 11. The children of a school gave a sleigh-ride | | | party. There were 270 children to go on the ride | | | and each sleigh held 30 children. How many sleighs | | | were needed? | | | |-----------+----| 12. In September there were 43 children in the | | | eighth grade of a certain school; by June there | | | were 59. How many children entered the grade | | | during the year? | | | |-----------+----| 13. A girl who lived 17 blocks away walked to | | | school and back twice a day. What was the total | | | number of blocks the girl walked each day in | | | going to and from school? | | | |-----------+----| 14. A boy who made 67¢. a day carrying papers, was | | | hired to run on a long errand for which he received | | | 50¢. What was the total amount the boy earned that | | | day? | | | |-----------+----| Total Right | | | +-----------+----+ (Two more similar problems follow.) Test 6 and Test 8 are from the Courtis Standard Test. Used by permission of S. A. Courtis. ARITHMETIC--Test No. 8. Reasoning In the blank space below, work as many of the following examples as possible in the time allowed. Work them in order as numbered, entering each answer in the "answer" column before commencing a new example. Do not work on any other paper. +--------+-+ | ANSWER | | |--------+-| 1. The children in a certain school gave a Christmas | | | party. One of the presents was a box of candy. In filling | | | the boxes, one grade used 16 pounds of candy, another 17 | | | pounds, a third 12 pounds, and a fourth 13 pounds. What | | | did the candy cost at 26¢. a pound? | | | |--------+-| 2. A school in a certain city used 2516 pieces of chalk | | | in 37 school days. Three new rooms were opened, each | | | room holding 50 children, and the school was then found | | | to use 84 sticks of chalk per day. How many more sticks | | | of chalk were used per day than at first? | | | |--------+-| 3. Several boys went on a bicycle trip of 1500 miles. | | | The first week they rode 374 miles, the second week 264 | | | miles, the third 423 miles, the fourth 401 miles. They | | | finished the trip the next week. How many miles did they | | | ride the last week? | | | |--------+-| 4. Forty-five boys were hired to pick apples from 15 | | | trees in an apple orchard. In 50 minutes each boy had | | | picked 48 choice apples. If all the apples picked were | | | packed away carefully in 8 boxes of equal size, how many | | | apples were put in each box? | | | |--------+-| 5. In a certain school 216 children gave a sleigh-ride | | | party. They rented 7 sleighs at a cost of $30.00 and paid | | | $24.00 for the refreshments. The party travelled 15 miles | | | in 2-1/2 hours and had a very pleasant time. What was | | | each child's share of the expense? | | | |--------+-| 6. A girl found, by careful counting, that there were | | | 2400 letters on one page of her history, and only 2295 | | | letters on a page of her reader. How many more letters | | | had she read in one book than in the other if she had | | | read 47 pages in each of the books? | | | |--------+-| 7. Each of 59 rooms in the schools of a certain city | | | contributed 25 presents to a Christmas entertainment for | | | poor children. The stores of the city gave 1986 other | | | articles for presents. What was the total number of | | | presents given away at the entertainment? | | | |--------+-| 8. Forty-eight children from a certain school paid 10¢. | | | apiece to ride 7 miles on the cars to a woods. There in a | | | few hours they gathered 2765 nuts. 605 of these were bad, | | | but the rest were shared equally among the children. How | | | many good nuts did each one get? | | | |--------+-| Total | | | +--------+-+ These proposed measures of ability to apply arithmetic illustrate very nicely the differences of opinion concerning what applied arithmetic and arithmetical reasoning should be. The thinker who emphasizes the fact that in life out of school the situation demanding quantitative treatment is usually real rather than described, will condemn a test all of whose constituents are _described_ problems. Unless we are excessively hopeful concerning the transfer of ideas of method and procedure from one mental function to another we shall protest against the artificiality of No. 3 of the Stone series, and of the entire Courtis Test 8 except No. 4. The Courtis speed-reasoning test (No. 6) is a striking example of the mixture of ability to understand quantitative relations with the ability to understand words. Consider these five, for example, in comparison with the revised versions attached.[3] [3] The form of Test 6 quoted here is that given by Courtis ['11-'12, p. 20]. This differs a little from the other series of Test 6, shown on pages 43 and 44. 1. The children of a school gave a sleigh-ride party. There were 9 sleighs, and each sleigh held 30 children. How many children were there in the party? REVISION. _If one sleigh holds 30 children, 9 sleighs hold .... children._ 2. Two school-girls played a number-game. The score of the girl that lost was 57 points and she was beaten by 16 points. What was the score of the girl that won? REVISION. _Mary and Nell played a game. Mary had a score of 57. Nell beat Mary by 16. Nell had a score of ...._ 3. A girl counted the automobiles that passed a school. The total was 60 in two hours. If the girl saw 27 pass the first hour how many did she see the second? REVISION. _In two hours a girl saw 60 automobiles. She saw 27 the first hour. She saw .... the second hour._ 4. On a playground there were five equal groups of children each playing a different game. If there were 75 children all together, how many were there in each group? REVISION. _75 pounds of salt just filled five boxes. The boxes were exactly alike. There were .... pounds in a box._ 5. A teacher weighed all the children in a certain grade. One girl weighed 70 pounds. Her older sister was 49 pounds heavier. How many pounds did the sister weigh? REVISION. _Mary weighs 70 lb. Jane weighs 49 pounds more than Mary. Jane weighs .... pounds._ The distinction between a problem described as clearly and simply as possible and the same problem put awkwardly or in ill-known words or willfully obscured should be regarded; and as a rule measurements of ability to apply arithmetic should eschew all needless obscurity or purely linguistic difficulty. For example, _A boy bought a two-cent stamp. He gave the man in the store 10 cents. The right change was .... cents._ is better as a test than _If a boy, purchasing a two-cent stamp, gave a ten-cent stamp in payment, what change should he be expected to receive in return?_ The distinction between the description of a _bona fide_ problem that a human being might be called on to solve out of school and the description of imaginary possibilities or puzzles should also be considered. Nos. 3 and 9 of Stone are bad because to frame the problems one must first know the answers, so that in reality there could never be any point in solving them. It is probably safe to say that nobody in the world ever did or ever will or ever should find the number of apples in a box by the task of No. 4 of the Courtis Test 8. This attaches no blame to Dr. Stone or to Mr. Courtis. Until very recently we were all so used to the artificial problems of the traditional sort that we did not expect anything better; and so blind to the language demands of described problems that we did not see their very great influence. Courtis himself has been active in reform and has pointed out ('13, p. 4 f.) the defects in his Tests 6 and 8. "Tests Nos. 6 and 8, the so-called reasoning tests, have proved the least satisfactory of the series. The judgments of various teachers and superintendents as to the inequalities of the units in any one test, and of the differences between the different editions of the same test, have proved the need of investigating these questions. Tests of adults in many lines of commercial work have yielded in many cases lower scores than those of the average eighth grade children. At the same time the scores of certain individuals of marked ability have been high, and there appears to be a general relation between ability in these tests and accuracy in the abstract work. The most significant facts, however, have been the difficulties experienced by teachers in attempting to remedy the defects in reasoning. It is certain that the tests measure abilities of value but the abilities are probably not what they seem to be. In an attempt to measure the value of different units, for instance, as many problems as possible were constructed based upon a single situation. Twenty-one varieties were secured by varying the relative form of the question and the relative position of the different phrases. One of these proved nineteen times as hard as another as measured by the number of mistakes made by the children; yet the cause of the difference was merely the changes in the phrasing. This and other facts of the same kind seem to show that Tests 6 and 8 measure mainly the ability to read." The scientific measurement of the abilities and achievements concerned with applied arithmetic or problem-solving is thus a matter for the future. In the case of described problems a beginning has been made in the series which form a part of the National Intelligence Tests ['20], one of which is shown on page 49 f. In the case of problems with real situations, nothing in systematic form is yet available. Systematic tests and scales, besides defining the abilities we are to establish and improve, are of very great service in measuring the status and improvement of individuals and of classes, and the effects of various methods of instruction and of study. They are thus helpful to pupils, teachers, supervisors, and scientific investigators; and are being more and more widely used every year. Information concerning the merits of the different tests, the procedure to follow in giving and scoring them, the age and grade standards to be used in interpreting results, and the like, is available in the manuals of Educational Measurement, such as Courtis, _Manual of Instructions for Giving and Scoring the Courtis Standard Tests in the Three R's_ ['14]; Starch, _Educational Measurements_ ['16]; Chapman and Rush, _Scientific Measurement of Classroom Products_ ['17]; Monroe, DeVoss, and Kelly, _Educational Tests and Measurements_ ['17]; Wilson and Hoke, _How to Measure_ ['20]; and McCall, _How to Measure in Education_ ['21]. TEST 1 National Intelligence Tests. Scale A. Form 1, Edition 1 Find the answers as quickly as you can. Write the answers on the dotted lines. Use the side of the page to figure on. #Begin here# 1 Five cents make 1 nickel. How many nickels make a dime? _Answer_ ..... 2 John paid 5 dollars for a watch and 3 dollars for a chain. How many dollars did he pay for the watch and chain? _Answer_ ..... 3 Nell is 13 years old. Mary is 9 years old. How much younger is Mary than Nell? _Answer_ ..... 4 One quart of ice cream is enough for 5 persons. How many quarts of ice cream are needed for 25 persons? _Answer_ ..... 5 John's grandmother is 86 years old. If she lives, in how many years will she be 100 years old? _Answer_ ..... 6 If a man gets $2.50 a day, what will he be paid for six days' work? _Answer_ ..... 7 How many inches are there in a foot and a half? _Answer_ ..... 8 What is the cost of 12 cakes at 6 for 5 cents? _Answer_ ..... 9 The uniforms for a baseball team of nine boys cost $2.50 each. The shoes cost $2 a pair. What was the total cost of uniforms and shoes for the nine? _Answer_ ..... 10 A train that usually arrives at half-past ten was 17 minutes late. When did it arrive? _Answer_ ..... 11 At 10¢ a yard, what is the cost of a piece 10-1/2 ft. long? _Answer_ ..... 12 A man earns $6 a day half the time, $4.50 a day one fourth of the time, and nothing on the remaining days for a total period of 40 days. What did he earn in all in the 40 days? _Answer_ ..... 13 What per cent of $800 is 4% of $1000? _Answer_ ..... 14 If 60 men need 1500 lb. flour per month, what is the requirement per man per day counting a month as 30 days? _Answer_ ..... 15 A car goes at the rate of a mile a minute. A truck goes 20 miles an hour. How many times as far will the car go as the truck in 10 seconds? _Answer_ ..... 16 The area of the base (inside measure) of a cylindrical tank is 90 square feet. How tall must it be to hold 100 cubic yards? _Answer_ ..... From National Intelligence Tests by National Research Council. Copyright, 1920, by The World Book Company, Yonkers-on-Hudson, New York. Used by permission of the publishers. CHAPTER III THE CONSTITUTION OF ARITHMETICAL ABILITIES THE ELEMENTARY FUNCTIONS OF ARITHMETICAL LEARNING It would be a useful work for some one to try to analyze arithmetical learning into the unitary abilities which compose it, showing just what, in detail, the mind has to do in order to be prepared to pass a thorough test on the whole of arithmetic. These unitary abilities would make a very long list. Examination of a well-planned textbook will show that such an ability as multiplication is treated as a composite of the following: knowledge of the multiplications up to 9 × 9; ability to multiply two (or more)-place numbers by 2, 3, and 4 when 'carrying' is not required and no zeros occur in the multiplicand; ability to multiply by 2, 3, ... 9, with carrying; the ability to handle zeros in the multiplicand; the ability to multiply with two-place numbers not ending in zero; the ability to handle zero in the multiplier as last number; the ability to multiply with three (or more)-place numbers not including a zero; the ability to multiply with three- and four-place numbers with zero in second or third, or second and third, as well as in last place; the ability to save time by annexing zeros; and so on and on through a long list of further abilities required to multiply with United States money, decimal fractions, common fractions, mixed numbers, and denominate numbers. The units or 'steps' thus recognized by careful teaching would make a long list, but it is probable that a still more careful study of arithmetical ability as a hierarchy of mental habits or connections would greatly increase the list. Consider, for example, ordinary column addition. The majority of teachers probably treat this as a simple application of the knowledge of the additions to 9 + 9, plus understanding of 'carrying.' On the contrary there are at least seven processes or minor functions involved in two-place column addition, each of which is psychologically distinct and requires distinct educational treatment. These are:-- A. Learning to keep one's place in the column as one adds. B. Learning to keep in mind the result of each addition until the next number is added to it. C. Learning to add a seen to a thought-of number. D. Learning to neglect an empty space in the columns. E. Learning to neglect 0s in the columns. F. Learning the application of the combinations to higher decades may for the less gifted pupils involve as much time and labor as learning all the original addition tables. And even for the most gifted child the formation of the connection '8 and 7 = 15' probably never quite insures the presence of the connections '38 and 7 = 45' and '18 + 7 = 25.' G. Learning to write the figure signifying units rather than the total sum of a column. In particular, learning to write 0 in the cases where the sum of the column is 10, 20, etc. Learning to 'carry' also involves in itself at least two distinct processes, by whatever way it is taught. We find evidence of such specialization of functions in the results with such tests as Woody's. For example, 2 + 5 + 1 = .... surely involves abilities in part different from 2 4 3 - because only 77 percent of children in grade 3 do the former correctly, whereas 95 percent of children in that grade do the latter correctly. In grade 2 the difference is even more marked. In the case of subtraction 4 4 - involves abilities different from those involved in 9 3, - being much less often solved correctly in grades 2 and 4. 6 0 - is much harder than either of the above. 43 1 21 2 33 13 is much harder than 35. -- -- It may be said that these differences in difficulty are due to different amounts of practice. This is probably not true, but if it were, it would not change the argument; if the two abilities were identical, the practice of one would improve the other equally. I shall not undertake here this task of listing and describing the elementary functions which constitute arithmetical learning, partly because what they are is not fully known, partly because in many cases a final ability may be constituted in several different ways whose descriptions become necessarily tedious, and partly because an adequate statement of what is known would far outrun the space limits of this chapter. Instead, I shall illustrate the results by some samples. KNOWLEDGE OF THE MEANING OF A FRACTION As a first sample, consider knowledge of the meaning of a fraction. Is the ability in question simply to understand that a fraction is a statement of the number of parts, each of a certain size, the upper number or numerator telling how many parts are taken and the lower number or denominator telling what fraction of unity each part is? And is the educational treatment required simply to describe and illustrate such a statement and have the pupils apply it to the recognition of fractions and the interpretation of each of them? And is the learning process (1) the formation of the notions of part, size of part, number of part, (2) relating the last two to the numbers in a fraction, and, as a necessary consequence, (3) applying these notions adequately whenever one encounters a fraction in operation? Precisely this was the notion a few generations ago. The nature of fractions was taught as one principle, in one step, and the habits of dealing with fractions were supposed to be deduced from the general law of a fraction's nature. As a result the subject of fractions had to be long delayed, was studied at great cost of time and effort, and, even so, remained a mystery to all save gifted pupils. These gifted pupils probably of their own accord built up the ability piecemeal out of constituent insights and habits. At all events, scientific teaching now does build up the total ability as a fusion or organization of lesser abilities. What these are will be seen best by examining the means taken to get them. (1) First comes the association of 1/2 of a pie, 1/2 of a cake, 1/2 of an apple, and such like with their concrete meanings so that a pupil can properly name a clearly designated half of an obvious unit like an orange, pear, or piece of chalk. The same degree of understanding of 1/4, 1/8, 1/3, 1/6, and 1/5 is secured. The pupil is taught that 1 pie = 2 1/2s, 3 1/3s, 4 1/4s, 5 1/5s, 6 1/6s, and 8 1/8s; similarly for 1 cake, 1 apple, and the like. So far he understands 1/_x_ of _y_ in the sense of certain simple parts of obviously unitary _y_s. (2) Next comes the association with 1/2 of an inch, 1/2 of a foot, 1/2 of a glassful and other cases where _y_ is not so obviously a unitary object whose pieces still show their derivation from it. Similarly for 1/4, 1/3, etc. (3) Next comes the association with 1/2 of a collection of eight pieces of candy, 1/3 of a dozen eggs, 1/5 of a squad of ten soldiers, etc., until 1/2, 1/3, 1/4, 1/5, 1/6, and 1/8 are understood as names of certain parts of a collection of objects. (4) Next comes the similar association when the nature of the collection is left undefined, the pupil responding to 1/2 of 6 is ..., 1/4 of 8 is ..., 2 is 1/5 of ..., 1/3 of 6 is ..., 1/3 of 9 is ..., 2 is 1/3 of ..., and the like. Each of these abilities is justified in teaching by its intrinsic merits, irrespective of its later service in helping to constitute the general understanding of the meaning of a fraction. The habits thus formed in grades 3 or 4 are of constant service then and thereafter in and out of school. (5) With these comes the use of 1/5 of 10, 15, 20, etc., 1/6 of 12, 18, 42, etc., as a useful variety of drill on the division tables, valuable in itself, and a means of making the notion of a unit fraction more general by adding 1/7 and 1/9 to the scheme. (6) Next comes the connection of 3/4, 2/5, 3/5, 4/5, 2/3, 1/6, 5/6, 3/8, 5/8, 7/8, 3/10, 7/10, and 9/10, each with its meaning as a certain part of some conveniently divisible unit, and, (7) and (8), connections between these fractions and their meanings as parts of certain magnitudes (7) and collections (8) of convenient size, and (9) connections between these fractions and their meanings when the nature of the magnitude or collection is unstated, as in 4/5 of 15 = ..., 5/8 of 32 = .... (10) That the relation is general is shown by using it with numbers requiring written division and multiplication, such as 7/8 of 1736 = ..., and with United States money. Elements (6) to (10) again are useful even if the pupil never goes farther in arithmetic. One of the commonest uses of fractions is in calculating the cost of fractions of yards of cloth, and fractions of pounds of meat, cheese, etc. The next step (11) is to understand to some extent the principle that the value of any of these fractions is unaltered by multiplying or dividing the numerator and denominator by the same number. The drills in expressing fractions in lower and higher terms which accomplish this are paralleled by (12) and (13) simple exercises in adding and subtracting fractions to show that fractions are quantities that can be operated on like any quantities, and by (14) simple work with mixed numbers (addition and subtraction and reductions), and (15) improper fractions. All that is done with improper fractions is (_a_) to have the pupil use a few of them as he would any fractions and (_b_) to note their equivalent mixed numbers. In (12), (13), and (14) only fractions of the same denominators are added or subtracted, and in (12) (13), (14), and (15) only fractions with 2, 3, 4, 5, 6, 8, or 10 in the denominator are used. As hitherto, the work of (11) to (15) is useful in and of itself. (16) Definitions are given of the following type:-- Numbers like 2, 3, 4, 7, 11, 20, 36, 140, 921 are called whole numbers. Numbers like 7/8, 1/5, 2/3, 3/4, 11/8, 7/6, 1/3, 4/3, 1/8, 1/6 are called fractions. Numbers like 5-1/4, 7-3/8, 9-1/2, 16-4/5, 315-7/8, 1-1/3, 1-2/3 are called mixed numbers. (17) The terms numerator and denominator are connected with the upper and lower numbers composing a fraction. Building this somewhat elaborate series of minor abilities seems to be a very roundabout way of getting knowledge of the meaning of a fraction, and is, if we take no account of what is got along with this knowledge. Taking account of the intrinsically useful habits that are built up, one might retort that the pupil gets his knowledge of the meaning of a fraction at zero cost. KNOWLEDGE OF THE SUBTRACTION AND DIVISION TABLES Consider next the knowledge of the subtraction and division 'Tables.' The usual treatment presupposes that learning them consists of forming independently the bonds:-- 3 - 1 = 2 4 ÷ 2 = 2 3 - 2 = 1 6 ÷ 2 = 3 4 - 1 = 3 6 ÷ 3 = 2 . . . . . . . . . . . . 18 - 9 = 9 81 ÷ 9 = 9 In fact, however, these 126 bonds are not formed independently. Except perhaps in the case of the dullest twentieth of pupils, they are somewhat facilitated by the already learned additions and multiplications. And by proper arrangement of the learning they may be enormously facilitated thereby. Indeed, we may replace the independent memorizing of these facts by a set of instructive exercises wherein the pupil derives the subtractions from the corresponding additions by simple acts of reasoning or selective thinking. As soon as the additions giving sums of 9 or less are learned, let the pupil attack an exercise like the following:-- Write the missing numbers:-- A B C D 3 and ... are 5. 5 and ... are 8. 4 and ... are 5. 4 and ... are 8. 3 and ... are 9. 3 and ... are 6. 5 and ... are 6. 1 and ... are 7. 4 and ... are 7. 4 and ... are 9. 6 and ... are 9. 6 and ... are 7. 5 and ... are 7. 2 and ... = 6. 1 and ... are 8. 8 and ... are 9. 6 and ... are 8. 5 and ... = 9. 3 and ... are 7. 3 + ... are 4. 4 and ... are 6. 2 and ... = 7. 1 + ... are 3. 7 + ... are 8. 2 and ... are 5. 3 and ... = 8. 1 + ... are 5. 4 + ... are 9. 2 and ... = 8. 1 and ... = 4. 4 + ... are 8. 2 + ... are 3. 3 and ... = 6. 2 and ... = 4. 7 + ... are 9. 1 + ... are 9. 6 and ... = 9. 3 and ... = 8. 2 + ... = 4. 3 + ... = 6. 4 and ... = 6. 6 and ... = 7. 3 + ... = 8. 5 + ... = 9. 4 and ... = 7. 2 and ... = 5. 4 + ... = 5. 1 + ... = 3. The task for reasoning is only to try, one after another, numbers that seem promising and to select the right one when found. With a little stimulus and direction children can thus derive the subtractions up to those with 9 as the larger number. Let them then be taught to do the same with the printed forms:-- Subtract 9 7 8 5 8 6 3 5 6 2 2 4 etc. - - - - - - and 9 - 7 = ..., 9 - 5 = ..., 7 - 5 = ..., etc. In the case of the divisions, suppose that the pupil has learned his first table and gained surety in such exercises as:-- 4 5s = .... 6 × 5 = .... 9 nickels = .... cents. 8 5s = .... 4 × 5 = .... 6 " = .... " 3 5s = .... 2 × 5 = .... 5 " = .... " 7 5s = .... 9 × 5 = .... 7 " = .... " If one ball costs 5 cents, two balls cost .... cents, three balls cost .... cents, etc. He may then be set at once to work at the answers to exercises like the following:-- Write the answers and the missing numbers:-- A B C D .... 5s = 15 40 = .... 5s .... × 5 = 25 20 cents = .... nickels. .... 5s = 20 20 = .... 5s .... × 5 = 50 30 cents = .... nickels. .... 5s = 40 15 = .... 5s .... × 5 = 35 15 cents = .... nickels. .... 5s = 25 45 = .... 5s .... × 5 = 10 40 cents = .... nickels. .... 5s = 30 50 = .... 5s .... × 5 = 40 .... 5s = 35 25 = .... 5s .... × 5 = 45 E For 5 cents you can buy 1 small loaf of bread. For 10 cents you can buy 2 small loaves of bread. For 25 cents you can buy .... small loaves of bread. For 45 cents you can buy .... small loaves of bread. For 35 cents you can buy .... small loaves of bread. F 5 cents pays 1 car fare. 15 cents pays .... car fares. 10 cents pays .... car fares. 20 cents pays .... car fares. G How many 5 cent balls can you buy with 30 cents? .... How many 5 cent balls can you buy with 35 cents? .... How many 5 cent balls can you buy with 25 cents? .... How many 5 cent balls can you buy with 15 cents? .... In the case of the meaning of a fraction, the ability, and so the learning, is much more elaborate than common practice has assumed; in the case of the subtraction and division tables the learning is much less so. In neither case is the learning either mere memorizing of facts or the mere understanding of a principle _in abstracto_ followed by its application to concrete cases. It is (and this we shall find true of almost all efficient learning in arithmetic) the formation of connections and their use in such an order that each helps the others to the maximum degree, and so that each will do the maximum amount for arithmetical abilities other than the one specially concerned, and for the general competence of the learner. LEARNING THE PROCESSES OF COMPUTATION As another instructive topic in the constitution of arithmetical abilities, we may take the case of the reasoning involved in understanding the manipulations of figures in two (or more)-place addition and subtraction, multiplication and division involving a two (or more)-place number, and the manipulations of decimals in all four operations. The psychology of these is of special interest and importance. For there are two opposite explanations possible here, leading to two opposite theories of teaching. The common explanation is that these methods of manipulation, if understood at all, are understood as deductions from the properties of our system of decimal notation. The other is that they are understood partly as inductions from the experience that they always give the right answer. The first explanation leads to the common preliminary deductive explanations of the textbooks. The other leads to explanations by verification; _e.g._, of addition by counting, of subtraction by addition, of multiplication by addition, of division by multiplication. Samples of these two sorts of explanation are given below. SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATION MULTIPLICATION is the process of taking one number as many times as there are units in another number. The PRODUCT is the result of the multiplication. The MULTIPLICAND is the number to be taken. The MULTIPLIER is the number denoting how many times the multiplicand is to be taken. The multiplier and multiplicand are the FACTORS. Multiply 623 by 3 OPERATION _Multiplicand_ 623 _Multiplier_ 3 ---- _Product_ 1869 EXPLANATION.--For convenience we write the multiplier under the multiplicand, and begin with units to multiply. 3 times 3 units are 9 units. We write the nine units in units' place in the product. 3 times 2 tens are 6 tens. We write the 6 tens in tens' place in the product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and 8 hundreds. The 1 thousand we write in thousands' place and the 8 hundreds in hundreds' place in the product. Therefore, the product is 1 thousand 8 hundreds, 6 tens and 9 units, or 1869. SHORT MULTIPLICATION WITHOUT CARRYING: INDUCTIVE EXPLANATION 1. The children of the third grade are to have a picnic. 32 are going. How many sandwiches will they need if each of the 32 children has four sandwiches? _Here is a quick way to find out_:-- 32 _Think "4 × 2," write 8 under the 2 in the ones column._ 4 _Think "4 × 3," write 12 under the 3 in the tens column._ -- 2. How many bananas will they need if each of the 32 children has two bananas? 32 × 2 or 2 × 32 will give the answer. 3. How many little cakes will they need if each child has three cakes? 32 × 3 or 3 × 32 will give the answer. 32 3 × 2 = .... _Where do you write the 6?_ 3 3 × 3 = .... _Where do you write the 9?_ -- 4. Prove that 128, 64, and 96 are right by adding four 32s, two 32s, and three 32s. 32 32 32 32 32 32 32 32 32 -- -- -- Multiplication You #multiply# when you find the answers to questions like How many are 9 × 3? How many are 3 × 32? How many are 8 × 5? How many are 4 × 42? 1. Read these lines. Say the right numbers where the dots are: If you #add# 3 to 32, you have .... 35 is the #sum#. If you #subtract# 3 from 32, the result is .... 29 is the #difference# or #remainder#. If you #multiply# 3 by 32 or 32 by 3, you have .... 96 is the #product#. Find the products. Check your answers to the first line by adding. 2. 3. 4. 5. 6. 7. 8. 9. 41 33 42 44 53 43 34 24 3 2 4 2 3 2 2 2 -- -- -- -- -- -- -- -- 10. 11. 12. 13. 14. 15. 16. 43 52 32 23 41 51 14 3 3 3 3 2 4 2 -- -- -- -- -- -- -- 17. 213 _Write the 9 in the ones column._ 3 _Write the 3 in the tens column._ --- _Write the 6 in the hundreds column._ _Check your answer by adding._ Add 213 213 213 --- 18. 19. 20. 21. 22. 23. 24. 214 312 432 231 132 314 243 2 3 2 3 3 2 2 --- --- --- --- --- --- --- SHORT DIVISION: DEDUCTIVE EXPLANATION Divide 1825 by 4 Divisor 4 |1825 Dividend -------- 456-1/4 Quotient EXPLANATION.--For convenience we write the divisor at the left of the dividend, and the quotient below it, and begin at the left to divide. 4 is not contained in 1 thousand any thousand times, therefore the quotient contains no unit of any order higher than hundreds. Consequently we find how many times 4 is contained in the hundreds of the dividend. 1 thousand and 8 hundreds are 18 hundreds. 4 is contained in 18 hundreds 4 hundred times and 2 hundreds remaining. We write the 4 hundreds in the quotient. The 2 hundreds we consider as united with the 2 tens, making 22 tens. 4 is contained in 22 tens 5 tens times, and 2 tens remaining. We write the 5 tens in the quotient, and the remaining 2 tens we consider as united with the 5 units, making 25 units. 4 is contained in 25 units 6 units times and 1 unit remaining. We write the 6 units in the quotient and indicate the division of the remainder, 1 unit, by the divisor 4. Therefore the quotient of 1825 divided by 4 is 456-1/4, or 456 and 1 remainder. SHORT DIVISION: INDUCTIVE EXPLANATION Dividing Large Numbers 1. Tom, Dick, Will, and Fred put in 2 cents each to buy an eight-cent bag of marbles. There are 128 marbles in it. How many should each boy have, if they divide the marbles equally among the four boys? ----- 4 |128 _Think "12 = three 4s." Write the 3 over the 2 in the tens column._ _Think "8 = two 4s." Write the 2 over the 8 in the ones column._ _32 is right, because 4 × 32 = 128._ 2. Mary, Nell, and Alice are going to buy a book as a present for their Sunday-school teacher. The present costs 69 cents. How much should each girl pay, if they divide the cost equally among the three girls? ---- 3|69 _Think "6 = .... 3s." Write the 2 over the 6 in the tens column._ _Think "9 = .... 3s." Write the 3 over the 9 in the ones column._ _23 is right, for 3 × 23 = 69._ 3. Divide the cost of a 96-cent present equally among three girls. How much should each girl pay? ------ 3|96 4. Divide the cost of an 84-cent present equally among 4 girls. How much should each girl pay? 5. Learn this: (Read ÷ as "_divided by_.") 12 + 4 = 16. 16 is the sum. 12 - 4 = 8. 8 is the difference or remainder. 12 × 4 = 48. 48 is the product. 12 ÷ 4 = 3. 3 is the quotient. 6. Find the quotients. Check your answers by multiplying. ---- ---- ---- ----- ----- ----- 3|99 2|86 5|155 6|246 4|168 3|219 [Uneven division is taught by the same general plan, extended.] LONG DIVISION: DEDUCTIVE EXPLANATION To Divide by Long Division 1. Let it be required to divide 34531 by 15. _Operation_ Divided Divisor 15)34531(2302-1/15 Quotient 30 -- 45 45 -- 31 30 -- 1 Remainder For convenience we write the divisor at the left and the quotient at the right of the dividend, and begin to divide as in Short Division. 15 is contained in 3 ten-thousands 0 ten-thousands times; therefore, there will be 0 ten-thousands in the quotient. Take 34 thousands; 15 is contained in 34 thousands 2 thousands times; we write the 2 thousands in the quotient. 15 × 2 thousands = 30 thousands, which, subtracted from 34 thousands, leaves 4 thousands = 40 hundreds. Adding the 5 hundreds, we have 45 hundreds. 15 in 45 hundreds 3 hundreds times; we write the 3 hundreds in the quotient. 15 × 3 hundreds = 45 hundreds, which subtracted from 45 hundreds, leaves nothing. Adding the 3 tens, we have 3 tens. 15 in 3 tens 0 tens times; we write 0 tens in the quotient. Adding to the three tens, which equal 30 units, the 1 unit, we have 31 units. 15 in 31 units 2 units times; we write the 2 units in the quotient. 15 × 2 units = 30 units, which, subtracted from 31 units, leaves 1 unit as a remainder. Indicating the division of the 1 unit, we annex the fractional expression, 1/15 unit, to the integral part of the quotient. Therefore, 34531 divided by 15 is equal to 2302-1/15. [B. Greenleaf, _Practical Arithmetic_, '73, p. 49.] LONG DIVISION: INDUCTIVE EXPLANATION Dividing by Large Numbers 1. Just before Christmas Frank's father sent 360 oranges to be divided among the children in Frank's class. There are 29 children. How many oranges should each child receive? How many oranges will be left over? _Here is the best way to find out:_ 12 and 12 _Think how many 29s there are in 36. 1 is right._ ______ remainder _Write 1 over the 6 of 36. Multiply 29 by 1._ 29 )360 _Write the 29 under the 36. Subtract 29 from 36._ 29 _Write the 0 of 360 after the 7._ --- _Think how many 29s there are in 70. 2 is right._ 70 _Write 2 over the 0 of 360. Multiply 29 by 2._ 58 _Write the 58 under 70. Subtract 58 from 70._ -- _There is 12 remainder._ 12 _Each child gets 12 oranges, and there are 12 left over. This is right, for 12 multiplied by 29 = 348, and 348 + 12 = 360._ * * * * * 8. _In No. 8, keep on dividing by 31 until you have ________ used the 5, the 8, and the 7, and have four 31)99,587 figures in the quotient._ 9. 10. 11. 12. 13. _____ _____ _____ ____ _______ 22)253 22)2895 21)8891 22)290 32)16,368 Check your results for 9, 10, 11, 12, and 13. 1. The boys and girls of the Welfare Club plan to earn money to buy a victrola. There are 23 boys and girls. They can get a good second-hand victrola for $5.75. How much must each earn if they divide the cost equally? _Here is the best way to find out_: $.25 _Think how many 23s there are in 57. 2 is right._ ----- _Write 2 over the 7 of 57. Multiply 23 by 2._ 23|$5.75 _Write 46 under 57 and subtract. Write the 5 of 575 46 after the 11._ ---- _Think how many 23s there are in 115. 5 is right._ 115 _Write 5 over the 5 of 575. Multiply 23 by 5._ 115 _Write the 115 under the 115 that is there and subtract._ ---- _There is no remainder._ _Put $ and the decimal point where they belong._ _Each child must earn 25 cents. This is right, for $.25 multiplied by 23 = $5.75._ 2. Divide $71.76 equally among 23 persons. How much is each person's share? 3. Check your result for No. 2 by multiplying the quotient by the divisor. Find the quotients. Check each quotient by multiplying it by the divisor. 4. 5. 6. 7. 8. _______ _______ ________ _______ _______ 23)$99.13 25)$18.50 21)$129.15 13)$29.25 32)$73.92 1 bushel = 32 qt. 9. How many bushels are there in 288 qt.? 10. In 192 qt.? 11. In 416 qt.? Crucial experiments are lacking, but there are several lines of well-attested evidence. First of all, there can be no doubt that the great majority of pupils learn these manipulations at the start from the placing of units under units, tens under tens, etc., in adding, to the placing of the decimal point in division with decimals, by imitation and blind following of specific instructions, and that a very large proportion of the pupils do not to the end, that is to the fifth school-year, understand them as necessary deductions from decimal notation. It also seems probable that this proportion would not be much reduced no matter how ingeniously and carefully the deductions were explained by textbooks and teachers. Evidence of this fact will appear abundantly to any one who will observe schoolroom life. It also appears in the fact that after the properties of the decimal notation have been thus used again and again; _e.g._, for deducing 'carrying' in addition, 'borrowing' in subtraction, 'carrying' in multiplication, the value of the digits in the partial product, the value of each remainder in short division, the value of the quotient figures in division, the addition, subtraction, multiplication, and division of United States money, and the placing of the decimal point in multiplication, no competent teacher dares to rely upon the pupil, even though he now has four or more years' experience with decimal notation, to deduce the placing of the decimal point in division with decimals. It may be an illusion, but one seems to sense in the better textbooks a recognition of the futility of the attempt to secure deductive derivations of those manipulations. I refer to the brevity of the explanations and their insertion in such a form that they will influence the pupils' thinking as little as possible. At any rate the fact is sure that most pupils do not learn the manipulations by deductive reasoning, or understand them as necessary consequences of abstract principles. It is a common opinion that the only alternative is knowing them by rote. This, of course, is one common alternative, but the other explanation suggests that understanding the manipulations by inductive reasoning from their results is another and an important alternative. The manipulations of 'long' multiplication, for instance, learned by imitation or mechanical drill, are found to give for 25 × _A_ a result about twice as large as for 13 × _A_, for 38 or 39 × _A_ a result about three times as large; for 115 × _A_ a result about ten times as large as for 11 × _A_. With even the very dull pupils the procedure is verified at least to the extent that it gives a result which the scientific expert in the case--the teacher--calls right. With even the very bright pupils, who can appreciate the relation of the procedure to decimal notation, this relation may be used not as the sole deduction of the procedure beforehand, but as one partial means of verifying it afterward. Or there may be the condition of half-appreciation of the relation in which the pupil uses knowledge of the decimal notation to convince himself that the procedure _does_, but not that it _must_ give the right answer, the answer being 'right' because the teacher, the answer-list, and collateral evidence assure him of it. I have taken the manipulation of the partial products as an illustration because it is one of the least favored cases for the explanation I am presenting. If we take the first case where a manipulation may be deduced from decimal notation, known merely by rote, or verified inductively, namely, the addition of two-place numbers, it seems sure that the mental processes just described are almost the universal rule. Surely in our schools at present children add the 3 of 23 to the 3 of 53 and the 2 of 23 to the 5 of 53 at the start, in nine cases out of ten because they see the teacher do so and are told to do so. They are protected from adding 3 + 3 + 2 + 5 not by any deduction of any sort but because they do not know how to add 8 and 5, because they have been taught the habit of adding figures that stand one above the other, or with a + between them; and because they are shown or told what they are to do. They are protected from adding 3 + 5 and 2 + 3, again, by no deductive reasoning but for the second and third reasons just given. In nine cases out of ten they do not even think of the possibility of adding in any other way than the '3 + 3, 2 + 5' way, much less do they select that way on account of the facts that 53 = 50 + 3 and 23 = 20 + 3, that 50 + 20 = 70, that 3 + 3 = 6, and that (_a_ + _b_) + (_c_ + _d_) = (_a_ + _c_) + (_b_ + _d_)! Just as surely all but the very dullest twentieth or so of children come in the end to something more than rote knowledge,--to _understand_, to _know_ that the procedure in question is right. Whether they know _why_ 76 is right depends upon what is meant by _why_. If it means that 76 is the result which competent people agree upon, they do. If it means that 76 is the result which would come from accurate counting they perhaps know why as well as they would have, had they been given full explanations of the relation of the procedure in two-place addition to decimal notation. If _why_ means because 53 = 50 + 3, 23 = 20 + 3, 50 + 20 = 70, and (_a_ + _b_) + (_c_ + _d_) = (_a_ + _c_) + (_b_ + _d_), they do not. Nor, I am tempted to add, would most of them by any sort of teaching whatever. I conclude, therefore, that school children may and do reason about and understand the manipulations of numbers in this inductive, verifying way without being able to, or at least without, under present conditions, finding it profitable to derive them deductively. I believe, in fact, that pure arithmetic _as it is learned and known_ is largely an _inductive science_. At one extreme is a minority to whom it is a series of deductions from principles; at the other extreme is a minority to whom it is a series of blind habits; between the two is the great majority, representing every gradation but centering about the type of the inductive thinker. CHAPTER IV THE CONSTITUTION OF ARITHMETICAL ABILITIES (CONTINUED): THE SELECTION OF THE BONDS TO BE FORMED When the analysis of the mental functions involved in arithmetical learning is made thorough it turns into the question, 'What are the elementary bonds or connections that constitute these functions?' and when the problem of teaching arithmetic is regarded, as it should be in the light of present psychology, as a problem in the development of a hierarchy of intellectual habits, it becomes in large measure a problem of the choice of the bonds to be formed and of the discovery of the best order in which to form them and the best means of forming each in that order. THE IMPORTANCE OF HABIT-FORMATION The importance of habit-formation or connection-making has been grossly underestimated by the majority of teachers and writers of textbooks. For, in the first place, mastery by deductive reasoning of such matters as 'carrying' in addition, 'borrowing' in subtraction, the value of the digits in the partial products in multiplication, the manipulation of the figures in division, the placing of the decimal point after multiplication or division with decimals, or the manipulation of the figures in the multiplication and division of fractions, is impossible or extremely unlikely in the case of children of the ages and experience in question. They do not as a rule deduce the method of manipulation from their knowledge of decimal notation. Rather they learn about decimal notation by carrying, borrowing, writing the last figure of each partial product under the multiplier which gives that product, etc. They learn the method of manipulating numbers by seeing them employed, and by more or less blindly acquiring them as associative habits. In the second place, we, who have already formed and long used the right habits and are thereby protected against the casual misleadings of unfortunate mental connections, can hardly realize the force of mere association. When a child writes sixteen as 61, or finds 428 as the sum of 15 19 16 18 -- or gives 642 as an answer to 27 × 36, or says that 4 divided by 1/4 = 1, we are tempted to consider him mentally perverse, forgetting or perhaps never having understood that he goes wrong for exactly the same general reason that we go right; namely, the general law of habit-formation. If we study the cases of 61 for 16, we shall find them occurring in the work of pupils who after having been drilled in writing 26, 36, 46, 62, 63, and so on, in which the order of the six in writing is the same as it is in speech, return to writing the 'teen numbers. If our language said onety-one for eleven and onety-six for sixteen, we should probably never find such errors except as 'lapses' or as the results of misperception or lack of memory. They would then be more frequent _before_ the 20s, 30s, etc., were learned. If pupils are given much drill on written single column addition involving the higher decades (each time writing the two-figure sum), they are forming a habit of writing 28 after the sum of 8, 6, 9, and 5 is reached; and it should not surprise us if the pupil still occasionally writes the two-figure sum for the first column though a second column is to be added also. On the contrary, unless some counter force influences him, he is absolutely sure to make this mistake. The last mistake quoted (4 ÷ 1/4 = 1) is interesting because here we have possibly one of the cases where deduction from psychology alone can give constructive aid to teaching. Multiplication and division by fractions have been notorious for their difficulty. The former is now alleviated by using _of_ instead of × until the new habit is fixed. The latter is still approached with elaborate caution and with various means of showing why one must 'invert and multiply' or 'multiply by the reciprocal.' But in the author's opinion it seems clear that the difficulty in multiplying and dividing by a fraction was not that children felt any logical objections to canceling or inverting. I fancy that the majority of them would cheerfully invert any fraction three times over or cancel numbers at random in a column if they were shown how to do so. But if you are a youngster inexperienced in numerical abstractions and if you have had _divide_ connected with 'make smaller' three thousand times and never once connected with 'make bigger,' you are sure to be somewhat impelled to make the number smaller the three thousand and first time you are asked to divide it. Some of my readers will probably confess that even now they feel a slight irritation or doubt in saying or writing that 16/1 ÷ 1/8 = 128. The habits that have been confirmed by every multiplication and division by integers are, in this particular of '_the ratio of result to number operated upon_,' directly opposed to the formation of the habits required with fractions. And that is, I believe, the main cause of the difficulty. Its treatment then becomes easy, as will be shown later. These illustrations could be added to almost indefinitely, especially in the case of the responses made to the so-called 'catch' problems. The fact is that the learner rarely can, and almost never does, survey and analyze an arithmetical situation and justify what he is going to do by articulate deductions from principles. He usually feels the situation more or less vaguely and responds to it as he has responded to it or some situation like it in the past. Arithmetic is to him not a logical doctrine which he applies to various special instances, but a set of rather specialized habits of behavior toward certain sorts of quantities and relations. And in so far as he does come to know the doctrine it is chiefly by doing the will of the master. This is true even with the clearest expositions, the wisest use of objective aids, and full encouragement of originality on the pupil's part. Lest the last few paragraphs be misunderstood, I hasten to add that the psychologists of to-day do not wish to make the learning of arithmetic a mere matter of acquiring thousands of disconnected habits, nor to decrease by one jot the pupil's genuine comprehension of its general truths. They wish him to reason not less than he has in the past, but more. They find, however, that you do not secure reasoning in a pupil by demanding it, and that his learning of a general truth without the proper development of organized habits back of it is likely to be, not a rational learning of that general truth, but only a mechanical memorizing of a verbal statement of it. They have come to know that reasoning is not a magic force working in independence of ordinary habits of thought, but an organization and coöperation of those very habits on a higher level. The older pedagogy of arithmetic stated a general law or truth or principle, ordered the pupil to learn it, and gave him tasks to do which he could not do profitably unless he understood the principle. It left him to build up himself the particular habits needed to give him understanding and mastery of the principle. The newer pedagogy is careful to help him build up these connections or bonds ahead of and along with the general truth or principle, so that he can understand it better. The older pedagogy commanded the pupil to reason and let him suffer the penalty of small profit from the work if he did not. The newer provides instructive experiences with numbers which will stimulate the pupil to reason so far as he has the capacity, but will still be profitable to him in concrete knowledge and skill, even if he lacks the ability to develop the experiences into a general understanding of the principles of numbers. The newer pedagogy secures more reasoning in reality by not pretending to secure so much. The newer pedagogy of arithmetic, then, scrutinizes every element of knowledge, every connection made in the mind of the learner, so as to choose those which provide the most instructive experiences, those which will grow together into an orderly, rational system of thinking about numbers and quantitative facts. It is not enough for a problem to be a test of understanding of a principle; it must also be helpful in and of itself. It is not enough for an example to be a case of some rule; it must help review and consolidate habits already acquired or lead up to and facilitate habits to be acquired. Every detail of the pupil's work must do the maximum service in arithmetical learning. DESIRABLE BONDS NOW OFTEN NEGLECTED As hitherto, I shall not try to list completely the elementary bonds that the course of study in arithmetic should provide for. The best means of preparing the student of this topic for sound criticism and helpful invention is to let him examine representative cases of bonds now often neglected which should be formed and representative cases of useless, or even harmful, bonds now often formed at considerable waste of time and effort. (1) _Numbers as measures of continuous quantities._--The numbers one, two, three, 1, 2, 3, etc., should be connected soon after the beginning of arithmetic each with the appropriate amount of some continuous quantity like length or volume or weight, as well as with the appropriate sized collection of apples, counters, blocks, and the like. Lines should be labeled 1 foot, 2 feet, 3 feet, etc.; one inch, two inches, three inches, etc.; weights should be lifted and called one pound, two pounds, etc.; things should be measured in glassfuls, handfuls, pints, and quarts. Otherwise the pupil is likely to limit the meaning of, say, _four_ to four sensibly discrete things and to have difficulty in multiplication and division. Measuring, or counting by insensibly marked off repetitions of a unit, binds each number name to its meaning as ---- _times whatever 1 is_, more surely than mere counting of the units in a collection can, and should reënforce the latter. (2) _Additions in the higher decades._--In the case of all save the very gifted children, the additions with higher decades--that is, the bonds, 16 + 7 = 23, 26 + 7 = 33, 36 + 7 = 43, 14 + 8 = 22, 24 + 8 = 32, and the like--need to be specifically practiced until the tendency becomes generalized. 'Counting' by 2s beginning with 1, and with 2, counting by 3s beginning with 1, with 2, and with 3, counting by 4s beginning with 1, with 2, with 3, and with 4, and so on, make easy beginnings in the formation of the decade connections. Practice with isolated bonds should soon be added to get freer use of the bonds. The work of column addition should be checked for accuracy so that a pupil will continually get beneficial practice rather than 'practice in error.' (3) _The uneven divisions._--The quotients with remainders for the divisions of every number to 19 by 2, every number to 29 by 3, every number to 39 by 4, and so on should be taught as well as the even divisions. A table like the following will be found a convenient means of making these connections:-- 10 = .... 2s 10 = .... 3s and .... rem. 10 = .... 4s and .... rem. 10 = .... 5s 11 = .... 2s and .... rem. 11 = .... 3s and .... rem. . . . 89 = .... 9s and .... rem. These bonds must be formed before short division can be efficient, are useful as a partial help toward selection of the proper quotient figures in long division, and are the chief instruments for one of the important problem series in applied arithmetic,--"How many _x_s can I buy for _y_ cents at _z_ cents per _x_ and how much will I have left?" That these bonds are at present sadly neglected is shown by Kirby ['13], who found that pupils in the last half of grade 3 and the first half of grade 4 could do only about four such examples per minute (in a ten-minute test), and even at that rate made far from perfect records, though they had been taught the regular division tables. Sixty minutes of practice resulted in a gain of nearly 75 percent in number done per minute, with an increase in accuracy as well. (4) _The equation form._--The equation form with an unknown quantity to be determined, or a missing number to be found, should be connected with its meaning and with the problem attitude long before a pupil begins algebra, and in the minds of pupils who never will study algebra. Children who have just barely learned to add and subtract learn easily to do such work as the following:-- Write the missing numbers:-- 4 + 8 = .... 5 + .... = 14 .... + 3 = 11 .... = 5 + 2 16 = 7 + .... 12 = .... + 5 The equation form is the simplest uniform way yet devised to state a quantitative issue. It is capable of indefinite extension if certain easily understood conventions about parentheses and fraction signs are learned. It should be employed widely in accounting and the treatment of commercial problems, and would be except for outworn conventions. It is a leading contribution of algebra to business and industrial life. Arithmetic can make it nearly as well. It saves more time in the case of drills on reducing fractions to higher and lower terms alone than is required to learn its meaning and use. To rewrite a quantitative problem as an equation and then make the easy selection of the necessary technique to solve the equation is one of the most universally useful intellectual devices known to man. The words 'equals,' 'equal,' 'is,' 'are,' 'makes,' 'make,' 'gives,' 'give,' and their rarer equivalents should therefore early give way on many occasions to the '=' which so far surpasses them in ultimate convenience and simplicity. (5) _Addition and subtraction facts in the case of fractions._--In the case of adding and subtracting fractions, certain specific bonds--between the situation of halves and thirds to be added and the responses of thinking of the numbers as equal to so many sixths, between the situation thirds and fourths to be added and thinking of them as so many twelfths, between fourths and eighths to be added and thinking of them as eighths, and the like--should be formed separately. The general rule of thinking of fractions as their equivalents with some convenient denominator should come as an organization and extension of such special habits, not as an edict from the textbook or teacher. (6) _Fractional equivalents._--Efficiency requires that in the end the much used reductions should be firmly connected with the situations where they are needed. They may as well, therefore, be so connected from the beginning, with the gain of making the general process far easier for the dull pupils to master. We shall see later that, for all save the very gifted pupils, the economical way to get an understanding of arithmetical principles is not, usually, to learn a rule and then apply it, but to perform instructive operations and, in the course of performing them, to get insight into the principles. (7) _Protective habits in multiplying and dividing with fractions._--In multiplying and dividing with fractions special bonds should be formed to counteract the now harmful influence of the 'multiply = get a larger number' and 'divide = get a smaller number' bonds which all work with integers has been reënforcing. For example, at the beginning of the systematic work with multiplication by a fraction, let the following be printed clearly at the top of every relevant page of the textbook and displayed on the blackboard:-- _When you multiply a number by anything more than 1 the result is larger than the number._ _When you multiply a number by 1 the result is the same as the number._ _When you multiply a number by anything less than 1 the result is smaller than the number._ Let the pupils establish the new habit by many such exercises as:-- 18 × 4 = .... 9 × 2 = .... 4 × 4 = .... 6 × 2 = .... 2 × 4 = .... 3 × 2 = .... 1 × 4 = .... 1 × 2 = .... 1/2 × 4 = .... 1/3 × 2 = .... 1/4 × 4 = .... 1/6 × 2 = .... 1/8 × 4 = .... 1/9 × 2 = .... In the case of division by a fraction the old harmful habit should be counteracted and refined by similar rules and exercises as follows:-- _When you divide a number by anything more than 1 the result is smaller than the number._ _When you divide a number by 1 the result is the same as the number._ _When you divide a number by anything less than 1 the result is larger than the number._ State the missing numbers:-- 8 = .... 4s 12 = .... 6s 9 = .... 9s 8 = .... 2s 12 = .... 4s 9 = .... 3s 8 = .... 1s 12 = .... 3s 9 = .... 1s 8 = .... 1/2s 12 = .... 2s 9 = .... 1/3s 8 = .... 1/4s 12 = .... 1s 9 = .... 1/9s 8 = .... 1/8s 12 = .... 1/2s 12 = .... 1/3s 12 = .... 1/4s 16 ÷ 16 = 9 ÷ 9 = 10 ÷ 10 = 12 ÷ 6 = 16 ÷ 8 = 9 ÷ 3 = 10 ÷ 5 = 12 ÷ 4 = 16 ÷ 4 = 9 ÷ 1 = 10 ÷ 1 = 12 ÷ 3 = 16 ÷ 2 = 9 ÷ 1/3 = 10 ÷ 1/5 = 12 ÷ 2 = 16 ÷ 1 = 9 ÷ 1/9 = 10 ÷ 1/10 = 12 ÷ 1 = 16 ÷ 1/2 = 12 ÷ 1/2 = 16 ÷ 1/4 = 12 ÷ 1/3 = 16 ÷ 1/8 = 12 ÷ 1/4 = 12 ÷ 1/6 = (8) _'% of' means 'hundredths times'._--In the case of percentage a series of bonds like the following should be formed:-- 5 percent of = .05 times 20 " " " = .20 " 6 " " " = .06 " 25% " = .25 × 12% " = .12 × 3% " = .03 × Four five-minute drills on such connections between '_x_ percent of' and 'its decimal equivalent times' are worth an hour's study of verbal definitions of the meaning of percent as per hundred or the like. The only use of the study of such definitions is to facilitate the later formation of the bonds, and, with all save the brighter pupils, the bonds are more needed for an understanding of the definitions than the definitions are needed for the formation of the bonds. (9) _Habits of verifying results._--Bonds should early be formed between certain manipulations of numbers and certain means of checking, or verifying the correctness of, the manipulation in question. The additions to 9 + 9 and the subtractions to 18 - 9 should be verified by objective addition and subtraction and counting until the pupil has sure command; the multiplications to 9 × 9 should be verified by objective multiplication and counting of the result (in piles of tens and a pile of ones) eight or ten times,[4] and by addition eight or ten times;[4] the divisions to 81 ÷ 9 should be verified by multiplication and occasionally objectively until the pupil has sure command; column addition should be checked by adding the columns separately and adding the sums so obtained, and by making two shorter tasks of the given task and adding the two sums; 'short' multiplication should be verified eight or ten times by addition; 'long' multiplication should be checked by reversing multiplier and multiplicand and in other ways; 'short' and 'long' division should be verified by multiplication. [4] Eight or ten times _in all_, not eight or ten times for each fact of the tables. These habits of testing an obtained result are of threefold value. They enable the pupil to find his own errors, and to maintain a standard of accuracy by himself. They give him a sense of the relations of the processes and the reasons why the right ways of adding, subtracting, multiplying, and dividing are right, such as only the very bright pupils can get from verbal explanations. They put his acquisition of a certain power, say multiplication, to a real and intelligible use, in checking the results of his practice of a new power, and so instill a respect for arithmetical power and skill in general. The time spent in such verification produces these results at little cost; for the practice in adding to verify multiplications, in multiplying to verify divisions, and the like is nearly as good for general drill and review of the addition and multiplication themselves as practice devised for that special purpose. Early work in adding, subtracting, and reducing fractions should be verified by objective aids in the shape of lines and areas divided in suitable fractional parts. Early work with decimal fractions should be verified by the use of the equivalent common fractions for .25, .75, .125, .375, and the like. Multiplication and division with fractions, both common and decimal, should in the early stages be verified by objective aids. The placing of the decimal point in multiplication and division with decimal fractions should be verified by such exercises as:-- 20 It cannot be 200; for 200 × 1.23 is much more than 24.6. ______ It cannot be 2; for 2 × 1.23 is much less than 24.6. 1.23 )24.60 246 ---- The establishment of habits of verifying results and their use is very greatly needed. The percentage of wrong answers in arithmetical work in schools is now so high that the pupils are often being practiced in error. In many cases they can feel no genuine and effective confidence in the processes, since their own use of the processes brings wrong answers as often as right. In solving problems they often cannot decide whether they have done the right thing or the wrong, since even if they have done the right thing, they may have done it inaccurately. A wrong answer to a problem is therefore too often ambiguous and uninstructive to them.[5] [5] The facts concerning the present inaccuracy of school work in arithmetic will be found on pages 102 to 105. These illustrations of the last few pages are samples of the procedures recommended by a consideration of all the bonds that one might form and of the contribution that each would make toward the abilities that the study of arithmetic should develop and improve. It is by doing more or less at haphazard what psychology teaches us to do deliberately and systematically in this respect that many of the past advances in the teaching of arithmetic have been made. WASTEFUL AND HARMFUL BONDS A scrutiny of the bonds now formed in the teaching of arithmetic with questions concerning the exact service of each, results in a list of bonds of small value or even no value, so far as a psychologist can determine. I present here samples of such psychologically unjustifiable bonds with some of the reasons for their deficiencies. (1) _Arbitrary units._--In drills intended to improve the ability to see and use the meanings of numbers as names for ratios or relative magnitudes, it is unwise to employ entirely arbitrary units. The procedure in II (on page 84) is better than that in I. Inches, half-inches, feet, and centimeters are better as units of length than arbitrary As. Square inches, square centimeters, and square feet are better for areas. Ounces and pounds should be lifted rather than arbitrary weights. Pints, quarts, glassfuls, cupfuls, handfuls, and cubic inches are better for volume. All the real merit in the drills on relative magnitude advocated by Speer, McLellan and Dewey, and others can be secured without spending time in relating magnitudes for the sake of relative magnitude alone. The use of units of measure in drills which will never be used in _bona fide_ measuring is like the use of fractions like sevenths, elevenths, and thirteenths. A very little of it is perhaps desirable to test the appreciation of certain general principles, but for regular training it should give place to the use of units of practical significance. [Illustration: FIG. 3. A ---------- B ------------------------------ C -------------------- D ---------------------------------------- I. If _A_ is 1 which line is 2? Which line is 4? Which line is 3? _A_ and _C_ together equal what line? _A_ and _B_ together equal what line? How much longer is _B_ than _A_? How much longer is _B_ than _C_? How much longer is _D_ than _A_?] [Illustration: FIG. 4. A ---------- B ------------------------------ C -------------------- D ---------------------------------------- II. _A_ is 1 inch long. Which line is 2 inches long? Which line is 4 inches long? Which line is 3 inches long? _A_ and _C_ together make ... inches? _A_ and _B_ together make ... inches? _B_ is ... ... longer than _A_? _B_ is ... ... longer than _C_? _D_ is ... ... longer than _A_?] (2) _Multiples of 11._--The multiplications of 2 to 12 by 11 and 12 as single connections should be left for the pupil to acquire by himself as he needs them. These connections interfere with the process of learning two-place multiplication. The manipulations of numbers there required can be learned much more easily if 11 and 12 are used as multipliers in just the same way that 78 or 96 would be. Later the 12 × 2, 12 × 3, etc., may be taught. There is less reason for knowing the multiples of 11 than for knowing the multiples of 15, 16, or 25. (3) _Abstract and concrete numbers._--The elaborate emphasis of the supposed fact that we cannot multiply 726 by 8 dollars and the still more elaborate explanations of why nevertheless we find the cost of 726 articles at $8 each by multiplying 726 by 8 and calling the answer dollars are wasteful. The same holds of the corresponding pedantry about division. These imaginary difficulties should not be raised at all. The pupil should not think of multiplying or dividing men or dollars, but simply of the necessary equation and of the sort of thing that the missing number represents. "8 × 726 = .... Answer is dollars," or "8, 726, multiply. Answer is dollars," is all that he needs to think, and is in the best form for his thought. Concerning the distinction between abstract and concrete numbers, both logic and common sense as well as psychology support the contention of McDougle ['14, p. 206f.], who writes:-- "The most elementary counting, even that stage when the counts were not carried in the mind, but merely in notches on a stick or by DeMorgan's stones in a pot, requires some thought; and the most advanced counting implies memory of things. The terms, therefore, abstract and concrete number, have long since ceased to be used by thinking people. "Recently the writer visited an arithmetic class in a State Normal School and saw a group of practically adult students confused about this very question concerning abstract and concrete numbers, according to their previous training in the conventionalities of the textbook. Their teacher diverted the work of the hour and she and the class spent almost the whole period in reëstablishing the requirements 'that the product must always be the same kind of unit as the multiplicand,' and 'addends must all be alike to be added.' This is not an exceptional case. Throughout the whole range of teaching arithmetic in the public schools pupils are obfuscated by the philosophical encumbrances which have been imposed upon the simplest processes of numerical work. The time is surely ripe, now that we are readjusting our ideas of the subject of arithmetic, to revise some of these wasteful and disheartening practices. Algebra historically grew out of arithmetic, yet it has not been laden with this distinction. No pupil in algebra lets _x_ equal the horses; he lets _x_ equal the _number_ of horses, and proceeds to drop the idea of horses out of his consideration. He multiplies, divides, and extracts the root of the _number_, sometimes handling fractions in the process, and finally interprets the result according to the conditions of his problem. Of course, in the early number work there have been the sense-objects from which number has been perceived, but the mind retreats naturally from objectivity to the pure conception of number, and then to the number symbol. The following is taken from the appendix to Horn's thesis, where a seventh grade girl gets the population of the United States in 1820:-- 7,862,166 whites 233,634 free negroes 1,538,022 slaves --------- 9,633,822 In this problem three different kinds of addends are combined, if we accept the usual distinction. Some may say that this is a mistake,--that the pupil transformed the 'whites,' 'free negroes,' and 'slaves' into a common unit, such as 'people' of 'population' and then added these common units. But this 'explanation' is entirely gratuitous, as one will find if he questions the pupil about the process. It will be found that the child simply added the figures as numbers only and then interpreted the result, according to the statement of the problem, without so much mental gymnastics. The writer has questioned hundreds of students in Normal School work on this point, and he believes that the ordinary mind-movement is correctly set forth here, no matter how well one may maintain as an academic proposition that this is not logical. Many classes in the Eastern Kentucky State Normal have been given this problem to solve, and they invariably get the same result:-- 'In a garden on the Summit are as many cabbage-heads as the total number of ladies and gentlemen in this class. How many cabbage-heads in the garden?' And the blackboard solution looks like this each time:-- 29 ladies 15 gentlemen -- 44 cabbage-heads So, also, one may say: I have 6 times as many sheep as you have cows. If you have 5 cows, how many sheep have I? Here we would multiply the number of cows, which is 5, by 6 and call the result 30, which must be linked with the idea of sheep because the conditions imposed by the problem demand it. The mind naturally in this work separates the pure number from its situation, as in algebra, handles it according to the laws governing arithmetical combinations, and labels the result as the statement of the problem demands. This is expressed in the following, which is tacitly accepted in algebra, and should be accepted equally in arithmetic: 'In all computations and operations in arithmetic, all numbers are essentially abstract and should be so treated. They are concrete only in the thought process that attends the operation and interprets the result.'" (4) _Least common multiple._--The whole set of bonds involved in learning 'least common multiple' should be left out. In adding and subtracting fractions the pupil should _not_ find the least common multiple of their denominators but should find any common multiple that he can find quickly and correctly. No intelligent person would ever waste time in searching for the least common multiple of sixths, thirds, and halves except for the unfortunate traditions of an oversystematized arithmetic, but would think of their equivalents in sixths or twelfths or twenty-fourths or _any other convenient common multiple_. The process of finding the least common multiple is of such exceedingly rare application in science or business or life generally that the textbooks have to resort to purely fantastic problems to give drill in its use. (5) _Greatest common divisor._--The whole set of bonds involved in learning 'greatest common divisor' should also be left out. In reducing fractions to lowest terms the pupil should divide by anything that he sees that he can divide by, favoring large divisors, and continue doing so until he gets the fraction in terms suitable for the purpose in hand. The reader probably never has had occasion to compute a greatest common divisor since he left school. If he has computed any, the chances are that he would have saved time by solving the problem in some other way! The following problems are taken at random from those given by one of the best of the textbooks that make the attempt to apply the facts of Greatest Common Divisor and Least Common Multiple to problems.[6] Most of these problems are fantastic. The others are trivial, or are better solved by trial and adaptation. 1. A certain school consists of 132 pupils in the high school, 154 in the grammar, and 198 in the primary grades. If each group is divided into sections of the same number containing as many pupils as possible, how many pupils will there be in each section? 2. A farmer has 240 bu. of wheat and 920 bu. of oats, which he desires to put into the least number of boxes of the same capacity, without mixing the two kinds of grain. Find how many bushels each box must hold. 3. Four bells toll at intervals of 3, 7, 12, and 14 seconds respectively, and begin to toll at the same instant. When will they next toll together? 4. A, B, C, and D start together, and travel the same way around an island which is 600 mi. in circuit. A goes 20 mi. per day, B 30, C 25, and D 40. How long must their journeying continue, in order that they may all come together again? 5. The periods of three planets which move uniformly in circular orbits round the sun, are respectively 200, 250, and 300 da. Supposing their positions relatively to each other and the sun to be given at any moment, determine how many da. must elapse before they again have exactly the same relative positions. [6] McLellan and Ames, _Public School Arithmetic_ [1900]. (6) _Rare and unimportant words._--The bonds between rare or unimportant words and their meanings should not be formed for the mere sake of verbal variety in the problems of the textbook. A pupil should not be expected to solve a problem that he cannot read. He should not be expected in grades 2 and 3, or even in grade 4, to read words that he has rarely or never seen before. He should not be given elaborate drill in reading during the time devoted to the treatment of quantitative facts and relations. All this is so obvious that it may seem needless to relate. It is not. With many textbooks it is now necessary to give definite drill in reading the words in the printed problems intended for grades 2, 3, and 4, or to replace them by oral statements, or to leave the pupils in confusion concerning what the problems are that they are to solve. Many good teachers make a regular reading-lesson out of every page of problems before having them solved. There should be no such necessity. To define _rare_ and _unimportant_ concretely, I will say that for pupils up to the middle of grade 3, such words as the following are rare and unimportant (though each of them occurs in the very first fifty pages of some well-known beginner's book in arithmetic). absentees account Adele admitted Agnes agreed Albany Allen allowed alternate Andrew Arkansas arrived assembly automobile baking powder balance barley beggar Bertie Bessie bin Boston bouquet bronze buckwheat Byron camphor Carl Carrie Cecil Charlotte charity Chicago cinnamon Clara clothespins collect comma committee concert confectioner cranberries crane currants dairyman Daniel David dealer debt delivered Denver department deposited dictation discharged discover discovery dish-water drug due Edgar Eddie Edwin election electric Ella Emily enrolled entertainment envelope Esther Ethel exceeds explanation expression generally gentlemen Gilbert Grace grading Graham grammar Harold hatchet Heralds hesitation Horace Mann impossible income indicated inmost inserts installments instantly insurance Iowa Jack Jennie Johnny Joseph journey Julia Katherine lettuce-plant library Lottie Lula margin Martha Matthew Maud meadow mentally mercury mineral Missouri molasses Morton movements muslin Nellie nieces Oakland observing obtained offered office onions opposite original package packet palm Patrick Paul payments peep Peter perch phaeton photograph piano pigeons Pilgrims preserving proprietor purchased Rachel Ralph rapidity rather readily receipts register remanded respectively Robert Roger Ruth rye Samuel San Francisco seldom sheared shingles skyrockets sloop solve speckled sponges sprout stack Stephen strap successfully suggested sunny supply Susan Susie's syllable talcum term test thermometer Thomas torpedoes trader transaction treasury tricycle tube two-seated united usually vacant various vase velocipede votes walnuts Walter Washington watched whistle woodland worsted (7) _Misleading facts and procedures._--Bonds should not be formed between articles of commerce and grossly inaccurate prices therefor, between events and grossly improbable consequences, or causes or accompaniments thereof, nor between things, qualities, and events which have no important connections one with another in the real world. In general, things should not be put together in the pupil's mind that do not belong together. If the reader doubts the need of this warning let him examine problems 1 to 5, all from reputable books that are in common use, or have been within a few years, and consider how addition, subtraction, and the habits belonging with each are confused by exercise 6. 1. If a duck flying 3/5 as fast as a hawk flies 90 miles in an hour, how fast does the hawk fly? 2. At 5/8 of a cent apiece how many eggs can I buy for $60? 3. At $.68 a pair how many pairs of overshoes can you buy for $816? 4. At $.13 a dozen how many dozen bananas can you buy for $3.12? 5. How many pecks of beans can be put into a box that will hold just 21 bushels? 6. Write answers: 537 Beginning at the bottom say 11, 18, and 2 (writing it in 365 its place) are 20. 5, 11, 14, and 6 (writing it) are 20, ? 5, 10. The number, omitted, is 62. 36 ---- 1000 _a._ 581 _b._ 625 _c._ 752 _d._ 314 _e._ ? 97 ? 414 429 845 364 90 130 ? 223 ? 417 ? 76 95 ---- ---- ---- ---- ---- 1758 2050 2460 1000 2367 (8) _Trivialities and absurdities._--Bonds should not be formed between insignificant or foolish questions and the labor of answering them, nor between the general arithmetical work of the school and such insignificant or foolish questions. The following are samples from recent textbooks of excellent standing:-- On one side of George's slate there are 32 words, and on the other side 26 words. If he erases 6 words from one side, and 8 from the other, how many words remain on his slate? A certain school has 14 rooms, and an average of 40 children in a room. If every one in the school should make 500 straight marks on each side of his slate, how many would be made in all? 8 times the number of stripes in our flag is the number of years from 1800 until Roosevelt was elected President. In what year was he elected President? From the Declaration of Independence to the World's Fair in Chicago was 9 times as many years as there are stripes in the flag. How many years was it? (9) _Useless methods._--Bonds should not be formed between a described situation and a method of treating the situation which would not be a useful one to follow in the case of the real situation. For example, "If I set 96 trees in rows, sixteen trees in a row, how many rows will I have?" forms the habit of treating by division a problem that in reality would be solved by counting the rows. So also "I wish to give 25 cents to each of a group of boys and find that it will require $2.75. How many boys are in the group?" forms the habit of answering a question by division whose answer must already have been present to give the data of the problem. (10) _Problems whose answers would, in real life, be already known._--The custom of giving problems in textbooks which could not occur in reality because the answer has to be known to frame the problem is a natural result of the lazy author's tendency to work out a problem to fit a certain process and a certain answer. Such bogus problems are very, very common. In a random sampling of a dozen pages of "General Review" problems in one of the most widely used of recent textbooks, I find that about 6 percent of the problems are of this sort. Among the problems extemporized by teachers these bogus problems are probably still more frequent. Such are:-- A clerk in an office addressed letters according to a given list. After she had addressed 2500, 4/9 of the names on the list had not been used; how many names were in the entire list? The Canadian power canal at Sault Ste. Marie furnished 20,000 horse power. The canal on the Michigan side furnished 2-1/2 times as much. How many horse power does the latter furnish? It may be asserted that the ideal of giving as described problems only problems that might occur and demand the same sort of process for solution with a real situation, is too exacting. If a problem is comprehensible and serves to illustrate a principle or give useful drill, that is enough, teachers may say. For really scientific teaching it is not enough. Moreover, if problems are given merely as tests of knowledge of a principle or as means to make some fact or principle clear or emphatic, and are not expected to be of direct service in the quantitative work of life, it is better to let the fact be known. For example, "I am thinking of a number. Half of this number is twice six. What is the number?" is better than "A man left his wife a certain sum of money. Half of what he left her was twice as much as he left to his son, who receives $6000. How much did he leave his wife?" The former is better because it makes no false pretenses. (11) _Needless linguistic difficulties._--It should be unnecessary to add that bonds should not be formed between the pupil's general attitude toward arithmetic and needless, useless difficulty in language or needless, useless, wrong reasoning. Our teaching is, however, still tainted by both of these unfortunate connections, which dispose the pupil to think of arithmetic as a mystery and folly. Consider, for example, the profitless linguistic difficulty of problems 1-6, whose quantitative difficulties are simply those of:-- 1. 5 + 8 + 3 + 7 2. 64 ÷ 8, and knowledge that 1 peck = 8 quarts 3. 12 ÷ 4 4. 6 ÷ 2 5. 3 × 2 6. 4 × 4 1. What amount should you obtain by putting together 5 cents, 8 cents, 3 cents, and 7 cents? Did you find this result by adding or multiplying? 2. How many times must you empty a peck measure to fill a basket holding 64 quarts of beans? 3. If a girl commits to memory 4 pages of history in one day, in how many days will she commit to memory 12 pages? 4. If Fred had 6 chickens how many times could he give away 2 chickens to his companions? 5. If a croquet-player drove a ball through 2 arches at each stroke, through how many arches will he drive it by 3 strokes? 6. If mamma cut the pie into 4 pieces and gave each person a piece, how many persons did she have for dinner if she used 4 whole pies for dessert? Arithmetically this work belongs in the first or second years of learning. But children of grades 2 and 3, save a few, would be utterly at a loss to understand the language. We are not yet free from the follies illustrated in the lessons of pages 96 to 99, which mystified our parents. LESSON I [Illustration: FIG. 5.] 1. In this picture, how many girls are in the swing? 2. How many girls are pulling the swing? 3. If you count both girls together, how many are they? _One_ girl and _one_ other girl are how many? 4. How many kittens do you see on the stump? 5. How many on the ground? 6. How many kittens are in the picture? One kitten and one other kitten are how many? 7. If you should ask me how many girls are in the swing, or how many kittens are on the stump, I could answer aloud, _One_; or I could write _One_; or thus, _1_. 8. If I write _One_, this is called the _word One_. 9. This, _1_, is named a _figure One_, because it means the same as the word _One_, and stands for _One_. 10. Write 1. What is this named? Why? 11. A figure 1 may stand for _one_ girl, _one_ kitten, or _one_ anything. 12. When children first attend school, what do they begin to learn? _Ans._ Letters and words. 13. Could you read or write before you had learned either letters or words? 14. If we have all the _letters_ together, they are named the Alphabet. 15. If we write or speak _words_, they are named Language. 16. You are commencing to study Arithmetic; and you can read and write in Arithmetic only as you learn the Alphabet and Language of Arithmetic. But little time will be required for this purpose. LESSON II [Illustration: FIG. 6.] 1. If we speak or write words, what do we name them, when taken together? 2. What are you commencing to study? _Ans._ Arithmetic. 3. What Language must you now learn? 4. What do we name this, 1? Why? 5. This figure, 1, is part of the Language of Arithmetic. 6. If I should write something to stand for _Two_--_two_ girls, _two_ kittens, or _two_ things of any kind--what do you think we would name it? 7. A _figure Two_ is written thus: _2._ Make a _figure two_. 8. Why do we name this a _figure two_? 9. This figure two (2) is part of the Language of Arithmetic. 10. In this picture one boy is sitting, playing a flageolet. What is the other boy doing? If the boy standing should sit down by the other, how many boys would be sitting together? One boy and one other boy are how many boys? 11. You see a flageolet and a violin. They are musical instruments. One musical instrument and one other musical instrument are how many? 12. I will write thus: 1 1 2. We say that 1 boy and 1 other boy, counted together, are 2 boys; or are equal to 2 boys. We will now write something to show that the first 1 and the other 1 are to be counted together. 13. We name a line drawn thus, -, a _horizontal line_. Draw such a line. Name it. 14. A line drawn thus, |, we name a _vertical line_. Draw such a line. Name it. 15. Now I will put two such lines together; thus, +. What kind of a line do we name the first (-)? And what do we name the last? (|)? Are these lines long or short? Where do they cross each other? 16. Each of you write thus: -, |, +. 17. This, +, is named _Plus_. _Plus_ means _more_; and + also means _more_. 18. I will write. _One and One More Equal Two._ 19. Now I will write part of this in the Language of Arithmetic. I write the first _One_ thus, 1; then the other _One_ thus, 1. Afterward I write, for the word _More_, thus, +, placing the + between 1 and 1, so that the whole stands thus: 1 + 1. As I write, I say, _One and One more_. 20. Each of you write 1 + 1. Read what you have written. 21. This +, when written between the 1s, shows that they are to be put together, or counted together, so as to make 2. 22. Because + shows what is to be done, it is called a _Sign_. If we take its name, _Plus_, and the word _Sign_, and put both words together, we have _Sign Plus_, or _Plus Sign_. In speaking of this we may call it _Sign Plus_, or _Plus Sign_, or _Plus_. 23. 1, 2, +, are part of the Language of Arithmetic. _Write the following in the Language of Arithmetic_: 24. One and one more. 25. One and two more. 26. Two and one more. (12) _Ambiguities and falsities._--Consider the ambiguities and false reasoning of these problems. 1. If you can earn 4 cents a day, how much can you earn in 6 weeks? (Are Sundays counted? Should a child who earns 4 cents some day expect to repeat the feat daily?) 2. How many lines must you make to draw ten triangles and five squares? (I can do this with 8 lines, though the answer the book requires is 50.) 3. A runner ran twice around an 1/8 mile track in two minutes. What distance did he run in 2/3 of a minute? (I do not know, but I do know that, save by chance, he did not run exactly 2/3 of 1/8 mile.) 4. John earned $4.35 in a week, and Henry earned $1.93. They put their money together and bought a gun. What did it cost? (Maybe $5, maybe $10. Did they pay for the whole of it? Did they use all their earnings, or less, or more?) 5. Richard has 12 nickels in his purse. How much more than 50 cents would you give him for them? (Would a wise child give 60 cents to a boy who wanted to swap 12 nickels therefor, or would he suspect a trick and hold on to his own coins?) 6. If a horse trots 10 miles in one hour how far will he travel in 9 hours? 7. If a girl can pick 3 quarts of berries in 1 hour how many quarts can she pick in 3 hours? (These last two, with a teacher insisting on the 90 and 9, might well deprive a matter-of-fact boy of respect for arithmetic for weeks thereafter.) The economics and physics of the next four problems speak for themselves. 8. I lost $15 by selling a horse for $85. What was the value of the horse? 9. If floating ice has 7 times as much of it under the surface of the water as above it, what part is above water? If an iceberg is 50 ft. above water, what is the entire height of the iceberg? How high above water would an iceberg 300 ft. high have to be? 10. A man's salary is $1000 a year and his expenses $625. How many years will elapse before he is worth $10,000 if he is worth $2500 at the present time? 11. Sound travels 1120 ft. a second. How long after a cannon is fired in New York will the report be heard in Philadelphia, a distance of 90 miles? GUIDING PRINCIPLES The reader may be wearied of these special details concerning bonds now neglected that should be formed and useless or harmful bonds formed for no valid reason. Any one of them by itself is perhaps a minor matter, but when we have cured all our faults in this respect and found all the possibilities for wiser selection of bonds, we shall have enormously improved the teaching of arithmetic. The ideal is such choice of bonds (and, as will be shown later, such arrangement of them) as will most improve the functions in question at the least cost of time and effort. The guiding principles may be kept in mind in the form of seven simple but golden rules:-- 1. Consider the situation the pupil faces. 2. Consider the response you wish to connect with it. 3. Form the bond; do not expect it to come by a miracle. 4. Other things being equal, form no bond that will have to be broken. 5. Other things being equal, do not form two or three bonds when one will serve. 6. Other things being equal, form bonds in the way that they are required later to act. 7. Favor, therefore, the situations which life itself will offer, and the responses which life itself will demand. CHAPTER V THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS An inventory of the bonds to be formed in learning arithmetic should be accompanied by a statement of how strong each bond is to be made and kept year by year. Since, however, the inventory itself has been presented here only in samples, the detailed statement of desired strength for each bond cannot be made. Only certain general facts will be noted here. THE NEED OF STRONGER ELEMENTARY BONDS The constituent bonds involved in the fundamental operations with numbers need to be much stronger than they now are. Inaccuracy in these operations means weakness of the constituent bonds. Inaccuracy exists, and to a degree that deprives the subject of much of its possible disciplinary value, makes the pupil's achievements of slight value for use in business or industry, and prevents the pupil from verifying his work with new processes by some previously acquired process. The inaccuracy that exists may be seen in the measurements made by the many investigators who have used arithmetical tasks as tests of fatigue, practice, individual differences and the like, and in the special studies of arithmetical achievements for their own sake made by Courtis and others. Burgerstein ['91], using such examples as 28704516938276546397 + 35869427359163827263 ---------------------- and similar long numbers to be multiplied by 2 or by 3 or by 4 or by 5 or by 6, found 851 errors in 28,267 answer-figures, or 3 per hundred answer-figures, or 3/5 of an error per example. The children were 9-1/2 to 15 years old. Laser ['94], using the same sort of addition and multiplication, found somewhat over 3 errors per hundred answer-figures in the case of boys and girls averaging 11-1/2 years, during the period of their most accurate work. Holmes ['95], using addition of the sort just described, found 346 errors in 23,713 answer-figures or about 1-1/2 per hundred. The children were from all grades from the third to the eighth. In Laser's work, 21, 19, 13, and 10 answer-figures were obtained per minute. Friedrich ['97] with similar examples, giving the very long time of 20 minutes for obtaining about 200 answer-figures, found from 1 to 2 per hundred wrong. King ['07] had children in grade 5 do sums, each consisting of 5 two-place numbers. In the most accurate work-period, they made 1 error per 20 columns. In multiplying a four-place by a four-place number they had less than one total answer right out of three. In New York City Courtis found ['11-'12] with his Test 7 that in 12 minutes the average achievement of fourth-grade children is 8.8 units attempted with 4.2 right. In grade 5 the facts are 10.9 attempts with 5.8 right; in grade 6, 12.5 attempts with 7.0 right; in grade 7, 15 attempts with 8.5 right; in grade 8, 15.7 attempts with 10.1 right. These results are near enough to those obtained from the country at large to serve as a text here. The following were set as official standards, in an excellent school system, Courtis Series B being used:-- SPEED PERCENT OF GRADE. ATTEMPTS. CORRECT ANSWERS. Addition 8 12 80 7 11 80 6 10 70 5 9 70 4 8 70 Subtraction 8 12 90 7 11 90 6 10 90 5 9 80 4 7 80 Multiplication 8 11 80 7 10 80 6 9 80 5 7 70 4 6 60 Division 8 11 90 7 10 90 6 8 80 5 6 70 4 4 60 Kirby ['13, pp. 16 ff. and 55 ff.] found that, in adding columns like those printed below, children in grade 4 got on the average less than 80 percent of correct answers. Their average speed was about 2 columns per minute. In doing division of the sort printed below children of grades 3 _B_ and 4 _A_ got less than 95 percent of correct answers, the average speed being 4 divisions per minute. In both cases the slower computers were no more accurate than the faster ones. Practice improved the speed very rapidly, but the accuracy remained substantially unchanged. Brown ['11 and '12] found a similar low status of ability and notable improvement from a moderate amount of special practice. 3 5 6 2 3 8 9 7 4 9 7 9 6 5 5 6 4 5 8 2 3 4 7 8 7 3 7 9 3 7 8 8 4 8 2 6 8 2 9 8 2 2 4 7 6 9 8 5 6 2 6 9 5 7 8 5 2 3 2 4 9 6 4 2 7 2 9 4 4 5 3 3 7 9 9 9 2 8 9 7 6 8 9 6 4 7 7 9 2 4 8 4 6 9 9 2 6 9 8 9 -- -- -- -- -- -- -- -- -- -- 20 = .... 5s 56 = .... 9s and .... _r_. 30 = .... 7s and .... _r_. 89 = .... 9s and .... _r_. 20 = .... 8s and .... _r_. 56 = .... 6s and .... _r_. 31 = .... 4s and .... _r_. 86 = .... 9s and .... _r_. It is clear that numerical work as inaccurate as this has little or no commercial or industrial value. If clerks got only six answers out of ten right as in the Courtis tests, one would need to have at least four clerks make each computation and would even then have to check many of their discrepancies by the work of still other clerks, if he wanted his accounts to show less than one error per hundred accounting units of the Courtis size. It is also clear that the "habits of ... absolute accuracy, and satisfaction in truth as a result" which arithmetic is supposed to further must be largely mythical in pupils who get right answers only from three to nine times out of ten! EARLY MASTERY The bonds in question clearly must be made far stronger than they now are. They should in fact be strong enough to abolish errors in computation, except for those due to temporary lapses. It is much better for a child to know half of the multiplication tables, and to know that he does not know the rest, than to half-know them all; and this holds good of all the elementary bonds required for computation. Any bond should be made to work perfectly, though slowly, very soon after its formation is begun. Speed can easily be added by proper practice. The chief reasons why this is not done now seem to be the following: (1) Certain important bonds (like the additions with higher decades) are not given enough attention when they are first used. (2) The special training necessary when a bond is used in a different connection (as when the multiplications to 9 × 9 are used in examples like 729 8 --- where the pupil has also to choose the right number to multiply, keep in mind what is carried, use it properly, and write the right figure in the right place, and carry a figure, or remember that he carries none) is neglected. (3) The pupil is not taught to check his work. (4) He is not made responsible for substantially accurate results. Furthermore, the requirement of (4) without the training of (1), (2), and (3) will involve either a fruitless failure on the part of many pupils, or an utterly unjust requirement of time. The common error of supposing that the task of computation with integers consists merely in learning the additions to 9 + 9, the subtractions to 18 - 9, the multiplications to 8 × 9, and the divisions to 81 ÷ 9, and in applying this knowledge in connection with the principles of decimal notation, has had a large share in permitting the gross inaccuracy of arithmetical work. The bonds involved in 'knowing the tables' do not make up one fourth of the bonds involved in real adding, subtracting, multiplying, and dividing (with integers alone). It should be noted that if the training mentioned in (1) and (2) is well cared for, the checking of results as recommended in (3) becomes enormously more valuable than it is under present conditions, though even now it is one of our soundest practices. If a child knows the additions to higher decades so that he can add a seen one-place number to a thought-of two-place number in three seconds or less with a correct answer 199 times out of 200, there is only an infinitesimal chance that a ten-figure column twice added (once up, once down) a few minutes apart with identical answers will be wrong. Suppose that, in long multiplication, a pupil can multiply to 9 × 9 while keeping his place and keeping track of what he is 'carrying' and of where to write the figure he writes, and can add what he carries without losing track of what he is to add it to, where he is to write the unit figure, what he is to multiply next and by what, and what he will then have to carry, in each case to a surety of 99 percent of correct responses. Then two identical answers got by multiplying one three-place number by another a few minutes apart, and with reversal of the numbers, will not be wrong more than twice in his entire school career. Checks approach proofs when the constituent bonds are strong. If, on the contrary, the fundamental bonds are so weak that they do not work accurately, checking becomes much less trustworthy and also very much more laborious. In fact, it is possible to show that below a certain point of strength of the fundamental bonds, the time required for checking is so great that part of it might better be spent in improving the fundamental bonds. For example, suppose that a pupil has to find the sum of five numbers like $2.49, $5.25, $6.50, $7.89, and $3.75. Counting each act of holding in mind the number to be carried and each writing of a column's result as equivalent in difficulty to one addition, such a sum equals nineteen single additions. On this basis and with certain additional estimates[7] we can compute the practical consequences for a pupil's use of addition in life according to the mastery of it that he has gained in school. [7] These concern allowances for two errors occurring in the same example and for the same wrong answer being obtained in both original work and check work. I have so computed the amount of checking a pupil will have to do to reach two agreeing numbers (out of two, or three, or four, or five, or whatever the number before he gets two that are alike), according to his mastery of the elementary processes. The facts appear in Table 1. It is obvious that a pupil whose mastery of the elements is that denoted by getting them right 96 times out of 100 will require so much time for checking that, even if he were never to use this ability for anything save a few thousand sums in addition, he would do well to improve this ability before he tried to do the sums. An ability of 199 out of 200, or 995 out of 1000, seems likely to save much more time than would be taken to acquire it, and a reasonable defense could be made for requiring 996 or 997 out of 1000. A precision of from 995 to 997 out of 1000 being required, and ordinary sagacity being used in the teaching, speed will substantially take care of itself. Counting on the fingers or in words will not give that precision. Slow recourse to memory of serial addition tables will not give that precision. Nothing save sure memory of the facts operating under the conditions of actual examples will give it. And such memories will operate with sufficient speed. TABLE 1 THE EFFECT OF MASTERY OF THE ELEMENTARY FACTS OF ADDITION UPON THE LABOR REQUIRED TO SECURE TWO AGREEING ANSWERS WHEN ADDING FIVE THREE-FIGURE NUMBERS ====================================================================== MASTERY OF |APPROXIMATE |APPROXIMATE |APPROXIMATE |APPROXIMATE THE |NUMBER OF |NUMBER OF |NUMBER OF |NUMBER OF ELEMENTARY |WRONG ANSWERS|AGREEING |AGREEING |CHECKINGS ADDITIONS |IN SUMS OF 5 |ANSWERS, |ANSWERS, |REQUIRED (OVER TIMES RIGHT |THREE-PLACE |AFTER ONE |AFTER A |AND ABOVE THE IN 1000 |NUMBERS PER |CHECKING, |CHECKING OF |FIRST GENERAL |1000 |PER 1000 |THE FIRST |CHECKING OF | | |DISCREPANCIES|THE 1000 SUMS) | | | |TO SECURE TWO | | | |AGREEING | | | |RESULTS -------------+-------------+-------------+-------------+-------------- 960 | 700 | 90 | 216 | 4500 980 | 380 | 384 | 676 | 1200 990 | 190 | 656 | 906 | 470 995 | 95 | 819 | 975 | 210 996 | 76 | 854 | 984 | 165 997 | 54 | 895 | 992 | 115 998 | 38 | 925 | 996 | 80 999 | 19 | 962 | 999 | 40 -------------+-------------+-------------+-------------+-------------- There is one intelligent objection to the special practice necessary to establish arithmetical connections so fully as to give the accuracy which both utilitarian and disciplinary aims require. It may be said that the pupils in grades 3, 4, and 5 cannot appreciate the need and that consequently the work will be dull, barren, and alien, without close personal appropriation by the pupil's nature. It is true that no vehement life-purpose is directly involved by the problem of perfecting one's power to add 7 to 28 in grade 2, or by the problem of multiplying 253 by 8 accurately in grade 3 or by precise subtraction in long division in grade 4. It is also true, however, that the most humanly interesting of problems--one that the pupil attacks most whole-heartedly--will not be solved correctly unless the pupil has the necessary associative mechanisms in order; and the surer he is of them, the freer he is to think out the problem as such. Further, computation is not dull if the pupil can compute. He does not himself object to its barrenness of vital meaning, so long as the barrenness of failure is prevented. We must not forget that pupils like to learn. In teaching excessively dull individuals, who has not often observed the great interest which they display in anything that they are enabled to master? There is pathos in their joy in learning to recognize parts of speech, perform algebraic simplifications, or translate Latin sentences, and in other accomplishments equally meaningless to all their interests save the universal human interest in success and recognition. Still further, it is not very hard to show to pupils the imperative need of accuracy in scoring games, in the shop, in the store, and in the office. Finally, the argument that accurate work of this sort is alien to the pupil in these grades is still stronger against _inaccurate_ work of the same sort. If we are to teach computation with two- and three- and four-place numbers at all, it should be taught as a reliable instrument, not as a combination of vague memories and faith. The author is ready to cut computation with numbers above 10 out of the curriculum of grades 1-6 as soon as more valuable educational instruments are offered in its place, but he is convinced that nothing in child-nature makes a large variety of inaccurate computing more interesting or educative or germane to felt needs, than a smaller variety of accurate computing! THE STRENGTH OF BONDS FOR TEMPORARY SERVICE The second general fact is that certain bonds are of service for only a limited time and so need to be formed only to a limited and slight degree of strength. The data of problems set to illustrate a principle or improve some habit of computation are, of course, the clearest cases. The pupil needs to remember that John bought 3 loaves of bread and that they were 5-cent loaves and that he gave 25 cents to the baker only long enough to use the data to decide what change John should receive. The connections between the total described situation and the answer obtained, supposing some considerable computation to intervene, is a bond that we let expire almost as soon as it is born. It is sometimes assumed that the bond between a certain group of features which make a problem a 'Buy _a_ things at _b_ per thing, find total cost' problem or a 'Buy _a_ things at _b_ per thing, what change from _c_' problem or a 'What gain on buying for _a_ and selling for _b_' problem or a 'How many things at _a_ each can I buy for _b_ cents' problem--it is assumed that the bond between these essential defining features and the operation or operations required for solution is as temporary as the bonds with the name of the buyer or the price of the thing. It is assumed that all problems are and should be solved by some pure act of reasoning without help or hindrance from bonds with the particular verbal structure and vocabulary of the problems. Whether or not they _should_ be, they _are not_. Every time that a pupil solves a 'bought-sold' problem by subtraction he strengthens the tendency to respond to any problem whatsoever that contains the words 'bought for' and 'sold for' by subtraction; and he will by no means surely stop and survey every such problem in all its elements to make sure that no other feature makes inapplicable the tendency to subtract which the 'bought sold' evokes. To prevent pupils from responding to the form of statement rather than the essential facts, we should then not teach them to forget the form of statement, but rather give them all the common forms of statement to which the response in question is an appropriate response, and only such. If a certain form of statement does in life always signify a certain arithmetical procedure, the bond between it and that procedure may properly be made very strong. Another case of the formation of bonds to only a slight degree of strength concerns the use of so-called 'crutches' such as writing +, -, and × in copying problems like those below:-- Add Subtract Multiply 23 79 32 61 24 3 -- -- -- or altering the figures when 'borrowing' in subtraction, and the like. Since it is undesirable that the pupil should regard the 'crutch' response as essential to the total procedure, or become so used to having it that he will be disturbed by its absence later, it is supposed that the bond between the situation and the crutch should not be fully formed. There is a better way out of the difficulty, in case crutches are used at all. This is to associate the crutch with a special 'set,' and its non-use with the general set which is to be the permanent one. For example, children may be taught from the start never to write the crutch sign or crutch figure unless the work is accompanied by "Write ... to help you to...." Write - to help you to Find the differences:-- remember that you must 39 67 78 56 45 subtract in this row. 23 44 36 26 24 -- -- -- -- -- Remember that you must Find the differences:-- subtract in this row. 85 27 96 38 78 63 14 51 45 32 -- -- -- -- -- The bond evoking the use of the crutch may then be formed thoroughly enough so that there is no hesitation, insecurity, or error, without interfering to any harmful extent with the more general bond from the situation to work without the crutch. THE STRENGTH OF BONDS WITH TECHNICAL FACTS AND TERMS Another instructive case concerns the bonds between certain words and their meanings, and between certain situations of commerce, industry, or agriculture and useful facts about these situations. Illustrations of the former are the bonds between _cube root_, _hectare_, _brokerage_, _commission_, _indorsement_, _vertex_, _adjacent_, _nonagon_, _sector_, _draft_, _bill of exchange_, and their meanings. Illustrations of the latter are the bonds from "Money being lent 'with interest' at no specified rate, what rate is charged?" to "The legal rate of the state," from "$_X_ per M as a rate for lumber" to "Means $_X_ per thousand board feet, a board foot being 1 ft. by 1 ft. by 1 in." It is argued by many that such bonds are valuable for a short time; namely, while arithmetical procedures in connection with which they serve are learned, but that their value is only to serve as a means for learning these procedures and that thereafter they may be forgotten. "They are formed only as accessory means to certain more purely arithmetical knowledge or discipline; after this is acquired they may be forgotten. Everybody does in fact forget them, relearning them later if life requires." So runs the argument. In some cases learning such words and facts only to use them in solving a certain sort of problems and then forget them may be profitable. The practice is, however, exceedingly risky. It is true that everybody does in fact forget many such meanings and facts, but this commonly means either that they should not have been learned at all at the time that they were learned, or that they should have been learned more permanently, or that details should have been learned with the expectation that they themselves would be forgotten but that a general fact or attitude would remain. For example, duodecagon should not be learned at all in the elementary school; indorsement should either not be learned at all there, or be learned for permanence of a year or more; the details of the metric system should be so taught as to leave for several years at least knowledge of the facts that there is a system so named that is important, whose tables go by tens, hundreds, or thousands, and a tendency (not necessarily strong) to connect meter, kilogram, and liter with measurement by the metric system and with approximate estimates of their several magnitudes. If an arithmetical procedure seems to require accessory bonds which are to be forgotten, once the procedure is mastered, we should be suspicious of the value of the procedure itself. If pupils forget what compound interest is, we may be sure that they will usually also have forgotten how to compute it. Surely there is waste if they have learned what it is only to learn how to compute it only to forget how to compute it! THE STRENGTH OF BONDS CONCERNING THE REASONS FOR ARITHMETICAL PROCESSES The next case of the formation of bonds to slight strength is the problematic one of forming the bonds involved in understanding the reasons for certain processes only to forget them after the process has become a habit. Should a pupil, that is, learn why he inverts and multiplies, only to forget it as soon as he can be trusted to divide by a fraction? Should he learn why he puts the units figure of each partial product in multiplication under the figure that he multiplies by, only to forget the reason as soon as he has command of the process? Should he learn why he gets the number of square inches in a rectangle by multiplying the length by the width, both being expressed in linear inches, and forget why as soon as he is competent to make computations of the areas of rectangles? On general psychological grounds we should be suspicious of forming bonds only to let them die of starvation later, and tend to expect that elaborate explanations learned only to be forgotten either should not be learned at all, or should be learned at such a time and in such a way that they would not be forgotten. Especially we should expect that the general principles of arithmetic, the whys and wherefores of its fundamental ways of manipulating numbers, ought to be the last bonds of all to be forgotten. Details of _how_ you arranged numbers to multiply might vanish, but the general reasons for the placing would be expected to persist and enable one to invent the detailed manipulations that had been forgotten. This suspicion is, I think, justified by facts. The doctrine that the customary deductive explanations of why we invert and multiply, or place the partial products as we do before adding, may be allowed to be forgotten once the actual habits are in working order, has a suspicious source. It arose to meet the criticism that so much time and effort were required to keep these deductive explanations in memory. The fact was that the pupil learned to compute correctly _irrespective of_ the deductive explanations. They were only an added burden. His inductive learning that the procedure gave the right answer really taught him. So he wisely shuffled off the extra burden of facts about the consequences of the nature of a fraction or the place values of our decimal notation. The bonds weakened because they were not used. They were not used because they were not useful in the shape and at the time that they were formed, or because the pupil was unable to understand the explanations so as to form them at all. The criticism was valid and should have been met in part by replacing the deductive explanations by inductive verifications, and in part by using the deductive reasoning as a check after the process itself is mastered. The very same discussions of place-value which are futile as proof that you must do a certain thing before you have done it, often become instructive as an explanation of why the thing that you have learned to do and are familiar with and have verified by other tests works as well as it does. The general deductive theory of arithmetic should not be learned only to be forgotten. Much of it should, by most pupils, not be learned at all. What is learned should be learned much later than now, as a synthesis and rationale of habits, not as their creator. What is learned of such deductive theory should rank among the most rather than least permanent of a pupil's stock of arithmetical knowledge and power. There are bonds which are formed only to be lost, and bonds formed only to be lost _in their first form_, being used in a new organization as material for bonds of a higher order; but the bonds involved in deductive explanations of why certain processes are right are not such: they are not to be formed just to be forgotten, nor as mere propædeutics to routine manipulations. PROPÆDEUTIC BONDS The formation of bonds to a limited strength because they are to be lost in their first form, being worked over in different ways in other bonds to which they are propædeutic or contributing is the most important case of low strength, or rather low permanence, in bonds. The bond between four 5s in a column to be added and the response of thinking '10, 15, 20' is worth forming, but it is displaced later by the multiplication bond or direct connection of 'four 5s to be added' with '20.' Counting by 2s from 2, 3s from 3, 4s from 4, 5s from 5, etc., forms serial bonds which as series might well be left to disappear. Their separate steps are kept as permanent bonds for use in column addition, but their serial nature is changed from 2 (and 2) 4, (and 2) 6, (and 2) 8, etc., to two 2s = 4, three 2s = 6, four 2s = 8, etc.; after playing their part in producing the bonds whereby any multiple of 2 by 2 to 9, can be got, the original serial bonds are, as series, needed no longer. The verbal response of saying 'and' in adding, after helping to establish the bonds whereby the general set of the mind toward adding coöperates with the numbers seen or thought of to produce their sum, should disappear; or remain so slurred in inner speech as to offer no bar to speed. The rule for such bonds is, of course, to form them strongly enough so that they work quickly and accurately for the time being and facilitate the bonds that are to replace them, but not to overlearn them. There is a difference between learning something to be held for a short time, and the same amount of energy spent in learning for long retention. The former sort of learning is, of course, appropriate with many of these propædeutic bonds. The bonds mentioned as illustrations are not _purely_ propædeutic, nor formed _only_ to be transmuted into something else. Even the saying of 'and' in addition has some genuine, intrinsic value in distinguishing the process of addition, and may perhaps be usefully reviewed for a brief space during the first steps in adding common fractions. Some such propædeutic bonds may be worth while apart from their value in preparing for other bonds. Consider, for example, exercises like those shown below which are propædeutic to long division, giving the pupil some basis in experience for his selection of the quotient figures. These multiplications are intrinsically worth doing, especially the 12s and 25s. Whatever the pupil remembers of them will be to his advantage. 1. Count by 11s to 132, beginning 11, 22, 33. 2. Count by 12s to 144, beginning 12, 24, 36. 3. Count by 25s to 300, beginning 25, 50, 75. 4. State the missing numbers:-- A. B. C. D. 3 11s = 5 11s = 8 ft. = .... in. 2 dozen = 4 12s = 3 12s = 10 ft. = .... in. 4 dozen = 5 12s = 6 12s = 7 ft. = .... in. 10 dozen = 6 11s = 12 11s = 4 ft. = .... in. 5 dozen = 9 11s = 2 12s = 6 ft. = .... in. 7 dozen = 7 12s = 9 12s = 9 ft. = .... in. 12 dozen = 8 12s = 7 11s = 11 ft. = .... in. 9 dozen = 11 11s = 12 12s = 5 ft. = .... in. 6 dozen = 5. Count by 25s to $2.50, saying, "25 cents, 50 cents, 75 cents, one dollar," and so on. 6. Count by 15s to $1.50. 7. Find the products. Do not use pencil. Think what they are. A. B. C. D. E. 2 × 25 3 × 15 2 × 12 4 × 11 6 × 25 3 × 25 10 × 15 2 × 15 4 × 15 6 × 15 5 × 25 4 × 15 2 × 25 4 × 12 6 × 12 10 × 25 2 × 15 2 × 11 4 × 25 6 × 11 4 × 25 7 × 15 3 × 25 5 × 11 7 × 12 6 × 25 9 × 15 3 × 15 5 × 12 7 × 15 8 × 25 5 × 15 3 × 11 5 × 15 7 × 25 7 × 25 8 × 15 3 × 12 5 × 25 7 × 11 9 × 25 6 × 15 8 × 12 9 × 12 8 × 25 State the missing numbers:-- A. 36 = .... 12s B. 44 = .... 11s C. 50 = .... 25s 60 = .... 12s 88 = .... 11s 125 = .... 25s 24 = .... 12s 77 = .... 11s 75 = .... 25s 48 = .... 12s 55 = .... 11s 200 = .... 25s 144 = .... 12s 99 = .... 11s 250 = .... 25s 108 = .... 12s 110 = .... 11s 175 = .... 25s 72 = .... 12s 33 = .... 11s 225 = .... 25s 96 = .... 12s 66 = .... 11s 150 = .... 25s 84 = .... 12s 22 = .... 11s 100 = .... 25s Find the quotients and remainders. If you need to use paper and pencil to find them, you may. But find as many as you can without pencil and paper. Do Row A first. Then do Row B. Then Row C, etc. __ __ __ __ __ __ Row A. 11|45 12|45 25|45 15|45 21|45 22|45 __ __ __ __ __ __ Row B. 25|55 11|55 12|55 15|55 22|55 30|55 __ __ __ __ __ __ Row C. 12|60 25|60 15|60 11|60 30|60 21|60 __ __ __ __ __ __ Row D. 12|75 11|75 15|75 25|75 30|75 35|75 ___ ___ ___ ___ ___ ___ Row E. 11|100 12|100 25|100 15|100 30|100 22|100 __ __ __ __ __ __ Row F. 11|96 12|96 25|96 15|96 30|96 22|96 ___ ___ ___ ___ ___ ___ Row G. 25|105 11|105 15|105 12|105 22|105 35|105 __ __ __ __ __ __ Row H. 12|64 15|64 25|64 11|64 22|64 21|64 __ __ __ __ __ __ Row I. 11|80 12|80 15|80 25|80 35|80 21|80 ___ ___ ___ ___ ___ ___ Row J. 25|200 30|200 75|200 63|200 65|200 66|200 Do this section again. Do all the first column first. Then do the second column, then the third, and so on. Consider, from the same point of view, exercises like (3 × 4) + 2, (7 × 6) + 5, (9 × 4) + 6, given as a preparation for written multiplication. The work of 48 68 47 3 7 9 -- -- -- and the like is facilitated if the pupil has easy control of the process of getting a product, and keeping it in mind while he adds a one-place number to it. The practice with (3 × 4) + 2 and the like is also good practice intrinsically. So some teachers provide systematic preparatory drills of this type just before or along with the beginning of short multiplication. In some cases the bonds are purely propædeutic or are formed _only_ for later reconstruction. They then differ little from 'crutches.' The typical crutch forms a habit which has actually to be broken, whereas the purely propædeutic bond forms a habit which is left to rust out from disuse. For example, as an introduction to long division, a pupil may be given exercises using one-figure divisors in the long form, as:-- 773 and 5 remainder ______ 7)5416 49 -- 51 49 -- 26 21 -- 5 The important recommendation concerning these purely propædeutic bonds, and bonds formed only for later reconstruction, is to be very critical of them, and not indulge in them when, by the exercise of enough ingenuity, some bond worthy of a permanent place in the individual's equipment can be devised which will do the work as well. Arithmetical teaching has done very well in this respect, tending to err by leaving out really valuable preparatory drills rather than by inserting uneconomical ones. It is in the teaching of reading that we find the formation of propædeutic bonds of dubious value (with letters, phonograms, diacritical marks, and the like) often carried to demonstrably wasteful extremes. CHAPTER VI THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF PRACTICE AND THE ORGANIZATION OF ABILITIES THE AMOUNT OF PRACTICE It will be instructive if the reader will perform the following experiment as an introduction to the discussion of this chapter, before reading any of the discussion. Suppose that a pupil does all the work, oral and written, computation and problem-solving, presented for grades 1 to 6 inclusive (that is, in the first two books of a three-book series) in the average textbook now used in the elementary school. How many times will he have exercised each of the various bonds involved in the four operations with integers shown below? That is, how many times will he have thought, "1 and 1 are 2," "1 and 2 are 3," etc.? Every case of the action of each bond is to be counted. THE FUNDAMENTAL BONDS 1 + 1 2 - 1 1 × 1 2 ÷ 1 1 + 2 2 - 2 2 × 1 2 ÷ 2 1 + 3 3 × 1 1 + 4 4 × 1 1 + 5 3 - 1 5 × 1 3 ÷ 1 1 + 6 3 - 2 6 × 1 3 ÷ 2 1 + 7 3 - 3 7 × 1 3 ÷ 3 1 + 8 8 × 1 1 + 9 9 × 1 4 - 1 4 ÷ 1 4 - 2 4 ÷ 2 11 (or 21 or 31, etc.) + 1 4 - 3 1 × 2 4 ÷ 3 11 " + 2 4 - 4 2 × 2 4 ÷ 4 11 " + 3 3 × 2 11 " + 4 4 × 2 11 " + 5 5 - 1 5 × 2 5 ÷ 1 11 " + 6 5 - 2 6 × 2 5 ÷ 2 11 " + 7 5 - 3 7 × 2 5 ÷ 3 11 " + 8 5 - 4 8 × 2 5 ÷ 4 11 " + 9 5 - 5 9 × 2 5 ÷ 5 6 - 1 1 × 3 6 ÷ 1 2 + 1 6 - 2 2 × 3 6 ÷ 2 2 + 2 6 - 3 3 × 3 6 ÷ 3 2 + 3 6 - 4 4 × 3 6 ÷ 4 2 + 4 6 - 5 5 × 3 6 ÷ 5 2 + 5 6 - 6 6 × 3 6 ÷ 6 2 + 6 7 × 3 2 + 7 8 × 3 2 + 8 7 - 1 9 × 3 7 ÷ 1 2 + 9 7 - 2 7 ÷ 2 7 - 3 7 ÷ 3 7 - 4 1 × 4 7 ÷ 4 12 (or 22 or 32, etc.) + 1 7 - 5 2 × 4 7 ÷ 5 12 " + 2 7 - 6 and so on 7 ÷ 6 7 - 7 to 9 × 9 7 ÷ 7 and so on to and so on and so on to 9 + 9 to 18 - 9 82 ÷ 9 19 (or 29 or 39, etc.) + 9 83 ÷ 9, etc. If estimating for the entire series is too long a task, it will be sufficient to use eight or ten from each, say:-- 3 + 2 13, 23, etc. + 2 7 + 2 17, 27, etc. + 2 " 3 " 3 " 3 " 3 " 4 " 4 " 4 " 4 " 5 " 5 " 5 " 5 " 6 " 6 " 6 " 6 " 7 " 7 " 7 " 7 " 8 " 8 " 8 " 8 " 9 " 9 " 9 " 9 3 - 3 7 - 7 9 × 7 63 ÷ 9 4 " 8 " 7 × 9 64 " 5 " 9 " 8 × 6 65 " 6 " 10 " 6 × 8 66 " 7 " 11 " 67 " 8 " 12 " 68 " 9 " 13 " 69 " 10 " 14 " 70 " 11 " 15 " 71 " 12 " 16 " TABLE 2 ESTIMATES OF THE AMOUNT OF PRACTICE PROVIDED IN BOOKS I AND II OF THE AVERAGE THREE-BOOK TEXT IN ARITHMETIC; BY 50 EXPERIENCED TEACHERS ====================================================================== | LOWEST | MEDIAN | HIGHEST |RANGE REQUIRED TO ARITHMETICAL FACT |ESTIMATE|ESTIMATE|ESTIMATE | INCLUDE HALF OF | | | | THE ESTIMATES -----------------------+--------+--------+---------+------------------ 3 or 13 or 23, etc. + 2| 25 | 1500 |1,000,000| 800-5000 " " 3| 24 | 1450 | 80,000| 475-5000 " " 4| 23 | 1150 | 50,000| 750-5000 " " 5| 22 | 1400 | 44,000| 700-5000 " " 6| 21 | 1350 | 41,000| 700-4500 " " 7| 21 | 1500 | 37,000| 600-4000 " " 8| 20 | 1400 | 33,000| 550-4100 " " 9| 20 | 1150 | 28,000| 650-4500 | | | | 7 or 17 or 27, etc. + 2| 20 | 1250 |2,000,000| 600-5000 " " 3| 19 | 1100 |1,000,000| 650-4900 " " 4| 18 | 1000 | 80,000| 650-4900 " " 5| 17 | 1300 | 80,000| 650-4400 " " 6| 16 | 1100 | 29,000| 650-4500 " " 7| 15 | 1100 | 25,000| 500-4500 " " 8| 13 | 1100 | 21,000| 650-3800 " " 9| 10 | 1275 | 17,000| 500-4000 | | | | 3 - 3 | 25 | 1000 | 100,000| 500-4000 4 - 3 | 20 | 1050 | 500,000| 525-3000 5 - 3 | 20 | 1100 |2,500,000| 650-4200 6 - 3 | 10 | 1050 | 21,000| 650-3250 7 - 3 | 22 | 1100 | 15,000| 550-3050 8 - 3 | 21 | 1075 | 15,000| 650-3000 9 - 3 | 21 | 1000 | 15,000| 700-2600 10 - 3 | 20 | 1000 | 20,000| 600-2500 11 - 3 | 20 | 1000 | 15,000| 465-2550 12 - 3 | 18 | 1000 | 15,000| 650-2100 | | | | 7 - 7 | 10 | 1000 | 18,000| 425-3000 8 - 7 | 15 | 1000 | 18,000| 413-3100 9 - 7 | 15 | 950 | 18,000| 550-3000 10 - 7 | 15 | 950 | 18,000| 600-3950 11 - 7 | 10 | 900 | 18,000| 550-3000 12 - 7 | 10 | 925 | 18,000| 525-3100 13 - 7 | 10 | 900 | 18,000| 500-2600 14 - 7 | 10 | 900 | 18,000| 500-3100 15 - 7 | 10 | 925 | 18,000| 500-3000 16 - 7 | 10 | 875 | 18,000| 500-2500 | | | | 9 × 7 | 10 | 700 | 20,000| 500-2000 7 × 9 | 10 | 700 | 20,000| 500-1750 8 × 6 | 10 | 750 | 20,000| 500-2500 6 × 8 | 9 | 700 | 20,000| 500-2500 | | | | 63 ÷ 9 | 9 | 500 | 4,500| 300-2500 64 ÷ 9 | 9 | 200 | 4,000| 100- 700 65 ÷ 9 | 8 | 200 | 4,000| 100- 600 66 ÷ 9 | 7 | 200 | 4,000| 100- 550 67 ÷ 9 | 7 | 200 | 4,000| 75- 450 68 ÷ 9 | 6 | 200 | 4,000| 87- 575 69 ÷ 9 | 6 | 200 | 4,000| 87- 450 70 ÷ 9 | 5 | 200 | 4,000| 75- 575 71 ÷ 9 | 5 | 200 | 4,000| 75- 700 | | | | _XX_ | 40 | 550 |1,000,000| 300-2000 _XO_ | 20 | 500 | 11,500| 150-2000 _XXX_ | 15 | 450 | 12,000| 100-1000 _XXO_ | 25 | 400 | 15,000| 150-1000 _XOO_ | 15 | 400 | 5,000| 100-1000 _XOX_ | 10 | 400 | 10,000| 100- 975 ====================================================================== Having made his estimates the reader should compare them first with similar estimates made by experienced teachers (shown on page 124 f.), and then with the results of actual counts for representative textbooks in arithmetic (shown on pages 126 to 132). It will be observed in Table 2 that even experienced teachers vary enormously in their estimates of the amount of practice given by an average textbook in arithmetic, and that most of them are in serious error by overestimating the amount of practice. In general it is the fact that we use textbooks in arithmetic with very vague and erroneous ideas of what is in them, and think they give much more practice than they do. The authors of the textbooks as a rule also probably had only very vague and erroneous ideas of what was in them. If they had known, they would almost certainly have revised their books. Surely no author would intentionally provide nearly four times as much practice on 2 + 2 as on 8 + 8, or eight times as much practice on 2 × 2 as on 9 × 8, or eleven times as much practice on 2 - 2 as on 17 - 8, or over forty times as much practice on 2 ÷ 2 as on 75 ÷ 8 and 75 ÷ 9, both together. Surely no author would have provided intentionally only twenty to thirty occurrences each of 16 - 7, 16 - 8, 16 - 9, 17 - 8, 17 - 9, and 18 - 9 for the entire course through grade 6; or have left the practice on 60 ÷ 7, 60 ÷ 8, 60 ÷ 9, 61 ÷ 7, 61 ÷ 8, 61 ÷ 9, and the like to occur only about once a year! TABLE 3 AMOUNT OF PRACTICE: ADDITION BONDS IN A RECENT TEXTBOOK (A) OF EXCELLENT REPUTE. BOOKS I AND II, ALL SAVE FOUR SECTIONS OF SUPPLEMENTARY MATERIAL, TO BE USED AT THE TEACHER'S DISCRETION The Table reads: 2 + 2 was used 226 times, 12 + 2 was used 74 times, 22 + 2, 32 + 2, 42 + 2, and so on were used 50 times. ====================================================================== | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | TOTAL ----------------+-----+-----+-----+-----+-----+-----+-----+----+------ 2 | 226 | 154 | 162 | 150 | 97 | 87 | 66 | 45| 12 | 74 | 53 | 76 | 46 | 51 | 37 | 36 | 33| 22, etc. | 50 | 60 | 68 | 63 | 42 | 50 | 38 | 26| | | | | | | | | | 3 | 216 | 141 | 127 | 89 | 82 | 54 | 58 | 40| 13 | 43 | 43 | 60 | 70 | 52 | 30 | 22 | 18| 23, etc. | 15 | 30 | 51 | 50 | 42 | 32 | 29 | 30| | | | | | | | | | 7 | 85 | 90 | 103 | 103 | 84 | 81 | 61 | 47| 17 | 35 | 25 | 42 | 32 | 35 | 21 | 29 | 16| 27, etc. | 30 | 23 | 32 | 29 | 24 | 23 | 25 | 28| | | | | | | | | | 8 | 185 | 112 | 146 | 99 | 75 | 71 | 73 | 61| 18 | 28 | 35 | 52 | 46 | 28 | 29 | 24 | 14| 28, etc. | 53 | 36 | 34 | 38 | 23 | 36 | 27 | 27| | | | | | | | | | 9 | 104 | 81 | 112 | 96 | 63 | 74 | 58 | 57| 19 | 13 | 11 | 31 | 38 | 25 | 14 | 22 | 11| 29, etc. | 19 | 17 | 27 | 20 | 32 | 32 | 19 | 18| | | | | | | | | | 2, 12, 22, etc. | 350 | 277 | 306 | 260 | 190 | 174 | 140 | 104| 1801 3, 13, 23, etc. | 274 | 214 | 230 | 209 | 176 | 116 | 109 | 88| 1406 | | | | | | | | | 7, 17, 27, etc. | 148 | 138 | 187 | 164 | 141 | 125 | 115 | 91| 1109 8, 18, 28, etc. | 266 | 183 | 232 | 185 | 126 | 136 | 124 | 102| 1354 9, 19, 29, etc. | 136 | 109 | 170 | 154 | 120 | 120 | 99 | 86| 994 | | | | | | | | | Totals |1164 | 921 |1125 | 972 | 753 | 671 | 687 | 471| ====================================================================== TABLE 4 AMOUNT OF PRACTICE: SUBTRACTION BONDS IN A RECENT TEXTBOOK (A) OF EXCELLENT REPUTE. BOOKS I AND II, ALL SAVE FOUR SECTIONS OF SUPPLEMENTARY MATERIAL, TO BE USED AT THE TEACHER'S DISCRETION ================================================================ | SUBTRAHENDS MINUENDS |----------------------------------------------------- | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 ----------+-----+-----+-----+-----+-----+-----+-----+-----+----- 1 | 372 | | | | | | | | 2 | 214 | 311 | | | | | | | 3 | 136 | 149 | 189 | | | | | | 4 | 146 | 142 | 103 | 205 | | | | | 5 | 171 | 91 | 92 | 164 | 136 | | | | 6 | 80 | 59 | 69 | 71 | 81 | 192 | | | 7 | 106 | 57 | 55 | 67 | 59 | 156 | 80 | | 8 | 73 | 50 | 50 | 75 | 50 | 62 | 48 | 152 | 9 | 71 | 75 | 54 | 74 | 48 | 55 | 55 | 124 | 133 10 | 261 | 84 | 63 | 100 | 193 | 83 | 57 | 124 | 91 | | | | | | | | | 11 | | 48 | 31 | 50 | 36 | 41 | 32 | 46 | 35 12 | | | 48 | 77 | 57 | 51 | 35 | 80 | 30 13 | | | | 35 | 22 | 40 | 29 | 35 | 28 14 | | | | | 25 | 37 | 36 | 49 | 32 15 | | | | | | 33 | 19 | 48 | 20 | | | | | | | | | 16 | | | | | | | 16 | 36 | 26 17 | | | | | | | | 27 | 20 18 | | | | | | | | | 19 | | | | | | | | | Total | | | | | | | | | excluding | | | | | | | | | 1-1, 2-2, | | | | | | | | | etc. |1258 | 755 | 565 | 713 | 571 | 558 | 327 | 569 | 301 ================================================================ TABLE 5 FREQUENCIES OF SUBTRACTIONS NOT INCLUDED IN TABLE 4 These are cases where the pupil would, by reason of his stage of advancement, probably operate 35-30, 46-46, etc., as one bond. ====================================================================== | SUBTRAHENDS |----+----+----+----+----+----+----+----+----+---- | 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | MINUENDS | 11| 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 10 | 21| 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 20 |etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc. --------------------+----+----+----+----+----+----+----+----+----+---- 10, 20, 30, 40, etc.| 11 | 29 | 16 | 52 | 32 | 51 | 7 | 30 | 22 | 60 11, 21, 31, 41, etc.| 42 | 14 | 22 | 32 | 12 | 26 | 19 | 52 | 17 | 10 12, 22, 32, 42, etc.| 47 | 97 | 5 | 13 | 9 | 21 | 11 | 24 | 19 | 17 13, 23, 33, 43, etc.| 7 | 40 | 7 | 14 | 15 | 13 | 19 | 19 | 22 | 3 14, 24, 34, 44, etc.| 8 | 28 | 14 | 58 | 13 | 16 | 14 | 26 | 19 | 7 15, 25, 35, 45, etc.| 21 | 28 | 29 | 54 | 51 | 15 | 21 | 12 | 24 | 8 16, 26, 36, 46, etc.| 5 | 18 | 12 | 27 | 35 | 69 | 13 | 17 | 19 | 2 17, 27, 37, 47, etc.| 5 | 9 | 12 | 40 | 32 | 54 | 24 | 12 | 12 | 1 18, 28, 38, 48, etc.| 2 | 16 | 10 | 23 | 22 | 36 | 18 | 47 | 16 | 0 19, 29, 39, etc. | 5 | 7 | 7 | 10 | 13 | 28 | 14 | 23 | 16 | 0 | | | | | | | | | | Totals |153 |286 |134 |323 |234 |329 |160 |262 |186 |108 ===================================================================== TABLE 6 AMOUNT OF PRACTICE: MULTIPLICATION BONDS IN ANOTHER RECENT TEXTBOOK (B) OF EXCELLENT REPUTE. BOOKS I AND II ====================================================================== | MULTIPLICANDS MULTIPLIERS |--------------------------------------------------------- | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |Totals ------------+----+----+----+----+----+----+----+----+----+----+------- 1 | 299| 534| 472| 271| 310| 293| 261| 178| 195| 99| 2912 2 | 350| 644| 668| 480| 458| 377| 332| 238| 239| 155| 3941 3 | 280| 487| 509| 388| 318| 302| 247| 199| 227| 152| 3109 4 | 186| 375| 398| 242| 203| 265| 197| 163| 159| 93| 2281 5 | 268| 359| 393| 234| 263| 243| 217| 192| 197| 114| 2480 6 | 180| 284| 265| 199| 196| 191| 168| 169| 165| 106| 1923 7 | 135| 283| 277| 176| 187| 158| 155| 121| 145| 118| 1755 8 | 137| 272| 292| 175| 192| 164| 158| 157| 126| 126| 1799 9 | 71| 173| 140| 122| 97| 102| 101| 100| 82| 110| 1098 | | | | | | | | | | | Totals |1906|3411|3414|2287|2224|2095|1836|1517|1535|1073| ====================================================================== TABLE 7 AMOUNT OF PRACTICE: DIVISIONS WITHOUT REMAINDER IN TEXTBOOK B, PARTS I AND II ====================================================================== | DIVISORS DIVIDENDS |---------------------------------------------- | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |Totals -----------------------+----+----+----+----+----+----+----+----+------ Integral | 397| 224| 250| 130| 93| 44| 98| 23| 1259 multiples | | | | | | | | | of 2 to 9 | 256| 124| 152| 79| 28| 43| 61| 25| 768 in sequence; | | | | | | | | | _i.e._, 4 ÷ 2 | 318| 123| 130| 65| 50| 19| 39| 19| 763 occurred | | | | | | | | | 397 times, | 258| 98| 86| 105| 25| 24| 34| 20| 650 6 ÷ 2 occurred | | | | | | | | | 256 times, | 198| 49| 76| 27| 22| 30| 33| 16| 451 6 ÷ 3, 224 times, | | | | | | | | | 9 ÷ 3, 124 times. | 77| 54| 36| 31| 28| 27| 16| 9| 278 | 180| 91| 50| 38| 17| 13| 22| 16| 427 | 69| 46| 37| 24| 12| 17| 16| 15| 236 | | | | | | | | | Totals |1753| 809| 817| 499| 275| 217| 319| 142| ====================================================================== TABLE 8 DIVISION BONDS, WITH AND WITHOUT REMAINDERS. BOOK B All work through grade 6, except estimates of quotient figures in long division. Dividend 2 3 4 5 Divisor 1 2 1 2 3 1 2 3 4 1 2 3 4 5 Number of Occurrences 41 386 27 189 240 26 397 66 185 23 136 43 53 135 Dividend 6 7 Divisor 1 2 3 4 5 6 1 2 3 4 5 6 7 Number of Occurrences 21 256 224 68 43 83 23 72 55 38 46 32 54 Dividend 8 9 Divisor 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 Number of Occurrences 17 318 30 250 22 28 39 91 19 50 124 49 25 15 18 30 38 Dividend 10 11 Divisor 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 Number of Occurrences 258 38 46 120 19 9 24 24 32 21 16 3 7 11 14 3 Dividend 12 13 Divisor 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 Number of Occurrences 198 123 152 29 93 9 16 7 45 16 15 11 7 4 5 3 Dividend 14 15 Divisor 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 Number of Occurrences 77 20 13 5 8 44 8 6 69 98 16 79 8 8 4 6 Dividend 16 17 Divisor 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 Number of Occurrences 180 19 130 14 6 9 98 3 61 9 15 14 6 6 12 3 Dividend 18 19 Divisor 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 Number of Occurrences 69 49 13 6 28 7 7 23 21 6 10 5 3 4 10 4 Dividend 20 21 Divisor 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Number of Occurrences 24 86 65 11 3 23 5 54 12 8 5 43 10 5 Dividend 22 23 Divisor 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Number of Occurrences 17 16 15 8 13 6 15 7 8 11 8 6 3 2 Dividend 24 25 Divisor 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Number of Occurrences 91 76 18 50 5 61 1 11 13 105 5 6 5 3 Dividend 26 27 Divisor 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Number of Occurrences 5 6 3 3 4 6 3 46 8 10 4 2 6 25 Dividend 28 29 Divisor 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Number of Occurrences 4 36 8 3 19 3 7 6 8 0 5 11 2 3 Dividend 30 31 32 Divisor 4 5 6 7 8 9 4 5 6 7 8 9 4 5 6 7 8 9 Number of Occurrences 21 27 25 6 7 13 4 3 1 1 4 2 50 11 3 6 39 5 Dividend 33 34 35 Divisor 4 5 6 7 8 9 4 5 6 7 8 9 4 5 6 7 8 9 Number of Occurrences 8 7 7 2 6 1 8 3 5 2 1 1 10 31 5 24 5 3 Dividend 36 37 38 Divisor 4 5 6 7 8 9 4 5 6 7 8 9 4 5 6 7 8 9 Number of Occurrences 37 16 22 2 6 19 12 8 7 5 3 9 7 8 7 1 1 5 Dividend 39 40 41 42 Divisor 4 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 Number of Occurrences 4 3 7 4 3 1 38 9 2 34 2 6 6 3 7 5 7 28 30 10 3 Dividend 43 44 45 46 Divisor 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 Number of Occurrences 7 5 10 3 2 7 6 4 5 0 24 6 7 10 20 3 3 2 2 2 Dividend 47 48 49 50 Divisor 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 6 7 8 9 Number of Occurrences 6 2 2 0 3 7 17 4 33 2 4 7 27 9 2 4 6 3 8 Dividend 51 52 53 54 Divisor 6 7 8 9 6 7 8 9 6 7 8 9 6 7 8 9 Number of Occurrences 2 3 1 2 5 5 5 3 4 3 2 2 12 5 1 16 Dividend 55 56 57 58 59 Divisor 6 7 8 9 6 7 8 9 6 7 8 9 6 7 8 9 6 7 8 9 Number of Occurrences 5 3 4 2 0 13 16 8 0 3 1 3 2 2 3 1 2 3 0 3 Dividend 60 61 62 63 64 65 Divisor 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 Number of Occurrences 3 9 1 1 2 5 4 6 1 17 5 9 5 22 0 1 10 1 Dividend 66 67 68 69 70 71 Divisor 7 8 9 7 8 9 7 8 9 7 8 9 8 9 8 9 Number of Occurrences 2 1 4 0 1 1 1 3 2 0 6 1 6 2 1 0 Dividend 72 73 74 75 76 77 78 79 Divisor 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 Number of Occurrences 16 10 7 5 3 3 5 3 3 2 3 0 4 1 0 2 Dividend 80 81 82 83 84 85 86 87 88 89 Divisor 9 9 9 9 9 9 9 9 9 9 Number of Occurrences 4 15 2 4 1 2 0 3 2 7 Tables 3 to 8 show that even gifted authors make instruments for instruction in arithmetic which contain much less practice on certain elementary facts than teachers suppose; and which contain relatively much more practice on the more easily learned facts than on those which are harder to learn. How much practice should be given in arithmetic? How should it be divided among the different bonds to be formed? Below a certain amount there is waste because, as has been shown in Chapter VI, the pupil will need more time to detect and correct his errors than would have been required to give him mastery. Above a certain amount there is waste because of unproductive overlearning. If 668 is just enough for 2 × 2, 82 is not enough for 9 × 8. If 82 is just enough for 9 × 8, 668 is too much for 2 × 2. It is possible to find the answers to these questions for the pupil of median ability (or any stated ability) by suitable experiments. The amount of practice will, of course, vary according to the ability of the pupil. It will also vary according to the interest aroused in him and the satisfaction he feels in progress and mastery. It will also vary according to the amount of practice of other related bonds; 7 + 7 = 14 and 60 ÷ 7 = 8 and 4 remainder will help the formation of 7 + 8 = 15 and 61 ÷ 7 = 8 and 5 remainder. It will also, of course, vary with the general difficulty of the bond, 17 - 8 = 9 being under ordinary conditions of teaching harder to form than 7 - 2 = 5. Until suitable experiments are at hand we may estimate for the fundamental bonds as follows, assuming that by the end of grade 6 a strength of 199 correct out of 200 is to be had, and that the teaching is by an intelligent person working in accord with psychological principles as to both ability and interest. For one of the easier bonds, most facilitated by other bonds (such as 2 × 5 = 10, or 10 - 2 = 8, or the double bond 7 = two 3s and 1 remainder) in the case of the median or average pupil, twelve practices in the week of first learning, supported by twenty-five practices during the two months following, and maintained by thirty practices well spread over the later periods should be enough. For the more gifted pupils lesser amounts down to six, twelve, and fifteen may suffice. For the less gifted pupils more may be required up to thirty, fifty, and a hundred. It is to be doubted, however, whether pupils requiring nearly two hundred repetitions of each of these easy bonds should be taught arithmetic beyond a few matters of practical necessity. For bonds of ordinary difficulty, with average facilitation from other bonds (such as 11 - 3, 4 × 7, or 48 ÷ 8 = 6) in the case of the median or average pupil, we may estimate twenty practices in the week of first learning, supported by thirty, and maintained by fifty practices well spread over the later periods. Gifted pupils may gain and keep mastery with twelve, fifteen, and twenty practices respectively. Pupils dull at arithmetic may need up to twenty, sixty, and two hundred. Here, again, it is to be doubted whether a pupil for whom arithmetical facts, well taught and made interesting, are so hard to acquire as this, should learn many of them. For bonds of greater difficulty, less facilitated by other bonds (such as 17 - 9, 8 × 7, or 12-1/2% of = 1/8 of), the practice may be from ten to a hundred percent more than the above. UNDERLEARNING AND OVERLEARNING If we accept the above provisional estimates as reasonable, we may consider the harm done by giving less and by giving more than these reasonable amounts. Giving less is indefensible. The pupil's time is wasted in excessive checking to find his errors. He is in danger of being practiced in error. His attention is diverted from the learning of new facts and processes by the necessity of thinking out these supposedly mastered facts. All new bonds are harder to learn than they should be because the bonds which should facilitate them are not strong enough to do so. Giving more does harm to some extent by using up time that could be spent better for other purposes, and (though not necessarily) by detracting from the pupil's interest in arithmetic. In certain cases, however, such excess practice and overlearning are actually desirable. Three cases are of special importance. The first is the case of a bond operating under a changed mental set or adjustment. A pupil may know 7 × 8 adequately as a thing by itself, but need more practice to operate it in 285 7 --- where he has to remember that 3 is to be added to the 56 when he obtains it, and that only the 9 is to be written down, the 5 to be held in mind for later use. The practice required to operate the bond efficiently in this new set is desirable, even though it is excess from a narrower point of view, and causes the straightforward 'seven eights are fifty-six' to be overlearned. So also a pupil's work with 24, 34, 44, etc., +9 may react to give what would be excess practice from the point of view of 4 + 9 alone; his work in estimating approximate quotient figures in long division may give excess practice on the division tables. There are many such cases. Even adding the 5 and 7 in 5/12 + 7/12 is not quite the same task as adding 5 and 7 undisturbed by the fact that they are twelfths. We know far too little about the amount of practice needed to adapt arithmetical bonds to efficient operation in these more complicated conditions to estimate even approximately the allowances to be made. But some allowance, and often a rather large allowance, must be made. The second is the case where the computation in general should be made very easy and sure for the pupil except for some one new element that is being learned. For example, in teaching the meaning and uses of 'Averages' and of uneven division, we may deliberately use 2, 3, and 4 as divisors rather than 7 and 9, so as to let all the pupil's energy be spent in learning the new facts, and so that the fraction in the quotient may be something easily understood, real, and significant. In teaching the addition of mixed numbers, we may use, in the early steps, 11-1/2 13-1/2 24 ------ rather than 79-1/2 98-1/2 67 ------ so as to save attention for the new process itself. In cancellation, we may give excess practice to divisions by 2, 3, 4, and 5 in order to make the transfer to the new habits of considering two numbers together from the point of view of their divisibility by some number. In introducing trade discount, we may give excess practice on '5% of' and '10% of' deliberately, so that the meaning of discount may not be obscured by difficulties in the computation itself. Excess practice on, and overlearning of, certain bonds is thus very often justifiable. The third case concerns bonds whose importance for practical uses in life or as notable facilitators of other bonds is so great that they may profitably be brought to a greater strength than 199 correct out of 200 at a speed of 2 sec. or less, or be brought to that degree of strength very early. Examples of bonds of such special practical use are the subtractions from 10, 1/2 + 1/2, 1/2 + 1/4, 1/2 of 60, 1/4 of 60, and the fractional parts of 12 and of $1.00. Examples of notable facilitating bonds are ten 10s = 100, ten 100s = 1000, additions like 2 + 2, 3 + 3, and 4 + 4, and all the multiplication tables to 9 × 9. In consideration of these three modifying cases or principles, a volume could well be written concerning just how much practice to give to each bond, in each of the types of complex situations where it has to operate. There is evidently need for much experimentation to expose the facts, and for much sagacity and inventiveness in making sure of effective learning without wasteful overlearning. The facts of primary importance are:-- (1) The textbook or other instrument of instruction which is a teacher's general guide may give far too little practice on certain bonds. (2) It may divide the practice given in ways that are apparently unjustifiable. (3) The teacher needs therefore to know how much practice it does give, where to supplement it, and what to omit. (4) The omissions, on grounds of apparent excess practice, should be made only after careful consideration of the third principle described above. (5) The amount of practice should always be considered in the light of its interest and appeal to the pupil's tendency to work with full power and zeal. Mere repetition of bonds when the learner does not care whether he is improving is rarely justifiable on any grounds. (6) Practice that is actually in excess is not a very grave defect if it is enjoyed and improves the pupil's attitude toward arithmetic. Not much time is lost; a hundred practices for each of a thousand bonds after mastery to 199 in 200 at 2 seconds will use up less than 60 hours, or 15 hours per year in grades 3 to 6. (7) By the proper division of practice among bonds, the arrangement of learning so that each bond helps the others, the adroit shifting of practice of a bond to each new type of situation requiring it to operate under changed conditions, and the elimination of excess practice where nothing substantial is gained, notable improvements over the past hit-and-miss customs may be expected. (8) Unless the material for practice is adequate, well balanced and sufficiently motivated, the teacher must keep close account of the learning of pupils. Otherwise disastrous underlearning of many bonds is almost sure to occur and retard the pupil's development. THE ORGANIZATION OF ABILITIES There is danger that the need of brevity and simplicity which has made us speak so often of a bond or an ability, and of the amount of practice it requires, may mislead the reader into thinking that these bonds and abilities are to be formed each by itself alone and kept so. They should rarely be formed so and never kept so. This we have indicated from time to time by references to the importance of forming a bond in the way in which it is to be used, to the action of bonds in changed situations, to facilitation of one bond by others, to the coöperation of abilities, and to their integration into a total arithmetical ability. As a matter of fact, only a small part of drill work in arithmetic should be the formation of isolated bonds. Even the very young pupil learning 5 and 3 are 8 should learn it with '5 and 5 = 10,' '5 and 2 = 7,' at the back of his mind, so to speak. Even so early, 5 + 3 = 8 should be part of an organized, coöperating system of bonds. Later 50 + 30 = 80 should become allied to it. Each bond should be considered, not simply as a separate tool to be put in a compartment until needed, but also as an improvement of one total tool or machine, arithmetical ability. There are differences of course. Knowledge of square root can be regarded somewhat as a separate tool to be sharpened, polished, and used by itself, whereas knowledge of the multiplication tables cannot. Yet even square root is probably best made more closely a part of the total ability, being taught as a special case of dividing where divisor is to be the same as quotient, the process being one of estimating and correcting. In general we do not wish the pupil to be a repository of separated abilities, each of which may operate only if you ask him the sort of questions which the teacher used to ask him, or otherwise indicate to him which particular arithmetical tool he is to use. Rather he is to be an effective organization of abilities, coöperating in useful ways to meet the quantitative problems life offers. He should not as a rule have to think in such fashion as: "Is this interest or discount? Is it simple interest or compound interest? What did I do in compound interest? How do I multiply by 2 percent?" The situation that calls up interest should also call up the kind of interest that is appropriate, and the technique of operating with percents should be so welded together with interest in his mind that the right coöperation will occur almost without supervision by him. As each new ability is acquired, then, we seek to have it take its place as an improvement of a thinking being, as a coöperative member of a total organization, as a soldier fighting together with others, as an element in an educated personality. Such an organization of bonds will not form itself any more than any one bond will create itself. If the elements of arithmetical ability are to act together as a total organized unified force they must be made to act together in the course of learning. What we wish to have work together we must put together and give practice in teamwork. We can do much to secure such coöperative action when and where and as it is needed by a very simple expedient; namely, to give practice with computation and problems such as life provides, instead of making up drills and problems merely to apply each fact or principle by itself. Though a pupil has solved scores of problems reading, "A triangle has a base of _a_ feet and an altitude of _b_ feet, what is its area?" he may still be practically helpless in finding the area of a triangular plot of ground; still more helpless in using the formula for a triangle which is one of two into which a trapezoid is divided. Though a pupil has learned to solve problems in trade discount, simple interest, compound interest, and bank discount one at a time, stated in a few set forms, he may be practically helpless before the actual series of problems confronting him in starting in business, and may take money out of the savings bank when he ought to borrow on a time loan, or delay payment on his bills when by paying cash he could save money as well as improve his standing with the wholesaler. Instead of making up problems to fit the abilities given by school instruction, we should preferably modify school instruction so that arithmetical abilities will be organized into an effective total ability to meet the problems that life will offer. Still more generally, _every bond formed should be formed with due consideration of every other bond that has been or will be formed; every ability should be practiced in the most effective possible relations with other abilities_. CHAPTER VII THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION OF BONDS The bonds to be formed having been chosen, the next step is to arrange for their most economical order of formation--to arrange to have each help the others as much as possible--to arrange for the maximum of facilitation and the minimum of inhibition. The principle is obvious enough and would probably be admitted in theory by any intelligent teacher, but in practice we are still wedded to conventional usages which arose long before the psychology of arithmetic was studied. For example, we inherit the convention of studying addition of integers thoroughly, and then subtraction, and then multiplication, and then division, and many of us follow it though nobody has ever given a proof that this is the best order for arithmetical learning. We inherit also the opposite convention of studying in a so-called "spiral" plan, a little addition, subtraction, multiplication, and division, and then some more of each, and then some more, and many of us follow this custom, with an unreasoned faith that changing about from one process to another is _per se_ helpful. Such conventions are very strong, illustrating our common tendency to cherish most those customs which we cannot justify! The reductions of denominate numbers ascending and descending were, until recently, in most courses of study, kept until grade 4 or grade 5 was reached, although this material is of far greater value for drills on the multiplication and division tables than the customary problems about apples, eggs, oranges, tablets, and penholders. By some historical accident or for good reasons the general treatment of denominate numbers was put late; by our naïve notions of order and system we felt that any use of denominate numbers before this time was heretical; we thus became blind to the advantages of quarts and pints for the tables of 2s; yards and feet for the tables of 3s; gallons and quarts for the tables of 4s; nickels and cents for the 5s; weeks and days for the 7s; pecks and quarts for the 8s; and square yards and square feet for the 9s. Problems like 5 yards = __ feet or 15 feet = __ yards have not only the advantages of brevity, clearness, practical use, real reference, and ready variation, but also the very great advantage that part of the data have to be _thought of_ in a useful way instead of _read off_ from the page. In life, when a person has twenty cents with which to buy tablets of a certain sort, he _thinks of_ the price in making his purchase, asking it of the clerk only in case he does not know it, and in planning his purchases beforehand he _thinks of_ prices as a rule. In spite of these and other advantages, not one textbook in ten up to 1900 made early use of these exercises with denominate numbers. So strong is mere use and wont. Besides these conventional customs, there has been, in those responsible for arithmetical instruction, an admiration for an arrangement of topics that is easy for a person, after he knows the subject, to use in thinking of its constituent parts and their relations. Such arrangements are often called 'logical' arrangements of subject matter, though they are often far from logical in any useful sense. Now the easiest order in which to think of a hierarchy of habits after you have formed them all may be an extremely difficult order in which to form them. The criticism of other orders as 'scrappy,' or 'unsystematic,' valid enough if the course of study is thought of as an object of contemplation, may be foolish if the course of study is regarded as a working instrument for furthering arithmetical learning. We must remember that all our systematizing and labeling is largely without meaning to the pupils. They cannot at any point appreciate the system as a progression from that point toward this and that, since they have no knowledge of the 'this or that.' They do not as a rule think of their work in grade 4 as an outcome of their work in grade 3 with extensions of a to a_1, and additions of b_2 and b_3 to b and b_1, and refinements of c and d by c_4 and d_5. They could give only the vaguest account of what they did in grade 3, much less of why it should have been done then. They are not much disturbed by a lack of so-called 'system' and 'logical' progression for the same reason that they are not much helped by their presence. What they need and can use is a _dynamically_ effective system or order, one that they can learn easily and retain long by, regardless of how it would look in a museum of arithmetical systems. Unless their actual arithmetical habits are usefully related it does no good to see the so-called logical relations; and if their habits are usefully related, it does not very much matter whether or not they do see these; finally, they can be brought to see them best by first acquiring the right habits in a dynamically effective order. DECREASING INTERFERENCE AND INCREASING FACILITATION Psychology offers no single, easy, royal road to discovering this dynamically best order. It can only survey the bonds, think what each demands as prerequisite and offers as future help, recommend certain orders for trial, and measure the efficiency of each order as a means of attaining the ends desired. The ingenious thought and careful experimentation of many able workers will be required for many years to come. Psychology can, however, even now, give solid constructive help in many instances, either by recommending orders that seem almost certainly better than those in vogue, or by proposing orders for trial which can be justified or rejected by crucial tests. Consider, for example, the situation, 'a column of one-place numbers to be added, whose sum is over 9,' and the response 'writing down the sum.' This bond is commonly firmly fixed before addition with two-place numbers is undertaken. As a result the pupil has fixed a habit that he has to break when he learns two-place addition. If _oral_ answers only are given with such single columns until two-place addition is well under way, the interference is avoided. In many courses of study the order of systematic formation of the multiplication table bonds is: 1 × 1, 2 × 1, etc., 1 × 2, 2 × 2, etc., 1 × 3, 2 × 3, etc., 1 × 9, 2 × 9, etc. This is probably wrong in two respects. There is abundant reason to believe that the × 5s should be learned first, since they are easier to learn than the 1s or the 2s, and give the idea of multiplying more emphatically and clearly. There is also abundant reason to believe that the 1 × 5, 1 × 2, 1 × 3, etc., should be put very late--after at least three or four tables are learned, since the question "What is 1 times 2?" (or 3 or 5) is unnecessary until we come to multiplication of two- and three-place numbers, seems a foolish question until then, and obscures the notion of multiplication if put early. Also the facts are best learned once for all as the habits "1 times _k_ is the same as _k_," and "_k_ times 1 is the same as _k_."[8] [8] The very early learning of 2 × 2, 2 × 3, 3 × 2, 2 × 4, 4 × 2, 3 × 3, and perhaps a few more multiplications is not considered here. It is advisable. The treatment of 0 × 0, 0 × 1, 1 × 0, etc., is not considered here. It is probably best to defer the '× 0' bonds until after all the others are formed and are being used in short multiplication, and to form them in close connection with their use in short multiplication. The '0 ×' bonds may well be deferred until they are needed in 'long' multiplication, 0 × 0 coming last of all. In another connection it was recommended that the divisions to 81 ÷ 9 be learned by selective thinking or reasoning from the multiplications. This determines the order of bonds so far as to place the formation of the division bonds soon after the learning of the multiplications. For other reasons it is well to make the proximity close. One of the arbitrary systematizations of the order of formation of bonds restricts operations at first to the numbers 1 to 10, then to numbers under 100, then to numbers under 1000, then to numbers under 10,000. Apart from the avoidance of unreal and pedantic problems in applied arithmetic to which work with large numbers in low grades does somewhat predispose a teacher, there is little merit in this restriction of the order of formation of bonds. Its demerits are many. For example, when the pupil is learning to 'carry' in addition he can be given better practice by soon including tasks with sums above 100, and can get a valuable sense of the general use of the process by being given a few examples with three- and four-place numbers to be added. The same holds for subtraction. Indeed, there is something to be said in favor of using six- or seven-place numbers in subtraction, enforcing the 'borrowing' process by having it done again and again in the same example, and putting it under control by having the decision between 'borrowing' and 'not borrowing' made again and again in the same example. When the multiplication tables are learned the most important use for them is not in tedious reviews or trivial problems with answers under 100, but in regular 'short' multiplication of two- and three- and even four-place numbers. Just as the addition combinations function mainly in the higher-decade modifications of them, so the multiplication combinations function chiefly in the cases where the bond has to operate while the added tasks of keeping one's place, adding what has been carried, writing down the right figure in the right place, and holding the right number for later addition, are also taken care of. It seems best to introduce such short multiplication as soon as the × 5s, × 2s, × 3s, and × 4s are learned and to put the × 6s, × 7s, and the rest to work in such short multiplication as soon as each is learned. Still surer is the need for four-, five-, and six-place numbers when two-place numbers are used in multiplying. When the process with a two-place multiplier is learned, multiplications by three-place numbers should soon follow. They are not more difficult then than later. On the contrary, if the pupil gets used to multiplying only as one does with two-place multipliers, he will suffer more by the resulting interference than he does from getting six- or seven-place answers whose meaning he cannot exactly realize. They teach the rationale and the manipulations of long multiplication with especial economy because the principles and the procedures are used two or three times over and the contrasts between the values which the partial products have in adding become three instead of one. The entire matter of long multiplication with integers and United States money should be treated as a teaching unit and the bonds formed in close organization, even though numbers as large as 900,000 are occasionally involved. The reason is not that it is more logical, or less scrappy, but that each of the bonds in question thus gets much help from, and gives much help to, the others. In sharp contrast to a topic like 'long multiplication' stands a topic like denominate numbers. It most certainly should not be treated as a large teaching unit, and all the bonds involved in adding, subtracting, multiplying, and dividing with all the ordinary sorts of measures should certainly not be formed in close sequence. The reductions ascending and descending for many of the measures should be taught as drills on the appropriate multiplication and division tables. The reduction of feet and inches to inches, yards and feet to yards, gallons and quarts to quarts, and the like are admirable exercises in connection with the (_a_ × _b_) + _c_ = .... problems,--the 'Bought 3 lbs. of sugar at 7 cents and 5 cents worth of matches' problems. The reductions of inches to feet and inches and the like are admirable exercises in the _d_ = (.... × _b_) + _c_ or 'making change' problem, which in its small-number forms is an excellent preparatory step for short division. They are also of great service in early work with fractions. The feet-mile, square-foot-square-inch, and other simple relations give a genuine and intelligible demand for multiplication with large numbers. Knowledge of the metric system for linear and square measure would perhaps, as an introduction to decimal fractions, more than save the time spent to learn it. It would even perhaps be worth while to invent a measure (call it the _twoqua_) midway between the quart and gallon and teach carrying in addition and borrowing in subtraction by teaching first the addition and subtraction of 'gallon, twoqua, quart, and pint' series! Many of the bonds which a system-made tradition huddled together uselessly in a chapter on denominate numbers should thus be formed as helpful preparations for and applications of other bonds all the way from the first to the eighth half-year of instruction in arithmetic. The bonds involved in the ability to respond correctly to the series:-- 5 = .... 2s and .... remainder 5 = .... 3s and .... remainder 88 = .... 9s and .... remainder should be formed before, not during, the training in short division. They are admirable at that point as practice on the division tables; are of practical service in the making-change problems of the small purchase and the like; and simplify the otherwise intricate task of keeping one's place, choosing the quotient figure, multiplying by it, subtracting and holding in mind the new number to be divided, which is composed half of the remainder and half of a figure in the written dividend. This change of order is a good illustration of the nearly general rule that "_When the practice or review required to perfect or hold certain bonds can, by an inexpensive modification, be turned into a useful preparation for new bonds, that modification should be made._" The bonds involved in the four operations with United States money should be formed in grades 3 and 4 along with or very soon after the corresponding bonds with three-place and four-place integers. This statement would have seemed preposterous to the pedagogues of fifty years ago. "United States money," they would have said, "is an application of decimals. How can it be learned until the essentials of decimal fractions are known? How will the child understand when multiplying $.75 by 3 that 3 times 5 cents is 1 dime and 5 cents, or that 3 times 70 cents is 2 dollars and 1 dime? Why perplex the young pupils with the difficulties of placing the decimal point? Why disturb the learning of the four operations with integers by adding at each step a second 'procedure with United States money'?" The case illustrates very well the error of the older oversystematic treatment of the order of topics and the still more important error of confusing the logic of proof with the psychology of learning. To prove that 3 × $.75 = $2.25 to the satisfaction of certain arithmeticians, you may need to know the theory of decimal fractions; but to do such multiplication all a child needs is to do just what he has been doing with integers and then "Put a $ before the answer to show that it means dollars and cents, and put a decimal point in the answer to show which figures mean dollars and which figures mean cents." And this is general. The ability to operate with integers plus the two habits of prefixing $ and separating dollars from cents in the result will enable him to operate with United States money. Consequently good practice came to use United States money not as a consequence of decimal fractions, learned by their aid, but as an introduction to decimal fractions which aids the pupil to learn them. So it has gradually pushed work with United States money further and further back, though somewhat timidly. We need not be timid. The pupil will have no difficulty in adding, subtracting, multiplying, and dividing with United States money--unless we create it by our explanations! If we simply form the two bonds described above and show by proper verification that the procedure always gives the right answer, the early teaching of the four operations with United States money will in fact actually show a learning profit! It will save more time in the work with integers than was spent in teaching it! For, in the first place, it will help to make work with four-place and five-place numbers more intelligible and vital. A pupil can understand $16.75 or $28.79 more easily than 1675 or 2879. The former may be the prices of a suit or sewing machine or bicycle. In the second place, it permits the use of a large stock of genuine problems about spending, saving, sharing, and the like with advertisements and catalogues and school enterprises. In the third place, it permits the use of common-sense checks. A boy may find one fourth of 3000 as 7050 or 75 and not be disturbed, but he will much more easily realize that one fourth of $30.00 is not over $70 or less than $1. Even the decimal point of which we used to be so afraid may actually help the eye to keep its place in adding. INTEREST So far, the illustrations of improvements in the order of bonds so as to get less interference and more facilitation than the customary orders secure have sought chiefly to improve the mechanical organization of the bonds. Any gain in interest which the changes described effected would be largely due to the greater achievement itself. Dewey and others have emphasized a very different principle of improving the order of formation of bonds--the principle of determination of the bonds to be formed by some vital, engaging problem which arouses interest enough to lighten the labor and which goes beyond or even against cut-and-dried plans for sequences in order to get effective problems. For example, the work of the first month in grade 2B might sacrifice facilitations of the mechanical sort in order to put arithmetic to use in deciding what dimensions a rabbit's cage should have to give him 12 square feet of floor space, how much bread he should have per meal to get 6 ounces a day, how long a ten-cent loaf would last, how many loaves should be bought per week, how much it costs to feed the rabbit, how much he has gained in weight since he was brought to the school, and so on. Such sacrifices of the optimal order if interest were equal, in order to get greater interest or a healthier interest, are justifiable. Vital problems as nuclei around which to organize arithmetical learning are of prime importance. It is even safe probably to insist that some genuine problem-situation requiring a new process, such as addition with carrying, multiplication by two-place numbers, or division with decimals, be provided in every case as a part of the introduction to that process. The sacrifice should not be too great, however; the search for vital problems that fit an economical order of subject matter is as much needed as the amendment of that order to fit known interests; and the assurance that a problem helps the pupil to learn arithmetic is as important as the assurance that arithmetic is used to help the pupil solve his personal problems. Much ingenuity and experimentation will be required to find the order that is satisfactory in both quality and quantity of interest or motive and helpfulness of the bonds one to another. The difficulty of organizing arithmetic around attractive problems is much increased by the fact of class instruction. For any one pupil vital, personal problems or projects could be found to provide for many arithmetical abilities; and any necessary knowledge and technique which these projects did not develop could be somehow fitted in along with them. But thirty children, half boys and half girls, varying by five years in age, coming from different homes, with different native capacities, will not, in September, 1920, unanimously feel a vital need to solve any one problem, and then conveniently feel another on, say, October 15! In the mechanical laws of learning children are much alike, and the gain we may hope to make from reducing inhibitions and increasing facilitations is, for ordinary class-teaching, probably greater than that to be made from the discovery of attractive central problems. We should, however, get as much as possible of both. GENERAL PRINCIPLES The reader may by now feel rather helpless before the problem of the arrangement of arithmetical subject matter. "Sometimes you complete a topic, sometimes you take it piecemeal months or years apart, often you make queer twists and shifts to get a strategic advantage over the enemy," he may think, "but are there no guiding principles, no general rules?" There is only one that is absolutely general, to _take the order that works best for arithmetical learning_. There are particular rules, but there are so many and they are so limited by an 'other things being equal' clause, that probably a general eagerness to think out the _pros_ and _cons_ for any given proposal is better than a stiff attempt to adhere to these rules. I will state and illustrate some of them, and let the reader judge. _Other things being equal, one new sort of bonds should not be started until the previous set is fairly established, and two different sets should not be started at once._ Thus, multiplication of two- and three-place numbers by 2, 3, 4, and 5 will first use numbers such that no carrying is required, and no zero difficulties are encountered, then introduce carrying, then introduce multiplicands like 206 and 320. If other things were equal, the carrying would be split into two steps--first drills with (4 × 6) + 2, (3 × 7) + 3, (5 × 4) + 1, and the like, and second the actual use of these habits in the multiplication. The objection to this separation of the double habit is that the first part of it in isolation is too artificial--that it may be better to suffer the extra difficulty of forming the two together than to teach so rarely used habits as the (_a_ × _b_) + _c_ series. Experimental tests are needed to decide this point. _Other things being equal, bonds should be formed in such order that none will have to be broken later._ For example, there is a strong argument for teaching long division first, or very early, with remainders, letting the case of zero remainder come in as one of many. If the pupils have been familiarized with the remainder notion by the drills recommended as preparation for short division,[9] the use of remainders in long division will offer little difficulty. The exclusive use of examples without remainders may form the habit of not being exact in computation, of trusting to 'coming out even' as a sole check, and even of writing down a number to fit the final number to be divided instead of obtaining it by honest multiplication. [9] See page 76. For similar reasons additions with 2 and 3 as well as 1 to be 'carried' have much to recommend them in the very first stages of column addition with carrying. There is here the added advantage that a pupil will be more likely to remember to carry if he has to think _what_ to carry. The present common practice of using small numbers for ease in the addition itself teaches many children to think of carrying as adding one. _Other things being equal, arrange to have variety._ Thus it is probably, though not surely, wise to interrupt the monotony of learning the multiplication and division tables, by teaching the fundamentals of 'short' multiplication and perhaps of division after the 5s, 2s, 3s, and 4s are learned. This makes a break of several weeks. The facts for the 6s, 7s, 8s, and 9s can then be put to varied use as fast as learned. It is almost certainly wise to interrupt the first half-year's work with addition and subtraction, by teaching 2 × 2, 2 × 3, 3 × 2, 2 × 4, 4 × 2, 2 × 5, later by 2 × 10, 3 × 10, 4 × 10, 5 × 10, later by 1/2 + 1/2, 1-1/2 + 1/2, 1/2 of 2, 1/2 of 4, 1/2 of 6, and at some time by certain profitable exercises wherein a pupil tells all he knows about certain numbers which may be made nuclei of important facts (say, 5, 8, 10, 12, 15, and 20). _Other things being equal, use objective aids to verify an arithmetical process or inference after it is made, as well as to provoke it._ It is well at times to let pupils do everything that they can with relations abstractly conceived, testing their results by objective counting, measuring, adding, and the like. For example, an early step in adding should be to show three things, put them under a book, show two more, put these under the book, and then ask how many there are under the book, letting the objective counting come later as the test of the correctness of the addition. _Other things being equal, reserve all explanations of why a process must be right until the pupils can use the process accurately, and have verified the fact that it is right._ Except for the very gifted pupils, the ordinary preliminary deductive explanations of what must be done are probably useless as means of teaching the pupils what to do. They use up much time and are of so little permanent effect that, as we have seen, the very arithmeticians who advocate making them, admit that after a pupil has mastered the process he may be allowed to forget the reasons for it. I am not sure that the deductive proofs of why we place the decimal point as we do in division by a decimal, or invert and multiply in dividing by a fraction, and the like, are worth teaching at all. If they are to be taught at all, the time to teach them is (except for the very gifted) after the pupil has mastered the process and has confidence in it. He then at least knows what process he is to prove is right, and that it is right, and has had some chance of seeing _why_ it is right from his experience with it. One more principle may be mentioned without illustration. _Arrange the order of bonds with due regard for the aims of the other studies of the curriculum and the practical needs of the pupil outside of school._ Arithmetic is not a book or a closed system of exercises. It is the quantitative work of the pupils in the elementary school. No narrower view of it is adequate. CHAPTER VIII THE DISTRIBUTION OF PRACTICE THE PROBLEM The same amount of practice may be distributed in various ways. Figures 7 to 10, for example, show 200 practices with division by a fraction distributed over three and a half years of 10 months in four different ways. In Fig. 7, practice is somewhat equally distributed over the whole period. In Fig. 8 the practice is distributed at haphazard. In Fig. 9 there is a first main learning period, a review after about ten weeks, a review at the beginning of the seventh grade, another review at the beginning of the eighth grade, and some casual practice rather at random. In Fig. 10 there is a main learning period, with reviews diminishing in length and separated by wider and wider intervals, with occasional practice thereafter to keep the ability alive and healthy. Plans I and II are obviously inferior to Plans III and IV; and Plan IV gives promise of being more effective than Plan III, since there seems danger that the pupil working by Plan III might in the ten weeks lose too much of what he had gained in the initial practice, and so again in the next ten weeks. It is not wise, however, to try now to make close decisions in the case of practice with division by a fraction; or to determine what the best distribution of practice is for that or any other ability to be improved. The facts of psychology are as yet not adequate for very close decisions, nor are the types of distribution of practice that are best adapted to different abilities even approximately worked out. [Illustration: FIG. 7.--Plan I. 200 practices distributed somewhat evenly over 3-1/2 years of 10 months. In Figs. 7, 8, 9, and 10, each tenth of an inch along the base line represents one month. Each hundredth of a square inch represents four practices, a little square 1/20 of an inch wide and 1/20 inch high representing one practice.] [Illustration: FIG. 8.--Plan II. 200 practices distributed haphazard over 3-1/2 years of 10 months.] [Illustration: FIG. 9.--Plan III. A learning period, three reviews, and incidental practice.] [Illustration: FIG. 10.--Plan IV. A learning period with reviews of decreasing length at increasing intervals.] SAMPLE DISTRIBUTIONS Let us rather examine some actual cases of distribution of practice found in school work and consider, not the attainment of the best possible distribution, but simply the avoidance of gross blunders and the attainment of reasonable, defensible procedures in this regard. Figures 11 to 18 show the distribution of examples in multiplication with multipliers of various sorts. _X_ stands for any digit except zero. _O_ stands for 0. _XXO_ thus means a multiplier like 350 or 270 or 160; _XOX_ means multipliers like 407, 905, or 206; _XX_ means multipliers like 25, 17, 38. Each of these diagrams covers approximately 3-1/2 years of school work, or from about the middle of grade 3 to the end of grade 6. They are made from counts of four textbooks (A, B, C, and D), the count being taken for each successive 8 pages.[10] Each tenth of an inch along the base line equals 8 pages of the text in question. Each .01 sq. in. equals one example. The books, it will be observed, differ in the amount of practice given, as well as in the way in which it is distributed. [10] At the end of a volume or part, the count may be from as few as 5 or as many as 12 pages. These distributions are worthy of careful study; we shall note only a few salient facts about them here. Of the distributions of multiplications with multipliers of the _XX_ type, that of book D (Fig. 14) is perhaps the best. A (Fig. 11) has too much of the practice too late; B (Fig. 12) gives too little practice in the first learning; C (Fig. 13) gives too much in the first learning and in grade 6. Among the distributions of multiplication with multipliers of the _XOX_ type, that of book D (Fig. 18) is again probably the best. A, B, and C (Figs. 15, 16, and 17) have too much practice early and too long intervals between reviews. Book C (Fig. 17) by a careless oversight has one case of this very difficult process, without any explanation, weeks before the process is taught! [Illustration: FIG. 11.--Distribution of practise with multipliers of the _XX_ type in the first two books of the three-book text A.] [Illustration: FIG. 12.--Same as Fig. 11, but for text B. Following this period come certain pages of computation to be used by the teacher at her discretion, containing 24 _XX_ multiplications.] [Illustration: FIG. 13.--Same as Fig. 11, but for text C.] [Illustration: FIG. 14.--Same as Fig. 11, but for text D.] [Illustration: FIG. 15.--Distribution of practice with multipliers of the _XOX_ type in the first two books of the three-book text A.] [Illustration: FIG. 16.--Same as Fig. 15, but for text B. Following this period come certain pages of computation to be used by the teacher at her discretion, containing 17 _XOX_ multiplications.] [Illustration: FIG. 17.--Same as Fig. 16, but for text C.] [Illustration: FIG. 18.--Same as Fig. 16, but for text D.] Figures 19, 20, 21, 22, and 23 all concern the first two books of the three-book text E. Figure 19 shows the distribution of practice on 5 × 5 in the first two books of text E. The plan is the same as in Figs. 11 to 18, except that each tenth of an inch along the base line represents ten pages. Figure 20 shows the distribution of practice on 7 × 7; Fig. 21 shows it for 6 × 7 and 7 × 6 together. In Figs. 20 and 21 also, 0.1 inch along the base line equals ten pages. Figures 22 and 23 show the distribution of practice on the divisions of 72, 73, 74, 75, 76, 77, 78, and 79 by either 8 or 9, and on the divisions of 81, 82 ... 89 by 9. Each tenth of an inch along the base line represents ten pages here also. Figures 19 to 23 show no consistent plan for distributing practice. With 5 × 5 (Fig. 19) the amount of practice increases from the first treatment in grade 3 to the end of grade 6, so that the distribution would be better if the pupil began at the end and went backward! With 7 × 7 (Fig. 20) the practice is distributed rather evenly and in small doses. With 6 × 7 and 7 × 6 (Fig. 21) much of it is in very large doses. With the divisions (Figs. 22 and 23) the practice is distributed more suitably, though in Fig. 23 there is too much of it given at one time in the middle of the period. [Illustration: FIG. 19.--Distribution of practice with 5 × 5 in the first two books of the three-book text E.] [Illustration: FIG. 20.--Distribution of practice with 7 × 7 in the first two books of text E.] [Illustration: FIG. 21.--Distribution of practice with 6 × 7 or 7 × 6 in the first two books of text E.] [Illustration: FIG. 22.--Distribution of practice with 72, 73 ... 79 ÷ 8 or 9 in the first two books of text E.] [Illustration: FIG. 23.--Distribution of practice with 81, 82 ... 89 ÷ 9 in the first two books of text E.] POSSIBLE IMPROVEMENTS Even if we knew what the best distribution of practice was for each ability of the many to be inculcated by arithmetical instruction, we could perhaps not provide it for all of them. For, in the first place, the allotments for some of them might interfere with those for others. In the second place, there are many other considerations of importance in the ordering of topics besides giving the optimal distribution of practice to each ability. Such are considerations of interest, of welding separate abilities into an integrated total ability, and of the limitations due to the school schedule with its Saturdays, Sundays, holidays, and vacations. Improvement can, however, be made over present practice in many respects. A scientific examination of the teaching of almost any class for a year, or of many of our standard instruments of instruction, will reveal opportunities for improving the distribution of practice with no sacrifice of interest, and with an actual gain in integrated functioning arithmetical power. In particular it will reveal cases where an ability is given practice and then, never being used again, left to die of inactivity. It will reveal cases where an ability is given practice and then left so long without practice that the first effect is nearly lost. There will be cases where practice is given and reviews are given, but all in such isolation from everything else in arithmetic that the ability, though existent, does not become a part of the pupil's general working equipment. There will be cases where more practice is given in the late than the earlier periods for no apparent extrinsic advantage; and cases where the practice is put where it is for no reason that is observable save that the teacher or author in question has decided to have some drill work at that time! Each ability has its peculiar needs in this matter, and no set rules are at present of much value. It will be enough for the present if we are aroused to the problem of distribution, avoid obvious follies like those just noted, and exercise what ingenuity we have. CHAPTER IX THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND GENERAL NOTIONS IN ARITHMETIC[11] [11] Certain paragraphs in this and the following chapter are taken from the author's _Educational Psychology_, with slight modifications. RESPONSES TO ELEMENTS AND CLASSES The plate which you see, the egg before you at the breakfast table, and this page are concrete things, but whiteness, whether of plate, egg, or paper, is, we say, an abstract quality. To be able to think of whiteness irrespective of any concrete white object is to be able to have an abstract idea or notion of white; to be able to respond to whiteness, irrespective of whether it is a part of china, eggshell, paper or whatever object, is to be able to respond to the abstract element of whiteness. Learning arithmetic involves the formation of very many such ideas, the acquisition of very many such powers of response to elements regardless of the gross total situations in which they appear. To appreciate the fiveness of five boys, five pencils, five inches, five rings of a bell; to understand the division into eight equal parts of 40 cents, 32 feet, 64 minutes, or 16 ones; to respond correctly to the fraction relation in 2/3, 5/6, 3/4, 7/12, 1/8, or any other; to be sensitive to the common element of 9 = 3 × 3, 16 = 4 × 4, 625 = 25 × 25, .04 = .2 × .2, 1/4 = 1/2 × 1/2,--these are obvious illustrations. All the numbers which the pupil learns to understand and manipulate are in fact abstractions; all the operations are abstractions; percent, discount, interest, height, length, area, volume, are abstractions; sum, difference, product, quotient, remainder, average, are facts that concern elements or aspects which may appear with countless different concrete surroundings or concomitants. Towser is a particular dog; your house lot on Elm Street is a particular rectangle; Mr. and Mrs. I.S. Peterson and their daughter Louise are a particular family of three. In contrast to these particulars, we mean by a dog, a rectangle, and a family of three, _any_ specimens of these classes of facts. The idea of a dog, of rectangles in general, of any family of three is a general notion, a concept or idea of a class or species. The ability to respond to any dog, or rectangle, or family of three, regardless of which particular one it may be, is the general notion in action. Learning arithmetic involves the formation of very many such general notions, such powers of response to any member of a certain class. Thus a hundred different sized lots may all be responded to as rectangles; 9/18, 12/27, 15/24, and 27/36 may all be responded to as members of the class, 'both members divisible by 3.' The same fact may be responded to in different ways according to the class to which it is assigned. Thus 4 in 3/4, 4/5, 45, 54, and 405 is classed respectively as 'a certain sized part of unity,' 'a certain number of parts of the size shown by the 5,' 'a certain number of tens,' 'a certain number of ones,' and 'a certain number of hundreds.' Each abstract quality may become the basis of a class of facts. So fourness as a quality corresponds to the class 'things four in number or size'; the fractional quality or relation corresponds to the class 'fractions.' The bonds formed with classes of facts and with elements or features by which one whole class of facts is distinguished from another, are in fact, a chief concern of arithmetical learning.[12] [12] It should be noted that just as concretes give rise to abstractions, so these in turn give rise to still more abstract abstractions. Thus fourness, fiveness, twentyness, and the like give rise to 'integral-number-ness.' Similarly just as individuals are grouped into general classes, so classes are grouped into still more general classes. Half, quarter, sixth, and tenth are general notions, but 'one ...th' is more general; and 'fraction' is still more general. FACILITATING THE ANALYSIS OF ELEMENTS Abstractions and generalizations then depend upon analysis and upon bonds formed with more or less subtle elements rather than with gross total concrete situations. The process involved is most easily understood by considering the means employed to facilitate it. The first of these is having the learner respond to the total situations containing the element in question with the attitude of piecemeal examination, and with attentiveness to one element after another, especially to so near an approximation to the element in question as he can already select for attentive examination. This attentiveness to one element after another serves to emphasize whatever appropriate minor bonds from the element in question the learner already possesses. Thus, in teaching children to respond to the 'fiveness' of various collections, we show five boys or five girls or five pencils, and say, "See how many boys are standing up. Is Jack the only boy that is standing here? Are there more than two boys standing? Name the boys while I point at them and count them. (Jack) is one, and (Fred) is one more, and (Henry) is one more. Jack and Fred make (two) boys. Jack and Fred and Henry make (three) boys." (And so on with the attentive counting.) The mental set or attitude is directed toward favoring the partial and predominant activity of 'how-many-ness' as far as may be; and the useful bonds that the 'fiveness,' the 'one and one and one and one and one-ness,' already have, are emphasized as far as may be. The second of the means used to facilitate analysis is having the learner respond to many situations each containing the element in question (call it A), but with varying concomitants (call these V. C.) his response being so directed as, so far as may be, to separate each total response into an element bound to the A and an element bound to the V. C. Thus the child is led to associate the responses--'Five boys,' 'Five girls,' 'Five pencils,' 'Five inches,' 'Five feet,' 'Five books,' 'He walked five steps,' 'I hit my desk five times,' and the like--each with its appropriate situation. The 'Five' element of the response is thus bound over and over again to the 'fiveness' element of the situation, the mental set being 'How many?,' but is bound only once to any one of the concomitants. These concomitants are also such as have preferred minor bonds of their own (the sight of a row of boys _per se_ tends strongly to call up the 'Boys' element of the response). The other elements of the responses (boys, girls, pencils, etc.) have each only a slight connection with the 'fiveness' element of the situations. These slight connections also in large part[13] counteract each other, leaving the field clear for whatever uninhibited bond the 'fiveness' has. [13] They may, of course, also result in a fusion or an alternation of responses, but only rarely. The third means used to facilitate analysis is having the learner respond to situations which, pair by pair, present the element in a certain context and present that same context with _the opposite of the element in question_, or with something at least very unlike the element. Thus, a child who is being taught to respond to 'one fifth' is not only led to respond to 'one fifth of a cake,' 'one fifth of a pie,' 'one fifth of an apple,' 'one fifth of ten inches,' 'one fifth of an army of twenty soldiers,' and the like; he is also led to respond to each of these _in contrast with_ 'five cakes,' 'five pies,' 'five apples,' 'five times ten inches,' 'five armies of twenty soldiers.' Similarly the 'place values' of tenths, hundredths, and the rest are taught by contrast with the tens, hundreds, and thousands. These means utilize the laws of connection-forming to disengage a response element from gross total responses and attach it to some situation element. The forces of use, disuse, satisfaction, and discomfort are so maneuvered that an element which never exists by itself in nature can influence man almost as if it did so exist, bonds being formed with it that act almost or quite irrespective of the gross total situation in which it inheres. What happens can be most conveniently put in a general statement by using symbols. Denote by _a_ + _b_, _a_ + _g_, _a_ + _l_, _a_ + _q_, _a_ + _v_, and _a_ + _B_ certain situations alike in the element _a_ and different in all else. Suppose that, by original nature or training, a child responds to these situations respectively by r_{1} + r_{2}, r_{1} + r_{7}, r_{1} + r_{12}, r_{1} + r_{17}, r_{1} + r_{22}, r_{1} + r_{27}. Suppose that man's neurones are capable of such action that r_{1}, r_{2}, r_{7}, r_{12}, r_{22}, and r_{27}, can each be made singly. Case I. Varying Concomitants Suppose that _a_ + _b_, _a_ + _g_, _a_ + _l_, etc., occur once each. We have _a_ + _b_ responded to by r_{1} + r_{2}, _a_ + _g_ " " r_{1} + r_{7}, _a_ + _l_ " " r_{1} + r_{12}, _a_ + _q_ " " r_{1} + r_{17}, _a_ + _v_ " " r_{1} + r_{22}, and _a_ + _B_ " " r_{1} + r_{27}, as shown in Scheme I. Scheme I _a_ _b_ _g_ _l_ _q_ _v_ _B_ r_{1} 6 1 1 1 1 1 1 r_{2} 1 1 r_{7} 1 1 r_{12} 1 1 r_{17} 1 1 r_{22} 1 1 r_{27} 1 1 _a_ is thus responded to by r_{1} (that is, connected with r_{1}) each time, or six in all, but only once each with _b_, _g_, _l_, _q_, _v_, and _B_. _b_, _g_, _l_, _q_, _v_, and _B_ are connected once each with r_{1} and once respectively with r_{2}, r_{7}, r_{12}, etc. The bond from _a_ to r_{1}, has had six times as much exercise as the bond from _a_ to r_{2}, or from _a_ to r_{7}, etc. In any new gross situation, _a_ 0, _a_ will be more predominant in determining response than it would otherwise have been; and r_{1} will be more likely to be made than r_{2}, r_{7}, r_{12}, etc., the other previous associates in the response to a situation containing _a_. That is, the bond from the element _a_ to the response r_{1} has been notably strengthened. Case II. Contrasting Concomitants Now suppose that _b_ and _g_ are very dissimilar elements (_e.g._, white and black), that _l_ and _q_ are very dissimilar (_e.g._, long and short), and that _v_ and _B_ are also very dissimilar. To be very dissimilar means to be responded to very differently, so that r_{7}, the response to _g_, will be very unlike r_{2}, the response to _b_. So r_{7} may be thought of as r_{not 2} or r_{-2}. In the same way r_{12} may be thought of as r_{not 12} or r_{-12}, and r_{27} may be called r_{not 22} or r_{-22}. Then, if the situations _a_ _b_, _a _g_, _a _l_, _a _q_, _a _v_, and _a_ _B_ are responded to, each once, we have:-- _a_ + _b_ responded to by r_{1} + r_{2}, _a_ + _g_ " " r_{1} + r_{not 2}, _a_ + _l_ " " r_{1} + r_{12}, _a_ + _q_ " " r_{1} + r_{not 12}, _a_ + _v_ " " r_{1} + r_{22}, and _a_ + _B_ " " r_{1} + r_{not 22}, as shown in Scheme II. Scheme II _a_ _b_ _g_ _l_ _q_ _v_ _B_ (opp. of _b_) (opp. of _l_) (opp. of _v_) r_{1} 6 1 1 1 1 1 1 r_{not 1} r_{2} 1 1 r_{not 2} 1 1 r_{12} 1 1 r_{not 12} 1 1 r_{22} 1 1 r_{not 22} 1 1 r_{1} is connected to _a_ by 6 repetitions. r_{2} and r_{not 2} are each connected to _a_ by 1 repetition, but since they interfere, canceling each other so to speak, the net result is for _a_ to have zero tendency to call up r_{2} or r_{not 2}. r_{12} and r_{not 12} are each connected to _a_ by 1 repetition, but they interfere with or cancel each other with the net result that _a_ has zero tendency to call up r_{12} or r_{not 12}. So with r_{22} and r_{not 22}. Here then the net result of the six connections of _a_ _b_, _a_ _g_, _a_ _l_, _a_ _q_, _a_ _v_, and _a_ _B_ is to connect _a_ with _r_, and with nothing else. Case III. Contrasting Concomitants and Contrasting Element Suppose now that the facts are as in Case II, but with the addition of six experiences where a certain element which is the opposite of, or very dissimilar to, _a_ is connected with the response r_{not 1}, or r_{-1} which is opposite to, or very dissimilar to r_{1}. Call this opposite of _a_, - _a_. That is, we have not only _a_ + _b_ responded to by r_{1} + r_{2}, _a_ + _g_ " " r_{1} + r_{not 2}, _a_ + _l_ " " r_{1} + r_{12}, _a_ + _q_ " " r_{1} + r_{not 12}, _a_ + _v_ " " r_{1} + r_{22}, and _a_ + _B_ " " r_{1} + r_{not 22}, but also - _a_ + _b_ responded to by r_{not 1} + r_{2}, - _a_ + _g_ " " r_{not 1} + r_{not 2}, - _a_ + _l_ " " r_{not 1} + r_{12}, - _a_ + _q_ " " r_{not 1} + r_{not 12}, - _a_ + _v_ " " r_{not 1} + r_{22}, and - _a_ + _B_ " " r_{not 1} + r_{not 22}, as shown in Scheme III. Scheme III _a_ opp. _b_ _g_ _l_ _q_ _v_ _B_ of _a_ (opp. of _b_) (opp. of _l_) (opp. of _v_) r_{1} 6 1 1 1 1 1 1 r_{not 1} 6 1 1 1 1 1 1 r_{2} 1 1 2 r_{not 2} 1 1 2 r_{12} 1 1 2 r_{not 12} 1 1 2 r_{22} 1 1 2 r_{not 22} 1 1 2 In this series of twelve experiences _a_ connects with r_{1} six times and the opposite of _a_ connects with r_{not 1} six times. _a_ connects equally often with three pairs of mutual destructives r_{2} and r_{not 2}, r_{12} and r_{not 12}, r_{22} and r_{not 22}, and so has zero tendency to call them up. - _a_ has also zero tendency to call up any of these responses except its opposite, r_{not 1}. _b_, _g_, _l_, _q_, _v_, and _B_ are made to connect equally often with r_{1} and r_{not 1}. So, of these elements, _a_ is the only one left with a tendency to call up r_{1}. Thus, by the mere action of frequency of connection, r_{1} is connected with _a_; the bonds from _a_ to anything except r_{1} are being counteracted, and the slight bonds from anything except _a_ to r_{1} are being counteracted. The element _a_ becomes predominant in situations containing it; and its bond toward r_{1} becomes relatively enormously strengthened and freed from competition. These three processes occur in a similar, but more complicated, form if the situations _a_ + _b_, _a_ + _g_, etc., are replaced by _a_ + _b_ + _c_ + _d_ + _e_ + _f_, _a_ + _g_ + _h_ + _i_ + _j_ + _k_, etc., and the responses r_{1} + r_{2}, r_{1} + r_{7}, r_{1} + r_{12}, etc., are replaced by r_{1} + r_{2} + r_{3} + r_{4} + r_{5} + r_{6}, r_{1} + r_{7} + r_{8} + r_{9} + r_{10} + r_{11}, etc.--_provided the_ r_{1}, r_{2}, r_{3}, r_{4}, etc., _can be made singly_. In so far as any one of the responses is necessarily co-active with any one of the others (so that, for example, r_{13} always brings r_{26} with it and _vice versa_), the exact relations of the numbers recorded in schemes like schemes I, II, and III on pages 172 to 174 will change; but, unless r_{1} has such an inevitable co-actor, the general results of schemes I, II, and III will hold good. If r_{1} does have such an inseparable co-actor, say r_{2}, then, of course, _a_ can never acquire bonds with r_{1} alone, but everywhere that r_{1} or r_{2} appears in the preceding schemes the other element must appear also. r_{1} r_{2} would then have to be used as a unit in analysis. The '_a_ + _b_,' '_a_ + _g_,' '_a_ + _l_,' ... '_a_ + _B_' situations may occur unequal numbers of times, altering the exact numerical relations of the connections formed and presented in schemes I, II, and III; but the process in general remains the same. So much for the effect of use and disuse in attaching appropriate response elements to certain subtle elements of situations. There are three main series of effects of satisfaction and discomfort. They serve, first, to emphasize, from the start, the desired bonds leading to the responses r_{1} + r_{2}, r_{1} + r_{7}, etc., to the total situations, and to weed out the undesirable ones. They also act to emphasize, in such comparisons and contrasts as have been described, every action of the bond from _a_ to r_{1}; and to eliminate every tendency of _a_ to connect with aught save r_{1}, and of aught save _a_ to connect with r_{1}. Their third service is to strengthen the bonds produced of appropriate responses to _a_ wherever it occurs, whether or not any formal comparisons and contrasts take place. The process of learning to respond to the difference of pitch in tones from whatever instrument, to the 'square-root-ness' of whatever number, to triangularity in whatever size or combination of lines, to equality of whatever pairs, or to honesty in whatever person or instance, is thus a consequence of associative learning, requiring no other forces than those of use, disuse, satisfaction, and discomfort. "What happens in such cases is that the response, by being connected with many situations alike in the presence of the element in question and different in other respects, is bound firmly to that element and loosely to each of its concomitants. Conversely any element is bound firmly to any one response that is made to all situations containing it and very, very loosely to each of those responses that are made to only a few of the situations containing it. The element of triangularity, for example, is bound firmly to the response of saying or thinking 'triangle' but only very loosely to the response of saying or thinking white, red, blue, large, small, iron, steel, wood, paper, and the like. A situation thus acquires bonds not only with some response to it as a gross total, but also with responses to any of its elements that have appeared in any other gross totals. Appropriate response to an element regardless of its concomitants is a necessary consequence of the laws of exercise and effect if an animal learns to make that response to the gross total situations that contain the element and not to make it to those that do not. Such prepotent determination of the response by one or another element of the situation is no transcendental mystery, but, given the circumstances, a general rule of all learning." Such are at bottom only extreme cases of the same learning as a cat exhibits that depresses a platform in a certain box whether it faces north or south, whether the temperature is 50 or 80 degrees, whether one or two persons are in sight, whether she is exceedingly or moderately hungry, whether fish or milk is outside the box. All learning is analytic, representing the activity of elements within a total situation. In man, by virtue of certain instincts and the course of his training, very subtle elements of situations can so operate. * * * * * Learning by analysis does not often proceed in the carefully organized way represented by the most ingenious marshaling of comparing and contrasting activities. The associations with gross totals, whereby in the end an element is elevated to independent power to determine response, may come in a haphazard order over a long interval of time. Thus a gifted three-year-old boy will have the response element of 'saying or thinking _two_,' bound to the 'two-ness' element of very many situations in connection with the 'how-many' mental set; and he will have made this analysis without any formal, systematic training. An imperfect and inadequate analysis already made is indeed usually the starting point for whatever systematic abstraction the schools direct. Thus the kindergarten exercises in analyzing out number, color, size, and shape commonly assume that 'one-ness' _versus_ 'more-than-one-ness,' black and white, big and little, round and not round are, at least vaguely, active as elements responded to in some independence of their contexts. Moreover, the tests of actual trial and success in further undirected exercises usually coöperate to confirm and extend and refine what the systematic drills have given. Thus the ordinary child in school is left, by the drills on decimal notation, with only imperfect power of response to the 'place-values.' He continues to learn to respond properly to them by finding that 4 × 40 = 160, 4 × 400 = 1600, 800 - 80 = 720, 800 - 8 = 792, 800-800 = 0, 42 × 48 = 2016, 24 × 48 = 1152, and the like, are satisfying; while 4 × 40 = 16, 23 × 48 = 832, 800 - 8 = 0, and the like, are not. The process of analysis is the same in such casual, unsystematized formation of connections with elements as in the deliberately managed, piecemeal inspection, comparison, and contrast described above. SYSTEMATIC AND OPPORTUNISTIC STIMULI TO ANALYSIS The arrangement of a pupil's experiences so as to direct his attention to an element, vary its concomitants instructively, stimulate comparison, and throw the element into relief by contrast may be by fixed, formal, systematic exercises. Or it may be by much less formal exercises, spread over a longer time, and done more or less incidentally in other connections. We may call these two extremes the 'systematic' and 'opportunistic,' since the chief feature of the former is that it systematically provides experiences designed to build up the power of correct response to the element, whereas the chief feature of the latter is that it uses especially such opportunities as occur by reason of the pupil's activities and interests. Each method has its advantages and disadvantages. The systematic method chooses experiences that are specially designed to stimulate the analysis; it provides these at a certain fixed time so that they may work together; it can then and there test the pupils to ascertain whether they really have the power to respond to the element or aspect or feature in question. Its disadvantages are, first, that many of the pupils will feel no need for and attach no interest or motive to these formal exercises; second, that some of the pupils may memorize the answers as a verbal task instead of acquiring insight into the facts; third, that the ability to respond to the element may remain restricted to the special cases devised for the systematic training, and not be available for the genuine uses of arithmetic. The opportunistic method is strong just where the systematic is weak. Since it seizes upon opportunities created by the pupil's abilities and interests, it has the attitude of interest more often. Since it builds up the experiences less formally and over a wider space of time, the pupils are less likely to learn verbal answers. Since its material comes more from the genuine uses of life, the power acquired is more likely to be applicable to life. Its disadvantage is that it is harder to manage. More thought and experimentation are required to find the best experiences; greater care is required to keep track of the development of an abstraction which is taught not in two days, but over two months; and one may forget to test the pupils at the end. In so far as the textbook and teacher are able to overcome these disadvantages by ingenuity and care, the opportunistic method is better. ADAPTATIONS TO ELEMENTARY SCHOOL PUPILS We may expect much improvement in the formation of abstract and general ideas in arithmetic from the application of three principles in addition to those already described. They are: (1) Provide enough actual experiences before asking the pupil to understand and use an abstract or general idea. (2) Develop such ideas gradually, not attempting to give complete and perfect ideas all at once. (3) Develop such ideas so far as possible from experiences which will be valuable to the pupil in and of themselves, quite apart from their merit as aids in developing the abstraction or general notion. Consider these three principles in order. Children, especially the less gifted intellectually, need more experiences as a basis for and as applications of an arithmetical abstraction or concept than are usually given them. For example, in paving the way for the principle, "Any number times 0 equals 0," it is not safe to say, "John worked 8 days for 0 minutes per day. How many minutes did he work?" and "How much is 0 times 4 cents?" It will be much better to spend ten or fifteen minutes as follows:[14] "What does zero mean? (Not any. No.) How many feet are there in eight yards? In 5 yards? In 3 yards? In 2 yards? In 1 yard? In 0 yard? How many inches are there in 4 ft.? In 2 ft.? In 0 ft.? 7 pk. = .... qt. 5 pk. = .... qt. 0 pk. = .... qt. A boy receives 60 cents an hour when he works. How much does he receive when he works 3 hr.? 8 hr.? 6 hr.? 0 hr.? A boy received 60 cents a day for 0 days. How much did he receive? How much is 0 times $600? How much is 0 times $5000? How much is 0 times a million dollars? 0 times any number equals.... 232 (At the blackboard.) 0 time 232 equals what? 30 I write 0 under the 0.[15] 3 times 232 equals what? ---- 6960 Continue at the blackboard with 734 321 312 41 20 40 30 60 etc." --- --- --- -- [14] The more gifted children may be put to work using the principle after the first minute or two. [15] 232 30 If desired this form may be used, with the appropriate --- difference in the form of the questions and statements. 000 696 ---- 6960 Pupils in the elementary school, except the most gifted, should not be expected to gain mastery over such concepts as _common fraction_, _decimal fraction_, _factor_, and _root_ quickly. They can learn a definition quickly and learn to use it in very easy cases, where even a vague and imperfect understanding of it will guide response correctly. But complete and exact understanding commonly requires them to take, not one intellectual step, but many; and mastery in use commonly comes only as a slow growth. For example, suppose that pupils are taught that .1, .2, .3, etc., mean 1/10, 2/10, 3/10, etc., that .01, .02, .03, etc., mean 1/100, 2/100, 3/100, etc., that .001, .002, .003, etc., mean 1/1000, 2/1000, 3/1000, etc., and that .1, .02, .001, etc., are decimal fractions. They may then respond correctly when asked to write a decimal fraction, or to state which of these,--1/4, .4, 3/8, .07, .002, 5/6,--are common fractions and which are decimal fractions. They may be able, though by no means all of them will be, to write decimal fractions which equal 1/2 and 1/5, and the common fractions which equal .1 and .09. Most of them will not, however, be able to respond correctly to "Write a decimal mixed number"; or to state which of these,--1/100, .4-1/2, .007/350, $.25,--are common fractions, and which are decimals; or to write the decimal fractions which equal 3/4 and 1/3. If now the teacher had given all at once the additional experiences needed to provide the ability to handle these more intricate and subtle features of decimal-fraction-ness, the result would have been confusion for most pupils. The general meaning of .32, .14, .99, and the like requires some understanding of .30, .10, .90, and .02, .04, .08; but it is not desirable to disturb the child with .30 while he is trying to master 2.3, 4.3, 6.3, and the like. Decimals in general require connection with place value and the contrasts of .41 with 41, 410, 4.1, and the like, but if the relation to place values in general is taught in the same lesson with the relation to /10s, /100s, /1000s, the mind will suffer from violent indigestion. A wise pedagogy in fact will break up the process of learning the meaning and use of decimal fractions into many teaching units, for example, as follows:-- (1) Such familiarity with fractions with large denominators as is desirable for pupils to have, as by an exercise in reducing to lowest terms, 8/10, 36/64, 20/25, 18/24, 24/32, 21/30, 25/100, 40/100, and the like. This is good as a review of cancellation, and as an extension of the idea of a fraction. (2) Objective work, showing 1/10 sq. ft., 1/50 sq. ft., 1/100 sq. ft., and 1/1000 sq. ft., and having these identified and the forms 1/10 sq. ft., 1/100 sq. ft., and 1/1000 sq. ft. learned. Finding how many feet = 1/10 mile and 1/100 mile. (3) Familiarity with /100s and /1000s by reductions of 750/1000, 50/100, etc., to lowest terms and by writing the missing numerators in 500/1000 = /100 = /10 and the like, and by finding 1/10, 1/100, and 1/1000 of 3000, 6000, 9000, etc. (4) Writing 1/10 as .1 and 1/100 as .01, 11/100, 12/100, 13/100, etc., as .11, .12, .13. United States money is used as the introduction. Application is made to miles. (5) Mixed numbers with a first decimal place. The cyclometer or speedometer. Adding numbers like 9.1, 14.7, 11.4, etc. (6) Place value in general from thousands to hundredths. (7) Review of (1) to (6). (8) Tenths and hundredths of a mile, subtraction when both numbers extend to hundredths, using a railroad table of distances. (9) Thousandths. The names 'decimal fractions or decimals,' and 'decimal mixed numbers or decimals.' Drill in reading any number to thousandths. The work will continue with gradual extension and refinement of the understanding of decimals by learning how to operate with them in various ways. Such may seem a slow progress, but in fact it is not, and many of these exercises whereby the pupil acquires his mastery of decimals are useful as organizations and applications of other arithmetical facts. That, it will be remembered, was the third principle:--"Develop abstract and general ideas by experiences which will be intrinsically valuable." The reason is that, even with the best of teaching, some pupils will not, within any reasonable limits of time expended, acquire ideas that are fully complete, rigorous when they should be, flexible when they should be, and absolutely exact. Many children (and adults, for that matter) could not within any reasonable limits of time be so taught the nature of a fraction that they could decide unerringly in original exercises like:-- Is 2.75/25 a common fraction? Is $.25 a decimal fraction? Is one _x_th of _y_ a fraction? Can the same words mean both a common fraction and a decimal fraction? Express 1 as a common fraction. Express 1 as a decimal fraction. These same children can, however, be taught to operate correctly with fractions in the ordinary uses thereof. And that is the chief value of arithmetic to them. They should not be deprived of it because they cannot master its subtler principles. So we seek to provide experiences that will teach all pupils something of value, while stimulating in those who have the ability the growth of abstract ideas and general principles. Finally, we should bear in mind that working with qualities and relations that are only partly understood or even misunderstood does under certain conditions give control over them. The general process of analytic learning in life is to respond as well as one can; to get a clearer idea thereby; to respond better the next time; and so on. For instance, one gets some sort of notion of what 1/5 means; he then answers such questions as 1/5 of 10 = ? 1/5 of 5 = ? 1/5 of 20 = ?; by being told when he is right and when he is wrong, he gets from these experiences a better idea of 1/5; again he does his best with 1/5 = _/10, 1/5 = _/15, etc., and as before refines and enlarges his concept of 1/5. He adds 1/5 to 2/5, etc., 1/5 to 3/10, etc., 1/5 to 1/2, etc., and thereby gains still further, and so on. What begins as a blind habit of manipulation started by imitation may thus grow into the power of correct response to the essential element. The pupil who has at the start no notion at all of 'multiplying' may learn what multiplying is by his experience that '4 6 multiplying gives 24'; '3 9 multiplying gives 27,' etc. If the pupil keeps on doing something with numbers and differentiates right results, he will often reach in the end the abstractions which he is supposed to need in the beginning. It may even be the case with some of the abstractions required in arithmetic that elaborate provision for comprehension beforehand is not so efficient as the same amount of energy devoted partly to provision for analysis itself beforehand and partly to practice in response to the element in question without full comprehension. It certainly is not the best psychology and not the best educational theory to think that the pupil first masters a principle and then merely applies it--first does some thinking and then computes by mere routine. On the contrary, the applications should help to establish, extend, and refine the principle--the work a pupil does with numbers should be a main means of increasing his understanding of the principles of arithmetic as a science. CHAPTER X THE PSYCHOLOGY OF THINKING: REASONING IN ARITHMETIC THE ESSENTIALS OF ARITHMETICAL REASONING We distinguish aimless reverie, as when a child dreams of a vacation trip, from purposive thinking, as when he tries to work out the answer to "How many weeks of vacation can a family have for $120 if the cost is $22 a week for board, $2.25 a week for laundry, and $1.75 a week for incidental expenses, and if the railroad fares for the round trip are $12?" We distinguish the process of response to familiar situations, such as five integral numbers to be added, from the process of response to novel situations, such as (for a child who has not been trained with similar problems):--"A man has four pieces of wire. The lengths are 120 yd., 132 meters, 160 feet, and 1/8 mile. How much more does he need to have 1000 yd. in all?" We distinguish 'thinking things together,' as when a diagram or problem or proof is understood, from thinking of one thing after another as when a number of words are spelled or a poem in an unknown tongue is learned. In proportion as thinking is purposive, with selection from the ideas that come up, and in proportion as it deals with novel problems for which no ready-made habitual response is available, and in proportion as many bonds act together in an organized way to produce response, we call it reasoning. When the conclusion is reached as the effect of many particular experiences, the reasoning is called inductive. When some principle already established leads to another principle or to a conclusion about some particular fact, the reasoning is called deductive. In both cases the process involves the analysis of facts into their elements, the selection of the elements that are deemed significant for the question at hand, the attachment of a certain amount of importance or weight to each of them, and their use in the right relations. Thought may fail because it has not suitable facts, or does not select from them the right ones, or does not attach the right amount of weight to each, or does not put them together properly. In the world at large, many of our failures in thinking are due to not having suitable facts. Some of my readers, for example, cannot solve the problem--"What are the chances that in drawing a card from an ordinary pack of playing-cards four times in succession, the same card will be drawn each time?" And it will be probably because they do not know certain facts about the theory of probabilities. The good thinkers among such would look the matter up in a suitable book. Similarly, if a person did not happen to know that there were fifty-two cards in all and that no two were alike, he could not reason out the answer, no matter what his mastery of the theory of probabilities. If a competent thinker, he would first ask about the size and nature of the pack. In the actual practice of reasoning, that is, we have to survey our facts to see if we lack any that are necessary. If we do, the first task of reasoning is to acquire those facts. This is specially true of the reasoning about arithmetical facts in life. "Will 3-1/2 yards of this be enough for a dress?" Reason directs you to learn how wide it is, what style of dress you intend to make of it, how much material that style normally calls for, whether you are a careful or a wasteful cutter, and how big the person is for whom the dress is to be made. "How much cheaper as a diet is bread alone, than bread with butter added to the extent of 10% of the weight of the bread?" Reason directs you to learn the cost of bread, the cost of butter, the nutritive value of bread, and the nutritive value of butter. In the arithmetic of the school this feature of reasoning appears in cases where some fact about common measures must be brought to bear, or some table of prices or discounts must be consulted, or some business custom must be remembered or looked up. Thus "How many badges, each 9 inches long, can be made from 2-1/2 yd. ribbon?" cannot be solved without getting into mind 1 yd. = 36 inches. "At Jones' prices, which costs more, 3-3/4 lb. butter or 6-1/2 lb. lard? How much more?" is a problem which directs the thinker to ascertain Jones' prices. It may be noted that such problems are, other things being equal, somewhat better training in thinking than problems where all the data are given in the problem itself (_e.g._, "Which costs more, 3-3/4 lb. butter at 48¢ per lb. or 6-1/2 lb. lard at 27¢ per lb.? How much more?"). At least it is unwise to have so many problems of the latter sort that the pupil may come to think of a problem in applied arithmetic as a problem where everything is given and he has only to manipulate the data. Life does not present its problems so. The process of selecting the right elements and attaching proper weight to them may be illustrated by the following problem:--"Which of these offers would you take, supposing that you wish a D.C.K. upright piano, have $50 saved, can save a little over $20 per month, and can borrow from your father at 6% interest?" A A Reliable Piano. The Famous D.C.K. Upright. You pay $50 cash down and $21 a month for only a year and a half. _No interest_ to pay. We ask you to pay only for the piano and allow you plenty of time. B We offer the well-known D.C.K. Piano for $390. $50 cash and $20 a month thereafter. Regular interest at 6%. The interest soon is reduced to less than $1 a month. C The D.C.K. Piano. Special Offer, $375, cash. Compare our prices with those of any reliable firm. If you consider chiefly the "only," "No interest to pay," "only," and "plenty of time" in offer A, attaching much weight to them and little to the thought, "How much will $50 plus (18 × $21) be?", you will probably decide wrongly. The situations of life are often complicated by many elements of little or even of no relevance to the correct solution. The offerer of A may belong to your church; your dearest friend may urge you to accept offer B; you may dislike to talk with the dealer who makes offer C; you may have a prejudice against owing money to a relative; that prejudice may be wise or foolish; you may have a suspicion that the B piano is shopworn; that suspicion may be well-founded or groundless; the salesman for C says, "You don't want your friends to say that you bought on the installment plan. Only low-class persons do that," etc. The statement of arithmetical problems in school usually assists the pupil to the extent of ruling out all save definitely quantitative elements, and of ruling out all quantitative elements except those which should be considered. The first of the two simplifications is very beneficial, on the whole, since otherwise there might be different correct solutions to a problem according to the nature and circumstances of the persons involved. The second simplification is often desirable, since it will often produce greater improvement in the pupils, per hour of time spent, than would be produced by the problems requiring more selection. It should not, however, be a universal custom; for in that case the pupils are tempted to think that in every problem they must use all the quantities given, as one must use all the pieces in a puzzle picture. It is obvious that the elements selected must not only be right but also be in the right relations to one another. For example, in the problems below, the 6 must be thought of in relation to a dozen and as being half of a dozen, and also as being 6 times 1. 1 must be mentally tied to "each." The 6 as half of a dozen must be related to the $1.00, $1.60, etc. The 6 as 6 times 1 must be related to the $.09, $.14, etc. Buying in Quantity These are a grocer's prices for certain things by the dozen and for a single one. He sells a half dozen at half the price of a dozen. Find out how much you save by buying 6 all at one time instead of buying them one at a time. Doz. Each 1. Evaporated Milk $1.00 $.09 2. Puffed Rice 1.60 .14 3. Puffed Wheat 1.10 .10 4. Canned Soup 1.90 .17 5. Sardines 1.80 .16 6. Beans (No. 2 cans) 1.50 .13 7. Pork and Beans 1.70 .15 8. Peas (No. 2 cans) 1.40 .12 9. Tomatoes (extra cans) 3.20 .28 10. Ripe olives (qt. cans) 7.20 .65 It is obvious also that in such arithmetical work as we have been describing, the pupil, to be successful, must 'think things together.' Many bonds must coöperate to determine his final response. As a preface to reasoning about a problem we often have the discovery of the problem and the classification of just what it is, and as a postscript we have the critical inspection of the answer obtained to make sure that it is verified by experiment or is consistent with known facts. During the process of searching for, selecting, and weighting facts, there may be similar inspection and validation, item by item. REASONING AS THE COÖPERATION OF ORGANIZED HABITS The pedagogy of the past made two notable errors in practice based on two errors about the psychology of reasoning. It considered reasoning as a somewhat magical power or essence which acted to counteract and overrule the ordinary laws of habit in man; and it separated too sharply the 'understanding of principles' by reasoning from the 'mechanical' work of computation, reading problems, remembering facts and the like, done by 'mere' habit and memory. Reasoning or selective, inferential thinking is not at all opposed to, or independent of, the laws of habit, but really is their necessary result under the conditions imposed by man's nature and training. A closer examination of selective thinking will show that no principles beyond the laws of readiness, exercise, and effect are needed to explain it; that it is only an extreme case of what goes on in associative learning as described under the 'piecemeal' activity of situations; and that attributing certain features of learning to mysterious faculties of abstraction or reasoning gives no real help toward understanding or controlling them. It is true that man's behavior in meeting novel problems goes beyond, or even against, the habits represented by bonds leading from gross total situations and customarily abstracted elements thereof. One of the two reasons therefor, however, is simply that the finer, subtle, preferential bonds with subtler and less often abstracted elements go beyond, and at times against, the grosser and more usual bonds. One set is as much due to exercise and effect as the other. The other reason is that in meeting novel problems the mental set or attitude is likely to be one which rejects one after another response as their unfitness to satisfy a certain desideratum appears. What remains as the apparent course of thought includes only a few of the many bonds which did operate, but which, for the most part, were unsatisfying to the ruling attitude or adjustment. Successful responses to novel data, associations by similarity and purposive behavior are in only apparent opposition to the fundamental laws of associative learning. Really they are beautiful examples of it. Man's successful responses to novel data--as when he argues that the diagonal on a right triangle of 796.278 mm. base and 137.294 mm. altitude will be 808.022 mm., or that Mary Jones, born this morning, will sometime die--are due to habits, notably the habits of response to certain elements or features, under the laws of piecemeal activity and assimilation. Nothing is less like the mysterious operations of a faculty of reasoning transcending the laws of connection-forming, than the behavior of men in response to novel situations. Let children who have hitherto confronted only such arithmetical tasks, in addition and subtraction with one- and two-place numbers and multiplication with one-place numbers, as those exemplified in the first line below, be told to do the examples shown in the second line. ADD ADD ADD SUBT. SUBT. MULTIPLY MULTIPLY MULTIPLY 8 37 35 8 37 8 9 6 5 24 68 5 24 5 7 3 -- -- 23 -- -- -- -- -- 19 -- MULTIPLY MULTIPLY MULTIPLY 32 43 34 23 22 26 -- -- -- They will add the numbers, or subtract the lower from the upper number, or multiply 3 × 2 and 2 × 3, etc., getting 66, 86, and 624, or respond to the element of 'Multiply' attached to the two-place numbers by "I can't" or "I don't know what to do," or the like; or, if one is a child of great ability, he may consider the 'Multiply' element and the bigness of the numbers, be reminded by these two aspects of the situation of the fact that '9 9 multiply' -- gave only 81, and that '10 10 multiply' -- gave only 100, or the like; and so may report an intelligent and justified "I can't," or reject the plan of 3 × 2 and 2 × 3, with 66, 86, and 624 for answers, as unsatisfactory. What the children will do will, in every case, be a product of the elements in the situation that are potent with them, the responses which these evoke, and the further associates which these responses in turn evoke. If the child were one of sufficient genius, he might infer the procedure to be followed as a result of his knowledge of the principles of decimal notation and the meaning of 'Multiply,' responding correctly to the 'place-value' element of each digit and adding his 6 tens and 9 tens, 20 twos and 3 thirties; but if he did thus invent the shorthand addition of a collection of twenty-three collections, each of 32 units, he would still do it by the operation of bonds, subtle but real. Association by similarity is, as James showed long ago, simply the tendency of an element to provoke the responses which have been bound to it. _abcde_ leads to _vwxyz_ because _a_ has been bound to _vwxyz_ by original nature, exercise, or effect. Purposive behavior is the most important case of the influence of the attitude or set or adjustment of an organism in determining (1) which bonds shall act, and (2) which results shall satisfy. James early described the former fact, showing that the mechanism of habit can give the directedness or purposefulness in thought's products, provided that mechanism includes something paralleling the problem, the aim, or need, in question. The second fact, that the set or attitude of the man helps to determine which bonds shall satisfy, and which shall annoy, has commonly been somewhat obscured by vague assertions that the selection and retention is of what is "in point," or is "the right one," or is "appropriate," or the like. It is thus asserted, or at least hinted, that "the will," "the voluntary attention," "the consciousness of the problem," and other such entities are endowed with magic power to decide what is the "right" or "useful" bond and to kill off the others. The facts are that in purposive thinking and action, as everywhere else, bonds are selected and retained by the satisfyingness, and are killed off by the discomfort, which they produce; and that the potency of the man's set or attitude to make this satisfy and that annoy--to put certain conduction-units in readiness to act and others in unreadiness--is in every way as important as its potency to set certain conduction-units in actual operation. Reasoning is not a radically different sort of force operating against habit but the organization and coöperation of many habits, thinking facts together. Reasoning is not the negation of ordinary bonds, but the action of many of them, especially of bonds with subtle elements of the situation. Some outside power does not enter to select and criticize; the pupil's own total repertory of bonds relevant to the problem is what selects and rejects. An unsuitable idea is not killed off by some _actus purus_ of intellect, but by the ideas which it itself calls up, in connection with the total set of mind of the pupil, and which show it to be inadequate. Almost nothing in arithmetic need be taught as a matter of mere unreasoning habit or memory, nor need anything, first taught as a principle, ever become a matter of mere habit or memory. 5 × 4 = 20 should not be learned as an isolated fact, nor remembered as we remember that Jones' telephone number is 648 J 2. Almost everything in arithmetic should be taught as a habit that has connections with habits already acquired and will work in an organization with other habits to come. The use of this organized hierarchy of habits to solve novel problems is reasoning. CHAPTER XI ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE SCHOOL THE UTILIZATION OF INSTINCTIVE INTERESTS The activities essential to acquiring ability in arithmetic can rely on little in man's instinctive equipment beyond the purely intellectual tendencies of curiosity and the satisfyingness of thought for thought's sake, and the general enjoyment of success rather than failure in an enterprise to which one sets oneself. It is only by a certain amount of artifice that we can enlist other vehement inborn interests of childhood in the service of arithmetical knowledge and skill. When this can be done at no cost the gain is great. For example, marching in files of two, in files of three, in files of four, etc., raising the arms once, two times, three times, showing a foot, a yard, an inch with the hands, and the like are admirable because learning the meanings of numbers thus acquires some of the zest of the passion for physical action. Even in late grades chances to make pictures showing the relations of fractional parts, to cut strips, to fold paper, and the like will be useful. Various social instincts can be utilized in matches after the pattern of the spelling match, contests between rows, certain number games, and the like. The scoring of both the play and the work of the classroom is a useful field for control by the teacher of arithmetic. Hunt ['12] has noted the more important games which have some considerable amount of arithmetical training as a by-product and which are more or less suitable for class use. Flynn ['12] has described games, most of them for home use, which give very definite arithmetical drill, though in many cases the drills are rather behind the needs of children old enough to understand and like the game itself. It is possible to utilize the interests in mystery, tricks, and puzzles so as to arouse a certain form of respect for arithmetic and also to get computational work done. I quote one simple case from Miss Selkin's admirable collection ['12, p. 69 f.]:-- I. ADDITION "We must admit that there is nothing particularly interesting in a long column of numbers to be added. Let the teacher, however, suggest that he can write the answer at sight, and the task will assume a totally different aspect. "A very simple number trick of this kind can be performed by making use of the principle of complementary addition. The arithmetical complement of a number with respect to a larger number is the difference between these two numbers. Most interesting results can be obtained by using complements with respect to 9. "The children may be called upon to suggest several numbers of two, three, or more digits. Below these write an equal number of addends and immediately announce the answer. The children, impressed by this apparently rapid addition, will set to work enthusiastically to test the results of this lightning calculation. "Example:-- 357 } 999 682 } A × 3 793 } ---- 2997 642 } 317 } B 206 } "Explanation:--The addends in group A are written down at random or suggested by the class. Those in group B are their complements. To write the first number in group B we look at the first number in group A and, starting at the left write 6, the complement of 3 with respect to 9; 4, the complement of 5; 2, the complement of 7. The second and third addends in group B are derived in the same way. Since we have three addends in each group, the problem reduces itself to multiplying 999 by 3, or to taking 3000 - 3. Any number of addends may be used and each addend may consist of any number of digits." Respect for arithmetic as a source of tricks and magic is very much less important than respect for its everyday services; and computation to test such tricks is likely to be undertaken zealously only by the abler pupils. Consequently this source of interest should probably be used only sparingly, and perhaps the teacher should give such exhibitions only as a reward for efficiency in the regular work. For example, if the work for a week is well done in four days the fifth day might be given up to some semi-arithmetical entertainment, such as the demonstration of an adding machine, the story of primitive methods of counting, team races in computation, an exhibition of lightning calculation and intellectual sleight-of-hand by the teacher, or the voluntary study of arithmetical puzzles. The interest in achievement, in success, mentioned above is stronger in children than is often realized and makes advisable the systematic use of the practice experiment as a method of teaching much of arithmetic. Children who thus compete with their own past records, keeping an exact score from week to week, make notable progress and enjoy hard work in making it. THE ORDER OF DEVELOPMENT OF ORIGINAL TENDENCIES Negatively the difficulty of the work that pupils should be expected to do is conditioned by the gradual maturing of their capacities. Other things being equal, the common custom of reserving hard things for late in the elementary school course is, of course, sound. It seems probable that little is gained by using any of the child's time for arithmetic before grade 2, though there are many arithmetical facts that he can learn in grade 1. Postponement of systematic work in arithmetic to grade 3 or even grade 4 is allowable if better things are offered. With proper textbooks and oral and written exercises, however, a child in grades 2 and 3 can spend time profitably on arithmetical work. When all children can be held in school through the eighth grade it does not much matter whether arithmetic is begun early or late. If, however, many children are to leave in grades 5 and 6 as now, we may think it wise to provide somehow that certain minima of arithmetical ability be given them. There are, so far as is known, no special times and seasons at which the human animal by inner growth is specially ripe for one or another section or aspect of arithmetic, except in so far as the general inner growth of intellectual powers makes the more abstruse and complex tasks suitable to later and later years. Indeed, very few of even the most enthusiastic devotees of the recapitulation theory or culture-epoch theory have attempted to apply either to the learning of arithmetic, and Branford is the only mathematician, so far as I know, who has advocated such application, even tempered by elaborate shiftings and reversals of the racial order. He says:-- "Thus, for each age of the individual life--infancy, childhood, school, college--may be selected from the racial history the most appropriate form in which mathematical experience can be assimilated. Thus the capacity of the infant and early childhood is comparable with the capacity of animal consciousness and primitive man. The mathematics suitable to later childhood and boyhood (and, of course, girlhood) is comparable with Archæan mathematics passing on through Greek and Hindu to mediæval European mathematics; while the student is become sufficiently mature to begin the assimilation of modern and highly abstract European thought. The filling in of details must necessarily be left to the individual teacher, and also, within some such broadly marked limits, the precise order of the marshalling of the material for each age. For, though, on the whole, mathematical development has gone forward, yet there have been lapses from advances already made. Witness the practical world-loss of much valuable Hindu thought, and, for long centuries, the neglect of Greek thought: witness the world-loss of the invention by the Babylonians of the Zero, until re-invented by the Hindus, passed on by them to the Arabs, and by these to Europe. "Moreover, many blunders and false starts and false principles have marked the whole course of development. In a phrase, rivers have their backwaters. But it is precisely the teacher's function to avoid such racial mistakes, to take short cuts ultimately discovered, and to guide the young along the road ultimately found most accessible with such halts and retracings--returns up side-cuts--as the mental peculiarities of the pupils demand. "All this, the practical realization of the spirit of the principle, is to be wisely left to the mathematical teacher, familiar with the history of mathematical science and with the particular limitations of his pupils and himself." ['08, p. 245.] The latitude of modification suggested by Branford reduces the guidance to be derived from racial history to almost _nil_. Also it is apparent that the racial history in the case of arithmetical achievement is entirely a matter of acquisition and social transmission. Man's original nature is destitute of all arithmetical ideas. The human germs do not know even that one and one make two! INVENTORIES OF ARITHMETICAL KNOWLEDGE AND SKILL A scientific plan for teaching arithmetic would begin with an exact inventory of the knowledge and skill which the pupils already possessed. Our ordinary notions of what a child knows at entrance to grade 1, or grade 2, or grade 3, and of what a first-grade child or second-grade child can do, are not adequate. If they were, we should not find reputable textbooks arranging to teach elaborately facts already sufficiently well known to over three quarters of the pupils when they enter school. Nor should we find other textbooks presupposing in their first fifty pages a knowledge of words which not half of the children can read even at the end of the 2 B grade. We do find just such evidence that ordinary ideas about the abilities of children at the beginning of systematic school training in arithmetic may be in gross error. For example, a reputable and in many ways admirable recent book has fourteen pages of exercises to teach the meaning of two and the fact that one and one make two! As an example of the reverse error, consider putting all these words in the first twenty-five pages of a beginner's book:--_absentees, attendance, blanks, continue, copy, during, examples, grouped, memorize, perfect, similar, splints, therefore, total_! Little, almost nothing, has been done toward providing an exact inventory compared with what needs to be done. We may note here (1) the facts relevant to arithmetic found by Stanley Hall, Hartmann, and others in their general investigations of the knowledge possessed by children at entrance to school, (2) the facts concerning the power of children to perceive differences in length, area, size of collection, and organization within a collection such as is shown in Fig. 24, and certain facts and theories about early awareness of number. In the Berlin inquiry of 1869, knowledge of the meaning of two, three, and four appeared in 74, 74, and 73 percent of the children upon entrance to school. Some of those recorded as ignorant probably really knew, but failed to understand that they were expected to reply or were shy. Only 85 percent were recorded as knowing their fathers' names. Seven eighths as many children knew the meanings of two, three, and four as knew their fathers' names. In a similar but more careful experiment with Boston children in September, 1880, Stanley Hall found that 92 percent knew three, 83 percent knew four, and 71-1/2 percent knew five. Three was known about as well as the color red; four was known about as well as the color blue or yellow or green. Hartmann ['90] found that two thirds of the children entering school in Annaberg could count from one to ten. This is about as many as knew money, or the familiar objects of the town, or could repeat words spoken to them. [Illustration: FIG. 24.--Objective presentation.] In the Stanford form of the Binet tests counting four pennies is given as an ability of the typical four-year-old. Counting 13 pennies correctly in at least one out of two trials, and knowing three of the four coins,--penny, nickel, dime, and quarter,--are given as abilities of the typical six-year-old. THE PERCEPTION OF NUMBER AND QUANTITY We know that educated adults can tell how many lines or dots, etc., they see in a single glance (with an exposure too short for the eye to move) up to four or more, according to the clearness of the objects and their grouping. For example, Nanu ['04] reports that when a number of bright circles on a dark background are shown to educated adults for only .033 second, ten can be counted when arranged to form a parallelogram, but only five when arranged in a row. With certain groupings, of course, their 'perception' involves much inference, even conscious addition and multiplication. Similarly they can tell, up to twenty and beyond, the number of taps, notes, or other sounds in a series too rapid for single counting if the sounds are grouped in a convenient rhythm. These abilities are, however, the product of a long and elaborate learning, including the learning of arithmetic itself. Elementary psychology and common experience teach us that the mere observation of groups or quantities, no matter how clear their number quality appears to the person who already knows the meanings of numbers, does not of itself create the knowledge of the meanings of numbers in one who does not. The experiments of Messenger ['03] and Burnett ['06] showed that there is no direct intuitive apprehension even of two as distinct from one. We have to _learn_ to feel the two touches or see the two dots or lines as two. We do not know by exact measurements the growth in children of this ability to count or infer the number of elements in a collection seen or series heard. Still less do we know what the growth would be without the influence of school training in counting, grouping, adding, and multiplying. Many textbooks and teachers seem to overestimate it greatly. Not all educated adults can, apart from measurement, decide with surety which of these lines is the longer, or which of these areas is the larger, or whether this is a ninth or a tenth or an eleventh of a circle. [Illustration] Children upon entering school have not been tested carefully in respect to judgments of length and area, but we know from such studies as Gilbert's ['94] that the difference required in their case is probably over twice that required for children of 13 or 14. In judging weights, for example, a difference of 6 is perceived as easily by children 13 to 15 years of age as a difference of 15 by six-year-olds. A teacher who has adult powers of estimating length or area or weight and who also knows already which of the two is longer or larger or heavier, may use two lines to illustrate a difference which they really hide from the child. It is unlikely, for example, that the first of these lines ______________ ________________ would be recognized as shorter than the second by every child in a fourth-grade class, and it is extremely unlikely that it would be recognized as being 7/8 of the length of the latter, rather than 3/4 of it or 5/6 of it or 9/10 of it or 11/12 of it. If the two were shown to a second grade, with the question, "The first line is 7. How long is the other line?" there would be very many answers of 7 or 9; and these might be entirely correct arithmetically, the pupils' errors being all due to their inability to compare the lengths accurately. _A_ ______________ ________________ ______________ ________________ _B_ |______________| |________________| _C_ |-|-|-|-|-|-|-| |-|-|-|-|-|-|-|-| __ __ __ __ __ __ __ _D_ |__|__|__|__|__|__|__| __ __ __ __ __ __ __ __ |__|__|__|__|__|__|__|__| _E_ .'\##|##/`. .'\##|##/`. /###\#|#/ \ /###\#|#/###\ |-----------| |-----------| \###/#|#\###/ \###/#|#\###/ `./##|##\.' `./##|##\.' The quantities used should be such that their mere discrimination offers no difficulty even to a child of blunted sense powers. If 7/8 and 1 are to be compared, _A_ and _B_ are not allowable. _C_, _D_, and _E_ are much better. Teachers probably often underestimate or neglect the sensory difficulties of the tasks they assign and of the material they use to illustrate absolute and relative magnitudes. The result may be more pernicious when the pupils answer correctly than when they fail. For their correct answering may be due to their divination of what the teacher wants; and they may call a thing an inch larger to suit her which does not really seem larger to them at all. This, of course, is utterly destructive of their respect for arithmetic as an exact and matter-of-fact instrument. For example, if a teacher drew a series of lines 20, 21, 22, 23, 24, and 25 inches long on the blackboard in this form--____ ______ and asked, "This is 20 inches long, how long is this?" she might, after some errors and correction thereof, finally secure successful response to all the lines by all the children. But their appreciation of the numbers 20, 21, 22, 23, 24, and 25 would be actually damaged by the exercise. THE EARLY AWARENESS OF NUMBER There has been some disagreement concerning the origin of awareness of number in the individual, in particular concerning the relative importance of the perception of how-many-ness and that of how-much-ness, of the perception of a defined aggregate and the perception of a defined ratio. (See McLellan and Dewey ['95], Phillips ['97 and '98], and Decroly and Degand ['12].) The chief facts of significance for practice seem to be these: (1) Children with rare exceptions hear the names _one_, _two_, _three_, _four_, _half_, _twice_, _two times_, _more_, _less_, _as many as_, _again_, _first_, _second_, and _third_, long before they have analyzed out the qualities and relations to which these words refer so as to feel them at all clearly. (2) Their knowledge of the qualities and relations is developed in the main in close association with the use of these words to the child and by the child. (3) The ordinary experiences of the first five years so develop in the child awareness of the 'how many somethings' in various groups, of the relative magnitudes of two groups or quantities of any sort, and of groups and magnitudes as related to others in a series. For instance, if fairly gifted, a child comes, by the age of five, to see that a row of four cakes is an aggregate of four, seeing each cake as a part of the four and the four as the sum of its parts, to know that two of them are as many as the other two, that half of them would be two, and to think, when it is useful for him to do so, of four as a step beyond three on the way to five, or to think of hot as a step from warm on the way to very hot. The degree of development of these abilities depends upon the activity of the law of analysis in the individual and the character of his experiences. (4) He gets certain bad habits of response from the ambiguity of common usage of 2, 3, 4, etc., for second, third, fourth. Thus he sees or hears his parents or older children or others count pennies or rolls or eggs by saying one, two, three, four, and so on. He himself is perhaps misled into so counting. Thus the names properly belonging to a series of aggregations varying in amount come to be to him the names of the positions of the parts in a counted whole. This happens especially with numbers above 3 or 4, where the correct experience of the number as a name for the group has rarely been present. This attaching to the cardinal numbers above three or four the meanings of the ordinal numbers seems to affect many children on entrance to school. The numbering of pages in books, houses, streets, etc., and bad teaching of counting often prolong this error. (5) He also gets the habit, not necessarily bad, but often indirectly so, of using many names such as eight, nine, ten, eleven, fifteen, a hundred, a million, without any meaning. (6) The experiences of half, twice, three times as many, three times as long, etc., are rarer; even if they were not, they would still be less easily productive of the analysis of the proper abstract element than are the experiences of two, three, four, etc., in connection with aggregates of things each of which is usually called one, such as boys, girls, balls, apples. Experiences of the names, two, three, and four, in connection with two twos, two threes, two fours, are very rare. Hence, the names, two, three, etc., mean to these children in the main, "one something and one something," "one something usually called one, and one something usually called one, and another something usually called one," and more rarely and imperfectly "two times anything," "three times anything," etc. With respect to Mr. Phillips' emphasis of the importance of the series-idea in children's minds, the matters of importance are: first, that the knowledge of a series of number names in order is of very little consequence to the teaching of arithmetic and of still less to the origin of awareness of number. Second, the habit of applying this series of words in counting in such a way that 8 is associated with the eighth thing, 9 with the ninth thing, etc., is of consequence because it does so much mischief. Third, the really valuable idea of the number series, the idea of a series of groups or of magnitudes varying by steps, is acquired later, as a result, not a cause, of awareness of numbers. With respect to the McLellan-Dewey doctrine, the ratio aspect of numbers should be emphasized in schools, not because it is the main origin of the child's awareness of number, but because it is _not_, and because the ordinary practical issues of child life do _not_ adequately stimulate its action. It also seems both more economical and more scientific to introduce it through multiplication, division, and fractions rather than to insist that 4 and 5 shall from the start mean 4 or 5 times anything that is called 1, for instance, that 8 inches shall be called 4 two-inches, or 10 cents, 5 two-cents. If I interpret Professor Dewey's writings correctly, he would agree that the use of inch, foot, yard, pint, quart, ounce, pound, glassful, cupful, handful, spoonful, cent, nickel, dime, and dollar gives a sufficient range of units for the first two school years. Teaching the meanings of 1/2 of 4, 1/2 of 6, 1/2 of 8, 1/2 of 10, 1/2 of 20, 1/3 of 6, 1/3 of 9, 1/3 of 30, 1/4 of 8, two 2s, five 2s, and the like, in early grades, each in connection with many different units of measure, provides a sufficient assurance that numbers will connect with relationships as well as with collections. CHAPTER XII INTEREST IN ARITHMETIC CENSUSES OF PUPILS' INTERESTS Arithmetic, although it makes little or no appeal to collecting, muscular manipulation, sensory curiosity, or the potent original interests in things and their mechanisms and people and their passions, is fairly well liked by children. The censuses of pupils' likes and dislikes that have been made are not models of scientific investigation, and the resulting percentages should not be used uncritically. They are, however, probably not on the average over-favorable to arithmetic in any unfair way. Some of their results are summarized below. In general they show arithmetic to be surpassed in interest clearly by only the manual arts (shopwork and manual training for boys, cooking and sewing for girls), drawing, certain forms of gymnastics, and history. It is about on a level with reading and science. It clearly surpasses grammar, language, spelling, geography, and religion. Lobsien ['03], who asked one hundred children in each of the first five grades (_Stufen_) of the elementary schools of Kiel, "Which part of the school work (literally, 'which instruction period') do you like best?" found arithmetic led only by drawing and gymnastics in the case of the boys, and only by handwork in the case of the girls. This is an exaggerated picture of the facts, since no count is made of those who especially dislike arithmetic. Arithmetic is as unpopular with some as it is popular with others. When full allowance is made for this, arithmetic still has popularity above the average. Stern ['05] asked, "Which subject do you like most?" and "Which subject do you like least?" The balance was greatly in favor of gymnastics for boys (28--1), handwork for girls (32--1-1/2), and drawing for both (16-1/2--6). Writing (6-1/2--4), arithmetic (14-1/2--13), history (9--6-1/2), reading (8-1/2--8), and singing (6--7-1/2) come next. Religion, nature study, physiology, geography, geometry, chemistry, language, and grammar are low. McKnight ['07] found with boys and girls in grades 7 and 8 of certain American cities that arithmetic was liked better than any of the school subjects except gymnastics and manual training. The vote as compared with history was:-- Arithmetic 327 liked greatly, 96 disliked greatly. History 164 liked greatly, 113 disliked greatly. In a later study Lobsien ['09] had 6248 pupils from 9 to 15 years old representing all grades of the elementary school report, so far as they could, the subject most disliked, the subject most liked, the subject next most liked, and the subject next in order. No child was forced to report all of these four judgments, or even any of them. Lobsien counts the likes and the dislikes for each subject. Gymnastics, handwork, and cooking are by far the most popular. History and drawing are next, followed by arithmetic and reading. Below these are geography, writing, singing, nature study, biblical history, catechism, and three minor subjects. Lewis ['13] secured records from English children in elementary schools of the order of preference of all the studies listed below. He reports the results in the following table of percents: =================================================================== | TOP THIRD OF | MIDDLE THIRD OF | LOWEST THIRD OF | STUDIES FOR | STUDIES FOR | STUDIES FOR | INTEREST | INTEREST | INTEREST ----------------+--------------+-----------------+----------------- Drawing | 78 | 20 | 2 Manual Subjects | 66 | 26 | 8 History | 64 | 24 | 12 Reading | 53 | 38 | 9 Singing | 32 | 48 | 20 | | | Drill | 20 | 55 | 25 Arithmetic | 16 | 53 | 31 Science | 23 | 37 | 40 Nature Study | 16 | 36 | 48 Dictation | 4 | 57 | 39 | | | Composition | 18 | 28 | 54 Scripture | 4 | 38 | 58 Recitation | 9 | 23 | 68 Geography | 4 | 24 | 72 Grammar | -- | 6 | 94 =================================================================== Brandell ['13] obtained data from 2137 Swedish children in Stockholm (327), Norrköping (870), and Gothenburg (940). In general he found, as others have, that handwork, shopwork for boys and household work for girls, and drawing were reported as much better liked than arithmetic. So also was history, and (in this he differs from most students of this matter) so were reading and nature study. Gymnastics he finds less liked than arithmetic. Religion, geography, language, spelling, and writing are, as in other studies, much less popular than arithmetic. Other studies are by Lilius ['11] in Finland, Walsemann ['07], Wiederkehr ['07], Pommer ['14], Seekel ['14], and Stern ['13 and '14], in Germany. They confirm the general results stated. The reasons for the good showing that arithmetic makes are probably the strength of its appeal to the interest in definite achievement, success, doing what one attempts to do; and of its appeal, in grades 5 to 8, to the practical interest of getting on in the world, acquiring abilities that the world pays for. Of these, the former is in my opinion much the more potent interest. Arithmetic satisfies it especially well, because, more than any other of the 'intellectual' studies of the elementary school, it permits the pupil to see his own progress and determine his own success or failure. The most important applications of the psychology of satisfiers and annoyers to arithmetic will therefore be in the direction of utilizing still more effectively this interest in achievement. Next in importance come the plans to attach to arithmetical learning the satisfyingness of bodily action, play, sociability, cheerfulness, and the like, and of significance as a means of securing other desired ends than arithmetical abilities themselves. Next come plans to relieve arithmetical learning from certain discomforts such as the eyestrain of some computations and excessive copying of figures. These will be discussed here in the inverse order. RELIEVING EYESTRAIN At present arithmetical work is, hour for hour, probably more of a tax upon the eyes than reading. The task of copying numbers from a book to a sheet of paper is one of the very hardest tasks that the eyes of a pupil in the elementary schools have to perform. A certain amount of such work is desirable to teach a child to write numbers, to copy exactly, and to organize material in shape for computation. But beyond that, there is no more reason for a pupil to copy every number with which he is to compute than for him to copy every word he is to read. The meaningless drudgery of copying figures should be mitigated by arranging much work in the form of exercises like those shown on pages 216, 217, and 218, and by having many of the textbook examples in addition, subtraction, and multiplication done with a slip of paper laid below the numbers, the answers being written on it. There is not only a resulting gain in interest, but also a very great saving of time for the pupil (very often copying an example more than quadruples the time required to get its answer), and a much greater efficiency in supervision. Arithmetical errors are not confused with errors of copying,[16] and the teacher's task of following a pupil's work on the page is reduced to a minimum, each pupil having put the same part of the day's work in just the same place. The use of well-printed and well-spaced pages of exercises relieves the eyestrain of working with badly made gray figures, unevenly and too closely or too widely spaced. I reproduce in Fig. 25 specimens taken at random from one hundred random samples of arithmetical work by pupils in grade 8. Contrast the task of the eyes in working with these and their task in working with pages 216 to 218. The customary method of always copying the numbers to be used in computation from blackboard or book to a sheet of paper is an utterly unjustifiable cruelty and waste. [16] Courtis finds in the case of addition that "of all the individuals making mistakes at any given time in a class, at least one third, and usually two thirds, will be making mistakes in carrying or copying." [Illustration: FIG. 25_a_.--Specimens taken at random from the computation work of eighth-grade pupils. This computation occurred in a genuine test. In the original gray of the pencil marks the work is still harder to make out.] [Illustration: FIG. 25_b_.--Specimens taken at random from the computation work of eighth-grade pupils. This computation occurred in a genuine test. In the original gray of the pencil marks the work is still harder to make out.] Write the products:-- A. 3 4s= B. 5 7s= C. 9 2s= 5 2s= 8 3s= 4 4s= 7 2s= 4 2s= 2 7s= 1 6 = 4 5s= 6 4s= 1 3 = 4 7s= 5 5s= 3 7s= 5 9s= 3 6s= 4 1s= 7 5s= 3 2s= 6 8s= 7 1s= 3 9s= 9 8s= 6 3s= 5 1s= 4 3s= 4 9s= 8 6s= 2 4s= 3 5s= 8 4s= 2 2s= 9 6s= 8 5s= 8 7s= 2 5s= 7 9s= 5 8s= 5 4s= 6 2s= 7 6s= 8 2s= 7 4s= 7 3s= 8 9s= 9 3s= D. 4 20s = E. 9 60s = F. 40 × 2 = 80 4 200s = 9 600s = 20 × 2 = 6 30s = 5 30s = 30 × 2 = 6 300s = 5 300s = 40 × 2 = 7 × 50 = 8 × 20 = 20 × 3 = 7 × 500 = 8 × 200 = 30 × 3 = 3 × 40 = 2 × 70 = 300 × 3 = 900 3 × 400 = 2 × 700 = 300 × 2 = Write the missing numbers: (_r_ stands for remainder.) 25 = .... 3s and .... _r_. 25 = .... 4s " .... _r_. 25 = .... 5s " .... _r_. 25 = .... 6s " .... _r_. 25 = .... 7s " .... _r_. 25 = .... 8s " .... _r_. 25 = .... 9s " .... _r_. 26 = .... 3s and .... _r_. 26 = .... 4s " .... _r_. 26 = .... 5s " .... _r_. 26 = .... 6s " .... _r_. 26 = .... 7s " .... _r_. 26 = .... 8s " .... _r_. 26 = .... 9s " .... _r_. 30 = .... 4s and .... _r_. 30 = .... 5s " .... _r_. 30 = .... 6s " .... _r_. 30 = .... 7s " .... _r_. 30 = .... 8s " .... _r_. 30 = .... 9s " .... _r_. 31 = .... 4s and .... _r_. 31 = .... 5s " .... _r_. 31 = .... 6s " .... _r_. 31 = .... 7s " .... _r_. 31 = .... 8s " .... _r_. 31 = .... 9s " .... _r_. Write the whole numbers or mixed numbers which these fractions equal:-- 5 4 9 4 7 - - - - - 4 3 5 2 3 7 5 11 3 8 - - -- - - 4 3 8 2 8 8 6 9 9 16 - - - - -- 4 3 8 4 8 11 7 13 8 6 -- - -- - - 4 5 8 5 6 Write the missing figures:-- 6 2 8 1 2 - = - - = - -- = - - = -- - = - 8 4 4 2 10 5 5 10 3 6 Write the missing numerators:-- 1 - = -- - -- - -- - -- 2 12 8 10 4 16 6 14 1 - = -- - -- - -- -- -- 3 12 9 18 6 15 24 21 1 - = -- -- - -- -- -- -- 4 12 16 8 24 20 28 32 1 - = -- -- -- -- -- -- -- 5 10 20 15 25 40 35 30 2 - = -- -- -- - -- -- - 3 12 18 21 6 15 24 9 3 - = - -- -- -- -- -- -- 4 8 16 12 20 24 32 28 Find the products. Cancel when you can:-- 5 11 2 -- × 4 = -- × 3 = - × 5 = 16 12 3 7 8 1 -- × 8 = - × 15 = - × 8 = 12 5 6 SIGNIFICANCE FOR RELATED ACTIVITIES The use of bodily action, social games, and the like was discussed in the section on original tendencies. "Significance as a means of securing other desired ends than arithmetical learning itself" is therefore our next topic. Such significance can be given to arithmetical work by using that work as a means to present and future success in problems of sports, housekeeping, shopwork, dressmaking, self-management, other school studies than arithmetic, and general school life and affairs. Significance as a means to future ends alone can also be more clearly and extensively attached to it than it now is. Whatever is done to supply greater strength of motive in studying arithmetic must be carefully devised so as not to get a strong but wrong motive, so as not to get abundant interest but in something other than arithmetic, and so as not to kill the goose that after all lays the golden eggs--the interest in intellectual activity and achievement itself. It is easy to secure an interest in laying out a baseball diamond, measuring ingredients for a cake, making a balloon of a certain capacity, or deciding the added cost of an extra trimming of ribbon for one's dress. The problem is to _attach_ that interest to arithmetical learning. Nor should a teacher be satisfied with attaching the interest as a mere tail that steers the kite, so long as it stays on, or as a sugar-coating that deceives the pupil into swallowing the pill, or as an anodyne whose dose must be increased and increased if it is to retain its power. Until the interest permeates the arithmetical activity itself our task is only partly done, and perhaps is made harder for the next time. One important means of really interfusing the arithmetical learning itself with these derived interests is to lead the pupil to seek the help of arithmetic himself--to lead him, in Dewey's phrase, to 'feel the need'--to take the 'problem' attitude--and thus appreciate the technique which he actively hunts for to satisfy the need. In so far as arithmetical learning is organized to satisfy the practical demands of the pupil's life at the time, he should, so to speak, come part way to get its help. Even if we do not make the most skillful use possible of these interests derived from the quantitative problems of sports, housekeeping, shopwork, dressmaking, self-management, other school studies, and school life and affairs, the gain will still be considerable. To have them in mind will certainly preserve us from giving to children of grades 3 and 4 problems so devoid of relation to their interests as those shown below, all found (in 1910) in thirty successive pages of a book of excellent repute:-- A chair has 4 legs. How many legs have 8 chairs? 5 chairs? A fly has 6 legs. How many legs have 3 flies? 9 flies? 7 flies? (Eight more of the same sort.) In 1890 New York had 1,513,501 inhabitants, Milwaukee had 206,308, Boston had 447,720, San Francisco 297,990. How many had these cities together? (Five more of the same sort.) Milton was born in 1608 and died in 1674. How many years did he live? (Several others of the same sort.) The population of a certain city was 35,629 in 1880 and 106,670 in 1890. Find the increase. (Several others of this sort.) A number of others about the words in various inaugural addresses and the Psalms in the Bible. It also seems probable that with enough care other systematic plans of textbooks can be much improved in this respect. From every point of view, for example, the early work in arithmetic should be adapted to some extent to the healthy childish interests in home affairs, the behavior of other children, and the activities of material things, animals, and plants. TABLE 9 FREQUENCY OF APPEARANCE OF CERTAIN WORDS ABOUT FAMILY LIFE, PLAY, AND ACTION IN EIGHT ELEMENTARY TEXTBOOKS IN ARITHMETIC, pp. 1-50. ================================================================ | A | B | C | D | E | F | G | H ----------------+-----+-----+-----+-----+-----+-----+-----+----- baby | | | | 2 | | 4 | | brother | 2 | | 6 | 1 | 1 | | 1 | family | | | 2 | | 2 | | 4 | father | 1 | | 3 | 5 | | 2 | 1 | help | | | | | | | | home | 2 | | 4 | 4 | 2 | 2 | 7 | 1 mother | 4 | 2 | 9 | 5 | | 5 | 1 | 7 sister | | | 1 | 2 | 2 | 9 | 1 | 1 | | | | | | | | fork | | | | | | | | knife | | | | | | | | plate | 4 | 2 | | 2 | | 1 | | spoon | | | | | | | | | | | | | | | | doll | 10 | 1 | 10 | 6 | | 10 | | 9 game | 1 | | | 3 | | | 5 | 5 jump | | | | | | | | 4 marbles | 10 | 4 | 10 | | 10 | | 1 | play | | | 1 | | | 3 | | run | | | | | | 1 | | 3 sing | | | | | | | | tag | | | | | | | | toy | | | | | | | | 1 | | | | | | | | car | | | 2 | 4 | | 2 | 3 | 1 cut | | | 10 | | 6 | 2 | | 8 dig | | | | | | | 2 | flower | 1 | | | 4 | 1 | 1 | 2 | grow | | | | 1 | | | | plant | | | 2 | | | | | seed | | | | 3 | | | 1 | string | | | | | 1 | 10 | 1 | 1 wheel | 5 | | | | | 10 | | ================================================================ The words used by textbooks give some indication of how far this aim is being realized, or rather of how far short we are of realizing it. Consider, for example, the words home, mother, father, brother, sister, help, plate, knife, fork, spoon, play, game, toy, tag, marbles, doll, run, jump, sing, plant, seed, grow, flower, car, wheel, string, cut, dig. The frequency of appearance in the first fifty pages of eight beginners' arithmetics was as shown in Table 9. The eight columns refer to the eight books (the first fifty pages of each). The numbers refer to the number of times the word in question appeared, the number 10 meaning 10 _or more_ times in the fifty pages. Plurals, past tenses, and the like were counted. _Help_, _fork_, _knife_, _spoon_, _jump_, _sing_, and _tag_ did not appear at all! _Toy_ and _grow_ appeared each once in the 400 pages! _Play_, _run_, _dig_, _plant_, and _seed_ appeared once in a hundred or more pages. _Baby_ did not appear as often as _buggy_. _Family_ appeared no oftener than _fence_ or _Friday_. _Father_ appears about a third as often as _farmer_. Book A shows only 10 of these thirty words in the fifty pages; book B only 4; book C only 12; and books D, E, F, G, and H only 13, 8, 14, 13, 10, respectively. The total number of appearances (counting the 10s as only 10 in each case) is 40 for A, 9 for B, 60 for C, 42 for D, 25 for E, 62 for F, 30 for G, and 37 for H. The five words--apple, egg, Mary, milk, and orange--are used oftener than all these thirty together. If it appeared that this apparent neglect of childish affairs and interests was deliberate to provide for a more systematic treatment of pure arithmetic, a better gradation of problems, and a better preparation for later genuine use than could be attained if the author of the textbook were tied to the child's apron strings, the neglect could be defended. It is not at all certain that children in grade 2 get much more enjoyment or ability from adding the costs of purchases for Christmas or Fourth of July, or multiplying the number of cakes each child is to have at a party by the number of children who are to be there, than from adding gravestones or multiplying the number of hairs of bald-headed men. When, however, there is nothing gained by substituting remote facts for those of familiar concern to children, the safe policy is surely to favor the latter. In general, the neglect of childish data does not seem to be due to provision for some other end, but to the same inertia of tradition which has carried over the problems of laying walls and digging wells into city schools whose children never saw a stone wall or dug well. * * * * * I shall not go into details concerning the arrangement of courses of study, textbooks, and lesson-plans to make desirable connections between arithmetical learning and sports, housework, shopwork, and the rest. It may be worth while, however, to explain the term _self-management_, since this source of genuine problems of real concern to the pupils has been overlooked by most writers. By self-management is meant the pupil's use of his time, his abilities, his knowledge, and the like. By the time he reaches grade 5, and to some extent before then, a boy should keep some account of himself, of how long it takes him to do specified tasks, of how much he gets done in a specified time at a certain sort of work and with how many errors, of how much improvement he makes month by month, of which things he can do best, and the like. Such objective, matter-of-fact, quantitative study of one's behavior is not a stimulus to morbid introspection or egotism; it is one of the best preventives of these. To treat oneself impersonally is one of the essential elements of mental balance and health. It need not, and should not, encourage priggishness. On the contrary, this matter-of-fact study of what one is and does may well replace a certain amount of the exhortations and admonitions concerning what one ought to do and be. All this is still truer for a girl. The demands which such an accounting of one's own activities make of arithmetic have the special value of connecting directly with the advanced work in computation. They involve the use of large numbers, decimals, averaging, percentages, approximations, and other facts and processes which the pupil has to learn for later life, but to which his childish activities as wage-earner, buyer and seller, or shopworker from 10 to 14 do not lead. Children have little money, but they have time in thousands of units! They do not get discounts or bonuses from commercial houses, but they can discount their quantity of examples done for the errors made, and credit themselves with bonuses of all sorts for extra achievements. INTRINSIC INTEREST IN ARITHMETICAL LEARNING There remains the most important increase of interest in arithmetical learning--an increase in the interest directly bound to achievement and success in arithmetic itself. "Arithmetic," says David Eugene Smith, "is a game and all boys and girls are players." It should not be a _mere_ game for them and they should not _merely_ play, but their unpractical interest in doing it because they can do it and can see how well they do do it is one of the school's most precious assets. Any healthy means to give this interest more and better stimulus should therefore be eagerly sought and cherished. Two such means have been suggested in other connections. The first is the extension of training in checking and verifying work so that the pupil may work to a standard of approximately 100% success, and may know how nearly he is attaining it. The second is the use of standardized practice material and tests, whereby the pupil may measure himself against his own past, and have a clear, vivid, and trustworthy idea of just how much better or faster he can do the same tasks than he could do a month or a year ago, and of just how much harder things he can do now than then. Another means of stimulating the essential interest in quantitative thinking itself is the arrangement of the work so that real arithmetical thinking is encouraged more than mere imitation and assiduity. This means the avoidance of long series of applied problems all of one type to be solved in the same way, the avoidance of miscellaneous series and review series which are almost verbatim repetitions of past problems, and in general the avoidance of excessive repetition of any one problem-situation. Stimulation to real arithmetical thinking is weak when a whole day's problem work requires no choice of methods, or when a review simply repeats without any step of organization or progress, or when a pupil meets a situation (say the 'buy _x_ things at _y_ per thing, how much pay' situation) for the five-hundredth time. Another matter worthy of attention in this connection is the unwise tendency to omit or present in diluted form some of the topics that appeal most to real intellectual interests, just because they are hard. The best illustration, perhaps, is the problem of ratio or "How many times as large (long, heavy, expensive, etc.) as _x_ is _y_?" Mastery of the 'times as' relation is hard to acquire, but it is well worth acquiring, not only because of its strong intellectual appeal, but also because of its prime importance in the applications of arithmetic to science. In the older arithmetics it was confused by pedantries and verbal difficulties and penalized by unreal problems about fractions of men doing parts of a job in strange and devious times. Freed from these, it should be reinstated, beginning as early as grade 5 with such simple exercises as those shown below and progressing to the problems of food values, nutritive ratios, gears, speeds, and the like in grade 8. John is 4 years old. Fred is 6 years old. Mary is 8 years old. Nell is 10 years old. Alice is 12 years old. Bert is 15 years old. Who is twice as old as John? Who is half as old as Alice? Who is three times as old as John? Who is one and one half times as old as Nell? Who is two thirds as old as Fred? etc., etc., etc. Alice is .... times as old as John. John is .... as old as Mary. Fred is .... times as old as John. Alice is .... times as old as Fred. Fred is .... as old as Mary. etc., etc., etc. Finally it should be remembered that all improvements in making arithmetic worth learning and helping the pupil to learn it will in the long run add to its interest. Pupils like to learn, to achieve, to gain mastery. Success is interesting. If the measures recommended in the previous chapters are carried out, there will be little need to entice pupils to take arithmetic or to sugar-coat it with illegitimate attractions. CHAPTER XIII THE CONDITIONS OF LEARNING We shall consider in this chapter the influence of time of day, size of class, and amount of time devoted to arithmetic in the school program, the hygiene of the eyes in arithmetical work, the use of concrete objects, and the use of sounds, sights, and thoughts as situations and of speech and writing and thought as responses.[17] [17] Facts concerning the conditions of learning in general will be found in the author's _Educational Psychology_, Vol. 2, Chapter 8, or in the _Educational Psychology, Briefer Course_, Chapter 15. EXTERNAL CONDITIONS Computation of one or another sort has been used by several investigators as a test of efficiency at different times in the day. When freed from the effects of practice on the one hand and lack of interest due to repetition on the other, the results uniformly show an increase in speed late in the school session with a falling off in accuracy that about balances it.[18] There is no wisdom in putting arithmetic early in the session because of its _difficulty_. Lively and sociable exercises in mental arithmetic with oral answers in fact seem to be admirably fitted for use late in the session. Except for the general principles (1) of starting the day with work that will set a good standard of cheerful, efficient production and (2) of getting the least interesting features of the day's work done fairly early in the day, psychology permits practical exigencies to rule the program, so far as present knowledge extends. Adequate measurements of the effect of time of day on _improvement_ have not been made, but there is no reason to believe that any one time between 9 A.M. and 4 P.M. is appreciably more favorable to arithmetical learning than to learning geography, history, spelling, and the like. [18] See Thorndike ['00], King ['07], and Heck ['13]. The influence of size of class upon progress in school studies is very difficult to measure because (1) within the same city system the average of the six (or more) sizes of class that a pupil has experienced will tend to approximate closely to the corresponding average for any other child; because further (2) there may be a tendency of supervisory officers to assign more pupils to the better teachers; and because (3) separate systems which differ in respect to size of class probably differ in other respects also so that their differences in achievement may be referable to totally different differences. Elliott ['14] has made a beginning by noting size of class during the year of test in connection with his own measures of the achievements of seventeen hundred pupils, supplemented by records from over four hundred other classes. As might be expected from the facts just stated, he finds no appreciable difference between classes of different sizes within the same school system, the effect of the few months in a small class being swamped by the antecedents or concomitants thereof. The effect of the amount of time devoted to arithmetic in the school program has been studied extensively by Rice ['02 and '03] and Stone ['08]. Dr. Rice ['02] measured the arithmetical ability of some 6000 children in 18 different schools in 7 different cities. The results of these measurements are summarized in Table 10. This table "gives two averages for each grade as well as for each school as a whole. Thus, the school at the top shows averages of 80.0 and 83.1, and the one at the bottom, 25.3 and 31.5. The first represents the percentage of answers which were absolutely correct; the second shows what per cent of the problems were correct in principle, _i.e._ the average that would have been received if no mechanical errors had been made." The facts of Dr. Rice's table show that there is a positive relation between the general standing of a school system in the tests and the amount of time devoted to arithmetic by its program. The relation is not close, however, being that expressed by a correlation coefficient of .36-1/2. Within any one school system there is no relation between the standing of a particular school and the amount of time devoted to arithmetic in that school's program. It must be kept in mind that the amount of time given in the school program may be counterbalanced by emphasizing work at home and during study periods, or, on the other hand, may be a symptom of correspondingly small or great emphasis on arithmetic in work set for the study periods at home. A still more elaborate investigation of this same topic was made by Stone ['08]. I quote somewhat fully from it, since it is an instructive sample of the sort of studies that will doubtless soon be made in the case of every elementary school subject. He found that school systems differed notably in the achievements made by their sixth-grade pupils in his tests of computation (the so-called 'fundamentals') and of the solution of verbally described problems (the so-called 'reasoning'). The facts were as shown in Table 11. TABLE 10 AVERAGES FOR INDIVIDUAL SCHOOLS IN ARITHMETIC KEY A: CITY B: SCHOOL C: Result D: Principle E: Percent of Mechanical Errors F: Minutes Daily =========================================================== | |6TH YEAR |7TH YEAR |8TH YEAR |SCHOOL AVERAGE | | |----+----+----+----+----+----+----+----+-----+----- A | B | C | D | C | D | C | D | C | D | E | F ---+---+----+----+----+----+----+----+----+----+-----+----- III| 1 |79.3|80.3|81.1|82.3|91.7|93.9|80.0|83.1| 3.7 | 53 I| 1 |80.4|81.5|64.2|67.2|80.9|82.8|76.6|80.3| 4.6 | 60 I| 2 |80.9|83.4|43.5|50.9|72.7|79.1|69.3|75.1| 7.7 | 25 I| 3 |72.2|74.0|63.5|66.2|74.5|76.6|67.8|72.2| 6.1 | 45 I| 4 |69.9|72.2|54.6|57.8|66.5|69.1|64.3|70.3| 8.5 | 45 II| 1 |71.2|75.3|33.6|35.7|36.8|40.0|60.2|64.8| 7.1 | 60 III| 2 |43.7|45.0|53.9|56.7|51.1|53.1|54.5|58.9| 7.4 | 60 IV| 1 |58.9|60.4|31.2|34.1|41.6|43.5|55.1|58.4| 5.6 | 60 IV| 2 |59.8|63.1| -- | -- |22.5|22.5|53.9|58.8| 8.3 | -- IV| 3 |54.9|58.1|35.2|38.6|43.5|45.0|51.5|57.6|10.5 | 60 IV| 4 |42.3|45.1|16.1|19.2|48.7|48.7|42.8|48.2|11.2 | -- V| 1 |44.1|48.7|29.2|32.5|51.1|58.3|45.9|51.3|10.5 | 40 VI| 1 |68.3|71.3|33.5|36.6|26.9|30.7|39.0|42.9| 9.0 | 33 VI| 2 |46.1|49.5|19.5|24.2|30.2|40.6|36.5|43.6|16.2 | 30 VI| 3 |34.5|36.4|30.5|35.1|23.3|24.1|36.0|42.5|15.2 | 48 VII| 1 |35.2|37.7|29.1|32.5|25.1|27.2|40.5|45.9|11.7 | 42 VII| 2 |35.2|38.7|15.0|16.4|19.6|21.2|36.5|40.6|10.1 | 75 VII| 3 |27.6|33.7| 8.9|10.1|11.3|11.3|25.3|31.5|19.6 | 45 =========================================================== High achievement by a system in computation went with high achievement in solving the problems, the correlation being about .50; and the system that scored high in addition or subtraction or multiplication or division usually showed closely similar excellence in the other three, the correlations being about .90. TABLE 11 SCORES MADE BY THE SIXTH-GRADE PUPILS OF EACH OF TWENTY-SIX SCHOOL SYSTEMS ================================================= SYSTEM | SCORE IN TESTS WITH | SCORE IN TESTS IN | PROBLEMS | COMPUTING -------+---------------------+------------------- 23 | 356 | 1841 24 | 429 | 3513 17 | 444 | 3042 4 | 464 | 3563 25 | 464 | 2167 22 | 468 | 2311 16 | 469 | 3707 20 | 491 | 2168 18 | 509 | 3758 15 | 532 | 2779 3 | 533 | 2845 8 | 538 | 2747 6 | 550 | 3173 1 | 552 | 2935 10 | 601 | 2749 2 | 615 | 2958 21 | 627 | 2951 13 | 636 | 3049 14 | 661 | 3561 9 | 691 | 3404 7 | 734 | 3782 12 | 736 | 3410 11 | 759 | 3261 26 | 791 | 3682 19 | 848 | 4099 5 | 914 | 3569 ================================================= Of the conditions under which arithmetical learning took place, the one most elaborately studied was the amount of time devoted to arithmetic. On the basis of replies by principals of schools to certain questions, he gave each of the twenty-six school systems a measure for the probable time spent on arithmetic up through grade 6. Leaving home study out of account, there seems to be little or no correlation between the amount of time a system devotes to arithmetic and its score in problem-solving, and not much more between time expenditure and score in computation. With home study included there is little relation to the achievement of the system in solving problems, but there is a clear effect on achievement in computation. The facts as given by Stone are:-- TABLE 12 CORRELATION OF TIME EXPENDITURES WITH ABILITIES Without Home Study { Reasoning and Time Expenditure -.01 { Fundamentals and Time Expenditure .09 Including Home Study { Reasoning and Time Expenditure .13 { Fundamentals and Time Expenditure .49 These correlations, it should be borne in mind, are for school systems, not for individual pupils. It might be that, though the system which devoted the most time to arithmetic did not show corresponding superiority in the product over the system devoting only half as much time, the pupils within the system did achieve in exact proportion to the time they gave to study. Neither correlation would permit inference concerning the effect of different amounts of time spent by the same pupil. Stone considered also the printed announcements of the courses of study in arithmetic in these twenty-six systems. Nineteen judges rated these announced courses of study for excellence according to the instructions quoted below:-- CONCERNING THE RATING OF COURSES OF STUDY Judges please read before scoring I. Some Factors Determining Relative Excellence. (N. B. The following enumeration is meant to be suggestive rather than complete or exclusive. And each scorer is urged to rely primarily on his own judgment.) 1. Helpfulness to the teacher in teaching the subject matter outlined. 2. Social value or concreteness of sources of problems. 3. The arrangement of subject matter. 4. The provision made for adequate drill. 5. A reasonable minimum requirement with suggestions for valuable additional work. 6. The relative values of any predominating so-called methods--such as Speer, Grube, etc. 7. The place of oral or so-called mental arithmetic. 8. The merit of textbook references. II. Cautions and Directions. (Judges please follow as implicitly as possible.) 1. Include references to textbooks as parts of the Course of Study. This necessitates judging the parts of the texts referred to. 2. As far as possible become equally familiar with all courses before scoring any. 3. When you are ready to begin to score, (1) arrange in serial order according to excellence, (2) starting with the middle one score it 50, then score above and below 50 according as courses are better or poorer, indicating relative differences in excellence by relative differences in scores, _i.e._ in so far as you find that the courses differ by about equal steps, score those better than the middle one 51, 52, etc., and those poorer 49, 48, etc., but if you find that the courses differ by unequal steps show these inequalities by omitting numbers. 4. Write ratings on the slip of paper attached to each course. The systems whose courses of study were thus rated highest did not manifest any greater achievement in Stone's tests than the rest. The thirteen with the most approved announcements of courses of study were in fact a little inferior in achievement to the other thirteen, and the correlation coefficients were slightly negative. Stone also compared eighteen systems where there was supervision of the work by superintendents or supervisors as well as by principals with four systems where the principals and teachers had no such help. The scores in his tests were very much lower in the four latter cities. THE HYGIENE OF THE EYES IN ARITHMETIC We have already noted that the task of reading and copying numbers is one of the hardest that the eyes have to perform in the elementary school, and that it should be alleviated by arranging much of the work so that only answers need be written by the pupil. The figures to be read and copied should obviously be in type of suitable size and style, so arranged and spaced on the page or blackboard as to cause a minimum of effort and strain. [Illustration: FIG. 26.--Type too large.] [Illustration: FIG. 27.--12-point, 11-point, and 10-point type.] _Size._--Type may be too large as well as too small, though the latter is the commoner error. If it is too large, as in Fig. 26, which is a duplicate of type actually used in a form of practice pad, the eye has to make too many fixations to take in a given content. All things considered, 12-point type in grades 3 and 4, 11-point in grades 5 and 6, and 10-point in grades 7 and 8 seem the most desirable sizes. These are shown in Fig. 27. Too small type occurs oftenest in fractions and in the dimension-numbers or scale numbers of drawings. Figures 28, 29, and 30 are samples from actual school practice. Samples of the desirable size are shown in Figs. 31 and 32. The technique of modern typesetting makes it very difficult and expensive to make fractions of the horizontal type (1 3 5 - - - 4, 8, 6) large enough without making the whole-number figures with which they are mingled too large or giving an uncouth appearance to the total. Consequently fractions somewhat smaller than are desirable may have to be used occasionally in textbooks.[19] There is no valid excuse, however, for the excessively small fractions which often are made in blackboard work. [19] A special type could be constructed that would use a large type body, say 14 point, with integers in 10 or 12 point and fractions much larger than now. [Illustration: FIG. 28.--Type of measurements too small. This is a picture of Mary's garden. How many feet is it around the garden?] [Illustration: FIG. 29.--Type too small.] [Illustration: FIG. 30.--Numbers too small and badly designed.] [Illustration: FIG. 31.--Figure 28 with suitable numbers.] [Illustration: FIG. 32.--Figure 30 with suitable numbers.] _Style._--The ordinary type forms often have 3 and 8 so made as to require strain to distinguish them. 5 is sometimes easily confused with 3 and even with 8. 1, 4, and 7 may be less easily distinguishable than is desirable. Figure 33 shows a specially good type in which each figure is represented by its essential[20] features without any distracting shading or knobs or turns. Figure 34 shows some of the types in common use. There are no demonstrably great differences amongst these. In fractions there is a notable gain from using the slant form (2/3, 3/4) for exercises in addition and subtraction, and for almost all mixed numbers. This appears clearly to the eye in the comparison of Fig. 35 below, where the same fractions all in 10-point type are displayed in horizontal and in slant form. The figures in the slant form are in general larger and the space between them and the fraction-line is wider. Also the slant form makes it easier for the eye to examine the denominators to see whether reductions are necessary. Except for a few cases to show that the operations can be done just as truly with the horizontal forms, the book and the blackboard should display mixed numbers and fractions to be added or subtracted in the slant form. The slant line should be at an angle of approximately 45 degrees. Pupils should be taught to use this form in their own work of this sort. [20] It will be still better if the 4 is replaced by an open-top 4. When script figures are presented they should be of simple design, showing clearly the essential features of the figure, the line being everywhere of equal or nearly equal width (that is, without shading, and without ornamentation or eccentricity of any sort). The opening of the 3 should be wide to prevent confusion with 8; the top of the 3 should be curved to aid its differentiation from 5; the down stroke of the 9 should be almost or quite straight; the 1, 4, 7, and 9 should be clearly distinguishable. There are many ways of distinguishing them clearly, the best probably being to use the straight line for 1, the open 4 with clear angularity, a wide top to the 7, and a clearly closed curve for the top of the 9. [Illustration: FIG. 33.--Block type; a very desirable type except that it is somewhat too heavy.] [Illustration: FIG. 34.--Common styles of printed numbers.] [Illustration: FIG. 35.--Diagonal and horizontal fractions compared.] [Illustration: FIG. 36.--Good vertical spacing.] [Illustration: FIG. 37.--Bad vertical spacing.] [Illustration: FIGS. 38 (above) and 39 (below).--Good and bad left-right spacing.] The pupil's writing of figures should be clear. He will thereby be saved eyestrain and errors in his school work as well as given a valuable ability for life. Handwriting of figures is used enormously in spite of the development of typewriters; illegible figures are commonly more harmful than illegible letters or words, since the context far less often tells what the figure is intended to be; the habit of making clear figures is not so hard to acquire, since they are written unjoined and require only the automatic action of ten minor acts of skill. The schools have missed a great opportunity in this respect. Whereas the hand writing of words is often better than it needs to be for life's purposes, the writing of figures is usually much worse. The figures presented in books on penmanship are also commonly bad, showing neglect or misunderstanding of the matter on the part of leaders in penmanship. _Spacing._--Spacing up and down the column is rarely too wide, but very often too narrow. The specimens shown in Figs. 36 and 37 show good practice contrasted with the common fault. Spacing from right to left is generally fairly satisfactory in books, though there is a bad tendency to adopt some one routine throughout and so to miss chances to use reductions and increases of spacing so as to help the eye and the mind in special cases. Specimens of good and bad spacing are shown in Figs. 38 and 39. In the work of the pupils, the spacing from right to left is often too narrow. This crowding of letters, together with unevenness of spacing, adds notably to the task of eye and mind. _The composition or make-up of the page._--Other things being equal, that arrangement of the page is best which helps a child most to keep his place on a page and to find it after having looked away to work on the paper on which he computes, or for other good reasons. A good page and a bad page in this respect are shown in Figs. 40 and 41. [Illustration: FIG. 40.--A page well made up to suit the action of the eye.] [Illustration: FIG. 41.--The same matter as in Fig. 40, much less well made up.] _Objective presentations._--Pictures, diagrams, maps, and other presentations should not tax the eye unduly, (_a_) by requiring too fine distinctions, or (_b_) by inconvenient arrangement of the data, preventing easy counting, measuring, comparison, or whatever the task is, or (_c_) by putting too many facts in one picture so that the eye and mind, when trying to make out any one, are confused by the others. Illustrations of bad practices in these respects are shown in Figs. 42 to 52. A few specimens of work well arranged for the eye are shown in Figs. 53 to 56. Good rules to remember are:-- Other things being equal, make distinctions by the clearest method, fit material to the tendency of the eye to see an 'eyeful' at a time (roughly 1-1/2 inch by 1/2 inch in a book; 1-1/2 ft. by 1/2 ft. on the blackboard), and let one picture teach only one fact or relation, or such facts and relations as do not interfere in perception. The general conditions of seating, illumination, paper, and the like are even more important when the eyes are used with numbers than when they are used with words. [Illustration: FIG. 42.--Try to count the rungs on the ladder, or the shocks in the wagon.] [Illustration: FIG. 43.--How many oars do you see? How many birds? How many fish?] [Illustration: FIG. 44.--Count the birds in each of the three flocks of birds.] [Illustration: FIG. 45.--Note the lack of clear division of the hundreds. Consider the difficulty of counting one of these columns of dots.] [Illustration: FIG. 46.--What do you suppose these pictures are intended to show?] [Illustration: FIG. 47.--Would a beginner know that after THIRTEEN he was to switch around and begin at the other end? Could you read the SIX of TWENTY-SIX if you did not already know what it ought to be? What meaning would all the brackets have for a little child in grade 2? Does this picture illustrate or obfuscate?] [Illustration: FIG. 48.--How long did it take you to find out what these pictures mean?] [Illustration: FIG. 49.--Count the figures in the first row, using your eyes alone; have some one make lines of 10, 11, 12, 13, and more repetitions of this figure spaced closely as here. Count 20 or 30 such lines, using the eye unaided by fingers, pencil, etc. ] [Illustration: FIG. 50.--Can you answer the question without measuring? Could a child of seven or eight?] [Illustration: FIG. 51.--What are these drawings intended to show? Why do they show the facts only obscurely and dubiously?] [Illustration: FIG. 52.--What are these drawings intended to show? What simple change would make them show the facts much more clearly?] [Illustration: FIG. 53.--Arranged in convenient "eye-fulls."] [Illustration: FIG. 54.--Clear, simple, and easy of comparison.] [Illustration: FIG. 55.--Clear, simple, and well spaced.] [Illustration: FIG. 56.--Well arranged, though a little wider spacing between the squares would make it even better.] THE USE OF CONCRETE OBJECTS IN ARITHMETIC We mean by concrete objects actual things, events, and relations presented to sense, in contrast to words and numbers and symbols which mean or stand for these objects or for more abstract qualities and relations. Blocks, tooth-picks, coins, foot rules, squared paper, quart measures, bank books, and checks are such concrete things. A foot rule put successively along the three thirds of a yard rule, a bell rung five times, and a pound weight balancing sixteen ounce weights are such concrete events. A pint beside a quart, an inch beside a foot, an apple shown cut in halves display such concrete relations to a pupil who is attentive to the issue. Concrete presentations are obviously useful in arithmetic to teach meanings under the general law that a word or number or sign or symbol acquires meaning by being connected with actual things, events, qualities, and relations. We have also noted their usefulness as means to verifying the results of thinking and computing, as when a pupil, having solved, "How many badges each 5 inches long can be made from 3-1/3 yd. of ribbon?" by using 10 × 12/5, draws a line 3-1/3 yd. long and divides it into 5-inch lengths. Concrete experiences are useful whenever the meaning of a number, like 9 or 7/8 or .004, or of an operation, like multiplying or dividing or cubing, or of some term, like rectangle or hypothenuse or discount, or some procedure, like voting or insuring property against fire or borrowing money from a bank, is absent or incomplete or faulty. Concrete work thus is by no means confined to the primary grades but may be appropriate at all stages when new facts, relations, and procedures are to be taught. How much concrete material shall be presented will depend upon the fact or relation or procedure which is to be made intelligible, and the ability and knowledge of the pupil. Thus 'one half' will in general require less concrete illustration than 'five sixths'; and five sixths will require less in the case of a bright child who already knows 2/3, 3/4, 3/8, 5/8, 7/8, 2/5, 3/5, and 4/5 than in the case of a dull child or one who only knows 2/3 and 3/4. As a general rule the same topic will require less concrete material the later it appears in the school course. If the meanings of the numbers are taught in grade 2 instead of grade 1, there will be less need of blocks, counters, splints, beans, and the like. If 1-1/2 + 1/2 = 2 is taught early in grade 3, there will be more gain from the use of 1-1/2 inches and 1/2 inch on the foot rule than if the same relations were taught in connection with the general addition of like fractions late in grade 4. Sometimes the understanding can be had either by connecting the idea with the reality directly, or by connecting the two indirectly _via_ some other idea. The amount of concrete material to be used will depend on its relative advantage per unit of time spent. Thus it might be more economical to connect 5/12, 7/12, and 11/12 with real meanings indirectly by calling up the resemblance to the 2/3, 3/4, 3/8, 5/8, 7/8, 2/5, 3/5, 4/5, and 5/6 already studied, than by showing 5/12 of an apple, 7/12 of a yard, 11/12 of a foot, and the like. In general the economical course is to test the understanding of the matter from time to time, using more concrete material if it is needed, but being careful to encourage pupils to proceed to the abstract ideas and general principles as fast as they can. It is wearisome and debauching to pupils' intellects for them to be put through elaborate concrete experiences to get a meaning which they could have got themselves by pure thought. We should also remember that the new idea, say of the meaning of decimal fractions, will be improved and clarified by using it (see page 183 f.), so that the attainment of a _perfect_ conception of decimal fractions before doing anything with them is unnecessary and probably very wasteful. A few illustrations may make these principles more instructive. (_a_) Very large numbers, such as 1000, 10,000, 100,000, and 1,000,000, need more concrete aids than are commonly given. Guessing contests about the value in dollars of the school building and other buildings, the area of the schoolroom floor and other surfaces in square inches, the number of minutes in a week, and year, and the like, together with proper computations and measurements, are very useful to reënforce the concrete presentations and supply genuine problems in multiplication and subtraction with large numbers. (_b_) Numbers very much smaller than one, such as 1/32, 1/64, .04, and .002, also need some concrete aids. A diagram like that of Fig. 57 is useful. (_c_) _Majority_ and _plurality_ should be understood by every citizen. They can be understood without concrete aid, but an actual vote is well worth while for the gain in vividness and surety. [Illustration: FIG. 57.--Concrete aid to understanding fractions with large denominators. A = 1/1000 sq. ft.; B = 1/100 sq. ft.; C = 1/50 sq. ft.; D = 1/10 sq. ft.] (_d_) Insurance against loss by fire can be taught by explanation and analogy alone, but it will be economical to have some actual insuring and payment of premiums and a genuine loss which is reimbursed. (_e_) Four play banks in the corners of the room, receiving deposits, cashing checks, and later discounting notes will give good educational value for the time spent. (_f_) Trade discount, on the contrary, hardly requires more concrete illustration than is found in the very problems to which it is applied. (_g_) The process of finding the number of square units in a rectangle by multiplying with the appropriate numbers representing length and width is probably rather hindered than helped by the ordinary objective presentation as an introduction. The usual form of objective introduction is as follows:-- [Illustration: FIG. 58.] How long is this rectangle? How large is each square? How many square inches are there in the top row? How many rows are there? How many square inches are there in the whole rectangle? Since there are three rows each containing 4 square inches, we have 3 × 4 square inches = 12 square inches. Draw a rectangle 7 inches long and 2 inches wide. If you divide it into inch squares how many rows will there be? How many inch squares will there be in each row? How many square inches are there in the rectangle? [Illustration: FIG. 59.] It is better actually to hide the individual square units as in Fig. 59. There are four reasons: (1) The concrete rows and columns rather distract attention from the essential thing to be learned. This is not that "_x_ rows one square wide, _y_ squares in a row will make _xy_ squares in all," but that "by using proper units and the proper operation the area of any rectangle can be found from its length and width." (2) Children have little difficulty in learning to multiply rather than add, subtract, or divide when computing area. (3) The habit so formed holds good for areas like 1-2/3 by 4-1/2, with fractional dimensions, in which any effort to count up the areas of rows is very troublesome and confusing. (4) The notion that a square inch is an area 1' by 1' rather than 1/2' by 2' or 1/3 in. by 3 in. or 1-1/2 in. by 2/3 in. is likely to be formed too emphatically if much time is spent upon the sort of concrete presentation shown above. It is then better to use concrete counting of rows of small areas as a means of _verification after_ the procedure is learned, than as a means of deriving it. There has been, especially in Germany, much argument concerning what sort of number-pictures (that is, arrangement of dots, lines, or the like, as shown in Fig. 60) is best for use in connection with the number names in the early years of the teaching of arithmetic. Lay ['98 and '07], Walsemann ['07], Freeman ['10], Howell ['14], and others have measured the accuracy of children in estimating the number of dots in arrangements of one or more of these different types.[21] Many writers interpret a difference in favor of estimating, say, the square arrangements of Born or Lay as meaning that such is the best arrangement to use in teaching. The inference is, however, unjustified. That certain number-pictures are easier to estimate numerically does not necessarily mean that they are more instructive in learning. One set may be easier to estimate just because they are more familiar, having been oftener experienced. Even if the favored set was so after equal experience with all sets, accuracy of estimation would be a sign of superiority for use in instruction only if all other things were equal (or in favor of the arrangement in question). Obviously the way to decide which of these is best to use in teaching is by using them in teaching and measuring all relevant results, not by merely recording which of them are most accurately estimated in certain time exposures. [21] For an account in English of their main findings see Howell ['14], pp. 149-251. It may be noted that the Born, Lay, and Freeman pictures have claims for special consideration on grounds of probable instructiveness. Since they are also superior in the tests in respect to accuracy of estimate, choice should probably be made from these three by any teacher who wishes to connect one set of number-pictures systematically with the number names, as by drills with the blackboard or with cards. [Illustration: FIG. 60.--Various proposed arrangements of dots for use in teaching the meanings of the numbers 1 to 10.] Such drills are probably useful if undertaken with zeal, and if kept as supplementary to more realistic objective work with play money, children marching, material to be distributed, garden-plot lengths to be measured, and the like, and if so administered that the pupils soon get the generalized abstract meaning of the numbers freed from dependence on an inner picture of any sort. This freedom is so important that it may make the use of many types of number-pictures advisable rather than the use of the one which in and of itself is best. As Meumann says: "Perceptual reckoning can be overdone. It had its chief significance for the surety and clearness of the first foundation of arithmetical instruction. If, however, it is continued after the first operations become familiar to the child, and extended to operations which develop from these elementary ones, it necessarily works as a retarding force and holds back the natural development of arithmetic. This moves on to work with abstract number and with mechanical association and reproduction." ['07, Vol. 2, p. 357.] Such drills are commonly overdone by those who make use of them, being given too often, and continued after their instructiveness has waned, and used instead of more significant, interesting, and varied work in counting and estimating and measuring real things. Consequently, there is now rather a prejudice against them in our better schools. They should probably be reinstated but to a moderate and judicious use. ORAL, MENTAL, AND WRITTEN ARITHMETIC There has been much dispute over the relative merits of oral and written work in arithmetic--a question which is much confused by the different meanings of 'oral' and 'written.' _Oral_ has meant (1) work where the situations are presented orally and the pupil's final responses are given orally, or (2) work where the situations are presented orally and the pupils' final responses are written or partly written and partly oral, or (3) work where the situations are presented in writing or print and the final responses are oral. _Written_ has meant (1) work where the situations are presented in writing or print and the final responses are made in writing, or (2) work where also many of the intermediate responses are written, or (3) work where the situations are presented orally but the final responses and a large percentage of the intermediate computational responses are written. There are other meanings than these. It is better to drop these very ambiguous terms and ask clearly what are the merits and demerits, in the case of any specified arithmetical work, of auditory and of visual presentation of the situations, and of saying and of writing each specified step in the response. The disputes over mental _versus_ written arithmetic are also confused by ambiguities in the use of 'mental.' Mental has been used to mean "done without pencil and paper" and also "done with few overt responses, either written or spoken, between the setting of the task and the announcement of the answer." In neither case is the word _mental_ specially appropriate as a description of the total fact. As before, we should ask clearly, "What are the merits and demerits of making certain specified intermediate responses in inner speech or imaged sounds or visual images or imageless thought--that is, _without_ actual writing or overt speech?" It may be said at the outset that oral, written, and inner presentations of initial situations, oral, written, and inner announcements of final responses, and oral, written, and inner management of intermediate processes have varying degrees of merit according to the particular arithmetical exercise, pupil, and context. Devotion to oralness or mentalness as such is simply fanatical. Various combinations, such as the written presentation of the situation with inner management of the intermediate responses and oral announcement of the final response have their special merits for particular cases. These merits the reader can evaluate for himself for any given sort of work for a given class by considering: (1) The amount of practice received by the class per hour spent; (2) the ease of correction of the work; (3) the ease of understanding the tasks; (4) the prevention of cheating; (5) the cheerfulness and sociability of the work; (6) the freedom from eyestrain, and other less important desiderata. It should be noted that the stock schemes A, B, C, and D below are only a few of the many that are possible and that schemes E, F, G, and H have special merits. PRESENTATION OF MANAGEMENT OF ANNOUNCEMENT OF INITIAL SITUATION INTERMEDIATE PROCESSES FINAL RESPONSE A. Printed or written Written Written B. " " Inner Oral by one pupil, inner by the rest C. Oral (by teacher) Written Written D. " " Inner Oral by one pupil, inner by the rest E. As in A or C A mixture, the pupil As in A or B or H writing what he needs F. The real situation As in E As in A or B or H itself, in part at least G. Both read by the pupil As in E As in A or B or H and put orally by the teacher H. As in A or C or G As in E Written by all pupils, announced orally by one pupil The common practice of either having no use made of pencil and paper or having all computations and even much verbal analysis written out elaborately for examination is unfavorable for learning. The demands which life itself will make of arithmetical knowledge and skill will range from tasks done with every percentage of written work from zero up to the case where every main result obtained by thought is recorded for later use by further thought. In school the best way is that which, for the pupils in question, has the best total effect upon quality of product, speed, and ease of production, reënforcement of training already given, and preparation for training to be given. There is nothing intellectually criminal about using a pencil as well as inner thought; on the other hand there is no magical value in writing out for the teacher's inspection figures that the pupil does not need in order to attain, preserve, verify, or correct his result. The common practice of having the final responses of all _easy_ tasks given orally has no sure justification. On the contrary, the great advantage of having all pupils really do the work should be secured in the easy work more than anywhere else. If the time cost of copying the figures is eliminated by the simple plan of having them printed, and if the supervision cost of examining the papers is eliminated by having the pupils correct each other's work in these easy tasks, written answers are often superior to oral except for the elements of sociability and 'go' and freedom from eyestrain of the oral exercise. Such written work provides the gifted pupils with from two to ten times as much practice as they would get in an oral drill on the same material, supposing them to give inner answers to every exercise done by the class as a whole; it makes sure that the dull pupils who would rarely get an inner answer at the rate demanded by the oral exercise, do as much as they are able to do. Two arguments often made for the oral statement of problems by the teacher are that problems so put are better understood, especially in the grades up through the fifth, and that the problems are more likely to be genuine and related to the life the pupils know. When these statements are true, the first is a still better argument for having the pupils read the problems _aided by the teacher's oral statement of them_. For the difficulty is largely that the pupils cannot read well enough; and it is better to help them to surmount the difficulty rather than simply evade it. The second is not an argument for oralness _versus_ writtenness, but for good problems _versus_ bad; the teacher who makes up such good problems should, in fact, take special care to write them down for later use, which may be by voice or by the blackboard or by printed sheet, as is best. CHAPTER XIV THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE Dewey, and others following him, have emphasized the desirability of having pupils do their work as active seekers, conscious of problems whose solution satisfies some real need of their own natures. Other things being equal, it is unwise, they argue, for pupils to be led along blindfold as it were by the teacher and textbook, not knowing where they are going or why they are going there. They ought rather to have some living purpose, and be zealous for its attainment. This doctrine is in general sound, as we shall see, but it is often misused as a defense of practices which neglect the formation of fundamental habits, or as a recommendation to practices which are quite unworkable under ordinary classroom conditions. So it seems probable that its nature and limitations are not thoroughly known, even to its followers, and that a rather detailed treatment of it should be given here. ILLUSTRATIVE CASES Consider first some cases where time spent in making pupils understand the end to be attained before attacking the task by which it is attained, or care about attaining the end (well or ill understood) is well spent. It is well for a pupil who has learned (1) the meanings of the numbers one to ten, (2) how to count a collection of ten or less, and (3) how to measure in inches a magnitude of ten, nine, eight inches, etc., to be confronted with the problem of true adding without counting or measuring, as in 'hidden' addition and measurement by inference. For example, the teacher has three pencils counted and put under a book; has two more counted and put under the book; and asks, "How many pencils are there under the book?" Answers, when obtained, are verified or refuted by actual counting and measuring. The time here is well spent because the children can do the necessary thinking if the tasks are well chosen; because they are thereby prevented from beginning their study of addition by the bad habit of pseudo-adding by looking at the two groups of objects and counting their number instead of real adding, that is, thinking of the two numbers and inferring their sum; and further, because facing the problem of adding as a real problem is in the end more economical for learning arithmetic and for intellectual training in general than being enticed into adding by objective or other processes which conceal the difficulty while helping the pupil to master it. The manipulation of short multiplication may be introduced by confronting the pupils with such problems as, "How to tell how many Uneeda biscuit there are in four boxes, by opening only one box." Correct solutions by addition should be accepted. Correct solutions by multiplication, if any gifted children think of this way, should be accepted, even if the children cannot justify their procedure. (Inferring the manipulation from the place-values of numbers is beyond all save the most gifted and probably beyond them.) Correct solution by multiplication by some child who happens to have learned it elsewhere should be accepted. Let the main proof of the trustworthiness of the manipulation be by measurement and by addition. Proof by the stock arguments from the place-values of numbers may also be used. If no child hits on the manipulation in question, the problem of finding the length _without_ adding may be set. If they still fail, the problem may be made easier by being put as "4 times 22 gives the answer. Write down what you think 4 times 22 will be." Other reductions of the difficulty of the problem may be made, or the teacher may give the answer without very great harm being done. The important requirement is that the pupils should be aware of the problem and treat the manipulation as a solution of it, not as a form of educational ceremonial which they learn to satisfy the whims of parents and teachers. In the case of any particular class a situation that is more appealing to the pupils' practical interests than the situation used here can probably be devised. The time spent in this way is well spent (1) because all but the very dull pupils can solve the problem in some way, (2) because the significance of the manipulation as an economy over addition is worth bringing out, and (3) because there is no way of beginning training in short multiplication that is much better. In the same fashion multiplication by two-place numbers may be introduced by confronting pupils with the problem of the number of sheets of paper in 72 pads, or pieces of chalk in 24 boxes, or square inches in 35 square feet, or the number of days in 32 years, or whatever similar problem can be brought up so as to be felt as a problem. Suppose that it is the 35 square feet. Solutions by (5 × 144) + (30 × 144), however arranged, or by (10 × 144) + (10 × 144) + (10 × 144) + (5 × 144), or by 3500 + (35 × 40) + (35 × 4), or by 7 × (5 × 144), however arranged, should all be listed for verification or rejection. The pupils need not be required to justify their procedures by a verbal statement. Answers like 432,720, or 720,432, or 1152, or 4220, or 3220 should be listed for verification or rejection. Verification may be by a mixture of short multiplication and objective work, or by a mixture of short multiplication and addition, or by addition abbreviated by taking ten 144s as 1440, or even (for very stupid pupils) by the authority of the teacher. Or the manipulation in cases like 53 × 9 or 84 × 7 may be verified by the reverse short multiplication. The deductive proof of the correctness of the manipulation may be given in whole or in part in connection with exercises like 10 × 2 = 30 × 14 = 10 × 3 = 3 × 44 = 10 × 4 = 30 × 44 = 10 × 14 = 3 × 144 = 10 × 44 = 20 × 144 = 10 × 144 = 40 × 144 = 20 × 2 = 30 × 144 = 20 × 3 = 5 × 144 = 30 × 3 = 35 = 30 + .... 30 × 4 = 30 × 144 added to 5 × 144 = 3 × 14 = Certain wrong answers may be shown to be wrong in many ways; _e.g._, 432,720 is too big, for 35 times a thousand square inches is only 35,000; 1152 is too small, for 35 times a hundred square inches would be 3500, or more than 1152. The time spent in realizing the problem here is fairly well spent because (1) any successful original manipulation in this case represents an excellent exercise of thought, because (2) failures show that it is useless to juggle the figures at random, and because (3) the previous experience with short multiplication makes it possible for the pupils to realize the problem in a very few minutes. It may, however, be still better to give the pupils the right method just as soon as the problem is realized, without having them spend more time in trying to solve it. Thus:-- 1 square foot has 144 square inches. How many square inches are there in 35 square feet (marked out in chalk on the floor as a piece 10 ft. × 3 ft. plus a piece 5 ft. × 1 ft.)? 1 yard = 36 inches. How many inches long is this wall (found by measure to be 13 yards)? Here is a quick way to find the answers:-- 144 35 ---- 720 432 ---- 5040 sq. inches in 35 sq. ft. 36 13 --- 108 36 --- 468 inches in 13 yd. Consider now the following introduction to dividing by a decimal:-- Dividing by a Decimal 1. How many minutes will it take a motorcycle, to go 12.675 miles at the rate of .75 mi. per minute? 16.9 ______ .75|12.675 7 5 --- 5 17 4 50 ---- 675 675 --- 2. Check by multiplying 16.9 by .75. 3. How do you know that the quotient cannot be as little as 1.69? 4. How do you know that the quotient cannot be as large as 169? 5. Find the quotient for 3.75 ÷ 1.5. 6. Check your result by multiplying the quotient by the divisor. 7. How do you know that the quotient cannot be .25 or 25? ____ 8. Look at this problem. .25|7.5 How do you know that 3.0 is wrong for the quotient? How do you know that 300 is wrong for the quotient? State which quotient is right for each of these:-- .021 or .21 or 2.1 or 21 or 210 ______ 9. 1.8|3.78 .021 or .21 or 21 or 210 ______ 10. 1.8|37.8 .03 or .3 or 3 or 30 or 300 ______ 11. 1.25|37.5 .03 or .3 or 3 or 30 or 300 ______ 12. 12.5|37.5 .05 or .5 or 5 or 50 or 500 ______ 13. 1.25|6.25 .05 or .5 or 5 or 50 or 500 ______ 14. 12.5|6.25 15. Is this rule true? If it is true, learn it. #In a correct result, the number of decimal places in the divisor and quotient together equals the number of decimal places in the dividend.# These and similar exercises excite the problem attitude in children _who have a general interest in getting right answers_. Such a series carefully arranged is a desirable introduction to a statement of the rule for placing the decimal point in division with decimals. For it attracts attention to the general principle (divisor × quotient should equal dividend), which is more important than the rule for convenient location of the decimal point, and it gives training in placing the point by inspection of the divisor, quotient, and dividend, which suffices for nineteen out of twenty cases that the pupil will ever encounter outside of school. He is likely to remember this method by inspection long after he has forgotten the fixed rule. It is well for the pupil to be introduced to many arithmetical facts by way of problems about their common uses. The clockface, the railroad distance table in hundredths of a mile, the cyclometer and speedometer, the recipe, and the like offer problems which enlist his interest and energy and also connect the resulting arithmetical learning with the activities where it is needed. There is no time cost, but a time-saving, for the learning as a means to the solution of the problems is quicker than the mere learning of the arithmetical facts by themselves alone. A few samples of such procedure are shown below:-- GRADE 3 To be Done at Home Look at a watch. Has it any hands besides the hour hand and the minute hand? Find out all that you can about how a watch tells seconds, how long a second is, and how many seconds make a minute. GRADE 5 Measuring Rainfall =Rainfall per Week= (=cu. in. per sq. in. of area=) June 1-7 1.056 8-14 1.103 15-21 1.040 22-28 .960 29-July 5 .915 July 6-12 .782 13-19 .790 20-26 .670 27-Aug. 2 .503 Aug. 3-9 .512 10-16 .240 17-23 .215 24-30 .811 1. In which weeks was the rainfall 1 or more? 2. Which week of August had the largest rainfall for that month? 3. Which was the driest week of the summer? (Driest means with the least rainfall.) 4. Which week was the next to the driest? 5. In which weeks was the rainfall between .800 and 1.000? 6. Look down the table and estimate whether the average rainfall for one week was about .5, or about .6, or about .7, or about .8, or about .9. Dairy Records =Record of Star Elsie= Pounds of Milk Butter-Fat per Pound of Milk Jan. 1742 .0461 Feb. 1690 .0485 Mar. 1574 .0504 Apr. 1226 .0490 May 1202 .0466 June 1251 .0481 Read this record of the milk given by the cow Star Elsie. The first column tells the number of pounds of milk given by Star Elsie each month. The second column tells what fraction of a pound of butter-fat each pound of milk contained. 1. Read the first line, saying, "In January this cow gave 1742 pounds of milk. There were 461 ten thousandths of a pound of butter-fat per pound of milk." Read the other lines in the same way. 2. How many pounds of butter-fat did the cow produce in Jan.? 3. In Feb.? 4. In Mar.? 5. In Apr.? 6. In May? 7. In June? GRADE 5 OR LATER Using Recipes to Make Larger or Smaller Quantities I. State how much you would use of each material in the following recipes: (_a_) To make double the quantity. (_b_) To make half the quantity. (_c_) To make 1-1/2 times the quantity. You may use pencil and paper when you cannot find the right amount mentally. 1. PEANUT PENUCHE 1 tablespoon butter 2 cups brown sugar 1/3 cup milk or cream 3/4 cup chopped peanuts 1/3 teaspoon salt 2. MOLASSES CANDY 1/2 cup butter 2 cups sugar 1 cup molasses 1-1/2 cups boiling water 3. RAISIN OPERA CARAMELS 2 cups light brown sugar 7/8 cup thin cream 1/2 cup raisins 4. WALNUT MOLASSES SQUARES 2 tablespoons butter 1 cup molasses 1/3 cup sugar 1/2 cup walnut meats 5. RECEPTION ROLLS 1 cup scalded milk 1-1/2 tablespoons sugar 1 teaspoon salt 1/4 cup lard 1 yeast cake 1/4 cup lukewarm water White of 1 egg 3-1/2 cups flour 6. GRAHAM RAISED LOAF 2 cups milk 6 tablespoons molasses 1-1/2 teaspoons salt 1/3 yeast cake 1/4 cup lukewarm water 2 cups sifted Graham flour 1/2 cup Graham bran 3/4 cup flour (to knead) II. How much would you use of each material in the following recipes: (_a_) To make 2/3 as large a quantity? (_b_) To make 1-1/3 times as much? (_c_) To make 2-1/2 times as much? 1. ENGLISH DUMPLINGS 1/2 pound beef suet 1-1/4 cups flour 3 teaspoons baking powder 1 teaspoon salt 1/2 teaspoon pepper 1 teaspoon minced parsley 1/2 cup cold water 2. WHITE MOUNTAIN ANGEL CAKE 1-1/2 cups egg whites 1-1/2 cups sugar 1 teaspoon cream of tartar 1 cup bread flour 1/4 teaspoon salt 1 teaspoon vanilla In many cases arithmetical facts and principles can be well taught in connection with some problem or project which is not arithmetical, but which has special potency to arouse an intellectual activity in the pupil which by some ingenuity can be directed to arithmetical learning. Playing store is the most fundamental case. Planning for a party, seeing who wins a game of bean bag, understanding the calendar for a month, selecting Christmas presents, planning a picnic, arranging a garden, the clock, the watch with second hand, and drawing very simple maps are situations suggesting problems which may bring a living purpose into arithmetical learning in grade 2. These are all available under ordinary conditions of class instruction. A sample of such problems for a higher grade (6) is shown below. Estimating Areas The children in the geography class had a contest in estimating the areas of different surfaces. Each child wrote his estimates for each of these maps, A, B, C, D, and E. (Only C and D are shown here.) In the arithmetic class they learned how to find the exact areas. Then they compared their estimates with the exact areas to find who came nearest. [Illustration] Write your estimates for A, B, C, D, and E. Then study the next 6 pages and learn how to find the exact areas. (The next 6 pages comprise training in the mensuration of parallelograms and triangles.) In some cases the affairs of individual pupils include problems which may be used to guide the individual in question to a zealous study of arithmetic as a means of achieving his purpose--of making a canoe, surveying an island, keeping the accounts of a Girls' Canning Club, or the like. It requires much time and very great skill to direct the work of thirty or more pupils each busy with a special type of his own, so as to make the work instructive for each, but in some cases the expense of time and skill is justified. GENERAL PRINCIPLES In general what should be meant when one says that it is desirable to have pupils in the problem-attitude when they are studying arithmetic is substantially as follows:-- _First._--Information that comes as an answer to questions is better attended to, understood, and remembered than information that just comes. _Second._--Similarly, movements that come as a step toward achieving an end that the pupil has in view are better connected with their appropriate situations, and such connections are longer retained, than is the case with movements that just happen. _Third._--The more the pupil is set toward getting the question answered or getting the end achieved, the greater is the satisfyingness attached to the bonds of knowledge or skill which mean progress thereto. _Fourth._--It is bad policy to rely exclusively on the purely intellectualistic problems of "How can I do this?" "How can I get the right answer?" "What is the reason for this?" "Is there a better way to do that?" and the like. It is bad policy to supplement these intellectualistic problems by only the remote problems of "How can I be fitted to earn a higher wage?" "How can I make sure of graduating?" "How can I please my parents?" and the like. The purely intellectualistic problems have too weak an appeal for many pupils; the remote problems are weak so long as they are remote and, what is worse, may be deprived of the strength that they would have in due time if we attempt to use them too soon. It is the extreme of bad policy to neglect those personal and practical problems furnished by life outside the class in arithmetic the solution of which can really be furthered by arithmetic then and there. It is good policy to spend time in establishing certain mental sets--stimulating, or even creating, certain needs--setting up problems themselves--when the time so spent brings a sufficient improvement in the quality and quantity of the pupils' interest in arithmetical learning. _Fifth._--It would be still worse policy to rely exclusively on problems arising outside arithmetic. To learn arithmetic is itself a series of problems of intrinsic interest and worth to healthy-minded children. The need for ability to multiply with United States money or to add fractions or to compute percents may be as truly vital and engaging as the need for skill to make a party dress or for money to buy it or for time to play baseball. The intellectualistic needs and problems should be considered along with all others, and given whatever weight their educational value deserves. DIFFICULTY AND SUCCESS AS STIMULI There are certain misconceptions of the doctrine of the problem-attitude. The most noteworthy is that difficulty--temporary failure--an inadequacy of already existing bonds--is the essential and necessary stimulus to thinking and learning. Dewey himself does not, as I understand him, mean this, but he has been interpreted as meaning it by some of his followers.[22] [22] In his _How We Think_. Difficulty--temporary failure, inadequacy of existing bonds--on the contrary does nothing whatsoever in and of itself; and what is done by the annoying lack of success which sometimes accompanies difficulty sometimes hinders thinking and learning. Mere difficulty, mere failure, mere inadequacy of existing bonds, does nothing. It is hard for me to add three eight-place numbers at a glance; I have failed to find as effective illustrations for pages 276 to 277 as I wished; my existing sensori-motor connections are inadequate to playing a golf course in 65. But these events and conditions have done nothing to stimulate me in respect to the behavior in question. In the first of the three there is no annoying lack and no dynamic influence at all; in the second there was to some degree an annoying lack--a slight irritation at not getting just what I wanted,--and this might have impelled me to further thinking (though it did not, and getting one tiptop illustration would as a rule stimulate me to hunt for others more than failing to get such). In the third case the lack of the 65 does not annoy me or have any noteworthy dynamic effect. The lack of 90 instead of 95-100 is annoying and is at times a stimulus to further learning, though not nearly so strong a stimulus as the attainment of the 90 would be! At other times this annoying lack is distinctly inhibitory--a stimulus to ceasing to learn. In the intellectual life the inhibitory effect seems far the commoner of the two. Not getting answers seems as a rule to make us stop trying to get them. The annoying lack of success with a theoretical problem most often makes us desert it for problems to whose solution the existing bonds promise to be more adequate. The real issue in all this concerns the relative strength, in the pupil's intellectual life, of the "negative reaction" of behavior in general. An animal whose life processes are interfered with so that an annoying state of affairs is set up, changes his behavior, making one after another responses as his instincts and learned tendencies prescribe, until the annoying state of affairs is terminated, or the animal dies, or suffers the annoyance as less than the alternatives which his responses have produced. When the annoying state of affairs is characterized by the failure of things as they are to minister to a craving--as in cases of hunger, loneliness, sex-pursuit, and the like,--we have stimulus to action by an annoying lack or need, with relief from action by the satisfaction of the need. Such is in some measure true of man's intellectual life. In recalling a forgotten name, in solving certain puzzles, or in simplifying an algebraic complex, there is an annoying lack of the name, solution, or factor, a trial of one after another response, until the annoyance is relieved by success or made less potent by fatigue or distraction. Even here the _difficulty_ does not do anything--but only the annoying interference with our intellectual peace by the problem. Further, although for the particular problem, the annoying lack stimulates, and the successful attainment stops thinking, the later and more important general effect on thinking is the reverse. Successful attainment stops our thinking _on that problem_ but makes us more predisposed later to thinking _in general_. Overt negative reaction, however, plays a relatively small part in man's intellectual life. Filling intellectual voids or relieving intellectual strains in this way is much less frequent than being stimulated positively by things seen, words read, and past connections acting under modified circumstances. The notion of thinking as coming to a lack, filling it, meeting an obstacle, dodging it, being held up by a difficulty and overcoming it, is so one-sided as to verge on phantasy. The overt lacks, strains, and difficulties come perhaps once in five hours of smooth straightforward use and adaptation of existing connections, and they might as truly be called hindrances to thought--barriers which past successes help the thinker to surmount. Problems themselves come more often as cherished issues which new facts reveal, and whose contemplation the thinker enjoys, than as strains or lacks or 'problems which I need to solve.' It is just as true that the thinker gets many of his problems as results from, or bonuses along with, his information, as that he gets much of his information as results of his efforts to solve problems. As between difficulty and success, success is in the long run more productive of thinking. Necessity is not the mother of invention. Knowledge of previous inventions is the mother; original ability is the father. The solutions of previous problems are more potent in producing both new problems and their solutions than is the mere awareness of problems and desire to have them solved. In the case of arithmetic, learning to cancel instead of getting the product of the dividends and the product of the divisors and dividing the former by the latter, is a clear case of very valuable learning, with ease emphasized rather than difficulty, with the adequacy of existing bonds (when slightly redirected) as the prime feature of the process rather than their inadequacy, and with no sense of failure or lack or conflict. It would be absurd to spend time in arousing in the pupil, before beginning cancellation, a sense of a difficulty--viz., that the full multiplying and dividing takes longer than one would like. A pupil in grade 4 or 5 might well contemplate that difficulty for years to no advantage. He should at once begin to cancel and prove by checking that errorless cancellation always gives the right answer. To emphasize before teaching cancellation the inadequacy of the old full multiplying and dividing would, moreover, not only be uneconomical as a means to teaching cancellation; it would amount to casting needless slurs on valuable past acquisitions, and it would, scientifically, be false. For, until a pupil has learned to cancel, the old full multiplying is not inadequate; it is admirable in every respect. The issue of its inadequacy does not truly appear until the new method is found. It is the best way until the better way is mastered. In the same way it is unwise to spend time in making pupils aware of the annoying lacks to be supplied by the multiplication tables, the division tables, the casting out of nines, or the use of the product of the length and breadth of a rectangle as its area, the unit being changed to the square erected on the linear unit as base. The annoying lack will be unproductive, while the learning takes place readily as a modification of existing habits, and is sufficiently appreciated as soon as it does take place. The multiplication tables come when instead of merely counting by 7s from 0 up saying "7, 14, 21," etc., the pupil counts by 7s from 0 up saying "Two sevens make 14, three sevens make 21, four sevens make 28," etc. The division tables come as easy selections from the known multiplications; the casting out of nines comes as an easy device. The computation of the area of a rectangle is best facilitated, not by awareness of the lack of a process for doing it, but by awareness of the success of the process as verified objectively. In all these cases, too, the pupil would be misled if we aroused first a sense of the inadequacy of counting, adding, and objective division, an awareness of the difficulties which the multiplication and division tables and nines device and area theorem relieve. The displaced processes are admirable and no unnecessary fault should be found with them, and they are _not_ inadequate until the shorter ways have been learned. FALSE INFERENCES One false inference about the problem-attitude is that the pupil should always understand the aim or issue before beginning to form the bonds which give the method or process that provides the solution. On the contrary, he will often get the process more easily and value it more highly if he is taught what it is _for_ gradually while he is learning it. The system of decimal notation, for example, may better be taken first as a mere fact, just as we teach a child to talk without trying first to have him understand the value of verbal intercourse, or to keep clean without trying first to have him understand the bacteriological consequences of filth. A second inference--that the pupil should always be taught to care about an issue and crave a process for managing it before beginning to learn the process--is equally false. On the contrary, the best way to become interested in certain issues and the ways of handling them is to learn the process--even to learn it by sheer habituation--and then note what it does for us. Such is the case with ".1666-2/3 × = divide by 6," ".333-1/3 × = divide by 3," "multiply by .875 = divide the number by 8 and subtract the quotient from the number." A third unwise tendency is to degrade the mere giving of information--to belittle the value of facts acquired in any other way than in the course of deliberate effort by the pupil to relieve a problem or conflict or difficulty. As a protest against merely verbal knowledge, and merely memoriter knowledge, and neglect of the active, questioning search for knowledge, this tendency to belittle mere facts has been healthy, but as a general doctrine it is itself equally one-sided. Mere facts not got by the pupil's thinking are often of enormous value. They may stimulate to active thinking just as truly as that may stimulate to the reception of facts. In arithmetic, for example, the names of the numbers, the use of the fractional form to signify that the upper number is divided by the lower number, the early use of the decimal point in U. S. money to distinguish dollars from cents, and the meanings of "each," "whole," "part," "together," "in all," "sum," "difference," "product," "quotient," and the like are self-justifying facts. A fourth false inference is that whatever teaching makes the pupil face a question and think out its answer is thereby justified. This is not necessarily so unless the question is a worthy one and the answer that is thought out an intrinsically valuable one and the process of thinking used one that is appropriate for that pupil for that question. Merely to think may be of little value. To rely much on formal discipline is just as pernicious here as elsewhere. The tendency to emphasize the methods of learning arithmetic at the expense of what is learned is likely to lead to abuses different in nature but as bad in effect as that to which the emphasis on disciplinary rather than content value has led in the study of languages and grammar, or in the old puzzle problems of arithmetic. The last false inference that I shall discuss here is the inference that most of the problems by which arithmetical learning is stimulated had better be external to arithmetic itself--problems about Noah's Ark or Easter Flowers or the Merry Go Round or A Trip down the Rhine. Outside interests should be kept in mind, as has been abundantly illustrated in this volume, but it is folly to neglect the power, even for very young or for very stupid children, of the problem "How can I get the right answer?" Children do have intellectual interests. They do like dominoes, checkers, anagrams, and riddles as truly as playing tag, picking flowers, and baking cake. With carefully graded work that is within their powers they like to learn to add, subtract, multiply, and divide with integers, fractions, and decimals, and to work out quantitative relations. In some measure, learning arithmetic is like learning to typewrite. The learner of the latter has little desire to present attractive-looking excuses for being late, or to save expense for paper. He has no desire to hoard copies of such and such literary gems. He may gain zeal from the fact that a school party is to be given and invitations are to be sent out, but the problem "To typewrite better" is after all his main problem. Learning arithmetic is in some measure a game whose moves are motivated by the general set of the mind toward victory--winning right answers. As a ball-player learns to throw the ball accurately to first-base, not primarily because of any particular problem concerning getting rid of the ball, or having the man at first-base possess it, or putting out an opponent against whom he has a grudge, but because that skill is required by the game as a whole, so the pupil, in some measure, learns the technique of arithmetic, not because of particular concrete problems whose solutions it furnishes, but because that technique is required by the game of arithmetic--a game that has intrinsic worth and many general recommendations. CHAPTER XV INDIVIDUAL DIFFERENCES The general facts concerning individual variations in abilities--that the variations are large, that they are continuous, and that for children of the same age they usually cluster around one typical or modal ability, becoming less and less frequent as we pass to very high or very low degrees of the ability--are all well illustrated by arithmetical abilities. NATURE AND AMOUNT The surfaces of frequency shown in Figs. 61, 62, and 63 are samples. In these diagrams each space along the baseline represents a certain score or degree of ability, and the height of the surface above it represents the number of individuals obtaining that score. Thus in Fig. 61, 63 out of 1000 soldiers had no correct answer, 36 out of 1000 had one correct answer, 49 had two, 55 had three, 67 had four, and so on, in a test with problems (stated in words). Figure 61 shows that these adults varied from no problems solved correctly to eighteen, around eight as a central tendency. Figure 62 shows that children of the same year-age (they were also from the same neighborhood and in the same school) varied from under 40 to over 200 figures correct. Figure 63 shows that even among children who have all reached the same school grade and so had rather similar educational opportunities in arithmetic, the variation is still very great. It requires a range from 15 to over 30 examples right to include even nine tenths of them. [Illustration: FIG. 61.--The scores of 1000 soldiers in the National Army born in English-speaking countries, in Test 2 of the Army Alpha. The score is the number of correct answers obtained in five minutes. Probably 10 to 15 percent of these men were unable to read or able to read only very easy sentences at a very slow rate. Data furnished by the Division of Psychology in the office of the Surgeon General.] It should, however, be noted that if each individual had been scored by the average of his work on eight or ten different days instead of by his work in just one test, the variability would have been somewhat less than appears in Figs. 61, 62, and 63. [Illustration: FIG. 62.--The scores of 100 11-year-old pupils in a test of computation. Estimated from the data given by Burt ['17, p. 68] for 10-, 11-, and 12-year-olds. The score equals the number of correct figures.] It is also the case that if each individual had been scored, not in problem-solving alone or division alone, but in an elaborate examination on the whole field of arithmetic, the variability would have been somewhat less than appears in Figs. 61, 62, and 63. On the other hand, if the officers and the soldiers rejected for feeblemindedness had been included in Fig. 61, if the 11-year-olds in special classes for the very dull had been included in Fig. 62, and if all children who had been to school six years had been included in Fig. 63, no matter what grade they had reached, the effect would have been to _increase_ the variability. [Illustration: FIG. 63.--The scores of pupils in grade 6 in city schools in the Woody Division Test A. The score is the number of correct answers obtained in 20 minutes. From Woody ['16, p. 61].] In spite of the effort by school officers to collect in any one school grade those somewhat equal in ability or in achievement or in a mixture of the two, the population of the same grades in the same school system shows a very wide range in any arithmetical ability. This is partly because promotion is on a more general basis than arithmetical ability so that some very able arithmeticians are deliberately held back on account of other deficiencies, and some very incompetent arithmeticians are advanced on account of other excellencies. It is partly because of general inaccuracy in classifying and promoting pupils. In a composite score made up of the sum of the scores in Woody tests,--Add. A, Subt. A, Mult. A, and Div. A, and two tests in problem-solving (ten and six graded problems, with maximum attainable credits of 30 and 18), Kruse ['18] found facts from which I compute those of Table 13, and Figs. 64 to 66, for pupils all having the training of the same city system, one which sought to grade its pupils very carefully. [Illustration: FIGS. 64, 65, and 66.--The scores of pupils in grade 6 (Fig. 64), grade 7 (Fig. 65), and grade 8 (Fig. 66) in a composite of tests in computation and problem-solving. The time was about 120 minutes. The maximum score attainable was 196.] The overlapping of grade upon grade should be noted. Of the pupils in grade 6 about 18 percent do better than the average pupil in grade 7, and about 7 percent do better than the average pupil in grade 8. Of the pupils in grade 8 about 33 percent do worse than the average pupil in grade 7 and about 12 percent do worse than the average pupil in grade 6. TABLE 13 RELATIVE FREQUENCIES OF SCORES IN AN EXTENSIVE TEAM OF ARITHMETICAL TESTS.[23] IN PERCENTS ============================================== SCORE | GRADE 6 | GRADE 7 | GRADE 8 ------------+-----------+-----------+--------- 70 to 79 | 1.3 | .9 | .4 80 " 89 | 5.5 | 2.3 | .4 90 " 99 | 10.6 | 4.3 | 2.9 100 " 109 | 19.4 | 5.2 | 4.4 110 " 119 | 19.8 | 18.5 | 5.8 120 " 129 | 23.5 | 16.2 | 16.8 130 " 139 | 12.6 | 17.5 | 16.8 140 " 149 | 4.6 | 13.9 | 22.9 150 " 159 | 1.7 | 13.6 | 17.1 160 " 169 | 1.2 | 4.8 | 9.4 170 " 179 | | 2.5 | 3.3 ============================================== [23] Compiled from data on p. 89 of Kruse ['18]. DIFFERENCES WITHIN ONE CLASS The variation within a single class for which a single teacher has to provide is great. Even when teaching is departmental and promotion is by subjects, and when also the school is a large one and classification within a grade is by ability--there may be a wide range for any given special component ability. Under ordinary circumstances the range is so great as to be one of the chief limiting conditions for the teaching of arithmetic. Many methods appropriate to the top quarter of the class will be almost useless for the bottom quarter, and _vice versa_. [Illustration: FIGS. 67 and 68.--The scores of ten 6 B classes in a 12-minute test in computation with integers (the Courtis Test 7). The score is the number of units done. Certain long tasks are counted as two units.] Figures 67 and 68 show the scores of ten classes taken at random from ninety 6 B classes in one city by Courtis ['13, p. 64] in amount of computation done in 12 minutes. Observe the very wide variation present in the case of every class. The variation within a class would be somewhat reduced if each pupil were measured by his average in eight or ten such tests given on different days. If a rather generous allowance is made for this we still have a variation in speed as great as that shown in Fig. 69, as the fact to be expected for a class of thirty-two 6 B pupils. [Illustration: FIG. 69.--A conservative estimate of the amount of variation to be expected within a single class of 32 pupils in grade 6, in the number of units done in Courtis Test 7 when all chance variations are eliminated.] The variations within a class in respect to what processes are understood so as to be done with only occasional errors may be illustrated further as follows:--A teacher in grade 4 at or near the middle of the year in a city doing the customary work in arithmetic will probably find some pupil in her class who cannot do column addition even without carrying, or the easiest written subtraction (8 9 78) (5 3 or 37) (- - --), who does not know his multiplication tables or how to derive them, or understand the meanings of + - × and ÷, or have any useful ideas whatever about division. There will probably be some child in the class who can do such work as that shown below, and with very few errors. Add 3/8 + 5/8 + 7/8 + 1/8 2-1/2 1/6 + 3/8 6-3/8 3-3/4 ----- Subtract 10.00 4 yd. 1 ft. 6 in. 3.49 2 yd. 2 ft. 3 in. ----- ---------------------- Multiply 1-1/4 × 8 16 145 2-5/8 206 ------ --- _______ _____ Divide 2)13.50 25)9750 The invention of means of teaching thirty so different children at once with the maximum help and minimum hindrance from their different capacities and acquisitions is one of the great opportunities for applied science. Courtis, emphasizing the social demand for a certain moderate arithmetical attainment in the case of nearly all elementary school children of, say, grade 6, has urged that definite special means be taken to bring the deficient children up to certain standards, without causing undesirable 'overlearning' by the more gifted children. Certain experimental work to this end has been carried out by him and others, but probably much more must be done before an authoritative program for securing certain minimum standards for all or nearly all pupils can be arranged. THE CAUSES OF INDIVIDUAL DIFFERENCES The differences found among children of the same grade in the same city are due in large measure to inborn differences in their original natures. If, by a miracle, the children studied by Courtis, or by Woody, or by Kruse had all received exactly the same nurture from birth to date, they would still have varied greatly in arithmetical ability, perhaps almost as much as they now do vary. The evidence for this is the general evidence that variation in original nature is responsible for much of the eventual variation found in intellectual and moral traits, plus certain special evidence in the case of arithmetical abilities themselves. Thorndike found ['05] that in tests with addition and multiplication twins were very much more alike than siblings[24] two or three years apart in age, though the resemblance in home and school training in arithmetic should be nearly as great for the latter as for the former. Also the young twins (9-11) showed as close a resemblance in addition and multiplication as the older twins (12-15), although the similarities of training in arithmetic have had twice as long to operate in the latter case. [24] Siblings is used for children of the same parents. If the differences found, say among children in grade 6 in addition, were due to differences in the quantity and quality of training in addition which they have had, then by giving each of them 200 minutes of additional identical training the differences should be reduced. For the 200 minutes of identical training is a step toward equalizing training. It has been found in many investigations of the matter that when we make training in arithmetic more nearly equal for any group the variation within the group is not reduced. On the contrary, equalizing training seems rather to increase differences. The superior individual seems to have attained his superiority by his own superiority of nature rather than by superior past training, for, during a period of equal training for all, he increases his lead. For example, compare the gains of different individuals due to about 300 minutes of practice in mental multiplication of a three-place number by a three-place number shown in Table 14 below, from data obtained by the author ['08].[25] [25] Similar results have been obtained in the case of arithmetical and other abilities by Thorndike ['08, '10, '15, '16], Whitley ['11], Starch ['11], Wells ['12], Kirby ['13], Donovan and Thorndike ['13], Hahn and Thorndike ['14], and on a very large scale by Race in a study as yet unpublished. TABLE 14 THE EFFECT OF EQUAL AMOUNTS OF PRACTICE UPON INDIVIDUAL DIFFERENCE IN THE MULTIPLICATION OF THREE-PLACE NUMBERS ==================================================================== | AMOUNT | PERCENTAGE OF | |CORRECT FIGURES |----------------+--------------- | Initial | | Initial | | Score | Gain | Score | Gain -----------------------------------+---------+------+---------+----- Initially highest five individuals | 85 | 61 | 70 | 18 next five " | 56 | 51 | 68 | 10 next six " | 46 | 22 | 74 | 8 next six " | 38 | 8 | 58 | 12 next six " | 29 | 24 | 56 | 14 ==================================================================== THE INTERRELATIONS OF INDIVIDUAL DIFFERENCES Achievement in arithmetic depends upon a number of different abilities. For example, accuracy in copying numbers depends upon eyesight, ability to perceive visual details, and short-term memory for these. Long column addition depends chiefly upon great strength of the addition combinations especially in higher decades, 'carrying,' and keeping one's place in the column. The solution of problems framed in words requires understanding of language, the analysis of the situation described into its elements, the selection of the right elements for use at each step and their use in the right relations. Since the abilities which together constitute arithmetical ability are thus specialized, the individual who is the best of a thousand of his age or grade in respect to, say, adding integers, may occupy different stations, perhaps from 1st to 600th, in multiplying with integers, placing the decimal point in division with decimals, solving novel problems, copying figures, etc., etc. Such specialization is in part due to his having had, relatively to the others in the thousand, more or better training in certain of these abilities than in others, and to various circumstances of life which have caused him to have, relatively to the others in the thousand, greater interest in certain of these achievements than in others. The specialization is not wholly due thereto, however. Certain inborn characteristics of an individual predispose him to different degrees of superiority or inferiority to other men in different features of arithmetic. We measure the extent to which ability of one sort goes with or fails to go with ability of some other sort by the coefficient of correlation between the two. If every individual keeps the same rank in the second ability--if the individual who is the best of the thousand in one is the best of the group in the other, and so on down the list--the correlation is 1.00. In proportion as the ranks of individuals vary in the two abilities the coefficient drops from 1.00, a coefficient of 0 meaning that the best individual in ability A is no more likely to be in first place in ability B than to be in any other rank. The meanings of coefficients of correlation of .90, .70, .50, and 0 are shown by Tables 15, 16, 17 and 18.[26] [26] Unless he has a thorough understanding of the underlying theory, the student should be very cautious in making inferences from coefficients of correlation. TABLE 15 DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .90 ====================================================================== |10TH |9TH |8TH |7TH |6TH |5TH |4TH |3D |2D |1ST ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+----- 1st tenth | | | | | .1 | .4 | 1.8 | 6.6 |22.4 |68.7 2d tenth | | | .1 | .4 | 1.4 | 4.7 |11.5 |23.5 |36.0 |22.4 3d tenth | | .1 | .5 | 2.1 | 5.8 |12.8 |21.1 |27.4 |23.5 | 6.6 4th tenth | | .4 | 2.1 | 6.4 |12.8 |20.1 |23.8 |21.2 |11.5 | 1.8 5th tenth | .1 | 1.4 | 5.8 |12.8 |19.3 |22.6 |20.1 |12.8 | 4.7 | .4 6th tenth | .4 | 4.7 |12.8 |20.1 |22.6 |19.3 |12.8 | 5.8 | 1.4 | .1 7th tenth | 1.8 |11.5 |21.2 |23.8 |20.1 |12.8 | 6.4 | 2.1 | .4 | 8th tenth | 6.6 |23.5 |27.4 |21.1 |12.8 | 5.8 | 2.1 | .5 | .1 | 9th tenth |22.4 |36.0 |23.5 |11.5 | 4.7 | 1.4 | .4 | .1 | | 10th tenth|68.7 |22.4 | 6.6 | 1.8 | .4 | .1 | | | | ====================================================================== TABLE 16 DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .70 ====================================================================== |10TH |9TH |8TH |7TH |6TH |5TH |4TH |3D |2D |1ST ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+----- 1st tenth | | .2 | .7 | 1.5 | 2.8 | 4.8 | 8.0 |13.0 |22.3 |46.7 2d tenth | .2 | 1.2 | 2.6 | 4.5 | 7.0 | 9.8 |13.4 |17.3 |21.7 |22.3 3d tenth | .7 | 2.6 | 5.0 | 7.3 |10.0 |12.5 |14.9 |16.7 |17.3 |13.0 4th tenth | 1.5 | 4.5 | 7.3 | 9.8 |12.0 |13.7 |14.8 |14.9 |13.4 | 8.0 5th tenth | 2.8 | 7.0 |10.0 |12.0 |13.4 |14.0 |13.7 |12.5 | 9.8 | 4.8 6th tenth | 4.8 | 9.8 |12.5 |13.7 |14.0 |13.4 |12.0 |10.0 | 7.0 | 2.8 7th tenth | 8.0 |13.4 |14.9 |14.8 |13.7 |12.0 | 9.8 | 7.3 | 4.5 | 1.5 8th tenth |13.0 |17.3 |16.7 |14.9 |12.5 |10.0 | 7.3 | 5.0 | 2.6 | .7 9th tenth |22.3 |21.7 |17.3 |13.4 | 9.8 | 7.0 | 4.5 | 2.6 | 1.2 | .2 10th tenth|46.7 |22.3 |13.0 | 8.0 | 4.8 | 2.8 | 1.5 | .7 | .2 | ====================================================================== TABLE 17 DISTRIBUTION OF ARRAYS OF SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .50 ====================================================================== |10TH |9TH |8TH |7TH |6TH |5TH |4TH |3D |2D |1ST ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+----- 1st tenth | .8 | 2.0 | 3.2 | 4.6 | 6.2 | 8.1 |10.5 |13.9 |18.0 |31.8 2d tenth | 2.0 | 4.1 | 5.7 | 7.3 | 8.8 |10.5 |12.2 |14.1 |16.4 |18.9 3d tenth | 3.2 | 5.7 | 7.4 | 8.9 |10.0 |11.2 |12.3 |13.3 |14.1 |13.9 4th tenth | 4.6 | 7.3 | 8.8 | 9.9 |10.8 |11.6 |12.0 |12.3 |12.2 |10.5 5th tenth | 6.2 | 8.8 |10.0 |10.8 |11.3 |11.5 |11.6 |11.2 |10.5 | 8.1 6th tenth | 8.1 |10.5 |11.2 |11.6 |11.5 |11.3 |10.8 |10.0 | 8.8 | 6.2 7th tenth |10.5 |12.2 |12.3 |12.0 |11.6 |10.8 | 9.9 | 8.8 | 7.5 | 4.6 8th tenth |13.9 |14.1 |13.3 |12.3 |11.2 |10.0 | 8.8 | 7.4 | 5.7 | 3.2 9th tenth |18.9 |16.4 |14.1 |12.2 |10.5 | 8.8 | 7.3 | 5.7 | 4.1 | 2.0 10th tenth|31.8 |18.9 |13.9 |10.5 | 8.1 | 6.2 | 4.6 | 3.2 | 2.0 | .8 ====================================================================== TABLE 18 DISTRIBUTION OF ARRAYS, IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .0 ====================================================================== |10TH |9TH |8TH |7TH |6TH |5TH |4TH |3D |2D |1ST ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+----- 1st tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 2d tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 3d tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 4th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 5th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 6th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 7th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 8th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 9th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 10th tenth|10 |10 |10 |10 |10 |10 |10 |10 |10 |10 ====================================================================== The significance of any coefficient of correlation depends upon the group of individuals for which it is determined. A correlation of .40 between computation and problem-solving in eighth-grade pupils of 14 years would mean a much closer real relation than a correlation of .40 in all 14-year-olds, and a very, very much closer relation than a correlation of .40 for all children 8 to 15. Unless the individuals concerned are very elaborately tested on several days, the correlations obtained are "attenuated" toward 0 by the "accidental" errors in the original measurements. This effect was not known until 1904; consequently the correlations in the earlier studies of arithmetic are all too low. In general, the correlation between ability in any one important feature of computation and ability in any other important feature of computation is high. If we make enough tests to measure each individual exactly in:-- (_A_) Subtraction with integers and decimals, (_B_) Multiplication with integers and decimals, (_C_) Division with integers and decimals, (_D_) Multiplication and division with common fractions, and (_E_) Computing with percents, we shall probably find the intercorrelations for a thousand 14-year-olds to be near .90. Addition of integers (_F_) will, however, correlate less closely with any of the above, being apparently dependent on simpler and more isolated abilities. The correlation between problem-solving (_G_) and computation will be very much less, probably not over .60. It should be noted that even when the correlation is as high as .90, there will be some individuals very high in one ability and very low in the other. Such disparities are to some extent, as Courtis ['13, pp. 67-75] and Cobb ['17] have argued, due to inborn characteristics of the individual in question which predispose him to very special sorts of strength and weakness. They are often due, however, to defects in his learning whereby he has acquired more ability than he needs in one line of work or has failed to acquire some needed ability which was well within his capacity. In general, all correlations between an individual's divergence from the common type or average of his age for one arithmetical function, and his divergences from the average for any other arithmetical function, are positive. The correlation due to original capacity more than counterbalances the effects that robbing Peter to pay Paul may have. Speed and accuracy are thus positively correlated. The individuals who do the most work in ten minutes will be above the average in a test of accuracy. The common notion that speed is opposed to accuracy is correct when it means that the same person will tend to make more errors if he works at too rapid a rate; but it is entirely wrong when it means that the kind of person who works more rapidly than the average person is likely to be less accurate than the average person. Interest in arithmetic and ability at arithmetic are probably correlated positively in the sense that the pupil who has more interest than other pupils of his age tends in the long run to have more ability than they. They are certainly correlated in the sense that the pupil who 'likes' arithmetic better than geography or history tends to have relatively more ability in arithmetic, or, in other words, that the pupil who is more gifted at arithmetic than at drawing or English tends also to like it better than he likes these. These correlations are high. It is correct then to think of mathematical ability as, in a sense, a unitary ability of which any one individual may have much or little, most individuals possessing a moderate amount of it. This is consistent, however, with the occasional appearance of individuals possessed of very great talents for this or that particular feature of mathematical ability and equally notable deficiencies in other features. Finally it may be noted that ability in arithmetic, though occasionally found in men otherwise very stupid, is usually associated with superior intelligence in dealing with ideas and symbols of all sorts, and is one of the best early indications thereof. BIBLIOGRAPHY OF REFERENCES MADE IN THE TEXT Ames, A. F., and McLellan, J. F.; '00 Public School Arithmetic. Ballou, F. W.; '16 Determining the Achievements of Pupils in the Addition of Fractions. School Document No. 3, 1916, Boston Public Schools. Brandell, G.; '13 Skolbarns intressen. Translated ['15] by W. 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McLellan, J.A., and Ames, A.F.; '00 Public School Arithmetic. Messenger, J.F.; '03 The Perception of Number. Psychological Review, Monograph Supplement No. 22. Meumann, E.; '07 Vorlesungen zur Einführung in die experimentelle Pädagogik. Mitchell, H.E.; '20 Unpublished studies of the uses of arithmetic in factories, shops, farms, and the like. Monroe, W.S., De Voss, J.C., and Kelly, F.J.; '17 Educational Tests and Measurements. Nanu, H.A.; '04 Zur Psychologie der Zahl Auffassung. National Intelligence Tests; '20 Scale A, Form 1, Edition 1. Phillips, D.E.; '97 Number and Its Application Psychologically Considered. Pedagogical Seminary, vol. 5, pp. 221-281. Pommer, O.; '14 Die Erforschung der Beliebtheit der Unterrichtsfächer. Ihre psychologischen Grundlagen und ihre pädagog. Bedeutung. VII. Jahresber. des k.k. Ssaatsgymn. im XVIII Bez. v. Wien. Rice, J.M.; '02 Test in Arithmetic. Forum, vol. 34, pp. 281-297. Rice, J.M.; '03 Causes of Success and Failure in Arithmetic. Forum, vol. 34, pp. 437-452. Rush, G.P.; '17 The Scientific Measurement of Classroom Products. (With J. C. Chapman.) Seekel, E.; '14 Ueber die Beziehung zwischen der Beliebtheit und der Schwierigkeit der Schulfächer. Ergebnisse einer Erhebung. Zeitschrift für Angewandte Psychologie 9. S. 268-277. Selkin, F. B.; '12 Number Games Bordering on Arithmetic and Algebra. Teachers College Record, vol. 13, pp. 452-493. Smith, D. E.; '01 The Teaching of Elementary Mathematics. Smith, D. E.; '11 The Teaching of Arithmetic. Speer, W. W.; '97 Arithmetic: Elementary for Pupils. Starch, D.; '11 Transfer of Training in Arithmetical Operations. Journal of Educational Psychology, vol. 2, pp. 306-310. Starch, D.; '16 Educational Measurements. Stern, W.; '05 Ueber Beliebtheit und Unbeliebtheit der Schulfächer. Zeitschrift für pädagogische Psychologie, VII, 267-296. Stern, C., and Stern, W.; '13 Beliebtheit und Schwierigkeit der Schulfächer. (Freie Schulgemeinde Wickersdorf.) Auf Grund der von Herrn Luserke beschafften Materialien bearbeitet. In: "Die Ausstellung zur vergleichenden Jungendkunde der Geschlechter in Breslau." Arbeit 7 des Bundes für Schulreform. S. 24-26. Stern, W.; '14 Zur vergleichenden Jugendkunde der Geschlechter. Vortrag. III. Deutsch. Kongr. f. Jugendkunde usw. Arbeiten 8 des Bundes für Schulreform. S. 17-38. Stone, C.W.; '08 Arithmetical Abilities and Some Factors Determining Them. Teachers College Contributions to Education, No. 19. Suzzallo, H.; '11 The Teaching of Primary Arithmetic. Thorndike, E.L.; '00 Mental Fatigue. Psychological Review, vol. 7, pp. 466-482 and 547-579. Thorndike, E.L.; '08 The Effect of Practice in the Case of a Purely Intellectual Function. American Journal of Psychology, vol. 10, pp. 374-384. Thorndike, E.L.; '10 Practice in the case of Addition. American Journal of Psychology, vol. 21, pp. 483-486. Thorndike, E.L., and Donovan, M.E.; '13 Improvement in a Practice Experiment under School Conditions. American Journal of Psychology, vol. 24, pp. 426-428. Thorndike, E.L., and Donovan, M.E., and Hahn, H.H.; '14 Some Results of Practice in Addition under School Conditions. Journal of Educational Psychology, vol. 5, No. 2, pp. 65-84. Thorndike, E.L.; '15 The Relation between Initial Ability and Improvement in a Substitution Test. School and Society, vol. 12, p. 429. Thorndike, E.L.; '16 Notes on Practice, Improvability and the Curve of Work. American Journal of Psychology, vol 27, pp. 550-565. Walsh, J.H.; '06 Grammar School Arithmetic. Wells, F.L.; '12 The Relation of Practice to Individual Differences. American Journal of Psychology, vol. 23, pp. 75-88. White, E. E.; '83 A New Elementary Arithmetic. Whitley, M. T.; '11 An Empirical Study of Certain Tests for Individual Differences. Archives of Psychology, No. 19. Wiederkehr, G.; '07 Statistiche Untersuchungen über die Art und den Grad des Interesses bei Kindern der Volksschule. Neue Bahnen, vol. 19, pp. 241-251, 289-299. Wilson, G. M.; '19 A Survey of the Social and Business Usage of Arithmetic. Teachers College Contributions to Education, No. 100. Wilson, G. M., and Hoke, K. J.; '20 How to Measure. Woody, C.; '10 Measurements of Some Achievements in Arithmetic. Teachers College Contributions to Education, No. 80. INDEX Abilities, arithmetical, nature of, 1 ff.; measurement of, 27 ff.; constitution of, 51 ff.; organization of, 137 ff. Abstract numbers, 85 ff. Abstraction, 169 ff. Accuracy, in relation to speed, 31; in fundamental operations, 102 ff. Addition, measurement of, 27 ff., 34; constitution of, 52 f.; habit in relation to, 71 f.; in the higher decades, 75 f.; accuracy in, 108 f.; amount of practice in, 122 ff.; interest in 196 f. Aims of the teaching of arithmetic, 23 f. AMES, A. F., 89 Analysis, learning by, 169 ff.; systematic and opportunistic stimuli to, 178 f.; gradual progress in, 180 ff. Area, 257 f., 275 Arithmetic, sociology of, 24 ff. Arithmetical abilities. _See_ Abilities. Arithmetical language, 8 f., 19, 89 ff., 94 ff. Arithmetical learning, before school, 199 ff.; conditions of, 227 ff.; in relation to time of day, 227 ff.; in relation to time devoted to arithmetic, 228 ff. Arithmetical reasoning. _See_ Reasoning. Arithmetical terms, 8, 19 Averages, 40 f.; 135 f. BALLOU, F. W., 34, 38 Banking, 256 f. BINET, A., 201 Bonds, selection of, 70 ff.; strength of, 102 ff.; for temporary service, 111 ff.; order of formation of, 141 ff. _See also_ Habits. BRANDELL, G., 211 BRANDFORD, B., 198 f. BROWN, J. C., xvi, 103 BURGERSTEIN, L., 103 BURNETT, C. J., 202 BURT, C., 286 Cardinal and ordinal numbers confused, 206 Catch problems, 21 ff. CHAPMAN, J. C., 49 Class, size of, in relation to arithmetical learning, 228; variation within a, 289 ff. COBB, M. V., 299 COFFMAN, L. D., xvi Collection meaning of numbers, 3 ff. Computation, measurements of, 33 ff.; explanations of the processes in, 60 ff.; accuracy in, 102 ff. _See also_ Addition, Subtraction, Multiplication, Division, Fractions, Decimal numbers, Percents. Concomitants, law of varying, 172 ff.; law of contrasting, 173 ff. Concrete numbers, 85 ff. Concrete objects, use of, 253 ff. Conditions of arithmetical learning, 227 ff. Constitution of arithmetical abilities, 51 ff. Copying of numbers, eyestrain due to, 212 f. Correlations of arithmetical abilities, 295 ff. Courses of study, 232 f. COURTIS, S. A., 28 ff., 43 ff., 49, 103, 291, 293, 299 Crutches, 112 f. Culture-epoch theory, 198 f. Dairy records, 273 Decimal numbers, uses of, 24 f.; measurement of ability with, 36 ff.; learning, 181 ff.; division by, 270 f. DE CROLY, M., 205 Deductive reasoning, 60 ff., 185 ff. DEGAND, J., 205 Denominate numbers, 141 f., 147 f. Described problems, 10 ff. Development of knowledge of number, 205 ff. DE VOSS, J. C., 49 DEWEY, J., 3, 83, 150, 205, 207, 208, 219, 266, 277 Differences in arithmetical ability, 285 ff.; within a class, 289 ff. Difficulty as a stimulus, 277 ff. Drill, 102 ff. Discipline, mental, 20 Distribution of practice, 156 ff. Division, measurement of, 35 f., 37; constitution of, 57 ff.; deductive explanations of, 63, 64 f.; inductive explanations of, 63 f., 65 f.; habit in relation to, 72; with remainders, 76; with fractions, 78 ff.; amount of practice in, 122 ff.; distribution of practice in, 167; use of the problem attitude in teaching, 270 f. DONOVAN, M. E., 295 Elements, responses to, 169 ff. Eleven, multiples of, 85 ELLIOTT, C. H., 228 Equation form, importance of, 77 f. Explanations of the processes of computation, 60 ff.; memory of, 115 f.; time for giving, 154 ff. Eyestrain in arithmetical work, 212 ff. Facilitation, 143 ff. Figures, printing of, 235 ff.; writing of, 214 f., 241 FLYNN, F. J., 196 Fractions, uses of, 24 f.; measurement of ability with, 36 ff.; knowledge of the meaning of, 54 ff. FREEMAN, F. N., 259, 261 FRIEDRICH, J., 103 Generalization, 169 ff. GILBERT, J. A., 203 Graded tests, 28 ff., 36 ff. Greatest common divisor, 88 f. Habits, importance of, in arithmetical learning, 70 ff.; now neglected, 75 ff.; harmful or wasteful, 83 ff.; 91 ff.; propædeutic, 117 ff.; organization of, 137 ff.; arrangement of, 141 ff. HAHN, H. H., 295 HALL, G. S., 200 f. HARTMANN, B., 200 f. HECK, W. H., 227 Heredity in arithmetical abilities, 293 ff. Highest common factor, 88 f. HOKE, K. J., 49 HOLMES, M. E., 103 HOWELL, H. B., 259 HUNT, C. W., 196 Hygiene of arithmetic, 212 ff., 234 ff. Individual differences, 285 ff. Inductive reasoning, 60 ff., 169 ff. Insurance, 256 Interest as a principle determining the order of topics, 150 ff. Interests, instinctive 195 ff.; censuses of, 209 ff.; neglect of childish, 226 ff.; in self-management, 223 f.; intrinsic, 224 ff. Interference, 143 ff. Inventories of arithmetical knowledge and skill, 199 ff. JESSUP, W. A., xvi KELLY, F. J., 49 KING, A. C., 103, 227 KIRBY, T. J., 76 f., 104, 295 KLAPPER, P., xvi KRUSE, P. J., 289, 293 Ladder tests, 28 ff., 36 ff. Language in arithmetic, 8 f., 19, 89 ff., 94 ff. LASER, H., 103 LAY, W. A., 259, 261 Learning, nature of arithmetical, 1 ff. Least common multiple, 88 f. LEWIS, E. O., 210 f. LOBSIEN, M., 209 f. MCCALL, W. A., 49 MCDOUGLE, E. C., 85 ff. MCKNIGHT, J. A., 210 MCLELLAN, J. A., 3, 83, 89, 205, 207 Manipulation of numbers, 60 ff. Meaning, of numbers, 2 ff., 171; of a fraction, 54 ff.; of decimals, 181 f. Measurement of arithmetical abilities, 27 ff. Mental arithmetic, 262 ff. MESSENGER, J. F., 202 Metric system, 147 MEUMANN, E., 261 MITCHELL, H. E., 24 MONROE, W. S., 49 Multiplication, measurement of, 35, 36; constitution of, 51; deductive explanations of, 61; inductive explanations of, 61 f.; with fractions, 78 ff.; by eleven, 85; amount of practice in, 122 ff.; order of learning the elementary facts of, 144 f.; distribution of practice in, 158 ff.; use of the problem attitude in teaching, 267 ff. NANU, H. A., 202 National Intelligence Tests, 49 f. Negative reaction in intellectual life, 278 f. Number pictures, 259 ff. Numbers, meaning of, 2; as measures of continuous quantities, 75; abstract and concrete, 85 ff.; denominate, 141 f., 147 f.; use of large, 145 f.; perception of, 205 ff.; early awareness of, 205 ff.; confusion of cardinal and ordinal, 206. _See also_ Decimal numbers _and_ Fractions. Objective aids, used for verification, 154; in general, 243 ff. Oral arithmetic, 262 ff. Order of topics, 141 ff. Ordinal numbers, confused with cardinal, 206 Original tendencies and arithmetic, 195 ff. Overlearning, 134 ff. Percents, 80 f. Perception of number, 202 ff. PHILLIPS, D. E., 3, 4, 205, 207 Pictures, hygiene of, 246 ff.; number, 259 ff. POMMER, O., 212 Practice, amount of, 122 ff.; distribution of, 156 ff. Precision in fundamental operations, 102 ff. Problem attitude, 266 ff. Problems, 9 ff.; "catch," 21 ff.; measurement of ability with, 42 ff.; whose answer must be known in order to frame them, 93 f.; verbal form of, 111 f.; interest in, 220 ff.; as introductions to arithmetical learning, 266 ff. Propædeutic bonds, 117 ff. Purposive thinking, 193 ff. Quantity, number and, 85 ff.; perception of, 202 ff. RACE, H., 295 Rainfall, 272 Ratio, 225 f.; meaning of numbers, 3 ff. Reaction, negative, 278 f. Reality, in problems, 9 ff. Reasoning, arithmetical, nature of, 19 ff.; measurement of ability in, 42 ff.; derivation of tables by, 58 f.; about the rationale of computations, 60 ff.; habit in relation to, 73 f., 190 ff.; problems which provoke false, 100 f.; the essentials of arithmetical, 185 ff.; selection in, 187 ff.; as the coöperation of organized habits, 190 ff. Recapitulation theory, 198 f. Recipes, 273 f. Rectangle, area of, 257 f. RICE, J. M., 228 ff. RUSH, G. P., 49 SEEKEL, E., 212 SELKIN, F. B., 196 f. Sequence of topics, 141 ff. Series meaning of numbers, 2 ff. Size of class in relation to arithmetical learning, 228 SMITH, D. E., xvi, 224 Social instincts, use of, 195 f. Sociology of arithmetic, 24 ff. Speed in relation to accuracy, 31, 108 SPEER, W. W., 3, 5, 83 Spiral order, 141, 145 STARCH, D., 49, 295 STERN, W., 210, 212 STONE, C. W., 27 ff., 42 ff., 228 ff. Subtraction, measurement of, 34 f.; constitution of, 57 f.; amount of practice in, 122 ff. Supervision, 233 f. SUZZALLO, H., xvi Temporary bonds, 111 ff. Terms, 113 f. Tests of arithmetical abilities, 27 ff. THORNDIKE, E. L., 34, 38 ff., 227, 294 Time, devoted to arithmetic, 228 ff.; of day, in relation to arithmetical learning, 227 f. Type, hygiene of, 235 ff. Underlearning, 134 ff. United States money, 148 ff. Units of measure, arbitrary, 5, 83 f. Variation, among individuals, 285 ff. Variety, in teaching, 153 Verification, 81 f.; aided by greater strength of the fundamental bonds, 107 ff. WALSH, J. H., 11 WELLS, F. L., 295 WHITE, E. E., 5 WHITLEY, M. T., 295 WIEDERKEHR, G., 212 WILSON, G. M., 24, 49 WOODY, C., 29 ff., 52, 287, 293 Words. _See_ Language _and_ Terms. Written arithmetic, 262 ff. Zero in multiplication, 179 f. TRANSCRIBER'S NOTES: 1. Passages in italics are surrounded by _underscores_. 2. Passages in bold are indicated by #bold#. 3. Mixed fractions are represented using forward slash and hyphen in this text version. For example, 5-1/2 represents five and a half. 4. Images and footnotes have been moved from the middle of a paragraph to the closest paragraph break. 5. Obvious errors in spelling and punctuation have been silently closed. 
744	----	This is the golden ratio, (1+sqrt(5))/2, with 1.000.000 digits. It is based on square root of 5 computed by Robert Nemiroff and Jerry Bonnell. The golden ratio = 1.6180339887498948482045868343656381177203091798057628621354486227052604628189 02449707207204189391137484754088075386891752126633862223536931793180060766726354 43338908659593958290563832266131992829026788067520876689250171169620703222104321 62695486262963136144381497587012203408058879544547492461856953648644492410443207 71344947049565846788509874339442212544877066478091588460749988712400765217057517 97883416625624940758906970400028121042762177111777805315317141011704666599146697 98731761356006708748071013179523689427521948435305678300228785699782977834784587 82289110976250030269615617002504643382437764861028383126833037242926752631165339 24731671112115881863851331620384005222165791286675294654906811317159934323597349 49850904094762132229810172610705961164562990981629055520852479035240602017279974 71753427775927786256194320827505131218156285512224809394712341451702237358057727 86160086883829523045926478780178899219902707769038953219681986151437803149974110 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37681	----	THE TEACHING OF GEOMETRY BY DAVID EUGENE SMITH GINN AND COMPANY BOSTON · NEW YORK · CHICAGO · LONDON COPYRIGHT, 1911, BY DAVID EUGENE SMITH ALL RIGHTS RESERVED 911.6 The Athenæum Press GINN AND COMPANY · PROPRIETORS BOSTON · U.S.A. PREFACE A book upon the teaching of geometry may be planned in divers ways. It may be written to exploit a new theory of geometry, or a new method of presenting the science as we already have it. On the other hand, it may be ultraconservative, making a plea for the ancient teaching and the ancient geometry. It may be prepared for the purpose of setting forth the work as it now is, or with the tempting but dangerous idea of prophecy. It may appeal to the iconoclast by its spirit of destruction, or to the disciples of _laissez faire_ by its spirit of conserving what the past has bequeathed. It may be written for the few who always lead, or think they lead, or for the many who are ranked by the few as followers. And in view of these varied pathways into the joint domain of geometry and education, a writer may well afford to pause before he sets his pen to paper, and to decide with care the route that he will take. At present in America we have a fairly well-defined body of matter in geometry, and this occupies a fairly well-defined place in the curriculum. There are not wanting many earnest teachers who would change both the matter and the place in a very radical fashion. There are not wanting others, also many in number, who are content with things as they find them. But by far the largest part of the teaching body is of a mind to welcome the natural and gradual evolution of geometry toward better things, contributing to this evolution as much as it can, glad to know the best that others have to offer, receptive of ideas that make for better teaching, but out of sympathy with either the extreme of revolution or the extreme of stagnation. It is for this larger class, the great body of progressive teachers, that this book is written. It stands for vitalizing geometry in every legitimate way; for improving the subject matter in such manner as not to destroy the pupil's interest; for so teaching geometry as to make it appeal to pupils as strongly as any other subject in the curriculum; but for the recognition of geometry for geometry's sake and not for the sake of a fancied utility that hardly exists. Expressing full appreciation of the desirability of establishing a motive for all studies, so as to have the work proceed with interest and vigor, it does not hesitate to express doubt as to certain motives that have been exploited, nor to stand for such a genuine, thought-compelling development of the science as is in harmony with the mental powers of the pupils in the American high school. For this class of teachers the author hopes that the book will prove of service, and that through its perusal they will come to admire the subject more and more, and to teach it with greater interest. It offers no panacea, it champions no single method, but it seeks to set forth plainly the reasons for teaching a geometry of the kind that we have inherited, and for hoping for a gradual but definite improvement in the science and in the methods of its presentation. DAVID EUGENE SMITH CONTENTS CHAPTER PAGE I. CERTAIN QUESTIONS NOW AT ISSUE 1 II. WHY GEOMETRY IS STUDIED 7 III. A BRIEF HISTORY OF GEOMETRY 26 IV. DEVELOPMENT OF THE TEACHING OF GEOMETRY 40 V. EUCLID 47 VI. EFFORTS AT IMPROVING EUCLID 57 VII. THE TEXTBOOK IN GEOMETRY 70 VIII. THE RELATION OF ALGEBRA TO GEOMETRY 84 IX. THE INTRODUCTION TO GEOMETRY 93 X. THE CONDUCT OF A CLASS IN GEOMETRY 108 XI. THE AXIOMS AND POSTULATES 116 XII. THE DEFINITIONS OF GEOMETRY 132 XIII. HOW TO ATTACK THE EXERCISES 160 XIV. BOOK I AND ITS PROPOSITIONS 165 XV. THE LEADING PROPOSITIONS OF BOOK II 201 XVI. THE LEADING PROPOSITIONS OF BOOK III 227 XVII. THE LEADING PROPOSITIONS OF BOOK IV 252 XVIII. THE LEADING PROPOSITIONS OF BOOK V 269 XIX. THE LEADING PROPOSITIONS OF BOOK VI 289 XX. THE LEADING PROPOSITIONS OF BOOK VII 303 XXI. THE LEADING PROPOSITIONS OF BOOK VIII 321 INDEX 335 THE TEACHING OF GEOMETRY CHAPTER I CERTAIN QUESTIONS NOW AT ISSUE It is commonly said at the present time that the opening of the twentieth century is a period of unusual advancement in all that has to do with the school. It would be pleasant to feel that we are living in such an age, but it is doubtful if the future historian of education will find this to be the case, or that biographers will rank the leaders of our generation relatively as high as many who have passed away, or that any great movements of the present will be found that measure up to certain ones that the world now recognizes as epoch-making. Every generation since the invention of printing has been a period of agitation in educational matters, but out of all the noise and self-assertion, out of all the pretense of the chronic revolutionist, out of all the sham that leads to dogmatism, so little is remembered that we are apt to feel that the past had no problems and was content simply to accept its inheritance. In one sense it is not a misfortune thus to be blinded by the dust of present agitation and to be deafened by the noisy clamor of the agitator, since it stirs us to action at finding ourselves in the midst of the skirmish; but in another sense it is detrimental to our progress, since we thereby tend to lose the idea of perspective, and the coin comes to appear to our vision as large as the moon. In considering a question like the teaching of geometry, we at once find ourselves in the midst of a skirmish of this nature. If we join thoughtlessly in the noise, we may easily persuade ourselves that we are waging a mighty battle, fighting for some stupendous principle, doing deeds of great valor and of personal sacrifice. If, on the other hand, we stand aloof and think of the present movement as merely a chronic effervescence, fostered by the professional educator at the expense of the practical teacher, we are equally shortsighted. Sir Conan Doyle expressed this sentiment most delightfully in these words: The dead are such good company that one may come to think too little of the living. It is a real and pressing danger with many of us that we should never find our own thoughts and our own souls, but be ever obsessed by the dead. In every generation it behooves the open-minded, earnest, progressive teacher to seek for the best in the way of improvement, to endeavor to sift the few grains of gold out of the common dust, to weigh the values of proposed reforms, and to put forth his efforts to know and to use the best that the science of education has to offer. This has been the attitude of mind of the real leaders in the school life of the past, and it will be that of the leaders of the future. With these remarks to guide us, it is now proposed to take up the issues of the present day in the teaching of geometry, in order that we may consider them calmly and dispassionately, and may see where the opportunities for improvement lie. At the present time, in the educational circles of the United States, questions of the following type are causing the chief discussion among teachers of geometry: 1. Shall geometry continue to be taught as an application of logic, or shall it be treated solely with reference to its applications? 2. If the latter is the purpose in view, shall the propositions of geometry be limited to those that offer an opportunity for real application, thus contracting the whole subject to very narrow dimensions? 3. Shall a subject called geometry be extended over several years, as is the case in Europe,[1] or shall the name be applied only to serious demonstrative geometry[2] as given in the second year of the four-year high school course in the United States at present? 4. Shall geometry be taught by itself, or shall it be either mixed with algebra (say a day of one subject followed by a day of the other) or fused with it in the form of a combined mathematics? 5. Shall a textbook be used in which the basal propositions are proved in full, the exercises furnishing the opportunity for original work and being looked upon as the most important feature, or shall one be employed in which the pupil is expected to invent the proofs for the basal propositions as well as for the exercises? 6. Shall the terminology and the spirit of a modified Euclid and Legendre prevail in the future as they have in the past, or shall there be a revolution in the use of terms and in the general statements of the propositions? 7. Shall geometry be made a strong elective subject, to be taken only by those whose minds are capable of serious work? Shall it be a required subject, diluted to the comprehension of the weakest minds? Or is it now, by proper teaching, as suitable for all pupils as is any other required subject in the school curriculum? And in any case, will the various distinct types of high schools now arising call for distinct types of geometry? This brief list might easily be amplified, but it is sufficiently extended to set forth the trend of thought at the present time, and to show that the questions before the teachers of geometry are neither particularly novel nor particularly serious. These questions and others of similar nature are really side issues of two larger questions of far greater significance: (1) Are the reasons for teaching demonstrative geometry such that it should be a required subject, or at least a subject that is strongly recommended to all, whatever the type of high school? (2) If so, how can it be made interesting? The present work is written with these two larger questions in mind, although it considers from time to time the minor ones already mentioned, together with others of a similar nature. It recognizes that the recent growth in popular education has brought into the high school a less carefully selected type of mind than was formerly the case, and that for this type a different kind of mathematical training will naturally be developed. It proceeds upon the theory, however, that for the normal mind,--for the boy or girl who is preparing to win out in the long run,--geometry will continue to be taught as demonstrative geometry, as a vigorous thought-compelling subject, and along the general lines that the experience of the world has shown to be the best. Soft mathematics is not interesting to this normal mind, and a sham treatment will never appeal to the pupil; and this book is written for teachers who believe in this principle, who believe in geometry for the sake of geometry, and who earnestly seek to make the subject so interesting that pupils will wish to study it whether it is required or elective. The work stands for the great basal propositions that have come down to us, as logically arranged and as scientifically proved as the powers of the pupils in the American high school will permit; and it seeks to tell the story of these propositions and to show their possible and their probable applications in such a way as to furnish teachers with a fund of interesting material with which to supplement the book work of their classes. After all, the problem of teaching any subject comes down to this: Get a subject worth teaching and then make every minute of it interesting. Pupils do not object to work if they like a subject, but they do object to aimless and uninteresting tasks. Geometry is particularly fortunate in that the feeling of accomplishment comes with every proposition proved; and, given a class of fair intelligence, a teacher must be lacking in knowledge and enthusiasm who cannot foster an interest that will make geometry stand forth as the subject that brings the most pleasure, and that seems the most profitable of all that are studied in the first years of the high school. Continually to advance, continually to attempt to make mathematics fascinating, always to conserve the best of the old and to sift out and use the best of the new, to believe that "mankind is better served by nature's quiet and progressive changes than by earthquakes,"[3] to believe that geometry as geometry is so valuable and so interesting that the normal mind may rightly demand it,--this is to ally ourselves with progress. Continually to destroy, continually to follow strange gods, always to decry the best of the old, and to have no well-considered aim in the teaching of a subject,--this is to join the forces of reaction, to waste our time, to be recreant to our trust, to blind ourselves to the failures of the past, and to confess our weakness as teachers. It is with the desire to aid in the progressive movement, to assist those who believe that real geometry should be recommended to all, and to show that geometry is both attractive and valuable that this book is written. FOOTNOTES: [1] And really, though not nominally, in the United States, where the first concepts are found in the kindergarten, and where an excellent course in mensuration is given in any of our better class of arithmetics. That we are wise in not attempting serious demonstrative geometry much earlier seems to be generally conceded. [2] The third stage of geometry as defined in the recent circular (No. 711) of the British Board of Education, London, 1909. [3] The closing words of a sensible review of the British Board of Education circular (No. 711), on "The Teaching of Geometry" (London, 1909), by H. S. Hall in the _School World_, 1909, p. 222. CHAPTER II WHY GEOMETRY IS STUDIED With geometry, as with other subjects, it is easier to set forth what are not the reasons for studying it than to proceed positively and enumerate the advantages. Although such a negative course is not satisfying to the mind as a finality, it possesses definite advantages in the beginning of such a discussion as this. Whenever false prophets arise, and with an attitude of pained superiority proclaim unworthy aims in human life, it is well to show the fallacy of their position before proceeding to a constructive philosophy. Taking for a moment this negative course, let us inquire as to what are not the reasons for studying geometry, or, to be more emphatic, as to what are not the worthy reasons. In view of a periodic activity in favor of the utilities of geometry, it is well to understand, in the first place, that geometry is not studied, and never has been studied, because of its positive utility in commercial life or even in the workshop. In America we commonly allow at least a year to plane geometry and a half year to solid geometry; but all of the facts that a skilled mechanic or an engineer would ever need could be taught in a few lessons. All the rest is either obvious or is commercially and technically useless. We prove, for example, that the angles opposite the equal sides of a triangle are equal, a fact that is probably quite as obvious as the postulate that but one line can be drawn through a given point parallel to a given line. We then prove, sometimes by the unsatisfactory process of _reductio ad absurdum_, the converse of this proposition,--a fact that is as obvious as most other facts that come to our consciousness, at least after the preceding proposition has been proved. And these two theorems are perfectly fair types of upwards of one hundred sixty or seventy propositions comprising Euclid's books on plane geometry. They are generally not useful in daily life, and they were never intended to be so. There is an oft-repeated but not well-authenticated story of Euclid that illustrates the feeling of the founders of geometry as well as of its most worthy teachers. A Greek writer, Stobæus, relates the story in these words: Some one who had begun to read geometry with Euclid, when he had learned the first theorem, asked, "But what shall I get by learning these things?" Euclid called his slave and said, "Give him three obols, since he must make gain out of what he learns." Whether true or not, the story expresses the sentiment that runs through Euclid's work, and not improbably we have here a bit of real biography,--practically all of the personal Euclid that has come down to us from the world's first great textbook maker. It is well that we read the story occasionally, and also such words as the following, recently uttered[4] by Sir Conan Doyle,--words bearing the same lesson, although upon a different theme: In the present utilitarian age one frequently hears the question asked, "What is the use of it all?" as if every noble deed was not its own justification. As if every action which makes for self-denial, for hardihood, and for endurance was not in itself a most precious lesson to mankind. That people can be found to ask such a question shows how far materialism has gone, and how needful it is that we insist upon the value of all that is nobler and higher in life. An American statesman and jurist, speaking upon a similar occasion[5], gave utterance to the same sentiments in these words: When the time comes that knowledge will not be sought for its own sake, and men will not press forward simply in a desire of achievement, without hope of gain, to extend the limits of human knowledge and information, then, indeed, will the race enter upon its decadence. There have not been wanting, however, in every age, those whose zeal is in inverse proportion to their experience, who were possessed with the idea that it is the duty of the schools to make geometry practical. We have them to-day, and the world had them yesterday, and the future shall see them as active as ever. These people do good to the world, and their labors should always be welcome, for out of the myriad of suggestions that they make a few have value, and these are helpful both to the mathematician and the artisan. Not infrequently they have contributed material that serves to make geometry somewhat more interesting, but it must be confessed that most of their work is merely the threshing of old straw, like the work of those who follow the will-o'-the-wisp of the circle squarers. The medieval astrologers wished to make geometry more practical, and so they carried to a considerable length the study of the star polygon, a figure that they could use in their profession. The cathedral builders, as their art progressed, found that architectural drawings were more exact if made with a single opening of the compasses, and it is probable that their influence led to the development of this phase of geometry in the Middle Ages as a practical application of the science. Later, and about the beginning of the sixteenth century, the revival of art, and particularly the great development of painting, led to the practical application of geometry to the study of perspective and of those curves[6] that occur most frequently in the graphic arts. The sixteenth and seventeenth centuries witnessed the publication of a large number of treatises on practical geometry, usually relating to the measuring of distances and partly answering the purposes of our present trigonometry. Such were the well-known treatises of Belli (1569), Cataneo (1567), and Bartoli (1589).[7] The period of two centuries from about 1600 to about 1800 was quite as much given to experiments in the creation of a practical geometry as is the present time, and it was no doubt as much by way of protest against this false idea of the subject as a desire to improve upon Euclid that led the great French mathematician, Legendre, to publish his geometry in 1794,--a work that soon replaced Euclid in the schools of America. It thus appears that the effort to make geometry practical is by no means new. Euclid knew of it, the Middle Ages contributed to it, that period vaguely styled the Renaissance joined in the movement, and the first three centuries of printing contributed a large literature to the subject. Out of all this effort some genuine good remains, but relatively not very much.[8] And so it will be with the present movement; it will serve its greatest purpose in making teachers think and read, and in adding to their interest and enthusiasm and to the interest of their pupils; but it will not greatly change geometry, because no serious person ever believed that geometry was taught chiefly for practical purposes, or was made more interesting or valuable through such a pretense. Changes in sequence, in definitions, and in proofs will come little by little; but that there will be any such radical change in these matters in the immediate future, as some writers have anticipated, is not probable.[9] A recent writer of much acumen[10] has summed up this thought in these words: Not one tenth of the graduates of our high schools ever enter professions in which their algebra and geometry are applied to concrete realities; not one day in three hundred sixty-five is a high school graduate called upon to "apply," as it is called, an algebraic or a geometrical proposition.... Why, then, do we teach these subjects, if this alone is the sense of the word "practical"!... To me the solution of this paradox consists in boldly confronting the dilemma, and in saying that our conception of the practical utility of those studies must be readjusted, and that we have frankly to face the truth that the "practical" ends we seek are in a sense _ideal_ practical ends, yet such as have, after all, an eminently utilitarian value in the intellectual sphere. He quotes from C. S. Jackson, a progressive contemporary teacher of mechanics in England, who speaks of pupils confusing millimeters and centimeters in some simple computation, and who adds: There is the enemy! The real enemy we have to fight against, whatever we teach, is carelessness, inaccuracy, forgetfulness, and slovenliness. That battle has been fought and won with diverse weapons. It has, for instance, been fought with Latin grammar before now, and won. I say that because we must be very careful to guard against the notion that there is any one panacea for this sort of thing. It borders on quackery to say that elementary physics will cure everything. And of course the same thing may be said for mathematics. Nevertheless it is doubtful if we have any other subject that does so much to bring to the front this danger of carelessness, of slovenly reasoning, of inaccuracy, and of forgetfulness as this science of geometry, which has been so polished and perfected as the centuries have gone on. There have been those who did not proclaim the utilitarian value of geometry, but who fell into as serious an error, namely, the advocating of geometry as a means of training the memory. In times not so very far past, and to some extent to-day, the memorizing of proofs has been justified on this ground. This error has, however, been fully exposed by our modern psychologists. They have shown that the person who memorizes the propositions of Euclid by number is no more capable of memorizing other facts than he was before, and that the learning of proofs verbatim is of no assistance whatever in retaining matter that is helpful in other lines of work. Geometry, therefore, as a training of the memory is of no more value than any other subject in the curriculum. If geometry is not studied chiefly because it is practical, or because it trains the memory, what reasons can be adduced for its presence in the courses of study of every civilized country? Is it not, after all, a mere fetish, and are not those virulent writers correct who see nothing good in the subject save only its utilities?[11] Of this type one of the most entertaining is William J. Locke,[12] whose words upon the subject are well worth reading: ... I earned my living at school slavery, teaching to children the most useless, the most disastrous, the most soul-cramping branch of knowledge wherewith pedagogues in their insensate folly have crippled the minds and blasted the lives of thousands of their fellow creatures--elementary mathematics. There is no more reason for any human being on God's earth to be acquainted with the binomial theorem or the solution of triangles, unless he is a professional scientist,--when he can begin to specialize in mathematics at the same age as the lawyer begins to specialize in law or the surgeon in anatomy,--than for him to be expert in Choctaw, the Cabala, or the Book of Mormon. I look back with feelings of shame and degradation to the days when, for a crust of bread, I prostituted my intelligence to wasting the precious hours of impressionable childhood, which could have been filled with so many beautiful and meaningful things, over this utterly futile and inhuman subject. It trains the mind,--it teaches boys to think, they say. It doesn't. In reality it is a cut-and-dried subject, easy to fit into a school curriculum. Its sacrosanctity saves educationalists an enormous amount of trouble, and its chief use is to enable mindless young men from the universities to make a dishonest living by teaching it to others, who in their turn may teach it to a future generation. To be fair we must face just such attacks, and we must recognize that they set forth the feelings of many honest people. One is tempted to inquire if Mr. Locke could have written in such an incisive style if he had not, as was the case, graduated with honors in mathematics at one of the great universities. But he might reply that if his mind had not been warped by mathematics, he would have written more temperately, so the honors in the argument would be even. Much more to the point is the fact that Mr. Locke taught mathematics in the schools of England, and that these schools do not seem to the rest of the world to furnish a good type of the teaching of elementary mathematics. No country goes to England for its model in this particular branch of education, although the work is rapidly changing there, and Mr. Locke pictures a local condition in teaching rather than a general condition in mathematics. Few visitors to the schools of England would care to teach mathematics as they see it taught there, in spite of their recognition of the thoroughness of the work and the earnestness of many of the teachers. It is also of interest to note that the greatest protests against formal mathematics have come from England, as witness the utterances of such men as Sir William Hamilton and Professors Perry, Minchin, Henrici, and Alfred Lodge. It may therefore be questioned whether these scholars are not unconsciously protesting against the English methods and curriculum rather than against the subject itself. When Professor Minchin says that he had been through the six books of Euclid without really understanding an angle, it is Euclid's text and his own teacher that are at fault, and not geometry. Before considering directly the question as to why geometry should be taught, let us turn for a moment to the other subjects in the secondary curriculum. Why, for example, do we study literature? "It does not lower the price of bread," as Malherbe remarked in speaking of the commentary of Bachet on the great work of Diophantus. Is it for the purpose of making authors? Not one person out of ten thousand who study literature ever writes for publication. And why do we allow pupils to waste their time in physical education? It uses valuable hours, it wastes money, and it is dangerous to life and limb. Would it not be better to set pupils at sawing wood? And why do we study music? To give pleasure by our performances? How many who attempt to play the piano or to sing give much pleasure to any but themselves, and possibly their parents? The study of grammar does not make an accurate writer, nor the study of rhetoric an orator, nor the study of meter a poet, nor the study of pedagogy a teacher. The study of geography in the school does not make travel particularly easier, nor does the study of biology tend to populate the earth. So we might pass in review the various subjects that we study and ought to study, and in no case would we find utility the moving cause, and in every case would we find it difficult to state the one great reason for the pursuit of the subject in question,--and so it is with geometry. What positive reasons can now be adduced for the study of a subject that occupies upwards of a year in the school course, and that is, perhaps unwisely, required of all pupils? Probably the primary reason, if we do not attempt to deceive ourselves, is pleasure. We study music because music gives us pleasure, not necessarily our own music, but good music, whether ours, or, as is more probable, that of others. We study literature because we derive pleasure from books; the better the book the more subtle and lasting the pleasure. We study art because we receive pleasure from the great works of the masters, and probably we appreciate them the more because we have dabbled a little in pigments or in clay. We do not expect to be composers, or poets, or sculptors, but we wish to appreciate music and letters and the fine arts, and to derive pleasure from them and to be uplifted by them. At any rate, these are the nobler reasons for their study. So it is with geometry. We study it because we derive pleasure from contact with a great and an ancient body of learning that has occupied the attention of master minds during the thousands of years in which it has been perfected, and we are uplifted by it. To deny that our pupils derive this pleasure from the study is to confess ourselves poor teachers, for most pupils do have positive enjoyment in the pursuit of geometry, in spite of the tradition that leads them to proclaim a general dislike for all study. This enjoyment is partly that of the game,--the playing of a game that can always be won, but that cannot be won too easily. It is partly that of the æsthetic, the pleasure of symmetry of form, the delight of fitting things together. But probably it lies chiefly in the mental uplift that geometry brings, the contact with absolute truth, and the approach that one makes to the Infinite. We are not quite sure of any one thing in biology; our knowledge of geology is relatively very slight, and the economic laws of society are uncertain to every one except some individual who attempts to set them forth; but before the world was fashioned the square on the hypotenuse was equal to the sum of the squares on the other two sides of a right triangle, and it will be so after this world is dead; and the inhabitant of Mars, if he exists, probably knows its truth as we know it. The uplift of this contact with absolute truth, with truth eternal, gives pleasure to humanity to a greater or less degree, depending upon the mental equipment of the particular individual; but it probably gives an appreciable amount of pleasure to every student of geometry who has a teacher worthy of the name. First, then, and foremost as a reason for studying geometry has always stood, and will always stand, the pleasure and the mental uplift that comes from contact with such a great body of human learning, and particularly with the exact truth that it contains. The teacher who is imbued with this feeling is on the road to success, whatever method of presentation he may use; the one who is not imbued with it is on the road to failure, however logical his presentation or however large his supply of practical applications. Subordinate to these reasons for studying geometry are many others, exactly as with all other subjects of the curriculum. Geometry, for example, offers the best developed application of logic that we have, or are likely to have, in the school course. This does not mean that it always exemplifies perfect logic, for it does not; but to the pupil who is not ready for logic, per se, it offers an example of close reasoning such as his other subjects do not offer. We may say, and possibly with truth, that one who studies geometry will not reason more clearly on a financial proposition than one who does not; but in spite of the results of the very meager experiments of the psychologists, it is probable that the man who has had some drill in syllogisms, and who has learned to select the essentials and to neglect the nonessentials in reaching his conclusions, has acquired habits in reasoning that will help him in every line of work. As part of this equipment there is also a terseness of statement and a clearness in arrangement of points in an argument that has been the subject of comment by many writers. Upon this same topic an English writer, in one of the sanest of recent monographs upon the subject,[13] has expressed his views in the following words: The statement that a given individual has received a sound geometrical training implies that he has segregated from the whole of his sense impressions a certain set of these impressions, that he has then eliminated from their consideration all irrelevant impressions (in other words, acquired a subjective command of these impressions), that he has developed on the basis of these impressions an ordered and continuous system of logical deduction, and finally that he is capable of expressing the nature of these impressions and his deductions therefrom in terms simple and free from ambiguity. Now the slightest consideration will convince any one not already conversant with the idea, that the same sequence of mental processes underlies the whole career of any individual in any walk of life if only he is not concerned entirely with manual labor; consequently a full training in the performance of such sequences must be regarded as forming an essential part of any education worthy of the name. Moreover, the full appreciation of such processes has a higher value than is contained in the mental training involved, great though this be, for it induces an appreciation of intellectual unity and beauty which plays for the mind that part which the appreciation of schemes of shape and color plays for the artistic faculties; or, again, that part which the appreciation of a body of religious doctrine plays for the ethical aspirations. Now geometry is not the sole possible basis for inculcating this appreciation. Logic is an alternative for adults, provided that the individual is possessed of sufficient wide, though rough, experience on which to base his reasoning. Geometry is, however, highly desirable in that the objective bases are so simple and precise that they can be grasped at an early age, that the amount of training for the imagination is very large, that the deductive processes are not beyond the scope of ordinary boys, and finally that it affords a better basis for exercise in the art of simple and exact expression than any other possible subject of a school course. Are these results really secured by teachers, however, or are they merely imagined by the pedagogue as a justification for his existence? Do teachers have any such appreciation of geometry as has been suggested, and even if they have it, do they impart it to their pupils? In reply it may be said, probably with perfect safety, that teachers of geometry appreciate their subject and lead their pupils to appreciate it to quite as great a degree as obtains in any other branch of education. What teacher appreciates fully the beauties of "In Memoriam," or of "Hamlet," or of "Paradise Lost," and what one inspires his pupils with all the nobility of these world classics? What teacher sees in biology all the grandeur of the evolution of the race, or imparts to his pupils the noble lessons of life that the study of this subject should suggest? What teacher of Latin brings his pupils to read the ancient letters with full appreciation of the dignity of style and the nobility of thought that they contain? And what teacher of French succeeds in bringing a pupil to carry on a conversation, to read a French magazine, to see the history imbedded in the words that are used, to realize the charm and power of the language, or to appreciate to the full a single classic? In other words, none of us fully appreciates his subject, and none of us can hope to bring his pupils to the ideal attitude toward any part of it. But it is probable that the teacher of geometry succeeds relatively better than the teacher of other subjects, because the science has reached a relatively higher state of perfection. The body of truth in geometry has been more clearly marked out, it has been more successfully fitted together, its lesson is more patent, and the experience of centuries has brought it into a shape that is more usable in the school. While, therefore, we have all kinds of teaching in all kinds of subjects, the very nature of the case leads to the belief that the class in geometry receives quite as much from the teacher and the subject as the class in any other branch in the school curriculum. But is this not mere conjecture? What are the results of scientific investigation of the teaching of geometry? Unfortunately there is little hope from the results of such an inquiry, either here or in other fields. We cannot first weigh a pupil in an intellectual or moral balance, then feed him geometry, and then weigh him again, and then set back his clock of time and begin all over again with the same individual. There is no "before taking" and "after taking" of a subject that extends over a year or two of a pupil's life. We can weigh utilities roughly, we can estimate the pleasure of a subject relatively, but we cannot say that geometry is worth so many dollars, and history so many, and so on through the curriculum. The best we can do is to ask ourselves what the various subjects, with teachers of fairly equal merit, have done for us, and to inquire what has been the experience of other persons. Such an investigation results in showing that, with few exceptions, people who have studied geometry received as much of pleasure, of inspiration, of satisfaction, of what they call training from geometry as from any other subject of study,--given teachers of equal merit,--and that they would not willingly give up the something which geometry brought to them. If this were not the feeling, and if humanity believed that geometry is what Mr. Locke's words would seem to indicate, it would long ago have banished it from the schools, since upon this ground rather than upon the ground of utility the subject has always stood. These seem to be the great reasons for the study of geometry, and to search for others would tend to weaken the argument. At first sight they may not seem to justify the expenditure of time that geometry demands, and they may seem unduly to neglect the argument that geometry is a stepping-stone to higher mathematics. Each of these points, however, has been neglected purposely. A pupil has a number of school years at his disposal; to what shall they be devoted? To literature? What claim has letters that is such as to justify the exclusion of geometry? To music, or natural science, or language? These are all valuable, and all should be studied by one seeking a liberal education; but for the same reason geometry should have its place. What subject, in fine, can supply exactly what geometry does? And if none, then how can the pupil's time be better expended than in the study of this science?[14] As to the second point, that a claim should be set forth that geometry is a _sine qua non_ to higher mathematics, this belief is considerably exaggerated because there are relatively few who proceed from geometry to a higher branch of mathematics. This argument would justify its status as an elective rather than as a required subject. Let us then stand upon the ground already marked out, holding that the pleasure, the culture, the mental poise, the habits of exact reasoning that geometry brings, and the general experience of mankind upon the subject are sufficient to justify us in demanding for it a reasonable amount of time in the framing of a curriculum. Let us be fair in our appreciation of all other branches, but let us urge that every student may have an opportunity to know of real geometry, say for a single year, thereafter pursuing it or not, according as we succeed in making its value apparent, or fail in our attempt to present worthily an ancient and noble science to the mind confided to our instruction. The shortsightedness of a narrow education, of an education that teaches only machines to a prospective mechanic, and agriculture to a prospective farmer, and cooking and dressmaking to the girl, and that would exclude all mathematics that is not utilitarian in the narrow sense, cannot endure. The community has found out that such schemes may be well fitted to give the children a good time in school, but lead them to a bad time afterward. Life is hard work, and if they have never learned in school to give their concentrated attention to that which does not appeal to them and which does not interest them immediately, they have missed the most valuable lesson of their school years. The little practical information they could have learned at any time; the energy of attention and concentration can no longer be learned if the early years are wasted. However narrow and commercial the standpoint which is chosen may be, it can always be found that it is the general education which pays best, and the more the period of cultural work can be expanded the more efficient will be the services of the school for the practical services of the nation.[15] Of course no one should construe these remarks as opposing in the slightest degree the laudable efforts that are constantly being put forth to make geometry more interesting and to vitalize it by establishing as strong motives as possible for its study. Let the home, the workshop, physics, art, play,--all contribute their quota of motive to geometry as to all mathematics and all other branches. But let us never forget that geometry has a _raison d'être_ beyond all this, and that these applications are sought primarily for the sake of geometry, and that geometry is not taught primarily for the sake of these applications. When we consider how often geometry is attacked by those who profess to be its friends, and how teachers who have been trained in mathematics occasionally seem to make of the subject little besides a mongrel course in drawing and measuring, all the time insisting that they are progressive while the champions of real geometry are reactionary, it is well to read some of the opinions of the masters. The following quotations may be given occasionally in geometry classes as showing the esteem in which the subject has been held in various ages, and at any rate they should serve to inspire the teacher to greater love for his subject. The enemies of geometry, those who know it only imperfectly, look upon the theoretical problems, which constitute the most difficult part of the subject, as mental games which consume time and energy that might better be employed in other ways. Such a belief is false, and it would block the progress of science if it were credible. But aside from the fact that the speculative problems, which at first sight seem barren, can often be applied to useful purposes, they always stand as among the best means to develop and to express all the forces of the human intelligence.--ABBÉ BOSSUT. The sailor whom an exact observation of longitude saves from shipwreck owes his life to a theory developed two thousand years ago by men who had in mind merely the speculations of abstract geometry.--CONDORCET. If mathematical heights are hard to climb, the fundamental principles lie at every threshold, and this fact allows them to be comprehended by that common sense which Descartes declared was "apportioned equally among all men."--COLLET. It may seem strange that geometry is unable to define the terms which it uses most frequently, since it defines neither movement, nor number, nor space,---the three things with which it is chiefly concerned. But we shall not be surprised if we stop to consider that this admirable science concerns only the most simple things, and the very quality that renders these things worthy of study renders them incapable of being defined. Thus the very lack of definition is rather an evidence of perfection than a defect, since it comes not from the obscurity of the terms, but from the fact that they are so very well known.--PASCAL. God eternally geometrizes.--PLATO. God is a circle of which the center is everywhere and the circumference nowhere.--RABELAIS. Without mathematics no one can fathom the depths of philosophy. Without philosophy no one can fathom the depths of mathematics. Without the two no one can fathom the depths of anything.--BORDAS-DEMOULIN. We may look upon geometry as a practical logic, for the truths which it studies, being the most simple and most clearly understood of all truths, are on this account the most susceptible of ready application in reasoning.--D'ALEMBERT. The advance and the perfecting of mathematics are closely joined to the prosperity of the nation.--NAPOLEON. Hold nothing as certain save what can be demonstrated.--NEWTON. To measure is to know.--KEPLER. The method of making no mistake is sought by every one. The logicians profess to show the way, but the geometers alone ever reach it, and aside from their science there is no genuine demonstration.--PASCAL. The taste for exactness, the impossibility of contenting one's self with vague notions or of leaning upon mere hypotheses, the necessity for perceiving clearly the connection between certain propositions and the object in view,--these are the most precious fruits of the study of mathematics.--LACROIX. =Bibliography.= Smith, The Teaching of Elementary Mathematics, p. 234, New York, 1900; Henrici, Presidential Address before the British Association, _Nature_, Vol. XXVIII, p. 497; Hill, Educational Value of Mathematics, _Educational Review_, Vol. IX, p. 349; Young, The Teaching of Mathematics, p. 9, New York, 1907. The closing quotations are from Rebière, Mathématiques et Mathématiciens, Paris, 1893. FOOTNOTES: [4] In an address in London, June 15, 1909, at a dinner to Sir Ernest Shackelton. [5] Governor Hughes, now Justice Hughes, of New York, at the Peary testimonial on February 8, 1910, at New York City. [6] The first work upon this subject, and indeed the first printed treatise on curves in general, was written by the famous artist of Nürnberg, Albrecht Dürer. [7] Several of these writers are mentioned in Chapter IV. [8] If any reader chances upon George Birkbeck's English translation of Charles Dupin's "Mathematics Practically Applied," Halifax, 1854, he will find that Dupin gave more good applications of geometry than all of our American advocates of practical geometry combined. [9] See, for example, Henrici's "Congruent Figures," London, 1879, and the review of Borel's "Elements of Mathematics," by Professor Sisam in the _Bulletin of the American Mathematical Society_, July, 1910, a matter discussed later in this work. [10] T. J. McCormack, "Why do we study Mathematics: a Philosophical and Historical Retrospect," p. 9, Cedar Rapids, Iowa, 1910. [11] Of the fair and candid arguments against the culture value of mathematics, one of the best of the recent ones is that by G. F. Swain, in the _Atti del IV Congresso Internazionale dei Matematici_, Rome, 1909, Vol. III, p. 361. The literature of this school is quite extensive, but Perry's "England's Neglect of Science," London, 1900, and "Discussion on the Teaching of Mathematics," London, 1901, are typical. [12] In his novel, "The Morals of Marcus Ordeyne." [13] G. W. L. Carson, "The Functions of Geometry as a Subject of Education," p. 3, Tonbridge, 1910. [14] It may well be, however, that the growing curriculum may justify some reduction in the time formerly assigned to geometry, and any reasonable proposition of this nature should be fairly met by teachers of mathematics. [15] Professor Münsterberg, in the _Metropolitan Magazine_ for July, 1910. CHAPTER III A BRIEF HISTORY OF GEOMETRY The geometry of very ancient peoples was largely the mensuration of simple areas and solids, such as is taught to children in elementary arithmetic to-day. They early learned how to find the area of a rectangle, and in the oldest mathematical records that have come down to us there is some discussion of the area of triangles and the volume of solids. The earliest documents that we have relating to geometry come to us from Babylon and Egypt. Those from Babylon are written on small clay tablets, some of them about the size of the hand, these tablets afterwards having been baked in the sun. They show that the Babylonians of that period knew something of land measures, and perhaps had advanced far enough to compute the area of a trapezoid. For the mensuration of the circle they later used, as did the early Hebrews, the value [pi] = 3. A tablet in the British Museum shows that they also used such geometric forms as triangles and circular segments in astrology or as talismans. The Egyptians must have had a fair knowledge of practical geometry long before the date of any mathematical treatise that has come down to us, for the building of the pyramids, between 3000 and 2400 B.C., required the application of several geometric principles. Some knowledge of surveying must also have been necessary to carry out the extensive plans for irrigation that were executed under Amenemhat III, about 2200 B.C. The first definite knowledge that we have of Egyptian mathematics comes to us from a manuscript copied on papyrus, a kind of paper used about the Mediterranean in early times. This copy was made by one Aah-mesu (The Moon-born), commonly called Ahmes, who probably flourished about 1700 B.C. The original from which he copied, written about 2300 B.C., has been lost, but the papyrus of Ahmes, written nearly four thousand years ago, is still preserved, and is now in the British Museum. In this manuscript, which is devoted chiefly to fractions and to a crude algebra, is found some work on mensuration. Among the curious rules are the incorrect ones that the area of an isosceles triangle equals half the product of the base and one of the equal sides; and that the area of a trapezoid having bases _b_, _b'_, and the nonparallel sides each equal to _a_, is 1/2_a_(_b_ + _b'_). One noteworthy advance appears, however. Ahmes gives a rule for finding the area of a circle, substantially as follows: Multiply the square on the radius by (16/9)^2, which is equivalent to taking for [pi] the value 3.1605. This papyrus also contains some treatment of the mensuration of solids, particularly with reference to the capacity of granaries. There is also some slight mention of similar figures, and an extensive treatment of unit fractions,--fractions that were quite universal among the ancients. In the line of algebra it contains a brief treatment of the equation of the first degree with one unknown, and of progressions.[16] Herodotus tells us that Sesostris, king of Egypt,[17] divided the land among his people and marked out the boundaries after the overflow of the Nile, so that surveying must have been well known in his day. Indeed, the _harpedonaptæ_, or rope stretchers, acquired their name because they stretched cords, in which were knots, so as to make the right triangle 3, 4, 5, when they wished to erect a perpendicular. This is a plan occasionally used by surveyors to-day, and it shows that the practical application of the Pythagorean Theorem was known long before Pythagoras gave what seems to have been the first general proof of the proposition. From Egypt, and possibly from Babylon, geometry passed to the shores of Asia Minor and Greece. The scientific study of the subject begins with Thales, one of the Seven Wise Men of the Grecian civilization. Born at Miletus, not far from Smyrna and Ephesus, about 640 B.C., he died at Athens in 548 B.C. He spent his early manhood as a merchant, accumulating the wealth that enabled him to spend his later years in study. He visited Egypt, and is said to have learned such elements of geometry as were known there. He founded a school of mathematics and philosophy at Miletus, known from the country as the Ionic School. How elementary the knowledge of geometry then was may be understood from the fact that tradition attributes only about four propositions to Thales,--(1) that vertical angles are equal, (2) that equal angles lie opposite the equal sides of an isosceles triangle, (3) that a triangle is determined by two angles and the included side, (4) that a diameter bisects the circle, and possibly the propositions about the angle-sum of a triangle for special cases, and the angle inscribed in a semicircle.[18] The greatest pupil of Thales, and one of the most remarkable men of antiquity, was Pythagoras. Born probably on the island of Samos, just off the coast of Asia Minor, about the year 580 B.C., Pythagoras set forth as a young man to travel. He went to Miletus and studied under Thales, probably spent several years in Egypt, very likely went to Babylon, and possibly went even to India, since tradition asserts this and the nature of his work in mathematics suggests it. In later life he went to a Greek colony in southern Italy, and at Crotona, in the southeastern part of the peninsula, he founded a school and established a secret society to propagate his doctrines. In geometry he is said to have been the first to demonstrate the proposition that the square on the hypotenuse is equal to the sum of the squares upon the other two sides of a right triangle. The proposition was known in India and Egypt before his time, at any rate for special cases, but he seems to have been the first to prove it. To him or to his school seems also to have been due the construction of the regular pentagon and of the five regular polyhedrons. The construction of the regular pentagon requires the dividing of a line into extreme and mean ratio, and this problem is commonly assigned to the Pythagoreans, although it played an important part in Plato's school. Pythagoras is also said to have known that six equilateral triangles, three regular hexagons, or four squares, can be placed about a point so as just to fill the 360°, but that no other regular polygons can be so placed. To his school is also due the proof for the general case that the sum of the angles of a triangle equals two right angles, the first knowledge of the size of each angle of a regular polygon, and the construction of at least one star-polygon, the star-pentagon, which became the badge of his fraternity. The brotherhood founded by Pythagoras proved so offensive to the government that it was dispersed before the death of the master. Pythagoras fled to Megapontum, a seaport lying to the north of Crotona, and there he died about 501 B.C.[19] [Illustration: FANCIFUL PORTRAIT OF PYTHAGORAS Calandri's Arithmetic, 1491] For two centuries after Pythagoras geometry passed through a period of discovery of propositions. The state of the science may be seen from the fact that Oenopides of Chios, who flourished about 465 B.C., and who had studied in Egypt, was celebrated because he showed how to let fall a perpendicular to a line, and how to make an angle equal to a given angle. A few years later, about 440 B.C., Hippocrates of Chios wrote the first Greek textbook on mathematics. He knew that the areas of circles are proportional to the squares on their radii, but was ignorant of the fact that equal central angles or equal inscribed angles intercept equal arcs. Antiphon and Bryson, two Greek scholars, flourished about 430 B.C. The former attempted to find the area of a circle by doubling the number of sides of a regular inscribed polygon, and the latter by doing the same for both inscribed and circumscribed polygons. They thus approximately exhausted the area between the polygon and the circle, and hence this method is known as the method of exhaustions. About 420 B.C. Hippias of Elis invented a certain curve called the quadratrix, by means of which he could square the circle and trisect any angle. This curve cannot be constructed by the unmarked straightedge and the compasses, and when we say that it is impossible to square the circle or to trisect any angle, we mean that it is impossible by the help of these two instruments alone. During this period the great philosophic school of Plato (429-348 B.C.) flourished at Athens, and to this school is due the first systematic attempt to create exact definitions, axioms, and postulates, and to distinguish between elementary and higher geometry. It was at this time that elementary geometry became limited to the use of the compasses and the unmarked straightedge, which took from this domain the possibility of constructing a square equivalent to a given circle ("squaring the circle"), of trisecting any given angle, and of constructing a cube that should have twice the volume of a given cube ("duplicating the cube"), these being the three famous problems of antiquity. Plato and his school interested themselves with the so-called Pythagorean numbers, that is, with numbers that would represent the three sides of a right triangle and hence fulfill the condition that _a_^2 + _b_^2 = _c_^2. Pythagoras had already given a rule that would be expressed in modern form, as 1/4(_m_^2 + 1)^2 = _m_^2 + 1/4(_m_^2 - 1)^2. The school of Plato found that [(1/2_m_)^2 + 1]^2 = _m_^2 + [(1/2_m_)^2 - 1]^2. By giving various values to _m_, different Pythagorean numbers may be found. Plato's nephew, Speusippus (about 350 B.C.), wrote upon this subject. Such numbers were known, however, both in India and in Egypt, long before this time. One of Plato's pupils was Philippus of Mende, in Egypt, who flourished about 380 B.C. It is said that he discovered the proposition relating to the exterior angle of a triangle. His interest, however, was chiefly in astronomy. Another of Plato's pupils was Eudoxus of Cnidus (408-355 B.C.). He elaborated the theory of proportion, placing it upon a thoroughly scientific foundation. It is probable that Book V of Euclid, which is devoted to proportion, is essentially the work of Eudoxus. By means of the method of exhaustions of Antiphon and Bryson he proved that the pyramid is one third of a prism, and the cone is one third of a cylinder, each of the same base and the same altitude. He wrote the first textbook known on solid geometry. The subject of conic sections starts with another pupil of Plato's, Menæchmus, who lived about 350 B.C. He cut the three forms of conics (the ellipse, parabola, and hyperbola) out of three different forms of cone,--the acute-angled, right-angled, and obtuse-angled,--not noticing that he could have obtained all three from any form of right circular cone. It is interesting to see the far-reaching influence of Plato. While primarily interested in philosophy, he laid the first scientific foundations for a system of mathematics, and his pupils were the leaders in this science in the generation following his greatest activity. The great successor of Plato at Athens was Aristotle, the teacher of Alexander the Great. He also was more interested in philosophy than in mathematics, but in natural rather than mental philosophy. With him comes the first application of mathematics to physics in the hands of a great man, and with noteworthy results. He seems to have been the first to represent an unknown quantity by letters. He set forth the theory of the parallelogram of forces, using only rectangular components, however. To one of his pupils, Eudemus of Rhodes, we are indebted for a history of ancient geometry, some fragments of which have come down to us. The first great textbook on geometry, and the greatest one that has ever appeared, was written by Euclid, who taught mathematics in the great university at Alexandria, Egypt, about 300 B.C. Alexandria was then practically a Greek city, having been named in honor of Alexander the Great, and being ruled by the Greeks. In his work Euclid placed all of the leading propositions of plane geometry then known, and arranged them in a logical order. Most geometries of any importance written since his time have been based upon Euclid, improving the sequence, symbols, and wording as occasion demanded. He also wrote upon other branches of mathematics besides elementary geometry, including a work on optics. He was not a great creator of mathematics, but was rather a compiler of the work of others, an office quite as difficult to fill and quite as honorable. Euclid did not give much solid geometry because not much was known then. It was to Archimedes (287-212 B.C.), a famous mathematician of Syracuse, on the island of Sicily, that some of the most important propositions of solid geometry are due, particularly those relating to the sphere and cylinder. He also showed how to find the approximate value of [pi] by a method similar to the one we teach to-day, proving that the real value lay between 3 1/7 and 3 10/71. The story goes that the sphere and cylinder were engraved upon his tomb, and Cicero, visiting Syracuse many years after his death, found the tomb by looking for these symbols. Archimedes was the greatest mathematical physicist of ancient times. The Greeks contributed little more to elementary geometry, although Apollonius of Perga, who taught at Alexandria between 250 and 200 B.C., wrote extensively on conic sections, and Hypsicles of Alexandria, about 190 B.C., wrote on regular polyhedrons. Hypsicles was the first Greek writer who is known to have used sexagesimal fractions,--the degrees, minutes, and seconds of our angle measure. Zenodorus (180 B.C.) wrote on isoperimetric figures, and his contemporary, Nicomedes of Gerasa, invented a curve known as the conchoid, by means of which he could trisect any angle. Another contemporary, Diocles, invented the cissoid, or ivy-shaped curve, by means of which he solved the famous problem of duplicating the cube, that is, constructing a cube that should have twice the volume of a given cube. The greatest of the Greek astronomers, Hipparchus (180-125 B.C.), lived about this period, and with him begins spherical trigonometry as a definite science. A kind of plane trigonometry had been known to the ancient Egyptians. The Greeks usually employed the chord of an angle instead of the half chord (sine), the latter having been preferred by the later Arab writers. The most celebrated of the later Greek physicists was Heron of Alexandria, formerly supposed to have lived about 100 B.C., but now assigned to the first century A.D. His contribution to geometry was the formula for the area of a triangle in terms of its sides a, b, and c, with s standing for the semiperimeter 1/2(_a_ + _b_ + _c_). The formula is [sqrt](_s_(_s_-_a_)(_s_-_b_)(_s_-_c_)). Probably nearly contemporary with Heron was Menelaus of Alexandria, who wrote a spherical trigonometry. He gave an interesting proposition relating to plane and spherical triangles, their sides being cut by a transversal. For the plane triangle _ABC_, the sides _a_, _b_, and _c_ being cut respectively in _X_, _Y_, and _Z_, the theorem asserts substantially that (_AZ_/_BZ_) · (_BX_/_CX_) · (_CY_/_AY_) = 1. The most popular writer on astronomy among the Greeks was Ptolemy (Claudius Ptolemaeus, 87-165 A.D.), who lived at Alexandria. He wrote a work entitled "Megale Syntaxis" (The Great Collection), which his followers designated as _Megistos_ (greatest), on which account the Arab translators gave it the name "Almagest" (_al_ meaning "the"). He advanced the science of trigonometry, but did not contribute to geometry. At the close of the third century Pappus of Alexandria (295 A.D.) wrote on geometry, and one of his theorems, a generalized form of the Pythagorean proposition, is mentioned in Chapter XVI of this work. Only two other Greek writers on geometry need be mentioned. Theon of Alexandria (370 A.D.), the father of the Hypatia who is the heroine of Charles Kingsley's well-known novel, wrote a commentary on Euclid to which we are indebted for some historical information. Proclus (410-485 A.D.) also wrote a commentary on Euclid, and much of our information concerning the first Book of Euclid is due to him. The East did little for geometry, although contributing considerably to algebra. The first great Hindu writer was Aryabhatta, who was born in 476 A.D. He gave the very close approximation for [pi], expressed in modern notation as 3.1416. He also gave rules for finding the volume of the pyramid and sphere, but they were incorrect, showing that the Greek mathematics had not yet reached the Ganges. Another Hindu writer, Brahmagupta (born in 598 A.D.), wrote an encyclopedia of mathematics. He gave a rule for finding Pythagorean numbers, expressed in modern symbols as follows: 1/4((_p_^2/_q_) + _q_)^2 = 1/4((_p_^2/_q_) - _q_)^2 + _p_^2. He also generalized Heron's formula by asserting that the area of an inscribed quadrilateral of sides _a_, _b_, _c_, _d_, and semiperimeter _s_, is [sqrt]((_s_ - _a_)(_s_ - _b_)(_s_ - _c_)(_s_ - _d_)). The Arabs, about the time of the "Arabian Nights Tales" (800 A.D.), did much for mathematics, translating the Greek authors into their language and also bringing learning from India. Indeed, it is to them that modern Europe owed its first knowledge of Euclid. They contributed nothing of importance to elementary geometry, however. The greatest of the Arab writers was Mohammed ibn Musa al-Khowarazmi (820 A.D.). He lived at Bagdad and Damascus. Although chiefly interested in astronomy, he wrote the first book bearing the name "algebra" ("Al-jabr wa'l-muq[=a]balah," Restoration and Equation), composed an arithmetic using the Hindu numerals,[20] and paid much attention to geometry and trigonometry. Euclid was translated from the Arabic into Latin in the twelfth century, Greek manuscripts not being then at hand, or being neglected because of ignorance of the language. The leading translators were Athelhard of Bath (1120), an English monk; Gherard of Cremona (1160), an Italian monk; and Johannes Campanus (1250), chaplain to Pope Urban IV. The greatest European mathematician of the Middle Ages was Leonardo of Pisa[21] (_ca._ 1170-1250). He was very influential in making the Hindu-Arabic numerals known in Europe, wrote extensively on algebra, and was the author of one book on geometry. He contributed nothing to the elementary theory, however. The first edition of Euclid was printed in Latin in 1482, the first one in English appearing in 1570. Our symbols are modern, + and - first appearing in a German work in 1489; = in Recorde's "Whetstone of Witte" in 1557; > and < in the works of Harriot (1560-1621); and × in a publication by Oughtred (1574-1660). The most noteworthy advance in geometry in modern times was made by the great French philosopher Descartes, who published a small work entitled "La Géométrie" in 1637. From this springs the modern analytic geometry, a subject that has revolutionized the methods of all mathematics. Most of the subsequent discoveries in mathematics have been in higher branches. To the great Swiss mathematician Euler (1707-1783) is due, however, one proposition that has found its way into elementary geometry, the one showing the relation between the number of edges, vertices, and faces of a polyhedron. There has of late arisen a modern elementary geometry devoted chiefly to special points and lines relating to the triangle and the circle, and many interesting propositions have been discovered. The subject is so extensive that it cannot find any place in our crowded curriculum, and must necessarily be left to the specialist.[22] Some idea of the nature of the work may be obtained from a mention of a few propositions: The medians of a triangle are concurrent in the centroid, or center of gravity of the triangle. The bisectors of the various interior and exterior angles of a triangle are concurrent by threes in the incenter or in one of the three excenters of the triangle. The common chord of two intersecting circles is a special case of their radical axis, and tangents to the circles from any point on the radical axis are equal. If _O_ is the orthocenter of the triangle _ABC_, and _X_, _Y_, _Z_ are the feet of the perpendiculars from _A_, _B_, _C_ respectively, and _P_, _Q_, _R_ are the mid-points of _a_, _b_, _c_ respectively, and _L_, _M_, _N_ are the mid-points of _OA_, _OB_, _OC_ respectively; then the points _L_, _M_, _N_; _P_, _Q_, _R_; _X_, _Y_, _Z_ all lie on a circle, the "nine points circle." In the teaching of geometry it adds a human interest to the subject to mention occasionally some of the historical facts connected with it. For this reason this brief sketch will be supplemented by many notes upon the various important propositions as they occur in the several books described in the later chapters of this work. FOOTNOTES: [16] It was published in German translation by A. Eisenlohr, "Ein mathematisches Handbuch der alten Aegypter," Leipzig, 1877, and in facsimile by the British Museum, under the title, "The Rhind Papyrus," in 1898. [17] Generally known as Rameses II. He reigned in Egypt about 1350 B.C. [18] Two excellent works on Thales and his successors, and indeed the best in English, are the following: G. J. Allman, "Greek Geometry from Thales to Euclid," Dublin, 1889; J. Gow, "A History of Greek Mathematics," Cambridge, 1884. On all mathematical subjects the best general history is that of M. Cantor, "Geschichte der Mathematik," 4 vols, Leipzig, 1880-1908. [19] Another good work on Greek geometry, with considerable material on Pythagoras, is by C. A. Bretschneider, "Die Geometrie und die Geometer vor Eukleides," Leipzig, 1870. [20] Smith and Karpinski, "The Hindu-Arabic Numerals," Boston, 1911. [21] For a sketch of his life see Smith and Karpinski, loc. cit. [22] Those who care for a brief description of this phase of the subject may consult J. Casey, "A Sequel to Euclid," Dublin, fifth edition, 1888; W. J. M'Clelland, "A Treatise on the Geometry of the Circle," New York, 1891; M. Simon, "Über die Entwicklung der Elementar-Geometrie im XIX. Jahrhundert," Leipzig, 1906. CHAPTER IV DEVELOPMENT OF THE TEACHING OF GEOMETRY We know little of the teaching of geometry in very ancient times, but we can infer its nature from the teaching that is still seen in the native schools of the East. Here a man, learned in any science, will have a group of voluntary students sitting about him, and to them he will expound the truth. Such schools may still be seen in India, Persia, and China, the master sitting on a mat placed on the ground or on the floor of a veranda, and the pupils reading aloud or listening to his words of exposition. In Egypt geometry seems to have been in early times mere mensuration, confined largely to the priestly caste. It was taught to novices who gave promise of success in this subject, and not to others, the idea of general culture, of training in logic, of the cultivation of exact expression, and of coming in contact with truth being wholly wanting. In Greece it was taught in the schools of philosophy, often as a general preparation for philosophic study. Thus Thales introduced it into his Ionic school, Pythagoras made it very prominent in his great school at Crotona in southern Italy (Magna Græcia), and Plato placed above the door of his _Academia_ the words, "Let no one ignorant of geometry enter here,"--a kind of entrance examination for his school of philosophy. In these gatherings of students it is probable that geometry was taught in much the way already mentioned for the schools of the East, a small group of students being instructed by a master. Printing was unknown, papyrus was dear, parchment was only in process of invention. Paper such as we know had not yet appeared, so that instruction was largely oral, and geometric figures were drawn by a pointed stick on a board covered with fine sand, or on a tablet of wax. But with these crude materials there went an abundance of time, so that a number of great results were accomplished in spite of the difficulties attending the study of the subject. It is said that Hippocrates of Chios (_ca._ 440 B.C.) wrote the first elementary textbook on mathematics and invented the method of geometric reduction, the replacing of a proposition to be proved by another which, when proved, allows the first one to be demonstrated. A little later Eudoxus of Cnidus (_ca._ 375 B.C.), a pupil of Plato's, used the _reductio ad absurdum_, and Plato is said to have invented the method of proof by analysis, an elaboration of the plan used by Hippocrates. Thus these early philosophers taught their pupils not facts alone, but methods of proof, giving them power as well as knowledge. Furthermore, they taught them how to discuss their problems, investigating the conditions under which they are capable of solution. This feature of the work they called the _diorismus_, and it seems to have started with Leon, a follower of Plato. Between the time of Plato (_ca._ 400 B.C.) and Euclid (_ca._ 300 B.C.) several attempts were made to arrange the accumulated material of elementary geometry in a textbook. Plato had laid the foundations for the science, in the form of axioms, postulates, and definitions, and he had limited the instruments to the straightedge and the compasses. Aristotle (_ca._ 350 B.C.) had paid special attention to the history of the subject, thus finding out what had already been accomplished, and had also made much of the applications of geometry. The world was therefore ready for a good teacher who should gather the material and arrange it scientifically. After several attempts to find the man for such a task, he was discovered in Euclid, and to his work the next chapter is devoted. After Euclid, Archimedes (_ca._ 250 B.C.) made his great contributions. He was not a teacher like his illustrious predecessor, but he was a great discoverer. He has left us, however, a statement of his methods of investigation which is helpful to those who teach. These methods were largely experimental, even extending to the weighing of geometric forms to discover certain relations, the proof being given later. Here was born, perhaps, what has been called the laboratory method of the present. Of the other Greek teachers we have but little information as to methods of imparting instruction. It is not until the Middle Ages that there is much known in this line. Whatever of geometry was taught seems to have been imparted by word of mouth in the way of expounding Euclid, and this was done in the ancient fashion. The early Church leaders usually paid no attention to geometry, but as time progressed the _quadrivium_, or four sciences of arithmetic, music, geometry, and astronomy, came to rank with the _trivium_ (grammar, rhetoric, dialectics), the two making up the "seven liberal arts." All that there was of geometry in the first thousand years of Christianity, however, at least in the great majority of Church schools, was summed up in a few definitions and rules of mensuration. Gerbert, who became Pope Sylvester II in 999 A.D., gave a new impetus to geometry by discovering a manuscript of the old Roman surveyors and a copy of the geometry of Boethius, who paraphrased Euclid about 500 A.D. He thereupon wrote a brief geometry, and his elevation to the papal chair tended to bring the study of mathematics again into prominence. Geometry now began to have some place in the Church schools, naturally the only schools of high rank in the Middle Ages. The study of the subject, however, seems to have been merely a matter of memorizing. Geometry received another impetus in the book written by Leonardo of Pisa in 1220, the "Practica Geometriae." Euclid was also translated into Latin about this time (strangely enough, as already stated, from the Arabic instead of the Greek), and thus the treasury of elementary geometry was opened to scholars in Europe. From now on, until the invention of printing (_ca._ 1450), numerous writers on geometry appear, but, so far as we know, the method of instruction remained much as it had always been. The universities began to appear about the thirteenth century, and Sacrobosco, a well-known medieval mathematician, taught mathematics about 1250 in the University of Paris. In 1336 this university decreed that mathematics should be required for a degree. In the thirteenth century Oxford required six books of Euclid for one who was to teach, but this amount of work seems to have been merely nominal, for in 1450 only two books were actually read. The universities of Prague (founded in 1350) and Vienna (statutes of 1389) required most of plane geometry for the teacher's license, although Vienna demanded but one book for the bachelor's degree. So, in general, the universities of the thirteenth, fourteenth, and fifteenth centuries required less for the degree of master of arts than we now require from a pupil in our American high schools. On the other hand, the university students were younger than now, and were really doing only high school work. The invention of printing made possible the study of geometry in a new fashion. It now became possible for any one to study from a book, whereas before this time instruction was chiefly by word of mouth, consisting of an explanation of Euclid. The first Euclid was printed in 1482, at Venice, and new editions and variations of this text came out frequently in the next century. Practical geometries became very popular, and the reaction against the idea of mental discipline threatened to abolish the old style of text. It was argued that geometry was uninteresting, that it was not sufficient in itself, that boys needed to see the practical uses of the subject, that only those propositions that were capable of application should be retained, that there must be a fusion between the demands of culture and the demands of business, and that every man who stood for mathematical ideals represented an obsolete type. Such writers as Finæus (1556), Bartoli (1589), Belli (1569), and Cataneo (1567), in the sixteenth century, and Capra (1678), Gargiolli (1655), and many others in the seventeenth century, either directly or inferentially, took this attitude towards the subject,--exactly the attitude that is being taken at the present time by a number of teachers in the United States. As is always the case, to such an extreme did this movement lead that there was a reaction that brought the Euclid type of book again to the front, and it has maintained its prominence even to the present. The study of geometry in the high schools is relatively recent. The Gymnasium (classical school preparatory to the university) at Nürnberg, founded in 1526, and the Cathedral school at Württemberg (as shown by the curriculum of 1556) seem to have had no geometry before 1600, although the Gymnasium at Strassburg included some of this branch of mathematics in 1578, and an elective course in geometry was offered at Zwickau, in Saxony, in 1521. In the seventeenth century geometry is found in a considerable number of secondary schools, as at Coburg (1605), Kurfalz (1615, elective), Erfurt (1643), Gotha (1605), Giessen (1605), and numerous other places in Germany, although it appeared but rarely in the secondary schools of France before the eighteenth century. In Germany the Realschulen--schools with more science and less classics than are found in the Gymnasium--came into being in the eighteenth century, and considerable effort was made to construct a course in geometry that should be more practical than that of the modified Euclid. At the opening of the nineteenth century the Prussian schools were reorganized, and from that time on geometry has had a firm position in the secondary schools of all Germany. In the eighteenth century some excellent textbooks on geometry appeared in France, among the best being that of Legendre (1794), which influenced in such a marked degree the geometries of America. Soon after the opening of the nineteenth century the _lycées_ of France became strong institutions, and geometry, chiefly based on Legendre, was well taught in the mathematical divisions. A worthy rival of Legendre's geometry was the work of Lacroix, who called attention continually to the analogy between the theorems of plane and solid geometry, and even went so far as to suggest treating the related propositions together in certain cases. In England the preparatory schools, such as Rugby, Harrow, and Eton, did not commonly teach geometry until quite recently, leaving this work for the universities. In Christ's Hospital, London, however, geometry was taught as early as 1681, from a work written by several teachers of prominence. The highest class at Harrow studied "Euclid and vulgar fractions" one period a week in 1829, but geometry was not seriously studied before 1837. In the Edinburgh Academy as early as 1885, and in Rugby by 1839, plane geometry was completed. Not until 1844 did Harvard require any plane geometry for entrance. In 1855 Yale required only two books of Euclid. It was therefore from 1850 to 1875 that plane geometry took a definite place in the American high school. Solid geometry has not been generally required for entrance to any eastern college, although in the West this is not the case. The East teaches plane geometry more thoroughly, but allows a pupil to enter college or to go into business with no solid geometry. Given a year to the subject, it is possible to do little more than cover plane geometry; with a year and a half the solid geometry ought easily to be covered also. =Bibliography.= Stamper, A History of the Teaching of Elementary Geometry, New York, 1909, with a very full bibliography of the subject; Cajori, The Teaching of Mathematics in the United States, Washington, 1890; Cantor, Geschichte der Mathematik, Vol. IV, p. 321, Leipzig, 1908; Schotten, Inhalt und Methode des planimetrischen Unterrichts, Leipzig, 1890. CHAPTER V EUCLID It is fitting that a chapter in a book upon the teaching of this subject should be devoted to the life and labors of the greatest of all textbook writers, Euclid,--a man whose name has been, for more than two thousand years, a synonym for elementary plane geometry wherever the subject has been studied. And yet when an effort is made to pick up the scattered fragments of his biography, we are surprised to find how little is known of one whose fame is so universal. Although more editions of his work have been printed than of any other book save the Bible,[23] we do not know when he was born, or in what city, or even in what country, nor do we know his race, his parentage, or the time of his death. We should not feel that we knew much of the life of a man who lived when the Magna Charta was wrested from King John, if our first and only source of information was a paragraph in the works of some historian of to-day; and yet this is about the situation in respect to Euclid. Proclus of Alexandria, philosopher, teacher, and mathematician, lived from 410 to 485 A.D., and wrote a commentary on the works of Euclid. In his writings, which seem to set forth in amplified form his lectures to the students in the Neoplatonist School of Alexandria, Proclus makes this statement, and of Euclid's life we have little else: Not much younger than these[24] is Euclid, who put together the "Elements," collecting many of the theorems of Eudoxus, perfecting many of those of Theætetus, and also demonstrating with perfect certainty what his predecessors had but insufficiently proved. He flourished in the time of the first Ptolemy, for Archimedes, who closely followed this ruler,[25] speaks of Euclid. Furthermore it is related that Ptolemy one time demanded of him if there was in geometry no shorter way than that of the "Elements," to whom he replied that there was no royal road to geometry.[26] He was therefore younger than the pupils of Plato, but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says.[27] Thus we have in a few lines, from one who lived perhaps seven or eight hundred years after Euclid, nearly all that is known of the most famous teacher of geometry that ever lived. Nevertheless, even this little tells us about when he flourished, for Hermotimus and Philippus were pupils of Plato, who died in 347 B.C., whereas Archimedes was born about 287 B.C. and was writing about 250 B.C. Furthermore, since Ptolemy I reigned from 306 to 283 B.C., Euclid must have been teaching about 300 B.C., and this is the date that is generally assigned to him. Euclid probably studied at Athens, for until he himself assisted in transferring the center of mathematical culture to Alexandria, it had long been in the Grecian capital, indeed since the time of Pythagoras. Moreover, numerous attempts had been made at Athens to do exactly what Euclid succeeded in doing,--to construct a logical sequence of propositions; in other words, to write a textbook on plane geometry. It was at Athens, therefore, that he could best have received the inspiration to compose his "Elements."[28] After finishing his education at Athens it is quite probable that he, like other savants of the period, was called to Alexandria by Ptolemy Soter, the king, to assist in establishing the great school which made that city the center of the world's learning for several centuries. In this school he taught, and here he wrote the "Elements" and numerous other works, perhaps ten in all. Although the Greek writers who may have known something of the life of Euclid have little to say of him, the Arab writers, who could have known nothing save from Greek sources, have allowed their imaginations the usual latitude in speaking of him and of his labors. Thus Al-Qif[t.][=i], who wrote in the thirteenth century, has this to say in his biographical treatise "Ta'r[=i]kh al-[H.]ukam[=a]": Euclid, son of Naucrates, grandson of Zenarchus, called the author of geometry, a Greek by nationality, domiciled at Damascus, born at Tyre, most learned in the science of geometry, published a most excellent and most useful work entitled "The Foundation or Elements of Geometry," a subject in which no more general treatise existed before among the Greeks; nay, there was no one even of later date who did not walk in his footsteps and frankly profess his doctrine. This is rather a specimen of the Arab tendency to manufacture history than a serious contribution to the biography of Euclid, of whose personal history we have only the information given by Proclus. [Illustration: EUCLID From an old print] Euclid's works at once took high rank, and they are mentioned by various classical authors. Cicero knew of them, and Capella (_ca._ 470 A.D.), Cassiodorius (_ca._ 515 A.D.), and Boethius (_ca._ 480-524 A.D.) were all more or less familiar with the "Elements." With the advance of the Dark Ages, however, learning was held in less and less esteem, so that Euclid was finally forgotten, and manuscripts of his works were either destroyed or buried in some remote cloister. The Arabs, however, whose civilization assumed prominence from about 750 A.D. to about 1500, translated the most important treatises of the Greeks, and Euclid's "Elements" among the rest. One of these Arabic editions an English monk of the twelfth century, one Athelhard (Æthelhard) of Bath, found and translated into Latin (_ca._ 1120 A.D.). A little later Gherard of Cremona (1114-1187) made a new translation from the Arabic, differing in essential features from that of Athelhard, and about 1260 Johannes Campanus made still a third translation, also from Arabic into Latin.[29] There is reason to believe that Athelhard, Campanus, and Gherard may all have had access to an earlier Latin translation, since all are quite alike in some particulars while diverging noticeably in others. Indeed, there is an old English verse that relates: The clerk Euclide on this wyse hit fonde Thys craft of gemetry yn Egypte londe ... Thys craft com into England, as y yow say, Yn tyme of good Kyng Adelstone's day. If this be true, Euclid was known in England as early as 924-940 A.D. Without going into particulars further, it suffices to say that the modern knowledge of Euclid came first through the Arabic into the Latin, and the first printed edition of the "Elements" (Venice, 1482) was the Campanus translation. Greek manuscripts now began to appear, and at the present time several are known. There is a manuscript of the ninth century in the Bodleian library at Oxford, one of the tenth century in the Vatican, another of the tenth century in Florence, one of the eleventh century at Bologna, and two of the twelfth century at Paris. There are also fragments containing bits of Euclid in Greek, and going back as far as the second and third century A.D. The first modern translation from the Greek into the Latin was made by Zamberti (or Zamberto),[30] and was printed at Venice in 1513. The first translation into English was made by Sir Henry Billingsley and was printed in 1570, sixteen years before he became Lord Mayor of London. Proclus, in his commentary upon Euclid's work, remarks: In the whole of geometry there are certain leading theorems, bearing to those which follow the relation of a principle, all-pervading, and furnishing proofs of many properties. Such theorems are called by the name of _elements_, and their function may be compared to that of the letters of the alphabet in relation to language, letters being indeed called by the same name in Greek [[Greek: stoicheia], stoicheia].[31] This characterizes the work of Euclid, a collection of the basic propositions of geometry, and chiefly of plane geometry, arranged in logical sequence, the proof of each depending upon some preceding proposition, definition, or assumption (axiom or postulate). The number of the propositions of plane geometry included in the "Elements" is not entirely certain, owing to some disagreement in the manuscripts, but it was between one hundred sixty and one hundred seventy-five. It is possible to reduce this number by about thirty or forty, because Euclid included a certain amount of geometric algebra; but beyond this we cannot safely go in the way of elimination, since from the very nature of the "Elements" these propositions are basic. The efforts at revising Euclid have been generally confined, therefore, to rearranging his material, to rendering more modern his phraseology, and to making a book that is more usable with beginners if not more logical in its presentation of the subject. While there has been an improvement upon Euclid in the art of bookmaking, and in minor matters of phraseology and sequence, the educational gain has not been commensurate with the effort put forth. With a little modification of Euclid's semi-algebraic Book II and of his treatment of proportion, with some scattering of the definitions and the inclusion of well-graded exercises at proper places, and with attention to the modern science of bookmaking, the "Elements" would answer quite as well for a textbook to-day as most of our modern substitutes, and much better than some of them. It would, moreover, have the advantage of being a classic,--somewhat the same advantage that comes from reading Homer in the original instead of from Pope's metrical translation. This is not a plea for a return to the Euclid text, but for a recognition of the excellence of Euclid's work. The distinctive feature of Euclid's "Elements," compared with the modern American textbook, is perhaps this: Euclid begins a book with what seems to him the easiest proposition, be it theorem or problem; upon this he builds another; upon these a third, and so on, concerning himself but little with the classification of propositions. Furthermore, he arranges his propositions so as to construct his figures before using them. We, on the other hand, make some little attempt to classify our propositions within each book, and we make no attempt to construct our figures before using them, or at least to prove that the constructions are correct. Indeed, we go so far as to study the properties of figures that we cannot construct, as when we ask for the size of the angle of a regular heptagon. Thus Euclid begins Book I by a problem, to construct an equilateral triangle on a given line. His object is to follow this by problems on drawing a straight line equal to a given straight line, and cutting off from the greater of two straight lines a line equal to the less. He now introduces a theorem, which might equally well have been his first proposition, namely, the case of the congruence of two triangles, having given two sides and the included angle. By means of his third and fourth propositions he is now able to prove the _pons asinorum_, that the angles at the base of an isosceles triangle are equal. We, on the other hand, seek to group our propositions where this can conveniently be done, putting the congruent propositions together, those about inequalities by themselves, and the propositions about parallels in one set. The results of the two arrangements are not radically different, and the effect of either upon the pupil's mind does not seem particularly better than that of the other. Teachers who have used both plans quite commonly feel that, apart from Books II and V, Euclid is nearly as easily understood as our modern texts, if presented in as satisfactory dress. The topics treated and the number of propositions in the plane geometry of the "Elements" are as follows: Book I. Rectilinear figures 48 Book II. Geometric algebra 14 Book III. Circles 37 Book IV. Problems about circles 16 Book V. Proportion 25 Book VI. Applications of proportion 33 ---- 173 Of these we now omit Euclid's Book II, because we have an algebraic symbolism that was unknown in his time, although he would not have used it in geometry even had it been known. Thus his first proposition in Book II is as follows: If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments. This amounts to saying that if _x_ = _p_ + _q_ + _r_ + ···, then _ax_ = _ap_ + _aq_ + _ar_ + ···. We also materially simplify Euclid's Book V. He, for example, proves that "If four magnitudes be proportional, they will also be proportional alternately." This he proves generally for any kind of magnitude, while we merely prove it for numbers having a common measure. We say that we may substitute for the older form of proportion, namely, _a_ : _b_ = _c_ : _d_, the fractional form _a_/_b_ = _c_/_d_. From this we have _ad_ = _bc_. Whence _a_/_c_ = _b_/_d_. In this work we assume that we may multiply equals by _b_ and _d_. But suppose _b_ and _d_ are cubes, of which, indeed, we do not even know the approximate numerical measure; what shall we do? To Euclid the multiplication by a cube or a polygon or a sphere would have been entirely meaningless, as it always is from the standpoint of pure geometry. Hence it is that our treatment of proportion has no serious standing in geometry as compared with Euclid's, and our only justification for it lies in the fact that it is easier. Euclid's treatment is much more rigorous than ours, but it is adapted to the comprehension of only advanced students, while ours is merely a confession, and it should be a frank confession, of the weakness of our pupils, and possibly, at times, of ourselves. If we should take Euclid's Books II and V for granted, or as sufficiently evident from our study of algebra, we should have remaining only one hundred thirty-four propositions, most of which may be designated as basal propositions of plane geometry. Revise Euclid as we will, we shall not be able to eliminate any large number of his fundamental truths, while we might do much worse than to adopt these one hundred thirty-four propositions _in toto_ as the bases, and indeed as the definition, of elementary plane geometry. =Bibliography.= Heath, The Thirteen Books of Euclid's Elements, 3 vols., Cambridge, 1908; Frankland, The First Book of Euclid, Cambridge, 1906; Smith, Dictionary of Greek and Roman Biography, article Eukleides; Simon, Euclid und die sechs planimetrischen Bücher, Leipzig, 1901; Gow, History of Greek Mathematics, Cambridge, 1884, and any of the standard histories of mathematics. Both Heath and Simon give extensive bibliographies. The latest standard Greek and Latin texts are Heiberg's, published by Teubner of Leipzig. FOOTNOTES: [23] Riccardi, Saggio di una bibliografia Euclidea, Part I, p. 3, Bologna, 1887. Riccardi lists well towards two thousand editions. [24] Hermotimus of Colophon and Philippus of Mende. [25] Literally, "Who closely followed the first," i.e. the first Ptolemy. [26] Menæchmus is said to have replied to a similar question of Alexander the Great: "O King, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all." [27] This is also shown in a letter from Archimedes to Eratosthenes, recently discovered by Heiberg. [28] On this phase of the subject, and indeed upon Euclid and his propositions and works in general, consult T. L. Heath, "The Thirteen Books of Euclid's Elements," 3 vols., Cambridge, 1908, a masterly treatise of which frequent use has been made in preparing this work. [29] A contemporary copy of this translation is now in the library of George A. Plimpton, Esq., of New York. See the author's "Rara Arithmetica," p. 433, Boston, 1909. [30] A beautiful vellum manuscript of this translation is in the library of George A. Plimpton, Esq., of New York. See the author's "Rara Arithmetica," p. 481, Boston, 1909. [31] Heath, loc. cit., Vol. I, p. 114. CHAPTER VI EFFORTS AT IMPROVING EUCLID From time to time an effort is made by some teacher, or association of teachers, animated by a serious desire to improve the instruction in geometry, to prepare a new syllabus that shall mark out some "royal road," and it therefore becomes those who are interested in teaching to consider with care the results of similar efforts in recent years. There are many questions which such an attempt suggests: What is the real purpose of the movement? What will the teaching world say of the result? Shall a reckless, ill-considered radicalism dominate the effort, bringing in a distasteful terminology and symbolism merely for its novelty, insisting upon an ultralogical treatment that is beyond the powers of the learner, rearranging the subject matter to fit some narrow notion of the projectors, seeking to emasculate mathematics by looking only to the applications, riding some little hobby in the way of some particular class of exercises, and cutting the number of propositions to a minimum that will satisfy the mere demands of the artisan? Such are some of the questions that naturally arise in the mind of every one who wishes well for the ancient science of geometry. It is not proposed in this chapter to attempt to answer these questions, but rather to assist in understanding the problem by considering the results of similar attempts. If it shall be found that syllabi have been prepared under circumstances quite as favorable as those that obtain at present, and if these syllabi have had little or no real influence, then it becomes our duty to see if new plans may be worked out so as to be more successful than their predecessors. If the older attempts have led to some good, it is well to know what is the nature of this good, to the end that new efforts may also result in something of benefit to the schools. It is proposed in this chapter to call attention to four important syllabi, setting forth briefly their distinguishing features and drawing some conclusions that may be helpful in other efforts of this nature. In England two noteworthy attempts have been made within a century, looking to a more satisfactory sequence and selection of propositions than is found in Euclid. Each began with a list of propositions arranged in proper sequence, and each was thereafter elaborated into a textbook. Neither accomplished fully the purpose intended, but each was instrumental in provoking healthy discussion and in improving the texts from which geometry is studied. The first of these attempts was made by Professor Augustus de Morgan, under the auspices of the Society for the Diffusion of Useful Knowledge, and it resulted in a textbook, including "plane, solid, and spherical" geometry, in six books. According to De Morgan's plan, plane geometry consisted of three books, the number of propositions being as follows: Book I. Rectilinear figures 60 Book II. Ratio, proportion, applications 69 Book III. The circle 65 ---- Total for plane geometry 194 Of the 194 propositions De Morgan selected 114 with their corollaries as necessary for a beginner who is teaching himself. In solid geometry the plan was as follows: Book IV. Lines in different planes, solids contained by planes 52 Book V. Cylinder, cone, sphere 25 Book VI. Figures on a sphere 42 ---- Total for solid geometry 119 Of these 119 propositions De Morgan selected 76 with their corollaries as necessary for a beginner, thus making 190 necessary propositions out of 305 desirable ones, besides the corollaries in plane and solid geometry. In other words, of the desirable propositions he considered that about two thirds are absolutely necessary. It is interesting to note, however, that he summed up the results of his labors by saying: It will be found that the course just laid down, excepting the sixth book of it only, is not of much greater extent, nor very different in point of matter from that of Euclid, whose "Elements" have at all times been justly esteemed a model not only of easy and progressive instruction in geometry, but of accuracy and perspicuity in reasoning. De Morgan's effort, essentially that of a syllabus-maker rather than a textbook writer, although it was published under the patronage of a prominent society with which were associated the names of men like Henry Hallam, Rowland Hill, Lord John Russell, and George Peacock, had no apparent influence on geometry either in England or abroad. Nevertheless the syllabus was in many respects excellent; it rearranged the matter, it classified the propositions, it improved some of the terminology, and it reduced the number of essential propositions; it had the assistance of De Morgan's enthusiasm and of the society with which he was so prominently connected, and it was circulated with considerable generosity throughout the English-speaking world; but in spite of all this it is to-day practically unknown. A second noteworthy attempt in England was made about a quarter of a century ago by a society that was organized practically for this very purpose, the Association for the Improvement of Geometrical Teaching. This society was composed of many of the most progressive teachers in England, and it included in its membership men of high standing in mathematics in the universities. As a result of their labors a syllabus was prepared, which was elaborated into a textbook, and in 1889 a revised syllabus was issued. As to the arrangement of matter, the syllabus departs from Euclid chiefly by separating the problems from the theorems, as is the case in our American textbooks, and in improving the phraseology. The course is preceded by some simple exercises in the use of the compasses and ruler, a valuable plan that is followed by many of the best teachers everywhere. Considerable attention is paid to logical processes before beginning the work, such terms as "contrapositive" and "obverse," and such rules as the "rule of conversion" and the "rule of identity" being introduced before any propositions are considered. The arrangement of the work and the number of propositions in plane geometry are as follows: Book I. The straight line 51 Book II. Equality of areas 19 Book III. The circle 42 Book IV. Ratio and proportion 32 Book V. Proportion 24 ---- Total for plane geometry 168 Here, then, is the result of several years of labor by a somewhat radical organization, fostered by excellent mathematicians, and carried on in a country where elementary geometry is held in highest esteem, and where Euclid was thought unsuited to the needs of the beginner. The number of propositions remains substantially the same as in Euclid, and the introduction of some unusable logic tends to counterbalance the improvement in sequence of the propositions. The report provoked thought; it shook the Euclid stronghold; it was probably instrumental in bringing about the present upheaval in geometry in England, but as a working syllabus it has not appealed to the world as the great improvement upon Euclid's "Elements" that was hoped by many of its early advocates. The same association published later, and republished in 1905, a "Report on the Teaching of Geometry," in which it returned to Euclid, modifying the "Elements" by omitting certain propositions, changing the order and proof of others, and introducing a few new theorems. It seems to reduce the propositions to be proved in plane geometry to about one hundred fifteen, and it recommends the omission of the incommensurable case. This number is, however, somewhat misleading, for Euclid frequently puts in one proposition what we in America, for educational reasons, find it better to treat in two, or even three, propositions. This report, therefore, reaches about the same conclusion as to the geometric facts to be mastered as is reached by our later textbook writers in America. It is not extreme, and it stands for good mathematics. In the United States the influence of our early wars with England, and the sympathy of France at that time, turned the attention of our scholars of a century ago from Cambridge to Paris as a mathematical center. The influx of French mathematics brought with it such works as Legendre's geometry (1794) and Bourdon's algebra, and made known the texts of Lacroix, Bertrand, and Bezout. Legendre's geometry was the result of the efforts of a great mathematician at syllabus-making, a natural thing in a country that had early broken away from Euclid. Legendre changed the Greek sequence, sought to select only propositions that are necessary to a good understanding of the subject, and added a good course in solid geometry. His arrangement, with the number of propositions as given in the Davies translation, is as follows: Book I. Rectilinear figures 31 Book II. Ratio and proportion 14 Book III. The circle 48 Book IV. Proportions of figures and areas 51 Book V. Polygons and circles 17 ---- Total for plane geometry 161 Legendre made, therefore, practically no reduction in the number of Euclid's propositions, and his improvement on Euclid consisted chiefly in his separation of problems and theorems, and in a less rigorous treatment of proportion which boys and girls could comprehend. D'Alembert had demanded that the sequence of propositions should be determined by the order in which they had been discovered, but Legendre wisely ignored such an extreme and gave the world a very usable book. The principal effect of Legendre's geometry in America was to make every textbook writer his own syllabus-maker, and to put solid geometry on a more satisfactory footing. The minute we depart from a standard text like Euclid's, and have no recognized examining body, every one is free to set up his own standard, always within the somewhat uncertain boundary prescribed by public opinion and by the colleges. The efforts of the past few years at syllabus-making have been merely attempts to define this boundary more clearly. Of these attempts two are especially worthy of consideration as having been very carefully planned and having brought forth such definite results as to appeal to a large number of teachers. Other syllabi have been made and are familiar to many teachers, but in point of clearness of purpose, conciseness of expression, and form of publication they have not been such as to compare with the two in question. The first of these is the Harvard syllabus, which is placed in the hands of students for reference when trying the entrance examinations of that university, a plan not followed elsewhere. It sets forth the basal propositions that should form the essential part of the student's preparation, and that are necessary and sufficient for proving any "original proposition" (to take the common expression) that may be set on the examination. The propositions are arranged by books as follows: Book I. Angles, triangles, parallels 25 Book II. The circle, angle measure 18 Book III. Similar polygons 10 Book IV. Area of polygons 8 Book V. Polygons and circle measure 11 Constructions 21 Ratio and proportion 6 ---- Total for plane geometry 99 The total for solid geometry is 79 propositions, or 178 for both plane and solid geometry. This is perhaps the most successful attempt that has been made at reaching a minimum number of propositions. It might well be further reduced, since it includes the proposition about two adjacent angles formed by one line meeting another, and the one about the circle as the limit of the inscribed and circumscribed regular polygons. The first of these leads a beginner to doubt the value of geometry, and the second is beyond the powers of the majority of students. As compared with the syllabus reported by a Wisconsin committee in 1904, for example, here are 99 propositions against 132. On the other hand, a committee appointed by the Central Association of Science and Mathematics Teachers reported in 1909 a syllabus with what seems at first sight to be a list of only 59 propositions in plane geometry. This number is fictitious, however, for the reason that numerous converses are indicated with the propositions, and are not included in the count, and directions are given to include "related theorems" and "problems dealing with the length and area of a circle," so that in some cases one proposition is evidently intended to cover several others. This syllabus is therefore lacking in definiteness, so that the Harvard list stands out as perhaps the best of its type. The second noteworthy recent attempt in America is that made by a committee of the Association of Mathematical Teachers in New England. This committee was organized in 1904. It held sixteen meetings and carried on a great deal of correspondence. As a result, it prepared a syllabus arranged by topics, the propositions of solid geometry being grouped immediately after the corresponding ones of plane geometry. For example, the nine propositions on congruence in a plane are followed by nine on congruence in space. As a result, the following summarizes the work in plane geometry: Congruence in a plane 9 Equivalence 3 Parallels and perpendiculars 9 Symmetry 20 Angles 15 Tangents 4 Similar figures 18 Inequalities 8 Lengths and areas 17 Loci 2 Concurrent lines 5 ---- Total for plane geometry 110 Not so conventional in arrangement as the Harvard syllabus, and with a few propositions that are evidently not basal to the same extent as the rest, the list is nevertheless a very satisfactory one, and the parallelism shown between plane and solid geometry is suggestive to both student and teacher. On the whole, however, the Harvard selection of basal propositions is perhaps as satisfactory as any that has been made, even though it appears to lack a "factor of safety," and it is probable that any further reduction would be unwise. What, now, has been the effect of all these efforts? What teacher or school would be content to follow any one of these syllabi exactly? What textbook writer would feel it safe to limit his regular propositions to those in any one syllabus? These questions suggest their own answers, and the effect of all this effort seems at first thought to have been so slight as to be entirely out of proportion to the end in view. This depends, however, on what this end is conceived to be. If the purpose has been to cut out a very large number of the propositions that are found in Euclid's plane geometry, the effort has not been successful. We may reduce this number to about one hundred thirty, but in general, whatever a syllabus may give as a minimum, teachers will favor a larger number than is suggested by the Harvard list, for the purpose of exercise in the reading of mathematics if for no other reason. The French geometer, Lacroix, who wrote more than a century ago, proposed to limit the propositions to those needed to prove other important ones, and those needed in practical mathematics. If to this we should add those that are used in treating a considerable range of exercises, we should have a list of about one hundred thirty. But this is not the real purpose of these syllabi, or at most it seems like a relatively unimportant one. The purpose that has been attained is to stop the indefinite increase in the number of propositions that would follow from the recent developments in the geometry of the triangle and circle, and of similar modern topics, if some such counter-movement as this did not take place. If the result is, as it probably will be, to let the basal propositions of Euclid remain about as they always have been, as the standards for beginners, the syllabi will have accomplished a worthy achievement. If, in addition, they furnish an irreducible minimum of propositions to which a student may have access if he desires it, on an examination, as was intended in the case of the Harvard and the New England Association syllabi, the achievement may possibly be still more worthy. In preparing a syllabus, therefore, no one should hope to bring the teaching world at once to agree to any great reduction in the number of basal propositions, nor to agree to any radical change of terminology, symbolism, or sequence. Rather should it be the purpose to show that we have enough topics in geometry at present, and that the number of propositions is really greater than is absolutely necessary, so that teachers shall not be led to introduce any considerable number of propositions out of the large amount of new material that has recently been accumulating. Such a syllabus will always accomplish a good purpose, for at least it will provoke thought and arouse interest, but any other kind is bound to be ephemeral.[32] Besides the evolutionary attempts at rearranging and reducing in number the propositions of Euclid, there have been very many revolutionary efforts to change his treatment of geometry entirely. The great French mathematician, D'Alembert, for example, in the eighteenth century, wished to divide geometry into three branches: (1) that dealing with straight lines and circles, apparently not limited to a plane; (2) that dealing with surfaces; and (3) that dealing with solids. So Méray in France and De Paolis[33] in Italy have attempted to fuse plane and solid geometry, but have not produced a system that has been particularly successful. More recently Bourlet, Grévy, Borel, and others in France have produced several works on the elements of mathematics that may lead to something of value. They place intuition to the front, favor as much applied mathematics as is reasonable, to all of which American teachers would generally agree, but they claim that the basis of elementary geometry in the future must be the "investigation of the group of motions." It is, of course, possible that certain of the notions of the higher mathematical thought of the nineteenth century may be so simplified as to be within the comprehension of the tyro in geometry, and we should be ready to receive all efforts of this kind with open mind. These writers have not however produced the ideal work, and it may seriously be questioned whether a work based upon their ideas will prove to be educationally any more sound and usable than the labors of such excellent writers as Henrici and Treutlein, and H. Müller, and Schlegel a few years ago in Germany, and of Veronese in Italy. All such efforts, however, should be welcomed and tried out, although so far as at present appears there is nothing in sight to replace a well-arranged, vitalized, simplified textbook based upon the labors of Euclid and Legendre. The most broad-minded of the great mathematicians who have recently given attention to secondary problems is Professor Klein of Göttingen. He has had the good sense to look at something besides the mere question of good mathematics.[34] Thus he insists upon the psychologic point of view, to the end that the geometry shall be adapted to the mental development of the pupil,--a thing that is apparently ignored by Méray (at least for the average pupil), and, it is to be feared, by the other recent French writers. He then demands a careful selection of the subject matter, which in our American schools would mean the elimination of propositions that are not basal, that is, that are not used for most of the exercises that one naturally meets in elementary geometry and in applied work. He further insists upon a reasonable correlation with practical work to which every teacher will agree so long as the work is really or even potentially practical. And finally he asks that we look with favor upon the union of plane and solid geometry, and of algebra and geometry. He does not make any plea for extreme fusion, but presumably he asks that to which every one of open mind would agree, namely, that whenever the opportunity offers in teaching plane geometry, to open the vision to a generalization in space, or to the measurement of well-known solids, or to the use of the algebra that the pupil has learned, the opportunity should be seized. FOOTNOTES: [32] The author is a member of a committee that has for more than a year been considering a syllabus in geometry. This committee will probably report sometime during the year 1911. At the present writing it seems disposed to recommend about the usual list of basal propositions. [33] "Elementi di Geometria," Milan, 1884. [34] See his "Elementarmathematik vom höheren Standpunkt aus," Part II, Leipzig, 1909. CHAPTER VII THE TEXTBOOK IN GEOMETRY In considering the nature of the textbook in geometry we need to bear in mind the fact that the subject is being taught to-day in America to a class of pupils that is not composed like the classes found in other countries or in earlier generations. In general, in other countries, geometry is not taught to mixed classes of boys and girls. Furthermore, it is generally taught to a more select group of pupils than in a country where the high school and college are so popular with people in all the walks of life. In America it is not alone the boy who is interested in education in general, or in mathematics in particular, who studies geometry, and who joins with others of like tastes in this pursuit, but it is often the boy and the girl who are not compelled to go out and work, and who fill the years of youth with a not over-strenuous school life. It is therefore clear that we cannot hold the interest of such pupils by the study of Euclid alone. Geometry must, for them, be less formal than it was half a century ago. We cannot expect to make our classes enthusiastic merely over a logical sequence of proved propositions. It becomes necessary to make the work more concrete, and to give a much larger number of simple exercises in order to create the interest that comes from independent work, from a feeling of conquest, and from a desire to do something original. If we would "cast a glamor over the multiplication table," as an admirer of Macaulay has said that the latter could do, we must have the facilities for so doing. It therefore becomes necessary in weighing the merits of a textbook to consider: (1) if the number of proved propositions is reduced to a safe minimum; (2) if there is reasonable opportunity to apply the theory, the actual applications coming best, however, from the teacher as an outside interest; (3) if there is an abundance of material in the way of simple exercises, since such material is not so readily given by the teacher as the seemingly local applications of the propositions to outdoor measurements; (4) if the book gives a reasonable amount of introductory work in the use of simple and inexpensive instruments, not at that time emphasizing the formal side of the subject; (5) if there is afforded some opportunity to see the recreative side of the subject, and to know a little of the story of geometry as it has developed from ancient to modern times. But this does not mean that there is to be a geometric cataclysm. It means that we must have the same safe, conservative evolution in geometry that we have in other subjects. Geometry is not going to degenerate into mere measuring, nor is the ancient sequence going to become a mere hodge-podge without system and with no incentive to strenuous effort. It is now about fifteen hundred years since Proclus laid down what he considered the essential features of a good textbook, and in all of our efforts at reform we cannot improve very much upon his statement. "It is essential," he says, "that such a treatise should be rid of everything superfluous, for the superfluous is an obstacle to the acquisition of knowledge; it should select everything that embraces the subject and brings it to a focus, for this is of the highest service to science; it must have great regard both to clearness and to conciseness, for their opposites trouble our understanding; it must aim to generalize its theorems, for the division of knowledge into small elements renders it difficult of comprehension." It being prefaced that we must make the book more concrete in its applications, either directly or by suggesting seemingly practical outdoor work; that we must increase the number of simple exercises calling for original work; that we must reasonably reduce the number of proved propositions; and that we must not allow the good of the ancient geometry to depart, let us consider in detail some of the features of a good, practical, common-sense textbook. The early textbooks in geometry contained only the propositions, with the proofs in full, preceded by lists of definitions and assumptions (axioms and postulates). There were no exercises, and the proofs were given in essay form. Then came treatises with exercises, these exercises being grouped at the end of the work or at the close of the respective books. The next step was to the unit page, arranged in steps to aid the eye, one proposition to a page whenever this was possible. Some effort was made in this direction in France about two hundred years ago, but with no success. The arrangement has so much to commend it, however, the proof being so much more easily followed by the eye than was the case in the old-style works, that it has of late been revived. In this respect the Wentworth geometry was a pioneer in America, and so successful was the effort that this type of page has been adopted, as far as the various writers were able to adopt it, in all successful geometries that have appeared of late years in this country. As a result, the American textbooks on this subject are more helpful and pleasing to the eye than those found elsewhere. The latest improvements in textbook-making have removed most of the blemishes of arrangement that remained, scattering the exercises through the book, grading them with greater care, and making them more modern in character. But the best of the latest works do more than this. They reduce the number of proved theorems and increase the number of exercises, and they simplify the proofs whenever possible and eliminate the most difficult of the exercises of twenty-five years ago. It would be possible to carry this change too far by putting in only half as many, or a quarter as many, regular propositions, but it should not be the object to see how the work can be cut down, but to see how it can be improved. What should be the basis of selection of propositions and exercises? Evidently the selection must include the great basal propositions that are needed in mensuration and in later mathematics, together with others that are necessary to prove them. Euclid's one hundred seventy-three propositions of plane geometry were really upwards of one hundred eighty, because he several times combined two or more in one. These we may reduce to about one hundred thirty with perfect safety, or less than one a day for a school year, but to reduce still further is undesirable as well as unnecessary. It would not be difficult to dispense with a few more; indeed, we might dispense with thirty more if we should set about it, although we must never forget that a goodly number in addition to those needed for the logical sequence are necessary for the wide range of exercises that are offered. But let it be clear that if we teach 100 instead of 130, our results are liable to be about 100/130 as satisfactory. We may theorize on pedagogy as we please, but geometry will pay us about in proportion to what we give. And as to the exercises, what is the basis of selection? In general, let it be said that any exercise that pretends to be real should be so, and that words taken from science or measurements do not necessarily make the problem genuine. To take a proposition and apply it in a manner that the world never sanctions is to indulge in deceit. On the other hand, wholly to neglect the common applications of geometry to handwork of various kinds is to miss one of our great opportunities to make the subject vital to the pupil, to arouse new interest, and to give a meaning to it that is otherwise wanting. It should always be remembered that mental discipline, whatever the phrase may mean, can as readily be obtained from a genuine application of a theorem as from a mere geometric puzzle. On the other hand, it is evident that not more than 25 per cent of propositions have any genuine applications outside of geometry, and that if we are to attempt any applications at all, these must be sought mainly in the field of pure geometry. In the exercises, therefore, we seek to-day a sane and a balanced book, giving equal weight to theory and to practice, to the demands of the artisan and to those of the mathematician, to the applications of concrete science and to those of pure geometry, thus making a fusion of pure and applied mathematics, with the latter as prominent as the supply of genuine problems permits. The old is not all bad and the new is not all good, and a textbook is a success in so far as it selects boldly the good that is in the old and rejects with equal boldness the bad that is in the new. Lest the nature of the exercises of geometry may be misunderstood, it is well that we consider for a moment what constitutes a genuine application of the subject. It is the ephemeral fashion just at present in America to call these genuine applications by the name of "real problems." The name is an unfortunate importation, but that is not a matter of serious moment. The important thing is that we should know what makes a problem "real" to the pupil of geometry, especially as the whole thing is coming rapidly into disrepute through the mistaken zeal of some of its supporters. A real problem is a problem that the average citizen may sometime be called upon to solve; that, if so called upon, he will solve in the manner indicated; and that is expressed in terms that are familiar to the pupil. This definition, which seems fairly to state the conditions under which a problem can be called "real" in the schoolroom, involves three points: (1) people must be liable to meet such a problem; (2) in that case they will solve it in the way suggested by the book; (3) it must be clothed in language familiar to the pupil. For example, let the problem be to find the dimensions of a rectangular field, the data being the area of the field and the area of a road four rods wide that is cut from three sides of the field. As a real problem this is ridiculous, since no one would ever meet such a case outside the puzzle department of a schoolroom. Again, if by any stretch of a vigorous imagination any human being should care to find the area of a piece of glass, bounded by the arcs of circles, in a Gothic window in York Minster, it is fairly certain that he would not go about it in the way suggested in some of the earnest attempts that have been made by several successful teachers to add interest to geometry. And for the third point, a problem is not real to a pupil simply because it relates to moments of inertia or the tensile strength of a steel bar. Indeed, it is unreal precisely because it does talk of these things at a time when they are unfamiliar, and properly so, to the pupil. It must not be thought that puzzle problems, and unreal problems generally, have no value. All that is insisted upon is that such problems as the above are not "real," and that about 90 per cent of problems that go by this name are equally lacking in the elements that make for reality in this sense of the word. For the other 10 per cent of such problems we should be thankful, and we should endeavor to add to the number. As for the great mass, however, they are no better than those that have stood the test of generations, and by their pretense they are distinctly worse. It is proper, however, to consider whether a teacher is not justified in relating his work to those geometric forms that are found in art, let us say in floor patterns, in domes of buildings, in oilcloth designs, and the like, for the purpose of arousing interest, if for no other reason. The answer is apparent to any teacher: It is certainly justifiable to arouse the pupil's interest in his subject, and to call his attention to the fact that geometric design plays an important part in art; but we must see to it that our efforts accomplish this purpose. To make a course in geometry one on oilcloth design would be absurd, and nothing more unprofitable or depressing could be imagined in connection with this subject. Of course no one would advocate such an extreme, but it sometimes seems as if we are getting painfully near it in certain schools. A pupil has a passing interest in geometric design. He should learn to use the instruments of geometry, and he learns this most easily by drawing a few such patterns. But to keep him week after week on questions relating to such designs of however great variety, and especially to keep him upon designs relating to only one or two types, is neither sound educational policy nor even common sense. That this enthusiastic teacher or that one succeeds by such a plan is of no significance; it is the enthusiasm that succeeds, not the plan. The experience of the world is that pupils of geometry like to use the subject practically, but that they are more interested in the pure theory than in any fictitious applications, and this is why pure geometry has endured, while the great mass of applied geometry that was brought forward some three hundred years ago has long since been forgotten. The question of the real applications of the subject is considered in subsequent chapters. In Chapter VI we considered the question of the number of regular propositions to be expected in the text, and we have just considered the nature of the exercises which should follow those propositions. It is well to turn our attention next to the nature of the proofs of the basal theorems. Shall they appear in full? Shall they be merely suggested demonstrations? Shall they be only a series of questions that lead to the proof? Shall the proofs be omitted entirely? Or shall there be some combination of these plans? The natural temptation in the nervous atmosphere of America is to listen to the voice of the mob and to proceed at once to lynch Euclid and every one who stands for that for which the "Elements" has stood these two thousand years. This is what some who wish to be considered as educators tend to do; in the language of the mob, to "smash things"; to call reactionary that which does not conform to their ephemeral views. It is so easy to be an iconoclast, to think that _cui bono_ is a conclusive argument, to say so glibly that Raphael was not a great painter,--to do anything but construct. A few years ago every one must take up with the heuristic method developed in Germany half a century back and containing much that was commendable. A little later one who did not believe that the Culture Epoch Theory was vital in education was looked upon with pity by a considerable number of serious educators. A little later the man who did not think that the principle of Concentration in education was a _regula aurea_ was thought to be hopeless. A little later it may have been that Correlation was the saving factor, to be looked upon in geometry teaching as a guiding beacon, even as the fusion of all mathematics is the temporary view of a few enthusiasts to-day.[35] And just now it is vocational training that is the catch phrase, and to many this phrase seems to sound the funeral knell of the standard textbook in geometry. But does it do so? Does this present cry of the pedagogical circle really mean that we are no longer to have geometry for geometry's sake? Does it mean that a panacea has been found for the ills of memorizing without understanding a proof in the class of a teacher who is so inefficient as to allow this kind of work to go on? Does it mean that a teacher who does not see the human side of geometry, who does not know the real uses of geometry, and who has no faculty of making pupils enthusiastic over geometry,--that this teacher is to succeed with some scrappy, weak, pretending apology for a real work on the subject? No one believes in stupid teaching, in memorizing a textbook, in having a book that does all the work for a pupil, or in any of the other ills of inefficient instruction. On the other hand, no fair-minded person can condemn a type of book that has stood for generations until something besides the mere transient experiments of the moment has been suggested to replace it. Let us, for example, consider the question of having the basal propositions proved in full, a feature that is so easy to condemn as leading to memorizing. The argument in favor of a book with every basal proposition proved in full, or with most of them so proved, the rest having only suggestions for the proof, is that the pupil has before him standard forms exhibiting the best, most succinct, most clearly stated demonstrations that geometry contains. The demonstrations stand for the same thing that the type problems stand for in algebra, and are generally given in full in the same way. The argument against the plan is that it takes away the pupil's originality by doing all the work for him, allowing him to merely memorize the work. Now if all there is to geometry were in the basal propositions, this argument might hold, just as it would hold in algebra in case there were only those exercises that are solved in full. But just as this is not the case in algebra, the solved exercises standing as types or as bases for the pupil's real work, so the demonstrated proposition forms a relatively small part of geometry, standing as a type, a basis for the more important part of the work. Moreover, a pupil who uses a syllabus is exposed to a danger that should be considered, namely, that of dishonesty. Any textbook in geometry will furnish the proofs of most of the propositions in a syllabus, whatever changes there may be in the sequence, and it is not a healthy condition of mind that is induced by getting the proofs surreptitiously. Unless a teacher has more time for the course than is usually allowed, he cannot develop the new work as much as is necessary with only a syllabus, and the result is that a pupil gets more of his work from other books and has less time for exercises. The question therefore comes to this: Is it better to use a book containing standard forms of proof for the basal propositions, and have time for solving a large number of original exercises and for seeking the applications of geometry? Or is it better to use a book that requires more time on the basal propositions, with the danger of dishonesty, and allows less time for solving originals? To these questions the great majority of teachers answer in favor of the textbook with most of the basal propositions fully demonstrated. In general, therefore, it is a good rule to use the proofs of the basal propositions as models, and to get the original work from the exercises. Unless we preserve these model proofs, or unless we supply them with a syllabus, the habit of correct, succinct self-expression, which is one of the chief assets of geometry, will tend to become atrophied. So important is this habit that "no system of education in which its performance is neglected can hope or profess to evolve men and women who are competent in the full sense of the word. So long as teachers of geometry neglect the possibilities of the subject in this respect, so long will the time devoted to it be in large part wasted, and so long will their pupils continue to imbibe the vicious idea that it is much more important to be able to do a thing than to say how it can be done."[36] It is here that the chief danger of syllabus-teaching lies, and it is because of this patent fact that a syllabus without a carefully selected set of model proofs, or without the unnecessary expenditure of time by the class, is a dangerous kind of textbook. What shall then be said of those books that merely suggest the proofs, or that give a series of questions that lead to the demonstrations? There is a certain plausibility about such a plan at first sight. But it is easily seen to have only a fictitious claim to educational value. In the first place, it is merely an attempt on the part of the book to take the place of the teacher and to "develop" every lesson by the heuristic method. The questions are so framed as to admit, in most cases, of only a single answer, so that this answer might just as well be given instead of the question. The pupil has therefore a proof requiring no more effort than is the case in the standard form of textbook, but not given in the clear language of a careful writer. Furthermore, the pupil is losing here, as when he uses only a syllabus, one of the very things that he should be acquiring, namely, the habit of reading mathematics. If he met only syllabi without proofs, or "suggestive" geometries, or books that endeavored to question every proof out of him, he would be in a sorry plight when he tried to read higher mathematics, or even other elementary treatises. It is for reasons such as these that the heuristic textbook has never succeeded for any great length of time or in any wide territory. And finally, upon this point, shall the demonstrations be omitted entirely, leaving only the list of propositions,--in other words, a pure syllabus? This has been sufficiently answered above. But there is a modification of the pure syllabus that has much to commend itself to teachers of exceptional strength and with more confidence in themselves than is usually found. This is an arrangement that begins like the ordinary textbook and, after the pupil has acquired the form of proof, gradually merges into a syllabus, so that there is no temptation to go surreptitiously to other books for help. Such a book, if worked out with skill, would appeal to an enthusiastic teacher, and would accomplish the results claimed for the cruder forms of manual already described. It would not be in general as safe a book as the standard form, but with the right teacher it would bring good results. In conclusion, there are two types of textbook that have any hope of success. The first is the one with all or a large part of the basal propositions demonstrated in full, and with these propositions not unduly reduced in number. Such a book should give a large number of simple exercises scattered through the work, with a relatively small number of difficult ones. It should be modern in its spirit, with figures systematically lettered, with each page a unit as far as possible, and with every proof a model of clearness of statement and neatness of form. Above all, it should not yield to the demand of a few who are always looking merely for something to change, nor should it in a reactionary spirit return to the old essay form of proof, which hinders the pupil at this stage. The second type is the semisyllabus, otherwise with all the spirit of the first type. In both there should be an honest fusion of pure and applied geometry, with no exercises that pretend to be practical without being so, with no forced applications that lead the pupil to measure things in a way that would appeal to no practical man, with no merely narrow range of applications, and with no array of difficult terms from physics and engineering that submerge all thought of mathematics in the slough of despond of an unknown technical vocabulary. Outdoor exercises, even if somewhat primitive, may be introduced, but it should be perfectly understood that such exercises are given for the purpose of increasing the interest in geometry, and they should be abandoned if they fail of this purpose. =Bibliography.= For a list of standard textbooks issued prior to the present generation, consult the bibliography in Stamper, History of the Teaching of Geometry, New York, 1908. FOOTNOTES: [35] For some classes of schools and under certain circumstances courses in combined mathematics are very desirable. All that is here insisted upon is that any general fusion all along the line would result in weak, insipid, and uninteresting mathematics. A beginning, inspirational course in combined mathematics has a good reason for being in many high schools in spite of its manifest disadvantages, and such a course may be developed to cover all of the required mathematics given in certain schools. [36] Carson, loc. cit., p. 15. CHAPTER VIII THE RELATION OF ALGEBRA TO GEOMETRY From the standpoint of theory there is or need be no relation whatever between algebra and geometry. Algebra was originally the science of the equation, as its name[37] indicates. This means that it was the science of finding the value of an unknown quantity in a statement of equality. Later it came to mean much more than this, and Newton spoke of it as universal arithmetic, and wrote an algebra with this title. At present the term is applied to the elements of a science in which numbers are represented by letters and in which certain functions are studied, functions which it is not necessary to specify at this time. The work relates chiefly to functions involving the idea of number. In geometry, on the other hand, the work relates chiefly to form. Indeed, in pure geometry number plays practically no part, while in pure algebra form plays practically no part. In 1687 the great French philosopher, Descartes, wishing to picture certain algebraic functions, wrote a work of about a hundred pages, entitled "La Géométrie," and in this he showed a correspondence between the numbers of algebra (which may be expressed by letters) and the concepts of geometry. This was the first great step in the analytic geometry that finally gave us the graph in algebra. Since then there have been brought out from time to time other analogies between algebra and geometry, always to the advantage of each science. This has led to a desire on the part of some teachers to unite algebra and geometry into one science, having simply a class in mathematics without these special names. It is well to consider the advantages and the disadvantages of such a plan, and to decide as to the rational attitude to be taken by teachers concerning the question at issue. On the side of advantages it is claimed that there is economy of time and of energy. If a pupil is studying formulas, let the formulas of geometry be studied; if he is taking up ratio and proportion; let him do so for algebra and geometry at the same time; if he is solving quadratics, let him apply them at once to certain propositions concerning secants; and if he is proving that (_a_ + _b_)^2 equals _a_^2 + 2_ab_ + _b_^2, let him do so by algebra and by geometry simultaneously. It is claimed that not only is there economy in this arrangement, but that the pupil sees mathematics as a whole, and thus acquires more of a mastery than comes by our present "tandem arrangement." On the side of disadvantages it may be asked if the same arguments would not lead us to teach Latin and Greek together, or Latin and French, or all three simultaneously? If pupils should decline nouns in all three languages at the same time, learn to count in all at the same time, and begin to translate in all simultaneously, would there not be an economy of time and effort, and would there not be developed a much broader view of language? Now the fusionist of algebra and geometry does not like this argument, and he says that the cases are not parallel, and he tries to tell why they are not. He demands that his opponent abandon argument by analogy and advance some positive reason why algebra and geometry should not be fused. Then his opponent says that it is not for him to advance any reason for what already exists, the teaching of the two separately; that he has only to refute the fusionist's arguments, and that he has done so. He asserts that algebra and geometry are as distinct as chemistry and biology; that they have a few common points, but not enough to require teaching them together. He claims that to begin Latin and Greek at the same time has always proved to be confusing, and that the same is true of algebra and geometry. He grants that unified knowledge is desirable, but he argues that when the fine arts of music and color work fuse, and when the natural sciences of chemistry and physics are taught in the same class, and when we follow the declension of a German noun by that of a French noun and a Latin noun, and when we teach drawing and penmanship together, then it is well to talk of mixing algebra and geometry. It is well, before deciding such a question for ourselves (for evidently we cannot decide it for the world), to consider what has been the result of experience. Algebra and geometry were always taught together in early times, as were trigonometry and astronomy. The Ahmes papyrus contains both primitive algebra and primitive geometry. Euclid's "Elements" contains not only pure geometry, but also a geometric algebra and the theory of numbers. The early works of the Hindus often fused geometry and arithmetic, or geometry and algebra. Even the first great printed compendium of mathematics, the "S[=u]ma" of Paciuolo (1494) contained all of the branches of mathematics. Much of this later attempt was not, however, an example of perfect fusion, but rather of assigning one set of chapters to algebra, another to geometry, and another to arithmetic. So fusion, more or less perfect, has been tried over long periods, and abandoned as each subject grew more complete in itself, with its own language and its peculiar symbols. But it is asserted that fusion is being carried on successfully to-day by more than one enthusiastic teacher, and that this proves the contention that the plan is a good one. Books are cited to show that the arrangement is feasible, and classes are indicated where the work is progressing along this line. What, then, is the conclusion? That is a question for the teacher to settle, but it is one upon which a writer on the teaching of mathematics should not fear to express his candid opinion. It is a fact that the Greek and Latin fusion is a fair analogy. There are reasons for it, but there are many more against it, the chief one being the confusion of beginning two languages at once, and the learning simultaneously of two vocabularies that must be kept separate. It is also a fact that algebra and geometry are fully as distinct as physics and chemistry, or chemistry and biology. Life may be electricity, and a brief cessation of oxidization in the lungs brings death, but these facts are no reasons for fusing the sciences of physics, biology, and chemistry. Algebra is primarily a theory of certain elementary functions, a generalized arithmetic, while geometry is primarily a theory of form with a highly refined logic to be used in its mastery. They have a few things in common, as many other subjects have, but they have very many more features that are peculiar to the one or the other. The experience of the world has led it away from a simultaneous treatment, and the contrary experience of a few enthusiastic teachers of to-day proves only their own powers to succeed with any method. It is easy to teach logarithms in the seventh school year, but it is not good policy to do so under present conditions. So the experience of the world is against the plan of strict fusion, and no arguments have as yet been advanced that are likely to change the world's view. No one has written a book combining algebra and geometry in this fashion that has helped the cause of fusion a particle; on the contrary, every such work that has appeared has damaged that cause by showing how unscientific a result has come from the labor of an enthusiastic supporter of the movement. But there is one feature that has not been considered above, and that is a serious handicap to any effort at combining the two sciences in the high school, and this is the question of relative difficulty. It is sometimes said, in a doctrinaire fashion, that geometry is easier than algebra, since form is easier to grasp than function, and that therefore geometry should precede algebra. But every teacher of mathematics knows better than this. He knows that the simplest form is easier to grasp than the simplest function, but nevertheless that plane geometry, as we understand the term to-day, is much more difficult than elementary algebra for a pupil of fourteen. The child studies form in the kindergarten before he studies number, and this is sound educational policy. He studies form, in mensuration, throughout his course in arithmetic, and this, too, is good educational policy. This kind of geometry very properly precedes algebra. But the demonstrations of geometry, the study by pupils of fourteen years of a geometry that was written for college students and always studied by them until about fifty years ago,--that is by no means as easy as the study of a simple algebraic symbolism and its application to easy equations. If geometry is to be taught for the same reasons as at present, it cannot advantageously be taught earlier than now without much simplification, and it cannot successfully be fused with algebra save by some teacher who is willing to sacrifice an undue amount of energy to no really worthy purpose. When great mathematicians like Professor Klein speak of the fusion of all mathematics, they speak from the standpoint of advanced students, not for the teacher of elementary geometry. It is therefore probable that simple mensuration will continue, as a part of arithmetic, to precede algebra, as at present; and that algebra into or through quadratics will precede geometry,[38] drawing upon the mensuration of arithmetic as may be needed; and that geometry will follow this part of algebra, using its principles as far as possible to assist in the demonstrations and to express and manipulate its formulas. Plane geometry, or else a year of plane and solid geometry, will probably, in this country, be followed by algebra, completing quadratics and studying progressions; and by solid geometry, or a supplementary course in plane and solid geometry, this work being elective in many, if not all, schools.[39] It is also probable that a general review of mathematics, where the fusion idea may be carried out, will prove to be a feature of the last year of the high school, and one that will grow in popularity as time goes on. Such a plan will keep algebra and geometry separate, but it will allow each to use all of the other that has preceded it, and will encourage every effort in this direction. It will accomplish all that a more complete fusion really hopes to accomplish, and it will give encouragement to all who seek to modernize the spirit of each of these great branches of mathematics. There is, however, a chance for fusion in two classes of school, neither of which is as yet well developed in this country. The first is the technical high school that is at present coming into some prominence. It is not probable even here that the best results can be secured by eliminating all mathematics save only what is applicable in the shop, but if this view should prevail for a time, there would be so little left of either algebra or geometry that each could readily be joined to the other. The actual amount of algebra needed by a foreman in a machine shop can be taught in about four lessons, and the geometry or mensuration that he needs can be taught in eight lessons at the most. The necessary trigonometry may take eight more, so that it is entirely feasible to unite these three subjects. The boy who takes such a course would know as much about mathematics as a child who had read ten pages in a primer would know about literature, but he would have enough for his immediate needs, even though he had no appreciation of mathematics as a science. If any one asks if this is not all that the school should give him, it might be well to ask if the school should give only the ability to read, without the knowledge of any good literature; if it should give only the ability to sing, without the knowledge of good music; if it should give only the ability to speak, without any training in the use of good language; and if it should give a knowledge of home geography, without any intimation that the world is round,--an atom in the unfathomable universe about us. The second opportunity for fusion is possibly (for it is by no means certain) to be found in a type of school in which the only required courses are the initial ones. These schools have some strong advocates, it being claimed that every pupil should be introduced to the large branches of knowledge and then allowed to elect the ones in which he finds himself the most interested. Whether or not this is sound educational policy need not be discussed at this time; but if such a plan were developed, it might be well to offer a somewhat superficial (in the sense of abridged) course that should embody a little of algebra, a little of geometry, and a little of trigonometry. This would unconsciously become a bait for students, and the result would probably be some good teaching in the class in question. It is to be hoped that we may have some strong, well-considered textbooks upon this phase of the work. As to the fusion of trigonometry and plane geometry little need be said, because the subject is in the doctrinaire stage. Trigonometry naturally follows the chapter on similar triangles, but to put it there means, in our crowded curriculum, to eliminate something from geometry. Which, then, is better,--to give up the latter portion of geometry, or part of it at least, or to give up trigonometry? Some advocates have entered a plea for two or three lessons in trigonometry at this point, and this is a feature that any teacher may introduce as a bit of interest, as is suggested in Chapter XVI, just as he may give a popular talk to his class upon the fourth dimension or the non-Euclidean geometry. The lasting impression upon the pupil will be exactly the same as that of four lessons in Sanskrit while he is studying Latin. He might remember each with pleasure, Latin being related, as it is, to Sanskrit, and trigonometry being an outcome of the theory of similar triangles. But that either of these departures from the regular sequence is of any serious mathematical or linguistic significance no one would feel like asserting. Each is allowable on the score of interest, but neither will add to the pupil's power in any essential feature. Each of these subjects is better taught by itself, each using the other as far as possible and being followed by a review that shall make use of all. It is not improbable that we may in due time have high schools that give less extended courses in algebra and geometry, adding brief practical courses in trigonometry and the elements of the calculus; but even in such schools it is likely to be found that geometry is best taught by itself, making use of all the mathematics that has preceded it. It will of course be understood that the fusion of algebra and geometry as here understood has nothing to do with the question of teaching the two subjects simultaneously, say two days in the week for one and three days for the other. This plan has many advocates, although on the whole it has not been well received in this country. But what is meant here is the actual fusing of algebra and geometry day after day,--a plan that has as yet met with only a sporadic success, but which may be developed for beginning classes in due time. FOOTNOTES: [37] _Al-jabr wa'l-muq[=a]balah_: "restoration and equation" is a fairly good translation of the Arabic. [38] Or be carried along at the same time as a distinct topic. [39] With a single year for required geometry it would be better from every point of view to cut the plane geometry enough to admit a fair course in solid geometry. CHAPTER IX THE INTRODUCTION TO GEOMETRY There are two difficult crises in the geometry course, both for the pupil and for the teacher. These crises are met at the beginning of the subject and at the beginning of solid geometry. Once a class has fairly got into Book I, if the interest in the subject can be maintained, there are only the incidental difficulties of logical advance throughout the plane geometry. When the pupil who has been seeing figures in one plane for a year attempts to visualize solids from a flat drawing, the second difficult place is reached. Teachers going over solid geometry from year to year often forget this difficulty, but most of them can easily place themselves in the pupil's position by looking at the working drawings of any artisan,--usually simple cases in the so-called descriptive geometry. They will then realize how difficult it is to visualize a solid from an unfamiliar kind of picture. The trouble is usually avoided by the help of a couple of pieces of heavy cardboard or box board, and a few knitting needles with which to represent lines in space. If these are judiciously used in class for a few days, until the figures are understood, the second crisis is easily passed. The continued use of such material, however, or the daily use of either models or photographs, weakens the pupil, even as a child is weakened by being kept too long in a perambulator. Such devices have their place; they are useful when needed, but they are pernicious when unnecessary. Just as the mechanic must be able to make and to visualize his working drawings, so the student of solid geometry must be able to get on with pencil and paper, representing his solid figures in the flat. But the introduction to plane geometry is not so easily disposed of. The pupil at that time is entering a field that is entirely unfamiliar. He is only fourteen or fifteen years of age, and his thoughts are distinctly not on geometry. Of logic he knows little and cares less. He is not interested in a subject of which he knows nothing, not even the meaning of its name. He asks, naturally and properly, what it all signifies, what possible use there is for studying geometry, and why he should have to prove what seems to him evident without proof. To pass him successfully through this stage has taxed the ingenuity of every real teacher from the time of Euclid to the present; and just as Euclid remarked to King Ptolemy, his patron, that there is no royal road to geometry, so we may affirm that there is no royal road to the teaching of geometry. Nevertheless the experience of teachers counts for a great deal, and this experience has shown that, aside from the matter of technic in handling the class, certain suggestions are of value, and a few of these will now be set forth. First, as to why geometry is studied, it is manifestly impossible successfully to explain to a boy of fourteen or fifteen the larger reasons for studying anything whatever. When we confess ourselves honestly we find that these reasons, whether in mathematics, the natural sciences, handwork, letters, the vocations, or the fine arts, are none too clear in our own minds, in spite of any pretentious language that we may use. It is therefore most satisfactory to anticipate the question at once, and to set the pupils, for a few days, at using the compasses and ruler in the drawing of geometric designs and of the most common figures that they will use. This serves several purposes: it excites their interest, it guards against the slovenly figures that so often lead them to erroneous conclusions, it has a genuine value for the future artisan, and it shows that geometry is something besides mere theory. Whether the textbook provides for it or not, the teacher will find a few days of such work well spent, it being a simple matter to supplement the book in this respect. There was a time when some form of mechanical drawing was generally taught in the schools, but this has given place to more genuine art work, leaving it to the teacher of geometry to impart such knowledge of drawing as is a necessary preliminary to the regular study of the subject. Such work in drawing should go so far, and only so far, as to arouse an interest in geometric form without becoming wearisome, and to familiarize the pupil with the use of the instruments. He should be counseled about making fine lines, about being careful in setting the point of his compasses on the exact center that he wishes to use, and about representing a point by a very fine dot, or, preferably at first, by two crossed lines. Unless these details are carefully considered, the pupil will soon find that the lines of his drawings do not fit together, and that the result is not pleasing to the eye. The figures here given are good ones upon which to begin, the dotted construction lines being erased after the work is completed. They may be constructed with the compasses and ruler alone, or the draftsman's T-square, triangle, and protractor may be used, although these latter instruments are not necessary. We should constantly remember that there is a danger in the slavish use of instruments and of such helps as squared paper. Just as Euclid rode roughshod over the growing intellects of boys and girls, so may instruments ride roughshod over their growing perceptions by interfering with natural and healthy intuitions, and making them the subject of laborious measurement.[40] [Illustration] The pupil who cannot see the equality of vertical angles intuitively better than by the use of the protractor is abnormal. Nevertheless it is the pupil's interest that is at stake, together with his ability to use the instruments of daily life. If, therefore, he can readily be supplied with draftsmen's materials, and is not compelled to use them in a foolish manner, so much the better. They will not hurt his geometry if the teacher does not interfere, and they will help his practical drawing; but for obvious reasons we cannot demand that the pupil purchase what is not really essential to his study of the subject. The most valuable single instrument of the three just mentioned is the protractor, and since a paper one costs only a few cents and is often helpful in the drawing of figures, it should be recommended to pupils. There is also another line of work that often arouses a good deal of interest, namely, the simple field measures that can easily be made about the school grounds. Guarding against the ever-present danger of doing too much of such work, of doing work that has no interest for the pupil, of requiring it done in a way that seems unreal to a class, and of neglecting the essence of geometry by a line of work that involves no new principles,--such outdoor exercises in measurement have a positive value, and a plentiful supply of suggestions in this line is given in the subsequent chapters. The object is chiefly to furnish a motive for geometry, and for many pupils this is quite unnecessary. For some, however, and particularly for the energetic, restless boy, such work has been successfully offered by various teachers as an alternative to some of the book work. Because of this value a considerable amount of such work will be suggested for teachers who may care to use it, the textbook being manifestly not the place for occasional topics of this nature. For the purposes of an introduction only a tape line need be purchased. Wooden pins and a plumb line can easily be made. Even before he comes to the propositions in mensuration in geometry the pupil knows, from his arithmetic, how to find ordinary areas and volumes, and he may therefore be set at work to find the area of the school ground, or of a field, or of a city block. The following are among the simple exercises for a beginner: [Illustration] 1. Drive stakes at two corners, _A_ and _B_, of the school grounds, putting a cross on top of each; or make the crosses on the sidewalk, so as to get two points between which to measure. Measure from _A_ to _B_ by holding the tape taut and level, dropping perpendiculars when necessary by means of the plumb line, as shown in the figure. Check the work by measuring from _B_ back to _A_ in the same way. Pupils will find that their work should always be checked, and they will be surprised to see how the results will vary in such a simple measurement as this, unless very great care is taken. If they learn the lesson of accuracy thus early, they will have gained much. [Illustration] 2. Take two stakes, _X_, _Y_, in a field, preferably two or three hundred feet apart, always marked on top with crosses so as to have exact points from which to work. Let it then be required to stake out or "range" the line from _X_ to _Y_ by placing stakes at specified distances. One boy stands at _Y_ and another at _X_, each with a plumb line. A third one takes a plumb line and stands at _P_, the observer at _X_ motioning to him to move his plumb line to the right or the left until it is exactly in line with _X_ and _Y_. A stake is then driven at _P_, and the pupil at _X_ moves on to the stake _P_. Then _Q_ is located in the same way, and then _R_, and so on. The work is checked by ranging back from _Y_ to _X_. In some of the simple exercises suggested later it is necessary to range a line so that this work is useful in making measurements. The geometric principle involved is that two points determine a straight line. [Illustration] [Illustration] 3. To test a perpendicular or to draw one line perpendicular to another in a field, we may take a stout cord twelve feet long, having a knot at the end of every foot. If this is laid along four feet, the ends of this part being fixed, and it is stretched as here shown, so that the next vertex is five feet from one of these ends and three feet from the other end, a right angle will be formed. A right angle can also be run by making a simple instrument, such as is described in Chapter XV. Still another plan of drawing a line perpendicular to another line _AB_, from a point _P_, consists in swinging a tape from _P_, cutting _AB_ at _X_ and _Y_, and then bisecting _XY_ by doubling the tape. This fixes the foot of the perpendicular. [Illustration] 4. It is now possible to find the area of a field of irregular shape by dividing it into triangles and trapezoids, as shown in the figure. Pupils know from their work in arithmetic how to find the area of a triangle or a trapezoid, so that the area of the field is easily found. The work may be checked by comparing the results of different groups of pupils, or by drawing another diagonal and dividing the field into other triangles and trapezoids. These are about as many types of field work as there is any advantage in undertaking for the purpose of securing the interest of pupils as a preliminary to the work in geometry. Whether any of it is necessary, and for what pupils it is necessary, and how much it should trespass upon the time of scientific geometry are matters that can be decided only by the teacher of a particular class. [Illustration] [Illustration] [Illustration] [Illustration] [Illustration] [Illustration] [Illustration] A second difficulty of the pupil is seen in his attitude of mind towards proofs in general. He does not see why vertical angles should be proved equal when he knows that they are so by looking at the figure. This difficulty should also be anticipated by giving him some opportunity to know the weakness of his judgment, and for this purpose figures like the following should be placed before him. He should be asked which of these lines is longer, _AB_ or _XY_. Two equal lines should then be arranged in the form of a letter T, as here shown, and he should be asked which is the longer, _AB_ or _CD_. A figure that is very deceptive, particularly if drawn larger and with heavy cross lines, is this one in which _AB_ and _CD_ are really parallel, but do not seem to be so. Other interesting deceptions have to do with producing lines, as in these figures, where it is quite difficult in advance to tell whether _AB_ and _CD_ are in the same line, and similarly for _WX_ and _YZ_. Equally deceptive is this figure, in which it is difficult to tell which line _AB_ will lie along when produced. In the next figure _AB_ appears to be curved when in reality it is straight, and _CD_ appears straight when in reality it is curved. The first of the following circles seems to be slightly flattened at the points _P_, _Q_, _R_, _S_, and in the second one the distance _BD_ seems greater than the distance _AC_. There are many equally deceptive figures, and a few of them will convince the beginner that the proofs are necessary features of geometry. It is interesting, in connection with the tendency to feel that a statement is apparent without proof, to recall an anecdote related by the French mathematician, Biot, concerning the great scientist, Laplace: Once Laplace, having been asked about a certain point in his "Celestial Mechanics," spent nearly an hour in trying to recall the chain of reasoning which he had carelessly concealed by the words "It is easy to see." A third difficulty lies in the necessity for putting a considerable number of definitions at the beginning of geometry, in order to get a working vocabulary. Although practically all writers scatter the definitions as much as possible, there must necessarily be some vocabulary at the beginning. In order to minimize the difficulty of remembering so many new terms, it is helpful to mingle with them a considerable number of exercises in which these terms are employed, so that they may become fixed in mind through actual use. Thus it is of value to have a class find the complements of 27°, 32° 20', 41° 32' 48", 26.75°, 33 1/3°, and 0°. It is true that into the pure geometry of Euclid the measuring of angles in degrees does not enter, but it has place in the practical applications, and it serves at this juncture to fix the meaning of a new term like "complement." The teacher who thus anticipates the question as to the reason for studying geometry, the mental opposition to proving statements, and the forgetfulness of the meaning of common terms will find that much of the initial difficulty is avoided. If, now, great care is given to the first half dozen propositions, the pupil will be well on his way in geometry. As to these propositions, two plans of selection are employed. The first takes a few preliminary propositions, easily demonstrated, and seeks thus to introduce the pupil to the nature of a proof. This has the advantage of inspiring confidence and the disadvantage of appearing to prove the obvious. The second plan discards all such apparently obvious propositions as those about the equality of right angles, and the sum of two adjacent angles formed by one line meeting another, and begins at once on things that seem to the pupil as worth the proving. In this latter plan the introduction is usually made with the proposition concerning vertical angles, and the two simplest cases of congruent triangles. Whichever plan of selection is taken, it is important to introduce a considerable number of one-step exercises immediately, that is, exercises that require only one significant step in the proof. In this way the pupil acquires confidence in his own powers, he finds that geometry is not mere memorizing, and he sees that each proposition makes him the master of a large field. To delay the exercises to the end of each book, or even to delay them for several lessons, is to sow seeds that will result in the attempt to master geometry by the sheer process of memorizing. As to the nature of these exercises, however, the mistake must not be made of feeling that only those have any value that relate to football or the laying out of a tennis court. Such exercises are valuable, but such exercises alone are one-sided. Moreover, any one who examines the hundreds of suggested exercises that are constantly appearing in various journals, or who, in the preparation of teachers, looks through the thousands of exercises that come to him in the papers of his students, comes very soon to see how hollow is the pretense of most of them. As has already been said, there are relatively few propositions in geometry that have any practical applications, applications that are even honest in their pretense. The principle that the writer has so often laid down in other works, that whatever pretends to be practical should really be so, applies with much force to these exercises. When we can find the genuine application, if it is within reasonable grasp of the pupil, by all means let us use it. But to put before a class of girls some technicality of the steam engine that only a skilled mechanic would be expected to know is not education,--it is mere sham. There is a noble dignity to geometry, a dignity that a large majority of any class comes to appreciate when guided by an earnest teacher; but the best way to destroy this dignity, to take away the appreciation of pure mathematics, and to furnish weaker candidates than now for advance in this field is to deceive our pupils and ourselves into believing that the ultimate purpose of mathematics is to measure things in a way in which no one else measures them or has ever measured them. In the proof of the early propositions of plane geometry, and again at the beginning of solid geometry, there is a little advantage in using colored crayon to bring out more distinctly the equal parts of two figures, or the lines outside the plane, or to differentiate one plane from another. This device, however, like that of models in solid geometry, can easily be abused, and hence should be used sparingly, and only until the purpose is accomplished. The student of mathematics must learn to grasp the meaning of a figure drawn in black on white paper, or, more rarely, in white on a blackboard, and the sooner he is able to do this the better for him. The same thing may be said of the constructing of models for any considerable number of figures in solid geometry; enough work of this kind to enable a pupil clearly to visualize the solids is valuable, but thereafter the value is usually more than offset by the time consumed and the weakened power to grasp the meaning of a geometric drawing. There is often a tendency on the part of teachers in their first years of work to overestimate the logical powers of their pupils and to introduce forms of reasoning and technical terms that experience has proved to be unsuited to one who is beginning geometry. Usually but little harm is done, because the enthusiasm of any teacher who would use this work would carry the pupils over the difficulties without much waste of energy on their part. In the long run, however, the attempt is usually abandoned as not worth the effort. Such a term as "contrapositive," such distinctions as that between the logical and the geometric converse, or between perfect and partial geometric conversion, and such pronounced formalism as the "syllogistic method,"--all these are happily unknown to most teachers and might profitably be unknown to all pupils. The modern American textbook in geometry does not begin to be as good a piece of logic as Euclid's "Elements," and yet it is to be observed that none of these terms is found in this classic work, so that they cannot be thought to be necessary to a logical treatment of the subject. We need the word "converse," and some reference to the law of converse is therefore permissible; the meaning of the _reductio ad absurdum_, of a necessary and sufficient condition, and of the terms "synthesis" and "analysis" may properly form part of the pupil's equipment because of their universal use; but any extended incursion into the domain of logic will be found unprofitable, and it is liable to be positively harmful to a beginner in geometry. A word should be said as to the lettering of the figures in the early stages of geometry. In general, it is a great aid to the eye if this is carried out with some system, and the following suggestions are given as in accord with the best authors who have given any attention to the subject: 1. In general, letter a figure counterclockwise, for the reason that we read angles in this way in higher mathematics, and it is as easy to form this habit now as to form one that may have to be changed. Where two triangles are congruent, however, but have their sides arranged in opposite order, it is better to letter them so that their corresponding parts appear in the same order, although this makes one read clockwise. [Illustration] 2. For the same reason, read angles counterclockwise. Thus [L]_A_ is read "_BAC_," the reflex angle on the outside of the triangle being read "_CAB_." Of course this is not vital, and many authors pay no attention to it; but it is convenient, and if the teacher habitually does it, the pupils will also tend to do it. It is helpful in trigonometry, and it saves confusion in the case of a reflex angle in a polygon. Designate an angle by a single letter if this can conveniently be done. 3. Designate the sides opposite angles _A_, _B_, _C_, in a triangle, by _a_, _b_, _c_, and use these letters in writing proofs. 4. In the case of two congruent triangles use the letters _A_, _B_, _C_ and _A'_, _B'_, _C'_, or _X_, _Y_, _Z_, instead of letters chosen at random, like _D_, _K_, _L_. It is easier to follow a proof where some system is shown in lettering the figures. Some teachers insist that a pupil at the blackboard should not use the letters given in the textbook, hoping thereby to avoid memorizing. While the danger is probably exaggerated, it is easy to change with some system, using _P_, _Q_, _R_ and _P'_, _Q'_, _R'_, for example. 5. Use small letters for lines, as above stated, and also place them within angles, it being easier to speak of and to see [L]_m_ than [L]_DEF_. The Germans have a convenient system that some American teachers follow to advantage, but that a textbook has no right to require. They use, as in the following figure, _A_ for the point, _a_ for the opposite side, and the Greek letter [alpha] (alpha) for the angle. The learning of the first three Greek letters, alpha ([alpha]), beta ([beta]), and gamma ([gamma]), is not a hardship, and they are worth using, although Greek is so little known in this country to-day that the alphabet cannot be demanded of teachers who do not care to use it. [Illustration] 6. Also use small letters to represent numerical values. For example, write _c_ = 2[pi]_r_ instead of _C_ = 2[pi]_R_. This is in accord with the usage in algebra to which the pupil is accustomed. 7. Use initial letters whenever convenient, as in the case of _a_ for area, _b_ for base, _c_ for circumference, _d_ for diameter, _h_ for height (altitude), and so on. Many of these suggestions seem of slight importance in themselves, and some teachers will be disposed to object to any attempt at lettering a figure with any regard to system. If, however, they will notice how a class struggles to follow a demonstration given with reference to a figure on the blackboard, they will see how helpful it is to have some simple standards of lettering. It is hardly necessary to add that in demonstrating from a figure on a blackboard it is usually better to say "this line," or "the red line," than to say, without pointing to it, "the line _AB_." It is by such simplicity of statement and by such efforts to help the class to follow demonstrations that pupils are led through many of the initial discouragements of the subject. FOOTNOTES: [40] Carson, loc. cit., p. 13. CHAPTER X THE CONDUCT OF A CLASS IN GEOMETRY No definite rules can be given for the detailed conduct of a class in any subject. If it were possible to formulate such rules, all the personal magnetism of the teacher, all the enthusiasm, all the originality, all the spirit of the class, would depart, and we should have a dull, dry mechanism. There is no one best method of teaching geometry or anything else. The experience of the schools has shown that a few great principles stand out as generally accepted, but as to the carrying out of these principles there can be no definite rules. Let us first consider the general question of the employment of time in a recitation in geometry. We might all agree on certain general principles, and yet no two teachers ever would or ever should divide the period even approximately in the same way. First, a class should have an opportunity to ask questions. A teacher here shows his power at its best, listening sympathetically to any good question, quickly seeing the essential point, and either answering it or restating it in such a way that the pupil can answer it for himself. Certain questions should be answered by the teacher; he is there for that purpose. Others can at once be put in such a light that the pupil can himself answer them. Others may better be answered by the class. Occasionally, but more rarely, a pupil may be told to "look that up for to-morrow," a plan that is commonly considered by students as a confession of weakness on the part of the teacher, as it probably is. Of course a class will waste time in questioning a weak teacher, but a strong one need have no fear on this account. Five minutes given at the opening of a recitation to brisk, pointed questions by the class, with the same credit given to a good question as to a good answer, will do a great deal to create a spirit of comradeship, of frankness, and of honesty, and will reveal to a sympathetic teacher the difficulties of a class much better than the same amount of time devoted to blackboard work. But there must be no dawdling, and the class must feel that it has only a limited time, say five minutes at the most, to get the help it needs. Next in order of the division of the time may be the teacher's report on any papers that the class has handed in. It is impossible to tell how much of this paper work should be demanded. The local school conditions, the mental condition of the class, and the time at the disposal of the teacher are all factors in the case. In general, it may be said that enough of this kind of work is necessary to see that pupils are neat and accurate in setting down their demonstrations. On the other hand, paper work gives an opportunity for dishonesty, and it consumes a great deal of the teacher's time that might better be given to reading good books on the subject that he is teaching. If, however, any papers have been submitted, about five minutes may well be given to a rapid review of the failures and the successes. In general, it is good educational policy to speak of the errors and failures impersonally, but occasionally to mention by name any one who has done a piece of work that is worthy of special comment. Pupils may better be praised in public and blamed in private. There is such a thing, however, as praising too much, when nothing worthy of note has been done, just as there is danger of blaming too much, resulting in mere "nagging." The third division of the recitation period may profitably go to assigning the advance lesson. The class questions and the teacher's report on written work have shown the mental status of the pupils, so that the teacher now knows what he may expect for the next lesson. If he assigns his lesson at the beginning of the period, he does not have this information. If he waits to the end, he may be too hurried to give any "development" that the new lesson may require. There can be no rule as to how to assign a new lesson; it all depends upon what the lesson is, upon the mental state of the class, and not a little upon the idiosyncrasy of the teacher. The German educator, Herbart, laid down certain formal steps in developing a new lesson, and his successors have elaborated these somewhat as follows: 1. _Aim._ Always take a class into your confidence. Tell the members at the outset the goal. No one likes to be led blindfolded. 2. _Preparation._ A few brief questions to bring the class to think of what is to be considered. 3. _Presentation of the new._ Preferably this is done by questions, the answers leading the members of the class to discover the new truth for themselves. 4. _Apperception._ Calling attention to the fact that this new fact was known before, in part, and that it relates to a number of things already in the mind. The more the new can be tied up to the old the more tenaciously it will be held. 5. _Generalization and application._ It is evident at once that a great deal of time may be wasted in always following such a plan, perhaps in ever following it consciously. But, on the other hand, probably every good teacher, whether he has heard of Herbart or not, naturally covers these points in substantially this order. For an inexperienced teacher it is helpful to be familiar with them, that he may call to mind the steps, arranged in a psychological sequence, that he would do well to follow. It must always be remembered that there is quite as much danger in "developing" too much as in taking the opposite extreme. A mechanical teacher may develop a new lesson where there is need for only a question or two or a mere suggestion. It should also be recognized that students need to learn to read mathematics for themselves, and that always to take away every difficulty by explanations given in advance is weakening to any one. Therefore, in assigning the new lesson we may say "Take the next two pages," and thus discourage most of the class. On the other hand, we may spend an unnecessary amount of time and overdevelop the work of those same pages, and have the whole lesson lose all its zest. It is here that the genius of the teacher comes forth to find the happy mean. The fourth division of the hour should be reached, in general, in about ten minutes. This includes the recitation proper. But as to the nature of this work no definite instructions should be attempted. To a good teacher they would be unnecessary, to a poor one they would be harmful. Part of the class may go to the board, and as they are working, the rest may be reciting. Those at the board should be limited as to time, for otherwise a premium is placed on mere dawdling. They should be so arranged as to prevent copying, but the teacher's eye is the best preventive of this annoying feature. Those at their seats may be called upon one at a time to demonstrate at the blackboard, the rest being called upon for quick responses, as occasion demands. The European plan of having small blackboards is in many respects better than ours, since pupils cannot so easily waste time. They have to work rapidly and talk rapidly, or else take their seats. What should be put on the board, whether the figure alone, or the figure and the proof, depends upon the proposition. In general, there should be a certain number of figures put on the board for the sake of rapid work and as a basis for the proofs of the day. There should also be a certain amount of written work for the sake of commending or of criticizing adversely the proof used. There are some figures that are so complicated as to warrant being put upon sheets of paper and hung before the class. Thus there is no rule upon the subject, and the teacher must use his judgment according to the circumstances and the propositions. If the early "originals" are one-step exercises, and a pupil is required to recite rapidly, a habit of quick expression is easily acquired that leads to close attention on the part of all the class. Students as a rule recite slower than they need to, from mere habit. Phlegmatic as we think the German is, and nervous as is the American temperament, a student in geometry in a German school will usually recite more quickly and with more vigor than one with us. Our extensive blackboards have something to do with this, allowing so many pupils to be working at the board that a teacher cannot attend to them all. The result is a habit of wasting the minutes that can only be overcome by the teacher setting a definite but reasonable time limit, and holding the pupil responsible if the work is not done in the time specified. If this matter is taken in hand the first day, and special effort made in the early weeks of the year, much of the difficulty can be overcome. As to the nature of the recitation to be expected from the pupil, no definite rule can be laid down, since it varies so much with the work of the day. In general, however, a pupil should state the theorem quickly, state exactly what is given and what is to be proved, with respect to the figure, and then give the proof. At first it is desirable that he should give the authorities in full, and later give only the essential part in a few words. It is better to avoid the expression "by previous proposition," for it soon comes to be abused, and of course the learning of section numbers in a book is a barbarism. It is only by continually stating the propositions used that a pupil comes to have well fixed in his memory the basal theorems of geometry, and without these he cannot make progress in his subsequent mathematics. In general, it is better to allow a pupil to finish his proof before asking him any questions, the constant interruptions indulged in by some teachers being the cause of no little confusion and hesitancy on the part of pupils. Sometimes it is well to have a figure drawn differently from the one in the book, or lettered differently, so as to make sure that the pupil has not memorized the proof, but in general such devices are unnecessary, for a teacher can easily discover whether the proof is thoroughly understood, either by the manner of the pupil or by some slight questioning. A good textbook has the figures systematically lettered in some helpful way that is easily followed by the class that is listening to the recitation, and it is not advisable to abandon this for a random set of letters arranged in no proper order. It is good educational policy for the teacher to commend at least as often as he finds fault when criticizing a recitation at the blackboard and when discussing the pupils' papers. Optimism, encouragement, sympathy, the genuine desire to help, the putting of one's self in the pupil's place, the doing to the pupil as the teacher would that he should do in return,--these are educational policies that make for better geometry as they make for better life. The prime failure in teaching geometry lies unquestionably in the lack of interest on the part of the pupil, and this has been brought about by the ancient plan of simply reading and memorizing proofs. It is to get away from this that teachers resort to some such development of the lesson in advance, as has been suggested above. It is usually a good plan to give the easier propositions as exercises before they are reached in the text, where this can be done. An English writer has recently contributed this further idea: It might be more stimulating to encourage investigation than to demand proofs of stated facts; that is to say, "Here is a figure drawn in this way, find out anything you can about it." Some such exercises having been performed jointly by teachers and pupils, the lust of investigation and healthy competition which is present in every normal boy or girl might be awakened so far as to make such little researches really attractive; moreover, the training thus given is of far more value than that obtained by proving facts which are stated in advance, for it is seldom, if ever, that the problems of adult life present themselves in this manner. The spirit of the question, "What is true?" is positive and constructive, but that involved in "Is this true?" is negative and destructive.[41] When the question is asked, "How shall I teach?" or "What is the Method?" there is no answer such as the questioner expects. A Japanese writer, Motowori, a great authority upon the Shinto faith of his people, once wrote these words: "To have learned that there is no way to be learned and practiced is really to have learned the way of the gods." FOOTNOTES: [41] Carson, loc. cit., p. 12. CHAPTER XI THE AXIOMS AND POSTULATES The interest as well as the value of geometry lies chiefly in the fact that from a small number of assumptions it is possible to deduce an unlimited number of conclusions. With the truth of these assumptions we are not so much concerned as with the reasoning by which we draw the conclusions, although it is manifestly desirable that the assumptions should not be false, and that they should be as few as possible. It would be natural, and in some respects desirable, to call these foundations of geometry by the name "assumptions," since they are simply statements that are assumed to be true. The real foundation principles cannot be proved; they are the means by which we prove other statements. But as with most names of men or things, they have received certain titles that are time-honored, and that it is not worth the while to attempt to change. In English we call them axioms and postulates, and there is no more reason for attempting to change these terms than there is for attempting to change the names of geometry[42] and of algebra.[43] Since these terms are likely to continue, it is necessary to distinguish between them more carefully than is often done, and to consider what assumptions we are justified in including under each. In the first place, these names do not go back to Euclid, as is ordinarily supposed, although the ideas and the statements are his. "Postulate" is a Latin form of the Greek [Greek: aitêma] (_aitema_), and appears only in late translations. Euclid stated in substance, "Let the following be assumed." "Axiom" ([Greek: axiôma], _axioma_) dates perhaps only from Proclus (fifth century A.D.), Euclid using the words "common notions" ([Greek: koinai ennoiai], _koinai ennoiai_) for "axioms," as Aristotle before him had used "common things," "common principles," and "common opinions." The distinction between axiom and postulate was not clearly made by ancient writers. Probably what was in Euclid's mind was the Aristotelian distinction that an axiom was a principle common to all sciences, self-evident but incapable of proof, while the postulates were the assumptions necessary for building up the particular science under consideration, in this case geometry.[44] We thus come to the modern distinction between axiom and postulate, and say that a general statement admitted to be true without proof is an axiom, while a postulate in geometry is a geometric statement admitted to be true, without proof. For example, when we say "If equals are added to equals, the sums are equal," we state an assumption that is taken also as true in arithmetic, in algebra, and in elementary mathematics in general. This is therefore an axiom. At one time such a statement was defined as "a self-evident truth," but this has in recent years been abandoned, since what is evident to one person is not necessarily evident to another, and since all such statements are mere matters of assumption in any case. On the other hand, when we say, "A circle may be described with any given point as a center and any given line as a radius," we state a special assumption of geometry, and this assumption is therefore a geometric postulate. Some few writers have sought to distinguish between axiom and postulate by saying that the former was an assumed theorem and the latter an assumed problem, but there is no standard authority for such a distinction, and indeed the difference between a theorem and a problem is very slight. If we say, "A circle may be passed through three points not in the same straight line," we state a theorem; but if we say, "Required to pass a circle through three points," we state a problem. The mental process of handling the two propositions is, however, practically the same in spite of the minor detail of wording. So with the statement, "A straight line may be produced to any required length." This is stated in the form of a theorem, but it might equally well be stated thus: "To produce a straight line to any required length." It is unreasonable to call this an axiom in one case and a postulate in the other. However stated, it is a geometric postulate and should be so classed. What, now, are the axioms and postulates that we are justified in assuming, and what determines their number and character? It seems reasonable to agree that they should be as few as possible, and that for educational purposes they should be so clear as to be intelligible to beginners. But here we encounter two conflicting ideas. To get the "irreducible minimum" of assumptions is to get a set of statements quite unintelligible to students beginning geometry or any other branch of elementary mathematics. Such an effort is laudable when the results are intended for advanced students in the university, but it is merely suggestive to teachers rather than usable with pupils when it touches upon the primary steps of any science. In recent years several such attempts have been made. In particular, Professor Hilbert has given a system[45] of congruence postulates, but they are rather for the scientist than for the student of elementary geometry. In view of these efforts it is well to go back to Euclid and see what this great teacher of university men[46] had to suggest. The following are the five "common notions" that Euclid deemed sufficient for the purposes of elementary geometry. 1. _Things equal to the same thing are also equal to each other._ This axiom has persisted in all elementary textbooks. Of course it is a simple matter to attempt criticism,--to say that -2 is the square root of 4, and +2 is also the square root of 4, whence -2 = +2; but it is evident that the argument is not sound, and that it does not invalidate the axiom. Proclus tells us that Apollonius attempted to prove the axiom by saying, "Let _a_ equal _b_, and _b_ equal _c_. I say that _a_ equals _c_. For, since _a_ equals _b_, _a_ occupies the same space as _b_. Therefore _a_ occupies the same space as _c_. Therefore _a_ equals _c_." The proof is of no value, however, save as a curiosity. 2. _And if to equals equals are added, the wholes are equal._ 3. _If equals are subtracted from equals, the remainders are equal._ Axioms 2 and 3 are older than Euclid's time, and are the only ones given by him relating to the solution of the equation. Certain other axioms were added by later writers, as, "Things which are double of the same thing are equal to one another," and "Things which are halves of the same thing are equal to one another." These two illustrate the ancient use of _duplatio_ (doubling) and _mediatio_ (halving), the primitive forms of multiplication and division. Euclid would not admit the multiplication axiom, since to him this meant merely repeated addition. The partition (halving) axiom he did not need, and if needed, he would have inferred its truth. There are also the axioms, "If equals are added to unequals, the wholes are unequal," and "If equals are subtracted from unequals, the remainders are unequal," neither of which Euclid would have used because he did not define "unequals." The modern arrangement of axioms, covering addition, subtraction, multiplication, division, powers, and roots, sometimes of unequals as well as equals, comes from the development of algebra. They are not all needed for geometry, but in so far as they show the relation of arithmetic, algebra, and geometry, they serve a useful purpose. There are also other axioms concerning unequals that are of advantage to beginners, even though unnecessary from the standpoint of strict logic. 4. _Things that coincide with one another are equal to one another._ This is no longer included in the list of axioms. It is rather a definition of "equal," or of "congruent," to take the modern term. If not a definition, it is certainly a postulate rather than an axiom, being purely geometric in character. It is probable that Euclid included it to show that superposition is to be considered a legitimate form of proof, but why it was not placed among the postulates is not easily seen. At any rate it is unfortunately worded, and modern writers generally insert the postulate of motion instead,--that a figure may be moved about in space without altering its size or shape. The German philosopher, Schopenhauer (1844), criticized Euclid's axiom as follows: "Coincidence is either mere tautology or something entirely empirical, which belongs not to pure intuition but to external sensuous experience. It presupposes, in fact, the mobility of figures." 5. _The whole is greater than the part._ To this Clavius (1574) added, "The whole is equal to the sum of its parts," which may be taken to be a definition of "whole," but which is helpful to beginners, even if not logically necessary. Some writers doubt the genuineness of this axiom. Having considered the axioms of Euclid, we shall now consider the axioms that are needed in the study of elementary geometry. The following are suggested, not from the standpoint of pure logic, but from that of the needs of the teacher and pupil. 1. _If equals are added to equals, the sums are equal._ Instead of this axiom, the one numbered 8 below is often given first. For convenience in memorizing, however, it is better to give the axioms in the following order: (1) addition, (2) subtraction, (3) multiplication, (4) division, (5) powers and roots,--all of equal quantities. 2. _If equals are subtracted from equals, the remainders are equal._ 3. _If equals are multiplied by equals, the products are equal._ 4. _If equals are divided by equals, the quotients are equal._ 5. _Like powers or like positive roots of equals are equal._ Formerly students of geometry knew nothing of algebra, and in particular nothing of negative quantities. Now, however, in American schools a pupil usually studies algebra a year before he studies demonstrative geometry. It is therefore better, in speaking of roots, to limit them to positive numbers, since the two square roots of 4 (+2 and -2), for example, are not equal. If the pupil had studied complex numbers before he began geometry, it would have been advisable to limit the roots still further to real roots, since the four fourth roots of 1 (+1, -1, +[sqrt](-1), -[sqrt](-1)), for example, are not equal save in absolute value. It is well, however, to eliminate these fine distinctions as far as possible, since their presence only clouds the vision of the beginner. It should also be noted that these five axioms might be combined in one, namely, _If equals are operated on by equals in the same way, the results are equal_. In Axiom 1 this operation is addition, in Axiom 2 it is subtraction, and so on. Indeed, in order to reduce the number of axioms two are already combined in Axiom 5. But there is a good reason for not combining the first four with the fifth, and there is also a good reason for combining two in Axiom 5. The reason is that these are the axioms continually used in equations, and to combine them all in one would be to encourage laxness of thought on the part of the pupil. He would always say "by Axiom 1" whatever he did to an equation, and the teacher would not be certain whether the pupil was thinking definitely of dividing equals by equals, or had a hazy idea that he was manipulating an equation in some other way that led to an answer. On the other hand, Axiom 5 is not used as often as the preceding four, and the interchange of integral and fractional exponents is relatively common, so that the joining of these two axioms in one for the purpose of reducing the total number is justifiable. 6. _If unequals are operated on by positive equals in the same way, the results are unequal in the same order._ This includes in a single statement the six operations mentioned in the preceding axioms; that is, if _a_ > _b_ and if _x_ = _y_, then _a_ + _x_ > _b_ + _y_, _a_ - _x_ > _b_ - _y_, _ax_ > _by_, etc. The reason for thus combining six axioms in one in the case of inequalities is apparent. They are rarely used in geometry, and if a teacher is in doubt as to the pupil's knowledge, he can easily inquire in the few cases that arise, whereas it would consume a great deal of time to do this for the many equations that are met. The axiom is stated in such a way as to exclude multiplying or dividing by negative numbers, this case not being needed. 7. _If unequals are added to unequals in the same order, the sums are unequal in the same order; if unequals are subtracted from equals, the remainders are unequal in the reverse order._ These are the only cases in which unequals are necessarily combined with unequals, or operate upon equals in geometry, and the axiom is easily explained to the class by the use of numbers. 8. _Quantities that are equal to the same quantity or to equal quantities are equal to each other._ In this axiom the word "quantity" is used, in the common manner of the present time, to include number and all geometric magnitudes (length, area, volume). 9. _A quantity may be substituted for its equal in an equation or in an inequality._ This axiom is tacitly assumed by all writers, and is very useful in the proofs of geometry. It is really the basis of several other axioms, and if we were seeking the "irreducible minimum," it would replace them. Since, however, we are seeking only a reasonably abridged list of convenient assumptions that beginners will understand and use, this axiom has much to commend it. If we consider the equations (1) _a_ = _x_ and (2) _b_ = _x_, we see that for _x_ in equation (1) we may substitute _b_ from equation (2) and have _a_ = _b_; in other words, that Axiom 8 is included in Axiom 9. Furthermore, if (1) _a_ = _b_ and (2) _x_ = _y_, then since _a_ + _x_ is the same as _a_ + _x_, we may, by substituting, say that _a_ + _x_ = _a_ + _x_ = _b_ + _x_ = _b_ + _y_. In other words, Axiom 1 is included in Axiom 9. Thus an axiom that includes others has a legitimate place, because a beginner would be too much confused by seeing its entire scope, and because he will make frequent use of it in his mathematical work. 10. _If the first of three quantities is greater than the second, and the second is greater than the third, then the first is greater than the third._ This axiom is needed several times in geometry. The case in which _a_ > _b_ and _b_ = _c_, therefore _a_ > _c_, is provided for in Axiom 9. 11. _The whole is greater than any of its parts and is equal to the sum of all its parts._ The latter part of this axiom is really only the definition of "whole," and it would be legitimate to state a definition accordingly and refer to it where the word is employed. Where, however, we wish to speak of a polygon, for example, and wish to say that the area is equal to the combined areas of the triangles composing it, it is more satisfactory to have this axiom to which to refer. It will be noticed that two related axioms are here combined in one, for a reason similar to the one stated under Axiom 5. In the case of the postulates we are met by a problem similar to the one confronting us in connection with the axioms,--the problem of the "irreducible minimum" as related to the question of teaching. Manifestly Euclid used postulates that he did not state, and proved some statements that he might have postulated.[47] The postulates given by Euclid under the name [Greek: aitêmata](_aitemata_) were requests made by the teacher to his pupil that certain things be conceded. They were five in number, as follows: 1. _Let the following be conceded: to draw a straight line from any point to any point._ Strictly speaking, Euclid might have been required to postulate that points and straight lines exist, but he evidently considered this statement sufficient. Aristotle had, however, already called attention to the fact that a mere definition was sufficient only to show what a concept is, and that this must be followed by a proof that the thing exists. We might, for example, define _x_ as a line that bisects an angle without meeting the vertex, but this would not show that an _x_ exists, and indeed it does not exist. Euclid evidently intended the postulate to assert that this line joining two points is unique, which is only another way of saying that two points determine a straight line, and really includes the idea that two straight lines cannot inclose space. For purposes of instruction, the postulate would be clearer if it read, _One straight line, and only one, can be drawn through two given points_. 2. _To produce a finite straight line continuously in a straight line._ In this postulate Euclid practically assumes that a straight line can be produced only in a straight line; in other words, that two different straight lines cannot have a common segment. Several attempts have been made to prove this fact, but without any marked success. 3. _To describe a circle with any center and radius._ 4. _That all right angles are equal to one another._ While this postulate asserts the essential truth that a right angle is a _determinate magnitude_ so that it really serves as an invariable standard by which other (acute and obtuse) angles may be measured, much more than this is implied, as will easily be seen from the following consideration. If the statement is to be _proved_, it can only be proved by the method of applying one pair of right angles to another and so arguing their equality. But this method would not be valid unless on the assumption of the _invariability of figures_, which would have to be asserted as an antecedent postulate. Euclid preferred to assert as a postulate, directly, the fact that all right angles are equal; and hence his postulate must be taken as equivalent to the principle of _invariability of figures_, or its equivalent, the _homogeneity_ of space.[48] It is better educational policy, however, to assert this fact more definitely, and to state the additional assumption that figures may be moved about in space without deformation. The fourth of Euclid's postulates is often given as an axiom, following the idea of the Greek philosopher Geminus (who flourished in the first century B.C.), but this is because Euclid's distinction between axiom and postulate is not always understood. Proclus (410-485 A.D.) endeavored to prove the postulate, and a later and more scientific effort was made by the Italian geometrician Saccheri (1667-1733). It is very commonly taken as a postulate that all straight angles are equal, this being more evident to the senses, and the equality of right angles is deduced as a corollary. This method of procedure has the sanction of many of our best modern scholars. 5. _That, if a straight line falling on two straight lines make the interior angle on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles._ This famous postulate, long since abandoned in teaching the beginner in geometry, is a remarkable evidence of the clear vision of Euclid. For two thousand years mathematicians sought to prove it, only to demonstrate the wisdom of its author in placing it among the assumptions.[49] Every proof adduced contains some assumption that practically conceals the postulate itself. Thus the great English mathematician John Wallis (1616-1703) gave a proof based upon the assumption that "given a figure, another figure is possible which is similar to the given one, and of any size whatever." Legendre (1752-1833) did substantially the same at one time, and offered several other proofs, each depending upon some equally unprovable assumption. The definite proof that the postulate cannot be demonstrated is due to the Italian Beltrami (1868). Of the alternative forms of the postulate, that of Proclus is generally considered the best suited to beginners. As stated by Playfair (1795), this is, "Through a given point only one parallel can be drawn to a given straight line"; and as stated by Proclus, "If a straight line intersect one of two parallels, it will intersect the other also." Playfair's form is now the common "postulate of parallels," and is the one that seems destined to endure. Posidonius and Geminus, both Stoics of the first century B.C., gave as their alternative, "There exist straight lines everywhere equidistant from one another." One of Legendre's alternatives is, "There exists a triangle in which the sum of the three angles is equal to two right angles." One of the latest attempts to suggest a substitute is that of the Italian Ingrami (1904), "Two parallel straight lines intercept, on every transversal which passes through the mid-point of a segment included between them, another segment the mid-point of which is the mid-point of the first." Of course it is entirely possible to assume that through a point more than one line can be drawn parallel to a given straight line, in which case another type of geometry can be built up, equally rigorous with Euclid's. This was done at the close of the first quarter of the nineteenth century by Lobachevsky (1793-1856) and Bolyai (1802-1860), resulting in the first of several "non-Euclidean" geometries.[50] Taking the problem to be that of stating a reasonably small number of geometric assumptions that may form a basis to supplement the general axioms, that shall cover the most important matters to which the student must refer, and that shall be so simple as easily to be understood by a beginner, the following are recommended: 1. _One straight line, and only one, can be drawn through two given points._ This should also be stated for convenience in the form, _Two points determine a straight line_. From it may also be drawn this corollary, _Two straight lines can intersect in only one point_, since two points would determine a straight line. Such obvious restatements of or corollaries to a postulate are to be commended, since a beginner is often discouraged by having to prove what is so obvious that a demonstration fails to commend itself to his mind. 2. _A straight line may be produced to any required length._ This, like Postulate 1, requires the use of a straightedge for drawing the physical figure. The required length is attained by using the compasses to measure the distance. The straightedge and the compasses are the only two drawing instruments recognized in elementary geometry.[51] While this involves more than Euclid's postulate, it is a better working assumption for beginners. 3. _A straight line is the shortest path between two points._ This is easily proved by the method of Euclid[52] for the case where the paths are broken lines, but it is needed as a postulate for the case of curve paths. It is a better statement than the common one that a straight line is the shortest _distance_ between two points; for distance is measured on a line, but it is not itself a line. Furthermore, there are scientific objections to using the word "distance" any more than is necessary. 4. _A circle may be described with any given point as a center and any given line as a radius._ This involves the use of the second of the two geometric instruments, the compasses. 5. _Any figure may be moved from one place to another without altering the size or shape._ This is the postulate of the homogeneity of space, and asserts that space is such that we may move a figure as we please without deformation of any kind. It is the basis of all cases of superposition. 6. _All straight angles are equal._ It is possible to prove this, and therefore, from the standpoint of strict logic, it is unnecessary as a postulate. On the other hand, it is poor educational policy for a beginner to attempt to prove a thing that is so obvious. The attempt leads to a loss of interest in the subject, the proposition being (to state a paradox) hard because it is so easy. It is, of course, possible to postulate that straight angles are equal, and to draw the conclusion that their halves (right angles) are equal; or to proceed in the opposite direction, and postulate that all right angles are equal, and draw the conclusion that their doubles (straight angles) are equal. Of the two the former has the advantage, since it is probably more obvious that all straight angles are equal. It is well to state the following definite corollaries to this postulate: (1) _All right angles are equal_; (2) _From a point in a line only one perpendicular can be drawn to the line_, since two perpendiculars would make the whole (right angle) equal to its part; (3) _Equal angles have equal complements, equal supplements, and equal conjugates_; (4) _The greater of two angles has the less complement, the less supplement, and the less conjugate._ All of these four might appear as propositions, but, as already stated, they are so obvious as to be more harmful than useful to beginners when given in such form. The postulate of parallels may properly appear in connection with that topic in Book I, and it is accordingly treated in Chapter XIV. There is also another assumption that some writers are now trying to formulate in a simple fashion. We take, for example, a line segment _AB_, and describe circles with _A_ and _B_ respectively as centers, and with a radius _AB_. We say that the circles will intersect as at _C_ and _D_. But how do we know that they intersect? We assume it, just as we assume that an indefinite straight line drawn from a point inclosed by a circle will, if produced far enough, cut the circle twice. Of course a pupil would not think of this if his attention was not called to it, and the harm outweighs the good in doing this with one who is beginning the study of geometry. With axioms and with postulates, therefore, the conclusion is the same: from the standpoint of scientific geometry there is an irreducible minimum of assumptions, but from the standpoint of practical teaching this list should give place to a working set of axioms and postulates that meet the needs of the beginner. =Bibliography.= Smith, Teaching of Elementary Mathematics, New York, 1900; Young, The Teaching of Mathematics, New York, 1901; Moore, On the Foundations of Mathematics, _Bulletin of the American Mathematical Society_, 1903, p. 402; Betz, Intuition and Logic in Geometry, _The Mathematics Teacher_, Vol. II, p. 3; Hilbert, The Foundations of Geometry, Chicago, 1902; Veblen, A System of Axioms for Geometry, _Transactions of the American Mathematical Society_, 1904, p. 343. FOOTNOTES: [42] From the Greek [Greek: gê], _ge_ (earth), + [Greek: metrein], _metrein_ (to measure), although the science has not had to do directly with the measure of the earth for over two thousand years. [43] From the Arabic _al_ (the) + _jabr_ (restoration), referring to taking a quantity from one side of an equation and then restoring the balance by taking it from the other side (see page 37). [44] One of the clearest discussions of the subject is in W. B. Frankland, "The First Book of Euclid's 'Elements,'" p. 26, Cambridge, 1905. [45] "Grundlagen der Geometrie," Leipzig, 1899. See Heath's "Euclid," Vol. I, p. 229, for an English version; also D. E. Smith, "Teaching of Elementary Mathematics," p. 266, New York, 1900. [46] We need frequently to recall the fact that Euclid's "Elements" was intended for advanced students who went to Alexandria as young men now go to college, and that the book was used only in university instruction in the Middle Ages and indeed until recent times. [47] For example, he moves figures without deformation, but states no postulate on the subject; and he proves that one side of a triangle is less than the sum of the other two sides, when he might have postulated that a straight line is the shortest path between two points. Indeed, his followers were laughed at for proving a fact so obvious as this one concerning the triangle. [48] T. L. Heath, "Euclid," Vol. I, p. 200. [49] For a résumé of the best known attempts to prove this postulate, see Heath, "Euclid," Vol. I, p. 202; W. B. Frankland, "Theories of Parallelism," Cambridge, 1910. [50] For the early history of this movement see Engel and Stäckel, "Die Theorie der Parallellinien von Euklid bis auf Gauss," Leipzig, 1895; Bonola, Sulla teoria delle parallele e sulle geometrie non-euclidee, in his "Questioni riguardanti la geometria elementare," 1900; Karagiannides, "Die nichteuklidische Geometrie vom Alterthum bis zur Gegenwart," Berlin, 1893. [51] This limitation upon elementary geometry was placed by Plato (died 347 B.C.), as already stated. [52] Book I, Proposition 20. CHAPTER XII THE DEFINITIONS OF GEOMETRY When we consider the nature of geometry it is evident that more attention must be paid to accuracy of definitions than is the case in most of the other sciences. The essence of all geometry worthy of serious study is not the knowledge of some fact, but the proof of that fact; and this proof is always based upon preceding proofs, assumptions (axioms or postulates), or definitions. If we are to prove that one line is perpendicular to another, it is essential that we have an exact definition of "perpendicular," else we shall not know when we have reached the conclusion of the proof. The essential features of a definition are that the term defined shall be described in terms that are simpler than, or at least better known than, the thing itself; that this shall be done in such a way as to limit the term to the thing defined; and that the description shall not be redundant. It would not be a good definition to say that a right angle is one fourth of a perigon and one half of a straight angle, because the concept "perigon" is not so simple, and the term "perigon" is not so well known, as the term and the concept "right angle," and because the definition is redundant, containing more than is necessary. It is evident that satisfactory definitions are not always possible; for since the number of terms is limited, there must be at least one that is at least as simple as any other, and this cannot be described in terms simpler than itself. Such, for example, is the term "angle." We can easily explain the meaning of this word, and we can make the concept clear, but this must be done by a certain amount of circumlocution and explanation, not by a concise and perfect definition. Unless a beginner in geometry knows what an angle is before he reads the definition in a textbook, he will not know from the definition. This fact of the impossibility of defining some of the fundamental concepts will be evident when we come to consider certain attempts that have been made in this direction. It should also be understood in this connection that a definition makes no assertion as to the existence of the thing defined. If we say that a tangent to a circle is an unlimited straight line that touches the circle in one point, and only one, we do not assert that it is possible to have such a line; that is a matter for proof. Not in all cases, however, can this proof be given, as in the existence of the simplest concepts. We cannot, for example, prove that a point or a straight line exists after we have defined these concepts. We therefore tacitly or explicitly assume (postulate) the existence of these fundamentals of geometry. On the other hand, we can prove that a tangent exists, and this may properly be considered a legitimate proposition or corollary of elementary geometry. In relation to geometric proof it is necessary to bear in mind, therefore, that we are permitted to define any term we please; for example, "a seven-edged polyhedron" or Leibnitz's "ten-faced regular polyhedron," neither of which exists; but, strictly speaking, we have no right to make use of a definition in a proof until we have shown or postulated that the thing defined has an existence. This is one of the strong features of Euclid's textbook. Not being able to prove that a point, a straight line, and a circle exists, he practically postulates these facts; but he uses no other definition in a proof without showing that the thing defined exists, and this is his reason for mingling his problems with his theorems. At the present time we confessedly sacrifice his logic in this respect for the reason that we teach geometry to pupils who are too young to appreciate that logic. It was pointed out by Aristotle, long before Euclid, that it is not a satisfactory procedure to define a thing by means of terms that are strictly not prior to it, as when we attempt to define something by means of its opposite. Thus to define a curve as "a line, no part of which is straight," would be a bad definition unless "straight" had already been explicitly defined; and to define "bad" as "not good" is unsatisfactory for the reason that "bad" and "good" are concepts that are evolved simultaneously. But all this is only a detail under the general principle that a definition must employ terms that are better understood than the one defined. It should be understood that some definitions are much more important than others, considered from the point of view of the logic of geometry. Those that enter into geometric proofs are basal; those that form part of the conversational language of geometry are not. Euclid gave twenty-three definitions in Book I, and did not make use of even all of these terms. Other terms, those not employed in his proofs, he assumed to be known, just as he assumed a knowledge of any other words in his language. Such procedure would not be satisfactory under modern conditions, but it is of great importance that the teacher should recognize that certain definitions are basal, while others are merely informational. It is now proposed to consider the basal definitions of geometry, first, that the teacher may know what ones are to be emphasized and learned; and second, that he may know that the idea that the standard definitions can easily be improved is incorrect. It is hoped that the result will be the bringing into prominence of the basal concepts, and the discouraging of attempts to change in unimportant respects the definitions in the textbook used by the pupil. In order to have a systematic basis for work, the definitions of two books of Euclid will first be considered.[53] 1. POINT. _A point is that which has no part._ This was incorrectly translated by Capella in the fifth century, "Punctum est cuius pars nihil est" (a point is that of which a part is nothing), which is as much as to say that the point itself is nothing. It generally appears, however, as in the Campanus edition,[54] "Punctus est cuius pars non est," which is substantially Euclid's wording. Aristotle tells of the definitions of point, line, and surface that prevailed in his time, saying that they all defined the prior by means of the posterior.[55] Thus a point was defined as "an extremity of a line," a line as "the extremity of a surface," and a surface as "the extremity of a solid,"--definitions still in use and not without their value. For it must not be assumed that scientific priority is necessarily priority in fact; a child knows of "solid" before he knows of "point," so that it may be a very good way to explain, if not to define, by beginning with solid, passing thence to surface, thence to line, and thence to point. The first definition of point of which Proclus could learn is attributed by him to the Pythagoreans, namely, "a monad having position," the early form of our present popular definition of a point as "position without magnitude." Plato defined it as "the beginning of a line," thus presupposing the definition of "line"; and, strangely enough, he anticipated by two thousand years Cavalieri, the Italian geometer, by speaking of points as "indivisible lines." To Aristotle, who protested against Plato's definitions, is due the definition of a point as "something indivisible but having position." Euclid's definition is essentially that of Aristotle, and is followed by most modern textbook writers, except as to its omission of the reference to position. It has been criticized as being negative, "which has _no_ part"; but it is generally admitted that a negative definition is admissible in the case of the most elementary concepts. For example, "blind" must be defined in terms of a negation. At present not much attention is given to the definition of "point," since the term is not used as the basis of a proof, but every effort is made to have the concept clear. It is the custom to start from a small solid, conceive it to decrease in size, and think of the point as the limit to which it is approaching, using these terms in their usual sense without further explanation. 2. LINE. _A line is breadthless length._ This is usually modified in modern textbooks by saying that "a line is that which has length without breadth or thickness," a statement that is better understood by beginners. Euclid's definition is thought to be due to Plato, and is only one of many definitions that have been suggested. The Pythagoreans having spoken of the point as a monad naturally were led to speak of the line as dyadic, or related to two. Proclus speaks of another definition, "magnitude in one dimension," and he gives an excellent illustration of line as "the edge of a shadow," thus making it real but not material. Aristotle speaks of a line as a magnitude "divisible in one way only," as contrasted with a surface which is divisible in two ways, and with a solid which is divisible in three ways. Proclus also gives another definition as the "flux of a point," which is sometimes rendered as the path of a moving point. Aristotle had suggested the idea when he wrote, "They say that a line by its motion produces a surface, and a point by its motion a line." Euclid did not deem it necessary to attempt a classification of lines, contenting himself with defining only a straight line and a circle, and these are really the only lines needed in elementary geometry. His commentators, however, made the attempt. For example. Heron (first century A.D.) probably followed his definition of line by this classification: { Straight Lines { { Circular circumferences { Not straight { Spiral shaped { Curved (generally) Proclus relates that both Plato and Aristotle divided lines into "straight," "circular," and "a mixture of the two," a statement which is not quite exact, but which shows the origin of a classification not infrequently found in recent textbooks. Geminus (_ca._ 50 B.C.) is said by Proclus to have given two classifications, of which one will suffice for our purposes: { Composite (broken line forming an angle) { Lines { { Forming a figure, or determinate. (Circle, { { ellipse, cissoid.) { Incomposite { Not forming a figure, or indeterminate and { extending without a limit. (Straight { line, parabola, hyperbola, conchoid.) Of course his view of the cissoid, the curve represented by the equation _y_^2(_a_ + _x_) = (_a_ - _x_)^3, is not the modern view. 3. _The extremities of a line are points._ This is not a definition in the sense of its two predecessors. A modern writer would put it as a note under the definition of line. Euclid did not wish to define a point as the extremity of a line, for Aristotle had asserted that this was not scientific; so he defined point and line, and then added this statement to show the relation of one to the other. Aristotle had improved upon this by stating that the "division" of a line, as well as an extremity, is a point, as is also the intersection of two lines. These statements, if they had been made by Euclid, would have avoided the objection made by Proclus, that some lines have no extremities, as, for example, a circle, and also a straight line extending infinitely in both directions. 4. STRAIGHT LINE. _A straight line is that which lies evenly with respect to the points on itself._ This is the least satisfactory of all of the definitions of Euclid, and emphasizes the fact that the straight line is the most difficult to define of the elementary concepts of geometry. What is meant by "lies evenly"? Who would know what a straight line is, from this definition, if he did not know in advance? The ancients suggested many definitions of straight line, and it is well to consider a few in order to appreciate the difficulties involved. Plato spoke of it as "that of which the middle covers the ends," meaning that if looked at endways, the middle would make it impossible to see the remote end. This is often modified to read that "a straight line when looked at endways appears as a point,"--an idea that involves the postulate that our line of sight is straight. Archimedes made the statement that "of all the lines which have the same extremities, the straight line is the least," and this has been modified by later writers into the statement that "a straight line is the shortest distance between two points." This is open to two objections as a definition: (1) a line is not distance, but distance is the _length_ of a line,--it is measured on a line; (2) it is merely stating a property of a straight line to say that "a straight line is the shortest path between two points,"--a proper postulate but not a good definition. Equally objectionable is one of the definitions suggested by both Heron and Proclus, that "a straight line is a line that is stretched to its uttermost"; for even then it is reasonable to think of it as a catenary, although Proclus doubtless had in mind the Archimedes statement. He also stated that "a straight line is a line such that if any part of it is in a plane, the whole of it is in the plane,"--a definition that runs in a circle, since plane is defined by means of straight line. Proclus also defines it as "a uniform line, capable of sliding along itself," but this is also true of a circle. Of the various definitions two of the best go back to Heron, about the beginning of our era. Proclus gives one of them in this form, "That line which, when its ends remain fixed, itself remains fixed." Heron proposed to add, "when it is, as it were, turned round in the same plane." This has been modified into "that which does not change its position when it is turned about its extremities as poles," and appears in substantially this form in the works of Leibnitz and Gauss. The definition of a straight line as "such a line as, with another straight line, does not inclose space," is only a modification of this one. The other definition of Heron states that in a straight line "all its parts fit on all in all ways," and this in its modern form is perhaps the most satisfactory of all. In this modern form it may be stated, "A line such that any part, placed with its ends on any other part, must lie wholly in the line, is called a straight line," in which the force of the word "must" should be noted. This whole historical discussion goes to show how futile it is to attempt to define a straight line. What is needed is that we should explain what is meant by a straight line, that we should illustrate it, and that pupils should then read the definition understandingly. 5. SURFACE. _A surface is that which has length and breadth._ This is substantially the common definition of our modern textbooks. As with line, so with surface, the definition is not entirely satisfactory, and the chief consideration is that the meaning of the term should be made clear by explanations and illustrations. The shadow cast on a table top is a good illustration, since all idea of thickness is wanting. It adds to the understanding of the concept to introduce Aristotle's statement that a surface is generated by a moving line, modified by saying that it _may_ be so generated, since the line might slide along its own trace, or, as is commonly said in mathematics, along itself. 6. _The extremities of a surface are lines._ This is open to the same explanation and objection as definition 3, and is not usually given in modern textbooks. Proclus calls attention to the fact that the statement is hardly true for a complete spherical surface. 7. PLANE. _A plane surface is a surface which lies evenly with the straight lines on itself._ Euclid here follows his definition of straight line, with a result that is equally unsatisfactory. For teaching purposes the translation from the Greek is not clear to a beginner, since "lies evenly" is a term not simpler than the one defined. As with the definition of a straight line, so with that of a plane, numerous efforts at improvement have been made. Proclus, following a hint of Heron's, defines it as "the surface which is stretched to the utmost," and also, this time influenced by Archimedes's assumption concerning a straight line, as "the least surface among all those which have the same extremities." Heron gave one of the best definitions, "A surface all the parts of which have the property of fitting on [each other]." The definition that has met with the widest acceptance, however, is a modification of one due to Proclus, "A surface such that a straight line fits on all parts of it." Proclus elsewhere says, "[A plane surface is] such that the straight line fits on it all ways," and Heron gives it in this form, "[A plane surface is] such that, if a straight line pass through two points on it, the line coincides with it at every spot, all ways." In modern form this appears as follows: "A surface such that a straight line joining any two of its points lies wholly in the surface is called a plane," and for teaching purposes we have no better definition. It is often known as Simson's definition, having been given by Robert Simson in 1756. The French mathematician, Fourier, proposed to define a plane as formed by the aggregate of all the straight lines which, passing through one point on a straight line in space, are perpendicular to that line. This is clear, but it is not so usable for beginners as Simson's definition. It appears as a theorem in many recent geometries. The German mathematician, Crelle, defined a plane as a surface containing all the straight lines (throughout their whole length) passing through a fixed point and also intersecting a straight line in space, but of course this intersected straight line must not pass through the fixed point. Crelle's definition is occasionally seen in modern textbooks, but it is not so clear to the pupil as Simson's. Of the various ultrascientific definitions of a plane that have been suggested of late it is hardly of use to speak in a book concerned primarily with practical teaching. No one of them is adapted to the needs and the comprehension of the beginner, and it seems that we are not likely to improve upon the so-called Simson form. 8. PLANE ANGLE. _A plane angle is the inclination to each other of two lines in a plane which meet each other and do not lie in a straight line._ This definition, it will be noticed, includes curvilinear angles, and the expression "and do not lie in a straight line" states that the lines must not be continuous one with the other, that is, that zero and straight angles are excluded. Since Euclid does not use the curvilinear angle, and it is only the rectilinear angle with which we are concerned, we will pass to the next definition and consider this one in connection therewith. 9. RECTILINEAR ANGLE. _When the lines containing the angle are straight, the angle is called rectilinear._ This definition, taken with the preceding one, has always been a subject of criticism. In the first place it expressly excludes the straight angle, and, indeed, the angles of Euclid are always less than 180°, contrary to our modern concept. In the second place it defines angle by means of the word "inclination," which is itself as difficult to define as angle. To remedy these defects many substitutes have been proposed. Apollonius defined angle as "a contracting of a surface or a solid at one point under a broken line or surface." Another of the Greeks defined it as "a quantity, namely, a distance between the lines or surfaces containing it." Schotten[56] says that the definitions of angle generally fall into three groups: _a._ An angle is the difference of direction between two lines that meet. This is no better than Euclid's, since "difference of direction" is as difficult to define as "inclination." _b._ An angle is the amount of turning necessary to bring one side to the position of the other side. _c._ An angle is the portion of the plane included between its sides. Of these, _b_ is given by way of explanation in most modern textbooks. Indeed, we cannot do better than simply to define an angle as the opening between two lines which meet, and then explain what is meant by size, through the bringing in of the idea of rotation. This is a simple presentation, it is easily understood, and it is sufficiently accurate for the real purpose in mind, namely, the grasping of the concept. We should frankly acknowledge that the concept of angle is such a simple one that a satisfactory definition is impossible, and we should therefore confine our attention to having the concept understood. 10. _When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands._ We at present separate these definitions and simplify the language. 11. _An obtuse angle is an angle greater than a right angle._ 12. _An acute angle is an angle less than a right angle._ The question sometimes asked as to whether an angle of 200° is obtuse, and whether a negative angle, say -90°, is acute, is answered by saying that Euclid did not conceive of angles equal to or greater than 180° and had no notion of negative quantities. Generally to-day we define an obtuse angle as "greater than one and less than two right angles." An acute angle is defined as "an angle less than a right angle," and is considered as positive under the general understanding that all geometric magnitudes are positive unless the contrary is stated. 13. _A boundary is that which is an extremity of anything._ The definition is not exactly satisfactory, for a circle is the boundary of the space inclosed, but we hardly consider it as the extremity of that space. Euclid wishes the definition before No. 14. 14. _A figure is that which is contained by any boundary or boundaries._ The definition is not satisfactory, since it excludes the unlimited straight line, the angle, an assemblage of points, and other combinations of lines and points which we should now consider as figures. 15. _A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another._ 16. _And the point is called the center of the circle._ Some commentators add after "one line," definition 15, the words "which is called the circumference," but these are not in the oldest manuscripts. The Greek idea of a circle was usually that of part of a plane which is bounded by a line called in modern times the circumference, although Aristotle used "circle" as synonymous with "the bounding line." With the growth of modern mathematics, however, and particularly as a result of the development of analytic geometry, the word "circle" has come to mean the bounding line, as it did with Aristotle, a century before Euclid's time. This has grown out of the equations of the various curves, _x_^2 + _y_^2 = _r_^2 representing the circle-_line_, _a_^2_y_^2 + _b_^2_x_^2 = _a_^2_b_^2 representing the ellipse-_line_, and so on. It is natural, therefore, that circle, ellipse, parabola, and hyperbola should all be looked upon as lines. Since this is the modern use of "circle" in English, it has naturally found its way into elementary geometry, in order that students should not have to form an entirely different idea of circle on beginning analytic geometry. The general body of American teachers, therefore, at present favors using "circle" to mean the bounding line and "circumference" to mean the length of that line. This requires redefining "area of a circle," and this is done by saying that it is the area of the plane space inclosed. The matter is not of greatest consequence, but teachers will probably prefer to join in the modern American usage of the term. 17. DIAMETER. _A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle._ The word "diameter" is from two Greek words meaning a "through measurer," and it was also used by Euclid for the diagonal of a square, and more generally for the diagonal of any parallelogram. The word "diagonal" is a later term and means the "through angle." It will be noticed that Euclid adds to the usual definition the statement that a diameter bisects the circle. He does this apparently to justify his definition (18), of a semicircle (a half circle). Thales is said to have been the first to prove that a diameter bisects the circle, this being one of three or four propositions definitely attributed to him, and it is sometimes given as a proposition to be proved. As a proposition, however, it is unsatisfactory, since the proof of what is so evident usually instills more doubt than certainty in the minds of beginners. 18. SEMICIRCLE. _A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle._ Proclus remarked that the semicircle is the only plane figure that has its center on its perimeter. Some writers object to defining a circle as a line and then speaking of the area of a circle, showing minds that have at least one characteristic of that of Proclus. The modern definition of semicircle is "half of a circle," that is, an arc of 180°, although the term is commonly used to mean both the arc and the segment. 19. RECTILINEAR FIGURES. _Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four, straight lines._ 20. _Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal._ 21. _Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute._ These three definitions may properly be considered together. "Rectilinear" is from the Latin translation of the Greek _euthygrammos_, and means "right-lined," or "straight-lined." Euclid's idea of such a figure is that of the space inclosed, while the modern idea is tending to become that of the inclosing lines. In elementary geometry, however, the Euclidean idea is still held. "Trilateral" is from the Latin translation of the Greek _tripleuros_ (three-sided). In elementary geometry the word "triangle" is more commonly used, although "quadrilateral" is more common than "quadrangle." The use of these two different forms is eccentric and is merely a matter of fashion. Thus we speak of a pentagon but not of a tetragon or a trigon, although both words are correct in form. The word "multilateral" (many-sided) is a translation of the Greek _polypleuros_. Fashion has changed this to "polygonal" (many-angled), the word "multilateral" rarely being seen. Of the triangles, "equilateral" means "equal-sided"; "isosceles" is from the Greek _isoskeles_, meaning "with equal legs," and "scalene" from _skalenos_, possibly from _skazo_ (to limp), or from _skolios_ (crooked). Euclid's limitation of isosceles to a triangle with two, and only two, equal sides would not now be accepted. We are at present more given to generalizing than he was, and when we have proved a proposition relating to the isosceles triangle, we wish to say that we have thereby proved it for the equilateral triangle. We therefore say that an isosceles triangle has two sides equal, leaving it possible that all three sides should be equal. The expression "equal legs" is now being discarded on the score of inelegance. In place of "right-angled triangle" modern writers speak of "right triangle," and so for the obtuse and acute triangles. The terms are briefer and are as readily understood. It may add a little interest to the subject to know that Plutarch tells us that the ancients thought that "the power of the triangle is expressive of the nature of Pluto, Bacchus, and Mars." He also states that the Pythagoreans called "the equilateral triangle the head-born Minerva and Tritogeneia (born of Triton) because it may be equally divided by the perpendicular lines drawn from each of its angles." 22. _Of quadrilateral figures a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral and not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another, but is neither equilateral nor right-angled. And let all quadrilaterals other than these be called trapezia._ In this definition Euclid also specializes in a manner not now generally approved. Thus we are more apt to-day to omit the oblong and rhomboid as unnecessary, and to define "rhombus" in such a manner as to include a square. We use "parallelogram" to cover "rhomboid," "rhombus," "oblong," and "square." For "oblong" we use "rectangle," letting it include square. Euclid's definition of "square" illustrates his freedom in stating more attributes than are necessary, in order to make sure that the concept is clear; for he might have said that it "is that which is equilateral and has one right angle." We may profit by his method, sacrificing logic to educational necessity. Euclid does not use "oblong," "rhombus," "rhomboid," and "trapezium" (_plural_, "trapezia") in his proofs, so that he might well have omitted the definitions, as we often do. 23. PARALLELS. _Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction._ This definition of parallels, simplified in its language, is the one commonly used to-day. Other definitions have been suggested, but none has been so generally used. Proclus states that Posidonius gave the definition based upon the lines always being at the same distance apart. Geminus has the same idea in his definition. There are, as Schotten has pointed out, three general types of definitions of parallels, namely: _a._ They have no point in common. This may be expressed by saying that (1) they do not intersect, (2) they meet at infinity. _b._ They are equidistant from one another. _c._ They have the same direction. Of these, the first is Euclid's, the idea of the point at infinity being suggested by Kepler (1604). The second part of this definition is, of course, unusable for beginners. Dr. (now Sir Thomas) Heath says, "It seems best, therefore, to leave to higher geometry the conception of infinitely distant points on a line and of two straight lines meeting at infinity, like imaginary points of intersection, and, for the purposes of elementary geometry, to rely on the plain distinction between 'parallel' and 'cutting,' which average human intelligence can readily grasp." The direction definition seems to have originated with Leibnitz. It is open to the serious objection that "direction" is not easy of definition, and that it is used very loosely. If two people on different meridians travel due north, do they travel in the same direction? on parallel lines? The definition is as objectionable as that of angle as the "difference of direction" of two intersecting lines. From these definitions of the first book of Euclid we see (1) what a small number Euclid considered as basal; (2) what a change has taken place in the generalization of concepts; (3) how the language has varied. Nevertheless we are not to be commended if we adhere to Euclid's small number, because geometry is now taught to pupils whose vocabulary is limited. It is necessary to define more terms, and to scatter the definitions through the work for use as they are needed, instead of massing them at the beginning, as in a dictionary. The most important lesson to be learned from Euclid's definitions is that only the basal ones, relatively few in number, need to be learned, and these because they are used as the foundations upon which proofs are built. It should also be noticed that Euclid explains nothing in these definitions; they are hard statements of fact, massed at the beginning of his treatise. Not always as statements, and not at all in their arrangement, are they suited to the needs of our boys and girls at present. Having considered Euclid's definitions of Book I, it is proper to turn to some of those terms that have been added from time to time to his list, and are now usually incorporated in American textbooks. It will be seen that most of these were assumed by Euclid to be known by his mature readers. They need to be defined for young people, but most of them are not basal, that is, they are not used in the proofs of propositions. Some of these terms, such as magnitudes, curve line, broken line, curvilinear figure, bisector, adjacent angles, reflex angles, oblique angles and lines, and vertical angles, need merely a word of explanation so that they may be used intelligently. If they were numerous enough to make it worth the while, they could be classified in our textbooks as of minor importance, but such a course would cause more trouble than it is worth. Other terms have come into use in modern times that are not common expressions with which students are familiar. Such a term is "straight angle," a concept not used by Euclid, but one that adds so materially to the interest and value of geometry as now to be generally recognized. There is also the word "perigon," meaning the whole angular space about a point. This was excluded by the Greeks because their idea of angle required it to be less than a straight angle. The word means "around angle," and is the best one that has been coined for the purpose. "Flat angle" and "whole angle" are among the names suggested for these two modern concepts. The terms "complement," "supplement," and "conjugate," meaning the difference between a given angle and a right angle, straight angle, and perigon respectively, have also entered our vocabulary and need defining. There are also certain terms expressing relationship which Euclid does not define, and which have been so changed in recent times as to require careful definition at present. Chief among these are the words "equal," "congruent," and "equivalent." Euclid used the single word "equal" for all three concepts, although some of his recent editors have changed it to "identically equal" in the case of congruence. In modern speech we use the word "equal" commonly to mean "like-valued," "having the same measure," as when we say the circumference of a circle "equals" a straight line whose length is 2[pi]_r_, although it could not coincide with it. Of late, therefore, in Europe and America, and wherever European influence reaches, the word "congruent" is coming into use to mean "identically equal" in the sense of superposable. We therefore speak of congruent triangles and congruent parallelograms as being those that are superposable. It is a little unfortunate that "equal" has come to be so loosely used in ordinary conversation that we cannot keep it to mean "congruent"; but our language will not permit it, and we are forced to use the newer word. Whenever it can be used without misunderstanding, however, it should be retained, as in the case of "equal straight lines," "equal angles," and "equal arcs of the same circle." The mathematical and educational world will never consent to use "congruent straight lines," or "congruent angles," for the reason that the terms are unnecessarily long, no misunderstanding being possible when "equal" is used. The word "equivalent" was introduced by Legendre at the close of the eighteenth century to indicate equality of length, or of area, or of volume. Euclid had said, "Parallelograms which are on the same base and in the same parallels are equal to one another," while Legendre and his followers would modify the wording somewhat and introduce "equivalent" for "equal." This usage has been retained. Congruent polygons are therefore necessarily equivalent, but equivalent polygons are not in general congruent. Congruent polygons have mutually equal sides and mutually equal angles, while equivalent polygons have no equality save that of area. In general, as already stated, these and other terms should be defined just before they are used instead of at the beginning of geometry. The reason for this, from the educational standpoint and considering the present position of geometry in the curriculum, is apparent. We shall now consider the definitions of Euclid's Book III, which is usually taken as Book II in America. 1. EQUAL CIRCLES. _Equal circles are those the diameters of which are equal, or the radii of which are equal._ Manifestly this is a theorem, for it asserts that if the radii of two circles are equal, the circles may be made to coincide. In some textbooks a proof is given by superposition, and the proof is legitimate, but Euclid usually avoided superposition if possible. Nevertheless he might as well have proved this as that two triangles are congruent if two sides and the included angle of the one are respectively equal to the corresponding parts of the other, and he might as well have postulated the latter as to have substantially postulated this fact. For in reality this definition is a postulate, and it was so considered by the great Italian mathematician Tartaglia (_ca._ 1500-_ca._ 1557). The plan usually followed in America to-day is to consider this as one of many unproved propositions, too evident, indeed, for proof, accepted by intuition. The result is a loss in the logic of Euclid, but the method is thought to be better adapted to the mind of the youthful learner. It is interesting to note in this connection that the Greeks had no word for "radius," and were therefore compelled to use some such phrase as "the straight line from the center," or, briefly, "the from the center," as if "from the center" were one word. 2. TANGENT. _A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle._ Teachers who prefer to use "circumference" instead of "circle" for the line should notice how often such phrases as "cut the circle" and "intersecting circle" are used,--phrases that signify nothing unless "circle" is taken to mean the line. So Aristotle uses an expression meaning that the locus of a certain point is a circle, and he speaks of a circle as passing through "all the angles." Our word "touch" is from the Latin _tangere_, from which comes "tangent," and also "tag," an old touching game. 3. TANGENT CIRCLES. _Circles are said to touch one another which, meeting one another, do not cut one another._ The definition has not been looked upon as entirely satisfactory, even aside from its unfortunate phraseology. It is not certain, for instance, whether Euclid meant that the circles could not cut at some other point than that of tangency. Furthermore, no distinction is made between external and internal contact, although both forms are used in the propositions. Modern textbook makers find it convenient to define tangent circles as those that are tangent to the same straight line at the same point, and to define external and internal tangency by reference to their position with respect to the line, although this may be characterized as open to about the same objection as Euclid's. 4. DISTANCE. _In a circle straight lines are said to be equally distant from the center, when the perpendiculars drawn to them from the center are equal._ It is now customary to define "distance" from a point to a line as the length of the perpendicular from the point to the line, and to do this in Book I. In higher mathematics it is found that distance is not a satisfactory term to use, but the objections to it have no particular significance in elementary geometry. 5. GREATER DISTANCE. _And that straight line is said to be at a greater distance on which the greater perpendicular falls._ Such a definition is not thought essential at the present time. 6. SEGMENT. _A segment of a circle is the figure contained by a straight line and the circumference of a circle._ The word "segment" is from the Latin root _sect_, meaning "cut." So we have "sector" (a cutter), "section" (a cut), "intersect," and so on. The word is not limited to a circle; we have long spoken of a spherical segment, and it is common to-day to speak of a line segment, to which some would apply a new name "sect." There is little confusion in the matter, however, for the context shows what kind of a segment is to be understood, so that the word "sect" is rather pedantic than important. It will be noticed that Euclid here uses "circumference" to mean "arc." 7. ANGLE OF A SEGMENT. _An angle of a segment is that contained by a straight line and a circumference of a circle._ This term has entirely dropped out of geometry, and few teachers would know what it meant if they should hear it used. Proclus called such angles "mixed." 8. ANGLE IN A SEGMENT. _An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined._ Such an involved definition would not be usable to-day. Moreover, the words "circumference of the segment" would not be used. 9. _And when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference._ 10. SECTOR. _A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them._ There is no reason for such an extended definition, our modern phraseology being both more exact (as seen in the above use of "circumference" for "arc") and more intelligible. The Greek word for "sector" is "knife" (_tomeus_), "sector" being the Latin translation. A sector is supposed to resemble a shoemaker's knife, and hence the significance of the term. Euclid followed this by a definition of similar sectors, a term now generally abandoned as unnecessary. It will be noticed that Euclid did not use or define the word "polygon." He uses "rectilinear figure" instead. Polygon may be defined to be a bounding line, as a circle is now defined, or as the space inclosed by a broken line, or as a figure formed by a broken line, thus including both the limited plane and its boundary. It is not of any great consequence geometrically which of these ideas is adopted, so that the usual definition of a portion of a plane bounded by a broken line may be taken as sufficient for elementary purposes. It is proper to call attention, however, to the fact that we may have cross polygons of various types, and that the line that "bounds" the polygon must be continuous, as the definition states. That is, in the second of these figures the shaded portion is not considered a polygon. Such special cases are not liable to arise, but if questions relating to them are suggested, the teacher should be prepared to answer them. If suggested to a class, a note of this kind should come out only incidentally as a bit of interest, and should not occupy much time nor be unduly emphasized. [Illustration] It may also be mentioned to a class at some convenient time that the old idea of a polygon was that of a convex figure, and that the modern idea, which is met in higher mathematics, leads to a modification of earlier concepts. For example, here is a quadrilateral with one of its diagonals, _BD_, _outside_ the figure. Furthermore, if we consider a quadrilateral as a figure formed by four intersecting lines, _AC_, _CF_, _BE_, and _EA_, it is apparent that this _general quadrilateral_ has six vertices, _A_, _B_, _C_, _D_, _E_, _F_, and three diagonals, _AD_, _BF_, and _CE_. Such broader ideas of geometry form the basis of what is called modern elementary geometry. [Illustration] The other definitions of plane geometry need not be discussed, since all that have any historical interest have been considered. On the whole it may be said that our definitions to-day are not in general so carefully considered as those of Euclid, who weighed each word with greatest skill, but they are more teachable to beginners, and are, on the whole, more satisfactory from the educational standpoint. The greatest lesson to be learned from this discussion is that the number of basal definitions to be learned for subsequent use is very small. Since teachers are occasionally disturbed over the form in which definitions are stated, it is well to say a few words upon this subject. There are several standard types that may be used. (1) We may use the dictionary form, putting the word defined first, thus: "_Right triangle_. A triangle that has one of its angles a right angle." This is scientifically correct, but it is not a complete sentence, and hence it is not easily repeated when it has to be quoted as an authority. (2) We may put the word defined at the end, thus: "A triangle that has one of its angles a right angle is called a right triangle." This is more satisfactory. (3) We may combine (1) and (2), thus: "_Right triangle_. A triangle that has one of its angles a right angle is called a right triangle." This is still better, for it has the catchword at the beginning of the paragraph. There is occasionally some mental agitation over the trivial things of a definition, such as the use of the words "is called." It would not be a very serious matter if they were omitted, but it is better to have them there. The reason is that they mark the statement at once as a definition. For example, suppose we say that "a triangle that has one of its angles a right angle is a right triangle." We have also the fact that "a triangle whose base is the diameter of a semicircle and whose vertex lies on the semicircle is a right triangle." The style of statement is the same, and we have nothing in the phraseology to show that the first is a definition and the second a theorem. This may happen with most of the definitions, and hence the most careful writers have not consented to omit the distinctive words in question. Apropos of the definitions of geometry, the great French philosopher and mathematician, Pascal, set forth certain rules relating to this subject, as also to the axioms employed, and these may properly sum up this chapter. 1. Do not attempt to define terms so well known in themselves that there are no simpler terms by which to express them. 2. Admit no obscure or equivocal terms without defining them. 3. Use in the definitions only terms that are perfectly understood or are there explained. 4. Omit no necessary principles without general agreement, however clear and evident they may be. 5. Set forth in the axioms only those things that are in themselves perfectly evident. 6. Do not attempt to demonstrate anything that is so evident in itself that there is nothing more simple by which to prove it. 7. Prove whatever is in the least obscure, using in the demonstration only axioms that are perfectly evident in themselves, or propositions already demonstrated or allowed. 8. In case of any uncertainty arising from a term employed, always substitute mentally the definition for the term itself. =Bibliography.= Heath, Euclid, as cited; Frankland, The First Book of Euclid, as cited; Smith, Teaching of Elementary Mathematics, p. 257, New York, 1900; Young, Teaching of Mathematics, p. 189, New York, 1907; Veblen, On Definitions, in the _Monist_, 1903, p. 303. FOOTNOTES: [53] Free use has been made of W. B. Frankland, "The First Book of Euclid's 'Elements,'" Cambridge, 1905; T. L. Heath, "The Thirteen Books of Euclid's 'Elements,'" Cambridge, 1908; H. Schotten, "Inhalt und Methode des planimetrischen Unterrichts," Leipzig, 1893; M. Simon, "Euclid und die sechs planimetrischen Bücher," Leipzig, 1901. [54] For a facsimile of a thirteenth-century MS. containing this definition, see the author's "Rara Arithmetica," Plate IV, Boston, 1909. [55] Our slang expression "The cart before the horse" is suggestive of this procedure. [56] Loc. cit., Vol. II, p. 94. CHAPTER XIII HOW TO ATTACK THE EXERCISES The old geometry, say of a century ago, usually consisted, as has been stated, of a series of theorems fully proved and of problems fully solved. During the nineteenth century exercises were gradually introduced, thus developing geometry from a science in which one learned by seeing things done, into one in which he gained power by actually doing things. Of the nature of these exercises ("originals," "riders"), and of their gradual change in the past few years, mention has been made in Chapter VII. It now remains to consider the methods of attacking these exercises. It is evident that there is no single method, and this is a fortunate fact, since if it were not so, the attack would be too mechanical to be interesting. There is no one rule for solving every problem nor even for seeing how to begin. On the other hand, a pupil is saved some time by having his attention called to a few rather definite lines of attack, and he will undoubtedly fare the better by not wasting his energies over attempts that are in advance doomed to failure. There are two general questions to be considered: first, as to the discovery of new truths, and second, as to the proof. With the first the pupil will have little to do, not having as yet arrived at this stage in his progress. A bright student may take a little interest in seeing what he can find out that is new (at least to him), and if so, he may be told that many new propositions have been discovered by the accurate drawing of figures; that some have been found by actually weighing pieces of sheet metal of certain sizes; and that still others have made themselves known through paper folding. In all of these cases, however, the supposed proposition must be proved before it can be accepted. As to the proof, the pupil usually wanders about more or less until he strikes the right line, and then he follows this to the conclusion. He should not be blamed for doing this, for he is pursuing the method that the world followed in the earliest times, and one that has always been common and always will be. This is the synthetic method, the building up of the proof from propositions previously proved. If the proposition is a theorem, it is usually not difficult to recall propositions that may lead to the demonstration, and to select the ones that are really needed. If it is a problem, it is usually easy to look ahead and see what is necessary for the solution and to select the preceding propositions accordingly. But pupils should be told that if they do not rather easily find the necessary propositions for the construction or the proof, they should not delay in resorting to another and more systematic method. This is known as the method of analysis, and it is applicable both to theorems and to problems. It has several forms, but it is of little service to a pupil to have these differentiated, and it suffices that he be given the essential feature of all these forms, a feature that goes back to Plato and his school in the fifth century B.C. For a theorem, the method of analysis consists in reasoning as follows: "I can prove this proposition if I can prove this thing; I can prove this thing if I can prove that; I can prove that if I can prove a third thing," and so the reasoning runs until the pupil comes to the point where he is able to add, "but I _can_ prove that." This does not prove the proposition, but it enables him to reverse the process, beginning with the thing he can prove and going back, step by step, to the thing that he is to prove. Analysis is, therefore, his method of discovery of the way in which he may arrange his synthetic proof. Pupils often wonder how any one ever came to know how to arrange the proofs of geometry, and this answers the question. Some one guessed that a statement was true; he applied analysis and found that he _could_ prove it; he then applied synthesis and _did_ prove it. For a problem, the method of analysis is much the same as in the case of a theorem. Two things are involved, however, instead of one, for here we must make the construction and then prove that this construction is correct. The pupil, therefore, first supposes the problem solved, and sees what results follow. He then reverses the process and sees if he can attain these results and thus effect the required construction. If so, he states the process and gives the resulting proof. For example: In a triangle _ABC_, to draw _PQ_ parallel to the base _AB_, cutting the sides in _P_ and _Q_, so that _PQ_ shall equal _AP_ + _BQ_. [Illustration] =Analysis.= Assume the problem solved. Then _AP_ must equal some part of _PQ_ as _PX_, and _BQ_ must equal _QX_. But if _AP_ = _PX_, what must [L]_PXA_ equal? [because] _PQ_ is || _AB_, what does [L]_PXA_ equal? Then why must [L]_BAX_ = [L]_XAP_? Similarly, what about [L]_QBX_ and [L]_XBA_? =Construction.= Now reverse the process. What may we do to [Ls] _A_ and _B_ in order to fix _X_? Then how shall _PQ_ be drawn? Now give the proof. [Illustration] [Illustration] The third general method of attack applies chiefly to problems where some point is to be determined. This is the method of the intersection of loci. Thus, to locate an electric light at a point eighteen feet from the point of intersection of two streets and equidistant from them, evidently one locus is a circle with a radius eighteen feet and the center at the vertex of the angle made by the streets, and the other locus is the bisector of the angle. The method is also occasionally applicable to theorems. For example, to prove that the perpendicular bisectors of the sides of a triangle are concurrent. Here the locus of points equidistant from _A_ and _B_ is _PP'_, and the locus of points equidistant from _B_ and _C_ is _QQ'_. These can easily be shown to intersect, as at _O_. Then _O_, being equidistant from _A_, _B_, and _C_, is also on the perpendicular bisector of _AC_. Therefore these bisectors are concurrent in _O_. These are the chief methods of attack, and are all that should be given to an average class for practical use. Besides the methods of attack, there are a few general directions that should be given to pupils. 1. In attacking either a theorem or a problem, take the most general figure possible. Thus, if a proposition relates to a quadrilateral, take one with unequal sides and unequal angles rather than a square or even a rectangle. The simpler figures often deceive a pupil into feeling that he has a proof, when in reality he has one only for a special case. 2. Set forth very exactly the thing that is given, using letters relating to the figure that has been drawn. Then set forth with the same exactness the thing that is to be proved. The neglect to do this is the cause of a large per cent of the failures. The knowing of exactly what we have to do and exactly what we have with which to do it is half the battle. 3. If the proposition seems hazy, the difficulty is probably with the wording. In this case try substituting the definition for the name of the thing defined. Thus instead of thinking too long about proving that the median to the base of an isosceles triangle is perpendicular to the base, draw the figure and think that there is given _AC_ = _BC_, _AD_ = _BD_, and that there is to be proved that [L]_CDA_ = [L]_BDC_. [Illustration] Here we have replaced "median," "isosceles," and "perpendicular" by statements that express the same idea in simpler language. =Bibliography.= Petersen, Methods and Theories for the Solution of Geometric Problems of Construction, Copenhagen, 1879, a curious piece of English and an extreme view of the subject, but well worth consulting; Alexandroff, Problèmes de géométrie élémentaire, Paris, 1899, with a German translation in 1903; Loomis, Original Investigation; or, How to attack an Exercise in Geometry, Boston, 1901; Sauvage, Les Lieux géométriques en géométrie élémentaire, Paris, 1893; Hadamard, Leçons de géométrie, p. 261, Paris, 1898; Duhamel, Des Méthodes dans les sciences de raisonnement, 3^e éd., Paris, 1885; Henrici and Treutlein, Lehrbuch der Elementar-Geometrie, Leipzig, 3. Aufl., 1897; Henrici, Congruent Figures, London, 1879. CHAPTER XIV BOOK I AND ITS PROPOSITIONS Having considered the nature of the geometry that we have inherited, and some of the opportunities for improving upon the methods of presenting it, the next question that arises is the all-important one of the subject matter, What shall geometry be in detail? Shall it be the text or the sequence of Euclid? Few teachers have any such idea at the present time. Shall it be a mere dabbling with forms that are seen in mechanics or architecture, with no serious logical sequence? This is an equally dangerous extreme. Shall it be an entirely new style of geometry based upon groups of motions? This may sometime be developed, but as yet it exists in the future if it exists at all, since the recent efforts in this respect are generally quite as ill suited to a young pupil as is Euclid's "Elements" itself. No one can deny the truth of M. Bourlet's recent assertion that "Industry, daughter of the science of the nineteenth century, reigns to-day the mistress of the world; she has transformed all ancient methods, and she has absorbed in herself almost all human activity."[57] Neither can one deny the justice of his comparison of Euclid with a noble piece of Gothic architecture and of his assertion that as modern life demands another type of building, so it demands another type of geometry. But what does this mean? That geometry is to exist merely as it touches industry, or that bad architecture is to replace the good? By no means. A building should to-day have steam heat and elevators and electric lights, but it should be constructed of just as enduring materials as the Parthenon, and it should have lines as pleasing as those of a Gothic façade. Architecture should still be artistic and construction should still be substantial, else a building can never endure. So geometry must still exemplify good logic and must still bring to the pupil a feeling of exaltation, or it will perish and become a mere relic in the museum of human culture. What, then, shall the propositions of geometry be, and in what manner shall they answer to the challenge of the industrial epoch in which we live? In reply, they must be better adapted to young minds and to all young minds than Euclid ever intended his own propositions to be. Furthermore, they must have a richness of application to pure geometry, in the way of carefully chosen exercises, that Euclid never attempted. And finally, they must have application to this same life of industry of which we have spoken, whenever this can really be found, but there must be no sham and pretense about it, else the very honesty that permeated the ancient geometry will seem to the pupil to be wanting in the whole subject.[58] Until some geometry on a radically different basis shall appear, and of this there is no very hopeful sign at present, the propositions will be the essential ones of Euclid, excluding those that may be considered merely intuitive, and excluding all that are too difficult for the pupil who to-day takes up their study. The number will be limited in a reasonable way, and every genuine type of application will be placed before the teacher to be used as necessity requires. But a fair amount of logic will be retained, and the effort to make of geometry an empty bauble of a listless mind will be rejected by every worthy teacher. What the propositions should be is a matter upon which opinions may justly differ; but in this chapter there is set forth a reasonable list for Book I, arranged in a workable sequence, and this list may fairly be taken as typical of what the American school will probably use for many years to come. With the list is given a set of typical applications, and some of the general information that will add to the interest in the work and that should form part of the equipment of the teacher. An ancient treatise was usually written on a kind of paper called papyrus, made from the pith of a large reed formerly common in Egypt, but now growing luxuriantly only above Khartum in Upper Egypt, and near Syracuse in Sicily; or else it was written on parchment, so called from Pergamos in Asia Minor, where skins were first prepared in parchment form; or occasionally they were written on ordinary leather. In any case they were generally written on long strips of the material used, and these were rolled up and tied. Hence we have such an expression as "keeping the roll" in school, and such a word as "volume," which has in it the same root as "involve" (to roll in), and "evolve" (to roll out). Several of these rolls were often necessary for a single treatise, in which case each was tied, and all were kept together in a receptacle resembling a pail, or in a compartment on a shelf. The Greeks called each of the separate parts of a treatise _biblion_ ([Greek: biblion]), a word meaning "book." Hence we have the books of the Bible, the books of Homer, and the books of Euclid. From the same root, indeed, comes Bible, bibliophile (booklover), bibliography (list of books), and kindred words. Thus the books of geometry are the large chapters of the subject, "chapter" being from the Latin _caput_ (head), a section under a new heading. There have been efforts to change "books" to "chapters," but they have not succeeded, and there is no reason why they should succeed, for the term is clear and has the sanction of long usage. THEOREM. _If two lines intersect, the vertical angles are equal._ This was Euclid's Proposition 15, being put so late because he based the proof upon his Proposition 13, now thought to be best taken without proof, namely, "If a straight line set upon a straight line makes angles, it will make either two right angles or angles equal to two right angles." It is found to be better pedagogy to assume that this follows from the definition of straight angle, with reference, if necessary, to the meaning of the sum of two angles. This proposition on vertical angles is probably the best one with which to begin geometry, since it is not so evident as to seem to need no proof, although some prefer to rank it as semiobvious, while the proof is so simple as easily to be understood. Eudemus, a Greek who wrote not long before Euclid, attributed the discovery of this proposition to Thales of Miletus (_ca._ 640-548 B.C.), one of the Seven Wise Men of Greece, of whom Proclus wrote: "Thales it was who visited Egypt and first transferred to Hellenic soil this theory of geometry. He himself, indeed, discovered much, but still more did he introduce to his successors the principles of the science." The proposition is the only basal one relating to the intersection of two lines, and hence there are no others with which it is necessarily grouped. This is the reason for placing it by itself, followed by the congruence theorems. There are many familiar illustrations of this theorem. Indeed, any two crossed lines, as in a pair of shears or the legs of a camp stool, bring it to mind. The word "straight" is here omitted before "lines" in accordance with the modern convention that the word "line" unmodified means a straight line. Of course in cases of special emphasis the adjective should be used. THEOREM. _Two triangles are congruent if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other._ This is Euclid's Proposition 4, his first three propositions being problems of construction. This would therefore have been his first proposition if he had placed his problems later, as we do to-day. The words "congruent" and "equal" are not used as in Euclid, for reasons already set forth on page 151. There have been many attempts to rearrange the propositions of Book I, putting in separate groups those concerning angles, those concerning triangles, and those concerning parallels, but they have all failed, and for the cogent reason that such a scheme destroys the logical sequence. This proposition may properly follow the one on vertical angles simply because the latter is easier and does not involve superposition. As far as possible, Euclid and all other good geometers avoid the proof by superposition. As a practical test superposition is valuable, but as a theoretical one it is open to numerous objections. As Peletier pointed out in his (1557) edition of Euclid, if the superposition of lines and figures could freely be assumed as a method of demonstration, geometry would be full of such proofs. There would be no reason, for example, why an angle should not be constructed equal to a given angle by superposing the given angle on another part of the plane. Indeed, it is possible that we might then assume to bisect an angle by imagining the plane folded like a piece of paper. Heath (1908) has pointed out a subtle defect in Euclid's proof, in that it is said that because two lines are equal, they can be made to coincide. Euclid says, practically, that if two lines can be made to coincide, they are equal, but he does not say that if two straight lines are equal, they can be made to coincide. For the purposes of elementary geometry the matter is hardly worth bringing to the attention of a pupil, but it shows that even Euclid did not cover every point. Applications of this proposition are easily found, but they are all very much alike. There are dozens of measurements that can be made by simply constructing a triangle that shall be congruent to another triangle. It seems hardly worth the while at this time to do more than mention one typical case,[59] leaving it to teachers who may find it desirable to suggest others to their pupils. [Illustration] Wishing to measure the distance across a river, some boys sighted from _A_ to a point _P_. They then turned and measured _AB_ at right angles to _AP_. They placed a stake at _O_, halfway from _A_ to _B_, and drew a perpendicular to _AB_ at _B_. They placed a stake at _C_, on this perpendicular, and in line with _O_ and _P_. They then found the width by measuring _BC_. Prove that they were right. This involves the ranging of a line, and the running of a line at right angles to a given line, both of which have been described in Chapter IX. It is also fairly accurate to run a line at any angle to a given line by sighting along two pins stuck in a protractor. THEOREM. _Two triangles are congruent if two angles and the included side of the one are equal respectively to two angles and the included side of the other._ Euclid combines this with his Proposition 26: If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides, and the remaining angle to the remaining angle. He proves this cumbersome statement without superposition, desiring to avoid this method, as already stated, whenever possible. The proof by superposition is old, however, for Al-Nair[=i]z[=i][60] gives it and ascribes it to some earlier author whose name he did not know. Proclus tells us that "Eudemus in his geometrical history refers this theorem to Thales. For he says that in the method by which they say that Thales proved the distance of ships in the sea, it was necessary to make use of this theorem." How Thales did this is purely a matter of conjecture, but he might have stood on the top of a tower rising from the level shore, or of such headlands as abound near Miletus, and by some simple instrument sighted to the ship. Then, turning, he might have sighted along the shore to a point having the same angle of declination, and then have measured the distance from the tower to this point. This seems more reasonable than any of the various plans suggested, and it is found in so many practical geometries of the first century of printing that it seems to have long been a common expedient. The stone astrolabe from Mesopotamia, now preserved in the British Museum, shows that such instruments for the measuring of angles are very old, and for the purposes of Thales even a pair of large compasses would have answered very well. An illustration of the method is seen in Belli's work of 1569, as here shown. At the top of the picture a man is getting the angle by means of the visor of his cap; at the bottom of the picture a man is using a ruler screwed to a staff.[61] The story goes that one of Napoleon's engineers won the imperial favor by quickly measuring the width of a stream that blocked the progress of the army, using this very method. [Illustration: SIXTEENTH-CENTURY MENSURATION Belli's "Del Misurar con la Vista," Venice, 1569] This proposition is the reciprocal or dual of the preceding one. The relation between the two may be seen from the following arrangement: Two triangles are congruent if two _sides_ and the included _angle_ of the one are equal respectively to two _sides_ and the included _angle_ of the other. Two triangles are congruent if two _angles_ and the included _side_ of the one are equal respectively to two _angles_ and the included _side_ of the other. In general, to every proposition involving _points_ and _lines_ there is a reciprocal proposition involving _lines_ and _points_ respectively that is often true,--indeed, that is always true in a certain line of propositions. This relation is known as the Principle of Reciprocity or of Duality. Instead of points and lines we have here angles (suggested by the vertex points) and lines. It is interesting to a class to have attention called to such relations, but it is not of sufficient importance in elementary geometry to justify more than a reference here and there. There are other dual features that are seen in geometry besides those given above. THEOREM. _In an isosceles triangle the angles opposite the equal sides are equal._ This is Euclid's Proposition 5, the second of his theorems, but he adds, "and if the equal straight lines be produced further, the angles under the base will be equal to one another." Since, however, he does not use this second part, its genuineness is doubted. He would not admit the common proof of to-day of supposing the vertical angle bisected, because the problem about bisecting an angle does not precede this proposition, and therefore his proof is much more involved than ours. He makes _CX_ = _CY_, and proves [triangles]_XBC_ and _YAC_ congruent,[62] and also [triangles]_XBA_ and _YAB_ congruent. Then from [L]_YAC_ he takes [L]_YAB_, leaving [L]_BAC_, and so on the other side, leaving [L]_CBA_, these therefore being equal. [Illustration] This proposition has long been called the _pons asinorum_, or bridge of asses, but no one knows where or when the name arose. It is usually stated that it came from the fact that fools could not cross this bridge, and it is a fact that in the Middle Ages this was often the limit of the student's progress in geometry. It has however been suggested that the name came from Euclid's figure, which resembles the simplest type of a wooden truss bridge. The name is applied by the French to the Pythagorean Theorem. Proclus attributes the discovery of this proposition to Thales. He also says that Pappus (third century A.D.), a Greek commentator on Euclid, proved the proposition as follows: Let _ABC_ be the triangle, with _AB_ = _AC_. Conceive of this as two triangles; then _AB_ = _AC_, _AC_ = _AB_, and [L]_A_ is common; hence the [triangles]_ABC_ and _ACB_ are congruent, and [L]_B_ of the one equals [L]_C_ of the other. This is a better plan than that followed by some textbook writers of imagining [triangle]_ABC_ taken up and laid down on _itself_. Even to lay it down on its "trace" is more objectionable than the plan of Pappus. THEOREM. _If two angles of a triangle are equal, the sides opposite the equal angles are equal, and the triangle is isosceles._ The statement is, of course, tautological, the last five words being unnecessary from the mathematical standpoint, but of value at this stage of the student's progress as emphasizing the nature of the triangle. Euclid stated the proposition thus, "If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another." He did not define "subtend," supposing such words to be already understood. This is the first case of a converse proposition in geometry. Heath distinguishes the logical from the geometric converse. The logical converse of Euclid I, 5, would be that "_some_ triangles with two angles equal are isosceles," while the geometric converse is the proposition as stated. Proclus called attention to two forms of converse (and in the course of the work, but not at this time, the teacher may have to do the same): (1) the complete converse, in which that which is given in one becomes that which is to be proved in the other, and vice versa, as in this and the preceding proposition; (2) the partial converse, in which two (or even more) things may be given, and a certain thing is to be proved, the converse being that one (or more) of the preceding things is now given, together with what was to be proved, and the other given thing is now to be proved. Symbolically, if it is given that _a_ = _b_ and _c_ = _d_, to prove that _x_ = _y_, the partial converse would have given _a_ = _b_ and _x_ = _y_, to prove that _c_ = _d_. Several proofs for the proposition have been suggested, but a careful examination of all of them shows that the one given below is, all things considered, the best one for pupils beginning geometry and following the sequence laid down in this chapter. It has the sanction of some of the most eminent mathematicians, and while not as satisfactory in some respects as the _reductio ad absurdum_, mentioned below, it is more satisfactory in most particulars. The proof is as follows: [Illustration: =Given the triangle ABC, with the angle A equal to the angle B.=] _To prove that_ _AC_ = _BC_. =Proof.= Suppose the second triangle _A'B'C'_ to be an exact reproduction of the given triangle _ABC_. Turn the triangle _A'B'C'_ over and place it upon _ABC_ so that _B'_ shall fall on _A_ and _A'_ shall fall on _B_. Then _B'A'_ will coincide with _AB_. Since [L]_A'_ = [L]_B'_, Given and [L]_A_ = [L]_A'_, Hyp. [therefore][L]_A_ = [L]_B'_. [therefore]_B'C'_ will lie along _AC_. Similarly, _A'C'_ will lie along _BC_. Therefore _C'_ will fall on both _AC_ and _BC_, and hence at their intersection. [therefore]_B'C'_ = _AC_. But _B'C'_ was made equal to _BC_. [therefore]_AC_ = _BC_. Q.E.D. If the proposition should be postponed until after the one on the sum of the angles of a triangle, the proof would be simpler, but it is advantageous to couple it with its immediate predecessor. This simpler proof consists in bisecting the vertical angle, and then proving the two triangles congruent. Among the other proofs is that of the _reductio ad absurdum_, which the student might now meet, but which may better be postponed. The phrase _reductio ad absurdum_ seems likely to continue in spite of the efforts to find another one that is simpler. Such a proof is also called an indirect proof, but this term is not altogether satisfactory. Probably both names should be used, the Latin to explain the nature of the English. The Latin name is merely a translation of one of several Greek names used by Aristotle, a second being in English "proof by the impossible," and a third being "proof leading to the impossible." If teachers desire to introduce this form of proof here, it must be borne in mind that only one supposition can be made if such a proof is to be valid, for if two are made, then an absurd conclusion simply shows that either or both must be false, but we do not know which is false, or if only one is false. THEOREM. _Two triangles are congruent if the three sides of the one are equal respectively to the three sides of the other._ It would be desirable to place this after the fourth proposition mentioned in this list if it could be done, so as to get the triangles in a group, but we need the fourth one for proving this, so that the arrangement cannot be made, at least with this method of proof. This proposition is a "partial converse" of the second proposition in this list; for if the triangles are _ABC_ and _A'B'C'_, with sides _a_, _b_, _c_ and _a'_, _b'_, _c'_, then the second proposition asserts that if _b_ = _b'_, _c_ = _c'_, and [L]_A_ = [L]_A'_, then _a_ = _a'_ and the triangles are congruent, while this proposition asserts that if _a_ = _a'_, _b_ = _b'_, and _c_ = _c'_, then [L]_A_ = [L]_A'_ and the triangles are congruent. The proposition was known at least as early as Aristotle's time. Euclid proved it by inserting a preliminary proposition to the effect that it is impossible to have on the same base _AB_ and the same side of it two different triangles _ABC_ and _ABC'_, with _AC_ = _AC'_, and _BC_ = _BC'_. The proof ordinarily given to-day, wherein the two triangles are constructed on opposite sides of the base, is due to Philo of Byzantium, who lived after Euclid's time but before the Christian era, and it is also given by Proclus. There are really three cases, if one wishes to be overparticular, corresponding to the three pairs of equal sides. But if we are allowed to take the longest side for the common base, only one case need be considered. Of the applications of the proposition one of the most important relates to making a figure rigid by means of diagonals. For example, how many diagonals must be drawn in order to make a quadrilateral rigid? to make a pentagon rigid? a hexagon? a polygon of _n_ sides. In particular, the following questions may be asked of a class: [Illustration] 1. Three iron rods are hinged at the extremities, as shown in this figure. Is the figure rigid? Why? 2. Four iron rods are hinged, as shown in this figure. Is the figure rigid? If not, where would you put in the fifth rod to make it rigid? Prove that this would accomplish the result. [Illustration] Another interesting application relates to the most ancient form of leveling instrument known to us. This kind of level is pictured on very ancient monuments, and it is still used in many parts of the world. Pupils in manual training may make such an instrument, and indeed one is easily made out of cardboard. If the plumb line passes through the mid-point of the base, the two triangles are congruent and the plumb line is then perpendicular to the base. In other words, the base is level. With such simple primitive instruments, easily made by pupils, a good deal of practical mathematical work can be performed. The interesting old illustration here given shows how this form of level was used three hundred years ago. [Illustration: EARLY METHODS OF LEVELING Pomodoro's "La geometria prattica," Rome, 1624] [Illustration] Teachers who seek for geometric figures in practical mechanics will find this proposition illustrated in the ordinary hoisting apparatus of the kind here shown. From the study of such forms and of simple roof and bridge trusses, a number of the usual properties of the isosceles triangle may be derived. THEOREM. _The sum of two lines drawn from a given point to the extremities of a given line is greater than the sum of two other lines similarly drawn, but included by them._ It should be noted that the words "the extremities of" are necessary, for it is possible to draw from a certain point within a certain triangle two lines to the base such that their sum is greater than the sum of the other two sides. [Illustration] Thus, in the right triangle _ABC_ draw any line _CX_ from _C_ to the base. Make _XY_ = _AC_, and _CP_ = _PY_. Then it is easily shown that _PB_ + _PX_ > _CB_ + _CA_. [Illustration] It is interesting to a class to have a teacher point out that, in this figure, _AP_ + _PB_ < _AC_ + _CB_, and _AP'_ + _P'B_ < _AP_ + _PB_, and that the nearer _P_ gets to _AB_, the shorter _AP_ + _PB_ becomes, the limit being the line _AB_. From this we may _infer_ (although we have not proved) that "a straight line (_AB_) is the shortest path between two points." THEOREM. _Only one perpendicular can be drawn to a given line from a given external point._ THEOREM. _Two lines drawn from a point in a perpendicular to a given line, cutting off on the given line equal segments from the foot of the perpendicular, are equal and make equal angles with the perpendicular._ THEOREM. _Of two lines drawn from the same point in a perpendicular to a given line, cutting off on the line unequal segments from the foot of the perpendicular, the more remote is the greater._ THEOREM. _The perpendicular is the shortest line that can be drawn to a straight line from a given external point._ These four propositions, while known to the ancients and incidentally used, are not explicitly stated by Euclid. The reason seems to be that he interspersed his problems with his theorems, and in his Propositions 11 and 12, which treat of drawing a perpendicular to a line, the essential features of these theorems are proved. Further mention will be made of them when we come to consider the problems in question. Many textbook writers put the second and third of the four before the first, forgetting that the first is assumed in the other two, and hence should precede them. THEOREM. _Two right triangles are congruent if the hypotenuse and a side of the one are equal respectively to the hypotenuse and a side of the other._ THEOREM. _Two right triangles are congruent if the hypotenuse and an adjacent angle of the one are equal respectively to the hypotenuse and an adjacent angle of the other._ As stated in the notes on the third proposition in this sequence, Euclid's cumbersome Proposition 26 covers several cases, and these two among them. Of course this present proposition could more easily be proved after the one concerning the sum of the angles of a triangle, but the proof is so simple that it is better to leave the proposition here in connection with others concerning triangles. THEOREM. _Two lines in the same plane perpendicular to the same line cannot meet, however far they are produced._ This proposition is not in Euclid, and it is introduced for educational rather than for mathematical reasons. Euclid introduced the subject by the proposition that, if alternate angles are equal, the lines are parallel. It is, however, simpler to begin with this proposition, and there is some advantage in stating it in such a way as to prove that parallels exist before they are defined. The proposition is properly followed by the definition of parallels and by the postulate that has been discussed on page 127. A good application of this proposition is the one concerning a method of drawing parallel lines by the use of a carpenter's square. Here two lines are drawn perpendicular to the edge of a board or a ruler, and these are parallel. THEOREM. _If a line is perpendicular to one of two parallel lines, it is perpendicular to the other also._ This, like the preceding proposition, is a special case under a later theorem. It simplifies the treatment of parallels, however, and the beginner finds it easier to approach the difficulties gradually, through these two cases of perpendiculars. It should be noticed that this is an example of a partial converse, as explained on page 175. The preceding proposition may be stated thus: If _a_ is [perp] to _x_ and _b_ is [perp] to _x_, then _a_ is || to _b_. This proposition may be stated thus: If _a_ is [perp] to _x_ and _a_ is || to _b_, then _b_ is [perp] to _x_. This is, therefore, a partial converse. These two propositions having been proved, the usual definitions of the angles made by a transversal of two parallels may be given. It is unfortunate that we have no name for each of the two groups of four equal angles, and the name of "transverse angles" has been suggested. This would simplify the statements of certain other propositions; thus: "If two parallel lines are cut by a transversal, the transverse angles are equal," and this includes two propositions as usually given. There is not as yet, however, any general sanction for the term. THEOREM. _If two parallel lines are cut by a transversal, the alternate-interior angles are equal._ Euclid gave this as half of his Proposition 29. Indeed, he gives only four theorems on parallels, as against five propositions and several corollaries in most of our American textbooks. The reason for increasing the number is that each proposition may be less involved. Thus, instead of having one proposition for both exterior and interior angles, modern authors usually have one for the exterior and one for the interior, so as to make the difficult subject of parallels easier for beginners. THEOREM. _When two straight lines in the same plane are cut by a transversal, if the alternate-interior angles are equal, the two straight lines are parallel._ This is the converse of the preceding theorem, and is half of Euclid I, 28, his theorem being divided for the reason above stated. There are several typical pairs of equal or supplemental angles that would lead to parallel lines, of which Euclid uses only part, leaving the other cases to be inferred. This accounts for the number of corollaries in this connection in later textbooks. Surveyors make use of this proposition when they wish, without using a transit instrument, to run one line parallel to another. [Illustration] For example, suppose two boys are laying out a tennis court and they wish to run a line through _P_ parallel to _AB_. Take a 60-foot tape and swing it around _P_ until the other end rests on _AB_, as at _M_. Put a stake at _O_, 30 feet from _P_ and _M_. Then take any convenient point _N_ on _AB_, and measure _ON_. Suppose it equals 20 feet. Then sight from _N_ through _O_, and put a stake at _Q_ just 20 feet from _O_. Then _P_ and _Q_ determine the parallel, according to the proposition just mentioned. THEOREM. _If two parallel lines are cut by a transversal, the exterior-interior angles are equal._ This is also a part of Euclid I, 29. It is usually followed by several corollaries, covering the minor and obvious cases omitted by the older writers. While it would be possible to dispense with these corollaries, they are helpful for definite reference in later propositions. THEOREM. _The sum of the three angles of a triangle is equal to two right angles._ Euclid stated this as follows: "In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles." This states more than is necessary for the basal fact of the proposition, which is the constancy of the sum of the angles. The theorem is one of the three most important propositions in plane geometry, the other two being the so-called Pythagorean Theorem, and a proposition relating to the proportionality of the sides of two triangles. These three form the foundation of trigonometry and of the mensuration of plane figures. The history of the proposition is extensive. Eutocius (_ca._ 510 A.D.), in his commentary on Apollonius, says that Geminus (first century B.C.) testified that "the ancients investigated the theorem of the two right angles in each individual species of triangle, first in the equilateral, again in the isosceles, and afterwards in the scalene triangle." This, indeed, was the ancient plan, to proceed from the particular to the general. It is the natural order, it is the world's order, and it is well to follow it in all cases of difficulty in the classroom. Proclus (410-485 A.D.) tells us that Eudemus, who lived just before Euclid (or probably about 325 B.C.), affirmed that the theorem was due to the Pythagoreans, although this does not necessarily mean to the actual pupils of Pythagoras. The proof as he gives it consists in showing that _a_ = _a_´, _b_ = _b_´, and _a_´ + _c_ + _b_´ = two right angles. Since the proposition about the exterior angle of a triangle is attributed to Philippus of Mende (_ca._ 380 B.C.), the figure given by Eudemus is probably the one used by the Pythagoreans. [Illustration] There is also some reason for believing that Thales (_ca._ 600 B.C.) knew the theorem, for Diogenes Laertius (_ca._ 200 A.D.) quotes Pamphilius (first century A.D.) as saying that "he, having learned geometry from the Egyptians, was the first to inscribe a right triangle in a circle, and sacrificed an ox." The proof of this proposition requires the knowledge that the sum of the angles, at least in a right triangle, is two right angles. The proposition is frequently referred to by Aristotle. There have been numerous attempts to prove the proposition without the use of parallel lines. Of these a German one, first given by Thibaut in the early part of the eighteenth century, is among the most interesting. This, in simplified form, is as follows: [Illustration] Suppose an indefinite line _XY_ to lie on _AB_. Let it swing about _A_, counterclockwise, through [L]_A_, so as to lie on _AC_, as _X'Y'_. Then let it swing about _C_, through [L]_C_, so as to lie on _CB_, as _X''Y''_. Then let it swing about _B_, through [L]_B_, so as to lie on _BA_, as _X'''Y'''_. It now lies on _AB_, but it is turned over, _X'''_ being where _Y_ was, and _Y'''_ where _X_ was. In turning through [Ls]_A_, _B_, and _C_ it has therefore turned through two right angles. One trouble with the proof is that the rotation has not been about the same point, so that it has never been looked upon as other than an interesting illustration. Proclus tried to prove the theorem by saying that, if we have two perpendiculars to the same line, and suppose them to revolve about their feet so as to make a triangle, then the amount taken from the right angles is added to the vertical angle of the triangle, and therefore the sum of the angles continues to be two right angles. But, of course, to prove his statement requires a perpendicular to be drawn from the vertex to the base, and the theorem of parallels to be applied. Pupils will find it interesting to cut off the corners of a paper triangle and fit the angles together so as to make a straight angle. This theorem furnishes an opportunity for many interesting exercises, and in particular for determining the third angle when two angles of a triangle are given, or the second acute angle of a right triangle when one acute angle is given. Of the simple outdoor applications of the proposition, one of the best is illustrated in this figure. [Illustration] To ascertain the height of a tree or of the school building, fold a piece of paper so as to make an angle of 45°. Then walk back from the tree until the top is seen at an angle of 45° with the ground (being therefore careful to have the base of the triangle level). Then the height _AC_ will equal the base _AB_, since _ABC_ is isosceles. A paper protractor may be used for the same purpose. Distances can easily be measured by constructing a large equilateral triangle of heavy pasteboard, and standing pins at the vertices for the purpose of sighting. [Illustration] To measure _PC_, stand at some convenient point _A_ and sight along _APC_ and also along _AB_. Then walk along _AB_ until a point _B_ is reached from which _BC_ makes with _BA_ an angle of the triangle (60°). Then _AC_ = _AB_, and since _AP_ can be measured, we can find _PC_. Another simple method of measuring a distance _AC_ across a stream is shown in this figure. [Illustration] Measure the angle _CAX_, either in degrees, with a protractor, or by sighting along a piece of paper and marking down the angle. Then go along _XA_ produced until a point _B_ is reached from which _BC_ makes with _A_ an angle equal to half of angle _CAX_. Then it is easily shown that _AB_ = _AC_. A navigator uses the same principle when he "doubles the angle on the bow" to find his distance from a lighthouse or other object. [Illustration] If he is sailing on the course _ABC_ and notes a lighthouse _L_ when he is at _A_, and takes the angle _A_, and if he notices when the angle that the lighthouse makes with his course is just twice the angle noted at _A_, then _BL_ = _AB_. He has _AB_ from his log (an instrument that tells how far a ship goes in a given time), so he knows _BL_. He has "doubled the angle on the bow" to get this distance. It would have been possible for Thales, if he knew this proposition, to have measured the distance of the ship at sea by some such device as this: [Illustration] Make a large isosceles triangle out of wood, and, standing at _T_, sight to the ship and along the shore on a line _TA_, using the vertical angle of the triangle. Then go along _TA_ until a point _P_ is reached, from which _T_ and _S_ can be seen along the sides of a base angle of the triangle. Then _TP_ = _TS_. By measuring _TB_, _BS_ can then be found. THEOREM. _The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side_. If the postulate is assumed that a straight line is the shortest path between two points, then the first part of this theorem requires no further proof, and the second part follows at once from the axiom of inequalities. This seems the better plan for beginners, and the proposition may be considered as semiobvious. Euclid proved the first part, not having assumed the postulate. Proclus tells us that the Epicureans (the followers of Epicurus, the Greek philosopher, 342-270 B.C.) used to ridicule this theorem, saying that even an ass knew it, for if he wished to get food, he walked in a straight line and not along two sides of a triangle. Proclus replied that it was one thing to know the truth and another thing to prove it, meaning that the value of geometry lay in the proof rather than in the mere facts, a thing that all who seek to reform the teaching of geometry would do well to keep in mind. The theorem might simply appear as a corollary under the postulate if it were of any importance to reduce the number of propositions one more. If the proposition is postponed until after those concerning the inequalities of angles and sides of a triangle, there are several good proofs. [Illustration] For example, produce _AC_ to _X_, making _CX_ = _CB_. Then [L]_X_ = [L]_XBC_. [therefore] [L]_XBA_ > [L]_X_. [therefore] _AX_ > _AB_. [therefore] _AC_ + _CB_ > _AB_. The above proof is due to Euclid. Heron of Alexandria (first century A.D.) is said by Proclus to have given the following: [Illustration] Let _CX_ bisect [L]_C_. Then [L]_BXC_ > [L]_ACX_. [therefore] [L]_BXC_ > [L]_XCB_. [therefore] _CB_ > _XB_. Similarly, _AC_ > _AX_. Adding, _AC_ + _CB_ > _AB_. THEOREM. _If two sides of a triangle are unequal, the angles opposite these sides are unequal, and the angle opposite the greater side is the greater._ Euclid stated this more briefly by saying, "In any triangle the greater side subtends the greater angle." This is not so satisfactory, for there may be no greater side. THEOREM. _If two angles of a triangle are unequal, the sides opposite these angles are unequal, and the side opposite the greater angle is the greater._ Euclid also stated this more briefly, but less satisfactorily, thus, "In any triangle the greater angle is subtended by the greater side." Students should have their attention called to the fact that these two theorems are reciprocal or dual theorems, the words "sides" and "angles" of the one corresponding to the words "angles" and "sides" respectively of the other. It may also be noticed that the proof of this proposition involves what is known as the Law of Converse; for (1) if _b_ = _c_, then [L]_B_ = [L]_C_; (2) if _b_ > _c_, then [L]_B_ > [L]_C_; (3) if _b_ < _c_, then [L]_B_ < [L]_C_; therefore the converses must necessarily be true as a matter of logic; for if [L]_B_ = [L]_C_, then _b_ cannot be greater than _c_ without violating (2), and _b_ cannot be less than _c_ without violating (3), therefore _b_ = _c_; and if [L]_B_ > [L]_C_, then _b_ cannot equal _c_ without violating (1), and _b_ cannot be less than _c_ without violating (3), therefore _b_ > _c_; similarly, if [L]_B_ < [L]_C_, then _b_ < _c_. This Law of Converse may readily be taught to pupils, and it has several applications in geometry. THEOREM. _If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle of the first triangle greater than the included angle of the second, then the third side of the first is greater than the third side of the second, and conversely._ [Illustration] In this proposition there are three possible cases: the point _Y_ may fall below _AB_, as here shown, or on _AB_, or above _AB_. As an exercise for pupils all three may be considered if desired. Following Euclid and most early writers, however, only one case really need be proved, provided that is the most difficult one, and is typical. Proclus gave the proofs of the other two cases, and it is interesting to pupils to work them out for themselves. In such work it constantly appears that every proposition suggests abundant opportunity for originality, and that the complete form of proof in a textbook is not a bar to independent thought. The Law of Converse, mentioned on page 190, may be applied to the converse case if desired. THEOREM. _Two angles whose sides are parallel, each to each, are either equal or supplementary._ This is not an ancient proposition, although the Greeks were well aware of the principle. It may be stated so as to include the case of the sides being perpendicular, each to each, but this is better left as an exercise. It is possible, by some circumlocution, to so state the theorem as to tell in what cases the angles are equal and in what cases supplementary. It cannot be tersely stated, however, and it seems better to leave this point as a subject for questioning by the teacher. THEOREM. _The opposite sides of a parallelogram are equal._ THEOREM. _If the opposite sides of a quadrilateral are equal, the figure is a parallelogram._ [Illustration] This proposition is a very simple test for a parallelogram. It is the principle involved in the case of the common folding parallel ruler, an instrument that has long been recognized as one of the valuable tools of practical geometry. It will be of some interest to teachers to see one of the early forms of this parallel ruler, as shown in the illustration.[63] If such an instrument is not available in the school, one suitable for illustrative purposes can easily be made from cardboard. [Illustration: PARALLEL RULER OF THE SEVENTEENTH CENTURY San Giovanni's "Seconda squara mobile," Vicenza, 1686] A somewhat more complicated form of this instrument may also be made by pupils in manual training, as is shown in this illustration from Bion's great treatise. The principle involved may be taken up in class, even if the instrument is not used. It is evident that, unless the workmanship is unusually good, this form of parallel ruler is not as accurate as the common one illustrated above. The principle is sometimes used in iron gates. [Illustration: PARALLEL RULER OF THE EIGHTEENTH CENTURY N. Bion's "Traité de la construction ... des instrumens de mathématique," The Hague, 1723] THEOREM. _Two parallelograms are congruent if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other._ This proposition is discussed in connection with the one that follows. THEOREM. _If three or more parallels intercept equal segments on one transversal, they intercept equal segments on every transversal._ These two propositions are not given in Euclid, although generally required by American syllabi of the present time. The last one is particularly useful in subsequent work. Neither one offers any difficulty, and neither has any interesting history. There are, however, numerous interesting applications to the last one. One that is used in mechanical drawing is here illustrated. [Illustration] If it is desired to divide a line _AB_ into five equal parts, we may take a piece of ruled tracing paper and lay it over the given line so that line 0 passes through _A_, and line 5 through _B_. We may then prick through the paper and thus determine the points on _AB_. Similarly, we may divide _AB_ into any other number of equal parts. Among the applications of these propositions is an interesting one due to the Arab Al-Nair[=i]z[=i] (_ca._ 900 A.D.). The problem is to divide a line into any number of equal parts, and he begins with the case of trisecting _AB_. It may be given as a case of practical drawing even before the problems are reached, particularly if some preliminary work with the compasses and straightedge has been given. Make _BQ_ and _AQ'_ perpendicular to _AB_, and make _BP_ = _PQ_ = _AP'_ = _P'Q'_. Then [triangle]_XYZ_ is congruent to [triangle]_YBP_, and also to [triangle]_XAP'_. Therefore _AX_ = _XY_ = _YB_. In the same way we might continue to produce _BQ_ until it is made up of _n_ - 1 lengths _BP_, and so for _AQ'_, and by properly joining points we could divide _AB_ into _n_ equal parts. In particular, if we join _P_ and _P'_, we bisect the line _AB_. [Illustration] THEOREM. _If two sides of a quadrilateral are equal and parallel, then the other two sides are equal and parallel, and the figure is a parallelogram._ This was Euclid's first proposition on parallelograms, and Proclus speaks of it as the connecting link between the theory of parallels and that of parallelograms. The ancients, writing for mature students, did not add the words "and the figure is a parallelogram," because that follows at once from the first part and from the definition of "parallelogram," but it is helpful to younger students because it emphasizes the fact that here is a test for this kind of figure. THEOREM. _The diagonals of a parallelogram bisect each other._ This proposition was not given in Euclid, but it is usually required in American syllabi. There is often given in connection with it the exercise in which it is proved that the diagonals of a rectangle are equal. When this is taken, it is well to state to the class that carpenters and builders find this one of the best checks in laying out floors and other rectangles. It is frequently applied also in laying out tennis courts. If the class is doing any work in mensuration, such as finding the area of the school grounds, it is a good plan to check a few rectangles by this method. An interesting outdoor application of the theory of parallelograms is the following: [Illustration] Suppose you are on the side of this stream opposite to _XY_, and wish to measure the length of _XY_. Run a line _AB_ along the bank. Then take a carpenter's square, or even a large book, and walk along _AB_ until you reach _P_, a point from which you can just see _X_ and _B_ along two sides of the square. Do the same for _Y_, thus fixing _P_ and _Q_. Using the tape, bisect _PQ_ at _M_. Then walk along _YM_ produced until you reach a point _Y'_ that is exactly in line with _M_ and _Y_, and also with _P_ and _X_. Then walk along _XM_ produced until you reach a point _X'_ that is exactly in line with _M_ and _X_, and also with _Q_ and _Y_. Then measure _Y'X'_ and you have the length of _XY_. For since _YX'_ is [perp] to _PQ_, and _XY'_ is also [perp] to _PQ_, _YX'_ is || to _XY'_. And since _PM_ = _MQ_, therefore _XM_ = _MX'_ and _Y'M_ = _MY_. Therefore _Y'X'YX_ is a parallelogram. The properties of the parallelogram are often applied to proving figures of various kinds congruent, or to constructing them so that they will be congruent. [Illustration] For example, if we draw _A'B'_ equal and parallel to _AB_, _B'C'_ equal and parallel to _BC_, and so on, it is easily proved that _ABCD_ and _A'B'C'D'_ are congruent. This may be done by ordinary superposition, or by sliding _ABCD_ along the dotted parallels. There are many applications of this principle of parallel translation in practical construction work. The principle is more far-reaching than here intimated, however, and a few words as to its significance will now be in place. The efforts usually made to improve the spirit of Euclid are trivial. They ordinarily relate to some commonplace change of sequence, to some slight change in language, or to some narrow line of applications. Such attempts require no particular thought and yield no very noticeable result. But there is a possibility, remote though it may be at present, that a geometry will be developed that will be as serious as Euclid's and as effective in the education of the thinking individual. If so, it seems probable that it will not be based upon the congruence of triangles, by which so many propositions of Euclid are proved, but upon certain postulates of motion, of which one is involved in the above illustration,--the postulate of parallel translation. If to this we join the two postulates of rotation about an axis,[64] leading to axial symmetry; and rotation about a point,[65] leading to symmetry with respect to a center, we have a group of three motions upon which it is possible to base an extensive and rigid geometry.[66] It will be through some such effort as this, rather than through the weakening of the Euclid-Legendre style of geometry, that any improvement is likely to come. At present, in America, the important work for teachers is to vitalize the geometry they have,--an effort in which there are great possibilities,--seeing to it that geometry is not reduced to mere froth, and recognizing the possibility of another geometry that may sometime replace it,--a geometry as rigid, as thought-compelling, as logical, and as truly educational. THEOREM. _The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides._ This interesting generalization of the proposition about the sum of the angles of a triangle is given by Proclus. There are several proofs, but all are based upon the possibility of dissecting the polygon into triangles. The point from which lines are drawn to the vertices is usually taken at a vertex, so that there are _n_ - 2 triangles. It may however be taken within the figure, making _n_ triangles, from the sum of the angles of which the four right angles about the point must be subtracted. The point may even be taken on one side, or outside the polygon, but the proof is not so simple. Teachers who desire to do so may suggest to particularly good students the proving of the theorem for a concave polygon, or even for a cross polygon, although the latter requires negative angles. Some schools have transit instruments for the use of their classes in trigonometry. In such a case it is a good plan to measure the angles in some piece of land so as to verify the proposition, as well as show the care that must be taken in reading angles. In the absence of this exercise it is well to take any irregular polygon and measure the angles by the help of a protractor, and thus accomplish the same results. THEOREM. _The sum of the exterior angles of a polygon, made by producing each of its sides in succession, is equal to four right angles._ This is also a proposition not given by the ancient writers. We have, however, no more valuable theorem for the purpose of showing the nature and significance of the negative angle; and teachers may arouse a great deal of interest in the negative quantity by showing to a class that when an interior angle becomes 180° the exterior angle becomes 0, and when the polygon becomes concave the exterior angle becomes negative, the theorem holding for all these cases. We have few better illustrations of the significance of the negative quantity, and few better opportunities to use the knowledge of this kind of quantity already acquired in algebra. [Illustration] In the hilly and mountainous parts of America, where irregular-shaped fields are more common than in the more level portions, a common test for a survey is that of finding the exterior angles when the transit instrument is set at the corners. In this field these angles are given, and it will be seen that the sum is 360°. In the absence of any outdoor work a protractor may be used to measure the exterior angles of a polygon drawn on paper. If there is an irregular piece of land near the school, the exterior angles can be fairly well measured by an ordinary paper protractor. The idea of locus is usually introduced at the end of Book I. It is too abstract to be introduced successfully any earlier, although authors repeat the attempt from time to time, unmindful of the fact that all experience is opposed to it. The loci propositions are not ancient. The Greeks used the word "locus" (in Greek, _topos_), however. Proclus, for example, says, "I call those locus theorems in which the same property is found to exist on the whole of some locus." Teachers should be careful to have the pupils recognize the necessity for proving two things with respect to any locus: (1) that any point on the supposed locus satisfies the condition; (2) that any point outside the supposed locus does not satisfy the given condition. The first of these is called the "sufficient condition," and the second the "necessary condition." Thus in the case of the locus of points in a plane equidistant from two given points, it is _sufficient_ that the point be on the perpendicular bisector of the line joining the given points, and this is the first part of the proof; it is also _necessary_ that it be on this line, i.e. it cannot be outside this line, and this is the second part of the proof. The proof of loci cases, therefore, involves a consideration of "the necessary and sufficient condition" that is so often spoken of in higher mathematics. This expression might well be incorporated into elementary geometry, and when it becomes better understood by teachers, it probably will be more often used. In teaching loci it is helpful to call attention to loci in space (meaning thereby the space of three dimensions), without stopping to prove the proposition involved. Indeed, it is desirable all through plane geometry to refer incidentally to solid geometry. In the mensuration of plane figures, which may be boundaries of solid figures, this is particularly true. It is a great defect in most school courses in geometry that they are entirely confined to two dimensions. Even if solid geometry in the usual sense is not attempted, every occasion should be taken to liberate boys' minds from what becomes the tyranny of paper. Thus the questions: "What is the locus of a point equidistant from two given points; at a constant distance from a given straight line or from a given point?" should be extended to space.[67] The two loci problems usually given at this time, referring to a point equidistant from the extremities of a given line, and to a point equidistant from two intersecting lines, both permit of an interesting extension to three dimensions without any formal proof. It is possible to give other loci at this point, but it is preferable merely to introduce the subject in Book I, reserving the further discussion until after the circle has been studied. It is well, in speaking of loci, to remember that it is entirely proper to speak of the "locus of a point" or the "locus of points." Thus the locus of a _point_ so moving in a plane as constantly to be at a given distance from a fixed point in the plane is a circle. In analytic geometry we usually speak of the locus of a _point_, thinking of the point as being anywhere on the locus. Some teachers of elementary geometry, however, prefer to speak of the locus of _points_, or the locus of _all points_, thus tending to make the language of elementary geometry differ from that of analytic geometry. Since it is a trivial matter of phraseology, it is better to recognize both forms of expression and to let pupils use the two interchangeably. FOOTNOTES: [57] Address at Brussels, August, 1910. [58] For a recent discussion of this general subject, see Professor Hobson on "The Tendencies of Modern Mathematics," in the _Educational Review_, New York, 1910, Vol. XL, p. 524. [59] A more extended list of applications is given later in this work. [60] Ab[=u]'l-'Abb[=a]s al-Fadl ibn H[=a]tim al-Nair[=i]z[=i], so called from his birthplace, Nair[=i]z, was a well-known Arab writer. He died about 922 A.D. He wrote a commentary on Euclid. [61] This illustration, taken from a book in the author's library, appeared in a valuable monograph by W. E. Stark, "Measuring Instruments of Long Ago," published in _School Science and Mathematics_, Vol. X, pp. 48, 126. With others of the same nature it is here reproduced by the courtesy of Principal Stark and of the editors of the journal in which it appeared. [62] In speaking of two congruent triangles it is somewhat easier to follow the congruence if the two are read in the same order, even though the relatively unimportant counterclockwise reading is neglected. No one should be a slave to such a formalism, but should follow the plan when convenient. [63] Stark, loc. cit. [64] Of which so much was made by Professor Olaus Henrici in his "Congruent Figures," London, 1879,--a book that every teacher of geometry should own. [65] Much is made of this in the excellent work by Henrici and Treutlein, "Lehrbuch der Geometrie," Leipzig, 1881. [66] Méray did much for this movement in France, and the recent works of Bourlet and Borel have brought it to the front in that country. [67] W. N. Bruce, "Teaching of Geometry and Graphic Algebra in Secondary Schools," Board of Education circular (No. 711), p. 8, London, 1909. CHAPTER XV THE LEADING PROPOSITIONS OF BOOK II Having taken up all of the propositions usually given in Book I, it seems unnecessary to consider as specifically all those in subsequent books. It is therefore proposed to select certain ones that have some special interest, either from the standpoint of mathematics or from that of history or application, and to discuss them as fully as the circumstances seem to warrant. THEOREMS. _In the same circle or in equal circles equal central angles intercept equal arcs; and of two unequal central angles the greater intercepts the greater arc_, and conversely for both of these cases. Euclid made these the twenty-sixth and twenty-seventh propositions of his Book III, but he limited them as follows: "In equal circles equal angles stand on equal circumferences, whether they stand at the centers or at the circumferences, and conversely." He therefore included two of our present theorems in one, thus making the proposition doubly hard for a beginner. After these two propositions the Law of Converse, already mentioned on page 190, may properly be introduced. THEOREMS. _In the same circle or in equal circles, if two arcs are equal, they are subtended by equal chords; and if two arcs are unequal, the greater is subtended by the greater chord_, and conversely. Euclid dismisses all this with the simple theorem, "In equal circles equal circumferences are subtended by equal straight lines." It will therefore be noticed that he has no special word for "chord" and none for "arc," and that the word "circumference," which some teachers are so anxious to retain, is used to mean both the whole circle and any arc. It cannot be doubted that later writers have greatly improved the language of geometry by the use of these modern terms. The word "arc" is the same, etymologically, as "arch," each being derived from the Latin _arcus_ (a bow). "Chord" is from the Greek, meaning "the string of a musical instrument." "Subtend" is from the Latin _sub_ (under), and _tendere_ (to stretch). It should be noticed that Euclid speaks of "equal circles," while we speak of "the same circle or equal circles," confining our proofs to the latter, on the supposition that this sufficiently covers the former. THEOREM. _A line through the center of a circle perpendicular to a chord bisects the chord and the arcs subtended by it._ This is an improvement on Euclid, III, 3: "If in a circle a straight line through the center bisects a straight line not through the center, it also cuts it at right angles; and if it cuts it at right angles, it also bisects it." It is a very important proposition, theoretically and practically, for it enables us to find the center of a circle if we know any part of its arc. A civil engineer, for example, who wishes to find the center of the circle of which some curve (like that on a running track, on a railroad, or in a park) is an arc, takes two chords, say of one hundred feet each, and erects perpendicular bisectors. It is well to ask a class why, in practice, it is better to take these chords some distance apart. Engineers often check their work by taking three chords, the perpendicular bisectors of the three passing through a single point. Illustrations of this kind of work are given later in this chapter. THEOREM. _In the same circle or in equal circles equal chords are equidistant from the center, and chords equidistant from the center are equal._ This proposition is practically used by engineers in locating points on an arc of a circle that is too large to be described by a tape, or that cannot easily be reached from the center on account of obstructions. [Illustration] If part of the curve _APB_ is known, take _P_ as the mid-point. Then stretch the tape from _A_ to _B_ and draw _PM_ perpendicular to it. Then swing the length _AM_ about _P_, and _PM_ about _B_, until they meet at _L_, and stretch the length _AB_ along _PL_ to _Q_. This fixes the point _Q_. In the same way fix the point _C_. Points on the curve can thus be fixed as near together as we wish. The chords _AB_, _PQ_, _BC_, and so on, are equal and are equally distant from the center. THEOREM. _A line perpendicular to a radius at its extremity is tangent to the circle._ The enunciation of this proposition by Euclid is very interesting. It is as follows: The straight line drawn at right angles to the diameter of a circle at its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further, the angle of the semicircle is greater and the remaining angle less than any acute rectilineal angle. The first assertion is practically that of tangency,--"will fall outside the circle." The second one states, substantially, that there is only one such tangent, or, as we say in modern mathematics, the tangent is unique. The third statement relates to the angle formed by the diameter and the circumference,--a mixed angle, as Proclus called it, and a kind of angle no longer used in elementary geometry. The fourth statement practically asserts that the angle between the tangent and circumference is less than any assignable quantity. This gives rise to a difficulty that seems to have puzzled many of Euclid's commentators, and that will interest a pupil: As the circle diminishes this angle apparently increases, while as the circle increases the angle decreases, and yet the angle is always stated to be zero. Vieta (1540-1603), who did much to improve the science of algebra, attempted to explain away the difficulty by adopting a notion of circle that was prevalent in his time. He said that a circle was a polygon of an infinite number of sides (which it cannot be, by definition), and that, a tangent simply coincided with one of the sides, and therefore made no angle with it; and this view was also held by Galileo (1564-1642), the great physicist and mathematician who first stated the law of the pendulum. THEOREM. _Parallel lines intercept equal arcs on a circle._ The converse of this proposition has an interesting application in outdoor work. [Illustration] Suppose we wish to run a line through _P_ parallel to a given line _AB_. With any convenient point _O_ as a center, and _OP_ as a radius, describe a circle cutting _AB_ in _X_ and _Y_. Draw _PX_. Then with _Y_ as a center and _PX_ as a radius draw an arc cutting the circle in _Q_. Then run the line from _P_ to _Q_. _PQ_ is parallel to _AB_ by the converse of the above theorem, which is easily shown to be true for this figure. THEOREM. _If two circles are tangent to each other, the line of centers passes through the point of contact._ There are many illustrations of this theorem in practical work, as in the case of cogwheels. An interesting application to engineering is seen in the case of two parallel streets or lines of track which are to be connected by a "reversed curve." [Illustration] If the lines are _AB_ and _CD_, and the connection is to be made, as shown, from _B_ to _C_, we may proceed as follows: Draw _BC_ and bisect it at _M_. Erect _PO_, the perpendicular bisector of _BM_; and _BO_, perpendicular to _AB_. Then _O_ is one center of curvature. In the same way fix _O'_. Then to check the work apply this theorem, _M_ being in the line of centers _OO'_. The curves may now be drawn, and they will be tangent to _AB_, to _CD_, and to each other. At this point in the American textbooks it is the custom to insert a brief treatment of measurement, explaining what is meant by ratio, commensurable and incommensurable quantities, constant and variable, and limit, and introducing one or more propositions relating to limits. The object of this departure from the ancient sequence, which postponed this subject to the book on ratio and proportion, is to treat the circle more completely in Book III. It must be confessed that the treatment is not as scientific as that of Euclid, as will be explained under Book III, but it is far better suited to the mind of a boy or girl. It begins by defining measurement in a practical way, as the finding of the number of times a quantity of any kind contains a known quantity of the same kind. Of course this gives a number, but this number may be a surd, like [sqrt]2. In other words, the magnitude measured may be incommensurable with the unit of measure, a seeming paradox. With this difficulty, however, the pupil should not be called upon to contend at this stage in his progress. The whole subject of incommensurables might safely be postponed, although it may be treated in an elementary fashion at this time. The fact that the measure of the diagonal of a square, of which a side is unity, is [sqrt]2, and that this measure is an incommensurable number, is not so paradoxical as it seems, the paradox being verbal rather than actual. It is then customary to define ratio as the quotient of the numerical measures of two quantities in terms of a common unit. This brings all ratios to the basis of numerical fractions, and while it is not scientifically so satisfactory as the ancient concept which considered the terms as lines, surfaces, angles, or solids, it is more practical, and it suffices for the needs of elementary pupils. "Commensurable," "incommensurable," "constant," and "variable" are then defined, and these definitions are followed by a brief discussion of limit. It simplifies the treatment of this subject to state at once that there are two classes of limits,--those which the variable actually reaches, and those which it can only approach indefinitely near. We find the one as frequently as we find the other, although it is the latter that is referred to in geometry. For example, the superior limit of a chord is a diameter, and this limit the chord may reach. The inferior limit is zero, but we do not consider the chord as reaching this limit. It is also well to call the attention of pupils to the fact that a quantity may decrease towards its limit as well as increase towards it. Such further definitions as are needed in the theory of limits are now introduced. Among these is "area of a circle." It might occur to some pupil that since a circle is a line (as used in modern mathematics), it can have no area. This is, however, a mere quibble over words. It is not pretended that the line has area, but that "area of a circle" is merely a shortened form of the expression "area inclosed by a circle." The Principle of Limits is now usually given as follows: "If, while approaching their respective limits, two variables are always equal, their limits are equal." This was expressed by D'Alembert in the eighteenth century as "Magnitudes which are the limits of equal magnitudes are equal," or this in substance. It would easily be possible to elaborate this theory, proving, for example, that if _x_ approaches _y_ as its limit, then _ax_ approaches _ay_ as its limit, and _x/a_ approaches _y/a_ as its limit, and so on. Very much of this theory, however, wearies a pupil so that the entire meaning of the subject is lost, and at best the treatment in elementary geometry is not rigorous. It is another case of having to sacrifice a strictly scientific treatment to the educational abilities of the pupil. Teachers wishing to find a scientific treatment of the subject should consult a good work on the calculus. THEOREM. _In the same circle or in equal circles two central angles have the same ratio as their intercepted arcs._ This is usually proved first for the commensurable case and then for the incommensurable one. The latter is rarely understood by all of the class, and it may very properly be required only of those who show some aptitude in geometry. It is better to have the others understand fully the commensurable case and see the nature of its applications, possibly reading the incommensurable proof with the teacher, than to stumble about in the darkness of the incommensurable case and never reach the goal. In Euclid there was no distinction between the two because his definition of ratio covered both; but, as we shall see in Book III, this definition is too difficult for our pupils. Theon of Alexandria (fourth century A.D.), the father of the Hypatia who is the heroine of Kingsley's well-known novel, wrote a commentary on Euclid, and he adds that sectors also have the same ratio as the arcs, a fact very easily proved. In propositions of this type, referring to the same circle or to equal circles, it is not worth while to ask pupils to take up both cases, the proof for either being obviously a proof for the other. Many writers state this proposition so that it reads that "central angles are _measured by_ their intercepted arcs." This, of course, is not literally true, since we can measure anything only by some thing, of the same kind. Thus we measure a volume by finding how many times it contains another volume which we take as a unit, and we measure a length by taking some other length as a unit; but we cannot measure a given length in quarts nor a given weight in feet, and it is equally impossible to measure an arc by an angle, and vice versa. Nevertheless it is often found convenient to _define_ some brief expression that has no meaning if taken literally, in such way that it shall acquire a meaning. Thus we _define_ "area of a circle," even when we use "circle" to mean a line; and so we may define the expression "central angles are measured by their intercepted arcs" to mean that central angles have the same numerical measure as these arcs. This is done by most writers, and is legitimate as explaining an abbreviated expression. THEOREM. _An inscribed angle is measured by half the intercepted arc._ In Euclid this proposition is combined with the preceding one in his Book VI, Proposition 33. Such a procedure is not adapted to the needs of students to-day. Euclid gave in Book III, however, the proposition (No. 20) that a central angle is twice an inscribed angle standing on the same arc. Since Euclid never considered an angle greater than 180°, his inscribed angle was necessarily less than a right angle. The first one who is known to have given the general case, taking the central angle as being also greater than 180°, was Heron of Alexandria, probably of the first century A.D.[68] In this he was followed by various later commentators, including Tartaglia and Clavius in the sixteenth century. One of the many interesting exercises that may be derived from this theorem is seen in the case of the "horizontal danger angle" observed by ships. [Illustration] If some dangerous rocks lie off the shore, and _L_ and _L'_ are two lighthouses, the angle _A_ is determined by observation, so that _A_ will lie on a circle inclosing the dangerous area. Angle _A_ is called the "horizontal danger angle." Ships passing in sight of the two lighthouses _L_ and _L'_ must keep out far enough so that the angle _L'SL_ shall be less than angle _A_. To this proposition there are several important corollaries, including the following: 1. _An angle inscribed in a semicircle is a right angle._ This corollary is mentioned by Aristotle and is attributed to Thales, being one of the few propositions with which his name is connected. It enables us to describe a circle by letting the arms of a carpenter's square slide along two nails driven in a board, a pencil being held at the vertex. [Illustration] A more practical use for it is made by machinists to determine whether a casting is a true semicircle. Taking a carpenter's square as here shown, if the vertex touches the curve at every point as the square slides around, it is a true semicircle. By a similar method a circle may be described by sliding a draftsman's triangle so that two sides touch two tacks driven in a board. [Illustration] Another interesting application of this corollary may be seen by taking an ordinary paper protractor _ACB_, and fastening a plumb line at _B_. If the protractor is so held that the plumb line cuts the semicircle at _C_, then _AC_ is level because it is perpendicular to the vertical line _BC_. Thus, if a class wishes to determine the horizontal line _AC_, while sighting up a hill in the direction _AB_, this is easily determined without a spirit level. It follows from this corollary, as the pupil has already found, that the mid-point of the hypotenuse of a right triangle is equidistant from the three vertices. This is useful in outdoor measuring, forming the basis of one of the best methods of letting fall a perpendicular from an external point to a line. [Illustration] Suppose _XY_ to be the edge of a sidewalk, and _P_ a point in the street from which we wish to lay a gas pipe perpendicular to the walk. From _P_ swing a cord or tape, say 60 feet long, until it meets _XY_ at _A_. Then take _M_, the mid-point of _PA_, and swing _MP_ about _M_, to meet _XY_ at _B_. Then _B_ is the foot of the perpendicular, since [L]_PBA_ can be inscribed in a semicircle. 2. _Angles inscribed in the same segment are equal._ [Illustration] By driving two nails in a board, at _A_ and _B_, and taking an angle _P_ made of rigid material (in particular, as already stated, a carpenter's square), a pencil placed at _P_ will generate an arc of a circle if the arms slide along _A_ and _B_. This is an interesting exercise for pupils. THEOREM. _An angle formed by two chords intersecting within the circle is measured by half the sum of the intercepted arcs._ THEOREM. _An angle formed by a tangent and a chord drawn from the point of tangency is measured by half the intercepted arc._ THEOREM. _An angle formed by two secants, a secant and a tangent, or two tangents, drawn to a circle from an external point, is measured by half the difference of the intercepted arcs._ These three theorems are all special cases of the general proposition that the angle included between two lines that cut (or touch) a circle is measured by half the sum of the intercepted arcs. If the point passes from within the circle to the circle itself, one arc becomes zero and the angle becomes an inscribed angle. If the point passes outside the circle, the smaller arc becomes negative, having passed through zero. The point may even "go to infinity," as is said in higher mathematics, the lines then becoming parallel, and the angle becoming zero, being measured by half the sum of one arc and a negative arc of the same absolute value. This is one of the best illustrations of the Principle of Continuity to be found in geometry. PROBLEM. _To let fall a perpendicular upon a given line from a given external point._ This is the first problem that a student meets in most American geometries. The reason for treating the problems by themselves instead of mingling them with the theorems has already been discussed.[69] The student now has a sufficient body of theorems, by which he can prove that his constructions are correct, and the advantage of treating these constructions together is greater than that of following Euclid's plan of introducing them whenever needed. Proclus tells us that "this problem was first investigated by Oenopides,[70] who thought it useful for astronomy." Proclus speaks of such a line as a gnomon, a common name for the perpendicular on a sundial, which casts the shadow by which the time of day is known. He also speaks of two kinds of perpendiculars, the plane and solid, the former being a line perpendicular to a line, and the latter a line perpendicular to a plane. It is interesting to notice that the solution tacitly assumes that a certain arc is going to cut the given line in two points, and only two. Strictly speaking, why may it not cut it in only one point, or even in three points? We really assume that if a straight line is drawn through a point within a circle, this line must get out of the circle on each of two sides of the given point, and in getting out it must cut the circle twice. Proclus noticed this assumption and endeavored to prove it. It is better, however, not to raise the question with beginners, since it seems to them like hair-splitting to no purpose. The problem is of much value in surveying, and teachers would do well to ask a class to let fall a perpendicular to the edge of a sidewalk from a point 20 feet from the walk, using an ordinary 66-foot or 50-foot tape. Practically, the best plan is to swing 30 feet of the tape about the point and mark the two points of intersection with the edge of the walk. Then measure the distance between the points and take half of this distance, thus fixing the foot of the perpendicular. PROBLEM. _At a given point in a line, to erect a perpendicular to that line._ This might be postponed until after the problem to bisect an angle, since it merely requires the bisection of a straight angle; but considering the immaturity of the average pupil, it is better given independently. The usual case considers the point not at the extremity of the line, and the solution is essentially that of Euclid. In practice, however, as for example in surveying, the point may be at the extremity, and it may not be convenient to produce the line. [Illustration] Surveyors sometimes measure _PB_ = 3 ft., and then take 9 ft. of tape, the ends being held at _B_ and _P_, and the tape being stretched to _A_, so that _PA_ = 4 ft. and _AB_ = 5 ft. Then _P_ is a right angle by the Pythagorean Theorem. This theorem not having yet been proved, it cannot be used at this time. A solution for the problem of erecting a perpendicular from the extremity of a line that cannot be produced, depending, however, on the problem of bisecting an angle, and therefore to be given after that problem, is attributed by Al-Nair[=i]z[=i] (tenth century A.D.) to Heron of Alexandria. It is also given by Proclus. [Illustration] Required to draw from _P_ a perpendicular to _AP_. Take _X_ anywhere on the line and erect _XY_ [perp] to _AP_ in the usual manner. Bisect [L]_PXY_ by the line _XM_. On _XY_ take _XN_ = _XP_, and draw _NM_ [perp] to _XY_. Then draw _PM_. The proof is evident. These may at the proper time be given as interesting variants of the usual solution. PROBLEM. _To bisect a given line._ Euclid said "finite straight line," but this wording is not commonly followed, because it will be inferred that the line is finite if it is to be bisected, and we use "line" alone to mean a straight line. Euclid's plan was to construct an equilateral triangle (by his Proposition 1 of Book I) on the line as a base, and then to bisect the vertical angle. Proclus tells us that Apollonius of Perga, who wrote the first great work on conic sections, used a plan which is substantially that which is commonly found in textbooks to-day,--constructing two isosceles triangles upon the line as a common base, and connecting their vertices. PROBLEM. _To bisect a given angle._ It should be noticed that in the usual solution two arcs intersect, and the point thus determined is connected with the vertex. Now these two arcs intersect twice, and since one of the points of intersection may be the vertex itself, the other point of intersection must be taken. It is not, however, worth while to make much of this matter with pupils. Proclus calls attention to the possible suggestion that the point of intersection may be imagined to lie outside the angle, and he proceeds to show the absurdity; but here, again, the subject is not one of value to beginners. He also contributes to the history of the trisection of an angle. Any angle is easily trisected by means of certain higher curves, such as the conchoid of Nicomedes (_ca._ 180 B.C.), the quadratrix of Hippias of Elis (_ca._ 420 B.C.), or the spiral of Archimedes (_ca._ 250 B.C.). But since this problem, stated algebraically, requires the solution of a cubic equation, and this involves, geometrically, finding three points, we cannot solve the problem by means of straight lines and circles alone. In other words, the trisection of _any_ angle, by the use of the straightedge and compasses alone, is impossible. Special angles may however be trisected. Thus, to trisect an angle of 90° we need only to construct an angle of 60°, and this can be done by constructing an equilateral triangle. But while we cannot trisect the angle, we may easily approximate trisection. For since, in the infinite geometric series 1/2 + 1/8 + 1/32 + 1/128 + ..., _s_ = _a_ ÷ (1 - _r_), we have _s_ = 1/2 ÷ 3/4 = 2/3. In other words, if we add 1/2 of the angle, 1/8 of the angle, 1/32 of the angle, and so on, we approach as a limit 2/3 of the angle; but all of these fractions can be obtained by repeated bisections, and hence by bisections we may approximate the trisection. The approximate bisection (or any other division) of an angle may of course be effected by the help of the protractor and a straightedge. The geometric method is, however, usually more accurate, and it is advantageous to have the pupils try both plans, say for bisecting an angle of about 49 1/2°. [Illustration] Applications of this problem are numerous. It may be desired, for example, to set a lamp-post on a line bisecting the angle formed by two streets that come together a little unsymmetrically, as here shown, in which case the bisecting line can easily be run by the use of a measuring tape, or even of a stout cord. A more interesting illustration is, however, the following: [Illustration] Let the pupils set a stake, say about 5 feet high, at a point _N_ on the school grounds about 9 A.M., and carefully measure the length of the shadow, _NW_, placing a small wooden pin at _W_. Then about 3 P.M. let them watch until the shadow _NE_ is exactly the same length that it was when _W_ was fixed, and then place a small wooden pin at _E_. If the work has been very carefully done, and they take the tape and bisect the line _WE_, thus fixing the line _NS_, they will have a north and south line. If this is marked out for a short distance from _N_, then when the shadow falls on _NS_, it will be noon by sun time (not standard time) at the school. PROBLEM. _From a given point in a given line, to draw a line making an angle equal to a given angle._ Proclus says that Eudemus attributed to Oenopides the discovery of the solution which Euclid gave, and which is substantially the one now commonly seen in textbooks. The problem was probably solved in some fashion before the time of Oenopides, however. The object of the problem is primarily to enable us to draw a line parallel to a given line. Practically, the drawing of one line parallel to another is usually effected by means of a parallel ruler (see page 191), or by the use of draftsmen's triangles, as here shown, or even more commonly by the use of a T-square, such as is here seen. This illustration shows two T-squares used for drawing lines parallel to the sides of a board upon which the drawing paper is fastened.[71] [Illustration] [Illustration] An ingenious instrument described by Baron Dupin is illustrated below. [Illustration] To the bar _A_ is fastened the sliding check _B_. A movable check _D_ may be fastened by a screw _C_. A sharp point is fixed in _B_, so that as _D_ slides along the edge of a board, the point marks a line parallel to the edge. Moreover, _F_ and _G_ are two brass arms of equal length joined by a pointed screw _H_ that marks a line midway between _B_ and _D_. Furthermore, it is evident that _H_ will draw a line bisecting any irregular board if the checks _B_ and _D_ are kept in contact with the irregular edges. Book II offers two general lines of application that may be introduced to advantage, preferably as additions to the textbook work. One of these has reference to topographical drawing and related subjects, and the other to geometric design. As long as these can be introduced to the pupil with an air of reality, they serve a good purpose, but if made a part of textbook work, they soon come to have less interest than the exercises of a more abstract character. If a teacher can relate the problems in topographical drawing to the pupil's home town, and can occasionally set some outdoor work of the nature here suggested, the results are usually salutary; but if he reiterates only a half-dozen simple propositions time after time, with only slight changes in the nature of the application, then the results will not lead to a cultivation of power in geometry,--a point which the writers on applied geometry usually fail to recognize. [Illustration] One of the simple applications of this book relates to the rounding of corners in laying out streets in some of our modern towns where there is a desire to depart from the conventional square corner. It is also used in laying out park walks and drives. [Illustration] The figure in the middle of the page represents two streets, _AP_ and _BQ_, that would, if prolonged, intersect at _C_. It is required to construct an arc so that they shall begin to curve at _P_ and _Q_, where _CP_ = _CQ_, and hence the "center of curvature" _O_ must be found. The problem is a common one in railroad work, only here _AP_ is usually oblique to _BQ_ if they are produced to meet at _C_, as in the second figure on page 218. It is required to construct an arc so that the tracks shall begin to curve at _P_ and _Q_, where _CP_ = _CQ_. [Illustration] The problem becomes a little more complicated, and correspondingly more interesting, when we have to find the center of curvature for a street railway track that must turn a corner in such a way as to allow, say, exactly 5 feet from the point _P_, on account of a sidewalk. [Illustration] The problem becomes still more difficult if we have two roads of different widths that we wish to join on a curve. Here the two centers of curvature are not the same, and the one road narrows to the other on the curve. The solutions will be understood from a study of the figures. The number of problems of this kind that can easily be made is limitless, and it is well to avoid the danger of hobby riding on this or any similar topic. Therefore a single one will suffice to close this group. [Illustration] If a road _AB_ on an arc described about _O_, is to be joined to road _CD_, described about _O'_, the arc _BC_ should evidently be internally tangent to _AB_ and externally tangent to _CD_. Hence the center is on _BOX_ and _O'CY_, and is therefore at _P_. The problem becomes more real if we give some width to the roads in making the drawing, and imagine them in a park that is being laid out with drives. It will be noticed that the above problems require the erecting of perpendiculars, the bisecting of angles, and the application of the propositions on tangents. A somewhat different line of problems is that relating to the passing of a circle through three given points. It is very easy to manufacture problems of this kind that have a semblance of reality. [Illustration] For example, let it be required to plan a driveway from the gate _G_ to the porch _P_ so as to avoid a mass of rocks _R_, an arc of a circle to be taken. Of course, if we allow pupils to use the Pythagorean Theorem at this time (and for metrical purposes this is entirely proper, because they have long been familiar with it), then we may ask not only for the drawing, but we may, for example, give the length from _G_ to the point on _R_ (which we may also call _R_), and the angle _RGO_ as 60°, to find the radius. A second general line of exercises adapted to Book II is a continuation of the geometric drawing recommended as a preliminary to the work in demonstrative geometry. The copying or the making of designs requiring the describing of circles, their inscription in or circumscription about triangles, and their construction in various positions of tangency, has some value as applying the various problems studied in this book. For a number of years past, several enthusiastic teachers have made much of the designs found in Gothic windows, having their pupils make the outline drawings by the help of compasses and straightedge. While such work has its value, it is liable soon to degenerate into purposeless formalism, and hence to lose interest by taking the vigorous mind of youth from the strong study of geometry to the weak manipulation of instruments. Nevertheless its value should be appreciated and conserved, and a few illustrations of these forms are given in order that the teacher may have examples from which to select. The best way of using this material is to offer it as supplementary work, using much or little, as may seem best, thus giving to it a freshness and interest that some have trouble in imparting to the regular book work. The best plan is to sketch rapidly the outline of a window on the blackboard, asking the pupils to make a rough drawing, and to bring in a mathematical drawing on the following day. [Illustration] It might be said, for example, that in planning a Gothic window this drawing is needed. The arc _BC_ is drawn with _A_ as a center and _AB_ as a radius. The small arches are described with _A_, _D_, and _B_ as centers and _AD_ as a radius. The center _P_ is found by taking _A_ and _B_ as centers and _AE_ as a radius. How may the points _D_, _E_, and _F_ be found? Draw the figure. From the study of the rectilinear figures suggested by such a simple pattern the properties of the equilateral triangle may be inferred. The Gothic window also offers some interesting possibilities in connection with the study of the square. For example, the illustration given on page 223 shows a number of traceries involving the construction of a square, the bisecting of angles, and the describing of circles.[72] [Illustration] The properties of the square, a figure now easily constructed by the pupils, are not numerous. What few there are may be brought out through the study of art forms, if desired. In case these forms are shown to a class, it is important that they should be selected from good designs. We have enough poor art in the world, so that geometry should not contribute any more. This illustration is a type of the best medieval Gothic parquetry.[73] [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES] Even simple designs of a semipuzzling nature have their advantage in this connection. In the following example the inner square contains all of the triangles, the letters showing where they may be fitted.[74] Still more elaborate designs, based chiefly upon the square and circle, are shown in the window traceries on page 225, and others will be given in connection with the study of the regular polygons. [Illustration] Designs like the figure below are typical of the simple forms, based on the square and circle, that pupils may profitably incorporate in any work in art design that they may be doing at the time they are studying the circle and the problems relating to perpendiculars and squares. [Illustration] Among the applications of the problem to draw a tangent to a given circle is the case of the common tangents to two given circles. Some authors give this as a basal problem, although it is more commonly given as an exercise or a corollary. One of the most obvious applications of the idea is that relating to the transmission of circular motion by means of a band over two wheels,[75] _A_ and _B_, as shown on page 226. [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES] The band may either not be crossed (the case of the two exterior tangents), or be crossed (the interior tangents), the latter allowing the wheels to turn in opposite directions. In case the band is liable to change its length, on account of stretching or variation in heat or moisture, a third wheel, _D_, is used. We then have the case of tangents to three pairs of circles. Illustrations of this nature make the exercise on the drawing of common tangents to two circles assume an appearance of genuine reality that is of advantage to the work. [Illustration] FOOTNOTES: [68] This is the latest opinion. He is usually assigned to the first century B.C. [69] See page 54. [70] A Greek philosopher and mathematician of the fifth century B.C. [71] This illustration and the following two are from C. Dupin, "Mathematics Practically Applied," translated from the French by G. Birkbeck, Halifax, 1854. This is probably the most scholarly attempt ever made at constructing a "practical geometry." [72] This illustration and others of the same type used in this work are from the excellent drawings by R. W. Billings, in "The Infinity of Geometric Design Exemplified," London, 1849. [73] From H. Kolb, "Der Ornamentenschatz ... aus allen Kunst-Epochen," Stuttgart, 1883. The original is in the Church of Saint Anastasia in Verona. [74] From J. Bennett, "The Arcanum ... A Concise Theory of Practicable Geometry," London, 1838, one of the many books that have assumed to revolutionize geometry by making it practical. [75] The figures are from Dupin, loc. cit. CHAPTER XVI THE LEADING PROPOSITIONS OF BOOK III In the American textbooks Book III is usually assigned to proportion. It is therefore necessary at the beginning of this discussion to consider what is meant by ratio and proportion, and to compare the ancient and the modern theories. The subject is treated by Euclid in his Book V, and an anonymous commentator has told us that it "is the discovery of Eudoxus, the teacher of Plato." Now proportion had been known long before the time of Eudoxus (408-355 B.C.), but it was numerical proportion, and as such it had been studied by the Pythagoreans. They were also the first to study seriously the incommensurable number, and with this study the treatment of proportion from the standpoint of rational numbers lost its scientific position with respect to geometry. It was because of this that Eudoxus worked out a theory of geometric proportion that was independent of number as an expression of ratio. The following four definitions from Euclid are the basal ones of the ancient theory: A ratio is a sort of relation in respect of size between two magnitudes of the same kind. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. Let magnitudes which have the same ratio be called proportional.[76] Of these, the first is so loose in statement as often to have been thought to be an interpolation of some later writer. It was probably, however, put into the original for the sake of completeness, to have some kind of statement concerning ratio as a preliminary to the important definition of quantities in the same ratio. Like the definition of "straight line," it was not intended to be taken seriously as a mathematical statement. The second definition is intended to exclude zero and infinite magnitudes, and to show that incommensurable magnitudes are included. The third definition is the essential one of the ancient theory. It defines what is meant by saying that magnitudes are in the same ratio; in other words, it defines a proportion. Into the merits of the definition it is not proposed to enter, for the reason that it is no longer met in teaching in America, and is practically abandoned even where the rest of Euclid's work is in use. It should be said, however, that it is scientifically correct, that it covers the case of incommensurable magnitudes as well as that of commensurable ones, and that it is the Greek forerunner of the modern theories of irrational numbers. As compared with the above treatment, the one now given in textbooks is unscientific. We define ratio as "the quotient of the numerical measures of two quantities of the same kind," and proportion as "an equality of ratios." But what do we mean by the quotient, say of [sqrt]2 by [sqrt]3? And when we multiply a ratio by [sqrt]5, what is the meaning of this operation? If we say that [sqrt]2 : [sqrt]3 means a quotient, what meaning shall we assign to "quotient"? If it is the number that shows how many times one number is contained in another, how many _times_ is [sqrt]3 contained in [sqrt]2? If to multiply is to take a number a certain number of times, how many times do we take it when we multiply by [sqrt]5? We certainly take it more than 2 times and less than 3 times, but what meaning can we assign to [sqrt]5 times? It will thus be seen that our treatment of proportion assumes that we already know the theory of irrationals and can apply it to geometric magnitudes, while the ancient treatment is independent of this theory. Educationally, however, we are forced to proceed as we do. Just as Dedekind's theory of numbers is a simple one for college students, so is the ancient theory of proportion; but as the former is not suited to pupils in the high school, so the latter must be relegated to the college classes. And in this we merely harmonize educational progress with world progress, for the numerical theory of proportion long preceded the theory of Eudoxus. The ancients made much of such terms as duplicate, triplicate, alternate, and inverse ratio, and also such as composition, separation, and conversion of ratio. These entered into such propositions as, "If four magnitudes are proportional, they will also be proportional alternately." In later works they appear in the form of "proportion by composition," "by division," and "by composition and division." None of these is to-day of much importance, since modern symbolism has greatly simplified the ancient expressions, and in particular the proposition concerning "composition and division" is no longer a basal theorem in geometry. Indeed, if our course of study were properly arranged, we might well relegate the whole theory of proportion to algebra, allowing this to precede the work in geometry. We shall now consider a few of the principal propositions of Book III. THEOREM. _If a line is drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally._ In addition to the usual proof it is instructive to consider in class the cases in which the parallel is drawn through the two sides produced, either below the base or above the vertex, and also in which the parallel is drawn through the vertex. THEOREM. _The bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides._ The proposition relating to the bisector of an exterior angle may be considered as a part of this one, but it is usually treated separately in order that the proof shall appear less involved, although the two are discussed together at this time. The proposition relating to the exterior angle was recognized by Pappus of Alexandria. If _ABC_ is the given triangle, and _CP__{1}, _CP__{2} are respectively the internal and external bisectors, then _AB_ is divided harmonically by _P__{1} and _P__{2}. [therefore]_AP__{1} : _P__{1}_B_ = _AP__{2} : _P__{2}_B_. [therefore]_AP__{2} : _P__{2}_B_ = _AP__{2} - _P__{1}_P__{2} : _P__{1}_P__{2} - _P__{2}_B_, and this is the criterion for the harmonic progression still seen in many algebras. For, letting _AP__{2} = _a_, _P__{1}_P__{2} = _b_, _P__{2}_B_ = _c_, we have _a_/_c_ = (_a_ - _b_)/(_b_ - _c_), which is also derived from taking the reciprocals of _a_, _b_, _c_, and placing them in an arithmetical progression, thus: 1/_b_ - 1/_a_ = 1/_c_ - 1/_b_, whence (_a_ - _b_)/_ab_ = (_b_ - _c_)/_bc_, or (_a_ - _b_)/(_b_ - _c_) = _ab_/_bc_ = _a_/_c_. This is the reason why the line _AB_ is said to be divided harmonically. The line _P__{1}_P__{2} is also called the _harmonic mean_ between _AP__{2} and _P__{2}_B_, and the points _A_, _P__{1}, _B_, _P__{2} are said to form an _harmonic range_. [Illustration] It may be noted that [L]_P__{2}_CP__{1}, being made up of halves of two supplementary angles, is a right angle. Furthermore, if the ratio _CA_ : _CB_ is given, and _AB_ is given, then _P__{1} and _P__{2} are both fixed. Hence _C_ must lie on a semicircle with _P__{1}_P__{2} as a diameter, and therefore the locus of a point such that its distances from two given points are in a given ratio is a circle. This fact, Pappus tells us, was known to Apollonius. At this point it is customary to define similar polygons as such as have their corresponding angles equal and their corresponding sides proportional. Aristotle gave substantially this definition, saying that such figures have "their sides proportional and their angles equal." Euclid improved upon this by saying that they must "have their angles severally equal and the sides about the equal angles proportional." Our present phraseology seems clearer. Instead of "corresponding angles" we may say "homologous angles," but there seems to be no reason for using the less familiar word. [Illustration] [Illustration] [Illustration] It is more general to proceed by first considering similar figures instead of similar polygons, thus including the most obviously similar of all figures,--two circles; but such a procedure is felt to be too difficult by many teachers. By this plan we first define similar sets of points, _A__{1}, _A__{2}, _A__{3}, ..., and _B__{1}, _B__{2}, _B__{3}, ..., as such that _A__{1}_A__{2}, _B__{1}_B__{2}, _C__{1}_C__{2}, ... are concurrent in _O_, and _A__{1}_O_ : _A__{2}_O_ = _B__{1}_O_ : _B__{2}_O_ = _C__{1}_O_ : _C__{2}_O_ = ... Here the constant ratio _A__{1}_O_ : _A__{2}_O_ is called the _ratio of similitude_, and _O_ is called the _center of similitude_. Having defined similar sets of points, we then define similar figures as those figures whose points form similar sets. Then the two circles, the four triangles, and the three quadrilaterals respectively are similar figures. If the ratio of similitude is 1, the similar figures become symmetric figures, and they are therefore congruent. All of the propositions relating to similar figures can be proved from this definition, but it is customary to use the Greek one instead. [Illustration] Among the interesting applications of similarity is the case of a shadow, as here shown, where the light is the center of similitude. It is also well known to most high school pupils that in a camera the lens reverses the image. The mathematical arrangement is here shown, the lens inclosing the center of similitude. The proposition may also be applied to the enlargement of maps and working drawings. The propositions concerning similar figures have no particularly interesting history, nor do they present any difficulties that call for discussion. In schools where there is a little time for trigonometry, teachers sometimes find it helpful to begin such work at this time, since all of the trigonometric functions depend upon the properties of similar triangles, and a brief explanation of the simplest trigonometric functions may add a little interest to the work. In the present state of our curriculum we cannot do more than mention the matter as a topic of general interest in this connection. It is a mistaken idea that geometry is a prerequisite to trigonometry. We can get along very well in teaching trigonometry if we have three propositions: (1) the one about the sum of the angles of a triangle; (2) the Pythagorean Theorem; (3) the one that asserts that two right triangles are similar if an acute angle of the one equals an acute angle of the other. For teachers who may care to make a little digression at this time, the following brief statement of a few of the facts of trigonometry may be of value: [Illustration] In the right triangle _OAB_ we shall let _AB_ = _y_, _OA_ = _x_, _OB_ = _r_, thus adopting the letters of higher mathematics. Then, so long as [L]_O_ remains the same, such ratios as _y_/_x_, _y_/_r_, etc., will remain the same, whatever is the size of the triangle. Some of these ratios have special names. For example, we call _y_/_r_ the _sine_ of _O_, and we write sin _O_ = _y_/_r_; _x_/_r_ the _cosine_ of _O_, and we write cos _O_ = _x_/_r_; _y_/_x_ the _tangent_ of _O_, and we write tan _O_ = _y_/_x_. Now because sin _O_ = _y_/_r_, therefore _r_ sin _O_ = _y_; and because cos _O_ = _x_/_r_, therefore _r_ cos _O_ = _x_; and because tan _O_ = _y_/_x_, therefore _x_ tan _O_ = _y_. Hence, if we knew the values of sin _O_, cos _O_, and tan _O_ for the various angles, we could find _x_, _y_, or _r_ if we knew any one of them. Now the values of the sine, cosine, and tangent (_functions_ of the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining them by actual measurement, using the protractor and squared paper. Some of those needed for such angles as a pupil in geometry is likely to use are as follows: ============================================================ ANGLE | SINE |COSINE|TANGENT|| ANGLE | SINE |COSINE|TANGENT ------+------+------+-------++-------+------+------+-------- 5° | .087 | .996 | .087 || 50° | .766 | .643 | 1.192 ------+------+------+-------++-------+------+------+-------- 10° | .174 | .985 | .176 || 55° | .819 | .574 | 1.428 ------+------+------+-------++-------+------+------+-------- 15° | .259 | .966 | .268 || 60° | .866 | .500 | 1.732 ------+------+------+-------++-------+------+------+-------- 20° | .342 | .940 | .364 || 65° | .906 | .423 | 2.145 ------+------+------+-------++-------+------+------+-------- 25° | .423 | .906 | .466 || 70° | .940 | .342 | 2.748 ------+------+------+-------++-------+------+------+-------- 30° | .500 | .866 | .577 || 75° | .966 | .259 | 3.732 ------+------+------+-------++-------+------+------+-------- 35° | .574 | .819 | .700 || 80° | .985 | .174 | 5.671 ------+------+------+-------++-------+------+------+-------- 40° | .643 | .766 | .839 || 85° | .996 | .087 |11.430 ------+------+------+-------++-------+------+------+-------- 45° | .707 | .707 | 1.000 || 90° | 1.00 | .000 |[infinity] ============================================================ It will of course be understood that the values are correct only to the nearest thousandth. Thus the cosine of 5° is 0.99619, and the sine of 85° is 0.99619. The entire table can be copied by a class in five minutes if a teacher wishes to introduce this phase of the work, and the author has frequently assigned the computing of a simpler table as a class exercise. Referring to the figure, if we know that _r_ = 30 and [L]_O_ = 40°, then since _y_ = _r_ sin _O_, we have _y_ = 30 × 0.643 = 19.29. If we know that _x_ = 60 and [L]_O_ = 35°, then since _y_ = _x_ tan _O_, we have _y_ = 60 × 0.7 = 42. We may also find _r_, for cos _O_ = _x_/_r_, whence _r_ = _x_/(cos _O_) = 60/0.819 = 73.26. Therefore, if we could easily measure [L]_O_ and could measure the distance _x_, we could find the height of a building _y_. In trigonometry we use a transit for measuring angles, but it is easy to measure them with sufficient accuracy for illustrative purposes by placing an ordinary paper protractor upon something level, so that the center comes at the edge, and then sighting along a ruler held against it, so as to find the angle of elevation of a building. We may then measure the distance to the building and apply the formula _y_ = _x_ tan _O_. [Illustration: A QUADRANT OF THE SIXTEENTH CENTURY Finaeus's "De re et praxi geometrica," Paris, 1556] It should always be understood that expensive apparatus is not necessary for such illustrative work. The telescope used on the transit is only three hundred years old, and the world got along very well with its trigonometry before that was invented. So a little ingenuity will enable any one to make from cheap protractors about as satisfactory instruments as the world used before 1600. In order that this may be the more fully appreciated, a few illustrations are here given, showing the old instruments and methods used in practical surveying before the eighteenth century. [Illustration: A QUADRANT OF THE SEVENTEENTH CENTURY] The illustration on page 236 shows a simple form of the quadrant, an instrument easily made by a pupil who may be interested in outdoor work. It was the common surveying instrument of the early days. A more elaborate example is seen in the illustration, on page 237, of a seventeenth-century brass specimen in the author's collection.[77] [Illustration: A QUADRANT OF THE SEVENTEENTH CENTURY Bartoli's "Del modo di misurare," Venice, 1689] Another type, easily made by pupils, is shown in the above illustration from Bartoli, 1689. Such instruments were usually made of wood, brass, or ivory.[78] Instruments for the running of lines perpendicular to other lines were formerly common, and are easily made. They suffice, as the following illustration shows, for surveying an ordinary field. [Illustration: SURVEYING INSTRUMENT OF THE EIGHTEENTH CENTURY N. Bion's "Traité de la construction ... des instrumens de mathématique," The Hague, 1723] [Illustration: THE QUADRANT USED FOR ALTITUDES Finaeus's "De re et praxi geometrica," Paris, 1556] The quadrant was practically used for all sorts of outdoor measuring. For example, the illustration from Finaeus, on this page, shows how it was used for altitudes, and the one reproduced on page 240 shows how it was used for measuring depths. A similar instrument from the work of Bettinus is given on page 241, the distance of a ship being found by constructing an isosceles triangle. A more elaborate form, with a pendulum attachment, is seen in the illustration from De Judaeis, which also appears on page 241. [Illustration: THE QUADRANT USED FOR DEPTHS Finaeus's "Protomathesis," Paris, 1532] [Illustration: A QUADRANT OF THE SIXTEENTH CENTURY De Judaeis's "De quadrante geometrico," Nürnberg, 1594] [Illustration: THE QUADRANT USED FOR DISTANCES Bettinus's "Apiaria universae philosophiae mathematicae," Bologna, 1645] The quadrant finally developed into the octant, as shown in the following illustration from Hoffmann, and this in turn developed into the sextant, which is now used by all navigators. [Illustration: THE OCTANT Hoffmann's "De Octantis," Jena, 1612] In connection with this general subject the use of the speculum (mirror) in measuring heights should be mentioned. The illustration given on page 243 shows how in early days a simple device was used for this purpose. Two similar triangles are formed in this way, and we have only to measure the height of the eye above the ground, and the distances of the mirror from the tower and the observer, to have three terms of a proportion. All of these instruments are easily made. The mirror is always at hand, and a paper protractor on a piece of board, with a plumb line attached, serves as a quadrant. For a few cents, and by the expenditure of an hour or so, a school can have almost as good instruments as the ordinary surveyor had before the nineteenth century. [Illustration: THE SPECULUM Finaeus's "De re et praxi geometrica," Paris, 1556] A well-known method of measuring the distance across a stream is illustrated in the figure below, where the distance from _A_ to some point _P_ is required. [Illustration] Run a line from _A_ to _C_ by standing at _C_ in line with _A_ and _P_. Then run two perpendiculars from _A_ and _C_ by any of the methods already given,--sighting on a protractor or along the edge of a book if no better means are at hand. Then sight from some point _D_, on _CD_, to _P_, putting a stake at _B_. Then run the perpendicular _BE_. Since _DE_ : _EB_ = _BA_ : _AP_, and since we can measure _DE_, _EB_, and _BA_ with the tape, we can compute the distance _AP_. There are many variations of this scheme of measuring distances by means of similar triangles, and pupils may be encouraged to try some of them. Other figures are suggested on page 244, and the triangles need not be confined to those having a right angle. A very simple illustration of the use of similar triangles is found in one of the stories told of Thales. It is related that he found the height of the pyramids by measuring their shadow at the instant when his own shadow just equaled his height. He thus had the case of two similar isosceles triangles. This is an interesting exercise which may be tried about the time that pupils are leaving school in the afternoon. [Illustration] Another application of the same principle is seen in a method often taken for measuring the height of a tree. [Illustration] The observer has a large right triangle made of wood. Such a triangle is shown in the picture, in which _AB_ = _BC_. He holds _AB_ level and walks toward the tree until he just sees the top along _AC_. Then because _AB_ = _BC_, and _AB_ : _BC_ = _AD_ : _DE_, the height above _D_ will equal the distance _AD_. Questions like the following may be given to the class: 1. What is the height of the tree in the picture if the triangle is 5 ft. 4 in. from the ground, and _AD_ is 23 ft. 8 in.? 2. Suppose a triangle is used which has _AB_ = twice _BC_. What is the height if _AD_ = 75 ft.? There are many variations of this principle. One consists in measuring the shadows of a tree and a staff at the same time. The height of the staff being known, the height of the tree is found by proportion. Another consists in sighting from the ground, across a mark on an upright staff, to the top of the tree. The height of the mark being known, and the distances from the eye to the staff and to the tree being measured, the height of the tree is found. [Illustration] An instrument sold by dealers for the measuring of heights is known as the hypsometer. It is made of brass, and is of the form here shown. The base is graduated in equal divisions, say 50, and the upright bar is similarly divided. At the ends of the hinged radius are two sights. If the observer stands 50 feet from a tree and sights at the top, so that the hinged radius cuts the upright bar at 27, then he knows at once that the tree is 27 feet high. It is easy for a class to make a fairly good instrument of this kind out of stiff pasteboard. An interesting application of the theorem relating to similar triangles is this: Extend your arm and point to a distant object, closing your left eye and sighting across your finger tip with your right eye. Now keep the finger in the same position and sight with your left eye. The finger will then seem to be pointing to an object some distance to the right of the one at which you were pointing. If you can estimate the distance between these two objects, which can often be done with a fair degree of accuracy when there are houses intervening, then you will be able to tell approximately your distance from the objects, for it will be ten times the estimated distance between them. The finding of the reason for this by measuring the distance between the pupils of the two eyes, and the distance from the eye to the finger tip, and then drawing the figure, is an interesting exercise. Perhaps some pupil who has read Thoreau's descriptions of outdoor life may be interested in what he says of his crude mathematics. He writes, "I borrowed the plane and square, level and dividers, of a carpenter, and with a shingle contrived a rude sort of a quadrant, with pins for sights and pivots." With this he measured the heights of a cliff on the Massachusetts coast, and with similar home-made or school-made instruments a pupil in geometry can measure most of the heights and distances in which he is interested. THEOREM. _If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse:_ 1. _The triangles thus formed are similar to the given triangle, and are similar to each other._ 2. _The perpendicular is the mean proportional between the segments of the hypotenuse._ 3. _Each of the other sides is the mean proportional between the hypotenuse and the segment of the hypotenuse adjacent to that side._ To this important proposition there is one corollary of particular interest, namely, _The perpendicular from any point on a circle to a diameter is the mean proportional between the segments of the diameter_. By means of this corollary we can easily construct a line whose numerical value is the square root of any number we please. Thus we may make _AD_ = 2 in., _DB_ = 3 in., and erect _DC_ [perp] to _AB_. Then the length of _DC_ will be [sqrt]6 in., and we may find [sqrt]6 approximately by measuring _DC_. [Illustration] Furthermore, if we introduce negative magnitudes into geometry, and let _DB_ = +3 and _DA_ = -2, then _DC_ will equal [sqrt](-6). In other words, we have a justification for representing imaginary quantities by lines perpendicular to the line on which we represent real quantities, as is done in the graphic treatment of imaginaries in algebra. It is an interesting exercise to have a class find, to one decimal place, by measuring as above, the value of [sqrt]2, [sqrt]3, [sqrt]5, and [sqrt]9, the last being integral. If, as is not usually the case, the class has studied the complex number, the absolute value of [sqrt](-6), [sqrt](-7), ..., may be found in the same way. A practical illustration of the value of the above theorem is seen in a method for finding distances that is frequently described in early printed books. It seems to have come from the Roman surveyors. [Illustration] If a carpenter's square is put on top of an upright stick, as here shown, and an observer sights along the arms to a distant point _B_ and a point _A_ near the stick, then the two triangles are similar. Hence _AD_ : _DC_ = _DC_ : _DB_. Hence, if _AD_ and _DC_ are measured, _DB_ can be found. The experiment is an interesting and instructive one for a class, especially as the square can easily be made out of heavy pasteboard. THEOREM. _If two chords intersect within a circle, the product of the segments of the one is equal to the product of the segments of the other._ THEOREM. _If from a point without a circle a secant and a tangent are drawn, the tangent is the mean proportional between the secant and its external segment._ COROLLARY. _If from a point without a circle a secant is drawn, the product of the secant and its external segment is constant in whatever direction the secant is drawn._ These two propositions and the corollary are all parts of one general proposition: _If through a point a line is drawn cutting a circle, the product of the segments of the line is constant_. [Illustration] If _P_ is within the circle, then _xx'_ = _yy'_; if _P_ is on the circle, then _x_ and _y_ become 0, and 0 · _x'_ = 0 · _y'_ = 0; if _P_ is at _P__{3}, then _x_ and _y_, having passed through 0, may be considered negative if we wish, although the two negative signs would cancel out in the equation; if _P_ is at _P__{4}, then _y_ = _y'_ and we have _xx'_ = _y_^2, or _x_ : _y_ = _y_ : _x'_, as stated in the proposition. We thus have an excellent example of the Principle of Continuity, and classes are always interested to consider the result of letting _P_ assume various positions. Among the possible cases is the one of two tangents from an external point, and the one where _P_ is at the center of the circle. Students should frequently be questioned as to the meaning of "product of lines." The Greeks always used "rectangle of lines," but it is entirely legitimate to speak of "product of lines," provided we define the expression consistently. Most writers do this, saying that by the product of lines is meant the product of their numerical values, a subject already discussed at the beginning of this chapter. THEOREM. _The square on the bisector of an angle of a triangle is equal to the product of the sides of this angle diminished by the product of the segments made by the bisector upon the third side of the triangle._ This proposition enables us to compute the length of a bisector of a triangle if the lengths of the sides are known. [Illustration] For, in this figure, let _a_ = 3, _b_ = 5, and _c_ = 6. Then [because] _x_ : _y_ = _b_ : _a_, and _y_ = 6 - _x_, we have _x_/(6 - _x_) = 5/3. [therefore] 3_x_ = 30 - 5_x_. [therefore] _x_ = 3 3/4, _y_ = 2 1/4. By the theorem, _z_^2 = _ab_ - _xy_ = 15 - (8 7/16) = 6 9/16. [therefore] _z_ = [sqrt](6 9/16) = 1/4 [sqrt]105 = 2.5+. THEOREM. _In any triangle the product of two sides is equal to the product of the diameter of the circumscribed circle by the altitude upon the third side._ This enables us, after the Pythagorean Theorem has been studied, to compute the length of the diameter of the circumscribed circle in terms of the three sides. [Illustration] For if we designate the sides by _a_, _b_, and _c_, as usual, and let _CD_ = _d_ and _PB_ = _x_, then (_CP_)^2 = _a_^2 - _x_^2 = _b_^2 - (_c_ - _x_)^2. [therefore] _a_^2 - _x_^2 = _b_^2 - _c_^2 + 2_cx_ - _x_^2. [therefore] _x_ = (_a_^2 - _b_^2 + _c_^2) / 2_c_. [therefore] (_CP_)^2 = _a_^2 - ((_a_^2 - _b_^2 + _c_^2) / 2_c_)^2. But _CP_ · _d_ = _ab_. [therefore] _d_ = 2_abc_ / [sqrt](4_a_^2_c_^2 - (_a_^2 - _b_^2 + _c_^2)^2). This is not available at this time, however, because the Pythagorean Theorem has not been proved. These two propositions are merely special cases of the following general theorem, which may be given as an interesting exercise: _If ABC is an inscribed triangle, and through C there are drawn two straight lines CD, meeting AB in D, and CP, meeting the circle in P, with angles ACD and PCB equal, then AC × BC will equal CD × CP._ [Illustration: FIG. 1] [Illustration: FIG. 2] [Illustration: FIG. 3] [Illustration: FIG. 4] Fig. 1 is the general case where _D_ falls between _A_ and _B_. If _CP_ is a diameter, it reduces to the second figure given on page 249. If _CP_ bisects [L]_ACB_, we have Fig. 3, from which may be proved the proposition given at the foot of page 248. If _D_ lies on _BA_ produced, we have Fig. 2. If _D_ lies on _AB_ produced, we have Fig. 4. This general proposition is proved by showing that [triangles]_ADC_ and _PBC_ are similar, exactly as in the second proposition given on page 249. These theorems are usually followed by problems of construction, of which only one has great interest, namely, _To divide a given line in extreme and mean ratio._ The purpose of this problem is to prepare for the construction of the regular decagon and pentagon. The division of a line in extreme and mean ratio is called "the golden section," and is probably "the section" mentioned by Proclus when he says that Eudoxus "greatly added to the number of the theorems which Plato originated regarding the section." The expression "golden section" is not old, however, and its origin is uncertain. If a line _AB_ is divided in golden section at _P_, we have _AB_ × _PB_ = (_AP_)^2. Therefore, if _AB_ = _a_, and _AP_ = _x_, we have _a_(_a_ - _x_) = _x_^2, or _x_^2 + _ax_ - _a_^2 = 0; whence _x_ = - _a_/2 ± _a_/2[sqrt]5 = _a_(1.118 - 0.5) = 0.618_a_, the other root representing the external point. That is, _x_ = about 0.6_a_, and _a_ - _x_ = about 0.4_a_, and _a_ is therefore divided in about the ratio of 2 : 3. There has been a great deal written upon the æsthetic features of the golden section. It is claimed that a line is most harmoniously divided when it is either bisected or divided in extreme and mean ratio. A painting has the strong feature in the center, or more often at a point about 0.4 of the distance from one side, that is, at the golden section of the width of the picture. It is said that in nature this same harmony is found, as in the division of the veins of such leaves as the ivy and fern. FOOTNOTES: [76] For a very full discussion of these four definitions see Heath's "Euclid," Vol. II, p. 116, and authorities there cited. [77] These two and several which follow are from Stark, loc. cit. [78] The author has a beautiful ivory specimen of the Sixteenth century. CHAPTER XVII THE LEADING PROPOSITIONS OF BOOK IV Book IV treats of the area of polygons, and offers a large number of practical applications. Since the number of applications to the measuring of areas of various kinds of polygons is unlimited, while in the first three books these applications are not so obvious, less effort is made in this chapter to suggest practical problems to the teachers. The survey of the school grounds or of vacant lots in the vicinity offers all the outdoor work that is needed to make Book IV seem very important. THEOREM. _Two rectangles having equal altitudes are to each other as their bases._ Euclid's statement (Book VI, Proposition 1) was as follows: _Triangles and parallelograms which are under the same height are to one another as their bases_. Our plan of treating the two figures separately is manifestly better from the educational standpoint. In the modern treatment by limits the proof is divided into two parts: first, for commensurable bases; and second, for incommensurable ones. Of these the second may well be omitted, or merely be read over by the teacher and class and the reasons explained. In general, it is doubtful if the majority of an American class in geometry get much out of the incommensurable case. Of course, with a bright class a teacher may well afford to take it as it is given in the textbook, but the important thing is that the commensurable case should be proved and the incommensurable one recognized. Euclid's treatment of proportion was so rigorous that no special treatment of the incommensurable was necessary. The French geometer, Legendre, gave a rigorous proof by _reductio ad absurdum_. In America the pupils are hardly ready for these proofs, and so our treatment by limits is less rigorous than these earlier ones. THEOREM. _The area of a rectangle is equal to the product of its base by its altitude._ The easiest way to introduce this is to mark a rectangle, with commensurable sides, on squared paper, and count up the squares; or, what is more convenient, to draw the rectangle and mark the area off in squares. It is interesting and valuable to a class to have its attention called to the fact that the perimeter of a rectangle is no criterion as to the area. Thus, if a rectangle has an area of 1 square foot and is only 1/440 of an inch high, the perimeter is over 2 miles. The story of how Indians were induced to sell their land by measuring the perimeter is a very old one. Proclus speaks of travelers who described the size of cities by the perimeters, and of men who cheated others by pretending to give them as much land as they themselves had, when really they made only the perimeters equal. Thucydides estimated the size of Sicily by the time it took to sail round it. Pupils will be interested to know in this connection that of polygons having the same perimeter and the same number of sides, the one having equal sides and equal angles is the greatest, and that of plane figures having the same perimeter, the circle is the greatest. These facts were known to the Greek writers, Zenodorus (_ca._ 150 B.C.) and Proclus (410-485 A.D.). The surfaces of rectangular solids may now be found, there being an advantage in thus incidentally connecting plane and solid geometry wherever it is natural to do so. THEOREM. _The area of a parallelogram is equal to the product of its base by its altitude._ The best way to introduce this theorem is to cut a parallelogram from paper, and then, with the class, separate it into two parts by a cut perpendicular to the base. The two parts may then be fitted together to make a rectangle. In particular, if we cut off a triangle from one end and fit it on the other, we have the basis for the proof of the textbooks. The use of squared paper for such a proposition is not wise, since it makes the measurement appear to be merely an approximation. The cutting of the paper is in every way more satisfactory. THEOREM. _The area of a triangle is equal to half the product of its base by its altitude._ Of course, the Greeks would never have used the wording of either of these two propositions. Euclid, for example, gives this one as follows: _If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle._ As to the parallelogram, he simply says it is equal to a parallelogram of equal base and "in the same parallels," which makes it equal to a rectangle of the same base and the same altitude. The number of applications of these two theorems is so great that the teacher will not be at a loss to find genuine ones that appeal to the class. Teachers may now introduce pyramids, requiring the areas of the triangular faces to be found. The Ahmes papyrus (_ca._ 1700 B.C.) gives the area of an isosceles triangle as 1/2 _bs_, where _s_ is one of the equal sides, thus taking _s_ for the altitude. This shows the primitive state of geometry at that time. THEOREM. _The area of a trapezoid is equal to half the sum of its bases multiplied by the altitude._ [Illustration] An interesting variation of the ordinary proof is made by placing a trapezoid _T'_, congruent to _T_, in the position here shown. The parallelogram formed equals _a_(_b_ + _b'_), and therefore _T_ = _a_ · (_b_ + _b'_)/2. The proposition should be discussed for the case _b_ = _b'_, when it reduces to the one about the area of a parallelogram. If _b'_= 0, the trapezoid reduces to a triangle, and _T_ = _a_ · _b_/2. This proposition is the basis of the theory of land surveying, a piece of land being, for purposes of measurement, divided into trapezoids and triangles, the latter being, as we have seen, a kind of special trapezoid. The proposition is not in Euclid, but is given by Proclus in the fifth century. The term "isosceles trapezoid" is used to mean a trapezoid with two opposite sides equal, but not parallel. The area of such a figure was incorrectly given by the Ahmes papyrus as 1/2(_b_ + _b'_)_s_, where _s_ is one of the equal sides. This amounts to taking _s_ = _a_. The proposition is particularly important in the surveying of an irregular field such as is found in hilly districts. It is customary to consider the field as a polygon, and to draw a meridian line, letting fall perpendiculars upon it from the vertices, thus forming triangles and trapezoids that can easily be measured. An older plan, but one better suited to the use of pupils who may be working only with the tape, is given on page 99. THEOREM. _The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles._ This proposition may be omitted as far as its use in plane geometry is concerned, for we can prove the next proposition here given without using it. In solid geometry it is used only in a proposition relating to the volumes of two triangular pyramids having a common trihedral angle, and this is usually omitted. But the theorem is so simple that it takes but little time, and it adds greatly to the student's appreciation of similar triangles. It not only simplifies the next one here given, but teachers can at once deduce the latter from it as a special case by asking to what it reduces if a second angle of one triangle is also equal to a second angle of the other triangle. It is helpful to give numerical values to the sides of a few triangles having such equal angles, and to find the numerical ratio of the areas. THEOREM. _The areas of two similar triangles are to each other as the squares on any two corresponding sides._ [Illustration] This may be proved independently of the preceding proposition by drawing the altitudes _p_ and _p'_. Then [triangle]_ABC_/[triangle]_A'B'C'_ = _cp_/_c'p'_. But _c_/_c'_ = _p_/_p'_, by similar triangles. [therefore] [triangle]_ABC_/[triangle]_A'B'C'_ = _c_^2/_c'_^2, and so for other sides. This proof is unnecessarily long, however, because of the introduction of the altitudes. In this and several other propositions in Book IV occurs the expression "the square _on_ a line." We have, in our departure from Euclid, treated a line either as a geometric figure or as a number (the length of the line), as was the more convenient. Of course if we are speaking of a line, the preferable expression is "square _on_ the line," whereas if we speak of a number, we say "square _of_ the number." In the case of a rectangle of two lines we have come to speak of the "product of the lines," meaning the product of their numerical values. We are therefore not as accurate in our phraseology as Euclid, and we do not pretend to be, for reasons already given. But when it comes to "square _on_ a line" or "square _of_ a line," the former is the one demanding no explanation or apology, and it is even better understood than the latter. THEOREM. _The areas of two similar polygons are to each other as the squares on any two corresponding sides._ This is a proposition of great importance, and in due time the pupil sees that it applies to circles, with the necessary change of the word "sides" to "lines." It is well to ask a few questions like the following: If one square is twice as high as another, how do the areas compare? If the side of one equilateral triangle is three times as long as that of another, how do the perimeters compare? how do the areas compare? If the area of one square is twenty-five times the area of another square, the side of the first is how many times as long as the side of the second? If a photograph is enlarged so that a tree is four times as high as it was before, what is the ratio of corresponding dimensions? The area of the enlarged photograph is how many times as great as the area of the original? THEOREM. _The square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the other two sides._ Of all the propositions of geometry this is the most famous and perhaps the most valuable. Trigonometry is based chiefly upon two facts of plane geometry: (1) in similar triangles the corresponding sides are proportional, and (2) this proposition. In mensuration, in general, this proposition enters more often than any others, except those on the measuring of the rectangle and triangle. It is proposed, therefore, to devote considerable space to speaking of the history of the theorem, and to certain proofs that may profitably be suggested from time to time to different classes for the purpose of adding interest to the work. Proclus, the old Greek commentator on Euclid, has this to say of the history: "If we listen to those who wish to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honor of his discovery. But for my part, while I admire those who first observed the truth of this theorem, I marvel more at the writer of the 'Elements' (Euclid), not only because he made it fast by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in Book VI. For in that book he proves, generally, that in right triangles the figure on the side subtending the right angle is equal to the similar and similarly placed figures described on the sides about the right angle." Now it appears from this that Proclus, in the fifth century A.D., thought that Pythagoras discovered the proposition in the sixth century B.C., that the usual proof, as given in most of our American textbooks, was due to Euclid, and that the generalized form was also due to the latter. For it should be made known to students that the proposition is true not only for squares, but for any similar figures, such as equilateral triangles, parallelograms, semicircles, and irregular figures, provided they are similarly placed on the three sides of the right triangle. Besides Proclus, Plutarch testifies to the fact that Pythagoras was the discoverer, saying that "Pythagoras sacrificed an ox on the strength of his proposition as Apollodotus says," but saying that there were two possible propositions to which this refers. This Apollodotus was probably Apollodorus, surnamed Logisticus (the Calculator), whose date is quite uncertain, and who speaks in some verses of a "famous proposition" discovered by Pythagoras, and all tradition makes this the one. Cicero, who comments upon these verses, does not question the discovery, but doubts the story of the sacrifice of the ox. Of other early writers, Diogenes Laertius, whose date is entirely uncertain (perhaps the second century A.D.), and Athenæus (third century A.D.) may be mentioned as attributing the theorem to Pythagoras, while Heron (first century A.D.) says that he gave a rule for forming right triangles with rational integers for the sides, like 3, 4, 5, where 3^2 + 4^2 = 5^2. It should be said, however, that the Pythagorean origin has been doubted, notably in an article by H. Vogt, published in the _Bibliotheca Mathematica_ in 1908 (Vol. IX (3), p. 15), entitled "Die Geometrie des Pythagoras," and by G. Junge, in his work entitled "Wann haben die Griechen das Irrationale entdeckt?" (Halle, 1907). These writers claim that all the authorities attributing the proposition to Pythagoras are centuries later than his time, and are open to grave suspicion. Nevertheless it is hardly possible that such a general tradition, and one so universally accepted, should have arisen without good foundation. The evidence has been carefully studied by Heath in his "Euclid," who concludes with these words: "On the whole, therefore, I see no sufficient reason to question the tradition that, so far as Greek geometry is concerned ..., Pythagoras was the first to introduce the theorem ... and to give a general proof of it." That the fact was known earlier, probably without the general proof, is recognized by all modern writers. [Illustration] Pythagoras had studied in Egypt and possibly in the East before he established his school at Crotona, in southern Italy. In Egypt, at any rate, he could easily have found that a triangle with the sides 3, 4, 5, is a right triangle, and Vitruvius (first century B.C.) tells us that he taught this fact. The Egyptian _harpedonaptae_ (rope stretchers) stretched ropes about pegs so as to make such a triangle for the purpose of laying out a right angle in their surveying, just as our surveyors do to-day. The great pyramids have an angle of slope such as is given by this triangle. Indeed, a papyrus of the twelfth dynasty, lately discovered at Kahun, in Egypt, refers to four of these triangles, such as 1^2 + (3/4)^2 = (1 1/4)^2. This property seems to have been a matter of common knowledge long before Pythagoras, even as far east as China. He was, therefore, naturally led to attempt to prove the general property which had already been recognized for special cases, and in particular for the isosceles right triangle. How Pythagoras proved the proposition is not known. It has been thought that he used a proof by proportion, because Proclus says that Euclid gave a new style of proof, and Euclid does not use proportion for this purpose, while the subject, in incomplete form, was highly esteemed by the Pythagoreans. Heath suggests that this is among the possibilities: [Illustration] [triangles]_ABC_ and _APC_ are similar. [therefore] _AB_ × _AP_ = (_AC_)^2. Similarly, _AB_ × _PB_ = (_BC_)^2. [therefore] _AB_(_AP_ + _PB_) = (_AC_)^2 + (_BC_)^2, or (_AB_)^2 = (_AC_)^2 + (_BC_)^2. Others have thought that Pythagoras derived his proof from dissecting a square and showing that the square on the hypotenuse must equal the sum of the squares on the other two sides, in some such manner as this: [Illustration: FIG. 1] [Illustration: FIG. 2] Here Fig. 1 is evidently _h_^2 + 4 [triangles]. Fig. 2 is evidently _a_^2 + _b_^2 + 4 [triangles]. [therefore] _h_^2 + 4 [triangles] = _a_^2 + _b_^2 + 4 [triangles], the [triangles] all being congruent. [therefore] _h_^2 = _a_^2 + _b_^2. The great Hindu mathematician, Bhaskara (born 1114 A.D.), proceeds in a somewhat similar manner. He draws this figure, but gives no proof. It is evident that he had in mind this relation: [Illustration] _h_^2 = 4 · _ab_/2 + (_b_ - _a_)^2 = _a_^2 + _b_^2. A somewhat similar proof can be based upon the following figure: [Illustration] If the four triangles, 1 + 2 + 3 + 4, are taken away, there remains the square on the hypotenuse. But if we take away the two shaded rectangles, which equal the four triangles, there remain the squares on the two sides. Therefore the square on the hypotenuse must equal the sum of these two squares. [Illustration] It has long been thought that the truth of the proposition was first observed by seeing the tiles on the floors of ancient temples. If they were arranged as here shown, the proposition would be evident for the special case of an isosceles right triangle. The Hindus knew the proposition long before Bhaskara, however, and possibly before Pythagoras. It is referred to in the old religious poems of the Brahmans, the "Sulvasutras," but the date of these poems is so uncertain that it is impossible to state that they preceded the sixth century B.C.,[79] in which Pythagoras lived. The "Sulvasutra" of Apastamba has a collection of rules, without proofs, for constructing various figures. Among these is one for constructing right angles by stretching cords of the following lengths: 3, 4, 5; 12, 16, 20; 15, 20, 25 (the two latter being multiples of the first); 5, 12, 13; 15, 36, 39; 8, 15, 17; 12, 35, 37. Whatever the date of these "Sulvasutras," there is no evidence that the Indians had a definite proof of the theorem, even though they, like the early Egyptians, recognized the general fact. It is always interesting to a class to see more than one proof of a famous theorem, and many teachers find it profitable to ask their pupils to work out proofs that are (to them) original, often suggesting the figure. Two of the best known historic proofs are here given. The first makes the Pythagorean Theorem a special case of a proposition due to Pappus (fourth century A.D.), relating to any kind of a triangle. [Illustration] Somewhat simplified, this proposition asserts that if _ABC_ is _any_ kind of triangle, and _MC_, _NC_ are parallelograms on _AC_, _BC_, the opposite sides being produced to meet at _P_; and if _PC_ is produced making _QR_ = _PC_; and if the parallelogram _AT_ is constructed, then _AT_ = _MC_ + _NC_. For _MC_ = _AP_ = _AR_, having equal bases and equal altitudes. Similarly, _NC_ = _QT_. Adding, _MC_ + _NC_ = _AT_. If, now, _ABC_ is a right triangle, and if _MC_ and _NC_ are squares, it is easy to show that _AT_ is a square, and the proposition reduces to the Pythagorean Theorem. The Arab writer, Al-Nair[=i]z[=i] (died about 922 A.D.), attributes to Th[=a]bit ben Qurra (826-901 A.D.) a proof substantially as follows: [Illustration] The four triangles _T_ can be proved congruent. Then if we take from the whole figure _T_ and _T'_, we have left the squares on the two sides of the right angle. If we take away the other two triangles instead, we have left the square on the hypotenuse. Therefore the former is equivalent to the latter. A proof attributed to the great artist, Leonardo da Vinci (1452-1519), is as follows: [Illustration] The construction of the following figure is evident. It is easily shown that the four quadrilaterals _ABMX_, _XNCA_, _SBCP_, and _SRQP_ are congruent. [therefore] _ABMXNCA_ equals _SBCPQRS_ but is not congruent to it, the congruent quadrilaterals being differently arranged. Subtract the congruent triangles _MXN_, _ABC_, _RAQ_, and the proposition is proved.[80] The following is an interesting proof of the proposition: Let _ABC_ be the original triangle, with _AB_ < _BC_. Turn the triangle about _B_, through 90°, until it comes into the position _A'BC'_. Then because it has been turned through 90°, _C'A'P_ will be perpendicular to _AC_. Then 1/2(_AB_)^2 = [triangle]_ABA'_, and 1/2(_BC'_)^2 = [triangle]_BC'C_, because _BC_ = _BC'_. [therefore] 1/2((_AB_)^2 + (_BC_)^2) = [triangle]_ABA'_ + [triangle]_BC'C_. [therefore] 1/2((_AB_)^2 + (_BC_)^2) = [triangle]_AC'A'_ + [triangle]_A'C'C_ [Illustration] (For [triangle]_ABA'_ + [triangle]_BC'A'_ + [triangle]_A'C'C_ is the second member of both equations.) = 1/2_A'C'_ · _AP_ + 1/2_A'C'_ · _PC_ = 1/2_A'C'_ · _AC_ = 1/2(_AC_)^2. [therefore] (_AB_)^2 + (_BC_)^2 = (_AC_)^2. The Pythagorean Theorem, as it is generally called, has had other names. It is not uncommonly called the _pons asinorum_ (see page 174) in France. The Arab writers called it the Figure of the Bride, although the reason for this name is unknown; possibly two being joined in one has something to do with it. It has also been called the Bride's Chair, and the shape of the Euclid figure is not unlike the chair that a slave carries on his back, in which the Eastern bride is sometimes transported to the wedding ceremony. Schopenhauer, the German philosopher, referring to the figure, speaks of it as "a proof walking on stilts," and as "a mouse-trap proof." An interesting theory suggested by the proposition is that of computing the sides of right triangles so that they shall be represented by rational numbers. Pythagoras seems to have been the first to take up this theory, although such numbers were applied to the right triangle before his time, and Proclus tells us that Plato also contributed to it. The rule of Pythagoras, put in modern symbols, was as follows: _n_^2 + ((_n_^2 - 1)/2)^2 = ((_n_^2 + 1)/2)^2, the sides being _n_, (_n_^2 - 1)/2, and (_n_^2 + 1)/2. If for _n_ we put 3, we have 3, 4, 5. If we take the various odd numbers, we have _n_ = 1, 3, 5, 7, 9, ···, (_n_^2 - 1)/2 = 0, 4, 12, 24, 40, ···, (_n_^2 + 1)/2 = 1, 5, 13, 25, 41, ···. Of course _n_ may be even, giving fractional values. Thus, for _n_ = 2 we have for the three sides, 2, 1 1/2, 2 1/2. Other formulas are also known. Plato's, for example, is as follows: (2_n_)^2 + (_n_^2 - 1)^2 = (_n_^2 + 1)^2. If 2_n_ = 2, 4, 6, 8, 10, ···, then _n_^2 - 1 = 0, 3, 8, 15, 24, ···, and _n_^2 + 1 = 2, 5, 10, 17, 26, ···. This formula evidently comes from that of Pythagoras by doubling the sides of the squares.[81] THEOREM. _In any triangle the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides by the projection of the other upon that side._ THEOREM. _A similar statement for the obtuse triangle._ These two propositions are usually proved by the help of the Pythagorean Theorem. Some writers, however, actually construct the squares and give a proof similar to the one in that proposition. This plan goes back at least to Gregoire de St. Vincent (1647). [Illustration] It should be observed that _a_^2 = _b_^2 + _c_^2 - 2_b'c_. If [L]_A_ = 90°, then _b'_ = 0, and this becomes _a_^2 = _b_^2 + _c_^2. If [L]_A_ is obtuse, then _b'_ passes through 0 and becomes negative, and _a_^2 = _b_^2 + _c_^2 + 2_b'c_. Thus we have three propositions in one. [Illustration] At the close of Book IV many geometries give as an exercise, and some give as a regular proposition, the celebrated problem that bears the name of Heron of Alexandria, namely, to compute the area of a triangle in terms of its sides. The result is the important formula Area = [sqrt](_s_(_s_ - _a_)(_s_ - _b_)(_s_ - _c_)), where _a_, _b_, and _c_ are the sides, and _s_ is the semiperimeter 1/2(_a_ + _b_ + _c_). As a practical application the class may be able to find a triangular piece of land, as here shown, and to measure the sides. If the piece is clear, the result may be checked by measuring the altitude and applying the formula _a_ = 1/2_bh_. It may be stated to the class that Heron's formula is only a special case of the more general one developed about 640 A.D., by a famous Hindu mathematician, Brahmagupta. This formula gives the area of an inscribed quadrilateral as [sqrt]((_s_ - _a_)(_s_ - _b_)(_s_ - _c_)(_s_ - _d_)), where _a_, _b_, _c_, and _d_ are the sides and _s_ is the semiperimeter. If _d_ = 0, the quadrilateral becomes a triangle and we have Heron's formula.[82] At the close of Book IV, also, the geometric equivalents of the algebraic formulas for (_a_ + _b_)^2, (_a_ - _b_)^2, and (_a_ + _b_)(_a_ - _b_) are given. The class may like to know that Euclid had no algebra and was compelled to prove such relations as these by geometry, while we do it now much more easily by algebraic multiplication. FOOTNOTES: [79] See, for example, G. B. Kaye, "The Source of Hindu Mathematics," in the _Journal of the Royal Asiatic Society_, July, 1910. [80] An interesting Japanese proof of this general character may be seen in Y. Mikami, "Mathematical Papers from the Far East," p. 127, Leipzig, 1910. [81] Special recognition of indebtedness to H. A. Naber's "Das Theorem des Pythagoras" (Haarlem, 1908), Heath's "Euclid," Gow's "History of Greek Mathematics," and Cantor's "Geschichte" is due in connection with the Pythagorean Theorem. [82] The rule was so ill understood that Bhaskara (twelfth century) said that Brahmagupta was a "blundering devil" for giving it ("Lilavati," § 172). CHAPTER XVIII THE LEADING PROPOSITIONS OF BOOK V [Illustration] Book V treats of regular polygons and circles, and includes the computation of the approximate value of [pi]. It opens with a definition of a regular polygon as one that is both equilateral and equiangular. While in elementary geometry the only regular polygons studied are convex, it is interesting to a class to see that there are also regular cross polygons. Indeed, the regular cross pentagon was the badge of the Pythagoreans, as Lucian (_ca._ 100 B.C.) and an unknown commentator on Aristophanes (_ca._ 400 B.C.) tell us. At the vertices of this polygon the Pythagoreans placed the Greek letters signifying "health." Euclid was not interested in the measure of the circle, and there is nothing in his "Elements" on the value of [pi]. Indeed, he expressly avoided numerical measures of all kinds in his geometry, wishing the science to be kept distinct from that form of arithmetic known to the Greeks as logistic, or calculation. His Book IV is devoted to the construction of certain regular polygons, and his propositions on this subject are now embodied in Book V as it is usually taught in America. If we consider Book V as a whole, we are struck by three features. Of these the first is the pure geometry involved, and this is the essential feature to be emphasized. The second is the mensuration of the circle, a relatively unimportant piece of theory in view of the fact that the pupil is not ready for incommensurables, and a feature that imparts no information that the pupil did not find in arithmetic. The third is the somewhat interesting but mathematically unimportant application of the regular polygons to geometric design. As to the mensuration of the circle it is well for us to take a broad view before coming down to details. There are only four leading propositions necessary for the mensuration of the circle and the determination of the value of [pi]. These are as follows: (1) The inscribing of a regular hexagon, or any other regular polygon of which the side is easily computed in terms of the radius. We may start with a square, for example, but this is not so good as the hexagon because its side is incommensurable with the radius, and its perimeter is not as near the circumference. (2) The perimeters of similar regular polygons are proportional to their radii, and their areas to the squares of the radii. It is now necessary to state, in the form of a postulate if desired, that the circle is the limit of regular inscribed and circumscribed polygons as the number of sides increases indefinitely, and hence that (2) holds for circles. (3) The proposition relating to the area of a regular polygon, and the resulting proposition relating to the circle. (4) Given the side of a regular inscribed polygon, to find the side of a regular inscribed polygon of double the number of sides. It will thus be seen that if we were merely desirous of approximating the value of [pi], and of finding the two formulas _c_ = 2[pi]_r_ and _a_ = [pi]_r_^2, we should need only four propositions in this book upon which to base our work. It is also apparent that even if the incommensurable cases are generally omitted, the notion of _limit_ is needed at this time, and that it must briefly be reviewed before proceeding further. There is, however, a much more worthy interest than the mere mensuration of the circle, namely, the construction of such polygons as can readily be formed by the use of compasses and straightedge alone. The pleasure of constructing such figures and of proving that the construction is correct is of itself sufficient justification for the work. As to the use of such figures in geometric design, some discussion will be offered at the close of this chapter. The first few propositions include those that lead up to the mensuration of the circle. After they are proved it is assumed that the circle is the limit of the regular inscribed and circumscribed polygons as the number of sides increases indefinitely. This may often be proved with some approach to rigor by a few members of an elementary class, but it is the experience of teachers that the proof is too difficult for most beginners, and so the assumption is usually made in the form of an unproved theorem. The following are some of the leading propositions of this book: THEOREM. _Two circumferences have the same ratio as their radii._ This leads to defining the ratio of the circumference to the diameter as [pi]. Although this is a Greek letter, it was not used by the Greeks to represent this ratio. Indeed, it was not until 1706 that an English writer, William Jones, in his "Synopsis Palmariorum Matheseos," used it in this way, it being the initial letter of the Greek word for "periphery." After establishing the properties that _c_ = 2[pi]_r_, and _a_ = [pi]_r_^2, the textbooks follow the Greek custom and proceed to show how to inscribe and circumscribe various regular polygons, the purpose being to use these in computing the approximate numerical value of [pi]. Of these regular polygons two are of special interest, and these will now be considered. PROBLEM. _To inscribe a regular hexagon in a circle._ That the side of a regular inscribed hexagon equals the radius must have been recognized very early. The common divisions of the circle in ancient art are into four, six, and eight equal parts. No draftsman could have worked with a pair of compasses without quickly learning how to effect these divisions, and that compasses were early used is attested by the specimens of these instruments often seen in museums. There is a tradition that the ancient Babylonians considered the circle of the year as made up of 360 days, whence they took the circle as composed of 360 steps or grades (degrees). This tradition is without historic foundation, however, there being no authority in the inscriptions for this assumption of the 360-division by the Babylonians, who seem rather to have preferred 8, 12, 120, 240, and 480 as their division numbers. The story of 360° in the Babylonian circle seems to start with Achilles Tatius, an Alexandrian grammarian of the second or third century A.D. It is possible, however, that the Babylonians got their favorite number 60 (as in 60 seconds make a minute, 60 minutes make an hour or degree) from the hexagon in a circle (1/6 of 360° = 60°), although the probabilities seem to be that there is no such connection.[83] The applications of this problem to mensuration are numerous. The fact that we may use for tiles on a floor three regular polygons--the triangle, square, and hexagon--is noteworthy, a fact that Proclus tells us was recognized by Pythagoras. The measurement of the regular hexagon, given one side, may be used in computing sections of hexagonal columns, in finding areas of flower beds, and in other similar cases. This review of the names of the polygons offers an opportunity to impress their etymology again on the mind. In this case, for example, we have "hexagon" from the Greek words for "six" and "angle." PROBLEM. _To inscribe a regular decagon in a given circle._ Euclid states the problem thus: _To construct an isosceles triangle having each of the angles at the base double of the remaining one._ This makes each base angle 72° and the vertical angle 36°, the latter being the central angle of a regular decagon,--essentially our present method. This proposition seems undoubtedly due to the Pythagoreans, as tradition has always asserted. Proclus tells us that Pythagoras discovered "the construction of the cosmic figures," or the five regular polyhedrons, and one of these (the dodecahedron) involves the construction of the regular pentagon. Iamblichus (_ca._ 325 A.D.) tells us that Hippasus, a Pythagorean, was said to have been drowned for daring to claim credit for the construction of the regular dodecahedron, when by the rules of the brotherhood all credit should have been assigned to Pythagoras. If a regular polygon of _s_ sides can be inscribed, we may bisect the central angles, and therefore inscribe one of 2_s_ sides, and then of 4_s_ sides, and then of 8_s_ sides, and in general of 2^{_n_}_s_ sides. This includes the case of _s_ = 2 and _n_ = 0, for we can inscribe a regular polygon of two sides, the angles being, by the usual formula, 2(2 - 2)/2 = 0, although, of course, we never think of two equal and coincident lines as forming what we might call a _digon_. We therefore have the following regular polygons: From the equilateral triangle, regular polygons of 2^_n_ · 3 sides; From the square, regular polygons of 2^_n_ sides; From the regular pentagon, regular polygons of 2^_n_ · 5 sides; From the regular pentedecagon, regular polygons of 2^_n_ · 15 sides. This gives us, for successive values of _n_, the following regular polygons of less than 100 sides: From 2^_n_ · 3, 3, 6, 12, 24, 48, 96; From 2^_n_, 2, 4, 8, 16, 32, 64; From 2^_n_ · 5, 5, 10, 20, 40, 80; From 2^_n_ · 15, 15, 30, 60. [Illustration: ROMAN MOSAIC FOUND AT POMPEII] Gauss (1777-1855), a celebrated German mathematician, proved (in 1796) that it is possible also to inscribe a regular polygon of 17 sides, and hence polygons of 2^_n_ · 17 sides, or 17, 34, 68, ..., sides, and also 3 · 17 = 51 and 5 · 17 = 85 sides, by the use of the compasses and straightedge, but the proof is not adapted to elementary geometry. In connection with the study of the regular polygons some interest attaches to the reference to various forms of decorative design. The mosaic floor, parquetry, Gothic windows, and patterns of various kinds often involve the regular figures. If the teacher uses such material, care should be taken to exemplify good art. For example, the equilateral triangle and its relation to the regular hexagon is shown in the picture of an ancient Roman mosaic floor on page 274.[84] In the next illustration some characteristic Moorish mosaic work appears, in which it will be seen that the basal figure is the square, although at first sight this would not seem to be the case.[85] This is followed by a beautiful Byzantine mosaic, the original of which was in five colors of marble. Here it will be seen that the equilateral triangle and the regular hexagon are the basal figures, and a few of the properties of these polygons might be derived from the study of such a design. In the Arabic pattern on page 276 the dodecagon appears as the basis, and the remarkable powers of the Arab designer are shown in the use of symmetry without employing regular figures. [Illustration: MOSAIC FROM DAMASCUS] [Illustration: MOSAIC FROM AN ANCIENT BYZANTINE CHURCH] PROBLEM. _Given the side and the radius of a regular inscribed polygon, to find the side of the regular inscribed polygon of double the number of sides._ [Illustration: ARABIC PATTERN] The object of this proposition is, of course, to prepare the way for finding the perimeter of a polygon of 2_n_ sides, knowing that of _n_ sides. The Greek plan was generally to use both an inscribed and a circumscribed polygon, thus approaching the circle as a limit both from without and within. This is more conclusive from the ultrascientific point of view, but it is, if anything, less conclusive to a beginner, because he does not so readily follow the proof. The plan of using the two polygons was carried out by Archimedes of Syracuse (287-212 B.C.) in his famous method of approximating the value of [pi], although before him Antiphon (fifth century B.C.) had inscribed a square (or equilateral triangle) as a basis for the work, and Bryson (his contemporary) had attacked the problem by circumscribing as well as inscribing a regular polygon. PROBLEM. _To find the numerical value of the ratio of the circumference of a circle to its diameter._ As already stated, the usual plan of the textbooks is in part the method followed by Archimedes. It is possible to start with any regular polygon of which the side can conveniently be found in terms of the radius. In particular we might begin with an inscribed square instead of a regular hexagon. In this case we should have _Length of Side_ _Perimeter_ _s__{4} = 1.414... = 1.41 5.66 _s__{8} = [sqrt](2 - [sqrt](4 - 1.414^2)) = 0.72 5.76 and so on. It is a little easier to start with the hexagon, however, for we are already nearer the circle, and the side and perimeter are both commensurable with the radius. It is not, of course, intended that pupils should make the long numerical calculations. They may be required to compute _s__{12} and possibly _s__{24}, but aside from this they are expected merely to know the process. If it were possible to find the value of [pi] exactly, we could find the circumference exactly in terms of the radius, since c = 2[pi]_r_. If we could find the circumference exactly, we could find the area exactly, since _a_ = [pi]_r_^2. If we could find the area exactly in this form, [pi] times a square, we should have a rectangle, and it is easy to construct a square equivalent to any rectangle. Therefore, if we could find the value of [pi] exactly, we could construct a square with area equivalent to the area of the circle; in other words, we could "square the circle." We could also, as already stated, construct a straight line equivalent to the circumference; in other words, we could "rectify the circumference." These two problems have attracted the attention of the world for over two thousand years, but on account of their interrelation they are usually spoken of as a single problem, "to square the circle." Since we can construct [sqrt]_a_ by means of the straightedge and compasses, it would be possible for us to square the circle if we could express [pi] by a finite number of square roots. Conversely, every geometric construction reduces to the intersection of two straight lines, of a straight line and a circle, or of two circles, and is therefore equivalent to a rational operation or to the extracting of a square root. Hence a geometric construction cannot be effected by the straightedge and compasses unless it is equivalent to a series of rational operations or to the extracting of a finite number of square roots. It was proved by a German professor, Lindemann, in 1882, that [pi] cannot be expressed as an algebraic number, that is, as the root of an equation with rational coefficients, and hence it cannot be found by the above operations, and, furthermore, that the solution of this famous problem is impossible by elementary geometry.[86] It should also be pointed out to the student that for many practical purposes one of the limits of [pi] stated by Archimedes, namely, 3 1/7, is sufficient. For more accurate work 3.1416 is usually a satisfactory approximation. Indeed, the late Professor Newcomb stated that "ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimal places would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope." Probably the earliest approximation of the value of [pi] was 3. This appears very commonly in antiquity, as in I Kings vii, 23, and 2 Chronicles iv, 2. In the Ahmes papyrus (_ca._ 1700 B.C.) there is a rule for finding the area of the circle, expressed in modern symbols as (8/9)^2_d_^2, which makes [pi] = 256/81 or 3.1604.... Archimedes, using a plan somewhat similar to ours, found that [pi] lay between 3 1/7 and 3 10/71. Ptolemy, the great Greek astronomer, expressed the value as 3 17/120, or 3.14166.... The fact that Ptolemy divided his diameter into 120 units and his circumference into 360 units probably shows, however, the influence of the ancient value 3. In India an approximate value appears in a certain poem written before the Christian era, but the date is uncertain. About 500 A.D. Aryabhatta (or possibly a later writer of the same name) gave the value 62832/20000, or 3.1416. Brahmagupta, another Hindu (born 598 A.D.), gave [sqrt](10), and this also appears in the writings of the Chinese mathematician Chang Hêng (78-139 A.D.). A little later in China, Wang Fan (229-267) gave 142 ÷ 45, or 3.1555...; and one of his contemporaries, Lui Hui, gave 157 ÷ 50, or 3.14. In the fifth century Ch'ung-chih gave as the limits of [pi], 3.1415927 and 3.1415926, from which he inferred that 22/7 and 355/113 were good approximations, although he does not state how he came to this conclusion. In the Middle Ages the greatest mathematician of Italy, Leonardo Fibonacci, or Leonardo of Pisa (about 1200 A.D.), found as limits 3.1427... and 3.1410.... About 1600 the Chinese value 355/113 was rediscovered by Adriaen Anthonisz (1527-1607), being published by his son, who is known as Metius (1571-1635), in the year 1625. About the same period the French mathematician Vieta (1540-1603) found the value of [pi] to 9 decimal places, and Adriaen van Rooman (1561-1615) carried it to 17 decimal places, and Ludolph van Ceulen (1540-1610) to 35 decimal places. It was carried to 140 decimal places by Georg Vega (died in 1793), to 200 by Zacharias Dase (died in 1844), to 500 by Richter (died in 1854), and more recently by Shanks to 707 decimal places. There have been many interesting formulas for [pi], among them being the following: [pi]/2 = 2/1 · 2/3 · 4/3 · 4/5 · 6/5 · 6/7 · 8/7 · 8/9 · .... (Wallis, 1616-1703) 4/[pi] = 1 + 1/2 + 9/2 + 25/2 + 49/2 + .... (Brouncker, 1620-1684) [pi]/4 = 1 - 1/3 + 1/5 - 1/7 + .... (Gregory, 1638-1675) [pi]/6 = [sqrt](1/3) · (1 - 1/(3 · 3) + 1/(3^2 · 5) - 1/(3^3 · 7) + ...). [pi]/2 = (log _i_) / _i_. (Bernoulli) [pi]/(2[sqrt](3)) = 1 - 1/5 + 1/7 - 1/11 + 1/13 - 1/17 + 1/19..., thus connecting the primes. [pi]^2/16 = 1 - 1/2^2 - 1/3^2 + 1/4^2 - 1/5^2 + 1/6^2 - 1/7^2 - 1/8^2 + 1/9^2 + .... [pi]/2 = _x_/2 + sin _x_ + (sin^2 _x_) / 2 + (sin^3 _x_) / 3 + .... (0 < _x_ < 2[pi]) [pi]/4 = 3/4 + 1/(2 · 3 · 4) - 1/(4 · 5 · 6) + 1/(6 · 7 · 8) - .... 2[pi]^2/3 = 7 - (1/(1 · 3) + 1/(3 · 6) + 1/(6 · 10) + ...). [pi] = 2^_n_[sqrt](2 - [sqrt](2 + [sqrt](2 + [sqrt](2 + [sqrt](2...))))). Students of elementary geometry are not prepared to appreciate it, but teachers will be interested in the remarkable formula discovered by Euler (1707-1783), the great Swiss mathematician, namely, 1 + _e_^{_i_[pi]} = 0. In this relation are included the five most interesting quantities in mathematics,--zero, the unit, the base of the so-called Napierian logarithms, _i_ = [sqrt](-1), and [pi]. It was by means of this relation that the transcendence of _e_ was proved by the French mathematician Hermite, and the transcendence of [pi] by the German Lindemann. [Illustration] There should be introduced at this time, if it has not already been done, the proposition of the lunes of Hippocrates (_ca._ 470 B.C.), who proved a theorem that asserts, in somewhat more general form, that if three semicircles be described on the sides of a right triangle as diameters, as shown, the lunes _L_ + _L'_ are together equivalent to the triangle _T_. [Illustration] In the use of the circle in design one of the simplest forms suggested by Book V is the trefoil (three-leaf), as here shown, with the necessary construction lines. This is a very common ornament in architecture, both with rounded ends and with the ends slightly pointed. The trefoil is closely connected with hexagonal designs, since the regular hexagon is formed from the inscribed equilateral triangle by doubling the number of sides. The following are designs that are easily made: [Illustration] It is not very profitable, because it is manifestly unreal, to measure the parts of such figures, but it offers plenty of practice in numerical work. [Illustration: CHOIR OF LINCOLN CATHEDRAL] [Illustration: PORCH OF LINCOLN CATHEDRAL] In the illustrations of the Gothic windows given in Chapter XV only the square and circle were generally involved. Teachers who feel it necessary or advisable to go outside the regular work of geometry for the purpose of increasing the pupil's interest or of training his hand in the drawing of figures will find plenty of designs given in any pictures of Gothic cathedrals. For example, this picture of the noble window in the choir of Lincoln Cathedral shows the use of the square, hexagon, and pentagon. In the porch of the same cathedral, shown in the next illustration, the architect has made use of the triangle, square, and pentagon in planning his ornamental stonework. It is possible to add to the work in pure geometry some work in the mensuration of the curvilinear figures shown in these designs. This form of mensuration is not of much value, however, since it places before the pupil a problem that he sees at once is fictitious, and that has no human interest. [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES] [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND SQUARES] [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND THE EQUILATERAL TRIANGLE] [Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND THE REGULAR HEXAGON] The designs given on page 283 involve chiefly the square as a basis, but it will be seen from one of the figures that the equilateral triangle and the hexagon also enter. The possibilities of endless variation of a single design are shown in the illustration on page 284, the basis in this case being the square. The variations in the use of the triangle and hexagon have been the object of study of many designers of Gothic windows, and some examples of these forms are shown on page 285. In more simple form this ringing of the changes on elementary figures is shown on page 286. Some teachers have used color work with such designs for the purpose of increasing the interest of their pupils, but the danger of thus using the time with no serious end in view will be apparent. [Illustration] In the matter of the mensuration of the circle the annexed design has some interest. The figure is not uncommon in decoration, and it is interesting to show, as a matter of pure geometry, that the area of the circle is divided into three equal portions by means of the four interior semicircles. [Illustration] An important application of the formula _a_ = [pi]_r_^2 is seen in the area of the annulus, or ring, the formula being _a_ = [pi]_r_^2 - [pi]_r'_^2 = [pi](_r_^2 - _r'_^2) = [pi](_r_ + _r'_)(_r_ - _r'_). It is used in finding the area of the cross section of pipes, and this is needed when we wish to compute the volume of the iron used. Another excellent application is that of finding the area of the surface of a cylinder, there being no reason why such simple cases from solid geometry should not furnish working material for plane geometry, particularly as they have already been met by the pupils in arithmetic. A little problem that always has some interest for pupils is one that Napoleon is said to have suggested to his staff on his voyage to Egypt: To divide a circle into four equal parts by the use of circles alone. [Illustration] Here the circles _B_ are tangent to the circle _A_ at the points of division. Furthermore, considering areas, and taking _r_ as the radius of _A_, we have _A_ = [pi]_r_^2, and _B_ = [pi](_r_/2)^2. Hence _B_ = 1/4_A_, or the sum of the areas of the four circles _B_ equals the area of _A_. Hence the four _D'_s must equal the four _C'_s, and _D_ = _C_. The rest of the argument is evident. The problem has some interest to pupils aside from the original question suggested by Napoleon. At the close of plane geometry teachers may find it helpful to have the class make a list of the propositions that are actually used in proving other propositions, and to have it appear what ones are proved by them. This forms a kind of genealogical tree that serves to fix the parent propositions in mind. Such a work may also be carried on at the close of each book, if desired. It should be understood, however, that certain propositions are used in the exercises, even though they are not referred to in subsequent propositions, so that their omission must not be construed to mean that they are not important. An exercise of distinctly less value is the classification of the definitions. For example, the classification of polygons or of quadrilaterals, once so popular in textbook making, has generally been abandoned as tending to create or perpetuate unnecessary terms. Such work is therefore not recommended. FOOTNOTES: [83] Bosanquet and Sayre, "The Babylonian Astronomy," _Monthly Notices of the Royal Asiatic Society_, Vol. XL, p. 108. [84] This and the three illustrations following are from Kolb, loc. cit. [85] This was in five colors of marble. [86] The proof is too involved to be given here. The writer has set it forth in a chapter on the transcendency of [pi] in a work soon to be published by Professor Young of The University of Chicago. CHAPTER XIX THE LEADING PROPOSITIONS OF BOOK VI There have been numerous suggestions with respect to solid geometry, to the effect that it should be more closely connected with plane geometry. The attempt has been made, notably by Méray in France and de Paolis in Italy, to treat the corresponding propositions of plane and solid geometry together; as, for example, those relating to parallelograms and parallelepipeds, and those relating to plane and spherical triangles. Whatever the merits of this plan, it is not feasible in America at present, partly because of the nature of the college-entrance requirements. While it is true that to a boy or girl a solid is more concrete than a plane, it is not true that a geometric solid is more concrete than a geometric plane. Just as the world developed its solid geometry, as a science, long after it had developed its plane geometry, so the human mind grasps the ideas of plane figures earlier than those of the geometric solid. There is, however, every reason for referring to the corresponding proposition of plane geometry when any given proposition of solid geometry is under consideration, and frequent references of this kind will be made in speaking of the propositions in this and the two succeeding chapters. Such reference has value in the apperception of the various laws of solid geometry, and it also adds an interest to the subject and creates some approach to power in the discovery of new facts in relation to figures of three dimensions. The introduction to solid geometry should be made slowly. The pupil has been accustomed to seeing only plane figures, and therefore the drawing of a solid figure in the flat is confusing. The best way for the teacher to anticipate this difficulty is to have a few pieces of cardboard, a few knitting needles filed to sharp points, a pine board about a foot square, and some small corks. With the cardboard he can illustrate planes, whether alone, intersecting obliquely or at right angles, or parallel, and he can easily illustrate the figures given in the textbook in use. There are models of this kind for sale, but the simple ones made in a few seconds by the teacher or the pupil have much more meaning. The knitting needles may be stuck in the board to illustrate perpendicular or oblique lines, and if two or more are to meet in a point, they may be held together by sticking them in one of the small corks. Such homely apparatus, costing almost nothing, to be put together in class, seems much more real and is much more satisfactory than the German models.[87] An extensive use of models is, however, unwise. The pupil must learn very early how to visualize a solid from the flat outline picture, just as a builder or a mechanic learns to read his working drawings. To have a model for each proposition, or even to have a photograph or a stereoscopic picture, is a very poor educational policy. A textbook may properly illustrate a few propositions by photographic aids, but after that the pupil should use the kind of figures that he must meet in his mathematical work. A child should not be kept in a perambulator all his life,--he must learn to walk if he is to be strong and grow to maturity; and it is so with a pupil in the use of models in solid geometry.[88] The case is somewhat similar with respect to colored crayons. They have their value and their proper place, but they also have their strict limitations. It is difficult to keep their use within bounds; pupils come to use them to make pleasing pictures, and teachers unconsciously fall into the same habit. The value of colored crayons is two-fold: (1) they sometimes make two planes stand out more clearly, or they serve to differentiate some line that is under consideration from others that are not; (2) they enable a class to follow a demonstration more easily by hearing of "the red plane perpendicular to the blue one," instead of "the plane _MN_ perpendicular to the plane _PQ_." But it should always be borne in mind that in practical work we do not have colored ink or colored pencils commonly at hand, nor do we generally have colored crayons. Pupils should therefore become accustomed to the pencil and the white crayon as the regulation tools, and in general they should use them. The figures may not be as striking, but they are more quickly made and they are more practical. The definition of "plane" has already been discussed in Chapter XII, and the other definitions of Book VI are not of enough interest to call for special remark. The axioms are the same as in plane geometry, but there is at least one postulate that needs to be added, although it would be possible to state various analogues of the postulates of plane geometry if we cared unnecessarily to enlarge the number. The most important postulate of solid geometry is as follows: _One plane, and only one, can be passed through two intersecting straight lines._ This is easily illustrated, as in most textbooks, as also are three important corollaries derived from it: 1. _A straight line and a point not in the line determine a plane._ Of course this may be made the postulate, as may also the next one, the postulate being placed among the corollaries, but the arrangement here adopted is probably the most satisfactory for educational purposes. 2. _Three points not in a straight line determine a plane._ The common question as to why a three-legged stool stands firmly, while a four-legged table often does not, will add some interest at this point. 3. _Two parallel lines determine a plane._ This requires a slight but informal proof to show that it properly follows as a corollary from the postulate, but a single sentence suffices. While studying this book questions of the following nature may arise with an advanced class, or may be suggested to those who have had higher algebra: How many straight lines are in general (that is, at the most) determined by _n_ points in space? Two points determine 1 line, a third point adds (in general, in all these cases) 2 more, a fourth point adds 3 more, and an _n_th point _n_ - 1 more. Hence the maximum is 1 + 2 + 3 + ... + (_n_ - 1), or _n_(_n_-1)/2, which the pupil will understand if he has studied arithmetical progression. The maximum number of intersection points of _n_ straight lines in the same plane is also _n_(_n_ - 1)/2. How many straight lines are in general determined by _n_ planes? The answer is the same, _n_(_n_ - 1)/2. How many planes are in general determined by _n_ points in space? Here the answer is 1 + 3 + 6 + 10 + ... + (_n_ - 2)(_n_ - 1)/2, or _n_(_n_ - 1)(_n_ - 2)/(1 × 2 × 3). The same number of points is determined by _n_ planes. THEOREM. _If two planes cut each other, their intersection is a straight line._ Among the simple illustrations are the back edges of the pages of a book, the corners of the room, and the simple test as to whether the edge of a card is straight by testing it on a plane. It is well to call attention to the fact that if two intersecting straight lines move parallel to their original position, and so that their intersection rests on a straight line not in the plane of those lines, the figure generated will be that of this proposition. In general, if we cut through any figure of solid geometry in some particular way, we are liable to get the figure of a proposition in plane geometry, as will frequently be seen. THEOREM. _If a straight line is perpendicular to each of two other straight lines at their point of intersection, it is perpendicular to the plane of the two lines._ If students have trouble in visualizing the figure in three dimensions, some knitting needles through a piece of cardboard will make it clear. Teachers should call attention to the simple device for determining if a rod is perpendicular to a board (or a pipe to a floor, ceiling, or wall), by testing it twice, only, with a carpenter's square. Similarly, it may be asked of a class, How shall we test to see if the corner (line) of a room is perpendicular to the floor, or if the edge of a box is perpendicular to one of the sides? In some elementary and in most higher geometries the perpendicular is called a _normal_ to the plane. THEOREM. _All the perpendiculars that can be drawn to a straight line at a given point lie in a plane which is perpendicular to the line at the given point._ Thus the hands of a clock pass through a plane as the hands revolve, if they are, as is usual, perpendicular to the axis; and the same is true of the spokes of a wheel, and of a string with a stone attached, swung as rapidly as possible about a boy's arm as an axis. A clock pendulum too swings in a plane, as does the lever in a pair of scales. THEOREM. _Through a given point within or without a plane there can be one perpendicular to a given plane, and only one._ This theorem is better stated to a class as two theorems. Thus a plumb line hanging from a point in the ceiling, without swinging, determines one definite point in the floor; and, conversely, if it touches a given point in the floor, it must hang from one definite point in the ceiling. It should be noticed that if we cut through this figure, on the perpendicular line, we shall have the figure of the corresponding proposition in plane geometry, namely, that there can be, under similar circumstances, only one perpendicular to a line. THEOREM. _Oblique lines drawn from a point to a plane, meeting the plane at equal distances from the foot of the perpendicular, are equal, etc._ There is no objection to speaking of a right circular cone in connection with this proposition, and saying that the slant height is thus proved to be constant. The usual corollary, that if the obliques are equal they meet the plane in a circle, offers a new plan of drawing a circle. A plumb line that is a little too long to reach the floor will, if swung so as just to touch the floor, describe a circle. A 10-foot pole standing in a 9-foot room will, if it moves so as to touch constantly a fixed point on either the floor or the ceiling, describe a circle on the ceiling or floor respectively. One of the corollaries states that the locus of points in space equidistant from the extremities of a straight line is the plane perpendicular to this line at its middle point. This has been taken by some writers as the definition of a plane, but it is too abstract to be usable. It is advisable to cut through the figure along the given straight line, and see that we come back to the corresponding proposition in plane geometry. A good many ships have been saved from being wrecked by the principle involved in this proposition. [Illustration] If a dangerous shoal _A_ is near a headland _H_, the angle _HAX_ is measured and is put down upon the charts as the "vertical danger angle." Ships coming near the headland are careful to keep far enough away, say at _S_, so that the angle _HSX_ shall be less than this danger angle. They are then sure that they will avoid the dangerous shoal. Related to this proposition is the problem of supporting a tall iron smokestack by wire stays. Evidently three stays are needed, and they are preferably placed at the vertices of an equilateral triangle, the smokestack being in the center. The practical problem may be given of locating the vertices of the triangle and of finding the length of each stay. THEOREM. _Two straight lines perpendicular to the same plane are parallel._ Here again we may cut through the figure by the plane of the two parallels, and we get the figure of plane geometry relating to lines that are perpendicular to the same line. The proposition shows that the opposite corners of a room are parallel, and that therefore they lie in the same plane, or are _coplanar_, as is said in higher geometry. It is interesting to a class to have attention called to the corollary that if two straight lines are parallel to a third straight line, they are parallel to each other; and to have the question asked why it is necessary to prove this when the same thing was proved in plane geometry. In case the reason is not clear, let some student try to apply the proof used in plane geometry. THEOREM. _Two planes perpendicular to the same straight line are parallel._ Besides calling attention to the corresponding proposition of plane geometry, it is well now to speak of the fact that in propositions involving planes and lines we may often interchange these words. For example, using "line" for "straight line," for brevity, we have: One _line_ does not determine One _plane_ does not determine a _plane_. a _line_. Two intersecting _lines_ Two intersecting _planes_ determine determine a _plane_. a _line_. Two _lines_ perpendicular to Two _planes_ perpendicular to a _plane_ are parallel. a _line_ are parallel. If one of two parallel _lines_ If one of two parallel _planes_ is perpendicular to a _plane_, the is perpendicular to a _line_, the other is also perpendicular to other is also perpendicular to the _plane_. the _line_. If two _lines_ are parallel, every If two _planes_ are parallel, _plane_ containing one of the every _line_ in one of the _planes_ _lines_ is parallel to the other is parallel to the other _plane_. _line_. THEOREM. _The intersections of two parallel planes by a third plane are parallel lines._ Thus one of the edges of a box is parallel to the next succeeding edge if the opposite faces are parallel, and in sawing diagonally through an ordinary board (with rectangular cross section) the section is a parallelogram. THEOREM. _A straight line perpendicular to one of two parallel planes is perpendicular to the other also._ Notice (1) the corresponding proposition in plane geometry; (2) the proposition that results from interchanging "plane" and (straight) "line." THEOREM. _If two intersecting straight lines are each parallel to a plane, the plane of these lines is parallel to that plane._ Interchanging "plane" and (straight) "line," we have: If two intersecting _planes_ are each parallel to a _line_, the _line_ of (intersection of) these _planes_ is parallel to that _line_. Is this true? THEOREM. _If two angles not in the same plane have their sides respectively parallel and lying on the same side of the straight line joining their vertices, they are equal and their planes are parallel._ Questions like the following may be asked in connection with the proposition: What is the corresponding proposition in plane geometry? Why do we need another proof here? Try the plane-geometry proof here. THEOREM. _If two straight lines are cut by three parallel planes, their corresponding segments are proportional._ Here, again, it is desirable to ask for the corresponding proposition of plane geometry, and to ask why the proof of that proposition will not suffice for this one. The usual figure may be varied in an interesting manner by having the two lines meet on one of the planes, or outside the planes, or by having them parallel, in which cases the proof of the plane-geometry proposition holds here. This proposition is not of great importance from the practical standpoint, and it is omitted from some of the standard syllabi at present, although included in certain others. It is easy, however, to frame some interesting questions depending upon it for their answers, such as the following: In a gymnasium swimming tank the water is 4 feet deep and the ceiling is 8 feet above the surface of the water. A pole 15 feet long touches the ceiling and the bottom of the tank. Required to know what length of the pole is in the water. At this point in Book VI it is customary to introduce the dihedral angle. The word "dihedral" is from the Greek, _di-_ meaning "two," and _hedra_ meaning "seat." We have the root _hedra_ also in "trihedral" (three-seated), "polyhedral" (many-seated), and "cathedral" (a church having a bishop's seat). The word is also, but less properly, spelled without the _h_, "diedral," a spelling not favored by modern usage. It is not necessary to dwell at length upon the dihedral angle, except to show the analogy between it and the plane angle. A few illustrations, as of an open book, the wall and floor of a room, and a swinging door, serve to make the concept clear, while a plane at right angles to the edge shows the measuring plane angle. So manifest is this relationship between the dihedral angle and its measuring plane angle that some teachers omit the proposition that two dihedral angles have the same ratio as their plane angles. THEOREM. _If two planes are perpendicular to each other, a straight line drawn in one of them perpendicular to their intersection is perpendicular to the other._ This and the related propositions allow of numerous illustrations taken from the schoolroom, as of door edges being perpendicular to the floor. The pretended applications of these propositions are usually fictitious, and the propositions are of value chiefly for their own interest and because they are needed in subsequent proofs. THEOREM. _The locus of a point equidistant from the faces of a dihedral angle is the plane bisecting the angle._ By changing "plane" to "line," and by making other obvious changes to correspond, this reduces to the analogous proposition of plane geometry. The figure formed by the plane perpendicular to the edge is also the figure of that analogous proposition. This at once suggests that there are two planes in the locus, provided the planes of the dihedral angle are taken as indefinite in extent, and that these planes are perpendicular to each other. It may interest some of the pupils to draw this general figure, analogous to the one in plane geometry. THEOREM. _The projection of a straight line not perpendicular to a plane upon that plane is a straight line._ In higher mathematics it would simply be said that the projection is a straight line, the special case of the projection of a perpendicular being considered as a line-segment of zero length. There is no advantage, however, of bringing in zero and infinity in the course in elementary geometry. The legitimate reason for the modern use of these terms is seldom understood by beginners. This subject of projection (Latin _pro-_, "forth," and _jacere_, "to throw") is extensively used in modern mathematics and also in the elementary work of the draftsman, and it will be referred to a little later. At this time, however, it is well to call attention to the fact that the projection of a straight line on a plane is a straight line or a point; the projection of a curve may be a curve or it may be straight; the projection of a point is a point; and the projection of a plane (which is easily understood without defining it) may be a surface or it may be a straight line. An artisan represents a solid by drawing its projection upon two planes at right angles to each other, and a map maker (cartographer) represents the surface of the earth by projecting it upon a plane. A photograph of the class is merely the projection of the class upon a photographic plate (plane), and when we draw a figure in solid geometry, we merely project the solid upon the plane of the paper. There are other projections than those formed by lines that are perpendicular to the plane. The lines may be oblique to the plane, and this is the case with most projections. A photograph, for example, is not formed by lines perpendicular to a plane, for they all converge in the camera. If the lines of projection are all perpendicular to the plane, the projection is said to be orthographic, from the Greek _ortho-_ (straight) and _graphein_ (to draw). A good example of orthographic projection may be seen in the shadow cast by an object upon a piece of paper that is held perpendicular to the sun's rays. A good example of oblique projection is a shadow on the floor of the schoolroom. THEOREM. _Between two straight lines not in the same plane there can be one common perpendicular, and only one._ The usual corollary states that this perpendicular is the shortest line joining them. It is interesting to compare this with the case of two lines in the same plane. If they are parallel, there may be any number of common perpendiculars. If they intersect, there is still a common perpendicular, but this can hardly be said to be between them, except for its zero segment. There are many simple illustrations of this case. For example, what is the shortest line between any given edge of the ceiling and the various edges of the floor of the schoolroom? If two galleries in a mine are to be connected by an air shaft, how shall it be planned so as to save labor? Make a drawing of the plan. At this point the polyhedral angle is introduced. The word is from the Greek _polys_ (many) and _hedra_ (seat). Students have more difficulty in grasping the meaning of the size of a polyhedral angle than is the case with dihedral and plane angles. For this reason it is not good policy to dwell much upon this subject unless the question arises, since it is better understood when the relation of the polyhedral angle and the spherical polygon is met. Teachers will naturally see that just as we may measure the plane angle by taking the ratio of an arc to the whole circle, and of a dihedral angle by taking the ratio of that part of the cylindric surface that is cut out by the planes to the whole surface, so we may measure a polyhedral angle by taking the ratio of the spherical polygon to the whole spherical surface. It should also be observed that just as we may have cross polygons in a plane, so we may have spherical polygons that are similarly tangled, and that to these will correspond polyhedral angles that are also cross, their representation by drawings being too complicated for class use. The idea of symmetric solids may be illustrated by a pair of gloves, all their parts being mutually equal but arranged in opposite order. Our hands, feet, and ears afford other illustrations of symmetric solids. THEOREM. _The sum of the face angles of any convex polyhedral angle is less than four right angles._ There are several interesting points of discussion in connection with this proposition. For example, suppose the vertex _V_ to approach the plane that cuts the edges in _A_, _B_, _C_, _D_, ..., the edges continuing to pass through these as fixed points. The sum of the angles about _V_ approaches what limit? On the other hand, suppose _V_ recedes indefinitely; then the sum approaches what limit? Then what are the two limits of this sum? Suppose the polyhedral angle were concave, why would the proof not hold? FOOTNOTES: [87] These may be purchased through the Leipziger Lehrmittelanstalt, Leipzig, Germany, which will send catalogues to intending buyers. [88] An excellent set of stereoscopic views of the figures of solid geometry, prepared by E. M. Langley of Bedford, England, is published by Underwood & Underwood, New York. Such a set may properly have place in a school library or in a classroom in geometry, to be used when it seems advantageous. CHAPTER XX THE LEADING PROPOSITIONS OF BOOK VII Book VII relates to polyhedrons, cylinders, and cones. It opens with the necessary definitions relating to polyhedrons, the etymology of the terms often proving interesting and valuable when brought into the work incidentally by the teacher. "Polyhedron" is from the Greek _polys_ (many) and _hedra_ (seat). The Greek plural, _polyhedra_, is used in early English works, but "polyhedrons" is the form now more commonly seen in America. "Prism" is from the Greek _prisma_ (something sawed, like a piece of wood sawed from a beam). "Lateral" is from the Latin _latus_ (side). "Parallelepiped" is from the Greek _parallelos_ (parallel) and _epipedon_ (a plane surface), from _epi_ (on) and _pedon_ (ground). By analogy to "parallelogram" the word is often spelled "parallelopiped," but the best mathematical works now adopt the etymological spelling above given. "Truncate" is from the Latin _truncare_ (to cut off). A few of the leading propositions are now considered. THEOREM. _The lateral area of a prism is equal to the product of a lateral edge by the perimeter of the right section._ It should be noted that although some syllabi do not give the proposition that parallel sections are congruent, this is necessary for this proposition, because it shows that the right sections are all congruent and hence that any one of them may be taken. It is, of course, possible to construct a prism so oblique and so low that a right section, that is, a section cutting all the lateral edges at right angles, is impossible. In this case the lateral faces must be extended, thus forming what is called a _prismatic space_. This term may or may not be introduced, depending upon the nature of the class. This proposition is one of the most important in Book VII, because it is the basis of the mensuration of the cylinder as well as the prism. Practical applications are easily suggested in connection with beams, corridors, and prismatic columns, such as are often seen in school buildings. Most geometries supply sufficient material in this line, however. THEOREM. _An oblique prism is equivalent to a right prism whose base is equal to a right section of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism._ This is a fundamental theorem leading up to the mensuration of the prism. Attention should be called to the analogous proposition in plane geometry relating to the area of the parallelogram and rectangle, and to the fact that if we cut through the solid figure by a plane parallel to one of the lateral edges, the resulting figure will be that of the proposition mentioned. As in the preceding proposition, so in this case, there may be a question raised that will make it helpful to introduce the idea of prismatic space. THEOREM. _The opposite lateral faces of a parallelepiped are congruent and parallel._ It is desirable to refer to the corresponding case in plane geometry, and to note again that the figure is obtained by passing a plane through the parallelepiped parallel to a lateral edge. The same may be said for the proposition about the diagonal plane of a parallelepiped. These two propositions are fundamental in the mensuration of the prism. THEOREM. _Two rectangular parallelepipeds are to each other as the products of their three dimensions._ This leads at once to the corollary that the volume of a rectangular parallelepiped equals the product of its three dimensions, the fundamental law in the mensuration of all solids. It is preceded by the proposition asserting that rectangular parallelepipeds having congruent bases are proportional to their altitudes. This includes the incommensurable case, but this case may be omitted. The number of simple applications of this proposition is practically unlimited. In all such cases it is advisable to take a considerable number of numerical exercises in order to fix in mind the real nature of the proposition. Any good geometry furnishes a certain number of these exercises. The following is an interesting property of the rectangular parallelepiped, often called the rectangular solid: If the edges are _a_, _b_, and _c_, and the diagonal is _d_, then (_a_/_d_)^2 + (_b_/_d_)^2 + (_c_/_d_)^2 = 1. This property is easily proved by the Pythagorean Theorem, for _d_^2 = _a_^2 + _b_^2 + _c_^2, whence (_a_^2 + _b_^2 + _c_^2) / _d_^2 = 1. In case _c_ = 0, this reduces to the Pythagorean Theorem. The property is the fundamental one of solid analytic geometry. THEOREM. _The volume of any parallelepiped is equal to the product of its base by its altitude._ This is one of the few propositions in Book VII where a model is of any advantage. It is easy to make one out of pasteboard, or to cut one from wood. If a wooden one is made, it is advisable to take an oblique parallelepiped and, by properly sawing it, to transform it into a rectangular one instead of using three different solids. On account of its awkward form, this figure is sometimes called the Devil's Coffin, but it is a name that it would be well not to perpetuate. THEOREM. _The volume of any prism is equal to the product of its base by its altitude._ This is also one of the basal propositions of solid geometry, and it has many applications in practical mensuration. A first-class textbook will give a sufficient list of problems involving numerical measurement, to fix the law in mind. For outdoor work, involving measurements near the school or within the knowledge of the pupils, the following problem is a type: [Illustration] If this represents the cross section of a railway embankment that is _l_ feet long, _h_ feet high, _b_ feet wide at the bottom, and _b'_ feet wide at the top, find the number of cubic feet in the embankment. Find the volume if _l_ = 300, _h_ = 8, _b_ = 60, and _b'_ = 28. The mensuration of the volume of the prism, including the rectangular parallelepiped and cube, was known to the ancients. Euclid was not concerned with practical measurement, so that none of this part of geometry appears in his "Elements." We find, however, in the papyrus of Ahmes, directions for the measuring of bins, and the Egyptian builders, long before his time, must have known the mensuration of the rectangular parallelepiped. Among the Hindus, long before the Christian era, rules were known for the construction of altars, and among the Greeks the problem of constructing a cube with twice the volume of a given cube (the "duplication of the cube") was attacked by many mathematicians. The solution of this problem is impossible by elementary geometry. If _e_ equals the edge of the given cube, then _e_^3 is its volume and 2_e_^3 is the volume of the required cube. Therefore the edge of the required cube is _e_[3root]2. Now if _e_ is given, it is not possible with the straightedge and compasses to construct a line equal to _e_[3root]2, although it is easy to construct one equal to _e_[sqrt]2. The study of the pyramid begins at this point. In practical measurement we usually meet the regular pyramid. It is, however, a simple matter to consider the oblique pyramid as well, and in measuring volumes we sometimes find these forms. THEOREM. _The lateral area of a regular pyramid is equal to half the product of its slant height by the perimeter of its base._ This leads to the corollary concerning the lateral area of the frustum of a regular pyramid. It should be noticed that the regular pyramid may be considered as a frustum with the upper base zero, and the proposition as a special case under the corollary. It is also possible, if we choose, to let the upper base of the frustum pass through the vertex and cut the lateral edges above that point, although this is too complicated for most pupils. If this case is considered, it is well to bring in the general idea of _pyramidal space_, the infinite space bounded on several sides by the lateral faces, of the pyramid. This pyramidal space is double, extending on two sides of the vertex. THEOREM. _If a pyramid is cut by a plane parallel to the base:_ 1. _The edges and altitude are divided proportionally._ 2. _The section is a polygon similar to the base._ To get the analogous proposition of plane geometry, pass a plane through the vertex so as to cut the base. We shall then have the sides and altitude of the triangle divided proportionally, and of course the section will merely be a line-segment, and therefore it is similar to the base line. The cutting plane may pass through the vertex, or it may cut the pyramidal space above the vertex. In either case the proof is essentially the same. THEOREM. _The volume of a triangular pyramid is equal to one third of the product of its base by its altitude, and this is also true of any pyramid._ This is stated as two theorems in all textbooks, and properly so. It is explained to children who are studying arithmetic by means of a hollow pyramid and a hollow prism of equal base and equal altitude. The pyramid is filled with sand or grain, and the contents is poured into the prism. This is repeated, and again repeated, showing that the volume of the prism is three times the volume of the pyramid. It sometimes varies the work to show this to a class in geometry. This proposition was first proved, so Archimedes asserts, by Eudoxus of Cnidus, famous as an astronomer, geometer, physician, and lawgiver, born in humble circumstances about 407 B.C. He studied at Athens and in Egypt, and founded a famous school of geometry at Cyzicus. His discovery also extended to the volume of the cone, and it was his work that gave the beginning to the science of stereometry, the mensuration part of solid geometry. THEOREM. _The volume of the frustum of any pyramid is equal to the sum of the volumes of three pyramids whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and the mean proportional between the bases of the frustum._ Attention should be called to the fact that this formula _v_ = 1/3 _a_(_b_ + _b'_ + [sqrt](_bb'_)) applies to the pyramid by letting _b'_ = 0, to the prism by letting _b_ = _b'_, and also to the parallelepiped and cube, these being special forms of the prism. This formula is, therefore, a very general one, relating to all the polyhedrons that are commonly met in mensuration. THEOREM. _There cannot be more than five regular convex polyhedrons._ Eudemus of Rhodes, one of the principal pupils of Aristotle, in his history of geometry of which Proclus preserves some fragments, tells us that Pythagoras discovered the construction of the "mundane figures," meaning the five regular polyhedrons. Iamblichus speaks of the discovery of the dodecahedron in these words: As to Hippasus, who was a Pythagorean, they say that he perished in the sea on account of his impiety, inasmuch as he boasted that he first divulged the knowledge of the sphere with the twelve pentagons. Hippasus assumed the glory of the discovery to himself, whereas everything belongs to Him, for thus they designate Pythagoras, and do not call Him by name. Iamblichus here refers to the dodecahedron inscribed in the sphere. The Pythagoreans looked upon these five solids as fundamental forms in the structure of the universe. In particular Plato tells us that they asserted that the four elements of the real world were the tetrahedron, octahedron, icosahedron, and cube, and Plutarch ascribes this doctrine to Pythagoras himself. Philolaus, who lived in the fifth century B.C., held that the elementary nature of bodies depended on their form. The tetrahedron was assigned to fire, the octahedron to air, the icosahedron to water, and the cube to earth, it being asserted that the smallest constituent part of each of these substances had the form here assigned to it. Although Eudemus attributes all five to Pythagoras, it is certain that the tetrahedron, cube, and octahedron were known to the Egyptians, since they appear in their architectural decorations. These solids were studied so extensively in the school of Plato that Proclus also speaks of them as the Platonic bodies, saying that Euclid "proposed to himself the construction of the so-called Platonic bodies as the final aim of his arrangement of the 'Elements.'" Aristæus, probably a little older than Euclid, wrote a book upon these solids. As an interesting amplification of this proposition, the centers of the faces (squares) of a cube may be connected, an inscribed octahedron being thereby formed. Furthermore, if the vertices of the cube are _A_, _B_, _C_, _D_, _A'_, _B'_, _C'_, _D'_, then by drawing _AC_, _CD'_, _D'A_, _D'B'_, _B'A_, and _B'C_, a regular tetrahedron will be formed. Since the construction of the cube is a simple matter, this shows how three of the five regular solids may be constructed. The actual construction of the solids is not suited to elementary geometry.[89] It is not difficult for a class to find the relative areas of the cube and the inscribed tetrahedron and octahedron. If _s_ is the side of the cube, these areas are 6_s_^2, (1/2)_s_^2[sqrt]3, and _s_^2[sqrt]3; that is, the area of the octahedron is twice that of the tetrahedron inscribed in the cube. Somewhat related to the preceding paragraph is the fact that the edges of the five regular solids are incommensurable with the radius of the circumscribed sphere. This fact seems to have been known to the Greeks, perhaps to Theætetus (_ca._ 400 B.C.) and Aristæus (_ca._ 300 B.C.), both of whom wrote on incommensurables. Just as we may produce the sides of a regular polygon and form a regular cross polygon or stellar polygon, so we may have stellar polyhedrons. Kepler, the great astronomer, constructed some of these solids in 1619, and Poinsot, a French mathematician, carried the constructions so far in 1801 that several of these stellar polyhedrons are known as Poinsot solids. There is a very extensive literature upon this subject. The following table may be of some service in assigning problems in mensuration in connection with the regular polyhedrons, although some of the formulas are too difficult for beginners to prove. In the table _e_ = edge of the polyhedron, _r_ = radius of circumscribed sphere, _r'_ = radius of inscribed sphere, _a_ = total area, _v_ = volume. ========================================================== NUMBER | | | OF FACES| 4 | 6 | 8 --------+-----------------+--------------+---------------- _r_ | _e_[sqrt](3/8) |(_e_/2)[sqrt]3| _e_[sqrt](1/2) | | | _r'_ | _e_[sqrt](1/24) | _e_/2 | _e_[sqrt](1/6) | | | _a_ | _e_^2[sqrt]3 | 6_e_^2 | 2_e_^2[sqrt]3 | | | _v_ |(_e_^3/12)[sqrt]2| _e_^3 |(_e_^3/3)[sqrt]2 ---------------------------------------------------------- ======================================================================== NUMBER | | OF FACES| 12 | 20 --------+----------------------------------+---------------------------- _r_ |(_e_/4)[sqrt]3([sqrt]5 + 1) |_e_[sqrt]((5 + [sqrt]5)/8) | | _r'_ |(_e_/2)[sqrt]((25 + 11[sqrt]5)/10)|(_e_[sqrt]3)/12([sqrt]5 + 3) | | _a_ |3_e_^2[sqrt](5(5 + 2[sqrt]5)) | (5_e_^2)[sqrt]3 | | _v_ |((_e_^3)/4)(15 + 7[sqrt]5) |((5_e_^3)/12)([sqrt]5 + 3) ------------------------------------------------------------------------ Some interest is added to the study of polyhedrons by calling attention to their occurrence in nature, in the form of crystals. The computation of the surfaces and volumes of these forms offers an opportunity for applying the rules of mensuration, and the construction of the solids by paper folding or by the cutting of crayon or some other substance often arouses a considerable interest. The following are forms of crystals that are occasionally found: [Illustration] They show how the cube is modified by having its corners cut off. A cube may be inscribed in an octahedron, its vertices being at the centers of the faces of the octahedron. If we think of the cube as expanding, the faces of the octahedron will cut off the corners of the cube as seen in the first figure, leaving the cube as shown in the second figure. If the corners are cut off still more, we have the third figure. Similarly, an octahedron may be inscribed in a cube, and by letting it expand a little, the faces of the cube will cut off the corners of the octahedron. This is seen in the following figures: [Illustration] This is a form that is found in crystals, and the computation of the surface and volume is an interesting exercise. The quartz crystal, an hexagonal pyramid on an hexagonal prism, is found in many parts of the country, or is to be seen in the school museum, and this also forms an interesting object of study in this connection. The properties of the cylinder are next studied. The word is from the Greek _kylindros_, from _kyliein_ (to roll). In ancient mathematics circular cylinders were the only ones studied, but since some of the properties are as easily proved for the case of a noncircular directrix, it is not now customary to limit them in this way. It is convenient to begin by a study of the cylindric surface, and a piece of paper may be curved or rolled up to illustrate this concept. If the paper is brought around so that the edges meet, whatever curve may form a cross section the surface is said to inclose a _cylindric space_. This concept is sometimes convenient, but it need be introduced only as necessity for using it arises. The other definitions concerning the cylinder are so simple as to require no comment. The mensuration of the volume of a cylinder depends upon the assumption that the cylinder is the limit of a certain inscribed or circumscribed prism as the number of sides of the base is indefinitely increased. It is possible to give a fairly satisfactory and simple proof of this fact, but for pupils of the age of beginners in geometry in America it is better to make the assumption outright. This is one of several cases in geometry where a proof is less convincing than the assumed statement. THEOREM. _The lateral area of a circular cylinder is equal to the product of the perimeter of a right section of the cylinder by an element._ For practical purposes the cylinder of revolution (right circular cylinder) is the one most frequently used, and the important formula is therefore _l_ = 2[pi]_rh_ where _l_ = the lateral area, _r_ = the radius, and _h_ = the altitude. Applications of this formula are easily found. THEOREM. _The volume of a circular cylinder is equal to the product of its base by its altitude._ Here again the important case is that of the cylinder of revolution, where _v_ = [pi]_r_^2_h_. The number of applications of this proposition is, of course, very great. In architecture and in mechanics the cylinder is constantly seen, and the mensuration of the surface and the volume is important. A single illustration of this type of problem will suffice. A machinist is making a crank pin (a kind of bolt) for an engine, according to this drawing. He considers it as weighing the same as three steel cylinders having the diameters and lengths in inches as here shown, where 7 3/4" means 7 3/4 inches. He has this formula for the weight (_w_) of a steel cylinder where _d_ is the diameter and _l_ is the length: _w_ = 0.07[pi]_d_^2_l_. Taking [pi] = 3 1/7, find the weight of the pin. The most elaborate study of the cylinder, cone, and sphere (the "three round bodies") in the Greek literature is that of Archimedes of Syracuse (on the island of Sicily), who lived in the third century B.C. Archimedes tells us, however, that Eudoxus (born _ca._ 407 B.C.) discovered that any cone is one third of a cylinder of the same base and the same altitude. Tradition says that Archimedes requested that a sphere and a cylinder be carved upon his tomb, and that this was done. Cicero relates that he discovered the tomb by means of these symbols. The tomb now shown to visitors in ancient Syracuse as that of Archimedes cannot be his, for it bears no such figures, and is not "outside the gate of Agrigentum," as Cicero describes. The cone is now introduced. A conic surface is easily illustrated to a class by taking a piece of paper and rolling it up into a cornucopia, the space inclosed being a _conic space_, a term that is sometimes convenient. The generation of a conic surface may be shown by taking a blackboard pointer and swinging it around by its tip so that the other end moves in a curve. If we consider a straight line as the limit of a curve, then the pointer may swing in a plane, and so a plane is the limit of a conic surface. If we swing the pointer about a point in the middle, we shall generate the two nappes of the cone, the conic space now being double. In practice the right circular cone, or cone of revolution, is the important type, and special attention should be given to this form. THEOREM. _Every section of a cone made by a plane passing through its vertex is a triangle._ At this time, or in speaking of the preliminary definitions, reference should be made to the conic sections. Of these there are three great types: (1) the ellipse, where the cutting plane intersects all the elements on one side of the vertex; a circle is a special form of the ellipse; (2) the parabola, where the plane is parallel to an element; (3) the hyperbola, where the plane cuts some of the elements on one side of the vertex, and the rest on the other side; that is, where it cuts both nappes. It is to be observed that the ellipse may vary greatly in shape, from a circle to a very long ellipse, as the cutting plane changes from being perpendicular to the axis to being nearly parallel to an element. The instant it becomes parallel to an element the ellipse changes suddenly to a parabola. If the plane tips the slightest amount more, the section becomes an hyperbola. While these conic sections are not studied in elementary geometry, the terms should be known for general information, particularly the ellipse and parabola. The study of the conic sections forms a large part of the work of analytic geometry, a subject in which the figures resemble the graphic work in algebra, this having been taken from "analytics," as the higher subject is commonly called. The planets move about the sun in elliptic orbits, and Halley's comet that returned to view in 1909-1910 has for its path an enormous ellipse. Most comets seem to move in parabolas, and a body thrown into the air would take a parabolic path if it were not for the resistance of the atmosphere. Two of the sides of the triangle in this proposition constitute a special form of the hyperbola. The study of conic sections was brought to a high state by the Greeks. They were not known to the Pythagoreans, but were discovered by Menæchmus in the fourth century B.C. This discovery is mentioned by Proclus, who says, "Further, as to these sections, the conics were conceived by Menæchmus." Since if the cutting plane is perpendicular to the axis the section is a circle, and if oblique it is an ellipse, a parabola, or an hyperbola, it follows that if light proceeds from a point, the shadow of a circle is a circle, an ellipse, a parabola, or an hyperbola, depending on the position of the plane on which the shadow falls. It is interesting and instructive to a class to see these shadows, but of course not much time can be allowed for such work. At this point the chief thing is to have the names "ellipse" and "parabola," so often met in reading, understood. It is also of interest to pupils to see at this time the method of drawing an ellipse by means of a pencil stretching a string band that moves about two pins fastened in the paper. This is a practical method, and is familiar to all teachers who have studied analytic geometry. In designing elliptic arches, however, three circular arcs are often joined, as here shown, the result being approximately an elliptic arc. [Illustration] Here _O_ is the center of arc _BC_, _M_ of arc _AB_, and _N_ of arc _CD_. Since _XY_ is perpendicular to _BM_ and _BO_, it is tangent to arcs _AB_ and _BC_, so there is no abrupt turning at _B_, and similarly for _C_.[90] THEOREM. _The volume of a circular cone is equal to one third the product of its base by its altitude._ It is easy to prove this for noncircular cones as well, but since they are not met commonly in practice, they may be omitted in elementary geometry. The important formula at this time is _v_ = 1/3[pi]_r_^2_h_. As already stated, this proposition was discovered by Eudoxus of Cnidus (born _ca._ 407 B.C., died _ca._ 354 B.C.), a man who, as already stated, was born poor, but who became one of the most illustrious and most highly esteemed of all the Greeks of his time. THEOREM. _The lateral area of a frustum of a cone of revolution is equal to half the sum of the circumferences of its bases multiplied by the slant height._ An interesting case for a class to notice is that in which the upper base becomes zero and the frustum becomes a cone, the proposition being still true. If the upper base is equal to the lower base, the frustum becomes a cylinder, and still the proposition remains true. The proposition thus offers an excellent illustration of the elementary Principle of Continuity. Then follows, in most textbooks, a theorem relating to the volume of a frustum. In the case of a cone of revolution _v_ = (1/3)[pi]_h_(_r_^2 + _r'_^2 + _rr'_). Here if _r'_ = 0, we have _v_ = (1/3)[pi]_r_^2_h_, the volume of a cone. If _r'_ = _r_, we have _v_ = (1/3)[pi]_h_(_r_^2 + _r_^2 + _r_^2) = [pi]_hr_^2, the volume of a cylinder. If one needs examples in mensuration beyond those given in a first-class textbook, they are easily found. The monument to Sir Christopher Wren, the professor of geometry in Cambridge University, who became the great architect of St. Paul's Cathedral in London, has a Latin inscription which means, "Reader, if you would see his monument, look about you." So it is with practical examples in Book VII. Appended to this Book, or more often to the course in solid geometry, is frequently found a proposition known as Euler's Theorem. This is often considered too difficult for the average pupil and is therefore omitted. On account of its importance, however, in the theory of polyhedrons, some reference to it at this time may be helpful to the teacher. The theorem asserts that in any convex polyhedron the number of edges increased by two is equal to the number of vertices increased by the number of faces. In other words, that _e_ + 2 = _v_ + _f_. On account of its importance a proof will be given that differs from the one ordinarily found in textbooks. Let _s__{1}, _s__{2}, ···, _s__{_n_} be the number of sides of the various faces, and _f_ the number of faces. Now since the sum of the angles of a polygon of _s_ sides is (_s_ - 2)180°, therefore the sum of the angles of all the faces is (_s__{1} + _s__{2} + _s__{3} + ··· + _s__{_n_} - 2_f_)180°. But _s__{1} + _s__{2} + _s__{3} + ··· + _s__{_n_} is twice the number of edges, because each edge belongs to two faces. [therefore] the sum of the angles of all the faces is (2_e_ - 2_f_)180°, or (_e_ - _f_)360°. Since the polyhedron is convex, it is possible to find some outside point of view, _P_, from which some face, as _ABCDE_, covers up the whole figure, as in this illustration. If we think of all the vertices projected on _ABCDE_, by lines through _P_, the sum of the angles of all the faces will be the same as the sum of the angles of all their projections on _ABCDE_. Calling _ABCDE_ _s__{1}, and thinking of the projections as traced by dotted lines on the opposite side of _s__{1}, this sum is evidently equal to (1) the sum of the angles in _s__{1}, or (_s__{1} - 2) 180°, plus (2) the sum of the angles on the other side of _s__{1}, or (_s__{1} - 2)180°, plus (3) the sum of the angles about the various points shown as inside of _s__{1}, of which there are _v_ - _s__{1} points, about each of which the sum of the angles is 360°, making (_v_ - _s__{1})360° in all. [Illustration] Adding, we have (_s__{1} - 2)180° + (_s__{1} - 2)180° + (_v_ - _s__{1})360° = [(_s__{1} - 2) + (_v_ - _s__{1})]360° = (_v_ - 2)360°. Equating the two sums already found, we have (_e_ - _f_)360° = (_v_ - 2)360°, or _e_ - _f_ = _v_ - 2, or _e_ + 2 = _v_ + _f_. This proof is too abstract for most pupils in the high school, but it is more scientific than those found in any of the elementary textbooks, and teachers will find it of service in relieving their own minds of any question as to the legitimacy of the theorem. Although this proposition is generally attributed to Euler, and was, indeed, rediscovered by him and published in 1752, it was known to the great French geometer Descartes, a fact that Leibnitz mentions.[91] This theorem has a very practical application in the study of crystals, since it offers a convenient check on the count of faces, edges, and vertices. Some use of crystals, or even of polyhedrons cut from a piece of crayon, is desirable when studying Euler's proposition. The following illustrations of common forms of crystals may be used in this connection: [Illustration] The first represents two truncated pyramids placed base to base. Here _e_ = 20, _f_ = 10, _v_ = 12, so that _e_ + 2 = _f_ + _v_. The second represents a crystal formed by replacing each edge of a cube by a plane, with the result that _e_ = 40, _f_ = 18, and _v_ = 24. The third represents a crystal formed by replacing each edge of an octahedron by a plane, it being easy to see that Euler's law still holds true. FOOTNOTES: [89] The actual construction of these solids is given by Pappus. See his "Mathematicae Collectiones," p. 48, Bologna, 1660. [90] The illustration is from Dupin, loc. cit. [91] For the historical bibliography consult G. Holzmüller, _Elemente der Stereometrie_, Vol. I, p. 181, Leipzig, 1900. CHAPTER XXI THE LEADING PROPOSITIONS OF BOOK VIII Book VIII treats of the sphere. Just as the circle may be defined either as a plane surface or as the bounding line which is the locus of a point in a plane at a given distance from a fixed point, so a sphere may be defined either as a solid or as the bounding surface which is the locus of a point in space at a given distance from a fixed point. In higher mathematics the circle is defined as the bounding line and the sphere as the bounding surface; that is, each is defined as a locus. This view of the circle as a line is becoming quite general in elementary geometry, it being the desire that students may not have to change definitions in passing from elementary to higher mathematics. The sphere is less frequently looked upon in geometry as a surface, and in popular usage it is always taken as a solid. Analogous to the postulate that a circle may be described with any given point as a center and any given line as a radius, is the postulate for constructing a sphere with any given center and any given radius. This postulate is not so essential, however, as the one about the circle, because we are not so concerned with constructions here as we are in plane geometry. A good opportunity is offered for illustrating several of the definitions connected with the study of the sphere, such as great circle, axis, small circle, and pole, by referring to geography. Indeed, the first three propositions usually given in Book VIII have a direct bearing upon the study of the earth. THEOREM. _A plane perpendicular to a radius at its extremity is tangent to the sphere._ The student should always have his attention called to the analogue in plane geometry, where there is one. If here we pass a plane through the radius in question, the figure formed on the plane will be that of a line tangent to a circle. If we revolve this about the line of the radius in question, as an axis, the circle will generate the sphere again, and the tangent line will generate the tangent plane. THEOREM. _A sphere may be inscribed in any given tetrahedron._ Here again we may form a corresponding proposition of plane geometry by passing a plane through any three points of contact of the sphere and the tetrahedron. We shall then form the figure of a circle inscribed in a triangle. And just as in the case of the triangle we may have escribed circles by producing the sides, so in the case of the tetrahedron we may have escribed spheres by producing the planes indefinitely and proceeding in the same way as for the inscribed sphere. The figure is difficult to draw, but it is not difficult to understand, particularly if we construct the tetrahedron out of pasteboard. THEOREM. _A sphere may be circumscribed about any given tetrahedron._ By producing one of the faces indefinitely it will cut the sphere in a circle, and the resulting figure, on the plane, will be that of the analogous proposition of plane geometry, the circle circumscribed about a triangle. It is easily proved from the proposition that the four perpendiculars erected at the centers of the faces of a tetrahedron meet in a point (are concurrent), the analogue of the proposition about the perpendicular bisectors of the sides of a triangle. THEOREM. _The intersection of two spherical surfaces is a circle whose plane is perpendicular to the line joining the centers of the surfaces and whose center is in that line._ The figure suggests the case of two circles in plane geometry. In the case of two circles that do not intersect or touch, one not being within the other, there are four common tangents. If the circles touch, two close up into one. If one circle is wholly within the other, this last tangent disappears. The same thing exists in relation to two spheres, and the analogous cases are formed by revolving the circles and tangents about the line through their centers. In plane geometry it is easily proved that if two circles intersect, the tangents from any point on their common chord produced are equal. For if the common chord is _AB_ and the point _P_ is taken on _AB_ produced, then the square on any tangent from _P_ is equal to _PB_ × _PA_. The line _PBA_ is sometimes called the _radical axis_. Similarly in this proposition concerning spheres, if from any point in the plane of the circle formed by the intersection of the two spherical surfaces lines are drawn tangent to either sphere, these tangents are equal. For it is easily proved that all tangents to the same sphere from an external point are equal, and it can be proved as in plane geometry that two tangents to the two spheres are equal. Among the interesting analogies between plane and solid geometry is the one relating to the four common tangents to two circles. If the figure be revolved about the line of centers, the circles generate spheres and the tangents generate conical surfaces. To study this case for various sizes and positions of the two spheres is one of the most interesting generalizations of solid geometry. An application of the proposition is seen in the case of an eclipse, where the sphere _O'_ represents the moon, _O_ the earth, and _S_ the sun. It is also seen in the case of the full moon, when _S_ is on the other side of the earth. In this case the part _MIN_ is fully illuminated by the moon, but the zone _ABNM_ is only partly illuminated, as the figure shows.[92] [Illustration] THEOREM. _The sum of the sides of a spherical polygon is less than 360°._ In all such cases the relation to the polyhedral angle should be made clear. This is done in the proofs usually given in the textbooks. It is easily seen that this is true only with the limitation set forth in most textbooks, that the spherical polygons considered are convex. Thus we might have a spherical triangle that is concave, with its base 359°, and its other two sides each 90°, the sum of the sides being 539°. THEOREM. _The sum of the angles of a spherical triangle is greater than 180° and less than 540°._ It is for the purpose of proving this important fact that polar triangles are introduced. This proposition shows the relation of the spherical to the plane triangle. If our planes were in reality slightly curved, being small portions of enormous spherical surfaces, then the sum of the angles of a triangle would not be exactly 180°, but would exceed 180° by some amount depending on the curvature of the surface. Just as a being may be imagined as having only two dimensions, and living always on a plane surface (in a space of two dimensions), and having no conception of a space of three dimensions, so we may think of ourselves as living in a space of three dimensions but surrounded by a space of four dimensions. The flat being could not point to a third dimension because he could not get out of his plane, and we cannot point to the fourth dimension because we cannot get out of our space. Now what the flat being thinks is his plane may be the surface of an enormous sphere in our three dimensions; in other words, the space he lives in may curve through some higher space without his being conscious of it. So our space may also curve through some higher space without our being conscious of it. If our planes have really some curvature, then the sum of the angles of our triangles has a slight excess over 180°. All this is mere speculation, but it may interest some student to know that the idea of fourth and higher dimensions enters largely into mathematical investigation to-day. THEOREM. _Two symmetric spherical triangles are equivalent._ While it is not a subject that has any place in a school, save perhaps for incidental conversation with some group of enthusiastic students, it may interest the teacher to consider this proposition in connection with the fourth dimension just mentioned. Consider these triangles, where [L]_A_ = [L]_A'_, _AB_ = _A'B'_, _AC_ = _A'C'_. We prove them congruent by superposition, turning one over and placing it upon the other. But suppose we were beings in Flatland, beings with only two dimensions and without the power to point in any direction except in the plane we lived in. We should then be unable to turn [triangle]_A'B'C'_ over so that it could coincide with [triangle]_ABC_, and we should have to prove these triangles equivalent in some other way, probably by dividing them into isosceles triangles that could be superposed. [Illustration] [Illustration] Now it is the same thing with symmetric spherical triangles; we cannot superpose them. But might it not be possible to do so if we could turn them through the fourth dimension exactly as we turn the Flatlander's triangle through our third dimension? It is interesting to think about this possibility even though we carry it no further, and in these side lights on mathematics lies much of the fascination of the subject. THEOREM. _The shortest line that can be drawn on the surface of a sphere between two points is the minor arc of a great circle joining the two points._ It is always interesting to a class to apply this practically. By taking a terrestrial globe and drawing a great circle between the southern point of Ireland and New York City, we represent the shortest route for ships crossing to England. Now if we notice where this great-circle arc cuts the various meridians and mark this on an ordinary Mercator's projection map, such as is found in any schoolroom, we shall find that the path of the ship does not make a straight line. Passengers at sea often do not understand why the ship's course on the map is not a straight line; but the chief reason is that the ship is taking a great-circle arc, and this is not, in general, a straight line on a Mercator projection. The small circles of latitude are straight lines, and so are the meridians and the equator, but other great circles are represented by curved lines. THEOREM. _The area of the surface of a sphere is equal to the product of its diameter by the circumference of a great circle._ This leads to the remarkable formula, _a_ = 4[pi]_r_^2. That the area of the sphere, a curved surface, should exactly equal the sum of the areas of four great circles, plane surfaces, is the remarkable feature. This was one of the greatest discoveries of Archimedes (_ca._ 287-212 B.C.), who gives it as the thirty-fifth proposition of his treatise on the "Sphere and the Cylinder," and who mentions it specially in a letter to his friend Dositheus, a mathematician of some prominence. Archimedes also states that the surface of a sphere is two thirds that of the circumscribed cylinder, or the same as the curved surface of this cylinder. This is evident, since the cylindric surface of the cylinder is 2[pi]_r_ × 2_r_, or 4[pi]_r_^2, and the two bases have an area [pi]_r_^2 + [pi]_r_^2, making the total area 6[pi]_r_^2. THEOREM. _The area of a spherical triangle is equal to the area of a lune whose angle is half the triangle's spherical excess._ This theorem, so important in finding areas on the earth's surface, should be followed by a considerable amount of computation of triangular areas, else it will be rather meaningless. Students tend to memorize a proof of this character, and in order to have the proposition mean what it should to them, they should at once apply it. The same is true of the following proposition on the area of a spherical polygon. It is probable that neither of these propositions is very old; at any rate, they do not seem to have been known to the writers on elementary mathematics among the Greeks. THEOREM. _The volume of a sphere is equal to the product of the area of its surface by one third of its radius._ This gives the formula _v_ = (4/3)[pi]_r_^3. This is one of the greatest discoveries of Archimedes. He also found as a result that the volume of a sphere is two thirds the volume of the circumscribed cylinder. This is easily seen, since the volume of the cylinder is [pi]_r_^2 × 2_r_, or 2[pi]_r_^3, and (4/3)[pi]_r_^3 is 2/3 of 2[pi]_r_^3. It was because of these discoveries on the sphere and cylinder that Archimedes wished these figures engraved upon his tomb, as has already been stated. The Roman general Marcellus conquered Syracuse in 212 B.C., and at the sack of the city Archimedes was killed by an ignorant soldier. Marcellus carried out the wishes of Archimedes with respect to the figures on his tomb. The volume of a sphere can also be very elegantly found by means of a proposition known as Cavalieri's Theorem. This asserts that if two solids lie between parallel planes, and are such that the two sections made by any plane parallel to the given planes are equal in area, the solids are themselves equal in volume. Thus, if these solids have the same altitude, _a_, and if _S_ and _S'_ are equal sections made by a plane parallel to _MN_, then the solids have the same volume. The proof is simple, since prisms of the same altitude, say _a_/_n_, and on the bases _S_ and _S'_ are equivalent, and the sums of _n_ such prisms are the given solids; and as _n_ increases, the sums of the prisms approach the solids as their limits; hence the volumes are equal. [Illustration] This proposition, which will now be applied to finding the volume of the sphere, was discovered by Bonaventura Cavalieri (1591 or 1598-1647). He was a Jesuit professor in the University of Bologna, and his best known work is his "Geometria Indivisilibus," which he wrote in 1626, at least in part, and published in 1635 (second edition, 1647). By means of the proposition it is also possible to prove several other theorems, as that the volumes of triangular pyramids of equivalent bases and equal altitudes are equal. [Illustration] To find the volume of a sphere, take the quadrant _OPQ_, in the square _OPRQ_. Then if this figure is revolved about _OP_, _OPQ_ will generate a hemisphere, _OPR_ will generate a cone of volume (1/3)[pi]_r_^3, and _OPRQ_ will generate a cylinder of volume [pi]_r_^3. Hence the figure generated by _ORQ_ will have a volume [pi]_r_^3 - (1/3)[pi]_r_^3, or (2/3)[pi]_r_^3, which we will call _x_. Now _OA_ = _AB_, and _OC_ = _AD_; also (_OC_)^2 - (_OA_)^2 = (_AC_)^2, so that (_AD_)^2 - (_AB_)^2 = (_AC_)^2, and [pi](_AD_)^2 - [pi](_AB_)^2 = [pi](_AC_)^2. But [pi](_AD_)^2 - [pi](_AB_)^2 is the area of the ring generated by _BD_, a section of _x_, and [pi](_AC_)^2 is the corresponding section of the hemisphere. Hence, by Cavalieri's Theorem, (2/3)[pi]_r_^3 = the volume of the hemisphere. [therefore] (4/3)[pi]_r_^3 = the volume of the sphere. In connection with the sphere some easy work in quadratics may be introduced even if the class has had only a year in algebra. For example, suppose a cube is inscribed in a hemisphere of radius _r_ and we wish to find its edge, and thereby its surface and its volume. If _x_ = the edge of the cube, the diagonal of the base must be _x_[sqrt]2, and the projection of _r_ (drawn from the center of the base to one of the vertices) on the base is half of this diagonal, or (_x_[sqrt]2)/2. Hence, by the Pythagorean Theorem, _r_^2 = _x_^2 + ((_x_[sqrt]2)/2)^2 = (3/2)_x_^2 [therefore] _x_ = _r_[sqrt](2/3), and the total surface is 6_x_^2 = 4_r_^2, and the volume is _x_^3 = (2/3)_r_^3[sqrt](2/3). FOOTNOTES: [92] The illustration is from Dupin, loc. cit. L'ENVOI In the Valley of Youth, through which all wayfarers must pass on their journey from the Land of Mystery to the Land of the Infinite, there is a village where the pilgrim rests and indulges in various excursions for which the valley is celebrated. There also gather many guides in this spot, some of whom show the stranger all the various points of common interest, and others of whom take visitors to special points from which the views are of peculiar significance. As time has gone on new paths have opened, and new resting places have been made from which these views are best obtained. Some of the mountain peaks have been neglected in the past, but of late they too have been scaled, and paths have been hewn out that approach the summits, and many pilgrims ascend them and find that the result is abundantly worth the effort and the time. The effect of these several improvements has been a natural and usually friendly rivalry in the body of guides that show the way. The mountains have not changed, and the views are what they have always been. But there are not wanting those who say, "My mountain may not be as lofty as yours, but it is easier to ascend"; or "There are quarries on my peak, and points of view from which a building may be seen in process of erection, or a mill in operation, or a canal, while your mountain shows only a stretch of hills and valleys, and thus you will see that mine is the more profitable to visit." Then there are guides who are themselves often weak of limb, and who are attached to numerous sand dunes, and these say to the weaker pilgrims, "Why tire yourselves climbing a rocky mountain when here are peaks whose summits you can reach with ease and from which the view is just as good as that from the most famous precipice?" The result is not wholly disadvantageous, for many who pass through the valley are able to approach the summits of the sand dunes only, and would make progress with greatest difficulty should they attempt to scale a real mountain, although even for them it would be better to climb a little way where it is really worth the effort instead of spending all their efforts on the dunes. Then, too, there have of late come guides who have shown much ingenuity by digging tunnels into some of the greatest mountains. These they have paved with smooth concrete, and have arranged for rubber-tired cars that run without jar to the heart of some mountain. Arrived there the pilgrim has a glance, as the car swiftly turns in a blaze of electric light, at a roughly painted panorama of the view from the summit, and he is assured by the guide that he has accomplished all that he would have done, had he laboriously climbed the peak itself. In the midst of all the advocacy of sand-dune climbing, and of rubber-tired cars to see a painted view, the great body of guides still climb their mountains with their little groups of followers, and the vigor of the ascent and the magnificence of the view still attract all who are strong and earnest, during their sojourn in the Valley of Youth. Among the mountains that have for ages attracted the pilgrims is Mons Latinus, usually called in the valley by the more pleasing name Latina. Mathematica, and Rhetorica, and Grammatica are also among the best known. A group known as Montes Naturales comprises Physica, Biologica, and Chemica, and one great peak with minor peaks about it is called by the people Philosophia. There are those who claim that these great masses of rock are too old to be climbed, as if that affected the view; while others claim that the ascent is too difficult and that all who do not favor the sand dunes are reactionary. But this affects only a few who belong to the real mountains, and the others labor diligently to improve the paths and to lessen unnecessary toil, but they seek not to tear off the summits nor do they attend to the amusing attempts of those who sit by the hillocks and throw pebbles at the rocky sides of the mountains upon which they work. * * * * * Geometry is a mountain. Vigor is needed for its ascent. The views all along the paths are magnificent. The effort of climbing is stimulating. A guide who points out the beauties, the grandeur, and the special places of interest commands the admiration of his group of pilgrims. One who fails to do this, who does not know the paths, who puts unnecessary burdens upon the pilgrim, or who blindfolds him in his progress, is unworthy of his position. The pretended guide who says that the painted panorama, seen from the rubber-tired car, is as good as the view from the summit is simply a fakir and is generally recognized as such. The mountain will stand; it will not be used as a mere commercial quarry for building stone; it will not be affected by pellets thrown from the little hillocks about; but its paths will be freed from unnecessary flints, they will be straightened where this can advantageously be done, and new paths on entirely novel plans will be made as time goes on, but these paths will be hewed out of rock, not made out of the dreams of a day. Every worthy guide will assist in all these efforts at betterment, and will urge the pilgrim at least to ascend a little way because of the fact that the same view cannot be obtained from other peaks; but he will not take seriously the efforts of the fakir, nor will he listen with more than passing interest to him who proclaims the sand heap to be a Matterhorn. INDEX Ahmes, 27, 254, 278, 306 Alexandroff, 164 Algebra, 37, 84 Al-Khowarazmi, 37 Allman, G. J., 29 Almagest, 35 Al-Nair[=i]z[=i], 171, 193, 214, 264 Al-Qif[t.][=i], 49 Analysis, 41, 161 Angle, 142, 155; trisection of, 31, 215 Anthonisz, Adriaen, 279 Antiphon, 31, 32, 276 Apollodotus (Apollodorus), 259 Apollonius, 34, 214, 231 Applied problems, 75, 103, 178, 186, 192, 195, 203, 204, 209, 215, 217, 242, 267, 295, 317 Appreciation of geometry, 19 Arab geometry, 37, 51 Archimedes, 34, 42, 48, 139, 141, 215, 276, 278, 314, 327, 328 Aristæus, 310 Aristotle, 33, 42, 134, 135, 137, 145, 154, 177, 209 Aryabhatta, 36, 279 Associations, syllabi of, 58, 60, 64 Assumptions, 116 Astrolabe, 172 Athelhard of Bath, 37, 51 Athenæus, 259 Axioms, 31, 41, 116 Babylon, 26, 272 Bartoli, 10, 44, 238 Belli, 10, 44, 172 Beltinus, 239, 241 Beltrami, 127 Bennett, J., 224 Bernoulli, 280 Bertrand, 62 Betz, 131 Bezout, 62 Bhaskara, 232, 268 Billings, R. W., 222 Billingsley, 52 Bion, 192, 239 Boethius, 43, 50 Bolyai, 128 Bonola, 128 Books of geometry, 165, 167, 201, 227, 252, 269, 289, 303, 321 Bordas-Demoulin, 24 Borel, 11, 67, 196 Bosanquet, 272 Bossut, 23 Bourdon, 62 Bourlet, 67, 165, 196 Brahmagupta, 36, 268, 279 Bretschneider, C. A., 30 Brouncker, 280 Bruce, W. N., 199 Bryson, 31, 32, 276 Cajori, 46 Calandri, 30 Campanus, 37, 51, 135 Cantor, M., 29, 46 Capella, 50, 135 Capra, 44 Carson, G. W. L., 18, 96, 114 Casey, J., 38 Cassiodorius, 50 Cataneo, 10, 44 Cavalieri, 136, 329 Chinese values of [pi], 279 Church schools, 43 Cicero, 34, 50, 259, 314 Circle, 145, 201, 270, 287; squaring the, 31, 32, 277 Circumference, 145 Cissoid, 34 Class in geometry, 108 Clavius, 121 Colleges, geometry in the, 46 Collet, 24 Commensurable magnitudes, 206, 207 Conchoid, 34 Condorcet, 23 Cone, 315 Congruent, 151 Conic sections, 33, 315 Continuity, 212 Converse proposition, 175, 190, 191 Crelle, 142 Cube, duplicating the, 32, 307 Cylinder, 313 D'Alembert, 24, 67 Dase, 279 Decagon, 273 Definitions, 41, 132 De Judaeis, 239, 241 De Morgan, A., 58 De Paolis, 67 Descartes, 38, 84, 320 Diameter, 146 Dihedral, 298 Diocles, 34 Diogenes Laertius, 259 Diorismus, 41 Direction, 150 Distance, 154 Doyle, Conan, 8 Drawing, 95, 221, 281 Duality, 173 Duhamel, 164 Dupin, 11, 217 Duplication problem, 32, 307 Dürer, 10 Educational problems, 1 Egypt, 26, 40 Eisenlohr, 27 Engel and Stäckel, 128 England, 14, 46, 58, 60 Epicureans, 188 Equal, 151, 153 Equilateral, 147 Equivalent, 151 Eratosthenes, 48 Euclid, 33, 42, 43, 44, 119, 125, 135, 156, 165, 167 ff., 201 ff., et passim; editions of, 47, 52; efforts at improving, 57; life of, 47; nature of his "Elements," 52, 55; opinions of, 8 Eudemus, 33, 168, 171, 185, 216, 309 Eudoxus, 32, 41, 48, 227, 308, 314, 317 Euler, 38, 280, 318 Eutocius, 184 Exercises, nature of, 74, 103; how to attack, 160 Exhaustions, method of, 31 Extreme and mean ratio, 250 Figures in geometry, 104, 107, 113 Finaeus, 44, 239, 240, 243 Fourier, 142 Fourth dimension, 326 Frankland, 56, 117, 127, 135, 159 Fusion, of algebra and geometry, 84; of geometry and trigonometry, 91 Gargioli, 44 Gauss, 140, 274 Geminus, 126, 128, 149 Geometry, books of, 165, 167, 201, 227, 252, 269, 289, 303, 321; compared with other subjects, 14; introduction to, 93; modern, 38; of motion, 68, 196; reasons for teaching, 7, 15, 20; related to algebra, 84; textbooks in, 70 Gerbert, 43 Gherard of Cremona, 37, 51 Gnomon, 212 Golden section, 250 Gothic windows, 75, 221 ff., 274, 282 Gow, J., 29, 56 Greece, 28, 40 Gregoire de St. Vincent, 267 Gregory, 280 Grévy, 67 Gymnasia, geometry in the, 45 Hadamard, 164 Hamilton, W., 14 Harmonic division, 231 Harpedonaptae, 28 Harriot, 37 Harvard syllabus, 63 Heath, T. L., 49, 56, 119, 126, 127, 135, 149, 159, 170, 175, 228, 261 Hebrews, 26 Henrici, O., 11, 14, 25, 164, 196 Henrici and Treutlein, 68, 164, 196 Hermite, 281 Herodotus, 28 Heron, 35, 137, 139, 141, 209, 259, 267 Hexagon, regular, 272 High schools, geometry in the, 45 Hilbert, 119, 131 Hipparchus, 35 Hippasus, 273, 309 Hippias, 31, 215 Hippocrates, 31, 41, 281 History of geometry, 26 Hobson, 166 Hoffmann, 242 Holzmüller, 320 Hughes, Justice, 9 Hypatia, 36 Hypsicles, 34 Hypsometer, 245 Iamblichus, 273, 309 Illusions, optical, 100 Ingrami, 128 Instruments, 96, 178, 236 Introduction to geometry, 93 Ionic school, 28 Jackson, C. S., 12 Jones, W., 271 Junge, 259 Karagiannides, 128 Karpinski, 37 Kaye, G. B., 232 Kepler, 24, 149 Kingsley, C., 36 Klein, F., 68, 89 Kolb, 222 Lacroix, 24, 46, 62, 66 Langley, E. M., 291 Laplace, 101 Legendre, 10, 45, 62, 127, 128, 152 Leibnitz, 140, 150 Leon, 41 Leonardo da Vinci, 264 Leonardo of Pisa, 37, 43, 279 Lettering figures, 105 Limits, 207 Lindemann, 278, 281 Line defined, 137 Lobachevsky, 128 Loci, 163, 198 Locke, W. J., 13 Lodge, A., 14 Logic, 17, 104 Loomis, 164 Ludolph van Ceulen, 279 Lycées, geometry in, 45 M'Clelland, 38 McCormack, T. J., 11 Measured by, 208 Memorizing, 12, 79 Menæchmus, 33, 316 Menelaus, 35 Méray, 67, 68, 196, 289 Methods, 41, 42, 115 Metius, 279 Mikami, 264 Minchin, 14 Models, 93, 290 Modern geometry, 38 Mohammed ibn Musa, 37 Moore, E. H., 131 Mosaics, 274 Müller, H., 68 Münsterberg, 22 Napoleon, 24, 287 Newton, 24 Nicomedes, 34, 215 Octant, 242 Oenopides, 31, 212, 216 Optical illusions, 100 Oughtred, 37 Paciuolo, 86 Pamphilius, 185 Pappus, 36, 230, 263 Parallelepiped, 303 Parallels, 149, 181 Parquetry, 222, 274 Pascal, 24, 159 Peletier, 169 Perigon, 151 Perry, J., 13, 14 Petersen, 164 Philippus of Mende, 32, 185 Philo, 178 Philolaus, 309 [pi], 26, 27, 34, 36, 271, 278, 280 Plane, 140 Plato, 25, 31, 41, 48, 129, 136, 137, 309, 310 Playfair, 128 Pleasure of geometry, 16 Plimpton, G. A., 51, 52 Plutarch, 259 Poinsot, 311 Point, 135 Polygons, 156, 252, 269, 274 Polyhedrons, 301, 303, 310 Pomodoro, 179 _Pons asinorum_, 174, 265 Posidonius, 128, 149 Postulates, 31, 41, 116, 125, 292 Practical geometry, 3, 7, 9, 44 Printing, effect of, 44 Prism, 303 Problems, applied, 75, 103, 178, 186, 192, 195, 203, 204, 209, 215, 217, 242, 267, 295, 317 Proclus, 36, 47, 48, 52, 71, 127, 128, 136, 137, 139, 140, 149, 155, 186, 188, 197, 212, 214, 253, 258, 310, 311 Projections, 300 Proofs in full, 79 Proportion, 32, 227 Psychology, 12, 20 Ptolemy, C., 35, 278; king, 48, 49 Pyramid, 307 Pythagoras, 29, 40, 258, 272, 273, 310 Pythagorean Theorem, 28, 36, 258 Pythagorean numbers, 32, 36, 263, 266 Pythagoreans, 136, 137, 185, 227, 269, 309 Quadrant, 236 Quadratrix, 31, 215 Quadrilaterals, 148, 157 Quadrivium, 42 Questions at issue, 3 Rabelais, 24 Radius, 153 Ratio, 205, 227 Real problem defined, 75, 103 Reasons for studying geometry, 7, 100 Rebière, 25 Reciprocal propositions, 173 Recitation in geometry, 113 Recorde, 37 Rectilinear figures, 146 Reductio ad absurdum, 41, 177 Regular polygons, 269 "Rhind Papyrus," 27 Rhombus and rhomboid, 148 Riccardi, 47 Richter, 279 Roman surveyors, 247 Saccheri, 127 Sacrobosco, 43 San Giovanni, 192 Sauvage, 164 Sayre, 272 Scalene, 147 Schlegel, 68 Schopenhauer, 121, 265 Schotten, 46, 135, 149 Sector, 154, 156 Segment, 154 Semicircle, 146 Shanks, 279 Similar figures, 232 Simon, 38, 56, 135 Simson, 142 Sisam, 11 Smith, D. E., 25, 37, 51, 52, 119, 131, 135, 159 Solid geometry, 289 Speusippus, 32 Sphere, 321 Square on a line, 257 Squaring the circle, 31, 32, 277 Stäckel, 128 Stamper, 46, 83 Stark, W. E., 172, 238 Stereoscopic slides, 291 Stobæus, 8 Straight angle, 151 Straight line, 138 Suggested proofs, 81 Sulvasutras, 232 Superposition, 169 Surface, 140 Swain, G. F., 13 Syllabi, 58, 60, 63, 64, 66, 67, 80, 82 Sylvester II, 43 Synthetic method, 161 Tangent, 154 Tartaglia, 153 Tatius, Achilles, 272 Teaching geometry, reasons for, 7, 15, 20; development of, 40 Textbooks, 32, 33, 41, 70, 80, 82 Thales, 28, 168, 171, 185, 210 Theætetus, 48, 310 Theon of Alexandria, 36 Thibaut, 185 Thoreau, 246 Trapezium, 148 Treutlein, 68, 164 Triangle, 147 Trigonometry, 234 Trisection problem, 31, 215 Trivium, 42 Universities, geometry in the, 43 Uselessness of mathematics, 13 Veblen, 131, 159 Vega, 279 Veronese, 68 Vieta, 279 Vogt, 259 Wallis, 127, 280 Young, J. W. A., 25, 131, 159, 277 Zamberti, 52 Zenodorus, 34, 253 ANNOUNCEMENTS BOOKS FOR TEACHERS List price Allen: Civics and Health $1.25 Brigham: Geographic Influences in American History 1.25 Channing and Hart: Guide to the Study of American History 2.00 Hall: Aspects of Child Life and Education 1.50 Hodge: Nature Study and Life 1.50 Johnson: Education by Plays and Games .90 Johnson: What to Do at Recess .25 Kern: Among Country Schools 1.25 Mace: Method in History 1.00 MacVicar: The Principles of Education .60 Moral Training in the Public Schools 1.25 Prince: Courses of Studies and Methods of Teaching .75 Scott: Social Education 1.25 Tompkins: Philosophy of School Management .75 Tompkins: Philosophy of Teaching .75 Wiltse: Place of the Story in Early Education, and Other Essays. A Manual for Teachers .50 FOR CLASS RECORDS Comings: Complete Record--Attendance and Scholarship Graded-School Edition .30 High-School Edition .30 Ginn and Company: Teacher's Class Books No. I .30 No. II .40 Twenty Weeks' Class Book .30 CIVICS AND HEALTH By WILLIAM H. ALLEN, Secretary of the Bureau of Municipal Research, New York City. With an Introduction by Professor WILLIAM T. SEDGWICK, Professor of Biology in the Massachusetts Institute of Technology List price, $1.25 _Adopted by the Teachers' Reading Circles of_ _Maryland_, _Kentucky_, _North Dakota_, _South Dakota_, _Oklahoma_, _New Mexico_, _South Carolina_, _Alabama_, _Arizona_, _Illinois_, _Michigan_, _Colorado_, _Texas_, _Virginia_, _Iowa_, _Arkansas_, _Wyoming_, _Missouri_, _Indiana_, _Nebraska_, _and Washington_ * * * * * For Dr. Allen prevention is a text and the making of sound citizens a sermon. In "Civics and Health" he sounds a slogan which should awaken every community in this country to its opportunities in municipal reform. Every teacher who reads this book will gain a new sense of duty in matters of hygiene and sanitation. =Civics and Health is enthrallingly interesting. It is humanized sociology.= Cleaning up children by scientific illumination will appeal to every father and mother, every child lover who has any patriotism or desire to learn how we as a people are to make moral-reform agitations fruitful through health of American children, and so establish health of national life.--_Boston Transcript._ This is one of the books =we wish the law required every citizen to have in his house and to know by heart=. Then, indeed, mankind would have made an immense stride forward.--_Chicago Medical Recorder._ =The book is alive from cover to cover.= It breathes reform but not of the platform variety. It abounds in ugly facts but superabounds in the statement of best methods of getting rid of this ugliness. As claimed by the publishers, it is preëminently a book on "getting things done."--_Hygiene and Physical Education_, Springfield, Mass. GINN AND COMPANY Publishers Transcribers notes On page 30: Pythagoras fled to Megapontum has been left as printed, though the author probably meant Metapontum. On page 269: 100 B.C. has been left as it was printed, though it is probably a typo for 100 A.D. 
812	----	See also: Catalan's Constant [Ramanujan's] PG#682 The algorithm used is the one presented by Greg Fee on ISSAC '90 (using only integer arithmetic in the main loop). Again, I have verified the previous 1000100-digit value using LiDIA and text comparison. Best Thomas Papanikolaou Catalan constant to 1500000 digits computed on October 26, 1996 by using a Sun Sparc20 in 7 day 7 hour 14 min 52 sec 71 hsec The algorithm used is the standard series for Catalan, accelerated by an Euler transform as shown by Greg Fee, ACM 1990, Proceedings of the ISAAC conference, 1990, p. 157 The algorithm was implemented using the LiDIA library for computational number theory and it will be part of the multiprecision floating-point arithmetic of the package in release 1.4. LiDIA is available from ftp://crypt1.cs.uni-sb.de/pub/systems/LiDIA/LiDIA-1.2.1.tgz http://www-jb.cs.uni-sb.de/LiDIA/linkhtml/lidia/lidia.html Here is the output of the program: Calculating Catalan's constant to 1500000 decimals Time required: 7 day 7 hour 14 min 52 sec 71 hsec Catalan = .91596559417721901505460351493238411077414937428167213426649811962176301977625\ 476947935651292611510624857442261919619957903589880332585905943159473748115840\ 699533202877331946051903872747816408786590902470648415216300022872764094238825\ 995774150881639747025248201156070764488380787337048990086477511322599713434074\ 854075532307685653357680958352602193823239508007206803557610482357339423191498\ 298361899770690364041808621794110191753274314997823397610551224779530324875371\ 878665828082360570225594194818097535097113157126158042427236364398500173828759\ 779765306837009298087388749561089365977194096872684444166804621624339864838916\ 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