University of California. FROM THE LIBRARY OF Dr. JOSEPH LeCONTE. GIFT OF MRS. LECONTE. No. Digitized by the Internet Archive in 2007 with funding from l\?lfcrosoft Corporation httb://www.archive.org/details/bourdonsarithmetOObourrich BOURDON'S ARITHMETIC: CONTAININQ A DISCUSSION OF THE THEORY OP NUMBERS. TRANSLATED FROM THE FRENCH OP M. BOURDON, AND ADAPTED TO THE USE OF THE COLLEGES AND ACADEMIES OF THE UNITED STATES, CHAELES S. YEI^TABLE, UOEHTUTE INSTRUCTOR IN THE UiaVERSITY OP VIRGINIA ; FORMER PROFESSOR OP MATHEMATICS IN HAMPDEN SIDNEY COLLEGE, VIR6INU ; FORMER PROFESSOR OF NATURAL PHILOSOPHY AND CHEMISTRY IN THE UNIVERSITY OP QEORGU. PHILADELPHIA : J. B. LIPPINCOTT & CO. 1858. V.' ^ Entered, according to Att of Congress, in the year 1857, by J. B. LIPPINCOTT & CO., in the Clerk's Office of the District Court of the United States for the Eastern District of Pennsylvania. PREFACE I AM led to offer the present translation to the public, from the conviction that such a work is very much needed in our Academies and Colleges. In fact, a long experience in teaching has convinced me that, one great difficulty which the young student has to encounter in the study of Algebra and the higher branches of analysis, results from the want of sound philosophical ideas on the fundamental properties of numbers, and from the fact that the funda- mental operations of Arithmetic are generally learned by rote, and not pursued as a system of close reasoning. Bourdon's treatise is the one adopted in the schedule of public instruction by the University of France. In pre- paring the translation, I have compared the seventh with the twenty-ninth Paris edition, and endeavoured to select the best methods of each. In this selection and arrange- ment, I have followed the outline of the lectures upon Arithmetic, delivered by the late Professors Bonny castle and Courtenay, in the University of Virginia. The tables \have been re-arranged, and a collection of examples an- nexed to the work. The portions of Bourdon's very complete treatise on the Extraction of Roots, Progressions, Logarithms, and their applications, I have left out, because they are very thoroughly discussed in the best treatises on Algebra adopted by our Colleges and Universities. I have followed (iii) IV PREFACE. the author in introducing some few of the signs and pre- liminary definitions of Algebra. This usage the author well defends, as follows : — "To attempt to make known even some of the simple properties of numbers without employing the signs of algebra, is to present them in a manner very incomplete and little methodical. To use these signs to some extent, enables us to establish the con- nexion between these properties and their most important applications. Moreover, the discussion of these properties, a knowledge of which is essential to a thorough knowledge of arithmetic, cannot properly enter into the elements of algebra, without breaking the chain of theories which con- stitute this other branch of mathematics. In fine, the work is designed for those who wish to make the first steps in the career of a scientific or liberal education in a sure and profitable manner." The translator hopes the present treatise will be a useful addition to the means of thorough instruction in the United States. C. S. V. LoNOwooD, Va., 1857. CONTENTS. PART FIRST. .INTRODUCTION. Aeticles 1-9. — Preliminaries — Numeration, spoken and vrritten — General Ideas on Fractions, and the Four Fundamental Opera- tions Page 9 CHAPTER I. Of the Four Fundamental Operations on Entire Numbers. Articles 10-41. — Addition — Subtraction — Multiplication — Division — Applications and Exercises on Chapter First 17 CHAPTER 11. Vulgar Fractions. Articles 41-65. — Introduction — Reduction of Fractions to a Com- mon Denominator — Of the Least Common Multiple — Simplifica- tion of Fractions — Of the Greatest Common Divisor of Two Numbers — The Four Fundamental Operations upon Fractions — Fractions with Fractional Terms — Fractions of Fractions — Ap- proximative Valuation of Fractions — General Observation on the Calculus of Fractions 59 CHAPTER III. Compound Numbers. Articles 65-81. — Systems of Weights, Coins, and Measures — Pre- liminary Operations — Compound Numbers — Addition, Subtrac- tion, Multiplication, and Division of Compound Numbers — Exer- cises 85 1* (V) VI CONTENTS. CHAPTER IV. Theory op Decimal Fractions — Decimal System of Weights, &c. Articles 81-109. — Introduction — Use of the Point or Comma — Fundamental Principles — Of the Four Operations on Decimal Fractions — Valuation of the Quotient of a Division in Decimals — Conversion of a Vulgar Fraction into Decimals — Decimal System of Weights and Measures — Divisions of Thermometers and Circumference — Conclusion of Part First — Exercises 100 PART SECOND. CHAPTER V. General Properties of Numbers. Articles 109-154. — Introduction — Use of Signs and Preliminary Definitions — Theory of Different Systems of Numeration — Some General Principles of Multiplication and Division — Divisibility of Numbers — Verification of Multiplication and Division by the Properties of 9 and 1 1 — All the Divisors of a Number — Remarks upon the Greatest Common Divisor of two Numbers — Prime Numbers — Greatest Common Divisor of Several Numbers — Least Common Multiple — Periodical or Repeating Decimals — Some of the Properties of Periodical Decimals — Exercises 128 CHAPTER VI. Op Ratios and Proportions — Resolution of Questions which depend UPON Proportional Quantities. Articles 154-188. — Introduction — Ratios and Proportions — Their Principal Properties — Equidifferences — Resolution of some Questions in Simple Rule of Three — Of Direct and Inverse Ratios — Employment of the Method of Reduction to Unity for all Questions of Compound Proportions — Remark upon the Use of Direct and Inverse Ratios in the Practical Solution of these Questions — Rule of Simple Interest — Rule of Discount — Rule of Fellowship — Rule of Alligation — Some Miscellaneous Ques- tions — Exercises 179 Collection of Examples on all the foregoing Chapters 220 j^ote A. — Different Systems of Numeration 237 Note B. — Abbreviated Methods of Division and Multiplication 240 Signs made use op in the Work. 1st. + plus, the sign of addition. 2d. — minus, the sign of subtraction. 3d. X multiplied hy, the sign of multiplication. 4th. -J- divided hy, " " division. 5th. = equal to, " " equality. (vii) " O^ THE DIVERSITY ELEMENTS OF ARITHMETIC FIRST PART INTRODUCTION. 1. We call magnitude, or quantity, every thing which admits of increase or diminution. For example, lines, surfaces, solids, intervals of time, weights, are magnitudes. We can only form an exact idea of a magnitude by comparing it with another mag- nitude of the same species, and this second magnitude is called unity, in as much as it is to serve as a term of comparison for all magnitudes of the same species. Thus, when we say that a wall is twenty yards long, we are understood to have already ac- quired the idea of the unit of length called yard, and we sup- pose that, after having laid down the yard twenty times upon the length of the wall, we have arrived at the end. Unify, in mathematics, is then a magnitude of any species whatever, taken arbitrarily or in nature, which serves as a term of comparison for all magnitudes of the same species. Whence it follows that there are as many species of units as of magni- tudes. The result of the comparison of any magnitude whatever with its unit, is called number. A number is called entire when it is (9) 10 NUMERATION. the assemblage of several units of the same species or denomina- tion. Thus, twenty dollars, thirty pounds, eight, twelve, fifteen units, of any species whatever, are entire numbers. A fraction is a part of a unit. A fractional or mixed number is an assemblage of several units of the same denomination, and of ^parts of this unit. 2. When, in enunciating a number, we add at the end of that number the species of magnitude taken for the unit, the number is called concrete. Thus, five feet, fifteen hours, six leagues, are concrete numbers. The first time we pronounce a number, the only sense we can attach to it, is the representing to ourselves a unit of a certain denomination, to which we compare another magnitude of the same denomination. But, by degrees, the mind, which accustoms itself to abstractions, represents to itself a collection of any like objects, of which each one is unity. In this case the collection is called an abstract number, because, in enunciating it, we make abstraction of the species of unit to which we refer it. It is in this last light that we are to consider numbers, in the discussion of the methods relating to the differ- ent operations which we have to perform upon them, if we wish to establish these methods so as to be able to apply them to all possible questions. NUMERATION. 8. The first researches on numbers should have, necessarily, for object, the giving them names easy to retain ; and, as there exists an infinity of numbers (since we can add to any number whatever, already formed, a new unit, which gives rise to a new number, also capable of being augmented by unity), it is neces- sary to find some means of expressing all numbers by a limited number of words, combined together in fit manner. Such is the object of spoken numeration. . Again, each one of the words which enter into the nomen- clature of numbers being expressed by several letters, it was found necessary to invent an abridged mode of writing these words and their combinations, in order that the mind might be NUMERATION. 11 able to seize with more facility the reasonings which we are obliged to make upon the numbers. This is the object of written numeration, which consists in representing, by a limited number of characters or ciphers, the numbers enunciated in the ordinary language. 4. Spoken Numeration. — Though the nomenclature of entire numbers is known, for the most part, to the young men for whom these elements are written, we think it best to give a succinct, yet methodical analysis of it ; for, the numeration which is adopted in nearly all countries, is founded upon this nomenclature. The first numbers are, one, two, three, four, Jive, six, seven, eight, nine. These numbers are called simple units, or, units of the first order. Adding a new unit to the number nine, we form the number ten, which we regard as a new denomination, or, spe- cies of unit called a ten, or, a unit of the second order. We count by tens in the same manner as we have counted by simple units. Thus, we say, one ten, two tens, &c., &c. ; ten, twenty, thirty, forty, fifty ^ sixty, &c. Between ten and twenty there are nine other numbers, which in English have the names, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nine- teen ; names established by usage, showing by their derivation, the addition of the preceding simple units successively to the unit of the second order. Between twenty and thirty, there are also nine numbers, which are enunciated, twenty-one, twenty-two, &c. And thus we can enunciate all the numbers up to ninety-nine. This last number, augmented by one, gives ten tens, or the number one hundred, which we regard as a new unit, or unit of the third order ; and we count by hundreds as we have counted by units and tens. Thus, one hundred, two hundred, &c. Placing successively between the words hundred and two hundred, two hundred and three hundred, eight hundred and nine hundred, and, after nine hundred, all the numbers comprised between one and ninety- nine, we form the names of all the numbers, from one hundred to nine hundred and ninety-nine. We can see that, in the enun- ciation of all these numbers, we have employed only the generic 12 NUMERATION. terras, one, two, three, four, five, six, seven, eight, nine, ten, hun- dred, and words easily derivable from these. Adding one .to the number nine hundred and ninety-nine, wc obtain a collection of ten hundreds, or the number thousand, which forms the unit of the fourth order. Having reached this number it is agreed, in order not to multiply words too much, to regard thousand as a new principal unit, before the name of which we place the names of the nine hundred and ninety-nine first numbers. Thus, we say, one thousand, two thousand, nine hundred and ninety-nine thousand. A ten thousand forms the unit of the fifth order ; a hundred thousand forms the unit of the sixth order. Now, placing between two consecutive numbers of the denomi- nation thousand, as twenty ihoviSdiud and twenty-one thousand, the names of all the numbers of lower denomination than thousands, it is clear that we can thus enunciate all the numbers up to nine hundred and ninety-nine thousand, nine hundred and ninety- nine. This last number, augmented by one, gives ten hundred thousand, or, a thousand thousand, to which collection the name million has been given ; in the same manner the collection of thousand millions is called billions; the collection of thousand billions is called trillions, and so on to infinity. We count by millions, billions, and trillions, as we have counted by thousands ; and it is easy to see that, by joining to the generic words indicated above, the words million, billion, trillion, quatr- illion, quintillion, we will form the nomenclature of all imagi- nable entire numbers. "VVe observe, in order to terminate this part of the subject, that the million is the unit of the seventh order, ten millions are units of the eighth order, hundred mil- lions, units of the ninth order. 5. Written Numeration. — Though the above nomenclature is very simple, still we would find much trouble in combining toge- ther two or more large numbers, unless we had some abridged mode of writing them. This is easily arrived at by reflecting a little upon the nomenclature. We observe at once, that, among the words employed to express numbers, the one part, as one, ten', NtMERATION. 13 hundredy thousand, ten thousand, &c., express the units of dif- ferent orders, while the words, one, two, three, nine, express how many times each of these sorts of units enter into a number. This being established, if we agree to represent the first nine numbers by the characters or ciphers, 123 4567 8 9 one, two, three, four, jive, six, seven, eight, nine^ the whole difficulty consists in finding a means of making these ciphers express the different orders of unity which compose the proposed number. Then, establishing this principle (purely con- ventional), that every figure placed to the left of another, expresses units of the order next higher to those of the other figure, or, in other words, that when several characters, signifying the first nine numbers, are written one after anotJier, then the first figure to the right expresses simple units, the next on the left, tens, the third figure counting from right to left, hundreds, the fourth, thou- sands ; it is easy to see that, in general, we can represent all num- bers by the aid of the preceding characters. Character 0. — While this is true in general, nevertheless, there are numbers which the preceding convention fails to repre- sent, unless we agree to use an additional character. If we un- dertake to write in figures the numbers, ten, twenty, thirty, &c., these numbers containing no simple units, we are compelled to adopt a character which has no value by itself but which serves to hold the place of the units of the order which is wanting in the number enunciated. This cipher is 0, and is called zero. By the aid of this cipher, the numbers, ten, twenty, &c., are expressed by 10, 20, 30, 40, &c. In the same manner, the numbers, one hundred, two hundred, &c., which contain neither simple units nor tens, are written thus : 100, 200, 300. In general, the zero is a cipher which has no value by itself, but which we employ to hold the place of the different orders of 9 14 NUMERATION. unity which may be wanting in the number to be written. The other characters are called significant figures, and have two values ', the one we call absolute, and is no other than that of the figure itself considered alone; the other, we call relative, which the figure acquires according to the place which it occupies to the left of other figures. Now, if we reflect that every number is composed of simple units, of tens, of hundreds, &c.; that the collection of units of each order is equal to nine; that, in the case where a number is deprived of certain orders of units, we have a character to hold their places, we will see at once that there is no entire number which cannot be expressed by the aid of a certain combination of the ten characters : — 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Take the example, thirty-six billions, five hundred millions, twenty thousand, four hundred and seven. This number contains seven simple units, no tens ; four hun- dreds, no ones of thousands ; two tens of thousands, no hundreds of thousands ; no ones of millions, no tens of millions, five hun- dreds of millions ; six ones of billions, and three tens of billions ; then the number will be represented by 365000 20 407. The system of numeration which we have just explained, has received the name of the decimal system, because we emploj' ten figures to express all numbers. Ten, or the number of characters employed is called the base of the system. 6. Let us make, now, an important observation : it results from the nomenclature, that every number can be divided into hun- dreds, tens, and simple units; into hundreds, tens, and ones of thousands ; into hundreds, tens, and ones of millions, etc. ; that is to say, into sets of simple units, thousands, millions, &c., each set expressed by three figures, except the last, which is that of the units of the highest order, and which cannot have more than two figures, and sometimes contains only one. When, then, we have become familiar with the manner of writing the numbers of three figures, it is sufiicient to write, successively, the different NUMERATION. 15 sets one to the left of the other; the set of units, the set of thou- sands, the set of millions, &c. We can even commence at the left; that is to say, write first the set of the units of the highest denomination, and, to the right of this, the other sets in the order of the magnitude of their units. It is thus that we ought to write a number dictated in ordinary language. But it is necessary to take care not to omit the zeros destined to replace the orders of units which are wanting ; and there can never be any difficulty, since we know that each set, except the first to the left, must always con- tain three figures. Suppose, for example, that we have to write, by aid of our characters, four hundred and six billions^ twenty- eight millions^ two hundred and fifty thousandj and forty-eight. Write in succession, each to the right of the other, the period of hill ions ; the period of millions; the period of thousands; and,\sist\j, thsit of simple units ; we will have 406, 028, 250, 048. 7. It is upon the preceding observation that the following means of translating into ordinary language, any number what- ever written in figures, is founded. • After having separated the number into periods of three figures each, commencing at the right, enunciate successively each period ^ setting out from the first period on the left, and taking care to give to each period the name which belongs to it. Example: 70345601. This number, being divided 70,345,601, is composed of seventy millions, three hundred and forty-five thousand, six hundred and one. 8. It remains for us still, in order to complete the theory of enumeration, to show the mode of writing fractions by means of figures. But we must first give a clear and precise idea of frac- tions, such as we consider them in arithmetic. Let us suppose that we have to determine the length of a piece of cloth. Taking the unit of length called yard, and applying it as many times as possible to the length of the piece, two cases may occur, either, after the unit has been applied a certain number of times — 15 times, for example, nothing will remain — or, we will obtain a remainder less than the yard. In the first case, the piece will contain an entire numher of yards. In the second case, it will 16 NUMERATION. be necessary in order to have the whole length of the piece, to add to these 15 yards the fraction or part of the yard which remains. But how value this part ? how compare it to the unit ? We can first conceive this unit separated into two equal parts or halves; and, if the remainder is exactly equal to one of these halves, we say that the piece of cloth is 15 yards and one half long. If the remainder is less or greater than a half^ yard, we con- ceive this half divided into two new equal parts, called quarters. Instead of dividing the unit into two or four equal parts, we can conceive it to be divided into three equal parts called thirdsj &c., &c. Whence, we see that in order to form a clear idea of a fraction of a unit of any denomination whatever, it is necessary to con- ceive that this unit be divided into a certain entire number of equal parts, and that we take one, two, three, &c., of these parts; these parts thus taken, constitute what is called vl fraction. Thus, the enunciation of a fraction involves necessarily two entire numbers, to wit: — th^t which denotes into how many parts the unit has lieen divided, called the denominator ; and that ichich denotes how many of these parts are necessary to form the frac- tion, called the numerator. For example, five-eighths of a yard, thirteen-twentieths of a pound, are fractions. In the first, we conceive the yard divided into eight parts, and that we take five of these parts to form the fraction, eight is the denominator, and five the numerator. . . . (We see that in the spoken numeration of fractions the numerator remains unchanged in name, while the denominator is generally changed by the addition of th.) It results, also, from the above, that a fraction is a magnitude referred to a part of the principal unit, which part we can con- sider itself as a particular species of unit. Thus, the fraction thirteen-twentieths of a yard, being composed of thirteen times the twentieth of a yard, this twentieth is a particular unit, which the proposed fraction contains thirteen times. This being estab- lished, two fractions are said to be of the same species when their denominator is the same, (the original or compound unit being likewise the same). For example, five-twelfths and seven-twelfths ADDITION. 17 of a yard are fractions of the same species ; but three-fourths and two-thirds of a pound are fractions of different species or deno- minations, because the denominators are different. In order to express a fraction in figures, we place the numera- tor above the denominator, with a line between. Thus, the frac- tion three-fourths is denoted by |, seven-twelfths by j'^^- Reciprocally, |, ||, represent the fractions, seven-eighths, thirteenth-fifteenths, that is to say, we enunciate the numerator, and then the denominator, and add the termination — th, to the latter. CHAPTER I. 9. Arithmetic has, for it special object, to establish fixed and and certain rules for performing all possible operations upon numbers. It embraces, besides, the study of a great number of properties which have been discovered during the researches made in order to arrive at these methods, or to facilitate the use of them. We will now explain these operations in their order, recollecting that, in order to render the methods independent of every sort of question, it is best to consider the numbers as abstract numhers. Nevertheless, in the applications designed to familiarize begin- ners with the methods, we can propose questions also relating to concrete or denominate numhers. OPERATIONS ON ENTIRE NUMBERS. ADDITION. 10. To add or sum up several numbers^ is to unite all these •numhers into a sinylc one; or, to form a number which contains in itself alone as many units as there are in the different numbers taken separately. The result of this operation is called the sum, or total. The addition of numbers of a single figure offers no difficulty. It is 18 ADDITION. done unit by unit. Children learn thus to make these additions by means of their fingers, and fix the results in their memory. In this way, for example, they find thirty to be the sum of 5, 7, 8, 4, and 6; or, that 42 is the sum of the numbers 7,9,6,5,8,7. OPERATION. Let now the numbers to he added be 7,453 7,453 and 1,534. 1,534 After having written the numbers, one under 8,987 another, with a line under them, we commence with the simple units, and say, 3 and 4 make 7, which we place under the units. Passing to the tens, 5 and 3 make 8, which we write under the tens. Then, 4 and 5 make 9, which we write under the hundreds. Lastly, 7 and 1 make 8, which we write in the column of thousands. The number, 8,987, found by this operation, is the sum of the two given numbers, since it contains their units, their tens, their hundreds, and their thousands, which we have summed up successively. OPERATION. Again, let it be proposed to add the four num- 5,047 bers,^ 5,047, 859, 3,507, 846. We write them 859 one under the other, and, commencing with the 3,507 units, 7 and 9 make 16, and 7 make 23, and 6 846 make 29. We place the nine simple units under 10,259 the first column, and retain the two tens, in order to add them to the figures of the next column, which are also tens. Passing to the next column, we say that the two reserved, and 4 make 6, and 5 make 11, and make 11, and 4 make 15. We write the 5 in the column of tens, and retain the 1 hundred which we carry to the column of hundreds. Operating upon this column, as upon the preceding, we find 22 hundreds, or 2 hundreds, which we write under the hundreds, and 2 thousands, which we retain in order to carry them to the column of thou- sands. Lastly, 2 reserved, and 5 make 7, and 3 make 10. We SUBTRACTION. 19 place the under the thousands and advance the 1 to the left, which gives 10,259 for the required sum. General Rule. — In order to add several numbers together, commence by writing them one under another, so that the units of the same order may be in the same column. Add then successively the figures which compose each one of the vertical columns, commencing with the column of simple units, passing to the columns which are on the left : write below the line the sum of the figures of each column, provided the sum is expressed by a single figure. But if it exceeds 9, in which case it is expressed by several figures, of which the last to the right represents the units of this column, and the others to the left tens of the same order, write only the figure of units below the column, and reserve the tens in order to add them to the figures of the column immediately to the left. When you have operated in this manner upon all the columns, you loill obtain the sum required, because it results from the union of the units, tens, hundreds, &c., which enter into the given numbers. 11. Remark. — If the sum of the figures in each column does not exceed nine, we could commence the operation equally well by the addition of the units of the highest order as by the addi- tion of the simple units. But as it happens oftenest that several of these sums exceed nine, if we commence on the left, we will often be obliged to return upon our steps, in order to correct a figure already written, and increase it by as many units as we shall have obtained from the tens of the following column in operating upon that column. For this reason it is best in all cases to commence on the right rather than on the left. SUBTKACTION. 12. To subtract one number from another is to seeh the excess of the greater number over the less. The result of this operation is called remainder, excess, or difference. So long as the numbers proposed consist only of a single figure, the subtraction is easy 20 SUBTRACTION. Thus, the difference between 9 and 6 is 3. "VVe can easily sub- tract a number of a single figure from a number which does not exceed twenty. Thus, take 7 from 13, there remains 6, since by what we have learned in addition, 7 and 6 make 13. In the same manner, 9 from 17 there remains 8, because 8 and 9 make 17. These operations, which suppose only the exercise of the memory upon the addition of numbers of a single figure, serve as a basis for the subtraction of numbers of several figures. Let it he required to subtract 5467 from 8789. OPERATION. After having placed the smaller number under the 8789 greater, and underlined the whole, we say, com- 5467 mencing with the simple units, 7 from 9 leave 2, which 3322 we place in the column of simple units; passing to the • tens, 6 from 8 leave 2, which we write in the column of tens ; the same operation finally upon the hundreds and thousands, 4 from 7 leave 3, and 5 from 8 leave 3, gives 3322 for the re- quired remainder. For by the nature of the operations which have just been performed, we see that the greater number con- tains more than the second, 2 simple units, 2 tens, 3 hundreds, 3 thousands, and consequently exceeds the smaller by 3322. Let us propose for a second example, to find the difference which exists between the two numbers, 83456 and 28784. OPERATION. Having arranged the numbers as in the preceding 83456 example, we say, first, 4 from 6 leave 2, which we 28784 write under the units. But when we pass to the 54672 column of tens, we meet with a difficulty : the lower figure, 8, is greater than the upper one, 5, and consequently cannot be subtracted. In order to overcome this difficulty, we borrow mentally from the hundreds figure 1 hundred, which equals 10 tens, and add it to the 5 tens which we have already, giving us 15 tens; we then say, 8 from 15 leave 7, which we write in the column of tens. Passing to the column of hundreds, we observe that the upper figure, 4, ought to be diminished by 1, SUBTRACTION. 21 since we have borrowed this unit in the preceding subtraction ; we say, then, 7 from 3, which is impossible ; but we borrow, as before, 1 thousand, which equals ten hundreds, giving 13 hun- dreds, and take 7 from 13, which gives 6, to be written in the column of hundreds. Passing to the thousands, 8 cannot be taken from 2 ; but 8 from 12 leave 4, to be written in the column of thousands. Lastly, as the figure 8, of tens of thousands, on account of the 1 just borrowed, ought to be replaced by 7, we say, 2 from 7 leave 5. Thus, the remamdevj or the excess of the greater number over the less, is 54672. In order to understand how, by this means, we arrive at the end proposed, it is sufficient to remark that, according to the artifices employed tft effect the partial subtractions, we can ar- range the two numbers in the following manner : — Tens of thousands, thousands, hundreds, tens, units. 1st number, 7 12 13 15 6 2d number, _2 8 7 8 4 5 4 6 7 2 From this we see that the upper number exceeds the lower one by two units, 7 tens, 6 hundreds, 4 thousands, and 5 tens of thousands — or exceeds it 54672 units. Let it he proposed, for example, to subtract 158429 from 300405. OPERATION. 99 9 As 9, the units figure of the lower number, is 300405 larger than 5, the corresponding figure of the greater, 158429 we have to borrow 1 ten from the first figure to the 141976 left ; but this figure being 0, it is necessary to have recourse to the figure 4, of hundreds, from which we borrow 1, which equals 10 tens; and since we have need of only a single ten, we leave 9 of them above the 0; we then add 1 ten to 5, which gives 15, and say, 9 from 15 leave 6, which we write under the units. Passing to the tens, we say, 2 from 9 leave 7. For the hundreds, as the upper figure, 4, has been diminished 22 SUBTRACTION. by the 1 whicli we borrowed, and as we cannot take 4 from 3, we have recourse to the next figure to the left ; but that and the figure which is to its left being zeros, we borrow 1 from the next significant figure, 3. This 1 equals 10 of the order following, and 100 units of the order thousands ; and since we have need of only 1 unit of this order, we leave 99 of them, which we place above the two zeros; adding 1 thousand to the 3 hundreds, it becomes 13 hundreds, and we say, 4 from 13 leave 9, which we place under the column of hundreds. In the two following subtractions, each one of the zeros being replaced by a 9, we say, 8 from 9 leave 1, and 5 from 9 leave 4. Passing to the first column to the left, we say, 1 from 2 (for the figure 3 is diminished by 1) leaves 1. Thus we have for the required remainder 141976. If we reflect upon the manner in which the greater number has been decomposed, we can arrange the operation thus : — hundreds of thous., tens of thous., thous., hundreds, tens, units. 1st number, 2 9 9 13 9 15 2d number, _1 5 8 4 2 9 1 4 19 7 6 Then the greater number exceeds the less by 6 units, 7 tens, 9 hundreds, 1 thousand, 4 tens of thousands, 1 hundred thou- sand, or by 141976. General Rule. — In order to perform the subtraction of two numbers, place the less number under the greater, so that the units of the same denomination fall in the same column ; then underline the two numbers; sid)tract then successively, units from, units, tens from lens, hundreds from hundreds, &c., and write the partial remainder's one to the left of another; the number formed by these remainders is the total remainder, or the result required. When a figure of the lower line is greater than the figure above it, augment mentally this last figure by \(} units, and diminish the figure to the left of it by one unit. SUBTRACTION. 28 Jf immediately to the left of an upper figure less than the one below, corresponding, there are one or more zeros, increase this figure above mentally always by 10 units ; but in the following subtractions replace the Os by 9s, and diminish by a unit the upper significant figure which is immediately to the left of these zeros. 13. First Remark. — If each one of the figures of the lower number is less than the corresponding figure of the greater, we could commence the operation indifi'erently at the right or left. But as it often happens that one of the figures of the less num- ber exceeds the figure of the greater above it, the partial subtrac- tion cannot be effected without borrowing from one of the figures to the left of that one with which we are operating; for this reason it is necessary to commence on the right, in order to bor- row when there is need of it. 14. Second Remark. — It is clear that instead of diminishing by one unit the figure from which we have borrowed it, we can leave this figure unchanged, provided we augment the corres- ponding figure below by one unit. This manner of operating is in general more convenient in practice. Thus, in the ^ast example, after having said for the simple units, 7 from 11 leave 4, instead of saying for the tens, 8 from 9 leave 1, we say, 9 from 10 leave 1 ; in the same manner, instead of saying for the hundreds, 7 from 13 leave 6, we say, 8 from 14 leave 6, and so on for the rest. But when we employ this modification, we must be careful to augment the lower figure only when we have been obliged to borrow in the subtraction of the preceding figures. This modifi- cation is used particularly in division. VERIFICATION OF ADDITION AND SUBTRACTION. 15. We call the verification of an arithmetical operation, another operation which we perform in order to convince our- selves of the accuracy of the first. 24 SUBTRACTION. The verification of addition is effected by adding anew, but commencing at the left hand. After having formed the sum of the figures in the first column on the left^ we subtract it from, that part which answers to it in the sum total ; we write down the remainder, which we reduce mentally into units of the order of the following figure, in order to join them to the units of this order in the sum total. In the same manner we sum up the second column on the left, and subtract this partial sum from the corresponding part of the sum total ; ice continue this operation to the last column ; the last subtraction leaves no remainder. Thus, after having found that the four numbers, 6047 859 3507 846 have for their sum 10259 in order to verify the result 2120 we add the same numbers commencing on the left. We say, 5 and 3 make 8 thousands, which we subtract from 10 thousands, leaving 2 thousands for remainder; which, with the figure 2 hundreds, make 22 hundreds ; then 8 and 5 make 13, and 8 make 21, which we take from 22, which gives for remainder 1 hundred, which, joined to 5 tens, forms 15 tens ; 4*and 5 make 9, and 4 make 13 ; 13 from 15, there remains 2, which, joined to the 9 units following, gives us 29; lastly, 7 and 9, and 7 and 6 make 29 ; 29 from 29 and nothing remains ; then the operation is exact. The verification of subtraction is effected by adding to the smaller number the remainder found by the operation ; and it is evident that we ought thus to reproduce the greater number, since this remainder is nothing more than the excess of the greater number over the less. SUBTRACTION. Thus, in the annexed examples, after having found that 54682 is the excess of the greater number over the less, if we add this excess to the number 28784, we ought to obtain the number 83466 — which we do in fact obtain. 25 83466 , 28784 Rem. 54682 Proo/83466 16. Here we give some examples of addition and subtraction, with their verifications. Additions. 83054 700548 256870 897597 748759 6588 90874 69764 130909 407300 8746 987846 1319212 1207047 2:^^^0 Subtractions. 4276690 4073050062 20004001003 2803767086 ' 8405128605 1269282976 11598872398 4073050062 20004001003 Problem. — A banker had in his chest a sum o/* $65, 750; he gave one person $13,259 ; to a second, $18,704 ; to a third, $22,050 ; to a fourth, $9850 j what was the state of his chest after these payments ? Solution. — After having summed up the four sums succes- sively paid, we subtract the sum total from that which he had, 3 26 MULTIPLICATION. and the result of the subtraction will be what ought to remain in his chest. Thus, 13259 65750 18704 63863 22050 $1887 what he has left. 9850 63863 We remark, that in effecting the preceding addition and sub- traction, we have considered the given numbers as abstract, al- thouoh they were denominate numbers according to the enuncia- tion of the question ; but, arrived at the result, 1887, we have given it the name of the species of unit which the numbers expressed in the enunciation. We must always perform the operations in this manner, when we wish to apply the results of the operations to questions in denominate numbers. ■ The results being altogether independent of the nature of the nuin- bers, we consider them in a point of view purely abstract, except in giving to the final result the name of the unit which the enunciation of the question indicates. MULTIPLICATIOlf. 17. To multiply one number hy another , is to compomid a third number with the first, as the second is compounded with unity. Then, if the two given numbers are entire numbers, to multiply them is, to take the first as many times as there arc units in the second. We call the result of multiplication, product; the number to be multiplied, multiplicand ; and the number by which we mul- tiply, multiplier ; which denotes how many times we are to take the first. The two numbers bear jointly the name of factors of the product. Properly speaking, multiplication is nothing else than addition ; for, in order to obtain the result, it would suffice to write the multiplicand as many times as there are units in the MULTIPLICATION. Z / multiplier, and then add all these numbers together. But this man- ner of operating would be very long, if the multiplier was composed of several figures ; we are then to seek a method of simplifying it, and it is in this abbreviation that multiplication consists. 18. As long as the two factors are expressed, each one by a single figure, their product is obtained by the successive addition of the multiplicand to itself; thus, in order to multiply 7 by 5, we say, 7 and 7 make 14, and 7 make 21, and 7 make 28, and 7 make 35 ; this last number being the result of the addition of five numbers equal to 7, expresses the product of 7 by 5. Beginners will do well to exercise themselves in this sort of multiplication ; for they ought to impress the results upon the memory, if they wish subsequently to obtain easily the product of numbers expressed by several figures. Nevertheless, for those who are suflBciently exercised, all that is necessary is to give a table called the multiplication tabUj or tahle of Pythagoras, from the name of its inventor, or at least from him who first brought it into public use. 1 1 2 •^ 1 4 5 1 « 7 1 « 1 9 2 , 1 4 6 1 8 10 |12 14 116 |18 3 1 6 9 |12 15 |18 21 |24 |27 4 1 8 12 |16 20 |24 28 |82 36 5 |10 15 |20 25 |30 35 140 |45 6 |12 18 |24 30 |36 42 |48 |54 7 |14 21 |28 35 42 49 1 56 1 63 8 |16 24 132 40 48 50 |64 |72 9 |18 27 36 45 54 63 |72 81 The first horizontal row of this table is formed by adding 1 to itself up to 9 ; the second, by adding 2 to itself; the third, by adding 3 ; and so on for the rest. We remark, moreover, that the same arrangement is made in the vertical columns. Each vertical column, taken in order, is composed of the same num- bers as each horizontal row. Thus, the sixth horizontal row is 28 MULTIPLICATION. composed of 6, 12, 18... 54, and the sixth vertical column is composed of the same numbers, 6, 12, 18... 54. That being established, when we wish to obtain the product of two numbers from this table, we seek the multiplicand in the first horizontal row, and go down from this number vertically, until we arrive at that one which is opposite to the multiplier, which we find in the first vertical column. This number, con- tained in the little square, is the product. For example, in order to find the product of 8 by 5, we descend from 8, taken in the first horizontal row opposite to 5 in the first vertical column, and the number 40 in the little square is the required product. 19. Suppose, now, that the multiplicand consists of several figures, and the multiplier of a single figure. OPIRATION. 8459 Let it be proposed to multiply 8459 by 7. We 8459 could (17) obtain the result by writing one under an- 8459 other seven numbers equal to 8459, and adding suc- 8459 cessively the simple units, the tens, hundreds, &c., 8459 together. We would thus find 59213 for a result. But 8459 it is evident that this is nothing more than taking 8459 7 times the 9 units of the multiplicand, 7 times the 5 59213 tens, &c., and then to take the sum of all the pro- ducts. 8459 Thus, after having placed the multiplier, 7, under 7 the multiplicand, we say, 7 times 9 make 63, (see 59213 table of multiplication), or 6 tens and 3 units; we place the 3 under the units, and reserve the 6 tens in order to add them to the product of the tens of the multiplicand by 7. We thus say, 7 times 5 make 35, and 6 make 41 tens, or 4 hundreds and 1 ten ; we place 1 in the column of tens, and reserve the 4 hundreds ; 7 times 4 make 28, and 4 make 32 hundreds, or 3 thousands and 2 hundreds; we place 2 in the column of hundreds, and retain the 3 ; lastly, 7 times 8 make 56, and 3 make 59 ; we write down the 9, and carry the 5 one place to the left, because there are no more figures in the multi- MULTIPLICATION. 29 plicand to be multiplied. We find thus, 59213 for the required product. Whence we see that, in order to multiply one number of several Jigures hy another of a single figure, we must multiply successively the units, tens, hundreds, dsc, of the midtiplicand hy the multiplier, and write these different palatial products in the columns to which they belong, taking care at each partial multi- plication, to reserve the tens in order to add them to the tens, the hundreds in order to add them to the hundreds, &c. OPERATION. Let it be proposed as a second example to multiply 37008 37008 by 9. We say, first, 9 times 8 make 72 ; we 9 write 2 in the column of units, and reserve the 7. 333072 Then 9 times give ; but we have reserved 7 from the preceding operation, so we write these 7 tens in the column of tens ; 9 times make ; we write in the rank of hundreds, since there are none, and since it is necessary to preserve the place of hundreds ; then 9 times 7 make 63 ; we set down 3 and reserve 6 ; lastly, 9 times 3 make 27, and 6 make 33 ; we set down 3, and advance 3 one place to the left. Thus, the required product is 333072. 20. Before passing to the case in which the multiplier is com- posed of two or more figures, we will explain the method of rendering a number 10, 100, 1000 times greater, or of multiply- ing it by 10, 100, 1000. It results from the fundamental principle of numeration (5), that if we place a to the right of a number already written, each one of the significant figures of the number being thus advanced one step towards the left, expresses units ten times greater than before. In the same manner, by placing two O's to the right, we render it 100 times as great, because each signifi- cant figure expresses units 100 times as great. Then, in order to multiply any entire number whatever by 10, 100, 1000, &c., it sufires to annex 1, 2, S,... zeros. 3 * " •SO MULTIPLICATION. Thus, the products of 439 by 10, 100, 1000, 10,000, &c., are 4390, 48,900, 439,000. 21. Let us consider now the case in which the multiplicand and multiplier are composed of several figures. OPERATION. We propose to multiply 87468 By :..... 5847 We commeDce by placing the multiplier under 612276 the multiplicand, so that the units of the same 3498720 order fall in the same column. This being ar- 69974400 ranged, we observe that, to multiply 87468 by 437340000 5847, is to take the multiplicand, 7 times, 40 511425396 times, 800 times, and 5000 times; then to add together these partial products. We can first find, by the rule of (19) the product of 87468 by 7, which gives 612276. But how obtain that of 87468 by 40 ? Let us conceive, for an instant, that we have written, one under another, 40 numbers equal to 87468, and that we make the addition of these numbers ; we will thus have the required product. But it is evident that these 40 numbers form ten divisions, each division containing 4 times 87468. We form this product by rule (19), and find it to be 349872. Multiplying this product by 10, which (20) is efi'ected by annexing a 0, we obtain 3498720 for the product of 87468 by 40. We see, then, that this second operation reduces itself to mul- tiplying the multiplicand by the figure 4, considered as express- ing simple units, in writing a to the right of the product, and in placing the result as we see above, below the first partial pro- duct. In like manner, in order to perform the multiplication of 87468 by 800, it suffices to multiply 87468 by 8, which gives 699744; then annex two O's to the right of this product; we thus have a third partial product, 69974400, which we place below the two preceding products. For 800 numbers, equal to 87468, and, placed one under another, form evidently 100 divi- sions of 8 numbers, each equal to 87468, or 100 numbers, equal to the product of 87468 by 8 ; that is to say, 6997400 We MULTIPLICATIOx\. 31 could prove by a similar course of reasoning, that, in order to multiply by 5000, it suffices to multiply by 5, to annex three zeros to the product, and write the result, 437340000, thus ob- tained, below the three first products. Performing now the addi- tion of these four partial products, we find at last the total pro- duct, 511425396. N. B. — In practice, we dispense ordinarily with adding the zeros to the right of the partial products, found by multiplying by the figures in the tens, hundreds, places; but we write each partial product below the preceding product, advancing it one place to the right with reference to this product ', that is to say, we make its last figure occupy the same column which the figure by which we multiply, occupies. General Rule. — In order to multiply a number of several figures by a number of several figures — Multiply first the multi- plicand hy the units figure of the inultiph'er, after the rule of (19) ) multiply in the same manner the whole multiplicand, suc- cessively by the tens figure, by that of hundreds, &c., considered as simple units, and write the partial products one under the other, so that each one is advanced one column to the left, with reference to the preceding ; then add these products; the respJt will be the total required produx^t. 22. Often some of the figures of the multiplier are zeros, and then it is necessary to make some modifications in the arrange- ment of the partial products. OPERATION. Multiply 870497 By 500407 We multiply, first, th« whole multiplicand by 6093479 7, which gives for a product 6093479. Now, 3481988 as there are no tens in the multiplier, we pass 43524S5 to the multiplication by 4, the hundreds figure, 435002792279 which gives the product, 3481988 ; and, since it is necessary to make it express hundreds, we place it under the first product, advancing it two columns to the left. In like man- 32 MULTIPLICATION. ner, as there are no thousands, nor tens of thousands in tlie mul- tiplier, we pass to the multiplication by 5, the figure in the place of hundreds of thousands, and write the product, 4352485, under the preceding, advancing it three places to the left, with reference to that one. In general, when there are one or more zer&& between two significant figures of the multiplier, we advance the product corresponding to the significant figure, which is to the left of these zeros, one more column to the left than there are zer'os between the figures. In fine, in order to avoid all error on this subject, we must take care at each operation that the last figure of each partial product falls in the column of units of the. same order as that of the figure by which we multiply. 23. If one of the two factors of the multiplication, or both, are terminated by zeros, we abridge the operation by multiplying them as if the zeros were not there ; but we place them at the end of the product. EXAMPLE. OPEBATIOir. Multiply 47000 By .' 2900 After having multiplied 47 by 29, according to 423 the known method, we annex 5 zeros to the right 94 of the product, and thus obtain 136300000 for 136300000 the required product. For, if we had at first to multiply 47000 only by 29, it would be necessary to make the product express thousands (i. e.) units of the same species as the multiplicand; thus we ought to add 3 zeros. But to multiply a number by 2900, is (21) to take 100 times the product by 29 ; then we must add two new zeros. The same reasoning applies to all similar cases. 24. But little reflection on the method of multiplication will convince us of the necessity of commencing the operation on the ( VNIVrR \HtLlT[PX.ICATION. 33 right, at hast in the partial muhipllcaf ion hy each one of the figures of the multiplier^ because of the reservations of figures which we frequently make in multiplying each figure of the multipli- cand by each figure of the multiplier. But nothing prevents us from inverting the order of the partial multiplications by the different figures of the multiplier, as we can see in the following example. "We have here commenced the multiplication with operation. the hundreds figure of the multiplier 3 but in the 5704 following operation we have taken care to advance 487 the product one column to the right. In the same 22816 manner the third product is advanced one place to 45632 the right with reference to the preceding. Usage 39928 alone requires us to form the products from right to 2777848 left; it is also the more natural and convenient method. 25. We will close the subject of multiplication by the ex- planation of several properties, of which we will often have to make use. • 1st. Let it he required to multiply, '^^b hy 12, equal to 8 mul- tiplied hy 9. We say, that to multiply 345 by 72, is to multiply 345 by 9, and the result by 8. In order to establish this proposition without performing i\\Q operations, we must employ a mode of reasoning analogous to that employed in (21). To multiply 345 by 72, is to sum up 72 numbers equal to 345. But these 72 numbers, written one under another, form evidently 8 divisions of 9 numbers, equal to 345 ; then, after having multiplied 345 by 9, we must take this product 8 times. Thus, to multiply 345 by the product 72 of the two factors, 9 and 8, is to multiply 345 by 9, and the new result by 8. As 9 is itself equal to the product of 3 and 3, we can say, that to multiply 345 by 72, is to multiply 345 by 3, the result obtained by 3, and finally the new result by 8. As we can apply this rea- soning to other numbers, this general proposition results from it : to multiply a numher hy a product of tico or more numhers already formed, amounts to the same tltivg as multiplying the numher hy each one of the factors successively. 34 . Divisiox. 26. — 2d. In a multiplication of two factors, we can talce in- differently the first number for multiplicand, the second for mul- tiplier, or reciprocally In other terms — the product of two numbers is the same in whatever order we perform the operation. Thus, the product of 459 by 237, is equal to the product of 237 by 459. For, let us conceive unity written 1, 1, 1, T, 1, 459 times in a horizontal line, and let 1, 1, 1, 1, 1, us form 237 of these lines ; it is clear 1, 1, 1, 1, 1, that the sum of the units contained in 1, 1, 1, 1, 1, such a figure is equal to as many times the 459 units of a horizontal row as there are units in a vertical column, or in 237, (i. e.) that this sum is equal to the product of 459 by 237. But we can say also, that this sum is equal to as many times the 237 units of the vertical column as there are units in a horizontal row, or in 459 ; that is to say, is equal to the product of 237 by 459. Then, &c. If the nature of a question con'ducts to the multiplication of 76 by 5672, according to the proposition which we have just demonstrated, we would prefer to take the product of 5672 by 76, because in that case we would only have two partial products to form, while in the other operation we would have to form four of them. This proposition will be demonstrated for any number whatever of factors. DIVISION. 27. To divide one num.ber by another, is to find a third num- ber, which, multiplied by the second, will reproduce the first ; or, in other terms, being given the product and one of the factors, to determine the other factor. As in the multiplication of entire numbers, the product is composed of as many times the multipli- cand as there are units in the multiplier, we can also say, that, to divide one entire number by another, is to seek how many times the first number, considered as a product, contains the se- DIVISION. 35 cond, considered as multiplicand ; the number of times is then the multiplier. Finally, we can also say, that, to divide a numher hi/ another, is to divide the first number into as many equal parts as there are units in the second. These last two points of view, under which we sometimes con- sider division, pertain only to entire numbers, while the two first pertain to all possible numbers, whether entire or fractional. Nevertheless, the names given to the terms of division have been drawn from these last two points of view. Thus, the first number is called dividend, the second is called divisor, and the third quotient, from the Latin word quoties ; because it expresses how many times the dividend contains the divisor. It results, obviously, from the first two definitions, that when we have obtained the quotient, in order to make the verification of the operation, it will suffice to multiply the divisor by tha quotient ; and, if the operation has been exact, we will thus re- produce the dividend.* Reciprocally in multiplication, the product may be considered as the dividend, the multiplicand as the divisor, and the multi- plier as the quotient ; thus, we make the verification of multipli- cation by dividing the product by one of the factors ', and if the operation is exact, we ought to reproduce the other factor. These ideas being established, we pass to the explanation of the method of division. 28. In the same manner as multiplication can be effected by the addition of a number several times to itself, we can also find the quotient of a division by a series of subtractions. For, let it be required to divide 60 by 12. As many times as we can subtract 12 from 60, so many times is 12 contained in 60. Thus, the quotient is equal to the number of subtractions which we can make before the dividend is exhausted. 30 DIVISION. 60 In this example, as we are obliged to make 5 1^ subtractions, it follows that the quotient is 5. But 1st rem. 48 this manner of performing the division would be 12 too long in practice, especially if the dividend was 2d rem. 36 very great in comparison with the divisor. It is 12 in the art of abridging the operation that the or- 3c? rem. 24 dinary method of division consists: 12 29. From the fact that we know by heart the pro- 4it7i 7'em. 12 ducts of two numbers of a single figure, we can 12 determine easily the quotient of the division of a bth rem. number of one or two figures by a number of a single figure. For example, 35 divided by 7, gives for a quotient 5. This we know, because we know that 7 times 5 give 35. We say, also, in this example, that the 7th of 35 is 5, because 7 times 5 make 35. Suppose, again, that we have to divide 68 by 9. As 7 times 9, or 63, and 8 times 9, or 72, comprise 68 between them, it follows that 68, divided by 9, gives for the quotient, 7, with a remainder, 5 ; or the 9th of 68 is 7, .with a remainder, 5. In like manner, 47 contains 8, 5 times, with a remainder 7 ; because 5 times 8 gives 40, and 6 times 8 gives 48. We will see farther on what is to be done with the remainder, when the divisor is not contained exactly in the dividend. 30. Let us consider the case in which the dividend is com- posed of any number of figures, the divisor containing but a single figure. Divide 6766453 by 8. 6766453)8 64 845806 36 32 46 Proof by multiplication. 845806 40 8 64 6766448 64 6 053 6766458 48 5 DIVISION. 37 After having written the divisor to the right of the dividend, and separated them by a vertical line, we draw below the divisor a horizontal line. This arranged, We see at once that, if we place (mentally) to the right of the divisor, 8, five zeros, (i. e. ) multi- ply it by 10,000, then six zeros, or multiply it by 100,000, the two products, 80,000 and 800,000, are the one smaller, the other greater than the dividend. Whence we conclude that the quo- tient demanded is comprised between 10,000 and 100,000 ; that is to say, is composed of six figures, and that thus the highest units of the quotient are hundreds of thousands, of which we must find the figure. Now, as the product of the divisor by the figure sought cannot give units of a lower order than hundreds of thousands, it fol- lows, that this product is contained wholly in the 67 hundreds of thousands of the dividend; and if we divide 67 by 8, which gives the quotient 8 for 64, and the remainder 3, we can affirm that the figure of hundreds of thousands in the quotient is 8. In fact, 800,000 times 8 gives 6,400,000, a number which can be subtracted from the dividend, 6766453 ; while 900,000 times 8, or 7,200,000 cannot be so subtracted. The figure 8 being thus determined, we place it under the divisor; then we subtract the product 8 by 8, or 64 from 67, and conceive the remaining figures of the dividend to be written to the right of the remain- der 3, which gives 366453 for the total remainder of this first operation. (This first operation is evidently nothing more than subtracting from the dividend 800,000 times the divisor, or is equivalent to 800,000 successive subtractions of the divisor 8.) It would seem necessary to write on the right of the quotient already obtained, five zeros, in order to give it its true value ; but we avoid this by the arrangement which we will make of the following figures of the quotient. We must now determine the figure of tens of thousands of the quotient. Since the product of the divisor by this figure cannot give units of an order inferior to tens of thousands, it is con- tained wholly in the 36 tens of thousands of the remaining divi- dend. It suffices then to bring down to the side of the remain- 4 38 DIVISION. der, 3, the following figure, 6, of the dividend ; then to divide 36 bj 8, which gives the quotient, 4, for 32, and the remainder, 4. We write this quotient, which expresses necessarily the tens of thousands of the whole quotient, on the right of the first quo- tient, 8 ; then, after having subtracted 4 times 8, or 32 from 36, we bring down to the right of the 4, the next figure of the divi- dend, which gives 64. (This new operation, which amounts to subtracting 40,000 times 8, or 320,000 from 366,453, is equiva- lent to 40,000 new successive subtractions of the divisor, 8.) In order to obtain the ones of thousands of the whole quo- tient, we divide 46 by 8 ; the quotient is 5 for 40, and the re- mainder, 6. We write this new quotient, 5, to the right of the first two ; then, after having subtracted 5 times 8, or 40, from 46, we bring down to the right of the remainder, 6, the next figure, 4, of the dividend, which gives 64. (This third operation is equivalent to 5000 successive subtractions of the divisor, 8.) In order to obtain the figure of hundreds of the total quotient, we divide 64 by 8, which gives 8, and 0, for remainder; we write the new quotient to the right of the three first ; then, after having subtracted 8 times 8, or 64 from 64, we bring down to the right of the remainder, 0, the next figure of the dividend, which gives 05, or simply 5. Here a particular case presents itself; as the new partial divi- dend, 05 or 5, which is to give the tens of the quotient, is less than the divisor, 8, we must conclude that the total quotient has no tens, (and in fact the remaining dividend is 53, a number less than 10 times 8, or 80.) We place, then, a in the quotient to the right of the four figures already obtained, in order to replace the tens which are wanting, and preserve the relative value of the preceding and following figures ; we then bring down to the right of the re- mainder, 5, the next and last figure of the dividend, and con- tinue the operation. The quotient of 53 divided by 8, being 6 for 48, we write this figure to the right of the first five quotients already found; we then subtract 48 from 53, which gives at last 5 for the remainder of the entire operation ; and the required DIVISION. 39 quotient is 845806, which we can easily verify by multiplying 8 by 845806, or rather 848806 by 8, and adding the remainder, 5, to the product thus obtained. (All the operations which have been performed in eifecting this division are equivalent, evidently, to 800,000, then 40,000 subtractions, then 5000, then 800, then 6, or 845806 successive subtractions, in which the divisor, 8, is constantly the number to be subtracted.) 31. We will not establish for the case of division which we have just discussed, a general rule founded on the preceding reasoning, because there exists (for this case only) a practical method, more convenient and more simple in reference to the arrangement of the calculations. Let us take again the above example : 6766453 to be divided by 8. Quotient, 845806; remainder, 5. We know already (No. 27) that to divide a number by 8, or to seek how many times 8 is contained in this number, amounts to dividing the number into 8 equal parts, or taking the eighth of it. This being fixed, taking the two first figures to the left of the dividend, 67, we say, the eighth of 67 is 8, with the re- mainder, 3. We write the quotient, 8, under the figure^ 7, of the dividend; then we place, mentally, the remainder, 3, ex- pressing 3 hundreds of thousands, or 30 tens of thousands, to the left of the figure, 6, of the dividend, which expresses also tens of thousands ; we say, as before, the eighth of 36 is 4, with remainder 4. We write the second quotient, 4, to the right of the first; placing again, mentally, the remainder, 4, expressing 4 tens of thousands, or 40 thousands, to the left of the thousands figure, 6, of the dividend ; we say, again, the eighth of 46 is 5, with the remainder, 6 ; we write the third quotient, 5, to the right of the preceding; continuing in the same manner, we say, again, the eighth of 64 is 8, with the remainder, 0, and we write the fourth quotient, 8, to the right of the third. The eighth of 05, or 5, is 0, with the remainder, 5 ; we write this fifth quotient to the right of the fourth. Finally, the eighth of 53 is 6, with 40 DIVISION. the remainder, 5 ; we write to the right of the fifth quotient the sixth and last partial quotient, which thus falls beneath the units figure of the dividend^ and we have for the result the quotient, 845806, with the remainder, 5. Second example : 8230200409 to be divided by 6. Quotient, 1371700068; remainder, 1. Here, the first figure on the left of the dividend being greater than the divisor, we see that the quotient ought to have units of the same order as those of the figure 8 ; and we say, the sixth of 8 is 1, which we write under the figure 8, with the remainder, 2 ; then the sixth of 22 is 3, which we place to the right of the figure 1, with the remainder, 4. The 6th of 43 is 7, with the remainder, 1. The 6th of 10 is 1, with the remainder, 1. The 6th of 42 is 7, with the remainder, 0. The 6th of is 0, with the remainder, 0. The 6th of is 0, with the remainder, 0. The 6th of 4 is 0, with the remainder, 4. The 6th of 40 is 6, with the remainder, 4. Finally, the 6th of 49 is 8, with the remainder, 1. The required quotient is then 1371700068, with the remain- der, 1. It is very important to understand thoroughly this method, because it finds its application in the case of division, which is yet to be discussed. We will observe, moreover, that when we know by heart the multiplication table as far as the number 12, we can obtain very easily, by the same method, the 10th, 11th, and 12th, of any number whatever. EXAMPLES. 1st. 897614708497, to be divided by 12. Quotient, 74801225708, remainder, 1. (The 12th of 89 is 7, with remainder, 5 ; the 12th of 57 is 4, with remainder, 9; the 12th of 96 is 8, remainder, 0; &c., &c.) DIVISION. 41 .2d. 23054273896, to be divided by 11. Quotient, 2095843081 ; remainder, 5. (The 11th of 23 is 2, with remainder, 1 ; the 11th of 10 is 0, with remainder, 10; the 11th of 105 is 9, with remainder, 6; &c., &c.) As to the division by 10, instead of applying the method, it is simpler to separate in thought the last figure to the right of the dividend. The part to the left expresses the quotient, and this last figure separated (which can be 0), is the remainder of the division. This is an evident consequence of the system of nu- meration. Thus, the 10th of 2710548 is 271054, and the remainder, 8 ; the 10th of 863005704 is 86300507, and remainder, 4; the 10th of 3805670 is exactly 380567 ; results which can be found also by the application of the method above. 32. Let us pass to the case in which the given numbers being both composed of several figures, the quotient is to have one only. This case deserves, of itself, particular attention ; and it will serve us, besides, as a basis for the development of the general case. Let it be given to divide 730465 by 87467. 87467 730465 699736 "^729 We remark, first, that the product of the divisor by 10, or 874670, is greater than the dividend ; thus, the quotient sought is less than 10, and can have only one figure. In the second place, the product of 8 tens of thousands of the divisor by the figure sought, as it cannot give units of an order inferior to tens of thousands, must be found wholly in the 73 tens of thousands of the dividend ) whence it follows, that the figure sought cannot exceed the quotient of the division of 73 by 8. We are then conducted to the division of the part, 73, on the left of the dividend, by the first figure, 8, of the divisor, 4* 42 DIVISION. wHcli gives the quotient, 9. But 9 is evidently too large; for, in the multiplication of the whole divisor by this figure, we find, in multiplying the thousands figure, 7, of the divisor, by 9, 63 units of this order, and, consequently, 6 tens of thousands, to be added to the 72 tens of thousands, product of the first figure, 8, of the divisor, by the same figure, 9 ; which would give 78 tens of thousands, a number greater than the dividend It is not necessary, then, to try any figure higher than 8, as figure of the quotient required. Effecting the multiplication of 87467, by 8 (which we have placed under the divisor), we ob- tain a product of 699736, less than the dividend ; which proves that the quotient, 8, is correct. On subtracting this product from the dividend, as the operation shows, we find for remainder, 80729. Again, divide 974065 by 189768. 974065 1 189768 948840 I 5 25225 As the dividend and the divisor are composed of the same number of figures, it is clear that the quotient ought to have only one figure; and in order to find it, we divide, first, the first figure on the left of the dividend, 9, by the first figure, 1, on the left of the divisor. The quotient is 9 ; but this figure, and the next lower, 8, 7, 6, are too large, if we consider the two first figures, 18, on the left of the divisor; for the products of 18, by 9, 8, 7, 6, being 162, 144, 126, and 108, all surpass the 97 tens of thousands of the dividend. This leads us to try the figure 5. On multiplying the divisor by 5, we have the product, 948840, which, subtracted from the dividend, gives for remainder, 25225, a number smaller than the divisor ; which proves that the quo- tient, 5, is not too small. 33. In the two preceding examples, we have been able to de- termine pretty easily what was the true figure of the quotient ; but as this is not always the case, it is important to have a me- DIVISION. 43 thod of ascertaining, without effecting tlie product of tlie divisor by the quotient, whether the trial figure is the true one. We will now develop this method. Particular method of trial. Given, 556428, to be divided by 69784. 556428 I 69784 488488 I 7 • 67940 The division of 55 (the two first figures on the left of the dividend), by 6 (the first figure on the left of the divisor), gives 9 for quotient, with the remainder, 1. In order that 9 may not be larger than the quotient sought, 9 times the divisor must be less, or, at most, equal to the dividend; or, which is the same thing, the 9th of the dividend must be greater, or at least equal to the divisor. We then commence to take the 9th of 556428, after the me- thod of (31). We find for the two first figures on the left, 61 tens of thousands, a number less than 69 tens of thousands of the divisor, which shows that the 9th of the dividend is less than the divisor; 9 ought then to be rejected. We next try 8. We find for the three first figures of the 8th of the dividend, 695 hundreds less than the 697 hundreds of the divisor; then 8 is too large. We now try 7. The first figure of the 7th of the dividend is 7, greater than 6, the first figure of the divisor. Whence it follows, that the 7th of the dividend is greater than the divisor ; or, in other terms, that the product of the divisor by 7, is less than the dividend. Thus, the figure 7 is the true one. Multiplying the divisor by 7, and writing the product, 488488, below the dividend, then effecting the subtraction, we obtain the remainder, 67940, a number smaller than the divisor. Another Example. Given, to divide 1148367 by 169987. 1148367 1019922 "128445 1 69987 6 44 DIVISION. The division of 11 by 1, would give 11 for a quotient; but the required quotient cannot be greater than 9, siq.ee the TS-JUgOU* After having formed the product of the five denominators, 8, 11, 13, 25, and 43, which gives for the common denominator of the transformed fractions, 1229800, we divide successively this product by each one of the denominators, and we obtain the five quotients, 153725, 111800, 94600, 49192, 28600, which we place respectively below the five proposed fractions ; after which, we multiply the numerator of each fraction by the quotient which corresponds to it ; and we obtain thus the difi"erent numerators. As to the common denominator, it is, as we have said above, equal to 1229800. The reason of this manner of proceeding is easily perceived, for the number, 1229800, being the product of the five denomi- nators, the quotient, 153725, of the division of 1229800 by 8, expresses necessarily the product of the four other denominators, 11, 13, 25, 43. In the same manner, 111800, being the quotient of the divi- sion of 1229800, by the second denominator, 11, is equal to the product of the four other denominators, 8, 13, 25, and 43 ; and the same reasoning applies to the other quotients. This method is, moreover, much more expeditious, than if, for each fraction, we performed the multiplication of the denominators of the four others. But it is only really advantageous when there are more than three fractions to be reduced to the same denominator. 64 FRACTIONS. 45. There is a case in which the reduction to the same deno- minator can be performed in a very simple manner^; that is, when the greatest of the denominators is exactly divisible by each one of the others. Let the fractions be, for example, 2 3 5 7 23 3 4 g 1'2 3 5 12 9 6 3 1 24 27 30 21 23 3BJ "3 6? 3 6? 3gJ 3B' It is easy to see that 36, divisible by itself, is also divisible by each one of the four other denominators, 3, 4, 6, and 12. This being fixed, we effect successively these divisions, and place the quotients, 12, 9, 6, 31, below the four first fractions; after which, we multiply the numerator of each one of them by the quotient which corresponds to it; the fraction, ||, remains as it was, and all the fractions are reduced to the denominator^ 36. Sometimes, although the greatest denominator is not divisible by all the others, we perceive that, by multiplying it by 2, 3, 4 we obtain a product exactly divisible by all the denominators. This afibrds us, likewise, a means of simplification. Let the fractions be, 2 4 i H 1 3 IB 1 7 24 II 18 9 G 4 3 2 H fl ^1 52 72- H H The denominator, 36, is divisible separately by 4, 12, and 18, but is not divisible by 8 nor by 24 ; but, if we double it, we ob- tain 72, a number exactly divisible by each one of the denomi- nators. This being fixed, we form the quotients of 72 by each one of the denominators, and place them respectively below the frac- tions ; we then multiply the numerator of each one of them by the quotient which corresponds to it; all these fractions will have 72 for common denominator. FRACTIONS. 65 Formation of the Least Common Denominator of Several Fractions. 46. The simplifications which we have just explained, require some practice to see when they can be applied ; but there is a direct means of obtaining, in all cases, the Least Common De- nominator of several fractions. To do this, we must find the least common multiple of the de- nominators; that is, the least number divisible by all of them. To do this, decompose the numbers into their smallest possible factors ; that is, prime factors, or factors divisible only by them- selves and unity. Then form the product of all these prime factors, common or not common, to the numbers. We obtain thus a result, evidently divisible by all the numbers ; and it is, besides, the smallest number so divisible ; for, any number con- taining one of the prime factors a smaller number of times than one of the given numbers, would not be divisible by that one of these numbers which contained thfs factor a greater num- ber of times. (A more thorough discussion of this we will give under the chapter on the properties of numbers). Applying the above to the last example, we have, 4 8 12 18 24 36 2.2 2.2.2 2.2.3 2.3.3 2.2.2.3 2.2.3.3 Having thus arranged the numbers and their prime factors, we see that 2.2.2.3.3 is evidently the least common multiple. Performing the multiplication, we obtain 72, as before. Let the fractions be for a new example, 11 if 1 7 2 5 37 83 29 233 450 1? 24 2F 44 T45 IIH the numerators of which do not contain, at least apparently, prime factors (as 2.3.5 ....), which may be, at the same time, con- tained in the corresponding denominators; otherwise, it would be necessary to suppress these factors in the two terms. 6* 66 FRACTIONS. Kecomposing the denominators, we find for results, 3.5 1 2.3.3 1 2.2.2.3 1 2.2.7 1 2.2.11 1 2.2.5.7 1 7.5.5 1 2^2.2.2.2.5.3; which gives for the least common multiple, 2.2.2.2.2.3.3.5.5.7.11, or 554400; which is the least common denominator to be given to all the fractions; a number far less than that which we would obtain by applying the general rule in No. (44). Nothing more remains now but to determine the numbers by which we are to multiply the numerators, in order to obtain the numerators of the new fractions ; and for this it is necessary, as we have already seen, to divide 554400 by each one of the given denominators. Kelations of Magnitude among Several Fractions. We have here some applications of the preceding transforma- tions. 47. Question 1st. — Of the two fractions^ | and ^^^ which is the greater ? "VVe cannot, at first sight, answer this question ; because, though on the one hand the unit in the second fraction is divided into a greater number of parts than in the first, on the other hand, we take more of these parts, since the numerator, 7, is greater than 3. But we remove the difiiculty by reducing them to the same denominator; for it is evident that of two fractions which have the same denominator, the greater is that which has the greater numerator. This reduction effected, we obtain |g for the first fraction, and |J for the second; the fraction, |, is the greater of the two. We find, in the same manner, that of the three fractions, ^, A' T^3' ^^ greatest is f.^, the smallest -{\) for, being reduced to the same denominator, they become, respectively, f^fj, tVijVj FRACTIONS. 67 We could equally well reduce the fractions to the same nume- rator (by applying to the numerators what has been said concern- ing the denominators) ; and of these fractions the greatest would be that which would have the smallest denominator; since, the parts being greater, we take the same number of them. But the first method has the advantage of making known, at the same time, the differences which exist between the fractions, compared two and two. 48. Question 2d. — What change do we produce in a fraction^ hy adding the same number to its two terms ? Let the fraction be -^^^ for example, to both terms of which we add 6 ; jf is the resulting fraction. If now we reduce these two fractions to the same denominator, the first becomes ^||, and the seeond ^f |. The proposed frac- tion is then increased in value. In order to give a reason for this fact, we observe that, unity being equal to i|, the excess of unity above ^^^ is expressed by f^ ; in the same manner, the excess of unity above || is expressed by f^. The numerators of these two differences are the same, which should be the case; for, 18 and 13, having been formed by the addition of 6 to the two terms, 7 and 12, it follows, that there is the same difference be- tween 18 and 13, as between 7 and 12. But the difference, -f^, is necessarily less than the difference, y^^, since the first denomi- nator is the greater, and the numerators are equal; then the fraction, ||, differs less from unity* than the fraction, -^^^ conse- quently, the first is greater than the second. We see, moreover, that the greater the number added to the two terms of the fraction, y^^, the smaller the difference between unity and the new fraction ; since the numerator of this differ- ence, being always 5, the denominator becomes greater and greater. As this same reasoning can be applied to every other fraction, we can draw the conclusion that if to the two terms of a fraction we add thjB same number, the resulting fraction is greater than the given fraction ; and it is greater, the greater the number added. 68 FRACTIONS. Conversely, by the same reasoning, a fraction is diminished in value when we subtract the same number from its two terms. N. B. The contrary would take place, if the fractional number was greater than unity, as i|. Adding 8 to the two terms, we would have | j less than ||. For, II exceeds unity by ^^ only, while i| surpasses unity by ■f^, greater than ■^^. We have thought it necessary to enter into some details upon this proposition, in order to prevent beginners from confounding this with (43), when we multiply or divide the two terms of a fraction by the same number. Keduction op a Fraction to its Simplest Terms. 49. It happens often, in the calculus of fractions, that we are led to a fraction expressed by large numbers ; now, the greater the numerator and denominator, the greater trouble we have to form a just idea of the fraction. For example, the fraction, ||, indicates, that we must divide unity into 15 equal parts, and take 12 of these parts. But 12 and 15 being, at the same time, divisible by 3, if we perform the divisions, there results |, a fraction equivalent to the one given ; then, in order to form an idea of it, it suffices to conceive the unit divided into 5 equal partg, and to take 4 of them, which is much simpler. When then we have a fraction, the terms of which are quite large, it is best to reduce it, if possible, to a fraction whose terms are smaller. The first method which presents itself is to divide the two terms by the numbers, 2, 3, 4 .... as long as that is possible. 1st. Let the fraction, |J|, he given. The two terms of this fraction are evidently divisible by 4 ; and, in effecting the divi- sion, we obtain |^ ; but the two terms of this are divisible by 9 ; and this new division gives for a result, |, which cannot be far- ther reduced. FRACTIONS. 69 This example presents no difficulty ; but this is not always the case, especially when the two terms of the given fraction are composed of three or more figures; for it can happen that a prime factor of two or three figures is common to the two terms of the fraction, without our being able to find it by mere inspec- tion. Hence, we see the necessity of having a general method of reducing a given fraction to the most simple expression possi- ble. This method we will now discuss. It is called the method of the greatest common divisor. 50. Wo commence by establishing several preliminary no- tions. A number is called the multiple of another number, when it contains it a certain number of times, as we have already seen. Reciprocally, the second number is called a suhmultiple, or an aliquot part, or simply a divisor of the first. We call a prime number a number which is only divisible by itself, and by unity, which is a divisor of every number. Thus, 2, e3, 5, 7, 11, 13 ... . are prime numbers; but 4, 6, 8, 9, 12, are not prime numbers; since they have the divisors, 2 and 3. Two numbers are said to be prime with respect to each other, when they have no other common divisor except unity ; thus, 4 and 9, 7, 1, and 12, are numbers prime with respect to each other; 8 and 12 are not, since they are divisible at the same time either by 2 or 4. First Principle. — Every number, which exactly divides an- other number, divides also any multiple whatever of this second number. For example, 24 being divisible by 8, and giving for a quo- tient 3, 5 times 24, or 120, divided by 8, will give (No. 40) for quotient, 5 times 3, or 15. In the same manner, 60 being divi- sible by 12, and giving for quotient 5, 7 times 60, or 420, divided by 12, will give for quotient 7 times 5, or 35. Second Principle. — Every number, decomposed into two parts, both divisible by a second number, is itself divisible by 70 FRACTIONS. this second number. For the quotient of the division of the total being equal to the sum of the two partial quotients, if these two partial quotients are entire, their sum, or the total quotient, must be entire. Third Principle. — Every number which divides exactly a whole, decomposed into two parts, and which divides one of these parts, ought to divide also the other part. For the total quotient being equal to the sum of the two partial quotients, if one of these partial quotients is fractional, it would follow that an entire number would be equal to a fractional number ; which would be absurd. 51. So much being established, let the two numbers., 360 and 276, be given, of which we propose to determine the greatest com- mon divisor, or the greatest number which will divide them both exactly. It is at once evident that this greatest common divisor cannot exceed the smaller number, 276; and as 276 divides itself, it follows, that if it will divide 360 also, it will be the greatest common divisor sought. Attempting this division of 360 by 276, we find for quotient, 1, and remainder, 84 ; then, 276 is not the greatest common divisor. We say, now, that the greatest common divisor of 360 and 276, is the same as that which exists between the smaller number, 276, and the remain- der of the division. For the greatest common divisor sought, since it ought to divide 360, and one of its parts, 276, divides necessarily the other part, 84, (50) ; whence, we can conclude at once, that the greatest common divisor of 360 and 276, cannot exceed that which exists between 276 and 84 ; since it must divide these two numbers. In the second place, the Gr. C. D. of 276 and 84, dividing the two parts, 276 and 84, of 360, divides necessarily this number; being the exact divisor of 360 and 276, it cannot exceed the greatest C. D. of 360 and 276. Whence, we see, that the G-. C. D. of 360 and 276, and the G. C. D. of 276 and 84, cannot be greater than each other; then they are equal. FRACTIONS. 71 Thus, tLe question is reduced to seeking the greatest C. D. of 276 and 84 ; numbers simpler than 360 and 276. "We now reason on 276 and 84, as we have about the primitive numbers; that is, we try the division of 276 by 84; because, if the division is exact, 84 will be the G. C. D. of 276 and 84, and consequently of 360 and 276. EiFecting this new division, we have 3 for quotient, and 24 for remainder; then 84 is not the Gr. C. D. sought. But, by analogous reasoning to that above, we can prove that the Gr. C. D. of 276 and 84. is the same as that of the first remainder, 84, and the second remainder, 24. The question is then reduced to finding the G. C. D. of 84 and 24; we divide 84 by 24, and obtain 3 for quotient, and 12 for remainder; then 24 is not the G. C. D. ; but, as this G. C. D. is the same as that of 24 and 12, we divide 24 by 12 ; we find an exact quotient, 2 ; thus, 12 is the greatest C. D. of 12 and 24, hence of 84 and 24, of 276 and 84, and, finally, of 360 and 276, or the G. C. D. sought. In practice, we arrange the operation thus : 13 3 2 360 84 276 24 84 12 24 12 After having divided 360 by 276, we place the quotient, 1, above the divisor, and a remainder, 84 ; we write this remainder to the right of the less number, 276, and we divide 276 by 84, placing the quotient, 3, above the divisor, and the remainder, 24, to the right of the 84, and so on for the rest. General Rule. — In order to find the G. C. D. of two num- hers, divide the greater number hy the less; if there is no remain- der, the smaller number is the G. C. D. If there is a remainder, divide the less number by this remain- der } and if this division is without remainder, the first remain- der is the G. 0. D. If this second division gives a remainder, divide the first re- mainder by thf i'wnd, and continu* always to divide the pre- 72 FRACTIONS. ceding remainder hy the last remainder, until the division becomes exact; then the last divisor employed will he the G. 0. D. sought. If the last divisor is unity, it is a proof that the two numbers are prime with respect to each other. Reciprocally, if two num- bers are prime with each other, and if we apply the method above, we will find necessarily a final remainder equal to unity. For, according to the nature of the method, the remainders go on diminishing ; besides, we cannot obtain a remainder, nothing, before having obtained a remainder, 1 ; since the divisor, different from unity, which gave this remainder nothing, would be the common divisor of the two numbers. Thus, we must, necessa- rily, after a smaller or greater number of operations, obtain unity for a remainder. 52. We give now the application of this method. Reduce the fraction, |||, to its simplest form. 2 18_5 I /, 00 37 16 00 We find, for the greatest common divisor, 37, and, dividing 999, and 592, by 37, we have ^f , for the value of |||, reduced to its least terms. If we can find no common divisor greater than unity for the terms of the fraction, the fraction is irreducible, its terms being prime with each other. Remark. — If, in the operation for the common divisor, we arrive at a prime number for a remainder, as, for example, 5 or 7, we can conclude at once that unless this prime remainder exactly divides the last divisor, the two primitive numbers have no common divisor greater than unity. The reason of this is obvious. We will return once more to the operation of the 1 1 999 592 407 407 185 37 999 37 592 259 27 222 FRACTlOxNS. 73 greatest common divisor, as it is one of the most important in the arithmetic. Second example, f |§g|. We find for G-. C. D. 1261, respective quotients, 29 and 23; thus, || is the fraction reduced to its simplest terms. 53. From what precedes, it results, that if from the two terms of a fraction, we subtract the same multiples of the two terms of the equivalent irreducible fraction, the resulting fraction is also equivalent to the given one. Let us take, for example, the fraction, ^|, which, reduced to its least terms, according to the method indicated in the preceding article, is equal to |. If, from the two terms, 18 and 24, of the given fraction, we subtract four times 3, or 12, and four times 4, or 16, we obtain a new fraction, |, which, expressed in simpler terms than those of the given fraction, is equal to it. For, in suppressing the factor, 2, common to 6 and 8, we find |, as for the first fraction, ||. It is easy to explain this result. For, if ^| is equal to |, an irreducible fraction, of which the two terms are prime with each other, 18 and 24 must be the same multiples (6 times 3, and 6 times 4), of the two terms of the fraction, | ; and when from 18 and 24 we subtract four times 3, and four times 4, we obtain difi'erences, twice 3, and twice 4, which are also the same multi- ples of 3 and 4; whence results a new fraction, |, equal to |. It would seem that this proposition ought to furnish a means of simplifying a fraction ; but we see that this means would be alto- gether illusory, since it supposes the irreducible fraction known, to which the given one is equivalent. N. B. We would remark here, that we subtract from the two terras of the fraction two different numbers, and not the same number as in (48). We pass, now, to the four fundamental operations upon frac- tions. 7 74 FKACTIONS. Addition of" Fractions. 64. The addition of fractions has for its object to find a single fraction which shall express the value of the sum of several fractions. There are two cases ; the fractions to be added are either of the same species j that is to say, have the same denominator ^ or of different species. In the first case, we sutti up the numerators^ and then give to this sian the common denominator. In the second case, we reduce the fractions to the same deno- minator ; after ichich we operate as in the first case. The reason is obvious, since the denominator is a sign indicating the value or species of the units to be added, and the numerator the num- ber of these units. ^. 2 ^3 ^4 2+3+4 9 Thus, n+n+n= iir-==iT In the same manner, 5_ . 1 , 1 , i _ 5+2+7+4 _18 23 "^23 "'■23 ■^23"" 23 "~23 Let it be given, now, to add the three fractions, 2 3 7 3 4 H 8 6 3 16 18 21 54 24 24 After having reduced these fractions to the least common de- nominator, 24, (No. 46), we add the numerators, 16, 18, and 21 ', we then give to the sum, 55, the denominator, 24. We have thus || for the result required. 55. This last example ISads to a fractional expression, ^|, greater than unity, which gives rise to a new operation. We have seen that unity is equivalent to ||, or twenty-four twenty-fourths; whence, it follows, that as many times as 55 contains 24, so many units there are in 4|. Now, dividing 55 FRACTIONS. 75 by 24, we have for a quotient, 2, with remainder, 7 ; thus, j-^ is a number composed of 2 units and ■^^. In general, when we obtain a fractional result, of which the numerator exceeds the denominator, in order to extract the whole number contained in this expression, ice must divide the numerator hy the denomina- tor. The quotient lohich loe obtain represents the entire number ^ and the remainder is the numerator of the fraction which is to be added to the entire number, (43). By this mode, we find, 17 1 5 . 153 10 3 ini • 65 4 — 73 1 Reciprocally, when we have an entire number joined to a frac- tion, in order to form a single fractional number, we must mul- tiply the entire number by the denominator, add the product to the numerator, and (jive to the sum the denominator of the fraction. For example, _2 3x5 + 2 17 ,, ^ 11x12+7 139 „ Subtraction of Fractions. 66. The subtraction of fractions has for its object to find the excess of a greater fraction over a smaller. If the two fractions have the same denominator, we subtract the smaller numerator from the greater, and give to the difference the common denominator. If they have not the same denominator, we reduce them to such as have ; after which, we proceed as in the first case. Given, to subtract -^^ from |i ; there remain y^^, or 1. In the same manner, ^| — ^'^^ = 1 J=-J'^. Given, to subtract | from |. These two fractions give ^| and §i) by reduction to the same denominator; and we have 21 16_ 21— 16 5 24 24 ~" 24 ~24* 19 13_ 19x 17 — 13x20 63 ' 20 17~ 20x17 ""340' 76 FRACTIONS. We can have an entire number joined to a fraction, to be sub- tracted from an entire number joined to a fractiom ; or, as they are called, a mixed numher, to be subtracted from a mixed number. Griven, for example, to subtract b\l from 12|. 12|=12f|=ll|< 5H= 5J|= 5JI 6i|. We commence by reducing the two fractions to the same de- nominators, which gives || for the first, and || for the second. Then, as we cannot subtract |4 from |J, we take from the entire part, 12 of the greater number, one unit, which we add to the II, making |i ) we then subtract || from |^, and have for a remainder, 4^. Passing to the subtraction of the entire numbers, we regard the greater number as diminished by unity, and subtract 5 from 11, which gives 6. We have thus 6|| for the required result. The same result could be obtained by reducing the mixed numbers to single fractions by last article, and then following the rule given for subtraction of fractions. Multiplication of Fractions. 57. Multiplication has for its object in general, two numher s being given to form a third number, which is compounded with the first number in the same manner as the second number is compounded with unity. This being established, we distinguish three principal cases in the multiplication of fractions. We can have, 1st. A fraction to be multiplied, hy an entire number. Given, for example, ^^^ to be multiplied by 5. According to the definition above, since the multiplier, 5, con- tains 5 times unity, it follows, that the product ought to be equal to 5 times y^^, or 5 times as great as -^^. Now, we have seen in FRACTIONS. 77 (43), that we render a fraction 5 times greater by multiplying its numerator by 5. We thus have =^ or — , for the required product. Then, in order to multipli/ a fraction hy an entire number, we must m,ultiply the numerator hy the entire number, and give to the product the denominator of the fraction. Given, to multiply jg by 9. We obtain f | for the product, or 5y®g, or 5^. This result can be obtained more simply thus. For, by (43), we can divide the denominator by 9, instead of multiplying the numerator. And we find thus, y , or 51, for the required product. We can only apply this last method, when the denominator is divisible by the number. The established rule is always appli- cable. Usage alone renders us familiar with these simplifications. 2d. To multiply an entire number by a fraction. Example. — 12 to be multiplied by ^. Since, in this case, the multiplier, ^, is equal to 4 times the 7th part of unity, the product ought to be equal to 4 times the 7th of 12. Now, the 7th of 12 is equal to ^^ ] and, in order to render this 4 times as great as L^^ -^^e must multiply the nume- rator by 4 ; we thus obtain %f, or 6f , for the required product. Then, to multiply an entire number by a fraction, we multiply the entire number by the numerator, and give to the prodtict the denominator of the fraction. Thus, 29 X ^=:i^ = 25|. We might find this last result by dividing 24 by 6, and multi- plying the result by 5. But, we repeat, these simplifications are not always possible. 7=^ 78 FRACTIONS. 3d. A fraction to be multiplied hy a fraction. Example. — Griven, to multiply | by |. The reasoning is analogous to that of the preceding case; since I is equal to 5 times the 8th part of unity, the product ought itself to be 5 times the 8th part of the multiplicand, |. Now, in order to take the 8th of |, we must (43) multiply the denominator by 8, which gives Z^-; and in order to obtain a fraction 5 times as great as ^^, we must multiply the numerator by 5; which gives ^j for the product required. Then, to multiply one fraction hy another, multiply numerator hy numerator J and denominator hy denominator ; then make the second product denominator of the first. We find, thus, 12^ 6 ~ 72* , ^ 8 3 24 2 ^^^r5^T = 6o = y- N. B. In the two preceding cases, the product is always less than the multiplicand ; and this ought to be the case, since the operation is really taking a part of the multiplicand indicated by the fractional multiplier. 58. Finally, one of the factors of the multiplication, or both of them, may be mixed numbers. These numbers are equivalent, respectively, to the improper fractions, (the fractions greater than unity being called improper fractions), ^^ and "^^ ; per- forming the multiDlication of these by the rule above, we obtain ^§fSor453V. We could effect this multiplication by parts ; that is to say, multiply, first, 7 by 5, | by 5, 7 by |, and | by | ; then add these four products ; but this method is much the longest. FRACTIONS. 79 Division op Fractions. 59. Division has for its object: Given.y the product of two factors, and one of the factors to determine the other. It results, obviously, from this definition, and from that of multiplication, that the first number, called dividend, is com- pounded with the third, called quotient, in the same manner that the divisor is compounded with unity. This established, in the division as in the multiplication of fractions, we distinguish three principal cases. 1st. To divide a fraction hy an entire number. Given, for example, |, to be divided by 6. Since the divisor is 6 times unity, it follows, that the dividend, f , is equal to 6 times the required quotient; then, reciprocally, the quotient ought to be the 6th part of f . Now, in order to take the 6th part of a fraction, or to obtain one 6 times as small, we must (43) 5 5 multiply the denominator by 6 ; thus, we obtain — -: -, or t-^, for the required quotient. Then, to divide a fraction by an entire number, multiply the denominator of the fraction hy tlie entire number, leaving the numerator the same. Thus, 11 divided by 8 = -^ = |1. 23 23 In the same manner, ^ divided by 12 = -^^. oU 360 The quotient of ^f by 6, is -^f^ ; but we can efiiect the divi- sion of Jl by 6, by taking the 6th of the numerator, which gives ■^j^; the same with -^f^, when we suppress the factor, 6, com- mon to the two terms. Then we add to the above rule, or divide the numerator hy the divisor, when that is possible. 2d. To divide an entire number by a fraction. Given, to divide 12 by J. 80 FRACTIONS. Since the divisor, |, is equal to 7 times the 9th part of unity, it follows, that the dividend is also equal to 7 times the 9th part of the required quotient. Then, taking the 7th of 12, which gives L2^ we will have the 9th of the quotient sought; and to obtain this quotient itself, we must take 9 times ^^f, which is done by multiplying the numerator by 9; we thus obtain 9 times 12 108 ,,-,., , or — , equal to 15|. Then, in order to divide an entire number hy a fraction, we must multiply the entire 7iumher hy the denominator, and divide the product hy the numerator. Or, we can say, as we have here multiplied 12 by |, multiply the entire numhcr hy the fraction inverted. 3d. To divide a fraction hy a fraction. Griven, to divide | by -f^. The reasoning is like the preceding. The divisor, -^j, being 8 times the 11th part of unity, the dividend, |, is also equal to 8 times the 11th of the quotient; then, the 8th of |, or -^^^ is the 11th of the quotient; and 11 times -f^, or ||, is the quotient sought. Then, to divide a fraction hy a fraction^ we must multiply the numerator of the dividend hy the denominator of the divisor, and the denominator of the dividend hy the numerator of the divisor ; then make the second, product the denominator of the first. Or, in simple terms, multiply the dividend hy the divisor vnth its terms inverted. Thus, 1-^ •f = 1 times 7 2 1 1 ft — 30 — - L3V In the same manner. 28 . 13 30 • r5~ 23 '30 15 ^r3 = 23 X 15 "30x13 345 ~390' (We could have suppressed the factor, 15, obviously common to both terms of the product in this last example, before perform- ing the multiplication). FRACTIONS. 81 N. B. Whenever, in the division of fractions, the divisor is less than unity, the quotient will be greater than the dividend. For this quotient results from the multiplication of the dividend by the divisor inverted, a number greater than unity. 60. Finally, if we have a mixed number, we reduce first the entire parts to fractions, and then proceed as in the case above. Given, 12| to be divided by 6|. We have 193_:_fi2 51_i_20— 51v 3 153 In the same manner, 7 8 _i_1 Pis 85_i.l25 — 85 V 8 680 Remarh. — The rules for the division and multiplication of fractions can be very readily deduced by regarding them as un- performed divisions. Remarh II. — It is evident that the division of fractions can give rise to fractions with fractional terms, or complex fractions^ as they are sometimes called. We can have, for example, 1, ^j:!!A, I ^^^^^ i ^ f 24 + 1 4f times f Which are reduced to fractions of two terms by performing all the operations indicated upon the separate fractions, according to known rules. Fractions op Fractions. 61. To the multiplication of fractions attaches itself another species of operation, known under the name of the ride for frac- tions of fractions, or compound fractions. In order to give an accurate idea of this operation, suppose, first, that we have to take a part of the fraction, |, indicated by the fraction, |, As this is the same thing as taking twice the third of f, or (57), multiplying | by |, we have for result, 5 times 2 10 7 times 3' 21* 82 FRACTIONS. Suppose, DOW, we wish to take a part of JJ, indicated by the fraction, -f.^ ; we would have, as above, — — — - =-——, and this last expression would represent -^^ of | of |. Hence, we see, that in order to take fractions of fractions, we must multiply all the numerators together j and all the denomi- nators together, and give the last product as denominator to the first. When we have to take fractions of fractions of a given entire number, we put this entire number under the form of a fraction, having 1 for denominator, and apply the rule which has just been established. Thus, the | of | of f of f of \^= ^ • 3 • | ; « ; ^^^ = M^, or, reducing, 3^ 44 ^ 3_3_. We can simplify these, and similar operations, by suppressing the factors common to both terms. Thus, in the example, | of | of -{^^ of ^ of \^, if we suppress the common factors, we have 3x5 or y, or 7^. Approximative Valuation of Vulgar Fractions. 62. Tn order to complete the general theory of fractions, we will resolve the following question, which has many useful appli- cations. Given, an irredueihle fraction, of luhich the terms are so large, that it is dijfficuH to form an accurate idea of its value, to replace it hy another ichich approaches it in value to within certain limits, but whose terms are much more simple; that is, which have for denominators, 2, 3, 4, 5, 6, &c. Take, for example, the fraction, |||. We propose to find its approximate value in twelfths, (i. e.) to replace it by a fraction having 12 for denominator. FRACTIONS. 83 We remark, first, that unity being equal to if, |f| of unity 523 X 12 are equal to f f j of j|, or equal to g^g^^- Multiplying 523 by 12, we obtain 6276 ; which, divided by 949, gives 6 for quo- tient, and 582 for remainder. Then, the fraction is y^^, with the 582 remainder, niQ-^o, less than j\. Hence, -j^^ is the value of the fraction to within less than j^^. 63. In general, in order to transform a fraction, -, into ano- ther having a denominator, n, at the same time differing from the first by less than - , we have the following rule. Multiply the numerator of the proposed fraction hy 7i, and divide the product hy the denominator. Form J then, a new fraction, having for numerator the entire part of the quotientj and n for denominator. General Observations on Fractions. 64. It results, obviously, from the nature of the methods established for the calculus of fractions, that the four fundamen- tal operations performed upon them, to wit : addition, subtrac- tion, multiplication, and division, are reduced always to the same operations performed on entire numbers. Thus, for example, the addition and subtraction of fractions is brought back, by the reduction to common denominators, to the addition and subtraction of their numerators. In the same manner, multiplication of fractions is effected by multiplying the numerators together, and the denominators. Division of fractions becomes multiplication, after inverting the divisor. We conclude from this, that the principles established in Nos. 25 and 26, upon the multiplication of entire numbers, are equally applicable to fractions ; that is to say, 1st, to multiply a fraction by the product of several others, is the same thing as 84 FRACTIONS. to multiply the first fraction successively hy each one of the factors of the product: 2d, the product of two or more fractions is the same in whatever order we perform their multiplication. In fine, we can apply to fractions all the propositions established in (40), concerning the changes which the product of a multiplica- tion, or the quotient of a division undergo, when we cause one of the terms of the operation to undergo certain changes. We can multiply or divide both terms of any fractional expression whatever by the same number, without altering its value; and so on of other principles. We can deduce from the definition of multiple and submultiple, or divisor of a number, that there exist fractions which are multiples and submultiples of other fractions, in the sense that the division of the multiple fraction by the submultiple gives an entire quotient. Thus, the fractions, J|, 3j^3, 2^3, • • • • are multiples of 3^^, since they contain the latter, 6, 4, 3 ... . times, without remainder. In general, every fraction has for divisors its half, its third, its fourth, &c. ; whence, it follows, that the number of its divisors is infinite, which is not true of entire numbers, if the divisors are to be entire. Two fractions can also have common divisors; thus, ||, -^^^ have for common divisor the fraction, -^^^ and all its submulti- ples ; for the quotients of || and -^^j divided by -^^^ are re- spectively 5 and 2, entire numbers. We can, then, generally establish, with relation to fractions, properties analogous to those which we have proved concerning the greatest common divisor, and the least common multiple of two or more numbers. Of Tvji 85 CHAPTER III. COMPOUND NUMBERS. 65. The theory of compound numbers we place here as an im- mediate application of the theory of vulgar fractions. The units of smaller denominations being fractions of the principal units, or units of higher denominations, and fractions being really no- thing more than units of lesser value than the principal unit with which they are compared. We can thus, in the number, 5|, regard the ninths as simple units, and 5 as a number made up of compound units, each one equal to 9 times the simple unit 3 and the 9 under the 4 is the sign or denominator, showing the relative value of the simple units expressed by the number, 4. Thus, we have seen, in (No. 8), that in order to value quantities smaller than the principal unit, we conceive this unit divided into a certain number of equal parts, which we regard as forming new units. In the theory which now occupies our attention, the principal unit is first divided into a small number of equal parts, then these are divided into others, and these new parts into others, &c., &c. Thus, for coin, the pound sterling, English, is divided into 20 parts, called shillings ; the shillings into 12 parts, called pence, &c. In the same manner, the unit of length, the yard, is divided into 3 parts, called feet ; the foot into 12 parts, called inches, &c. 66. Every art, each trade, each country, subdivides the prin- cipal unit, according to its own method. The following tables give for the most important o^ these quan- tities, the principal units and their subdivisions ; that is to say, those which follow the analogies of vulgar fractions. The deci- mal divisions of the principal units we reserve for the chapter on decimal fractions. 8 86 COMPOUND NUMBERS. TABLES In the estimation of time, the year is adopted as the principal unit j the subdivisions being, months, weeks, days, hours, minutes, peconds. The year is divided into 365 days. The day 24 hours. The hour 60 minutes. The minute 60 seconds. (The minutes and seconds are generally indicated by ' and ".) Or we may write the table thus : One second = ^^ of a minute. One minute = g*^ of an hour. One hour = ^^ of a day. One day = ^ of a week. Ex. — 5 days, 6 hours, 25 minutes, and 36 seconds, may be written either in columns, bd, 6h, 25', 36", or thus : 6 + ^4 + U of 5\ + IS of 5*0 of 3S\- COINS. Of coins, we give only the chief divisions of the English cur- rency; the American and French coming under the decimal systems. English Money. One pound sterling = £ is divided into 20 shillings. One shilling =s 12 pence. One penny = c? 4 farthings. Or we may write it thus : One farthing = ^J of a penny. One penny = y^^j of a shilling. One shilling = .^^ of a pound. Ex. — 5 pounds sterling, 6 shillings, and 10 pence, may bo written £5, 6s., 10^., or £5 -\-^ + |? of -^-q. COMPOUND NUMBERS. 87 WEIGHTS. The standard avoirdupois pound of the United States, is the weight of 27-7015 cubic inches of distilled water weighed in air, at a fixed temperature. This gives us a fixed unit of com- parison, or a principal unit of weight, of which the other divi- sions of the table are either multiples or submultiples. TABLE OF AVOIRDUPOIS WEIGHT. The ton is divided into 20 hundreds = cwt. The hundred weight 4 quarters = qrs. The quarter 28 pounds = lbs. The pound 16 ounces = oz. The ounce 16 drams = dr. The cwt. in this table contains 112 Ihs., but the cwt. of one hundred pounds is very generally adopted in commerce, as more convenient, and much better adapted to the decimal system of the Federal money. TROY WEIGHT. The standard Troy pound of the United States is the weight of 22-794377 cubic inches of distilled water, weighed in air at a given temperature. , TABLE. The pound (flb) is divided into 12 ounces = oz. The ounce 20 pennyweights = dwt. The pennyweight 24 grains = grs. (7000 grains Troy make 1 lb. avoirdupois.) The Apothecaries' weight for mixing medicines has the same principal unit as the Troy weight, but difiers only in its subdivi- sions. COMPOUND NUMBERS. TABLE. The pound (lb) is divided into 12 ounces = ^ . The ounce 8 drams =5. Thedram 3scruples = 9. The scruple 20 grains =r gr. (The English pound, Avoirdupois and Troy, differ a little from those of the United States). MEASURES OF LENGTH, AREA AND VOLUME. Long Measure. The principal unit of length is the yard, which is determined on the principle in physics that the pendulum which vibrates once in a second at the same place on the earth's surface, under the same surrounding circumstances, has a fixed and invariable length. This pendulum, or metal rod, is then divided off accu- rately, and a certain number of these subdivisions is called a yard. For the United States, the length of the pendulum is determined in New York city. TABLE. 12 inches make 1 foot. 3 feet 1 yard. 6 feet 1 fathom. 5^ yards 1 pole or perch. 40 poles 1 furlong. 8 furlongs 1 mile. 3 miles 1 league. MEASURE OP AREA, OR SQUARE MEASURE. The principal unit here, with which surfaces are compared, is a square whose side is 1 yard, or square yard. COMPOUND NUMBERS. 89 TABLE. 144 square inches make = 1 sq. foot. 9 sq. feet = 1 sq. yard. 30| sq. yards = 1 sq. pole or percA. 40 perches = 1 rood. 4 roods = 1 acre. The acre then contains 4840 sq. yards. For larger areas, we have the square, one of whose sides is a mile. This square mile contains 640 acres, (called a section in the public lands of the United States). CUBIC, OR SOLID MEASURE. The unit of volume, or solid measure, is a cube having one yard for its side, the other divisions being either multiples or subdivisions of this. TABLE. 1728 cubic inches = 1 cubic foot. 27 cubic feet = 1 cubic yard, &c., &c. The relations between the three tables of long measure, square and cubic measure, depend upon simple geometrical principles, which the student will find developed in any elementary work upon that subject. LIQUID MEASURE. The standard gallon of the United States is the wine gallon^ which is equal to 231 cubic inches. The gallon is divided into 4 quarts. The quart 2 pints. The pint 4 gills. For the higher measures, 63 gallons = 1 hogshead. 2 hogsheads =^ 1 pipe or butt. 4 hogsheads = 1 tun. 8* 90 COMPOUND NUMBERS. DRY MEASURE. The principal unit is the bushel. The standard bushel of the United States measures 21504 cubic inches. The names of the subdivisions, though the same as in liquid measure, do not repre- sent the same volumes. The gallons^ quarts, and pints, in liquid measure, measure respectively, 231, 57|, and 28| cubic inches; while in dri/ measure, they measure 268|, 67 J, and 33| cubic inches respectively. TABLE. The bushel is divided into 4 pecks. The peck 2 gallons. The gallon 4 quarts. The quart 2 pints. (The English imperial gallon measures 277*274 cubic inches.) We see from these tables the great importance of determining accurately the standard of length, as all the other principal units of commerce depend upon this. Thus, the standard of dry and liquid measure is a certain number of cubic inches. The standard weight is a certain number of cubic inches of water. The standard of money is a coin containing a given weight of metal. 67. We call a compound number every concrete or denominate number, which contains, at the same time, one or more principal units of a certain species, and one or more subdivisions of this unit, or simply one or more subdivisions of the principal unit alone. Thus, £10 12s. Sd., 2b mis. 4: fur. 1yds., 70 days, 23 hours, 10 min., or simply 12s. lOc/., 4 A. iOmin., &c., &c., are compound numbers. But, £10, or 10s., or 23 hours, are not compound numbers, considered thus isolated. The resolution of the following ques- tions serves as a basis for the four fundamental operations on compound numbers. COMPOUND NUMBERS. 91 68. Question first. — A compound number being given^ to re- duce this number^ or express it in units of the smallest subdivi- sion of the principal unit. Given, for example, 2lb. 4toz. 17 dwfs. bgrs., to be converted into grains. It results from the tables, that the pound equals 12 ounces. Therefore, 2 lb. 4 oz..= 2 X 12 + 4 = 28 02., or, 2/^ lb = f | lb. In the same manner, the ounce equals 20 dwt. Hence, 28 oz., 17 dwt. = 28 X 20 + 17 pennyweights = 577 dwt., or, 281^ = ^y oz. Again, 577 dwt. bgrs. = 577 X 24 + ^grs. = 13853 grains, or, 5773j\ = ^ \%^ ^ dwt. GrENERAL RuLE. — Multiply j first, the number of principal units by the number of units of the first subdivision which the principal unit contains, and add to the product the units of this first division, which are contained in the given number. Then multiply the result thus obtained by the number of units of the second subdivision which the first contains, and add to this second product the units of the second subdivision, which enter into the given compound number ; and thus, in succession, until we arrive at the last subdivision or denomination. We will find, by this method, 1st. — 59 ft). 13 dwts. 5 gr. = 340157 gr. 2d. — 121 lb. Os. d^d. = 58099 halfpence. 3d. — 23 h. 55 min. 19'' = 26119 seconds. 69. Second question. — Reciprocally, given a number of units of a certain division of the principal unit, to be converted into a compound number. The rule to be followed is evident from what precedes, and can be enunciated thus : Divide, first, the proposed number, by the number which ex- presses how many times the given subdivision is contained in the subdivision next higher ; v:e obtain thus for quotient, a certain number of units of this next higher division, and for remainder the units of the given denomination which are to enter into the compound number sought. Divide, then, the quotient obtained 92 COMPOUND NUMBERS. hy the numher whicJi expresses liow many times the suhdivision next higher is contained in the denomination higher hy txoo than the given one ; we obtain a new quotient^ which contains a certain numher of units of the third denomination^ of which we have just spoken, and a new remainder, expressing the units of the denomination next to the given one, which make part of the com- pound numher sought. Continue thus, until the quotients cease to he divisible by the numher expressing the relation between the value of two successive denominations. N. B. If we obtain for any one of the remainders, this proves that the denomination corresponding is wanting in the number sought. Let us apply this rule to the first example of (68). 13853 1 190 24 20 185 173 577 177 17 28 1 12 4 12 Or thus : 13853 ^r. 577 dwt. 28 oz. Result, 21b. 4loz. 11 dwt. bgr. 24: = 677 dwt. + 5gr. 20 = 28 oz. + 17 dwt. --12= 21b -f 4:0Z. 70. Question third. — To convert a given compound numher into a fraction of the principal unit. This is also a consequence of (68). Take, for example, 21b. 4:oz. 17 dwt. 5gr. This, reduced to grains, gives 13853 grs. ; and, by the tables of (66), 1 gr. is ^^^j of ^^ of -f\j of a pound, or j^^^^ of a pound ; the required frac- tion is obviously then VWo^ ^^ ^ pound. COMPOUND NUMBERS. 93 Rule. — Commence hy reducing the given compound number into units of the lowest denomination which it contains; then form a fractional number which has for numerator the number thus obtained, and for denominator the ntimber of units of this lowest denomination which the principal unit contains. We will find, by this method, 23 h. 55' 9"= J^^ =^^ of ail hour. 71. Question fourth. — Reciprocally j given, any fractional number of the principal unit of a certain denomination, to con- vert it into a compound number. OPERATION. Given, for example, | of a mile to be converted 5 into furlongs, poles, &c. Since each mile equals eight 8 furlongs, f of a mile is f of 8 furlongs ; equals "^^ of 40 | 7 1 furlong. We then divide 40 by 7 ; the quotient, 5 5 5, expresses obviously the furlongs, and the remain- 40 der, 5, with the divisor, 7, for denominator, is a 200 | 7 fraction of a furlong which it is necessary to reduce 60 28 to poles. Now, 1 furlong equals 40 poles; hence, ^ 4 of a fur. = f of 40 poles, equals — - — of 1 pole. Per- — - ^ 7 Z2i I 7 forming the operations here indicated, we have 28 for 1 | 3 quotient, and 4 of a pole, for the fraction correspond- ing to the remainder ; ^ o^ ^ P^^^ = 4 ^^ ^i yards = ^ of a yard, which is equal to 3^. Hence, the required compound num- ber is 5 fur. 28 pis. 3| yds. General Rule. — In order to convert a fractional number of any principal unit into a compound number, obtain first the entire number, if there be one, contained in the fraction; you obtain thus a certain number of the principal units. Multiply, then, the renfiainder of this division, by the number which expresses how often the principal unit contains the next lower subdivision, and divide this product by the denominator of the given number ; ice thus obtain a certain number of units 94 COMPOUND NUMBERS. of this next lower suhdiviswn, and a second remainder. Pro- ceed with this remainder in the same manner j until you arrive at a result with no remainder ^ or exhaust the suhdivisions-of the principal unit. N, B. Principal unit can apply to any one of the denomina- tions of the tables (66), which has itself been subdivided ; that is to say, every subdivision can be principal unit to the subdivi- sions below it. 72. Remark I. — The operations of the two last rules can serve as verification for each other. Thus, in applying (71), to the fractional numbers of (70), we ought to reproduce the four com- pound numbers which correspond. In the same manner, we can verify the result of (71), by means of the rule in (70). 73. Remark IL — The principles which have just been deve- loped would be, properly speaking, sufficient to permit us to per- form the four fundamental operations of arithmetic upon com- pound numbers. We would thus pursue the following method : 1st. Transform the compound numbers, each one into a frac- tion of the principal unit corresponding. 2d. Perform upon these fractional numbers the operation pro- posed (^according to the rules of the calculus of fractions^ which icill give for a result a fractional number. 3d. Convert this fractional number into a compound number of the species indicated by the nature of the question. Nevertheless, since the direct methods of performing the four fundamental operations upon compound numbers give rise to im- portant observations, and offer for the theories which we shall develop later, useful applications, we will proceed to discuss them, as simply as possible, with very simple examples. ADDITION AND SUBTRACTION. 74. — 1st. Addition. — Place (as in abstract entire numbers), the given numbers, one under another, so that the units of the same denomination fall under each other ; after whichy make COMPOUND NUMBERS. 95 the addition of the units contained in each column ^ commencing on the right. If the sum of the units contained in a column exceeds the number which expresses how many times the unit of the denomi- nation corresponding is contained in the unit of the next higher denomination, we divide the sum obtained by this number (69) ; we obtain thus a remainder, (^possibly 0^) which we write below the horizontal line drawn under all the columns, and a quotient which we carry to the units of the following column ; we operate in the same manner upon this column, and upon each successive column. (This rule is obviously established by the same reason- ing which was given for the rule in simple addition of abstract numbers). 2d Subtraction. — Write the smaller number under the greater, so that the units of the same denomination fall under each other ; then subtract successively, one from the other, the units of each denomination, commencing with the loivest. When, in any one of the columns, the number of units to be subtracted is greater than the number from which it is to be taken, we add to this latter (14), a unit of the denomination next higher, converted into units of the denomination on which we are operating ; the partial subtraction becomes possible. We must take care, however, to augment the next number to be subtracted by the one unit borrowed from this denomination. (This rule is obviously founded on the reasoning for subtrac- tion of simple numbers). We give below some examples : £ s. d. K>. oz. dwt. gr. 17 13 4 14 10 13 20 13 10 2 13 10 18 21 10 17 3 14 10 10 10 8 8 7 1 4 4 4 3 3 4 45 • 7 7 8 8 13 2 12 54 1 4 Proof 23 32 Proof 96 COMPOUND NUMBERS. SUBTRACTION. £. s. d. mis. fur. pol. yd. ft. in. 67 13 8 14 3 17 1 2 1 49 17 11 10 7 30 2 10 7 15 9 3 3 26 ^~l 3 57 13 8 Proof 14 3 17 1 2 1 Proof The methods of verification are the same as in abstract num- bers, taking care to preserve the relative values of the units carried from the columns of higher denominations to lower, as in the verification of addition, and from lower to higher, as in sub- traction. MULTIPLICATION. 75. To multiply a compound number by a simple factor, we consider the multiplication of each denomination of the compound number as a separate question ; then reduce the partial products to compound numbers by (69), and add these compound numbers by last article. Or, what is the same thing, commence on the right hand, and proceed with the multiplication as in simple numbers, taking care to preserve the proper relative values be- tween the successive columns. Thus, £4 13s. M. to be multiplied by 9. £ .9. d. 4 13 3 9 41 19 3 9 times 3 gives 27, which we reduce to shillings, giving 2 for quotient, with 3 remainder; set down the 3, and carry the 2 to the next multiplication; 9 times 13 gives 119 ~ £5 19s. We set down the remainder, and add the 5 to the product of 9 by £4, giving £41. If we have one denominate number to be multiplied by another, we reduce multiplicand and multiplier to fractional numbers of their principal units by (70); then multiply, and reduce the result back to the compound number required by the COMPOUND NUMBERS. 97 question ; or we may simply reduce the multiplier to such a frac- tion, and proceed as in the first example. Example. — £2 5s. to be multiplied by 101b boz. avoirdupois. We may either multiply |J by Le_5^ and reduce theiresult to pounds and shillings, or we may multiply £2 5s. by ^g^, reducing each result separately. 76. Remarh. — It xesults obviously from this mode of pro- ceeding, 1st. That although the multiplier is a denominate number, yet we consider the principal unit of this factor and its subdivisions as abstract numbers, which express the number of times we must take the multiplicand, and what parts of it we must take, in order to obtain the required result \ but we preserve always in the mul- tiplicand its essential quality of concrete number. 2d. That all the partial products and the total product are always of the same nature as the multiplicand. Certain questions of Geometry, however, namely : those which have for their object the measure of surfaces and volumes, give rise to operations which form exceptions to this general principle. The considerations on which these are founded do not belong to arithmetic. DIVISION. We will dwell but little on this operation^ in effecting which, in general, it is better to apply the method established in (73). Nevertheless, we will consider the two principal cases which can present themselves. 77. Ca&e I. — In which the dividend and divisor are com- pound numbers of the same species. For example : — Required, How many yards of a certain work can we have executed for £75 19s. 6d., if one yard cost £8 15s. Qd. ? It is clear that, for the resolution of this question, we must determine how many times the smaller of these two compound numbers is contained in the greater. This is effected, 1st, by 98 COMPOUND NUMBERS. reducing the two numbers to tlie lowest denomination which enter them ; 2d, by then dividing the entire numbers thus ob- tained one by the other. The quotient is at first an abstract number, which, according to the enunciation of the question, can then be expressed in yards, feet, inches, &c. Converting the two given numbers to pence, we find 18,233£7. and 210Qd. The frac- tional number then will be ^^Wg , which can be converted into yards, &c., by rule in Art. (71). 78. Case II. — That in which the dividend and divisor are of different sjjecies. In this case, whatever be the question proposed, the quotient must express principal units of the same species as the dividend ; since it is necessary that the dividend, considered as a product, must be of the same species as one of its factors. Eut then, the compound divisor, being converted into a fractional number of the principal unit, becomes an abstract number, by which we must divide the dividend, which is done by multiplying the dividend by this fraction inverted (60). 79. Remark I. — We conclude from the above, 1st. That in every division of compound numbers, if the two numbers are of the same species, the quotient is considered first as an abstract number, which we make express the units and subdivisions of units, fixed by the enunciation of the question. This quotient is to be the multiplier in the verification of the operation by multi- plication. 2d. That if, on the contrary, the two terms of the division are of difi"erent species, the quotient expresses necessarily units of the same species as the dividend ; while the divisor, though com- pound at first, is to be regarded as an abstract number, which plays the part of multiplier in the verification of the operation. 80. Remark II. — So far, we have only ^iven one method of verifying multiplication, viz : the method by division, and reci- procally. But in the practice of the operations upon compound numbers, it is generally more convenient to verify, 1st, Multipli- cation, by doubling one of the two factors, and taking the half COMPOUND NUMBERS. 99 of tlie other; then performing the operation anew with the re- sulting numbers. 2d, Division, by doubling the two terms of the division. We avoid thus the difficulties arising from the vulgar fractions, which ordinarily accompany the results ar- rived at. It is evident that this means of verification can also be em- ployed with entire abstract numbers. EXERCISES. 1. Find a nifmber, the 'i, the |, |, and | of which, added to- gether, form a sum which, diminished by 139, gives 1289 for remainder. 2. A reservoir is filled by four difi'erent pipes. The first can fill it alone in 5 hours ; the second in 7 hours ; the third in 9 hours; the fourth in 11 houi*s. Required, the time of filling the reservoir, all four pipes being opened at onc6. 3. The population of Asia is estimated at 390,257,000 inhab- itants : Required the population of Europe, Africa and America; knowing that the population of Europe is -^^ of the population of Asia; that of Africa, -j^y of that of Europe; and that of America, j\ of the same. 4. The sea covers i| of the whole surface of the globe. The surface of Asia is equal to ^^V ^^ ^^^^ ^^ Europe ; that of Africa is ^^ of the same; that of America, y^^ ; and that of Oceanica, 1^ ; we know, besides, that Africa has a superficies of 13,450,000 square miles. Calculate the superficies of the other parts of the world, and deduce the number of square miles in the whole surface. 5. Demonstrate that, by adding the same number to the two terms of a fractional number, we obtain a result which approaches unity more as the number added is greater. Show that the dif- ference between the result and unity can become less than any given quantity. 100 DECIMAL FRACTIONS. 6. Find the method of obtaining the greatest common divisor of two or more fractions. Apply to the fractions, 3 5 7 19 7. Demonstrate the method for obtaining the least common multiple of several fractions. Apply to the fractions, -f^, IJ, !§. 8. What is the greatest common multiple of the fractions, ||, -/p and If, less than 100,000. 9. What will be the price of a piece of stuif, 23^'^ yards long, each yard costing £5 10s. Qd. ? 10. 87 R). 10 0^. 5(7r., of a certain material, was bought for 50£ lis. del. What is the price per pound ? CHAPTER IV. » 0/ Decimal Fractions, and their Principal Properties — Of tlie Decimal Systems of Compound Numbers. I. -DECIMAL FRACTIONS. 81. Introduction. — In the ordinary system of numeration, the most simple method, and the most convenient one of subdividing unity, is the suhdivision into successive 2Jarfs, decreasing in a ten- fold ratio. From this mode of subdivision result fractions which have for denominators unit?/, followed hy one or more 7:eros^ and these fractions we call decimal fractions. This mode of subdividing unity oflers great advantages, inas- much as it reduces immediately, or at least by very simple trans- formations, all the operations upon fractional numbers, to simple operations upon entire numbers. These methods we will develop f fter having made known the numeration of decimal fractions; DECIMAL FE ACTIONS. 101 that is, their nomenclature, and the manner of writing them in figures. 82. Numeration of Decimals. — As, by increasing unity ten- fold, one hundred-fold, &c., successively, we form new units, to which we give the name of tens, hundreds, thousands, and so- forth, in the same manner we conceive unity to be divided into 10 equal parts, which we call tenths, each tenth divided into 10 equal parts, which we call hundredths, (because the principal unit contains 10 times 10, or 100 of these new parts or units) ; then each hundredth divided into 10 equal parts, called thou- sandths, and so on; thus giving ten thousandths, hundred thou- sandths, &c. In the second place, it results, (5), from the fundamental principle of the written numeration of entire numbers, that the figures, proceeding from right to left, have their relative value increased tenfold for each place to the left, and decreased ten- fold, going from left to right. Whence it follows, that if to the right of an entire number written in figures we place new figures, taking care always to distinguish them by any sign whatever, a comma or point for example, from the entire number, we shall thus represent successive parts of unity, decreasing tenfold to the right ; that is, tenths, hundredths, thousandths, &c. Thus, the collection of figures, 24,75, expresses 24 luiits, 7 tenths, and 5 hundredths; 5,478 equals 5 units, 4 tenths, 7 hun- dredths, and 8 thousandths. 83. Let it be required to enunciate in ordinary language the number 56,3506. This number can at first be enunciated 56 units^ 3 tenths, 5 hundredths, thousandths, and 6 ten thousandths. But 3 tenths are equal to 30 hundredths, or 300 thousandths, or 3000 ten thousandths; in the same manner, 5 hundredths are equal to 50 thousandths, or 500 ten thousandths The number can then be enunciated 56 units, and 3506 ten thousandths. Thus, in order to enunciate in ordinary language a decimal fractional number written in figures, we must enunciate sepa^ ratelij the entire part, and then enunciate the part ichich is to * 102 DECIMAL FRACTIONS. tlie right of tlie comma, as an entire rnnnhcr, aiving at the close the name of the unit of the last decimal aubdivisicrii. Thus, 7,49305 represents 7 uniu and 49305 hundred thou- sandths. In the same manner, 249,007,056 represents 249 units and 7056 millionths. We can also, if we wish, include in one single enunciation the entire as well as the decimal part Take, for example, the number 56,3506. As one unit equals 10 tenths, or 100 hundredths, 1000 thousandths, &c., it follows, that 56 units are equal to 560000 ten thousandths ; and, conse- quently, 56,3506 represents 563506 ten thousandths. That is, we must, after enunciating the number as if it had no comma, place at the end of the number thus enunciated, the name of the last subdivision. It is customary, however, to enunciate the entire part separately. We will indicate a method for enunciating the decimal part, which, in general, is more convenient in practice. After an- nouncing the entire part, as we have just said, separate mentally the decimal part into periods of three figures, beginning at the comma, (the last period having often only one or two figures) ; enunciate then each period or division separately, and place at the end of each partial enunciation the name of the last unit of the period. Example. — The number, 2,74986329, is enunciated; 2 units, 749 thousandths, 863 millionths, 29 hundred millionths. 84. Reciprocally, we propose to write in figures a decimal fraction enunciated in ordinary language. Required to write in figures the number; twenty-nine units, three hundred and fifty four thousandths. Write first the entire part, 29 ; then, as 300 thousandths are equal to 3 tenths, and 50 thousandths equal 5 hundredths, place a comma to the right of 29, and write successively the numbers 3, 5, and 4 ; we thus have 29,354. Iq like manner, one hundred and' nine units, two thousand and three ten thousandths, are written 109,2003. DECIMAL FRACTIONS. 103 Required, again, to write the number eight units, thirty-seven thousandths. As thirty thousandths make 3 hundredths, and as there are no tenths in the number enunciated, we write 8,087; that is to say, we make the same use of the in both these last cases as in whole numbers, placing it here to the right of the comma, to take the place of the tens which are wanting, and to give the figures which follow their true value. GrENERAL KuLE. — In order to write, in figures, a decimal enunciated in ordinary language, commence hy writing the entire part, and after it a comma or point; then write successively, to the right of this point, the figures luhich represent the tenths, hun- dredths, &c., included in the number, taking care to replace hy zeros the different orders of unit^s which are wanting. If there is no entire part, lorite a to take the place of it, and proceed v:ith the decimal part as before. Thus, seventeen hundredths are represented by 0,17; one hundred and twenty-five ten thousandths by 0,0125. It may happen that, in the enunciation of the number, the entire part is not distinguished from the decimal part. We must then write the number as if it expressed entire units, and then place a point so that the last figure to the right shall express the units of the last subdivision of the number enunciated. For example, in order to write the number four thousand. Wo hundred and fourteen hundredths, write first 4214 ; and, as the last figure must express hundredths, place the comma between the 2 and 1, giving 42,14. Two hundred and fifty-three thou- sand and twenty-nine ten thousandths, are represented by 25,3029. 85. Decimal fractions placed under the form of vulgar frac- tions. A fraction being composed of two terms, the numerator and the denominator, the comma serves, in the method which we have just developed, to indicate the denominator, which is equal to unity, followed by as many zeros as there are decimal figures ; that is, figures to lAe right of the comma. The numerator, we 104 DECIMAL FRACTIONS. have seen, is composed of the collection of figures to the right of the comma. Or, if we consider the entire part" as reduced to a fraction, the numerator is then the number given, with the comma stricken out. Thus, the number, 23,6037, put under the form of a vulgar fraction, is 23fgO_3j7_^ qj.^ 2_35__o^3J7^ The number, 2,00409, is equal to 2^/oVoo. or, f gg-J-oa. Finally, 0,0002154, is equal to _^§f54^^. Reciprocally, 2y§-3_o, or, f-Q§-3, is equal to 2,053 ; VVoVcf is equal to 17,2049. These two transformations are of continual use in the calculus of decimal fractions. 86. CJianging the place of the point. — If, in a decimal frac- tion, we advance the point one or more places to the right, we multiply the number by 10, 100, 1000, &c. ; and if, on the con- trary, we place it one or more places farther to the left, we divide the number by 10, 100, 1000, &c. For, let the number be 153-07295. Suppose we advance the point three places to the right, which gives 153072-95. The two numbers are now ^f^J2§§^, and *^ Wo^^^- ^ow, the denominator of the second number is 1000 times smaller than that of the first, while the numerator is the same. Then, the second fraction is 1000 times greater than the first. On the contrary, remove the point two places towards the left, it becomes 1-5307295, or, 1550129.5^ a fraction evidently 100 times smaller than the given one. We could establish the same thing by reasoning thus : — By changing the place of the point, the value of each figure becomes 10, 100, 1000, &c., times greater or smaller. Thus, in comparing 153072-95, with 153-07295, we see that the figure 3, which expresses in the latter simple units, expresses now thousands; the figure 5, to the left of the figure 3, which expressed tens, represents now tens of thousands ; and the same with the other figures. 87. Zeros placed to the right of a decimal fraction. By annexing any number lahatevcr of zeros to the right of a decimal fraction, v-e do not change its value. DECIMAL FRACTIONS 105 Thus, 3-415 is equivalent to 3-4150, 3-41500 . . . . ; for these numbers can be (85), put under the form, 3415 3 415 3 4_15 00 1000' Toooo> 100000? • • • •; Now, the last two fractions are nothing more than the first, with its two terms, multiplied by 10, 100, which (43), does not change its value. Then, &c Or, we may observe that zeros, placed to the right of decimal figures already written, do not change their value ] and, as these zeros have no value of themselves, the fraction remains always the same. As the value of a figure in a decimal fraction depends entirely on the number of places it is distant from the point, it is obvious that we do alter this value by prefixing zeros between the decimal point and the first decimal figure. 88. Reduction of several decimal fractions to the same deno- minator. The principle which has just been established, gives us a me- thod of reducing several decimal fractions to the same number of decimal fgures, without changing their value ; or, in other terms, to the same denominator. For example, the fractions 12-407 I 0-25 I 7-0456 | 23-4 are equal to 12-4070 | 0-2500 | 7-0456 | 23-4000. They have 10000 for common denominator. These prelimi- nary ideas being established, we pass to the four fundamental operations upon decimal fractions. Addition and Subtraction. 89. We perform the addition of decimal fractions in the same manner as ice do that of entire vumhers, offer reducing them all to the same denominator, and we pfoint off in the result as man?/ decimal places as there are in any one of the reduced numbers, or the greatest number which any one of the given numbers con- tains. lOG DECIMAL FRACTIONS. A single example will suffice to illustrate and make plain this rule. Given, to add the numbers 32-4056 I 245-379 | 12-0476 | 9-38 | and 459-2375. 32-4056 245-3790 12-0476 9-3800 459-2375 758-4497 121-2210 Verification. We write, first, one zero to the right of the second number, and two to the right of the fourth ; we then place the numbers thus prepared, one under another, so that the units of the same order correspond, and then make the addition in the ordinary manner. We find for result, 7584497; or, separating the four figures to the right, 758-4497; because the numbers added ex- press units of the order of ten thousandths. In practice, we can dispense with writing the zeros to the right of the numbers, which contain fewer decimal places than the others, provided we take care to arrange the units of the same order in the same column. Subtra-ction is 'performed in the same manner as in entire numbers, after ice have reduced the decimals to the same deno- minator (88). Example. — G^'wi^w, to subtract 23-0784 from 62-09. 62 0900 23-0784 39-0116 62-0900 Verification. We write two zeros to the right of the 62-09, which gives 62-0900; we then perform subtraction in the usual manner, DEC13IAL FRACTIONS. 107 taking care to separate four decimal figures to the right of the result. These methods are obviousl3' founded upon the fact that the units of different orders, in decimal fractions, having the same relations of magnitude, one to the other, as in entire numbers, we have the same operations to be performed with the figures to be carried as in entire numbers. Multiplication. 90. In order to perform this operation, multipli/ the two given numbers one hi/ the other, without regarding the comma or point ivhich they contain; then separate hy a point, from the right of the product thus ohtained, as many decimal figures as there are in both factors. Required, for example, to multiply 85-407 by 12-54. We find first for the product of the two numbers, the points being disre- garded, 44400378. Pointing ofi", then, on the right of the pro- duct, 3 -f 2, or 5 figures, we obtain for the required product, 444-00378. In order to see the reason of this method, we re- mark, that the two given numbers are equal to (160), jVqV^ ^^^ Wo"*- Whence we deduce the product by the rule in (57), 35407x1254 ,, , . , ., . , , . , , -TKc^ — TKcT ) t"^* IS to say, it IS necessary, 1st, to multiply the two numbers, disregarding the point ; 2d, to divide this product by 100000, or unity, followed by as many zeros as there are de- cimal figures in the two factors, which is equivalent to separating 5 decimal figures on the right of the product. The method is thus justified. Or, we may reason thus : by removing the point from the multiplicand, we multiply it by 1000 ; since, at first, it expresses thousandths, but after the multiplication, principally units; then, the product is 1000 times too great. In the same manner, by removing the point from the multiplier, we render it 100 times greater. Thus, by the suppression of both points, the product is rendered 100000 times too great ; then, in order to bring it back to its just value, it must be divided by 100000, or 108 DECIMAL FRACTIONS. five figures must be pointed off for decimals on the right. The reasoning would obviously be the same, whateve/be the number of decimals in the two factors. It can happen that one of the two numbers, only contains decimals. In this case, we point off, on the right of the product, as many decimal figures as there are in this number. The demonstration is too easy and obvious to detain us. We will find, according to these rules, 1st. The product of 4-057 by 9-503, is 38-553671. 2d. The product of 4-0015 by 29, is 116-0435. 3d. The product of 0-03054 by 0-023, is 0-00070242. N. B. This last example deserves some attention. Suppress- ing the point in the two factors, and performing the multiplica- tion, we find for a product, 70242 ; but, as there are five decimals in the multiplicand, and three in the multiplier, there must be eight of them in a product which contains only five figures. In order to remove the difficulty, we observe, that as the product ought to express units of the 8th order of decimals, it suffices to write, on the left of 70242, zeros in such number that, the point being placed on the left of them, the last figure to the right shall occupy the 8th decimal rank. We write three zeros then on the left, besides one for the entire number, and obtain 0-00070242. Division. 91. Two principal cases present themselves. Either the divi- dend and divisor have the same number of decimals, or this number is different. In the first case, suppress the point in the dividend and in the divisor ; then operate upon the entire num- bers which result from it. according to the ordinary ride of division. In the second, commence by reducing the two given numbers to the same number of decimal places, or to the same denomina- tor. The second case thus becomes the first. First Case.— Required to divide 47-359 by 8-234. These two numbers can be put under the forms (85), VoVu? IMJ- I^ividing DECIMAL FRACTIONS. 109 them one by the other, according to rule for the division of frac- 47359 1000 47359x1000 47359 tions (59), we have -looo" ^ 8 234 = 8234x1000 = ^234' suppressing the factor, 1000, common to the two terms. We see, then, that the quotient required is equal to that of the two given numbers with the point removed ; and the rule above is proved. We can also say, the two decimal fractions having the same denominator, if we suppress the point, we mul- tiply the two terms of the division by the same number, 1000; then, the value of the quotient remains the same. The division of 47359 by 8234, gives for the entire part of the quotient, 5, and for remainder, 6189; thus the total quotient is, 5|^||. 92. Valuation of the quotient in decimals. — The vulgar frac- tion, which accompanies the entire part of the quotient, having terms pretty large, it is difficult to value it in its present state ; moreover, it is natural to endeavour to express it in parts of the same species as the given numbers. We arrive at this now by the rule in (63) : 4735918234 61890 5-7516395 42520 13500 ~52660 ~^5'60 "78580 ~uuo "3570" After obtaining the entire part, 5, of the quotient, in order to make the remainder, 6189, express tenths, we multiply it (63), by 10 ; this we effect by placing a on its right ; then we divide 61890 by 8234; the quotient, 7, expresses then tenths; and we 110 DECIMAL FRACTIONS. write it to the right of the figure 5, with a point before it. To the right of the new remainder, 4252, we place a 0, in order to convert it into hundredths ; we then divide 42520 by 8, which gives the quotient, 5 ; this we place on the right of 7, and annex another to the remainder, 1350 ; and so on, until we have ob- tained the number of decimal places which the enunciation ques- tion giving rise to the decision demands. GrENERAL KuLE. — In order to express, in decimals, the quo- tient of the division of two decimal numbers of the same denom minator, or (which is the same thing after the suppression of the point), of any two entire numhers ivhatever, Commence hy determining the entire part of the quotient ^ (which can be 0), and lorite a point after it. Annex a zero to the right of the remainder ; divide the num- ber thus formed by the divisor ; then place the quotient on the right of the point. Annex aO to the right of the new remainder, and perform the division by the same divisor ; write the quotient on the right of the two first. Continue thus until you have the number of decimals requi7-ed. 93. Remark on these approximations. — In the preceding ex- ample, we have carried the operation as far as the seventh deci- mal figure, in order to establish some principles upon the different degrees of approximation which can be obtained by the develop- ment of a number into decimals. By taking at first only the two first decimal figures, we have 5-75 for the value of the quotient, to within less than 0*01, since the part neglected is obviously less than the unit of this order of decimals. Again, as this neglected part is less than 0-002, oy—XqjOT -gj^, it follows, that 5-75 expresses the value of the quotient to within less than ^i^. Now, if we take the three first decimal figures, we have 5*751 for the value of the quotient, to ivithin less than 0*001, since the part which we neglect, 0*00063 .... is less than 0*001. But here we must make an important observation. As the figure 6 DECIMAL FRACTIONS. Ill exceeds 5, it follows that 0-0006 exceeds 0-0005, or a Jialfunit of the order tJiousandths ; then, by taking 5-752, instead of 5-751, for the value of the quotient, we commit an error in thus taking more than the true value, less than is committed when we take 5-751 for this value; and we can say that 5*752 expresses the quotient, not only to within less than 0-001, but to within less than the half of O'OOl. Generally, whenever the figure which follows that one at which we wish to stop in the divisioUj is less than 5, tee preserve the figure obfainedj and we then have the value of the quotient to within less than a half unit of the denomination at which we stop. If on the contrary, the figure which follows is equal or greater than 5, it is best to increase hy one unit the last figure obtained, in order to obtain a value nearer the quotient; the error committed is an error of excess, but it is less than a half unit of the order at which we stop the operation. Thus, in the example above, we have successively for the quo- tient of the proposed division, 5-752, too great by less than a half thousandth ; 5-7516, too small by less than a half ten thou- sandth; 5-75164, too great by less than a half hundred thou- sandth; 5,751640, too great by less than a half millionth. "We will add, that when we have arrived at any decimal figure what- ever, in the operation performed, the last remainder obtained shows whether the following figure of the quotient is greater or less than 5, without necessarily calculating this figure. If the remainder is less than half the divisor, the following figure of the quotient will necessarily be less than 5. If this remainder is equal to, or greater than the half of the divisor, the next figure of the quotient will be equal to, or greater than 5. Thus, in the example which we have just discussed, the eighth figure of the quotient must be less than 5 ; for, the remainder at which we stopped, 3570, is obviously less than the half of the divisor, 8234. 112 DECIMAL FRACTIONS. We have here given the whole theory of approximations in the valuation of fractional numbers in decimals. 94. Case Second. — This divides itself into two others : Firstly, — The dividend contains fewer decimal figures than the divisor. We write on the right of the dividend the number of zeros necessary to reduce the two terms of the division to the same number of decimal places ; and the question is solved by Case First without farther modification. For example : — Required, to divide 2-405 by 0-03497. Placing two zeros to the right of the dividend, which gives 2-40500; then, suppressing the comma in both numbers, we perform the division of the two resulting numbers, 240500, and 3497, ac- cording to the rules in (91 and 92). We find thus the value of the quotient to within less than -0001, to be 68-7732. Secondly, — IVie dividend has more decimal figures than the divisor ; ive can then employ two methods. 1st. Required to divide 3*470456 by 1-027. If we suppress the point in the divisor, thus rendering it 1000 times as great, and if we advance the point in the dividend three places to the right, rendering it thus also 1000 times as great as at first, the quotient of the division of these two numbers resulting, will be the same as that of the given numbers. The question is thus reduced to dividing 3470456 by 1027. 3470-45611027 3894 13-379217 8135 "9466 ~2230 1760 T3'30 "Hi DECIMAL FRACTIONS. 113 After finding the entire part, 3, of tlie quotient, and the re- mainder, 389, instead of placing, as in (92), a zero to the right of this remainder, we bring down the figure 4, which expresses tenths, and perform the division, obtaining for quotient, 3, which we place on the right of the first, separating .them by a point; we then bring down to the remainder, 813, the figure 5, which expresses hundredths; and we continue thus, until we have brought down all the decimal figures which are contained in the dividend. When we reach the remainder, 223, we place a zero on the right of it, and operate as in case first. We see that this method consists in suppressing the point in the divisor, taking care to remove it in the dividend as many places to the right as there are decimals in the divisor; then, in operating upon the resulting numbers, as in the first case, with this difi"erence, that instead of annexing at first zeros to the right of the diff"erent re- mainders, we commence by bringing down successively all the decimal figures of the dividend. 2d. We take the same example, and commence by writing to the right of the divisor three zeros ; that is to say, the number of zeros necessary to reduce the two terms to the same number of decimal places. We have then to divide 3470456 by 1027000. 347056 1 1027(000 389456 1 3-379217 ~81356 9466 2230 "1760 7330 lil In order to determine the entire part of the quotient, we com- mence by applying the. rule of (38), for the division of entire 114 DECIMAL FRACTIONS. numbers, when the divisor is terminated by zeros, We obtain thus the quotient, 3, and the remainder, 389456. Now, in order to find the tenths figure, we remark, that instead of multiplying the remainder by 10, (i. e.) placing a to ttie right of it, wc can divide the divisor by 10 ', that is, suppress one on its right. Performing then the division, we have 3 for quotient, expressing tenths, and the remainder, 81356. In the same manner, instead of putting a to the right of this remainder, we suppress a se- cond on the right of the divisor, and divide 81356 by 10270 ; applying still, if we wish, the rule of (38). We obtain thus the new quotient, 7, and the remainder, 9466. Suppressing the last on the right of the divisor, we divide 9466 by 1027 ) this gives the quotient, 9, and the remainder, 223. Setting out from this remainder, we follow the rule in (92), in order' to obtain the remaining decimal figures. This second me- thod is obviously less simple than the first; and we mention it, because it gives us the opportunity of showing how to operate when we have zeros to annex to the remainders of a division, of which the divisor is terminated by one or more zeros. 95. Particular Cases. — When there are no decimal places in one of the terms of the division — For example, we can have 51-47876 to be divided by 849, or 3145 to be divided by 23-479. In the first of these examples, we would proceed according to the first method indicated in (94), under the head secondly. In the second, we suppress the point in the divisor, and annex to the dividend as many zeros as there are decimals in the divi- sor. This is the same thing as multiplying both terms by the same number. These cases are too simple to demand farther de- velopment. Conversion op Vulgar Fractions into Decimals. 96. We have seen in (92) how we are led to convert a vulgar fraction into a decimal. This operation forms an essential part of the theory of the division of decimal fractions. But we will make here an important observation, which shall serve us in the DECIMAL FRACTIONS. 115 exposition of tlie properties of decimal fractions, whicli we have to establish hereafter. This observation consists in this, that instead of placing zeros to the right of the different remainders which we obtain hy apply- ing the rule of (92), we can place at once these zeros on the right of the dividend, and perform the division of the resulting num- ber by the divisor, taking care to place the point in the place to which it belongs in the quotient. In order to establish this second method of proceeding, we take the example, \^, and write out both methods. 130 47 13000000 47 360 0-276595 360 0-276595 310 310 280 280 450 450 270 270 35 35 In the first method, after writing a zero in the quotient, to take the place of the entire number, we annex a zero to the numera- tor, 13, of the fraction, in order to obtain the tenths; we then place another zero to the right of the remainder of this division, in order to obtain hundredths, and so on, until the total number of zesos thus successively brought down is six. In the second method, we multiply the numerator 13 by 1000-000 first, and then perform the division. It is obvious that the quotient thus obtained difi'ers from that obtained by the first method of pro- ceeding, in being 1000000 times greater, and that we reduce it to its true value by dividing it by 1000000, or by pointing off six decimal figures on the right. 116 DECIMAL SYSTEM OF WEIGHTS, &C. DECIMAL SYSTEM OF WEIGHTS, MEASimES, AND COINS. Having now discussed the four fundamental operations of arithmetic in their application to decimal fractions, we can ap- preciate the advantages which the calculus of decimal fractions presents over that of vulgar fractions, and are prepared to judge how important it is to establish a decimal system of weights, coins, and measures. In the United States we have the decimal system of coins in the Federal money. In France, the decimal system of weights, coins, and measures, has, after many efforts, been established, in spite of the obstacles occasioned by ignorance and prejudice. We give these decimal systems, with a few ex- amples, in order to illustrate their advantages over the ordinary systems, with their irregular subdivisions. 97. The denominations of the currency of the United States are Eagles, Dollars, Dimes, Cents, and Mills, (the last three terms expressing their relative values to the dollar by their deri- vation). TABLE. The Eagle is divided into 10 dollars. The Dollar 10 dimes. The Dime 10 cents. The Cent 10 mills. The dollar sign being ($), we would, for example, write 56 dollars, 57 cents, and 5 mills, simply $56-575. In order to make the comparison, if we wished to write £15 10s. 6f7. in parts of a pound, we would have to write £15 -f J jj + j^^ of -^^. And in order to express this decimally, we would have to reduce tlie compound fraction to a simple one, and then the vulgar fractions to decimals by last article. French Coins. The franc is the principal unit of the new French system of coins, its divisions being the decime and renflme. The Napoleon DECIMAL SYSTEM OP WEIGHTS, AC. 117 contains 20 francs. The sou, or piece of 5 centimes, is still re- tained, but all calculations are made with the franc and its deci- mal divisions. TABLE. The Franc is divided into 10 decimes. The Decime 10 centimes. Thus, we would write 545 francs, 8 decimes, (16 sous), and 4 centimes, 545-84/r. We will now explain the nomenclature of the French system of weights and measures, to which the name metrical system has been given, the metre being the principal unit. 98. The unit of length, to which we give the name metrej is the ten millionth part of the distance from the pole to the equa- tor, measured on the meridian of Paris. According to measure- ments made and verified with the utmost precision, the metre^ valued in old French feet and inches, is equal to 3 feetj inches, 11-296 line, to within less than y^L- of a line, or equal to 39-3809171 of our inches.* In order to designate measures smaller or larger than the metre, it is agreed upon to employ the following prefixes, (taken from the Greek and Latin). Myria, Kilo, Hecto, Deca, Bed, Centi, Milli, which signify ten thousand, thousand, hundred, ten, tenth of, hundredth of, thou- sandth of, (the multiples being indicated by the Greek prefix, the submultiples by the Latin). These prefixes are placed before the word metre ; and the following table is formed. For conve- nience of comparison, we convert the divisions and subdivisions into parts of our inch. * This measurement of the arc of the meridian was made under the auspices of Arago and Biot. Several degrees, measured with great accu- racy, served as a basis for the calculation of the length of the whole meridian. Myriametre, or Milometre, (C Hectometre, Decametre, Metre, Decimetre, = Centimetre =; Millimetre = "118 • DECIMAL SYSTEM OF WEIGHTS, &C. 10,000 metres = 393809-171 inches. 1000 metres = 39380-9171 " • 100 metres = 3938-09171 " 10 metres = 393-809171 " principal unit = 39-3809171 " -^\ of a metre = 3-93809171 " yi^ of a metre = 0-393809171 " -j-o^^j^ of a metre = 0-0393809171 " N. B. The myriametre, and the Mlometrey are the itinerary measures at present adopted in France. The myriametre is 6-22 miles. Measures of Superficies; or. Square Measure. 99. The natural unit of surface is the square metre; that is, a square which has a metre for its side. The decimetre squared, or the square which has a decimetre for its side, is jj^ of the metre squared; the square centimetre is j^j^^^, and so on, for the rest. The square decametre is equal to 100 square metres. This measure we take for the principal unit in all field measures; and this unit is called are. The multiples and subdivisions of the are are also designated by the aid of the prefixes, myria, hectOj deci, centi .... Thus, The Myriare = 10,000 ares = Kilare = 1000 ares Hectare = 100 « Decare = 10 " Are = the principal unit = 100 sq. ms. = 119-665 sq. yds. = ^ acre, about. Declare — ^^ of an are. Centiare = -j^^ of an are. Milliare = jj^^-q of an are. DECIMAL SYSTEM OF WEIGHTS, AC. 119 N. B. The myriarej the hectare, are, and centiare, are the only measures used. The centiare is the square metre."^ Measures of Volume. 100. The unit of volume is the cuhic metre ; that is, a cube, (solid, of the form of a die), which has a metre for its side. The multiples and submultiples of the cubic metre have as yet re- ceived no particular names. The 1000th of the cubic metre is called the cubic decimetre, because it is a cube with a decimetre for its side, &c., for the cubic centimetre .... When the mea- sures of volume are applied to wood for burning, or to materials of building, the principal unit or cubic metre is called stere. We then have the decastere, or measure of ten steres. The stere = 3 5 "3 7 5 cuhic feet. Measures of Capacity, both Dry and Liquid. 101. The unit of capacity is the cuhic decimetre, which is called litre. As to the decimal multiples and submultiples, we give those which are chiefly used. Hectolitre..., = 100 litres. Decalitre = .10 litres. [cub. in. Litre = principal unit = 1-057 U. S. qts. = 61.074 Decilitre = t o ^^ ^ ^^t^Q- Centilitre = jj^ of a litre. Weights. 102. The unit of weight is the weight of a cuhic centimetre of distilled water, at the temperature of maximum density, viz., 39-5° Fahrenheit. The name given to this unit \b gramme. The gramme is equal to 0-002204737 pounds avoirdupois. * A partial decimal square measure has been introduced among sur- veyors in the United States. The surveyor's chain, 66 feet in length, is divided into 100 equal links ; and we have 10,000 square links = 1 sq. chain. 10 square chains = 1 acre. 120 DECIMAL SYSTEM OP WEIGHTS, &C. TABLE. lbs. The Myriagramme is =10,000 grammes= 22-04737 Kilogramme = 1000 grammes = 2-204737 Hectogramme = 100 grammes= 0-2204737 Decagramme = 10 grammes = 02204737 Gramme = principal unit= 1 002204737 Decigramme = -j-'^ of a gramme = 0-0002204737 Centigramme = j-ioOfagramme= 0-00002204737 Milligramme = ^J^, of a gra. =0-000002204737 N. B. The half kilogramme is about equal to the old French pound, nearly equal to our pound avoirdupois. 103. Such is the nomenclature of the measures which compose the metrical system. We can now judge of the advantages which this system possesses over the ordinary measures. 1st. It is uniform and simple, inasmuch as its principal units and their subdivisions follow the law of the decimal system of numeration. 2d. It is fixed, invariable, and susceptible of being adopted in all countries, since it is equally adapted to any climate or lati- tude. All these measures have for their base one primitive measure, the metre, which is taken from the dimensions of the earth itself. We will dwell but little upon the application of the four fun- damental operations of arithmetic to the decimal system of weights and measures, since every collection of principal units and their subdivisions, according to the nomenclature, can be expressed by a decimal fraction ; and, therefore, these operations become opera- tions upon decimal fractions, considered as abstract numbers. For these last operations we have already established fixed rules. Nevertheless, we will propose some questions in multiplication and division, because they will afibrd opportunity for some im- portant remarks upon approximate calculations. DECIMAL SYSTEM OF WEIGHTS, &C. 121 Examples under the different tables illustrating the above. 1st. — 56 kilometres, 25 decametres, 5 metres, and 9 milli- metres, are written, 56255-009 metres. 2d. — 25 hectares, 4 ares, and 6 centiares, are written 2504-06 ares. 3d. — 34 hectolitres, and 6 centilitres, are written 340-06 litres, 4th. — 54 myriagrammes, 4 decagrammes, 7 decigrammes, and 3 milligrammes, are written 540040-703 grammes Multiplication. 104. Question first. — Required, the price of 35 metres, 429 millimetres of a certain stuff, one metre of which costs $19 and 76 cents. Here, if we multiply 35-429 w. by $19-76, we will obtain a product which, expressed in dollars, cents, and mills, will be the price required. The abstract product of these numbers (93), is 700-07704; then, $700-07, or, more exactly, $700-08 is the price of 35-429 m. Sometimes the fraction of the metre is ex- pressed by a vulgar fraction. In this case, the operation can be performed in two ways. Question second. — What is the price of 23| m. of a piece of stuff, at $8-25 cts. per metre ? 1st. The reduction of | to decimals, gives 0-75; the question is then reduced to multiplying 8-25 by 23-75, which gives 195-9375; then, $195-94 c^s. is the price of the 23| metres, to within less than ^ cent (93). 2d. We could also operate as follows : 8-25 23| 24-75 1650 189-75 i = 4-125 \ = 2-0625 195-9375 11 122 DECIMAL SYSTEM OF WEIGHTS, &G In this operation, after forming the product of ^he two entire parts, we have added the two partial products, and placed the point where it properly belongs,, in order to avoid all error.in the final result. We have then multiplied 8-25 by | (A + |) by taking first the half of 8-25, which gives 4-125; then the half of this half, which gives 2-0625. Now, taking the sum, we get 195-8375, as by the first method. This last method of proceeding is preferable, when the vulgar fraction cannot be converted into a limited number of decimal figures. Third Question. — To find the price of 89 j^ metres, supposing one metre to cost $47*19. 1st operation, 4719 424-71 3775-2 4199-91 = 23-595 = 11-7975 = 7-8650 42431675 Then, 89 {^ metres cost M243-17, to within less than one cent. Otherwise, commencing by converting i^ into decimals, we find (>-91666 . . . .; and we must multiply 89-916666 .... by 47-19. 89-91 47-19 89-916 47-19 89-9166 4719 80919 8991 62937 35964 809244 89916 629412 359664 8092494 899166 6294162 3596664 4242-8529 424313604 4243164354 This table gives three distinct operations. 1st, with two de- cimal figures of the multiplicand; 2d, with three; 3d, with four; DECIMAL SYSTEM OP WEIGHTS, &C. 123 and we see it is the last only wliich gives the approximation to within less than one cejit. The difficulty here is to know how many of the decimal figures of the multiplicand we must take, in order to be assured that we have the degree of approximation required by the nature of the question ; while by the first method we obtain a complete result, of which we can, according to choice, neglect more or less of the decimal figures. N. B. We could also reduce 89|-J to a single fraction ; then multiply 47 '19 by this fraction; an operation longer than the first method which we have used. Division. 105. Question Fourth. — A piece of land containing 23 hec- tares, 9 aresj 25 centiares, (23 A., 0925 c), loas bought for $83,719-25. Required the value of the hectare? We must here divide 83719-25 by 23-0925; and the quotient, valued in dollars and cents, will represent the price per hectare. We obtain, by simple division of decimals, $3625-38. Question Fifth. — 28^| kihgrammes, of a certain material, cost $519-35. What is the price per kilogramme f Here we may use two methods. 1st, Reduce 28^| (o a single fractional number, giving ^^^^ Then multiply 519-35 by ^J, inverted, (Art. -59); we thus find for result, 18-038. 2d. We convert ^| to decimals, which gives 0-79166 . . . .; then we divide 519-35 by 28-79, taking only two decimal places of the divisor; we obtain thus, 18-039. Then, $18-04 is the price per kilogramme of the stufi". These examples suffice to show how we must proceed in the multiplication and division of denominate numbers of the decimal systems, and to show how much simpler these operations are than in the ordinary systems of compound numbers. 124 DECIMAL SYSTEM OF WEIGHTS, &C. We will add here, as belonging properly to tlic preceding theories, some notions upon the different divisions of the circle and thermometer. 106. 0/ the two divisions of the Circle. — The circumference of a circle is defined in geometry a recutiant line, all the points of which are equally distant from a point within, called the centre. In all the scientific works in this country, the circumference is divided into 360 equal parts, called degrees (°) ; each degree into 60 equal parts, called miimte.s ('); each minute into 60 equal parts, called seconds ("). This is called the sexagesimal division. When the French reformed their system of weights and measures, they adopted also a centesimal division of the circumference of the circle, the use of which is becoming very general among the scientific men of Europe. In this new centesimal system, the circumference is divided into 400 equal parts, called degrees (°) ; each degree into 100 equal parts, called minutes (') ,' each minute into 100 parts, called seconds ("); each second into 100 equal parts, called thirds ('''), &c. Example of Sexagesimal Division. — 45 degrees, 38 minutes, 25 seconds, are written 45° 38' 25". Example of Centesimal Division. — 28 degrees, 56 minutes, and 23 seconds, are written 28-5623°, in the decimal form. In order to reduce the divisions of the sexagesimal system to a com- pound number of the centesimal, we observe that the quarter of the circumference, called a quadrant, is in one system 90°, and the other 100°. Then, 1° sexagesimal = ^-^^^ or y> of a degree centesimal, and vice versa; 1° centesimal = f^ of a degree sexa- gesimal. We are thus led to the two following rules : 1st. To convert a compound number sexagesimal to a com- pound centesimal. Reduce, first, to a fractional number of de- grees (JQ')] then multiply this number by ^-^j and convert the result into decimals. The entire part will express the centesimal degrees; the decimal part, divided into periods of two figures each, the minutes, seconds, &c. 2d. Reciprocally, to convert a compound centesimal number AC. 12. J into a compound sexagesimal. Subtract from the yiccn number , expressed in decimal form^ j'^ of this number, (o?- simpli/ take y^Q of if). The entire part of the result will represent the num- ber of sexagesimal degrees. The decimal part we convert into minutes and seconds by the known rules for converting fractions of a higher denomination into units of a lower. Examples. — 1st. Convert 34° 69' 17" sexagesimal, into de- grees, minutes, and seconds, centesimal. — 34° 59' 17", converted to seconds, give 125957", or ^*§|J^''' of a degree; this, multi- plied by Y? g^v^s ^lilB^- Finally, the division of 125957 by 3240, gives 38 -875617 or 38° 87' 56" 17'" centesimal. Reciprocally, 2d. — To convert 38° -875617 centesimal, into degrees, minutes, and seconds, sexagesimal. 38-8756170 -j-V 3-8875617 34-9880553 60 59-283318 60 16 99908 or, 34° 69' 17". Op the Principal Divisions of the Thermometer. 107. The thermometers mostly used on the continent of Eu- rope are, the thermometer of Reaumur, and the Centigrade. In England and the United States, the use of Fahrenheit's thermo- meter is almost universal. These all differ in their scales of sub- division only. In Reaumur's, the interval between the freezing and boiling points of water is divided into 80 equal parts, called degrees of Reaumur; in the Centigrade, this same interval is divided into 100 parts, called centesimal degrees. It follows, that each degree of Reaumur's is equal to "^^^^ or |, of the Centigrade degree; and, reciprocally, each Centigrade degree is equal to | of the degree of Reaumur. Moreover, the fractions of the de- gree are expressed generally in both by decimal fractions. Thus, it is a very simple matter to transform one into the other. 11* 126 DECIMAL SYSTEM OF WEIGHTS, &C. 1st. In order to convert a decimal number of de2;rees of R&iu* mur into Centigrade degrees, we add to the number one-fourth of itself. The result of the addition is the number sought. 2d. In order to convert a decimal, number of centesimal de- grees into degrees of Reaumur, subtract from the given number one-fifth of itself, and you have the number sought. Thus, for example : 39°-4716 R. == 39-4716 + 9-8679 = 49°-3395 0. Reciprocally, 49°-3395 C- = 49-3395 — 98679 = 39°-4716 C. In Fahrenheit's thermometer, the freezing point of water is 32°, instead of 0°, and the interval between that and the boiling point (212°) is 180°. Then, the degree of Fahrenheit is }§§ = ig, or i of the degree Centigrade; and, reciprocally, the de- gree Centigrade is |, or |§, of the degree of Fahrenheit. In the actual reduction from one of these scales to the other, we must always keep account of the different start point, both for negative and positive temperatures. Thus, 1st. To convert a decimal number of degrees Fahrenheit (-f ) into centesimal degrees, we must first subtract 32° ; then remove the decimal point one place farther to the right, and divide by 18, (or multiply by 5 and divide by 9). 2d. To convert a decimal number of degrees Centigrade into degrees of Fahrenheit, remove the decimal point one place to the left, and multiply by 18, (or multiply by 9, and divide by 5) ; then add 32° to the result. Example 1st. To convert 56° -259 Fahrenheit into Centigrade degrees. 56°-259— 32°=24°-259....24°-259xig=24_2^59=i3o.477c, 2d. To convert 13°-48 C. to degrees Fahrenheit. 13-48 X 18=1-348 X 18=24-259 .... 24-259-f 32° = 56°-259 F. DECIMAL SYSTEM OF WEIGHTS, AC. 127 The rules for the conversion of the (minus) degrees, and also for conversion of Fahrenheit into Reaumur, are too obvious to discuss them farther. 108. General Conclusion. — This first part of our work includes all which constitutes elementary arithmetic, the princi- pal object of which is the exposition and development of the methods to be followed, in order to perform upon numbers all possible operations. These operations are to the number of four fundamental ones, addition, suhtraction, multiplication, and division. All the others, such as the reduction of fractions to the same denominator, to their simplest form, the conversion of a vulgar fraction into a decimal, &c., are nothing more than com- binations of those which we have just given. There are two other species of operation, or rather two parti- cular cases of the last two fundamental operations, of which we have not spoken , because, in order to be developed in a complete manner, they require some knowledge of algebra. These are the formation of powers, and the extraction of roots of numbers. The powers of a number are the products which arise from the continued multiplication of a number by itself. Thus, 4x4x4 X 4 X 4 = the 5th power of 4. The formation of powers is evi- dently then a particular case of multiplication. The roots of a number are those numbers whose continued products, each by itself, will produce the given number. Then, the extraction of roots proposes the solution of the problem — Given a mimher, to find the tioo equal factors which form it, or the three equal factors, &c. ; evidently a particular case of division. We will not discuss these, because they are fully treated in all of the good text-books on algebra. In the next chapter, we propose to consider numbers in a general manner, independently of every system of numeration, and to develop the properties belonging to any given system. This will be in some sort Arithmetic Generalized. SECOND PART. CHAPTER V. GENERAL Pl^OPEETIES OF NUMBERS. 109. Introduction. — Before going farther into tlie science of numbers, and in order to investigate their properties with more facility, we must borrow from algebra some of its materials, such as letters and signs (some of which we have used already), by the aid of which we indicate, in a general and abridged man- ner, the operations and the reasoning which the resolution of a question requires. 1st. The letters, which we employ instead of iSgures, in order to represent numbers. Their use affords at once a mode of writing, more concise and more general than that of figures. 2d. The sign + plus (already used), to indicate the addition of two or more numbers. 3d. The sign minus — (already used), to indicate the subtrac- tion of one number from another. 4th. The sign of multiplication is X , or a point, which we place between the two numbers, read multiplied hy. Thus, aXh, or, a. hj mean a multiplied by h. N. B. We have already used both these signs. Now, when the numbers, the multiplication of which we wish to indicate, (128) GENERAL PROPERTIES OF NUMBERS. 129 are expressed by letters, then this multiplication will be indicated also by simply writing one of the letters after the other, with no sign between ; thus, ah signifies a multiplied by h. But this method cannot be employed when the numbers are indicated by figures ; for, if we wrote the product of 5 by 6, 56, this notation would be confounded with fifty-six. In the case of figures, then, X , or some such sign between the numbers, is necessary. An- other sign of multiplication is the parenthesis ( ). 5th. The sign of division, either a bar ( — ^), as already used in vulgar fractions, or a bar with two points, thus (-^-), or simply two points. Thus, \4 == 24 -- 6 = 24 : 6 = 24 divided by 6 6th. The Coefficient is the sign which we employ, when a number denoted by a letter is to be added to itself several times. Thus, instead of writing a-\-a-\-a-\-a-\-a, which represents the number a added to itself four times, we write 5a. We say, then, the coefficient is the number written on the left of another numher, denoted hy one or more letters, to show how many times this number is taken, or the number of times plus one it is added to itself. 7th. The exponent is the sign which we employ, when a num- ber denoted by a letter is multiplied several times by itself. Thus, instead of writing aXaXaXaxa, or aaaaa, we write simply a^ , which signifies that a is multiplied 4 times by itself. The exponent is then a number written to the right, and a little above another number, or letter expressing a number, showing the number of times plus one that this number or representative letter is multiplied by itself 8th. The sign which expresses that two numbers are equal, already used (=), read is equal to, or simply equals. The axioms applied in the operati<.ns on equations we annex to this definition, viz : If equals be added to, subtracted from, multiplied, or divided by equals, the results will be equal. These preliminaries being established, we will take up again some of the subjects of which we have treated in the first part, in order to investigate them more thoroughly. We will arrive 130 GENERAL PROPERTIES OP NUMBERS. thus at new properties, and at means of simplifying -or modifying the methods in the different operations of arithmetic. [In order to give some idea of the use of these different signs, and of the simplicity of the algebraic language, we will make a few applications. Let us suppose, first, that we wish to express that a number, represented by a, is to be multiplied 3 times by itself; that the product thus resulting is to be multiplied 3 times successively by h; and, finally, the new product is to be multiplied twice by c; we will simply- write a*iV. If we wish to express that it is necessary to add this last result 6 times to itself, or multiply it by 7, we write Ta^tV. In the same manner, 6a®6* is the abridged expression of 6 times the product of the 5th power of a by the second power of h. 3 a — 55 is the abridged expression of the difference between the triple of a and the quintuple of h. '±0^ — 3a6-j-45* is the abridged expression of the double of the square of a, diminished by the triple product of a and i, and augmented by four times the square of h. Let us now see how we can effect, upon quantities expressed algebraically, the fundamental operations of arithmetic. We will limit ourselves to the most simple cases — those to which we will have to refer in the latter part of this treatise. Addition. — In order to add two numbers, a and 5, we write simply a-f 6. In the same manner, a-\-h-\-c indicates the addi- tion of the numbers a^ h, c : that results from the notation we have established. In the same manner, a — 6 and c -J- r7 — /, added together, form the single quantity, a — h-\-c -\- d — f. If we had to add a — h and h — r, we would write a — h -{-h — c. But as, on the one hand, h is added, and on the other subtracted, it follows that these two operations balance each other, and the expression is reduced to a — c. GENERAL PROPERTIES OF NUMBERS. 131 Subtraction. — In order to subtract h from a, we write a — h. In the same manner, if we wish to subtract c from a — 6, we write a — h — c. Let it be required to subtract the expression e — d from the expression a — h. We can first indicate the subtraction thus: a — h — (c — c7). But if we wish to reduce the result to a single expression, we must reason as follows : If we had to subtract c alone from a — 6, the result would be a — h — c. Now, as it is not o, but c diminished by J, which is to be subtracted, the result, a — h — c, is too small by the num- ber of units in d ; thus, the result will be brought back to its just value by adding d to a — h — c, or writing a — h — c-{-d. That is to say, in order to subtract one algebraic expression from another, we must write the one to be subtracted with the signs of all its terms changed, after the other ; thus forming one single expression. We find by this rule, and analogous reasoning, 3a— (26— 3c) = 3a— 25 + 3c. 5a— 46 — (6c/— /-f ^7) = 5a— 46— 6c7 +f—g. Multiplication. — Eequired to multiply a* by h^. We write a^ x 6', or simply a^J?. But if we have a^ to be multiplied by a', we observe that the number a, being 5 times a factor in the multiplicand, and 3 times a factor in the multiplier, ought to be 5 + 3, or 8 times a factor in the product. Thus, we have a^Xa^^=a^; that is to say, when the same letter enters into both factors of the multiplication, we lorite it once in the product y and give it for exponent the sum of its exponents in the two factors. Required, now, to multiply a — 6 by c. We can first indicate the product in this manner (a — h) c. But, if wc wish actually to perform the operation, we remark, that to multiply a — 6 by c (27), is to multiply c by a — h ; that is, to take c a — b times, or as many times as there are units in a, diminished by the units in b. If, then, we multiply first c by a, which gives ac or ca, the product is too great by the product 132 GENERAL PROPERTIES OP NUMBERS. of c by h, or he. Thus, we must subtract he from ac, and we obtain ac — he for the required product, (a — h)c = ac — he. Required, again, to multiply a — 5 by c — d. The product can first be indicated thus : (a — h) (c — d). But, in order to obtain a single expression, we commence by multiplying a — h hj c, which gives ac — he; and we observe, then, that it is not by c alone that we have to multiply a — 6, but by c diminished by d. Thus, the product ae — 6c is too large by the product of a — b by a; that is, by ad — hd. Then, in order to reduce the pro- duct to its just value, we must subtract ad — 5c? from ac — be; which gives, by the rule of subtraction, ac — he — ad -\- hd. Examining this product, we deduce the following rule : In order to effect the multiplication of two algebraic expressionSj. multiply successively each term of the multiplicand by each term, of the multiplier ; observing, that if two terms of the multiplicand and multiplier are affected with the same sign, their product is affected with the sign -j- (^plus") ; hut if they are affected with different signs, their product is affected loith the sign — (minus). Division. — We will consider only a single case of this opera- tion, in which the two terms of the division contain the same letters. Required to divide a? by a^. We can first indicate the quotient in this manner : g, or CT^-f- a'. But a' is the product of which a^ and the quotient are the two factors ; hence, the exponent, 7, of the dividend, ought to be equal to the sum of the exponent of the factor known, 3, and of the unknown exponent of the quotient; then, recipro- cally, this last is equal to the difference between the exponent of the dividend and the exponent of the divisor ; that is, to 7 — 3 or 4. Thus, -r, = a\ -j-==ab . . . . &c. ' a-* a^b Such are the general notions of algebra, of which we will have to make use in the fifth and following chapters.] GENERAL PROPERTIES OF NUMBERS. 133 Theory op Different Systems op Numeration. 110. We have seen (Art. 5), how, by the aid of ten characters or figures we can represent all numbers, setting out with the conventional principle, jthat every figure placed on the left of another, expresses units ten times greater than those of the first figure. We now propose to show that we can write all numbers with more or less than ten characters, provided we do not use less than two, (zero, 0, being always one of these characters). We call, in general, the number of figures employed, the base of the system. The system in which two figures are used, viz : (10), is the binary system, and 2 is the base. The ternary system, of which 3 is the base, makes use of 8 figures, 1, 2, ; the quaternary has four figures, 1, 2, 3, 0; the quinary, five, 1, 2,3, 4, 0; &c., &c. The base may be greater than ten ; we must then have recourse to additional characters. Thus, in the system of which twelve is the base, the duodenary or dvodecimal, we will have to use two new signs, a and jS, to express ten and eleven numbers less than the base. In every system analogous to the decimal system, the conven- tional principle holds that every figure placed on the left of an- otlierj expresses units as many times greater than those of the first figure as there are units in the base of the system. Thus, in the binary system, each figure acquires a value twofold greater for each place that it is removed to the left. In the ternary system, they increase in a threefold ratio ; and, in general, in a system of which b is the base, a figure goes on increasing in a b-f old proportion, as it is removed one or more places to the left. When a number is written in a system whose base is b, the first figure on the right . expresses the units of the first order ; the figure immediately on its left the units of the second order j the next figure on the left the units of the third order ; and so on. It requires b units of the first order to make one of the second ; b units of the second to form one of the third, &c. 134 GENERAL PROPERTIES OF NUMBERS. 111. We pass, now, to the manner of expressing in figures any entire number, whatever be the system which we adopt. In order to fix our ideas, we will consider the septenary system, which makes use of the seven characters, 1, 2, 3, 4, 5, 6, 0. Adding unity to six, we obtain seven, or the unit of the second order; which, according to the principle enunciated above, can be expressed by 10 ; since the 0, having no value of itself, makes the figure 1 at its left express one unit of the second order, or seven simple units. Placing, successively, all the figures of the system in the first and second place, we will evidently form all the consecutive numbers comprised between 10 = seven and the number expressed by 66. For example, 11, 12, 13, 14, 15, 16, represent the numbers eight, nine, ten, eleven, twelve, thirteen, 20 = fourteen, 21 = fifteen, &c. After reaching the number 66, if we add to it a new unit, there will result 6 units of the second order, plus seven units of the first order; that is to say, seven units of the second order, or a single one of the third order, which can be expressed by 100. Placing, successively, in the first, second, and third places, the difi'erent figures of the system, we will form all the consecu- tive numbers comprised between 100 and the number expressed by 666. Reasoning on this last number, as upon 66, we shall arrive at the unit of the fourth order, which is expressed by 1000 ', then we obtain, successively, all the consecutive numbers comprised between 1000 and the number expressed by 6666; and so on, to infinity; whence we see that all possible entire numbers can be written in this system. The same reasoning applies to any other system. Whatever system be adopted, the units of the different orders are respectively represented by 1, 10, 100, 1000, 10000, &c., as in the decimal system. 112. N. B. We have said (110), that the character was in- dispensable in every system analogous to the decimal system; that is to say, in a system where the relative value of a figure depends upon the place which it occupies on the left of several GENERAL PROPERTIES OP NUMBERS. 135 others. To speak rigorously, we could do without it; but the •system would be less regular, as we shall see. Let it be proposed, for example, to establish the ternary sys- tem, using the three significant figures, 1, 2, 3. The first three numbers are expressed by these figures. In order to represent four,jjivej and six, it would suffice to write 11, 12, 13. In order to express seven, eighty nine, ten, eleven, twelve, we would write 21 22 23 31 32 33 In the same manner, 111 112 113 121 122 123 would express thirteen, fourteen, fifteen, sixteen, seventeen, eighteen. It is not necessary to go farther, in order to see the inconve- niences of this system. Its principal fault consists in this, that units of the same order are expressed in a different manner. Thus, in 13 and 23, the figure 3 expresses a unit of the second order, the same with the figures 1 and 2 on its left. In 123, 23 express nine, or units of the third order, the same with the figure 1 to the left of them. (The same process might be applied to the decimal system as is here applied to the ternary, by using a single character for ten, and dropping the 0.) In making use of 0, it suffices to determine the number of units of different orders which enter into the proposed number, and to write, one after the other, the figures which express these units. 113. The perfect adaptation of the nomenclature of numbers, and the manner of writing them in figures, in the decimal system, permits us to write them easily from dictation in ordinary lan- guage. The same thing would be true of every system of numeration which had a special nomenclature appropriate to the system ; in other words, a spoken numeration corresponding to its written one. But other systems do not present this immediate * connexion with the nomenclature now in use. 136 GENERAL PROPERTIES OF NUMBERS. Let it be proposed, for example, to express the number, three hundred and sixtj/'inne, referred to tlie decimal system in the septenary system. It is difficult to see, d priori, which are the figures proper to express units of the first, second, third .... order which it contains. Now, since this number, written in figures in the decimal sys- tem, is three, six, nine, it follows that the question above depends on the following, which is much more general : — A number being enunciated in ordinary language, or icritten in the decimal system, required to express this same number in the system whose base is b. In order to resolve it, we remark, that since it takes b units of the first order to make one unit of the second order, as many times as the proposed number contains the number b, so many units of the second order of the system, whose base is b, will it contain ; that is to say, that if we divide this number by b, the quotient will express units of the second order, and the remain- der, which will necessarily be less than b, will express the units of the first order of the number written in the system whose base is b. In the same manner, since b units of the second order in the system whose base is b, form one unit of the third order in the same system, if we divide the quotient which expresses units of the second order by b, the new quotient which we thus obtain shall express units of the third order, and the remainder, always less than b, shall represent the units of the second order written in the system whose base is b, and so on for the rest. Whence we see, that in order to pass from the decimal system to the system whose base is b, we must, 1st, divide the given number by the base of the new system written in the decimal sys- tem, and write the remainder of this division apart, as expressing the units of the first order in the new system ; 2d, divide the quo- tient obtained by the same base, and write tJie second remainder to the left of the first, as expressing the units of the second order ; 3d, divide the second quotient by the same base, and write tJie third remainder on the left of the two preceding, because it ex- GENERAL PROPERTIES OF NUMBERS. 137 presses units of the third order ; continue this series of operations until we arrive at a quotient smaller than the base of the new system ; this last quotient expresses the units of the highest order, and is written on the left of all the remainders successively ob- tained. Let us apply this rule to the number, 369, which we wish to express in the septenary system. 7 I 369 7 I 52 (5 1st rem. 7 1" (3 2d rem. 1 (0 3d rem. • Dividing 369 by 7, we obtain for quotient, 52, with remain- der, 5, which we write apart, in order to express the units of the Jirst order in the new system. Dividing 52 by 7, we find 7 for quotient, and 3 for remainder, which we write to the left of 5, as it expresses units of the se- cond order. Dividing 7 by 7, we have 1 for quotient, and for remainder, which indicates that there are no units of the third order; but we write a to take the place. Finally, as the quotient 1 is smaller than 7, it expresses the units of the fourth order, and the number in the septenary sys- tem is (1035). On examining this operation, we shall find that we have ob- tained the three hundred and forty threes, the forty nines, the sevens, and the units, which the given number, 369, contains Hence, we might also proceed by the following rule : Find, by inspection, the highest denomination of the new sys- tem which the given number contains; divide the given number by the number expressing the highest order of units written in the decimal system. Set the quotient apart, as expressing the highest order of units of the required number in the new system. Divide the remainder by the number expressing the value of the 'next lower order, and place the quotient on the right of the first 12* 138 GENERAL PROPERTIES OP NUMBERS. one to express the units of the next highest order-. Divide the remainder hy the next lower j placing the quotient on the right of the two preceding, &c., &c. Thus, in the same example, 369 to be converted into its equi- valent number in the septenary system. We see that three hun- dred and forty-three is the highest order of unit of the septenary system which it contains. We divide by 343 ; the remainder, 26, 343)369 ("1 by 49 ; the remainder, by 7 : the last re 343 mainder, 5, being necessarily less than 4q\9fi7o seven, the base of the system expresses units. ^0 we write the quotients from 7)26(3 left to right, commencing with the first 5 j^gt rem. obtained, and write the last remainder on the right of the last quotient. We obtain, as before, (1035). The first method given is, however, the best, especially for large numbers. We find, by this method, the number 5347 of the decimal system, equal to (12343) of the system which has eight for its base. 8 I 5347 1 1 668 (3 1st rem. 8 1 83 (4 2d rem. 8 1 10 (3 3d rem. 1 (2 4th rem. (12343) Remarh. — It can happen that the base of the new system is greater than ten, the base of the decimal system. In this case, we proceed as follows : — Required, for example, to convert the number 8423 of the decimal system into its equivalent number in the duodenary system. The figures of this system are 1, 2, 3, 4, 5, 6, 7, 8, 9, a, j3, 0. (The two Greek letters, a and /3, being employed to designate ten and eleven in the new system.) GENERAL PROPERTIES OP NUMBERS. 139 12 I 8423 12 |"701 (p 1st rem. 12 I 58 (5 2d rem. 4 (o 3d rem. The base twelve being expressed (4a5i3) by 12 in the decimal system, we divide 8423 by 12, which gives 701, and remainder, /3 = 11, in the decimal system. We write this ^ apart, as ex- pressing units of the first order. Likewise, in the third division, we obtain for a remainder ten, which, in the new system, is ex- pressed by a; we then write a to the left of the two figures already found. We obtain thus (4a5/3) for the equivalent of the given number in the new system. 114. Reciprocalli/j a number being written in a system whose base is b, required to enunciate it in the spoken numeration of the decimal system ; that is, to convert it into its equivalent in that system. In general, let ... . hgfdca be a number expressed in the system of which b is the base; a, c, d,f, &c., expressing units of the first, second, third .... order, (and not being an indicated product), as in (4°, Art. 109). It results from the fundamental principle established in (110), that the figure denoted by c, ex- presses units b times as great as the same figure standing alone would express; then, its relative value can be represented by cxb, or simply by cb (109). In the same manner, the figure a expresses units b times as great as those of the figure c : hence, its relative value is equal to the product of d hy bxb or b^, and can be expressed db^. We could show, in like manner, that/6^ g¥, hb^ .... are the relative values of the other figures. Then, the given number is expressed by a^cb-\-db^+/b^-}-gb'+hb'+ Giving to the base b and to the figures a, c, d, /, particular values, we effect all the operations indicated in this expression, and we shall obtain the number corresponding to the particular data, converted into the decimal system. 140 GENERAL PROPERTIES OP NUMBERS. Required, for example, to convert the number 4867, written in the system of eiyht figures, back into the decimal system. This number can, according to the expression above, be placed under the form 7 + 6x8-1-3x8^4x8'. We have at once, then, 7 =7 6x8 =48 3x8' =192 4x8=^ = 2048 2295 Adding these numbers, we have 2295 for the value of (4867) in the decimal system. We can verify the accuracy of this ope- ration by the rule of (113). 8 I 2295 8 I 286 (7 1st rem. Sjii" (6 2d rem. ~4 (3 3d rem. (4867). And, reciprocally, this last operation can be verified by the preceding one, which we will enumerate generally thus : Form, first, the different powers of the hase, h, written in the decimal system; multiply then all the figures of the number, written likewise in the decimal system, as a, c, d, f g, h, re- spectively, hy 1, h, h^, h^, h*, V". Adding the partial products, we shall have the number required. Given, for example, the number (4a5)3) in the duodenary system, to be converted into its equivalent in the decimal system. Since a and |3, written in the decimal system, are 10 and 11 re- spectively, this number can be placed under the form 11 + 5x12 + 10x122+4x12'. GENERAL PROPERTIES OF NUMBERS. 141 11 = 11 5x12 = 60 10x12^ =1440 4xl2» =6912 8423 Then, (4a5|3) equals 8423, written in the decimal system. 115. The two preceding rules lead to a third, more general, which has for its object to convert any number from a system whose base is 6, into its equivalent in a system whose base is c. Convert the number front the system h to the decimal system^ hy (114); then from the decimal system to tlie system c, by (113). Required, for example, to convert the number (23104) of the system whose base is 5, to its equivalent in the duodenary sys- tem. We obtain, first, for this number, transformed into the decimal system, 1654 ; then, for this last, transformed into the duodenary system, ()35a). We can verify this operation by making the transformations in an inverse order. N. B. The above transformation from the quinary to the duo- denai-y system, could be effected directly, without the intervention of the decimal system, by performing all the operations required in the quinary system ; the only difficulty of this mode of ope- rating being the want of agreement between the written numera- tion of this system and the spoken numeration, so universally in use. 124. The methods of performing the four fundamental opera- tions of arithmetic, upon numbers written in any system whatever, do not differ from those which have been established for the decimal system. We must only recollect the law which exists between the units of different orders, in order to be able to con- vert the units of any order into units of the order next higher or next lower. In order to familiarize beginners with the different systems of numeration, we will propose an example of each of the four operations in the duodenary system. 142 GENERAL PROPERTIES OF NUMBERS. 1st. Required to add 3704a, i32956, 27i3a5, 48a/3.. We find for the sum of the simple units thirty-two ; 3704a that is to say, 2 twelves and 8 units; we then write 8 132956 in the units column, and carry 2 to the column of units 27f3a5 of the second order. The sum of the units contained 48a/3 in this second column is thirty-one, or 2 units of the n- g^o third order, and 7 of the second ; we write the 7, and carry the 2 to the next column. Operating in the same manner on the other columns, we obtain (I5a678) for result. 2d. Required to subtract from 5a0046 The number, 47a68p 121577 As we cannot subtract jS from 6, we borrow one unit of the second order from the 4, and say, |3 from eighteen leave 7. Pass- ing to the next subtraction, as we cannot subtract 8 from 3, we borrow one unit from the first significant figure to the left. As there are two zeros between, we say, this unit thus borrowed equals twelve, or |3 + one of the next lower, which one equals J3 -f one of the next lower, which one equals twelve of the same order with the 3. We then Subtract 8 from fifteen, giving 7. In the two following subtractions, we regard the zeros as re- placed by |3, and continue the operation to the end. 3d. Required to multiply 3407a by 5a68 228528 180360 294664 148332 177608828 We premise here a table of multiplication as far as the figure |3, the highest figure of the system, after the manner of the table of Pythagoras. 177608828 5a68 l^a08 • 3a082 3407a 4a968 0000 GENERAL PROPERTIES OP NUMBERS. 143 This being premised, we multiply 3407a by 8, and say; 8 times a make eighty, or (68) of duodenary system ; we write the 8 and carry the 6. Then 8 times 7 make fifty-six, and 6 make sixty-two, or (52) of the duodenary. We write the 2 and re- serve 5 for the next column. Continiiinu; this operation, we obtain for a partial product, 228528. As to the products of the multiplicand by the other figures of the multiplier, the same reasoning applies, and we use the same processes as in the deci- mal system. Summing up the products, we obtain 177608828. 4th. Let us verify this operation by divi- sion. We simply divide the product ob- tained, by one of the factors. In order to obtain the number of units of the highest order in the quotient, we take the first five figures on the left of the dividend, and divide 17760 by 5a68. For, thus, we see that 17 contains 5 three times, with a remainder. Multiplying the divisor by 3, and subtracting the product from the first partial dividend, we obtain for a remainder, l/3aO. ' We bring down 8 and divide l|3a08 by 5a68, obtaining 4 for quotient, and 3a0 for remainder. AVhen the following figure 8 is brought down, the new divi- dend does not contain the divisor ; we then place in the quo- tient and bring down 2, which gives 3a082 for the next partial dividend. Proceeding in the .same manner with the rest, we obtain 3407a for the required quotient. We can now see how we can pass at once from the number (23104) of the quinary system, to its equivalent in the duode- nary (115). We must divide 23104 by 22, or twelve written in the quinary system, and perform the division in that system, we would thus obtain a remainder which would express the units of the first order in i\\Q duodenary system, and a quotient which we would divide again by 22, or twelve expressed in the quinary system, in order to get the units of the second order, &c., &c. 117. General RemarJc. — The duodecimal system ofiers some 144 GENERAL PROPERTIES OF NUMBERS, advantages over the decimal, inasmucKas its base twelve coniBins a greater number of factors than ten. For twelve is divisible by 2, ^, 4, 6 ; while the only factors of 10 are 2 and 5. Nevertheless, we could not substitute the duodenary system, or any other, for the decimal, without replacing the ancient nomen- clature by a new one, which was more appropriate to the system adopted, that is, which made the enunciation of written numbers easier. We shall perceive, moreover, that the greater part of the pro- perties of numbers which have been discovered are true, what- ever be the system of numeration which we adopt, and some, which shall seem to belong to the decimal system in particular, have their analogous properties in the other systems. The em- ployment of the letters of the alphabet in order to represent numbers, is well calculated to make the generality of these pro- perties appear, as they can express numbers enunciated in any system of numeration whatever. Principles of Multiplication and Division. Divisi- bility OF Numbers. 118. We have already demonstrated (25) and (26). 1st. That to multiply a number hy the product of several factors, is tlie same thinff as multiplying the number successively by each one of the factors. 2d. That the product of two numbers is the same in whatever order we effect their multiplication. Though the reasoning were developed upon particular num- bers, they are not the less rigorous on that account; and in order to convince ourselves, it suffices to go through it again, denoting the numbers by the letters a, h, c, &c. We propose ta verify only the accuracy of the second of the above propositions, whatever be the number of factors to be mul- tiplied together. We commence by remarking, that if we had to multiply a number N by 6, and then to multiply the product obtained by r, it will amount to the same thing to multiply N first GENERAL PROPERTIES OF NUMBERS. 145 by c, and then the product by b. In other terms (8), in a mul- tiplication of more than two factors, we can invert the order of the last two multiplications without changing their product j or, NxJxc = Nxcx &.* For it results from the first principle above, that N X 6 X c = N X 6c ; but in virtue of the second principle, we have he = ch ; then, Nx J X c = N xc6, or, NxiXc= N x cx6. Q. E. D. From this proposition, and the proposition that the product of two numbers is the same in whatever order we take them, it is easy to deduce the same proposition for three numbers. Let a, h, c, be the numbers proposed. . We say that ahc = hac =s bca = cha = cab = acb. For the second product is equal to the first, in virtue of pro- position 2d ; the third is equal to the second, in virtue of pro- position 3d ; the fourth is equal to the third, in virtue of 2d ; the fifth is equal to the fourth, in virtue of the 3d ; finally, the sixth is equal to the fifth, in virtue of 2d. Then all the pro- ducts are equal. From this demonstration for three factors, and from the incidental proposition (3), we deduce with the same faculty the proposition for four factors. Let a, 6, c, d, be the numbers proposed. We say then, that abed = bacd = bead := cbad = acbd = cabd =: abdc = = bcda = = cadb = Firstly, the six products of the first horizontal line are equal to each other, in virtue of the proposition for three factors, since they result from the multiplication of ahc, bac, &c., &c., by the same number, d. The first product of the second line is equal to the first of the first line by reason of (3); as to the other pro- ducts of this line, we dispense with writing them ; they can be * We here for convenience sake, shall give diflFerent significations to N X ^ X c and N X ^c, regarding the last as the product performed of b and c. 13 146 GENERAL PROPERTIES OF NUMBERS. found easily, keeping c in the last place in each ; they are all eqnal to the first by the proposition for three factors. We could thus proceed with the other two lines, applying alternately the incidental proposition (3), and the proposition for three factors. We thus prove the proposition for all possible products of a, h, c, d, since we cannot form more than 6 products terminated by the same letter. The same mode of demonstration can obviously be easily extended to any number of factors; 119. The demonstration which we have given of the pre- ceding principle, supposes that the numbers upon which we are reasoning are entire numbers (Arts. 25 and 26) ; but if we re- flect a little upon the rules established for the multiplication of fractions, we perceive that the property is equally applicable to fractional numbers. Moreover, this proposition completes the demonstration of the method established for the reduction of fractions to a common denominator given in the chapter on fractions. Divisihility of Numbers. 120. The property which certain numbers possess of being exactly divisible by others, and the investigation of the divisors of a number, form one of the most important theories of arith- metic. This theory depends upon a series of principles, which we proceed now to develop successively. We will first repeat some preliminary definitions which we have already given. We say that every entire number, which divides exactly another entire number, is called z. factor, divisor , or suhmultiple of this number, and this last is called a multiple of the first. Every entire number which has no other divisor except itself and unity, is called an absolute prime number, or simply a prime number. Two entire numbers are prime with each other when they have no other common divisor besides unity, which is a divisor of every number. It follows from this, that a prime number which does not exactly divide another number, is prime with the latter, as they can have no common divisor greater than unity. GENERAL PROPERTIES OF NUMBERS. 147 121. First Principle. — Every number, P, which divides exactly/ one of the factors of the product A X B, divides neces- sarily the product ; or, what amounts to the same thing, every entire number which divides another exactly, divides necessarily the multiples of this number. For, let Q be the quotient supposed exact of the division of A by P; we have then A = P X Q, whence, multiplying both sides by B, A X B = P X Q X IB = P X QB ; we see then that P is a factor of the product AB. 122. Second Principle. — Every number which divides exactly the product of two factors, and which is prime with one of them, divides necessarily the other factor. Let A X B be the given product, P the number which divides this product exactly ; we say that if P is prime with A, it will divide B. For A and B being by hypothesis prime with each other, if we apply to them the rule of the greatest common divisor, we will be led to a remainder equal to 1 ; that is to say, denoting by r, r'j r" , .... 1, the successive remainders, we will have the series of numbers. APr, /, r", ... 1, A being greater than P; or, PAr, /, /',... 1, if P is greater than A, for the different terms of the divisions to be performed. But, suppose that, before perform- ing the operations, we commence by multiplying A and P by B, there will result the new series, AxB, PxB, rxB, /xB .... 1 X B. Now, all these terms are divisible by P, since P is the common divisor of the two first terms. Then 1 X B, or B is divisible by P. Q. E. D. N. B. It is important to remark that the proposition is only true when P is prime with one of the factors of the product. For, if we have, for example, on the one hand 28 X 15, and on the other 12, which is not prime with either of the two factors of the product, the quotient of the division of (28 X 15), or 420 by 12, is exact and equal to 35, though 12 divides neither 28 148 GENERAL PROPERTIES OP NUMBERS. nor 15. It is obvious in tliis case, that the two factors contain together all the prime factors which compose the divisor. Thus we have 28 X 15 -^ 4x 7x3x 5 = (4x3)x 7x5. -=12x7x5=12x35 Consequence of Second Principle. — Any number whatever y P, j)rime with all the factors except one of a product^ A X B X C . . . ., can only divide the product^ when it divides exactly the remaining factor. This is too obvious for discussion. 123. Third Principle. — Every prime number, P, which divides Exactly the product of two factors, divides one of them necessarily. For, suppose that P does not divide A, it is necessarily prime with A (120); then it must divide B (122). From this result the following consequences. 124. 1st. If a prime number, P, divides the product A X B X C X of any number of factors, it divides one of the factors at least. 2d. Every prime number which divides the powers, K^, A^, A*, &G., of any number. A, divides A itself. For A^, A'', &c., being equal to A X A, A X A X A P, can only divide these diflferent products when it divides one of the factors. 3d. If two numbers, A and B, are prime with each other, their powers, A^ and B^, A^ and B^, &c., are also prime with each other. For any number, a, which is the common divisor of A^ and B*, for example, must divide A and B, which is, by hypo- thesis, impossible. 125. 4th. Every number, P, prime with each one of the fac- tors of a product, A X B X C X . . . ., is also prime with the product. For suppose that a prime number, d, differing from 1, can divide at once P and the product AxBxC , asc? ought to divide one of the factors of the product, P would not be prime with this factor, which is contrary to the hypothesis. GENERAL PROPERTIES OF NUMBERS. 149 126. 5th. Wheyi a nvmhcr, N, lias heen formed hy the multi- plication of several others. A, B, C, D, . . . ., this inimher can have no other prime factors except those which 'already enter into A, B, C, Dj &c. For every prime number which divides the pro- duct, A X B X C X D, and does not divide J), must divide A X B X C (123) ; in the same manner, every prime number which divides A X B X C, and does not divide C, must divide A X B, and, consequently, A or B. Thus, we can say in other terms, a number heimj formed hy the multiplication of several others, we cannot obtain it anew by multiplying numbers which contain prime factors different from those which enter into the numbers already multiplied. 127. Fourth Principle. — Every number, N, divisible by two or more numbers, d, d', d", , prime with each other, is divisible by their product. For, since d divides N, we have N = c? x g', q being an entire number; but by hypothesis, -f (10^-iy+(10^-l)/+ I U +6 +c +d +/ + j by adding and subtracting 6, c, d, &c., from the last number at the same time. Now, according to what was premised above, 10 — 1, 10^ — 1, 10^ — 1 .... being divisible by 3 or by 9, the first horizontal line is composed of a succession of numbers divisible by 3 or by 9. Thus, this first part of the number, N, is divisible by 9. Then, if the second part, which is nothing more than the sum of the figures of the given number, is divisible by 3 or by 9, the number itself is divisible by 3 or by 9 ; and, if this last part is not divisible by 3 or by 9, the remainder of this division will necessarily (129), be the remainder of the division of the num- ber itself, by 3 or by 9. -N. B. In practice, instead of determining the sum total of the figures, in order to divide it by 9, we subtract 9 from the partial sum so soon as it exceeds or equals 9, as we proceed with the 154 GENERAL PROPERTIES OF NUMBERS. summing up, and continue the operation to the last figure. These partial subtractions do not obviously change the remainder, which we seek. Example. — Given, the number 74683056743. "We say, 7 and 4 make 11 ; 9 from 11 leave 2 ; 2 and 6 make 8, and 8 make 16 ; 9 from 16 leave 7 ; 7 and 3 make 10 ; 9 from 10 leave 1 ; 1 and and 5 make 6, and 6 make 12 ; 9 from 12 leave 3 ; 3 and 7 make 10 ; 9 from 10 leave 1 ; 1 and 4 and 3 make 8. Then, 8 is the remainder of the division of the num- ber by 9. 132. Property of the number 11. — Every number is divisible by 11, when the difference between the sum of the figures in the odd places J counting from the right, and the sum of the figures in the even places, is equal to 0, or divisible by 11. Before demonstrating this property, it is necessary to re- mark, 1st. That every power of 10 of an even degree dimi^iished by unity, gives a result divisible by eleven. For this result is necessarily composed of an even number of 9's, written one after another. Now, each division of two figures, taken separately, forms 99, or 9 X 11, divisible by 11 ; then, the numbers themselves are divisible by 11 ; or, in general, 10^" — 1 is divisible by 11, (2n expressing the even numbers). 2d. Every uneven power of 10, augmented by unify, gives a result divisible by 11. For a power of an even degree of the number 10 can be ex- pressed by 10^"+' (130). Now, 10=^"+'=102"x 10, or 10^-+'= lO^" X 10 + 10 — 10 = 10 (10 2^—1) + 10; adding 1 to both members, lO^^+' + l = 10(10^"— 1) + 11. But,- according to (1) 10^" — 1 is divisible by 11; moreover, 11 is divisible by itself. Hence, 10^"+' + 1 is also divisible by 11. GENERAL PROPERTIES OP NUMBERS. 155 This being established, let ... . lujfdcha be the given number, which we will call N ; we have N==a-flO^* + 10\-+10'^i4-10y. . . ., an equation which we can put under the form 1 -\-a —h +c —d +/. ... ) Now, according to the two preceding remarks, the first line is composed of numbers essentially divisible by 11, and forms, con- sequently, a first part, which is divisible by 11. Then, if the second part, which is nothing more than the difference between the sum a + c-f-f+h-f- . . . . of (he figures in the odd places, and the sum of tJie figures b-4-d + g+ . . . . of the figures in the even places, is divisible by 11, as we have supposed; the number, N, is also divisible by 11. Q. E. D. 133. When the difi'erence between the sum of the figures in the odd places, and of those in the even places, is neither nor a multiple of 11, the number itself is not divisible by 11, since one of its parts is divisible, and the other is not. But, then, there are two cases to be considered with reference to the manner of obtaining the remainder of the division. 1st. If the sum of the figures of the odd places is greater than the second sum^ the difference is to be added to the first horizon- tal line of the value of N. Denoting then this first line by B, and the difference to be added by C, we will have, N = B -f C ; and if C is not divisible by 11, the remainder of the division of C % 11 will he the same as that which toe icould obtain hy dividing N hy 11 (129). 2d. If on the contrary, the sum of the figures of the odd orders is less than that of the figures of the even orders, the dif- ference will have to be subtracted from the first line, and we shall have N = B — C ; C designating always the numerical value of the difference. In order to determine in this case the remainder of the divi- sion of N by 11, let us observe that we have B = 11 x Q, 156 GENERAL PROPERTIES OP NUMBERS. Q being an entire number, and C = 11 x Q' ■hJR-'} then, N = 11 X Q — 11 X Q' — R, or, subtracting and adding 11, N=llxQ— 11x0^—11+ 11— 11=11 (Q—Q'—l) + ll—R. Whence we see in this case the remainder of the division of N by 11, is equal, tiot to the remainder R, of the division of C b?/ 11, but to the difference between II and 11. In order to fix these ideas, let the number be 47356708. Adding up the figures in the odd places, we obtain (setting out from the right), 27 ; adding uj) the figures of the even orders, we obtain 13. Now, the first sura is greater than the second. Then, if we take the diff'erence, which gives 14, the remainder, 3, of the division of this diff'erence by 11, is equal to that of the divi- sion of the number itself. But, if we had the number 370546345, since the sum of the figures of the odd orders is 15, and that of figures of the even orders, 22, it follows that if we take the diff'erence between the two sums, which gives 7, the re- mainder of the division of the number itself is not 7, but 11 — 7, or 4. 134. Verification of multi-plication and division, by the pro- perties of 9 and 11. We cannot pass over a simple and very convenient means of verifying the multiplication and division of entire numbers. We enunciate this method as follows : Add the figures of the multiplicand , and divide the "sum by 9 ; add. the figures of the multiplier , and divide this sum also by 9. We thus obtain two remainders, which (131), are nothing more than the remainders of the division of these numbers by 9. Mul- tiply these two remainders together,, and divide their product by 9; this gives a third remainder. Finally, add the figures of the product, and divide the sum by 9. We obtain thus a fourth remainder, which is equal to the third when the multiplication has been accurate. Let the two numbers be, for example, 5786 and 475, to be multiplied one by the other. The multiplication being performed, we add the figures of the multiplicand, rejecting the 9s by partial subtractions, as in (131). We thus obtain 8 for GENERAL PEOPERTIES OF NUMBERS. 157 the first remainder. We operate in the 5786 8 2 same manner on the multiplier which 475 7 2 gives 7 for remainder. This 7 we write 28930 under the 8, as in table. We then multiply 40502 8 by 7, giving 56, which we divide by 9, 23144 giving 2 for remainder (or we can say 5 and 6 make 11, and 9 fronj 11 leave 2). Finally, 2748350 we operate upon the product as upon the factors, which gives 2 for a fourth remainder. This being equal to the third, we con- clude that the operation is exact. In order to establish this method of verification by 9 in a general manner, let us denote by A and B the two factors, by Q, Q', R, and R', the quotients and the remainders of the division of the multiplier and multiplicand by 9 ; we have the following equations, A = 9 X Q + R, B = 9 X Q' + R'. Multiplying these two, member by member, we obtain AB = 9x9xQxQ'4-9xQ'xR+9xQxR'+RxR'. Now, the three first terms of the second number of this new equation, are evidently multiples of 9; then (129), the re- mainder of the division of the product AB by 9, must be that which the division of Rx R' by 9 gives. And this is what we wished to demonstrate. If one of the two factors of the multi- plication is divisible by 9, the product ought to be so also ; it is the same if the product R x R' is divisible by 9. Or, we may express it thus : if one of the first remainders is Oj the third must also be 0. Hence, the fourth must be 0. Again, when the first two remainders are equal to 3, in which case the third remainder is equal to 0. Hence, the fourth must be 0. As to the verifi- cation of division, two cases can occur; either there will be a re- mainder after the ordinary operation is performed, or there will be none. 1st. If there is no remainder, the dividend is regarded as the product exact of the quotient and divisor ; and we can apply the 14 158 GENERAL PROPERTIES OF NUMBERS. preceding nile regarding the divisor and quotient, as the two factors of a multiplication. 2d. If we obtain a remainder, we commence by subtracting this remainder from the dividend. The result of this subtrac- tion will be the exact product of the quotient and divisor, and we operate upon these three as before. N. B. The verification hy 9 is liable to several causes of error , of which the following are the principal. 1st. It is possible that either in the partial products or in the total product, we may have written a for a 9, or reciprocally ; or, in the one, a figure too small or too great by a certain number of units, and in the other, a figure too great or too small by the same number of units. 2d. I.t is possible, also, when there are zeros in the multi- plier, that we may not have written the partial products far enough to the left. We perceive at once, in these difi'erent cases, that the errors committed have no influence upon the remainders of the division by 9, of the terms of the operation to be verified. The verification by 9 is only then, properly speaking, a half proof, to which we can have recourse when pressed for time ) it being certain when the third and fourth remainders are not equal, that the operation is incorrect. But if they are equal, there is only a great prohahility that the product is the required one. The verification by 11, which does not differ from that by 9, except in the manner of obtaining the remainder of the division of a number by 11, is preferable, though itself subject to some errors; but these errors occur much less often than in the method by 9. These verifications can be applied equally to the multiplication and division of decimal fractions, since these ope- rations are performed in the same manner as in whole numbers. 135. There exist also, characteristics by which we can tell whether a number is divisible by the prime numbers, 7, 13, 17, . . . . ; but the rules which it is necessary to follow, are longer in practice than the division of the number by 7, 13 GENERAL PROPERTIES OP NUMBERS. 159 These questions demand, moreover, a greater knowledge of alge- bra than the questions heretofore discussed. We will, however, give the following question as an exercise for the pupil ; to determine in any system of numeration what- ever, whose base is b, what numbers enjoy properties analogous to the properties of 9 and 11 in the decimal system, and to de- monstrate these properties. This can be solved very readily ac- cording to the principle, that in every system of numeration, any power whatever of the base can be expressed by unity, followed by as many zeros as there are units in the exponent of the power. 136. As to the characteristics of the divisibility of a number by the multiples, 6, 12, 15, 18, 86, 45, of the prime numbers, 2, 3, 5, they are sufficiently simple to find a place here. 1st. An even number is divisible by 6 or 18, when the sum of its figures is divisible by 3 or 9. For this number is then divisible by 2 and 3, or 9 ; now 2 and 3, 2 and 9, are prime with each other; then (127), the number is divisible by 6 or 18. 2d. A number is divisible by 12 or 36, when the two last figures form a number divisible by 4, the sum of the figures of the number being at the same time divisible by 3 or 9. For then, &c. 3d. Finally, a number is divisible by 15 or 45, when the last figure is or 5, and in addition to this, the sum of the figures is divisible by 3 or 9. We pass now to the method of finding all the divisors of a number, both prime and multiple. ' 137. We will divide this question into two distinct parts : The first has for its object to determine all the prime factors which enter into any given number, and the number of times that each prime factor enters. The second has for its object to obtain all the divisors, prime or multiple, which the number contains. 2820 2 1410 2 705 3 235 5 47 47 1 160 GENERAL PROPERTIES OF NUMBERS. First Part. — To decompose a number into ^11 its prime factors. Let the number be for example, 2820. 2820=22x3x5x47. We draw first a vertical line, to the left of which we place the number, the divisors to be written to the right of the same line : 2820 being divisible by 2, which we write opposite it on the right of the vertical line. We perform the division of 2820 by 2, and write the quotient, 1410, below the 2820. As 1410 is divisible by 2, we place this second divisor below the first ; then the resulting quotient, 705, below the preceding, and we have 2820=2^x705. Now, we say, that the search for the prime divisors of 2820, other than 2, is now reduced to finding the prime divisors of 705. For, 1st. Every divisor of 705 must divide its multiple 2^x705. 2d. Eeciprocally, every prime divisor of 2820, other than 2, must divide 705. We are then to operate upon 705 as upon the given number; 705 is divisible by 3 ; we write this new divisor under the pre- ceding; then we place the corresponding quotient, 235, under the last already obtained, and from this results the new equality 2820=2^x3x235. 235 not being divisible by 3, the question is reduced to finding the prime divisors of 235. Now, this number is divisible by 5, which we write in the column of divisors. The quotient of 235 by 5, 47, we place in the column of quotients. We have then the equation 2820=2^x3x5x47. GENERAL PROPERTIES OF NUMBERS. 161 "We arc now led to seek the prime divisors of 47. But 47 is obviously itself prime ; for the simplest prime number, after 5, is 7 ; and 7 will not divide it. Moreover, 7 X 7=49, a number greater than 47 ; whence we conclude that 47 is a prime number. Dividing it by itself, we set the quotient, 1, below the others. Here the operation ceases, and we have 2820=2^x3x5x47 for the number 2820, decomposed into its prime factors. 138. Important Remark. — Before going farther, let us gene- ralize what has just been said, in order to prove that 47 is a prime number \ we will thus establish for every number a limit above which it is useless to go in the search for its prime divisors. Let N be the given number, and suppose that we have tried in vain, as divisors, all the prime numbers up to ascertain number, a, the corresponding quotient of which is g-, a fractional number less than a. We say, that the trial of any other number would be useless, and that N is a prime number For we have, according to the supposition, N = axg' {q being fractional and less than a). Now, if there existed a number a' greater than a, which could exactly divide N, we would have, denoting the quotient by g', N=a'X2' {^ being an entire number). Whence, a X q = a' x ^. Now, a' being greater than a, ^ must, to compensate, be less than q, which is itself less than a. Hence, the number N would have an entire divisor less than a, which is contrary to our hypothesis. Take, for example, the number 263. No one of the prime numbers, 2, 3, 5, 11, 13, will divide this number. But, trying 17, we find a fractional quotient, 15 + 1%, a number less than 17 ; whence, we conclude, that 263 is a prime number. In general, the limit of the trials in the search for the prime divisors of a number, is the smallest prime number which gives a fractional number less than this number taken for the divisor. There are other limits which we will not investigate. 14* 162 GENERAL PROPERTIES OF NUMBERS. Let us now render general the method which we' have, for the sake of clearness, commenced, by developing upon a particular example. Let a be the smallest prime number, commencing with 2, which divides N. We divide N by a, the quotient by a, the second quotient by a, as long as the exact division is possible. Calling n the number of divisions which we have found it possi- ble to perform, we have the equation N=a"xN' (N' being entire). We pursue the same course of reasoning as in (137), to show that the question is now reduced to operating upon N' in the same manner ; fcalling h the simplest prime number which divides N', and jp the number of successive divisions which can be performed, we have N = a"xZ>PxN", (N" being entire), admitting that c and d are the only factors of N", so that we have N'' = c-i X N'", and N"'c?», we obtain N = a" X Jp X ci X (7% and the number N is thus decomposed into its prime factors ; and we know, too, the number of times that each one of these factors enters into it. It results, moreover, from the general proposition (126), that these prime factors, raised to the powers denoted hy the exponents, n, p, q, s, respectively, form the only system of prime factors into which the number, N, can he decomposed. 140. Second Part. — To determine all the divisors, hoth simple and multiple, of any number whatever. From the same form under which we have just represented the number N, results a method of resolving this question. Let us write 1, a, a^, a^, a'', .... a" (n + 1 terms). 1, h, b\ b% b\ . . . . b^ {p-^l terms). 1, c, c^, c^, c*, . . . . c« (2 + 1 terms). 1, d, d", d% d\ . . . . d' (s-\-l terms). GENERAL PROPERTIES OP NUMBERS. 163 It is evident that we would obtain all the divisois of N, unity included, by multiplying all the terms of the first line by all of the second, then all the terms of the product, by all the terms of the third line, and, finally, all the terms of« the new product by those of the fourth line, since the different terms of this last product, are the products 1 and 1, 2 and 2, 3 and 3 . . ., of a, hj c, . . ., raised to powers whose degrees do not exceed n,^, q, and s. Now, the number of this last product is (w-f-l) X (p+1) X (g' 4- 1) X (s 4- 1). From this we deduce the following rule, also, for the total number of divisions of any number. Increase hy unity the exponents^ n, p, q, s, . . . of the differ- ent prime factors which enter into the number, N. Then multi- ply together these exponents, thus augmented by unity ; the pro- duct expresses the total number of divisors of N, unity being comprised among the number. Let N, for example, be equal to 2^ X 3^ X 5^ X 7 X 13^ The expression above becomes, in this case, 4x3x6x2x 3, or 432; thus, the number N has 432 divisors. 141. The method which we have just indicated for determin- ing all the divisors, prime and multiple, of a number, being not very convenient in practice, we will explain upon a new ex- ample, a more expeditious process. 1 5880 2 2940 2,4 1470 2,8 735 3, 8, 12, 24 245 5, 10, 20, 40 15, 49 7, 14, 28, 56 21, 30, 60, 120 42, 84, 168 I 35, 70, 140 | 280 | 105, 210, 420, 840 7, 49, 98, 196 | 147, 294, 588, 1176 | 245, 490 980, 1960 I 735, 1470, 2940, 5880. 1 In all, 4 X 2 X 2 X 3, or 48 divisors. 164 GENERAL PROPERTIES OP NUMBERS. ' , Explanation of the Table, After having determined the prime divisors of 5880, by the method of (137), 'we write 1 above the factor 2, in the column of divisors. We pass to the second divisor, 2, by which we mul- tiply the preceding; this gives the new divisor, 4, which we place on the right of the second divisor. Passing to the third divisor, 2, we multiply 4 only by 2, and place the product on the right of the third divisor. Passing to the divisor, 3, we multiply it by all the divisors which precede, viz : 2, 4, 8 ; which gives the new divisors, 6, 24, 48, which we place on the right of the divisor 3. In a word, when we descend to a new divisor, we multiply all the divisors which precede by this divisor, taking care not to repeat, however, the products already obtained. It is certain that the products to which this mode of proceeding leads, com- prehend all the divisors of the given number ; since they are the combinations of the factors, 2, 3, 5, 7, raised respectively to powers whose exponents do not exceed 3 for 2, 1 for 3, 1 for 5, and 2 for 7. 142. The search for the prime factors of every number, is one of the most important questions of arithmetic, and one of the most useful in practice. One of the applications we have seen in finding the least common multiple of several numbers. We may also apply it in finding the greatest common divisor of two numbers, this being obviously the product of all the prime fac- tors common to the two numbers. Thus, for example : We find for the prime divisors of the number 2150, 1 x 2 x 5 X 5 X 43, and for the number 3612, 1 X 2 x 2 x 3x7x43. Hence, the G. C. D. of these two numbers is 2 x 43=86. The reasons are obvious from the preceding articles. Formation of a Table op Prime Numbers. 143. The principles which we have established concerning prime numbers, and the application which have been made of GENERAL PROPERTIES OF NUMBERS. 165 them, show sufficiently the utility of a table of this sort of num- bers, as extended as possible. There are several tables of this sort, some of them compie- hending all the prime numbers from 1 to 3036000. To give some idea of the manner in which such tables are made, suppose that we wished to form a table of prime numbers from 1 to 1000. The first thousand numbers are written one after another in the most convenient form possible ; for example, in ten columns, containing one hundred numbers each. We then proceed as follows : We draw lines across, 1st, all the even numbers except 2 ; 2d, all the multiples of 3 except 3, which remain after the first operation; 3d, and, in the same manner, the multiples of 5, other than 5, which have not been crossed in the first two opera- tions. This done, we can affirm that all the numbers which have not been thus marked, from 1 to 7 X 7, or 49, are prime numbers, since all the multiples of 2, 3, and 5, as well as the multiplies of 7, below this limit, have necessarily been marked ; and we have thus the prime numbers from 1 to 47. In the same manner, if we mark all the multiples of 7, from 49 up to 121, or 11 X 11 (11 being the prime number which comes directly after 7), we are then certain that the numbers preceding 121, which are not marked, are prime numbers; we thus obtain all the prime numbers from 47 to 113, inclusive. Without carrying the details of this operation any farther, it is easy to see that we are thus led. to suppress, successively, all the multiples not yet suppressed, of the prime numbers already found, 11, 13, 17, . . , until we arrive at the number 997, the last one remaining of the first thousand numbers, after the suppres- sion already made of 998, 999, and 1000, as multiples of 2 and 3. We find, thus, the succession of 169 prime numbers comprised between 1 and 1000, the table of which we subjoin, adding the six prime numbers which follow them. 1G6 GENERAL PROPERTIES OF NUMBERS. Tahle of Prime Numbers from 1 to 1033. 1 97 229 379 541 691 863 2 101 233 383 547 701 877 3 103 239 389 557 709 881 5 109 241 397 563 719 883 7 113 251 401 569 727 887 11 127 257 409 571 733 907 13 131 263 419 577 739 911 17 137 269 421 587 743 919 19 139 271 431 593 751 929 23 149 277 433 599 757 937 29 151 281 439 601 761 941 31 157 283 443 607 769 947 37 163 293 449 613 773 953 41 167 307 457 617 787 967 43 173 311 461 619 797 971 47 179 313 463 631 809 977 63 181 317 467 641 811 983 59 191 331 479 643 821 991 61 193 337 487 647 823 997 67 197 347 491 653 827 1009 71 199 349 499 659 829 1013 73 211 353 503 661 839 1019 79 223 359 509 673 853 1021 83 227 367 521 677 857 1031 89 373 523 683 859 1033 144. Remark upon the greatest common divisor. We may find it necessary sometimes to find the greatest common divisor of several numbers. For this we give the following rule. We find first the Gr. C. D. of two of the num- bers, then the G. C D. of the one already found and a third number y then the G. C. D. of this last common divisor and a fourth number Let A, B, C, E, F, . . . be the given numbers, and call D the G. C. D. of A and B, D' the G. C. D. of C and D. Then we say that D' is the G. C. D. of A, B and C. For the G. C. I). of A, B and C, must divide D, and moreover must divide C. Hence, the greatest number which divides both C and D, is the greatest common divisor of A, B and C, and D' is that number. The same course of reasoning will apply to the rest of the numbers. GENERAL PROPERTIES OP NUMBERS. 167 We see that there is some advantage in operating first upon the two simplest numbers, since the Gr. C. D. sought cannot exceed that which exists between these two numbers. We could also decompose the numbers into their prime factors, and proceed as in the method proposed in (142). 145. Remark upon tlie least common multiples. We have already given a method of finding the least common multiple of several numbers in the chapter on vulgar fractions, which is rendered complete by the method of obtaining ihQ prime factors of any number whatever j given in (137, 138). We give here another method founded upon the preceding theo- ries. We consider, first, the two numbers, A and B. Denoting their greatest common divisor by D, and by q^ (( the quotients of the division of A and B by D, we have the two equations, A=Dx 2", B==D X ((; q and ^ being prime with each other. Now, we say, that the least common multiple required is equal to D X ^ X 2'. For this product is obviously a multiple of A and B, since it is divisible by D X g- and D X q'; it remains to be proved that it is the least multiple which we can obtain. Let us call Mamj multiple whatever of A and B. In order to be divisible by A or D X g', M must contain all the factors which enter into each one of the numbers, D and q; for the same reason it must contain all the factors of each of the numbers D and g' ; and since q and q^ are prime with each other, M can- not be less than J) X q X q'- We have then the following rule : Determine the Gr. C. D. of A and B ; then divide A and B by the Gr. C. D. ; multiply the Gr. C. D. and the product of the two quotients ; this gives M for the least common multiple of A and B. Find the least common multiple in the same manner of M and C. This will be the L. C. M. for A, B, C. Operate in the same man- ner on all the rest of the numbers in succession. The method given under the head of reducing fractions to the least common 168 GENERAL PROPERTIES OF NUMBERS. denominator, can be reduced to practice thus, (now that we know the method of jBnding all the prime factors of any number). For example, take the numbers, 6, 9, 4, 14, and 16. 2)6, 9, 4, 14, 16 2)3 9 2 7 8 3)3 9 1 7 4 13 17 4 2x 2x3x3 X7x4=least com. mult. We place the numbers in a horizontal line, and commence with the prime number, 2, as a divisor. We divide all those numbers which are divisible by 2, and bring down the quotients, together with the numbers not divisible. We proceed in the same manner with the quotients, until there are no two which 2 will divide. We then divide the last quotients and numbers brought down by the prime number, 3, and continue the operation until there are no two numbers left divisible by any number greater than unity. We then multiply the divisors and the numbers thus remaining together for the least common multiple. It is evident that we thus form the least number divisible by the given numbers. Or Periodical Decimal Fractions. 146. The valuation of vulgar fractions by decimals, that is to say, by tenths, hundredths .... of the principal unit, gives rise to singular circumstances which merit an examination. But, be- fore entering upon the discussion of them, we must return to the method for converting a vulgar fraction into a decimal. We have seen that, in order to effect this reduction, we must 1st. Annex a to the numerator', and divide the resultimj number hy the denominator ; this gives the tenths of the quotient and a remainder. 2d. Write a new on the right of the re- mainder , and divide hy the denominator , obtaining thus the hundredths of the quotient. We continue this operation until we have reached the degree of approximation required. This pro- GENERAL PROPERTIES OP NUMBERS. 169 cess is evidently the same as multiplying the numerator hy unity ^ followed hy as many zeros as we wish decimal figures in the re- sult; then dividing the result hy the denominator, and pointing off in the quotient the numher of decimal figures required. 147. This enables us to demonstrate the two following pro- perties : 1st. Every vulgar fraction whose denominator does not contain any prime factors other than 2 and 5, is reducible to a limited numher of decimal figures ; that is to say, after a certain num- ber of operations, we must arrive at a remainder equal to ; in which case the decimal fraction obtained expresses the exact value of the given fraction. Besides, if the fraction is reduced to its simplest form, the total riumher of operations to he per- formed in order to reduce it to its equivalent decimal is always equal to the greatest of the two exponents of 2 and 5, which enter into the denominator. Thus, the fractions 7 13 11 317 5J 3Tf? 4TJJ T3S0> which can be placed under the forms 2? 52' 2\b 2.5*' are reducible to a limited number of decimal figures. The first gives rise to three operations, the second to two, the third to three, and the fourth to 4. We find, in fact, for their values, 0-875; 0-52; 0-275; 0-2536. In order to prove this property generally, we remark, that 10, 100, 1000 .... being equal to 2x 5, 22x b\ 2'x 5^ • • ., if, in order to effect the reduction to a decimal fraction, we mul- tiply the numerator by 10, 100, 1000 . . . ., the resulting product will necessarily be divisible by 2 X 5, 2^ X 5^ .... ; then, in 15 170 GENERAL PROPERTIES OF NUMBERS. multiplying this numerator by unity, followed by as many zeros as there are units in the greatest of the exponents of 2 and 5, "which the denominator contains, the resulting product will ne- cessarily be a multiple of this denominator. Then, the number of operations to be performed is equal to the greatest of the two exponents of 2 and 6, which enter the denominator of the given vulgar fraction. 148. Every irreducible vulgar fraction^ whose denominator contains one or more prime factors different from 2 and 6, gives rise to an infinite number of decimal figures. Moreover j the decimal fraction resulting from it is periodical ; that is to say, after a certain number of operations, the same decimal figures recur again. For the multiplication of the numerator by 10, 100, 1000, can only cause the introduction of the two factors, 2 and 5, raised to certain powers ; thus, the prime factor which we suppose to be in the denominator, and not in the numerator, will not be in the latter, after this multiplication by 10, 100, .... Then, whatever number of zeros we add, we shall never obtain a product exactly divisible by the denominator; thus, the operations can be carried on to infinity. We say, moreover, that the decimal fraction will be periodical. For, as each remainder is always less than the divisor, it follows that when we shall have performed as many divisions as there are units less one in the divisor, we will necessarily arrive at a remainder already obtained, (if, in fact, this remainder does not recur sooner). Now, annexing a to this remainder, we will have a partial dividend exactly the same with one of the pre- ceding ; whence it follows, that we will have a series of quotients and remainders equal to the preceding j recurring periodically ^ setting out from the first partial dividend, which is equal to any of the preceding. Let us make some applications of this. GENERAL PROPERTIES OF NUMBERS. 171 149. Required to reduce the fraction, |, to decimals. 60)7 '40(0-857142 | 857142 60 To Yo lo Here the period shows itself after the 6th partial division. Second Example. — Let the fraction be ||. 130) 37 ~19b(0-331 I S'SI "l3 In this example, the period commences with the fourth partial division. Third Example, f|. 290 ) 84 380(0-34523809 | 623809 lio 200 "328 "680 "800 44 The period is manifest here after the 8th operation. But the two first decimal figures form no part of the period, while in the 172 GENERAL PROPERTIES OF NUMBERS. first two examples the period commences with the first decimal figure. The periodical decimal fractions, whose period com- mences with the first decimal figure, are called simple periodical fractions; and those whose period commences after a certain number of decimal places already written, are mixed periodical fractions. 150. We have just seen that certain vulgar fractions, reduced to decimals, give rise to periodical decimal fractions. Reciprocally, every periodical decimal fraction, simple or mixed, arises from, a vulgar fraction, which can easily he found from any given periodical fraction. This question presents two distinct cases ; either the periodical fraction is simple or it is mixed. Let us consider the first case. Take, for example, the periodical fraction 0-513513513513 and let us designate by N the fraction which has given rise to it. We have N=0-513513513 .... (1) Multiplying the two members of this equation by 1000, which is done in the second member by removing the point three places towards the right, we obtain Nx 1000=513-513513513 Or (2) NX 1000=513 + 0-513513513 Subtracting (1) from (2) we have NX 999 = 513. Then N=fi|. Let the fraction be N=0-714285714285 (1) Multiply both members by 1000000, we have NX 1000000=714285-714285 (2) GENERAL PROPERTIES OP NUMBERS. 173 Subtracting the first from the second, Nx 999999=714285 714285 N: 999999 Reducing the fractions |^| and m||f to their simplest terms, we get What we have shown proves that a simple periodical fraction is equivalent to a vulgar fraction which has for numerator the- figures of the period, and for denominator a number composed of as m,any 9'« as there are figures in the period. Thus, for an additional example, the fraction 0'351351351 .... is equivalent to the fraction |f^=yVT=i|. Again, the fraction 0-03960396 .... is equivalent to g|||, or simply ^W^=TtfT=7^T- In general, if a; = 0, ahcde .... ahcde .... ahcde (where abcde .... represent decimal figures with their relative values and not products^, we shall have ahcde .... "^^99999 N. B. If the periodical fraction contains an entire part, we do not regard it in forming the vulgar fraction ; but we add it to the vulgar fraction found after it is reduced to its simplest terms. Thus, given the periodical fraction 4162162 We have, first, 0-162162 . . . . = if|= JA=3^7. Then, 4-162162 = 4 + /^ = y^*- 151. Second Case. — Required to find the equivalent vulgar fraction, or generatrix^ as it is sometimes called, of a periodical mixed fraction. 15* 174 GENERAL PROPERTIES OP NUMBERS. Given, for example, the fraction 3-45891891 Multiplying this fraction by 100, we obtain 345-891891 ; and, according to (N. B.) of preceding article, this expression has for its value Q.. , 891 345 X 999+891 ^^^ + 999'^^ 999 ' 345 X (1000—1) + 891 345546 ^"■^ 999 ' ^^ '~9W~' But, as we have multiplied the fraction by 100, in order to reduce the result to its true value, we must divide it by 100 ; we thus obtain y^^^Yo^, a fraction which, reduced to its simplest form, becomes f|f§, the generatrix of the mixed periodical fraction, 3-45891891 .... If the fraction were under the general form, OfPqrSj abcde, abode .... its value would be abcde ^^'■^ + 99999' after multiplying it by 10000, or pqrs X 99999 + abcde 99999 ~' or, reducing the result to its true value, pqrs X 99999 + abcde 999990000 ' We say, then, ani/ mixed periodical fraction whatever is equi- valent to a vulgar fraction which has for its numerator the period, augmented by the product of the part which precedes the period by a number composed of as many 9s as there are figures in the period, and for denominator this same number of 9.S, followed by as many zeros as there are figures in the part which precedes the period. GENERAL PROPERTIES OP NUMBERS. 175 Take, for another example, 0-3193069306. The preceding rule gives for its value, 93 06 + 31 X 9999 _ 9306 + 31 (10000—1) __ 999900 . ~ 999900 309969 + 9306 319275 _ 129 999900 ~ 999900 ~ 404* We give here as examples of simple and mixed periodical decimals, 1st. 0-9999 =1 = 1 2d. 0-012345679012345679 = ^V 3d. 0-987654320987654320 = ffi 4th. 16-285714285714 = 5th. 4-9428571428571 = 6th. 5-52027027 = , r-o mi • P7^s X 99999 + ahcde , , ^ 152. The expression ^-^ leads to some re- markable consequences. It can be put under the form j?grs (100000 — 1) + ahcde 999990000 ' equal to pg-j-sOOOOO — pqrs + abcde 999990000 This being established, it is obvious from this last form, that if the calculations which are indicated in the numerator are effected, the result cannot be terminated by one or more zeros ; for, in order that this should be the case, it would be necessary that some of the last figures of pqrs should be the same as the last figures of ahcde; and, in this case, the period would not commence after the 4th decimal figure, as we have supposed. (For example, if we had s = e, r = d, the primitive fraction would be 0, pqdeabcdeabc . . . .) We s^e, then, that after the reduction of the expression above to its simplest terms, the result 176 GENERAL PROPERTIES OP NUMBERS. must be a fraction, wliose denominator contains the two factors, 2 and 5, or at least one of the two, to the 4th power ; that is to say, to a power whose degree is denoted by the number of figures which form no part of the period. We can infer from this the two following propositions : 1st. Every fraction whose denominator does not contain either of the two factors J 2 and 5, or is prime with 2 and 5, gives riscy when reduced to decimals, to a simple periodical fraction. For, if we could obtain a mixed periodical fraction, it should follow, that the equivalent vulgar fraction, which we obtain by the rule in (151), being reduced to its simplest terms, should be equal to the given fraction. Now, that is impossible, (for in order that one irreducible fraction be equal to another fraction, the terms of this last must be the same multiples of the terms of the first).* It results, then, that the denominator of the proposed fraction would be a multiple of 2 or of 6 ; which is contrary to the hypothesis. 2d. Every irreducible fraction, whose denominator contains one of the factors, 2 and 5, or both, raised to a certain power, gives rise to a mixed periodical fraction, whose period must commence after we have found as many decimal figures as there are units m the greater of the ttvo exponents of 2 and 5, which enter into the denominator. First, the periodical fraction cannot be simple ] for the formula for these sorts of fractions being „ „„ j it is impossible that this fraction, whose denominator does not contain either of * The terms of every irreducible fraction are prime with each other, and every fraction whose terms are prime with each other is an irredu- cible fraction. This is obvious, as this reduction depends upon suppress- ing the common divisor of the two terms. Hence, it is obvious, that no two irreducible fractions can be equal, unless the terms are identical in both, nor can an irreducible fraction be equal to any other fraction whose terms are not the same mvltiplen of the terms of the first fraction. GENERAL PROPERTIES OF NUMBERS. 177 the factors, 2 and 5, should be equal to the given fraction whose denominator contains these factors. In the second place, the period must commence after n figures, if n express the greater of the two exponents of 2 and 5, which is found in the denominator; for suppose, for example, that it commences after n — 1 figures ; the equivalent to this periodical fraction would have a denominator which would only contain the two factors, 2 and 5, or one of them to the (n — l)th power, and could not be equal to the given fraction, since these two fractions are supposed to be irreducible. For example, the fractions |, -l|, (149), gave simple periodical fractions, because 7 and 37 are prime with 2 and 5 ; but the fraction, ||, gave a mixed periodical fraction, whose period com- mences after the second figure, because 84 is equal to 2^X 21. Finally, the fraction, j4|^ which can be put under the form 145 ^^^— pr, should give a periodical fraction whose period commences after the 4th decimal figure. We find, in fact, for the value of this fraction in decimals, 0-8238636636 153. We will not carry farther the examination of the proper- ties of periodical decimal fractions, but close by observing that properties analogous to the preceding manifest themselves in any system of numeration whatever. The fractions in any other system, which enjoy these analogous properties, are those whose denominators are powers of the Base of the«6ystem. Let this base be h. First, in order to reduce a vulgar fraction into subdivisions h times smaller than unity, and into other subdivisions h times smaller than the first, &c., it would be necessary to multiply the numerator by 6 or 10 ; that is to say, to annex a 0, and divide the result by the denominator ; which should give in the quo- tient units h times smaller than the principal unit, and a certain remainder ; to write a new on the right of the remainder, and 178 GENERAL PROPERTIES OF NUMBERS. divide the result by the denominator, giving in the quotient units h times smaller than the preceding, and h^ times smaller than the principal unit, and so on. This being established, we deduce from it by reasoning precisely the same as that which served to establish the properties of decimal fractions which arise from vulgar fractions, that the vulgar fractions in a system whose base is b, being converted into subdivisions b, b^, c&c, smaller than unity, give rise to fractions (analogous to decimals') of a limited or infinite number of figures, simple or mixed periodical, and that the composition of the denominator of the vulgar fraction ivith reference to the prime factors which enter into the base b, suffices to characterize these different sorts of fractions. We propose as an exercise for the pupil the investigation of the enunciations and demonstrations of these properties. Exercises. 1. Prove that every entire even number is the sum of several powers of 2, and that every entire odd number is the sum of several powers of 2, augmented by unity. Examples, 876, 2539, 6750. 2. Every entire number, which is not prime, has at least one prime divisor other than unity. 3. The remainder of the division by 9 of the product of any number of factors, is equal to the remainder which the product of the remainders of the division of each factor by 9 gives. Prove that this property belongs to every number, and not to 9 alone. 4. The product of any three entire consecutive numbers is always divisible by 6. 5. Convert the numbers, 345 and 225, of the decimal system, into their equivalents in the binary system. Add these last in the binary system, and convert the sum back to the decimal system. THEORY OF RATIOS AND PROPORTION. 179 6. All the prime numbers, except 2 and 3, augmented or diminished by unity, are divisible by 6 ; that is, they are com- prised in the general formula, Qn ± 1, (read plus or minus), n being any entire number. 7. If the sum of the figures of any number be subtracted from the Bumber itself, the remainder will be divisible by 9. 8. The expression rr(n-f l)(2w+l) is always divisible by 6. CHAPTER VI. APPLICATION OF THE RULES OF ARITHMETIC. - THEORY OF RATIOS AND PROPORTION. 154. Introduction. — We have seen, in the course of the explanation of the different operations of arithmetic, that these operations give rise to two principal species of questions. 1st. Those which have for their object to demonstrate the existence of certain properties of certain numbers known and given. 2d. Those in which it is proposed to find certain numbers from the knowledge of other numbers having fixed relations with the first. The first are theorems, properly speaking ; but we have generally called them Principles and Propositions. The questions of the second species, which are not particular applications of the rules and principles, are called Problems. The Problems, which we have hitherto solved, have been easy of solution, because the data were simple, and the relations be- tween the known and unknown quantities very obvious. But this is not generally the case -, as very often, in order to arrive at a solution, we have a considerable difficulty to overcome, which consists in discovering and determining the series of operations to be executed upon the numbers known and given, in order to arrive at a knowledge of the numbers sought. 180 THEORY OF RATIOS AND PROPORTION. Nevertheless, there exists a certain class of questions, the re- solution of which can be subjected to fixed and certain rules ; these are particularly those in which we consider Proportional Magnitudes. The greater part of these questions are precisely those which the general necessities of society give rise to, in that which relates to its commercial, industrial, and financial interests; they are generally known as the Rule of Three^ the Rules for the calcula- tion of Interest, Discount, the Rule of Fellowship, Excharige, &c. To arrive easily at the solution of these questions, we will commence by explaining the theory of ratios and proportions. § I. — Of Ratios and Proportions, and of their Prin- cipal Properties. 155. We have already said (1), that in order to form an idea of any magnitude whatever, we must compare it with some other magnitude agreed upon, of the same species, which can be taken arbitrarily or in nature. The result of this comparison is what we have called number. Number, then, expresses the relation between any magnitude and its unit. Now, if we wish to com- pare any two magnitudes whatever, of the same species, or what is the same thing, to compare the numbers which express them, the result of this comparison is a relation between these two numbers. When we thus compare two magnitudes with each other, we may either wish to know hoio much the greater exceeds the less, or how many times the greater contains the less. From this results two sorts of relations between the numbers compared, one which is sometimes called an Arithmetical ratio, and another called a Geometrical ratio. But these names, which are but little significant, are well replaced by the word difference, in order to express the result of the comparison by subtraction, and Ratio to express the result of the comparison by division. Thus, let 24 and 6 be the two numbers which we wish to compare. We have 24 — 6 = 18 for the difference, and ^^ = 4 for the Ratio. THEORY OF RATIOS AND PROPORTION. 181 The relations of magnitudes by division or Ratios will chiefly occupy the present chapter, as by far the most important of the two classes of relations ; but we will first give one or two leading properties of the Relations hi/ Subtraction or Differences. 156. In every Difference or Ratio, the two terms are thus dis- tinguished. The one first written is the antecedent ; the second term is the consequent. Thus, in the expressions 24 — 6, \^, 24 is the antecedent in both cases, and 6 is the consequent. When the difference between two numbers is equal to the difference be- tween two other numbers, the four numbers taken together form an Equi-difference, For example, let the four numbers, 12, 5, 24, 17 ; the differ- ence of 12 and 5 is 7 ; the difference of 24 and 17 is also 7. These, then, form an equi-difference which We write thus : 12.5 : 24.17. Placing one point between 1st and 2d terms, two points between 2d and 3d, and one between the 3d and 4th. We enunciate it 12 is to 5 as 24 is to 17; that is, 12 exceeds 5 by as many units as 24 exceeds 17. We can also write it 12 — 5 = 24 — 17; 12 and 24 are the antecedents ; 5 and 17 the consequents. The first and last term are moreover called the extremes ; the second and third the means. This established, we say that, in every equi-difference^ the sum of the extremes equals the sum of the means. Let 11.7: 19.15; We have obviously 11 + 15 = 7 -f 19. To prove this generally, we observe that if the consequents were equal to their antecedents, as for example, 11.11 : 19.19, 16 182 THEORY OF RATIOS AND PROPORTION. the proposition would be manifestly true. Now, in order to place the first equi-difference under this form, we have simply to add 4 to each of its consequents ; that is, the sum of the means and sum of the extremes are augmented by the same number. Hence, if these sums are equal now, they must have been so before. Then, &c. As a consequence of this property, knowing three terms of an equi-difference, we can find the fourth. Thus, let 23.11 : 49.x, (x being the unknown term), be the equi-difference, we have 23 -i- a; = 49 -f 11 ; whence x is known. Sometimes two of the terms of the equi- difference are the same as 27.39 : 39.51. Here the double of one of the means is equal to the sum of the two extremes, or the mean itself is equal to half the sum of the extremes. Thus, in the equi-difference, 23..X : a:.49, ^^ 49+23 ^g^^ and this number is called the average or arithmetical mean of the two numbers. It is useless to proceed farther with the properties of equi- differences, as they are of very little use. We will merely add, that no transformation executed upon an equi-difference destroys this equi-difference, so long as the sum of the extremes remains equal to the sum of the means. We pass to the discussion of Ratios and Proportions, properly so called. 157. The ratio of two magnitudes, we have seen, is the quo- tient of the division of the numbers which express these magni- tudes. This ratio can be an entire number or a fractional THEORY OF RATIOS AND PROPORTION. 183 number, greater or less than unity. For example, the ratio of 24 to 6 is \S or 4 ; that of 6 to 24 is g^, or | ; that of 75 to 18 is 7 5 nr 2 5 It is in the sense of Ratio that we have hitherto understood the comparison of any magnitude whatever with its unit (No. 1). In the theory of compound numbers, the relation of the princi- pal unit to its subdivisions, or between two subdivisions, is the number of times which the one contains the other. 158. The comparison of two concrete numbers supposes always that these magnitudes are of the same species, since we cannot compare magnitudes of diflferent species (No. 2). The ratio is itself, by its very definition, essentially an abstract number, expressing how many times one of the numbers contains the other, or is contained in it. The antecedent and consequent, which form the latio, are, we have seen, the numerator and de- nominator of a fractional expression which we obtain, in indi- cating the division of the two magnitudes which we are com- paring. 159. When the ratio of two numbers is equal to the ratio of two other numbers, we say, that the four numbers or magnitudes which they represent, are in proportion, or proportional. A proportion is then the expression of the equality of two ratios. For example, the ratio of 48 to 12 being 4, and of 86 to 9 being also 4, we have the equation 48 = 3^6^ or 48 : 12 = 36 : 9. It is sometimes more convenient to present the proportion under the form 48 : 12 : : 36 : 9, which is thus enunciated : 48 is to 12 as 36 is to 9. The terms 48 and 36 are antecedents ; 12 and 9 are conse- quents. The first and fourth are extremes ; the second and third are means. 184 THEORY or RATIOS AND PROPORTION. 160. Fundamental Properti/. — All proportions possess a pro- perty which may serve as a basis for the resolution of the pro- blems whose enunciations contain j^roportional quantities. This property consists in this : In every proportion the 'product of the extremes is equal to that of the means. Let the proportion be (1) 24 : 18 T: 20 : 15, of which the ratios f | and f § each equals |. We say, that we must have 24 X 15 = 18 X 20. For the property would be evident if we had the proportion 24 : 24 : : 20 : 20, (2) (which we call an identical proportion). Now, to render the proportion (1) the same as (2), it suffices obviously to multiply each consequent by | ; but by this, we multiply the product of the extremes and the product of the means by the same number, and make the same change in both. Hence, if equal after the multiplication, they must have been equal at first. Hence the property is proved. 161. Reciprocally. — If the product of two numbers is equal to the product of two other numbers, these four numbers form a proportion of which either pair of factors will constitute the means, the other pair constituting the extremes. For, if no proportion existed among these four numbers, it would be necessary, in order to render the second and fourth respectively equal to the first and third, to multiply each one by a different number, expressing in the one case the ratio of the first term to the second, in the other of the third to the fourth ; and as the two products would thus become equal by the multi- plication of each by a different mimber, it would result that they were not equal before the multiplication ; which would be con- trary to the enunciation of the proposition. Then, &c., &c. THEORY OF RATIOS AND PROPORTION. 185 162. Another demonstration of the fundamental property and its reciprocal. (We employ letters in order to render the reason- ing more concise and general). Let a, h, c, d, be four numbers in proportion, so as to give a : b '. : c : a, or -j-= -j. o a If we multiply the two members of this equality hj h x d, product of the two consequents, we obtain a X b xd _c X b xd b ^ d • Suppressing in each member the factor common to the numera- tor and denominator, we have a X d — c X b. Then the product of the extremes is equal to that of the means. Reciprocally, let the four numbers, a, b, c, c?, be such, that we have a xd = b X c. Let us divide the two members of this equality hy b X d, product of one factor of the first member by one factor of the second, we have thus a X d _b X c b X d~ bx d^ or, simplifying, -^ = -7, or a : 6 : : c : a. a Thus, the /our numbers form a proportion of which the factors of the first product constitute the extremes, the factors of the second product the means. 163. First Consequence. — In every proportion we can cause, 1st, the tivo means to exchange places; 2d, the two extremes to change places J 3d, the means to exchange places with the extremes witJiout destroying the proportion between the four numbers thus written. 16* 186 THEORY OF RATIOS AND PROPORTION. For it is evident that these changes do not alier the equality of the two products which the extremes and means of the primi- tive proportion give. And since, in the new expressions, the product of the first number by the last always remains equal to the product of the second by the third., there will always exist a proportion between the four numbers after the changes are effected. Let the proportion be, for example, 48 : 36 : : 72 : 54. (1) We have, by changing the means for each other, 48 : 72 : : 36 : 54. (2) By exchanging the extremes, 54 : 36 : : 72 : 48. (3) By placing extremes in the places of the means, and the means in the places of the extremes, 36 : 48 : : 54 : 72. (4) In the expressions (2), (3), (4), the product of the second number by the third, is 36 X 72, or 48 X 54; and the product of the first by the fourth, 48 X 54, or 36 X 72. Now, these products are equal by virtue of proportion (1) ; then the expressions (2), (3), and (4), are also proportions. The common ratio of (1) is |, of (2), |, of (3), |, and | for the proportion (4). N. B. It is obvious that inverting the order of the terms in each ratio does not destroy the proportion, since it amounts to the same change as is exhibited in (4). 164. Second Consequence. — We can in every froportion multiply or divide one extreme and one mean by the same num- ber, without destroying the proportion. THEORY OF RATIOS AND PROPORTION. 187 For the products of the extremes and means of the given proportion being equal, the new products which result from the multiplication or division of these products by the same number will also be equal; and the proportion will still exist. There are many other properties of proportions ; but those which we have just developed are the only ones of which we shall have need for the resolution of the problems which depend on this theory. § II. — Resolution op Questions dependent on the Theory of Proportional Quantities. Rule of Three. 165. A great number of problems in commerce, banking, &c., contain in their enunciation numbers bearing relations to each other susceptible of being expressed by proportions. Of these numbers some are given and known, the others unknown, to be determined. We designate, under the title, the Rule of Three, the process by which we find the fourth term of a proportion when three terms are given. Now, from the property of every proportion that the product of the extremes is equal to the product of the means, it results necessarily that, in order to obtain the value of the unknown term, we must, if it is an extreme, divide the product of the means hy the known extreme. And if it is a mean, we must divide the product of the ex- tremes hy the known mean. Thus, let the two proportions be 24 :9 : :32 :cc; 45 :36 : ::« :24; (we denote the unknowns by the last letters of the alphabet). Since the first gives 24 X cc = 9 X 32, there results 9x32 ^^ ^=--24 ^2' 188 THEORY OF RATIOS AND PROPORTION. we have also for the second, SQx X = 4:5x24:. Whence, :r = l^^ = 15|? = 30. The proportions become then 24 :9 : : 32 : 12; 45 : 36 : : 30 : 24. The common ratio is | for the first, and | for the second. We pass now to the resolution of some problems, of which those in (41) may be considered particular examples. 166. Problem First. — Required, the price of 384 lbs. of a certain commodity , 2b lbs. of which cost ^650? Analysis. — Since 25 lbs. cost $6*50, it is clear that 2, 3, 4 times 25 lbs. must cost 2, 3, 4 ... . times as much ; thus, the two given numbers of pounds bear to each other the same rela- tion as their respective prices. Then, if we designate by x the unknown price of 384 lbs., and if we consider for the moment the three given numbers and x as abstract numbers, we have the proportion (1) 25 : 384 : : 650 : x. Whence (165), . == ''%''' = ^ = 9984 ; and we conclude that the 384 lbs. of the commodity ought to cost $99-84. N. B. Before seeking the value of x by means of the propor- tion (1), we can simplify that proportion in observing that the antecedents, that is, one extreme and one mean, are divisible by 25. We then suppress this factor (164), and obtain 1 : 384 : : 26 : a: ; whence, cc = 384 X 26 = 9984. Another method of resolution. — If 2b lbs. cost $6-50, one pound must cost 25 times less, or -^^ of $6-50; that is, — ijr-- THEORY OF RATIOS AND PROPORTION. 189 Then, SS4:lbs. will cost 384 times as much as 1 lb., or X 384; which gives 899-84. Second Prohlem. — It takes 135 men 20 days to do a certain piece of work ; how many days would 300 men require to per^ form the same labour ? Analysis. — If a certain number of men have employed 20 days in accomplishing the work, it is clear that a number of men 2, 3, 4 ... . times as great must occupy 2, 3, 4 ... . times shorter period to do the same work, other things being equal ; then, as many times as the first number of men, 135, is contained in the second number, 300, so many times the number of days necessary for the second number of men, or the number sought, Xj will be contained in the number of days necessary for the first number of men. Thus, we have the proportion 135 :300: : a; : 20; whence, (165), x = — 390" ^ ^' Then, it takes 300 men 9 days to do the work. We could have suppressed in this proportion the factor, 15, common to the two first terms, and the factor, 20, common to the two consequents. We should then have 1 : 9 : : 1 : a:; whence, x = 9. Another mode of resolution. — If 135 men took 20 days to do the work, it would have taken one man 135 times as much time, or 135 X 20 days, and 300 men would have required a number of days 300 times as small as 20 x 135 ; that is to say, 20x135 2700 _, 190 THEORY OF RATIOS AND PROPORTION. Ratios, Direct and Inverse. 1G7. Before treating more complicated problems, we must make known certain terms which the consideration of propor- tional quantities give rise to. In every question, the enunciation of which contains four numbers in proportion, two of these numbers are of a certain species, and the two others of another species ; but each term of the second species is closely connected by the conditions of the question with one of the terms of the first. It is thus in the first problem (166), two of the four numbers express the weights of a certain commodity ; the other two, the respective prices of these weights. In the same manner, in the second problena, we had two num- bers of men, and two numbers of days; and the latter expressed the respective periods employed by the two numbers of men to do the same work. It is agreed, for this reason, to call the two terms of difi"erent species, thus connected by the enunciation of the question, Correspondents. For example, in the first problem, the prices are the corre- spondents of the pounds ; and vice versa, the numbers of pounds are the correspondents of the prices. This established, we say, that there is a Direct Relation be- tween the numbers of the first species and the numbers of the second ; or that these numbers are directly proportional, when, the proportion having been established, we see that as each number increases or diminishes, its correspondent increases or diminishes; and that, on the contrary, the Relation is Inverse, or the four numbers are inversely (or reciprocally) proportional, when, as each number increases or diminishes, its correspondent diminishes or increases. The enunciation of the first problem oifcrs the example of a direct relation; for the greater the number of pounds, the greater the price. THEORY OP RATIOS AND PROPORTION. 191 The second problem gives rise to an inverse relation ; for the more men there are to do the work, the shorter period required. If the relation is direct^ and if we wish to write the propor- tion under the form a '. h '. \ c ', dj one of the numbers, and its correspondent^ must form the two antecedents, and the other two the consequents. On the contrary, if the relation is inverse, one of the numbers and its correspondent must form the extremes, while the other two form the two means. When we write the proportion under its equivalent form of two equal fractions, a c T'^d' it is necessary, in the case of the direct relation, that one of the numbers and its correspondent form the two terms of the first fraction, or the numerators of the two fractions, while the other numbers form the two terms of the second fraction, or the deno- minators of the two fractions ; and, in the case of the inverse relation, each number and its correspondent must form the numerator of the first fraction and the denominator of the second, or the denominator of the first fraction and the numera- tor of the second. N. B. All these distinctions in the manner o^. writing the proportions furnished by the enunciations of the problems are of importance, and should be carefully retained in the memory. 168. We say, also, when the relation is direct, that one quan- tity of each species is in direct proportion with its correspondent ; and if the relation is inverse, that each quantity is in inverse proportion with its correspondent. Thus, for example. Two fractions of the same denominator are in direct propor- tion with their numerators. For we have seen that if the numerator is rendered double, triple, quadruple, .... or one half, one quarter, one third of 192 THEORY OP RATIOS AND PROPORTION. what it is, the fraction will be rendered two, three, four .... times greater or less than it was. By an analogous process, we could prove, that two fractions, having the same numerators, are in the inverse proportion of their denominators. When the fractions have different numerators and denomina- toi'S, we commence by reducing them to the same denominator or to the same numerator, and the question is thus reduced to one of the two preceding cases. We are then led to a new mode of expression, which consists in saying that the given fractions are in Compound Proportion, direct or inverse, of the two products of the numerator of the Jlrst by the denominator of the second, and of the numerator of the second by the denominator of the first. In order to justify this mode of expression, let us consider, for example, the two fractions, | and -^^. Reducing them to the same denominator, we obtain 3 X 11 J 4 X 7 and 7 X 11 7 X 11' and these two fractions are in the direct ratio of 3 X 11 to 4 X 7, or of 33 to 28. If, on the contrary, we reduce them to the same numerator, they bcQome 3x4 ,3x4 and 7x4 3 x 11' and in this case the two fractions are in the inverse ratio of the denominators, or the first is to the second as 3 x 11 is to 7 X 4, or as 33 is to 28, the same as before. But we see that the two terms of this ratio are the one, the product of the numerator of the first fraction by the denomina- tor of the second ; the other, the product of the numerator of the second by the denominator of the first. This compound ratio is in some sort the result of the multipli- cation of two simple ratios, which are either direct or inverse with regard to each other. THEORY OF RATIOS AND PROPORTION. 193 169. Ap2)licat{ons. — As application of what has just been said, we will indicate the method of bringing into a proportion certain surfaces and volumes or solids, because there is a number of questions in which we have need of these numerical valua- tions. Let it be required to compare the superficial extent of two pieces of stuff, one of which is 24 yards long by | yard wide ; the other, 17 yards long by | wide. By a process of reasoning analogous to that used in (166), we see that 24 yards long by | yards wide.j is the same thing as 24 X I yards loiig by 1 yard wide. In the same manner, 17 yards long by j of a yard wide, is equal to 17 X | long by 1 yard wide. Then, since the breadth of the two pieces is the same, the ratio of the two superficial extents is equal to that of the two lengths, and we lind this ratio to be 94x^-17x= „,24x2.17x5 24x2x4. 3x17x5 . -4X,.17x„or 3 . ^ '"' 3xT- 3x4 ' or, simplifying, as 64 : 85. Again, let there be two rolls of paper-hangings, one of which is 15 yards long, by | of a yard wide ; the other 19 yards long, by I of a yard wide ; we would find in the same manner for the ratio of the superficial extents of the two rolls, -.r / in n 15x4 19x7 15x4x8 19x7x5 15xf :19xi, or -^:-g-, or -^_^:_^-^; or, simplifying, 96 : 133. We conclude from this, that, whenever the enunciation of a question gives rise to a comparison of superficial extents, in order to reduce them to the unit of length, we must form the product of the length hy the breadth, and then compare the resultitig quan- tities. As to volumes or solids, it will suffice to take one example, in order to determine the steps to be followed. 17 194 THEORY OF RATIOS AND PROPORTION. Required to determine the ratio in cubic yards of the solid contents of two pieces of masonry ? We suppose that the first piece is 60 yards long, by | of a yard thick, and 3 yards high; and the second, 125 yards long, by I of a yard thick, and 4^ yards high. Reasoning, as in the preceding case, we find that for the first wall it is as if it was 60 x | X 3 yards long, by 1 yard thick, and 1 yard high ; and for the second, 125 X | X | yards long, by 1 yard thick, and 1 yard high. In other words, the two walls must contain, respectively, 60 X | X 3 cubic yards, and 125 X | X | cubic yards. Then, the ratio of the two volumes is equal to that of these two products, or of 60x3x3x4^ 125x7x9 . .^ , ,^. T7^ to :r5 , or of 48 to 175. lb lb Whence we see, that in order to obtain the two pieces of work expressed in cubic yards, it suffices to form for each one of them the product of the length by the thickness and by the height. After which we .easily find the ratio of the two. Compound Rule of Three. — General Method of Reduction to Unity. 170. The enunciation of a question often contains more than four numbers, between which it becomes necessary to establish either direct or inverse proportions ; and thus arise the distinc- tions, Single Rule of Three, and Compound or Double Rule of Three. These names arise from the mode of resolution, which is an application of the theory of proportions. But this mode has been generally replaced by the method called the method of Reduction to Unity, which we will now develop, remarking that the second mode of resolution of the problems iu (166) is a par- ticular case of this general method. 171. Third Problem. — It requires 1800 yards of cloth, | of a yard wide, to clothe 500 men. Required the number of yards of cloth, \of a ya.rd wide, which shall clothe 960 men ? THEORY OF RATIOS AND PROPORTION. 195 Table of Calculation. 1800 yards long, | wide, 500 men. X I " 960 1800 x| '' 1 " 500 " XX I " 1 '^ 960 " 1800x5 4x500 xxl 960x8 la la i " 1 " Then ^X7 _ 1800x5 ■" ^ 960 X 8 4 X 500 * Analysis. — After arranging upon two horizontal lines, the six numbers which the enunciation contains, and of which the num- ber of yards required forms part, we reason in the following manner : 1800 yards long^ by | wide, and x yards long, by | 1800x5 .xxl ^ , "Wide, are the same thing as ^ , and — ^ yards long, by 1 yard wide. We write, then, these numbers upon two new lines, preserving the numbers 960 and 500 in their respective places in the two new lines. Since, with ^ yards long, by 1 yard wide, we can clotbe 500 men, one man could be clothed with —. — ^777^- 4 X 500 X X 1 In the same manner, if — ^ yards can clothe 960 men, one X X 7 man could be clothed with r: — ^77777, which eives a^ain two new 8x960 ® ^ lines, which we place below the preceding. Now, the two last expressions which we have just obtained, representing both the quantit}'- of cloth necessary to clothe one man, are necessarily equal. We have then XX 7 _ 1800 X 5 8 X 960 ~~ 4x500 ' 196 THEORY OP RATIOS AND PROPORTION. or, reducing to the same denominator, and then suppressing this denominator, X X 7 X 4 X 600 = 1800 X 5 X 8 X 960. Dividing the two members of the equation by the multiplier of Xj we have _ 1800x5x8x960 _3 6x960 _ 34560 _ ^" 7x4x500 - T~ 7--49^7^; that is, it would require 4937^ yards to clothe 960 men. Verification. 1800x5 , v . 1 . ifi .1 4^5QQ reduces, obviously, to Ls^ or 4^; on the other hand, 34560 X 7 8 X 960' reduces also to 4^. The number 4^, or 4 yards and a half, ex- presses in the two cases the quantity of cloth necessary to clothe one man. 172. Problem Fourth. — 500 men, icorldng 12 Jiours a day, employed 57 days in excavating a canal 1800 yards long, hy 7 yards wide, hy 3 yards deep ; required in how many days 860 men, working 10 hours a day, can dig another canal 2900 yards long, hy 12 wide, and 5 deep, in an earth 3 times as difficult to excavate as the first. (This is one of the most complicutcd ques- tions which can be given in this Compound Proportion, or Rule of Three.) Tahle of Calculations. 500 men. 12 hours. 57 days. (1800x 7x3x 1) cubic yards. 860 " 10 " X " (2900x12x5x3) '' _ ,, 51800x7x3x11 Iman 1 hour 1 day \ 500^1^^57 } 1 u 1 . ^ u ^ 2900X12X5X3 | ,, ^ ^ ^ \ 860x10 i THEORY OF RATIOS AND PROPORTION. 197 2900x12x5x3 ^. ., ^^ 1800x7x3x1 ,,^ ^^^^°^ ^= 860x10 - ^'^'^'^ ^y 500x12x57 ' ^^^ _ 2900x12x5x3 x500x1 2x57 or, X — ^^,^ ^^,^^ ^g^^ ^-^ ^g ^ . Analysis. — It is necessary, first, according to what has been laid down in (169), to convert into cubic yards the two pieces of work; the one already executed, and the other to be per- formed. This we do by multiplying together the length, breadth, and depth in each case. Besides, since, according to the enun- ciation, the earth of the second is three times more difficult to excavate than the first, if we express by 1 and 3 the relative difficulties, we must introduce into the two products, of which we have just spoken, the factors 1 and 3. This established, after having placed, as in the preceding pro- blem, all the numbers comprised in the enunciation upon two difi'erent lines, we are led, by a course of reasoning entirely similar to that which we pursued in the solution of the third problem, to form two new lines representing, — the one, the work done by 1 man in one hour and in one day ; the other, the work done by 1 man in one hour and in x days. Now, it is clear, that these two quantities of work must bear to each other the direct proportion of the two periods employed to perform them. We have then the equality (1) given in the table of the calculations, whence we deduce the final equation there given ; and, effecting all the operations indicated, this equa- tion gives, finally, a: = 5493VT; that is to say, it would require 549 days, and ^^j, or about J of a day, for 860 men to excavate the second canal. 173. The problems which precede, suffice to exhibit the steps to be followed when the method of Reduction to Unity is em- ployed. But it may be useful, perhaps, to consider the results furnished by the last two problems, in order to deduce from them some new consequences concerning the use of direct and inverse ratios. 17* 198 THEORY OF RATIOS AND PROPORTION. The analysis of the problem in (171) led to an expression for the number of cubic yards sought, 1800x5x8x960 7x4x500 Now, if we go back to the enunciation of the question, in order to distinguish the correspondents of each species, and if we separate by means of the sign of multiplication (X) the dif- ferent ratios of each term and its correspondent, we shall be able to place the preceding expression under the form X , ^ 960 1800 ~ 4 -^ ^ "^ 500 ' or again, under this, X 4 960 1800 I 500 Examining the product in the second member, we see that the second factor, which is the ratio of the two numbers of men to be clothed, is direct with that of the numbers of yards of cloth, X ; while the first factor, or the ratio of the two breadths, is X inverse with the same ratio, ; thus, this last ratio, called loOO compound (168), is equal to the product of the ratios of the two numbers of men, and of the two breadths, direct for the men, and inverse for the breadths. And, in fact, the more men there are to clothe, the more cloth necessary ; but, the wider the cloth, the smaller number of yards necessary to make a given quantity. The expression obtained in the problem of (172), 2900x12x5x3x500x12x57 ^^ 860x10x1800x7x3x1 ' can be put under the form X 2900 ., 5 3 V ^^^ V ^2 57T800 ^ ^ ^ -'^ • seo"" 10' THEORY OF RATIOS AND PROPORTION. 199 and we see also, in this case, that the ratio of the two numbers of days necessary for the performance of the two pieces of work is equal to the product of the ratios of the correspondents of each species; direct in the case of the dimensions of the canals and the difficulties of the excavation; but inverse for the num- bers of workmen employed, and the numbers of hours per diem which they laboured. • Whence we can give this sort of General Rule for the resolu- tion of every question whose enunciation contains proportional quantities : Form a product of all the ratios^ direct or inverse, of the correspondents of each species, excepting the ratio of which the quantity sought forms one part ; then equal this product to the ratio of the quantity sought to the quantity of the same species with itself We obtain thus the expression of the equality of two ratios, from which we easily deduce the value of the unknown. Rule of Simple Interest. 174. The Simple Interest on a sum of money is the profit arising from the loan of this sum for a certain time. The sum lent, or placed out at interest, is called the Principal or Capital. The interest upon a sum of money depends upon the amount of the Principal, upon the time for which it is lent, and upon what is called the rate of interest, or the interest which a certain fixed sum bears for a given fixed period. Ordinarily, the rate is, in the United States, the interest which the sum of one hundred dollars bears in one year, and hence is called the rdiio, per cent. This rate, which we consider a sort of unit of interest, is purely conventional, and depends generally on the abundance or scarcity of capital. Nevertheless, there are, in commerce and banking, certain limits (in most countries fixed by law), beyond which the rate becomes usury. 200 THEORY OF RATIOS AND PROPORTION. It is evident that the interest on two principals for the same period must be proportional to the principals, (the rate being constant), and the interest on the same principal for two different periods, are proportional to the lengths of the periods. Whence it follows, that the rule of interest is only a particular case of the Rule of Three. Thus, the questions which arise under it can be treated in the same manner as the preceding. 175. Example. — Required^ the Interest on $4500 for 2 year% and 5 months, at the rate ofWl for every $100 ; or, hy ahhrevia- tion^ at the rate of 7 per cent, per annum. This enunciation can be thus rendered : $100 bring $7 in one year, or l2 months ; how much ought $4500 to bring in 2 years and 5 months, or 29 months ? The numbers can be thus arranged : 100 12 months 7 4500 29 '< oc The quantities. 1 I month 11'^ "7 and 7 100x12 X 4500x29* 100x12 4500x29' express each what one dollar brings in one month, apd must therefore be equal, and we have, X ^ 7 4500x29 ""100X12^ whence, 4500 X 29 X 7 ^ 15 X 29 X 7 ^~ 100x12 ~ 4 Reducing to decimals, a: = $761-25, THEORY OP RATIOS AND PROPORTION. 201 the interest on $4500 for 2 years and 5 months, at 7 per cent, per annum. 176. Generally, let us denote the principal by a, the time by tj the rate per cent, per annum by i and by g, the interest on the capital, by a. We shall have, $100 1 year 1 dollar. a t 9 1 1 100 ^^*''''^ 1 1 aXt ThoH, 9 ** ax<~100' and, consequently, aXi 9= innX'- (1) The time t can be a fractional number of the unit, year haying for denominator the number of months or of days in the year. If we place (1) under the form aXi ^ 100 it can be translated into the following rule : In order to determine the interest (/, multiply the given prin- cipal hy the rate of interest for one year, and divide the product hy 100; then multiply the result hy the number of years, frac- tional or entire. Example. — Required, the interest on $2524 65, at 4^ per cent, per annum for 2 years and 7 months. 202 THEORY OF RATIOS AND PROPORTION. We have, first, 2524-65 x4-5 = 11360-925. Dividing by 100, 113-60925 For two years, 2 yrs. 7 mos. 21^7-21850 6 months, 56-804625 1 month, 9.467437 293-490562; or, $293-49. It is obvious that this division by 100 can be performed on the rate before the first multiplication, thus converting that into a decimal fraction, by which the principal is to be multiplied. Example. — Required j the interest on $365-874, at 5-^ per cent, for one year ? This rate, 5 J per cent., divided by 100, gives 0-055. We then multiply 365-874 by 0-055. 365-874 •055 1829370 1829370 $20-12307 $20-12. Ans. 177. This second method, which we have applied in the last two examples, is always to be preferred, especially when we wish to determine the interest for a certain number of days. Required, for example, to find the interest on $1748-19, for 113 days, at 4| per cent, per annum. (We suppose the year to contain 360 days, 30 days for each month). We multiply 1748-19 by 4|, divide by 100; we then divide 113 into 60 + 30 + 20+3 days, and find the interest for each one of these parts separately. Summing these parts, we have the interest required. THEORY OF RATIOS AND PROPORTION. -Oo Table of Calculationi 1748-19 6992-76 I 874-095 \ 437-0475 8303-9025 ;. by 100, 83-039025 for CDC year's interest. For 60 days, 13-839837 " 30 *^ 6-919918 half of the above. " 20 " 4-613279 \ (C it a 3 a 0-691992 J^ of the int. for 30. 26-065026 Thus, the interest on $1748-19 for 113 days, is $26-06.* 178. The equation (1) of (176), contains the solutions of four different problems. 1st. Knowing the Principal, time and rate, to find the In- terest. This we have discussed in several examples. 2d. Knowing the Interest, time, and rate, to find the Principal. 3d. Knowing the Interest, Principal, and time, to find the rate. 4th. Knowing the Principal, Interest, and rate, to find the time. All these admit readily of solution ; but we will limit ourselves here to an example of the fourth problem, treating it by both of the methods explained in a preceding article. * The rjito of 6 per cent, per annum admits of the following abbrevia- tion of the above rules when applied to a given number of months ; 6 per cent, per annum is J per cent, per month, or 1 per cent, for two month?. Then we can say, in order to find the interest on a certain principal for a given number of months, at the rate of 6 per cent, per annum, we multiply the principal by J the number of months, and divide by 100. 204 THEORY OP RATIOS AND PROPORTION. A sum of $2524-65 brought $293-49, at the-rate of 4^ per cent, per an num. Required the length of time the sum was placed at interest ? First Mode of Proceeding. 100 1 year 4J 2524-65 t 293-49 1 1 1 t 4-50 100 293-49 2524-65 L ~ ^Q^'^^ 12? __ 29349 _ 29349000 r ~ 2524-65 ^ 4T ■"252-465x45 " 11360925' Effecting the division, we obtain 2 years and 7 months, ne- glecting a fraction less than 0-001 of a month. Second Method. 2524-65 4-1 10098-60 1262-325 or, dividing by 100, 11360-925 113-60925 interest for one year. And as $293-49 is the interest for t years, we must divide 29349000 by 11360925, in order to obtain the time required, t. Rule of Discount. 179. Discount is the deduction which is made from an amount payable at the end of a certain timCj when we wish to mahe it payable at the present time, or before it falls due by agreement. It is usually, in bankers' terms, the deduction which we make from the face (amount of a promissory note, in order to get its cash value. This reduction is usually made at so much in the THEORY OF RATIOS AND PROPORTION. 205 hundred per annum; and this is the rate per cent, of discount. The discounter is he who cashes the note by anticipation. It is easy to see that the rule of discount is the same with the rule of interest, with this diflference, that, in the latter case, the horrower is obliged to restore to the lender the sum lent, in- creased by its interest ; while, in the case of discount, the pos- sessor or maker of the note receives only the diflference between the amount of the note and the discount which is made by reason of the anticipation of its payment. Example First. — Required^ the discount on a note q/'$875'49, payable in 18 months, at the rate of 4l'% per cent, per annum. First Method. $100 12 months 4-80 875-49 18 " X ^ 1 " Torm <^is^o^°* 0^ ^1 ^^^ 1 y^*^- 1 1 Then, X 875-49x18 4-80 875-49x18 1200' 4-80 X 875-49 x 18 40 x 87549 x 18 whence, x = j^OO == 1000000 > or, performing the calculations, a: = 63035280 = 63-04. Amount of the note, $875*49. Discount, .... 63-04. Difference, . . . $812-45, the amount which the discounter pays. 18 206 THEORY OF RATIOS AND PROPORTION. Second Method. Amount of note, $875-49 Rate of discount per an. 4-8 "700392 850196 4202-352 dividing by 100, 42-02352 1 year, 6 months. 1 year, 42-02352 6 months, 2101176 63-03528 as above. This example suffices to show the identity of the calculations under the Rules of Interest and Discount. Example Second. — Required , the discount on a note of $3478-19, payable in 286 days, the rate being 6*25 per cent, for 360 days. We commence by decomposing the number 286 into its parts, 180+90 + 10 + 5 + 1. We then make the following table of calculations : 347819 6-25 1739095 695638 2086914 217-386875 discount for 360 days. 108-693437 54-346719 6-038524 3-019262 0-603852 180 90 10 5 1 172-701794 discount for 286 days. $347819 172-70 $3306-49 cash value. THEORY OF RATIOS AND PROPORTION. 207 180. The generalization of the rule of discount would lead to the equation (^) ^ = -100-' in which e E would designate the discounts on $100, and on the amount of the note respectively. These letters would simply replace ff and t of (176). We could, according to the equation (2), establish the enun- ciations of four general problems analogous to those of (178). 181. There is another rule of discount which we cannot pass by ; for although it is not generally employed, it appears more rational and more just. One example will suflfice to give an idea of this second mode of discounting. A note o/SlSOO, 'payable at the end of 15 montJiSj is presented to a hanker J who agrees to cash it at a discount of 4 6 per cent, per annum. Required j what the holder of the note must re- ceive f Analysis. — Admit, that 4*60, the rate of discount, is at the same time the rate of interest of a sum put out at interest. It is clear that the possessor of the note ought to receive now a sum which, placed at interest at the rate of 4-6 per cent, per annum for 15 months, would give him, capital and interest added, the amount of his note. Now, the interest of $100 for one year, being 4-60, becomes, for 15 months, 4-60 + \ of $4-60, or $5-75. This proves that $100, placed out at interest, would, at the end of 15 months, become $105-75, capital and interest. Consequently, $105*75, payable in 15 months, are equivalent to $100 payable now ; then $1, payable in 15 months, is equal to , payable now ; and, consequently, $1500, payable in 15 105*75 months, can be represented by 100 X 1500 15000000 ^i . -, q . o n^r -10575-' '' -10575-' ^^ $1418*43*97, payable now. 208 THEORY OP RATIOS AND PROPORTION. Whence it follows, that the holder of the note ought to receive from the banker a sum of $1418-44. In fact, if we calculate by the Rule of Interest, what $1418-44 ought to bring at the end of 15 months, at the rate of 4-60 per cent, per annum, we obtain ^ = 81-5603, which, added to 1418-4397, gives $1500-0000, the amount of the note. Now, instead of following this method, the banker determines the interest on $1500 for 15 months, at 4-6 per cent., which gives $86-25; and this he subtracts from $1500-00, $1413-75, the difference which he gives the possessor of the note. N. B. It is to be remarked, that the excess of $86-25 over $81-56, or $4-69, which the banker gains by the last operation, is nothing more than the interest on $81-56 for 15 months. For, multiplying $81-56 by 5-75, (rate for 15 months,) and dividing by 100, we obtain $4-6897, or $4-69. This advantage which the banker gains, independently of the profit which belongs to him of right, is a sheer loss on the part of the holder of the note. There is a way of operating, according to the first rule, with- out injury to the interests of the possessors of notes. This would be to establish a rate of discount a little lower than the legal rate of interest ; but the difiiculty would be to proportion the one to the other fairly under all circumstances. We give the two rules or enunciations of the two methods which we have given above. 1st. (179). Calculate the interest on the amount named in the note J from the present time to the date at which it falls due ; then THEORY OF RATIOS AND PROPORTION. 209 subtract this interest from the amount named in the note. This will be the cash value of the note. 2d. (181). Find what $100, placed out at interest for the (jiven time will brine/, capital and interest added ; then multiply the amount named in the note by the ratio of $100 to this sum ; the quotient will be the present value of the note. The first rule is genferally received in commerce, because it is more expeditious and convenient with regard to the calculations. It is, moreover, a matter of agreement between the banker and holder of the note. The Questions of Compound Interest and Discount, and the subject of Annuities, require a knowledge of the use of Loga- rithms, in order to be thoroughly discussed. Hence, we pass them by here, merely adding, that, in Compound Interest, the interest is added to the principal at the end of the year, or period chosen as unit; and then this sum is regarded as a new principal, on which the interest is calculated for the given period, and again added, &c., &c. There are a great number of questions, such as Insurances, Rents, &c., &c., which come under the rule of per centage, but they present no difficulty to the student who understands tho- roughly the preceding discussions of proportional quantities. They are generally given in full in the Commercial Arithmetics. Rule of Fellowship. 182. The Rule of Fellowship has for its object, To divide among several persons associated in a partnership business the profit or loss which results from their enterprise. It is generally admitted, (and it is moreover conformable to equity,) that the part of gain or loss of each partner is — 1st, proportional to the amount of capital he has placed in the busi- ness, when the times are equal ; 2d, proportional to the time when the amounts invested are the same. 18* 210 THEORY OF RATIOS AND PROPORTION. From this it results that, for different capitals and different times, the parts are proportional to the products of the capital stocks by the times; since, by multiplying the stocks by the tiuies respectively, we bring them back to amounts invested for the same time. Thus, the question, considered under the most general point of view, is, to divide a given number into parts directly proportional to other numbers also given. Problem First. — Three persons are associated in trade. The first puts $15,000 in the common stock; the second^ $22,540; and the third, $25,600. At the end of one year, the profits of the enterprise are $12,000. Required, the share of each one of the partners f Analysis. — The sum of the three amounts invested in trade being $63,140, we reason in the following manner : $63,140 have given a profit of $12,000 ; then $1 has produced dollars profit. Then, for 15000 .... we have 1|?^ x 15000 = '-^^^ = 2850-807. 63140 6314 ....... .. s^^-'-^r-""- 11999-998. Thus, the first person must receive $2850-81; the second, $4383-81; and the third, $4865-38. And these three sums, added, reproduce the total gain, $12,000. Problem Second. — A capitalist commences an enterprise with a stock q/ $25,000. Five months later, a second capitalist joins the enterprise, and furnishes an additional capital of $40,000. Six months after this first addition, a third capitalist adds $60,000. At the end ff two years the partnership is dissolved, THEORY OF RATIOS AND PROPORTION. 211 after having realised a profit of $76,000. Required^ the share of each partner? The $76,000 are to be divided among tlie partners proportion- ally to the products of their respective investments, by the num- bers of months during which these funds were in the enterprise. • Now, 1st, $25,000, invested for 24 months, equal 25000 X 24, or $600,000 vested for 1 month; 2d, $40,000 invested for 19 months, are equivalent to $760,000 for 1 month; 3d, $60,000 for 13 months, are equivalent to $780,000 invested for 1 month. The question is then the same as the first. Having formed the sum of the three amounts invested = $2140000, we obtain suc- cessively for the three parts or shares of the profit, First share, ^^^ X 600000 = 21308-411. ' 2140000 Second share, —jjr^^ X 760000 = 26990-654. ' 2140000 Third share, J^^^ X 780000 = 27700-934. 2140000 75999-999. The shares are, respectively, $21308-42; $26990-65; $27700-93. 183. In general, let it be required to divide any number, a, into parts proportional to the given numbers, m, n, p, q . . . . Form, first, the sum of the numbers, m^ n^ p, q . . . . then, multiply each one of these numbers by the ratio m-f-n+p + q-j-.... We obtain, thus, a X m a xn a Xp m. -j- n -{-p -^ . . . / m + n +p + . . . . m -\- n + p + ' fractions, which have the same denominator, and are necessarily in the direct proportion of their numerators, or because of the common factor, a, in the direct proportion of m, n, p, q . . . . 212 THEORY OP RATIOS AND PROPORTION. When the numbers, m, n^ p, q . . . . are fractional, we com- mence by reducing them to the same denominator, and then the question becomes the same as the preceding. Divide 360 into four parts, proportional to the numbers 2 7 11 17 These fractions, reduced to the least common denominator, be- COI^G 6 4 8 4 8 8 5 1 • "se? "ggj -ggj -gg- Then, the four parts must be respectively proportional to the numbers 64, 84, 88, 51. The sum of these numbers being 287, we have, successively, For the first part. Ifo X 64= 80-28. '' second, 3 fi-o X 84 = 105-37. " third. 36^ X 88 = 110-38. " fourth, If? X 5T== 63-97. 36000. 184. The following questions belong also to the same rule : Prohlem Third. — Required, to divide a sum of $36,000 among four persons, so that the second shall have twice as much as the first; the third as much as the first two together; the fourth three times as much as the third. We can make the first share a principal unit, with which we compare the rest. Calling, then, the first part 1, the second part will be 2, the third 3, and the fourth 9, by the conditions of the question. The question is then to divide ^36,000 into four parts, propor- tional to the numbers 1, 2, 3, 9. We obtain for the four parts, 36000 , 36000 ^,^^ First part, ^-j-^-^-^-^^ X 1 or -^^ = 2400 Second part, X 2 " = 4800 Third part, ^^ X 3 " = 7200 Fourth part, -^^ X 9 ^' = 21600 THEORY OF RATIOS AND PROPORTION. 213 Problem Fourth. — A person leaves $40,000, to he divided among four heirs, so that the first shall have | of the whole ; the second |; the third | ; the fourth |. Required , the share of each heir. If the sum of the four fractions was exactly equal to 1, the conditions of the bequest would be fulfilled by taking successively I, |j I, and \, of $40,000. But, if we reduce these fractions to the same denominator, we find yo> "5^5? ^0? "go? the sum of which is greater than 1. Hence, the bequest would be more than absorbed by the three first parts. But if the $40,000 is to be divided proportionally to the four numbers, \, ^, I, \, we would simply have to divide it into parts propor- tional to the numbers 15, 36, 40, and 30, the same as the pro- blem in (183). 185. We add here a rule which has for its object to determine the relative value of the coins of two countries, knowing the proportions between these coins and those of other countries. It consists in reducing to a single proportion, by multiplication, several given proportions. It is really nothing more than an application of the rule of compound fractions, or fractions of fractions. A single example will suffice to give an idea of the rule and the mode of applying it. Example. 48 francs ... are equal to 39 English shillings. 13 English shillings " 8 German florins. 50 German florins " 9 ducats of Hamburg. 15 ducats of Hamburg ^^ 43 roubles, Russian. How many Russian roubles are equal in value to 2500 francs ? If 48 francs are worth 39 shillings, then 1 franc is worth || of a shilling. In the same manner, if 13 shillings are worth 8 florins, 1 shilling is worth -f^ of a florin ; and, consequently, 1 214 THEORY OF RATIOS AND PROPORTION. franc is worth || of j\ of a florin. Again, if. 50 florins are worth 9 ducats, then 1 florin equals -^^ of a ducat. Continuing this reasoning, we find that 2500 francs = 2500 times || of j% of J^ of f | of a rouble. rri, or^nn ^ 39x8x9x43x 2500 , , Then, 2500 francs 48 X 13 x 50 X 15 ''''^^''' Rule of Alligation. 186. The questions which come under this rule are of two sorts : We may either wish to find the mean value of several sorts of things J knowing the number and particular value of each sort, or it may he required to determine the quantities of several sorts of things which must enter into a mixture, knowing the price or value of each sort, and the price or total value of the mixture. We will discuss only the questions of the first nature ; the se- cond belonging to the province of algebra. Example First. — A wine merchant has mixed wines of different qualities, viz., 250 pints, at 60 cents the pint; 180 pints, at 75 cents ; and 200, at 80 cents. Required, the price of one pint of the mixture 9 We observe, first, that 250 pints, at 60 cents, bring $150 180 " at 75 " " US5 200 " at 80 (' " $160 $445 Giving $445 for the total price of the three quantities of 250 wine mixed. 180 If, now, we form the sum 630 of the three numbers, 200 250, 180, and 200, the question will obviously be reduced "7^ to the following : 630 pints of wine cost $445 ; what is the cost of each pint ? 71 cents is the price required. THEORY OP RATIOS AND PROPORTION. 216 General Rule. — In order to find the price of the principal unit of a mixtvre — 1st. Multiply the price of this principal unit of each sort of thing hy the number of units of this sort, and add all the products. 2d. Sum up the numbers of units of these different sorts. 3d. Divide the sum of the products or the total price by the sum of the numbers of units. Or, more briefly — .Find the total price of the mixture by summing up the prices of its parts. Then divide this total price by the number of principal units in the mixture. We thus obtain the price of one principal unit. Example Second. — We wish to melt together 23 kilogrammes of silver, 826 thousandths fine j 14 kilogrammes 910 thousandths fine ; and 19 kilogrammes 845 thousandths fine. Required, how many thousandths fine the mixture will be ? That is, how many parts of pure silver each 1000 parts of the new coin will contain ? (We say an ingot of gold or silver is -^-q, or 880 thousandths, &c., fine, when -f^, or 880 thousandths of it is pure silver or gold.) It results, then, from the enunciation, that 1st. 23 k. at -825 = 23 X -825, or 18-975 A;, of pure silver. 2d. 14 k. at -910 == 14 X -910, or 12-740 k. " 3d. 19 k. at -845 = 19 X -845, or 16-055Ar. " 56 47-770 « Then, the 56 kilogrammes of the mixture contain 47-770 kilogrammes of pure silver. Thus, the fineness of the new ingot will be expressed by — ^^ — , or 0-853; that is, it is 853 thou- sandths fine. 187. Mean or average values. — The determination of the m^an values of several things of difi'erent values, is a particular case of the rule of alligation of the first sort. We call the mean value of several things whose particular values are already known, the sum of the values of these things divided by their number. Thus, in the case of two things, the mean value is the half sum of the values of these things. 216 THEORY OF RATIOS AND PROPORTION. Example Third. — The leDgth of a park was measured four dififerent times. The first measurement gave 250-439 metres; the second, 250-695 metres; the third, 249-750 metres; finally, the fourth, 251-158 metres. Required, the length of the park? As none of the measurements agree, it is clear that the only- means of answering the question is to find the average or mean value of all these measurements. We find for their sum, 1002-042 ; dividing this result by 4, we obtain 250-5105 metres for the mean. Problems which j without depending on fixed or General RuJeSy can nevertheless he resolved arithmetically. 188. In the preceding questions, the methods of arriving at the required solution are fixed and general ; that is to say, sus- ceptible of being applied to all questions of the same nature. But an infinite number can be proposed which come only in part under these methods, or do not in any manner depend upon them. In these cases, algebra alone furnishes sure and direct methods of resolution. Nevertheless, we will show how these sorts of questions can be resolved arithmetically. We have seen, (154), that, in order to analyse or resolve a problem, we must, hy reflecting upon the enunciation, endeavour to discover in the relations established among the numbers which enter it, the suc- cession of operations to be performed upon the known quantities, in order to deduce from them the values of the unknown. Problem First — Required, a number, of which the half, third, fourth, and ^ths, added together, form the number 575 ? We commence by remarking that, to take the ^, \, \, and |, of any number, and add them together, is the same thing as multiplying this number by the sum of the fractions ^, \, \, and |, or by ^4°. Now, since the product of the number sought by y^^ must be equal to 575, it results from the definition of division, that this required number is equal to the quotient of 675 divided by y/ ; and consequently equal to 575 x j\^g. r VN IVfeli THEORY OF RATIOS AND PROPORTION. 217 Performing tlie operations indicated, we find 420 for the number. Verification, 420 The half, =~210 One-third, = 140 One-fourth, = 105 One-seventh, = 60 One-seventh, = 60 Total, ~575 Problem Second. — Required, three numbers whose sum is equal to 96, and such that the second exceeds the first by 2, and the third exceeds the sum of the other tioo by 4. It is evident that, if we diminished the second number by 2, it would become equal to the first ; and that if we diminished the third by 2 -j- 4, or by 6 units, it would become equal to double the first; thus, the sum of the three numbers would be, after these two subtractions, four times the first number. Now, the difierence between 96 and 2 -f- 4 -f 2, or 8, is 88 ; whence, we see, that the first number is equal to one-fourth of 88 = 22 Then the second is .... 22 + 2 = 24 And the third 22 X 2 + 6 = 50 Verification, 96 Problem- Third. — Three workmen are employed to do a piece of work; the first could do it alone in 12 days, working 10 hours a day ; the second in 15 days, working 6 hours a day ; the third in 9 days, icorking 8 hours a day. Required, \st. In what number of hours the three men working together can do the work; 2d. What part of it each one will do ; 3d. How much each one ought to get for his labour, the price of the whole work being 8108? Solution. — We observe that, according to the enunciation, the first workman could do the work in 12 X 10, or 120 hours ; then, in 1 hour, he could do y^^ of the work. The second could 19 218 THEORY OF 'ratios AND PROPORTION. do it in 15 X 6, or 90 hours; thus, in one hour, he could do -g^^ of it. The third would do it in 9 x 8, or 72 hours ; then in one hour he would do 77^ of it. These three workmen labouring to- gether would then, in 1 hour, do 750 + iJo + 72 = -BE^ or J^ of the work. Now, if in one hour they do 3^^ of the work, they would do the whole in 30 hours. Again, since in one hour the first workman does j|^, in 30 hours, he will do -f^^j X 30, or | = j^^. In the same manner, the second, in 30 hours, performs -^q x 30, or -i = j\. Finally, the third does t^^^xSO^ or j\. Then, to find the amount to be paid to each man, we must divide $108 into parts proportional to the three fractions, -j^^, j*^, y\, or the three numbers, 3, 4, 5 ; which gives $27, $36, and $45, for the respective wages of the labourers. Exercises. 1. A vessel has provisions for only 19 days; yet, by calcula- tions, 25 days must elapse before she can reach a port. Required, how much the ordinary rations must be reduced ? 2. Twenty workmen, working 15 days, 10 hours a day, exca- vated a ditch 65 yards long, by 2-30 yards wide, and -75 of a yard deep. Required, how many days it would take 36 men, working 12 hours a day, to dig a ditch 200 yards long, by 3 yards wide, by 1-25 yards deep; the difficulty of excavating the first earth being to that of the second as 3 to 4. 3. For what period must $3000 be placed out at interest at 6 per cent, per annum, in order to bring $1325-50 ? 4. What is the rate of discount on a note of $2500, payable in 18 months, for which the sum of $1860-45 was paid in cash? 5. Four partners invested the same sum in an enterprise; the funds of the first were in the business for 8 months ; the THEORY OF RATIOS AND PROPORTION. 219 second for 7 months; the third for 10 months; and the fourth for 1 year. Divide the profit of $1800 proportionally to the in- vestments augmented by the interest on each, at the rate of 4 per cent, per annum ? 6. We wish to divide $60,000 among three persons, so that the second shall have twice as much as the first, less $2500 ; that the third shall have three times as rnuch as the first, less $5000. What is the share of each person ? 7. Two pounds of copper, at 45 cents ; 7 pounds of zinc, at 70 cents ; 9 pounds of antimony, at 50 cents, are melted toge- ther. What is the price of one pound of the allojr ? 8. A person was asked how much money he had in his purse. He answered, If you add to the sum which I have, |, f , and | of that sum, I would then have 175 dollars. What sum of money has he ? EXAMPLES. For the convenience of teachers, we annex the following ex- amples for practice, as but few are embodied in the work itself. These are chiefly selected from difierent practical compilations on arithmetic. Addition. Add together, 1225, 3473, 7581, 9064, and 6060. Ans. Add together, 3004, 523, 8710, 6345, and 784. Ans. 19366. Add together, 7500, 234, 646, and 19760. Ans. 28140. Add together, 182796, 143274, 32160, 47047. Ans. 405277. Add together, 66947, 46742, and 132684. Ans. 246373. Subtraction. 16844 9786 103034 69845 5987432 278459 7058 33189 7896600 5403257 5403257 4250268 5789232 410204 Multiplication. 1st. Multiply 328 by 2. Multiply 745 by 3. Multiply 20508 by 5. Multiply 3605023 by 6. Multiply 9097030 by 9. 2d. Multiply 725 by 300. Multiply 35012 by 2000. Multiply 9120400 by 90. Multiply 4890000 by 36000. Ans. 756. Ans. 2235. Ans. 102540. Jlns. Ans. Ans. 217500. Ans. 70024000. Ans. 820836000. Ans. (220) DIVISION — VULGAR FRACTIONS. 221 3d. Multiply 793 by 345. Multiply 471493475 by 4395. Multiply 89999000 by 97770400. Multiply 17204774 by 125. Multiply 3768 by 4230. Multiply 9648 by 6137. Ans. 273585. Ans. 2072213822625. Ans. . Ans. 2150596750. Ans. . Ans. 49561776. Division. 1st. Divide 3788 by 2. Divide 4736511 by 9. Divide 78920 by 5. Divide 364251 by 3. Divide 34300 by 7. 2d. Divide 1203033 by 3679. Divide 49561766 by 5137 Divide 2150596750 by 125. Divide 71900715708 by 57149. Ans. Divide 78674 by 200. Divide 32500000 by 520. Divide 36000000 by 3600. Divide 27489000 by 350. Ans. 1894. Ans. 526279. Ans. 15784. Ans. 121417. Ans. 4900. Ans. 327. Ans. 9648. Ans. 17204774. 1258127. Rem. 15785. Ans. 393 + 74 Rem. Ans. 62500. Ans. . Ans. 7854. Vulgar Fractions. Reduction of Vulgar Fractions to a Common Denominator. Reduce | and | to a common denominator. Ans. 3g, ^g. Reduce ^, |, and | to a common denominator. ^ns. 18, If, If. Reduce -f^, J, ^j and |, to a common denominator. >4?)« _63_0 18911 1800 1750 ^^^' 3T5(J' 3T50? 3T^UJ 3T5IJ- Reduce y, I, j^^j, and /^, to a common denominator. Ai 19 222 EXAMPLES. Finding the Least Common MultipU. Find the least common multiple of 13, 12, and 4. Ans. 156. What is the least common multiple of 11, 17, 19, 21, and 7 ? Ans. . Find least common multiple of 6, 9, 4, 14, and 16. Ans. 1008. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8, 9 ? Ans. 2520. Reduction of Fractions to the Least Common Denominator. Reduce ^, |, |, and |, to the least common denominator. Ay,(i 6 8 9 10 Reduce j^g, ^^, and |, to the least common denominator. J^o 72 60_ 320 , ^Ai&. 3g^, 3gQ, 3g^. Reduce 4> I? I? i? il? ^^^ hh *^ *^^ ^^^'^^ common denomina- tor And, and in most cases the 8yuoiiyiu8 of the words defined are added, a grw«t advuntagt- to persfins engaged in literary compositions " Leeds Thnea. 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