GIFT OF .Professor Hat field HOWARD'S ANGLO-AMERICAN ft of THE STANDARD TEACHER AND REFEREE 01? IUSINESS BRITHMETIC, BY C. FRUSHER HOWARD, THE CALIFORNIA IALCULATOR. ONE SHILLING. HOWARD'S ' \v ANGLO-AMERICAN it of IBeckoning i GiiwiS^B V^ THE STANDARD TEACHER AND REFEREE. OF lUSINESS ARITHMETIC, FOR SCHOOLS, SELF-CULTURE, BUSINESS COLLEGES AND OFFICE USE, BY THE AUTHOR OF THE CALIFOBNIA CALCULATOR, C. FRUSHER/HOWARD, LONDON AND 1884. fc Entered according to Act of Congress, in the year of our Lord 1882, ly C. Frusher Howard, in the office of the Libra- rian of Congress, Washington, D. C. ALL RIGHTS SECURED. Entered at Stationers' Hall, London. 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. Beckoning Interest on a Basis of one per cent. on a Basis of five per cent. on a Basis of six per cent. Howard's Lightning Rule. Reckoning Annuities. Squaring numbers by their base and difference. California Calendar for thirty centuries. The original Tables and their arrangement. Square and Cube Root, finding and proving. Howard's Copyright Perpetual Calendar. This Book will be sent, post free, to any address, by any respectable Bookseller, or by the Author. C. FRUSHER HOWARD, San Francisco, California, and 61 Washington Street, Chicago, 111., and American Exchange, 449 Strand, London, England. Cloth Boards, Two Shillings ; Paper Covers, One Shilling. Cloth Boards, One Dollar ; Paper Covers, Fifty Cents* A very Liberal Discount to Schools, Business Colleges and Booksellers. INTRODUCTION, 44 9J r TlE perfection of Art will be the most apt and \f_J efficient system of Bules." KarslaJce. "Every Science is evolved out of its corre- sponding Art." "There must be practice and an accruing experience, with its empirical generalisations before there can be Science." " Progress from the simple to the complex, from the concrete to the ab- stract v from the empirical to the rational." Herbert Spencer. The ability to make business calculations with ease, accuracy, and speed, is an invaluable acquisition.. The methods of Arithmetic used in schools are too tedious and complex for practical uses, they are weighted with superfluous elements that needlessly encumber the operations and distract and confuse the learner : the Bules in this New Art of Beckoning omit the needless work, 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 drudgery of calculation into a pleasing pastime, and qualifying persons of ordinary intellect to surpass the perform- ances of the " Lightning Calculators " who have astonished mankind. INTRODUCTION. fO Accountants, Brokers, Farmers, Tradesmen and persons engaged in the ruder mechanical pur- suits, a knowledge of the SCIENCE or NUMBERS is of minor importance; SKILL in the ART OF BECKONING is absolutely indispensable; the business of this book is by new, original, and easily -acquired methods, to teach that ART in accord with, yet distinct from, the SCIENCE ; its province is to qualify the learner either for practical Business pursuits, or to study the science; he will master the higher Arithmetic with greater facility if he is first an exact and Rapid Reckoner. The phenomenal success achieved by the Author's former works two hundred and fifty thousand copies have been sold encourages the hope that this new Art of Reckoning will soon be in every School, making ALL the boys " Quick at Figures." As a SCHOOL BOOK, its aim is to make the learner a GOOD CALCULATOR, with the greatest possible economy of time and study. As a MANUAL for Business Men, Bookkeepers, Teachers, etc., to give the maximum of useful infor- mation in the briefest form consistent with clearness and completeness. The REFERENCE TABLES are very comprehensive, and their arrangement simple and original. The miscellaneous section is unique ; it embraces al- most every variety of BUSINESS CALCULATION, the work of finding 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." "Exact! 'Clear! Brief! Brilliant!" INDEX TO CONTENTS. Page. Addition, 16 Acoustics, 82 Assayer's Rules, 119 Annuities , 64 Aliquot Parts, 32 Averaging Accounts,... 70 Alligation, 119 63 Barter, 63 British Money, 58 Bonds and Stocks, 60 Cancellation, 40 Cental System, 75 Comp'ndlnterest,..52, 65 Cash Balances, 71, 73 Cube Root, 89,92 Coin, Gold, Silver, 100 Calendar-30 centuries, 110 Calendar perpetual; 127 Counterfeit Notes, 121 Decimals, 35 Division, 27, 34,37 Discount, 54, 73 Duties, 61 Definitions and Signs, 7 Exchange, 100, 57 Equation of Payments 66 Page. Fractions, 29 Formulas-Geometric, 95 Foreign Weights and Measures 103, 114 French Calendar, 108 German Calendar, 109 Gold and Silver,...100, 118 Gravity, 82 Interest 46,50 " Legal Rates,... 98 " Comp'd,....52, 65 Insurance,....; 62 Latitude & Longitude, 96 Measuring Land,77,79,112 " Lumber, 83 " Cylinders, 87 " Bricklays'Wk, 88 " Painters' Wk, 88 " Superfices and Solids, 8 " Telegraph Poles, 82 " Tanks & Wells, 116 " Gangers' Work,116 " Plasterers' Wk, 88 " Grain, 76 u Freight, 87 6 INDEX TO CONTENTS. Page. Multiplication, ..19, 31, 36 Meas. Hay Stacks, 76 Momentum, 82 Metric System, 103 Miscellaneous 122 Marking Goods, ...62, 106 Notation, 14 Numeration, 15 Nine; uses of the num- ber Nine, 74 Paper Table, 99 Partnership, 56 Partial Payments, 72 Percentage, 44 Present worth, 55 Pounds in a Bush el,... 117 Proportion, 39 Profit and Loss 62 Bapid Eeckoning for Farmers, 75 Rapid Reckonirg for Money and Bullion Brokers, ..119,57 .Rapid Eeckoning for Mechanics, 81 Rapid Eeckoning to find the cost of Hay, 76 Eailroad Freight, 115, 126 Silver, value of,.. 126, 102 Page, Subtraction, 19 Specific Gravity, 94 Square Root, 89 Squaring Numbers,.... 24 Stocks and Bonds, 60 Time Table,.... 110, 46, 120 Taxes and Duties, 61 Tables for Business Reference, 93 Tables of Standard Weights & Measures, 111 To lay off a Square Corner, 81 To find the value of Grain per cental, 75 To find the value of Gold and Currency, 57 To find the greatest Common Divisor,.... 29 To find the least Common Multiple,... 29 To find the difference of time between two Dates, 46, 120 To find the height of an object by the length of its shadow, 95 To repay a loan in equal payments, 64 Water Power, 82 HOWARD'S ART OF RECKONING, DEFINITIONS AND SIGNS. ARITHMETIC is the Science of Numbers, and the ART OF BECKONING by the use of figures. ACCOUNT SALES is a written account of goods sold, their price, expenses, and the net proceeds. AGENT. An Agent, or Commission Merchant, transacts business for another person, who is called the Principal. ALIQUOT. An aliquot part of a number is such a part as will exactly divide that number. ANGLE, a corner, or point where two lines meet; a square corner is called a right angle\ an angle greater or less than a right angle is called an oblique angle : The difference in the direction of two lines proceeding from a common point. AREA, the surface included within any given lines. ASSESSMENT, a specific sum charged against each share of a stock company, or against property for the purpose of Taxation. ASSETS. the available property of a Person, or a Company. Note. These definitions are limited to the sense in which the words are used in Practical Business Arithmetic. For Definitions omitted here see refated subjects in the Book. 8 HOWARD'S ART or RECKONING. ARITHMETICAL SIGNS are characters indicating operations to be performed, and are indispensable for briefly and clearly stating a problem : -{-,plus, and more, signifying addition; , minus, less, signifying subtraction; X, multiplied &?/, as 2 X 2 = 4 ; -i- signifies Division, or divided by; 6-^-3, or | means 6 divided by 3 ; ^-=4 means 6 multiplied by 2, divided by 3, equals 4. =, equality, or is equal to, as 6 + 2 X 2 = 16, and is read thus, " 6 plus 2, multiplied by 2, equals 16 "; the vinculum ; or ( ) the Parenthesis, are used to show that the numbers to which they are applied, are to be considered as one quantity, thus (6x4) -f 3 X 2 -f- 4 -f- 2, means, the sum of the products of 6 multiplied by 4, and 3 multiplied by 2 is to be divided by the sum of 4 and 2. \/ 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. BASE, the side upon which a figure is supposed to stand, the foundation of a calculation. BROKER, a person who buys or sells stocks, bills of ex- change, real estate, etc., for another on commission. BULLS are Brokers who aim to increase the price of stocks etc.; the opposite of Bears A CALL is the privilege to de- mand the delivery of shares of Stock within a certain time agreed upon; the privilege to deliver shares of Stock with- in a certain time is called a Put; the sum of money de- posited with a Broker by a Speculator in Stocks to secure the Broker against loss is called a Margin. DEFINITIONS AND SIGNS. 9 CAPITAL. The money and stock employed in trade ; the Principal. COMMISSION is the fee allowed an agent, usually at some rate per cent. CONSIGNMENT. Goods sent to an agent to be sold ; the person who sends the goods is called the Consignor, or Shiver; the person to whom they are sent is the Consignee. 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. CUBE, a solid body with six equal square sides. A product formed by multiplying any number twice by itself, as 4 x 4 x 4 = 64, 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, lawful money; coin or notes, or both ; gold and silver coins are sometimes called Specie. A currency whose denominations increase and diminish in a tenfold ratio is called Decimal Currency. DIAMETER, a right line passing from one side to the other and through the center of a circle, a sphere, or any object. 10 DIVIDEND. A sum divided. The pro rata division of assets among creditors, or profits among stockholders. ELLIPSE, an oval figure bounded by a regular curve. FACTORS. Numbers from the multiplication of which proceeds the product ; thus 3 and 4 are the factors of 12 ; the divisor and quotient of a num- ber are its factors. FIGURE. A figure is a written sign representing a number. FORMULA, a rule or principle concisely ex- pressed by the use of signs. GRAVITY, the force by which all bodies are attracted to the centre of the earth. INSOLVENT, one who is unable to pay his debts. INTEREST is the price, or sum charged for the use of money lent; the sum of money bearing Interest, or invested, is called the Principal. Simple Interest is that which arises from the principal sum only. The sum of Principal and Interest is called the Amount. INVENTORY, a catalogue of property on hand. INVOICE, an account in detail of goods sold. MANIFEST, a detailed list of a ship's cargo. MATHEMATICS is the science of quantities. MENSURATION is the art of measuring lengths, surfaces, and solids ; lineal measure relates to length only; superficial measure to length and breadth ; cubic or solid measure to length, breadth and thickness. 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 commom multiple is the least number that will do this ; 12 is the least common midtiple of 3 and 4. MOMENTUM, the quantity of motion in a moving body ; impetus. NET PEOCEEDS, the sum remaining after all ex- penses are paid. NUMBER, a number is a unit, or a collection of units ; a Whole Number ', or Integer, is a complete sum having no fractions. A Mixed Number con- sists of an integer and a fraction, as 7f, etc. A Prime Number is one that cannot be separated into two or more integral factors, except unity and itself; an abstract number is a number used with- out reference to any particular object, as 9, 184,etc. A number used with reference to some particular object or quantity is called a Concrete Number. PER CENT., from -per centum, by the hundred; any per cent of a number is so many hundredths of that number ; one per cent, is one of each hundred ; the number of hundreths taken is called the rate per cent. PER CENTAGE, the Per Centage is the sum total obtained by taking any number of hundredths of any given number. POWER A power is the product arising from multiplying a number by itself, or repeating it several times as a factor ; thus, 3 X 3 X 3, the product, 27, is the third power of 3. 12 HOWARD'S ART OF RECKONING. 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 2 , 2 3 . 2 == 2 1 = 2, the first ^w*r of 2. 2 X 2 = 2 2 = 4, the second /ow^r of 2. 2 X 2 X 2 == 2 3 == 8, the third power of 2. PYRAMID, a solid body uniformly tapering to a point at the top, standing on a plane, or flat surface ; if the base is round, the body is called a Cone ; the part that remains of a Pyramid, or of a Gone, after cutting off the top parallel with the base, is called its Frustrum. QUADRANGLE, a plane figure with four angles, and consequently four sides. QUANTITY is any thing that can be increased, diminished or measured ; therefore, all numerical calculations are made by one of two operations, viz : Addition or Subtraction. RADIUS, half the diameter of a circle. A right line passing from the center to the circumference. 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 f is i, of f is f , of 3-J- is T 3 . RECTANGLE, a plane figure bounded by four sides, having all its angles right angles. REDUCTION is the act of changing numbers from one denomination to another without altering their value, as the reduction of fractions to other terms, the reduction of acres to yards, bushels to gallons,etc. RULE A rule is a prescribed method of per- forming an operation. DEFINITIONS AND SIGNS. 18 SCALE A scale is a series of numbers regularly ascending or decending. A SOLID or BODY has length, breadth and thickness SPHERE, a body in which every part of the sur- face is equally distant from the center. SURFACE or SUPERFICES, the exterior part of anything that has length and breadth. SQUARE, a figure having four equal sides, and four right angles. The product of a number mul- tiplied by itself; thus 16 is the square of 4. SQUARE ROOT is the number which multiplied into itself, produces the number of which it is the root. 4 is the square root of 16. 4x4=1 6. TERMS, the terms of a fraction are numerator and denominator taken together. The terms of a Proportion are its members TRIANGLE, a figure with three sides and angles. UNIT. A unit is one thing ; one taken in the abstract is called an Abstract Unit, in distinction from a concrete or Denominate Unit. A Fractional Unit is the unit of a fraction; thus \ is the unit of j. UNITY, any definite quantity or number taken as one. USURY is a higher rate of interest than is allowed by law. VALUE is the estimated price or equivalent of anything ; in trading, money is the measure of value. VOLUME is the solid contents of a body the space included in the surfaces that bound it. WEIGHT is the measure of gravity ZERO, 0, naught, the starting point of a scale or reckoning. 14 HOW.VRD'S AET OP BECKONING. NOTATION is the act of expressing numbers by figures. All numbers are represented by the ten following figures : 1, 2, 3, 4, 5, 6, 7, 8, -9, 0. To establish their significance clearly in the minds of beginners it will be of great advantage occasionally to write and read them in the following manner : gi l 1 ! .al 1 ^9 ^q fi A a ^ K Q * r * xf f f f '^ } * ? 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. Each removal of a figure one place to the left in- creases its value ten times. NUMERATION is reading numbers expressed by figures : I 1 1 1 i f 1 I 1 s s | -3 HQ&Q^OrawO'O'HSSHt) 121,227,198,497,321,415,716,219,304,196,218,316,415,207,120. To read numbers expressed by figures : Point them off into periods of three figures each, commencing at the right hand ; then, beginning at the 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. NOTE. By the English method of numeration, the periods from millions upward have the same name, but consist of six figures each. HOWARD'S ART or RECKONING. 15 ADDITION. ADDITION is the act of adding numbers. The answer is called the Sum. Various suggestions have been made referring to improved methods of addition. In nearly every case 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, 16, 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. 1(> HOWARD'S ART OF RECKONING. g Iii adding long columns of figures, write in 4 2 the margin, lightly with pencil, opposite the 9 last figure added, the unit figure of the sum 6 immediately exceeding 100. By doing this the g mind is never burdened with numbers beyond ? 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, 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 equal number of nines in each column ; under these again, write another row of miscella- neous figures. EXAMPLE 4987 4736 2187 5012 one 9. 5 2 6 3 two 9's. 7812 three 9's. 4986 34983* RULE. Bring down the last row, less the num- ber of nines in each column, and prefix the number of nines. *This examine lias three nines in each column. 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 3 7 48 76 95 128 134 Add from the left, and say three seven, four eight, twelve eight, &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 41 47 74 83 22 32 36 21 41 183 250 6802 14640 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 PRACTICE 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; 39 + 9 = 48; 63 4- 9 = 72. B 2 2147 3472 * 1463 4614 2634 2123 7843 1785 4679 2183 6823 18 HOWARD'S ART OF RECKONING. SUBTRACTION is the process of finding the difference of two num- bers by taking one number called the Subtrahend from another number called the Minuend. The answer is called the Difference or Remainder. RULE. Write the numbers so that the units in the subtrahend shall be directly under the units of^ the same order in the minuend ; under, and in the same order, write the difference. 1694 Subtract 473 from 1694. 4T3 1221 To prove Subtraction, add the difference to the subtrahend', if correct, their sum = the minuend. MULTIPLICATION. MULTIPLICATION is the addition of several numbers in one act by adding to zero, one number called the Multiplicand^ as many times as there are units in another number called the Multiplier. The answer is called the Product. Note. The multiplier must toe an abstract number. 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. MULTIPLICATION. 19 In multiplying be careful always to write the units, tens, etc., of the multiplier under the units, tens, etc., of the multiplicand, and the figures of the product in the same order. To find the product of two numbers, each expresed by two figures only. Multiply 54 by 32. 54 32 1 T 28 Process. First multiply the units figure of the multiplicand by the units figure of the multiplier, thus : 4 X 2 = 8; put the 8 in the units place in the product, then 5x2 -f- 4x3=22, put the units 2 on the left of the 8 and carry the other 2 ; then, 5x3+2=17, which, put down, making a total of 1728, the answer. The same method can be applied when the mul- tiplicand has three or more figures. 163 24 3912 The steps are: 3 X 4 = 12, set down the 2 and carry the 1 ; (6 x 4) + (8x2) + 1 r= 31; set down the 1, and carry the 3. (1x4) + (6 X 2) + 3 = 19; set down 9 and carry I;lx2-fl=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. 163 EXAMPLE, multiply 163 by 24. 8 X 3 = 24. 1304 O 3912 Ans. 20 HOWARD'S ART OF RECKONING. When 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 When 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. Add one number to the units of another ; call the sum tens, and add the product of the units. 18 X 14=18+4 tens +8x4=252. The Area of a Circle=the square of the diameter X.7854. During the Author's visit to England in 1878, Mr. J. M. Gray, of Peckham, suggested the following easy rule : To multiply any number ly .7854. RULE. Multiply by 7, repeat, double and repeat, writing each successive product one place to the right. .7 =the product of 1 X .7. 7 =repeat one place to the right. 14 =double " " " " 14=repeat " " " " ?7854=1 X.7854. MULTIPLICATION. 21 When the multiplier is any number between 1 1 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 ; or multiply units by units, and then add to. each succeeding product, the next figure to the right of the figure multiplied, and the figure carried. EXAMPLE, multiply 1496 by 17. 1496 1496 10472 orthus: 17 25432. Ans, 25432 The process in the last example is : 6x7 = 42, set down 2 and carry 4. 9 X 7 + 6 -f 4 = 73; carry 7. 4x7 -f 9 -f 7 = 44; carry 4. 1 X 7 -f 4 -f 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 4 and 3 is 7 ; place the seven between the 4 and 3, for the product. NOTE. 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 X 11=135802458. 22 HOWARD'S ART OF RECKONING. 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 of the preceding figures is an odd number, add half 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 when the 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. 6 7 6 3 4221 To multiply two numbers when either has one or more ciphers on the right, as 26 by 20, 244 by 200, etc. 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 26 by 20. Ans. 520. Process. 260 x 2 = 520. 2. Multiply 244 by 200. Ans. 48800, 24400 x 2 = 48800. 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 with 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. 24 HOWARD'S ART OF RECKONING. RAPID METHOD OF SQUARING NUMBERS. BY THE DIFFERENCE OF A NUMBER AND ITS BASE. . For squaring a number greater than its base. KULE. 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 + HJX 10 + I 2 = 121. (22) 8 =484. Taking 20 for the Base the difference is two. (22+2) X 20+ 2 2 =484. (104) 2 =10,816. (104+4)X100+(4) 2 =10,816. (322) 2 .=103,684. (322+22) X 300+ (22) 2 =103,684. (813) 2 =660,969. (813+13) X 800+ (13) 2 =660,969. For squaring numbers less than the base. RULE. From the number to be squared subtract the difference, multiply the result by the base, to the product add the square of the difference. (9) 2 =81. Taking 10 for the Base the difference is one. (9-l)xlO+(l) 8 =81. (96) 2 =9216; (96-4) X 100+ (4) 2 =9216. (27) 8 = 729. (273) X 30+ (3) 2 =729. (99,946-54) X 100,000+ (54) 2 =9,989,202,916=(99,946) s . Multiply 19,,19,,llf d by X20+ or 19,,19,,llf X 20 - ^0= 399,,19,,2 ^ of a farthing. SQUARING NUMBERS. 25 NOTE. In squaring numbers between 50 and 60, take 50 for the base; to 25 add the difference, call the sum hundreds, to this add the square of the difference. 1. (51) 2 = 2601. 2._(52) 3 = 2704. KOTB. 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. l._(41)2 = 1681. Process. (15-f 1) XlOO-{-9 2 =1681. 2. (42) 2 = 1764. 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 -f- tne square of the num- ber of hundreds X 10,000+625. (3+6 8 )XlO,000+25 2 =390,625=625 2 In squaring very high 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 NOTE. Until this rule is thoroughly understood, the learner should limit his exercises to numbers near 10, 100, 1000, etc. ; and then operate with more complex numbers. 2fi HOWARD'S ART OF RECKONING. 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 called the Divisor is contained in another number called the Dividend. The answer is called the Quotient. 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. DIVISION. 27 Divide 49654809 by 4. 4)49654809 Ans. 124137021 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" Process. 79 23x3, the remainder is 10; the next unused figure in the dividend 8, annexed to 10=108; 10823x4, 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 =7 X 6. 6504 84 12 Ans. HOWARD'S ART OF RECKONING. FRACTIONS. A Fraction is a part or parts of a unit, a quanti- ty, or a whole number. Common fractions are written with figures below the line called the Denominator, and figures above the line called the Numerator, thus f, three-fourths, -, nine-fifths, etc., the Denominator shows into how many parts Unity is divided ; the Numerator shows how many parts are taken. When the numerator exceeds the denominator it is called an Improper Fraction. Multiplying or Dividing both terms of a fraction by the same number does not change its value. Multiplying the numerator, multiplies the fraction. Dividing the numerator,, divides the fraction. Multiplying the denominator, divides the fraction. Dividing the denominator, multiplies the fraction. Fractions are called similar when they have a common denominator, as f , -f , f, ^. To find a common denominator for any two fractions, multiply both terms of either fraction by the denominator of the other ; or if the denomina- tor of one fraction will exactly divide the denomi- nator of the other, multiply both terms of the one by the quotient. To reduce a fraction to its simplest form. RULE. Divide both terms by their greatest com- mon divisor or its factors, the simplest form, or lowest term of Jf , is obtained by dividing both terms by 12, ff = . COMMON FRACTION. 29 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 9)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 each 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, 4)30, 0, 0, 10 9, & 4X9X10=360 Ans. ADDITION OP FEACTIONS. RULE. Reduce the fractions to a common denominator; add the numerators and place the sum over the common denominator, or multiply either denominator by the other numerator, and place the sum of the products over the common denominator. add_f and i; 1= T %, J=, &+=& or 3x l-f4X2=ll the numerator, 4x 3=12 the denominator, or 5X1+2X3=11, the numerator. 5X2=10, the denominator. 30 HOWARD'S ART OF BECKONING. SUBTRACTION OF FRACTIONS. RULE. Reduce the Fractions to a common denomina- tor, and write the difference of the numerators over the common denominator? From | take J. PROCESS : |=f ; f f=^ Ans. From. ( Jl take 4|. Ans. 4-|. From 18 take 3. Ans. MULTIPLICATION OF FRACTIONS. RULE. Multiply the ivhole numbers tor/ether, then multiply the upper whole number by the lower frac- tion, and the lower luhole number by the upper fraction ; multiply the fractions together, and add all the products; or reduce mixed numbers to improper fractions, and multiply the numerators by each other, and the denominators by each other, cancelling as shown on pages 40 and 42. Multiply 91 by 2J. Ans. 21. 9 01 _ 28 - 18 or a -r 728x93 21 MULTIPLICATION OF ENGLISH MONET. Multiply 9 7s, 5|d. by 7. 9 7s. 5|d. 65 12s. 4|d. PROCESS. Say seven times 3 r= 21 farthings ; put down |d. and carry 5d. ; 7 times (od. + 5d.^3s. 4d. j put down 4d. and carry 3s. ; 7 times 7s.+3s.=2 12s. ; put down 12s. and carry 2 j 7 times ^9 + 2= 65. Total 65 12s. 4d. FJ? ACTIONS. 31 THE UNIT m- one thing is the idea of number in ita simplest form. UNITY is the basis of every number, the primary base of every fraction, the unit of six months in one month, the unit of a fraction is the reciprocal of the denominator, thus J is the unit of ; every step from the unit in- creases the complexity of numbers, and consequently demands an increase of mental power and energy in dealing with them; therefore, when the price per unit is in pence, find the amount at one penny each, and multiply by the given number of pence each. Find the cost of 381bs. of beef at 7|d. per Ib. : 381bs. @ ld.=38d.=3s. 2d., 3s. 2d.x7|d.=Xl 3s. 9d. Find the cost of 42|yds. of cloth at 8d. per yd. 4l%ds. @ Id, 3s. O'id.; 3s. 6|d.x8=l 8s. .4d. The price per unit being in shillings, find the amount at one shilling, and multiply by the price of one. Find the cost of 40 articles at 3s. 6d. each. 40 @ Is. = 2; X2X3J = J7. or, multiply the given number of units by half the price of one in shillings, and point off: one place to the left. Thus: 4-Oxl|=7. The price per ton, in , regarded as shillings, equals the price per cwt. The price of 1 cwt. in shillings, regarded as farthings, X|-=the price of lib. in farthings ; thus, the price of 21 x3 lib. in farthings, at 21s. per cwt.= =-- =9 farthings. The price of lib. in shillings, regarded as farthings, X3=the price of loz. ; thus, the price of loz. at 4s. per Ib.=4x3=12 farthings=3d. The price of loz. in farthings, regarded as shillings, -r-3 equals the price of lib. ; thus, the price of lib. at 2|d. per oz. = = 3s. When the price is an aliquot part of a <, a florin, or a shilling, multiply the given quantity by the aliquot part. Find the cost of 2793 articles at 3s. 4cl. each. 3s. 4d. = i of the 1, 1793-j-6 = 465 10s. NOTE. To attain greater skill in the Art of Reckoning extend and learn by heart Tables of Multiplication, pence and aliquot parts. See next page. 32 JTOWABD'S ABT OF BECKONING. An aliquot part of a number is a measure of that number, or such a part as will exactly divide it ; thus, 12| is an aliquot part of 100 because it is contained 8 times exactly in 100 ; Is. 8d. is an aliquot part of ! because it is contained in 1 12 times. The aliquot parts of 1 are 10s., 6s. 8d., 5s., 4s., 3s. 4d., 2s. 6d., 2s., Is. 8d., Is. 4d., Is. 3d., Is., 10d., 8d., 6d., 4d., 3d., 2d., &c. To find the cost when the price of one is the aliquot part of a . DECIMALS. 35 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 Eo, -007 = TSOO, 47.8 = 47 T 8 o, &c., Ac. Where common fractions occur the calculation may be often simplified by reducing them to decimals. To reduce a common fraction to a decimal. RULE. Divide the numerator by the denominator. J = .5 | = .25 j =.125 ^ = .0625. | = .75 i = .33 33 f = .66 66 l--=.2 * = .4 I = .8 .6 \ = .16 66 $ = .II 11 T C = -083 33 Decimals of ct. Of what number is 48, 8 ^ ct.? 48-^.08=600. Sold a horse for 60, made 25 $p ct., what did it cost? l+.25=H=i 5 I PERCENTAGE. . 45 Sold a horse for 40, lost 20 ^p ct. What did it cost? The population of a village increased from 900 to 1200 at what rate per cent, did it increase ? 300 Or, 1200 cent. --=1.33^. The sales of a firm fell off from 12000 to 9000, what was the rate per cent, of decline ? 300 -p7=2o per cent. ^ Bought a horse for $80, sold it for $105. What per cent, profit? 250 Or, 105 . -- cent. -==1.31 Bought a piano for $300, sold it for $250. What per cent, loss ? 50 -g-=I6 per cent. Bought a horse for $40. What must it be sold for to gain 20 per cent? 40 X. 20+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 =400 X. 01=4. Profit $4, or Vvf the profit is .1 the cost. Vof 4=2 X 10=20 Cost $20. Profit 20 per cent. How many dollars will earn 1 cent a day at 9 per cent per annum? Find the commission at 2 per cent., and the net proceeds on .147 15s. * 147-75 X'Q2=2'955=2 19s. ld.=the commission, <147 15s. 2 19s.ld.=144 15s. 1 Id, =net proceed. , 46 HOWARD'S ART OP RECKONING. INTEREST. INTEREST is the price or sum charged for the use of money lent; the sum of money bearing interest is called the Principal; Simple interest arises from the use of the Principal only. The Rate per cent, is the number of units charged for the use of each hundred units. The Common Method of reckoning interest in the United States is based on a year of 360 days ; in England and U. S. Courts interest is reckoned on the basis of a year of 365 days ; when using the common method, count thirty- days only for each entire month and the difference, if any, will be unimportant. Find the interest by both methods on any example from March 3rd to July 27th ; iff exactly equals ^|-. COMMON METHOD, GENERAL RULE TO RECKON INTEREST. The Principal X the number of days X the Rate -T- 360X100= the Interest. ENGLISH METHOD, GENERAL KULE TO RECKON INTEREST. The Principal X the EXACT number of days X the Rate -i-365 X 100 = the interest. TO FIND THE DIFFERENCE OF TIME BETWEEN TWO DATES. Rule. Subtract the earlier from the later date. EXAMPLE. For what time must Interest be charged on a debt due April 12th, 1882 and settled June 24th, 1883? 83 6 24 Process, 82 4 12 Ans. 1 yr. 2 mos. 12 days. ;t 2 12 TO FIND THE NUMBER OF DAYS BETWEEN TWO DATES. Common Methods-Multiply the number of entire months by three ; call the Product tens and add the extra days ; for the English Method add one day for each month of 31 days ; when February occurs, deduct two days for the Common year, one day for Leap year. To RECKON INTEREST ON STERLING AT 5 PER CENT. PER ANNUM. One tenth of the principal is the interest for 2 years, at 5 per cent. Multiply 'I of the principal by half the given number of years. Find the interest on 240 10s. for 8yrs. 8mos. at 5 per cent. yrs mo3. yrs. 8 : 8 -r- 2 = 4. 24'05 X 4 = 104 '216 = 104 4s. 4d. NOTE. If the exact number of days in each month is taken for a multi- plier, and 360 used for a divisor, the difference or excess will be -\^ ; about 1 J coats to be taken off each 100, or one penny off each six shttiirxjs of interest. INTEREST. . 47 Any number of sterling, regarded as pence, is the interest for one month, at 5 percent, per annum: '1 of the principal in sterling, regarded as pence, is the interest for three days ; hence the following Eules : Multiply the principal by the given number of months, and parts of a month ; or multiply '1 of the principal by one-third the given number of days ; the answer will be in pence. Or multiply '1 of the principal by half the number of months, and divide by 12, the answer will oe in sterling. Find the interest on 428 from July 27th, 1882, to March 3rd, 1883. 12 16s. 9^d. Ans. yrs. mo. dys. d. 8. d. 83 3 3 ] =216dys. 42'8_xj216 42-8 x7'2 428=! 15 8 82__7 27 I 7-2inos. 3 x 12 x 20 or 2x12 _ 7J 7 J See note page 46. 12 16 9^ Interest at 5 per cent, x '2= 1 per cent. ; x '4 = 2 per cent. ; X k = 2 per cent. ; x '6 = 3 per cent. ; = 3^ per cent. ; X '7 3 per cent. ; i=3| per cent. ; x "8 = 4 per cent. ; X -9 == 4| per cent. ; + ] = 5$ per cent ; +'2 = 6 per cent. ; + -3 6^ per cent. ; + -4=; 7 per cent., &c., c., &c. TO 11ECKON INTEREST ON STERLING AT 6 PER CENT. PER ANNUM : One hundredth of the principal is the interest for two months at 6 per cent. ; '001 of the principal is the interest for six days ; '1 of the principal in is the interest in shil- lings for one month ; '01 of the principal in is the interest in shillings for three clays ; hence the following Eultv Multiply '01 of the principal by half the given number of months; or multiply '001 of the 'principal by \, the number of days ; to have the answer in shillings multiply r l of the principal by the number of months; or multiply "01 of the principal by ?, the number of days. Find the interest on 428 10s., from March 3rd to July 27th. m. dys. See note oil pages 46, also 5!). 7 3 ^ } =144dy, 424 J 4 ' 8mos - I of the interest at 6 per cent.=l per cent. ; =2 per cent. |=3 per cent. ; ^ = 4 per cent. ; ^ = 4| per cent. ; = 5 per cent. ; T \ = 5 per cent. ; + T V = 6^ per cent. ; + e = 7 per cent. ; + | = 7^ per cent. ; + i = 8 per cent., &c. 48 HOWARD'S ART OF RECKONING. TO RECKON INTEREST AT 1 PER CENT PER MONTH. Rule No. 1. Multiply the Principal by the given number of days and remove the decimal point three places to the left. Find the interest on $143 for 33 days at \% per month. $143 X ll-s-1000=$1.573 Ans. Find the interest on 1428.50 from March 5th to July 29th. 4 24 =144 days. $428.50x48-f-1000=$20.568 Ans. Rule No. 2. Multiply the Principal by the time in months and fractions of a month and remove the decimal point two places to the left. Find the interest on the above examples by this Kule. $143xl.l-3-100=$1.573 Ans. $428.50 X 4.8-4-100=420.568. Ans. Interest at \% per month is equal Jto 12$ per annum; interest at 12$ perannum-4-4==3$ ; H-3=4$ ; x ^=5$ ; 7$; X|=7|$; X=8$; Xf=9$;etc. TO RECKON INTEREST BY CANCELLATION. 1st. On the right of an upright line write the Prin- cipal, the time, in days and the Rate per cent. 2nd.-0n 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 $428.50 at 5$ per annum of 360 days, from March 3rd to July 27th. Ans. $8.57 MOS. Days. 7 27 m 428.5 4 24=144 days 2 ; Find the interest on $99 at 4$ for 72 days. 99X72X4 36X10X100 -- Find the interest on <428,,10 Stg. at 5$ per annum of 365 days from March 3rd to July 27th. Ans. 8,,11,,5. 428.5 MOS. Bays. 7 27 3 3 4 24-f 2 days=146 days. See note ou pages 46 and 69. INTEREST, 49 When the given rate is not a convenient part oi: five, or six per cent., find the interest for the given time at one per cent., and multiply by the given rate. TO BECKON INTEREST AT ONE PER CENT. PER ANNUM. COMMON METHOD. EULE. To find the interest for one year, divide the Princi- pal by 100; to find the interest for 36 days remove the decimal point three places to the left; for any other time or Rate, in- crease or diminish in the manner shown in the following ex- amples. Find the interest on $1000 for 11 years, 1 month and 6 days, at 1 per cent, per annum. Ans. $111.00 YT. MO. Da. $ 10.00 int. for 1 at 1 % = the Principal -f- 100 100.00 " 10 = first line X 10 1.00 " 1 6= " " X .1 111.00 " 11 1 6 at 1% per annum. Find the interest on 124,,10,, Stg., from March 6th, 1882 to May 18th, 1883, at 7 % per annum. Ans. 10,,9,,2. yr. MO. Da. 1.245 = int. for 1 at one per cent. .2075 = 20 1st line X %, 2 mos. = % of 1 year .0415 = 12^ 2nd line X .2 12 days = .2 of 60 days 1.494 " 1 2 12 at one per cent. 10.458 = 4th line X 7 = Interest at 1$ = 10,,9,,2 The interest is found on all sums at 1 per cent, a month by removing the decimal point to the left, 3 places for 3 days, and 2 places for 30 days- Find the interest on 143 for 1 mo. 3 da. at 1 per cent per month. Ans. l,,ll,,5i. 1.43 int. for 1 mo. .143 " 3 days 1st line X . 1 1.573 NOTE 1. The Decimal expression of values has the same signifi- cance whether the examples are stated in terms of the Pound Sterling, the Dollar, or any other Standard Coin. (See page 59.) NOTE 2. Only a sufficient number of examples to clearly illustrate the working of the several rules are presented ; the Teacher or the Student may furnish additional examples for exercises to any extent. NOTE 3. The answer to three decimal places is sufficiently exact; the ti;ne being less than one year, use only two decimal places in the principal- D 2 50 HOWARD'S ART OF RECKONING. For Reckoning Interest at 5 per cent per year of 365 days. Txi H- 73X100 = 1X1X5 -f- 365X100 EULE. Multiply the Principal by the given number of days, remove the decimal point two places to the left and divide by 73. Find the interest on 100 Stg. for 365 days at 5^ a year. 100X365 100X73 =5 Ans. EULE No. 2. To Reckon Interest at 5 per cent per annum. (1X365-B)+ T V and yfo of H-10,000=.05 nearly; the excess is exactly -nnriro> hence the following. RULE. Multiply the Principal by the given number of days, Divide the Product by 3, and to it add the Quotient plus .1, plus .01 and remove 'the decimal point four places to the left. Find the Interest on 100 for 365 days at 5% per annum. 3)36500=100X365 121666=the quotient. 12166=.! Do. 1216=.Q1 Do. 5.000^=The Answer. To Reckon Interest on the basis of a year of 365 days. lXl2Vr-1000=lx42-^365xlOO nearly. The deficiency is about .0006 or T ^j- ; six cents to be added to each $100; one penny to each 7 of interest. RULE. Multiply the Principal by 1 %, remove the point three places to the left and the interest will be shown for 42 days at 1 per cent, 12 days at 3 per cent, 21 " 2 " 7 " 6 " 14 "3" 6 " 7 divide this interest by the number of days opposite the given rate and the Quotient is the interest for one day, multiply by the given number of days and add .0006 to the product. NOTE. To Multiply by 1^ add -fa and J of -fa of any number to itself. EXAMPLE 1. Find the interest on 100 for 7 days at 6 per cent per annum. Ans. 2s,,3|d. 100= the Principal, 10=.l=^j of the Principal, 5= $ of 3*0 =*V of the Principal, .115=the interest for 7 days at 6 per cent. INTEREST. 51 HOWARD'S LIGHTNING RULE FOR RECKONING INTEREST. Divide 36 by any given Rate' and the Quotient is the time, in days, in which the Interest on any given sum is equal to .001 of the Principal ; in 10 times the Quotient the interest equals .01 of the Principal; in 100 times the interest equals .1 ; and in 1000 times the Quotient, the in- terest equals the principal. Rule. Divide 36 by the given Rate, Multiply the Principal by the given number of days, divided by the Quotient, and re- move the decimal point three places to the left. Find the interest on $1000 for 9 days at k% per annum. 9 9X1000 ' The interest for 9 days is $1.000; for 90 days, $10.00; for 900 days, $100.00; and for 9000 days, $1000.00, or the interest is equal to the Principal. If millions of examples were written together in a col- umn, the same denominations being placed exactly un- der each other; a straight line drawn three decimal places to the left, from the top of the column to the bot- tom, would show the interest on each, and every one of these millions of examples for 9 days at 4 per cent, per annum without altering one figure of the Principal; simi- lar lines drawn two places, and one place to the left would show the interest for 90 days, and 900 days; thus doing the work of a long life in a moment of time. 36-4-3=12 the divisor at %%. 36-^-9=4 the divisor at 9% 36-r4= 9 " " 4#. 36-s-12=3 " " 12# 36-s-6= 6 " " 6f . 36-r-18=2 " " 18 # Find the interest on 428 Stg. at 9f per annum from July 27th, 1882 to March 3rd, 1883. Yr. MOS. Days. 83: 3: 3 82: 7; 2 7 428X^54^23.112^23,2,3. Ans. 7: 6=216 days. ^XlOOO See note on pages 16 and 59. 52 HOWARD S ART OF BECKONING. COMPOUND INTEEEST. Compound interest is interest on the principal, and also on the interest added to the principal, each time it becomes due. EULE. 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 1, or SI, 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. Eind the amount of 864 10s. Od. for six years at 8%; Ans. 1371 17s. Ojd. School Book Method, 184 Figures. 864 5 NOTE. Persons having frequent occasion to compute 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 other 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. (See page 65.) Howard's Method, 74 Figures. 864.5 69.16 933.66 74.693 1008.353 80.668 1089.021 87.122 1176.143 94.091 1270.234 101.619 1371.855 8 69160 8645 933.660 8_ 746928 93366 1008.3528 8_ 80668224 10083528 1089.021024 8_ 8712168192 1089021024 1176.14270592 8_ 940914164736 1176 14270592 _ 1270.2341223936 8_ 101618729791488" 1270.2341223936 1371.85285.2185088 COMPOUND INTEREST. 53 Paying Simple Interest for fractions of any given single period is usual, but it involves a loss to the Payer, because Simple Interest is more than Compound interest for any portion of any single period. Estimating our National debt at $2,000,000,000 and the average Inter- est at 5 per cent, the Bondholders gain and the Nation looses $1,890,673 a year by computing the quarterly payments at simple interest. Every day's true Compound Interest differs, increasing as the day is distant from one. Every day's true discount differs, decreasing aa the day is distant from one, The product of the amount for any two periods=the amount for the sum of the two given periods. The Square of the amount at Compound Interest for any given num- ber of terms = the amount for twice that number of terms, The Square Root of the amount for any given number of terms = the amount for half that number of terms. The Cube of the amount for any given number of terms=the amount for three times that number. The Cube Root of the amount for any given number of terms=thc amount for one-third that number. Example. $10,000 invested at 10 per cent, per annum true Com- pound Interest is to be divided so that each of three sons on becom- ing 21 years old is to receive an equal sum; A is 17H, B 13}^ and C 10 years old ; find how much each will receive and the present value of each son's share. 1st. Find the present worth of $1 due in S 1 /^ 7% and 11 years, true Compound Interest, then find the sum of the results. 2nd. Divide $10,000 by this sum and the ,|>/>//V/ will be the amount due when each son comes of age. 3rd. Multiply this result by the present worth of $1, as before found, for each son, and the products will be the present worths re- quired. ;=present worth of $1 for 3^ years=$.8155104 1.226226 1 =present worth of $1 for 7?^ years =$.63971 76 j=present worth of $1 for 11 years =$.5267875 Sum of results, $1.9820155 2nd. $10,000-7-1.9820155=$5045.37=sum each son will receive. 3rd. $5045.37X.8155104=$4114.55=present worth of A's share. " X. 6397176= 32:27.01= " - k B's " " X.52ii7875= 2657.84= " " C's " Another method. 1st. Find, by Proportion, what sum invested for 3 L /2, and 7% years respectively will equal the amount of $1 for 11 years. 2nd. Multiply each such investment by the $10,000 and divide the product by the sum of the investments; the quotient is the present value of each son's share. NOTE. By computing simple, instead of true Compound Interest for the fractions of a year, the present values would be $2658 62 $3227.36 and $4114.02, and the amount when of age, $5046.86. 54 HOWARD'S ART OF RECKONING. DISCOUNT. DISCOUNT is a certain per cent, deducted from, or allow- ance made for the payment of a debt or other obligation, before it is due. The PRESENT WORTH of any sum is a sum which if put at interest now at a given rate will amount to the required sum when due. TRUE DISCOUNT is the difference between the Present Worth and the amount. BANK DISCOUNT is simple interest on the Principal for a specified time, with three days added, called Days of Grace; a note for 3 months is due 3 months 3 days from date ; a note for 90 days is due in 93 days ; a possible difference of 2 days. In reckoning Bank Discount the sum on which interest is to be paid, is known, but in reckoning True Discount we have to find what sum must be placed at interest so that the sum together with its interest may amount to the given Principal. To find the present worth of any sum, and the true dis- count for any time at any rate per cent. RULE. Divide the given sum by the amount of $1 for the given time and rate; the quotient will be the present worth, and the difference will be the discount. Find the present worth and the true discount on $1000 for 1 year at 10 per cent. 1000=$909.09 present worth, 1000-909;09=$90.90true dis. 1.10 Find the Bank Discount on a note for $1000 for 1 year at \% per annum. $100+ .83=$100.83. Ans. $100=interest for 1 year. .83=interest for 3 days. TO RECKON TRUE DISCOUNT, NEW METHOD. RULE. Write a common fraction that shoivs the part re- quired, * take the Denominator, plus 1, for a Divisor and the Principal for a Dividend, the Quotient is^the true Discount. * The Quotient of the Reciprocal of the Rate, that is the Rate inverted, is the denominator of the fraction that shows the part to be taken, 5$=^, ^-=20, $% =^. DISCOUNT. 55 EXAMPLE. An agent receives 10 # on the net sum paid to his Principal, find his commission on $1000. =~= 10+1= 11, = W.90. Ans. COMMKECIAL DISCOUNT is a given Rate per cent. allowed off a Debt or part of a Debt for Cash, that is, "ready money ;" and is reckoned the same as interest, A bill of goods is bought, amounting to 960 dollars at a year's credit, the merchant offers to deduct 10 % for ready cash, what amount is to be deducted? $960-r-100 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 011 any article of merchandise, and the 10% be deducted, the gain at first sight would appear to be 20%, but is in reality only 17 % . 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%. The net amt. of a bill, less 10 per cent, discount, will be shown by multiplying by .9. Example. 100X.9=90.0 To find the net. amt. less discount at 5 per cent X 9^. 30 per cent X7. 50 per cent X 5. 15 " " XS. 35 " " X6^. 55 " " X4. 20 " " X8. 40 " " X6. 60 " " X4. 25 " " X7 45 " " X5. 70 * ' X3. and remove the point 1 place to the left. 56 HOWARD S AKT OF BECKONING. PARTNERSHIP. A Partnership or Firm is an association of two or more persons for the purpose of transacting business with an agreement to share the profits and losses accord- ing to the amount of capital furnished by each, and the time it is employed. Capital or Joint Stock is the amount of money or property used in the business; the amount due together with the property of all kinds belonging to the firm, is sometimes called the Assets. TliQNet Capital is the excess of assets over liabilities. The Liabilities of a firm are its debts. To find each partner's share of the profit or loss. RULE. Multiply the whole profit or loss by the ratio of the whole Capital to each man's share of the Capital. Example, A and B engage in trade, A furnishes 300 and B $400, they gain $91 ; what is each man's share of the profits? Capital $300, fH=f. Gain, $9!xf=$39=A's Share. " _400, |=f, " 91X|=_52=B's " Whole stock,$700. Whole profit, $91 Another method : Find the rate per cent, gained or lost, and multiply each person's share of the capital by the rate per cent. tffr=13#. 300x. 13=39 \_ c , qi 400X- 13=52/- When the respective capital of each partner is invested for unequal periods of time. RULE. Multiply each man's capital by the time it is employed, and regard each product as his capital, and tlie sum of the products as the entire capital. Take the above example, A's capital being invested for four months, andB's for three months, and find each man's share of profits. $300x4=1200 91xM=$45.50=A's sh?re. 400X3=1200 91 XM- 45.50=B's " Capital 2400 Profits $91.00 EXCHANGE. 67 EXCHANGE. Exchange, in Arithmetic, is a method of finding the value of one denomination of money in the terms of an- other. Exchange, in Commerce, is the paying, or receiving any sum in one kind of money for its value in another ; when the parties are distant from each other this is done by means of an Order or Draft called a Bill of Exchange. Bills drawn in one Country and made payable in another are called Foreign Bills; when drawn and payable in the same country they are called Inland Bills. PAR OF EXCHANGE is the established value of the Stand- ard Coin of one country when expressed in terms of the Standard Coin of another; the value of 1 Stg. in IT. S. gold coin is $4.8665. See page 100. Exchange is at Par when a Bill in New York, for the payment of 100 Stg in London can be bought for $486.65. Exchange is in favor of a place when it can be bought there at or above par ; Exchange diverges from Par by the difference in the amount of the indebtedness between one country and another, called the Balance of Trade. Find the value in currency of a gold dollar, the market price of currency being 75 cents. Ans. 133J cents. 100 4 Process = = 133^ 75 3 2. Find the value of currency, the price of gold being 1331 Ans. 75 cents. 100 3 Process = =.75 133 4 $500 in gold at 8 per cent, premium will buy how much currency ? $500X1.08=1540. $500 in currency will buy how much gold at 8 per cent, premium ? 500-^10S=$462.96. $1000 in gold is worth how much currency at 80 cents? $1000-r-.80=1250. 58 HOWARD'S ART OF RECKONING. What is the face value of a bill of Exchange cost ing 1000. Commission f per cent ? 1000-5-1.0075= 992.55 What is the cost of a bill of Exchange for $1000 Premium f per cent. $1000X1.003=$1007.50. Find the par value of 473 5 9 St'g. in Amer- ican gold coin. 473.2875X4.8665=42303.25. 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 28.39^^ 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" Pounds Sterling X4.866563=the Par value of U. S. Dollar. U. S. Dollars X.2054838=the Par value of Pounds Stg. 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 currency. The cost and temporary inconvenience incident to the change would be trifling, almost m7, 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 T ^o- of a pound, and the thing is done. See page 35. BRITISH MONEY. 59 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 -fa of a pound stg. 2. For every 1 shilling, write .05, because one shilling is T 6 7 of a florin, or T f ^ of a pound stg. 3. For every ninepence, write .0375, because ninepence is S of a pound stg. 4. For every sixpence, write .025, because six- pence is jfo of a florin or ^m of a pound stg. 5. For the ponce multiply -1, the given number of pence, by ^ of J. 6. For the farthings multiply !, the given number of farthings, by ^ of $. s. d. far. 1 =20 =240 = 960 .1 =2 =24 = 96 .051 =12 = 48 .0375 = f =9 =36 .025 = J =6 =24 .0125 = J =3 =12 004^ = ^ = 1 . = 4 .001041. ji = i - = i 19,,2 = 19.1 27,,12,,6 =27.625 19,,3 = 19.15 19,,19,,2J=19.96 19,,5 = 19.25 19,,18,,0|=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 indefinitely, the essentials to remember are thatjtfonns, shillings, [ninepence, sixpence and threepence are decimally expressed absolutely correct. The answer to three decimal places is suffi- ciently correct. 60 HOWARD'S ART OF RECKONING. STOCKS AND BONDS. A BOND is a duly certified instrument showing the in- debtedness, with the limits and conditions of the debt, of a Corporation or a Government. Government Bonds are sometimes called Consols. The capital of a company in transferable Shares, each of a certain amount, is called STOCKS. The value expressed on the face of any certificate of value, as Stocks, Commercial Paper, etc., is called the Par Value. When the market value is greater or less than par value, the Stock is said to be above or below Par, or is said to be at a Premium, or Discount, as the case may be. 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, a? 20 per cent premium! 120 | 800=6#er cent. What per cent will be gained by investing in 6 pe> cent stock at 1 per cent discount. 100 10=90. 90 | 600=63 per cent. To find at w hat price stock paying a given rate per cent, dividend can be purchased, so that the money in- vested shall produce a given rate of interest. RULE. Divide the rate per unit of dividend by the rate per unit of interest. What must be paid for stock paying 6 per cent did' dend } in order to realize on the investment 8 per cent? 8 [ 600=75. DUTIES. 61 TAXES. A Tax is a sum of money assessed on persons or property for the purpose. of defraying public expenses. Real Estate is fixed property, such as houses and lands. Personal Estate consists of money, cattle, ships, fur- niture and other movable property. To find the rate of taxation, the required tax and the value of the taxable property being known: EULE. Divide the required tax by the value of the taxable property, the quotient is the rate of taxation. Example. The taxable property of a township is valued at $4,835,000, the required tax is $96,700 ; what is the rate? r 4 835 OOO' The required tax divided by the rate=the valuation. To find the amount of any person's tax. BULE. Multiply the value of the property by the Rate. Example. The assessed value of Oscar Wilde's property in Dublin is 48,500; the Kate is of 1$; what is the amount of his Tax ? 48,500 X.00i=424;375=424,,7,,6 Ans. - DUTIES. Duties are taxes paid on many kinds of goods im- ported from abroad and are collected by the Custom house officers ; when the duty is a certain per cent, on the value of the goods it is called Advalorem duty; when the duty is on a certain quantity, as agreed, a pound, a gallon etc., it is called a Specific Duty. A Tariff is a schedule showing, the rates of duties fixed by law on all kinds of imported merchandise. Gross weight or value is the weight or value of the goods before any allowance is made. Net weight or value is the weight or value of the goods after all allowances have been deducted. 62 HOWARD'S ART OP RECKONING. INSURANCE. Insurance is a contract of indemnity against loss or damages. Fire Insurance is indemnity for loss of property by fire. Marine Insurance is indemnity for loss of vessels or cargo- Life Insurance is an agreement to pay a certain sum in case of the death of the insured. The Insurer or Underwriter is the party who takes the risk ; the written contract between the two parties is called the Policy. The Premium is the sum paid for Insurance. To find the Premium, the sum insured and the Rate being given. RULE. Multiply the sum invested by the rate. To find what sum must be insured to cover both the Property and Premium, the Rate being given. RULE. Divide the value of the Property by 1 minus the Hate. PROFIT AND LOSS. To find the gain or loss per cent. RULE. Divide the gain or loss by the cost. To find the selling price to gain a given per cent. RULE. Multiply 100, plus the gain per cent, by the cost and divide by 100. To mark goods so that a given per cent, may be deducted and yet make a given per cent profit. RULE. Divide the real setting price by 1 minus the given per cent, to be deducted, the quotient is the marking price. Example. Bought hats at 82.55 each, at what price must they be marked so that 15 per cent, may be deduct- ed, and yet be sold at 20$ profit. $2.55+20^=$3.06=the selling price. 3.06-j-l 15=$3.60=the asking price. To mark goods to gain a given per cent, on the selling price : Divide the cost by 1, minus the required Rate per cent See page 106. BARTER. 63 ALLIGATION. Alligation treats of mixing or compounding two or more ingredients of different values or quantities ; the process of finding the mean value or quantity of several ingredients, is called Alligation Medial. 'RvLE.Find the entire cost, or value of the ingredients and divide it by the sum of the simples. Alligation Alternate is the process of finding the pro- portional quantities to be used in any required mixture. RULE. Arrange the ingredinents in pairs, one of less and the other of greater value than the required value, the difference of one member of a pair and the required value is the required quantity of the other member. To prove the answer, multiply each value by its quantity, and divide the sum of the products by the sum of the quantities. Example. Having four qualities of tea worth 1, 2, 3 and 4 dollars a pound, how much of each must be used to make a mixture worth 24 dollars a pound. 1 X 4= 4 or 2X1^=3 4X 4=2 4x1^=6 2X i=l 3X 4=1 4) 10=2* 4) 10=24 By the first arrangement we get 4, 14> 14 an a. Zero date 4 2U4y )11532(5.63 Equated term 5 19 3 Rule Multiply each debt by the term of credit, plut the time between the date of the transaction and the zero date divide the sum of the products by the sum of the debts t 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 of months from the zero date, these together with the day of the month are the multipliers. mos. mos. da* 1st item 6 Cr. plus 0, 10 from date 6 1- mos. 2d " 2 " " 1, 21 " " " r=r3.7 " 3d " 4 " " 2, 1 " " " =63^ cc 4th " 3 " " 3, 8 " " " =6.1^- u Note. The use of the beginning of the month, instead of the date of the first transaction for the starting point, makes no difference in the ultimate result, and avoids the continual labor of finding on each item, the time between two dates, each date as written, itself representing the time. *3<>3 70 HOWARD'S ART OF RECKONING. AVERAGING ACCOUNTS. Find the equated time of paying the balance of the following accts ls ~8 Dr. 1878 Cr. Mar. Mar ft 15 Smos. 600X3*= j 1800 2 May 10 By Cash 300X2K=J 600 lApr. 3 4 " 700X5.1= j 3500 4 July 1 " 400X4>^ = j 1600 2 May 10 6 " 1000 X 8;Hf= j 8 00 5 Aug 15 41 500X5K = | 2 ^J 2300 14003 L200 5063 1200 5063 1100 )8940(8.13 3 Balance due Nov. 4th, 8 mos. 4 days after zero date. 1877 Cr. 1877 Dr. June !15S June C329 Uuly 4 By Note 16 ** 20 To Goods 9S6X^ = I 329 5 C760 BDcc. 18 " Md'c 228X6.6 = <> 1368 6 Nov. 16 1878 I 137 1878 ' * 6 (30 3 Mar 5 450X9V 6 = 1 40 g 8 Feb. 28 c 110x8.7^= \ 7? / TO '836 5809 V io 2474 1248 2474 412)3335(8.09 **& 2ero date Minus 412 8 mo. 3da before zero date. Balance due Sept. 27th, 1S76. 76, 9, 2 NOTE. The Dr. and Cr. sides are here transposed for convenience. 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 ter?n. AVERAGING ACCOUNTS. 71 CASH BALANCES. By the use of the following Eule the final desired re- sult, the Cash Balance, may be found in less time than is required to find the Date on which the Balance is due. RULE. Multiply each item by the number of months and fractions of a month between its date and the date on which the Cash Balance is required; the difference of the sums of the Products X .01 is the interest on the Balance at one per cent. per month. See note page 46. Find the Cash Balance on the lower example page 70, March 5th, 1878 ; interest 1 % per month. Ans. $483.13. 1877 Dr. 1877 Cr. June 20 To Goods. Q8fivi J 7888 Jul ? 4 By Cash $158 X 8^0=1269 Nov. 10 " $986X5i \ 493 Dec. 18 ( 456 1878 Feby 26 ( 456 1878 152x3.3J=J 46 1 51 Mar. 5 228x2.4! j 91 450 110 X- 3= 33 $836 1854 $1248 8967 836 1854 Balance $412 Interest 71.13 MONTHLY STATEMENTS. Find the Cash Balance 011 the following account, at the end of the month, Interest 6% per annum. Ans. $5345.73. Jany. 3 To Goods. $841.28X-9 = 757.1 " 5 u 730.75X5*^3 === 609.0 (C 6 u 815.00X-8 = 652.0 R 10 a 660.00X1 = 440.0 15 " 786.20 x 393.0 18 (t 1000.00 X. 4 = 400.0 11 27 " 496.00 X.I = 49.6 $33.00-j-2=Int. at 6#.= $5329.23 Int at 12% $33.007 16.50 $5345.73 Interest at \% per month is equal to interest at 12% perannum-4-4^=3 per annum ; $ ; XT%=5% ; -l=9#;etc. 72 HOWARD'S ART OF RECKONING. PARTIAL PAYMENTS. Bankers make a Business of loaning Money for Profit; they therefore find the Amount due atthe date on which a Payment is made, deduct the Payment, and regard the Balance as a new Principal. Merchants and Traders charge and allow the same Eate of Interest on both sides of the account. BANKERS' RULE AT ONE PER CENT. PER MONTH INTEREST. RULE. Multiply the Principal by the number of months and fractions of a month between its date and the date of a Partial Payment, add the product X .01 to the Principal and deduct the Payment; the Balance is the new Principal. A note is made for 1000 Stg. March 3rd, 1882, endorsed May 15th, 100, July 27th, 200, find the Cash Balance March 3rd. 1883. Ans. 799,,1S,,0. 1000.00X2.4X.01=24.00. 1000+24.00-100=924= Bal/ May 15 924,,0,,0 924.00X 2.4X -01=22.176. 924+22.176 200=746.176=Bal. July 27, 746,,3,,6 746.176x7.2x.01=53.724 746.176+53.724= 799.9= Cash Bal. Mar. 3,799,,18 MERCHANTS AND TRADERS' RULE AT ONE PER cr. PER MONTH. RULE. Multiply each item by the number of months and fractions of a month between its date and the date of settlement; the Balance of the JVoduc&X-Ol is the Interest on the Balance. Find the Cash Balance on the above Note, Merchants and Traders Rule. Ans. 796,,0,,0. May 15th, 100X9.6 =960 1000X12= 12000 July 27th, 200X7.2=1440 300 2400 2400 Bal. 700 Int. on Bal. 9600 700+96=796,,0,,0 Cash Balance March 3rd, 1883. MERCHANTS AND TRADERS' RULE AT FIVE PER CT. PER ANNUM. RULE. Multiply each item by the number of days between its date and the date of settlement; divide the Balance of the Products by 3; to the Balance add the Quotient plus .Iplus .01, and remove the decimal point four places to the left. Find the Cash Balance on the same acct. interest 5$ May 15th, 100x292=29200 .1000x365=36.5000 July 27th, 200X219=43800 7.3000 73000 3)29.2000 9.7333 .9733 .0973 700+40=740,,0,,0. Ans. * See page 50. 40.00*0*' PARTIAL PAYMENTS. 73 COMMERCIAL DISCOUNT is a given rate per cent, allowed off a Debt, or part of a Debt, not off the money paid, in consideration for Cash. The terms of the following transaction are 6 months' credit ; 7/ discount for cash in ten days ; 6/ in one month ; 4/ in two months. A discount of 7/o enables 93 to pay 100, con- sequently every 100 paid discharges 107.5269 of the Debt ; discounts at the otner rates named result in like manner. GEORGE SMITH & SONS. Dr. In account with BROWN & Co. Cr. 1883. 1883. T an....l4 To Mdse. 6 mo. 2,147 583 Jan.,.24 By Cash Discount 7 p. c. 1,000 75 260 Feb. . 13 Cash 500 Mch..l3 Discount 6 p. c. Cash 31 300 915 Discount 4 p. c. 12 5 July . 14 BaL due without discount 227 Son 083 2,147 583 2,147 July... 14 To Balance 227 son On what date is the following balance due? Rule> page 70. 1883. Dr. 1883. Cr. Jan. ...1 To goods, 6mos.... 1500 March 1 May....l By Cash .... 300 400 800 Jan. ...1 1500 x 6 = 9000 IMarch 1(300x2 = .... May 1400x4 600 2200 800)6800 1600 2200 Ans 8 months 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 G times 1500 or $9000 for 1 month on paying $ 300 in 2 months, the nse ot which for that time is =$600 for 1 month 400 4 = If.Op 1 he has used the equivalent of $2200 for 1 month, and is consequently entitled to the use of the balance for a time equal to the use one month. 74 HOWARD'S ART OF RECKONING. CASTING OUT THE NINES. The number nine has many peculiar properties in our system of notation. Any number is divisible by nine when the sum of its digits is divisible by nine. 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 exeeds 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 remainder 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 excesses 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 the excesses in Quotient and Divisor plus the excess of the remainder. Subtraction, by Addition, by the use of the Number Nine. RULE. Write nine times the subtrahend under the minuend, add each figure of the upper number to the figure of the same order, and all the inferior places, of the lower number, carrying as in addition, and stopping at the last carrying figure. SELLING GRAIN BY THE CENTAL. 75 BAPID RULES FOE 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 old practice of selling grain by 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 when the price per bushel is given. RULE. Set down the price per bushel; remove ihe 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. 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. In dealing with English market quotations write the given price per cental in pence, and divide by 20, the answer will be in shillings. EXAMPLE. If wheat is quoted at 8s. 9d. per cental, what is the value of a bushel ? 8s. 9d.=105d. =:63d., or =5'2o=-.5s.3d. The price per cental in U.S. Dollars, multiplied by 4-11, equals the value per cental in English shillings, thus : wheat at 83 per cental=4'llx3=12-33=12s. 4d. NOTE The number of pounds estimated to the bushel must conform to the local uscige; in the above examples the bushel is assumed to be equal to GOlbs. 76 HOWARD'S ART OF RECKONING. To find the number of cubic feet in a Hay Stack. If the Stack is round, add the height to the eaves in feet to the height from the eaves to the top, Multiply this sum by the square of the diameter Multiplied by .7854; or Multiply by the square of the circumference Multiplied by .07958. If the Stack is square find the height in the same way and Multiply the height by the square 'of one side. If the Stack is rectangular with gable ends add the height to the eaves to \ the height from the eaves to the top, Multiply the sum by the length of the stack Multi- plied by the width. The number of cubic feet to be reckoned for a ton, depends upon the character of the hay and local usuage. 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 thaee places to the left. EXAMPLE. W^hat is the cost of 764 Ibs. of hay at $14' per ton? 764x7-7-1000=5.348. NOTE, The above rule applies to anything of which 2,000 pounds is a ton. To find the number of trees required to plant an acre. RULE. Divide 43560 by the number of square feet occupied by one tree. The trees being eight feet apart, how many are required to plant an acre? 43560 43500ft. = 1 acre. = 778 trees ' Ans ' 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. 77 Process 20 x 10 X 5x8-^10= 800. Or, 1000 ft. 800.0 N"ote. Exactness requires the addition of 44.5 bushels to every ten thousand U. S. Bushels. Cubic feetX.?79=Imperial Bushels nearly. 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; and, in walking, keep these objects constantly in line. 78 HOWARD'S ART OF RECKONING. Farmers and others by adopting the following sim- ple and ingenious contrivance, may ahvays 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 bonders of the left arm, pricking in the marks with indelible ink. To find the area of a four-sided figure, the opposite sides being 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. 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. Only a sufficient number of examples to clearly illus- trate the working of the Rules are presented; the Teacher or Student may furnish additional examples for exercises, RULES FOR FARMERS. 79 EXAMPLE 3. How may yards in a piece of land, one side is 40 yards long, and the other side 00 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. 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 -~ 4 40 rods Ans. Note. In measuring irregular plots of land divide it into rectang'ea and triangles, and take the sum of the measurements. 80 HOWARD'S AKT OF BECKONING. 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 ? 42f acres Ans. Process.- 8)6840 "2)855 ~~42?75 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 1JX3J. Find the length in yards of & circular fence to enclose 10 acres. ^48400=220. 220 X 1J X 3 {=780 yards. A square plot of the same area requires a fence 880 yards long. The largest area enclosed within the shortest fence, in a rectangular plot, is a square. 393 J yards of fence will enclose a square plot of two acres; it would require 2 miles and 2 rods of fence to en' close the same area in a rectangular plot 1 rod wide. RULES FOR MECHANICS. 81 Base, 4 feet. RAPID RULES FOR MECHANICS To lay off a square corner. Take a measure and lay off with it a triangle, / one side of which is four ft long, anoth- er three feet, and the remaining side five ft., this triangle will be right-an- gled, and the two shorter sides will serve to lay off an exact square. TRIANGLES. The J.rea=the BaseX half the Altitude. The Areai/of the product of half the sum of the three sides X by the three remainders of each side sub- tracted from the half sum. The Hypothenuse=^/of the sum of the squares of the base and the perpendicular. Or, divide the square of the Base by the sum of the Hypothenuse and the Perpendicular. Half the sum of the Divisor and the Quotient, equals the Hypothenuse. A Diagonal line from the upper, to the opposite lower corner of a room = the square root of the sum of the squares of the length, breadth and height of the room. Either Base or Perpendicular= v /of the difference of the squares of the Hypothenuse and the given side. The Altitude = twice the quotient of the area -4- the given base, or the Hypothenuse being the base, divide the product of the two other sides by the Hypothenuse. The number of board feet in a Telegraph pole, or any frustrum of a Pyramid,=four times the sum of the areas of the two ends and the mean* in feet X the height. The number of board feet in a wedge = twice the sum of the three parallel edges, in feet X the width of the Matt X the length. The Area of a square constructed upon the Hypothenuse of a triangle is equal to the sum of the areas of the squares constructed upon the other two sides. t > 82 HOWARD'S ART OF RECKONING. A WEDGE, the solidity=4 the sum of the three parel- lel edges x by the breadth of the butt X by the length. The Frustrum of a Cone, a Pyramid, or a Wedge, the SoUdity=$ the sum of the areas of the two ends and the mean proportional X the height. The Mean Proportional of the Frustrum of a Cone, or the Frustrum of a Pyramid=the product of the diameters of the two ends ; of the Frustrum of a Wedge=i the sum of the products of either of the different edges of the butt x the other edge of the top. The height of a Pyramid^the height of its Frustrum X the diameter of the Base-s-the difference of the two end diameters. To find the number of BOARD feet in a telegraph pole, the diameters being given in inches, and the height in feet. If the two ends are square, multiply the sum of the squares of the two end diameters, plus the product of the two end diameters by the height, and divide by 12. If the pole is round, multiply the squares of the two end diameters, plus the product of the two end diameters, by the heightxby.0218. For cubic feetx .001818. ACOUSTICS. The time, in seconds, multiplied by 1125, gives, in feet, the distance of sound. GRAVITY. The square of the number of seconds x IGj^the distance, in feet, a body will fall in a given time. MOMENTUM. The weight, in pounds X the velocity in feet, per second, gives the momentum of bodies, ATMOSPHERE. The weight of the atmosphere, in pounds, at the surface of the ocean=the given area, in square inches X 15. WATER POWER. The weight or pressure, in pounds, of water at any given depth, on a square foot=the depth, in feetx 62 . THE LATERAL pressure, in pounds=the area of the reservoir, x half its average depth X 62 J- RULES FOR MECHANICS. 83 MEASURE OF SUPERFICES AND SOLIDS. Lineal Measure relates to length only, Superficial Measure to length and breadth ; Cubic or Solid Measure to length, breadth and thickness. 1. If the floor of a room be 20 feet long by 18 feet wide, how many square feet are contained in it? 180 X 2 = 360. Ans. 360 feet. 2. If a board be 4 inches wide, how much in length will make a foot square ? Ans. 36 inches. 144 divided by the width, thus, 1 | 4 =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. 2 Or thus, 18 inches equals 1J ft.; 21 x H = 31J. 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. 84 HOWARD'S ART OF RECKONING. 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. 16J- sq. ft. Process 9 + 11 20 x 10 10 in., mean width; - = 16|. 2 12 To find the board measure of planks and joists. EULE. Multiply the length in feet, by the Product of the thickness and the wdth, in inches; and divide by 12. 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 To find how many board feet, one inch in thickness, can be sawed from a round log of any given length. EULE. Subtract four inches from the given diameter, square the difference,Multiply by the length in feet, and divide by 16. How many board feet can be cut from a log 24 inches in diameter, 18 feet long ? 24-4=20 20X20X18] 8X2 To find the cost of any number of feet of Lumber. EULE. Multiply the given number of feet by the price per 1000 and remove the point three places to the left. RULES FOR MECHANICS. 85 To find how many solid feet a round stick of timber of the same thickness throughout, will contain when squared. RULE. Multiply the length by the Product of the diameter and the Radius, all in feet. Find how many solid feet when squared, in around log 2} feet wide and 10 feet long. 5X5X10 2X4 =31.25 feet. Ans. General rule for measuring timber to find the solid contents 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. 2 X 6 5xl2 =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 J feet in diameter at the base and 24 feet long. 8X8X22X8 =14 ' 4X4X7 How many cubic feet in a square block of marble, truly tapering to a point, 24 inches on each side at the base, and twelve feet high. 24X24X4 Qr 2x2X 4 == i6 feet, Ans. 144 86 HOWARD'S ART OF RECKONING. The diameter being given, to find the circumference. RULE. Multiply the diameter by 3. EXAMPLE. What is the circumference of a wheel the diameter of which is 42 inches? Ans. 11 ft. 42X3J- or ^X22_ 11 feet. 12 2X7 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. Process W X_7___oi ~~1~ 2 22 What is the width of a circular pond, 154 rods in circumference? Ans. 49 rods. Process 7 1H X 7 _ 4 g i n 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. 8X8X22 Q7 f 2X2X 7 The length of a cylinder is equal to the capacity -r-thc square of the radius-^3}. Find the depth of a circular cistern, 7 feet wide, containing 2400 U. S. gallons. 15X7X7X22 RULES FOR MECHANICS. 87 To find the volume of a Cylinder. RULE. Multiply the square of the radius by the thickness, both in feet, or fractions of a foot, and the product by 3i[- ; or, Multiply the square of the diameter by the thick- ness, both in inches, and divide by 2200, the answer is in cubic feet ; or, Multiply the square of the diameter by .7854, and that product by the length. EXAMPLE. How many feet in a grindstone 24 inches in diameter and 4 inches thick? 1st Method. 2nd Method. 1X1X22 24X24X4 3X7 2X11 A Cylindrical foot is the volume of a cylinder, one foot in depth and diameter, and is equal to 1728 cylindrical inches. Cylindrical inches, X .7854 cubic inches. A Cylinder, the surface = the circumference X the length. TO MEASURE FREIGHT, ETC. HOLE. Multiply together the length, breadth and depth of one package in feet, and the largest fractions of a foot and multiply by the given number of packages of the same dimensions. Find the number of cubic feet in six packages, each 1ft. by 1ft. 2m. by 1 jft. 1X7X7XG 6x4 =12|ft. AJIS. Find the charges on 1000 cases, each 16x12x6 inches, at 16s. per ton of 40ft. 1000x4x1x1x16 3 x 2x40~ . = 13 6s. 8d. 88 HOWARD'S ART or RECKONING. 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 : 4-J in. wall, per ft., superficial, (^ brick) 7 bricks 9 " " " (1 brick) 14 " 13 " " (1 brick) 21 " 18 " " " (2 bricks) 28 " 22 (2^ bricks) 35 NOTE. For eacli half brick added to the thickness of the wall, add seven 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. SQUARE AND CUBE BOOT. 89 SQUARE AND CUBE EOOT. 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 thgysquare root of a number is a composite number, the square itself may be divided into inte^ 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 shall 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 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. 90 HOWARD'S ART OF RECKONING. TABLE For comparing the natural numbers with the unit figure of their squares and cubes. By the use oi this, many roots may be extracted by observation s Numbers... 123456789 10 Squares.... 14 9 16 25 86 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 oi 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. 4 2 =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. SQUAKE HOOT. 91 To find the square root of a number. Use, to find the length of one side of a given square. 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 ported is found, annex periods of ciphers, and continue the root to as many decimal places as are required. Example. Find the square root of 643204. 64 N 32 V 04( 802 Square Boot. 64 1602) 3204 3204 or, Divide the number into two parts, take twice the Root of the larger part for a Divisor, and the other part for a Dividend, the sum of the two Roots is the Root required. Ex. Required the length of a ladder standing 80 feet from the Base, to reach the top of a cliff 798 feet high. 798x2+4=1600 ) 6400 ( 4 80 2 =6400. 6400 798+4=802 ft. Ans. 92 HOWARD'S ART OF RECKONING. To find the cube root of a number. Use, to find the length of one edge of a given cube. 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 term in the left hand column; multiply the last result by the new figure in the root and add the prey- 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. Eepeat 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 coin, divisor. 300 64 364 4,320,000 14,416 4,334,416 1,745,337,664(1204, cube 1 root. ~~745 728 17,337,664 17,337,664 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 is a remainder, after the root of the lust period is found, annex periods of ciphers and proceed as before to as many decimal places as the answer requires. 3. The cube root of a fraction may be found by extracting the cube root of the numerator and denominator, or reduce the fraction to a decimal and extract tue root. REFERENCE TABLES, MULTIPLICATION TABLE. 1 2 3 4 5 6 7 8 9 10 11 12 2 4 4 8 12 16 5 10 15 20 25 6 12 18 24 30 36 T 14 21 28 35 42 49 8 16 24 32 40 48 56 64 9 18 27 36 45 54 63 72 81 70 20 30 40 50 60 70 80 90 100 11 22 33 44 55 GG 77 88 99 110 121 12 24 36 48 60 72 84 96 108 120 132 144 ABBREVIATIONS A I, First Class. @ At. a/ c or Acc't . .Account. Ain't Amount. Ass'd Assorted. Bal Balance. Bbl Barrel. B. L ..Bill of Lading. (fo Per cent. Co Company. C. 0. D. ..Collect on Deli very. Cr Creditor. Com Commission. Cons't Consignment. Cwt Hundred Weight. Dft.. Draft. Disc't Discount. Do The same Dox Dozen. Dr...., Debtor. E. E Errors cxccptcd. Ea ..Each. Exch Exchange. Exps Expenses, Fol Folio. Fvr'd Forward. "Fr't Freight. USED IN BUSINESS. $, Dollar C, Cents. Guar Guarantee. Gal Gallon. Had ....Hogshead. Ins Insurance. Inst This month. Invt 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. Pros Next month. Ps Pieces. KecM Received. R. R Railroad. Ship't Shipment. Sund's Sundries. S. S Steamship. Ult Last month. 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 cubic foot of rain water weighs 1000 ounces Avoirdupois. NOTE. To find the weight, in ounces, of one cubic foot of any substance here named, remove the 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 1.035 Alum 1.724 Aluminum, 2.560 Amber 1.064 Amethyst, 2.750 Ammonia, 875 Ash, 800 Blood, Human, .1.054 Brass, (about) 8.000 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 240 Diamond, 3.530 Earth (mean of the Globe) 5.210 Elm, 671 Emerald, 2.678 Ether 632 Fat of Beef, 923 Fir, 550 Glass plate, 2.760 Gold, hammered, 19.362 " Coin, 17.647 Granite 2.625 Graphite, 1.987 Gunpowder, 900 Gum Arabic, 1.452 Gypsum, 2.288 Hazel, 600 Hematite Ore,.., 4.705 Honey, 1.456 Ice,.... , 930 Iodine 4.948 Iridium, , 23.000 Iron, 7.645 " Ore, 4.900 Ivory, 1.917 Lard, 94? Lead, cast 11.350 41 white, 7.235 Lignum Vitae, 1.333 Lime, 804 " 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, 2.114 Opium, 1.337 Pearl, 3.510 Pewter, 7.471 Platinum Wire, 21.041 Poplar, 383 Porcelain, 2.385 Quartz, 2.500 liosin, 1.100 Salt 2.130 Sand, 1.750 Silver coin, 10.534 Slate, 2.110 Steel 7.816 Stone 2.500 . Tallow, .941 Tin, 7.291 Turpentine, spirits of, 870 Walnut 671 Water, distilled, 1.000 Wax 897 Willow 585 Wine, 993 Zinc, cast, 7,1'JO FORMULAS, 96 The Diameter of a Circle. i 3 '3183 } =The Circumference. X .8862 \ =The side of an equal -5-1.1284 I Square. X.866 \=The side of an inscribed -5-. 1547 J Equilateral Triangle. X .707=The side of an inscribed square Xthe Radius=The area of do. f X.2821 -4-3.545 The Circumfer- X .2756 ence of a Circle. -4-3.6276 X.2251 -4-4.4428 X. 15915 -4-6.28318 : -4-3.1416 The Area of a X 1.2732 Circle vx i o K.aaxl6 FLAT CAP PAPERS. Law Blank, 13x16 Medium 18x23 Flat Cap, 14x17 Royal, 19x24 Crown, 15x19 Super Royal, 20x28 Demy, 16x21 Imperial 22x30 Folio Post, 17x23 Elephant 22^x27% Check Folio, 17x24 Columbia, 23x33^ DoubleCap, 17x28 Atlas, 26x33 Extra Size Folio, 19x23 Double Elephant 26x40 SIZE OF PRINTING PAPERS. Medium, 19x24 Double Medium, 24x38 Royal 20x2-5 Doable Royal, 26x40 Super Royal 22x28 Double Super Royal, 28x42 Imperial, 22x32 " " 23x43 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. When a sheet i The Book ( 1 sheet of Vhen a sheet ) The Book 5 1 . sheet of is folded into j is called ( Paper makes is folded into > is called { Paper makes 2 leaves. A Folio. 4 pages 16 leaves. A 16mo. 32 pages' 4 " A Quarto or 4to. 8 " 18 " AnlSmo. 36 " 8 " An Octavo or 8vo. 16 " 24 " A 24mo, 43 " 12 " A Duodecimo or 12mo. 24 " 32 " A32mo. 64 " Clerks and Copyists are often paid by the Folio ior making copies ;>f 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, computed by C. FRUSHER HOWARD. Country. Denomination. Wo Grains. ighfc Ounces. Fine lOOOths nass. Carats. Value s. d. . U.S. $ Austria, Union Crown, 171.36 0.357 900. 21.00 1,, 7,, V/ 2 6.6419 Belgium, 25 Francs, 121.92 0.254 899. 21.57 19,, V/ 2 4.7203 Bolivia, Doubloon, 41G.1G' 0.807 870. 20.88 3,, 4,, 1 15.5925 Brazil, 20 Milries, 27G.OO 0.575 917.5 22.02 2,, 4,, 10 10.9057 Chili, Doubloon, 416.16 0.807 870. 20.88 3,, 4,, 1 15.5925 Denmark, 10 Thaler, 214.96 0.427 895. 21.48 1,,12,, 5^ 7.9000 England, Sovereign, 123.27 0.2568 916.6 22.00 1,, 0,, 4.8065 France, 20 Francs, 99.00 0.2075 899. 21.57 ,15,,10>4 3.8562 Germany, 20 Marks, 122.90 0.256 900. 21.00 19,, VA 4.7027 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,, 9J4 3.8426 Japan, 5 Yen, 128.30 0.267 900. 21.60 1,, 0,, 5 4.9674 Mexico, Doubloon, 410.16 0.8075 870.5 20.89 s,, 4,, iy z 15.6105 " 20 Pesos, 518.88 1.081 873. 20.95 4,, 0,, 2 19.5083 Nethcrl'ds. 10 Guilders, 103.72 0.216 899. 21.57 16,, 5 3.9956 Peru, Doubloon, 410.16 0.807 868. 20.83 3,, 3,,11^ 15.55G7 a 20 Soles, 490.80 1.035 898. 21.55 3,,18,,11% 19.2130 Portugal, Gold Crown, 147.84 0.308 912. 21.88 1, 3,,10K 5.80G6 Rome, 2^ Scudi, 07.20 0.140 900. 21.00 10,, 8 2.6047 Piussia, 5 Roubles, 100.80 0.210 916. 22.00 1C,, 4 3.9764 Spain, 100 Reales, 128.64 0.268 896. 21.50 1,, 0,, 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,,HK 4.3693 United j 20 Dollars, 516.00 1.075 900. 21.60 4,, 2,, 2^ 20.0000 States, j One Dollar. 25.80 .05375 900. 21.60 .2054838 l.OCOO The Gold Talent of Scripture= 5464 " Silver " " 341 Exactly the existing ratio between U. , 5,,8 = $26592.809. ,10. ,4 $ 1662.025. S. Gold nnd Silver Coins- -16 to 1. SILVER COINS.' 101- Table of various Silver Coins, showing their weight, fineness and quota of pure silyf, f commuted from-- U. S. Mint assays, by C. FRUSHER HOWARD. Country. Denomination. Fine- ness. ei 'Ounces. ght. Grains. Pure S 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 0.806 415.68 353.328 .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.GO 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, Scndo, .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 0.7G6 367.68 334.221 .696294 Rome, Scudo, .900 0.8G4 414.72 373.248 .777600 Russia, Rouble, .875 O.C67 320.16 280.140 .583625 Spain, New Pistareon, .899 0.166 79.68 71.632 .149234 Sweden, Rix Daler, .750 1.092 524.16 393.120 ,319000 Turkey, 20 Piasters, .830 0.770 369.60 306.765 .639100 Tuscany, Florin, .925 0.220 105.60 97.680 .203500 United States. Dollar .900 0.8594 412.50 371.25 .7734375 Trade " .900 0.875 420.00 378.00 .787500 103 GOLD VALUE OP U.S. 'SILVER AND RUPEES:, TABLE showing the value in U.S. Gold Coin of an oapoe of 'Silver (480 gr.), and a Standard dollar (412^ gr.), each 9-10 line, at London quotations for Silver bullion .925 fine, calculated at the par of exchange, $4.8665, to the pound sterling, by C. FRUSHER HOWARD. Londo Pence. 1 Price. Ster'g Value of Ounce. 480 Grn's, Value of Stand. $ 4121 G . Londoi Pence. i Price. Ster'g Value of Ounce. 480 Grs. Value of Stand. $ 4121 Grs. 60 .2083 1 0.986 1 0.847 551 .2302 1.089 1 .936 50i .2094 0.991 0.851 551 .2312 1.094 .940 50 .2104 O.C96 0.855 55f .2323 1.099 .945 60f .2115 1.001 0.860 56 .2333 1.104 .949 51 .2125 1.006 0.864 561 .2343 1.109 .953 51| .2135 1.011 0.868 56 J .2354 1.114 .958 5H .2146 1.016 0.873 56| .2365 1.119 .962 Blf .2156 ^1.020 0.877 57 .2375 1.124 .967 62 .2167 *1.025 0.881 571 .2385 1.129 .971 52j .2177 1.030 0.886 571 .2396 1.134 .975 521 .2187 1.035 0.890 57f .2406 1.139 .979 52| .2198 1.040 0.894 58 .2417 1.144 .984 53 .2208 1.045 0.898 58 .2427 1.149 .988 531 .2219 1.050 0.903 58 .2437 1.154 .992 531 .2229 1.055 0.907 58f .2448 2.159 .996 63f .2239 1.060 0.911 59 .2458 1.164 1.000 54 .2250 1.065 0.915 59i .2469 1.169 1.004 541 .2260 1.070 0.920 59j .2479 1.173 J.009 54| .2271 1.075 0.924 59f .2489 1.178 1.013 64f .2281 1.080 0.928 60 .2500 1.183 L017 55 .2292 1.085 0.932 The Trade dollar is worth two-tenths of a cent, more than the Mexican. Is cents more than the U.S. Standard Dollar. The value in U. S. Gold Coin of 1 ounce of silver. .9 fine = .01972931 X the given number of pence per ounce in the London market. The value of the U.S. Standard silver dollar = .01695 X the market price in pence. The Standard weight and fineness, respectively cf ths Indian Rupee is 180 grains, and 916.6 millesimal fineness. The value of the Rupee in pence =-37162 X the number of pence per ounce. Francs X JTJQ = sterling ; x -4- = Francs. METKICT SYSTEM. 103 THE METKIC SYSTEM of Weights and Measures is based upon the decimal scale ; its paramount simplicity insures its early adoption by all civilised nations. The Meter is the base of the system, and = 39.37079 in. The Are (air) is the unit of surface, the Stere (stair) is the unit of volume, the Litre (leeter) is the unit of capacity, the Gram is the unit o weight; these constitute the primary units of the system. The Multiple Units, or higher denomina- tions, are named by prefixing to the name of the primary units the Greek numerals Deka (10), Hecto (100), Kilo (1,000), and Myra (10,000). The siibmultiple units, or lower denornina- J tions, are named by prefixing to the names of J the lower denominations the Latin numerals, | Deci ( T \) ), Centi ( T fo\ Milli ( 117 Vo)- The Name of a unit indicates whether it is ]j f/reater or less than the standard units, and g also how many times. MEASURES OF EXTENSION. The Meter is the unit of the length, and ,-, = 39,37079 inches, and is used in measuring || cloths and short distances. The Kilometer is commonly used for moa- 6 sunng long distances, and is about five-eights | of an English mile. TABLE. Metric Denomination?. English or U.S values. 1 Millimeter ^ .03937079 in. 10 Millimeters, mm.-l Centimeter = .3937079 10 Centimeters, on. =1 Decimeter = 3.937079 10 Decimeters, dm. -I Meter =39.37079 10 METERS, M. -\ Dekameter =32.808992 ft. - J 10 Decameters, Dm. -\ Hectometer =19.927817 rd. 10 Hectometers, Ilm. -I Kilometer = .6213824 mi 10 Kilometers, Km. -I Myrameter - 6.213824 HOWARDS ART OF RECKONING. The Are is the unit of land measure, and is a square whose side is 10 meters, equal to a square dekameter, or 119.6 sq. yards. TABLE. ICentiare, ca. =(1 Sq. Meter) = 1.106034 sq. yd. 100 Centaires, ca. = 1 Are = 119.6034 sq. yd. 100 ARES, A. = 1 Hectare (Ha.) = 2.47114 acres. The Square Meter is the unit for measuring ordinary surfaces ; as flooring, ceilings, etc. TABLE. 100 Sq. Millimeter, sq. mm.=\ Sq. Centimeter = .155+ sq. in. 100 Sq. Centimeters, sg. cm.=\ Sq. Decimeter, =15.5+ sq. in. 100 Sq. Decimeters, sg. dm.=l Sq. Meter (Sg. M.~)= 1.196+ sq. yd. The Stere is the unit of wood or solid measure, and is equal to a cubic meter, or .2759 cord. TABLE. 1 Dicistere = 3.531+ cu. ft. 10 Dicisteres, dst. =1 Stere =35.316+ cu. ft. 10 STERES, St-. =1 Dekastere (DSt.) =13.079+ cu. yd. The Cubic Meter is the unit for measuring ordinary solids ; as excavations, embankments, etc. TABLE. 1000 Cu. Millimeters, cu. ;>.=! Cu. Centimeter= .061+ cu. in. 1000 Cu. Centimeters, cu. cm. =1 Cu. Decimeter =61.026+ cu. in. 1000 Cu. Decimeters, cu. dm. =1 Cu. Meter =35.316+ cu. ft. MEASURES OF CAPACITY. The Liter is the unit of capacity, both of Liquid and Dry Meas- ures, and is a vessel whose volume is equal to a cube whose edge is one-tenth of a meter, equal to 1.05673 Liquid Measure, and .9081 quart Dry Measure. TABLE. 10 Milliliters, ml =1 Centiliter. 10 Dekaliters, />/.=! Hectoliter. 10 Centiliters, cl. =1 Deciliter. 10 Hectoliters, Hl.=\ Kiloliter. (Stere) 10 Deciliters, dl. =1 Liter. 10 Kiloliters, A7.=l Myrialiter. 10 LITERS, L. =1 Dekaliter. (Ml.) The Hectoliter is the unit in measuring liquids, grain, fruit and roots in large quantities METRIC SYSTEM. 105 EQUIVALENTS IN U. S. AND IMPERIAL MEASURES. Metric Denominates. Cubic Measure. Dry Measure. Wine Measure. Imp. Measure. 1 Myrialiter=10 Cubic Meters=283.72ibu.=2641.4igal.=2200.9G711gal. IKiloliter = 1 Cubic Meter =28.372Jbu.=264.17 gal.= 220.09G71gal. 1 Hectoliter=i^Cubic Meter =2.8372Jbu.=2G.417 gal.= 22.00967gal. 1 Dekaliter =10 Cu.Decimet 1 r=9.08 qts. =2.6417 gal. = 2.20096gal. 1 Liter = 1 Cu.Decimet 1 ^. 908 quart=1.0567 qt. = 7.0430gilla 1 Deciliter = T 1 oCu.Decimet 1 r=6.1022c.in=. 845 gill = .7043 gill. lCentiliter=10Cu.C 1 ntim 1 t'r=.6102cu.in = .338fl'doz.= .0704 gill. 1 Milliliter = 1 Cu. C'ntim't'r=.OGl cu. in=.27 fl'd dr. = .0070 gill. MEASURES OP WEIGHT. The Gram is the unit of weight, and equal to the weight of a cube of distilled water, the edge of which is one hundredth of a me- ler, equal to 15.432 Troy grains. The Kilogram, or Kilo, is the unit of common weight in trade, and is a trifle less than 2 1 Ibs. Avoirdupois. The Tonneau is used for weighing very heavy articles, and ia about 204 Ibs. more than a common ton. TABLE. .15432+ oz. Troy. 1.54324+ ' " 15.43248+ " " .35273+ oz. Avoir. 3.52739+ " 10 Milligrams, 10 Centigrams 10 Decigrams, 10 GRAMS, 10 Dekagrams, 10 Hectograms, 10 Kilograms, 10 Myriagrams, 100 Kilograms 10 Quintals, or 1000 KILOS, mg. <&> dg. G. Dg. Kg. or Mg. =1 Centigram = 1 Decigram =1 Gram =1 Dekagram = 1 Hectogram =1 Myriagram 1=1 Quintal | j Tonueau, t~ l \ or Ton, = 22.04621+ " " = 220.46212+ " " =2204.62125 " *' COMPARISON OF THE COMMON AND METRIC SYSTEMS. 1 Inch, = 2. 54 Centimeters 1 Foot, = 30.48 Centimeters 1 Yard, = 9144 Meters 1 Rod, = 5.029 Meters IMile, = 1.6093 Kilometers 1 Sq. in. =6.4528 Sq. CentimTrs 1 Sq. ft., = 929 Sq. Centimeters 1 k> yard, = 8.361 Sq. Meters 1 " rod = 25.29 Centairs 1 Acre. = 40.47 Ares. 1 Sq. mile, = 2(59 Hectares 1 Cu. in. =16.39 Cu. CenthnVrs 1 fci ft. .=28320 " 1 " yd., =.7646" Meiers. 1 Cord, = 3.625 Stores 1 PI. ounce. = .2.958 Centiliters 1 Gallon, 1 Bushel, 1 Troy gr. = 1 " lb. = 1 Av. lb. 1 Ton, 3.786 Liters .3524 Hectoliter = 64.8 Milligrams .373 Kilo .4536 " .907 Tonuean 106 HOWARD'S ART OF RECKONING. 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 FOB MARKING ALL GOODS BOUGHT BY THE DOZEN. To make 20^ remove the point 1 place to left. " " Add oV itself . 26^ " " " " "' -fa " " " tt a . " " ' " -]- " " " " ^ " " "' " " -J- " " " "subtract^ " u u u tt ^ 4t " " " a " See page 62. POPULATION OF CITIES. POPULATION OF CITIES. 107 Table showing the Population of the principal Cities and Towns of the United States as shown by Census of 1880. Philadelphia, Pa Brooklyn, N. Y Chicago 111 847,452 556,930 503 298 Dallas, Tex Springfield, Mass Savannah, Ga 33,486 33,149 32,916 St. Louis, Mo 375,000 363 938 Manchester, N. H Grand Rapids, Mich,.. 82,458 32 037 330000 Peoria 111 31 789 Cincinnati, O San Francisco, Cal New Orleans, La Washington D C 255,809 233,066 215,239 161 111 Mobile, Ala Wheeling, W. Va Harrisburg, Pa Omaha Neb 31,295 81,186 30,728 80642 Cleveland O 159 504 Trenton, N. J 29 938 Pittsburgh Pa 153,883 Evansville, Ind 29,366 Buffalo N Y 149 500 Erie Pa 28346 Newark N J. 137,162 Quincy, 111 27,428 Louisville, Ky 126,566 Salem, Mass 27,347 Jersey City, N. J Milwaukee, Wis Detroit Mich 122,207 115,712 115,007 Terre Haute, Ind. . . . Lancaster, Pa Leadville, Colo 26,522 25,846 15,000 Providence R.I.' 104,760 Sacramento, Cal 25,000 Albany N Y 87584 Des Moines Iowa 22 900 Rochester NY 78,087 Galveston, Texas 22,308 Allegheny City, Pa. . . . Indianapolis, Ind Richmond, V& 78,472 76,200 63,243 Dubuque, Iowa Holyoke, Mass Davenport, Iowa 22,276 24,926 21,812 New Haven Ct 62,861 Portland, Oregon . . . 21,000 Lowell, Mass Worcester, Mass Troy, N. Y 59,340 58,040 57,000 Springfield, Ohio .... Elmira, N. Y San Antonio, Texas . . 20,727 20,646 20,594 Kansas City Mo 56,964 Springfield 111 19 500 Toledo, 0. ..'. Cambridge, Mass Syracuse, N. Y Columbus, O 53,635 52,800 52,168 51,650 Leavenworth, Ks Burlington, Iowa Council Bluffs, Iowa Bloomington, 111 18,800 19,000 18,400 17,700 Paterson N J. 50,950 Houston, Texas . 16,632 Denver Col 35 718 Akron O 16 462 Charleston, S. C Fall River Mass 49,027 48909 Jackson, Mich 16,121 16000 Minneapolis Minn 48.323 Oshkosh Wis 15 753 Scranton, Pa 45,925 Newport, R. I 15,698 Atlanta Ga 45000 Topeka Kan 15433 Nashville. Tenn Reading, Pa 43,543 43.230 Atchison, Kan Little Rock, Ark 15^130 15,000 Hartford, Conn Wilmington, Del St. Paul Minn 42,560 42,000 41,639 La Crosse, Wis Knoxville, Tenn Rock Island 111 14,470 13.928 13 699 Lawrence, Mass Dayton Ohio 39,068 38 751 Lincoln, Neb San Jose Cal 13,697 12 615 Lynn, Mass Memphis Tenn 38,376 35000 Cedar Rapids, la Keokuk la 12>00 12 176 St. Joseph Mo 35000 Kalamazoo, Mich 12078 Oaklancf Cal 34 700 Pueblo Colo 7000 Utica, N. Y. . . 33.927 108 HOWARD S ART OF RECKONING. DE CALCUL POUR L'ESPACE DE TREKTE SlECLES. REGLE. Des deux derniers chiffres del'an, rejetez tons les sept, tout en retenant ]e restant ; divisez les deux derniers chiffres de 1'an par quatre, re- tenant le quotient, sans tenir compte du restant, s'il y en a ; puis prenez le jour du mois, ensuite le chiffre donne' pour le mois, et finalement celui pour le siecle. Aycz 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 etDec. 3, Jan. et Oct. 5, AoAt. 2, Avril et Juillet. 4, Mai. 6, F6v., Mars, Nov. 0, Juin. NOTA. Dans I'ann6e bissexiile le chiffre pour Janvier eat 2, et celul pour F6vrier 5. TABLE DES CHIFFRES POUR LES SlECLES. 1, est le chiffro pour les 26me, 96me, et 166me, siecles. [sidcles. ler, Seme, 156me, 18eme, 22eme, 26eme, 30enie, 7eme, 14eme siecles. [siecles. 66me, 136nae, 176me, 21emc, 2fieme, 29dme, 6eme, lleme, 206me, 24eme, 286me, si^clee. 4eme, 11 erne siecles. Seme, lOeme, 196me, 236me, 27eme, sidcles. EXEMPLE. Quel futle jour de la semaine au 31 Aout, 1873 ? Ke*ponse, Dimanche. Proced6 Deux derniers chiffres de 1'an, 73 70=3 Quotient de 73 di vise" par quatre, 18 + 321 = Jour du mois, 3128=3 Chiffre pour le mois, 5 + 3 7 = 1 Apres avoir rejete' tous les s&pt il reste le chiffre 1 ; ce futdonc, le premier jour de la semaine, Dimanche. N. B. Lea Bifides pairs non-divisibles par le chiffre 400 no sont pa^ des amieea bissextiles. CALENDAR. 109 fceu <*$ in bet SJMfjobe. Streid) bie ieben au Don bte beiben le^ten ^ummern auf ba3 Saljr, bet TOnuent won ben betben le&ten 9lummern im 3afire, btoftirt Bet bier gebraudje nidjt ben SReft ben Saturn cmf ben 9ftonat, unb bte gignr ouf bag 3afyr. 2Ba itberbleibt tft bet Sag in bet SSodje, ber crftc onntag, bet stueite tag u f. ID. SDte gigitren oor bic donate. 1 wor ept. it. ecbr. 3 not 3an. it. Oct. 5 or 9tugufl. Dor %un\. 2 or 8tpril unb Suit. 4 uor s JJ?at. 6 oor get)., jDatum im 3 a ttuar unb gctruar tft ein irentger im cfyaltjaljr. Saturn auf bte jl b e ^tgur cor bA-3 2te, 9te unb 16te S Ite, 8te, 15te, I8te, 22te, 2Gte unb 30te 3a 7te, I4te 3a[)v^mbert. 6te, I3te, 17te, 21te, 25te, 5te, 12te, 20te, 24te, unb 28te 4tc nnb 11 Sa^rtjunbett. 3'e, lOte, late, 23te uno27te Sa ^empet SBelc^er Xag in ber 2Bo$e iuar ber 31. STuguft, 1873? Slnttoort, oitntag. ^)te (e^ten betben gigitren im ^a^re, 73 70 = 3 aJttimertt ouf bo. ~ bei mer, 18 + 3 21 = Saturn im Sftonat,' 31 28 = 3 gtgur auf ben Sftonat, 54- 3 7 = 1 SReft 1 jeigt @nc^ ben erften Xag in ber ift @onntag To find the figure for any century from the 1st to the letn, mul- tiply the figures expressing the hundreds in the given year by 6, add 2, and divide by 7 ; the remainder is the figure for the Century. To find the figure for the 17th and succeeding Centuries, subtract 16 from the number of hundreds in the given year, multiply by 5^, to the product less the fraction add 4, and divide by 7; the remainder is the figure for the Century. NOTE 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 1588 and in England in 1753. 110 HOWARD'S ART OF RECKONING. Howard's California Calendar for Thirty Centuries. BULE. Cast all the sevens out of the last two figures of the year ; add the remainder to tiie 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. 3, Jan. and Oct. 5, August. 0, June. , April and July. 4, May. 6, Feb., March, Nov. NOTE. The figure for January is 2, and February 5 in leap year. TABLE OF FIGURES FOR THE CENTURIES.* 1, is the figure for the 2d, 9th, and 16th centuries. o " ' * 1st, 8th, 15th, 18th, 22d, 26th, 30th centuries. 7th, 14th centuries. 6th, 13th, 17th, 2l8t, 25th, 29th centuries. 5th, 12th, 20th, 24th, 28 centuries. 4th, llth centuries. 3d, 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 ~ by four, 18 -f 3 21 =0 Day of months 31 28 3 Figure for the month, 5 -f 3 7 = 1 *Pay no attention to the figure for this, the 19th century, as it is 0; for the last century, add 2 ; for the coming century, add 5, After casting out the sevens the remainder is 1 : hence it was on the first day of the week, Sunday. H. B. The even centuries not divieable by 400 are not leap years. STANDARD WEIGHTS AND MEASURES. Ill HOWARD'S Tables of Standard Weights and Measures, A Standard Measure is a fixed unit established by lato, by which quantity, as extent, dimension, capacity or value is measured. The English Standard units are the YARD, the IMPERIAL GALLON, the TUOT POUND, and the GOLD SOVEREIGN. The TT. S. Standard units arc the YAKD, the GALLON, the BT/SHEL, the TROT 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 02 Fahrenheit, Barometer 30= 252.458 grains. 6760 grains = 1 Troy pound. In the Treasury at Washington is a brass scale vhich, at a tem- perature of 62 Fahrenheit, is 82 inches long; all U.S. weights and measures are referred to this unit. LONG MEASURE. SURVEYORS' LONG MEASURE. IX. FT. YD. RD. FUR 12 ....1 ..1 Foot. _ 36 3 1 1 Yard 7 02 1 1 Link 198 16/2 5!4 .1 ..1 Rod. 198 25 ..1 1 Rod. 7920 6GO 220 40 . 1 ..1 Furl'iig 792 100 ..4 .1 1 Ch'n. 633GO 5280 1760 320 ..8 ..1 Mile. 633GO 8000 320 SO 1 Mile. The Geographical Mile equals 1.15 Statute Miles, COMPARISON OF STANDARD MEASURES -OF DISTANCES. Country. Austria, . . . China, East Indies Egypt England, . . France, . . . Japan, Mexico, ... tJ. 3. Mile. .IMile, =4.98 .ILi, = .35 1 Coss, = 1.14 .IMili, = 1.15 .IMilc, = 1.00 .1 Kilomet'r.= .62 .1 Ri, =2.562 .1 Silio, - 6.76 Country. Persia, . Portugal Prussia, Russia, . Spain, . . Sweden Switzerland, Turkey, U.S Mile, .1 Farsang, = 4.17 .IMilha, = 1.28 . 1 Meile, = 4.93 .IVerst, = .66 .1 League, = 4.15 .IMil, = 6.64 1 Lieue, = 2.98 .IBern, = 1.04 H 112 SQUARE MEASURE. For measuring Land) Boards, Painting, Paving, Plastering, etc. SQ. INCH. SQ. FOOT. SQ. YARD. SQ. ED. SQ. 11. SQ. A. 144 1 ...1 SQ FT. 1296 9 1 ...1 YAKD. 39204 1568160 6272640 2T214 10890 43560 30^ 1210 4840 1 40 160 ...1 4 ...1 ....1 ....1 ...'.1 ROD. ROOD. ACRE. 4014489600 27878400 3097600 102400 2560 640 ...1 MILE. In 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 sectiou=-640 acj es. 36 square miles (6 miles square)=l township. The sections are all numbered 1 to 36, commencing at the north- cast corner, thus: The sections arc all divided info quarters, which are named by the cardinal points, as in section 1. The quarters are divided in the same way. The description of a forty-acre lot would read: The south half of the west half of the south-west quarter of section 1 in township 24, north of range 7 west, or as the case might be ; and some- times will fall short, and some- times overrun the mimbet of acres it is supposed to contain. *Re serve for school purposes. Gunter's Chain is a unit of measure, and is four Rods, or 60 feet long ; it consists of 100 links. It is also common to use a chain, or measuring tape, 100 feet long, each foot divided into tenths. In the Pacific Coast States and Territories the divisions of Land are frequently expressed by the old Mexican Measurements : A fifty Vara lot is 13714 feet square. 1 Vara=33-36=ll-12 of a yard. VarasXll-J-12= yards, Yards Xl2-f-ll=Varas. 6 5 4 3 2 NW | NK SW I SK 7 8 9 10 11 12 18 17 16* 15 14 13 19 20 21 22 23 24 30 29 28 27 26 25 31 32 33 34 35 36 CUBIC MEASURE, 118 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 \GG56 27 1 1 Cubic Yard 27648 16 16-27 1 1 Cord Foot 221184 128 4 20-27 8 1 1 Cord of Wood 42768 24% 1 1 69120 40 1 U S Ton ShipCargp 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 ciibic 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 FOO;T and SQUARE YARD. CUBIC FEETX4-i-99 = PERCHES. Five courses of bricks in the height of a wall arc called a foot, In board and lumber measure, estimates are made on 1 inch in thickness ; one-fourth the price is added for every J 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. 56 " 1 Firkin of Butter. 14 " 1 Stone, Avoir. 28 " 1 Quarter, " 21 y z Stones, 1 Pig of Iron. 8 Pigs, '. 1 Fother. 2 Weys (328 Ib) 1 Sack of Wool. 12 Sacks, (4368 Ib.) 1 Last. 3 Inches, 1 Palm. 4 " IHand. 9 " 1 Span, 3 ft 1 common pace. 6 " 1 Fathom. 3Miles, 1 League. 360 Degrees 1 Circle. H2 TROY WEIGHT. For Gold, Silver, Jewels, etc. AVOIRDUPOIS WEIGHT, For Groceries, Provisions, etc. Gr. Pwt. Oz. Gr. Oz. Lb. 24 1 1 437 Vi 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. = 112 Ibs. The Short Ton=2000 Ibs. Long Tonsxl.l2=Short Tons. To compare Troy weights with Avoirdupois, reduce both to grains. Pounds Avoirdupois X 100x7H-48= ounces Troy. Troy ounces X.06 6-7= Pounds avoirdupois: that is, ounces multiplied by .06+1-7 of the product. APOTHECARIES' WEIGHT. APOTHECARIES' MEASURE. GKS. sc. ^ DR. 3 OZ. I CO Minims = 1 Fluid Drachm. 20 1 1 SCRUPI.E. 8 Fl. Drms = 1 Fluid Ounce. 60 480 5760 3 24 288 ..1 8 96 ..1 12 ..1 ..1 ..1 mtAM. OUNCE. rouxo. 16 Fl. Ozs. = 1 Pint. 8 Pints = 1 Gallon. "Used iu compounding liquid medicines. The grain, ounce and pound are the same as Troy Weight. Drugs are bought and sold in quantities by Avoirdupois Weight. 1 Teaspoon = 45 Drops. 1 Tablespoon = y z Fluid Ounce. COMPARISON OF LIQUID MEASURES. Country. U.S. Gals. Country. U. S. Gats' England, . . .1 Gallon. =1.2 Switzerland, 1 Pot, = .40 France, . . . .1 Dekaliter, =2.64 Turkey, 1 Almud, = 1.38 Prussia, . , . .1 Quart, = .30 Mexico, 1 Fasco, = .63 Austria, . . . . 1 Maas, == .37 Brazil, 1 Medida. = .74 Sweden, . . ,1 Kanna, = .09 Cuba 1 Arroba, = 4 01 Denmark, . .IKaude, = .51 South Spain, 1 Arroba, = 4.25 COMPARISON OF GRAIN MEASURES. Country. U. S. Bushels. Country. C. 8. Bushels. England, .1 Bushel, = 1.031 Germany, ...1 Schef.=>1.5 to 3 France, . .1 Hectoliter=2,84 Persia, 1 Artaba, =1.85 Prussia, . .1 Schefiel. = 1.56 Turkey, 1 Kilo, = 1 .03 Austria, .IMetze. '= 1.75 Brazil 1 Fan. == 1.5 Russia .. .lChetverik= .74 Mexico, lAlque, =1.13 Greece, . .IKailon, =2.837 Madras, 1 Parah, =1.743 RAILROAD FREIGHT. 115 COMPARATIVE TABLE OP POUNDS IN DIFFERENT COUNTRIES Austria, 100 Ibs 123.50 II. S. Bavaria, " 123.50 " Belgium, " 103.35 " Brcmcu, " 110.12 " Berlin, " ....103.11 " Denmark, " ....110.00 u Gcr. Zoll. States, . .110.25 " Hamburg, .. .110.04 " Nederland, 100 Ibs . . 108.93 U. S. Portugal, Prussia, Russia, Spain, St. Domingo, Trieste, .101.19 .110.25 . 90.00 .101.44 .107.98 .123.60 COMPARISON OF COMMERCIAL WEIGHTS. Country. Austria Arabia, \Veight- U. S. U>. 1 Pfuncl. = 1.23 1 Maund. = .3 Country. Mexico, . Madras, . Weight. ...1 Libra, ...IVis. U. 8. Lbs. = 1.03 = 3.125 Brazil 1 Arratel, 1.02 Persia, .. . ..1 Rattel, 2.116 China, 1 Catty, = 1.33 Russia, , .. .1 Flint, = .1X1 Denmark. . . East Indies. 1 Fund. = 1.10 1 Seer. = 2.06 Sweden, Spain, ...IPund, 1 Libra, .98 = 1.01(5 Egypt, France Germany. . , 1 Rottoli, = 1.008 1 Kilogram.=2.20 1 Pfuud. = 1.10 Sicily. .. Turkey. . Japan, 1 ' '.!!l Oka, 1 Kin, .7 = 2.82 .62 Troy. Apothecaries. Avoirdupois. 1 Pound = 5TGO grains = 5700 grains = 7000 grains. 1 Ounce =480 " = 480 " = 437.5 " 175Pouuds= 175pounds= 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, Apples, dried, " green. Beef, 320 Ib. 170 u 100 " 24 " 50 " 150 u .320 li ^ per bbl. " bu . " bbl. u u " bu. " do/. " bbl. " bu. ' bbl. Lime, 200 Ib. per bbl. " bu. u n kt keg. " bbi. " bu. 14 bbl. " bu. " bbl. . 2000 Ib. Malt 38 " Millet, 45 " Nails, 108 " Oil, 400 " Peaches, dried, 33 " Pork . 320 " Potatoes (com.) 150 " Salt, Fine,.... 300 u " Coarse, 350 " " in Sack, 200 " Turnips, 56 " Viuegar, 350 " Whiskey, 350 " Brooms, Cider, . . 40 " 350 " Charcoal, Eggs, ...22 " 200 " Fish, 300 " Flour, Highwines,... ..200 " ...350 " One Ton Weight, 116 LIQUID, OR WINE, MEASURE. CU.FT. CU. IN . CU.FT. CU. IN. .0107 .0334 .13368 28.8T5 57.75 231 4 Gills,.. [1 Pint. 2 Pints.. 1 Quart. 4 Qts., .. 1 Gallon. 11.22ft 4.2109 8.421 19404 7276.5 14553 2Tiecs..lPunsh'n. 31 Y 2 Gals IBbl. 2 Bbls 1 Hhd 1.3368 2.406 2310 4158 10 Gals... 1 Anker. 18 Gals... 1 Runlet. 16.84 33.68 29106 58212 2Hhds IPipe. 2 Pipes 1 Tun. 5.614 9702 42 Gals. . . 1 Tierce. The U. S. Standard Gallon contains 231 cubic in.=-8^ Ibs. avoirdup's. " Imperial " 277.274 " =1.2 U. S. gallons. l old Beer Measure " " 282 " In measuring tanks, reservoirs, etc., it will be sufficiently accu- rate to regard one cubic foot=7}4 U. S. or 6% Imperial gallons. The contents of a circular tank, in barrels of 31J4 gallons, =the square of the diameter (in ft.) multiplied by the depth, mul. by .1865. The number of U. S. gallons in a round tank=the square of the diameter, in feet,Xby the depthX5%. The number of U. S. gallons in a round tank, wider at one end than the other, = % the sum of the squares, plus the product of the two end diameters, in feet,Xthe depthX5%. For Imperial gallonsX4.9 or 4.89469. GAUGERS' 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, all in inches, for U.S. Gallons X .0034, for Imperial gallons X .0029, for Ale gallons X '0028. NOTE. If the staves are but little curved, add .6 instead of . How many U.S. gallons in a cask, length 40 in., head diameter 21 in. and bung diameter 30 in. ? . .21+(30 21 x | )=27 in. mean diameter. ...27 2 X40x.0034=99.144 gallons, Ans. Or27 2 x40x.0029=84.564 Imperial gallons. DRY MEASURE. 11 DRY MEASURE, IMP. AND U. S. STANDARD. For measuring Grain, Fruit, Roots, Coal, etc. IMP. CTJ. IX. u. s. CU. IN. PT. QT. GAL. PK. BU. CM. QR. 34.659 33.60 1 1 Pint. 69.318 67.20 2 ....1 1 Quart, 277.274 268.80 8 4 ....1 1 Gallon. 554.548 537.60 16 8 2 1 1 Peck. S218.192 2150.42 64 32 8 4 ..1 ..1 Bushel. 8872.768 8601.68 256 128 32 16 4 ..1 ..1 Coomb. 17645.536 17203.36 512 256 64 32 8 2 ..1 ..1 Quarter. 70582.144 68813.44 2048 1024 256 128 32 8 4 ..1 Chaldron. 77415.12 2304 1152 288 144 36 OF C 30AL ..1 Chaldron. The TJ. S. Standard Bushel contains 2150.42 cubic inches. The Imperial English " " 2218.192 " " A cylinder 18 l / 2 inches in diameter, 8 inches deep= 1 Bushel. 5 Stricken measures= 4 heap measures. Cubic feet X.8 = U. S. bushels nearly; add 44.5 for every 10,000 bushels. Cubic feet X.779 = Imperial bushels nearly. Imperial BushelsXl.03152 the Product=U. S. Bushels. U. S. Bushels X. 969444 the Product=Imperial 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 12.X7X11 inches will contain 4 gallons. An open box made with the greatest economy of material ; the al- titnde= the radius of the Base ; if with a cover the altitude= the base. The number of bushels -f J4 = the number of cubic feet. The number of cubic feet l-5=the number of Bushels. U. S. The price per ce'ntal=the price per bushel xlOO-j-the number of pounds in the bushel. See page 75. It is usual with some local exceptions to estimate the number of pounds to th^e bushel, as follows : Bran, 20 Ibs. ; Oats, 32 Ibs ; Bar- ley, 48 Ibs. ; DLaize and Rye, 56 Ibs. ; Wheat, Beans, Clover Seed and Potatoes. 60 Ibs. 118 HOWARD'S ART OF BECKONING. 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- ntsfine." 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 nnd 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 cither ; 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 .910 2-3 fine 40 29-40 Sovereigns are made, each weighing 123.27448 grains = 113.001005 grains of fine gold = $4.866563. 1 ounce of U. S. Standard Gold = $18.60465 = 3.8230 = 3,,16,, 5 l / 2 1 " k ' British " k * = 18.94918 = 3.8938 = 3,,17,,10M 1 " k - Pure lk = 20.67184 = 4.248 = 4,, 4,,11& Thousandths of an ounce -f- 100 X 48 = grains. Grains X 100 -j- 48 = thousandths of an ounce. U. S. Standard ounces of Gold -i- .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. . How many ounces in one U. S. Gold dollar? Ans. .05375 ozs. The weight of gold, in ounces, and iho fineness beJng given, to findits val- ve in U. S. Gold Coin. RULE. Multiply the weight by twice the fineness, multiply by 10 find 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 18.60405 Aus. 818.60405. GOLD AND SILVER. 119 Or multiply the given weight by the fineness x 1000 X 8, and divide the product by 387. IX. 9X1000X8-7-387 = 18.60-465. The fineness and weight of Silver being given, to find its value in U. S. Silver dollars 9-10 fine, 412% grains weight. RULE. For piire silver, if in grains, divide by 9x10x11X3 and multiply by 8, or divide by .9x412.5. EXAMPLE. Pure silver, grains 371.25X8-r-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. lX.9Xl.28 = 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 bo alloyed; the quotient is the weight required to make a compoiind 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 -r-22 = .63636 gold + .36384 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 igSjJT = 3 ^' 3 ^ oz> 12 fine ~*~ L oz ' 21 fiue = 4 ^ oz ' 14 Carat fine< The silver dollar weighs 4l2 l / 2 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 o unce, 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. poo 11 ozs., 2 pwts. = -irrpr= .925 of an ounce. Ans. 89^ cts. ^4U 54 pence = 0.225. .225X4.86 = 1.0935. 1.0935 X 412.5 X .9 480 X .925 120 MEASURE OF TIME TABLE. SEC. MIN. HRS. PA. WK. GO 1 1 3GOO GO 1 1 86400 1440 24 ....1 ..1 Day, G04800 10080 168 1 ..1 ..1 Week, 31536000 ' 31622400 525GOO 527040 8760 8784 365 36G 52 ..1 ..1 Common Year, Leap Year. TIME i's a measured portion of duration, tfce unit of which is tne mean solar day. 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 365% days, nearly, we reckon three years in every four as containing 365 days, and the fourth, leap year, as con- taining 306 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. + 1-2+1 + 1 +1+1 +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 mouths 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. COUNTERFEIT NATIONAL BANK NOTES. 121 United States Treasury List of Counterfeit Bank Notes. New York, N. Y Mechanics' New York, N. Y Merchants' New York, N. Y. N. B. of Com'rc New York, N. Y. N. B. State N. Y New York, N. Y Union Philadelphia, Pa First Philadelphia, Pa Third Poughkeepsie, N. Y First Poughkeepsie, N. Y City Poughkeepsie, N. Y. F 1 mr's & Mf ' Red Hook. N. Y First Richmond, Ind Richmond Rochester, N. Y. Flour City Rome, N. Y Central 4 Syracuse, N. Y Syracuse Troy, N. Y Mutual 4 Waterford,N,Y. Saratoga County 3 Wutkins, N. Y Watkins TWENTIES. Tndianapoll s, Ind First New York, N. Y First New York, N. Y Market New York, N. Y Merchants' New York. N. Y. N. B. of Cornr' New York, N. Y. N Shoe & Leath New York, N. Y Tradesmen's 3 Philadelphia, Pa Fourth Portland, Conn First 1 Utica, N. Y City Utica, N. Y Oneida FIFTIES. Buffalo, N. Y Third* New York, N. Y Central New York, N. Y Mechanics' New York, N. Y. . .Metropolitan New York, N.Y.N. B. of Com'rc New York, N. Y. Nat. Broadway New York, N. Y Tradesmen's New York, N. Y Union ONE HUNDREDS. Central German .Merchants' ...Traders' ONES. Boston, Mass 1 . . . .Nat. Eagle TWOS. Kinderhook, N. Y. Nat. Un'n 1 Linderpark, N. Y. Nat. union Newport, R. I. N. B. of R. T. New York, N. Y Ninth New York, N. Y Marine New York, N. Y Market New York, N. Y. St. Nicholas Peekskill, N.Y.Westchst'r Co FIVES. 2 Amsterdam, N. Y. . .Manuf's 2 Aurora, 111 First Boston, Mass Globe Boston, Mass Pacific 2 Canton, 111 First 1 Cecil, 111 First 2 Chicago, 111 First 8 Chicago, 111.... 8 Chicago, 111..., 2 Chicago, 111 4 Chicago, 111.... Chicago, 111 Union Dedham, Mass. ... Dedham Fall River, Mass....Pocasset 1 Galena, 111 First 2 Hanover, Pa First Jackson, Mich People's 4 JewettCity, Conn Jew't City Montpelier, Vt. . .Montpelier 2 New Bedford, Mass, Mer. 2 Northampton, Mass First 2 Pawling, N. Y. National B. of 4 Paxton, 111 First 2 Peru, 111 First Rome, N. Y. . .Fort Stanwix Southbridge, Mass..Sonthbr 2 Tamaqua, Pa First 2 Troy, N. Y. . .National State Virginia. 111... Farmers' 2 Westfleld, Mass.. .Hampden TENS. Albany, N, Y. . . .Albany City 4 Auburn, N. Y. ..Auburn City 1 Buffalo, N.Y. Farmer &Mf's LaFayette. Ind...LaFayette Lockport, N.Y First Muncie, Ind Muncie Newburg, N. Y Highland New York, N. Y First 4 New York, N. Y... American 3 New York, N.Y Croton New York, N.Y Marine New York, N.Y. ...Market 2 Baltimore. Md,. ..Nat. Exchange Boston, Mass First 2 Boston, Mass Nat. Revere 4 Cincinnati, Ohio Ohio 2 New Bedford, Mass. .Merchants' New York, N.Y Central 2 Pittsburg, Pa. Pittsb. N. B. Comr' 2 Pittsfield, Mass Pittsfield 2 Wilkcs Barre, Pa Second 1 No such bank. 2 All notes of this bank of this de- nomination are replaced by notes of other denominations. 3 Failed, and notes being retired. 4 Notes are being retired. 122 HOWARD'S ART OF RECKONING. MISCELLANEOUS. How many strokes does a clock strike in 12 hours ? "12+1X12 7Q - 4> - <8 strokes. a How many barrels in a triangular pile, 49 bar- rels at the base and 1 at the top ? "49+1X49 j'. 122o barrels. 2 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 ? 16-s-2=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 140x^90) 14Q 9^9 9 140x A =50 I A Bin 9 ft. 6 in. long, 6 ft. wide, 4 ft. 3 in. deep, will hold how many Imperial bushels. XfXY-X&-4.845= 188. 955 bushels. Ans. NOTE. The imperial bushel is 2218.192 Inches, ten eighths of a foot,nearly, deduct 2^ from every 100 bushels in the product, this result multiplied by 8 will be the number of Imp. gallons, What is the cost of 732 Ibs. of Coal at $14. per ton, 2240 Ibs. to the ton? 732X14 . . . - =$4.0/0. Ans 8X4X7 MISCELLANEOUS. A bin 9 ft, 6 in. long, 6 ft. wide, and 4 ft. 3 in. deep is full of wheat, what is its value at $2.05 a. bushel? 7. Ans. Note. The standard bushel is 2150.42 inches; ten-eighths of a foot, nearly, the difference is .44 bu. in each 100. ^.259, Divide 1 into 3 parts in the proportion of A, , B, J, C, J. 6+4+3=13. 12 Ans. A, A, A. How many cubic feet in a case 3 ft. 6 in. by 2 fU 8 in. by 1 ft. 10 in? |X |X V 17 i ft. Ans. If 7 cats, kill 7 rats, in 7 minutes, how many cats will kill 100 rats in 50 minutes? 7X7X10014. 7X50 Ans. 14 cats. If it cost $24 to carry 6 tons 20 miles, what will it cost to carry 12 tons 120 miles? 24X12X120=288. 6X20 Ans. How many bricks will pave a walk 200 ft. long, by 16 feet; bricks 8 in., by 4 in? 200X16X3X3 2X1 = ' Ans. 14400 bricks. Multiply 19 19s. llf d by 19i J& v l. 399 _ or*19,,19,,llj X 20- v fa = 399,,19,,2 V J V of a farthing. 124: HOWARD'S AKT OF BECKONING. Multiply 66 by f : .22 ^~j~^ = 44. Divide 66 by : 33 ^ ^y^ = 99. 56 Divide 168 X 2 x 7 by 7 X 3: 7x2x# = n2> 7XjJ Divide .99 amongst 3 persons, A to have T \> B T \, and C T \. A45,B36,C18. j * | i Two merchants load a ship with goods worth 5000, A owns 3500, and B the rest; the goods suffer damage valued at 1000, what is each man's share of the loss ? \m 3500 A loses .700. 1500 B 300. B and C gain by trade 182; B put in 300, and C 400fwhat is the gain of each ? 3 ^ T0M 4 ^ B - ' 78 - 182 r 182 C 104. A person owning f of a mine sells | of his share for 1710, what is the value of the whole mine ? ;7/0 x 4 x 5 190 - X fi How much money will buy | of | of a mine worth 3800 ? - 9 L 1710. If J of 6 be 3, what will J of 20 be ? 3 X MISCELLANEOUS. 125 A compositor can set 20 pages in 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 work? 4 5 23 23 6 ~-j = inverted = of a day. 32 66 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 13T. 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 ? 7X6 = = Z\ pairs. A How many bricks, 8x^X2 inches, in a wall 160x20x2 feet? 160x20x2x3x3x6 OQ ^.^ ^xixi How many shingles for a roof 60 ft. long, rafters 20 feet, two sides, shingles to show 6x^ inches. 1 60X20X2X2X8 14>40Q * /\ -^ 126 HOWARD'S ART OF RECKONING. If 21 1 bushels of oats will seed 91 acres, how many bushels will seed 100 acres? 87x3x100 , 4X29 5 bushels * How many 16ths are there in .85 ? .85X16 -TOOT $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 ? $150 78=472. 6X2 ^ 1Q ^16| per cent. ( 2i Find the value of an ounce of silver, gold being worth 3 ,,18,, 7 per ounce, ratio 15Jto l.also 16 to 1. 3,,18,,7 -5-15J=60jjjd. 8,,18,,7-s-l6=68tJd- To find the amount in 365 days of any given number of pence per day : Multiply the given number of pence by 1J, call it pounds, and add the product of the pence multiplied by 5. DICE. The number of different combinations that can bo made with any given number of dice is equal to a power of 6, equal to the given number of Dice. How much is due to a man working 22 days, at S39 per month of thirty days ; also 26 working days to the month ? -- =$28-60. " $ 33 - Find the charges on 1000 cases, each 16x12x6 inches, at 1 6 shillings per ton of 40 feet. Hovvard'e Copyright Perpetual Calendar. 6 v LUJLlliXi 1XUJ9H s 1300 to 1899, 3 56 i 57 3 73 74 j6p; 61 !64i65i66i67 1 "W 2 Th 3 F. - S> Th 11 Sa 12 S. . " C.i } " Th F. - I a \\ 12 Tl' ru!18 Af. ; 11 F. 12 Sa 13 S. 14 M. R5 69 70 71! 75 I 76^ 77' 80 81 ! 82 83 86 87' 88 89 ^4 Tl: - Th . . 17 Tli :?.?.. 91; I 921 93 94 ,gS 96 97 98J99I : 80 M. :. - .;. \?,i s. SO F. 30 Th Sa|31 F..J31 Tl. 2 1 7 6 ; Vii_^! h l ei '