V i " '> V- 1 INQ TQR \ i Si 1) .CCOuH'prS MSSiST/\HT University of California • Berkeley The Theodore P. Hill Collection of Early American Mathematics Books .^:3^-7c --^ -iifi^-'j^-z^^f yftA^ % T i-. V-i. ^, a^^/ ORTON'S LIGHTNING CALCULATOR, AND THE SHORTEST, SIMPLEST, AND MOST RAPID METHOD OF COMPUTING >'IJMBP:RS, ADAPTED TO EVERT KIND OF^PSINESS, AND WITHIN THE COMPREHENSION OF EVERT ONE HAVING THE SLIGHTEST KNOWLEDGE OF FIG0RB3. BY HOY D. ORTON. ENTIRELY WE^S^V EDITIOW, WITH EXTEKSIVE MODIFICATIONS AND IMPROVEMENTS. N. B.— Any Infl-lncement upon the eopyrlsht of this book will prosecuted to the fullest extent of the law. PHILADELPHIA: COLLINS, PRINTER, 705 JAYNE STREET. Entered according to Act of Congress, in the year 1871, by HOY D. ORTON, in the Office of the Librarian of Congress, at Washington. N. B. — It gives me pleasure to state that, in the revision of this book, I have been deeply indebted to S. J. Donaldson, Jr., of Baltimore, a gentleman favorably known as the author of "Lyrics, and Other Poems." COLLINS, PRINTER. INTRODUCTION. Quantity is that which can be increased or diminished by augments or abatements of homo- geneous parts. Quantities are of two essential kinds, Geometrical and Physical, 1. Geometrical quantities are those which occupy space ; as Unes^ surfaces^ solids, liquids, gases, etc. 2. Physical quantities are those which exist in the time, but occupy no space ; they are known by their character and action upon geometrical quan- tities, as attraction, light, heat, electricity and mag^ netism, colors, force, power, etc. To obtain the magnitude of a quantity we com- pare it with a part of the same ; this part is im- printed m our mind as a unit, by which the whole is measured and conceived. No quantity can be measured by a quantity of another kind, but any quantity can be compared with any other quantity, and by such comparison arises what we call calcu' lation or Mathematics. lY INTRODUCTION. MATHEMATICS. Mathematics is a science by which the com- parative value of quantities are investigated ; it is divided into : 1. Arithmetic, that branch of Mathematics which treats of the nature and property of num- bers ; it is subdivided into Addition^ Subtraction^ Multiplication^ Division^ Involution^ Evolution and Logarithms. 2. Algebra, that branch of Mathematics which employs letters to represent quantities, and by that means performs solutions without knowing or noticing the value of the quantities. The subdi- visions of Algebra are the same as in Arithmetic. 3. Geometry, that branch of Mathematics which investigates the relative property of quantities that occupies space; its subdivisions are Longemetry^ Planemetry^ Stereometry^ Trigonometry and Conic Sections. 4. Differential-calculs, that branch of Math- ematics which ascertains the mean effect produced by group of continued variable causes. 5. Integral-calculs, the contrary of Differen- tial, or that branch of Mathematics which investi- gates the nature of a continued variable cause that has produced a known effect. PREFACE Mathematical laws are tlie acknowledged basis of all science. Ever since the streets of Athens resounded with that historical cry of "Eu- reka," emanating from one of antiquity's greatest mathematicians, the science has been steadily pro- gressing. It is not our purpose, in this small work, to in- troduce any of the higher branches of mathematics, VIZ.: Algebra, Conic Sections, Calculus, etc. Our object is merely to present to the public a system of calculation that is practical to every business man. It consists of the addition of numbers on a principle entirely different from the one ordinarily used. In the practical application of this new prin- ciple of addition, scarcely any mental labor is re- quired, compared with the principle of addition set forth in standard works. The superiority we claim for this principle above all others, is this, that it requires no great mental exertion, affording the Tl PREFACE. preatest facilities to tbe calculator in the addition of numbers, enabling him to add a whole day with- out any mental fatigue ; whereas, by the ordinary way, it is very laborious and fatiguing. Our system of calculation also embraces a concise, rapid, and at the same time practical method of Multiplication, by which one is enabled to arrive ftt the product of any number of figures, multiplied by any number, immediately, without the use of partial products. This small work also embraces the shorteyt and most concise method for the computation of Interest ever introduced to the public. Our system for com- puting interest is entirely diflferent from any rule ever introduced, for the computation of either Sim- ple or Compound Interest. A student having gone no further than Long Division in Arithmetic, can, by our rule, calculate Simple or Compound Interest at any given rate per cent., for any given time, in one-tenth of the time that the best calculators will compute it by the rules laid down in other books. By using our rules, you can entirely avoid the use of fractions, and save the calculation of 75 to 100 figures, where years, months and days are given oa a note. ADDITION. N B. The above process of addition is only re- commended for beginners. Process. — For adding the above example, com- mence at the bottom of the right-hand column. Add thus ; 12, 16, 22 ; then carry the 2 tens to the second column, then add thus, 8, 10, 18, 22^ carry the two hundreds to the third column, and add the same way, 9, 13, 16, 23. Never permit yourself for once to add up a column in this man- ner, 3 and 9 are 12 and 4 are 16, and 6 are 22 ; it is just as easy to name the sum at once, without naming the figures you add, and three times as rapid, 9 10 orton's lightning calculator, addition of short columns of figures. Addition is the basis of all numerical opera- tions, and is used in all departments of business, To aid the business man in acquiring facility and accuracy in adding short columns of figures, the following method is presented as the best : Process. — Commence at the bottom of 274 oj^g the right-hand column, add thus: 16, 22, 134 ^2 ; then carry the 3 tens to the second 342 column ; then add thus : 7, 14, 25 ; carry ^27 the 2 hundreds to the third column, and ^^^ add the same way: 12, 16, 21. In this 2152 'vvay you name tli3 sum of two figures at once, which is quite as easy as it is to add one figure at a time. Never permit yourself /or once to add up a column in this manner : 9 and 7 are 16, and 2 are 18 and 4 are 22, and 6 are 28, and 4 are 32. It is just as easy to name the result of two figures at once and four times as rapid. The following method is recommended for the addition OF LONG COLUMNS OF FIGURES. In the addition of long columns of figures which frequently occur in books of accounts, in order to add them with certainty, and, at the same time, with ease and expedition, study well the following method, which practice will render familiar, easy, rapid, and certain. ADDITION. 11 THE EASY WAY TO ADD. EXAMPLE 2— EXPLANATION. Commence at 9 to add, and add as near 20 as pos- sible, thus: 94-2-1-4+3=18, place the 8 to the right of the 3, as in example ; commence at 6 to 7' add 6-f 4-i-8=18 ; place the 8 to the right of 4 the 8, as in example j commence at 6 to add G 6+4-f7=17 ; place the 7 to the right of the 3« 7. as in example ; commence at 4 to add 4-|- 9 9-f 3=16 ; place the 6 to the right of the 3, 4 as in example; commence at 6 to add 6-f4 7' -f-7=17 ; place the 7 to the right of the 7, 4 as in example; now, having arrived at the 6 top of the column, we add the figures in the 8** new column, thus: 7-}-6-|-7-f8-J-8=36; place 4 the right hand figure of 36, which is a 6, 6 under the original column, as in example, and 3' add the left hand figure, which is a 3, to the 4 number of figures in the new column; there 2 are 5 figures in the new column, therefore 9 3-[-5=8 ; prefix the 8 with the 6, under the — original column, as in example ; this makes 86 86, which is the sum of the column. Remark 1. — If, upon arriving at the top of the column, there should be one, two or three figures whose sum will not equal 10, add them on to the Bum of the figures of the new culumn, never plyciug 12 orton's lightning calculator. an extra figure in the new column, unless it be an excess of units over ten. Remarh 2. — By this system of addition you can stop any place in the column, where the sum of the figures will equal 10 or the excess of 10 ; but the addition will be more rapid by your adding as near 20 as possible, because you will save the form- ing of extra figures in your new column. EXAMPLE— EXPLANATION. 2+6+7=15, drop 10, place the 5 to the right of the 7; 6+5+4=15, drop 10, place the 5 to the right of the 4, as in example; 8+3+7=18, drop 10, place the 8 to the right of the 7, 4 as in example ; now we have an extra figure, 7^ which is 4 ; add this 4 to the top figure of the 3 new column, and this sum on the balance of 8 the figures in the new column, thus: 4+8+ 4* 5+5=22 ; place the right hand figure of 22 5 under the original column, as in example, and 6 add the left hand figure of 22 to the num- 7' ber of figures in the new column, which are 6 three, thus : 2+3=5 ; prefix this 5 to the 2 figure 2, under the original column ; this — makes 52, which is the sum of the cokmn. 52 ADDITION. 13 "Rule.-— 2^ or adding two or more coIumnSy com- mence at the right handj or units* column; proceed in the sam^e manner as in adding one column; after the sum of the first column is obtained, add all except the right hand figure of this sum to the second column, adding the second column the same wag you added the first; proceed in like manner with all the columns, always adding to each successive column the sum of the column in the next lower order, minus ilie right hand figure. N. B. The small figures wliicli we place to tlie riglit of the column when adding are called integers. The addition by integers or by forming a new column, as explained in the preceding examplea should be used only in adding very long columns of figuies, say a long ledger column, where the foot- ings of each column would be two or three hundred, in which case it is superior and much more easy than any other mode of addition ; but in adding short columns it would be useless to form an extra column, where there is only, say, six or eight fig- ures to be added. In making short additions, the following suggestions will, we trust, be of use to the accountant who seeks for information on this subject. In the addition of several 3olumns of figures, where they are only four or five deep, or when their respective sums will range from twenty ^ve '14 orton's lightning calculator. to forty, the accountant should commence with the unit column, adding the sum of the first two figures to the sum of the next two, and so on, naming only the results, that is, the sum of every two figures. In the present example in adding the unit 346 column instead of saying 8 and 4 are 12 and 235 5 are It and 6 are 23, it is better to let the T24 eye glide up the column reading only, 8, 12, 598 It, 23; and still better, instead of making a separate addition for each figure, group the figures thus: 12 and 11 are 23, and proceed in like man- ner with each column. For short columns this is a very expeditious way, and indeed to be preferred ; but for long columns, the addition by integers is the most useful, as the mind is relieved at intervals and the mental labor of retaining the whole amount, as you add, is avoided, which is very important to any person whose mind is constantly employed in various commercial calculations. In adding a long column, where the figures are of a medium size, that is, as many 8s and 9s as there are 2s and 3s, it is better to add about three figures at a time, because the eye will distinctly see that many at once, and the ingenious student will in a short time, if he adds by integers, be able to read the amount of three figures at a glance, or as quick, we might say, as he would read a single figure. ADDITION. 15 Here we begin to add at the bottom of the *26« unit column and add successively three fig- "• ures at a time, and place their respective 004 sums, minus 10, to the right of the last fig- 95^ are added; if the three figures do not make 62 10, add on more figures; if the three figures 87^ make 20 or more, only add two of the fig- ^^^ ures. The little figures that are placed to ^^ 4 the right and left of the column are called 877 integers. The integers in the present ex- 33 ample, belonging to the units column, are 84* 4, 4, 5, 4, 6, which we add together, making ^^ 23; place down 3 and add 2 to the number of integers, which gives 7, which we add to 803 the tens and proceed as before. Reason. — In the above example, every time wo placed down an integer we discarded a ten, and when we set down the 3 in the answer we dis- carded two tens; hence, we add 2 on to the num- ber of integers to ascertain how many tens were discarded; there being 5 integers it made 7 tens, which we now add to the column of tens; on the same principle we might add between 20 and 30, always setting down a figure before we got to 30; then every integer set down would count for 2 tens, being discarded in the same way, it does in the present instance for one ten. When we add be- tween 10 and 20, and in very long columns, it 16 orton's ligutninq calculator. would be much better to go as near 30 as possible, and count 2 tens for every integer set down, in which case we would set down about one-half as many integers as when we write an integer for every ten we discard. When adding long columns in a ledger or day- book, and where the accountant wishes to avoid the writing of extra figures in the book, he can place a strip of paper alongside of the column he wishes to add, and write the integers on the paper, and in this way the column can be added as convenient almost as if the integers were written in the book. Perhaps, too, this would be as proper a time as any other to urge the importance of another good habit; I mean that of making ylain figures. Some persons accustom themselves to making mere scrawls, and important blunders are often the result. If letters be badly made you may judge from such as are known; but if one figure be illegible, its value can not be inferred from the others. The vexation of the man who wrote for 2 or 3 monkeys, and had 203 sent him, was of far less importance than errors and disappointments sometimes result- ing from this inexcusable practice. We will now proceed to give some methods of proof. Many persons are fond of proving the cor- rectness of work, and pupils are often instructed to do so, for the double purpose of giving them ADDITION. 17 exercise in calculation and saving their teacher the trouble of reviewing their work. There are special modes of proof of elementary operations, as by casting out threes or nines, or by changing the order of the operation, as in add- ing upward and then downward. In Addition, some prefer reviewing the work by performing the Addition downward, rather than repeating the ordinary operation. This is better, for if a mis- take be inadvertently made in any calculation, and the same routine be again followed, we are very liable to fall again into the same error. If, for instance, in running up a column of Addition you should say 84 and 8 are 93, you would be liable, ia going over the same again, in the same way to slide insensibly into a similar error; but by begin- ning at a different point this is avoided. This fact is one of the strongest objections to the plan of cutting off the upper line and adding it to the sum of the rest, and hence some cut off the lower line by which the spell is broken. The most thoughtless can not fail to see that adding a line to the sum of the rest, is the same as adding it in with the rest. The mode off proof by casting out the nines and threes will be fully explained in a following chapter. A very excellent mode of avoiding error in add- 18 okton's ligctnino calculator. ing long columns is to set down tlie result of each column on some waste spot, observing to place the numbers successively a place further to the left each time, as in putting down the product figures in multiplication; and afterward add up the amount. In this way if the operator lose his count, he is not compelled to go back to units, but only to the foot of the column on which he is op- erating. It is also true that the brisk accountant, who thinks on what he is doing, is less liable to err than the dilatory one who allows his mind to wander. Practice too will enable a person to read amounts without naming each figure, thus instead of saying 8 and 6 are 14, and 7 are 21 and 5 are 2G, it is better to let the eye glide up the column, read- ing only 8, 14, 21, 26, etc.; and, still further, it is quite practicable to accustom one's self to group 87 the figures in adding, and thus proceed very rap- 23 idly. Thus in adding the units' column, instead 45 of adding a figure at a time, we see at a glance 62 that 4 and 2 are 6, and that 5 and 3 are 8, then 24 6 and 8 are 14; we may then, if expert, add — constantly the sum of two or three figures at a time, and with practice this will be found highly advan- tageous in long columns of figures; or two or three columns may be added at a time, as the practiced eye will see that 24 and 62 are 86 almost as readily as that 4 and 2 are 6. ADDITION. 19 Teachers will find the following mode of matx3h- ing lines for beginners very convenient, as they can inspect them at a glance : Add 7G54384 8786286 3408698 2345615 1213713 23408696 In placing the above the lines are matched in pairs, the digits constantly making 9. In the above, the first and fourth, second and fifth are matched; and the middle is the hey line^ the result being just like it, except the units' place, which is as many less than the units in the key line as there are pairs of lines; and a similar number will oc- cupy the extreme left. Though sometimes used aa a puzzle, it is chiefly useful in teaching learners ; and as the location of the key line may be changed in each successive example, if necessary, the arti« fice could not be detected. The number of lines is necessarily odd. SHORT METHODS OF MULTIPLICATION. Rule. — Set down the smaller factor under the larger, units under units, tens under tens. Begin with the unit figure of the multiplier, multiply by it, first the units of the multiplicand, setting the units of the product, and reserviny the tens to be added to the next product; now multiply the tens of the multiplicand by the unit figure of the multijMer, and the units of the multiplicand by tens figure of 20 MULTIPLICATION. 21 the multiplier ; add these two products together^ set- ting down the units of their sum, and reserving the tens to he added to the next 'product ; now multiply the tens of the multiplicand hy the tens figure of the multiplier, and set down the whole amount. This will he the complete product. Remark. — Always add in the tens that are re- served as soon as you form the first produeb. EXAMPLE 1.— EXPLANATION. 1. Multiply the units of the multiplicand 24 by the unit figure of the multiplier, thus: 31 1X4 is 4 ; set the 4 down as in example. 2. Multiply the tens in the multiplicand by 744 the unit figure in the multiplier, and the units in the multiplicand by the tens figure in the multi- plier, thus : 1X2 is 2; 3X4 are 12, add these two products together, 2-|-12 are 14, set the 4 down as in example, and reserve the 1 to be added to the next product. 3. Multiply the tens in the multi- plicand by the tens figure in the multiplier, and add in the tens that were reserved, thus : 3x2 are 6, and 6-f 1=7 ; now set down the whole amount, which is 7. EXAMPLE 2.— EXPLANATION. 1. Multiply units by units, thus: 4x3 53 are 12, set down the 2 and reserve the 1 to 84 carry. 2. Multiply tens by units, and units by tens, and add in the one to carry on the 4152 22 ORTON & fullee's arithmetic. first product, then add these two products together, thus: 4X5 are 20+1 are 21, and 8X3 are 24, and 21-f 24 are 45, set down the 5 and reserve the 4 to carry to the next product. 3. Multiply tens by tens, and add in what was reserved to carry, thus: 8X5 are 40-f-4 are 44, now set down the whole amount, which is 44. EXAMPLE 3.— EXPLANATION. 5X3 are 15, set down the 5 and carry the 43 1 to the next product; 5X4 are 20=1 25 are 21; 2x3 are 6, 21+6 are 27, set down the 7 and carry the 2; 2X4 are 8+2 are 1075 10 ; now set down the whole amount. When the multiplicand is composed of three fig- ures, and there are only two figures in the multi- plier, we obtain the product by the following lluLE. — Set down the smaller factor under the larger, units under units, tens under tens ; now muU tiply the first upper figure hy tlie unit figure of the multiplier, setting down the units of the product, and reserving the tens to he added to the next product; now multiply the second upper hy units, and the first upper hy tens, add these two products together, set- ting down the units figure of their sum, and reserv- ing the tens to carry, as hefore; now multiply the third upper hy units, and the sezond upper hy tens, add these two products together, setting down the units figure of their sum, and reserving the tens to MULTIPLICATION. 23 carry ^ as usual ; now multiply the tJiird upper hy tens, add in the reserved figure, if there is one, and set down tlie whole amount. This will he the com- plete product. Remark. — One of the principal errors with the beginner, in this system of multiplication, is neglecting to add in the reserved figure. The stu- dent must bear in mind that the reserved figure is added on to the first product obtained after the set- ting down of a figure in the complete product. EXAMPLE 1.— EXPLANATION. Multiply first upper by units, 5x3 are 123 15, set down the 5, reserve the 1 to carry 45 to the next product; now multiply second upper by units and first upper by tens, 5X2 5535 are 10-|-1 are 11,4X3 are 12, add these products together; ll-j-12 are 23, set down the 3, reserve the 2 to carry ; now multiply third upper by units, and second upper by tens, add these two products together, always adding on the reserved "figure to the first product; 5x1 are 5-|-2 are 7, 4X2 are 8, and 7-}-8 are 15, set down the 5, re- serve the 1 ; now muitiply third upper by tens, and set down the whole amount; 4X1 are 4-|-l are 5, set down the 5. This will give the comple^« product. 24 orton's lightning calculator. Multiply 32 by 45 in a single line. Here we multiply 5X2 and set down and 32 carry as usual ; then to what you carry add 45 5X3 and 4X 2, which gives 24; set down 4 and carry 2 to 4x3, which gives 14 and 1440 completes the product. Multiply 123 by 456 in a single line. Here the first and second places are 123 found as before; for the third, add 6Xl> 456 5X2, 4x3, with the 2 you had to carry, making 30 ; set down and carry 3 ; then 56088 drop the units' place and multiply the hundreds and tens crosswise, as you did the tens and units, and you find the thousand figure ; then, dropping both units and tens, multiply the 4X1, adding the 1 you carried, and you have 5, which completes the product. The same principle may be extended to any number of places; but let each step be made perfectly familiar before advancing to another. Begin with two places, then take three, then four, but always practicing some time on each number, for any hesitation as you progress will, confuse you. N. B. The following mode of multiplying num- bers will only apply where the sum of the two last or unit figures equal ten, and the other figures in both factors are the same. MULTIPLICATION. 25 CONTRACTIONS IN MULTIPLICATION. To multiply when the unit figures added equal (10) and the tens are alike as t2 by IS, &c. 1st. Multiply the units and set down the result. 2d. Add 1 to either number in tens place and multiply by the other, and you have the complete product. EXAMPLE FIRST — PROCESS. Here because the sum of the units 4 and 6 86 are ten and the tens are alike ; we simply say 84 4 times 6 are 24, and set down both figures of the product ; then because 4 and 6 make ten we '^^24 add 1 to 8, making 9, and 9 times 8 are 72, which completes the product. Note. — If the product of units do not contain ten the place of tens must be filled with a cipher The above rule is useful in examples like the fol- 2. What will 93 acres of land cost at 97 dollars lowing : per acre ? Ans. $9021. 3. What will 89 pounds of tea cost at 81 cents per pound ? Ans. $72.09. I7i the above the product of 9 by 1 did not amount to ten, therefore is placed in tens place. 4. Multiply 998 by 992. Ans. 990016. In the above, because 2 and 8 are 10, we add 1 to 99, maJcing 100; then 100 times 99 are 9900. 26 orton's lightning calculator. EXAMPLE EIGHTEENTH. Multiply 79 by 71 in a single line. Here we multiply IX^ and set down the 79 result, then we multiply the 7 in the mul- 71 tiplicand, increased by 1 by the 7 in the multiplier, 7X8, which gives 56 and com- 5609 pletes the product. EXAMPLE NINETEENTH. Multiply 197 by 193 in a single line. Here we multiply 3x7 and set down the 197 result, then we multiply the 19 in the 193 multiplicand, increased by 1 by the 19 in the multiplier, 19X20, which gives 380 38021 and completes the product. EXAMPLE TWENTIETH. Multiply 996 by 994 in a single line. Here we multiply 4x6 and set down 996 the result, then we multiply the 99 in 994 the multiplicand, increased by 1 by the 99 in the multiplier, 99X100, which 990024 gives 9900 and completes the product. EXAMPLE TWENTY-FIRST. Multiply 1208 by 1202 in a single line. Here we multiply 2x8 and set down 1208 the result, then we multiply the 120 in 1202 the multiplicand, increased by 1 by the 120 in the multiplier, 120X121, which 1452016 gives 14520 and completes the product. MULTIPLICATION. 27 CURIOUS AND USEFUL CONTRACTIONS. To multiply any number, of two figures, by 11, HuLE. — Write the sum of the figures between them. 1. Multiply 45 by 11. Ans. 495. Here 4 and 5 are 9, which write between 4 & 5 2. Multiply 34 by 11. Ans. 374. N. B. When the sum of the two figures is over 9, increase the left-hand figure by the 1 to carry. 3. Multiply 87 by 11. Ans. 957. To square any number of 9s instantaneously, and without multiplying. Rule. — Write down as many 9s less one as there are 9s in the given number, an 8, as many Os a& 9«, and a 1. 4. What is the square of 9999? Ans. 99980001. Explanation. — We have four 9s in the given number, so we write down three 9s, then an 8, then three Os, and a 1. 5. Square 999999. Ans. 999998000001. To square any number ending in 5, KuLE. — Omit the 5 and mnltijyiy the number, as it will then stand by the next higher number, and annex 25 to the product. 6. What is the square of 75 ? Ans. 5625. Explanation. — We simply say, 7 times 8 are 56, to which we annex 25. 7. What is the square of 95? Ans. 9025. C 28 ORTON'S LIGHTNING CALCULATOR. Mental Operations in Fractions. To square any number containing J, as 6^, 9^, KuLE. — Multiply the whole number by the next higher whole number ^ and annex \ to the product. Ex. 1. What is the square of 7^? Ans. hQ{. We simply say, 7 times 8 are 56, to which we addi. 2. What will ^ lbs. beef cost at 9^ cts. a lb.? 3. What will 121 yds. tape cost at \2\ cts. a yd. ? 4. What will 5^ lbs. nails cost at 5^ cts. a lb. ? 5. What will 11^ yds. tape cost at 11^ cts. a yd.? 6. What will 19^ bu. bran cost at 191 cts. a bu.? Reason. — We multiply the whole number by the uext higher whole number, because half of any number taken twice and added to its square is the same as to multiply the given number by one moie than itself. The same principle will multiply any two like numbers together, when the sum of the fractions is one, as 8^ by 8|, or 11| by 11|, etc It is obvious that to multiply any number by any two fractions whose sum is one, that the sum of the products must be the original number^ and adding the number to its square is simply to multiply it by one more than itself; for instance, to multiply 7J by 7|, we simply say, 7 times 8 are 56, and then, to complete the multiplication, we add, of course, the product of the fractions (| times \ arc 3^), maJcing ^^-^^ the answer. MULTIPLICATION. 29 Wliere tlie sum of tlie Fractions is One. To multiply any two like numbers together when the sum of the fractions is one. IluLE. — Multiply the whole numher hy the next higher whole numher; after which^ add the product of the fractions. N. B. In the following examples, the product of the fractions are obtained y^rs^ for convenience. PRACTICAL EXAMPLES FOR BUSINESS MEN. Multiply 3| by 3^ in a single line. Here we multiply |X|, which gives y^^, 3J and set down the result; then we multiply 3-| the 3 in the multiplicand, increased by unity, by the 3 in the multiplier, 3x4, 12^^ which gives 12 and completes the product. Multiply 7f by 7f in a single line. Here we multiply |Xf) which gives -^^^ 7f and set down the result; then we multiply 7| the 7 in the multiplicand, increased by unity, by the 7 in the multiplier, 7x8, which gives 66^5 66, and completes the product. Multiply 11^ by 11| in a single line. Here we multiply f X^, which gives f , and 11 J set down the result; then we multiply the 11 11 1 in th« multiplicand, increased by unity, by the 11 in the multiplier, 11x12. which gives 1325 132, and completes the product. 30 orton's lightning calculator. EXAMPLE THIRTY-THIRD. Multiply 16f by 16J in a single line. Here we multiply JXf which gives I, and IGJ Bet down the result, then we multiply the 16J 16 in the multiplicand, increased by unity by the 16 in the multiplier, 16X17, 272f which gives 272 and completes the product. EXAMPLE THIRTY-FOURTH. Multiply 29 J by 29 J in a single line. Here we multiply JXj which gives J, 29 J and set down the result, then we multiply 29J the 29 in the multiplicand, increased by unity by the 29 in the multiplier, 29 x 87 OJ 30, which gives 870 and completes the pro- duct. EXAMPLE THIRTY-FIFTH. Multiply 999f by 999§ in a single line. Here we multiply fXfj which gives 999f J|, and set down the result, then we 999f multiply the 999 in the multiplicand, increased by unity by the 999 in the 999000j| multiplier, 999x1000, which gives 999000 and completes the product. Note. — The system of multiplication introduced in the preceding examples, applies to all numbers. Where the sum of the fractions is one, and the whole numbers are alike, or differ by one^ the learner is requested to study well these useful properties of numbers. orton's lightninq calculator. hi WJiere the sum of the Fractions is One. To multiply any two numbers whose difference is onej and the sum of the fractions is one, Rule. — Multiply the largernumher, increased hy ONE, hy the smaller number; then square the frac- tion of the larger number, and subtrac* its square from ONE. rRACTICAL EXAMPLES FOR BUSINESS MEN. 1. What will 9:^ lbs. sugar cost at 8J cts. a lb.? Here we multiply 9, increased by 1, by 8, 91 thus, 8X10 are 80, and set down the result; gs then from 1 we subtract the square of }, thus, \ squared is j\, and 1 less ^ is |-|. 8()fJ 2. What will 82 bu. coal cost at 7J cts. a bu.? Here we multiply 8, increased by 1, by S| 7. thus, 7 times 9 are 63, and set down the 7^ result ; then from 1 we subtract the square ~~^ of |, thus, I squared is *, and 1, less ^, is |. » 3. What will lly2^ bu. seed cost at $10]^ a bu.? Here we multiply 11, increased by 1, by 10, thus, 10 times 12 are 120, and set l^A down the result; then from 1 we subtract ^^il the square of -^3, thus, ^^ squared is y|^, ^J^^TTT and 1 less , I, is ill. -^ ^'^ 4. IIow many square inches in a floor 99| in wide and 98| in. long? Ans. 9800^5. c* 32 orton's lightning calculator.- METHOD OF OPERATION. EXAMPLE FIRST. Multiply 6 J by 6 J in a single line. Here we add 6J-|-i) "which gives 6J ; this 6} multiplied by the 6 in the multiplier, 6| 6x6i, gives 39, to which we add the pro- duet of the fractions, thus JXi gives y'g, added 39Jg to 39 completes the product. EXAMPLE SECOND. Multiply lljbylljina single line. Here we would add HJ+J, which gives 11 J 12; this multiplied by the 11 in the multi- il| plier gives 132, to which we add the product of the fractions, thus f X J gives -j?g, which 132-^^^ added to 132 completes the product. EXAMPLE THIRD. Multiply 12 J by 12f in a single line. Here we add 12J-["i) which gives 13 J ; 12J this multiplied by the 12 in the multiplier, 12 J 12X13J, gives 159, to which add the pro duct of the fractions, thus fxj gives f, 159| which added to 159 completes the product. orton's lightninci calculator. 33 Where the Fractions have a Lihe Denominator. To multiply any two lihe numbers together, each of which has a fraction with a lihe denominator, as H ^1 4J, or 11^ by llf, or 10| by lOJ, etc. Rule. — Add to the multiplicand the fraction of the multiplier^ and multiply this sum hy the whok number; after which^ add the product of the fractions. PRACTICAL EXAMPLES FOR BUSINESS MEN. N. B. In the following example, the sum of the frac- tions is ONE. 1. What will 9| lbs. beef cost at 9J cts. a lb.? The sum of 9| and \ is ten, so we simply 9| say, 9 times 10 are 90; then we add the ^ product of the fractions, \ times | are j^^. 90y*- N. B. In the following example, the sum of the frao tions is less than one. 2 What will 8J yds. tape cost at 8| cts. a yd. ? The sum of 8| and | is 8J, so we simply ^\ B&j, 8 times 8J are 70; then we add the _i product of the fractions, J times ^ are -^ or J. 70-J N. B. In the following example, the sum of the frac- tions is greater than one. 3. What will 4| yds. cloth cost at $4| a yd.? The sum of 4f and | is 5^, so we simply 4| say, 4 times 5J are 21 ; then we add the » product of the fractions, J times | are |^J. 21|^J N. B. Where the fractions have different denominatora, reduce them to a commoc denominator. S4 ORTONS LIGnTNINO CALCUI ATOR. Rapid Process of Multiplying Mixed ITumhers. A valuable and useful rule for the accountant in the practical calculations of the counting-room. To multiply any two numbers together, each of which involves the fraction ^, as 7^ by 9^, etc., Rttle. — To the product of the whole numbers add half their sum plus \, EXAMPLES FOR MENTAL OPERATIONS. 1. What will 3^doz. eggs cost at 7^ cts. a doz.? Here the sura of 7 and 3 is 10, and half this 31 sum is 5, so we simply say, 7 times 3 are 21 l\ and 5 are 2G, to which we add 4-. ^ 264 N. B. If the sum be an odd number, call it one less to make it even, and in such cases the fraction must be |. 2. What will \\^ lbs. cheese cost at 9|^ cts. a lb. ? 3. What will 8^ yds. tape cost at 15^ cts. a yd.? 4. What will 7^ lbs. rice cost at 13^ cts. a lb.? 5. What will lOi bu. coal cost at 121 cts. a bu.? Reason. — In explaining the above rule, we add half their sum because half of either number added to half the other would be half their sum, and we add \ because ^ by ^ is \. The same principle will multiply any two numbers together, each of which has the same fraction; for instance, if the fraction was -J^, we would add one-third their sum ; if |, we would add three-fourths their sum, etc.; and then, to complete the multiplication, we would add, of course, the product of the fractions. MULTIPLICATION. 35 GENERAL RULE For multiplying any two numbers together, each of which involves the same fraction. To the product of the whole numbers^ add the product of their sum hy either fraction ; after which, add the product of their fractions. EXAMPLES FOR MENTAL OPERATIONS. 1. What will 11} lbs. rice cost at 9| cts! a lb.? Here the sum of 9 and 11 is 20, and three- Jif fourths of this sum is 15, so we simply say, 9| 9 times 11 are 99 and 15 are 114, to which 111 s we add the product of the fractions (^). ^* 2. What will 7f doz. eggs cost at 8| cts. a doz. ? 3. What will 6| bu. coal cost at 6 j cts. a bu. ? 4. What will 45| bu. seed cost at 3f dol. a bu.? 5. What will 3| yds. cloth cost at 5f dol. a yd. ? 6. What will 17f ft. boards cost at 13| cts a ft.? 7. What will 18} lbs. butter cost at 18| cts. a lb. ? N. B. If the product of the sum by either frac- tion is a whole number with a fraction, it is better to reserve the fraction until we are through with the whole numbers, and then add it to the product of the fractions ; for instance, to multiply 3} by 7 J , we find the sum of 7 and 3, whicli is 10, and one- fourth of this sum is 2|; setting the ^ down in some waste spot, we simply say, 7 times 3 are 21 and 2 are 23 ; then, adding the J to the product of tho fractions (jq), gives j^, making 23^, Ans. 36 ORTON's LIGHTNINa CALCULATOR. Rapid Process of Multiplying all Mixed Numhen, N. B. Let the student remember that this is a general and universal rule. GENELAL RULE. To multiply any two mixed numbers together, 1st. Multiply the whole numhers together. 2d. Multiply the upper digit hy the lower fraction. 3d. Multiply the lower digit by the upper fraction. 4th. Multiply the fractions together. 5th. Add these FOUR products together. N. B. This rule is bo simple, so useful, and so true that every banker, broker, merchant, and clerk should post it up for reference and use. PRACTICAL EXAMPLES FOR BUSINESS ME?< N. B. The following method is recommended to begin- ners: Example.— Multiply 12| by 9|. 12| 1st. We multiply the whole numbers. 9| 2d. Multiply 12 by f and write it down. jqS 3d. Multiply 9 by | and write it down. 9 4th. Multiply f by J and write it down. 6 5th. Add these four products together, ___£^ and we have the complete result. 123 ,4^ N. B. When the student has become familiar with the above process, it is better to do the inter- mediate work in the head, and, instead of setting down the partial products, add them in the mind as you pass along, and thus proceed very rapidly. MULTIPLICATION. 37 Multiply 8i by lOf Here we simply say 10 times 8 are 80 8J and J of 8 is 2, making 82, and | of 10 is lOJ 2, which makes 84; th«n ^ times -J- is T^\y making 8^2^^ the answer. 81;^ PRACTICAL BUSINESS METHOD For Multiplying all 3Iixed Nmnhers. Merchants, grocers, and business men generally, in multiplying the mixed numbers that arise in the practical calculations of their business, only care about having the answer correct to the near- est cent; that is, they disregard the fraction. When it is a half cent or more, they call it an- other cent ; if less than half a cent, they drop it. And the object of the following rule is to show the business man the easiest and most rapid process of finding the product to the nearest unit of any two numbers, one or both of which involves a fraction. GENERAL RULE. To multiply any two numbers to the nearest unit, 1st. Multiply the whole number in the multiplicand by the fraction in the multiplier to the nearest unit. 2d. Multiply the whole number in the multiplier by the fraction in the m^ultiplicand to the nearest unit 3d. Multiply the whole numbers together and add the three products in your mind as you proceed. N. B. In actual business the work can generally be done mentally for only easy fractions occur iti business. 38 ORTON*S LIGHTNINQ CALCULATOR. N B. This rule is so simple and so true, according to all business usage, that every accountant should make himself perfectly familiar with its application. There being no such thing as a fractian to add in, there is scarcely any liability to error or mistake. By no other arithmetical process can the result be obtained by so few figures. EXAMi'LES FOR MENTAL OPERATION. EXAMPLE FIRST. Multiply 11^ by 8| by business method. 11 J Here J of 11 to the nearest unit is 3, and ^ of 8 J 8 to the nearest unit is 3, making 6, so we sim- ' ply say, 8 times 11 are 88 and 6 are 94, Ans. 94 Reason. — \ of 11 is nearer 3 than 2, and J of 8 is nearer 3 than 2. Make the nearest whole number the quotient. EXAMPLE SECOND. Multiply 7| by 9| by business method. Here | of 7 to the nearest unit is 3, and J 7| of 9 to the nearest unit is 7 ; then 3 plus 7 9| is 10, so we simply say, 9 times 7 are 63 and 10 are 73, Ans. 73 EXAMPLE THIRD. Multiply 23^ by 19^ by business method. Here ^ of 23 to the nearest unit is 6, and 23J ^ of 19 to the nearest unit is 6 ; then 6 plus 19 j 6 is 12, so we simply say, 19 times 23 are 437 and 12 are 449, Ans. "^'^^ N. B. In multiplying the whole numbers together, al- ways use the single-line method, ? MULTIPILCATION. 39 EXAMPLE rOURTII. Multiply 128| by 25 by businesb method. Here | of 25 to the nearest unit is 17, so 12^ J wc simply say, 25 times 128 are 3200 and ^^ 17 are 3217, the answer. 3217 PRACTICAL EXAMPLES FOR BUSINESS MEN. 1. What is the cost of 17^ lbs. sugar at 18| cts. per lb.? Here I of 17 to the nearest unit is 13, 17 J and ^ of 18; is 9 13 plus 9 is 22, so we 18| simply say, 18 times 17 are 306 and 22 arc 328, the answer. ^-^"^ 2. What is the cost of 11 lbs. 5 oz. of butter at 33 J cts. per lb.? Here ^ of 11 to the nearest unit is 4, 11^^ and ^ of 33 to the nearest unit is 10 ; 33J then 4 plus 10 is 14, so we simply say, 33 times 11 are 3G3, and 14 are 377, Ans. ^'^'^'^ 3. What is the cost of 17 doz. and 9 eggs at 12^ cts. per doz.? Here J of 17 to the nearest unit is 9, 17^^ and ^Sj of 12 is 9 ; then nine plus 9 is 18, 12^ so we simply say, 12 times 17 are 204 and 18 are 222, the answer. ^^^^ 4. What will be the cost of 15J yds. calico at 12.^ cts. per yd.? Ans. $1.97. N. B. To multiply by aliquot parts of 100, see page 44., D 40 orton's ligiitninu calculator. RAPID PROCESS OF MARKING GOODS. ▲ VALUABLE HINT TO MERCHANTS ANE ALL RETAIL DKALKRS IN FOREIGN AND DOMESTIC DRY GOODS. Retail merchants, in buying goods by wbolc- sale, buy a great many articles by the dozen, such as boots and shoes, hats and caps, and notions of various kinds. Now, the merchant, in buying, for instance, a dozen hats, knows exactly what one of those hats will retail for in the market where he deals ; and, unless he is a good accountant, it will often take him some time to determine whether he can afford to purchase the dozen bats and make a living profit in selling them by the single hat ; and in buying his goods by auction, as the merchant often does, he has not time to make the calculation before the goods are cried off. He therefore loses the chance of making good bargains by being afraid to bid at random, or if he bids, and the goods are cried off, he may have made a poor bar- gain, by bidding thus at a venture. It then be- comes a useful and practical problem to determine instantly what per cent, he would gain if he re- tailed the hats at a certain price. To tell what an article should retail for to make a profit of 20 per cent., Rule. — Divide what the articles cost per dozen hy 10, which is done hy removing the decimal point one place to the left. / MOLTIPLICATION. 41 For instance, if hats cost $17.50 per dozen, re- move the decimal point one place to the left, mak- ing §1.75, what they should be sold for apiece to gain 20 per cent, on the cost. If they cost $31.00 per dozen, they should be sold at $3.10 apiece, etc. We take 20 per cent, as the basis for the following reasons, viz. : because we can determine instantly, by simply removing the decimal point, without changing a figure; and, if the goods would not bring at least 20 per cent, profit in the home mar- ket, the merchant could not afford to purchase and would look for goods at lower figures. Reason. — The reason for the above rule is ob- vious : For if we divide the cost of a dozen by 12, we have the cost of a single article ; then if we wish to make 20 per cent, on the cost, (cost being { or f,), we add the 20 per cent., which is ^, to the |, making f or |J ; then as we multiply the cost, divided by 12, by the |^ to find at what price one must be sold to gain 20 per cent., it is evident that the 12s will cancel, and leave the cost of a dozen to be divided by 10, which is done by re- moving the decimal point one place to the left. .1. If I buy 2 doz. caps at $7.50 per doz., what shall I retail them at to make 20%? Ans. 75 cis. 2. "When a merchant retails a vest at $4.50 and makes 20%, what did he pay per doz.? Ans. $45. 3. At what price should I retail a pair cf boots that cost $85 per doz., to make 20%? Ans. $8.50. 42 orton's lightning calculator. I RAPID PROCESS OF MARKING GOODS AT DIFFERENT PER CENTS. Now, as removing the decimal point one place to the left, on the cost of a dozen articles, gives the selling price of a single one with 20 per cent, added to the cost, and, as the cost of any article is 100 per cent., it is obvious that the selling price would be 20 per cent, more, or 120 per cent.; hence, to find 50 per cent, profit, which would make the selling price 150 per cent., we would first find 120 per cent., then add 30 per cent., by increasing it one-fourth itself; to make 40 per cent., add 20 per cent., by increasing it one-sixth itself; for 35 per cent., increase it one-eighth itself, etc. Hence, to mark an article at any per cent, profit, we have the following GENERAL RULE First find 20 per cent, profit^ hy removing the decimal point one place to the left on the price the articles cost a dozen; then, as 20 per cent, profit is 120 per cent.y add to or subtract from this amount the fractional part that the required per cent, added to 100 is more or less than 120. Merchants, in marking goods, generally take a per cent, that is an aliqot part of 100, as 25^, 33^%, 50%, etc. The reason they do this is be- cause it makes it much easier to add such a per cent, to the cost; for instance, a merchant could MULTIPLICATION. 4Z mark almost a dozen articles at 50 per cent, profit in the time it would take him to mark a single one at 49 per cent. For the benefit of the student, and for the convenience of business men in mark- ing goods, we have arranged the following table : TABLE For Marking all Articles hoiight hy the Dozen. N. B. Most of these are used in business. To make 20% remove the point one place to the left. (( (( 80% (( a and add ^ : itself. t( (t 60% (( a a a a t( a 50% u a a a i( « « 44% (( a (( a u (( t( 40% (( (t a a t( u i( 37^% (( a a a (t u it 35% (( a a i( (i il « 33 J % (( a a a a « « 32% (( a a a A i( u (( 30% a a i( a A i( n <( 28% (( a (( a A <( (< (( 26% u a a a A (( u (( 25% « a a (( A it (t (i 121% a a subtract yV a 11 (( 10|% a (( a A a tt (( 18|% u a a a A a If I buy 1 doz. shirts for $28.00, what shall I retail them for to make 50%? Ans. $3.50. Explanation. — Remove the point one place to the left, and add on \ itself. 44 ORTON S LIGHTNING CALCULATOR. Where the Multiplier is an Aliquot Part of 100. Merchants in selling goods generally make the price of an article some aliquot part of 100, as in selling sugar at 12J cents a pound or 8 pounds for 1 dollar, or in selling calico for 16| cents a yard or 6 yards for 1 dollar, etc. And to be- come familiar with all the aliquot parts of 100, so that you can apply them readily when occasion requires, is perhaps the most useful, and, at the same time, one of the easiest arrived at of all the computations the accountant must perform in the practical calculations of the counting-room. TABLE OF THE ALIQUOT PARTS OF 100 AND 1000 N. B. Most of these are used in business. l^i is \ part of 100. H is ^^ part of 100. 25 8 1 or ^ of 100. IGf is ^^ovloi 100. 37i s 1 part of 100. 33| is i*^or|of 100. 50 18 1 or ^ of 100. 66| is ^j or 1 of 100. 62^ 8 1 part of 100. 83 i isfgorfof 100. 75 ] 8 f or f of 100. 125 is I part of 1000. 87| is I part of 100. 250 is 1 or ^ of 1000. Gi^ 8 tV part of 1<^<^- 375 is 1 part of 1000. 18| s -^^ part of 100. 625 is f part of 1000. '6\\. 8 T^g. part of 100. 875 is I part of 1000. To multiply by an aliquot part of 100, Rule — Add two ciphers to the multiplicand, thp.n take such part of it as the rmdtiplier is part of 100. N. B. If the multiplicand is a mixed number reduce the fraction to a decimal of two places before dividing. COUNTIXG-ROOM EXERCISES. Examples. — 1. Multiply 424 by 25. As 25 = J of 100, divide 42400 by 4 = 10600. N. B. If the multiplicand is a mixed number, reduce the fraction to a decimal of two places before dividing. 2. Give the cost of 12|- yds. cloth @ 18|c. per yd. Process. — 12J = ^; changing 18J to a decimal, we have 18.75 ^ 8 = $2.34f. Note. — Aliquot pnrts may be conveniently used when the mul- tiplier is but little more or less than an aliquot part. 3. Multiply 24 by ITf. 1st. Multiply 24 by 16f (the one-sixth of 100). Thus 24 X 16f = 2400 ^ 6 = 400 As 17| = 16| -f 1 multiply 24 by 1 = 24 Kcncc 24 X 17§ = the two products, 424 45 46 orton's liohtning calculator. Rationale. — As in the last case, by annexing two ciphers, we increase the multiplicand one hun- dred times ; and by dividing the number by 3, we only increase the multiplicand thirty-three and one-third times, because 33 J is one-third of 100. 4. To multiply any number by 333J add three ciphers, and divide by 3. Multiply 4797 by 333J. Product, 1599000. 3)4797000 1599000 5. To multiply any numbpr by 6§ add two ci- phers, and divide by 15 ; or add one cipher and multiply by §. Multiply 156G by 6f . 15)156600 10440 First method. 15660 2 3)31320 10440 Second method. 6. To multiply any number by 66f add three ciphers, and divide by 15 j or add two ciphers and multiply by §. MULTirLICATION. 47 Multiply 36G3 by CG|. 15)3663000 244200 First method. 366300 2 3)732600 244200 Second method. 7. To multiply any number by 8J add two ci- phers, and divide by 12. Multiply 2889 by 8J. Product, 24075. 12^288900 24075 8. To multiply any number by 83J add three ciphers, and divide by 12. Multiply 7695 by 83J. Troduct, 641250. 12)7095000 641250 9. To multiply any number by 6J add two ci- phers, and divide by 16 or its factors — 4X4. Multiply 7696 by 6J. Product, 48100. 4)769600 4)192400 48100 MEASUREMENT OF LUMBER. The unit of board measure is a square foot 1 inch thick. To measure inch boards. Rule. — Multiply the length of the board in feet by its breadth in inches, and divide the product by 12 ; the quotient is the contents in square feet. Note. — ^When the board is wider at one end than the other, add the width of the two ends together, and take half the sum for a mean width. Example. — How many square feet in a board 10 feet long, 13 inches wide at one end, and 9 inches wide at the other ? Process. — (13 + 9) -I- 2 = 11 (mean width) then 10 length X 11 = 110 -f. 12 = 9^ feet. Ans. 48 MEASUREMENT OP LUMBER. 49 Sawed lumber, as joists, plank, and scantlings, are now generally bought and sold by board mea- sure. The dimensions of a foot of board measure are 1 foot long, 1 foot wide, and 1 inch thick. To ascertain the contents (board measure) of boards, scantling, and plank. Rule. — Multiply the width in inches by the thickness in inches, and that product by the length in feet, which last product divide by 12. Example. — How many feet of lumber in 14 planks 16 feet long, 18 inches wide, and 4 inches thick ? Process. — 16 feet X 18 inches X 4 inches = 1152, then 1152 -T- 12 = 96 feet == (one plank) X 14 = 134 feet. Ans. 2. To ascertain the quantity of lumber in a log, or its board measure. Rule.* — Multiply the diameter in inches at the small end by one-half the number of inches, and this product by the length of the log in feet, which last product divide by 12. Example. — How many feet of lumber can be made from a log which is 36 inches in diameter, and 10 feet long? Solution.— 36 X 18 = 648 ; 648 X 10 = 6480 ; 6480 -T- 12 = 540. Ans. * The above rule is the one used in the great pineries of Northern Michigan, Wisconsin, and Minnesota. 50 ORTON S LIGHTNING CALCULATOR. o O O O O ^ o 5* o — ■ >-' F t>0 J' CO » t^ % Oi s •"l 1 •^ 3 "^ 3 •^ D •^ • 3 •^ t £ ■a (I t -l „. «. 1 — . B — O —. 5' .^ n w. B 1— 5* a o" p "a a — = o B 5* a o* 3 f-i a 3 M 3 3 s "2. S 04 s (K 3 oq 3 05 3 0? ft o ,-(■ n ^4- n o S" 3* s- B" 3* p p 3* 3- o o r cn m 15 w 15 15 108 15 15 m 135 90 99 117 126 15 IG 100 16 110 16 120 16 130 16 140 16 150 17 125 17 137 17 150 17 1()0 17 175 17 187 18 155 18 170 18 186 18 201 18 216 18 232 19 105 19 176 19 198 19 214 19 230 10 247 20 172 20 189 20 206 20 2(W 20 246 20 258 21 1H4 21 202 21 220 21 238 21 256 21 276 22 194 22 212 22 232 22 263 22 294 22 291 2;i 219 23 24(t •i:i 278 23 315 23 ;W2 23 3;J3 24 250 24 276 24 3t)0 24 325 24 350 24 375 25 280 25 308 25 336 25 3<^ 25 392 25 420 2(i 299 2(i 323 26 34(i 26 375 26 404 26 448 27 327 27 367 27 392 27 425 27 457 27 490 28 3(i0 28 396 28 4.32 28 462 28 604 28 540 2'J 370 29 414 29 451 29 488 29 526 29 6(^ 1 :w 412 3(J 452 30 494 30 .5.35 30 670 30 618 31 428 31 471 31 513 31 558 31 602 31 642 32 451 32 496 32 641 32 587 32 631 32 676 33 490 3,3 5.39 3;i 588 33 6:37 33 686 33 735 34 5;{2 34 585 |34 638 .34 691 34 744 34 798 35 682 .io 640 35 698 35 7.52 35 805 45 8ta ,30_ 593 36 657 36 717 36 821 36 836 36 889 A Log Table. — Showing the number of feet of boards any log will make whose diameter is from 15 to 36 inches at the smallest end, and from 10 to 15 feet in length. MEASUREMENT OF WOOD. Wood is measured by the cord, which contains 128 cubic feet. Wood is bought and sold by the cord and frac- tions of a cord. Pine and spruce spars from 10 to 4 inches in diameter inclusive, are measured by taking the dia- meter, clear of bark, at one-third of their length from the large end. Spars are usually purchased by the inch diame- ter ; all under 4 inches are considered poles. Spruce spars of T inches and less, should have 5 feet in length for every inch in diameter. E 61 52 orton's lightning calculator. Note. — A pile of wood that is 8 feet long, 4 feet higL, and 4 feet wide, contains 128 cubic feet, or a cord, and every cord contains 8 cord-feet; and as 8 is y'g of 128, every cord-foot contains 16 cubio feet ; therefore, dividing the cubic feet in a pile of wood by 16, the quotient is the cord-feet ; and if cord-feet be divided by 8, the quotient is cords. Note. — If we wish to find the circumference of a tree, which will hew any given number of inches square, we divide the given side of the square by .225, and the quotient is the circumference re- quired. What must be the circumference of a tree that will make a beam 10 inches square ? Note. — When wood is " corded" in a pile 4 feet wide, by multiplying its length by its hight, and dividing the product by 4, the quotient is the cord- feet ; and if a load of wood be 8 feet long, and its hight be multiplied by its width, and the product divided by 2, the quotient is the cord-feet. How many cords of wood in a pile 4 feet wide, 70 feet 6 inches long, and 5 feet 3 inches high ? Note. — Small fractions rejected. To find how large a cube may be cut from any given sphere, or be inscribed in it. B-ULE. — Square the diameter of the sphere, divide that product hy 3, and extract the square root of the quotient for the answer^ MENSURATION OR PRACTICAL GEOMETR?". 53 I have a piece of timber, 30 inches in diameter ; how large a square stick can be hewn from it? Rule. — Multiply the diameter hy .7071, and the product is the side of a square inscribed, I have a circular field, 360 rods in circumference; what must be the side of a square field that shali contain the same quantity? Rule. — Multiply the circumference hy .282, and the product is the side of an equal square. I have a round field, 50 rods in diameter; what is the side of a square field that shall contain the same area? Ans. 44.31 135-{- rods. Rule. — Multiply the diameter hy .886, and th« product is the side of an equal square. There is a certain piece of round timber, 30 inches in diameter ; required the side of an equi- lateral triangular beam that may be hewn from it. Rule. — Multij^ly the diameter hy .866, and the product is the side of an inscribed equilateral tri- angle. To find the area of a globe or sphere. Definition. — A sphere or globe is a round solid body, in the middle or center of which is an imag- inary point, from which every part of the surface is equally distant. An apple, or a ball used by children in some of their pastimes^ may be called a sphere or globe. ROUND TBIBER. Round timber, when squared, is estimated to lose one-fifth ; hence (50 cubic feet, or) a ton of round timber is said to contain only 40 cubic feet. Round, sawed, and hewn timber is bought and sold by the cubic foot. To measure round timber. Rule.* — Take the girth in feet, at both the large and small ends, add them, and divide their sum by two for the mean girth ; then multiply the length in feet by the square of one-fourth of the mean girth, and the quotient will be the contents in cubic feet, according to the common practice. * This rule gives ti\io\ii fmir-ffths of the true contents, one- fifth being allowed to the buyer for waste in hewing. 54 TIMBER MEASURE. 55 Example. — What are the cubic contents of a round log 20 feet long, 9 feet girth at the large end, and t feet at the small end ? Solution. — 9 + 7 = 16-7-2 = 8 mean girth. Then 20 length x 4 feet (the square of ^ mean girth) = 80 cubic feet. Aiis. Note. — If the girth be taken in inches, and the length in feet, divide the last product by 144. Example. — What are the cubic contents of a round log 12 feet long, 50 inches girth at the large end, 38 inches at the small end? Work.— 50 + 38 = 88 -^ 2 = 44 mean girth. Then 12 length X 121 inches (the square of i mean girth) = 1452 -^ 144 = lOjV cubic feet. To measure round timber as the frustum of a cone: that is, to measure all the timber in the log. Rule. — Multiply the square of the circumference at the middle of the log in feet by 8 times the length, and the product divided by 100 will be the contents. Extremely near the truth. Note. — The above rule makes 1 foot more timber in every 190 cubic feet a log contains if ciphered out by the long and tedious rules of Geometry. It is therefore suflScicntly correct for all practical pur- poses, and this rule being so short and simple in comparison with all others, every lumberman, ship- builder, carpenter, inspector or surveyor of timber, should post it up for reference and use. E* 56 ORTON S LIOHTNINQ CALCULATOR. A TABLE FOR MEASURING TIMBER. Quarter Girt. Area. Quarter Girt. Area. Quarter Girt. Area. Incbea. Feet. Inches. Feet. luches. Feet. 6 .250 12 1.000 18 2.250 6} .272 12} 1.042 18^ 2.376 6^ .294 12^ 1.085 19 2.506 6f .317 121 1.129 19^ 2.640 7 .340 13 1.174 20 2.777 n .364 13} 1.219 20^ 2.917 7J .390 13J 1.265 21 3.062 7| .417 13f 1.313 21J 3.209 8 .444 14 1.361 22 3.362 8} .472 14} 1.410 22^ 3.516 8i .501 14* 1.460 23 3.673 81 .531 141 1.511 23J 3.835 9 .562 15 1.562 24 4.000 n .594 15} 1.615 24^ 4.168 n .626 15J 1.668 25 4.340 9| .659 151 1.722 25J 4.516 10 .694 16 1.777 26 4.694 10} .730 16} 1.833 26J 4.876 m .766 16* 1.890 27 5.062 101 .803 161 1.948 27J 5.252 11 .840 17 2.006 28 5.444 lU .878 17} 2.066 28^ 5.640 m .918 17^ 2.126 29 5.840 111 .959 17f 2.187 29J 30 6.044 6.250 To measure round timber by the table. Multiply the area corresponding to the quarter- girt in inches by the length of the log in feet. TIMBER MEASURE. 57 Note. — If the quarter-girt exceed the table, take half of it, and four times the contents thus formed will be the answer. EXAMPLE 1. If a piece of round timber be 18 feet long, and the quarter girt 24 inches, how many feet of timber are contained therein? 24 quarter girt. 24 96 48 By the Tahle. 576 square. 18 Against 24 stands 4 00 Length, 18 4608 576 Product, 72.00 Anc? 72 fppt 144)10368(72 feet 1008 XJlIIO* 1 ^ Iv/VfUs 288 288 This table gives the customary, but only a.hout four Jifths of the true contents, one-fifth being allowed the buyer for waste in hewing or sawing to make the timber square. The following rule gives the true contents : — Multiply square of girth by .08 times length. In the above Example the whole girth is 8 feet, s.iuared is 64 x (.08 X 18 length) = 92.16 feet. 68 orton's lightning calculator. 1. Of Flooring. Joists are measured by multiplying tlieir breadtli b^ their depth, and that product by their length. They receive various names, according to the posi- tion in which they are laid to form a floor, such as trimming joists, common joists, girders, binding joists, bridging joists and ceiling joists. Girders and joists of floors, designed to bear great weights, should be let into the walls at each end about two-thirds of the wall's thickness. In boarded flooring, the dimensions must be taken to the extreme parts, and the number of squares of 100 feet must be calculated from these dimensions. Deductions must be made for stair- cases, chimneys, etc. Example 1. If a floor be 57 feet 3 inches long, and 28 feet 6 inches broad, how many squares of flooring are there in that room ? By Decimals. 57.25 28.5 28G25 45800 11450 By Duodccimah F. I. 57 : 3 28 : 6 456 114 28 : 7 : 6 7 ; : 100)1631.625 feet. Squares 16.31625 16:31 : 7 : 6 Am. IQ squares and 31 feet. SQUARE TIMBER. To measure square timber. Rule. — Multiply the breadth in feet by the depth in feet, and that by the length in feet, and the quotient will be the contents in cubic feet. Example. — How many cubic feet in a square log 12 feet long by 2 feet broad and IJ feet deep ? Explanation. — 2 feet breadth x 1^ feet depth X 12 feet length = 36 cubic feet. Ans. Note. — If the breadth and depth be taken in inches, divide the last product by 144. Example. — How many cubic feet in a square log 24 feet long, 30 inches broad, and 20 inches deep ? Solution. — 30 inches breadth x 20 inches depth X 24 feet length = 14400 -k- 144 = 100 cubic feet. 59 60 obion's lightninq calculatoe. PROBLEM III. To find the solid contents of squared or four-sided Timber. By the Carpenters^ Rule, As 12 on D : length on c : Quarter girt on D : solidity on c. Rule I. — Multiply the breadth in the middle by the depth in the middle^ and that product by the length for the solidity. Note. — If the tree taper regularly from one end to the other, half the sum of the breadths of tho two ends will be the breadth in the middle, and half the sum of the depths of the two ends will bo the depth in the middle. Rule II. — Multiply the sum of the breadths of the two ends by the sum of (he depths, to which add the product of the breadth and depth of each end; one-sixth of this sum multiplied by the length, will give the correct solidity of any piece of squared tim- ber tapering regularly. PROBLEM IV. To find how much in length will make a solid foot, or any other asssigned quantity, of squared timber^ of eqtial dimensions from end to end. Rule. — Divide 1728, the solid inches in a foot, or the solidity to be cut off, by the area of the end in inches, and the quotient will be the length in inches. TIMBER MEASURE. 61 Note. — To answer the purpose of the above rule, some carpenters' rules have a little table upoii them, in the following form, called a iahle of tun- her measure. r^ |0|9|0 11 |3 9 inches. 144 36 16 |9|5|4 2 |2 1 feet. 1 1 2 3 |4|5|6 7 |8 9 side of the square. This table shows, that if the side of the square be 1 inch, the length must be 144 feet ; if 2 inches be the side of the square, the length must be 36 feet, to make a solid foot. MEASUKEMENT OF HAY. The only correct mode of measuring hay is to weigh it. This, on account of its bulk and cha- racter, is very diflScult, unless it is baled or other- wise compacted. This difficulty has led farmers to estimate the weight by the bulk or cubic con- tents, a mode which is only approximately correct. Some kinds of hay are light, while others are heavy, their equal bulks varying in weight. But for all ordinary farming purposes of estimating the amount of hay in meadows, mows, and stacks, the following rules will be found sufficient : — As nearly as can be ascertained, 25 cubic yards of average meadow hay, in windrows, make a ton. 62 MEASUREMENT OF HAY. 63 When loaded on wagons, or stored in barns, 20 cubic yards make a ton. When well settled in mows, or stacks, 15 cubic yards make a ton. Note. — These estimates are for medium-sized mows or stacks ; if the hay is piled to a great height, as it often is where horse hay-forks are used, the row will be much heavier per cubic yard. When hay is baled, or closely packed for ship- ping, 10 cubic yards will weigh a ton. To find the number of tons in long square stacks. Rule.— Multiply the length in yards by the width in yards, and that by half the altitude in yards, and divide the product by 15. Example. — How many tons of hay in a square stack 10 yards long, 5 wide, and 9 high ? Solution.— 10 X 5 X 4J = 225 -?- 15 == 15 tons. Ans. To find the number of tons in circular stacks. Kule. — Multiply the square of the circumference in yards by 4 times the altitude in yards, and di- vide by 100 ; the quotient will be the number of cubic yards in the stack j then divide by 15 for the number of tons. Example. — How many tons of hay in a circular stack, whose circumference at the base is 25 yards, and height 9 yards ? F 64 orton's lightning calculator. Solution. — 25 x 25 = 625, the square of the circumference ; then 625 x 36 (four times the length), = 225000 ^ 100 = 225 (the number of cubic yards), then 225 H- 15 = 15, the number of tons. An easy mode of ascertaining the value of a given number of lbs. of hay, at a given price per ton of 2000 lbs. Rule. — Multiply the number of pounds of hay (coal, or anything else which is bought and sold by, the ton) by one-half the price per ton, pointing off three figures from the right hand ; the remain- ing figures will be the price of the hay (or any article by the ton). Example. — What will 658 lbs. of hay cost, @ $1 60 per ton ? Solution. — $T 50 divided by 2 equals $3 75, by which multiply the number of pounds, thus : 658 x $3 16 = 246. Y50, or $2 46. Ans. Kote. — The principle in this rule is the same as in interest — dividing the price by two gives us the price of half a ton, or 1000 lbs. ; and pointing off three figures to the right is dividing by 1000. A truss of hay, new, is 60 lbs. ; old, 56 lbs. ; straw, 40 lbs. A load of hay is 36 trusses. A bale of hay is 300 lbs. RULES FOR DETERMINING THE WEIGHT OF LIYE CATTLE. For cattle of a girth of from 5 to T feet, allow 23 lbs. to the superficial foot. For cattle of a girth of from t to 9 feet, allow 31 lbs. to the superficial foot. For small cattle and calves of a girth of from 3 to 5 feet, allow 16 lbs. to the superficial foot. For pigs, sheep, and all cattle measuring less than 3 feet girth, allow 11 lbs. to the superficial foot. Measure in inches the girth round the breast, just behind the shoulder-blade, and the length of the back from the tail to the forepart of the shoul- der-blade. Multiply the girth by the length, and divide hj 144 for the superficial feet, and then mul- 65 66 orton's lightning calculator. tiply the superficial feet by the number of lbs. allowed for cattle of different girths, and the pro- duct will be the number of lbs. of beef, veal, or pork, in the four quarters of the animaL To find the number of stone, divide the number of lbs. by 14. Example. — What is the estimated weight of beef in a steer, whose girth is 6 feet 4 inches, and length 5 feet 3 inches ? Solution. — *IQ inches girth, x 63 inches length, = 4188 -4- 144 = 33J square feet, X 23 = Y64| lbs., or 54f stone. Ans. Note. — When the animal is but half fattened, a deduction of one lb. in every 20 must be made ; and if very fat, one lb. for every 20 must be a^ded. Where great numbers of cattle are anaually bought and sold under circumstances that forbid ascertaining their weight with positive accuracy, the estimated weight may be thus taken with ap- proximate exactness — at least with as much accu- racy as is necessary in the aggregate valuation of stock. No rules or tables can, however, be at all times implicitly relied on, as there are many cir- cumstances connected with the build of the animal, the mode of fattening, its condition, breed, &c., that will influence the measurement, and conse- quently the weight. A person skilled in estimat- ing the weight of stock soon learns, however, to make allowance for all these circumstances. TO MEASURE CORN ON THE COB IN CRIBS. Corn is generally put up in cribs made of rails ; but the rule will apply to a crib of any size or kind, whether equilateral, or flared at the sides. IVhen the crib is equilateral. Rule. — Multiply the length in feet by the breadth in feet, and that again by the height in feet, which last product multiply by .63 (the frac- tional part of a heaped bushel in a cubic foot), and the result will be the heaped bushels of ears. For the number of bushels of shelled corn multiply by .42 (two-thirds of .63), instead of .63. i* 67 68 orton's lightning calculator. Example. — Required the number of bushels of shelled corn contained in a crib of ears, 15 feet long, by 5 feet wide, and 10 feet high ? 15 length X 5 width, x 10 height = T50 cubic feet. Then "750 X .63 = 4^2.50 heaped bushels of ears. Also T50 X .42 = 315 bushels of shelled corn. In measuring the height, of course, the height of the corn is intended. And there will be found to be a difference in measuring corn in this mode, be- tween fall and spring, because it shrinks very much in the winter and spring, and settles down. Wlien the crib is flared at the sides. Rule. — Multiply half the sum of the top and bot- tom widths in feet by the perpendicular height in feet, and that again by the length in feet, which last product multiply by .63 for heaped bushels of ears, and by .42 for the number of bushels of shelled corn. Note. — The above rule assumes that three heap- ing half bushels of ears make one struck bushel of shelled corn. This proportion has been adopted upon the authority of the major part, of our best agricultural journals. Nevertheless, some journals claim that two heaping bushels of ears to one of shelled corn is a more correct proportion, and it is the custom in many parts of the country to buy MULTIPLICATION AND DIVISION. 69 and sell at that rate. Of course much will de- pend upon the kind of corn, the shape of the ear, the size of the cob, &c. Some samples are to be found, three heaping half bushels of which will even overrun one bushel shelled ; while others again are to be found, two bushels of which will fall short of one bushel shelled. Every farmer must judge for himself, from the sample on hand, whether to allow one and a half or two bushels of ears to one of shelled corn. In either case, it is only an approximate measurement, but sufficient for all ordi- nary purposes of estimation. The only true way of measuring all such products is by weight. Multiplication and Division, To multiply one-half, is to take the multiplicand one-half of one time; that is, take one-half of it, or divide it by 2. To multiply by J, take a third of the multipli* cand, that is, divide it by 3. To multiply by f , take J, first, and multiply that by 2 ; or, multiply by 2 first, and divide the pro- duct by 3.* ^Sometimes one operation is preferable, and soinetimeg the other; good judgment alone can decide when the case is before us. 70 ORTON's LIQHTINa CALCULATOR. EXAMPLES. 1 . What will 360 barrels of flour come to at 5 J dollars a barrel. At 1 dollar a barrel it would be 360 dollars ; at 5 J dollars, it would be SJ times as much, 360 5 times, 1800 J of a time, 90 Ans. $1890 Before we attempt to divide by a mixed number, such as 2J, 3J, 5f , etc., we must explain, or rather observe the principle of division, namely: That the quotient will he the same if we multiply the divi- dend and divisor hy the same niimher. Thus 24 divided by 8, gives three for a quotient. Now, if we double 24 and 8, or multiply them by any num- ber wnatever, and then divide, we shall still have 3 for a quotient. 16)48(3; 32)96(3, etc. Now, suppose we have 22 to be divided by 5J ; we may double both these numbers, and thus be clear of the fraction, and have the same quotient. 5J)22(4 is the same as 11)44(4. How many times is IJ contained in 12? An^. Just as many times as 5 is contained in 48. The 5 is 4 times 1 J, and 48 is 4 times 12. From these observations, we draw the following rule for divid- ing by a mixed number. MULTIPLICATION AND DIVISION. 71 Rule. — Multiply the whole number hy the lower term of the fraction; add the upper term to the prO' duct for a divisor ; then multiply the dividend by the lower term of thefraction^ and then divide. How many times is li contained in 36? Arts. 30 times. N. B. If we multiply both these numbers by 5, they will have the same relation as before, and a quotient is nothing but a relation between two numbers. After multiplication, the numbers may be considered as having the denomination of fifths. How many times is \ contained in 12 ? Ans. 48 times. One-fourth multiplied by 4, gives 1 ; 12, multi- plied by 4, gives 48. Now, 1 in 48 is contained 48 times. Divide 132 by 2J. Ans. 48. Divide 121 by \^. Am. 8 How many times is f contained in 3 ? Ans. 4 times. By a little attention to the relation of numbers, we may often contract operations in multiplication. A dead uniformity of operation in all cases indi- cates a mechanical and not a scientific knowledge of numbers. As a uniform principle, it is much easier to multiply by the small numbers, 2, 3, 4, 5, than by 7, 8, 9. 72 orton's lightning calculator. Multiply 4532 by 639 (63:::z9<7.) 40788 285516 Multiply 4532 by 963 40788 285516 Product, 2895948 Product, 4364316 In both the foregoing examples we multiply the product of 9 by 7, because 7 times 9 are equal to 63. Because 9 is in the place of hundreds in exam- ple 2, the product for the other two figures is set two places toward the right. In this last example we may commence with the 3 units in the usual way ; then that 'product by 2, because 2 times 3 are 6 ; then the product of 3 by 3, which will give the same as the multiplicand by 9. The appearance of the work would then be the same as by the usual method, but would be easier, as we actually multiply by smaller numbers. Multiply 40788 by 497 285516 1998612 20271636 Product of the 7 units. As 7X7=49, multiply the product of 7 by 7. Every fact of this kind, though extremely sim- ple, should be known by all who seek for knowl- edge in figures. MULTIPLICATION AND DIVISION. 73 First multiply by 12, then that product by 12. Multiply by 576 186 (6X3 18.) 3456 10368 107136 Multiply by Commence with 6. (6X3=18.) Multiply 785460 by 14412 9425520 113106240 11320049520 Multiply this last number, 3456, (which is 6 times 576,) by 3, and place the product in the place of tens, and we have 180 times 576. Observe the same principle in the follow- ing examples : 576 Multiply 40788 618 by 497 3456 (7X7=49.) 285516 10368 1998612 355968 Multiply 61524 by 7209 20271636 f^- 553716 4429728 Product, 443646516 Multiply this pro- duct of 9 by 8, because 9 times 8 are 72, and place the product in the place of 100, be- cause it is 7200. 74 orton's lightning calculator. Multiply 1243 by 636 7458 First by 600. 44748 Multiply 7458 by Product, 790548 Multiply 7 8 6 4 by 24 6 This may be done by commencing with the 2; then that product by 2 and 3 ; or we may com- mence with the 6 units, and then that product by 4 ; because 4 times 6 are 24. Multiply 3764 by 199. Take 3764 200 times, and from that product sub- tract 3764. Multiply 764 by 498J. Take 764 500 times, and from that product sub- tract 1^ times 764. Multiply 396 by 21f , or, (which is the same,) 99X87=8700 —86 8613. N. B.— Ninety-nine is J of 396, and 87 is 4 times 2 If. How many times is 125 contained in 2125 ? Same as 250 in 4250 Same as 25 in 425 Same as 50 in 850 Same as 5 in 85 Same as 10 in 170 j that is, 17 times. MULTIPLICATION AND DIVISION. 75 The object of these changes is to give the learner an accurate and complete knowledg;e of numbers and of division ; and the result is not the only object sought for, as many young learners suppose. How many times is 75 contained in 575 ? or di- vide 575 by 75. Am. 7f . Divide 800 by 12J. Quotient, 64. Divide 27 by 16§. Quo. 3 j%%, or If J. A person spent 6 dollars for oranges, at 6 J cents a-piece; how many did he purchase? Ans. 96. "When two or more numbers are to be multiplied together, and one or more of them having a cipher on the right, as 24 by 20, we may take the cipher from one number and annex it to the other with- out affecting the product; thus, 24X20 is the same as 240X2; 286X1300^28600X13; and 350X 70x40=35x7x4X1000, etc. Every fact of this Idnd, though extremely simple, will be very useful to those who wish to he skillful in operation. Note. — If there are ciphers at the right hand either of the multiplier or multiplicand, or of both, they may be neglected to the close of the opera- tion, when they must be annexed to the product. Remarks. —We now give a few examples, for the pur- pose of teaching the pupil how to use his judgment; he will then have learned a rule more valuable than all others. G 76 orton's lightning calculator. Multiplication and Division Combined. When it becomes necessary to multiply two or more numbers together, and divide by a third, or by a product of a third and fourth, it must be lit- erally done if the numbers are prime. For example : Multiply 19 by 13 and divide that product by 7. This must be done at full length, because the numbers are prime ; and in all such cases there will result a fraction. But in actual business the problems are almost all reduceable by short operations ; as the prices of articles, or amount called for, always corresponds with some aliquot part of our scale of computation. And when two or more of the numbers are composite numberSy the work can always be contracted. Example : Multiply 375 by 7, and divide that product by 21. To obtain the answer, it is suffi- cient to divide 375 by 3, which gives 125. The 7 divides the 21, and the factor 3 remains for a divisor. Here it becomes necessary to lay down a plan of operation. Draw a perpendicular li*ie and place all numbers that are to be multiplied together under each other, on the right hand side, and all numbers that are divisors under each other, on the left hand side. MULTIPLICATION AND DIVISION. 77 EXAMPLES. Multiply 140 by 36, and divide that produot by 84. We place the numbers thus : 140 84 36 We may cast out equal factors from each side of the line without affecting the result. In this case 12 will divide 84 and 36 j then the numbers will stand thus : 140 But 7 divides 140, and gives 20, which, multi- plied by 3, gives 60 for the result. Multiply 4783 by 39, and divide that product by 13. ^^ n 3 Three times 4783 must be the result. Multiply 80 by 9, that product by 21, and di- vide the whole by the product of 60x6X14. 00 6 u 9 u In the above divide 60 and 80 by 20, and 14 and 21 by 7, and those numbers will stand canceled as above, with 3 and 4, 2 and 3, at their sides. * Now, the product 3X^X2, on the divisor side, is equal to 4 times 9 on the other, and the remain- ing 3 is the result. 78 orton's lightning calculator. General Rules for Cancellation. Rule 1st. Draw a perpendicular line ; observe this line represents the sign of equality. On the right hand side of this line place dividends only, on the left hand side place divisors only ; having placed dividends on the right and divisors on the left, as above directed. 2d. Notice whether there are ciphers both on the right and left of the line j if so, erase an equal number from each side. 3d. Notice whether the same number stands both on the right and left of the line ; if so, erase them both. 4th. Notice again if any number on either side of the line will divide any number on the opposite side without a remainder ; if so, divide and erase the two numbers, retaining the quotient figure on the side of the larger number. 5th. See if any two numbers, one on each side, can be divided by any assumed number without a remainder j if so, divide them by that number, and retain only their quotients. Proceed in the same manner, as far as practicable, then, 6th. Multiply all the numbers remaining on the rignt hand side of the line for a dividend, and those remaining on the left for a divisor. 7th Divide, and the quotient is the answer. INTEREST, DISCOUNT AND AVERAGE. 79 Note. — If only one number remain on either side of the line, that number is the dividend or divisor, according as it stands on the right or left of the line The figure 1 is net regarded in the operation, because it avails nothing, either to mul- tiply or divide by. Remarks. — This method may not work a great many problems, as they are found in some books, but it will work 90 out of every 100 that ought to be found in books. In a book we might find a problem like this : What is the cost of 21b. 7oz. 13pwt. of tea, at 7s. 5d. per pound. But the person who should go to a store and call for 31b. 7oz. and ISpwt. of tea would be a fit subject for a mad-house. The above problem requires downright drudgery, which every one ought to be able to perform ; but such drudgery never occurs in business. interest; discount, and average. Before entering upon an investigation of the difterent modes of calculating interest, it may be interesting to bestow some attention upon the his- tory of the subject, that we may be better prepared to understand it. 80 ORTON's LIQHTNIXG CALCULATOR. Among the Jews a law existed that they should not take interest of their brethren, though they wero permitted to take it of foreigner**. " Thou shalt not lend upon usury to thy brother: usury of money, usury of victuals, usury of any thing that is lent upon usury ; unto a stranger thou may- est lend upon usury ; but unto thy brother thou Bhalt not lend upon usury." (Deuteronomy xxiii, 19, 20.) After the dispersion of the Jews they wandered through the earth, but they yet remain a distinct people, mixing, but not becoming assim- ilated with the people among whom they reside. Still looking to the period when they shall return to the promised land, they seldom engage in per- manent business, but pursue traflBc, and especially dealing in money ; and if their national policy for- bids their taking interest of each other, they show no backwardness in taking it unsparingly of the rest of mankind. For ages they have been the money lenders of Europe, and we may safely at- tribute to this circumstance the prejudice, in some measure, that still exists even in our own country against such as pursue this business as a profession. The prejudice of the Christian against the Jew has been transferred to his occupation, and from the days of Shakspeare, who painted the inexorable Bhylock contending for his pound of flesh, down to the present time, the grasping money lender, no INTERESl, DISCOUNT, AND AVERAGE. 81 less than the grinding dealer in other matters, has been sneeringly called a Jew. For ages the taking of any compensation what- ever for the use of money was called usury, and was denounced as unchristian ; and we find Aris- totle, the heathen philosopher, gravely contending that as money could not beget money, it was bar- ren, and usury should not be charged for its use. The philosopher forgot that with money the bor- rower could add to his flocks and his fields, and profit by the produce of both. Definition of Terms. Interest is premium paid for the use of money, goods, or property. It is computed by percentage — a certain per cent, on the money being paid for its use for a stated time. The money on which interest is paid is called the principal. The per cent, paid is called the rate ; the prin- cipal and interest added together is called the AMOUNT. When a rate per cent, is stated, without the mention of any term of time, the time is under- stood to be 1 year. The first important step in the calculation of simple interest is the arranging of the time for which it is computed. The student must study the 82 ORTON^S LIGHTNING CALCULATOR. following Propositions carefully, if Le would be expert in this important and useful branch of bus- iness calculations : PROPOSITION 1. If the time consists of years, multiply the principal by the rate per cent., and that product by the number of years. Example 1. — Find the interest of $75 for 4 yejvrs at 6 per cent. Operation. $75 The decimal for 6 per cent, is .06 06. There being two places of decimals in the multiplier, wo 4.50 point oflf two in the product. 4 $18.00 Ans. PROPOSITION 2. If the time consists of years and mx)nths, reduce the time to months, and multiply the principal by the rate per cent, and number of months together, and divide the result by 12. Note. — The work can always be abbreviated at 4, 6, 8, 9, 12, and 15 per cent, by canceling the per cent., or time, or principal, with the common d'Tisor 12. INTEREST, DISCOUNT, AND AVERAGE, 83 Example 2.— Find the interest of $240 for 2 years and 7 months at 8 per cent. First method. Second method : Principal, $240 Per cent., .08 In. for lyr., 19.20 2yrs.-|-7mos., 31mns. 12)595.20 by cancellation. ^40—20 8 rate. 31 time. n 49.60 Ans. $49.60 ^ns. The operation by canceling is^much more brief. "We simply place the principal, rate, and time, on the right of the line, and 12 on the left ; then we cancel 12 in 240, and the quotient 20 multiplied with 8 and 31 gives the interest at once. Note. — After 12 is canceled the product of the remaining numbers is always the interest. PROPOSITION 3. Jf the time consists of years^ montJis, and daySy re- (luce the years to months^ add in the given months^ and 'place one-third of the days to the right of this number y which we multiply by the principal and rate per cent.^ and divide by 12, as before ; or cancel and divide by 12 before multiplying. Example 3. — Find the interest of $231 for 1 year, 1 month, and 6 days, at 5 per cent. 84 orton's lightning calculator. First method. Principal, Per cent., $231 .05 In. for lyr., 11.55 lyr.-f lmo.+6da., 13.2mo. 12)152.460 Second method: by cancellation. 231 prin. 5 rate. m—n n $12,705 Am. $12,705 An%. By the second method we cancel 12 in 132, and multiply the quotient 11 by 5 and 231. Note. — When the principal is $, and the time is in years or months, the interest is in cents ; if the time is in years, months, and days, the interest is in mills, unless the days are less than 3, in which case it would be in cents, as before. Note. — The reason we divide the days by 3 is because we calculate 30 days for a month, and di- viding by 3 reduces the days to the tenth of months. Note. — The three preceding propositions will work any note in interest for any time and at any given rate per cent. Mow to Avoid Fractions in Interest. PROPOSITION 4. If^ when the time consists of years^ months^ and days, are not divisible by 3, you can divide the days hy 3, and annex the mixed number as in Proposition INTEREST, DISCOUNT, AND AVERAOE. 85 3; or if you wish to avoid fractioTis^ you can reduce the time to interest days^ and multiply the principal^ rate and days togethery and divide the result by 36 or its factors^ 4X 9. Note. — la this case as in the preceding, the work can almost always be contracted by dividing the rate or time or principal with the divisor 36. Note. — We use the divisor 36, because we cal- culate 360 interest days to the year. We discard the 0, because it avails nothing to multiply or di- vide by. Example 4. — !Pind the interest of $210 for 1 year, 4 months, and 8 days, at 9 per cent. Year. Months. Days. 1 4 Operation By Prop. 3. $210 .9 18.90 16.2§ 12)307440 8=16.21 months or 488 days. Operation By Prop. 4. $210 9 18.90 488 $25,620 Ans, 36)922320 $25,620 ^rw. We will now work the example by cancellation to show its brevity. 86 orton's lighting calculatoe. Operation hy Cancellation, Time 488 days. 210 m 122 122 210 325.620 Now cancel 9 in 36 goes 4 times, then 4 into 488 goes 122. Now multiply remaining numbers to- gether, thus, 210X-22 and we have the interest at once. When the days are not divisible by 3 we reduce the whole time to days ; then we place the princi- pal rate and time on the right of the line. Now, because the time is in days, we place 36, on the left of the line for a divisor. (Jf the time was months we would place 12 on the left.") Note. — A very short method of reducing time to interest days is to multiply the years by 36 ; add in 3 times the number of months and the tens' figure of the days, and annex the unit figure; but if the days are less than 10 simply annex them. Example 1. — Reduce 1 year, 2 months, and 6 days, to days. Tears. Months. Days. 36Xl-j-''^X2==42 annex 6=426 Am. SIMPLE INTEREST Br CANCELLATION. 87 Example 2. — Reduce 2 years, 3 montlia and 17 days, to interest days. Years. M'ths. Days. Days. 36x2-1-3x34-1=82. annex 7=827 days Aiu, Note. — The student should commit to mei*:ory the multiplication of the number 36 up as far as 9 times 36, and then he can reduoe almost in- stantly years, months, and days, to days. SIMPLE INTEREST BY CANCELLATION, Rule. — Place tJie principal, time, and rate per cent, on the right hand side of the line. If the time consists of years and months, reduce them to months^ and place 12 (the number of months in a year') on the left hand side of the line. Should the iim^ con- iist of months and days, reduce them to days or deci- mal parts of a month. If reduced to days, place 36 071 the left. If to decimals parts of a Trwnthy place 12 only as before. Point off two decimal places when the time is in months, and three decimal places when the time is in days. Note. If the principal contains cents, point oflf four decimal places when the time is in months, and five decimal places when the time is in days. H 88 OETON'S LiaHTNINQ CALCULAIOR. ^OTE. — We place 36 on the left because there ^^ 360 interest days in a year. (^Custom has made this lawful.^ Example 1. — ^What is the interest on $60 for 117 days at 6 per cent? Operation. Here 117X0 must be the $0 answer. 00 Both sixes on tho right cancels 36 on 117 the left, and we have nothing left $1,170 Ans. to divide by. In this case we point off three decimal places be- cause the time is in days. If the time had been 117 months, we would have pointed off but two deci- mal places. Example 2. — What is the interest of $96.50 for 90 days at 6 per cent? Operation. 96.50 9650 0—^0 00—15 15 1.44.750 Ans. Now cancel 6 in 36 and the quotient 6 into 90, and we have no divisor left. Hence 15X96.50 must be the answer. Note — As there are cents in the principal, we point off five decimals ; three for days and two for cents. Pay no attention to the decimal point tintiJ the close of the operation. SIMPLE INTEREST BY CANCELLATION. 89 Example 3.— What is the interest of $480 for 361 days at 6 per cent? ^^0—80 361 ^-H 361 80 $28,880 Ans. Now cancel 6 in 36 and the quotient 6 into 480, and we have no divisor left. Hence 80X361 must be the answer. Example 4.— What is the interest of $720 for 9 months at 7 per cent? n0—eo 60 1^ 9 9 7 540 7 $37.80 Ans. Now cancel 12 in 720 there is nothing left to divide by. Hence 60x9x7 must be the answer. N. B. When interest is required on any sum for days only, it is a universal custom to consider 30 days a month, and 12 months a year ; and, as the unit of time is a year, the interest of any sum for one day is g^^, what it would be for a year. For 2 days, 3|q, etc.; hence if we multiply by the days, we must divide by 360, or divide by 36 and save labor. The old form of this method was to place 360, or 12 and 30, on the left of the line, but using 36 is much shorter. 90 ORTONS LIGHTNING CALCULATOR. WHEN THE DAYS ARE NOT DIVISIBLE BY THREE. Note. — When the time consists of months and days, and the days are not divisible by three, re- dace the time to days. Example 5. — What is the interest of $960 for 1 1 months and 20 days at 6 per cent? Months. Days. Operation. 11 20=350 days. 000—160 350 —36 350 160 6 $56,000 Now cancel 6 in 36 and the quotient 6 into 960, and we have no divisor left. Hence 160X 350 must be the answer. Example 6. — What is the interest of $173 for 8 months and 16 days at 9 per cent? Months. Days. Operation. 8 16=256 days. 173 173 4r-n 64 ^00^64 $11,072 Ans. Now cancel 9 in 36 and the quotient 4 into 256, and we have no divisor left. Hence 64X173 must be the answer. N. B. Let the puj il remember that this is a gen- eral and universal method, equally applicable to any per cent, or any required time, and all other rules must be reconcilable to it; and, in fact, all other rules are but modifications of this. SIMPLE INTEREST BY CANCELLATION. 91 Example 7. — What is the interest on $1080 for 7 months and 11 days at 7 per cent? Months. Days. 7 11=221 days. Operation. ^0^0—30 221 221 30 7 6630 7 H Now $46,410 Ans. cancel 36 in 1080 and we have no divisor lefl, hence 30X221X7 must be the answer. WITH MORE DIFFICULT TIME AND RATE PER CENT. Example 8.— What is the interest of $160 for 19 months and 23 days at4J per cent? Opera Months. Days. 19 23=593 days, tion. 160—20 593 593 20 H $11,860 A71S. Now cancel 4J in 36 and the quotient 8 into 160 we have no divisor left, hence 20x593 must be the interest. WHEN THE DAYS ARE DIVISIBLE BY THREE. Kule. — Place one-third of the days to the rigid 'if the months, and place 12 on the left of the line. 92 ORTCN S LIGHTNING CALCULATOR. Example 11. — What is the interest of $350 for 3 years 7 months and 6 days at 10 per cent? Years. Months. Days. Operation. 350 It 10 6=43.2 months. 350 36 12600 10 $126,000 Am, Now cancel 12 in 432 and \fe have no divisor left. Hence 350X36X10 equals the interest. Example 12.— What is the interest of $241 for 13 months and 9 days at 8 per cent? Months. Days. 13 9=13. 3 months. Operation. 241 z-n 13.3 241 133 32053 2 3)64106 $21.368f^«5. In this example I canceled 8 and 12 by 4, and then multiplied all on the right of the line and di- SIMPLE INTEREST BY CANCELLATIO.V. 93 vided by 3. If I could have divided by 3 before multiplying I would have saved labor, but when the numbers are prime the whole work must be liter' ally done. Closing Remarks. — We have now fully ex- plained the canceling system of computing inter- est. Any and every problem can be stated by this method, and the beauty and simplicity of the system ranks it high among the most important abbreviations ever discovered by man. As we have before remarked, at 6, 4, 8, 9, 12, 15, and 4^ per cents., every problem in interest can be canceled, besides a great many can be abbreviated at 5, 7, and other per cents.; and after the problem has been stated and we find that we can not cancel, what have we done? We have simply stated the prob- lem in its simplest and easiest form for working it by any other method. Hence we have a decided advantage of all notes that will cancel, and if we can not cancel we have stated the problem in its correct and proper form for going through the whole work ; but it is only when the principal, time, and rate per cent, are all prime, that the WHOLE work must be literally done. At 6 per cent, we can cancel through, and 6 is the rate most commonly used 91 orton's lightning calculator. SHORT PRACTICAL RULES, DEDUCED FROM THE CANCELING SYSTEM, For calculating interest at 6 per cent.j either for months, or months and days. To find the interest for months at 6 per cent. Rule. — Multiply the principal hy half the num- ber of months, expressed decimally as a per cent., that is, for 12 months, multiply hy .06 ; for 8 months, multiply hy .04. Note 1. — It is obvious that if the rate per cent, were 12, it would be 1 per cent, a month ; if, there- fore, it be 6 per cent., it will be a half per cent, a month J that is, half the months will be the per cent. Note 2. — If any other per cent. U wanted you can proceed as above, and then multiply by the given rate per cent, and divide by 6, and the quo- tient is the interest. 1. WhLt is the interest of $368 for 8 months? $368 .04=half the months. %14:.72=Ans. Note 3. — When the months are not even; that is, will not divide by 2, multiply one-half the principal SHORT PRACTICAL RULES. 95 by tlie whole number of months, expressed deci* mally. To find the interest of any sum at 6 per cent, per annum for any number of months and days. Rule. — Divide the days hy 3 and place the quo- lient to the right of the months; one-half of the nwm- her thus formed multiplied hy the principal^ or one- half of the principal multiplied hy this number J will give tlie interest — 'pointing off three decimal places when the principal is $. 2. What is the interest of $76 for 1 year. 6 months, and 12 days, at 6 per cent? Years. Months. Days. 1 6 12=18.4months— half 9 2. $76 Or, 184 9.2 38=rhalf prin. $6,992 Ans. $6,992 Ans. Note. — Dividing the days by 3 reduces them to the tenth of months. To find the the interest of any sum at 6 per cent, per annum for any number of days. Rule. — Divide the principal hy 6 aiid mtdtiply the quotient hy the number of days; or divide the days by 6 and multiply the quotient hy the principal^ pointing off three decimal places when the principal is $. Note. — Always divide 6 into the number that 96 orton's lightning calculator. will divide without a remainder; if neither one will divide, multiply the principal and days to- gether and divide the result by 6. 3. What is the interest of S240 for 18 days at 6 per cent? 18h-6=3 240-f-6=40 $240 Or, $40=J of prin. 3=:J of the days. 18 $0,720 ^715. $0,720 Ans. 4. What is the interest of $1800 for 72 days at 6 per cent. $1800 Or, $300=J of prin. 12=rJ of the days. 72 $21,600 Ans. $21,600 Ans. Useful Suggestions to the Accountant in Computing Interest at Q 'per cent. If the principal is divisible by 6, always reduce the time to days; then multiply the number of days by one -sixth of the principal. EXAMPLE. 5. Find the interest of $240 for 1 year, 5 months, and 17 days, at 6 per cent. 6)240 lyr, 5raos., 17da.=:527 days. Multiplied by 40 J of prin.=40 $21,080 Ans. SHORT PRACTICAL RULES. 97 If the days are only divisible by 3, multiply one-third of the principal by one-half of the days, 6. What is the interest of $210 for 80 days at 6 per cent. ? $70=J of the principal. 40=1 of the days. $2,800 Ans. When the Rate of Interest is 4 per cent Rule. — Multiply the principal hy one-third the number of months, or hy one-ninth the number of days, and the product is the interest. Note. — This principle is also deduced from the canceling method of computing interest ; the stu- dent can readily see that 4 is J of 12 and ^ of 36, When the Bate of Interest is 9 per cent KuLE. — Multiply the principal by three-fourths the number of months, or one-fourth the number of days, or vice versa. BANKS AND BANKING. A bank is an institution or corporation for the purpose of trafficking in money. Banks receive money on deposit, loan money on interest, and issue bank-notes, i. e., notes payable in specie to the bearer on demand, and which cir- culate as money. A promissory note, nfore commonly called a notej is a written promise to pay a specified sum of money. A person who indorses a note incurs all the obli- gations of such an indorsement, even though he may be ignorant of them at the time.* * Many a man has been reduced from affluence to poverty, by merely writing his name on the back of a note "just to accom- modate a friend." 98 BANKERS METHOD CP COMPUTING INTEREST. 99 BANKERS' METHOD or COMPUTING INTEREST, AT 6 PER CENT. FOR ANT NUMBER OP DAYS. Rule. — Draw a perpendicular UnCj cutting off the two right hand figures of the $, and you have the interest of the sum for 60 days at Qper cent. Note. — The figures on the left of the line arc S, and those on the right are decimals of $. Example 1. — What is the interest of 8423 60 days at 6 per cent ? $423=the principal. $4 I 23 cts.=:interest for 60 days. Note. — When the time is more or less than 60 days, first get the interest for 60 days, and from that to the time required. Example 2. — What is the interest of $1 24 for 15 days at 6 per cent.? Days. Days, 15=i of 60 $124:=principal. 4)1 I 24 cts.=interest for 60 days. I 31 cts.=rintercst for 15 days. I 100 orton's lightning calculator. Example 3. — What is the interest of $123.40 for 90 days at 6 per cent.? Days. Days. Days. 90=60-L30 $123.40=principal. 2)1 2340=interest for 60 days. 6170=interest for 30 days. An!f. U I 851z=interest for 90 days. Example 4. — AVhat is the interest of $324 for 75 days at 6 per cent.? Days. Days. Days. $324=:principal. 75=60+ 1 5 4)3 24 cts. interest for 60 days. 81 cts. interest for 15 days. A?is. $4 I 05 cts. interest for 75 days. Kemarks. — This system of Computing Interest is very easy and simple, especially when the days are aliquot parts of 60, and one simple division will suffice. It is used extensively by a large ma- jority of bur most prominent bankers ; and, indeed, is taught by most all Commercial Colleges as the shortest system of computing interest. Method of Calculating at Different Per Cents. This principle is not confined alone to 6 percent, as many suppose who teach and use it. It is their custom first to find the interest at 6 per cent., and from that to other per cents. But it is equally ap- plicable foi a?Z per cents., from 1 to 15 inclusive. bankers' method of computing interest. 101 The following table shows the diiFerent per cents., with the time that a given number of* $ will amount to the same number of cents when placed at interest. Rule. — Draw a perpendicular line, cutting off tlie two right hand figures of $, and you have the interest at the following percents. Interest at 4 per cent, for 90 days. Interest at 5 per cent, for 72 days. Interest at 6 per cent, for 60 days. Interest at 7 per cent, for 52 days. Interest at 8 per cent, for 45 days. Interest at 9 per cent, for 40 days. Interest at 10 per cent, for 36 days. Interest at 12 per cent, for 30 days. Interest at 7-30 per cent, for 50 days. Interest at 5-20 per cent, for 70 days. Interest at 10-40 per cent, for 35 days. Interest at 7 J per cent, for 48 days. Interest at 4J per cent, for 80 days. Note. — The figures on the left of the perpen- dicular line are dollars, and on the right decimals of $. If the $ are less than 10 prefix a 0. Example 1. — What is the interest of ^120 for 15 days at 4 per cent? Days. Havs. $120=:principal. 15= J of 90. 6)1 I 20 cts.-int for 90 days. I 20 cts.— int. for 15 days. 102 orton's lightning calculator. Example 2.— What is the interest of $132 for 13 days at 7 per cent. ? Days. Days. $132=principal. 13=J of 52. 4)1 32 cts.=rint. for 52 days. 33 cts.rziint. for 13 days. Example 3. — What is the interest of $520 for 9 days at 8 per cent. ? Days. Days. $520=principal. 9=^ of 45. 5)5 I 20 cts.=int. for 45 days. $1 I 04 cts.=int. for 9 days. Example 4. — What is the interest of $462 for for 64 days at 7 J per cent. ? Days. Days. Days. $462=principal. 64=4^+16. 3)4 1 62 cts. —int. for 48 days. 54 cts. ir^int. for 16 days. $6 I 16 cts.=int. for 64 days. Remark. — We have now illustrated several ex- amples by the different per cents. ; and if the stu- dent will study carefully the solution to the above examples, he will in a short time be very rapid in this mode of computing interest. Note. — The preceding mode of computing in- terest is derived and deduced from the canceling system ; as the ingenious student will readily see. It is a short and easy way of finding interest for days when the days are even or aliquot parts ; but when they are not multiples, and three or four dV bankers' method op computing interest. 103 visions are ncessary, the canceling system is much more simple and easy. We will here illustrate an example to show the diflference : Required the in- terest of $420 for 49 days at 6 per cent. 2)4 2)2 5)1 3) Bankers' method. 20 cts.=int. for 60 days. 10 cts.=rint. for 30 days. 05 cts.=int. for 15 days. 21 cts.=int. for 3 days. 7 cts.^int. for 1 day. Canceling moth. ^^0—70 ^-H 49 70 $3,430 Ans. $3 I 43 cts.=int. for 49 days. The canceling method is much more brief; we simply cancel 6 in 36, and the quotient G into 420 ; there is no divisor left; hence 70X49 gives the in- terest at once. If the time had been 15 or 20 days, the Bankers' Method would have been equally as short, because 15 and 20 are aliquot parts of 60. The superiority the canceling system has above all others is this : it takes advantage of the principal as well as the time. For the benefit of the student, and for the con- venience of business men, wo will investigate this system to its full extent and explain how to take advantage of the jirincipal when no advantage can be taken of the dai/s. This is one of the most im- portant characteristics of interest, and very oftea saves much labor. It sliould he used when the dayt are not even or aliquot parts. I* 104 ORTON'S LIGHTNING CALCULATOR. The following table shows the different sums of money (at the different per cents.) that bear 1 cent interest a day ; hence the time in days is always the interest in cents ; therefore, to find the interest on any of the following notes at the per cent, at- tached to it in the table, we have the following rule : Rule. — Draw a perpendicular line, cutting off tJie two right hand figures of the days for centSj arid you have the interest for the given time. Interest of $90 at 4 per cent, for 1 day is 1 cent. Interest of $72 at 5 per cent, for 1 day is 1 cent. Interest of $60 at 6 per cent, for 1 day is 1 cent. Interest of $52 at 7 per cent, for 1 day is 1 cent. Interest of $45 at 8 per cent, for 1 day is 1 cent. Interest of $40 at 9 per cent, for 1 day is 1 cent. Interest of $36 at 10 per cent, for 1 day is 1 cent. Interest of $30 at 12 per cent, for 1 day is 1 cent. Interest of $50 at 7.30 per ct. for 1 day is 1 ct. Interest of $70 at 5.20 per ct. for 1 day is 1 ct. Interest of $35 at 10.40 per ct. for 1 day is 1 ct. Interest of $48 at 7 J per. cent, for 1 day is 1 cent. Interest of $80 at 4J per cent, for 1 day is 1 cent. Interest of $24 at 15 per ct. for 1 day is 1 cent. Note. — The 7.30 Government Bonds are calcu- lated on the base of 365 days to the year, amd the 5.20's and 10.40's on the base of 364 days to the year. BANKERS METHOD OF COJirUTirfO INTEREST. 105 Note. — This table should be committed to mem- 01 y, as it is very useful when the days are not even or aliquot parts. If the days are less than 10 pre- fix a before drawing the line. Example 1. — Required the interest of $60 for 117 days at 6 per cent. llT^the days. Here we cut oflF the two $1 I 17 cts. Ans. right hand figures for cents. The student should bear in mind that the inter- est on $60 for 117 days is just the same as the interest on $117 for 60 days. By looking at the table we see that the interest for $60 at 6 per cent, is 1 cent a day ; hence the time in days is the answer in cents. If this note was $120, instead of $60, we would first find the interest for $60, and then double it; if it was $180, we would multiply by 3, etc. Example 2. — Required the interest of $45 for 219 days at 8 per cent. 219=the days. Here we cut off the two $2 I 19 cts. Ans. right hand figures for cents. The student should bear in mind that the inter- est on $45 for 219 days is just the same as the interest on $219 for 45 days. By looking at the table we see that the interest on $45 at 8 per cent, is 1 cent a day ; hence the time in days is the answer in cents. If this note 106 orton's lightning calculator. was $22.50, instead of $45, we would first get the interest for $45, and then divide by 2 ; if it was 875, we would add on f ; if $60, add on i etc. Example 3.— Required the interest of $48 for 115 days at 9 per cent. 115— the days. $48=$40-|-$8. 5)$1 I 15 cts.z=the int. of $40 for 115 days. I 23 cts.=rthe int. of $8 for 115 days. Ans. $1 I 38 cts.=the int. of $48 for 115 days. Here we first find the interest of $40, because the days is the interest in cents; then we divide by 5 to find the interest for $8 ; then by adding both we find the interest for $48, as required. Example. 4 — Required the interest of $260 for 104 days at 7 per cent. $52X5=:$260. 104=the days. $1 04 ctszzzthe int. of $52 for 104 days. A71S. $5 20 cts. Multiply by 5. Here we first find the interest of $52, because the days is the interest in cents ; then we multiply by 5 to get it for $260. We could have worked this note by the Bankers' Method, just as well, by cutting off two figures in the principal, making $2.60 cts. the interest for 52 days, and then multi- ply by 2 to get it for 104 days. %\e student must remember that the interest of $260 for 104 days is just the same as the interest of $104 for 260 days. bankers' aiETnoi) of computing interest. 107 Prohlems Solved hy Both Mctlwds. "We will now solve some examples by both metli- ods, to further illustrate this system, and for the purpose of teaching the pupil how to use his judg- ment. He will then have learned a rule more val- uable than all others. Example 5. — What is the interest S180 for 75 days at 6 per cent.? Operation by taking advantage of the $. 75=the days. $60x3r=S180, $0 I 75 cts.=:the int. of $60 for 75 days. I 3 Multiply by 3. Ans. §2 I 25 cts.=the int. of $180 for 75 days. Operation by the Bankers' Method. 6180=the principal. 60da.-f 15da.=75da. 4)§1 80 cts.=the int. for 60 days. 45 cts.=the int. for 15 days. An$. $2 I 25 cts.r^the int. for 75 days. By the first method we multiplied by 3, because 3X160=^180; by the second method we added on J-, because 60da.-[-'V°da.=:75da, N. B, — When advantage can be taken of both time and principal, if the student wishes to prove his work, he can first work it by the Bankers* Method, and then by taking advantage of the prin- cipal, or vice versa. And as the two operations are entirely difierent, if the same result is obtained by each, he may fairly conclude that the work is correct. 108 ORTON'S LIGHTNINa JALCULATOB. LIGHTNING METHOD OP COMPUTING INTEREST On all notes that hear $12 j)er annum, or any ali^ quotpart or multiple of $12. If a note bears $12 per annum, it will certainly bear $1 per month ; hence the time in months would be the interest in $ ; and the decimal parts of a month would be the interest in decimal part« of a $; therefore when the note bears $12 per annum we have the following rule : Rule. — Reduce tlie years to months, add in the given months, and place one-third of the days to the right of this number, and you have the interest in dimes. Example 1. — Required the interest of $200 for 3 years, 7 months, and 12 days, at 6 per cent. 200 J of 12 days=4. 6 T>. Mo. Da. $12.00=:int. for I yr. 3 7 12rrr43.4mo. Hence 43.4 dimes, or $43.40cts., Ans. "We see by inspection that this note bears $12 interest a year; hence the time reduced to mouths, HGQTNING METHOD OF COMPUTING INT, 109 with one-third of the days to the right, is the in- terest in dimes. If this note bore $6 a year, in- stead of $12, we would take one-half of the above interest; if it bore $18, instead of $12, we would add one-half; if it bore $24, instead of $12, we would multiply by 2, etc. Example 2. — Required the interest of $150 for 2 years, 5 months, and 13 days, at 8 per cent. 150 J of 13 days=:4J. 8 Yr. Mo. r>a. $12.00r=int. for 1 yr. 2 5 13=:r29.4Jmos. Hence $29.4J dimes, or $29.43J cts., A7is. We see by inspection that this note bears $12f interest a year; hence the time reduced to months, with one-third of the days placed to the right, gives the interest at once. Example 3. — Required the interest of $160 for 11 years, 11 months, and 11 days, at 7J per cent. IGO J of 11 days=3|. " Tr. Mo, Da $]2.00=:int.forlyr. 11 11 ll=143.3|mos. Hence $143.3J dimes, or $143.3(;f cts., Ans. When the Interest is more or less than $12 a Year. Rule. — First find the interest for the given time on the hase o/ $12 interest a year; then, if the in- terest on the note is only $G a year^ divide by 2 ; if 110 ORTON's LIGHTNINa CALCULATOR $24 a year, multiply ly 2j if $18 a year, add on one-half, etc. Example 1. — What is the interest of $300 for 4 years, 7 months, and 18 days, at 6 per cent. J of 18 days=6. 300 4yr. 7mo. 18da.=:55.6mo. 6 $] 8.00=int. for 1 year. 2)55.6, int. at 812 a year. US=ll times $12. 278 $83.4 Ans. If the interest was $12 a year, $55.<50 would be the answer ; because 55.6 is the time reduced to months ; but it bears $18 a year, or 1 J times 12 j hence 1^ times 55.6 gives the interest at once. Example 2. — Kequired the interest of $150 for 3 year^, 9 months, and 27 days, at 4 per cent. J of 27 days=9. 150 3yr. 9mo. 27da.— 45.9mo. 4 2)45.9, int. at $12 a year. $6.00=int. for 1 year. $22.95 Ans. $6=^ times $12. If the interest was $12 a year, $45.90 would be the answer j because 245.9 is the time reduced to months; but it bears $6 a year, or J times 12,* hence J- times 45.9 gives the interest at once. MERCHANTS METHOD OF COMPUTING xNT. Ill MERCHANTS' METHOD OP COMPUTING INTEREST. FOR YEARS, MONTHS, AND DATS. The computation of simple interest, where the time consists of years, months, and days, is quite difficult. Taking the aliquot parts for the differ- ent portions of time almost invariably involves the calculator in fractions, and, unless he is well versed in vulgar fractions he will not be able to arrive at the correct result. We have three bases by which we compute interest at different rates per cent, and by which we are enabled to entirely avoid the use of fractions. These three bases are each obtained different from the other, and consequently we have three rules for computing interest : one at a base of one per cent., a second at a base of twelve per sent., and a third at a base of thirty-six per cent. KuLE for computing interest at 1 per cent. : Take one-tJurd of the number of days and annex to the numher of months ; divide the nnmher thm formed hy \2\ annex the quotient thus obtained to the number of years^ and multiply tJie principal by this number ; if the principal contains cents, point off five decimal pluccs ; if not^ point off three deci- 112 ORTONS' LIGHTNING CALCULATOR. mal places; this will give the interest at one per cent. For any other rate per cent.^ multiply the in- terest at one per cent, hy the required rate per cent. Remark. — This rule applies to all problems in interest where the days are divisible by 3, and this number, annexed to the number of months, divisi- ble by 12. EXAMPLE. Required the interest on $112, at 1 per cent., for 3 years, 3 months and 18 days. SOLUTION. Take one-third of the number of days, J of 18 =6, annex this number to the months given, 36, divide this number by 12, 36-i-12=3, annex this number to the year gives, 33, multiply the princi- pal by 33, $112X33=3.G9 6, point off three deci- mal places, and we have the required interest, $3.69 6. EXAMPLE. Required the interest on $125 12, at 7 per cent., for 2 years, 8 months and 12 days. SOLUTION. Take one-third of the number of days, J of 12=::4, annex this number to the number of months we have 84, divide this number by 12, MEIICUANTS' METUOD OF COMPUTING INT. 113 8-1 T- 12=7, annex this number to the $125 12 number of years we have 27, multiply 27 the principal by this number, and point off five decimal places, and you 3.37824 have the interest at one per cent.; mul- 7 tiply this interest by 7, and you have the interest at 7 per cent., the required $23 .6-1768 rate. EXAMPLE. Required the interest on $1,023, at 8 per cent., for 1 year, 9 months and 18 days. SOLUTION. Take one-third the number of days and annex to the number of months, J of 18=6, we have 96-^12=8, annex this number to the years $1023 we have 18, multiply the principal by 18 this number, and point of three decimal places, which gives the interest at 1 per $18 .414 cent.; multiply the interest at one per 8 cent, by 8, and you have the required in teiest. $147 .312 Remarh. — This rule will apply to all problems in interest if one-third of the number of the days be taken decimally and annexed to the number of months, and this number, divided by 12, carried out decimally. But this makes the multiplier very large ; hence, to avoid this large number in 114 orton's lightning calculatcr. the multiplier, where the days are divisible by 3, and this number, annexed to the months, is not divisible by 12, we use the following rule, called our base at 12 per cent. : lluLE. — Reduce the years to months^ add in the months, taJce one-third of the number of days and annex to this number, midtiply the principal by the number thus formed; if there are cents in the prin- cipal, point off five decimal places ; if there are no cents in the principal, point off three decimal places ; this gives the interest at 12 per cent. For any other rate per cent., take such part of the base before mul- tiplying as the required rate is a part of 12. EXAMPLE. Required the interest on S123, at 12 per cent., for 2 years, 2 months and six days. SOLUTION. Reduce the 2 years to months gives us 24 months, add on the 2 months gives us 26 months, take one-third of the days, J of $123 6=2, annexed to the 26 months gives 262 262, which constitutes the base ; multiply the principal by this base, and you have $32 .226 the intcresi at 12 per cent. EXAMPLE. Required the interest on $144, at 6 per cent., for 4 years, 5 months and 12 days. MEECUANTS* METHOD OP COMPUIINQ INT. 115 SOLUTION. Reduce the 4 years to months gives 48 months^ add in the 5 months gives 53 months, take one- third of the days and annex to the number of months, J of 12=4. annex to the 53 months, 534 ; this number multiplied into the principal would give the interest at 12 per cent. But wc want it at 6 per cent. We will now take such part of either principal or base as 6 is a part of 12 ; 6 is J of 12, therefore we will take ^ of 144=72 one-half of the principal, and mul- «» 534 tiply it by the base, which will give the interest at 6 per cent. $38,448 EXAMPLE. Required the interest on $347 25, at 8 per cent., for 2 years, 3 months and 9 days. SOLUTION. Reduce the 2 years to months, 24 months, add the 3 months, 27 months, take one-third of the days, J of 9=3, annex to the months, 273, the base; this, multiplied into the principal, would give the interest at 12 per cent. But we want the interest at 8 per cent ; we will take two -thirds of the base before multiply- $347 25 ing: f of 273=182; the principal 182 multiplied by this number gives the interest at 8 per cent. $63.19950 Remark. — This base is used where the days are divisible by 3, and the number formed by annex- 116 ©ETON'S LIGHTNING CALCULATOR. ing one-third of the days to the months not divisi- ble by 12. We now come to time in which neithei days nor months are divisible. Where such time as this occurs, we uSe a base at 36 per cent. Rule. — Reduce the time to days^ hy multiplying the years hy 12, adding in the months^ if any, and multiplying this number hy 30, adding in the days, if any; multiply the principal hy this number, pointing off 5 decimal places, where cents are given in the principal, and 3 places where no cents are given. This will give the interest at 36 per cent. EXAMPLE. Required the interest on $144, at 36 per cent., for 3 years, 2 months and 2 days. SOLUTION. Reduce the time to days gives 1142 $144 days ; multiply the principal by this base, 1142 and you have the interest at 36 per cent $164,448 EXAMPLE. Required the interest on $144, at 9 per cent., lor 5 years, 7 months and 5 days. SOLUTION. Reduce the time to days gives 2,015 days ; if we multiply the principal by this base, we would get the interest at 36 per cent.; but we want it at 9 per cent. We can take such part of either merchants' method of computing int. 117 principal oi 6ase as 9 is a part of 36 before multi- plying ; 9 is J- of 36 ; we will take J^ of tlie prin- cipal, it being divisible by 4 ; J of 144=36, 2015 which, multiplied into the base, will give 36 the interest at 9 per cent., by pointing off 3 decimal places. $72,540 EXAMPLE. Bequired the interest on $875 15, at 6 per cent., for 5 years, 7 months and 12 days. SOLUTION. Reduce the time to days gives 2022 days ; 6 h J of 36 ; take one sixth of the base, J of 2022=337; multiply the prin- $875 15 cipal by this number, point off" 5 dec- 337 imal places, and you have the interest at 6 per cent., the required rate. $294.92555 Remark. — We have now fully explained our method of computing interest at the three different bases. Any and every problem in interest can be solved by one of these three bases. Some prob- lems can be solved easier by one base than another. Where the days are divisible by 3, and their num- ber, annexed to the months, divisible by 12, it is the shortest and best method to use the base at 1 per cent. By using one or the other of these three bases, the student can avoid the use of vulgar fractions. The student must study these three principles carefully, and learn to adopt readily the base best suited to the problem to be solved. 118 orton's LianTiNQ calculator. PARTIAL PAYMENTS To compute interest on notes, bonds, and mort- gages, on whicli partial payments have been made, two or three rules are given. The following is called the common rule, and applies to cases where the time is short, and payments made within a year of each other. This rule is sanctioned by custom and common law; it is true to the principles of simple interest, and requires no special enactment. The other rules are rules of law, made to suit such cases as require (either expressed or implied) an- nual interest to be paid, and of course apply to no business transactions closed within a year. E-ULE. — Compute the interest of the ^principal sum for the whole time to the day of settlement, and find the amount. Compute the interest on the several pay- ments, from the time each was pfC6,5x6,5)+C13x6,5)X2400 =1858,1 154- 382 And 1858,115H-144=12,903~i-ffc, Am. 154 orton's liqhtnino calculator. To find tbe content of timber in a tree, multiply the square of ^ of the circumference at the middle of the tree, in inches, by twice the length in feet^ and the product divided by 144 will be the content, extremely near the truth. In oak, an allowance of j*Q or y'2 must be made for the bark, if on the tree; in other wood, less trees of irregular growth, must be measured in parts. To find the solid content of a frustum or segment of a globe. Definition. — The frustum of a globe is an}^ part cut off by a plane. Rule. — To three times the square of the semi-di- ameter of the hase^ add tJie square of the hight ; multiply this sum by the hight, and the product again by .523G ; the last product will be the solid content. EXAMPLE. If the hight of a coal-pit, at the chimney, be 9 feet, and the diameter at the bottom be 24 feet, how many cords of wood does it contain^^ allowing nothing for the chimney? 24-f-2=12r:^h'fdiam. 12X12X3=432. 9x9=81 And432+81x9Xr5236 18,886+cords, Ans. 128=:rsclid feet in a cord^ SHORT RULES FOR THE MECHANIO. Question. — A stick of timber is carried by three men, one carries at the end, and the other two with a lever. How far should the lever be placed from the other end, that each man may carry equally ? Rule. — Divide the length of the stick by 4, and the quotient is the answer. There is a stick of timber, 30 feet long, to be carried by 3 men : one carries at the end, the other two carry by a lever ; how far must the lever be placed from the other end, that each may carry equally ? Ans. 7| feet from the end. 155 CASK-GAUGING. Gauging is Ihe art of measuring the capacity of casks and vessels of any form. In commerce, most of the gauging is done by the use of (he diagonal rod, which gives only approximate results, but suf- ficiently accurate for ordinary purposes. Ullage is the difference between the actual con- tents of a vessel and its capacity, or that part which is empty. To measure small cylindrical vessels. Rule. — Multiply the square of the diameter, in inches, by 34, and that by the height, in inches, and point off four figures ; the result will be the capacity, in wine gallons and decimals of a gallon. For beer gallons multiply by 28 instead of 34. 156 CASK-GAUlilNG. 157 Example. — A can measures 15 inches in diame- ter, and is 2 feet 2 inches in height. How many gal- lons will it contain ? 15x15 = 225 X 26 height = 5850 ; 5850 x 34 = 19.8900. Ans. ldj%% galls. Casks are usually regarded as the two equal frus- tums of a cone, and are very accurately gauged by three dimensions as follows : — To measure a cask by three dimensions. 1st. Add the bung and head diameters in inches, and divide by 2 for the mean diameter. 2d. Multiply the square of the mean diameter "by the length of the cask in inches. 3d. Multiply the last product by .0034 for wine gallons, .0028 for beer gallons. Example, — How many wine gallons in a cask, the bung diameter of which is 22 inches, the head diameter 20 inches, and the length 32 inches ? Work.— 22 + 20 = 42 h- 2 = 21 (mean diame- ter) : then 21 x 21 = 441 (square of mean diame- ter), X 32 length = 14112 x -0034 = 47.9808. Ans. Note. — If the cask is not full, stand it on the end, and multiply by the height of the liquid, in- stead of the length of the cask, for actual contents. When the cask is much bilged or rounded from the bung to the head, a more accurate way is to gauge by four dimensions, as follows : — 158 orton's lightning calculator. To measure a cash by four dimensions. 1st. Add the bung and head diameters in inches, and the diameter in inches between bung and head. 2d. Divide their sum by 3 for the mean diameter. 3d. Multiply the square of the mean diameter by the length of the cask in inches. 4th. Multiply the last product by .0034 for wine gallons, .0028 for beer gallons. Example. — What are the contents in gallons of a cask, the bung diameter of which is 24 inches, the middle diameter 20 inches, the head diameter 16 inches, and its length 40 inches? Work.— 24 + 20 + 16 = 60 -f- 3 = 20 (mean diameter), then 20 X 20 = 400 (square of mean dia- meter) X 40 length = 16000 X .0034 = 54. 4 gallons. 1. The ale gallon contains 282 cubic inches. 2. The wine gallon contains 231 cubic inches. 3 The bushel contains 2150.4 cubic inches. 4. A cubic foot of pure water weighs 1000 ounces = 62^ pounds avoirdupois. 5. To find what weight of water may be put into a given vessel. Multiply the cubic feet by 1000 /or the ounces, or by 62J /or the pounds avoirdupois. 6. What weight of water can be put into a cis- tern YJ feet square ? Ans. 26,361 lbs. 3 oz. MENSURATION OF PRACTICAL GEOMETRY. 159 To find the contents of a round vessel, wider at one end than th** other. Rule, — Multiply the great diameter hy the less; to this product add J of the square of their differ- ence, then multiply hy the hight, and divide as in the last rule. Having the diameter of a circle given, to find the area. Rule. — Multiply half the diameter hy half the circumference, and the product is the area ; or, which is the same thing, multiply the square of the diameter hy .7854, and the product is the area. To find the solidity of a sphere or globe. Rule. — Multiply the cuhe of the diameter by .5236. To find the convex surfiice of a sphere or globe. Rule. — Multiply iis diameter hy its circumfer- ence. To find the solidity of a prism. Rule — Multiply the area of the hase, or end, hy the hight. How many wine gallons will a cubical box con- tain, that is 10 long, 5 feet wide, and 4 feet high? Rule. — Take the dimensions in inches; then mid- tiply the length, hreadth, and hight together; di- vide the product hy 282 for ale gallons, 231 for idnc gallons, and 2150 for bushels. 160 ORTON'S LIGHTNING CALCULATOR. In estimating the capacity of cisterns, resep- VoIps, &c., the following table is used : — 31^ gallons . . .1 barrel. 63 " . . .1 hogshead. Note. — The barrels used in commerce vary from 30 to 45 gallons, and the hogshead from 40 to 60 gallons, &c. Hogshead of claret . 46 gallons. " of brandy 55 to 60 Puncheon of brandy 1 1 to 1 20 " " rum. 100 to 110 " Pipe of port . . 115 " " " Madeira . 92 " " Teneriffe . . 100 Butt of sherry . 108 " " Malaga . . 105 Physicians and druggists, in compounding medi- cines, divide the gallon diflferently, and according to the following : — APOTHECAKIES' FLUID MEASURE. Minima. Drachms. Ounces. 60 minims (a drop) 8 drachms 1 ounce 16 ounces 1 pint 8 pints 1 gallon . Note. — The gallon of the above measure is the same capacity as the gallon of liquid measure. 1 fluidrachm. . 480 . 9600 = 128 16800=1024 = 128 MEASURES OF CAPACITY. 161 TROY WEIGHT. Grains. Dwt. 480 5760 = 240 24 grains 1 dwt. 20 dwt. 1 ounce. 12 ounces 1 pound. Note. — Troy weight contains 5760 grains to the pound. Therefore, it will be seen that avoirdupois contains 1240 more grains to the pound than Troy weight. Apothecapjes' weight is used by physicians and druggists in dispensing medicines. APOTHECARIES' WEIGHT. 20 grains 1 scruple. 3 scruples 1 drachm. 8 drachms 1 ounce. 12 ounces 1 pound. Note. — The pound, ounce, and grain of this weight is the same as that of Troy weight. SHOEMAKERS' MEASURE. Number one, children's measure, is 4| inches, and that every additional number calls for an in- crease of J of an inch in length. Number one adults' measure is 8J inches long, with a gradual increase of J of au inch for additional numbers, so that, for example, number ten measures 11 J inches. This measure corresponds to the number of the last, and not to the length of the sole. Grains. Scruples. Drachms. 60 480 = 94 5760 = 218 = 96 BRICK BUILDING. A perch of stone is 24. 15 cubic feet ; when built in the wall, 22 cubic feet make 1 perch, 2 j cubic feet being allowed for the mortar and filling. Three pecks of lime and four bushels of sand to a perch of wall. To find the number of perches of stone in walls. Rule. — Multiply the length in feet by the height in feet, and that by the thickness in feet, and divide the product by 22. Example. — How many perches of stone con- tained in a wall 40 feet long, 20 feet high, and 18 inches thick ? Solution. — 40 feet length X 20 feet height x IJ feet«thick = 1200 -j- 22 = 54.54 perches. Ans. 162 bricklayer's work. 163 Note. — To find the perches of masonry, divide the cubic feet by 24.75, instead of 22. Brick-work. The dimensions of common bricks are from Tf to 8 inches long, by 4;^ wide, and 2J thick. Front bricks are 8;J^ inches long, by 4jwide, and 2J thick. The usual size of fire-bricks is 9^ inches long, by 4| wide, by 2| thick. Twenty common bricks to a cubic foot when laid ; 15 common bricks to a foot of 8-inch wall when laid. To find the number of common bricks in a wall. Rule. — Multiply the length of the wall in feet by the height in feet, and that by its thickness in feet, and that again by 20. Example. — IIow many common bricks in a wall 40 feet long by 20 feet high, and 12 inches thick? Solution. — 40 feet length x 20 f.et height, x 1 foot thick, X 20 = 16000. Ans. Note. — For walls 8 inches thick, multiply the length in feet by the height in feet, and that by 15. When the wall is perforated by doors and win- dows, deduct the sum of their cubic feet from the cubic contents of the wall, including the openings, before multiplying by 20 or 15 as before. Laths. Laths are 1 J to 1^ inch wide, by 4 feet long, are usually set J inch apart, and a bundle contains 100. o* 164 orton's lightning calculatoe. IV. OF bricklayers' work. The principal is tiling, slating, walling and chim- ney work. 1. Of Tiling or Slating, Tiling and slating are measured hy the square of 100 feet, as flooring, partitioning and roofing were in the Carpenters' work ; so that there is not much difi'erence between the roofing and tiling; yet the tiling will be the most ; for the bricklayers sometimes will require to have double measure for hips and valleys. When gutters are allowed double measure, the way is to measure the length along the ridge-tile, and add it to the content of the roof: this makea an allowance of one foot in breadth, the whole length of the hips or valleys. It is usual also to allow double measure at the eaves, so much as the projector is over the plate, which is commonly about 18 or 20 inches. Sky-lights and chimney shafts are generally de- ducted, if they be large, otherwise not. Example 1. There is a roof covered with tiles, whose depth on both sides (with the usual allow- ance at the eaves) is 37 feet 3 inches, and the length 45 feet ; how many squares of tiling are contained therein ? bricklayers' work. 165 BY DUODE CIMALS, FEET. INCHg^i 37 3 45 185 148 11 3 BY DECIMALS. 37.25 45 18625 14900 16 76.25 16 76 3 2. Of Walhng. Bricklayers commonly measure tlieir work by iLe rod of 16 J feet, or 272 J square feet. In some places it is a custom to allow 18 foet to the rod : that is, 324 square feet. Sometimes the work is measured by the rod of 21 feet long and 3 feer high, that is, 63 square feet ; and then no regard is paid to the thickness of the wall in measuring* but the price is regulated according to the thick- ness. When you measure a, piece of brick -work, tho first thing is to inquire by which of these ways it must be measured ; then, having multiplied the length and breaxith in feet together, divide the pro- duct by the proper divisor, viz.: 272.25, 324 or 63, according to the measure of the rod, and the quo- tient will be the answer in square rods cf that measure. But, commonly, brick walls that are measured by the rod are to be reduced to a standard thick- 166 orton's lighting calculator. ness of a brick and a-half, wliicli may be done by the following Rule. — Multiply the number of snperjicwLl feet that are contained in the wall hy the numher of half bricks which that wall is in thickness; one- third part of that product will be the content in feet. The dimensions of a building are generally taken by measuring half round the outside and half round the inside, for the whole length of the wall ; this length, being multiplied by the hight, gives the superficies. And to reduce it to the standard thickness, etc., proceed as above. All the vacuities, such as doors, windows, window backs, etc., must be deducted. To measure any arched way, arched window or door, etc., take the hight of the window or dooi from the crown or middle of the arch to the bot- tom or sill, and likewise from the bottom or sill to the spring of the arch ; that is, where the arch begins to turn. Then to the latter hight add twice the former, and multiply the sum by the width of the window, door, etc., and one-third of the pro- duct will be the area, sufficiently near for practice. Example 1. If a wall be 72 feet 6 inches long, and 19 feet 3 inches high, and 5 J bricks thick, how many rods of brick work are contained therein, when reduced to the standard ? GLAZIERS WORK. 167 VII. GLAZIERS WORK. Glaziers take their dimensions ir. feet, inches and eights or tenths, or else in feet and hundredth parts of a foot, and estimate their work by the square foot. Windows are sometimes measured by taking the dimensions of one pane, and multiplying its super- ficies by the number of panes. But, more gen- erally, they measure the length and breadth of the window over all the panes and their frames for the length and breadth of the glazing. Circular or oval windows, as fan lights, etc., are measured as if they were square, taking for their dimensions the greatest length and breadth, as a compensation for the waste of glass and labor in cutting it to the necessary forms. Example 1. If a pane of glass be 4 feet 8| inches long, and 1 foot 4J- inches broad, how many (ect of glass are in that pane ? BY DECIMALS. 4.729 1.354 BY DUODECIMALS. FT. IN. p. 4 8 9 1 4 3 4 8 9 1 6 11 1 2 2 3 6 4 10 2 3 18916 23645 14187 4729 6.403066 Am. 6 feet 4 inches. 168 orton's lightning calculator. VIII. PLUMBER8 WORK. Plumbers' work is generally rated at so much per pound, or by tlie hundred weight of 112 pounds, and the price is regulated according to the value of lead at the time when the work is per- formed. Sheet lead, used in roofing, guttering, etc., weighs from G to 12 pounds per square foot, ac- cording to the thickness, and leaden pipe varies in weight per yard, according to the diameter of its bore in inches. The following table shows the weight of a square foot of sheet lead, according to its thickness, reck- oned in parts of an inch, and the common weight of a yard of leaden pipe corresponding to the diameter of its bore in inches: Thickness of Lead. Pounds to a Square foot. Bore of Leaden Pipe. Pounds per yard. sV 5.899 f 10 J 6.554 1 12 HOC 7.373 H 16 >, 8.427 H 18 A 9.831 If 21 _L _ 11.797 2 21 i MASON S WORK. 169 Example 1. A piece of sheet lead measures 16 Feet 9 iuclics in length, and 6 feet 6 inches in breadth; what is its weight at 8^ pounds to a square foot ? BY DUODECIMALS FEET. 16 6 INCHES. 9 6 100 8 6 4 108 10 6 BY DECIMALS FEET. 16.75 6.5 8375 10050 108.875 ftet. Then 1 foot : 8J pounds : : 108.875 feet 898.21875 poundsi=8 cwt. 2 J pounds nearly. IX. MASON S WORK Masons measure their work sometimes by the foot solid, sometimes by the foot superficial, and sometimes by the foot in length. In taking dimensions they girt all their moldings as joiners do. The solids consist of blocks of marble, stwnc pillars, columns, etc. The superficies are pave- ments, slabs, chimney-pieces, etc. 170 obton's lightning calculator. V. PLASTERERS WORK. Plasterers' work is principally of two kinds; namely, plastering upon laths, called ceiling^ and plastering upon walls or partitions made of framed timber, called rendering. In plastering upon walls, no deductions are mado except for doors and windows, because cornice?, festoons, enriched moldings, etc., are put on after the room is plastered. In plastering timber partitions, in large ware- houses, etc., where several of the braces and larger timbers project from the plastering, a fifth part is commonly deducted. Plastering between their timbers is generally called rendering between quarters. Whitening and coloring are measured in the same manner as plastering ; and in timbered par- titions, one-fourth, or one-fifth of the whole area is commonly added, for the trouble of coloring the sides of the quarters and braces. Plasterers' work is measured by the yard square, consisting of nine square feet. In arches, the girt round them, multiplied by the length, will give the superficies. Example 1. — If a ceiling be 59 feet 6 inches long, and 24 feet 6 inches broad ; how many yards does that ceiling contain ? CISTERNS. 17J PROBLEM L To find tlie solid content of a Dome, having tJic hight and the dimensions of its base given, lluLE. — Multiply the area of the base by the hightj and f of the product will be the solidity. Example 1. — What is the solidity of a dome, in the form of a hemisphere, the diameter of the cir- cular base being 60 feet ? GO'X. 7854=2827.44 area of the base. Then f (2827.44x30)=56548.8 cubic feet. Am. PROBLEM IL To find the superficies of a dome, having the hight and dimensions of its base given. Rule. — Multiply the area of the base by 2, and the product will be the superficial content required ; or, multiply the sq^aare of the diameter of the base by 1.5708. For an Elliptical Dome. — Multiply the two diameters of the base together, and that product by 1.5708, the last product will be the area, sufficiently correct for practical purposes. XI. CISTERNS. Cisterns are large reservoirs constructed to hold water, and to be permanent, should be made either of brick or masonry. P 172 orton's lightnin*! calculator. It frequently occurs that they are to be so con- structed as to hold given quantities of water, and it then becomes a useful and practical problem to calculate their exact dimensions. How do you find the number of hogsheads which a cistern of given dimensions will contain ? 1st. Find the solid content of the cistern in cubic inches. 2d. Divide the content so found by 14553, and the quotient will be the number of hogsheads. If the hight of a cistern be given, how do you find the diameter, so that the cistern shall con- tain a given number of hogsheads ? 1st. Reduce the hight of the cistern to inches, and the content to cubic inches. 2d. Multiply the hight by the decimal .7854. 2. Divide the content by the last result, and extract the square root of the quotient, which will be the diameter of the cistern in inches. EXAMPLE. If the diameter of a cistern be given, how do yon find the hight, so that the cistern shall contain a given number of hogsheads ? 1st. Reduce the content to cubic inches. 2d. Reduce the diameter to inches, and then mul- tiply its square by the decimal .7854. MEASURING GRAIN. By the United States standard, 2150 cubic inches make a bushel. Now, as a cubic foot contains 1Y28 cubic inches, a bushel is to a cubic foot as 2150 to 1Y28; or, for practical purposes, as 4 to 5. Therefore, to convert cubic feet to bushels, it is necessary only to multiply by |. To measure the bushels of grain in a granary/. Rule. — Multiply the length in feet by the breadth in feet, and that again by the depth in feet, and that again by J. The last product will l>e the number of bushels the granary contains. Example. — How many bushels in a bin 10 feet long, 4 feet wide, and 4 feet deep. Work. — 10 feet length X 4 feet breadth x 4 feet depth = 160 cubic feet ; then 160 x | = 128. Ans. It3 174 orton's lightning calculator. 3d. Divide the content by tlie last result, and the quotient will be the hight in inches. XII. BINS FOR GRAIN. Having any number of bushels, how then will you find the corresponding number of cubic feet ? Increase the number of bushels one -fourth itself, and the result will be the number of cubic feet. How will you find the number of bushels which a bin of a given size will hold ? Find the content of the bin in cubic feet ; then diminish the content by one-fifth, and the resuU will be the content in bushels. How will you find the dimensions of a bin which shall contain a given number of bushels ? Increase the number of bushels one -fourth itself, and the result will show the number of cubic feet which the bin will contain. Then, when two dimensions of the bin are known, divide the last result by their product, and the quotient will be the other dimension. WEIGHTS AND MEASURES. 176 From To OQ Jan.... Feb.... March April .. May... June... July... Aug ... Sept ... Oct.... Nov.... Dec... 365 334 306 275 245 214 184 153 122 92 61 31 31 365 337 306 276 245 215 184 153 123 92 62 59 28 365 334 304 273 243 212 181 151 120 90 90 59 31 365 335 304 274 243 212 182 151 121 120 89 61 30 365 334 304 273 242 212 181 151 151 120 92 61 31 365 335 304 273 243 212 182 181 150 122 91 61 80 365 334 303 273 242 212 212 181 153 122 92 61 31 365 334 304 273 243 243 212 184 153 123 92 62 31 365 335 304 274 273 242 214 183 153 122 92 61 30 365 334 304 304 273 245 214 184 153 123 92 61 31 365 335 331 303 275 244 214 183 153 122 91 61 30 365 TABLE SHOWINa DIFFERENCE OF TIME AT 12 o'clock (noon) at new YORK. Boston 12.12 p. m. Quebec 12.12 " Portland 12.15 " London 4.55 " Paris 5.05 " Rome 5.45 " Constantinople 6.41 " Vienna 6 00 « St. Petersburg.. 6.57 " Pekin, night... 12.40 a. m. New York 12.00 n. Buffalo 11.40 A. M. Cincinnati 11.18 " Chicago 11.07 " St. Louis 10.55 " San Francisco... 8.45 " New Orleans 10.56 " Washington 11.48 " Charleston 11.36 " Havana 11.25 " TROT WEIGHT. By this weight gold, silver, platina and precious stones, except diamonds, are estimated. 20 Mites 1 Grain. I 20 P^jnnywts 1 Ounce. 20 Grains.... 1 Pennywt. j 12 Ounces 1 Pound. Any quantity of gold i» supposed to be divided p* 176 orton's lightning calculator. into 24 parts, called carats. If pure, it is said to be 24 carats fine; if there be 22 parts of pure gold and 2 parts of alloy, it is said to be 22 carats fine The standard of American coin is nine-tenths pure gold, and is worth $20.67. What is called the new standard J used for watch cases, etc., is 18 carats fine. The term carat is also applied to a weight of 3J grains troy, used in weighing diamonds ; it is divided into 4 parts, called grains ; 4 grains troy are thus equal to 5 grains diamond weight. apothecaries' weight USED IN MEDICAL PEESCRIPTIONS. The pound and ounce of this weight are the same as the pound and ounce troy, but diflferently divided. 20 Grains Troy... 1 Scruple. I 8 Drachms...! Ounce Troy. 8 Scruples 1 Drachm. | 12 Ounces....l Pound Troy. Druggists huy their goods by avoirdupois weight. AVOIRDUPOIS WEIGHT. By this weight all goods are sold except those named under troy weight. 27|J Grains 1 Dram. 16 Drams 1 Ounce. 16 Ounces 1 Pound. 28 Pounds 1 Quarter. 4 Quarters or 100 pounds 1 Hundred Weight. 20 Hundredweight 1 Ton. The grain avoirdupois, though never used, is the same as the grain in troy weight. 7,000 grains make the avoirdupois pound, and 5,760 grains the WEIGHTS AND MEASURES. 177 troj pound. Therefore, tlie troy pound is loss iLan the avoirdupoi3 pound in the proportion of 14 to 17, nearly; but the troy ounce is greater than the avoirdupois ounce in the proportion of 79 to 72, nearly. In times past it was the custom to allow 112 pounds for a hundred weight, but usage, as well as the laws of a majority of the States, at tho present time call 100 pounds a hundred weight. apothecaries' fluid measurk. 60 Minims 1 Fluid Drachm. 8 Fluid Drachms 1 Ounce (Troy). 16 Ounces (Troy) 1 Pint. 8 Pints 1 Gallon. MEASURE OP CAPACITY FOR ALL LIQUIDS. 6 Ounces Avoirdupois of water make 1 Gill. 4 Gills 1 Pint = 34| Cubic Inches (nearly). 2 Pints 1 Quart = 69| do 4 Quarts 1 Gallon =277J do 31J Gallons 1 Barrel, 42 Gallons 1 Tierce. 63 Gallons, or 2 bbls 1 Hogshead. 2 Hogsheads IPipeorButt^ 2 Pipes 1 Tun. The gallon must contain exactly 10 pounds avoir- dupois, of pure water, at a temperature of 62°, the barometer being at 30 inches. It is the standard unit of measure of capacity for liquids and dry goods of every description, and is ^ larger than the old wine measure, -j^^ larger than the old 178 ORTON A LIGHTNING CALCULATOR. dry measure, and 5*5 less than the old ale measure. The wine gallon must ccntain 231 cubic inches. MEASURE OP CAPACITY FOR ALL DRY GOODS. 4 Gills 1 pint =» 34f cubic inclis( nearly) 2 Pints 1 quart = 69| cubic inches. 4 Quarts 1 gallon «= 277J cubic inches. 2 Gallons 1 peck = 654J cubic inches. 4 Pecks, or 8 gals. 1 bushel =2150^ cubic inches. 8 Bushels 1 quarter = 10^ cubic feet (nearly). When selling the following articles a barrel weighs as here stated : For rice, 600 lbs.; flour, 196 lbs.; powder, 25 lbs.; corn, as bought and sold in Kentucky, Ten- nessee, etc., 5 bushels of shelled corn — as bought and sold at New Orleans, a flour-barrel full of ears: potatoes, as sold in New York, a barrel contains 2J bushels; pork, a barrel is 200 lbs., distinguished in quality by "clear," "mess," "prime;" a barrel of beef is the same weight. The legal bushel of America is the old Win- chester measure of 2,150.42 cubic inches. The imperial bushel of England is 2,218.142 cubic inches, so that 32 English bushels are about equal to 33 of ours. Although we are all the time talking about the price of grain, etc., by the bushel, we sell by weight, as follows : Wheat, beans, potatoes, and clover- seed, 60 lbs. WEIGHTS AND MEASURES. 179 to tlio bushel ; corn, rye, flax-seed, and onions, 56 lbs.; corn on the cob, 70 lbs.; buckwheat, 52 lbs.; barley 48 lbs.; hemp-seed, 44 lbs.; timothy -seed, 45 lbs.; castor beans, 46 lbs.; oats, 35 lbs.; bran, 20 lbs.; blue-grass seed, 14 lbs.; salt — the real weight of coarse salt is 85 lbs.; dried apples, 24 lbs.; dried peaches, 33 lbs., according to some rules, but others are 22 lbs. for a bushel, while in Indiana, dried apples and peaches are sold by the heaping bushel; so are potatoes, turnips, onions, apples, etc., and in some sections oats are heaped. A bushel of corn in the ear is three heaped half bushels, or four even full. In Tennessee a hundred ears of corn is some- times counted as a bushel. A hoop 18J inches diameter, 8 inches deep, holds a Winchester bushel. A box, 12 inches sauare, 7 and 7^5 deep, will hold half a bushel. A heaping bushel is 2,815 cubic inches. CLOTH MEASURE. 2^ Inches 1 nail. 4 Nails 1 quarter of a yard 4 Quarters 1 yard. FOREIQH CLOTH MEASURE. 2J Quarters 1 Ell Hamburgh. 3 Quarters 1 Ell Flemish. 5 Quarters 1 Ell English 6 Quarters I Ell French. 180 ORTON*S LiaHTNlNQ CALCULATOR. MEASURE OP LENGTH. 12 Inches 1 foot. 3 Feet 1 yard. 5J Yards 1 rod, pole, or porch. 40 Poles 1 furlong. 8 Furlongs, or 1,760 yds, 1 mile. ««-AM''- j^'TyLLlT/"- By scientific persons and revenue officers, tho inch is divided into tenths, hundredths, etc. Among mechanics, the inch is divided into eighths. The division of the inch into 12 parts, called lines, is not now in use. A standard English mile, which is the measure that we use, is 5,280 feet in length, 1,760 yards, or 320 rods. A strip, one rod wide and one mile long, is two acres. By this it is easy to calculate the quantity of land taken up by roads, and also how much is wasted by fences. qunter's chain. USED FOB LAND HEASURB 7,-^^ Inches 1 Link. 100 Links, or 66 feet, or 4 poles 1 Chain. 10 Chains long by 1 broad, or 10 ) .. . square chains | 80 Chains 1 Mile. WEIGHTS AND MEASURES. 181 SURFACE MEASURE. 144 Sq. inches 1 sq. foot [ 40 Sq. perches I rood 9 Sq. feet 1 sq. yard | 4 Roods 1 acre 80J Sq. yards 1 sq.rdorprch ^ 640 Acres 1 sq. mile Measure 209 feet on each side, and you have a square acre, within an inch. The following gives the comparative size, in square yards, of acres in different countries : English acre, 4,840 square yards ; Scotch, 6,150; Irish, 7,840; Hamburgh, 11,545; Amsterdam, 9,722; Dantzic, 6,650; France (hectare), 11,960; Prussia (morgen), 3,053. This difference should be borne in mind in read- ing of the products per acre in different countries. Our land measure is that of England. GOVERXMENT LAND MEASURE. A Township — 36 sections, each a mile square. A section — 640 acres. A quarter section, half a mile square — 160 acres. An eighth section, half a mile long, north and south, and a quarter of a mile wide — 80 acres. A sixteenth section, a quarter of a mile square — 40 acres. 132 ORTON S LIGHTNING CALCULATOE,. The sections are all numbered 1 to 36, com- mencing at the north-east corner, thus : 6 5 4 3 2 mr.NE 7 8 9 10 11 12 18 17 16* 15 14 13 19 20 21 22 23 24 30 29 28 27 26 25 31 32 33 34 35 36 The sections are all divided in quarters, which are named by the cardinal points, as in section 1. The quarters are divided in the same way. The description of a forty-acre lot would read: The south half of the west half of the south-west quarter of section 1 in township 24, north of range 7 west, or as the case might be ; and sometimes m\\ fall short, and sometimes overrun the number of acres it is supposed to contain. SQUARE MKASUBE — FOE CARPENTERS, MASONS, ETC 144 Sq Inches 1 Sq Foot. Sq Ft, or 1,296 Sq In. 1 Sq Yard. 100 Sq Feet 1 Sq of Flooring, Roofing, eto 30| Sq Yards 1 Sq Rod. 86 Sq Yards 1 Rood of Building. bcbool section. WEIGHTS AND r.lEASURES. 183 GEOOBAPHICAL OR NAtTTICAL MEASURE. 6 Feet 1 Fathom. 110 Fathoms or 660 ft. 1 Furlong. 6075f Feet 1 Nautical Mile. 3 Nautical Miles 1 League. 20 Leagues or 60 | . j. Geo. Miles...; ^ degree, o.^n Tk f The earth's circumference abO Degrees | ^24,855^ miles, nearly. The nautical mile is 795| feet longer than tho common mile. MEASURE OF SOLIDITY. 1728 Cubic Inches 1 Cubic Foot. 27 Cubic Feet 1 Cubic Yard. 16 Cubic Feet 1 Cord Foot, or a ft of wood. 8 Cord ft or 128 Cubic ft.. 1 Cord. 40 ft of round or 60 ft \ of hewn timber... j ITon. 42 Cubic Feet 1 Ton of Shipping. ANGULAR MEASURE, OR DIVISIONS OF THE CIRCLE. GO Seconds 1 Minute. I 30 Degrees 1 Sign. 60 Minutes 1 Degree. | 90 Degrees 1 Quadrant 360 Degrees 1 Circumference. MEASURE OF TIME. 60 Seconds 1 Minute 60 Minutes 1 Hour. 24 Hours 1 Day. 7 Days 1 Week. 28 Days 1 Lunar Month. 28, 29, 30 or 31 Days 1 Cal. Month. 12'Cal. Months 1 Year. 365 Days 1 Com. Year. 3G6 Days 1 Leap Year. 365|^ Days 1 Julian Year. .365 D., 5 H., 48 it., 49 s 1 Solar Year. 365 D., G n., 9 ai., 12 s 1 Siderial Year. Q 184 orton's lightning calculator. ROPES AND CABLES. 6 Feet 1 Fathom 120 Feet 1 Cable Length. Miscellaneous Important Facts about Weights and Measures, BOARD MEASURE. Boards are sold by superficial measure, at so much per foot of one inch or less in thickness, adding one-fourth to the price for each quarter- inch thickness over an inch. GRAIN MEASURE IN BULK. Multiply the width and length of the pile to- gether, and that product by the hight, and divide by 2,150, and you have the contents in bushels. If you wish the contents of a pile of ears of corn, or roots, in heaped bushels, ascertain the cubic inches and divide by 2,818. A TON WEIGHT. In this country a ton is 2,000 pounds. In most places a ton of hay, etc., is 2,240 pounds, and in some places that foolish fashion still prevails of weighing all bulky articles sold by the tun, by the "long weight," or tare of 12 lbs. per cwt. A tun of round timber s 40 feet ; of square timber, 54 cubic feet. WEIGHTS AND MEASURES. 185 A quarter of corn or other grain sold by the bushel Is eight imperial bushels, or quarter of a tun. A ton of liquid measure is 252 gallons. BUTTER Is sold by avoirdupois weight, which compares with troy weight as 144 to 175 ; the troy pound being that much the lightest. But 175 troy ounces equal 192 of avoirdupois. A firkin of butter is 56 lbs.; a tub of butter is 84 lbs. THE KILOGRAMME OF FRANCE Is 1000 grammes, and equal to 2 lbs. 2 oz. 4 grs. avoirdupois. A BALE OF COTTON, In Egypt, is 90 lbs.; in America a commercial bale is 400 lbs.; though put up to vary from 280 to 720, in difi"erent localities. A bale or bag of Sea Island cotton is 300 lbs. WOOL. In England, wool is sold by the sack or boll, of 22 stones, which, at 14 lbs. the stone, is 308 lbs. A pack of wool is 17 stone, 2 lbs., which is rated as a pack load for a horse. It is 240 lbs. A tod of wool is 2 stones of 14 lbs. A wey of wool is 6 J tods. Two weys, a sack. A clove of wool is half a stone. MEASUREMENT OF TIME. TIME IS THE MEASURE OF DUEATIOS^. We have in this engraving a representation of the magnificent transit instrument used in the Paris Observatory. It is made on the same model as the celebrated one in the Observatory at Green- wich. These, and the one at the National Observa- tory at Washington, are the finest in the world. The instrument is used for the purpose of determin- ing the instant of time a heavenly body passes, or makes a transit across the meridian. 186 MEASUREMENT OF TIME. 18T From the time a star appears on the meridian, uutil its reappearance the next day, is just 24 hours of sidereal time. So accurate is the apparent movement of the star, that it has never been known to vary the one-hundredth part of a second for over a thousand years. It enables us to correct chro- nometers to the tenth part of a second, so that sail- ors are enabled to tell the precise time of day, and exactly in what part of the world they are. (Much in little about measuring time.) One of the first inventions for measuring time was the clepsydra, or water-clock, which was a contrivance of the Assyrians, and was in use among them as early as the reign of the second Sardanap- alus. Clepsydra, or water-stealer, it was called, from two words which have that meaning. The instrument was of various materials ; sometimes transparent, but generally of brass, and in the shape of a cylinder, holding several gallons. In any case, the principle on which it operated was the sa ne. There was a very small hole, either in the side or bottom, through which the water slowly trickled, or as the name expresses it, stole away, into another vessel below. In the lower one a cork floated, showing the rise of the water. By calcu- lating how many times a day the, water was thus emptied from one to another, they gained a general 188 orton's lightning calculator. idea of the time. The Chinese and Egyptians used this ; so, also, did the Greeks and Romans ; and it is stated that something of the kind was found among the ancient Britons. It seems to have been one of the earliest rude attempts, in many nations, to keep a record of the hours. The idea of the hour-glass must have grown out of this. Instead of two large vessels, there were devised the pear- shaped glasses, joined by what may be called the stem ends ; and a delicate fine sand was used in- stead of water. It was the invention of a French monki THE PENDULUM AND TELESCOPE— HOW INVENTED. In 1682, Galileo, then a youth of eighteen, was seated in church, when the lamps suspended from the roof were replenished by the sacristan, who, in doing so, caused them to oscillate from side to side, as they had done hundreds of times before, when similarly disturbed. He watched the lamps, and thought he perceived that while the oscillations were diminishing, they still occupied the same time. The idea thus suggested never departed from his mind ; and fifty years afterward he constructed the first pendulum, and thus gave to the world one of the most important 'instruments for the measurement of time. Afterward, when living at Venice, it was MEASUREMENT OF TIME. 189 reported to him one day that the children of a poor spectacle-maker, while playing with two glasses, had observed, as they expressed it, that things were brought nearer by looking through them in a certain position. Everybody said, How curious I but Galileo seized the idea, and invented the first telescope. THE EQUATORIAL TELESCOPE. The Equatorial telescope is an instrument used for the purpose of viewing stars that appear on the horizon in the east, ascend to their highest declina- tion, and descend to the western horizon. The Transit telescope is stationary, and the star can only be viewed while it is passing the disk of the instrument. It is necessary to have other instru- ments that will follow the stars through the regions of the heavens in which they are carried by the daily movement of the earth. By means of com- plete machinery the Equatorial accomplishes all this, carrying the instrument in one direction as fast as the earth takes it in another. The observer can thus view the star for hours without changing his position. 190 orton's lighining calculator. ASTRONOMICAL CALCULATIONS. A scientific method of telling immediately what day of the week any date transpired or will transpire^ from the commencement of the Christian Era, for the term of three tnousand years. MONTHLY TABLE. The ratio to add for each month will be found in the following table: Ratio of June is Ratio of September is 1 Ratio of December is 1 Ratio of April is 2 Ratio of July is 2 Ratio of January is 3 Ratio of October is 3 Ratio of May is 4 Ratio of August is 5 Ratio of March is 6 Ratio of February is 6 Ratio of November is 6 Note. — On Leap Year the ratio of January is 2, and the ratio of February is 5. The ratio of the other tea months do not change on Leap Years. CENTENNIAL TABLE. The ratio to add for each century will be found in the following table: Q 200, 900, 1800, 2200, 2600, 3000, ratio is ...0 I 800, 1000, ratio is 6 I 400, 1100, 1900, 2300, 2700, ratio is 5 ^ 600 1200, 1600, 2000, 2400, 2800, ratio is 4 3 600 1300, ratio is 3 •)00, 700, 1400, 1700, 2100, 2500, 2900, ratio is 2 loo! 800, 1500 ratio is 1 ASTRONOMICAL CALCULATIONS. 191 Note. — The figure opposite each century is iU ratio; thus the ratio for 200, 900, etc., is 0. To find the ratio of any century, first find the century in tl»e above table, then run the eye along the line until you arrive at tho end; the small figure at the end is its ratio. METHOD OP OPERATION. Rule.* — To the given year add its fourth part^ rejecting the fractions ; to this sum add the day of the month; then add the ratio of the month and the ratio of the century. Divide this svm hy 7 ; the remainder is the day of the week, counting Suiiday OS the first, Monday as the second, Tuesday as the third, Wednesday as the fourth, Thursday as the fifth, Friday as the sixth, Saturday as the seventh; the remainder for Saturday will he or zero. Example 1. — Required the day of the week for the 4th of July, 1810. To the given year, which is 10 Add its fourth part, rejecting fractions 2 Now add the day of the month, which is 4 Now add the ratio of July, which is 2 Now add the ratio of 1800, which is.., Divide the v^hole sum by 7. 7 | 18 — 4 2 We have 4 for a remainder, which signifies the fourth day of the week, or Wednesday. •When dividing the year by 4, always leave off tte ccutuno!*. We diviile by 4 to find the number of Leap Year*. 192 orton's lightning calculator. Note. — In finding the day of the week for the present century, no attention need be paid ».o the centennial ratio^ as it is 0. Example 2. — Required the day of the week for the 2d of June, 1805, To the given year, which is 5 Add its fourth part, rejecting fractions 1 Now add the day of the month, which is 2 Now add the ratio of June, which is Divide the whole sum by 7. 7 | 8—1 T We have 1 for a remainder, which signifies the first day of the week, or Sunday. The Declaration of American Independence was signed July 4, 1776. Required the day of the week. To the given year, which is 76 Add its fourth part, rejecting fractions 19 Now add the day of the month, which is 4 Now add the ratio of July, which is 2 Now add the ratio of 1700, which is 2 Divide the whole sura by 7. 7 | 103 — 5 14 "We have 5 for a remainder, which signifies the fifth day of the week, or Thursday. The Pilgrim Fathers landed on Plymouth Rock Dec. 20, 1620. Required the day of the week. ASTRONOMICAL CALCULATIONS. 193 To tlie given year, which is 20 Add its fourth part, rejecting fractions 6 Now add the day of the month, which is 20 Now add the ratio of December, which is 1 Now add the ratio of 1600, which is 4 Divide the whole sum by 7. 7 | 50- -1 ~7 We have 1 for a remainder, wliicli signifies the first day of the week, or Sunday. On what day will happen the 8th of January, 1815? Ans. Sunday. On what day will happen the 4th of May, 1810? On what day will happen the 3d of December, 1423? Ans. Friday. On what day of the week were you born? The earth revolves round the sun once in 365 days, 5 hours, 48 minutes, 48 seconds; this period is, therefore, a Solar year. In order to keep pace with the solar year, in our reckoning, we make every fourth to contain 366 days, and call it Leap Year. Still greater accuracy requires, howevei, that the leap day be dispensed with three times m every 400 years. Hence, every year (except the centennial years) that is divisible by 4 is a Leap YeaVj and every centennial year that ia divisible by 400 is also a Leap Year. The next centennial year that will be a Leap Year h 2000. 194 orton's lightning calculator. tor the practical convenience of those who have occasion to refer to mensuration, we have arranged the following useful -able of multiples. It covers the whole ground of practical geometry; and should be studied carefully by those who wish to be skilled in this beautiful branch of mathematics : TABLK or MULTIPLES.' Diameter of a circle X 3-1416 — Circumference. Radius of a circle X 6.283185 — Circumference. Square of the radius of a circle X 3.1416 — Area. Square of the diameter of a circle X 0-7854 — Area. Square of the circumference of a circle X 0.07958 == Area. Half the circumference of a circle X by half its diameter = Area. Circumference of a circle X 0.159155 — Radius. Square root of the area of a circle X 0.56419 =- Radius. Circumference of a circle X 0.31831 — Diameter. Square root of the area of a circle X 112838 — Diameter. Diameter of a circle X 0-86 — Side of inscribed equilateral triangle. Diameter of a circle X 0.7071 — Side of an inscribed square. Circumference of a circle X 0.225 — Side of an inscribed square. Circumference of a circle X 0.282 — Side of an equal square. Diameter of a circle X 0.88G2 — Side of an equal square. Base of a triangle X by )4 the altitude — Area. Multiplying both diameters and .7854 together — Area of an ellipse. Surface of a sphere X by 3^ of its diameter— Solidity. Circumference of a sphere X by its diameter •= Surface. Square of the diameter of a sphere X 3.1416 — Surface. Square of the circumference of a sphere X 0.3183 = Surface. Cube of the diameter of a sphere X 0.5236 — Solidity. Cube of the radius of a sphere X 4-1888 — Solidity. Cube of the circumference of a sphere X 0.016887 — Solidity. Square root of the surface of a sphere X 0.56419 — Diameter. Square root of the surface of a sphere X 1-772454 — Circumference. Cube root of the solidity of a sph< re X 1-2407— Diameter. Cube root of the solidity of a sphere X 3.8978 — Circumferenco. Radius of a sphere X 1-1547 — Side of inscribed cube. Square root of (>^ of the square of) the diameter of a sphsre — Side of inscribed cube. Area of its base X by 3^ of its altitude — Solidity of a cone or pyr- amid, whether round, squaie, or triangular. Area of one of its sides X 6 — Surface of a cube. Altitude of Tapezoid X J^ the sum cf its par-ullel sides — Area O7(o