LIBRARY OF THE UNIVERSITY OF CALIFORNIA. UtM/Sr^l " Class ART OF COMPUTATION AND :R/U:L,:E / FOR EQUATION OF PAYMENTS. ONE DOLLAR. - OF THE UNIVERSITY OF HOWARD'S JIRT OF COMPUTATION OF THE UNIVERSITY OF AND GOLDEN RULE FOR EQUATION OF PAYMENTS FOR SCHOOLS, BUSINESS COLLEGES AND SELF-CULTURE. A NEW, CONCISE AND COMPREHENSIVE TEACHER AND MANUAL OP BUSINESS ARITHMETIC. ;Y TKK AUTHOR OP THK CALIFORNIA CAJLCOI*ATOK> C. FKUSHER HOWARD, SAN FRANCISCO. 1880. Entered according to Act of Congress, in the year of our Lord 1879, by C. Frusher Howard, in the office of the Libra- rian of Congress, Washington, D. C. ALL EIGHTS SECURED. The Author specially cautions all Book Pirates that his sole rights and title to the following original Rules and Tables are legally secured, and will be maintained against all infringements. HOWARD'S Golden Rule for Equation of Payments. Averaging Accounts. " " " Partial Payments. Computing Interest on a Basis of one per cent. " by dividing the year by the rate* " " Bank of England Rule. Compound Interest. Squaring numbers by their base and difference. California Calendar for thirty centuries. The original Tables and their arrangement. "HE MIVERSITY PREFACE. ?HE ability to make business calculations with ease, accuracy and rapidity, is an all-important acquisition to every class of the community. The methods of Arithmetic hitherto taught have been so abstruse and difficult as to deter all, but a small per centage, from giving the weary months and years of time, labor and study necessary to master its mysteries. WONDERFUL and STARTLING discoveries have RECENTLY been made and em- bodied in the following rules, simplify ing- and short- ening all the operations of numbers, so as to make RAPID CALCULATION easy to all; The rules taught in Schools are needlessly weight- ed with superfluous elements, that only serve to en- cumber the operations, and distract, and confuse the learner ; the Rules here taught avoid all this, and by an*easily learned, simple, and natural arrangement lead directly to the required answer. They are especially adapted to that large class of persons who find it difficult, or impossible, mentally to grasp, and retain complex numbers ; such persons will find in this book "A Complete Teacher of Business Arithmetic" all the examples being worked out, and explained so as to be readily understood, transforming the drudg- ery of calculation, into a pleasing pastime, and qual- ifying persons of ordinary intellect, to surpass the performances of the "Lightning Calculators" who have astonished mankind. 153721 PREFACE. The success achieved by the CALIFORNIA CALCULATOR encourages the hope that the ART OF COMPUTATION will soon be in every school, making ALL the Boys "quick at figures" The more gifted and ambitious will become expert Mathematicians with greater facility if they are FIRST good calculators. A knowledge of the SCIENCE of numbers is an invalu- able acquisition to those who are capable of acquiring it; to do this not more than one person in a hundred has either the time, necessity, or mental capacity. To Accountants, Brokers, Farmers, Traders and person? engaged in the ruder mechanical pursuits, a knowledge of the SCIENCE OF NUMBERS is of minor importance ; SKILL in the ART OF COMPUTATION is absolutely indispensable; the business of this Book is by new, original and easily ac? quired methods to teach that ART, in accord with, yet dis- tinct from the SCIENCE. As a SCHOOL BOOK, its aim is to make the learner a GOOD CALCULATOR with the greatest possible economy of time and study; its preeminence consists in the brevity and clearness of the rules ; by their use, Interest and other calculations may be made, easier than they can be copied from ordinary Tables. The REFERENCE TABLES are very comprehensive and their arrangement simple and original. The miscellaneous section is unique ; it embraces almost every variety of BUSINESS CALCULATION, the work of find- ing the answer to each question is so expressed that it constitutes a formula for all similar examples. One Reviewer of these Rules and Tables says: " Students, Teachers and Business Men can no more afford to be without them than they can afford to travel by OX-TEAMS, now the RAILWAY spans the Continent." TABLE OF CONTENTS: Addition, 16 Aliquot Parts, 32 British Money, 70 Cancellation, 40 Compound Interest, 65 Definitions and Signs, 7 Decimals, 35 Division, 27 Discount, 66 Exchange, 69 Fractions, 29 Gold and Silver, 100, 101, 102, 111 Multiplication, 19 Measuring Land, 47, 104 " Timber, 56,105 Marking Goods, 92 Miscellaneous, 114 Notation, 14 Numeration, 15 Percentage, 72 Subtraction, 19 Rapid Rules for Farmers, 45 " Reckoning for Mechanics, 51 " Method for Squaring Numbers, 24 " Rules for COMPUTING INTEREST, 59 Rules for Money and Bullion Brokers,... 100, 111 Tables for Business Reference, 93 Tables of Standard Weights and Measures, 10S 6 TABLE OF CONTENTS. PAGE Proportion 39 Rapid Bale for reckoning the cost of Hay 46 Subtraction of Fractions 31 Square and Cube Root 87 Stocks and Bonds 74 To find the Greatest Common Factor or Divisor.. 30 To find the value of Grain per Cental, or Bushel, the price of either being given 45 To Measure Grain 46 To Measure Land without Instruments 47 To lay oif a Square Corner 51 To Measure Grindstones 52 To Measure Superfices and Solids 52 Bricklayers' Work 57 Plasterers' Work 58 Painters' Work 58 Gangers' Work 57 To find the Difference of Time between any two Dates ..110, 83 Howard's New Rule for Interest on a basis of 1/ 59 Howard's California Calendar for Thirty Centuries 86 To find the value of Gold or Currency, the price of either being given 69 The Number Nine 91 Percentage < . 72 Partial Payments .' . . 82 Averaging Accounts 81 Cash Balances 83 Golden Bule for Equation of Payments 75 HOWARD'S ART OF COMPUTATION. DEFINITIONS AND SIGNS. ARITHMETIC is the science of numbers, and the art of computing by figures. ABSTRACT NUMBER. An abstract number is a number used without reference to any particular object, as 9, 745, 9764. ADDITION, the act of adding, opposed to subtrac- tion. AMOUNT. The sum of principal and interest. ALIQUOT. An aliquot part of a number is such a part as will exactly divide that number. AREA, the surface included within any given lines. ARITHMETICAL SIGNS are characters indicating operations to be performed, and are indispensable for briefly and clearly stating a problem : -j-,pfaS, and more, signifying addition; , minus, less, signifying subtraction; 8 HOWARD'S ART OF COMPUTATION. X, multiplied by, as 2 X 2 = 4 ; ~ or : divided by, as 6 -f- 3 = 2, or 6 : 3 = 2, orf = 2; =, equality, or is equal to, as 6 -f 2 X 2 = 16, and is read thus, " 6 plus 2, multiplied by 2, equals 16 "; , or ( ) &c. T the vinculum ; used to shew that all the numbers united by it are to be considered as one ; thus, 6x4 + 3x2 + 1 means the product of 6 X 4 is to be added to the product of 3 X 2, and the sum of the products to be added to 1. V 9, sign of the square root, read "the square root of 9 " ; 4 2 , sign of the square, read "the square of 4"; ^8, the cube root of 8. 8 3 , the cube of 8. AN ANGLE is the corner formed by two lines where they meet. BASE, the lower, or side upon which a figure stands; the foundation of a calculation. CONCRETE NUMBER, used with reference to some particular object or quantity, as 640 acres, 500 dollars. CIRCLE, a plane figure comprehended by a single curved line, called its circumference, every part of which is equidistant from its center. CIRCUMFERENCE, the line that goes around a circle or sphere. CYLINDER, a body bounded by a uniformly curved surface, its ends being equal and parallel circles. DEFINITIONS AND SIGNS. 9 CUBE, a solid body with six equal square sides. A product formed by multiplying an}' number twice by itself, as4x 4 X 4 = 6 4, the cube of 4. CUBE ROOT is the number or quantity which twice multiplied into itself produces the number of which it is the root, thus 4 is the cube root of 64. CURRENCY, the current medium of trade author- ized by government. DIVISION determines how many times any one number is contained in another. DISCOUNT, the sum deducted from an account,, note, or bill of exchange, usually at some rate per cent. DENOMINATOR, the number placed below the line in fractions, thus, in j- (seven-eights) 8 is the de- nominator. DECIMAL, a tenth ; a fraction having some power of 10 for its denominator. DECIMAL CURRENCY is a currency whose denom- inations increase or decrease in a ten-fold ratio. DIVIDEND, the number to be divided. DIVISOR, the number by which the dividend is to- be divided. A common divisor, is a number that will divide two or more numbers without a remain- der. DIAMETER, a right line passing through any object. DUODECIMALS are the divisions and subdivisions. 10 HOWARD'S ART OF COMPUTATION. of a unit, resulting from continually dividing by 12, as 1, T V, T i , TT V , etc. EXCHANGE, the receiving or paying of money in one place for its value in another, by order, draft, or bill of exchange. FRACTION, part or parts of a whole number or unit, thus J, three-fourths, |, one-fifth. An improper fraction is a fraction whose numera- tor exceeds its denominator. FACTORS, numbers, from the multiplication of which proceeds the product ; thus, 3 and 4 are the factors of 12. FIGURE A figure is a written sign representing & number. INTEGER An integer is a whole number or sum. INTEREST, the price or sum per cent, derived from the use of money lent. Simple interest is that which arises from the principal sum only. Compound interest is that which arises from the principal and interest added interest on interest. MATHEMATICS, the science of quantities. MULTIPLICATION, adding to zero any given num- ber as many times as there are units in the mul- tiplier. MULTIPLIER, the number that multiplies ; the multiplier must be an abstract number. MULTIPLICAND, the number multiplied. MENSURATION is the art of measuring lengths, eurfaces, and solids. DEFINITIONS AND SIGNS. 11 MULTIPLE, a quantity which contains another a certain number of times without a remainder. A common multiple of two or more numbers contains each of them a certain number of times, exactly. The least common multiple is the least number that will do this; 12 is the least common multiple of 3 and 4. NUMBER, a number is a unit, or a collection of units. A prime number is one that cannot be re- solved, or separated into two or more integral factors. NOTATION, writing numbers. NUMERATION, reading numbers. NUMERATOR, the number placed above the line, in fractions; thus, j; (five-ninths), five is the numerator. POWER A power is the product arising from multiplying a number by itself, or repeating it several times as a factor; thus, 3x3x3, the product, 27, is a power of 3. The exponent of a power is the number denoting how many times the factor is repeated to produce the power, and is written thus: 2 1 , 2 3 , 2 3 . 2 1 = 2 1 = 2, the first power of 2. 2x2=2 2 =4, the second power of 2. 2 X 2 X 2 = 2 3 = 8, the third power of 2. PRINCIPAL, the sum lent on interest, or invested. PER CENT., from per centum, signifying by the hundred ; hence, 1 per cent, of anything is one-hun- dredth part of it, 2 per cent, is one-fiftieth, etc. 12 HOWARD'S ART OF COMPUTATION. QUADRANGLE, the name of a figure with four sides. QUANTITY is anything that can be increased, diminished, or measured. RATIO, the quotient of one number divided by another. RECIPROCAL is a unit divided by any number. The reciprocal of any number or fraction, is that number or fraction inverted ; thus the reciprocal of is i, off is ,f of 3 is T 3 T . RATE PER CENT., the rate per hundred. RULE A rule is the prescribed method of per- forming an operation. RADIUS, half the diameter of a circle. A right line passing from the center to the circumference. SUBTRACTION is the process of finding the differ- ence of two numbers by taking one number called the subtrahend from another number called the minuend. SURFACE or SUPERFICES, the exterior part of anything that has length and breadth. SUPPLEMENT, the difference of a number and some particular number below it; thus 13, taking 10 as :he base, the supplement is 3, because the difference of 13 and 10 is 3. SQUARE, a figure having four equal sides, and four right angles. The product of a number DEFINITIONS AND SIGNS. 13 multiplied by itself; thus 16 is the square of 4. 4 X 4 16. SQUARE ROOT is the number which multiplied into itself, produces tlie number of which it is the root. 4 is the root of 16; 4 x 4 = 16. SPECIE, coin. SCALE A scale is a series of numbers regularly ascending or descending. A SOLID or BODY has length, breadth and thick- ness. SPHERE, a body in which every part of the sur- face is equally distant from the center. TRIANGLE, a figure with three sides. TERM The terms of a fraction are numerator and denominator taken together. UNIT A unit is one thing. VERTEX, the top of a pyramid or cone. ZERO, a cipher, or nothing. In arithmetic, the answer in each operation has a distinctive name. In addition it is called the sum; in subtraction, difference or remainder; in multiplication, the product; in division, the quo- tient. 14 HOWARD'S ART OF COMPUTATION. NOTATION. All numbers are represented by the ten follow- ing figures : s 1 g js I I 1 I O+j+3Scacea5flo 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. To establish their significance clearly in the mind of the pupil it will be of great advantage occasionally to write and read them in the follow- ing manner : i 1 5 ?i ^ si so ice, co GO " so cr> 02 cn o o o g o> s o> o o> i!| o u o *3v o> S 3> S a jt fl Ja fl S-g ^ .8 a S a.SP S .9 ^ o S 1234567 TTTT T T T The different values which the same figures have, are called simple and local values. The -simple value of a figure is the value it expresses when it stands alone, or in the right hand place. The local value of a figure is the increased value which it expresses by having other figures placed on its right. Ten is expressed by combining one and cipher, thus, 10; two and cipher combined make twenty, thus, 20, etc. A hundred is expressed by combin- ing the one and two ciphers, thus, 100; two NUMERATION. ] 5 hundred thus, 200, etc. Ten ones make a ten ; ten tens make a hundred ; ten hundreds make one thou- sand; that is, numbers increase from right to left in a ten-fold ratio. Each removal of a figure one place to the left increases its value ten times. NUMEEATION. ! 1 I . .9 .2 s s a ^3'!.^Z2 GoC5ScQ 1 j 1 1 1 I 1 1 I 1 I 1 HQPQ!z;Occc/2O > O > HW^H> 121,227,196,497,821,415,716,219,304,196,218,316,415,207,126. To read numbers expressed by figures : Point them off into periods of three figures each, com- mencing at the right hand ; then, beginning at left hand, read the figures of each period in the same manner as those of the right hand period are read, and at the end of each period pronounce its name; thus, 121 tredecillions, 227 duodecil- lions, 196 undecillions, 497 decillions, 321 nonil- lions, 415 octillions, 716 septillions, 219 sextillions, 304 quintillions, 196 quadrillions, 218 trillions, 31B billions, 415 millions, 207 thousands, 126. 16 HOWARD'S ART OF COMPUTATION. ADDITION. Variotas suggestions have been made referring to improved methods of addition. In nearly every ease the proposed improvement has been more fanciful than real. In practice, I have found no better or quicker method than the following: 3746 8743 6978 1256 3021 23744 Commence at the bottom of the right hand col- umn; add thus, 7, 15, 18, 24; set down the 4 in unit's place, and carry the two tens to the second column; then add thus, 4, 9, 1C, 24; set down the 4 in ten's place, and carry the two hundreds to the third column, and so on to the end. Never add in this manner: 1 and 6 are seven, and 8 are 15, and 3 are 18, and 6 are 24. It is just as easy to name the sum at once, omitting the name of each separate figure, and saves two thirds of time and labor. Book-keepers and others who have long columns of figures to add will find the following methods and suggestions acceptable. ADDITION. 17 Rule of addition for two columns at once : first practice adding two columns of two figures each, until you are able to grasp at a glance, and pronounce their sum. 23 15 32 38 87 56 14 33' 44 57 41 78 37 48 76 95128134 Add from the left, and say three seven, four eight, twelve eicfht-, &c., &c., instead of thirty-seven, forty-eight, one hundred and twenty-eight, &c., &c. ; this habit is readily acquired and saves half the time. When you can instantly, at sight, name the sum of two pairs of figures, practice with gradually increasing columns of pairs, then take examples consisting of two or more columns of pairs. 36 2147 41 3472 47 74 * 1463 83 22 4614 2634 32 36 2123 7843 1785 21 41 4679 2183 6823 183 250 6802 ]4640 18324 * The process is twelve six, one four naught ; the 40 is put down and the 1 carried to the units column in the next pair, then ten naught, one four six. Any person, who will PBACTICE this method, may add two columns with perfect ease: there is no royal road to this accomplishment : speed with precision can be attained only by persistent PRACTICE. .Fives are always easy to add ; so are 9's, when it is borne in mind that adding 9 to a sum places it in the next higher ten with the unit 1 less; thus, 17 + 9 = 26 : 89 -f 9 ^ 48 ; 63 + 9 = 72. 18 HOWARD'S ART OF COMPUTATION. In adding long columns of figures, write in e margin, lightly with pencil, opposite the st figure added, the unit figure of the sum immediately exceeding 100. By doing this the g mind is never burdened with numbers beyond 7 100; and if interrupted in the work, it can be 4 resumed at the stage at which the interruption 6 occurred. The example in the margin shows 7 the method; opposite the figure 7; the 2 indi- 9 eating the column, so far, with the 7 included, o 9 amounts to 102. INSTANTANEOUS ADDITION BY COMBINATION. Write two, three, four, or more rows of miscel- laneous figures, then write such figures as will make an equul number of nines in each column; under these again, write another row of miscella- neous figures. EXAMPLE 4987 47 3 G 2187 5012 one 9. 5263 two 9's. 7812 three 9's. 4986 RULE. Bring down the last row, less the num- ber of nines in each column, and prefix the number of nines. *This example has threo nines In each column. SUBTRACTION. 19 RULE. Write the numbers so that the units in the subtrahend shall be directly under the units of the same value in the minuend ; under, and in the same order, write the difference. Subtract 473 from 1694. 1694 473 .ia;i To prove Subtraction, add the difference to the subtrahend; if correct, their sum = the minuend. MULTIPLICATION. The base of our system of notation is 10; there- fore numbers increase and diminish in a tenfold ratio ; increasing from the decimal point to the left, and decreasing from the decimal point to the right ; hence to multiply any number by 10, annex a cipher, or remove the point one place to the right. To multiply any number by 100, annex two ci- phers, or remove the point two places to the right. To multiply any number by 1000, annex three ciphers, or remove the point three places to the right. To find the product of two numbers, when the multiplicand and the multiplier each contain but two figures. EXAMPLE 1 3 3 22 726 20 HOWARD' H ART OF COMPUTATION. EXPLANATION set down the smaller factor under the larger, units under units, tens under tens. Multiply the units of the multiplicand by the unit figure of the multiplier; thus, 2x3 = 6, set the 6 down in unit's place; multiply the tens in the multiplicand by the unit figure in the multiplier, and the units in the multiplicand by the tens figure in the multiplier; thus, 3x2 = 6, and 3x2 = 6, add these two products together; 6 and 6 are 12; set down 2, carrying the ten to the next product, then multiply the tens in the multiplicand by the tens in the multiplier; thus, 3x2 = 6; add the one carried from the last product, making the whole product 726. The same method can be applied when the in al- tiplicand has three or more figures. EXAMPLE 2 I 6 3 2 4 3912 The steps are: 3 X 4 =12, set dow'n the 2 and earry the 1 ; (6 x 4) + (3x2) + 1 = 31; set down the 1, and carry the 3. (1 X 4) + (6 X 2) -K 3 = 19; set down 9 and carry 1 ; 1 X 2 + 1 = 3, which place at the head of the line, making a total of 3912. When the multiplier can be resolved into two factors, it is sometimes shorter to multiply by each factor, than by the whole number. EXAMPLE, multiply 163 by 24. 8 X 3 = 24. MULTIPLICATION. 21 1 6 3 1304 3 3912. Ans. When the multiplier is any number between 11 and 20, the process is simply to multiply by the unit of the multiplier, set down the product under, and one place to the right o/, and then add to the multiplicand. EXAMPLE, multiply 1496 by 17. 1496 10472 254 3 2. Ans. or thus : 1496 1 7 25432 The process in the last example is : 6x7= 42, set down 2 and carry 4. 9x7 + 6 + 4 = 73; carry 7. 4x7 + 9+7= 44; carry 4. 1X7 + 4 + 4 = 15; carry 1. 1+1=2. To multiply two figures by 11. RULE. Between the two figures write their sum : thus: multiply 43 by 11. Ans. 473. The sum of 22 HOWARD'S ART OF COMPUTATION. 4 and 3 is 7 ; place the seven between the 4 and 3, for the product. NOTK. Add one to the hundreds when the sum exceeds 9. To multiply any number by 11. RULE Bring down the extreme right hand figure, then add the right hand figure to the next, and bring down the sum ; then add the second figure to the third and bring down the sum, adding in the figure carried, in each case, and so on to the end. EXAMPLE 12345678 11 135802458 To multiply any two numbers ending with 5. RULE. Add J the sum of the figures preceding the 5 in each number to the product of the same figures, and annex 25. NOTE. When the sum oi' the preceding figures Is an odd number, add hall 1 the number next smaller than the sum and annex 75. Multiply 85 by 65 and 105 by 35. 85x65=7+8X6 with 25 annexed =5525 105x35=6+10x3 " 75 " =3675 To multiply ivJien t/ie unit figures added, equal 10, and the tens are alike, as 67 X 63. RULE. Multiply the units and set down the result, then add one to the upper number in tens place, and multiply by the lower. MULTIPLICATION. 23 To multiply unlike numbers greater than a com- mon base. RULE. To the common base add the differences ; multiply the sum by the base and add the product of the differences. EXAMPLE. Multiply 603 by 612 603+12 x 600+3x12-369,036. To multiply unlike numbers less than a common base. RULE. To the multiplicand add the tens and units of the multiplier, less the last 1 to carry, mul- tiply the sum by the common base and add the product of the differences. EXAMPLE. Multiply 93 by 89 and 293 by 289. 89 293 93 89 8277 282x300+11x7=84,677. The product of any two numbers the square of their mean, diminished by the square of half their difference. EXAMPLE. Multiply 22 by 18. 20 2 2 2 = 396. To multiply two numbers having a common base, one ending ivith 25, the other ending with 75. RULE. Multiply the common base by one more than itself and annex 1875. EXAMPLE. Multiply 675 by 625. 6X7 with 1875 annexed=421,875. To multiply two numbers when either has one or more ciphers on the right,*as 26 by 20, 244 by 200, etc. 24 HOWARD'S ART OF COMPUTATION: RULE. Take the cipher or ciphers from one number and annex it, or them, to the other, multi- ply by the number expressed by the remaining figures. EXAMPLE 1. Multiply 20 by 20. Ans. 520. Process. 260 x 2 = 520. 2. Multiply 244 by 200. Ans. 48800. 24400 x 2 = 48800. RAPID METHOD OF SQDARIM NUMBERS, BY THE DIFFEREXCE OF A NUMBER AXD ITS BASJ;. For squaring a number greater, than Us base. RULE. To the given number add the differ- ence, multiply the sum by the base ; to the pro- duct add the square of the difference. NOTE. Take the nearest convenient multiple of ten for the base. EXAMPLE 1. What is the square of 11 ? Ans. 121. Process. Taking 10 for the base, the difference is one (1 + lljx 10 + I 2 = 121. NOTE. Until this rule is thoroughly understood, the learner should limit his exercises to numbers near 10, 100, 1000, Ac. ; and then operate with more complex numbers. 1. (22) 2 = 484. Process. Taking 20 for the base, the difference (2+ 22) x 20 + 2 2 = 484. 2._ (33)2 - 1089 SQUARING NUMBERS. 95 For squaring numbers less 'than the 'base. RULE. From the number to be squared subtract tie difference, multiply the result by the base, to the product add the square of the difference, 1. (9)8 = 81. Process. Taking 10 for the base, the difference or complement is 1. then (91) X 10 + I 2 = 81. NOTB. Iii squaring numbers between 60 and 60, take 50 for the base; to 25 add the difference, call the stun hundreds, to this add the square of the difference. s - 2601. Process. 25 + 1 = 2600 + I 3 x - 2601. 2. (52) 2 = 2704 NOTB. In squaring numbers between 40 and 50; to 15 add the unit figure, call the number hundreds, to the sum add the square of the difference, taking 50 for the base, 1. (41)* = 1681. Process. 15 + 1 = 1600 + 9 2 .= 1681. 2. (42) 2 = 1764. 3.__(43)s 1849. By this rule the squares of all numbers up to 1000, and larger numbers near the multiples of 10 may be found with less labor than is required to find them in tables; The square of any number ending with 25=half the number of hundreds -j- the square of the num- ber of hundreds X 10,000+625. 3+6 2 XlO,000+25 2 =390,625:=625'* 26 HOWARD'S ART OF COMPUTATION. In squaring very hi'gh numbers, use the foregoing rule in connection with the following formula: "The square of any number = the sum of the squares of its parts, plus twice the product of each part by the sum of all the others." EXAMPLE. Find the square of 823,732 823,000 2 ^677,329,000,000 823,000x732x2= 1,204,872,000 732 2 = 535,824 678,534,407,824 WJien either the tens or the units are alike. RULE. Multiply the units, set down the unit figure of the product; multiply the sum of the un- like figures by one of the like figures, then multiply the tens figures together, adding the carrying fig- ures as you proceed. Multiply 92 by 97 and 74 by 24. 97 74 _92 _24 ~8924 1776 W7ien the units are alike and the sum of the tens is ten. RULE. Add one of the units to the product of the tens, and annex the product of the units. Multiply 74 by 34. 7X3+4 with 16 annexed -2516. To multiply any two numbers between 10 and 20. RULE. To the product of the units prefix 1, and add the sum of the units calling it tens. Multiply 18 by 14. 8x4 with 1 picfixed = 132. 132+12 tens,=252. DIVISION. 27 When the multiplier is a number near, and less, than a multiple of 10. RULE. Annex to the multiplicand as many ci- phers as there are in the next order of tens higher than the multiplier, subtract the product of the mul- tiplicand by the complement. Multiply 222 by 93. 22,200222x7=20,646. When both numbers have a cipher in the tens place. RULE. Write the product of the units, then the sum of the products of the upper hundreds by the lower units, and the lower hundreds by the upper units, prefix the product of the hundreds. Multiply 409 by 704. 704 409 287936 DIVISION. DIVISION is the process of finding how many times one number or quantity is contained in another. RULE. To the left and in a line with the divi- dend, write the divisor, separated by an arc. Take so much of the dividend as contains a number less than ten times the divisor ; the number of times the divisor is contained in that part of the dividend is the first figure in the quotient; annex the next unused figure of the dividend to the remainder to find the second figure of the quotient, and so on to the end. 28 HOWARD'S ART OF COMPUTATION. Divide 49654809 by 4. 4)49654809 Ans. 12413702^ Process The divisor 4 is contained in the first figure of the dividend once, therefore 1 is the first figure in the quotient: 4 is contained twice and 1 re- mainder in 9 ; 2 is then the second figure in the quotient : the next unused figure 6 annexed to the remainder 1=16: 4 is contained in 16 four times, and so on to the end. Divide 7983204 by 23. 23)7983204(347095^ 108" 163 220_ "134 19 Process. 79 23x3, the remainder is 10; the next unused figure in the dividend 8, annexed to 10=10&; 108 23x4, the remainder is 16; to this remainder annex the next unused figure in the dividend, and so on until the quotient is complete. When the divisor is a composite number, divide by its factors. EXAMPLE. Divide 504 by 42. 42=7x6. 6504 7 JB 12 Ans. FRACTIONS. FKACTIONS. GENERAL PRINCIPLES OF FRACTIONS. % Multiplying the numerator, multiplies the frac- tion. Dividing the numerator, divides the fraction. Multiplying the denominator, divides the frac- tion. Dividing the denominator, multiplies the frac- tion. Multiplying or dividing both terms of the frac- tion by the same number, does not ghange its value. Fractions are 'called similar when they have a common denominator, as |, |- , f , |. Dissimilar fractions are fractions that are not alike, as |, *, f, -J. The numerators of similar fractions only can be added. The common denominator is written under the sum or difference. To reduce a fraction to its simplest f own. RULE. Divide both terms by their greatest com- mon divisor or its factors, the simplest form, or lowest term of f, is obtained by dividing both terms by 12, ff = f. 30 HOWARD'S ART OF COMPUTATION. To find the greatest common divisor of two numbers : RULE. Divide the greater by the less, arid the previous divisor by the remainder, and so on until there is no remainder ; the last divisor is the answer. Find the greatest common divisor of 18 and 27. 18)27(1 18 0)18(2 18 Ans. 9. To find the least common multiple: RULE. Cancel all the numbers that are con- tained in any of the others; divide all those not canceled by any number, or the greatest of its fac- tors, that will exactly divide any one of them, bring down eaeh quotient with the undivided numbers and proceed as before, until no two numbers have a common divisor; the product of all the divisors and the remaining numbers is the answer. Find the least common multiple of 36, 8, 9, 10, 12,25,84,75. Ans. 12x3x2x7x25=12600. 12)36, S, 9, 10, 19, gg, 84, 75 3, 2, ?, 7, 25 ADDITION OF FRACTIONS. RULE. Make the fractions similar by reducing them to the same denominator; add the numera- tors, and place the sum over the common denomi- nator. 1. What is the sum of -| and ? Ans. fj, 2. What is the sum of f and ? Ans. 1 T V FRACTIONS. 31 SUBTRACTION OF FRACTIONS. RULE. Make the fractions similar by reducing them to the same denominator, and write the dif- ference of the numerators over the common de- nominator. 1. From f take -^ Ans. j> Process, = f , f f = . 2. From 9J- take 4. Ans. 4J-. 3. From 8 take 3J. Ans. 5^. 4. From 18| take 3$. Ans. 15 T 8 7 . MULTIPLICATION OF FRACTIONS. RULE. Multiply the numerators together for a new numerator, and the denominators together for a new denominator. EXAMPLE. Multiply by f. General rule for multiplying fractions and all mixed numbers. RULE. Multiply the whole numbers together, then multiply the upper whole number by the lower fraction, then multiply the uppei fraction by the lower whole number, then multiply the frac- tions together, and add all the products together. 1. Multiply 8 by 4. Ans. 36|. 32 HOWARD'S ART OF COMPUTATION. When the multiplier, or divisor is an aliquot part o/lOO or 1000, the process may be shortened by the use of the TABLE OF ALIQUOT PARTS. 12 is I part of 100. 8 is T ^ part of 100 25 is f or J of 100. 16f is T * y or \ of 100 37| is f part of 100. 33 is T % or \ of 100 50 is | or | of 100. 66f is T ^ or f of 100 62| is | part of 100. 83$ is {f or f of 100 75 is | or f of 100. 125 is part of 1000 87 is part of 100. 250 is f or of 1000 -6 is T V part of 100. 375 is | part of 1000 18f is T 3 g-partof 100. 625 is f part of 1000 100. 875 is \ part of 1000 To multiply by the aliquot part of 100. NOTE. If the multiplicand is a mixed number, reduce the fraction to a decimal. RULE. Multiply by 100, by annexing two ci- phers; such part of the product as the multiplier is part of 100 will be the answer. EXAMPLE. Multiply 86 by 12|. Ans. 1075. To divide by the aliquot part of 100 or 1000. RULE. Reduce the fraction, if any, to a decimal, remove the point two places to the left for 100, three places for 1000 and multiply the quotient by the part the divisor is of 100 or 1000. 47825---100x8=47825-^12i=3826. FRACTIONS. 33 To multiply any two numbers together, ending with I, as 9i by 3. RULE. To the product of the whole numbers, add half their sum, plus ^. NOTE. When the sum is an odd number take half the next num- ber below it, and the fraction in the answer will be %. 1. What will 9J Ibs. of rice cost, at 3^- cts. per Ib? Ans. 33 J cents. Process. The sum of 9 and 3 is 12; half this sum is 6; then we say 9 times 3 is 27, and 6 = 33; to this add . 2. What will 9J doz. buttons cost, at 8 cts. per doz ? Ans. 80f cts. 3. What will 11| Ibs. of beef cost, at 9 cents $er Ib? Ans. $1.09^. 4. What will 7-J- doz. eggs cost, at 13 J cents per doz? Ans. $1.01 J. To multiply any two numbers together having the same fraction. RULE. To the product of the whole numbers, add the product of their sum by the fraction; to this add the product of the fractions. 1. What will 13f Ibs. of beef cost, at 7J cents per Ib? Ans. $1.06-^. Process. The sum of 13 and 7 is 20, three-fourths of this sum is 15, so we say, 7 times 13 is 91, and 15 = 106. to which add the product of the fractions, ( T \) and the result js the Ans. $1.06 T V 34 HOWARD'S ART OF COMPUTATION. In actual business calculations, any fraction less than a cent is reckoned as one cent; therefore in dealing with such questions as 13J pounds of beef at 7J cents a pound, it is sufficiently accurate to say : 1 6 oM3=3. of 7=2. 13><7+3+2 =96 cents; Or 17 J Ibs. of cheese at 9 J cents per pound. J of 17=6. i of 9=2. ffx9+6+2=$1.61. When the whole numbers are alike, and the sum of the fractions is a unit. RULE. Take the product of the whole numbers, to this add the integer in the multiplicand, then add the product of the fractions, and the result will be the answer. 1. Multiply 2| by 2. Ans. 6J. * Process 2 x2 + 2 = 6+|Xi = 6. 2. 31 X by 3f = 12f. 3. 7| X ?i = 56^- 4. 9f X 9f = 90|f. 5. 19| X 19f = 380tf 6. 101 1 X 101-i = 10302^. 7. 109 T V X 109 T 4 3 = 11990 T \V 8. 98 T V X 98^ = 9702 T W 9. 96} X 96f = 9312|f 10. 9947|| X 9947 T 6 T = 98952756^. 11. 9995714 X 99957JV = 9,991,501,806 T V^. DECIMALS. 35 DIVISION OF FRACTIONS. RULE. Reduce whole and mixed numbers to the form of an improper fraction. Multiply the divi- dend by the divisor inverted. Divide 8 by 1J. Ans. 65. Process 1^ inverted is 5. 5X1=^=65. To divide by any number expressed by 1 and any number of ciphers, remove the decimal point as many places to the left as there are ciphers in the divisor. 74864-1000=74.864 DECIMALS. The system of Decimal fractions is so pre-eminently simple, that when it is generally understood it will entirely displace the clumsy system of common frac- tions. In harmony with our system of notation, it is a fraction always having some power of ten for a denominator: thus .1 = T V, .03 = T 3 648 126486568 by 99000. - 12 g _ 1277.64208 Ans. The answer to this exarnple may be found by removing the point three places to the left and dividing by 9x11. 126486.568 14054.0631 1277.6421 The labor of finding the answer to valueless decimals may be saved by cutting off a figure from the right hand of the divisor, as each new figure in the quotient is found, carrying what would have been obtained by the multi- plication of the figure cut off, 1 if the multiplication pro- duces more than 5 and less than 15, 2 if more than 15 and less than 25, etc. 73.412)648.7654386(8.8373 587.296 58729 4 6 83~ 36 2739 2202 537 513 23 22 478 884 5946 0236 1 5740 73.412)648.7654386(8.8373 587.296 "61469 58730 2739 2202 537 514 .23 22 1 PROPORTION. 39 PROPORTION. Proportion is the equality of ratios. Ratio is the relation which one quantity bears to another of the same kind, with reference to the number of times that the one is contained in the other. Thus, the ratio of 7 to 21 is 3, because 7 is con- tained 3 times in 21, or 21 is 3 times seven. The same result is obtained if we divide 7 by 21, .for we then find 7 T = , which means that 7 is of 21, and this expresses the very same relation as before, to say that 7 is -J- of 21 is precisely the same as to say that 21 is 3 times 7. The ratio of 9 to 27 is 3, but we have seen that the ratio of 7 to 21 is also 3, therefore, the ratios of 7 to 21 and 9 to 27 are the same, 21 -~- 7 = 27 -f- 9, and these quantities are thefore called proportionals. In any proportion, as 7: 21:: 9: 27 the product of the middle numbers, 21 and 9, equals the product of the extremes, 7 and 27 : hence the rule, that when the fourth proportional is unknown, Multiply the second and third terms, and divide the product by the first. EXAMPLE. If 7 sheep cost 21 dollars, what will 9 cost at the same rate ? 27 dollars, Ana. 40 HOWARD'S ART OF COMPUTATION. 2d term, 21 3 3d term, 9 Or thus, #L X 9 = 27 1st term, 7)189 f 27 Proportion is so much used in business, and may be simplified and shortened so much by the fore- going process of cancellation, that the pupil must learn both before he can hope to be expert with business calculations. CANCEL ING IN CALCULATION. Whenever it is re- quired to multiply two or more numbers together, and divide by a third, the first step is to state the problem in its most manageable form ; this can only be done by the use of the arithmetical signs. The statement 28 X 1 2 14 is to be read, 28 multiplied by 12 is to be divided by 14. Stating the problem as above we see at a glance if the divisor is contained, and how many times, in either of the multipliers. In the foregoing example the divisor, 14, is con- tained twice in the multiplier, 28 ; then cancel the 14 and substitute 2 for the 28, and say, twice 12 is 24 the answer. Process, 2 n x 12 24. 14 PROPORTION. 41 EXAMPLE. It 9 turkeys cost $18, what will be the cost of 27? 3 is x n = $54, Answer. $ If the divisor is not contained evenly in either of the multipliers, there may be a common divisor for the divisor itself and one of the multipliers ; if so, the common divisor may be used in cancel ing,, thus: 7 0? X 8 = 18*, Ans. A glance shows that -9 is the common divisor for 63 and 27. When a common divisor has been used to change the expression of the divisor and one of the mul- tipliers, the new divisor may be cancel ed when it is contained an even number of times in the other multiplier. EXAMPLE 7 2 0? X $ Process 36 and 63 divided by 9, the common divisor, becomes 4 and 7 respectively, the 4 into 8, 2 times, cancel 4 and 8, and twice 7 is 14, the answer^ 42 HOWARD'S ART OF COMPUTATION. Summary of the rapid process for cancel ing. 1. Draw a horizontal line ; above the line write dividends only ; below the line write divisors only. 2. If there are ciphers above and below the line, erase an equal number on either side; 1 standing alone may be disregarded. 3. If the same number stands above and below the line, erase them both. 4. If any number on either side of the line will divide any number on the other side of the line without a remainder, divide, and erase the two numbers, retaining the quotient figure on the side of the larger number. 5. If any two numbers on either side have a common divisor, divide them by that number, and retain the quotients only. 6. Multiply all the numbers above the line for a dividend, and those below the line for a divisor j divide, and the quotient is the answer. 7. Write all the terms of the same kind in units, or fractions, of the same denomination* i. e., feet, or fractions of a foot ; yards, or fractions of a yard. EXAMPLE. If 7 inches of velvet cloth cost 2 dollar*, what will be the cost of 7 yards? $90, Ans. 18 5 K H Process, - X - X = 90. % 1 H XOTE. 2 dollars = f , 7 yards = |, 7 inches = r of a yard, ^\ inverted is \ 6 -. CANCELLATION. 43 If an upright line is used put dividends on the right, and divisors on the left. In stating a question put the term of the same kind as the required term first, at the top, on the right of the line ; then the other terms in pairs of the same kind ; if the effect is to increase- the answer, put the larger term on the right, and vice versa. EXAMPLE: If 5 compositors, in 16 days of 14 hours long, can compose 20 sheets of 24 pages in each sheet,, 50 lines in a page, and 40 letters in a line, in how many days of 7 hours long may 10 compositors com- pose a volume containing 40 sheets, 16 pages in a. sheet, 60 lines in a page, and 50 letters in a line, 1 of the second set of compositors being equal tt> 2 of the first ? Ans. 16 days. Days Compositors . 10 Hours 7 Sheets 20 Pages 24 Lines 50 Letters 40 Ratio 2 16 required term. 5 less time with 10 than 5 men. 14 more days with 7 than 14 hours a day.. 40 more time to set 40 than 20 sheets. 6 less time to set 16 than 24 pages. fiO more time to set 60 than 50 lines. 50 more time to set 50 than 40 letters. 1 NOTE. Excepting the upper term 16, the numbers on one side ex- actly balance the numbers on the other, and may all be canceled. This method acts like a pair of scales, we use known; to find the value of unknown quantities ; the arrange- ment of the terms is so very plain and natural as to be easily apprehended ; by its use the most complex problems are simplified, and all business calculations made with very few figures, and very little mental? effort ; it is accurate, and free from the risk of error. 44 HOWARD'S ART OF COMPUTATION. To compute Interest by Cancellation. 1st, on the right of the line write the principal, the time in days, and the rate per cent. 2nd, on the left the number of days, or its factors, in the year, and remove the decimal point two places to the left. Find the interest on 428.10 for 146 days at 5 per cent, per annum -of 365 days. tt 42.85 ) 140 2 > =8.57 = <8,,11,,5. 99 ) Find the interest on $99 for 23 days at 4 per cent, per annum of 360 days. 10 4 V =.253 = 25 cents, 3 mills. 23 In computing interest at rates per cent, per month, write principal, time and rate as above ; write 3 on the left of the line and remove the decimal point three places to the left. Find the interest on $348 for 24 days at 1J per cent, per month. 348) 24 V =3.480 = $3.48. 5 I Legal Interest is reckoned on the basis of 3G5 days to the year, when this is required, and the cal- culation is made on the basis of 360 days, subtract 7*3 for the common year, or %\ for a leap year, and the legal interest will be shewn; about 1J cents for each dollar of interest. RULES FOB FARMERS. 45 RAPID RULES FOR FARMERS. The practice of buying or selling grain by the 100 pounds, or the cental system, is becoming almost universal, and has many advantages over the bushel. The following rules for finding the relative values of the bushel and the cental are easy to learn, and true and rapid in execution. To find the value per cental tvhen the price per bushel is given,. RULE. Set down the price per bushel; remove the decimal point two places to the right, and divide by the number of pounds in the bushel. EXAMPLE. If wheat is $1.80 per bushel, what is its value per cental ? Ans. $3. Process 60)180 To find the value per bushel when the price per cental is given. RULE. Set down the price per cental ; multiply by the number of pounds in the bushel, and remove the decimal point two places to the left. 46 HOWARD'S ART OF COMPUTATION. EXAMPLE. If wheat is $3.00 per cental, what is the value of a bushel? . Ans. 1.80. Process 3.0 6 1 .8 RAPID RULE FOR RECKONING THE COST OF HAY. RULE Multiply the number of pounds by half the price per ton, and remove the decimal point three places to the left. EXAMPLE. What is the cost of 764 Ibs. of hay at $14 per ton? Ans. $5.348. Process 764 = 7 5.3 48 NOTE. 'The above rule applies to anything of which 2,000 pounda is a ton. To Measure Grain. RULE. Level the grain; ascertain the space it occupies in cubic feet; multiply the number of cubic feet by 8, and point off one place to the left. EXAMPLE. A box level full of grain is 20 feet long, 10 feet wide, and 5 feet deep. How many bushels does the box contain ? Ans. 800 bush. RULES FOR FARMERS. 47 Process 20 X 10 X 5x8 10= 800. Or, 1 Oft. 8 8 00.0 NOTE. Exactness requires the addition to every one hundred bnshela of .44 of a bushel The foregoing rule may be used for finding the number of gallons, by multiplying the number of bushels by 8. If the corn in the box is in the ear, divide the answer by 2, to find the number of bushels of shelled corn, because it requires two bushels of ear corn to make one of shelled corn. RAPID RULES FOR MEASURING LAND WITHOUT INSTRUMENTS. In measuring land, the first thing to ascertain is the contents of any given plot in square yards; then, given, the number of yards, find out the number of rods and acres. The most ancient and simple measure of dis- tance is a step. Now, an ordinary-sized man can train himself to cover 1 yard at a stride, on the average, with sufficient accuracy for ordinary pur- poses. To make use of this means of measuring dis- tances, it is essential to walk in a straight line ; to do this, fix the eye on two objects in a line straight ahead, one comparatively near, the other remote; 48 HOWARD'S ART OF COMPUTATION. and, in walking, keep these objects constantly in line. Farmers and others by adopting the following sim- ple and ingenious contrivance, may always carry with them the scale to construct a correct yard measure. Take a foot rule, and commencing at the base of the little finger of the left hand, mark the quarters of the foot on the outer borders of the left arm, pricking in the marks with indelible ink. To find the area of a Jour-sided figure, two of ivhicJi sides are parallel. RULE. Multiply the length and the breadth to- gether, and the product is the area. To find the area of a square, square one of its sides. RULE. When the length of two opposite sides is unequal, add them together, and take half the sum and multiply by the breadth. EXAMPLE 1. How many square yards in a square piece of land, 101 yds. on each side? Process 101 2 = Ans. 10,201 yards. EXAMPLE 2. How many yards in a piece of land 60 yards long and 20 yards wide? Ans. 1200. Process 600 x 2 = 1200. RULES FOR FARMERS. 49 EXAMPLE 3. How may yards in a piece of land, one side is 40 yards long, and the other side 60 yards long, parallel sides being 10 yards apart? 40 + 60 X 10 Process, = 500. 2 500 yards, Ans. To find the area of any three-sided figure. RULE. Multiply the longest side into one-half the distance from this side to the opposite angle. EXAMPLE. What is the area of a triangular plot of land, the longest side of which is 80 yards, and the shortest distance from this side to the opposite angle 40 yards ? 40x80 Process, 1600 yds. Ans. 2 To find how many rods in length will make an acre, the width being given. RULE. Divide 160 by the width, and the quo- tient will be the answer. EXAMPLE. If a piece of land be 4 rods wide, how many rods in length will make an acre ? 160 -7- 4 = 40 rods Ans. 50 HOWARD'S ART OF COMPUTATION. To find the number of acres in any plot of land, the number of rods being given. RULE. Divide the number of rods by 8, and the quotient by 2, and remove the decimal point one place to the left. EXAMPLE. In 6840 rods, how many acres ? 42 f acres Ans. Process. 8)6840 ~42?T5 To find the number of acres, the number of yards being given. Divide the number of yards by 4840 or its factors. EXAMPLE. Find how many acres in 21,780 yds. 21,780 A circle encloses the largest area within the short- est fence. The length of a circular fence = the square root of the area X 1JX3}. Find the length in yards of a circular fence to enclose 10 acres. ^48-100=220. 220xlfcX3J=780 yards. A square plot of the same area requires a fence 880 yards long. The largest area enclosed within the shortest fence, hi u rectangular plot, ia u square. 93 J yard* offence will enclose a square plot of two acres; it would require 2 mile; and 2 rods* offence to en- same area in a rectangular plot 1 rod wide. RULES FOR MECHANICS. 51 RAPID RULES FOE MECHANICS. To LAY OFF A SQUARE CORNER. Measure off eight feet from the end of one sill, and there make a. mark ; then measure off six feet on the sill lying at right angles with the first, and make another mark; then lay on a ten foot pole, one end of it squarely with the first mark. Move the sill in or out until it exactly squares with it. The figure thus made in marking off the sills, and in the laying down the ten foot pole is a right angle triangle. 8 ft. Another method for laying off a square corner. Take a measure and lay off with it a triangle, one side of which is lour feet long, another three feet, and the remaining side five feet. This trian- gle will be right angled, and the two shorter sides will serve to lay off an exact square. 52 HOWARD'S ART OF COMPUTATION. To Measure Grindstones, or any Cylinder. KTTLE. Multiply the square of the radius by the thickness, both in feet, or fractions of a foot, and the product by 3| ; or Multiply the square of the diameter by the thick- ness, both in inches, and divide by 2200, the answer is in cubic feet. EXAMPLE. How many feet m a grindstone 24 inches in diameter and 4 inches thick ? 1st Method. 2nd Method. 1 X 1 X 22 24 X 24 X 4 = 1-04 = 1-04 ft. 3x7 2x11 Measure of Superfices and Solids. Superficial measure is that which relates to length and breadth only, not regarding thickness. It is made up of squares, either greater or less, according to the different measures by which the dimensions of the figure are taken or measured. Land is measured in this way, its dimensions being taken in inches, feet and yards, or links, rods and acres. The contents of boards also, are found in this way, their dimensions being taken in feet and inches. The standard of measure is as follows: 12 inches in length make one foot of long measure ; therefore, 12 x 12 = 144, the square inches in a superficial foot. RULES FOR MECHANICS. 53 1. If the floor of a room be 20 feet long by 18 feet wide, how many square feet are contained in it? Ans. 360 feet. Process 180 X 2 360. 2. If a board be 4 inches wide, how much in length will make a foot square ? Ans. 36 inches. Process 144 divided by the width, thus, ij = 36. 3. If a board be 21 feet long and 18 inches broad, how many square feet are contained in it ? Ans. 31 sq. ft. Process Multiply the length in feet by the breadth in inches, and divide the product by 12. 3 21 X = 31*. n 2 Or thus, 18 inches equals 1 ft.; 21 x H = 31. To measure a board wider at one end than the other, of a true taper. RULE. Add the widths of both ends together; halve the sum for the mean width, and multiply the mean width by the length. 54 HOWARD'S ART OF COMPUTATION. EXAMPLE. How many square feet in a board 20 feet long, 9 inches in width at one end, and 11 inches at the other? Ans. 16f sq. ft. Process 9 + 11 20 X 10 = 10 in., mean width; 16| k 2 12 To find the board measure of planks and joists. RULE. Find the contents of one side of the plank or joist by the preceding rule, and multiply the result by the thickness in inches. EXAMPLE. What is the board measure of a plank 18 feet long, 10 inches wide, and 4 inches thick? Ans. 60 ft. 18 X 10 X 4 Process = 60. 12 TJie diameter being given, to find tJie circumference. RULE. Multiply the diameter by 3|. EXAMPLE. What is the circumference of a wheel the diam'eter of which is 42 inches ? Ans. 1 1 ft. RULES FOR MECHANICS. .% To find the diameter when the circumference is given, RULE. Divide the circumference by 3^. EXAMPLE. What is the diameter of a wheel, the circumference of which is 11 feet? Ans. 3 feet II 1 Process X = 3J 1 n 2 What is the width of a circular pond, 154 rods in circumference? 49 rods Ans. 1?4 7 Process X 49. 1 2 The diameter being given, to find the area. RULE. Multiply the square of the radius by 87. Find the area of a circle 36 inches in diameter. ' =7.07 feet. 2X2X 7 The length of a cylinder is equal to the capacity -the square of the radius-^-3 7. Find the depth of a circular cistern, 7 feet wide, containing 2400 U. S. gallons. 2400X2X2X2X 7 15X7X7X22 =8.31 feet. 56 HOWARD'S ART OF COMPUTATION. To find how many solid feet a round stick of tim- ber of the same thickness throughout, will contain when squared. RULE. Square half the diameter, in feet, mul- tiply by the depth, and then by 2. Find how many solid feet, when squared, in a round log 2J feet wide and 10 feet long. General rule for measuring timber to find the solid CQntents in feet. RULE. Multiply the depth, in feet, or fractions of a foot, by the breadth, multiplied by the length. How many solid feet in a piece of timber 2 feet wide, 10 inches thick and 12 feet long. 2x5x12 6 .=20 feet. To find the contents of a true tapered pyramid, whether round, square, or triangular. RULE. Multiply the area of the base by J the height. How many cubic feet in a round stick of timber, truly tapering to a point, 1| feet in diameter at the base and 24 feet long. How many cubic feet in a square block of RULES FOR MECHANICS. 57 marble, truly tapering to a point, 24 inches on each side at the base, and twelve feet high. 24X24X4 ^ 2X2X4=16 feet, Ans. 144 Gangers' Work. To find the contents of a cask in gallons. RULE. Add two -thirds the difference of the head and bung diameters to the head diameter, to find the mean diameter; then multiply the product of the square of the mean diameter into the length by .0034. NOTE. If the staves are but little curved, add six-tenths instead of two-thirdi. How many gallons in a cask, length 40 in., head diameter 21 in, and bung diameter 30 in.? . .21-f(30 21X)=27 in. mean diameter. ...272X40X.0034 = 99.144 gallons. Bricklayers' Work Is sometimes measured by the perch, but more frequently by the 1000 bricks laid in the wall. The following scale will give a fair average for estimating the quantity of brick required to build a given amount of wall : 58 HOWARD'S ART OF COMPUTATION. 4^ in. wall, per ft., superficial, (4^ brick) 7 bricks. 9 (1 brick) H " 13 (l| brick) 21 18 " (2 bricks) 28 " 22 " " (24- bricks) 35 NOTE.- For eac'a half brick added to the thickness of the wall, add eeveu bricks. A bricklayer's hod measuring 1 ft. 4 in. x 9 in. X 9 in., equals 1,296 inches in capacity, and will contain 20 bricks. A load of mortar measures 1 cubic yard, or 27 cubic feet; requires 1 cubic yard of sand, and 9 bushels of lime, and will fill 30 hods. Plasterers' Work Is measured by the square yard, for all plain work- by the foot, superficial, for plain cornices; and by foot, lineal, for enriched or carved mouldings in cornices. Painters' Work Is computed by the superficial yard ; every part is measured that is painted, and an allowance is added for difficult cornices, deep mouldings, carved sur- faces, iron railings, etc. Charges are usually made for each coat of paint put on, at a certain price per yard per coat. Howard's New Rule 5y FOR COMPUTING INTEREST On a Dasis of One Per Cent for all Rates- Interest, in the various forms under which jt accrues, has so large a place in every day business transactions, that a rapid and accurate method of computing interest is one of the most indispensable items of business knowledge. The one that I present here, in all respects, is, without ex- ception, the jieicest, the easiest to learn, and use, the quickest and most correct in existence, adapted to all sums, all periods, and all rates per cent. Some of the reasons that suggested the construction of this rule 'will assist the learner in its acquirement A child first con- ceives the idea of ONE thing, by and by, it is able to count six, but it is a long time before it apprehends that the six counted is $ix ONES. THE UNIT ^or one thing is the idea of number in its sim- plest form, it is the basis of every number, the primary base of every fraction, the unit of six months is one month, the unit of a fraction is the reciprocal of the denominator, thus, | is the unit of | ; every step from the unit, increases the complexity of numbers, and consequently demands an increase of mental power and energy in dealing with them. The most popular Interest rule is the "six per cent" method, by this rule, on removing the decimal point two places to the left, the interest on any sum is shewn for ono-sixth of a year at six per cent. The interest on any sum is shown for one year at one per cent by the same act ; the latter, which retains the unit of both denominations, unchanged must be the most natural and simple basis of calculation, and by consequence, the easiest to learn and use, hence the following : Rule, Multiply .01 of the principal by the given time and the product is the interest at one per cent. Multiply the interest at one per cent by tJie given rate. NOTE 1, Multiply by easy fractions of a year, or month, and the result will be uniformly correct, and requires less than half the mental labor demanded by other methods, a little practice, and careful study of the details of the following examples will enable the learner to select instantaneously the easiest multipliers, 60 HOWARD'S ART OF COMPUTATION. NOTE 2. To multiply by .1 remove the decimal point out place to thek/J; by .01 too places; by .001 three places. What is the interest on 1000 for 11 yrs., lino and 6 days at 1 per cent per annum ? Yr. Mo. Da. 10.00 Int. for 1 " u 100.00 " " 10 " " first lineX 10 1.00 " " 16 " " X .1 111.00 " " 11 1 6 What is the interest on $846 for 1 yr. 7 mo., 12 da. at 1 per cent? Yr. Mo. Da. $8.46 Int. for 1 " " 4.23 " " " 6 " 1st line X because 6 mo. is a year. .705" " " 1 " 2d " X " 1 " " I of 6 mo. .282" " " " 12 3d " X.4 " 12 da " .4 of 1 mo. 13.677 " "~~1 7 12. What is the int. on $427.20 for 2 yr., 5 mo., 27 da at 6 pj )laces to the 35 days at 28 6 per cent. 7 99 10* 14 21 left, and the 12 per cent. 15 Example : What is the interest on 100 for fourteen days at 3 per cent, per annum ? 100 10 5 .115 NOTE. To multiply by to itself. Ans. 25. 3|d. k, add T \j and J of T \y of any number OF THE COMPOUND INTEREST. 65 Compound interest is interest on the principal, and also on the interest added to the principal, each time it becomes due. RULE. Multiply the principal by the rate, setting the product under, and two decimal places to the right of the principal ; the sum of principal and interest will be the amount. Or, find the amount of !, or $1, for the given time and rate, and multiply by the given principal. NOTE. To avoid writing decimals of no value, begin at the third decimal, adding in the figure carried, if any, from the right hand figures. Find the amount of 864 10s. Od. for six years at 8%. Ans. 1371 17s. 0|d. School Book Method, 184 Figures. 864]5 8 69160 8615 933.060 8_ 746928 _93366 1008.3528" Howard's Method, 74 Figures. 864.5 69.16 74.693 1008.353 80.668 1089.021 87.122 1176.143 94.091 1270.234 101.619 11371.853 80668224 10083528 1089.021024 8_ 8712168192 1089021024 1176.14270592 8_ 940914164736 117614270592 _ 1270.2341223936 8 1016l872979148a 1270.2341223936 1371.85285.2185088 66 HOWARD'S ART OF COMPUTATION. To repay a loan, principal and compound inter- est in a given number of equal qnnual payments. RULE. Multiply the amount of one pound, or one dollar for the given time and rate, by the inter- est for one year, and divide the product by the com- pound interest on a pound, or dollar, for the given time and rate EXAMPLE. What must the be one of six equal annual payments to discharge a loan of 864,, 10,, for six years at 8 per cent. 1.5869x69.16 .5869 NOTE 1. Persons having frequent occasion to com- pute compound interest may save time and labor by the use of a table showing the amount of one pound, or one dollar, for a series of years, or <5ther stated periods ; the amount of one pound, or one dollar, for the given time and rate, multiplied by the given number of pounds, or dollars, will be the amount sought. NOTE 2. To prove interest, divide the computed in- terest by the interest for one day, and the quotient should be the number of days in the example ; or divide by the interest for one month, and the quotient should be the number of months. DISCOUNT. Discount, being of the same nature as interest, is, strictly speaking, the use of money before it is due. The term is also applied to a deduction of so much per cent, from the face of a bill, or the deducting of interest from the face of a note before any interest has accrued. Banks generally include DISCOUNT. 67 in their reckoning both the day when the note is discounted and the day on which the time specified in it expires, which, with three days of grace, makes the time for which discount is taken four e the discount. EXAMPLE 1. Find the present worth of a note for 228 dollars, due 2 years from date at 7 per cent. Ans. 2. Find the bank discount on .a note for 1200, for 60 days at 6 per cent. 68 HOWARD'S ART OF COMPUTATION. 60 -|- 4 64 days time for which discount must be reckoned. of 64 lOf X 1200 = 12 800. , Ans. 12.80. Merchants are in the habit of deducting a certain percentage from invoices of goods sold. This is- reckoned in the same manner as interest. A bill of goods is bought, amounting to 960 dol- lars at a year's credit, the merchant offers to de- duct 10$ for ready cash, what amount is to be deducted ? 9.60 x 10 = $96.00, Ans. By discounting the face of bills, a loss may be sustained without suspecting it; this arises from the fact that the discount is not only made on the first cost of the goods, but also on the profits ; for instance, if a profit of 30% be made on any article of merchandise, and the 10$ be deducted, the gain at first sight would appear to be 20$, but is in reality only 11%. If a profit of 60$ be added to the first cost, and then a discount made of 45$, the apparent profit would be 15$; instead of this, an actual loss is made of 12$, as will be seen by the following examples : Example 1. Example 2. Cost of goods, $100 Cost, $100 Add 30$ profit, 30 Profit 60$, 60 Selling price, 130 Selling price, 160 Deduct 10$ discount, 13 Discount 45$, 72 Cash price, $117 Cash price, $88 Gain 17$. Loss .12$. DISCOUNT. 69 The net amt. of a bill, less 10 per cent discount, will be shewn by multiplying by 9, and removing the decimal point one place to the left. Example. 100X9=^90.0 To find the net. amt. less discount at 5 per cent X 9^. 30 per cent X7. 50 per cent X5. 15 " " X8. 35 " X6. 55 " " X4. 20 " " X8. 40 " " X6. 60 " " X4. 25 " " X7 45 " X5i- 70 " " X3. -. and remove the point 1 place to the left. EXCHANGE. EXCHANGE is the giving or receiving of any sum in one kind of money for its value in another. EXAMPLE 1. Find the value of gold, the price of greenbacks being 75 cents Ans. 100 4 Process = = 1.38J 75 3 2. Find the value of currency, the price of gold being 133J, Ans. 75 cents, 100 . 3 Process - = =.75 ___ $500 in gold at 8 per cent, premium will buy how much currency? $500xl.08==$54G $500 in currency will buy how much gold at 8 per cent premium? 5QO~108=$462.96. $1000 in gold is worth how much currency at 80 cents? 70 HOWARD'S ART OF COMPUTATION. What is the face value of a bill of Exchange cost- ing 1000. Commission f per cent ? 1000-^1.0075=992.55 What is the cost of a bill of Exchange for $1 000 Premium f per cent. $1000X1. Find the par value of 473 5 9 St'g. in Amer- ican gold coin. ' 473.2875 X 4.8665=$2303.25. 473.2875 Note. To avoid encumbering the operation with 56.684 valueless decimals, reverse the multiplier, and begin 1893.150 each line of the partial products with the product of 378.630 the multiplying figure and the figure directly above 28.397 it, adding what otherwise would have been carried. 2839 The par value of 1 st'g is fixed by act of Con- .237 gress 1873, at $4.8665. ~2303.254 BRITISH MONEY. Howard's new rules for INTEREST, EQUATION OF PAYMENTS, &c., may be used with equal facility in dealing with British and other foreign money. The British people would simplify all their mon- etary operations, and save millions every year in labor alone, by adopting the decimal system of coinage. The cost and temporary inconvenience incident to the change would be trifling, almost nil, in view of the advantage to be gained. The pound, the florin, the shilling and the sixpence might be retained. Make the smallest coin, the farthing, equal to the T17 Vo of a pound, and the thing is done. BRITISH MONEY. 71 NOTE. By carefully observing and practicing the following in- structions, the converting of shillings, pence and farthings into decimals of a pound, and vice versa, will become a purely mental and instantaneous operation. 1. For every two shillings, or florin, write .1, because two shillings is ^ of a pound stg. 2. For every 1 shilling, write .05, because one shilling is T 5 ^ of a florin, or ^fa of a pound stg. 3. For every ninepence, write .0375, because ninepenee is ^So of a pound stg. 4. For every sixpence, write .025, because six- pence is 100 of a florin or ^ of a pound stg. 5. For every threepence, write .0125, because threepence is j^w of a pound stg. 6. For the farthings, write the product of .00104 multiplied by the number of farthings. s. 1 = 20 .1 = 2 .05 = 1 .0375 = | .025 = i .0125 = j d. far. = 240 = 960 = 24 =96 = 12 =48 = 9 =36 = 6 =24 = 3 =12 .00104= ^ = 4" == 1 19,,2 = 19.1 27,,12,,6 =27.625 19,,3 = 19.15 19,,19,,2i=19.96 19,,5 = 19.25 19,,18,,0f=19.903 19,,19 = 19.95 19,,16,,lf=19.807 19,,18 = 19.9 24,, 1,,H=24.056 The learner may extend the exercises indefinite- ly, the essentials to remember are 72 HOWARD'S ART OF COMPUTATION. 1st. Each unit of the first figure to the right of the decimal stands for two shillings. 2d. Each 5 in the second figure to the right of the decimal, stands for one shilling. 3d. j^ach unit above or below c in the second figure, stands for 2 pence. 4th. Each unit of the third figure to the right of the decimal, stands for 1 farthing. NOTE. The exact value of each unit in the sec- ond figure to the right of the decimal is 2 T \- ol a penny, and of each unit in the third figure to the right ol the decimal, -f-^ of a penny, the difference of the assumed and the real value is too trifling to affect any actual business operation. The^/forms, shillings, ninepence, sixpence and threepence are decimally expressed absolutely correct. PERCENTAGE. The following examples embrace most of the conditions under which percentage occurs in busi- ness, and the mode of solution in each case applies to all similar examples. How many of 500 sheep will be left, if 20 per cent, of them are- sold ? 600X.20=100. 500100=400 sheep. What per cent of 300 is 75? 75-f-300=25 $> ct. Of what number is 48, 8 $> ct.? 48-^.08=600. Sold a horse for 60, made 25 $> ct., what did it cost? 1+.25=H{}=S I a * =i48 PERCENTAGE. 73 Sold a horse for 40, lost 20 $> ct. What did it cost? The population of a village increased from 900 to 1200, at what rate per cent, did it increase ? 1200-^-900=1.33^ 1=33 J per cent. The sales of a firm fell off from 12000 to 9000, what was the rate per cent of decline? 9000-^-12000=.75. 1 .75=25 per cent. Bought a horse for $80, sold it for $105. What per cent profit? 105-^-80=1.311 1=311 per cent. Bought a piano for $300, sold it for $250. What per cent, loss? 300250 -f- 300 = .16| per cent. Bought a horse for $40. What must it be sold for to gain 20 per cent? 40 X- 20=8+40=48 dollars. A horse was sold for $24; the rate per cent profit was the same as the number of dollars it cost. What was the cost, and what the gain per cent? Cost $20. 20 2 =400X.01=4. Profit $4, or Vvf the profit is .1 the cost. Vof 4=2x10=20 Cost $20. Profit 2Qpcr cent. How many dollars will earn 1 cent a day at 9 per cent per annum? 360-7-9=40. AnsMO. How many dollars will earn 1 cent a day at 1^ per cent per month? 30-f-lJ=24. ' 74 HOWARD'S ART OF COMPUTATION. STOCKS AND BONDS, Stocks and bonds are quoted in New York by so much on the hundred, premium or discount; in Philadelphia at their actual price. That is, if the par value of a stock is $50, and it is 6% above par, the New York quotation would be 106, the Phila- delphia quotation 53. When the premium is known, the par value plus the premium equals the market value. When at a discount, the par value minus the discount equals the market value. To find to what rate of interest a given dividend cor- responds. RULE. Divide the rate per unit of dividend by 1 plus or minus the rate per cent., premium or dis- count, according as the stocks are above or below par. What per cent will be gained by investing in 8 pet cent stock, at 20 per cent premium? 120 | 800=;62HT cent. What per cent will be gained by investing in 6 per ent stock at 10 per cent discount. 100 10=90. 90 | 600=63 per cent. To find at what price stock pay ing a given rate per cent, dividend can be purchased, so that the money in- vested shall produce a given rate of interest. Hi .- I.E. Divide the rate per unit of dividend by the rate per unit of interest. What must be paid for stock paying 6 per cent divi- dend t in order to realize on the investment 8 per cent? 8 | 600=75. EQUATION or PAYMENTS. 7 a HOWARD'S GhOXJDHiT DE^-CnLiE. FOR EQUATION OF PAYMENTS, AVERAGING ACCOUNTS and PARTIAL PAYMENTS,, is so called, not only because it is absolutely cor- rect, and consequently equally just to both Debtor and Creditor, but also because it is exceedingly sim- ple, and easy to learn and use. The methods hitherto in use are intricate, tedious, and perplexing, and more or less inaccurate: the PRO- DUCT methods requiring, with each item, the find- ing the number of days between two dates, and the use of difficult multipliers. The INTEREST methods introduce a superfluous element, and needlessly increase the complexity of the operation. Interest, really has nothing to do with finding when a balance is due. The object sought is a certain date, HOWARD'S GOLDEN RULE seeks and finds this and this only directly, accurately and easily. By its use the CASH BALANCE of the most complex Dr. and Cr. accounts may be easily found, without reference to interest, except where it properly belongs; viz, on the balance. The novel and special excellence of this rule consists in multiplying by months,and easy fractions of a month, and also in the simple and natural ar- rangement of the parts of the problem, the dates. themselves representing the multipliers. Experienced Accountants say, " it very much lessens the drydgery of the counting house." 76 HOWARD'S ART OF COMPUTATION. EQUATION OF PAYMENTS is the process of find- ing the EQUATED TIME, or the date when the sum of several debts due at different times may be paid and includes, Bills bought ou unequal time on the same date. Bills bought oil equal time on different dates. Bills bought 011 unequal time on different dates, and MONTHLY STATEMENTS. AVERAGING ACCOUNTS is the process of finding the date on which the BALANCE is due, and applies to all Dr. and Cr. accounts. PARTIAL PAYMENTS are parts of a debt paid at different times ; usually written on the back of notes and other interest bearing obligations, and called indorsements. The term also includes pay- ments made on account of a debt before it is due. TERM OF CREDIT is the time to elapse before a Dill becomes due. The AVERAGE TERM of credit is the time at the end of which the sum of several debts due at differ- ent dates may be paid at once. EQUATED TERM is the average time for which in- terest is due on an account, or balance, and is al- ways reckoned from the zero date. Interest is reckoned on accounts, and balances from the date on which they are due. AN ACCOUNT is a statement of business transac- tions between Debtor and Creditor. A BALANCE is the difference of two sides of an ac- count. A CASH BALANCE is the same, with the interest due. THE ZERO DATE is the date, or starting point, from which all the other dates are reckoned, in this rule it is always the beginning or starting point of the month in which the first debt in the acct. occurs. EQUATION OF PAYMENTS. 77 BILLS BOUGHT ON UNEQUAL TIME ON THE SAME DATE. 1878, Jan. 1st, Bought goods on 8 mos. 100 tc a " " u a b 7 i 4 100 100 On what date may the whole 300 be paid? Term of CrMo.. 8 6 7 Jan. 1. 100X8=800 100X6=600 100X1=700 300 )2100(7mo.fr.Jan.l,orAug.l, Under the terms of this transaction the Debtor is entitled to the use of 1st, 100 for 8 months, =8 times loo or-Soo for i mo. 2d, 100 " 6 " =6 " 100 " 600 " i " 3d. 100 "7 " =7 " 100 " Too " i " ' a credit equal to 2100 for 1 month ; this will evidently entitle the debtor to the use of 300 for as many months as 300 is contained in 2100. The product of any number of pounds multiplied by any number of months, and fractions of a month, a Debtor is entitled to use them, is the number of pounds he is entitled to use for 1 month under the same terms, hence the following ; Rule. Multiply each debt by its term of credit^ divide the sum of the products by the sum of the debts, and the quotient is the equated term. First study this very simple example thoroughly, make yourself -familiar with each operation, the reason for its use, and the causes of the results, and you will then have no difficulty in comprehending the most complex Debtor and Creditor accounts. 78 HOWARD'S ART OF COMPUTATION. BILLS BOUGHT OX EQUAL TIME AT DIFFERENT DATES. Required the equated time of paying the follow- ing bills each bought on 8 months credit. 1878 No of months, June Zero date from zero date Tune 9 X180 X .3 54 1 July 15 84 X 11 l 84 X 2 j 42 3 Sept. 14 240X3. Qi (720 8 H n 4 Oct. 10 96X 4J=ji 600 )1428(2.38 $ mo. da. yr. mo. da. 11.4 Equated term 2, 1 1 ) after 78, 6, zero date. Plus term of Cr. 8, j :=__ 10, 11, Equated time ~79, 4, if, or April llth, 1879. Rule, Multiply each debt by the time in months and fractions of a month, between its occurrence and the zero date, divide the sum of the products, by the sum of the debts i and the quotient is the equated term in months and hundredths of a month, counting from the zero date, add the term of credit^ and the sum is the equated time. NOTE 1. To reduce hundredths of months to days, multiply by 3, and point off the right hand figure, when the right hand figure in the product is 5 or more add 1 day, otherwise disre- gard it NOTE 2, When the figures representing the day of the month are multiples of 3, such as the 3d, 9th, 27th, &c. &c., multiply by tenths, because 3 days is .1 of a month ; when they are not multiples of 3 then multiply by the simplest fraction, or fractions of a month. In the above example, Sept. 14th,. 3 months 14 days from zero date, we multiply by 3.3 \ , 3 months, plus 9 days, plus 5 days. Facility in selecting the simplest fractions for multipliers is easily acquired by practice. . E t ssb EQUATION OP PAYMENTS. 79 BILLS BOUGHT ON UNEQUAL TIME AT DIFFERENT DATES. Required the equated time of paying the follow- ing bills of goods. Term of -i A Cr.Moa. April 6f " 10 To Mdse. 310X6J [1SGO 103 o 1 May 21 u i' 468X3.7 = 1404 328 4 2 June 1 u 520X6^ - 3120 IT 4500 3 3 July 8 u 750X6. 1 75 125 MO. Da. OAtQ Zero date 4 20 * 8 )11532{5.63 Equat id term 5 1 9 3 18.9 Rule Multiply each debt by the term of credit, plut the time "between the date of the transaction and the sere date-, divide the sum of the products by the sum of the debts, and the quotient is the equated term. The figures, on the extreme left represent the terms of credit; the figures on the left of the month represent the number cf months from the zero date, these together with the day of the month are the multipliers. moe. meg. da- 1st item 6 Cr. plus 0, 10 from datc=6J mos. 2d " 2 " " 1, 21 " " " =3,7 " 3d " 4 " " 2,' 1 " " " =6^ (> 5809 2474 1248 2474 836 412)3335 , 6, 8, 3 412 o. 3da before zero date. 76, 9, 27 Balance due Sept. 27th, 1876. In this example the balance of the products is on the smaller side of the account; when this happens the eqated term is deducted from the zero date to find the equated time. The credit side has the ADVANTAGE of the use of the equivalent of "3335 for one month, then the other side is entitled to interest on the balance for as many months as 412 is contained in 3335. Rule. Multiply each item "by the time between its occur- rence and the zero date, added to the term of credit if any divide the balance of the products by the balance of the account and the quotient is the equated term. Note, This rule applies to Partial Payments and all Dr. and Cr. accounts. R JI4. E 24J 82 HOWARD'S ART or COMPUTATION PAUTI A L PA Y M EXTS. A note is made March 15th, 1878 for T20. Endorsed April 3d, pd. on account 170 May 20th, " " " . 246 June 18th " " " 87 from what date must interest be computed on the balance. Apr t 8|170Xl.l = 170 17 Mar. 15 720 Xi 360 2 May 20 245 X2 = 490 164 502 3 Ju. 17 S7X3Ak=\*ll 218 I 14 502 1151 360 2 1 ) 7 9 1(3 . C 2 Sraos. 19 da. before date. Zero date 78, 3, Minus 3, 19 Equated time 7, 11, 11 Nov. llth, 1877. PAYM'TS MADE OX ACC'T OF A DEBT BEFORE IT IS DUE [878 Cr. 1878 Dr. March l By Cash $306i Jan. 1st, to goods on 6 mos. $1500 May 1 " ki 4001 On what date is the balance $800 due. Jan 1500X69000 2200 2 March l! 300X2= 600 800)6800 ~ 4 May 400X^=1600 700 2200 Ans. 8J mos. after Jan. 1st, Sept. 15th. Explanation,/ under the terms of this transaction the debtor is entitled to the use of $1500 for 6 months, equal to 6 time* 1509 or $9000 for 1 month on paying $300 in 2 moe. the use of which for that time is=$GOO for 1 mo. 400 * 4 " " ' " " " " " "= 1600 1 " he has used the equivalent of $2900 for 1 month and is conse- quently entitled to the use of the balance for a time equal to the use of $6800 for one month. T. 2bq jR. /or CASH BALANCES. 83 The Creditor is entitled to interest on the Balance from the date on which it is due, to the date of set- tlement. The Debtor is entitled to discount off the Balance for the time he pays it before it is due. Find the Cash Balance on each of the four preceding acc'ts. 1st, 1100 clue Nov. 4th, settled Aug. 22d, int. at 6 per cent. mo. da. Balance due 11, 4 Am'tofK Date of settlement, 8, 22 mo. da. (1.012 Difference, ~2, 12 for 2 12 ) Cash Balance 1100=1086.95. Ans. 2d, .412 due 27 | | 76, date of settlement 27 | 7 | 78.^^,. Yr. Mo. Da. 78 7 27 76 9 27 Yr. Mo. ~1 10 Int. for 1, 10, 45.32+ 4l2=f457.32. Ans. 3d, 218 due 11 | 11 | 77, dale of settlement 5 | 9 | 78 J"^ 68 ^ 78 9 5 77 11 11 MO. Da. 9 24 Int. for 9 24, 10.68+ 2l8=228-68. Ans. 4th, $800 due 78 | 9 | 15 | date of settlement 78 | 12 | 19 7 * ct Mo. Da. 12 19 9 15 MO. Da. 3 4 Int. for 3, 4, 14.62+S800=$S14.G2. Ans. TO FIND THE DIFFERENCE OF TIiME BETWEEN TWO DATES. Rule. Subtract the earlier Jom the latter date. Example. For what time must interest be charged on a debt due the first of May, 1873, and settled on the ninth of March, 1875. Process, 75 : 3 : 9 73 : 5 : 1 1 : 10 : 8 Ans. 1 yr. 10 mo. 8 days. NOTE. To compute on a basts of 365 days to the year, add one day for each month of 31 days ; deduct 2 days in the common year, and one day in leap year, for February. HOWARD'S ART OF COMPUTATION. 33 DE CALCUL POUR L'ESPACE DE TREKTE SIECLES. BEGLE. Des deux derniers chiffres de 1'an, rejetez tous ]es sept, tout en retenant le restant; divisez les deux derniers chiffres de Tan par quatre, re- tenant le quotient, suns tenir coinpte du restant, s'il y en a ; puis prcnez le jour du mois, ensuite le chiffre donne pour le mois, et finalement celui pour le siccle. Ayez toujours soin de rejeter les sept oil il y en a. Le chiffre 1 (un) restant represente le premier ; 2, le second ; &c., et (zero) le dernier jour de la semaine. TABLE DES CHIFFRES POUR LES MOIS. 1, Septembre et Dec. 3, Jan. et Oct. 5, Aotit. 2, Avril et Juillet. 4, Mai. 6, Fev., Mars, Nov. 0, Juin. NOTA. Dans l'ance bissextile le chiffre pour Janvier est 2, et celui pour Fevrier 5. TABLE DES CHIFFRES POUR LES SIECLES. 1, est le chiffro pour les 2e"me, 9'dme, et 166me, sidcles. [sie~cleg. ler, Seme, 15dme, 18eme, 22eme, 266me, 3Ueme, 76me, 14eine siecles. [sidclea. 66me, 13dme f 17eme, ?16me, 2ff6me, 296me, 6eme, 12eme, 20dme, 24eme, 28dme, siecles. 46me, lldme sidcles. 3eme, lOeme, 196me, 23dme, 27eme, sleclea. EXEMPLE. Quel futle jour de la semaine au 31 Aoiit, 1873 ? R^ponse, Dimanche. Procedd Deux derniers chiffres de Tan, 73 70=3 Quotient de 73 di vise par quatre, 18 + 3 21 = Jour du mois, 31 28=3 Chiffre pour le mois, 5 + 3 7=1 Apres avoir rejetcS tous les sept il reste le chiffre 1 ; ce fut done, le premier jour de la semaine, Dimanche* N. B. Les eiecles pairs non-divisibles par le chiffre 400 ne sont pa*- des ann6es bissextiles. CALENDA.ll. 85 an fcen aa <*wf Me 9Jtet!jobe. Streid) bie ieben au Don bie beiben Ie|ten ftummern auf bag 3>afjr, bet SKinuent on ben beiben le^ten 5ftummern im Saljre, bit)toirt bet bier gebraucfye ni^t ben tReft ben atum auf ben SDJtonat, unb bie $tgur auf ba Sa^r. 28a ilberbleibt ift bet 2ag inber ZBodje, ber crfte onntag, bet jmeite tag u* f, m. oor 3 U ^ giguren t)or btc donate. ^ or @cpt. it. ecf>r. 3 oor 3att. u. Oct. 5 wor 3luguft. 2 oor Stpril itnb Sitlu 4 oor 3tfai. ' 6 oor eb., 3)er Saturn im S^uar unb ^c&riiar ift ein wentger SDatum auf bie 3afjre. l,t|ibt % e unb 30te Sal) igur nor bag 2te, 9te unb 16te 3a&' Ite, 8te, 15te, I8te, 22 3te, 7te, I4te Sa^ui 6te ; I3te, 17te, 21te, 25te, 29te 3a^ 5te, 12te,20te, 24te, unb 28te 3<4 4tc nnb 11 3ol)r^unbeii. Stc, lOte, I9te, 23te unb 27te ^a^r^unbert. jempeL 2BeId)er Sag in ber 2Bod)e tDar ber 31. Sluguft, 1873 ? Slntmort onntag. ffitc le^ten beiben SiQuren im 3a^re, 73 70 == 3 5Jlinuent auf bo. -f bei bier, 18 + 3 21=0 Datum im 2Monat, 3 1 28 = 3 Stgur auf ben SJlonat, 5 + 3 7 = i )er 9feft 1 geigt @ud) ben erften Sag in ber 2Bod)e, tt>el$er ift onntag* NOIE Between the Julian and the Augustan Calendars there was a difference of ten days in 1583 and of eleven days in 1753. At the present time the difference is twelve days. The latter came into use in Catholic countries in J683 and in England ia 1753. 86 HOWARD'S ART OF COMPUTATION. Howard's California Calendar for Thirty Centuries. EULE. Cast all the sevens out of the last two figures of the year ; add the remainder to ihe quotient* of the last two figures of the year, divided by four ; take this sum with the day of the month, the figure for the month, and the figure for the century, dropping all the sevens as they occur, one remainder will be the the first day of the week, Sunday ; 2, the second, &c. ; 0, last day of the week, Saturday. , * Disregard the fraction, if any, in the quotient. TABLE OF FIGURES FOR THE MONTHS. 1, Sept. and Dec. 8, Jan. and Oct. 5, August. 0, June. 2, April and July. 4, May. 6, Feb., March, Nov. NOTE. The figure for January is 2, and February 6 in leap year. TABLE OF FIGURES FOR THE CEKTURIES. 1, IB the figure for the 2d, Oth, and 16th centuries. 2, i4 l 1st, 8th, 15th, 18th, 23d, 26th, 30th centuries, 3, " ' 7th, I4th centuries. 4, " ' 6th, 13th, 17th, 21st, 25th, 29th centuries. 5, %k * 5th, 12th, 20th, 24th, 28 centuries. 6, " ' 4th, llth centuries. 0, " " 8d, 10th, 19th, 23th. 27th centuries. EXAMPLE. What day of the week was the 31st August, 1873? Sunday, Ans. Process Last two figures of the year, 73 70 = 3 Quotient of 73 -5- by four, 18+3 21=0 Day of month, 31 28 = 3 Figure for the month, 5 + 3 7 = 1 After casting out the sevens the remainder is 1 : hence it was on the first day of the week, Sunday. iq. B. The even centuries not divisable by 400 are not leap years. SQUARE AND CUBE ROOT, 87 SQUARE AND CUBE ROOT. 1. A square number multiplied by a square num- ber, the product will be a square number. 2. A square number divided by a square num- ber, the quotient is a square. 3. A cube number multiplied by a cube, the product is a cube. 4. A cube number divided by a cube, the quo- tient will be a cube. 5. If the square root of a number is a composite number, the square itself may be divided into inter ger square factors ; but if the root is a prime num- ber, the square cannot be separated into square factors without fractions. 6. If the unit figure of a square number is 5, we may multiply by the square number 4, and we shell have another square, whose unit period will be ciphers. 7. If the unit figure of a cube is 5, we may mul- tiply by the cube number 8, and produce another cube, whose unit period will be ciphers. 8. If a supposed cube, whose unit figure is 5, be multiplied by 8, and the product does not give 3 ciphers on the right, the number is not a cube. To prove cube root : from a cube number subtract its root; the remainder will be a multiple of 6. From a number that is not a cube, subtract the ascer- tained part of its cube root; divide the difference by 6; then divide the remainder in the example by 6 ; the ex- cess, if any, should in each case be the same. 88 HOWARD'S ART OF COMPUTATION. TABLE For comparing the natural numbers with the unit figure ol their squares and cubes. By the use ot this, many roots may be extracted by observation: Numbers. ..123456789 10 Squares.... 14 9 16 25 36 49 64 81 100 Cubes 1 8 27 64 125 216 343 512 729 1000 The product of a number taken any number of times as a factor, is called a power of the number. A root of a number is such a number as taken some number of times as a factor, will produce a given number. If the root is taken twice as a factor to produce the number, it is the square root; if three times, the cube root; if four times, the fourth root. By observing the above table, it will be seen that the square of any one of the digits is less than 100, and the cube of any one of the digits is less than 1000; therefore, the square root of two figures cannot be more than one figure. The square of any number equals its root, plus the preceding square and root of a consecutive series. 48=16. 4 + 9+3=16. The units figure in the. cube root of a perfect cube is the units figure in the product of the units figure of the cube multiplied twice into itself. Find the cube root of 343. The units figure 3 X 3 X 3=27. Ans. 7. The difference of the squares of two numbers equals their sum multiplied by their difference. SQUARE BOOT. 89 To find the square root of a number. EULE 1. Separate the given number into periods of two figures each, beginning at the unit's place. The number of figures in the root equals the number of periods. 2. Find the greatest number whose square is con- tained in the period on the left ; this will be the first figure in the root. Subtract the square of this figure from the period on the left; to the remainder annex the next period to form a dividend. .3. Divide this dividend, omitting the figure on the right, by double the part of the root already found, and annex the quotient to that part, and also to the divisor ; then multiply the divisor thus completed by the figure of the root last obtained, and subtract the product from the dividend. 4. If there are more periods to be brought down, continue the operation in the same manner as be- fore. NOTE 1. If a cipher occurs in the root, annex a cipher to the trial di- visor, and another period to the dividend, and proceed as before. 2. If there is a remainder after the root of the last period is found, annex periods of ciphers, and continue the root to as many decimal places as are required. .EXAMPLE. Find the square root of 1016064. 1,01,60,64(1008 1 2008) 016064 16064 NOTE. The square root of a fraction may be found by extracting the square root of the numerator and denominator separately. 90 HOWARD'S ART or COMPUTATION. To find the cube root of a number. RULE 1. Beginning at the units' place, separate the given number into periods of three figures each; the number of figures in the root will be equal to the num- ber of periods. 2. Find the greatest number whose cube is contained in the left-hand period ; this will be the first figure in the root ; subtract its cube, and to the remainder annex the next period. 3. Multiply the ascertained part of the root by 3, then multiply that result by the first figure in the root, the product with two ciphers annexed is the first trial divisor. 4. Find how many times the divisor is found in the div- idend and place the result in the root, and also to the right of the first terrain the left hand column; multiply the last result by the new figure in the root and add the pro^ duct to the trial divisor; the sum is the complete divisor. 5. Multiply the complete divisor by the second figure in the root, subtract the product from the dividend and bring down the next period. 6. To find the next trial divisor add the square of the last found figure in the root to the preceding divisor and its smaller part; to the sum annex two ciphers, complete the divisor as before. 7. Repeat the foregoing process with each period until the exact root, or a sufficient approximation to it is found. EXAMPLE. Find the length of one edge of an excava- tion from which a cubic mass of earth = 1,745,337,664 cubic feet is to be taken. Ans. 1204 feet. 32 1st complete divisor, 3604 2nd com. divisor. 300 64 1,745,337,664(1204, cube 1 root. 364 4,320,000 14,416 745 728 17,337,664 17,337,664 4,334,416 NOTE 1. If a cipher occurs in the root, annex two ciphers to the trial divisor and another period to the dividend, and then proceed as before. 2. If there f s a remainder, after the root of the last period is found, annex periods of ciphers and proceed as before to as many decimal places as the answer requires. 8. The cube root of a fraction may be found by extracting the cube root of the numerator and denominator. THE NUMBER .NINE. ui CASTING OUT THE NINES. The number nine has many peculiar properties in our system of notation. Any number is divisi- ble by 9 when the sum of its digits is divisible by 9. Any remainder left after dividing a number by 9, will be left after dividing the sum of its digits by 9. This peculiarity may be used with advantage in proving the four fundamental rules, by casting out the nines, that is, dropping 9 whenever the sum reaches or exceeds that number, thus to cast the 9s out of 846732, we say 8+4 less 9 leaves 3; 3+6 less 9 leaves ; 7+5 less 9 leaves 3 ; hence the following. To prove ADDITION, cast out the nines from the example, and from the ascertained sum, if correct the excess in each will be the same. To prove SUBTRACTION, the excess of the re- mainder should equal the excess in the minuend less the excess in the subtrahend. NOTE. If the excess in the minuend is less than, the excess in the subtrahend, it must be increased by nine. To prove MULTIPLICATION. The excess of the product, must equal the product of the excess of the factors. Note. If the multiplier or multiplicand is a multiple of nine, the product will have no excess. To prove DIVISION. The excess of the dividend must equal the product of tho excesses in Quotient and Divisor, plus the excess of the remainder. 92 HOWARD'S ART OF COMPUTATION. MARKING GOODS. Removing the decimal point one place to the left on the cost of a dozen articles, gives the cost of one article with 20 per cent, added. We remove the point one place to the left, because 12 tens make 120. Hence, to find the selling price, to gain the required percentage of profit, we have the fol- lowing general rule : RULE. Remove the decimal point one place to the left on the cost per dozen, to gain 20 per cent. ; increase or diminish to find the percentage, as per following table : TABLE FOR MARKING ALL GOODS BOUGHT BY THE DOZEN. To make 20$ remove the point 1 place to left. " 25$ " 44 4i Add yV foself. " 26$ u 4i 4i 44 4i fa " 28$ 44 ' 4 44 fa 4i 44 30$ 4i " 44 - T V " 44 32$ " 44 44 44 44 fa 44 4:0% " " " u " -i- " 44^'' " u u 4 ' 4t .i u 50$ " " " 4 - 4i ^ . " 60$ " " a " 4i ^- 44 80$ 4< 4t 44 4< a | 4i 12$ " " 44 44 subtract^ " 16|$ 4C " 44 44 fa " 18f$ 4t " 44 44 44 fa u REFERENCE TABLES, MULTIPLICATION TABLK. 1 2 3 4 5 6 7 8 g 10 11 12 2 4 a 6 9 4 8 12 16 5 10 15 20 25 6 12 18 24 30 36 7 14 21 28 35 42 49 8 16 24 32 40 48 56 64 9 18 27 36 45 54 63 72 81 10 20 30 40 50 60 70 80 90 100 11 22 33 44 55 66 77 88 99 110 121 12 24 36 48 60 72 84 96 108 120 132 144 ABBREVIATIONS rssu IN BUSINESS. @ At. % or Acc't . .Account. Ain't. Amount. Ass'd Assorted. Bal Balance. Bbl.. .; Barrel. B. L Bill of Lading. % Per cent. Co Company. C. 0. D . . . Collect on Delivery. Cr ....Creditor. Com Commission. Cous't Consignment. Cwt Hundred Weight. Dft Draft. Disct Discount. Do The same Dox.. Dozen. Dr Debtor. E. E Errors excepted. Ea Each. Exch Exchange. Exps Expenses. Fol Folio. Fw'd Forward. Fr't Freight. Gtiar Guarantee. Gal Gallon. Hhci Hogshead. Ins Insurance. Inst This month. Iiivt Inventory Int Interest. Mdse Merchandise. Mo Month. Net Without disc't. No Number. Pay *t Payment. Pd Paid. Per An By the year. Pk'gs Packages. Per By. ,,s,,d, Pounds, shil'gs, pence. Prem Premium. Prox Next month. Ps Pieces. Rec'd Received. R. R Railroad. Ship't Shipment. Sund's Sundries. S. S Steamship. Ult Last month. 94 SPECIFIC GRAVITY. Specific Gravity is the weight of a body compared with another of the same bulk taken as a standard. The exact weight of a cubic inch of gold, compared with a cubic inch of water, is called its SPE- CIFIC GRAVITY. Water is the standard for solids and liquids. A cu- bic foot of rain water weighs 1000 ounces. NOTE. To find the weight, in ounces, of one cubic foot of any substance here named, remove tha decimal point three places to the right. Acid, Acetic 1.008 Acid, Arsenic . . . 3.391 Acid, Nitric 1.271 Air 001 Alcohol, of Commerce, . . . .885 " Pure, 794 Alderwood, 800 Ale, 1.085 Alum, 1.724 Aluminum, 2.560 Amber 1.064 Amethyst 2.750 Ammonia, 875 Aeh, 800 Blood, Human, 1.054 Brass, (about) 6.000 Brick, 2.000 Butter, 942 Cherry, 715 Cider, 1.018 Coal, bituminous, (about) 1.250 " anthracite, 1.500 Copper, 8.788 Coral, 2.540 Cork, 340 Diamond 3.530 Earth (mean of the Globe) 5.210 Elm, 671 Emerald, 2.878 Ether. 632 Fat of Beef, 923 Fir, 550 Glaeo plate, 2.760 Gold, hammered, 19.362 " Coin, 17.647 Granite 2.625 1.987 Gunpowder, 900 Gum Arabic 1.462 Gypeum, 2.288 Hazel 600 Hematite Ore, 4.705 Honey, 1.456 Ice, 930 Iodine, 4.948 Iridium, 28.000 Iron, 7.645 " Ore, 4.900 Ivory, 1.917 Lard, 947 Lead, cast, 11.350 " white, 7.235 Lignum Vitao 1.333 Lime 804 * k stone, 2.386 Mahogany, 1.063 Malachite 3.700 Maple, 750 Marble, 2.716 Men (Living.) 891 Mercury, pure, 14.000 Mica, 2.750 Milk 1.032 Naptha 700 Nickel 8.279 Nitre 1.900 Oak, 1.170 Oil, Castor 970 Opal,.. 2.314 Opium, 1.337 Pearl, 2.510 Pewter, 7.471 Platinum Wire, 21.041 Poplar, 383 Porcelain 2.385 Quartz, 2.500 Rosin, 1.100 Salt 2.130 Sand 1.750 Silver coin, 10.534 Slate, 2.110 Steel, 7.816 Stone 2.500 Tajlow, 941 Tin 7.291 Turpentine, spirits of, 870 Walnut 671 Water, distilled, 1.000 Wax. 897 Willow, 585 Wine, 992 Zinc, cast, 7.190 FORMULAS. 95 The Diameter of a Circle. 1 f X.8862 1.1284 X.866 -H.1547 X.707 -7-1.4142 X.3183 -7-3.1416 X.2821 -7-3.545 The Circumfer- J x .2756 ence of a Circle. S _^3.6276 X.2251 -j-4.4428 X. 15915 X 3< if Jo 1 =The Circumference. ;- .OiO.J J \ =The side of an equal f Square. The Area of a Circle. The Surface of a Sphere The Volume of a Sphere The Diameter / of a Sphere ~~ \ The Circum- of a Sphere -i-3.1416 X 1.2732 -i- .7854 X 12.5663 -r- .07958 \ =The side of an inscribed J Equilateral Triangle. \ =The side of an inscribed I Square. j =The Diameter. \ =The side of an equal / Square. "I =The side of an inscribed J Equilateral Triangle. ) =The side of an inscribed / Square. -M5.2831S }=The Radius. =The square of Radius. > =The square of Diameter, \ =The square of Circum- / ference. Circumference X its Diameter. (Radius)2 X 12.5664 (Diameter) 2 X 3.1416 (Circumference) 2 X .3183 Surface X 1-6 its Diameter. (Radius) 3 X 4.1888 (Diameter) 3 X .5236 (Circumference) 3 X .0169 v/ of Surface X .5642 ^/ of Volume X 1.2407 v/ of Surface X 1 .77255 & of Volume X .38978 The Radius of__ j \f of Surface X .2821 a Sphere = \ & of Volume X .6204 The Side of an^ j Radius X 1.1.547 inscrib'dCube \ Diameter X .5774 TRIANGLES. The area=the base X half the altitude. The hypothenuse= x/of the sum of the squares of the base and the perpendicular. The base, or perpendicular=thev/of the difference be- tween the square of the hypothenuse and the square of the given side. 96 LATITUDE AND LONGITUDE. Longitude reckoned from the Meridian of Greenwich. NORTH AND SOUTH AMERICA. Place. Lat. Long. Place. Lat. Long. Albany, N. Y.. AnnArbXMich Annapolis, Mel. Augusta, Me. . . Austin, Texas. Baltimore, Md. Bangor, Me Boston, Mass.. Brooklyn, N. Y. Buffalo, N. Y.. Burlington, Vt. Buenos Ayres. Cambr'ge, Mass Cape May, N. J. Cape Horn Charleston, S.C Chicago, 111 Cincinnati, O. .. Columbia, S. C. Concord, N. H. Council Bluffs. DesMoines, lo. Detroit, Mich. . Dover, Del Dubuque, lo. . . Fred'csb'rg, Va Fort Laramie. . Ft. L'v'wth, Ks. Frankfort, Ky.. Galveston, Tex Georgetown, Bermuda, W.I. Guayaquil Havana o / 42 40 N 42 17 38 59 44 19 30 13 39 18 44 48 42 21 40 42 42 50 44 27 34 36s 42 23 N 38 56 55 59 s 32 47 N 41 54 3906 34 43 12 41 30 41 35 4220 39 10 4230 38 18 42 12 3921 38 14 29 18 32 22 2 13 s 23 9N 44 39 40 16 41 46 39 55 38 36 N 24 33 / 73 45w 83 43 76 29 69 50 97 39 76 37 68 46 71 03 73 58 7859 73 10 5822 71 08 7457 07 16 79 56 87 38 8430 81 02 71 29 93 48 9340 83 2 75 30 9040 77 27 104 48 94 44 84 40 94 47 64 37 79 53 82 21 63 35 76 50 72 41 86 5 92 8 81 47 Lima, o / 12 3 8 34 40 N 38 3 19 26 43 2 30 41 45 31 41 18 29 58 40 43 45 23 39 57 37 14 43 39 41 49 46 40 37 32 43 8 22 56 s 32 5N 38 35 29 48 38 37 44 53 40 46 37 48 35 41 3948 39 40 43 3 43 31 40 13 42 44 33 2s 38 53 N 41 23 40 7 34 14 42 16 37 13 / 77 6 92 12 85 30 99 5 8754 88 1 73 33 72 55 90 2 74 75 42 75 9 7724 70 15 71 24 71 12 7726 77 51 43 9 81 5 121 28 81 5 90 15 95 5 1126 122 47 106 1 89 33 94 52 76 9 7923 74 45 73 41 71 41 77 73 57 8042 77 57 71 48 76 34 Little Rock, Ark... Louisville, Ky Mexico, Mexico Milwaukee, Wis. . . Mobile Ala Montreal, C. E New Haven, Conn.. New Orleans, La... New York, N. Y... Ottawa C W Philadelphia, Pa... Petersburg, Va Portland Me Providence, R. I... Quebec. C. E Richmond, Va Rochester, N. Y... Rio Janeiro Savannah, Ga Sacramento, Cal. . . St. August'e, Fla.. St Loui** Mo St. Paul. Minn Salt Lake City San Francisco Santa Fe, N. Mex.. Springfield, 111 St. Joseph's, Mo... Syracuse, N. Y Toronto, C. W Trenton, N. J Troy N Y Valparaiso, WASHINGTON West Point, N. Y.. Wheeling, W. Va.. Wilmington. Del... Worcester, Mass... Yorktown, Va Halifax Harrisburg, Pa. Hartford, Conn Ind'nap'lis, Ind Jeffer 1 City, Mo Key West, Fla, A difference of 15 degrees of Longitude equals a, difference of one hour of time. The degrees of Longitude between two cities, multiplied by 4, equals , in minutes, the difference of time. For difference of There \a ft difference of For a difference A There is difference of 15 in Long. 1 hr. in Time. 1 " " 4min." " 15' " " 1 min." " 1' " " 4 sec. " u 15" " " 1 sec. " " V " " 1-15 sec. in tim* LATITUDE AND LONGITUDE. 97 EUROPE, ASIA, AFRICA, AND THE OCEANS. Place. Lat. Long. Place. Lat. Long. Antwerp -Alexandria o / 51 13 N 31 12 6432 37 58 36 11 36 47 52 22 5 34 2 41 23 18 56 53 5 52 30 50 51 51 26 50 58 41 1 23 7 59 59 55 41 33 56 s 2234N 37 54 30 3 9 49 53 23 51 8 55 57 17 41 s 43 46 N 51 29 / 424E 29 53 40 33 23 44 37 10 3 4 4 53 115 151 13 2 11 72 54 8 49 13 24 4 22 9 29w 1 51 E 28 59 113 14 29 47 12 34 18 29 88 20 E 22 52 31 18 80 23 6 20w 1 19 E 3 12w 17853E 11 16 Leghorn / 4332 51 20 3842 55 40 35 54 38 12 13 20 23 37 43 18 14 36 14 4 40 25 36 43 84 24 s 1528 3436N 4050 46 28 39 54 38 8 48 50 41 54 51 54 38 26 1 17 14 55 8 30 15 55 s 29 59 N 59 21 59 56 43 07 34 54 36 47 35 47 4550 48 13 52 13 6 28s / 10 18K 1222 9 9\v 35 33 E 14 30 15 35 43 12 58 35 5 22 121 2 80 16 3 42w 4 26 173 IE 1677 138 51 14 16 30 44 116 28 13 22 2 20 12 27 4 29 27 7 103 50 100 13 18w 5 45 32 34 K 18 6 30 19 5 22 13 11 10 6 554 12 26 1623 21 2 3933 Leipsic Archangel Lisbon Athens Moscow Aleppo Malta Algiers Messina . * Amsterdam Mocha Muscat Borneo Botany Bay Marseilles . . . Barcelona Bombay Manilla Madras Bremen Madrid Berlin. Malaga Brussels New Zealand New Hebrides.. . Cape Clear Calais Constantinople . . Canton Naples Odessa Cronstadt Pekin Copenhagen Cape of G. Hope. Calcutta Palermo Paris Rome Corinth Rotterdam.. Cairo Smyrna Ceylon Singapore Dublin Siam Dover Sierra Leone St. Helena Edinburgh Feejee Group.. . . ^Florence Suez Stockholm GREENWICH . . . St. Petersburgh . Toulon Geneva 46 12 55 52 36 7 44 24 21 19 53 33 49 29 31 48 N 53 25 6 9 4 16w 5 22 8 53 E 157 52w 9 58 E 6 37 20w 3 Glasgow Gibraltar Tripoli Tunis Genoa Tangier Honolulu Venice Hamburg Vienna Havre Warsaw Jerusalem . Zanzibar Liverpool MEASURE OF CIRCLES, OR ANGLES. The UNIT is the degree, which is 1-360 part of the circumference of any circle. 60 Seconds (") = 1 Minute. ' 60 Minutes = 1 Degree. 30 Degrees = 1 Sign. S 12 Signs, or 360 = 1 Circle. C 98 LEGAL RATES OF INTEREST And Statute Limitations in the different States. In some States there are exceptions, and any of the data are liable to change by the action of the State Legislatures. The English legal rate is 5 per cent. States and Territo- ries. o P g . ^0 =: 2 1! OJ K^ gj^ Penalties for Usury. Forfeiture of Statute Limitat'n. Open Acc'ts Yrs. Note. Yrs. Judg- ment, Yrs. Alabama, 8 10 G 10 10 G 7-10 6 G 8 7 10 G 6 6 7 6 5 G 6 6 7 7 6 6 10 10 10 6 7 6 7 6 6 10 G 6 7 6 8 10 6 G 10 G 7 12 8 Any. Any. Any. Any. Any. 10 Any. 10 Any. 10 10 10 12 10 8 Anv. G Anv. 10 12 10 10 Any. 12 Anv. G 7 8 8 12 Any. Any. Any. 10 Any. Any. 6 12 Any. G 10' Entire interest Entire interest Principal Entire interest Excess Entire interest Excess Entire interest Entire interest Excess Excess Entire interest Entire interest Thrice excess Entire interest Excess Entire interest Excess Excess Excess Excess Entire interest 3 3 2 2 6 G 3 3 5 3 5 G 5 3 2 3 6 3 G G G 3 5 4 G 6 G 3 6 G 6 6 G 6 2 6 5 5 10 G 4 4 6 15 6 3 5 3 G 20 10 5 7 5 G 3 G 6 G 6 10 5 6 16 6 3 15 6 6 G G G 4 G 5 5 6 20 10 10 5 ir 6> 20 12 10 20 20 10 14 10 20 12 20 20 10 20 20 5 20 20 20 10 20 10 20 20 20 10 10 10 10 10 Alaska Arizona, Arkansas, California Colorado Connecticut, Dakota Delaware Dist. of Columbia. Florida Georgia, Idaho Illinois Indiana, Indian Territory,.. Iowa, Kansas, Kentucky, Louisiana, Maine, Maryland, Massach usetts, Michigan Minnesota Mississippi, Missouri, Montana Nebraska, Nevada New Hampshire,.. New Jersey, New Mexico, New York, North Carolina,.... Ohio Oregon Pennsylvania, Rhode Island South Carolina, Tennessee, Texas Utah Vermont Virginia, Washington West Virginia, .... Wisconsin, Wyoming, PAPER TABLE. 99 Paper is bought at wholesale by the bale, bundle and ream; and at re- tail by the ream, quire and sheet. 24 Sheets 1 Quire, 2 "Reams = 1 Bundle, 20 Quires = 1 Ream, 5 Bundles 1 Bale. The names generally define the sizes. "Writing and Drawing Papers differ in size from Printing Papers of the same name. English sizes differ from American. SIZE OF FOLDED TAPERS, IN INCHES. BilletNote 6x8 Letter, 10x16 Octavo Note, 7x9 Commercial Letter, 11x17 Commercial Note, 8x10 Packet Post, 11^x18 Packet Note < 9x11 Extra Packet Post...... 11^x18^ Bath Note, 8^x14 Foolscap, , 12>xl6 FLAT CAP PAPERS. Juaw Blank, 13x16 Medium, 18x23 Flat Cap, 14x17 Royal, 19x24 Crown, 15x19 Super Royal, 20x28 Demy, 16x21 Imperial, 22x30 Folio Post, 17x22 Elephant, 22%x27% Check Folio, 17x24 Columbia, 23x33,^ DoubleCap 17x23 Atlas, 26x33 Extra Size Folio, 19x23 Double Elephant 26x40 SIZE OF PRINTING PAPERS. Medium, 19x24 Double Medium, 24x38 Royal, 20x25 Double Royal, 26x40 Super Royal 22x28 Double Super Royal, 28x42 Imperial, 22x32 " " 2i)x43 Medium-and-half, 24x30 Broad Twelves, 23x41 Small Double Medium,. . . . 24x36 Double Imperial 32x46 BOOKS. The terms folio, quarto, octavo, duodecimo, etc,, indicate the number of leaves into which a sheet of paper is folded. y hen a sheet > The Book $ 1 sheet of When a sheet ) The Book $ 1 . sheet of js folded into $ is called j Paper makes is folded into ) is called i Paper makes 2 leaves. A Folio. 4 pages 16 leaves. A 16mo. 32 pages* 4 " A Quarto or 4to. 8 " 18 " AiilSmo. 36 " 8 " An Octavo or 8vo. 16 " 24 " A 24mo. 43 " 12 " A Duodecimo or 12mo. 24 " 32 " A 32mo. 64 Clerks and Copyists are of ten paid by the Folio lor making copies of legal papers, records and documents. 72 words make 1 folio or sheet of Common Law. 90 " " * " " * " Chancery. A Folio varies in different States and Countries but usually contains from 75 to 100 words. 100 GOLD COINS. GOLD COINS their weight, fineness, and value in. British and United States money, based on U. S. Mint assays, 1879, computed by C. FRUSHER HOWARD. Country. Denomination. We Grains. ight. Ouncei. Fine leOOths ness. Carats. Value s. d. >. U.S. $ Austria, Union Crown, 171.36 0.357 900. 21.60 1 7,, 3^ 6.6410 Belgium, 25 Francs, 121.92 0.254 899. 21.57 19,, 4^ 4.7205 Bolivia, Doubloon, 41G.16 0.867 870. 20.88 3,, 4,, 1 15.5925 Brazil, 20 Milries, 276.00 0.575 917.5 22.02 2,, 4,, 10 10.905T Chili, Doubloon, 416.16 0.867 870. 20.88 3,, 4,, 1 15.5925- Denmark, 10 Thaler, 214.96 0.427 895. 21.48 M2,, $y 2 7.9000 England, Sovereign, 123.21 0.2567 916.6 22.00 i,, o,, o 4.SGG5- France, 20 Francs, 99.60 0.2075 899. 21.57 15,,10^ 3.8562 Germany, 20 Marks, 122.90 0.256 900. 21.60 19,, 6^ 4.7627 Greece, 20 Drachms, 88.80 0.185 900. 21.60 14,, 1% 3.4419 India, Mohur, 179.52 0.374 916. 22.00 1 9,, 1 7.0818 Italy, 20 Lire, 99.36 0.207 898. 21.55 15,, 9*4 3.8420 Japan, 5 Yen, 128.30 0.267 900. 21.60 1,, 0,, 5 4.9674 Mexico, Doubloon, 416.16 0.8675 870.5 20.89 3,, 4,, Ifc 15.6105 " 20 Pesos, 518.88 1.081 873. 20.95 4,, 0,, 2 19.5083. Netherl'ds. 10 Guilders, 103.72 0.216 899. 21.57 16,, 5 3.995S Peru, Doubloon, 416.16 0.867 868. 20.83 3,, 3,,11% 15.556T 20 Soles, 496.80 1.035 898. 21.55 3,,18,,11^ 19.2130 Portugal, Gold Crown, 147.84 0.308 912. 21.88 1, 3,,10H 5.806S Rome, 2J Scudi, 67.20 0.140 900. 21.60 10,, 8 2.6047 Russia, 5 Roubles, 100.80 0.210 916. 22.00 16,, 4 3.9764 Spain, 100 Reales, 128.64 0.268 896. 21.50 1 5 4.9639> Sweden, Ducat, 53.28 0.111 975. 23.40 9,, 2 2.2372 Turkey, 100 Piasters, 110.88 0.231 915. 21.96 17,,UH 4.3695 United ( 20 Dollars, 516.00 1.075 900. 21.60 4,, 2,, zy 2 20.0000 States. ) One Dollar. 25.80 .05375 900. 21.60 .205486 1.0000 The Gold Talent of Scripture 5 8 64 " Silver " " 341 Exactly the existing ratio between U. 5,,8 $26592.809. ,,10 ,4 $ 1662.025. S. Gold and Silver Coins-16 to 1. SILVER COINS. 101 Table of various Silver Coins, showing their weight, fineness and quota of pure silver, computed from U. S. Mint assays, by C. FRUSHER HOWARD. Country. Denomination. Pine- ness. Ve Ounces. ght. Grains. Pure! Grains. ilver. Ounces. Austria, New Florin, .900 0.397 190.56 171.504 .357300 " Dollar, .900 0.596 286.08 257.472 .536400 Belgium, 5 Francs, .897 0.803 385.44 345.739 .720291 Bolivia, New Dollar, .9035 0.643 308.64 278.856 .580950 Brazil, Double Milries, .9185 0.820 393.60 361.521 .753170 Canada, 20 Cents, .925 0.150 72.00 66.666 .138750 Cen. America. Dollar, .850 0.866 415.68 353.328 .736100 Chili, New Dollar, .9005 0.801 384.48 346.224 .721300 China, Hong K. English Dollar, .901 0.866 415.68 374.527 .780266 Denmark, Two Rigsdaler, .877 0.927 444.96 390.230 .812979 England, New Shilling, .9245 0.1825 87.60 80.986 .168721 France, 5 Franc, .900 0.800 384.00 345.6 .720000 Germany, Mark, .900 0.1785 85.70 77.13 .160650 Greece, 5 Drachms, .900 0.719 345.12 310.608 .647100 East Indies, Rupee, .916 .0.374 179.52 164.44 .342584 Japan, New Dollar, .900 0.875 420.00 378.000 .787500 Mexico, < .903 0.8675 416.40 376.009 .783352 Naples, Seudo, .830 0.844 405.12 336.249 .700520 Holland, 2} Guildeis, .944 0.804 385.92 364.308 .758976 - Norway, Specie Daler, .877 0.927 444.96 390.229 .812979 Peru, Dollar 1858, .909 0.766 367.68 334.221 .696294 Rome, Scudo, .900 0.864 414.72 373.248 .777600 Russia, Rouble, .875 0.667 320.16 280.140 .583625 Spain, New Pistareen, .899 0.166 79.68 71.632 .149234 Sweden, Rix Daler, .750 1.092 524.16 393.120 .819000 Turkey, 20 Piasters, .830 0.770 369.60 30G.765 .639100 Tuscany, Florin, .925 0.220 105.60 97.680 .203500 United States. Dollar .900 0.8594 412.50 371.25 .7734375 i< Trade " .900 0.875 420.00 378.00 .787500 102 GOLD VALUE OF SILVER. Table showing the value in U. S. Gold Coin of an ounce of silver, (480 gr.) a trade dollar (420 gr.), and a Standard dollar (412J gr.), all 9-10 fine, at London quotations for Silver bullion .9245 fine, calculated at the par of exchange, $4.8665, to the pound sterling, by C. FRUSHER HOWARD. London Pence. Quotation Ster'g Value of Ounce. 480 Gr'ns Value of Trade $ 420 Gr- Value of Stand $ 412% G $ 0.848 London Pence. Quotation Ster'g Value of Ounce. 480Gr's. Value of Trade $ 420 Gr Value of Stand. $ 412^ G 50 .2083 * 0.9869 0.864 55% .2302 $ 1.0905 $ .954 % .937 50% .2094 0.9918 0.868 0.852 55^ .2312 1.0955 .958 .941 50K .2104 0.9968 0.872 0.856 55% .2323 1.1005 .963 .946 50% .2115 1.0018 0.877 0.861 56 .2333 1.1055 .967 .950 51 .2125 1.0067 0.881 0.865 56% .2343 1.1104 .971 .954 51% .2135 1.0117 0.885 0.869 56% .2354 1.1153 .976 .959 WA .2146 1.0166 0.889 0.874 56% .2365 1.1202 .980 .963 51% .2156 1.0215 0.893 0.878 57 .2375 1.1252 .985 .968 52 .2167 1.0264 0.898 0.882 57% .2385 1.1301 .989 .972 52% .2177 1.0314 0.902 0.887 57^ .2396 1.1350 .993 .976 52% .2187 1.0362 0.907 0.891 57% .2406 1.1399 .997 .980 62% .2198 1.0412 0.911 0.895 58 .2417 1.1449 1.002 .985 53 .2208 1.0461 0.915 0.899 58% .2427 1.1498 1.006 .989 53% .2219 1.0511 0.919 0.904 58% .2437 1.1548 1.010 .993 53K .2229 1.0560 0.924 0.908 58% .2448 1.1597 1.015 .997 53% .2239 1.0610 0.928 0.912 59 .2458 1.1646 1.019 1.001 54 .2250 1.0659 0.932 0.916 59% .2469 1.1696 1.023 1.005 54% .2260 1.0709 0-937 0.921 59^ .2479 1.1745 1.028 1.010 54% .2271 1.0758 0.941 0.925 59% .2489 1.1794 1.032 1.014 54% .2281 1.0807 0.945 0.929 60 .2500 1.1844 1.036 1.018 55 .2292 1.0857 0949 2.933 London price per ounce, multiplied by 4.S66-'>, multiplied by .9, divided by .&245, equals the price per ounce, in United states Gold coin. The Trade dollar is worth two-tenths ol a cent more than the Mexican. STANDARD WEIGHTS AND MEASURES. 103 HOWARD'S Tables of Standard Weights and Measures. A Standard Measure is a fixed ur.it established by law, by which quantity, as extent, dimension, capacity or value is measured. The U. S. Standard units arc the YARD, the GALLON, the BLSHEL, the TROY POUND, and the GOLD DOLLAR. The Standard unit of weight must be of definite dimensions, and of definite gravity, of some substance, a certain volume of which, under certain conditions, will always have a certain weight. One cubic inch of pure water weighed in vacuo, thermometer 62 Fahrenheit, Barometer 30= 252.458 grains. 5760 grains = 1 Troy pound. In the Treasury at Washington is a brass scale \\hich, at a tem- perature of 62' J Fahrenheit, is 82 inches long; all our weights and measures are referred to this unit. LONG MEASURE. SURVEYORS' II LONG MEASURE. IN. FT. YD. UD FUR 12 I 1 Foot 3 .,..1 ..1 Yard. 7.92 ..1 1 Link. 36 198 16 l / 5'/ 2 .1 ..1 Rod. 198 25 ..1 1 Rod. 7920 660 220 40 . 1 ..1 Furl'ng 792 100 ..4 .1 1 Ch'n. 63360 5280 1760 320 '.'.S ..1 Mile. 63360 8000 320 80 ] Mile. The Geographical Mile equals 1.15 Statute Miles, COMPARISON OF STANDARD MEASURES OF DISTANCES. Country. U. S. Mile. Country. U.S Mile. Austria, 1 Mile, = 4.<)8 Persia, .1 Farsang, = 4.17 China 1 Li, = .35 Portugal,.... .IMilha, = 1.28 East Indies, 1 Coss, = 1.14 Prussia, .IMeile, = 4.93 Egypt 1 Mili, = 1.15 Russia, .1 Verst. = .66 England, ...IMile, = 1.00 Spain . 1 League, = 4.15 France, . .. .1 Kiloinet r.= .62 Sweden, .... .1MU, = 6.64 Japan, .. ...1 Hi. =2.562 Switzerland, 1 Lieue, = 2.98 Mexico, . ...1 Silio, 6.76 Turkey, .... .IBern, = 1.04 104 SQUARE MEASURE. For measuring land, Boards, Painting, Paving, Plastering, etc. SQ. INCH. SQ. FOOT. SQ. YARD, SQ. RD. SQ. 11. SQ. A. 144 1 1 SQ FT 1296 9 1 1 YAHD 39204 272*4 30J4 1 1 KOI) 15681GO 10890 1210 40 . . 1 1 HOOD 6272640 43560 4840 160 4 ...1 ....1 ACRE. 4014489GOO 27878400 3097600 102400 2560 640 ...1 311 LE. lu measuring Roofing, Paving, etc., 100 square feet -= one square. One thousand shingles, averaging 4 inches wide, and laid 5 inches to the weather, are estimated to be a square. One mile squarc=l section 640 acres. 36 square miles (6 miles square) =>! township. The sections are all numbered 1 to 36, commencing at the north- east corner, thus: The sections arc all divided into quarters, which arc 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 some- times will fall short, and some- times overrun the number of acres it is supposed to contain. 6 5 4 a 9 NW | NK SW | SK 7 8 9 10 11 12 18 17 16* 15 14 13 19 20 21 22 23 24 80 29 28 27 26 25 31 32 33 34 35 36 COMPARISON OF THE CoMMON T AND METRIC SYSTEMS. 1 Inch, = 2.54 Centimeters 1 Cu. in.= 1 Foot. = 30.48 Centimeters 1 " ft..= 1 Yard, = 9144 Meters 1 ik yd., 1 Rod, = 5.029 Meters 1 Cord, 1 Mile, = 1.6093 Kilometers 1 Fl. oun 1 Sq. in. =6.4528 Sq. CentimTrs 1 Gallon, 1 Sq. ft.. = 929 Sq. Centimeters 1 Bushel, 1 yard, = 8.361 Sq. Meters 1 Troy gr, 1 ik rod = 25.29 Centairs 1 " ID. 1 Acre. = 40.47 Ares. 1 Av. Ib. 1 Sq. mile, = 269 Hectares 1 Ton, 16.39 Cn. CentimVrs 28320 * k .7646" Meters. = 3.625 Stores ce. = .2.958 Centiliters = 3.786 Liters = .3524 Hectoliter = 64.8 Milligrams = .373 Kilo = .4536 " = .907 Tonneau CUBIC MEASURE, 105 For measuring timber, stone, boxes, packages, capacity of rooms, etc. CU. IN. CU. FT. CU. YD. CD. FT CD. PCH. 1728 1 ....1 Cubic Foot '- 46656 27 1 ....1 Cubic Yard 27648 16 16-27 ....1 1 Cord Foot 221184 128 420-27 ....8 ...1 ....1 Cord of "Wood 42768 24% ....1 ...1 Perch of Stone 69120 40 ....1 TJ S Ton Ship Cargo One ton of square timber = 50 cubic feet. The English shipping ton = 42 cu. ft. The Register ton = 100 cu. ft. A cord of wood is a pile 4 ft. high, 4 ft. wide, and 8 ft. long. A cord foot is one foot in length of such a pile. A cubic yard of common earth is called a load. In Board measure all boards are assumed to be 1 inch thick. A board foot is 1 ft. long, 1 ft. wide and 1 in. thick, hence 12 board feet make 1 cubic foot. Board feet are changed to cubic feet by dividing by 12. Cubic feet are changed to Board feet by multiplying by 12. Masonry is estimated by the CUBIC FOOT and PERCH ; also by the SQUARE FOOT and SQUARE YARD. Five courses of bricks in the height of a wall are called a foot, In board and lumber measure, estimates are made on 1 inch in thickness; one-fourth the price is added for every & inch in thick- ness over one inch. MISCELLANEOUS WEIGHTS AND MEASURES. 12 Units, 1 Dozen. 12 Dozen, 1 Gross. 12 Gross, 1 Great Gross. 20 Things, 1 Score. 196 Ibs 1 Barrel of Flour. 200 " ..1 Bbl. Beef, Pork, Fish. 66 " 1 Firkin of Butter. 14 " 1 Stone, Avoir. 28 " 1 Quarter, " 2iy t Stones, 1 Pig of Iron. SPigs, IFother. 2 Weys (328 Ib) 1 Sack of WooL 12 Sacks, (4368 Ib.) 1 Last 3 Inches, 1 Palm. 4 " 1 Hand. 9 " 1 Span, 3 ft 1 common pace. 6 " 1 Fathom. 3 Miles, 1 League. 360 Degrees, 1 Circle. 106 TROY WEIGHT. For Gold, Silver, Jewels, etc. AVOIRDUPOIS WEIGHT. For Groceries, Provisions, etc. Gr. Pwt. Oz. Gr. Oz. Lb. 24 ...1 ..1 Pennyweight 437 1 / 2 1 ..1 Ounce 480 20 ..1 ..1 Ounce. 7000 16 ...A ..1 Pound. 5760 240 12 ..1 Pound. 14000000 32000 2000 ..1 Ton. The Standard unit is the Troy Pound. The Long Ton = 2240 Ibs. 1 cwt. = 1121bs. To compare Troy weights with Avoirdupote, reduce both to grains. Pounds Avoirdupois X 100x7-j-48= ounces Troy. Troy ounces X. 06 6-7= Pounds avoirdupois; thafis, ounces multiplied by .06+1-7 of the product. APOTHECARIES' WEIGHT. APOTHECARIES' MEASURE. GRS. sc. 3 DR. 3 OZ. z 3 GO Minims = 1 Fluid Drachm. 8 Fl. Drms = 1 Fluid Ounce. 20 1 .1 SCRUPLE. 16 Fl. Ozs. 1 Pint.. GO 3 1 1 DIIAM. 8 Pints = 1 Gallon. 480 v4 8 1 1 OUNCE. Used in compounding liquid. 5760 288 % 12 ..1 POUND. medicines. The grain, ounce and pound arc the same as Troy Weight. Drugs are bought and sold in quantities by Avoirdupois Wei 1 Teaspoon = 45 Drops. 1 Tablespoon = l / 2 Fluid Ounce. COMPARISON OF LIQUID MEASURES. Weight. Country. U.S. Gals. Country. U. 8. Gals' England, . . France, . . . ..1 Gallon. 1.2 . .1 Dekaliter. =2.64 Switzerland, Turkey, 1 Pot, - .40 1 Almud, 1.38 Prussia, . ,. Austria, . .. ..1 Quart, .30 ..IMaas, -= .37 Mexico, Brazil, 1 Fasco. = .63 1 Medida. = .74 Sweden. . . . . 1 Kanna, = .09 Cuba 1 Arroba. = 4 01 Denmark, . . .1 Kande, = .51 South Spain, 1 Arroba. = 4.25 COMPARISON OF GRAIN MEASURES. Country. U. S. Bushels. Country U. S. Busheli. England, . . France, ... ..1 Bushel. 1.031 . .1 Hectoliter =2.84 Germany, . . . Persia 1 Schef.= 1.5to3 1 Artaba. 1.85 Prussia, . .. ..1 Scheffel. 1.56 Turkey, 1 Kilo, =1.03 Austria, . . ..IMetze. <= 1.75 Brazil 1 Fan. =- 1.5 Russia ..IChetverik .74 Mexico, 1 Alque. 1.13 Greece, . . . . .1 Kailou, -2.837 Madras, .iParah. =1.743 RAILROAD FREIGHT. 107 COMPARATIVE TABLE OP POUNDS IN DIFFERENT COUNTRIES Austria, 100 Ibs 123.50 U. S. Nedcrland, 100 Ibs . . 108.93 U. Bavaria, kk Belgium, u Bremen, " Berlin, " Denmark, kk Ger. Zoll. States, . Hamburg, .123.50 ,103,35 .110.12 .103.11 .110.00 .110.25 .110.04 Portugal, kk Prussia, Russia, kk Spain, kk St. Domingo, kk Trieste, kk ..101.19 ..110.25 .. 90.00 ..101.44 ..107.93 ..123.60 COMPARISON OF COMMERCIAL WEIGHTS. Country. Austria, Arabia, Brazil China. Denmark. . . East Indies. Egypt, France Germany, . . Weight- U. 8. Lbs. .1 Pfund. 1.23 .1 Mannd. .3 .1 Arratel. .1 Catty. = .IPund. .1 Seer. = 1.02 1.33 1.10 2.06 .1 Rottoli. = 1.0 .1 Kilogram. =2.20 .1 Pfund, = 1.10 Country. Mexico, Madras, .. Persia, . Russia, Sweden, Spain. . Sicily. . Turkey. . . Japan, Weight. U. B. Lb~ ...1 Libra, l.Ofr ...IVis. =3.125 ...1 Rattcl. 2.116 ...1 Funt, .90- ...IPund. .93 ...1 Libra, => 1.016 ...1 - k - .7 ...1 Oka, = 2.82 ...IKin, = .6$ Troy. Apothecaries. Avoirdupois. 1 Pound = 57GO grains = 57GO grains = 7000 grains. 1 Ounce =480 k> = 480 " = 437.5 kk 1 Ounce =480 k> = 480 >% = 437.5 kk 175Pounds= lt5 pounds^ 144 pounds. RAILROAD FREIGHT. TABLE OF GROSS WEIGHTS. When the actual weights are not known, the articles are billed as per the following table. Ale and Beer, 320 Ib. per bbl. Lime, 200 Ib. ]>er bbl 170 ; > " 1 A i% Malt, 38 k; kk bu. H tt tt 100 " " J4 " Millet, 45 kk Apples, dried, " green, 24 " 50 lk " bn Nails, ..108 kk Oil, 400 kk kl keg fc * ; bbl. Beef ... 150 fc * 320 u kk bbl. Peaches, dried, 33 kk Pork 320 " kk bu. " bbl Bran, ..20 ' l " bn. Potatoes (com ) 150 k; Brooms, ..40 u ki dox Salt Fine 300 k * .1 U Cider, .350 " kk bbl kk Coarse 350 kk it 4t Charcoal, . 22 ik ik bu kt in Sack 200 ki , kk It Eggs, . 200 lv kk bbl. Turnips 56 kk kk bu. Fish, .300 ** Vinegar 350 kk kk bbl Flour, ..200 " U Whiskey, 350 " Highwines, . . . ..350 tk U U One Ton Weight, .20001b. 108 LIQUID, OR WINE, MEASURE. CU.PT. CU. IN. CU.FT. CU. IN. .0167 28.875 4 Gills,.. ? Pint. n.ae i9404 2Tiecs..lPunsh'n. .0334 57.75 2 Pints.. 1 Quart. 4.2109 7276.5 31 Vi Gals IBbl. .1331 231 4 Qts., . . 1 Gallon. 8.421 14553 2Bbls 1 Hhd. 1.331 2310 10 Gals... 1 Anker. 16.84 29106 2 Hhds 1 Pipe. 2.406 4158 18 Gals... 1 Runlet. 33.68 58212 2 Pipes 1 Tun. 5.614 9702 42 Gals. . . 1 Tierce. The U. S. Standard Gallon contains 231 cubic in. 8^ Ibs. avoird tip's. " Imperial ik * 277.274 " =1.2 U. S. gallons. " old Beer Measure " " 282 " In measuring tanks, reservoirs, etc., it will be sufficiently accu- rate to regard one cubic foot=7H U. S. or 6J4 Imperial gallons. The contents of a circular tank, in barrels of 31 y z gallons,=the square of the diameter (in ft.) multiplied by the depth, mul. by .1865. DRY MEASURE, U. S. STANDARD, For measuring Grain, Fruit, Roots, Coal, etc. CU. FT. CU. IN. PT. QT. GAL. PK. BU. CM. QR. .01944 33.60 ....1 1 Pint. .03888 67.20 2 ....1 1 Quart. .1555 268.80 8 4 ....1 1 Gallon. .3111 537.60 16 8 2 1 1 Peck. 1.2444 2150.42 64 32 8 4 1 1 Bushel. 4.9778 8601.68 256 128 32 16 4 ..1 ..1 Coomb. 9.9556 17203.36 512 256 64 32 8 2 ..1 ..1 Quarter. 39.8225 68813.44 2048 1024 256 128 32 8 4 ..1 Chaldron. 44.8004 77415.12 2304 1152 288 144 36 OF C OAL ..1 Chaldron. The U. S. Standard Bushel contains 2150.42 cubic inches. The Imperial English " " 2218.192 " ' " A cylinder 18^ inches in diameter, 8 inches deep= 1 BusheL 5 Stricken measures= 4 heap measures. POUNDS IN A BUSHEL. 109 U. S. Bushels X 1.03152; the product will be Imperial Bushels. Imperial Bushels -i- 1.03152; the quotient will be U. S. Bushels. Any three factors that will produce the number of inches in a given quantity, will be the inside dimensions of a box to hold that quantity ; hence a box 11.2x16x12 in., will contain 1 Standard Bushel. 924 cu. inches = 4 Liquid Gallons ; therefore a box 12X7X11 inches will contain 4 gallons. An open box made with the greatest economy of material ; the al- titude = the radius of the Base ; if with a cover the altitude= the base. A cubic foot = 8-10 of a bushel, nearly ; add .44 of a bushel for each 100 bushels. The number of bushels -j- * = the number of cubic feet. The number of cubic feet l-5=the number of Bushels. TABLE OF AVOIRDUPOIS POUNDS IN A BUSHEL, As prescribed by statute in the several States named. Commodities. I cj 1 4 d =5 3' cs t-q S % S sg 3 C3 C3 ^ 1 S fe; >H" ^ o C? C5 $ ^ OS* E^ fet Barley, 50 48 48 48 48 32 46 48 48 48 48 48 48 46 47 4645 Beans, 60 60 60 GO 60 62 Blue SrassS'd 14 14 14 14 14 Buckwheat, 40 45 40 50 52 52 46 42 42 52 50 48 42 48 4642 Castor Beans, 46 46 46 46 Clover Seed, 60 60 60 GO 60 60 60 64 60 60 60 60 Dried Apples, 24 25 24 28 28 24 28 28 Dried Peaches, 33 33 33 28 28 33 28 28 Flax Seed, 56 56 56 56 56 55 55 56 Eemp Seed, 44 44 44 44 44 Indian Corn, 52 56 52 56 56 56 56 56 56 56 52 56 58 56 56 56 56 56 Corn, in ear, 70 68 68 Corn Meal, 48 50 50 50 50 50 Oats, 32 28 32 32 35 33^ 32 30 30 32 32 35 30 52 32 54 32 3236 Onions, 57 48 57 57 52 57 50 50 Potatoes, 60 60 60 60 GO 60 60 60 60 60 60 GO GO Rye, ,54 56 .54 56 56 56 32 56 56 56 56 56 56 56 56 ~>G 5G 56 Rye Meal, .50 50 50 Salt, 50 50 50 50 ->G Timothy Seed, 45 45 45 45 45 44 Wheat, 60 56 60 60 60 GO 60 60 60 GO 60 60 60 GO 60 60 6060 Wheat Bran. 20 20 20 20 The price per cental = the price per bushel X 100-7-the number of pounds in. the bushel. See page 45. 110 MEASURE OF TIME TABLE. SEC. MIX. HRS. DA. WK. 60 1 1 Minute 3GOO 60 ....1 1 Hour 86400 1440 24 ....1 ..1 Day, 604800 10080 168 7 ..1 ..1 Week, 31536000 31622400 525600 527040 8760 8784 365 366 52 ..1 ..1 Common Year, Leap Year. 12 Calender months = 13 lunar months = 1 year. 365 days, 5 hrs. 48 minutes, 50 seconds = 1 Solar year. 10 years = 1 decade. 10 decades = 1 century. 400 years = 146,097 days, a number exactly divisible by 7. The civil day begins and ends at 12 o'clock. Midnight. The Astronomical day begins and ends at 12 o'clock, Noon. As the year contains 365J4 days, nearly, we reckon three years in every four as containing 365 days, and the fourth, leap year, as con- taining 366 days; the leap year is always a multiple of 4. The even centuries not divisable by 400 are not leap years. Formerly the new year began on the 25th of March and was so reckoned in England until 1753. In ordinary business computations, 1 year = 12 mos. = 360 ds. 1 month = 30 days. -fl-2-fl 4-1 -f-l-fl +1 +1 Jan~ Feby. Mar. Apl. May, June, July, Aug. Sept. Oct. Nov. Dec. In the common year February has two days less than 30, in leap year 1 day less ; seven months have one day more. To find the exact number of days between two dates. Multiply the number of entire months by 3, call the product tens; add the extra days, and 1 day for each month of 31 days ; when Feb'y occurs, deduct 2 days for the common, and 1 day for Leap year. How many days from 1st of the 4th month to 9th of the llth month. 11 mo. 4 mo. = 7 mo. 7x30+9+4=223 days. GOLD AND SILVER. Ill DIAMOND WEIGHT. ASSAYERS' WEIGHT. 16 Parts = 1 Grain. 240 Grains = 1 Carat. 4 Grains = 1 Carat. 2 Carats = 1 Ounce. 1 Carat = 31-5 Troy grs. (nearly.) 24 Carats = 1 Pound. The term Carat is also employed in estimating the fineness of Gold and Silver; when perfectly pure the metal is said to be "24 Car- ats fine." English Gold coin is 22 carats fine, that is, it consists of 22-24 pure gold, and 2-24 alloy. To compute the fineness in thousandths, and the weight in ounces and thousandths is simpler, and admits of very minute subdivisions with great facility. The coining of gold or silver docs not change the REAL value of either; it stamps each piece of metal with a national, official certificate of its weight and fineness. From one Troy pound of gold 22 carats, or .916 2-3 fine 46 29-40 > Sovereigns are made, each weighing 123.27448 grains = 113.001605 grains of fine gold = 84.866563. 1 ounce of U. S. Standard Gold = $18.60465 = 3.8230 = 3,,16,, 5H 1 kk ^ British " k ' = 18.94918 = 3.8938 = 3,,17,,10^ 1 " - " Pure k> = 20.67184 = 4.248 = 4,, 4,,llfc Thousandths of an ounce -j- 100 X 48 = grains. Grains X 100 -f- 48 = thousandths of an ounce. U. S. Standard ounces of Gold -j- .05375 = U. S. Dollars. U. S. Gold Dollars X .05375 = Standard ounces. To multiply by .05375, remove the point one place to the left and divide by 2, divide this quotient by 20, and the second quotient by 2; the sum of the quotients is the answer. EXAMPLE. How many ounces in one U. S. Gold dollar? A _ 20 .05 .0025 Alls. . 05375 ozs. . Oo375 The weight of gold, in ounces, and the fineness being given, to find its val- ue in U. S. Gold Coin. RULE. Multiply the weight by twice the fineness, multiply by 10 and divide the product by 30, and the quotient by 129; the sum of the product and the quotients is the answer. EXAMPLE. Find the value of one ounce of gold 9-10 fine. 3018. 129 .6 _ .00465 18760465 Ans. $18.60465. 112 HOWARD'S ART OF COMPUTATION. Or multiply the given weight by the fineness X 1000 X 8, and divide the product by 387. IX. 9X1000X8-7-387 = 18.60465. The fineness and weight of Silver being given, to Snd its value in U. S. Silver dollars 9-10 fine, 4i2y z grains weight. RULE. For pure silver, if in grains, divide by 9x10x11x3 and multiply by 8, or divide by .9x412.5. EXAMPLE. Pure silver, grains 371.25x8-j-9XlOxllX3= $1. If in ounces, divide the weight and fineness by .9 X .895375. Or multiply the given weight by the fineness and by 1.28; repeat the figures in the product, under, and two places to the right, as often, and to as many decimal places as the answer requires ; the sum is the answer. EXAMPLE. Find the value in silver dollars of 1 oz. of silver 9-10 fine. 1X.9X1.38 = 1.152 1152 1152 $1.1636352 Ans. To make a compound of any weight and fineness. RULE. Divide the fineness sought by the fineness to be alloyed; the quotient is the weight required to make a compound of one ounce of the desired fineness. EXAMPLE. Required to make a compound of one ounce 14 carats fine by alloying gold 22 carats fine. 14-4-22 = .63636 gold + .36364 alloy = 1 ounce. To find how many ounces of a lower fineness must be added to one ounce of a higher fineness to make a compound of any given fineness. RULE. Divide the difference of the two higher by the difference of the two lower finenesses. EXAMPLE. Required a compound of 14 Carats fine by mixing 12 carat fine with 21 carat fine. 14 zl. = 3%. 3 l / t oz. 12 fine + 1 oz. 21 fine = 4J4 oz. 14 Carat fine. 14 122 The silver dollar weighs 412V4 grains, nine-tenths of which is pure silver. At the English mint, a mixture of 11 ozs., 2 pwts. of pure silver, with 18 pwts. of |alloy, is coined into 66 shillings. When English coin silver is worth 54 pence an ounce, in gold, and the pound stg. .(gold) is worth $4,86 in United States gold, what is the value in U. S. gold coin of the silver contained in the dollar ? (The value of the alloy in the English silver is not to be considered. 222 11 ozs., 2 pwts. = o7Q-= - 925 of an ounce * Ans - 89 ^ cts< 54 pence = 0.225. .225X4.86 = 1.0935. 1.0935 X 412.5 X .9 __ 480 X .925 GOLD AND SILVER. 113 Estimate, in Millions, from the latest official data, of the Popu- lation, Imports and Exports, National Debts, and present stock of Gold and Silver Coin and Bullion in the world, in U. S. dollars : Country. Pop. Impts Expts Debt. Gold. Silver. United States, 45 466 739 2256 245 85 Other American States, . . . France, 40 37 243 892 268 961 1250 3750 50 1300 50 350 Great Britain and Colonies Germany, 40 40 2109 918 1397 608 4308 86 650 225 100 175 Other European States, . . . China, 200 400 1790 105 1429 114 9418 11 300 50 300 800 British India, 240 244 325 694 100 500 Japan 33 24 27 349 40 10 Other Asiatic States, 65 45 75 50 200 Africa and the Islands, . . . 65 85 100 450 50 20 Total, 1,205 6,921 6,043 22,572 3,060 2,590 Municipal and other public debts are not here included. The city of Paris owes $459,000,000 ; United States cities, $550,000,000. The quantity of Gold and Silver in the form of Plate is perhaps equal to that in the form of Coin and Bullion. The product of the Gold and Silver mines of the world last year was about $170,000,000 ; the mines of the United States furnished : Gold, $47,000,000; Silver, $46,000,000. APPROXIMATE VALUE OF VARIOUS METALS, PER POUND AVOIRDUPOIS. Indium, $2522 Vanadium 2510 Ruthenium, 1400 Rhodium 700 Palladium, 653 Uranium 576 Titanium, 500 Osmium 325 Iridium, 317.44 Gold 301.46 Platinum 115.20 Thallium 108.77 Chromium 58.00 Magnesium, 46.50 Potassium 23.00 518 4 9 Silver $1885 3 17 , 6 515,. 15,, 5 287,,13,, 7 143.,16,,10 134;, 3,, 8 118., 7,, 3 Cobalt, Cadmium, . Bismuth, .. Sodium, . . . Nickel, .... ... 7.75 ... 6.00 ... 3.63 . .. 3.20 ... 2.50 l" 4" 8 0,,15,, 0,,13,, 0,,10,, 3 Mercury, . . . ... 1.35 0,, 5,, 6 66,.15,, 9 1 65,. 4,, 6 Antimony. . Tin, ... .36 ... .33 0,, 1,, 6 0,, 1,, 4 3 61 .18,, 11 Copper ... .25 0,, 1., ) 23.,13,, 5 1 22., 7,, } 11., 18,, 3 ) 9..11,, 3 4,,14,. 6 Arsenic, ... Zinc Lead, Iron, ... .15 ... .11 ... .07 ... .02 0,, 0,, 7 0,, 0,, 5 0,, 0,, 3 0,, 0,, 1 114 HOWARD'S ART OF COMPUTATION. MISCELLANEOUS. How many strokes does a clock strike in 12 hours ? "12+1X12 7Q , , - n - To strokes. A How many barrels in a triangular pile, 49 bar- rels at the base and 1 at the top? O'Leary with ten tramps have two days start, and make 8 miles a day ; how long will it take Row- ell with 5 trampers travelling 10 miles a day to overtake O'Leary and his men? 162=8 days. The sum of two numbers is 140 ; the larger is to the smaller as 1 to 9, what are the numbers ? 9 5 14 99 9 140x i 5 4 =50 | A Bin 9 ft. 6 in. long, 6 ft. wide, 4 ft. 3 in. deep, will hold how many Imperial bushels. -XfX-Y-X T 8 400 ' Ans. 14400 bricks. Multiply 19 19s. 11 f d by 19i* V* *. 399 J Tjf or*19,,19,,llf X 20 - ^ = 899,,19,,2 ^ of a farthing. 116 HOWARD'S ART OF COMPUTATION. Multiply 66 by f : 22 -^^ = 44. Divide 66 by f: 33 Divide IG8x2x7\)y7x3: / ^ r Divide 99 amongst 3 persons, A to have T \, B T 4 T , and C T \. , Two merchants load a ship with goods worth 5000, A owns .3500, and B the rest ; the goods suffer damage valued at XI 000, what is each man's share o the loss ? 1 WP wm 1 PW A loses .700. 3500 r 1500 B 300. B and C gain by trade 182; B put in 300, and C 400, what is the gain of each ? * I W B 78. 'W I 182 W I 182 C 104. A person owning f of a mine sells | of his share for X1710, what is the value of the whole mine ? 190 -___ = ^3800. f X p How much money will buv | of f of a mine worth 3800 ? 1X1 = A - - 1710. If J of 6 be 3, what will J of 20 be ? 3 X ? 2 (3 X MISCELLANEOUS. 117 A compositor can set 20 pages in f of a day, an- other could set 20 pages in f of a day, how long 'will it take the two men working together to do the ivork? 4 5 23 23 6 -H = inverted = of a day. 326 6 23 A cistern has 5 faucets ; the first will fill it in 1 hour, the second in two, the third in 3, the fourth in 4, and the fifth in 5 hours ; in what time will the cis- tern be filled, all the faucets running at once ? 60+30+20+15+12 137 137. 60 A says to B, give me $7 and I shall have as much money as you ; B replies, give me $7 and I shall have twice as much as you ; how much money had each ? 7x5-35 7X7=49 A $35, B $49. How many different pairs can be made with 7 units ? ^ = 21 pairs. tU How many bricks, 8x4x2 inches, in a wall 160x20x2 feet? 160X20X2X3X3X6 ^ 2xlXl How many shingles for a roof 60 ft. long, rafters 20 feet, two sides, shingles to show 6x4 inches. 1 60 X 20x2x2x8 =14>400 J. X -*- 118 HOWARD'S ART OF COMPUTATION. If 21 f bushels of oats will seed 9^ acres, how many bushels will seed 100 acres? .87x3x100 4x29 = 225 bushels - How many 16ths are there in .85 ? .85X16 __ ^oo- $150 is due Jan. 1st., $78 is paid down, on July 1st., the account is. settled by paying $78. What rate per cent is paid for the accomodation ? fc79 6 X 2X100_ 1 72 Find the value of an ounce of silver, gold being worth 3,,18,, 7 per ounce, ratio 15to 1. also 16 to 1. 3,,18,,7 -s-15=60$$d. 3,,18,,7-=-l6=58^d. What is the interest on 980 dollars for six days at 7 per cent, per annum ? 980 98 49 $150 78=472. ^ ^ =165 l^r cent. 1.127 Ans. SI. 127. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. OCT ^3 JS36 SFKfT ON ILL AMP A ^(\t\"^% All b 1 4 ?007 U.U. BERKELEY I LJ I H \