UC-NRLF $B 532 146 mjjt. HENRY RAND HATFIELD 2695 L.E CONTE AVENUE BERKELEY. CALIFORNIA COMPLETE TREATISE ON PRACTICAL ARITHMETIC AND BOTH BY 'SINGLE "AND DOUBLE ENTRY, ADAPTED TO THE USE OF SCHOOLS, BY CHARLES HUTTON, LL.D. AND F.R,S. & c . A NEW EDITION, CORRECTED AND ENLARGED v *BY ALEXANDER INGRAM, MATHEMATICIAN. EDINBURGH: PRINTED BY G. ROSS; TiND SOLD BY ALL THE BOOKSELLERS* 1807. PREFACE. Has" HFO adapt this work to the easy use of Preceptors, P ■-*■ have every where delivered the definitions and- rules in as brief a manner as possible, to make them, ?.t the same time, general, and free from the absurdity and nonsense whjch commonly attend definitions in works of this kind •, and have added such notes after them, as de- scribe some particulars not essential to the general rules themselves, but tending either to explain them, or to facilitate the operations ; also, to each rule there is annex- ed a great variety of the best examples, with their an- swers j and where the common method of setting down the operations did not please me, I have there inserted the work of an example, at large, in the method which I think most convenient. The operations of Addition, Subtraction, Multiplica- tion, and Division, are delivered, first in simple numbers, and afterwards in compound j thinking that the properest order in which they can be taught. But I would not be understood to mean, that it is always necessary to have these, or any other of the rules, copied into the Pupil's book in the very order in which they are inserted ; for they are all delivered in such a manner as to have little or no dependence on each other, that they may be taught in what order every master chooses : Nor that it is ever ne- cessary to copy out the first simple rules; but if any Master choose to do it, I would advise him to make his Scholars run over these first upon their slate or waste paper ; then begin them again, and write them, with an example or two to each, in their book : And thus he may with ease PREFACE.. mix the simple and compound rules together. AIs%, though I have inserted little or nothing superfluous in any of the problems or general rules, the judicious Teacher may omit any notes, or particular cases he may think fit. And if all the Pupils have printed books, as they ought, then they may eVrher write all the rules and examples in their books, or they may omit the rules and notes, and set down only the work of their examples, especially where the rules are long and tedious to write. These advantages, together with that of having the book always about them, to get off any precepts or tables when they ;*re out of school, will fully repay them the small expence of the book. It is also a good practice in gome Precep- tors, to procure parcels of these books interleaved with Writing paper, and bound up, for their Pupils to enter their examples in, after they are calculated ; as by giving to every one of them a copy so bound, the expence or a iarge paper book is saved, as well as the trouble of wri- ting out the rules, &c. I must here., also caution some. Masters against that foolish method of writing down the rules and examples for their Pupils, especially in Addi- tion and Subtraction, both of integers and decimals 5. for, in these rules, this is often the only difficulty, and the best part of his exercise ; and on this account I have given most of the questions in a form different from that in which they commonly are proposed. Reduction is placed before the rules of Addition* Subtraction, Multiplication, and Division of compound*, numbers 5 because, in my opinion, Compound Addition is best performed by it •, and- in Compound Division it is absolutely necessary. But every one may teach them in, what order he shall think fit v for the tables of money, weights, and measures, may be as conveniently inserted in^ Compound Addition as in Reduction, if they be copied at all in any part; which, however, I think-is needless, and only wasting time, especially if the Scholar has a printed book : Also, if any Masters choose to make their Pupils wrije PREFACE; ft th*ir tables, but not in the form they here stand, they may easily cause them to turn these tables into the com- mon form upon a slate or waste paper, and copy them from thence into their books ; and this will be no bad exercise for them. In the Rule-of-three, I have neglected :he common distinction of it into direct and inverse j and have given one general rule for the stating and operation, perfectly easy in every respect : Which rule is so comprehensive, that it also includes the questions of the Compound Rule, or Rule-of-Five ; and not only 'hose which are commonly given in books, wherein the statings are either both direct, or one direct and the other inverse, but also those in which the statings are both indirect. The fractions are pretty largely treated of, and parti- cularly the abbreviating part; because it is of the greatest use, by serving to abridge the operations in all the other rules. The advantage of fractions is so great, that I dare affirm it, a . person who is well acquainted with them, will in many cases, perform as many calculations as four or five who are not. In decimals the separating points are placed against the upper part of the figures j., which prevents them from being mistaken for stops or pauses in the reading ; the hint of which I had from some tables in Sir Isaac Newton's optics. In Compound Interest, when the time, at which th£' interest is supposed to be payable, is some part of a year, I have, in the calculations, accounted the rate correspond- ing to that time, the same part of the rate for the whole year: Thus, at 5 per cent, per annum, the rate for half- yearly payments I make 2^, and for quarterly payments, !£, &c. I say this here, to shew upon what supposition- those examples are calculated, as it is contested, whether this method ought to be used* or not* A a •ft PREFACE. In the extraction of roots, I have given a new, gene— ' sral, and very expeditious method, by which the third and higher roots may be found, without the intolerable labour attending the common methods. — ->Q4— # ft * In this new edition, the whole of the examples and tables have been computed anew; and corrected, where any errors had crept into them. And an Appendix is added, containing many useful rules and examples,, tending to illustrate the connection and dependence of the rules upon one another, and to render the opera- tions more easy and expeditious. Interminate decimals are introduced, and general and easy rules are given for working them ; by which means, many questions may be performed, with less than half of the figures that would be sufficient by the rules formerly given : And at the end there is given a new rule for extracting the Cube Root, and the method of measuring and cal- culating the works of Artificers. Likewise, many er- rors of the press that had escaped in former editions, are here corrected : So that it is hoped nothing is omit- ted, that can contribute to render this book worthy oi s* continuance of the public favour, CONTENTS. NOTATION - A Synocfis of the Roman Notation Simple Addition ■■ Subtraction ■ Multiplication — " Diviiion Reduction Compound Addition ■ ■ Subtraction — ■■ ■' " — , Multiplication » ■ Divifion Rule-of-Three,or Golden-Rule Rule-of-Five, or Compound Proportion Practice, or Rules of Practice Bills of Parcels, Book-Debts, &c. Tare and Tret - Vulgar Fraclions - Reduction of Vulgar Fractions Addition of Vulgar Fractions Subtraction of Vulgar Fraclions - Multiplication of Vulgap Fractions Divifion of Vulgar Fractions Rule-of-Three in Vulgar Fractions Rule-of-Five in Vulgar Fraclions Decimal Fractions - - - - " Addition and Subtraction of Decimals Multiplication of Decimals Divifion of Decimals Reduction of Decimals Rule-of-Three in Decimals — Ruh -of-Five in Decimals Simple Intereft • » • Page* i V4U CONTENTS, Compound Intcreft • DTfcount Equation of Payments - Single Feliowfhip ... Double Feliowfhip - Barter • Lofs and Gain - • - Exchange - Arbitration of exchanges - Alligation - Involution - Evolution - Gf Proportion in general Arithmetical Progreffion Geometrical Progreffion Single Pofition - - • Double Polition - A Collection of Promifcuous Queftions Book-keeping by Single Entry Book-keeping by Double Entry - Appendix - Artificers' Works - Page 86 88 89 90 92 94 96 97 101 no IJ 3 122 124 I2 7> I3O *35 141 J 73 2)S SIGNS USED IN ARITHMETIC. 4- Plus : Signifies Addition , or, added to. —~ Minus : ■ ■ Subtraction ; or, leflened by. X - ' - : . Multiplication ; or, multiplied by* ^ „ — . , ,, Divifion -, or, divided by. ... . „ , Proportion. jpj. ,„ Equality $ or, is equal to» ^. m - ■ ■».■ Xfac Squaxe Root* A COMPLETE SYSTEM ON PRACTICAL ARITHMETIC Practical Arithmetic is the art of numbering, or of performing calculations by numbers. NOTATION. "VTOTATION is the ex pre fling of any propofed number, •^ either by words or chnra&ers. All numbers are expreflible by thefe ten character?, or. figures, j, 2, 3, 4, 5, 6, 7, 8, 9, o, or cipher,.- and the ufual method of notation by thefe figures is fo contrived, that, any character is increafed in its value in a tenfold propor- tion for every place it is removed towards the left, among the other figures with which it is connected •, fo in thefe fi- gures 333, the firfr 3 (reckoning from the right to the left) is 3 ones, but the fecond is 3 tens, and the third is 3. hun* dreds j aifo in thefe 2759, the 9 reprefents 9 ones, but the 5 reprefents 5 tens, the 7 is 7 hundreds, and the 2 is 2 thou* fand. And although the cipher fignify nothing by itfelf, yet when fet on the right of any of the other figures, it in- creafeth their value in the fame tenfold proportion above defcribed ; thus, though 2 ftanding alone, or in the firfl place, reprefents only 2 ones, yet when a cipher is fet on the right of it thus, 20, it reprefents 2. tens, or twenty ^ and if another cipher be affixed thus, 200, it will reprefent 2 hundreds, &c. For the more eafy reading of large numbers, when they are exprefFed by figures, they are divided from the right- hand towards the left, into periods and half periods, each half period confilting of three figures j the common name of the flrft period being units, or ones •, of the fecond, mil* lions ; of the third, billions; of the fourth, trillions, &c. Alfo the flrft half of any period is fo many ones of it,, but ttc latter half is fo manv.thoufand.s-of.it. Z^ * NOTATION* The following, example exhibits a fummary of this wholes do&rine : Quintillions Quadrillions Trillions Billions Millions Uniti th. un. th. un. th. urn th. un. th, un. c. x.t.c.x.u, 373. 8zp. 759,-l 6 *- 235,871. 205,473. 918,651. 4 3 7> 2 5°'« Note. The firft nine characters are called figmfieant figures, to diftin- guiih them from the cipher, which of itfelf is infignificant. Alfo a num- ber exprefling a quantity of one name or denomination, is called a (imple number, as 20 pounds, or feventeen gallons, or five days; and that re- prefeoting- a quantity of feverai names, is called a compound number, as 13 pounds 5 (hillings and 6" pence, or 17 gallons and 2 pints, or 3 hours. and 50 minutes. ^ I". Having any number propofed in words, to exprefs the fame " in figures* RULE. Write down ciphers to (o many periods and places as are named in the given number ; then, beginning at the left, ob- ferve at each place what fignificant figure is named, and, taking away the cipher, write the fignificant figure in its place. EXAMPLES 1. Exprefj in figures, four thoufand, one hundred and feventy-three* 2. Write down in figures, Twenty-three millions, two* hundred and fixty thoufand, nine hundred and thirty. 3. Write in figures, Four thoufand and twenty-five mil- lions, one hundred and three thoufand, and fix 4. Exprefs in figures, Two hundred feventeerv thoufand- and fifty millions, eight thoufand, feven hundred, and fix- teen. 5. Write down in figures, Seventy thoufand billions, oner hundred three thoufand and fifty millions, three thoufand and eight. 6. Exprefs in figures, Eight hundred trillions, one hun- dred feventy-five thoufand feven hundred and forty-eight billions, three hundred thoufand millions, five thoufand and feventy. II. Having any number expreffed in figures, to read the fame t or to exprefs it in words, RULE. Divide the figures in he given number, as in the general exam-pie above, into periods and half periods, .by any coa- ROMAN NOTATION, .3 lenient marks \ then beginning at the left, the figures are thus read, viz. the firfl: figure of each half period is named "by itfelf with the word hundreds, but the other two are na- jned together ; and at the end of the firft half of each period, the word thoufands is named \ but at the end of the other half, the common name of the whole period, except it be the units period, whofe name is not expreffed. 'EXAMPLES. 1. Let it be required to exprefs in words, 17359* 2. Write down in words, 7301462. 3. Write down in words, 3920500706. 4. Exprefs in words, 102003000400. 5. Write down in words, 2073000091630702, 6. Write down in words, 503002786940003. *t ^» ■ ■ ' 1 A SYNOPSIS OF THE ROMAN NOTATION. x zz I * zz II : As often as any character is repeated, fo many times its 3 ZZ III value is repeated. 4 zz IlllorlV: A lefs character before a greater diminimes its 5 zz V value. "6 zz VI : A lefs character after a greater increafes its value* 7 zz VII 8 zz VIII 9 zzlX 30 zz .X 50 = L - 100 zz C 500 zz D or 13: For every 3 affixed, this becomes 10 times as many o 1000 zz M or 03 : For eveiy C and 3, fet one at each end. it be* 2000 ZZ MM __ comes 10 times as much. 5000 zr Iqd or V : A line over any number, increafes it 1000 fold* 6000 zz: VI 10000 zz Xor CGIoo 3CC00 zz ijp^ tfecoo zz LX 200000 = Cjor CCCIoo^ *oooooo zz: Mo r CCCCI3033 is 000000 zz MM &C. 1 &Co X 4 ] SIMPLE ADDITION. '■CIMPLE Addition is the finding, of one fimple number, ^ equal to feveral fimple numbers taken altogether. The number which is' equal to feveral taken together, is called their fum. Simple Addition may be performed by this RULE. 1. Place the feveral numbers, to be added, underneath each other, fo that the figures of the fame name, with re- fpect to unite, tens, &c. may be ftraight under each other. 2. Draw a line under the lowed number; then add up the column of units, and confider how many tens arc in the fum, for which you muft carry fo many ones to the next column, writing down only the excefs over and above the tens, below the line, ft might under its proper column. 3. Add all the columns in the fame manner, and tb^ figures below the line will exprefs the fum required. To prove Addition. Cut off the uppermoft number, by d awing a line below it. Add all the reft of the lines of numbers together, and fet their fum below the fum to be proved. Then add thvs.laft-found number and the uppermoft line together, and their fum will be the fame as that found by the firft addition when the work is all right. EXAMPLES. 1. Whatisthe fumof 37, 509, 7126, 17630, and 459273? 2. Required the fum of 3579, 41, 96120, 725, II, 1820, 5, and 720139. 3. What is the fum of 2591, 720396, 14, 259, 6, 370214* 974°5 53> l6 9 2 > a „ nd x 37 ? 4. How many days are in the twelve calendar months ? 5. Suppofe that from London to Hatfield is 20 miles f from thence to Stilton 57 miles, thence to Newark 48 miles, thence to Doncafter 37 miles, thence to Northallerton §2 miles, thence to Durham 34 miles, and from thence to New- caftle 15 miles j how many miles are between London and Newcattle ? 6. A perfon dying, left to his widow 1500 pounds, to his eldeft fon he left 30500, and to each of his other two fons 3406 ' 7 alfo 2700 to each or his three daughters, befides 751 pounds in other fmall legacies > what did he die poifeffedof i X 5 J ~ SIMPLE SUBTRACTION. Ci'MPLE Subtraction is the finding how much onefimp ^ number exceeds another, or the taking a lefs fimple num- ber out of a greater. The number to be fubtra&ed is called the fubtrahend '; and that out of which it is to be taken, is called the minuend: Alfo the number remaining after the one is taken out of the other, is named their difference. Simple Subtraction is performed by the following RULE. I. Place the fubtrahend under the minuend, units -under units, &c. according to the directions given in Addition, and p draw a line below them. . 2. Begin at the right, and fubtracl each undt r figure from that which (lands above it. writing the remainder liraight under them below the line j fo (hall all the remainders toge- ther exprefs the difference required 3. But when any under figure exceeds that which h above it, conceive 10 to be added to the upper, and fubtract the under from the fum 5 but in this cafe, you rnuit. add I to the next under figure be&re you fubtraft it. To prove Sub traBion. Add the difference and fubtrahend together, and the ;fum will be equal to the minuend, when the operation is right. EXAMPLES. 1. What is the difference between 1735 and 1=897348 ? 2. How much does 5403 12 exceed 7953 ? 3. How much 13 30491 lefs than 57321469? 4. Suppofe that from London to Edinburgh (by way cf Newcaftle) is 393 miles, and that from London to Newcaftle is 273 miles j how many miles are between Newcaftle and Edinburgh ? 5. How much is A older than B, A being born in the year 1701, and B in 1739 ? 6. How much isC, whofe age is 71, older than P, whofc age is 34 ? B E p j SIMPLE MULTIPLICATION'. CIMPLE Multiplication is the finding of a fimple nuio ~ ber, which fhall contain any given f:mp!e number a cer- tain propofed number of times; and it is therefore a com- pendious method cf Addition. The two propofed numbers are, in general, termed the fa&ors of the multiplication \ but, in particular, that which is to be multiplied, is called the multiplicand ; and that you multiply by, the multiplier ; alio the number found from the operation is named the product of the two factor?. Before proceeding to any operations in this rule, the fol* lov/ing table of products mull be learnt very perfectly : MULTIPLICATION TABLE.' J 2 3 4 5 6 7 8 9 IC i j 12 2 4 6 8 IC 12 '4 16 18 2C 22 24 3 6 9 12 '5 18 21 27 30 •33 36 4 12 16 2C 24 28 32 3^ 40 44 48 5 IC x 5 2C 2 i 3c 35 4^ 45 5C 55 60 -6 12 18 30 36 42 4* 54 6c 66 '72 7 2; "58 35 42 49 *1 70 77 84 b 16 24 40 48 5^ 64 72 8c 881 9«| iS 2/ 3* 45 54 6 ^ 7- 81 _9c 99 108 JC 20 3 C 4^ 5c 6c 7c 8c 9c ICC lie I2C I ) 22 _3i : 44 55 6( 77 A 8 99 IIC 121 132 12 24 36 48 6c 72 84 96 108 F2C 132 144 Simple Multiplication may be performed by the two fol. lowing RULES. I. To multiply by a Jingle figure, or by any number in thefirfl line of the foregoing table cf produBs. Begin at the right-hand fide of the multiplicand, and mul- tiply each figure in it by the multiplier, fetting down the whole of fuch products as are lefs than ten ', but for fuch as are juft equal to a certain number of tens, write down o, 2nd carry 1 for each ic to the next produft j and far fuch ?5 iPLE MULTIPLICATION. f eed a certain number of tens, fej down the excefs, and carry for the tens as before. LEX. Multiply 1234567890 by each number feparately from 2 to 12 II. To multiply by a number conjxjlirtg of fever al figur es. 1. Write it below the multiplicand, and fir.d the product: for each £gure in it as in the firft cafe, not regarding in what order the lines are found, provided the firft figure in each ftand ftraight below its refpeclive multiplier. 2. Add all the lines of produces together in the fame order as they ftand, and the fum will be the whole product required. To prvV~ Multiplication. Make the former multiplicand the rnulti- i, ind the multiplier the multiplicand, and proceed as before; and the new product will be the fame 23 before when the wok is right. — Otherwise, add together the figuresineach factor, calling ci ; all the in the fains as often as they amount to 9. Multiply the two remainders together, and the nine? cad out of their product will leave the fame re-- xaainder as the nines caft out of the sofwec when the work is light. example!* 1. Required the product of 273580961 and 23. Anf. 6292362103. 2. Required the product of 6241578309 and 37. Anf 230938397433. 3. What is the product of 5318625074 and 43 ? A.if. 228700878182. 4. What is the product of 751900368 and ci t Anf 38346918768. 5. What is the product of 402097316 and 195 ? Anf. 7K408976620. 6. What is the product of 82164973 and 3027 ? Anf 248713373271. 7. What is the product of 16358724 and 704006 ? Anf 1 15 16639848344, 8*. What is the product of 921760035 nd 520007091 ? Anf. 479321754400408185. 9. What is the product of 3801 ^^2 and 400700065 ? Anf 15232906283422580. CONTRACTIONS. I. When there are ciphers at the right of one or both fac- tors, proceed as before, neglecting the ciphers ; and to &* B 2 8r SIMPLE MULTIPLICATION. right of the product place as many ciphers as are in bot& factors. So to multiply 390720400 by 406000 ;S Having written the terms as are here annexed, 390720400 and multiplied by the two figniiicant figures of the 406000 multiplier, to the fum of the two produces fubjoin " "23 44-11 24" '"' the five ciphers whleh-are on the right-hand of 'the t 5628816 two factors. ' . ■■■ ■ " ■■ ■ 1 580,1 148^400000 .EXAMPLES. 1. Multiply 718603400 by 57. Anf. 409603938000 2. Required the product of 7010a and 9001635-. Anf. 631014613500. 3. What is the product of 9030100 and 21000 ? Anf. 1S963 2100000. 4. What is the product of 7030 and 815036000 ? t Anf. 527970308000a. So that, if any number is to be. multiplied by 1 with ci- phers annexed, the product will be found by only annexing the ciphers to the figures of the multiplicand. EXAMPLES. 1. The product of 71 and 10 is 710. 2. The product of 2103 and 100 is 3. The product of 5030700 and 1000 is III. When the multiplier is the product of 2 or more- numbers in the table, it is often of advantage to multiply continually by thofe numbers, inftead of it. Thus, to multiply 160430330 by iocoo Becaufe the component parts of 10B are 13 and 160430800 o,»thHt is 9 times 12 produce 108; therefore muU 12 tiply firft by 12, and that producl by o; and to tj^e 19251696 1 aft product affix the tour ciphers which are on the 9 right of the two factors. 17^6326400^ EXAMPLES. 1. Required the product of 5 1 307 298 and 56, or 7 times ;8. Anf. 2873208688. 2. What is the product of 31704592 and 36 ? Anf. 1141365312. 3. What is the product of 29753804 and 72 ? Anf. 2142273888. 4. What is the product of 7128368 and 96 ? Anf. 684323328. SIMPLE MULTIPLICATION. p j, What is the produft of 61835720 and 1320 ? Anf. 8162315040©, IV. When fome of the figures of the multiplier may be produced by multiplying fome others of them by any num- ber, it is much eafier and more concife, after having ob* tained the product of the lefs, to multiply that product by the fame number for the product of the greater, than to pro* ceed by the common method. Note. This holds as well when the lefs numbers are on the left, as when they are on the rightof the greater; for, by the general rule, the products ©f the figures of the multiplier may be taken in any order. So, in the ift of the following examples, firft multiply by the 3, and then its pro- duct by 3 again, for the product of 9, becaufe 3 times 3 are 9. — In the fecond example, firft multiply by the 4, and then double its product, for the product of 8, becaufe 2 times 4 are 8. — In the 3d example, firft mul- tiply by the right hand 8, then 6 times its product will be the product of 48 the next two figures, and again two times this laft will be the product for 95, the other two figures; fo that there are only three lines of pro- ducts to add up. — In the 4th example, firit multiply by the 5 on the left- hand, then 5 times its product will be the product for the 25 on the right of it, and" three times this laft will be the product for the remaining 75, becaufe 3 times 25 are 75 And in the 5th example, begin with the 3 in the middle, then 9 times its product will be the product for the 27 on the left, and again 2 times this laft will be the product for the 54 on the right. — So that none of thefe examples have more than three lines of products. — But in operating this way, be careful to attend to the general rule ot placing the firft figure of each product ftraight be- low the firft figure of its own. multiplier. 71380164.. 27354 The operation for- the 5th or lafc example is here 214140492. • annexed. 1927264428 . • 38545Z8856 i95 2 533 oo6c 5 6 li Multiply 35802916 by 93. - Anf. 3329671188, 2. Required the produft of 910738060 and 48. Anf. 43715426880. 3. What is the produft of 61370913 and 96488 ? ■ Anf. 59215,6653544. 4. What 18 the produft of "13861470 and 52575 ? Anf 728766785250. 5s What is the produft of 71380164 and 27354 I Anf, J95 2533006056.. »3 C w 3 t SIMPLE DIVISION. CIMPLE Divifion is the finding how often one finiple ~ number is contained in another \ or the dividing of any- given firople number into any propofed number of equal parts. The containing number, or number to be divided, is called the dividend. The contained number, or the number of parts into which the dividend is divided, is called the dim/or. The number of times the dividend contains the divifor, or the number which expreffes one of the equal parts, is call- ed the quotient. Thus : Dividend Divifor 3) 12 (4 Quotient Note. Divifon is a compendious fubtraction, the quotient being the -tiumber of fubtraelions in the operation. Simple Divifion may be performed by the following RULE. li Having written down the divifor and dividend,- as in the form above, confider if the divifor be lefs than, or equal to, the fame number of the left-hand figures of the dividend-, if fo, write the figure exprefiing the number of times it is contained in the quotient j but if not, take one place more of the dividend figures than are in the divifor, and write the number of times they contain it in the quotient as before. 2. Multiply the divifor by the quotient figure. 3. Sufetraft the product from the fame dividend figures. 4. To the remainder affix the next dividend figure^ and write in the quotient the number of times the divifor is-con- 1 tained in this number ; multiply the divifor by the laft quo- tient figure, and fubtract the product from the laft mentioned . number ; then proceed as before from the beginning of this article, till all the dividend figures are ufed* Note t. It is fometimes troublefome to find how often the di vifbr is con- tained in the feveral dividuals ; but part of the trouble will be faved by obferving, that when any product exceeds its dividual, the quotient fi- gure belonging to fox: h product mud be leflened till the produdt be equal to or lefs than its dividual; again, if,- after fubti acting the product from its dividual, the remainder be equal to, or exceed the divifor, the quo- tient figure rouft be increafed till the remainder be lefs than it 3. To complete the quotient, fet the laft remainder (if any) at the end of, it> above a fraaUiiae with the -divifor below it, simple division*- %i To prove Divijion. Multiply the quotient by the diyifor, to the produft add the remainder, and the fum will be equal to the dividend when the work is right, EXAMPLES. 1. Divide 73146085 by 4. - Anf. 18286*521!, 2. What is the quotient of 531 7986027 divided by 7 ? Anf. 759712289^ 3. What is the quotient of 570196382 by 12 ? Anf. 475*6365-^, 4* How often is 37 contained in 74638105 ? Anf 20 1 7 2463V times, 5. How often does 137896254 contain 97 ? Anf. J421610I4 times, Anf. 320-|§|44« Wiat is the quotient of 2304109 by 5800 ? Anf. 397^-|^§-. III. Hence to divide by 1 with any number of ciphers annexed, you need only ftrike off from the right of the divi- dend fo many figures as the divifor contains ciphers 5 which figures, fo ftruck off, will be the remainder, and thofe or the left the quotient, EXAMPLES. 5138602 divided by 100 is equal to Ji386 r §Q. 3701483 by 1000 is 3702140 by 100 is IV When the divifor is the produft of two or morefmaH numbers, it is much cafier to divide continually by thofe. numbers, than by the whole divifor at once. Note. If there be any remainders after fuch divisions, multiply the laft remainder by the preceding divifor, and to the product add the remain, der belonging to the fame divifor; then multiply the fum by the next preceding divifor* and to the produdl add its correfponding remainder; proceed in the fame manner thorough all the divifors and remainders; fo fhall the lad fum be the remainder, the fame as if the divifion had been performed at once. After the operation defcribed in this note is begun, it muft be conti- nued according to the defciiption. though fome of the preceding divifions ■ Should happen to have 110 remainder, KEDUCT; ^f*3i' S&, to divide 42901685 by 96, whofe component S) 4200:685 JWrts are 8 and 12; divide the firit by 8, and 12) 536271c J this quotient by the 12, and the' remainders Quotient 446S9 i.|^- are 5 and 6; then 6 times S are 48, to" which add the 5, and the (urn 53 13 thtr whole remainder to the whole divi for EXAMPLES. I. Divide 310468^5 by ^ or 7 times 8. Quot. 55440744. Z. Divide 7014596 hty 72. - - Quot. 97424ft. 3. Divide 5130652 by 132. - - Quot. 3 BS6S^. 4. Divide 83016572 by 240, - Qaot. 345902/^-. V. When you are pretty ready in divifion, you may, even in the largeft divifions fubtra& each figure of the product as you produce it, and write down only the re- f mainders. EXAMPLES* I. Divide 3104679 by 833. 8 33) 3 I0 4679 (37*7 A£- 6056 2257 • 59*9 83 5. Divide 79165238 by 238. - Quot. 332627/^, 3. Divide 29137062 by 5317. - Qaot. 5479IIH. 4, Divide 62015735 by 7803. - Qaot. 7947ft#f* REDUCTION. T> EDUCTION is the converfion of numbers from one *-^- name to another, but ft ill retaining the fame value. If the reduction be to a lefs name, it is commonly called reduction defcending ; but if to a greater, reduction cfcend* ing. RULE. Confider how many of the lefs name concerned make 1 of the greater, and by that number multiply the given 14 REDUCTION. number ; if the redu&ion be defcending, but divide if afcen A; f. 3511135 C |d. 5. In 35 guineas, how many iarthings ? Anf. 35280 farthings, 6. In 3528© farthings, how many guineas ? Anf. 2S guineas, 7. How many crowns, (hillings, it groats, and pence, are in 50 pounds ? Anf. 200 cr 1000s 3000 gr i2oqod. 8. Reduce I20OO pence to groats, [hillings, crowns, and pounds, Anf. 3000 gr 1c.00s.200 cr 50], OF T ROY WI 1 :ight. Grains | Pennyweights 24 j 1 j Ounces 480 | 20 1 1 Found 57 6 ° 1 240 I 12 x Note. By this weight are weigh- ed jewel*, gold, fiiver, corn, bread, and liquors. One grain of Troy weight is equal to one grain and a. half of found dry wheat. ^EXAMPLES. 1. How many ounces, pennyweights, and grains are in 37ID ? - Anf 444 cz, 8880 dwts, 213 1 20 grs. 2. Reduce 213120 grains to lbs. Anf. 371b. 3. In 59 lb J 3 dwt 5 gr, how many grains ? Anf. 340157 grs, 4. In 540157 grains, how many lbs, Sec. &c. > Anf. 59 lb 13 dwts 5 gr, E.EDCCXSOS. OF APOTHECARIES WEIGHT, Grams s cruples 1 20 | 1 1 D rams | 60 | 3 1 1 1 Ounces 480 | 24 ! 8 i » Pound 576o | 288 1 Q6 1 12 1 « 1 1 Nut?. This weight is fo called becaufe the apothecaries ufe it in com- pounding their medicines ; but they buy and fell their drugs by avoirdu- pois weight Apothecaries is the fame as troy weight, having only fome different divifions. EXAMPLES. 1. In 37 lb how many ounce?, drams, and fcruples ? Anf. 204 oz 1632 dr 4896 fcr. 2. How many lbs are in 4896 fcruples r Anf. 17 lb, 3. In 231 lb 3 oz and 5 gr, how many grains ? Anf. 1332005 grs. 4. In 1332005 grains, how many lb? Anf. 231 lb 3 oz 5 gr» OF AVOIRDUPOIS WEIGHT. Drams | Ounces 16 | 1 t Pounds | 256 j 16 1 1 | Quarters | 7168 | 448 i 28 1 1 Hundreds j 28672 i 1792 1 112 i 4 | 1 J Ton 573440 i 35840 1 2240 1 80 | 20 | I Note. By this weight are weighed all things of a coarfe ordrofiy nature; fuch as grocery and chandler's wares, and all metals except gold and filver. Note alfo, 1 lb Avoirdupois makes 14 oz 1 £ dwt 15-^ gr Troy. 1 oz - - - o 18 5-! I dr - - - o 1 3! EXAMPLES. 1. In 15 toss, how many c, qrs, and lb ? Anf. 300 c, I20O qrs, 33600 lb. 2. Reduce 33600 lb to tons ? - Anf. 15 tons. 3. In 9 c 5 lb, how many ounces ? Anf. 16208 oz. 4. How many cwt are in 16208 oz ? Anf. 9 cwt 5 lb. 5. In 35 ton 17 c 1 qr 23 lb 7 oz 13 dr, how many drams ? - . Anf. 20371005 dr. DEDUCTION. 17 hS. Reduce 20571005 drams to tons. Anf. 35 t 17 c 1 qr 23 lb 7 oz 13 dr. OF LONG ME ASURE. f Inches j Feet 1 12 1 1 | Yards | 36 1 3 1 1 1 Poles 198 1 164 i si\ 1 j Furlongs 1 7920 1 660 | 220 | 40 I * | Mile 63360 1 5280 1 i7&> 1 320 1 8 1 < Note. An inch is fuppofed equal to 3 barley-corns iu length. 4 inches — a hand. 6 feet, or 2 yards — a fathom. 3 miles — a league. 60 nautical or geographical miles— a degree, or 6g\ ftatute miles nearly. Alfo 360 degrees, or 25000 miles nearly, is the circumference of the earth. EXAMPLES. 2. How many inches are between London and NewcafUe, or in 273 miles ? - - Anf. 17297280 inch. 2. In 17297280 inches, how many miles ? Anf. 273 miles. 3. Reduce 5 mis 6 furl 3 yds into inches. Anf. 364428 inches* -4. In 364428 inches, how many miles ? Anf. 5 mis 6 furl 3 yds. 5. Reduce 2 mis 1 furl -8 pis 3 yds. 2 inc, into inches. Anf.*i36334 inches* •6. In 136334 inches, how many miles, &c. ? Anf. 2 mis 1 furl 8 pis 3 yds 2 inches. OF CLOTH MEASURE. Note. 3 qrs =: 1 ell Flemifli. 5 — — Engliih. 6 — — French. 4 qrs 1 J inch. Scotch. EXAMPLES. X. In 37 yds, how many qrs and nails i Anf. 148 qrs 59 2 nails* Inches | Nails | 2i | 1 | Quarters | 9 1 4 1 1 Yard 36 | 16 | .4 1 I," *8 REDUCTION, 2. How many yards are in 592 nails > 3. Reduce 15 yds 3 qrs 1 nail to nails. 4. How many yards are in 253 nails ? Anf. 5. In 73 ells Flemiih, how many qrs ? 6. How many ells Flem. are in 219 qrs? 7. Reduce 17 ells Eng. 3 qrs to nails. 8. In 352 nails, how many ells Englifti ? Anf. 17 elisEng OF SQUARE or XAND MEASURE, Anf. 37*vyds e Anf. 253 nails o 15 yds 3 qrs I nl. Anf.. 219 qrs. Anf. 73 ells Fl. Anf. 352 nhe 3 4 1- * Weeks | 1 I<58 i 7 j t " | Months 40320 | 672 | 28 4 I" 1 Note, rhs minute is divided into 60 (eco:ttls, and the fecond may bs (Vippofed to be divided into 6othirds, and thefe agai* into 60 fourths, &c» EXAMPLES. 1. IIow many minutes are in 1763 months ? Anf. 71084160 min. 2. In 71084160 minutes, how many months ? Anf. 1763 months. 3. How many feconds are in a folar year, or 365 days 5. brs 48 min 58 fee ? Anf. 31556938 fee 4. In 31556938 feconds, how many days, &c. ? Anf. 365 ds 5 hr8 48 min, 58 fee, 5. In a lunar month* or 29 ds 12 hrs 45 min, how man£ econds I Anf. 2551500 fee, 6. Reduce 2551500 feconds to days. Anf. 29 ds 12 hrs 45 min. COMPOUND) ADDITION. COMPOUND Addition is the finding the fum of fev?ral compound numbers. RULE. I. Place the numbers of the fame denomination under each other, according to the directions given in Simple Addition. COMPOUND ADDITION. fcg 2. Add up the figures in the lowed denomination alfo f as in Simple Addition. 3. Find how many units of the next higher denomination are contained in the fum, by dividing it by fo many, as of this name make one of the next, or any other way, 4. Write the remainder or overplus underneath, and carry the ones or units to the figures in the next denomi- nation ; whofe fum you muft find and proceed with as before ; and fo on, through all the denominations, to the highefc, whofe fum muft be all fet down, which, to- gether with the feveral remainders, will exprefs the total required. Note. Addition of money may be performed by the general rule, or bf help of the following tables. d s d a d 2D — — 1.8- 2 — — 24 30 26 3 3 6 4° 3 4 4 48 50 42 5 60 60 *— 50 6 — — 72 7 o — 5 10 7 84. 80 6 8 8 96 90 76 9 — — 108 100 — 84 10 — — — 120 no 9 2 11 ■ ■ ■ J32 120 ' IO O.. ~ 12 — — I44. EXAMPLES OF MONET. 1 s d 1 s d 1 s d i s d 7 13 3 14 7 5 ij *7 10 53 '4 8 3 5 1 of 8 J 9 *J 3 14 6 5 10 2* 6 18 7 5 3 44- 23 6 2| 93 M 6 2 C-2- 21 2 9 8 3 J 7 5 4 3 r 16 8f 15 6 4 2 9 n *5 4f 4 3 6 12 9l 18 7 L C3 %% COMPOUND ADDITION, 1 s d 1 6 d 1 8 d 1 s d 14 1i 37 15 8 61 3 2f 47^ 15 3 5 *3 6 14 12 91 7 16 8 9 2 2f 62 4 7 5 6 ii 29 13 »°i 27 12 «*■ 4 *7 8 2? ID 9i 8 14 O 370 16 2f «3 4l 8 6 7 Jt 25 3 8 6 6 7 14 5i 24 13 6 10 Ji- 91 lOj 54 2 n 5 I<4 30 "i Suppofe that A is indebted to B, 34I 13s 7d, and to C 9 1730I, to D, 9I 19s 2d, to E, 134I 7d, to F, 17s 2d, and to G, 9d : What is A's whole debt ? Anf. 1909I us 3d. Suppofe that B owes A 75I 17s ^ C owes 15s $d; D owes rzil 13s 6fd ; E owes 9! d 3 F owes 796I 3d 5 and G owes 17I 13s iod : What is due to A by all of them j Anf. 912I os iod. A owes to B, for tea, 13I 10s \ for cheefe, 17I 13s jd ^ for cotton, 208I 17s \ for Indian chintz, 861 7d ; for his ac- ceptance of a bill, 300I; for factorage, 15! 17s 3^d ^ alfo for infurance and other charges, 30I 10s 4-Jd : How much is A's whole debt to B ? Anf. 67 2l 8s 8jd. A corn-fadlor pays for wheat, 37I 15s 8d 5 for rye, ill 16s 3d ; for oats, 96I 7§d \ for barley, 53I 12s •, alfo for peas and beans, 10I ', he has alfo paid for carriage and other petty charges, 3I 17s 5^d •, and for infurance, ill 3|d : Now, fuppofing his commifiion on the whole is 7I 3s o-Jd, for how much muft he draw upon his employer to clear the account ? - - Anf 23 il 5s 4fd. A nobleman going out of town, is informed by his Reward, that his butcher's bill comes to 197I 13s 7-Jd y his baker's to 59I 5s 2|d , his brewer's to 8.5I \ his wine- merchant's to 1C3I 13s ; to his lonifhip's corn-chandler is due 75I 3d ; to his tallow-chandler and cheefemonger, £71 15s njd y and to his tailor, 55I 3s 5|d ; alfo for rent, fervants' wages, and other charges, 127I 3s: Now, fuppofing he would take loolwith him, to defray his charges on the road, {qx what fum muft he fend to his banker ? Anf. 830I 14s djd*-. compound ABmrros*. EXAMPLES OF WEIGHTS, MEASURES, bV. TROY WErGHT. *Ib oz dwt oz dwt gr APOTHECARIES WEIGHTV lb oz dr fc oz dr fc gr I 7 3 J 5 37 9 3 3 5 7 2 3 5 1 17 4 6. 3 9 5 3 13 7 3 7 3 2 5 o io 7 3 16 21 9 11 1 16 7 12 9 5o 17 7 8 9 I 2 9 5 1 5 176 2 17 5 9° 36 3 5 4 1 2 18 23 11 12 3 19. 5 8 6 1 36 4 ASURE. 1 14, AVOIRDUPOIS WEIGHT, LONG ME lb oz dr CWt qr lb mis furl pis yds feet inc 17 10 19 I? 2 15. -9 3 J 4 127 1 5 5 14 8 6 3 24 *9 6 29 12 2 9 8 6 15 70IO 5 4 20 2 6 27 1 6 9 * J 7 9 1 37 54 1 11 040 io 2 6 7 3 5- 2 7 6 14 10 3 3 4 5 9 2 3 5 CLOTH MEASURE* LAND MEASURE. yds qr nls el en qrs nls- ac ro p ac ro p 26 3 I 270 I o- 225 3 37' J 9 16 I'3 1 2 57 4 3 16 1 *$ 270 3 29 620 8 2 1 9 13 9 1 3 217 3 032 4 2 9 23 ° 34 9IO 10 1 42 I 19 7 2 16 55 3 * 4 4 1 7 6 75 23 WINE MEASURE, T hdsgal hdsgal pts *3 3 J5 8 1 37 4 2 26 25 O 12 3 1 9 72 3 21 15 61 7 16 29 23 3 *5 16 8 4 3& 5 3 7 1 o 6 ALE AND BEER MEASURE. lids gal pts hdsgalptg l l 37 3 ^9 43 5 4 13 3 6 5 M 5 2 7 9 .14 16 6 8 2 6 r 12 9 8 4.2 6 4 57 l 3 -J 6 4 &4 COMPOUND ADDITION DRY MEASURE. TIM! :. L qr bu q r bu pe mo we da hrs in s *3 5 2 25 7 3 < r *? 3 4 27 15 37 7 1 3 5 3 1 26 1 6 12 26 14 41 7 4 *7 5 2^ 7 2 3 7 16 3 7 6 2 3 19 2 3 35 42 59 24 3 33 2. 8 3 694 5 2 1 7 4 1 12 1 6 31 16 3,2 A gentleman bought of a filver-fmith, difhes to the weight of 23 lb 6 oz 5-dwt 5. plates 41 lb 7 oz 17 dwt 5 fpoons 12 lb 15 dwts ; falts 2 lb 7-oz.; waiters 13 lb -, and tankards 7 lb 17 dwt : What weight of plate did he buy in all ? Anf 99 lb 10 oz 14 dwt. An Apothecary made a compofition of 5 ingredients, the ift of which weighed .13 lb 7 oz ; the 2d, 1 1 oz 7 dr 13 gr; the 3d, 7 lb 2 fcr j the 4th, 11 lb 3 dr 1 fcr ; and the 5th weaned 15 lb, 5 oz 7 gr : What was the weight of the whole ? Anf. 48 lb 3 dr 1 fcr. A country (hop-keeper buys of a merchant in London, teas weighing 3 qrs 14 lb; coffee, 1 qr 23 lb; fugars 3 cwt 2 qr 5. lb ; fpices, 2 qr 3 lb 13 oz ; hops, 13 cwt 1 qr 24 lb ; and feveral other things to the weight of 3 cwt 17 lb 7 < z 5 For what weight has he to pay carriage on bringing them home ? Anf. 22 cwt 3 lb 4 oz From A to B is 3 mis 2 furl 7 pis ; rrorn B to C is 17 mis 13 pis - 7 from C to D is 7 fur ; and from D to -E is 5 mis 33 pis : What is the diftance between A and E. ? Anf. 26 mis 2 fur 13 pis* Bought four parcels of cloth, the ift of which contains 25 yds 3 qrs ; the 2d,, 37 yds 2 qrs 3 nls ; the 3d, 14 yd* I nl ; and the 4th, 23 yds : How many yards a:e in them all ? - - - Anf. ico yds 2 qrs. There are five pieces of ground, the ift of which mea- sures 13 ac 3 r 14 p ) the 2d, 27 ac 29 p ; the 3d, 19 ac 1 r? the 4th, 3 r 34 p ; and the 5th, 45 ac 2 r up : What is the fum of their meafures ? - Ani. 106 ac 3 ro 8 p. A gentleman bought of a wine-merchant, of port wine, X ton 3 hhds j of claret, 3 hnds 47 gal j of mountain^ 1 hint COMPOUND SUBTRACTION, ZJ 5 gal ; and of Liibon 2 hhds 23. gal \ What quantity did he buy in all ? * - Anf. 3 tons 2 hhds 12 gal. A beer-brewer has fent into the country, ale, as follows^ viz. at one time three hhds 14 gal ; at another 2 hhds 17 gal y at another 14 hhds 27 gal 5 and at another 6 hhds 47 gal : How much was fent at all the times ? Anf. 27 hhda 3 gal. A corn-merchant fends over the fea, of wheat, 13 laits 3 qr 5 bufh j of oats, 29 lads 7 qr 5 of rye, he has lent 3 lafts 7 bufh - y of peas, 8 qrs 3 bufli j and of beans, 5 qr : For what has he freight to pay ? Anf. 47 lfts 4 qr 7 bufh. When B was born, A's age was 113 rnths 2 wks •, when C was born, B's age was 97 mo 1 we 5 ds ? when D was born, C's age was 107 mo 3 ds 14 hs *, and when E was born > . D's age was 75 mo 3 we 19 hs : What was A's age when E was born ? Anf. 393 mo 3 we 2 ds 9 hs. COMPOUND SUBTRACTION. /COMPOUND Subtraaion is the finding Ue difference ^-* between two numbers, of which one or both are compound. RULE. 1. Set the lefs number under the greater, as directed in Compound Addition. 2. Thc^ beginning at the lcafl denomination, fubtraft the under number of each from the upper, writing their re« fpedlive remainders below them. 3. But if the under number of any of the denominations be greater than the upper, ^dd fo many to the upper as make one of the ne&t higher denomination ; then take the under from the fum, writing down the remainder as before, and carry os add one to the* under number of the next higher denomination before you fubtra.it it» 26 COMPOUND SUBTRACTION EXAMPLES OF MONET, 1 s d 1 s d 1 8 d 1 s d From 79 17 8£ 103 3 2f SI 10 251 13 Take 35 12 4{ 7* ^ Ji 29 *3 3i 35 4-7i Rem. Proof ~ What is the difference bcrween 73I 5fd and 19I 13s lod ? AnC 53I 6s 7£d. A lends to B iool ; how much is B in his debt, after A has taken- goods of him to the amount of 73I 12s 4^d. Anf. 26I 7s 7jd. Suppofe that my rent for half a year is icl 12s ^ and that I have laid out for the land-tax, 14s 6d, and for feveral re- pairs, il 3s 3^d : What have I to pay of my half year's rent ? Anf. 81 14s 2|d. A trader, failing, owes to A, 55I 7s 6d ; to B, 91I 13s 3|dj to C, 53I 7^-dj to D, 87I $s ' y and to E, nil 3s $%d> When this happened, he had by him in cadi, 23! 7s jd ; in wares, 53I us lojd 5 in houfehold furniture,* 63I 17s 7^-d ; and in recoverable book-debts, 25I 7s $d : What will his creditors lofe by bim, fuppofing thefe thingsr delivered to them ? - ' Anf, 21 2I 5s 6|d u EXAMPLES OF WEIGHTS, MEASURES, bV. TROY WEIGHT. APOTHECARIES WEIGHT*. lb ozdwtgr lb osdwtgr lb oz dr fc gr From 7 3 14 n 4 9 1 13 73 4 7^ *4 Take 3 7 5 19 3 7 16 12 26 7 2 1 16' Rem, ' . __ Proof ~ ~ "" 4:0M?0tT^D SUSTEACTIONp 1ZJ AVOIRDUPOIS WEIGHT. c qrs lb : From £ 017 Take 3 211 Rem. -Proof ^ LONG MEASURE, m fu pi yd ft in. 14. 3 17 96 1 4 3 7 9 4i 2 7 CLOTH MEASURE. yd qr nl yd qr ni From 17 2 1 902 Take 521 612 •Hem. ~"~~""~ ~ Proof XAND MEASURE. ac ro p ac ro p 17 1 14 57 1 16 936 24 2 25 WINE MEASURE. t hd gal hd gal pt From 17 2 23 504 Take 4 3 39 3 2 7 ALE AND BEER TREASURE. hd gal pt hd gal p 14 29 3 71 16 5 7 34 5 17 3 * Rem. Proof" la From 9 Take 3 DRY MEASURE. qr bu bu gal pt 4 7 *3 7 * 7 2 7 3 4 TIME. mo we da ds hrs mirk 71 2 5 114 17 26 ■ M 3 75 12 33 Rem. * Proof — um Iin ■■■»■ I 28 1 COMPOUND MULTIPLICATION. •f^OMPOUND Multiplication is the finding of a number ^ which (hall contain a given compound number any pro- pofed number of times, RULE. i. Set the multiplier under the loweft-denomination of the multiplicand. 2. Multiply the number of the loweft denomination by the multiplier, and find how many units of the next higher denomination are contained in the product, as in Compound Addition. 3. Write down the excefs, and carry the ones to the pro- duel of the next higher denomination ; with which proceed ■as before ; and in like manner with all the other denomina« lions to the higheft. L EXAMPLES OF MONET. I s d 1 s d lad o 9 4! 7 9 H l8 6 *i S 6 4 I, 17 7 2 14 6 7 O 17 2*. *4- 10 m 17 3 7f 6 12 of 8 9 o 10 2 6 of II 12 COMPOUND MULTIPLICATION. *9 XL EXAMPLES OF WEIGHTS, MEASURES, &c. TROY WEIGHT. lb oz dwt gv 3 o 14 9 3 AVOIRDUPOIS WEIGHT. cwt qr lb 17 3 2 3 5 LONG MEASURE. miles fur poles 3 1 3 *7 7 CLOTH MEASDRE. yd qr nl 53 * 3 9 WINE MEASURE. tuns hds gal l 9 3 17 11 hd fir pts Mult. 35 47 by 18. Anf. 71 hds 30 pints, laf qrs bus Mult. 5 3 7 by 72. Anf. 387 laf 9 qrs. mon we da Mult. 7 3 5 by 26. Anf. 206 mo 4 da. cwt qr lb Mult. 3 1 14 by 53. Anf. 178 cwt 3 qrs 14 lb. APOTHECARIES WEIGHT. lb oz dr fc gr *3 5 3 * *4 4_ AVOIRDUFOIS WEIGHT. lb oz dr 21 II 15 6 LONG MEASURE. yd feet inc 171 1 10 8 LAND MEASURE. ac ro pol 15 ^ 29 10 ALE MEASURE. hds gal pin S3 33 3 12 bar kild gal Mult. 21 1 13 by $6. Anf. 787. bar 1 kil 9 gal. buf pec gal Mult. 71 3 I by T32. Anf. 9487 bu 2 pec. days hre min Mult. 14 13 27 by 47. Anf. 684 da 8 hr 9 m. cwt lb Mult. 17 12 by 75. Anf. r 283 cwt 4 lb. D 30 COMPOUND DIVISION. See more of Compound Multiplication under Rule I. of Rules of Pr attic e. COMPOUND DIVISION. /"COMPOUND Divifion is the dividing compound num* ^-^ bers into any propofed number of equal parts. RULE* 1. Place the divifor and dividend as in Simple Divifion. 2. Begin at the higheft denomination, and divide each of them by the divifor, writing the quotients under their re- Ipe&ive dividends. 3. Bat if there be a remainder after dividing any of the denominations except the leaft, you may find how many of the next lower denomination it is equal to, and add it to the frazil number (if any) which was in this denomination before j then divide the fum. I. EXAMPLES OF MONET. 3) d 6 J) 6 6 7) o 11 ni 9) o 1 2 9 11) 19 10 6 4) 6) s 10 d 8 4 9 8) I I 10) 28 iS 4 12) 40 8 GOLDEN-RULE. 3* II. EXAMPLES OF WEIGHTS, MEASURES, fcV. TROY WEIGHT, lb oz dwt 4) 13 * I 5 ( AVOIRDUPOIS WEIGHT. c qr lb 7) 75 1 12 ( LONG MEASURE. miles fur pis 11) 58 5 12 ( CLOTH MEASURE. yds qrs nls 17) 3 1 ^ 3 ( APOTHECARIES WEIGHT. lb oz dr fc gr 6) 2 5 3 o 19 ( AVOIRDUPOIS WEIGHT. lb oz dr 9) 5 3 *4 ( LONG MEASURE. yds feet inc 12) 150 1 7 ( LAND MEASURE. ac ro pis 26) 17 3 17 ( See more of Compound Divifion under Rule II. of Rules sf ' Praclice. GOLDEN-RULE, or RULE-OF-THREE. r "PHE Rule-of-Three is that by which a number is found, **- having to a given number the fame proportion which is between two other given numbers. For this reafon it is fometimes named the Rule of Proportion. It is called the Rule-of-Three, becaufe in each of its quellions there are given three numbers at leaft. And be- caufe of its excellent and extenfiye ufe, it is often named the Golden-Rule. Fcr the flating, or rightly placing down the three given numbers, cbferve the following RULE. 1 Write down the number which is of the fame kind with the arfwrr or number required. 2. Confidcr whether the anfwer ought to be greater or lefs than this number ; then write refpe&ively the greater or lefs of the two remaining numbers on the light of it for the third, and the other on the left for the firft number or term. 3. Multiply the fecond ^nd third terms together, divide the produd by the firft, and the quotient will be the anfwer. D 2 32 " GOLDEN-RUI-E. Note i. When you can conveniently multiply and divjde as in Cora- pound Multiplication and Divifion, it i$ beft fo to do. 2. But if not, reduce the compound terms to the lowed name mention- ed in them, and the firft and third to the fame name, if they be not fo al- ready ; then will the anfwer be of the-fame name with the 2d term. 3. When there happens to be a remaisder after diviiion, reduce it to the name next below the lad quotient, and divide by the fame divifor, fo ihall the quotient be fo many of the faid next name ; do this as long as there is any remainder, or till you have reduced it to the leaft name, and all the quotients together will be the anfwer. 4. If the 1 ft term, and either the 2d or 3d, can be divided by any num- ber, without remainder, let them be divided, and the quotients uied in. itead of them. 5. There are four other methods of operation befides the general one above delivered, any of which, when pcffible, performs the work much ihorter than it. They are thus : Firft, Divide the 2d term by the 6rft, multiply the quotient by the 3d, and the product will be the anfwer. " Second, Divide the 3d term by the ift, multiply the quotient by the ad, and the product will be the anfwer. Third, Divide the ift term by the 2d, divide the 3d by the quotient, and the laft quotient will be the anfwer. Fourth* Divide the ift term by the 3d, divide the 2d by the quotient, tnd the laft quotient will be the anfwer. ior an example, let it be propofed to find the value of 14 03 8 dwt of gold, at 3I 19s 1 id an ounce. EXPLANA. Having ftated the three terms by the general rule, as here annexed, the ad terra is reduced to pence, and the 3d to dwts, thefe being their lowed denominations, as directed in the ad note. The ift term is alfo re- duced to dwts, that it may agree with the third, by the fame note The ad term is then multiplied by the 3d, and the product divided by the ift, according to the gene- ral rule, when the anfwer comes out 13809 pence, and 12 remain- ing over ; which remainder being reduced to farthings, and thefe divided by the fame divifor ao, by the 3d note, the quotient is i,~o)«V,o! > od aLq a farthings, and 8 remaining. — A „ f „„« „ *°j „ » Laftly, the pence are divide! by Anf> «' ,os * d * TO * xa to reduce them to (hillings, and thefe again by 20 for pounds ; when the final fura comes out 57 1 xcs od 2q, for the anfwer. 02 1 S d oz dwt 1 : 3 *9 n : : 14 8 20 20 ao ■-.-»■- 20 79 288 12 9S9 2S8 7672 7672 1918 2,0)27619,2 13809* *. pence, or I2)i38c9d 2u^qr oz 1 S d oz dwt i : 3 iq it : : 14 8; 20 8 20 GOLDEN- RVLE. 33. Bat this queftion will alfo ferve to illuftrate focne more of the notes, ' by means of which it can be eafier folved than by the general rule as — — . above given; for having dated it 4)2031 19 4 4)288 as before, and as here again annex- 5 9 7 Z ed, and reduced the ill and 3d terras to dwts, divide them each by 5) 28 7 x 4 ° 4» and ufe the quotients 5 and 72 Anf. 57I 10s 9d 2j.q. inftead of them, as by the 4th note: then multiply and divide, as directed by the t ft note, by Compound Mul* tiplication and Divifion, multiplying by 8 and 9 inftead of 72, its compo- nent paits; after which the anfwer comes out the fame as before, but with much let's trouble. Several of the following large collection of examples will afford occa- fions to illuftrate the different parts of the 5th note, as well as the other notes. EXAMPLES. 1. If 8 yards of cloth coft 24s, what will 96 yards coil ? Anf. 14I 8«. 2. How many yards of cloth may be bought for 14I 8s, when 8 yards coft 24s ? - - Atif. 96 yards. 3. What will be the piice of 72 yards of cambric, of which 9 yards coft 5I 12s ? - - Anf, 44I 16s. 4. What will 9 yards of cambric coft, at the rate of 44I 1 6s for 72 yards ? - - - Anf. 5I 12s. 5. If 7 cwt 1 qr of fugar coft 26I 10s 4c!, what will be the price of 43 cwt 2 qrs ? - - Anf. 1591 2s. 6. What quantity of fugar may be bought for 159L 2s, at the rate of 7 cwt 1 qr for 26I 10s 4d ? Anf. 43 cwt 2 qrs„ 7. How many yards of mnflin may be bought for 44I i6>, whereof 9 yards coft 5I 12s ? - Anf. 72 yards. 8. How much muft be paid for 26 bags of hops, at 21I 4s for 8 bags ? - - - Anf. 681 18s. 9. How many men muft be employed to flnifli a piece of work in 15 days, which 5 men can do in 24 days ? Anf. 8 men. iq~ How many yards of broad cloth may be bought for 5I 12s, of which 72 yards coft 44I 16s ? Anf. 9 yds. 11. What muft be paid for $-$ ells Eng. 1 qr of holiand, at the fate of 7s 9 [d per yard ? - *Anf, 25I 18s J £4, E>3 " 34 GOLDEN-RULE, 12. How many yards of matting, which is 2 feet 6 inches broad, will cover a floor which is 27 feet long and j20 broad ? - - Anf. 216 feet or 72 yards. 13. What quantity of fugar may be bought for 26I 10s 4d, when the price of 43 cwt 2 qr is 159I 25 ? Anf? 7 cwt 1 qr. 14. In how many days will 8 men finifn a piece of work, which 5 men can do in 24 days > - Anf. i$ days. 15. What mull be paid for 8 bags of hops, when the price of 26 bags is 681 18s 1 - - Anf. 21I 4s. 16. A perfon failing in trade, owes in all, 977I, and has in money, goods, and recoverable debts, 420I 6s 33d j fup- pofing thefe things delivered to his creditors, what will they get per pound ? Anf. 8s 7^d. 17. What muft be given for a piece of filver weighing 73 lb 5 cz 15 dwts, at the rate of 5s 93 per ounce ? Anf. 25 3I 1 os o|d. 18. Bought 3 cafks of raifins, each weighing 3 cwt 1 qr 7 lb neat, what will they cofl at 2l 6s 6d per cwt ? Anf. 23I 2s i|d. 19. A garrifon, being bedeged, has 5 months provilion in it, at the rate of 12 ounce* a day for each man $ but be- ing informed that it cannot be- relieved till after 9 months, how much per day muft each man have, that the provisions may laft that time ? - - Anf. 6\ ounces. 20. What will the tax upon 763I 15s be, at the rate of 3s 6d per pound ? - - Anf. 133I 13s J-Jd. 21. How much filver may I have for 253I 10s o|d, at 5s 9d per ounce ? - - Ar;f. 73 lb 5 oz 15 dwte. 22. What will 7 cwf i qr of fu^ar cot!, at the rate of 43 cwt 2 qr for 159 s 2> ? - - Anf. 26! 108 4}. 23. What mutt be paid for 1 cwt 3 qr 17 lb of wool, at 7s 4d, the ftone of 14 lb ? - Anf 5! us 6 J 3^q. 24. What quantity of hops may be bought for 684 18s, of which 8 baus cod 21 1 4% ? - - Anf. 26 bags. 25. How many elb ling, of holland may be bought lor 25I 18s i|d, at 7s 9Jd per yard ? Anf. 53 ells Eng. 1 qr. 26. A perfon flopping payment, owes to feveral 977I, but compounds with them for 8s 7 Jd per pound : What muit he pay them in all? ~ - Anf. 420! 6* 3£d. COLDBN-MJLE. 3$ 27. A borrowed of B 730I for 8 months \ afterwards A would requite B's kindnefs by lending him 375I \ required the time it muft be lent > Anf. 15 mo 2 we 2^%- ds. 28. What will 1 qr 1 nail of velvet coft, at 18s 6d per yard ? * Anf. 5s 9d if q, 29. What muft be given for 7 03 qr 14 lb of cheefe, at ll 14s 2d per cwt ? - - Anf. 13I 9s o|d. 30. What is the price of 2 c 1 qr 12 lb of beef, at 2$ 8d per done, or 14 lb ? - Anf. 2I 10s 3d i^q* 31. A tradefman failing, compounds with his creditors for 8s 7^d per pound, and at that rate he pays them in all 420I 6# 3|d : What was his debt ? - Anf. 977I. 32. Bought 73 lb 5 oz 15 dwts of filver, for 2J3I 10s o|d : What did it coft per ounce ? - - Anf. 5s pd. 33. If the tax upon 763I 15s be 133I 13s i£d \ at what rate is it per pound ? - - Anf, 3s 6d per 1. 34. What muft I pay for three-eighths ofafhip, which is valued at 700I ? Anf. 262I ios* 35. What will the carriage of 8 c 3 qrs 7 lb coft, at iod per ftone ? Anf. 2I 1 8s 9d. 36. If the carriage of 5 c 14 lb for 96 miles be il 12s 6d 5 how far may I have 3 c 1 qr carried for the fame money ? Anf 151 m 3 fur^poL 37. What coft 30 pieces of lead) each weighing 1 c 12 lb$ at the rate of 16s 4d per cwt ? - Anf 27I 2s 6d. 38. Bought a filver tankard, weighing 1 lb 7 oz 14 dwts, what will it coft me at 6s 4d per ounce ? Anf. 61 4s 9fd, 39. What muft be paid for 7 cafks of prunes, each weigh- ing 2 c I qr 14 lb, at 2l 19s 8d per cwt Anf. 49I us ii^d, 40. Bought 14 pockets of hops, each weighing 1 c 1 qr 18 lb, at 4I 2s 6d per cwt, what do they come to ? Anf. 81I 9? 4id. 41. What muft be given for 75 chaldrons 7 bufhels of coals, at the rate of il 139 6d per chaldron ? Anf. 125I 19s o£d.- 42. How much muft be paid for 17 qrs 1 peck of corn, at 3s iod per bufhel ? - - Anf. 26I 2s 34d. 43. What coft 43 qrs 5 bum of corn, at i\ 8s 6d the quarter ? ■» * * Anf. 62I 3s 3^d» 3& COMPOTJNt) PROPORTION, ' 44. How much a year will 173 acres 2 10 14 pis of land give, at the rate of il 7s 8d per acre ? Anf. 240I 2s 7to^« 45. What will 139 pigs of lead, weighing in all 243 cwt 2qrs, come to, at 9I i8s per fother of 21 cwt? Anf. 114I 15s iod irTjq* 46. What mud be paid for 73 pieces of lead, each weigh* ing 1 c 3 qr 7 lb, at iol 4s per fother of 19^ cwt ? Anf. 69I 4s 2d i^lq. 47. If .5 yards of cloth coft 14s 2d, what muft be given for 9 pieces, containing each 21 yds x qr ? Anf. 27I is ioid. 48. If a gentleman's eftate be worth 2107I 12s a year, what may he fpend a day to fave 500I in the year ? Anf. 4 l 8s i T V>3-d. COMPOUND PROPORTION, or RULE-OF-F1VE. THIS rule is called The RuIe-of-Five, becaufe that in it there are five numbers or terms given to find afixth.— It is often named the Double Ru/e-of-Three, becaufe its queftions are fometimes performed.by two operations of the Rule-of-Three. It is alfo called Compound Proportion. Note. Of the five given wumhers, three contain a fuppofirion, and the ether two a demand ; one of the term3 of fuppoOiior^ being of the fame kind with the number required, and the other two of the fame kind as the demanding terms. • RULE FOR STATING. r.i Write down the term of fuppoiition which* is of the fame kind with the anfwer, for the middle term. 2. Take one of the other two terms of fuppofnibn, and of the demanding terms, both of a kind 5 and from the di- rections given in the Rule-of-Three> confider which places they would poffefs if a ilating were made of theni and the middle term only, and place them accordingly ; do the fame ■with the other term of fuppoiition and its correfpondent- demanding one, writing the terms under each other which fall on the right and left of 4he middle termr COMPOUND PROPORTION. J7 METHOD OF OPERATION. 1. By two operations. — Take the two upper terms and the middle term, in the fame order as they (land, for the firft dating of the Rule-of-Three ; then take the fourth number refulting from the fir ft dating, for the middle term, and the two under terms in the general ftating, in the fame order as they (land, for the extreme terms of the fecond ftating \ and the fourth term refulting from it will be the anfwer. 2. By one operation.— Multiply together the terms o£ which the one is above the other, on both fides of the middle term ; then account the two producls and the middle term, as they Hand, the three terms of a Rule-4>f- Three ftating, and the fourth term thence refulting will be the anfwer. Note. It is generally beft to work by the htter method, viz.. by one operation. And after the ftating, and before commencing the opera- tion, if one of the two firft terms, and either the middle term or one of the two laft terms will exactly divide by one and the fame number, let them be divided, ind the quotients ufed inftead of them ; which will much (horten the work. Thus, if the queftion were to find how many men can complete a trench of 135 yards long in 8 days, it being known that 16 m*»n can dig 54 yards in 6 days: Having ftated the terms according to the rule, the 54 yards will ftand above the 8 days in the firft place, the 16 men in the middle, and the 135 above 6 in the laft place. Then divide the 54 and 1 7,5, each by 27, and place their quotients a and 5 after them : aJfo divide the 8 and 6, each by 2 t and place their quotients 4 and 3 after them. Multiply then the quo- tients of the firft two terms together, and their product is 8; and mul- tiply the quotients of the laft two together, and their product is 15 ; fo fhall 8, 16, and 15, be the 3 terms of a (ingle ftating. Of thefe di- vide the 2d term id by the ill or 8, and the quotient is 2, by which multiply 15 the 3d term, and the product is 30, the number of men required. The whole operation is here fubjoincd. yards 54, or 2 1 men C 135 yards, or 5I days 8, or 4 3 : 16* : : \ 6 days, or 3 J 8) i« (t "~*7 % AnC 30 menu 38 tOMPOVND *ROPORTiON* EXAMPLES. 1. If 100I in one year gain 5I intereft, what will be the intereft of 750I for 7 years? - - Anf. 262I 10s. 2. If 27s be the wages of 4 men for 7 days, what will be the wages of 14 men for 10 days ? - Anf. 61 15s. 3 What principal will gain 262I 10s In 7 years, at 5I per Cent, per annum ? - - Anf. 750I. 4. If a footman travel 130 miles in 3 days when the days are 12 hours long, in how many days of 10 hours each, may he travel 360 miles > - - Anf* 9I-3- days. 5. A wall which was to be built to the height of 27 feet, iva? raifed to the height 019 feet by 1 2 men in 6 days j how many men muft be employed to finilh the wall in 4 days, at the fame rate of working ? - - Anf, 36 men. 6. If the price of 10 ounces of bread; when the com is at 4s 3d per bufhel, be 3 4 d , what muft be paid for 2 lb 3 oz when the corn is 5s per bufhel ? - Anf 1 1 -/-yd. 7. What is the intereft of 340I for 2-J years, at 4^] per cent, per annum ? Anf 3 81 5s. 8. If 120 bufhels of corn can ferve i4horfes 56 days, how many days will 94 bufhels ferve 6 horfes ? Anf. I0 2^| days. 9. If 7 oz 5 dwts of bread be bought at 43d, when corn is at 4s 2d per budiel, what w-eight of it may be bought for is 2d, when the price of the bufhel is 5s 6d ? Anf. 1 lb 4 oz 3|44 dwts. 10. If the carriage of 13 cwt 1 qr for 72 miles be 2l 108 6d, what will be the carriage of 7 cwt 3 qrs for 112 miles i - - ■ Anf. 2I 5s nd i r VVq. 11. What is the intereft of 300I for 5 weeks, at 5I per cent, per annum ? - - Anf. il 8s io/^d. 12 If 3000 lb of beef ferve 340 feamen 15 days, how snany lb will ferve 120 feamen 25 days ? Anf. 1764 lb iiJ4 °Z' [ 39 ] RULES of PRACTICE. BY Rules of PraSrice are meant certain expeditious methods of calling up accounts : and they coniift of the moft general contractions of the Rule-of-Three, when either the fit ft or third term is i. Note i. One number is faid to be an aliquot part of another, when the former divides the lauer without a remainder. a. When the quantity concerned is not very large, and of one denomina- tion, it is commonly bed to work by the firft or fecond of the following rules; but if very large, or of feveral denomination?, ufe fome other of the Rules of Practice. RULE I. To find the price of any integer number of things. When that number is not very great, multiply the price of i, or the integer, by the given number whofe price is to be found, as in Compound Multiplication, and the pioduft will be their price required. EXAMPLES. Queflions. 1 s d 3 lb of green tea, at o 9 6 4 lb of bohea tea, at o 7 8 5 lb of fugar, at o i 3 6 lb of flax, at o 9i 7 lb of tobacco, at o i 8; 8 ftones of beef, at o 2 n 9 lb of galls, at o I 5 10 cwt cf cheefe, at 2 *7 IO li cwt of cheefe, at i *5 6 12 cwt of fugar, at 3 7 4 r 1 LIUVV s era. d I 8 6 I TO 8 6 3 4 9 ii "i I i o 12 9 28 18 4 J9 10 6 40 8 o If the multiplier exceed 12, it is commonly beft to multi- ply fucceflively by its component part?, as in Simple Multi- plication. EXAMPLES. Queftions. 1 ' s d Anfwers. 1 s d 14 moidore?, 15 piftoles, 16 cwt of cheefe, at 1 7 at 17 6 ati 18 8 — 18 18 13 2 6 30 18 8 40 PRACTICE. 1 s d 1 8 cwt of tobacco, at 511 4 20 cwt of hops, at 4 7 a 21 cwt hemp, - at 1 12 o 22 tons of hay, - at 1 2 o 25 yds of broad cloth, at o 9 2 28 yds fuperf. bd. cloth, at o 19 4 32 yds German ferge, at o 3 7 35 yds Irifh cloth, at o 2 54 40 ells of holland, at o 5 6 44 ells of dowlas, at o 1 4 48 ac of arable ground, at 2 3 o • 50 ac pafiure ground, at 1 4 6 j j tuns of wine, at 83 10 O 60 gallons of wine, at o 5 8 (4 gallons of brandy, at o 9 6 70 barrels of ale, at 1 4 o 77 firkins of beer, at o 11 7 81 firkins of foap, at 1 8 9 88qrsofoats, at on 6 96 qrs of rye, at 1 3 4 100 ftones of wool, at 7 3 j 20 doz of candles, at o 5 9 132 days wages, at o 2 4 — 1580 24 tons of hay, at 3 7 6 — 9 81 o o 27 yds fine broad cloth, at o 15 7 — 21 o 9 30 yds of fhalloon, at o 2 2-J •— 363 33 yds of flannel, at 013 — 2x3 36 yds of Scotch cloth, at o 2 9 ' — - 4 19 o 42 ells of holland, at 0.6 9} - 14 5 3 45 ells of dowlas, at o t 6 — 3 7 6 49 acr meadow, at 1 7 10 — 68 3 10 54 acr land, - at 1 13 o — 89 2 o 36 pipes of wine, at 37 7 6 — 2093 o O 63 gall cf oif, -at 023 — 719 66 gall of rum, at o 8 10 — 29 3 O 72 hbds - at 1 14 4 — 123 12 O 80 firkins of butter, at 1 5 6 — 102 O O 84 qrs of wheat, at 1 12 8 — 137 4 O 90 qrs of barley, at O 17 XO — 80 j Q 1 s d 100 4 a 87 3 4 33 12 24 4 11 9 2 27 1 4 5 *4 8 4 6 oi 11 2 18 8 103 4 61 5 459* 10 *7 30 8 84 44 n 11 116 8 9 5° 12 112 36 5 34 10 PRACTICE. ''Queflions. 1 S d s 99 buHiels of malt, at 6 3i no fheep, - . at 12 8 • 121 week's wages, •at 7 6 144 reams of paper, at 13 4 Anfwers.' 1 s d 3* 2 105 69 *3 4 4:? 7 6 96 s d 112 lb or 1 cwt at 3 4! per lb 224 lb or 2 cwt 33 6 at 7 at 1 3i P er lb 5 each 1 cwt 336 lb or 3 cwt 3J° at 1 at 3 9 per 3b 2^ per lb 2~ each 1 s 18 18 81 8 23 16 9 16 87 17 55 15 Note. The following examples require the continual product of three numbers. Queftions. ■ Anfwerf. d o 8 o o o 71 But if the multiplier cannot be exa&Iy produced by the multiplication of fmall numbers, find the neareit to it, either greater or lefs, which can be fo produced \ then after ha- ving multiplied continually by the component parts of this number, to or from the lad product, add or fubtracl: the produce of fo many as it is lefs or greater than the given ^number. EXAMPLES. Queftions. Anfwers. 1 8 d 2 s d 17 at o 5 6 ■ "" ■ ' "« 413 6 23 at o 1 6f •* ■ 115 $i 29- at 1 5 3i 65 12 ioj 34 at 19 7 33 5 10 38 at 1 11 si 59 1 S 5 41 at o 3 1 ■■■ 6 6$ 46 at o 4 7f — 10 11 9f 51 at o 6 7i ■ 16 18 n't 68 at o 9 11-J ■ ■ 33 15 9 79 at on sh 45 6 io£ 94 at o 12 2 57 3 8 ic6 at o 14 7J ■ 77 8 04 ,59 8ft o 7 13 > 23 2 2* 42 PRACTICE. Queftions. Anfwers* 1 s d 1 s d 117 at 1 2 3 ■ 230 3 3 19 at O 13 2 ■ ■ 12 IO 2 26 at O 3 0| •' ' 3 19 7f 3iato 17 si ' 2 7 * 2 * 37ato 12 ioi 23 15 7i 39 at o 7 y| — 14 5 2{ 43 at o 2 10 ■ 6 I 10 47 at o 5 2j 12 4 9? 53 at 3 15 2 199 3 10 62 at o 8 5 — 26 1 10 74 at o 10 o§- 37 3 1 86 at O O 7 — — — 2 IO 2 104 at o 13 si *>9 19 8 114 at 015 3i 87 5 74 127 at 3 o 2 — — 382 1 2 When the number is very large, as many hundreds or ihoufands \ multiply the price of 1 continually by 10 till it| come to the higheil denomination, namely, twice by 10 for hundreds, thrice by 10 for thoufands, See. \ then multiply thefe feveral produces by their reipeclive local digits in the given number 5 which laft produces place orderly under each other, and add them together for the anfwer. Ex. To find the price of 7985 at 7s lo|d. 1 s d t> 7 10J X 5 10 3 18 n£ x 8 10 39 394 9 H 1 7x9 IO 10 x 7 7 2763 3i 1 IO 6 11 *9 10 7000 3 9°° 8 80 ii 5 Anf. 3152 8 ^---7985 PRACTICE. 43 RULE IT. When there is given the price of fome certain number, to find the price of the integer, or i. Divide the given price by its number, as in Compound Divifion, and the quotient will be the price of i as required. Anfwers. 1 s d o 18 % EXAMPLES. Qucftions. 1 s d If 3 lb coil 2 14 6, what is 1 '? If 4 coft 7 * 2, what is 1 ? If 5 coft 2 17 IO, what is 1 ? If 6 coft J 7 3 0, what is 1 ? If 7 coft 2 13 4h what is 1 ? If 8 coft 57 * 2, what is 1 ? It 9 coft J 9 3 <%> what is 1 ? If IO coft 121 5 8, what is 1 ? If ii coft 6 14 * what is I ■? If 12 coft 27 18 7, what is 1 ? When the number or divifor exceeds 12, it is beft to divide fuccefrively by its component parts, as in Simple Di- vifion. Queflions. Anfwers. 1 s d If 16 cwt coft 30I 18s 8d, what is 1 ? - 1 18 8 If 22 cwt coft 24I 4s, what is 1 ? - - 120 Divide 5I 14s 8d by 32. - - - o 3 7 I)ivide 2l 1 8s 8d into 44 equal parts. - 014 Divide 459 2l ios equally among $5 pcrfons. 83 10 o If 70 be 84I, what is 1 ) - - 140 Ef 88 coft 50I 1 2s, what will ibe? - 0116 )ivide 34I ios by 120. - - O 5 9 f 18 cwt coft iool 4s, what is 1 ? - 5 11 4 ff 27 cwt coft 21I 9d, what is 1 ? - - o 15 7 Divide 4I 19s by 36. - - - 029 Divide 681 3s iod into 49 equal parts. - 1 7 10 Divide 7I is 9d equally among 63 perfons. 023 it 80 be 1021, what is 1 ? - - 156 f 99 coft 31I 2s io§d, what will 1 be ? o 6 34 Divide 45I 78 6d by 121. • - 076 E 2 44 PRACTICE, Note. Tho following examples require three ^ivifions. At 18I 18s per cwt, how much per lb f Divide 8il 8s 8d by 224. At 9I 1 6s per cwt, how much per lb ? Divide 55I 15s 7^d by 350. But if the divifor cannot be produced by the multiplica- tion of fmall numbers, you muft divide by it after the man- ner of lon£ divifion. 1 o o o o d 9 2* EXAMPLES. 1 s d 1 s d 17} 4 13 6 (o 5 6 39) 65 12 io£ (2 5 si 41) 6 6 5 (o 3 1 86) 2 10 2 (o o 7 1 s d 1 s d 19") 12 10 2 (o 13 2 37) 23 15 7i (° 12 I0 I 53)199 3 10 (3*15 2 127)382 1 2 (3 o z KULE III. If the given fir tee of 1 or the integer he an aliquot pert of a penny, pAUing, or pound, take the fame part of the given, quantity whofe price is to be found (by dividing it by the number of times which the given price of 1 is contained in a penny, (hilling, or pound) for the anfwer in pence, (hil- lings, or pounds, refpeftively. A Table of the Aliquot parts of Money. 8 d i IO is f 6 8isf 5 is -J 4 ois t 3 4,»t 2 6 is f 2 is T V 1 8 is T V 1 4" Vt d 3 o 10 8 7i 6 5 4 3* 1 r T(J 1 IS -5-5- 18 4V *^ S ori or f 3 2i I or t 1 X '« t4^ or £ |s t?^ or f [ 8 7^5 ° r T-2T | s ^S"S or T3| 5 ?£■ ■ - ■■■* 2 8 10* 2cwt2qr - at 3 14 3 ■ > 9 5 7i 2 qr 2 lb - at o 19 7i . ■■ ■ ■ o 10 if 3 cwt 22§ lb at O 13 5I V ■■ 232 2qri8£lb - at o 15 2^ o 10 oi 3 qr i^i lb at 2 16 10 ■ 2 8 nf 7 cwt 2 qr 15^ lb at 3 o 7 ■■ 23 2 9 3 ton 5 c 2qr «. at) 9 3 per ton 24 8 9t 17 lb 5 oz 14 dwts at 3 6 9 per lb 58 6 5$ 15 lb 2 oz 5 dwts at 4 7 o — — — 66 1 3^ 7 oz 15 dwt 12 gr at o 6 3 per oz 287 5 oz 6 dwt 17 gr at o 5 10 * 1 11 if 3 yds 1 qr - at o 17 6 per yd 2 16 io£ 4 yds 2 qrs 3 nls at 1 24 — — — 5 4 8£ 1 qr 2 nls - at 1 12 6 012 l\ 32 ac 1 ro 14 pis at 1 16 o per ac 58 4 ij 14 ac 3 ro 5 pi - at 2 12 10 -*— — 39 o 11J 3 gal 5 pts - at o 7 6 per gal 1 7 2j I2gal3pts - at o 5 8 — «— - 3 10 14 ;?2XE VII. If the price he any even number of Jhiilings : multiply the quantity by half the number^ doubling the firft figure of the product for fhillingsj the reft are pounds. EXAMPLES. Queftions. Anfwers, s Is 173 at 2 >• * 17 6 259 at 4 51 16 703 at 6 21 c 18 5013 at 8 i — 2005 4 £72 at 10 — s — 436 o 460 at 12 276 o v 627 at 14 438 18 598 at 16 478 8 214 at 18 — 4-4 192 33 PRACTICE. 49 RULE VIIL When the price is any odd number of fmllings: work for the greateft even tuimber contained in it, by the lafl rule ; and for the other {hilling, take T yh of the given quantity, as in xule I. Or, multiply by the number of (hilling?, and divide the product by 20 to reduce it to pounds. EXAMPLES. Que (lions. Anfwers- s 1 s 732 at 3 ■ ■■ ■■ ■ ■■ » 1:9 16 HI at 7 5 r 9 371 at 9 ■ ■■■■■- ■ 166 19 586 at 11 — 322 6 240 at 13 ■ 156 a 6j2at 15 — ■ ■ " 489 o 897 at 17 762 9 IC46 at 19 ■■ 993 14 RULE IX. If there be a fra&ion in the given quantity: after having worked for the integral part by any of the former rules, find the produce of the fraction, by multiplying the price by the numerator, and dividing the producl by the denominator 5 then add them together for the anfwer. EXAMPLES. QuefUons. Anfwers. 1 8 d 1 S d 273J at o 3 &c. 53 £80 *£ Sir Jeffery Slingftone, Sept. 8th, 1783. A Punch Bowl, \vt A Tankard, A Tea Pot and Lamp, 6 Plates, 18 Spoons, Bought of Samuel Silverfraith. oz dwt gr 23 4 o at 5 10 3° 73 4* 6 12 5 10 d 10 per 02 2 - - 3 - - 1 - - 3 - - £56 * 4 Mr. George Davies, Bought of Champion Chcefemonger* Sept. 6th, 1783. cwt qr lb 13 Chefhire Cheefes, wt 5 3 12 at 1 15 Gloucefter ditto, 3 o 18 - I 47 Stilton, ditto, - I 2 5 - 2 17 lb of Cream ditto, o q Flitches of Bacon, wt 53 ft 3 lb o Firkins of Butter, - ' - 1 I5i s 12 8 4 o 4 8 d 6 per cwt o - - 8 - - 7§ per lb 8 - ft o each £5* ° 9? [ 54 3 TARE and TRET. rf^ROSS weight of any commodity is its own weight, ^-^ together with that of its package, whether it be calk, eheft, or any thing elfe. Tare is the weight of the package, or an allowance made inflead of it. — What remains after the tare is taken from the grofs, may be called tare-futtle, if there be more deductions. Tret is an allowance of 4H) upon every 104 lb of tare- futtle, on account of dull or other wafte. — What remains after tret is deducted, may be called tret-futtle, if there be any following deduction. Cioffis an allowance of 2 lb for every 3 cwt, and fome fay for every 100 lb of tret-futtle, to make the weight hold good when fold by retail. When all the deductions are made, the lad remainder is called neat, or net weight. Note 1. When the tare is at fo much per cwt, it will be beft to divide it into aliquot parts of it, like as in the Rule of Practice. 2. The tret being 4 to 104, or 1 to 36, will be found by taking the 36*th part of the tare futtle. 3. In calculating oil and fpirit?, 7^ lb neat are allowed to the gallon. EXAMPLES. I. Grofs 17 cwt 3 qr 14 lb, tare 12 lb per cwt, tret 4 to J04, and cloff 2 to 100, or 1 to 50. How much neat ? lb I 1 26 cwt *1 qr lb 3 J 4 g rofs 8 = 4 = I O I 1 3 2 isi 3 i8f tare 15 O 3 23f tare-futtle 2 I2| tret 15 1 1 of tret-futtle 1 6J cloff 15 4f net TARE AND TRET. fig 2. What is the neat produce of 30 barrels of anchovies^ weighing each 36 lb grofs, allowing 8 lb per cwt tare > Anf. 1002I lb. 3. Grofs 12 cwt 14 lb, tare 1 cwt 2 qr 18 lb, how much neat ? - Anf. 10 cwt 1 qr 24 lb. 4. Suppofe 3 cwt 1 qr 5 lb tare were allowed on 71 cwt 3 qr of tobacco, what would the neat weight be ? Anf. 68 cwt 1 qr 23 life 5. In five chefts of fugar, weighing 112 cwt 1 qr grofs, how much neat, allowing 121 lb tare ? Anf. in cwt 19 lb. 6. In 26 bags of hops, containing 73 cwt 3 qr grofs, tare 10 lb per bag, how much neat ? Anf. 71 cwt x qr 20 lb. 7. What is the neat weight of 20 barrels of figs, each 3 cwt I qr 5 lb grofs, tare 14 lb per barrel > Anf. 63 cwt 1 qr 16 lb, 8. In 15 hhds of tobacco, each 2 cwt I qr 12 lb grofs. tare 1 qr 4 lb per hhd, how much neat i Anf. 3 1 cwt 8 lb. 9. What is the neat weight of 3 barrels of indigo, each 3 cwt 2 qr grofs, tare iof lb per cwt ? Anf. 9 cwt 2 qr i£ lb. 10. What is the neat weight of 4 hhds of fugar, weighing as under, qr lb f3 ^ 14^ 72 3 1 ish U 3 263 S rof * 'il 2 tS r » tarc °* ^ c w h° lc I cwt 3' qr 5 lb. Anf. 12 cwt 4 lb. 1. 2. 3- 4- £.132©. r 4 J Anf, 16 cwt 3 qr 24 lb* F 2 II. Five calks of raifins, wt. viz. cwt qr lb lb 3 2 1 2 tare 18] 1 2 4 3 1 9 * '7 • 16 23 1 ■•, ho w much neat ? 5 1 3 8 . 29 - 27 r '4j \ Anf, if $6 YULGAR FRACTIONS. 12* What is the neat weight of the three following lots af wormfeed ? viz. cwtqr lb N i. 32 8— tare 12 lb each. 2. 2 3 26 3. 3 1 15 Anf 9 cwt 2 qr 13 lb. 13. In 15 cwt 3 qr 14 lb grofs, tare 13 lb per cwt, and tret ^lb per 104 lb, how much neat ? Anf. 13 cwt 1 qr 27-5- lb. 14. Suppofe 17* lb per cwt tare, and 4 lb per 1041b tret, were allowed on feven cafes of prunes, each 3 cwt 1 qr 5 lb grofs y what would the neat weight be ? i Anf. 18 cwt 2 qr 24 lb. 15. What is the neat weight of 3 hhds of fugar, weigh- iag as follows ; the Brit^ 4 cwt 5 lb grofs, tare 73 lb ; the fecond, 3 cwt 2 qr grofs, tare $6 lb \ and the third, 2 cwt 3 qr 17 lb grofs, tare 47 lb, allowing alfo 4 lb per IO4 lb tret ? - - Anf. 8 cwt 2 qr 4 lb. 1-6. In four calks of currants, each 7 cwt 1 qr 12 lb grof3, tare 2 qr 10 lb per cafk, tret 4 ib per 104 lb, and cloff 2 lb per 100 lb •> how much neat ? Anf. 25 cwt 2 qr ij lb. 1.7. In 23 cwt 3 qr 7 lb grofs, how much neat, allowing X qr 3 lb per cwt tare, 4 lb per 104 lb tret, and 2 lb pe* 3C0 lb cloff? - - Anf. 16 cwt 1 qr 22J lb. 18. In 17 cwt 17 lb gfofs weight of galls, how much neat, allowing 18 lb per cwt tare, 4 lb per 104 lb tret, and 2 lb per 3 cwt cloff? - Anf. 13 cwt 3 qr j-J- lb. 19. In three cafks of oil weighing as follows : N° 1. 3 cwt 17 lb, N° 2. 2 cwt 3 qr 5 lb, N* 3. 4 cwt 1 qr 17 lb, how many gallons, allowing 18 lb per cwt tare, and 7 1 lb neat to a gallon i - • Anf. 1 29^ gal.. 20. In 7 cafts of oil, each weighing 3 cwt 1 qr grofs, how Stony neat gallons, allowing 20 lb per cwt tare, and 7^ lb jer gallon ? - * - Anf. 279/5- lb. VULGAR FRACTIONS. A FRACTION, or broken number, is an expreflion of one or more parts of any number. The number of parts into which the number is fuppofed to be divided, is called the denominator; and the number of thofe parts cxpreffed by the fra&ion, is called the numerator* Alfo VULGAR FRACTIONS. 57 thefe two numbers are in general named the terms of the fraction. If the number of which the fraction is part, or parts, be I, it is called zjimpie fraction \ and is denoted by the nu- merator written above the denominator with a fmall line be- tween them : So, \ denotes one-fourth of I j £ denotes three- fifths of I. But if the number be different from I, the fraction is called a compound one, and is denoted by the word o/i and the number fubjoined to the numerator and denominator ex- preffed as before : So, J of 6, denotes one-fourth of 6 j §• of 8, denotes three-fifths of 8 $ and f of £, denotes two- thirds of three-fourths of 1. Simple fractions whofe numerators are lefs than their de- nominators are called proper fractions.— And thofe whofe numerators are equal to or greater than their denominators, are called improper fractions; The expreflion formed from an integer and a fraction join- ed together, is called a mixt number. Note 1. A fraction having a fraction or mixed number for its nume- rator or denominator, or both, is by fome called a complex Fraction. 2. A whole or integer number may be espreffed like a fraction by writ- ing 1 under k for a denominator: So 3 maybe denoted by .£-, and laby x ^?. 3. A fraction denotes divifion, and its value is equal to the quotient obtained by dividing the nunfreratdr by the denominator : Thus * 2 is e- qual to 3, and 2 _° equal to 4. 4. Therefore, if the numerator be lefs than the denominator, the value of the fraction is lefs than 1. If the numerator be the fame as the deno- minator, the' fraction is juft equal to 1. And if the numerator be great- er than the denominator, the fraction is greater than i. Befides thefe, 00 is written between two numbers, to denote their differ* ence when it does not appear whetherofthera is the greater; as !£ ^ i%V* Note 6. If both the numerator and denominator of a fraction be mul- tiplied or divided by the fame number, the fraction will itill retain its original value. Let |. and T 8 T *be two fractions propofed : then ix|== 4s » and A'-f- ^—T* That is, if the numerator 3, and denominator 5, of the firft fraction, be each multiplied by the fame number 2, the produced frac- 5« REDUCTION OF VULGAR FRACTIONS, tion t^", is equal to the propofed one |- : For the numerator and dfr- initiator of the produced fraction, are in the fame proportion as the nu« rator and denominator of the propofed one. Alfo, ff the numerator S, and the denominator n, of the fecond fra&ion, be each divided by the fame number 4, the fractions £ and ^, are equal for the fame reafon. By this ufeful note, feveral fractions of different denominators may be readily reduced to a common denominator. Thus $• may be reduced to the fame denominator as * by multiplying its terms by 3 by which it Becomes A. Alfo .*., £, a ^ _* may be reduced to a common deno- minator, by multiplying the terms of the firft fraction by 6, of the fecond by 3, and dividing thole of the laft by 5. REDUCTION OF VULGAR FRACTIONS. I. To abbreviate or reduce fraBions to lefs terms. RULE I. IVIDE the terras of the given fradion by any number D which will divide them without a remainder, fo (hall the quotients be the terms of a new fra&ion, equal in value to the former j and this you may abbreviate again, and the next again, and fo on, till it appear that there is no number great- er than I that will divide them> in which cafe the fraction is faid to be in its leajl terms. EXAMPLES. Let 44 be propofed to be abbreviated. A 2 21 *7 — n-gzr— , by dividing frrft by 2 and then by 3. Reduce 44 to its leaft terras. Reduce y/ to its kail terms. Reduce £££ to its lead terms. Note 2, Any- number ending with 5 or 0, is divifible by 5. Note I. Any number ending with an even number, or a cipher, may be divided by 2. 9 EXAMPLES. — — 3 — jL — % 24~~"i2~~ 6 ~~ 3 I20__ 236 "^ EXAMPLES. 40 8 100 REDUCTIOH OF f VLGAR FRACTIONS. 59 Note 3. Any number is divifible J by 3, if the fum of its digits be fo : Thus 417 is divifible by 3 ; becaufe 12, which is the fum of 4, i, tnd 7, isfo. EXAMPLES. 120 40 8 116 Note 4. If there be any ciphers at the end of each, cut off as many as are common to both* EXAMPLES. 200 _ 20 ^ 10 1200 __ 18000 ""■ 411 534 = 67a Note 5. When any number which is expreffed by feverai others with % the fign of addition or fubtraction between them, is to be divided by any number ; then all the parts of it mult be divided by this number. Thus 4 + 6 — 8 24*3 — 4 = 5 — 4=*< Note 6. But if the given number be expreffed by others with the fign of multiplication between them, only one of them rauft be divided: So 7X3X8XIO _ 3X4XIO __ 1X4X10 ^ 1X2X10 " 20 __ 7x2x6 ~~ 1X6 1X2 "~ IXI *" I -" And in this cafe, when the fame number h in both the numerator and denominator, it may be left out of them N. B. But inft^ad of writing down the whole fraction anewafter every c divifion in the abbreviation, as is here done in this laft example in five" different values one after another in the whole length of the line, the fliorteft method, and which is conftantl^praclited by thofe who belt know the value and ufe of this part of the abbreviation, is thus ; what- ever numbers are abbreviated, or which cancel one another, draw a fmall line through them with the pen, and place the quotients of the: numbera above them or below them, according as they are in the numerator o* denominator. So the foregoing, fraction being here again written down, da(h or draw a fmall line through the two 7*s; becaufe they cancel each other, as being equal ; then dafh the 3 and the 6*, 2 and write tl?e quotient 2 below the 6; next ^ dafh the 8 and a 2, writing the quotient 4 2®» aam me o ana a 3, wriung sap quuuenc 4 * s/ 6 \/ above the 8; laftly, dafh the 4 and thee- * x 3* x ? X I g . ther 2, writing the quotient 2 ibove the 4: 4 x i X # there being then no more figures to divide v fey, multiply together the numbers a and ^ 10, which are not dafhed, and the product 30 is the value of the fraction required. — And in this manner let the pu» ptt be exercifed in many examples in this rule till he is perfect in it? be* tfo aEBUCTION OP YULGAR FRACTIONS. caufe it is of the greateft ufc of any by abbreviating all arithmetical ©* perationa wherever multiplications and divisions are concerned. Where this method is deviated from, in any of the following pages of this book, it is not to be underftood as done through choice, but only to avoid the trouble and difficulty of procuring the types to exprefs thefe operations properly ie print. EXAMPLES. 3X 7X9 __ 3X2X14"" 5X2X6 __. 3x5x2 ~~ 7 x 18 X4Q X9 _ 10x9x7x6 ~" RULE Ti Xfthefra&ion mujl be brought to its leqfi terms at one du vifion, divide its terms by their greateft common meafure, which common meafure is found by dividing the greater term by the lefs, and this divifor by the remainder 5 and fo on, always dividing the laft divifor by the laft remainder^ till o remain $ then is the laft divifor the greateft common meafure fought. EXAMPLES. Reduce |4* t0 lts leaft terms at one divifion. Firft, 246)372(1 246-7-6 41 126)246(1 Then =— 120)126(1 372-7-6 62- ■' The common meafure 6)120(20' Reduce -J?|- to its Jeaft terms. Reduce 4if to its leaft terms. Reduce J4ff to its leaft terms, II,' To reduce an improper fraclion to its equivalent whole 0f mixt number* RULE. Divide the numerator by the denominator, and the quo* ftient will be the integer or mixed number required. EXAMPLES. V=4* I V =2f J 4 — 2 S t — ,. •3" — I FT — REDUCTION OF VOLGA* FRACTIONS* 6i III. To reduce an integer to an equivalent fraB'ton of a given denominator. RULE. Multiply the integer by the given denominator, and the product will be the numerator required. EXAMPLES. Reduce 7 to a fraction whofe denominator (hall be 4. _7 X 4__ 28 7 ~7"4 - 4 - Reduce 5 to a fraction whofe denominator (hall be 9. Reduce 13 to a fraction whofe denominator ihali bs 12. IV. To reduce a mixt number to an equivalent improper fratlion. RULE. Multiply the integer by the denominator of the fraction ; to the product add the numerator ; then the fum written a $ove the denominator will form the fraction required. EXAMPLES. Reduce 2y to a fraction. ^i.~ 12L1±JL-L±±£~11 7 . 7 r Reduce 12-J to a fraction. Reduce 14-^ t0 a fraction. V. To reduce a compound fraBion to an equivalent Jimple one* RULE. Multiply all the numerators together for the numerator, and all the denominators together for the denominator of the iimple fraction required. Note. If part of the compound fraction be an integer or a mixt num- ber, reduce it to a fraction by one of the former cafes. EXAMPLES. Reduce \ of -| of % of 5 to a fimple fraction. * r2 '3 -5 1x2x3x5 ,, . . . — of — of-2-of-^-n 2 — i = (fay omitting th& 2 3 4 1 2x3 X4X 1 v • 6 common terms 1, 2, and 3) £• Reduce | of 4 of 3* to a fimple fraction. Reduce «§- of | of i of 4 to a fimple fractions 6% REDUCTION OF VULGAR FRACTIONS. VI. To reduce fra&ions of different denttmnQtcrt to equina- lent froclions of a common one* RULE I. If the fra&ions can be conveniently reduced to a common denominator, by multiplying or dividing their terms, ac- cording to note 6, page 57, proceed by that method \ but if not, multiply each numerator continually into all the de- nominators, except its own, for each new numerator: and multiply all the denominators together for the common de- nominator. Note. It is evident, that in this and feveral other operations, when any of the proposed quantities are integers, mixt numbers or compound frac- tions, they mull be reduced by their proper rules, to the form of fimple- fractions. EX4MPLKS Reduce *-, f, and J to a common denominator* Thus — , — , and — — — , — , and — . jx y 4 24 24 24 6 8 9 or= — , — , and — -. 12 12 12 Reduce 4 and J- to a common denominator. Reduce £, J-, and 5-$-' to fra&ions of a com. denom* Reduce 4, 2^, and 4 to fraclions of a com. denora. RULE II, If the denominators of two given fraclions have a com- rnon meafure, conceive them to be divided by their grcatevV common me afore } then multiply the terms of each given fraction by the quotient arifrng from the other's denominator. EXAMPLES, Reduce £ and •£? to a common denominator. tx 7 j 4 7X5 j 4 X 3 35 j I2 Here -*- and — = -* — ± and -? — ^ =^ and — . 9 iS 9*5 ^5X3 45 45 Note. In this laft example, and thofe of the two following rales, the forms [1 - and-- ■■) are printed only to ftiew which numbers or ^9X5 15X3/ ^ quotients are ufed in multiplying the terms of the fra&ions ; but I think it quite needlefs for the pupil to write down his example! in this way ; and I would advife him barely to write down fuch an example as the above; thu« : «-£- and — — — and — - ; and fo of others. 9 *-5 45 45 REDUCTION OF VULGAR FRACTION*. *J Reducs T %- and -3%" to a common denominator. Reduce T 5 ^ and -£ r to a common denominator. RULE III. If the lefs denominator of two fractions divide the great- er, multiply the terms of that which hath the lefs denonwnau tor by the quotient. EXAMPLES. Reduce -| and T 5 ^ to a common denominator. „ 2 .5 2x4 .5 8 .5 Here — and — = and — = — and — J 12 3X4 12 12 12 Reduce £ and -3%- to a common denominator. Reduce £ and TfV to a common denominator. When more than two fractions are propofed, it is fome- times convenient firft to reduce two of them to a common denominator, and then thefe and a third, and fo on till they be all reduced to their leafl common denominator. EXAMPLES. Reduce T ' T , ^, and T 5 ^ to their lead common denom* By rule 3, we fhall have — , -~, and — , =: J ° 12 8' 24 1x2 7x3 5 2 21 5 nr? 8 x^ and Tf-Tf tj .^4 Reduce J- VV> a!1 ^w to tnc * r * ea ft common denom. By rule 3 , we have J-A and -2- 55 3M J., and £- /k, i , ^4X3 4X4 j 9X3 36 16 .27 (by rule 2) ? — 2 — 3 J! — Z an d ii—^ — ^ an a -£. 5x4x3 15x4 20x3 6060 60 Reduce -f> t^> and -/-g- to a common denominator. Reduce -|, J^ and T ^ to a common denominator. Reduce £, £, and 5J to a common denominator. H REDUCTION OF VULGAR FRACTIONS. VII. To find the value of proper fractions in numbers of in* ferior denominations* RULE. Multiply the numerator by the integer, and divide by the denominator. EX 4 MPLS S. t* What is the £of 21 6s ? 2l 6s 4 5)9 4 Anf. il 1 6s pd 2lq. 2. Requiredthenralueof|ofiI, 2 20 3 ) 4Q ( *3 S 4<*, Anf. 1 12 12 (4d. Required the value of -fl. What is the value of T %1 ? What is the value of -| of a guinea ? What is the value of ■£ of a (hilling ? What is the value of ■£? of 9s iofd ? 8. What is the value of ^ of a lb troy ? 9 10 Anf. 7s 6d. Anf. 6s id 3tt^* Anf. 4> 8d. Anf. 9d i|q. Anf. is 3d 3fq. Anf 9 oz. What is the value of |^>f a lb avoirdupois ? Anf. 12 oz. What is the value of T \ of a cwt ? Anf. 1 qr 7 lb. II. What is the value of £ of 3 cwt 1 qr 14 lb ? Anf 3 qr 24 lb. What is the value of St of a mile ? Anf. 1 furlong 16 pis 2 yds 1 f 9^- in. What is the value of £ of a yard ? Anf. 3 qr 14 nl. What is the value of ^ of an acre ? Anf. 1 rood 2^§ pis. What is the valus of -£ of a ton of wine ? Anf. 3 hhd 31 gal 2 qrts. 12. *3- 14. 16, What is the value of ^ of a hhd of ale ? *7< 18. *9 Anf. 6 gal 3f qr. What is the value of 4 of a quarter of corn ? Anf. 4 bum 1 pec 1 gal 2^qr. What is the value of T V of a day ? Anf. 7 hrs 12 min. What is the value of 4- of a month ? Anf. 2 we 6 ds# 20. What is the value of 3 of an ell Englim ? Anf, I qj nl*. REDUCTION OF VULGAR FRACTIONS. 6$ VIII. To reduce fra&ions to other equivalent ones of a different integer ; a certain number of the lefs integer being contained in one of the greater. JIULE. Coniideriiow many of the lefs integer make one of the greater \ and by that number multiply the numerator, if the xedu&icn be to a lefs integer,, or the denominator, if to a greater. EXAMPLES. Reduce -Jl to the fraclion of a (hilling. 2 2X 20 _40 Reduce ^°s to the fraction of a 1. i2 s= _4^ 1= i. 1 . 9 9 x 20 9 Reduce T 2 T 1 to the fradion of a penny. Reduce 32d or yd to the fraction of a I. Reduce -^fl to the fraction of a farthing. Reduce x %f° of a farthing to the fraction of a 1. Reduce 4 cwt to the fraction of a lb. Reduce 32 lb or V 1° t0 tne fraction of a cwt. Note. If a compound whole number be propofed, reduce it all to the lowed denomination mentioned in it, and proceed as before. EXAMPLES. Reduce 7s 3d to the fraction of a 1. - Anf. |-g-l. Reduce 2|d to the fraction of a (hilling. Anf. Jf s. Reduce 3 qr 14 lb to the fraction of a cwt. Anf. •£ cwt, IX. To reduce fr a B ions to equivalent ones of a different integer >• when a certain number of the lefs is not exaftly contained in the greater. RULE. 1. By the laft, reduce the given fra&ion to an equivalent me of fuch an integer, of which a certain number are con- ained in the integer to which the fraction muft be brought, yv which, (hall contain a certain number of this, G 66 ADDITION OF VULGAR FRACTIONS, 2. By the laft alfo, reduce this fraction to an equivalent one of the integer required. EXAMPLES. Reduce \ of a 1 to the fraction of a guinea. 2 . 2 X 2Q 2 X 20 . AO — Irr ■ s=: ^ui ~ - — guinea* 7 7 7***; *47 6 Reduce |- of a crown to the iraclion of a guigea. .tt.ni. T6ff a Reduce V5V of a guinea tr the fraction of a 1. Anf. -* -I. Reduce -£ of a half-crown to the fraction of a (hilling. Anf. |4 or 2iV?« Reduce 2 r T T s to the fraction of a half-crown. Anf. £ of a half-crown. ADDITION OF VULGAR FRACTIONS, RULE. REDUCE, compound fractions to fimple ones, and all to the fame integer and denominator, if they be dif- ferent ^ then add all the numerators together, and fet.the fum over the common denominator, for the fum of the frac« lions required. Note. 'When feveral fra i + 7* + * of i = 4 + 7t 4- i = i + 7f + t = ^y ss 8| the fum. 2. What is the fum of £ and £? - - Anf. » 3. What is the fum of \ and ^ ? - Anf. ^ 4. What is the fum off, -j, and * ? - Anf. l||j 5. What ie the fum of -|, t> an d 2f ? - Anf. 3^ 6. What is the fum of £, £, of f , and cj^V ? Anf. 10^3 7. What is the fum of £ of a pound, and 4 of a milling > Anf. *| 5 s, or 13s lod 2|q SUBTRACTION OF VULGAR FRACTIONS. ^ ft What is the fum of |s and * 4 7 d ? Anf. Vx a d, or 7 d ij|q. 9. What is the fum of f 1, f s and T ^d ? An! >l 3s id |4§q. 10. Suppofe that I have | of a (hip worth 15O0I, and tfcat I buy another perfon's (hare ef her, which 11 y-- ; what part of her belongs to me then, and what is it worth ? Anf. 1 have i£, and it is worth 103 tl 5s. SUBTRACTION OF VULGAR FRACTIONS. RULE. 'I^HE fame preparations being made here as in Addition, **• take the difference of the numerators and fet it over nmon denominator, for the difference of the fractions . In fubflrt&ing mixt number*, when the fraction in thf. \ hw<\ it] inuend, fubtratt the numerator of the r*d from the denominator, uueia- (6r of tiie miuuend, and cai in tlie lubtrahtnd. What is the difference between | aiul - C - , =- 4 - = -l Anf. b 6 " 6 6 3 X W r hat is the difference between H and \\ > ii te ii = *ii-!i2 Anf. 7 1 17 17 X 22 374 5, W hit is the difference between yV *"& ft ' Anf. !• 4. What is the dirt*, bttwecfc ,', a,,ti A ? Anl 51 Wli.it it the diff. between Ani 0. What is the ditV. between $\ aid ? of^ ? Anf 4, 7. A* hat is the difT. betwc 1 1 and j Of 4 of 1 ihiU Anf. V^t| or ics 7d i ;q. 8. What is the diff. between \ of 5J] and j of a (hill ii Am 68 MULTIPLICATION OF VULGAR FRACTION. 9. Suppofe that I have £ of a (hip which is worth 900!, and that I fell f of my fhare \ what part of her have I left, and what is it worth ? - AnX -i % 5 and worth 187I 10s. MULTIPLICATION OF VULGAR FRACTIONS. REDUCE mixt cumbers, if there be any, to fractions ) then multiply all the numerators together for the nume- rator, and multiply all the denominators together for the denominator of the product required ? ■ Note. A fra&ion is Wit multiplied by an integer, by dividing the de- nominator by it if poflible, but if. tbai cannot be done, multiply the nu- merator by it. EXAMPLES. 1. What is the product of -J, 3 J, 5, and | of £ > 2 3 r3 P^X ISX/X^X 3 .. _ : — X3IXCX- 1 o£-2- = - 3 * ■ * — ?:=£?2=4J Anf. 3 4 5 2x4x4 x/ 8 2 2. What is the product of t and | ? " . Anf. -/ T . 3. What is the prpdoct of VV an ^ t?z • - Anf. T V 4. What is the product of £, f, and 44 ? - Anf. Z 8 T * 5. What is the product of 4, f, and 3 ? - Anf. r. 6 What is the product of T * and 7 ? - Anf. t£i 7. What is the product of if I, and 4/^? - Anf. 2/< ie product of -J, |, and 4/^? e product of J, and •§, of ~ > *3o* 8. W^hat IS the pruuuci oi -^ *nu -5-, ui -j r . • nni, T 9. What is the product of 5J- and 9 ? - Anf. 48. 10. What is the product of 6, and •§- of 5 ? - Anf. 20* 11. What is the product of -J, of £, and ■§• of 3^ ? Anf. f J. 12. What. is tta product of 34* and 4I4 ? Anf. 14^! £. 13. What is the product of 5, £• t* of i> aod 4 J ? Aaf. 2 ff r- H e 69 1 DIVISION OF VULGAR. FRACTIONS. RULE. AVING prepared the terras as in Multiplication, take the quotient of the numerators, and of the denomina- tors, if they will exa&ly divide, for the numerator and de- nominator of the fraction required ; but if that cannot be done, multiply the dividend by the reciprocal of the divifor, for the quotient required. Note 1. By the reciprocal of a fraction, is meant the fraction got by inverting its terms ; So the reciprocal of * j s ^ 9 and of 5 or ^- is «. Note z. A Ara&oii is divided by an integer, by dividing the numerator by it, if poiiible ; but if it will act exactly divide* then multiply the de- nominator by it* . EXAMPLES. 1. What is the quotient of V by | ? * 9 3 9*3 3 2. What is the quotient of £ by T 2 r ? 9*3 9 2 # x * 6 3d 3. What is- the quotient of 4f kj T.? - - Anf. f. 4. What is the quotient of T V by I ? - Anf. T V, 5., What is the quotient of V 4 by I ? - Anf. if. 6. What is the quotient of £ by V ? - Anf. T V. 7* What is the quotient of ]4 by | I - Anf, £ 8.' What is the quotient of | by =f ? - Anf 4^. 9. What is the quotient of ^ by 3 ? - Anf T \ % 10. What is the quotient of f by 7? - Anf. T 3 r . SI. What is the quotient of 5 by -^ ? * Anf. 74. 12. What is die quotient of 7^ by p| ? - Anf. j£. aj>. What is the quotient of f of J by 4 of jf ? Anf. T ^ T . G3 t 10 1 RULE-OF-THREE IN VULGAR FRACTIONS. RULE. TJTAVING made the neceffary preparations for Multipli- ■*--■■ cation, multiply continually together the 2d and 3d terms, and the reciprocal of the lft, for the anfwer. EXAMPLES. 1. If 4 of a yard of velvet coil Jl, what will T *y of a yard colt ? JL : ±. :: 4 : £^ii = J-l = 6 s 8dAnf. 8 s 16 ^x/x 3 3 Note. In this folution, having represented the 4th terra in the form of a compound fraction, by erpreffing the continual product of the ad term, and 3d term, and reciprocal of the ift, as muft always be done, then let the terms be always abbreviated as much as pomble, before the actual multiplication and divifion ; fo here the one 5 cancels the other, and the 35 cancels the 2 and the S, leaving only one third for the fimple value of t the 4th term required. 2. What will 3! oz of fiiver coft at 6s $d an ounce ? Anf. il is 44 d. 3. If T ' 7 of a fhip be worth 273I 2s 6d, what is ^ of her 'Worth? - Anf. 227I X2s id, 4. What will I3flb coft at the rate of 17H per cwt ? Anf. 2I 3?i+$* 5. What is the purchafe of 1230I bank-ftock, at 108 J per cent. ? Anf 1336I is 9d. 6. What is the intereft of 273I 15s for a year, at 3^ per cent.? - - - Anf. 81 17s 1 i|d. 7. If -J of a (hip be worth 73I is 3d, what part of her may I buy for 25 ol 10s ? - - Anf. 4 of her. 8. What mull be paid for 5f oz, at the rate of 5^s per lb troy ? , - - Anf. 2s 6^. 9. How much India-flock may be bought for 30411 2s 3d, at 172^1 per cent, ? . Anf. 1760I 8s 2d sUH* 10. What does the commiffion of 530I 2s 9d amount to at as 6d per cent. ? - * Anf. 13s $J&d* RULE-OF-FIVE IN VULGAR FRACTIONS, 7 1 11. How much Flemith money mud be given for 273I 6s 8d fterling, at the rate of 34s 6d Fiemifh per pound fteriing > Anf. 47 il ios, 12. How much South-fea-ftock, at ili|- per cent, will iooool purchafe ? - . - Anf. 8978I 13s 6-^ 6 T d* 13. How much fterling money muii be given for 47 il 105 "Flemim, at the rate of 34s 6d Flemifh for each pound fter- ling > - Anf. 273 1 6s 8d. RULE-OF-FIVE IN VULGAR FRACTIONS. RULE. HHAKE the continual product of the three laft, and the ■*■ reciprocals of the two firft terms, for the anfwer re* quired. EXAMPLES. 1. If 2I 10s be the wages of 15 men for 6 days, what will be the wages of 12 men for i8| days. - 1 j men 7 .Xi : : I I2men 7 :l2X SA x ± x 6 days 5 2 \ V days 3 3 2 — X -r- = , JJ . — ~; = ~ 1 =? 61 2S 2d 2jq Anf. 3 Note. In this folution, having put the fourth term required into the form of a compound fraction, it is abbreviated thus : The 12 cancels the 2 and the 6, and the 5 divides the 15, the quotient 3 being placed below it; then there is only 55 left for the numerator, and 3 to multiply by 3, producing 9 for the denominator of the fimple fraction required. And thus let every fuch queftion be managed, viz. put the terms of it into the form of a compound fraction, the multipliers all in the numerator, and the divifors in the denominator, after which let this compound fraction be abbreviated as much as pofiible. 2. What is the intereft of 350I for 18 months, at 5 per cent, per annum ? Anf. 26I 5$» 3. If I pay 16s 4d for the carriage of 5^ cwt 20 miles, what mull be paid for the carriage of iy| cwt j\ miles ? Anf. 1 1 8id. 4. If a footman travel 273 miles in 6\ days of 12 hours long, in how many days of 9^ hours each may he travel 13 a miles? - . « Anf. 4/ ¥ V day* [ 7a 1 DECIMAL FRACTIONS. A DECIMAL is a fraction whofe denominator is I with •^ *• iome number of ciphers annexed ; as T \ri or twW* Decimals are written down without their denominators, the numerators being fo diftinguifhed as to evince what the denominators are ; which is done, by feparating, by a point, fo many of the right-hand figures from the reft as there are ciphers in the denominator^ the figures on the left- hand fide oi the point being integers, and thole on the right decimals : So i£ is written 1*3, and named 1 and 3 tenths ; VoVz> 9 ls written 15*769, and named 15 and '769 thoufandth* pa' ts ; and T Vo% * s written '25, and named 25 hundredths^ or hundred parts But if there be not a fufficient number of figures in the numerator, ciphers are pitfixed to fupply the defect : So y^-i? written *oi, that is, 1 hundredth , and •xVoW t ^ us > #COI 5- that is, 15 ten \ thoufandlhs. So that the denominator of a decimal is, always a r, xdth , as many ciphers z% there Are figures in the decimal. Net? 1. The ift, '*J, 3d, 4th, &c, places of decimals, counting from the left hand towards the right* are called the places oi primes, iecoads a thirds, and fourths, &-c. refpectively. 2. Ciphers on the right-hand .of decimals do not alter their value,, AMH1ION and SUBTRACTION ©f DECIMALS. WRITE, the propofed numbers under each other, accord- ing to the value of their places, as in whole numbers; m which order the decimal points will Hand dire&iy below each other : Then add or fubtra£ as in whole numbers, plat. cing a decimal point in ths fum or difference fuaight below , the other points, EXAMPLES IN* AUDITION. 1. What is the fum of 276 39*213, 720149, 417, 5032, and' 2214*298 ? 2. What is the fum of 7530, i6'20i, 3*0142, 95713* £•7-281-9, and '03014? 3. What is the furn of 312*09, 3'5T 1J > V95* 6 > 7 1 42^ 9739 2* Sr*** m l - MULTIPLICATION OF DECIMALS, 73 4*. What is the fum of '014, '9816, '32, '15914, 72913, and '0047 ? 5. What is the fum of 27-143, 9*8'73> 14016, 294304, •713826, and 221*7? EXAMPLES IN SUBTRACTION. I* What is the difference between '9173 and '2138 ? 2. What is the difference of 1 91S5 and 2*73 ? 3. What is the difference of 214*81 and 4*90142? 4. What is the difference o£ 92*713 and 407 ? 5. What is the difference of 2714 and '916 ? MULTIPLICATION OF DECIMALS. TT^RITE down the factors, and multiply exactly a3 in ■ " integers, placing the decimal point in the product, fo as to make juft as many decimals in it as there are in both factors y and if there be not as many figures in the product a8 there ought to be decimals, prefix ciphers to fupply the defeft. EXAMPLES. 1. What is the product of -417 and 520*3 ? 2. What is the product of 9178 and -381 ? 3. What is the product of '217 and '0431 ? 4. What is the product of 51*6 and 21 I 5. What is the product of 314 and '029 '? 6. What is the product of '051 and -091 * CONTRACTIONS. I. When decimals are to be multiplied hy 1 with any num* her of ciphers ; it is done by only removing the decimal point fo many places farther to the tight-hand as there are ciphers in th« multiplier, and fubjoining ciphers if need be. EXAMPLES, f . The product of 51*3 and 1000 is 51330-9 2* The product of 2714 and ico.-is 74 MULTIPLICATION OF PEClMAlK 3. The product of '9163 and tooo is 4. The product of 21-31 'and 10000 is 2. When the produEi will contain many more decimals thati are necejpiry fjr the prefent purpofe y the work may be con* trcBed thus ; Write the units figure of the multiplier ftraight under fuch decimal place of the multiplicand as you intend the laft of your product (hall be, writing the other figures of the multiplier in an inverted order j then, in multiplying, reject ail the figures in the multiplicand which are on the right of the figure you are multiplying by ; writing the products down fo, that their right-hand figures fall ft raight below each other j and carrying to fuch right-hand figures from the product of the two preceding figures in the multiplicand thus ; viz. 1 from 5 to 15, 2 from 15 to 25, 3^from 25 to 35, &c and the fum of the lines will be the product to the number of the decimals required, and will be feldom wtong in the lali figure. EXAMPLES'. 1. Multiply 27*14986 by 92-41055, fo as to retain only* four places of decimals in the product. ntra&ed. Common way.' 27*14986 27*14986 53014-29 92-41035 2 44348/4 13 574930 542997 • 8l 44958 108599 2714 986 2715 108599 44 81 i42997 2 14 2443487-4 2508-9280 2508*9280 650510 2. Multiply 480*14936 by 272416, retaining four deci- mals in the product. 3. Multiply 2490*3048 by '573286, retaining five deci- mals in the product. 4. Multiply 325-701428 by 7218393, retaining three de- cimals in Uk product* I -75 DIVISION OF DECIMALS. DIVIDE as in integers: and to know how many de<= .cimals muft be in the quotient, obferve the following rules : RULE I. The fir ft figure of the quotient mull poffefs the fame place of decimals, or integers, as doth that figure of thedlvidend which Hands over the units place of the firft product. RULE II. The decimal places of the divifor and quotient together^ iriuft be equal in number to thofe of the dividend. — Whence, if the number of decimals in the divifor be taken from the number in the dividend, the remainder will be the number in the quotient. Note. If, in any cafe, there be a remainder after all- the dividend figures are ufed, the quotient may be continued to what number of deci- duals you pleafe, by fubjoining a cipher continually to tha latt remain- der. And whenever the number of figures in the quotient is lefs than the required number of decimal, prefix ciphers to fuppiy the defedt. EXAMPLES. •14 > 72Q*'93 ( 3 # 75 ) 3' 1 5 ( 7'i3 ) *8 ( -215 ) -iop ( CONTRACTIONS. 1. If the divifor he an integer with any number . of ciphers €t the end. Cut them off, and remove the decimal point ipi the dividend as many places farther to the left as there were ciphers cut off, .prefixing ciphers, if need be - y then proceed as before* EXAMPLES. 217 ) 45\5 ( 3 2 ° OG ) 4*020 ( 21000 ) 953 ( 79000 )6i ( 2. Whence, if the divifor he 1 with ciphers, the quotient will be the fame figures with the dividend, having the deci- mal point as many places farther to the left a* there are ci« phers in the divifor. J(5 DIVISION OF DECIMALS. EXAMPLES. 2173 -f- 100= 2*173 4*9 ky I0=2 51 6 by 1000 r± # 2i by 1000 = 3. When the number of figures in the divifor is great, the divifion at large will be very troublefome, but may be con- tracted thus : Having, by the firft general rule, found what place of deci- mals or integers the firft figure of the quotient will poffefs ^ confider how many figures of the quotient w T ill ferve the pre- fent purpole ' y then take the fame number of the left-hand figures of the divifor, -and as many of the dividend figures as will contain them (lefs than 10 times); by thefe find the firft figure of the quotient, and for each following figure divide the lail remainder by the divifor, wanting one figure to the right more than before, but obferving what muft be carried to the firft product for fuch omitted figures, as in thefecond contraction of multiplication } and continue the operation till the divifor is e>:hau£ed. Note. When there are not a? many figures in the divifor as are required to be in the quotient, begin the divifion with all the figures as ufual, and continue it till the number of figures in the divifor, and thofe remaining to be found in the quotient be equal, after which ufe the contraction. EXAMPLES. I. Divide 250892806 by ^92-41035, fo as to have four decimals in the quotient, in which cafe the quotient will -contain fix figures. ContraBed. 9*'4*~3>S) 25 o8 '92S,c6 (27-1438 660721 1^49 4608 go 6 Common way. 66072 106 13S48610 4 6o 7575° piiio'ioo 79467850 5539570 2. Divide 4109' 235 1 by 230*409, fo that the quotient may contain four decimals. 3. Divide 37*10438 by Sl l 3*9&> *h&t the quotient may contain five decimals. 4. Divide 913*08 by 2137*2, that the quotient may con* tain three decimals. C 77 ] REDUCTION OF DECIMALS. "* I. To reduce a vulgar fraclion to an equivalent decimal* RULE. Divide the numerator by the denominator as in divifion of decimals, and the quotient will be the decimal required. EXAMPLES. Reduce £, ^, i> f, £, $> h and h t0 decimals. Reduce \ to a decimal. Reduce T '-gV to a decimal. Reduce T 2 T V t0 a decimal. * Since to throw any vulgar fraction, whofe denominator is a prime number, greater than what it is common to divide by in one line, into a decimal, confiding of a great number of figures, has engaged the atten- tion of many eminent perfons, I (hall here fet down the method which Mr. Colfon has given in page 162 of Sir Ifaac Newton r s fluxions; which -method performs the work much fooner than any other that I know of. The method will be bell explained by an example, thus : " Suppofe {for inftance) I would find the reciprocal of the prime number 20, or the value of the fraction ^Vin decimal numbers. I divide 10000 by 29, in the common way, fo far as to find two or three of the firft figures, or till the remainder becomes a fmgie figure, and then I aflume the fupplement to complete the quotient. Thus I (hall have ^L ~ 0*03448 * for the complete quotient; which equation, if I multiply by the numerator 8, it will give T 8 ^ — 0-27584! £, or rather *^ — 0-27586,^. I fubftitute this inftead of the fraction in the firft equation, and I (hail have * — » 0*0344827586^. Again, I multiply this equation by 6, and it will give t zz 0*2068965517 ' , and then by fubftitution » 21 0*034482- 75862068965517 '. Again, I multiply this equation by 7, and it be- comes ^ rr 0*24137931034482758620^^ and then, by fubftitution, «/■£ rr o*03448275862o68p655i724E3793i0344827862o|° where eve- ary operation will at lea ft double the number of figures found by the preceding operation. And this will be an eafy expedient for converting Divifion into Multiplication in all cafes. For this reciprocal of the divi- for being thus found, it may be multiplied iuto the dividend to produce the quotient.** H 7$ REDUCTION OF DECIMALS* II. To reduce a decimal of a fuperior denomination to its va* lue in the inferior ones. RULE. Multiply the given decimal by fuch a number as- will re« duce it to the next inferior name, and point off in the pr». du& as many places of decimals as are in the given number \ then reduce thefe in the fame manner to the next name j and fo continue the reduction to the lowed name required, or till the decimals pointed off be all ciphers \ then the num- bers on the left of the points will exprefs the value of Jthe decimal. . EXAMPLES. I. What is the value of 775I ? - Anf. 15s 6d. :2. What is the value of -625 (hill ? - Anf. >j\di> 3. What is the value of '8635I ? • Anf. 17s 3*24d. 4. What is the value of '0125 lb troy ? Anf. 3 dwts, 5. What is the value of '4694 lb troy ? Anf. 5 oz 12 dwt 15744 gr. 6. What is the value of '625 cwt ? Anf. 2 qr 14 lb. 7. What is the valae of '009943 miles ? Anf. 17 yd 1 ft 5*98848 inc. 8. What is the value of '6875 yd ? Anf. 2 qr 3 nls. 9. What is the value of '3375 acr ? Anf. 1 rd 14 poles, ao. What is the value of ^083 hhd of wine ? Anf. 13*1229 gal. ,21. What is the value of '40625 qr of corn > Anf. 3 bum 1 peck, 12. What is the value of ^42857 month ? Anf. 1 we 4 ds 23*99904 hrs. till. To reduce integers or decimals to equivalent decimals 0) fuperior denominations* CASE L Jf a ftmpde* number or decimal be propofed, reduce it to the .pame required by dividing as in redu&ion of integers. REDUCTION OF BECI&ALS; 7£ EXAMPLES. t* Reduce i dwt to the decimal of a lb. Anf -004166, &c. lb. a. Reduce pd to the decimal of a pound. Anf ^0375 1. 3. Reduce 7 drams to the decimal of a lb avoird. Anf. -02734375- lb. 4. Reduce *26d to the decimal of a 1. Anf. '0010833, &c f 1, 5. Reduce 2*15 lb to the decimal of a cwt. Anf. '019196+cwt. 6. Reduce 24 yards to the decimal of a mile. Anf. '013636, &c. miles. 7. Reduce ^056 poles to the decimal of an acre. Anf. '00035 ac. 8. Reduce r2 pints of wine to the decimal of a hhd. Anf. •002384-hhd. 9. Reduce 14 minutes to the decimal of a day. Anf. '009722, &c. da. xo. Reduce # 2i pints to the decimal of a peck. Anf. '013125 pec. CASE II. A compound number may be reduced to a fuperior name by reducing each of its parts, and taking the fum of the de- cimals \ the beft way to do which is thus : Write the given numbers under each other, proceeding orderly from the leaft to the greater! name, for dividends ; draw a perpendicular line on the left of thefe, and on the left of it write oppolite to each dividend fuch a number, for a divifor, as will reduce it to the next fuperior name \ then begin with the upper divifion, and affix the quotient of each to the next dividend, as a decimal rart of it, before it be divided, and the laft fum will be the anfwer. EXAMPLES. jl Reduce 3I 1 2s 6|d to the denomination of pounds, 4 12 20 3 6 1S 12*5625 3*628125 Anf. H 2 SO RTJLE-OF-THREE IN DECIMALS. 2. Reduce 19I i7s3JdtoI. Anf. I9'86354i66, &c. I- 3. Reduce 15s 6d to the decimal of a 1. Anf. ' 775I. 4. Reduce 7^d to the decimal of a (hill. Anf. *625s. 5. Reduce 5 oz 12 dwts 16 gx to lbs. Anf. '46944, 8cc, lb. 6. Reduce 3 cwt 2 qr 14 lb to cwt*. Anf. 3*625 cwt. 7. Reduce 17 yd 1 ft 6 inc to the decimal of a mile. Anf. -0099431818, &c. roil. 8. Reduce 2* qrs 3 nls to the decimal of a yard. Anf. -6875 yd. 9. Reduce 13 ac 1 ro 14 pol to acres. Anf. I3'3375 acr. 10. Reduce 13 gal 1 pint of wine to the decimal of a hhd. Anf. -20833, &c. hhd. 11. Reduce 3 bufh I peck to the decimal of a quarter. Anf. -40625 qr. 12. Reduce 3 mo 1 we 5 da to months. Anf. 3-42857+11*0^ RULE-QF-THREE IN DECIMALS. REDUCE vulgar fractions to decimals, and compound numbers either to decimals of the higher names, on to integers of the lower, as alfo the firft and third to the fame name : Then ftate the quefiion, and proceed aa in in- tegers. Note. Any of the convenient examples in the Rule.of- Three or Rule-of- Five in iutegecs, or vulgar fra&iors, may be- taken as proper examples to the fame rales in decimals ; for it would be- filling the Book to ill purpofe to give different examples here. — The following example, which is the jfjrft in vulgar fraclions, is wrought here to fliew the method. If I of a yard of velvet coft \ \ what will Y 5 7 yd coil f i= •375 •375 * i •4 : 1 yd # 3 I2 5 : •4 12500 { 1250 125s. 1 '333* & c - «* s d 6 8 T — * - •4 -375)' = •3125 Anf. 6s 8d £'93333$ &c. 20 6-66666, &c. 12 d. 7*999<99» & c - == 8d, SIMPLE INTEREST. Si Note. The remainder in the divifion being always the fame, the quo- tient figure muft be to likewife ; fo that if the quotient were infinitely continued, it would be equal to Jl, as in vulgar fractions. RULE-OF-FIVE IN DECIMALS. HPHE fame preparations muft be ufed here as in the Rule- •*- of-Three, before the ftating j after which, proceed as in integers. EXAMPLE?. 1. If 2I 10s be the wages of 15 men for 6 days, what will be the wages of 12 men for 18 j days ? Here al 10s = apjlj and i8| =3 18*333, &c. Then J e\ l *St l { l8 ' 3 ?!} i 6'iu,&c.or6 2 2* 50 ;■ ■ 220 2\$.. no 44 90)5500 Anf. 61 2s 2^d. X* 6viirn,&c. 20 S. 2'2222 12 d. 2*6666 4 q. 2'6666 For more examples-, take any of thofe in the Eule-of-Five in vulgar frac- tions, or whole numbers. SIMPLE INTEREST. TNTEREST is the premium allowed for the loan of mo* ney. The fum lent is called the principal. The fum of the principal and inteieft is called the a* mount H3. '■:- $Z SIMPLE INTEREST. Intereft is allowed at fo much per cent, per annum, which premium per cent, per annum, or interefl: of iool for a year, is called the rate of intereft. Intereft is of two forts, Jimple and compound. Simple Intereft is that which is allowed for the principal lent only. Note. The rules for Simple Interefl ferve alfo to calculate commiflion, brokage, infur&nce, (locks, or any thing elfe rated at lb much per cent. RULES I. To find the intereft for a year, multiply the principal by the rate, and divide the product by iqq-. Note. When the rate is a convenient aliquot part of ioo, or can be divided into feveral fuch parts of ioo, take the fame pan or parts of the principal for the intereft of a year. EXAMPLES. 1. What is the intereft of 450I for a year, at 5. per cento per annum ? Anf. 22I ics. 2. What is the intereft of 230I 10s, at 4 per cent, per annum ? Anf 9I 4s 4d 3£q. 3. What is the intereft of 715I 12s 6d, at 4J per cent, per annum ? ... Anf. 3.2I 4s o|d. II. To find the intereft for feveral years, multiply the in* tereft of one year by the number of them. Note. When there are feveral years, or feveral years with fome parts of a year, it is commonly beft to multiply them by the rate, and divide the product into aliquot parts of 100, taking the fame parts of the prin* cipal for the anfwer. EXAMPLES. 1. What is the intereft of 720I for 3 years, at 5 percent* per annum ? - Anf. 108L 2. What is the intereft of 355I 13s for 4 years, at 4 per cent, per annum? - - Anf. 56I 18s 4c 33-q, 3. What is the intereft of 32I 5s 8d for 7 years, at 4J per Cfcat. per annum ? - Anf. 9! I2s i T |o-d. III. If there be any parts of a year, as f, -§-, |> &c. the intereft for them is found by taking the fame parts of one year's intereft, when it is not convenient 10 ufe the note in* the laft cafe, SIMPLE INTEREST. % EXAMPLES. 1. What is the altered of 170! for i£ year, at 5 per cent, per annum ? ... Anf 12I 15s. 2. What is the intereft of 205I 15s for ~ of a year, at 4 per cent, per annum ? Anf. 2l is id 33-q. 3. What is the intereft of 319! 6d for 5^ years, at 3^ percent, per annum ? - Anf. 681 15s 9a 1 2^q^ IV. For any number of days ^ multiply the intereft of a year by them, and divide by $65. A TABLE fhewing the number of days from any day of one month to the fame day of any other month. From any day of 3 Jan. Feb. Ma. Apr. May June July Aug Sept Odt. No^ Dec. Jan 3 1 59 90 I2C I CI I»X 212 243 273 3^4 ! 334 Feb. 33fi 365 28 39 % 120 181 212 242 ?23 3°3 Ma. 306 337 3 6 5 3 1 61 92 122 •53 184 214 245 U75 Apr 275 3°' 334 365 3° 61 9' 122 l S3 '83 214 244 May 245 276 304 335 3^5 3* 61 92 123 l 53 £?# 214 June 214 245 273 3°4 334 365 3° 61 9 l 122 *53 183 July 184 215 243 273 3°4 335 365 3 1 62 92 P3 153I Aug 153 184 212 243 273 3°4 334 365 3* 61 92 122 Sept 122 "*53 181. 212 242 2 73 3 C 3 334 3_ 6 > 3& 61 9 1 92 **$ 182 212 243 273 504 335 365 3< 61 Nov 6j 92 120 '5* 181 212 242 2 73 If* 334 & 3c. 3« 62 90 121 151 ib2 212 243 274 3?4 335 3 6 5 Note. In leap year, if the end of the month of February be in the time, one day muit be added on that account. EXAMPLES. 1. What is the intereft of 107I for 117 days, at 4! per cent, per annum ? - Anf. il 12* 7y^W^» 2. What is the amount of 120I from Jan. 7, to bept. I2 f J 5$7, at 4, per cent, per annum ? Anf, 123 1 js ad 2%&fa.* SIMPLE INTEREST. 3. What is the interelt of 213! from Feb. 12, to June 5* 1788, it being leap year, at 3^ per cent, per annum ? Anf. 2l 6s 6d StWtI* N, B. The following is a neat and concife method for working Simple Intereit for any number of day.% at any rate of intereft. Nam:> 1 S d q Nu. i 8 d q Nu. d 1 I OOO000 2 739 H 6 0-99 3000 $ 4 4 2-41 5 3 v*s OOOO..O 2465 *5 3-29 2000- 5 9 7 0-27 4 z 2-52 800000 219c 15 7 i*59 1000 z 14 9 2T 4 3 I 3^9 700000 1917 itf 1 y*9 900 z 9 3 3-12 2 I I'2 8 2-Ss 7^ 3 10 CII CO9 0*24 4OCOO 109 11 9 1*48 60 % 3 181 008 0*21 " 3OOOO 82 3 10 O II 50 2 8 3'S 1 O'07 O 18 200OC 54 *5 10 274 40 2 2 1*21 o*o6 0'i6 I OOOO *7 7 n i-37 : 30 * 7 2*90 0-05 13 9OOO 24 13 1 y*3 20 1 1 o # 6o 0*04 Oil 8coo 21 18 4 I'lC 10 6 2-30 °*°3 0*08 7000 19 3 6 2-96 Q ° 5 3*67 # 02 0-05 6000 16 8 9 0'S2 8 5 I-6 4 c»ci 0*03 5000 13 13 11 2'6S 7 4 2-4. 4000 IO 10 2 o'SS 60 3 V7S RULE. Multiply the principal by the rate, both in pounds y multiply the produft by the number of days, and divide^ this laft* product by 100 ; then take from the table the fe- veral fums which ftand oppofite the feveral parts of the quo- tient, and add them together for the intereft required. EXAMPLES. 2. What i* rhe intereft of 3:251 103 for 33 days, at 4i.per scat, per annum I SIMPLE INTEREST, g$ PHnc 225*5 1 * d q Rate 45 u again!! 200 is 9 10 11 2*03 3 30—0 1 7 2- 9 q IOI 475 Z 3—° o 1 3 89 days 23*3 0'3— o o o 079 - — ■ rj 0*09—0 O O 0'24 100) 23339-25 '2 — ■ ■ J£ " Anf. o 12 9 1*85 true in 233*3925 H the laft place of decimals. 2. What is the intereft of 17I 5s for 117 days, at 4J per cent, per annum ? - Anf. 5s 3d 0'iaq-. 3. What is the intereft of X12I r2s 6d from the 8th of May to the 3d of November, at 4 per cent, per annum ? Anf. 2I 4s 2d o"92q. QUESTIONS Concerning Brokage f CotnmiJpon y Infurance, and Stocks. Brokage is the allowance made to brokers for affiiling others in buying or difpofing of their goods. FaBorage^ provtjion^ or commiflton, is an allowance made to factors or agents beyond fea, for buying or felling goods for their employers. Infurance is fecurity given, in coniideratton of a premium paid down, to reftore, to a certain value, for which the premium is advanced, th« lofs or damage on fhips, houfes, goods, &c. by ftorma, fire, &c. But in the calculations* the word infurance is commonly written for premium. Stocks are the public funds of the nation ) the {hares oi which being transferable from one perfon to another, pcc&- fion the extenfive bufinefs of (lock-jobbing. EXAMPLES. 1. What is the brokage of 610I, at 5s or \ per cent. > Anf, il 10s 6&> 2. What is the brokage of 37 2l- 73 4d, at 4s 6d per cent. ? Anf. 16s 93%^d. 3. What is the factorage of 920I, at 3^ per cent. ? Anf. 32I 4?, 4. What Is the comraiffion of 50SI 17s iod, at 14 per cent,? . Anf. 7I 12s 8 T fsd. .86 COMPOUND INTEREST. 5. What is the infurance of 900!, at lof per cent. ? Anf. 96I 1559 • 6. What is the infurance of 712I 6s for 8 months, at *ji per cent, per annum? - Anf. 35I 12s 3d 2j-q* 7. What is the purchafe of 1200I South-fea tlock, at 103 f- percent.? - - Anf. 1243I I0?« 8. What is the purchafe of 912I 14s bank-ftock, at* 127! per cent. > - • Anf. 1165I 19s $4 3 7 Vq. 9. What is the purchafe of 2380I India ftock, at 147^ percent.? - - . Anf, 3504I us, 10. What 18 the purchafe of 8x61 12s bank annuities, at 89 1 per cent. ? - - Anf. 729I x6s 8d 2$^, COMPOUND INTEREST. /COMPOUND Interefl is that which is allowed, not on- ^* iy for the fum lent, but alfo for its intereft j as it be- comes due at the end of each ftated time of payment. RULES. I. Find the amount of the given principal for the time of the firft payment, by fimple intereft ; then confider this a- mount as the principal for the fecond payment, whofe amount calculate in the fame manner, and fo on through all the payments, ftiil accounting the laft amount as the principal of the next payment : Or, II. Find the amount of one pound for the time of the firft payment, and multiply it by itfelf fo often as are the number of payments wanting 1, that is, twice by itfelf if there be three payments, thrice if there be four, &c. ; then the laft product multiplied by the principal, gives the whole amount. Note t. The following table, adapted for the ufe of the fecond rule, contains the amount of 1 pound for each of the firft ten years or pay- ments, at feven ieveral rates of intereft, from 2 and a half to 6* per cent. ;.. and therefore any une of thde numbers multiplied by a given funa #v pro* 4uces its amount for the correfnonding rate and tjrae. COMPOUND INTEREST. 8 7 !n1 475 2 •18769 '22926 •27228 •3168 1 •36290 41060 04000 08160 124S6 16986 '21665 •26532 •31593 •36857 •42331 •48024 1 '04500 1*09202 1T4117 1*19251 1*24618 1*30226 [•36086 [•42210 [•48610 *'55 2 97 t -05000 [•10250 [•15762 I-2I5SI 1*27628 i*j340io [•40710 [•47746 f55i33 [•62" o6cocj 12360J '191021 '2624SJ •33823! 4185 •503 6 3 59385 '6894$ 79085 Note 2. It is not neceffary that the payments (hould be yearly ; fox the rule will hold whether they be yearly, half-yearly, quarterly, month- ly, or any other aliquot part of a year ; but there muft be a complete in- teger number of the times of payments, not a certain number of times and part of another, for the rule takes no notice of fuch parts, nor will it be juft to calculate for a complete time, and take the fame part of the refult as is the part of the time ; though in this manner have fome au- thors falfely calculated fome of their examples. It is poffible to perform ,all fuch calculations, both parts of times and whole ones, without loga- rithms, though the trouble is, in fome cafes, intolerable ; but by the lo- garithms it is as eafy to perform the calculations with parts ef times of payments, as with whole ones. EXAMPLES. x. What will 50I amount to in five years, at 5 per cent," per annum, compound intereft > ^ - Anf. 6$\ 16s 3^d. 2. What will 50), fuppofing the intereft payable half- yearly, amount to in 5 years, or 10 half years/at 5 per cent. per annum, compound btereft > - Anf, 64I id. 3. What will 50I, the intereft payable quarterly, amount to in 5 years, at 5 per cent, per annum, compound intereft ? Anf 64I 28 o^d. 4. What is the compound intereft of 370I forborn 6 years, at 4 per cent, per annum ? - Anf 98I 3s 43d. j. What is the compound intereft of 410I forborn ±\ years, at 4I- per cent, per annum, fuppofing the intereft pay- able half-yearly ? ... Anf. 48I 4s njd. 6. What is the amount of 21 7I, forborn 2J years, at 5 $>er cent, per annum, fuppofing the intereft payable quar- terly ? - 1 z • Anf, 2421- 13 s 4|d* t ss I DISCOUNT. T* EBATE or Difcount is the difference between a Turn df •*-^- money due at a certain time to come, and its prefent worth. The prefent worth of any fum or debt, due fome time hence, is fuch a fum, as if put to intereft, would in the time and at the rate for which the difcount is to be made, amount to the fum or debt then due, RULE. As the amount of iool for the given rate and time : Is to the given fum or debt : : So is i col to the prxfent worth, or So is the intereil of iool for the given time : : To the difcount of the given fum. Note I. The meaning of four things written in the form above, is that they are the four terms of a R.ule of-Three queftion. ■z. " The method ufed among bankers in difcounting bills, is to find the intereft of the fum drawn for, from the time the bill is difc 6. C and D would barter : C has $$ quarters 5 bufhels of corn, at li 10s per quarter - y for which D would give i^cwt 16 lb of fugar, at 4I 12s per cwt, and the balance in raiiins, at 6|d per lb : How many lb of rainns muft be given ? Anf 737$; lb. 7. E and F barter : E gives to F 90 gallons of brandy, at 7$ 8d per gallon; for which F gives to E 10 -guineas in mo- ney, and 500 lb of cotton : What is it valued at per lb ? Anf. 1 id 2 7 2 yq. 8. G and H bartered; G had 1^ cwt 5 lb of fugar, worth 3l 15s per cwt, but bartered it with H at 2I 4s per cwt, for wine worth 4s Sd per gallon : What was the barter price of the wine, and how much of it was given for the fugar ? Anf. 5s io-}d per gal, and 97?* 4; gal equal the fugar* 9. K and L barter : K has woollen cloth worth 8s per yard, which he barters at 9s 3d, with L, for linen cloth, at 3s per yard, which is worth only 2s 7d per yard : Whether has the advantage in barter, and how much linen does L give K for 70 yards of woollen ? Anf. 215!- yds of linen $ and L has the advantage, his proportional barter price being,, only 2s lid 3|o. C 96 3 LOSS AND GAIN. QUESTIONS in this rule are fuch whofe flotations de- termine the lofs or gain upon commodities ; of which queftions there is a great variety j but they may be alieafily folved from -a little confideration, and the following propor. tion, vi%. That the gains or loJfes % are in proportion as the quantities of goods, &c» EXAMPLES. 1. Bought 5 c 3 qr 14 lb of cheefe, at li 12s per cwt, and fold it again for 2\ os 8d per cwt 5 what was the gain upon the whole ? - - Anf, 2I 10s nd. 2. If 5 c 3 qr 14 lb be bought for 9I 8s, and fold for III 18s 1 id 5 what is the rate of gain per cwt ? Anf. 8s 8d. 3. If 8 c 18 lb be bought for 451 ; at what rate per lb ihall I fell it to gain 10I upon the whole i Anf. is 2d illrq. 4. If 8 c 18 lb coft 45I ', at what rate per lb mult it be fold, that the lof9 upon the whole may be iol ? Anf. 9^3-Vk 5. Bought hops at 4! 1 6s per cwt ; at what rate per cwt muft I fell them, to gain 15I per cent. ? Anf. 5I 10s 4d 3yq* 6. Bought hops at 4I 16s per cwt \ at what rate per cwt muft I fell them, to lole 15I per cent. > Anf. 4I is 7^-d. 7. * If, when I fell cloth at 7s 1 per yard, I gain iol per cent. : what will be the gain per cent., when it is fold for 83 6d per yard ? - - Anf. 33I us 5^d» * Queftions of this fort arc feldont rightly underftood or folved. Thus the queftion beginning at the 7th line of page 33 of the 4th edition of Webfter's arithmetic is wrong, the true anfwer being 15I, and not iol. And in the fame manner has Stonehoufe falfely folved the fecond example in page 8p of the third edition of his arithmetic, the anfwer to which que ftioa ihould be 23I 12s 6d, and not 81 13s c"d. Alfo in a manner fimilar tothefe has Dil worth falfely calculated his 7th example- in lofs and gain, -making fthe anfwer al 16s 3d, inftead of 3I ts ojd, though this queftion has Lowe copied into his book of arithmetic with the falfe anfwer ; as he has done feverai others from different authors, whereof one is from Malcolm, who brought out his anfwer wrong, not- by a falfe method of folution, but by fome wrong figures flipping from his pen in the procefs, orin the propofition. . Hill has alfo run into the fame miftake. — The error confifts in making the gain or lofs of ioolthe 3d term'of the ftating. inftead of the amount of* 100I. For the ftating fliould be thus. As 73 : 8s 6d :: nol : 133I ix&.-- 5^.d, from which taking away the. icol, there remains the anfwes. exchange, 97 8. If, when I fell cloth at 7s a yard, I gain icl per cent. ; what is the gain or lofs per cent., when it is fold at 6s a yard ? Anf. 5I 14s 34^ ]o:i>. 9. If, when I fell fugar at Ss 6d a ftoae, I Jofe 5I per cent. \ what do I gain or lofe per cent., when I fell it at 9s a ftone ? Anf. us p y 3 T d gain. 10. Bought for 17s 8d, and fold for iSs 4d, what was the gain per cent. ? - - Anf. 3I 15s jd 2ff <*, 11. Bought 12 yds of cloth for ct 9s, and ibid them again at 9s 6d a yard > what was the gain or lots per cent, r* Anf. 4I us Sd SvVVl g a * n » 12. At i£d per fhilling profit, how much per cent. ? Anf. 12I I05a J 3. At 3s 6d to the pound profit, how much per cent. ? Anf. 17I 10s. 14. Having fold 12 yards of cloth for 5I 14s, and there- by gained 81 per cent. > what is the prime coft of a yaid ? Anf. 8s 9d 2|q. EXCHANGE. X> Y exchange is meant the bartering or exchanging of the *" money of one place for that of another ; and, like the bartering of wares, it commonly confifts in finding what quantity of the money of one place will be equal to a given fum of another, according to a given courfe of exchange. By courfe of exchange is meant the variable fum of the money of one place which is propofed to be given for a con- itant piece or fum of that of another, to ferve for the pre. Tent, as a rate or proportion by which to exchange other fums \ and it is fometimes above and fometimes below the /tar. By the par of exchange is meant anintrinfic equality be- tween two pieces cr fums of money, one of which is the eonflant piece or fum to which the courfe is compared. The money in the banks of foreign places is liner or purer ihan that which is current in them % and the difference be- tween any fum ? as valued in the one, and its value in the other, is called agio. Note. It is by comparing the bank money with ours that the par is sfcertained. Alfo the exchange is always fuppofed to be made in bank money ; and if there be a necctnty tor taking currency in cafe of a defeat of the bank.. to anfwer the bills, the more of it mull be receivtd, acd that 10 proportion to tjxe agio. 9* EXCHANGE* I. With Ireland, America, and tbe West Indies. Accounts are kept in Ireland, America, and the Weft-In- dies, in pounds, (hillings, and pence, as in England ; and tbe exchange per cent, fterling; the par being io%\ 6s 8d Irifft. per iool fterling, or il is 8d per pound : alfo 5I fterling ia accounted worth 7I of the currency of the Weft-Indies, be- caufc of the great plenty of foreign coins there, EXAMPLES. I. Loadon remits to Dublin 375I 15s : What muft be re- ceived there, exchange at no per cent. ? Anf. 413I63 6d« 2* Dublin remits to London 413I 6s 6d : What muft be received there, exchange at no per cent. ? Anf. 375I 15s. 3. London remits to Jamaica for 21 2l 12s 6d fterling: What muft be received for it, exchange at 135 per cent. > Anf. 2871 iojd. 4. Jamaica remits to London for 287I iajd currency ? What muft be received for it, exchange at 1*3.5 P cr cent. ? Anf. 21 2l 12s 6i* II. With Holland, Flanders, and Germany. In thefe places accounts are kept fometimes in pounds, ihillings, and pence, as in England : and fometimes in guil- ders, ftivers, and pennings. The money of Holland and Flanders is diftinguifhed by the name fiemifh^ and they ex- change by the pound fterling, the par being 33s 44 flemifti per pound fterling. Note* 1 5 pennings m*ake 1 (liver. so (livers or 40 pence — 1 guilder or florin. 8 pennings — 1 grote or penny. 12 grotes or pence — 1 (killing. •20 (killings — 1 pound. And in Germany, iz pennings — x (hilling labsi 16 lubifti (hilling — 1 mark. , 6 pennings — t grote flem. 6 lubifti (hill.— 1 (kill flem. 7 \ marks lubs — x pound floW EXAMPLES. 1. To how much rlemifh will 700I fterling amount, ex- change at 34s rlem. per 1 fterling > - Anf. 1190L 2. To how much fterling will 1190I flemifh amount, ex- change at 34s per 1 fterling ? • • Anf. 70CU ■EXCHANGE. #9 g. How much flemifh muft be given for 314I 5s fterling, exchange at 33s 8d flcm. per 1 fter ? Anf. 528I 19s gd. 4. How much fterling mull be given for 528I 19s pd flernifti, exchange at 33s 8d per 1 fter ? Anf. 314I 5s. 5. How many guilders may 1 kave for 173I 143 2d fter- ling, exchange at 35s 3fd per 1 fterling ? Anf. 1839 gu 2 ft nf pew. 6. How much fterling mull I have for 3714 guil 15 ft, exchange at 35s 6d flemrfh per 1 fterling > Anf. 25 4! 1 8s id i|4q* 7. What quantity of flemifh currency mult I have for 290I lis iod fterling, exchange at 33* iod flem. per 1 fterling t .and agio at 44 per cent. ? Anf. 513I 14s id 1-jVoVh 8. How much -fterling muft I receive for 805I 15s flemifh currency, the agio being 4 per cent., and exchange 34$ 6d fern, per 1 fterling ? - Anf. 449I 2s sd 2|£fq. 9. To how much fterling will 7310 marks 8 fh 9 pn amount, exchange at 36s 4d flem. per 1 fterling ? Anf. 53 61 HS3^q. 10. How many marks muft be received for 536I fterling, exchange at 36s 4d flem. per 1 fter ? Anf. 7303 marks. III. With FrancEo In France, accounts are kept in livres, fols, and deniers, exchange being made by the French crown, the par of whicb h 2$ 6d fterling, or 2s j^d more nearly. Note, 1 a deniers — a fol or fou, value 30 fols — a livre Pi} * 43 3 livres — icrownDrecu a 53- EXAMPLES. 2. How many livres, &c. will 121I 18s 6d amount to, ex- change at 32^d per ecu ? • Anf. 2670 li 5 fol lid, 2. To how much fterling will 3956 livres amount, ex- change 3 id per ecu ? - Anf. 170I 6s 6f d. 3. How many French crowns may I have for 102I 13s Xifd fterling, exchange at 3i£d per crown ? Anf. 785 cr 34 fols. 4. To how much fterling will 1978 cr 25 fols amount, exchange at 3 if d per crown ? Anf. 260! 13s nfd, 10© EXCHANGfio IV. With Spatn, 6v. In Spain they keep their accounts in piaftres, rials, and mervadies j reckoning 372 mervadies to a rial, and 8 rials to a piaftre, by which they exchange, and its par is 4s 6d fterling. Not?, In Genoa and Leghorn they keep their account? in livrey. foI% and denicrs, as in France, but exchange by the piaftre, as in Spain, which in Genoa is accounted 5 livres, and at Leghorn 6. At Venice -too, accounts are by fome kept in the fame manner, and by others in du- cats and grofs, reckoning 24 grofs to a ducat, upon which they exchange, and its par is accounted 43 4,d fterling. EXAMPLES. 1. How- -many piaftres, &c. (hall I receive in Spain for 510I fterling, exchange at jod fterling per piaftre ? Anf 2448 piaftres. 2. Spain draws upon London for 2448 piaftres, exchange at jod per piaftre: How much fterling will the draught amount to ? - - Anf. 510I. ■ 3. How many livres, &c. muft be given at Genoa for 175I 15s fterling, exchange at 5 2d fterling per piaftre ? Anf. 4055 liv 15 fol 4i 8 T dens. 4. Genoa draws upon London for 3000 livres : How much fterling will fatisfy this draught, exchange at jo^d per piaftre > - - - Anf. 126I 5?. 5. How many livres, &c. muft be received at Leghorn for 7051 16s 4d fteiiing, exchange at 51-^d fterling per piaftre ? Anf. 19735 liv 9 fol i T y T den. 6. Leghorn draws upon London for 12000 liv 14 fol, ex- change at 5od fterling per piaftre : How much muft be paid at London for this draught ? Anf. 416I 13s 9d 3|q. 7. How many ducats at Venice will a draught of 427I fterling amount to, exchange at 4pd fterling per ducat ? Anf. 2091 due iOy gro^ 8. Venice draws upon London for 2091 due 10 gro, ex- change at 49d fter per ducat: To how much fterling does it amount ? - - Anf. 426I 19s ud ifq. V. With Portugal. In Portugal accounts are kept in milreas and reas, reckon- ing 100O reas to a milrea, as its name imports } and they ex- change by the milrea, the par of which is about 6s 8^d 3 ox 6s 90! fterling. ARBITRATION OF EXCHANGES. ICI' EXAMPLES. 1. To how many milreas will 7-1 5I amount, exchange at 5s 8d per milrea ? - Anf. 2523 mil. 529^- reas. 2. To how many I, Sec. will a draught of 2523 mil. 528 reas amount, exchange at 5s 8d per milrea ? Anf. 714I 19s 1 id 3x^q. 3. How many milreas mud be given for 213! 7s icd> exchange at 5s 9^d per milrea ? Anf. 736 mlr. 892-^ reas c 4. To how much fterling will 736 milreas amou change at 5s 9*d per milrea ? - Anf. 213! 7 - * d y -mount, ex~. Anf. 213I 2S 8d e ARBITRATION OF EXCHANGES. AS the courfe or rate of exchange between one nation and another, is almoft continually varying, either by riling or failing, from the variations in the circumstances and balance of trade*, fo the delign of arbitration is to remit or draw upon foreign places in fuch a manner, as (hall turn out the moft profitable. Arbitration is generally divided into two parts, Simple and Compound, I. SIMPLE ARBITRATION. In Simple Arbitration, the exchanges among three places only are concerned. The par of arbitration, or arbitrated trice, is fuch a rate of exchange between two places, as fhall pe in proportion with the rates affigned between each of them *nd a third place. After this par of arbitration is compu- ;ed, by comparing it'with the prefent courfe of exchange, 1 perfon can judge which way to remit or draw to the moflt advantage, and determine what the advantage (hall be, EXAMPLES. 1. If the exchange between London and Amfterdam be 3s 9d per 1 fterling, and the exchange between London and *aris be 3 2d per crow r n ; required the par of arbitration be- reen Amfterdam and Paris. Anf. 54d flem. per crown. 2. If the exchange between Amfterdam and Parrs be C4d K 102 ARBITRATION OF EXCHANGES. per crown, and between Amfterdam and London 33s $& per 1 fterling \ required the arbitrated price between Paris and London. - - Anf. 3 2d per crown* 3. If the exchange from London to Paris be 3 2d per crown, and to Amfterdam 40jd per 1 fterling \ and if by advice from Holland or France, the courfe of exchange be- tween Amfterdam and Paris be fallen to 5 2d per crown 5 "what may be gained per cent., by London drawing on the one place and remitting to the other ? Anf. By drawing on Paris and remitting to Amfterdam may be gained 3I 16s n^d per cent. 4. London is indebted to Peterfburgh 3000 rubles : Now the exchange between Peterfburgh and England is at 5©d per ruble, between Peterfburgh and Holland at c)od per ru- ble, and between Holland and England at 36s 4d : Which will be the more advantageous method for London to be drawn 4 upon ? Anf. London may gain 9I us i|d by making payment by way of Holland. 5. London was ordered to remit 500 ducats to Venice, at 50d per ducat, and to draw upon Spain for the value, at 4od per piafttve \ but when the order came to hand, bills on Venice were at 52-£d : Now if London can draw upon Spain at 42^d, whether will it gain or lofe, and how much ? Anf. 7 T y piaftre8 loft/ 6. London was ordered to remit 800 crowns to Paris, at 3i|d per crown, and to draw upon Amfterdam for the va- lue, at 36s 9d per 1 ; but when the order came up, bills on Paris were at 3ifd: What mud be the rate of exchange' with Amfterdam to compenfate the advance on the remit-. tance? - - - Anf. 36s 5U§d. 7. A merchant in London had 6000 guilders in the bank at Amfterdam, and was offered 22d fterling a piece for them j but not liking the offer, he indorfes a bill for the whole to* his factor at Paris, who foon brought the money to France,] by exchanging at 55d flemifh per crown \ he allows the fac-J tor -I per cent, commiflion for his trouble, and then draws! upon him for the whole, exchange at 3 2d per crown : Hov .much was this better than the offer of 22d per guilder ? Anf* *81 *8s h ARBITRATION OF IXCHANGES. T03 II. COMPOUND ARBITRATION. Comnound Arbitration refpects the cafes in which the ex- changes between three, four, or more places, are concerned. A perfon who knows at what rate he can draw or remit directly, and alio hath advice of the courfe of exchange in foreign places, may trace out a path for circulating his mo- ney, through more or fewer of fuch places, and alfo in fuch order, as to make a benefit of his fkill and credit : And herein lies the great art of fuch negociations. But to determine in what order, and through how many places to circulate a bill, no general rule can be given, as it depends entirely upon a perfon's judgment, and a clofe attention to the refults of former cafes of the like kind. The directions neceffary for determining whether a direct or an afligned circular draft fhall be preferable, are contained in the following KULtS. 1. Diftihguifh the given rates or prices in the circular courfe, into the antecedents and consequents ; and place the antecedents in one column, and the confequents in ano- ther, on the right, fronting one another by way of equa- tion. And in this diftribution into antecedents and confe- quents, each confequent rauft be of the fame kind with the next antecedent, and the firft antecedent of the fame kind with the laft confequent, which rauft be the fum whofe va- lue in exchange is required. 2. Multiply the antecedents continually for a divifor, and the confequents continually for a dividend j and the quotient of the products will be the value of the fum required by fuch exchange. 3. Then compute its value by the direct exchange, or by any other circular exchange, and by comparing the values together, you will perceive the mod advantageous method. EXAMPLES. I. If London would remit ioooi fterling to Spain, the direct exchange being 42^d per paiftre of 272 mervadies, it is required whether will be more profitable, the direct 1.Q4 ARBITRATION OF EXCHANGES. remittance, or by remitting firft to Holland, at 35s per 1 ; thence to France, at 58c! per crown ; thence to Venice, at ioc crowns per 60 ducats; and thence to Spain, at 360 mervadies per ducat. Antecedents, Confequents. il fterling r= 35s or 42od flemifh ^8d flemirh rr 1 crown 300 crowns = 60 ducats 1 ducat rz: 360 mervadies 272 mervadies £± 1 piaftre How many piaftres =: ioool fterling T1 420 X 60 X 360 X 1000 __ 210 X 30 X 45 X 10 • j8 x 100 x 272 29 x 17 2835000 „ _ . . mm ±L = 2J$ ||0 p ia ^ rcs —- t Jj e y a l ue Q f IC00 1 D y the circular exchange. But 42^d : 1 piaftre : : iocol, or 24000od : « — :=z z=: 5^47rV piaftres, the value by the direct exchange. So that the circular exchange is the more advantageous, and it produces 103 1^ piaftres more than the other. 2. A banker at Amfterdam remits to London 40olfle- mifh, thus j viz. firft to France at 56d per crown 5 thence to Venice at 100 crowns per 60 ducats ; thence to Ham- burgh at iood flemifh per ducat*, thence to Lifbon at 5od percrufade of 400 reasj and laftly, from Lifbon to London at 64d fterling per milrea. How much fterling money will the remittance amount to; and how much will be gained or iaved, fuppofing the direct exchange from Holland to Lon- don at 36s iod per I fterling ? Anf. 2l 4s 8 r 2 / 3 8 T d gained. 3. A merchant at London has credit for 680 piaftres at Leghorn, for which he can draw directly at jod per pi- aftre 5 but chufing to try the circular way, they are by his order remitted, firft to Venice at 94 piaftres per 100 du- cats; thence to Cadiz at 320 mervadies per ducat *, thence to Lifbon at 630 reas per piaftre of 272 mervadies ; to Am- fterdam at 5od per crufade of 40a reas y thence to Paris at j;6d per crown ; and thence to London at 3i£d per crown* ALLIGATION. Io$ How much is the circular remittance better than the direft draft, rekoning £ P er cent, for commiflion ? Anf. iol 14s 3||d- nearly. Note. The allowance for commiflion is made by deducting £ per cent,, from each of the coniequents xoo, 320, 630, 50, and 1, in the ftating; Che beft way of doing which, is to deduct the 200th part of each from itfelf, or to diminifh each in the ratio of 200 to 199. - ALLIGATION. A LLIGATION is the method of mixing together feve- ■ "^ ^ ral fimples of different qualities, fo that the compofition may be of a middle quality: And it is commonly diiiinguifh* ed into two principal cafes, denominated Alligation Medial, and Alligation Alternate. Case I. ALLIGATION MEDIAL. Alligation Medial is the method of finding the rate of the compound, from having the rates and quantities of the feveral fimples given.. Note. That by the rates are meant the numbers which determine or define the proportions of the quantities of the fimples and the compound ; fuch as the given prices of their integers or numbers expreffing their de- grees of finenefs, or any other numbers proportional to them. And if any one of the fimples be of little or no value with refptcl to the reft, its rate is fuppofed to be To ; as water, mixed with. wine, or alloy with gold and fllver. RULE. Multiply each quantity by its rate 5 then divide the fum of the produces by the fum of the quantities, or the whole compofition ^ and the quotient will be the rate of the com- pound required. EXAMPLES. ti A compofition being made of 5 lb of tea at 7s per lb, 9 lb at 8s 6d per lb, and 144 lb at 5s iod per lb ; what is a lb of it worth ?■ - - Anf. 6s iod 2ffq. 2. What is a gallon of a compofition of wine worth* which is made by mixing 4. gallons of 4s iod per gallon, with 7 gallons at 5s 3d, and £| gallons at 5s 8d per gallon ? Anf. js 4d iffq. K3 105 ALLIGATION. 3. Having mixed together 17 gallons of ale at gd per gallon, 14 at 7fd, 5 at gf d, and 21 at 4^ d - 9 how much per gallon is. the mixture worth I : - - Anf. 7 3-1^. 4. A. mixture being made of 12 bufhels of oats at xs 4d per bufliel, 9 bufhels of peas at is 7d, and 4 bufhels 2 pecks of beans at is 2d per bufhel ; what will it be worth per bufliel? . - - Anf, is 4d 2TT1* 5. A compofition being made by mixing 8 gallons of wine, worth 5s gd per gallon, with 7 gallons worth 5s nd, and :2 gallons of water - 7 what is a gallon of it worth ? Anf. 5s id 2^4q, 6. Having melted together 7 oz of gold of 22 caracts lane, 12J oz of 21 caracls fine, and 17 oz of 19 cara£b fine $ I would know the finenefs of the compofition ? Anf. 20yy cara&s fine. 7. Of what finenefs is that compofition, which ia made by mixing 3 lb of filver of 9 oz fine, with 5 lb 8 oz of 10 oz fine, and 1 lb 10 oz of alloy ? - Anf. 7-g-J- oz fine. Case II. ALLIGATION ALTERNATE. Alligation Alternate is the method of finding what quan- tity of each of the fimples, whofe rates are given, will com- pofe a mixture of a given rate % 7 (o that it is the reverfe of Alligation Medial, and may therefore be proved by it. RULE. 1. Write the rates of the fimples in a column under each other. * 2. Connect or link with a continued line, the rate of each fimple which is lefs than that of the compound, with one or any number of thofe which are greater than the com- pound •> and each greater rate with one or any number of the lefs. 3. Write the difference between the mixture rate and that of each of the fimples, oppofite the rates with which thefe are linked. 4. Then if only one difference ftand again ft .any rate, it will be the quantity belonging to that rate } but if there be ?vera! ; their fura will be the quantity, ALLIGATIOK. I07 Note. It appears from the rule, that many of the queftions of this cafe will admit of various anfwers each ; but from an algebraic procefs it ap- pears that they will all have infinite varieties of anfwers; nay, if the expreflion may be allowed, that they will admit of infinite varieties of infinite varieties of anfwers. After one or more anfwers are found by the rule, as many more are found as you pleafe by inci eating or decrea- fing the quantities in any proportion, or by only increafing or decreafing any one or more fingle pairs of yoke-fellows in any proportion, and lea- ving the other rates as they are ; but as that anfwer is commonly defired which gives the rates in the lead integer numbers, and thofe the neareft to each other, I have to each of the following queftions fet down fuch anfwers as I found by linking the rates together the moft poflible, and then,- where no limitation was propofed, dividing the refulting quantities by their greateft common meafure. EXAMPLES, 1. How much wine at 6s per gallon, and 4s per gallon-, muft be mixed together, that the compofition may be worth 5s per gallon ? Anf. 1 qr or; gal, or any one equal quantity of each fort. 2. How much fugar at /\d 9 at 6d, and at 1 id per pound, muft be mixed together, that the compofition may be worth *d per pound ? - Anf. 1 lb or 1 ftone, or 1 cwt, or any other equal quantity of each fort. 3. How much corn at 2s 6d, at 3s 8d, at 4s, and at 4s £d, per bulhel, muft be mixed together, that the compound may be worth 3s iod per bufhel ? Anf. 2 at 2s 6d, 2 at 38 8d, 3 at 4s, and 3 at 4s 8d» 4. A compofition whofe rate may be 78 6d, being to be made, by mixing together fimples whofe rates are 4s, 5s 8d, 6s, 7s 4d, and 8b ; how much of each muft be ufed ? Anf. An equal quantity of the firft four forts, and 14 times the fame quantity of the laft fort. 5. To mix gold of 19 caracls fine, with gold of 23, of 21, of 18, and of 17 caracls fine, that the compound may be 20 carafts fine j what quantity muft: be taken of each ? Anf. 2 at 17, 18, 19 5 and 3 at 21, 23. 6. Whatare the proportions of the quantities of alloy, and gold of 22 carafts fine 5 which, when mixed together, will make the comnofition of 20 caradls fine ? Anf. There muft be 10 times as much gold as alloy. Sometimes one or more of the ingredients, and fometimes the whole compofition is limited to a certain quantity, which I divide into the three following cafes or limitations, 103 ALLIGATION, LIMITATION L When the whole compofition is limited to a certain quan- tity, and that quantity is not found from the method of linking, and taking the differences $ then you may augment or diminifti the quantity of each ingredient, in the fame pro- portion as the given quantity is greater or lefs than the to- tal quantity found from the linking, by faying, As the total quantity fo found, is to the given quantity, fo is the quanti- ty of each ingredient, found by linking, to the required quantity of each. EXAMPLES. r. How much wine at 4s, at 5s, at 5s fid, and at 6s a gallon, mutt be mixed together, to form a compofition of 1.8 gallons, worth 5s 4d a gallon ? Anf. 3 gal at 4s and 5s, and 6 gal at $s 6d and 6s. 2. How much gold of 15, of 17, of 18, and of 22 caracta fine, muft be mixed together, to form a compofition of 40 ounces, of 20 caracls fine ? Anf, 5 of 15, of 17, and of 18 ; and 25 o% of 22 carafts fine. N. E. To this cafe belongs the queftioa concerning king Hiero's crown, \phich the workmen had debafed with filver, or copper ; and to find what quantity of gold or copper was in it, the famous Archimedes, it is faid, made two other crowns, ot the fame weight with the former, the one of gold, and. the other. of filver, or copper; and by putting each into a veflel full of water, the quantity of water expelled by them determined their fpecific bulks: From which, and their given weight, it is eafier to deter- mine the quantities of gold and copper in the crown, by this cafe of aU ligation, than by an algebraic procefs. Suppofe the weight of each crown to be iolb, and the water expelled by the copper or filver was 02lb, by the gold 521b, and by the compound grown was 641b ; that is, their fpecific bulks were as 92, 52* and 64. Here then the rates of the fimples are 92 and 52, and of the compound 64; therefore , [92^12 of copper, C The fum of thefe is 12 -{- 28 zz 40, which fliould* 4»52\28 of gold, {.have been but 10 j therefore, by our rule, •ALLIGATION. lOp LIMITATION IL When one of the ingredients is limited to a certain quan- tity, and that quantity is not found by the method of link- ing, you may either augment or diminifh the quantities of all the reft, in the fame proportion as the given quantity is greater or lefs than the quantity of the limited fimple found by linking, by dating, as in the -fir ft limitation*: Or, you may only augment, or diminifh, in the above proportion, that part of the quantity of the ingredients with which the limited one is linked, which is the difference of the mixture late and the rate of the limited fimple, and add the refulting quantity to other parts, inftead of the faid difference \ keep- ing the quantities of the other fimpies unaitertd. EXAMPLES 1. How much wine at 5s, at 53 6d t and at 6s the gallon, tnufl be mixed with 3 gallons at 4s the gallon, that the mixture may be worth 5s 4d a gallon, Anf.^d g ! 1 T fs 6dl h ? V'opotthmng all the quanti, C 10 gal at 5s "1 by proportioning only the differ- Or>} 8|- - - 5s 6d > ence of the mixture and limit- £ 8£ - - 6s J ed rate3 - 2. How much gold of 15, of 17, and of 22 caracls fine, muft be mixed with five ounces of 18 caradls fine, that the composition may be 20 cara£h fine ? C C oz of 1 5 caradls fine 1 , . . ,. , Anf l r n . L by proportioning all the '£25 - 22 - J quantities. * Rene© we may ohferve that Mr. Malcolm has inadvertantly given a rule in page 569 of his arithmetic, for queftiens of this fort, when the limited fimple is only once linked, which will net always give tnre an- fwers; he fays, "If the fimple whofe quantity is limited, is only once linked, we need do no more than raife or diminifh the quantity of that one fimple with which it is linked, and leave the reft as they are." In- ftead of which, if he had confidered that the fimple with which tfye li- mited one is linked, may alfo be linked with fome one or more of the reft, X apprehend he would have faid, Raife or diminifh that part of the fimple with which the limited one is linked, which is the difference be» twist the mixture rate and the rate of the limited fimple, 1 10 INVOLUTION C 2 oz of 15 caradls fine ~) by proportioning only the Or,-< 2 - 17 --- > difference of the mix- C 3 • 22 - • • 3 ture alu * limited rate*. LIMITATION III. If more than one of the fimples be limited, find, by Cafe 1, what will be the rate of a mixture made of the given quantities of the limited fimples only 5 then confider this as the rate of a limited fimple whofe quantity is the fum of the fir ft given limited fimples , from which, and the rates of the limited fimples, by the fecond limitation, calculate the quan-* tity of each. EXAMPLES. 1. How much wine at 5s 6d, and at 6s a gallon, muftbe mixed with 3 gallons at 4s, and 3 gallons at 5s a gallon,- that the mixture may be worth 5s 4d a gallon ? Anf. 6 gal at $s 6d, and 6 gal at 6s a gallon. 2. How much gold at 15 and 17 cara&s fine, muft be mixed with five ounces of 18, and 13 ounces of 22 caracts fine, that the compofition may be of 20 Caracas fine ? Anf. 2 oz of each fort. INVOLUTION. A POWER is a number produced by multiplying any •£*- given number continually by itfelf a certain number of times. Any number is called the firft power of itfelf, if it be multiplied by itfelf, the product is called the fecond power, and fometimes the fquare j if this be multiplied by the firft power again, the product is called the third power j and fometimes the cube 5 and if this be multiplied by the firft power again, the product is called the fourth power, and fo on j that is, the power is denominated from the number which exceeds the multiplications by 1. Thus : 3 is the firft power of 3. 3X3= 9 is the fecond power of 3. 3 X 3 X 3 = 27 ia the third power of 3. 3X-3X3X3 = 81 is the fourth power of 3. &c. &c. And in this manner may be calculated the following ta- ble of powers. INVOLUTION. Ill TABLE sfthejirft 12 powers of Numbers. sr 9 n 3" O O "5" O 3 n VO tr -a 3 00 sr •0 re O On 5* O Ov tr 13 O 4^ U 00 0- O to P- T3 O n -O ; O n - 1 4* O O 4* CO O to 4* On M to t* Oi ON to CO On go to Pn ■JO 4* to 4* «*1 -J 4? Cn vo O 4*> VO On CO Co on On on to 00 vo to 4^ Oo CO vo . 00 0. 4* V O IO 4* 1 ON CO 1 to On 1 M -a J 4* On 1 4* On On On Go On OO OO 4*. 4* O vo on O to 4* to on On On 4^ On 4^ to 4* 4* 4* O On lO 4» 00 00 to 00 to Co VO -J ON to Cn H VO Oi It* to On Oo VO £ M to On M to O. On to On to On to Cn o> -J 1 on on 1 *J 00 1 Vo to 1 -J oo 1 O OJ | On On 1 On on 4^ on On On M O O *^l on vo Ov On vo ON OS to vo vo u> On 4* On On Ox On »*4 "•J On M to vo O to On Oo On On Oo OO 4* to OO >0 vo "J £ Os •J 4* Go V* CO to 4*. -a On to 4^ vo 4* O Oo Ov 00 On O Ot CO OO to Co Oi 4* Oo M M «*4 vo M Ov OO O to 4* O Oo 4^ 4^ vo «-J On OO vo 4* 00 On 00 VO vo 00 -^ Cn vo to »a OO 4^ M OO to -&■ M OO 4* to t>t -4 N CO ON -a to On to On to 19 to 4* OO •J On CO 4>» O vo ON 1 J m | On to f 4^ ce to 00 to £ to VO Oi 00 00 cs s 00 | 00 s S In 1 00 VO | 4* 0\ 4^ O O 1 00 | 4^ «o 1 Oo to ' 4*. O I ON 4* 1 *** 00 | to vo 1 " 4* ^1 eo to vo 0\ vo Ot Oo »-« 4> 4^ On vo J- vo 1 1 ON I On « -vj On 1 to M 1 vo CO vo HZ INVOLUTION. Note r. The number which exceeds the multiplications by i, is called the index, or exponent of the power : So the index of the fiift power is I, that of the fecond power is 2, and that of the third is 3, See. 2. Powers are commonly denoted by writing their indices above the firft power : So the fecond power of 3 may be denoted thus 3 s , the third power thus 33, the fourth power thus 3*-, &.c. and the ilxth power of 503 thus 503 6 . Involution is the finding of powers, to do which, from their definition there evidently comes this RULE. Multiply the given number, or firft power, continually by itfelf, till the number of multiplications be x lefs than the index of the power to be found, and the laft product will be the power required* Note T. Whence becaufe fractions are multiplied by taking the pro- ducts of their numerators and of their denominators, they will be involved by railing each of their terms to the power required. And ifamixt num- ber be propofed, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal, and proceed by the rule. 2. The raifmg of powers will be fometimes fliottened by working, ac- cording to this ohfervation, vrz. whatever two or more powers are mul- tiplied together, their product is the power whofe index is the fum of the indices of the factors ; or if a power be multiplied by itfelf, the product will be the power whofe index is double of that which is multiplied : So if I would find the fixth power, I might multiply the given number twice Ivy itfelf for the third power, then the third power into itfelf would give the fixth power ; or if I would find the feventh power. I might firft find the third and fourth, and their product will be the feventh ; or laft- ly, if I would find the eighth power, I might firft find the fecond, then the fecond into itfelf would be- the fourth, and this into itfelf would be the eighth. EXAMPLES. J. What is the fecond power of 45? Anf. 2025. 2. What is the fquare of 4*16? - Anf. 17*3056. 3. What is the fquare of '027 ? - Anf. '000729. 4. What is the third power of 3*5 ? Anf. 42*875. 5. What is the fourth power of 71*8 ? Anf. 26576499'4576. 6. What is the fifth power of '029 ? Anf. '000000020511149. 7. What rs the fixth power of 5-03 ? Anf. 16196*005304479729. 8. What is the fecond power of f ? - Anf. £ . 9. What ia the third power of f ? - Anf. 444. 10. What is the fquare of g| ? Anf. Vy or lr 5 6 * I "3 1 EVOLUTION. THE root of any given number, or power, is fuch a number as being multiplied by itfelf a certain number of times, v/ill produce the power \ and it is denominated the firft, fecond, third, fourth, &c. root, refpectively, as the number of multiplications made of it to produce tbe given power is o, I, 2, 3, &c. \ that is, the name of the root is taken from the number which exceeds the multi- plications by 1, like the name of the power in Involution. Note r. The index of the root, like that of the power in Involution, i<* 1 more than the number of multiplications neceffary to produce the power or given number. 1. Roots are fometimes denoted by writing y/ before the power, with 3 the index of the root againft it : So the third root of 50 is V 50, and the fecond root of it is y/ 50", the index 2 being omitted ; which index is al- ways underftood when a root is named or written without one. But if the power be expreffed by feveral numbers with the fign + or — &c. between them, then a line is drawn from the top of the fign of the root, or radical lign, over all the parts of it : So the 3d root of 47 — 15 is {/ 47 — 15. And fometimes roots are defigned like powers, with the reciprocal of the index of the root of the given number. So the root of XT I 3 is 3"*' the root of 50 is 50" 2 "' and the third root of it is 5o"'■ , alfo the , s [third root of 47 — 15 is 47 — 15"^' And this method of notation has juic- ily prevailed in the modern algebra ; becaufe fuch roots, being confidered las fractional powers, need no other directions for any operations to be jade with them, than thofe for integral powers. 3. A number is called a complete power of any kind, when its root of the fame kind caa be accurately extracted ; but if not, the number i« :alled an imperfect power, and its root a furd or irrational quantity ; fo is a complete power of the fecond kind, its root being ?.; but an im- erfect power of the third root kind, its third root being a furd quantity. Evolution is the finding of the roots of numbers, either: accurately, or in decimals to any propofed extent. The power is firft to be prepared for extraction, or evolu- tion, by dividing it from the place of units, to the left-hand in integers, and to the right in decimal fractions, into peri- ads, containing each as many places of figures as are denominated by the index of the root, if the power con- lain a complete number of fuch periods: if it do not, he defect will be eitheron the right-hand or left, or both j U4 EVOLUTION. if the defect: be on the right-hand, it may be fupplied by annexing ciphers, and after this whole periods of ciphers may be annexed to continue the extraction with, if neceffary ; but if there be a defect on the left, fuck defective period muft remain unaltered, and is accounted the firft period of the given number, juft the fame as if it were complete. Now this divifion may be conveniently made by making a point over the place of unite, and alfo over the laft figure of every period on both fides of it *, that is, over every fe- cond figure if it be the fecond root, over every third if it be the third root, &c. Thus, to point this number . 21035896*12735 ; for the fecond root, it will be 21035896*1273505 but for the third root, thus 21035896*1 27350; • • • . and for the fourth, thus 21035896*12735000$ Note. The root will contain juft as many places of figures as there are periods or points in the given power ; and they will be integers or decii mals refpe&ively, as the periods are fo from which they are found, or to which they correspond ; that is, there will be as many integer or decimal figures in the root, as there are periods of integers or decimals in the gi« ven number. TO EXTRACT THE SQUARE ROOT. 1. Having pointed the given number into periods of two figures each, find, from the table of powers in page ur, or otherwife, a fquare number either equal to, or the next left than the fit ft period, which fubtrac/fc from it, and fet the root of the fquare on the right-hand fide of the given number, after the manner of a quotient in Divifion, for the firft figure of the root required. 2. To the remainder annex the fecond period for a divi- dend; and on the left-hand of it write the double of the root I already found, after the manner of a divifor. 3. Confider what figure, which, if annexed to the divifor, j and the refult multiplied by it, the product may be equal to or the next lefs than the dividend, and it will be the nex figure of the root. 4. From the dividend fubtraft the product, and to the remainder bring down the next period, for a new divj EVOLUTION. "5 ; dend : to which, as before, find a divifor by doubling the figures already found in the root *, and from thefe find the next figure of the root as id the laft article ; and continue the operation flill in the fame manner, till all the periods be ufed, or as far as you pleafe. Note. When the root is to be extracted to a great num- ber of places, the work may be much abbreviated thus s Having proceeded in the ex- traction after the common me- thod, till you have found half the required number of figures in the root, the reft may be id by dividing the laft re- mainder by its correfponding divifor, annexing a cipher to every dividual, as in divifion of decimals; or rather, with- out annexing ciphers, by omit- ting continually the right- hand figure of the divifor, after the manner of the third contraction in divifion of deci- mals in pag( So the operation for the root of 2 to i a or 13 pJacea, I may be thus: 2 ( 1-41421356237+1:001 I 24 4 JCO 96 281 1 400 281 2824 4 1 1900 1 1 296 28282 2 604CO 56564 282841 1 383600 282841 2828423 3 10075900 8485269 2828426 ) 1590631 ( 56237 + 176418 6712 *°55 206 8 What is What is What is What is What is What is What is 8. What is 9. What is 1. 2. 3- 4- 5- 6. 7 EXAMPLES. the root of 2025 > the root of 17*3056? the root of '000729 ? the root of 3 ? the root of 5 ? the root of 6 ? the root of 7 > the root of 10 ? the root of 11 ? 1*2 Anf. 45. Ant 4* 16. Anf. '027. Anf. 1732050. Anf. 2*236068. Anf. 2*449489. Anf. 2*645751. Ant 3-162278. Ant 3-316625. li«$ EVOLUTION. Rules for the Square Roots of Vulgar Fractions and 'Mixt Numbers. Firft prepare all vulgar fra£tions by reducing them to their leaft terms, both for this and alL other roots. Then 1. Take the root of the numerator and of the denomina- tor for the refpective terms of the root required. And this is the beft way if the denominator be a complete power, but if it be not, then 2. Multiply the numerator and denominator together \ take the root of the product : this root being made the nu- merator to the denominator of the given fraction, or made the denominator to the numerator of it, will form the frac- tional root required. That is, V -? : b And this rule will ferve whether the root be finite or infinite. 3. Or reduce the vulgar fraction to a decimal, and extract fcs root. 4. Mixt numbers may be either reduced to improper frac- tions, and extracted by the firft or fecond rule % or the vul- gar fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted. EXAMPLES. Ti What is the root of |4 ? Anf. f . 2. What is the root of -£& ? Anf. 4. s- What is the root of ■& > Anf. 866025. 4. What is the root of ^ ? Anf. 0*6454972. 5- What is the root of i^f ? Anf. 4U68333. ly means of the fquare root, alfo, we readily find the 4th* soot, or the 8th root, or the 1 6th root, &c. j that is, the; soot of any power whofe index is fome power of the num- ber 2 \ namely, by extracting fo often the fquare root as if] denoted by that power ,of 2 ; that ie, two extractions for fchej fourth rootj three for. the 8th root, and fo ©n> EVOLUTION, I If 11035*8000 I *45'°37 2 37 ( 12-043x407 root : 1 24 no 96 285 *435 1425 22 2404 45 44 10372 9616 29003 6 108000 24083 87009 6 75<537 72249 20991 ( 7237 687 107 20 3388 ( 1407 980 *7 TO EXTRACT THE CUBE ROOT. I. By the common rule. 1. Having divided the given number into periods of three figures each, find the neareft lefs cube to the firft period, in the third line of the table of powers in page in 5 fet its root in the quotient, and fubtracl: the faid cube from the firft period ; to the remainder bring down the fecond pe- jiod, and call this the rejohend. 2. To three times the fquare of the root juft found, add three times the root itfelf, fetting this one place more to the right than the former, and call this fum the divifor. Then divide the refolvend, wanting the laft figure, by the divifor, for the next figure of the root, which annex to the former $ calling this laft figure e, and the part of the root before found called a. 3. Add altogether thefe three produ&s, namely, thrice a fquare multiplied by e, thrice a multiplied by * fquare, and e cube, fetting each of them one place more to the right than the former, and call the fum the fubtrahend % . which muft not exceed the refolvend 3 and if it does, then make the laft figure e lefs, and repeat the operation for find* l»g the fubtrahend. L 3 1 18 EVOLUTION, 4. From the refolvend take the fabtrahend, and to the remainder join the next period of the given number for a new refolvend r, to which form a new divifor from the whole root now found ; and from thence another figure of the ioot 9 as directed in article 2, &c. EXAMPLE. To extraft the Cube Root of 48228*544. 3 X 3 2 = 27 I 48228-544 ( 36-4 root. 3x3 = 09 27 Divifor 279 I 21228 refolvend 3 X3 2 x6 = 162 3X3X6 2 = 324 J-add x6 = i62 "1 62 = 3 2 4 H 6 3 ~ 216 J 3X36 2 = 3888 3X36 = 108 38988 19656 fubtrahend 1572544 refolvend 3x36^4 = 15552 -) 3X36 X4 2 = 1728 Ladd 4 3 = 64J 1572544 fubtrahend 0000000 remainder. EVOLUTION, ilj II. To exlra& the Cube Root by a Jhort way. 1. By trials take the neareft rational cube to the given cube or number, whether it be greater or left, and call it the affumed cube. 2. Then the fum of the given number and double the af- fumed cube, will be to the fum of the affumed cube and double the given number, as the root of the affumed cube is to the root required nearly. Or, as the firft fum is to the difference of the given and affumed cube, fo is the affumed root to the difference of the roots nearly. 3. Again, by ufing, in like manner, the cube of the root laft found as a new affumed cube, another root will be got ftill nearer. And fo on, as far as we pleafe ; ufmg always the cube of the lafl found root, for the affumed cube. EXAMPLES. To find the Cube Root of 21035*8. Here we foon find that the root lies between 20 and ^o f and then between 27 and 28. Taking therefore 27, its cube is 19683 the affumed cube. Then J9683 21035-8 2 2 393 66 42071-6 21035-8 j 9683 As 60401*8 : 61754-6 27 4322822 1235092 27 : 27*6047 6o4Ci«8)i667374 , 2(27 , 6o47 the root nearly. 459338 36j25 284 -42 I 20 EVOLUTION. Again, for a fecond operation, the cube of this root is 21035-318645155823, and the procefs by the latter method will be thus : 21035-318645, &c. 2 42070*637290 21035*8 21035-8 21035-318645, &c. As 63106*43729 : diff. '481355 : : 27-6047 : the diflf. '000210834 confeq. the root req. is 27*604910834 TO EXTRACT ANT ROOT WHATEVER. Let G be the given power or number, n the index of the power, A the affumed power, r its root, R the required root of G. Then as the fum of n + 1 times A and 8—1 times G, is to the fum of n 4- 1 times G and n — 1 times A, fo is the 3ffumed root r, to the required root of R. Or, as half the faid fum of n -f* 1 times A and n — 1 times G, is to the difference between the given and affumed pow- ers, fo is the affumed root r, to the difference between the true and affumed roots : which difference, added or fubtraft- ed, gives the true root nearly. That is, »+i. A-}-*— -i. G : 0+1. G-f-*— 1. A : : r : R* Or, «TiT i A-f-/z— 1. iG:AcoG::r:Rcor. And the operation may be repeated as often as we pleafe, by ufing always the laft found root for the affumed root, and . its nth power for the affumed power A. Ex. To extraft the 5th root of 21035-8. Here it appears that the 5th root is between 7-3 and 7-4.^ Taking 7-3, its 5th power is 20730-71593. Hence thea we have EVOLUTION. 121 G => 21035-8; r=7'3> « = 5; i«*lc=S5 i*-— 1=2- A =3 20730*716 G — A = 305*084- 1 A = 20730716 G = 21035*8 3 I 3 A ==62192-148 420716 2 G rrr 420*71*6 As 1042637 : 305-084 : : 73 : '0213605 £2 9M252 2135588 104263-7 ) 2227-1 £32 (0213605 the diff. 14184 7-3 rrr add 3758 7*321360 = R the root true 630 to the laft figure. 5 OTHER EXAMPLES. 1. What is the 3d root of 2 ? 2. What is the 4th root of 2 ? 3. What is the 4th root of 97*41 ? 4. What is the 5th root of 2 ? 5. What is the^6th root of 21035-8 ? 6. What is the 6ih root of 2 ? 7. What is the 7th root of 210358 ? 8. What is the f.h root of 2 ? 9. What is the 8th root of 21035*8 ? jo. What is the 8th root of 2 ? 11. What is the 9th root of 21035*8 ? 12. What is the 9th root of 2 ? Anf. 1-259921. Anf 1-189207, Ani 3 !* I 4 I 5999- Anf i* 1 486^9, Anf. 5*254037. Anf. 1*122462. Anfi 4'i45392. Anf. 1-104089, Anf. 3*4703 23- Anf. 1-090508. Anf. 3*022239, Anf. 1*080059, Gihef.al Rules for extracting any root out of a Vulgar Fra&ion, or Mixt Number. 1. If the given fraction have a finite Voot of the kind re- quired, it is belt to extraft the root out of the numerator and denominator, for the terms of the root required- 2, But if the fraction be not a complete power, it may be thrown into a decimal, and then extracted. Or, *W GENERAL PROPORTION. 3. Take either of the terms of the given fra&ion for ite correfponding term-of the root •> and for the other term of the root, extraft the required root of the produft, arifing from the multiplication of fuch a power of the firft afligned term of the root whofe index is lefs by 1 than that of the given power, by the other term of the given number. This rule will do when the root is either finite or infinite. That is, for any root n in general. n a vi=^^?^ 4. Mixt numbers may be reduced either to improper frac- tions or decimals, and then extracted. EXAMPLES. 1. What is the cube root of -£t ? - Anf. -§. 2. What is the fourth root of ^° T ? - Anf. J-. 3. What is the cube root of -J- ? - Anf. '7937005. 4. What is the cube root of 2^ ? - Anf. ^ or if. 5. What is the third root of 7^ ? - Anf. 1*930979. OF PROPORTION IN GENERAL. *VTUMBERS are compared together to difcover the rela- ■*-^ tions they have to each other. There muft be two numbers to form a comparifon : The mumber which is compared, being written firft, is called the antecedent 5 and that to which it is compared, the confe- quent. Numbers are compared to each other two different ways : The one comparifon confiders the difference of the two num- bers, and is called arithmetical relation, the difference be- ing fometimes named the arithmetical ratio ; and the other confiders their quotient, which is termed geometrical rela- tion, and the quotient the geometrical ratio. So of thefe numbers 6 and 3, the difference or arithmetical ratio, is 6 — 3 or 3 - } and the geometrical ratio is •£ or z* GENERAL PROPORTION* I 2g Note. Ratios are here always confidered as the refult of the greater term of companion diminifhed, or divided, by the lefs, not regarding whether of them be the antecedent. If two or more couplets of numbers have equal ratios, or differences, the equality is named proportion y and their term?, fimilarly pofited, that is, either all the greater, or all the lefs taken as antecedents, and the reft as confequents, are called proportionals. So the two couplets 2, 4, and 6, 8, taken thus, 2, 4, 6, 8, or thus, 4, 2, 8, 6, are arithmetical proportionals j and the two couplets 2, 4, and 8, 16, taken thus, 2, 4, 8, 16, or thus, 4, 2, 16, 8, are geometrical pro- portionals. To denote numbers as being geometrically proportional, the couplets are feparated by a double colon, and a colon is written between the terms of each couplet : we may alfo de- note arithmetical proportionals by feparating the couplets with a double colon, and writing a colon turned horizontal- ly between the terms of each couplet. So the above arith- meticals may be written thus, 2 •• 4 : : 6 •• 8, and 4 •• 2 : : -8 •• 6 y where the firft antecedent is lefs or greater than its confequent, by juft as much as the fecond antecedent is lefs or greater than its confequent \ and the geometricals thus, 2:4: : 8 : 16, and 4 : 2 : : 16 : 8 j where the fir ft ante* cedent is contained in or contains its confequent, juft as often as the.fecond is contained in or contains its confequent. It is common to read the geometricals 4:2:: 16 : 8, thus, 4 is to 2 as 16 to 8. Proportion is diftinguifhed into continued and difconti- nued. If, of feveral couplets of proportionals placed in a ferie?, the difference or ratio of each confequent and the antecedent of the next following couplet, be the fame as the common difference or ratio of the couplets, the proportion is faid to be continued, and the numbers themfelves a feries of conti- nued proportionals, or an arithmetical or geometrical pro- grefTion. So 2, 4, 6, 8, form an arithmetical progreflion ; for 4 — 2 = 6 — 4=8 — 6=2 , and 2, 4, 8, 16, a geo- metrical progreflion 3 for 4 = 1 = V ** 2t 124 ARITHMETICAL PROGRESSION* But if the difference or ratio of the confequent of one couplet and the antecedent of the next couplet, be not the fame as the common difference or ratio of the couplets, the proportion is faid to be difcontinued. So 4, 2, 8. 6, are in difcontinued arithmetical proportion ; for 4 — 2 = 8 — 6=2, but 8 — • 2 =r 6', alfo 4, 2, 16, 8 are in difcontinued geome- trical proportion ; for £ = V* *= 2 » DUt V 6 = 8. If the fucceeding terms of a progrerlion exceed each o- theiy it is called an afcending progreftion or feries; if the contrary, a defcending feries. g fo, 1, 2, 3, 4, &c. is an afcending arithmetical feries. \ 1, 2, 4, 8, 16, &c. is an afcending geometrical feries. and i 4' 3' 2> *'* °> &c» is a defcending arithmetical Teries. \i6, 8, 4, 2, 1, &c. is a defcending geometrical feries. The full and laft terms of a progrefRon are called the ex- tremes j and the other terms, the means. ARITHMETICAL PROGRESSION. A N arithmetical progreflion is a feries, of which the fuc- •*- ^ ceeding terms are either all greater, or all lefs than their adjacent preceding terms, by the fame number or dif- ference. Note. The fundamental property of an arithmetical progreflion from which almoft all its other properties are deducible, and which evidently follows from its cor.ft ruction, i?, that the fum of any two of its terms, is equal to the fum of any other two terms, taken at an equal diitance, but co contrary (ides of the former ; or that the double of any one term is equal to the fum of any two terms taken at an equal diftance from it on each fide. And of any two couplets in difcontinued arithmetical propor- tion, the two fums are equal which are made by adding the antecedent of each to the confequent of the other. PROBLEM L Given one of the extremes, the common difference, and. the number of terms of an arithmetical feries j to find 1. The extrerae. Rule. Multiply the common difference by 1 lefs than the number of terms ^ then add the product to the lead term, and the fum will be the greateft, or fubttacl it from the] greateft term, to give leaft. ARITHMETICAL PROGRESSION. I2£ 2. The fum of all the terms of the feries. Rule. Multiply the fum of the extremes by the number ^f terms, and half the product will be the (urn of the feries. Fhus, if a reprefent the lefs extreme, % the greater, d the common difference, n the number of terms, and j the fum of the feries ; then. -,~a + dxn— i. f f _ ia+dn — d jzzz — dXr. — i EXAMPLES. 1. Given the leaft term 3, the common difference 2, and :he number of terms 9 : to 6nd the greateft term and the "am of the feries. Anf. I he greateft term is 19, and the fum of the feries is 99. 2. If the greateft term be 70, the common difference 3, md the number of terms 21 $ what is the leaft term, and the fum of the feries i Anf. The leaft term is 10, and the fum is 840. 3. A debt can be difcharged in a year, by paying 1 (hil- ling the firft week, 3 (hillings the fecond, and fo on, always 2 (hillings more every week \ what is the debt, and what will the laft payment be ? Anf. The laft payment will be 5I 3s, and the debt is 13 5I 4$, 4. One hundred ftones being difpofed on the ground in a ftraight line, at the diftance of a yard from each other; how many yards will a perfon travel who (hall bring them all, one by one, to a bafket placed one yard from the firft ftone ? Anf, 10100 yds, or 5 mis 1300 yds, or nearly 5 J mis, PROBLEM II. Given the extremes and the common difference, to find 1. The number of terms. Rule. Divide the difference of the extremes by the com- mon difference, add 1 to the quotient, and the fum will be the number of terms.. M ,126 ARITHMETICAL PROGRESSION. 2. The fum of the feries. Having found the number of terms, the fum of the fenca mil be had by the fecond cafe of problem i. Thus ufing the fame fymbols as before, n = —j- +i, and s == a + % X- — -3- d 2d EXAMPLES. 1. If the extremes be 3 and 19, and the common difference 2 j what is the number of terms, and the fum of the feries * Anf. The number of terms is 9, and the fum is 99 2. If the extremes be 10 and 70, and the common differ ence 3 •, what is the number of terms, and the fum of the feries ? Anf. The number of terms is 21, and the fum is 84c 3. What debt can be difcharged, and in what time, fup pofing the firft week the payment be is, and the payment every week following to increafe by 2s, till the lad paymer be 5I 3s ? - Anf. The debt is 135I 4s, and it will be difcharged in a year, ox 52 weeks. PRC&LEM III. Given the extremes and the number of terms, to find j. The common difference. Rule* This is found by dividing the difference of the ej tremesby 1 lefs than the number. of terms. 2. The fum of the feries. This is had from the 2d cafe of problem 1. Thus, d-=z , and srz X n. EXAMPLES. 1. If the extremes be 3 and 19, and* the number of terms M ; what is the common difference, and fum of ihe feries ^ Anf, The difference is 2, and the fum is 99. GEOMETRICAL PROGRESSION. 12 ^ 2. If the extremes be 10 and 70, and the number of term's 2i j what is the common difference, and the fum of the feV ries ? Anf. The difference is 3, and the fum is 840. 3. What debt can be difcharged in a year, by weekly payments in arithmetical progreflion, whereof the firft term or payment is is, and the laft 5I 3s \ and what is the com- ; xnon difference of the fe ries of payments I Anf. The difference is 2$, and the debt is 13 5I 4s. GEOMETRICAL PROGRESSION. . A GEOMETRICAL Progreflion is a feries of numbers, •"- of which the fucceeding terms are either all greater or all lefs than their adjacent preceding terms, ia fuch fort, that the ratio or quotient of every two adjacent terms is the fame. Note. The fame thing is true with refpedl to the produces of the terms of a geometrical proportion, as was obferved of the turns of the ttrms of an arithmetical propor>* >n, in the note in page 124. That is, the pro- duct of any two terms is equal to the product of any other two terms, ta- ken at an equal diftance, but on contrary (ides of the former ; or that th« fquare of any one term is equal to the product of any two term?, taken at (an equal difrance from it on each fide. And th-e fame analogy holds good tin mod of their problems ; fo that many of their rules are almoft verbally [the fame, and differ only in this, that inftead of the operations of addi- tion, fubtra&ion, multiplication and divifion in arithmetical progreflion, are required refpectively thofe of multiplication, divilion, involution, and evolution, in geometrical progreflicii. PROBLEM I Given one of the extremes, the ratio, and the number of the term? of a geometrical feries } to find 1. The other extreme. Rule, Raife the ratio to the power whofe index is one lefs than the number of terms 5 by which multiply the laft lerm to give the greateft, or divide the greateil term to find the ieaft. 2. The fum of the feries. Rule. Divide the difference of the extremes by the ra- tio lefs 1, to the quotient add the greater extreme, and it I will give the fum of the feries. Or multiply the greateft jllcrra by the ratio, from the product fubtraft the Jeaft term* M Z 128 GEOMETRICAL PROGRESSION. then divide the difference by the ratio lefs I, and the quo- tient will be the fum of the feries. Thus, if a reprefent the leaft term, % the greateft, r the ratio, n the number of the terms, and / the fum of the feries , then EXAMPLES. ti Given the leaft term i, the ratio 2, and the number of terms 10 ; what is the greateft term, and the fum of the feries ? Anf. The greateft term is 512, and the fum 1023. 2. If the greateft term be 885735, the ratio 3, and the Bumber of terms 1 2 ; what is the leaft term, and the fum o£ f the feries ? Anf The leaft term is 5, an^i the fum 13 28600*. 3. What debt will be difcharged in a year or 12 months, by paying il the firft month, 2l the fecond, 4I the third, and fo on, each fucceeding payment being double the laft,, and what will the laft payment be ? Anf. The debt is 4095I, and the laft payment 2048I. PROBLEM IT. Given the extremes and the ratio, to find 1. The fum of the feries. This is found by the fecond cafe of the laft problem. 2. The number of terms. Rule. Divide the greateft term by the leaft ; find what power of the ratio is equal to the quotient ; then add 1 to the index of that power, and the fum will be the number of terms. Or, divide the difference of the logarithms of the extremes by the logarithm of the ratio, add 1 to the quo- tient, and the fum will be the number of term* Thus, s tt - + 2 = . and — log** — log.*. . _ log % — log, a -f log, r* n ~ log. r. + * ~ log. r. GEOMETRICAL PROGRESSION. 1 29 EXAMPLES. 1. If the extremes be 1 and 512, and the ratio 2 5 what is the fum of the feries, and the number of terms ? Anf. The fum is 1023, and the number of terms 10. 2. If the extremes be 5, and 885735, and the ratio 3 - 7 what is the fum of the feries, and the number of terms > Anf. The fum is 1328600, and the number of terms 12. 3. What debt will be difcharged by monthly payments in geometrical progreflion, the firft of which is il, and the laft 2048I, the ratio being 2 ; and in what time will it be difcharged ? Anf. The debt is 40951, and it will be difcharged in a year. PROBLEM III. Given the extremes and the number of terms, to find 1. The ratio. This is found as in problem 2, by dividing the greater extreme by the lefs, and extracting that root of the quotient whofe index is one iefs than the number of terms. 2. The fum of the feries. This found as in problem 1. Thus, r=z V _, andj=: — 1T rr « V z — V a EXAMPLES. 1. Given the extremes 1 and 512, and the number of terms 10 \ to find the ratio, and the fum of the feries. Anf. The ratio is 2, and the fum is 1023. 2. If the extremes of a feries confifting of 12 terms, be 5 and 885735 j what is the ratio, and the fum of the feries ? Anf. The ratio is 3, and the fum is 132860^. 3. What debt can be difcharged in a year by monthly payments, in geometrical progreflion, of which the firfY payment is il, and the laft 2048 3 and what will the ratio of the feries be ? Anf. The ratio will be 2, and the debt 4095I M'3 C 130 3 SINGLE POSITION. T^HIS rule is alfo called Falfe Pofition, or Falfe Suppofi- ■*■ tion, becaufe it makes a fuppofition of falfe numbers, as if they were the true ones, and by their means, difcovers the true numbers fought. The llngle rule ufes only one fuppofkion, but the double rule two j whence come their names. To the rule of Pofition belong fuch queftions as cannot be refolved by the direft proce fs by any of the former rules ; and in which the required number or numbers do not afcend above the firft power : Such, for example, as rnoft of the queftions ufually brought to exercife the reduction of fimple equations in Algebra, But it will not bring out true anfwers when the numbers fought afcend above the firft power ; for then the refults are not proportional to their portions, nor the errors to the difference of the true number and each po- fition-, yet in all fuch cafes, it is a very good approximation, and in exponential equations, as well as many other things, fucceeds better than perhaps any other method. Thofe queftions, in which the refults are proportional to their fuppofitions, belong to Single Pofition } fuch are thofe which require the multiplication or divifion of the number fought by any number, or in which it is to be increafed or diminifhed by itfelf any number of times, or by any part or parts of it. But thofe in which the refults are not propor- tional to their pofitions, belong to the double rule ; fuch are thofe, in which the number fought is increafed or dimi- nilhed by fome given number, which is no known part of the number required. To work ^ueflhns in Single 'P-fiiion. Take any number, and perform the fame operations with it as, in the queftion, are defcribedto be performed with the number fought \ then if the refult be the fame with that in the queftion, the fuppofed number is the number fought ; but if it be not, fay, r\s the refult of the operation is to the pofition fo is the refult in the queftion to the number re- ♦"mredv DOUBLE POSITION. *3* EXAMPLES. X. A perfon after Tpending £ and J of his money f has yet remaining 6ol : What had he at firfl ? Suppofe he had at firft 120I. Proof, Now -J of 120 is 40 J of 144 is 48 i of it is 30 £ of 144 is 36 their fum is 70 their fura 84 Which taken from 120 , taken from 144 leaves 50 leaves 69 as per que ft ion. Then 50 : 120 : : 60 : 144, the anfwer. 2. What number is that, whrch multiplied by 7, and the product divided by 6, the quotient may be 14 ? Anf. 12. 3. What number is that, which being increafed by 4> ft and £ of itfelf, the fum (hall be 125 ? . Anf. 60. 4. A general, after fending out a foraging 4 an d |- of his men, had yet remaining 700 j what number had he in com- mand ? - Anf. 4200. 5. A gentleman diftributed 78 pence among a number of poor people, confining of men, women, and children ; to each man he gave 6d, to each woman 4d, and to each child 2d : Moreover, there were twice as many women as men, and thrice as many .children as women. How many were there of each ? Anf. 3 men, 6 women, and 18 children. 6. One being siked his age, faid if £ of the years I have lived be multiplied by 7, and J- of them be added to the product, the fum will be 292. What was his age ? Anf. 60 years* DOUBLE POSITION. TTAVING taken any two convenient numbers, for the po* •*"*■ fitions, proceed with each, according to the conditions of the queftion, as if they were the true numbers fought $ and find how much the refults are different from the refuit hi the queflion. Next multiply each of thefe errors or dif- ferences by the other's pofuion, then if the errors be of the fame affection, that is, if the refults be both either too great or too little, divide the difference of the produces by >ja DOUBLE POSITION* the difference of the errors, and the quotient will be the an- fwer 5 but if the errors be of different affe&ioas, that is, i£ one refult be too great and the other too little, divide the fum of the products by the fum of the errors, and the quo- tient will be the anfwer. Or having found the errors, fay, As the fum of the errors when they are of different kinds, or as the difference of the errors, when they are of the fame kind, is to the difference of ihe fuppofitions, fo is the Icaft error to the correction of the fuppofition belonging to this error j which muft be added to, or fubtracted from it, according to the following condi- tions, viz. if the errors be of the fame kind, add the cor- rection to this fuppofition if it is greater than the other fup- pofition, or fubtracl when iz is the lefs ; but if the errors be of different kinds, do the contrary, viz. add when that fuppofi- tion is the lefs, and fubtrafl when it is the greater of the twoj and the fum or difference will be the number fought. EXAMPLES. I. What number is that, which being multiplied by 6, the produ6l incieafed by 18, and the fum divided by 9, the quo- tient will be 20. Fit ft, fuppofe 30 to be the number fought. ,_, 30x6+18 Then - = 10 X 2 + 2 - 2a + 2 = 22 ; but 9 ought to be 20 y therefore the error is 2 in excefs. A eain, fuppofe 18 to be the number fought. _, °i8x6 + i8 /■ Ihen = 2X6 + 2 = 12 +2 ■= 14; but 3 ought to be 20} therefore the error is 6 in defeft. And the errors are of different kinds or affe&ions. ixTL u i £ * i 3 ox6 + l8 >< 2 J 5><3+9 Whence, by the fir ft rule, — 5-— = ~ 2 Z. r ; 2 + 6 ■= 8 2 St 5 T 4 = 27, the number fought. 2X12 And- by. the fecond rule, 2 + 6 : 30 — < 18 : : 2 8 * 3, the correction 5 then 30 — 3 == 27, t'he_number fought. 2. A fon afking his father how old he was, receives the fol- lowing anfwer: Your age is now \ of mine $ but 5 years ago your age was only \ of mine at that time. What are their ages I ■ » : Anf» So and zq* PROMISCUOUS QUESTIONS. I33 3. A workman was hired for 30 days, at 2$ 6d per day, for every day he worked \ but with this condition, that for every day he played, he (hould forfeit is. Now it fo hap- pened, that upon the whole he had 2\ 14s to receive : How many of the days did he work ? ' - Anf. 24. 4. A and B began to play together with equal fums of money : A firft won 20 guineas, but afterwards loft back f of what he then had ; after which B had 4 times as much as A : What fum did each begin with ? Anf. 100 guineas. 5. Two perfons, A and B, have both the fame income, A faves 3- of his ; but 13, by fpending 50I per annum more than A, at the end of 4 years finds himfelf loo) in debt : What doth each receive and fpend per annum ? Anf. They receive 125I per aanum \ alio A fpends J0©1, and B fpends 150I per annum ? PROMISCUOUS QUESTIONS. 1. A Was born when B was 21 years of age : How -^*- old will A be when B is 47 ; and what will be the age of B when A is 60 ? - . Anf. A 26, K 81. 2. What difference is there between twice five, and twen- ty, and twice twenty-five ? - - Anf. 20. 3. What number taken from the fquarc of 48, will leave 16 times 54 ? - • - Anf. 1440. 4. What number added to the thirty-full part of 381.3,, will make the fum 200 ? - - Anf. 77. 5. What number deducted from the 23d part of 29440, will leave the 64th part of the fame ? - Anf. 820* 6. The remainder of.a divifion is 325, the quotient 467, the divifor is 43 more than the fum of both ; What is the dividend ? - - Anf. 390270. 7. A perfon at the time of out-fetting in trade, owed 350I \ and had in cafh 5307I 108, in wares 713I 7d, and in good debts 210I 58 iod. Now after having traded a year, he owed 703I 17a, and had in cafh 4874I 9s 4d, in bills 350!* in wares 1075I 14s 34-d, and in recoverable debts 613I 13s lo£d : What was his real gain that year ? Anf. 329I 4s ld» 8. Two perfons depart from the fame place at the fame time, the one travels 30, the other 35 miles a day : How fat 134 PROMISCUOUS QUESTIONS. are they diftant after 7 days, if they travel both the fame road, and how far if they travel in contrary directions ? Anf. 35 and 455 miles. 9. A gentleman's daily expence is 4I 8s i^/yd, and he faves 500! in the year : What is his yearly income r Anf. 2107I 125. 10. Having a piece of land 11 poles in breadth, I demand what length of it muft be taken to contain an acre, when four poles in breadth require 40 poles in length to contain the fame ? - Anf. 14 pis 3 yds. 11. If a gentleman whcfe annual income is ioool, fpends 23 guineas a week, whether will he fave or run in debt, and how much in the year r - - Anf. 92I debt. 12. in the latitude of London, the diftance round the earth, m-afuring in the parallel of latitude, is about 15550 miles : No v as the earth turns roand in 23 hours 56 minutes, at what rate per hour is the city of London carried by this motion from weft to eaft ? Anf 6493-j-f miles an hour, 13. In order to raife a joint ft ock of ioocol, A, B, and C together, fubferibe 7950], and D the reft: Now A and B are known together to have fet their hands to 5800!, and A has been heard to fay that he had undertaken for'5501 more than B, Whit did each proprietor advance ? Anf. A 3175, B 2625, C 2150, D 2050. 14. A tradefman increafed his eftate annually by iool more than £ part of it, and at the end of 4 years found that his eftate amounted to 10342! 3 4 9d : Wtiat had he at out- fetting ? Anf. 4000I. I J. Pa'd 1012I 10s for 750I, taken in 7 years ago 5 at what rate per cent, per annum did I pay iutereft ? Anf. 5I. 16. What is the intereft of 720I for 73 days, or f of a year, at 3I per cent, per annum ? Anf. 4I 6s 4d 3 jq. 17. Part 1200 acres of land among A, B, and C, fo that B may have 100 more than A, and C 64 more than B. Anf. A 312, B 412, and C 476. 18. Divide 1000 crowns, give A 120 more, and B95 lefs than C. - Anf. A 445, B 230, and C 32J, 19. To how much amounts the ordex for which my factor, at the rate of zj per cent, receives 22' 103 ? Anf 900I. 20. What fum of money will amount to 13-2I 16s 3d, in 15 months, at 5 per cent, per annum fimple intereft I Anf* 125L PROMISCUOUS QUESTIOKS. 13$ 21. Laid Ot»t 165I 15s in wine at 4s 3d a gallon ; fome -of which receiving damage in carriage, I fold the reft at 63 4d a gallon, which produced only nol 16s 8d : What quan- tity was damaged ? • . Anf. 430 gal. 22. A father divided his fortune among his fons, giving A 4 as often as B 3, and C 5 as often as B 6 : What was the whole legacy, fuppoiing A's (hare were 5000I ? Anf. 11875I. 23. A ftationer fold quills at 10s 6d a thoufand, by which he cleared £ of the money ; but growing fcarce, he raifed them to 12s a thoufand : What did he clear per cent, by the latter price ? - - Anf. 71I 8s 6%d. 24. Jf 1000 m«n beiieged in a town, with provifions for 5 weeks, allowing each man 16 oz a day, were reinforced with 500 men more, and hearing that they cannot be re- lieved till the end of 8 weeks: How many ounces a day muft each man have, that the provision may laft that time ? Anf. 6| oz, 25. If a quantity of provifions ferve 1500 men 12 weeks, at the rate of 20 ounces a day for each man ; how rnary men will the fame provifions maintain for 20 weeks, at the rate of 8 oz a day for each man ? - ArX 2250 men, 26. In what time will the intereft of 72I I2s equal that of j jl 5s for 64 days, at any rate of intereft ? Anf. 13-^-5- days. 27. A perfon poffefled of £ of a {hip, fold f of hrs (hare for 1260I : What was the reputed value of the whole at the fame rate ? - * Anf. 5040I. 28. What fum of money at <\\ per cent, will clear 29I .15$ in a year and a half's time ? Anf. 440I 14s 9-Jd. 29. What number is that, to which if -§• of 4 be added, the fum will be 1 ? - - - Anf. |^« 30. A father dying, left His fon a fortune, \ of which he ran through in 8 months; \ of the remainder lafted him a tweiyemonth longer, after which he had bare 410I left : What did his father bequeath him ? Anf. 936I 13s 4d. 31. Bought a quantity of goods for 250I, and three months after fold it for 275I : How much per cent, per annum did 1 gain by them ? Anf. 40I, 32. A guardian paid his ward 350 ol for 250c 1 which he had in his hand 3 years : What rate of intereft did he allow him ? - - - - Am. 5 per' cent. 33. Bcught a quantity cf goods for jjoi ready money, 1 3*5 PROMISCUOUS QUESTIONS. and fold it again for 200], payable at the end of 9 months : What was the gain in ready money, fuppoiing rebate to be made at 5 per cent. ? *- - Anf. 42I 15s 5vVd« 34. A perfon being afked the hour of the day, faid, The time paft noon is equal to £ths of the time till midnight: What was the time ? . Anf 20 min. paft 5. 35. A perfon looking on hia watch, was afked what was the time of the day, who anfwered, It is between 4 and 5 ; but a mo r e particular anfwer being required, he faid that the hour and minute hands were then exactly together : What was the time ? - Anf 2It q t rnin. paft 4. 36. With 12 gal of canary, at 6s 4d a gal, I mixed 18 gal of white wine, at 4s iod a gal, and 12 gal of cyder, at 3s id a gal : At what rate muft 1 fell a quart of this com- pofition fo as to clear 10 per cent. ? - Anf, is 34d. 37. Suppofe that I have -^ of a fhip worth 1270], what part of her have I left after felling J- of f of my (hare ; and what is it worth P - •• Anf. 7 \V? worth 185J. 38. What length muft be cut off a board 8| inches broad, to contain a fquare foot, or as much as 12 inches in length and 12 in breadth? - - Anf. i7fy inches. 39. What fum of money will produce as much intereft in 3~ years, as 210I 3s can produce in 5 years and 5 months > Anf. 350I 5s. 40. There is gained by trading with a (hip 120I 14s : Now fuppofe that -J of her belongs to S, -| to T, f to V, and the itii to W : What muft each have of the gain ? Anf. S 30I 3s 6d, T 45I gs 3d, V 15I is pd, W 30I 3s 6d. 41. If iool in 5 years be allowed to gain 20I iq>, in what time will any fum of money double itfelf at t'he fame rate of intereft ? - - - Anf. 24J-J years. 42. What difference is therebetween the intereft of 350I at 4 per cent, for 8 years, and the difcount of the fame fum at the fame rate, 2nd for the fame time ? . Anf. 27I ^Tl Sm 43. If, by felling goods at 50s per cwt, I gain 20 per cent., what do I gain or lofe per cent., by felling at 45s per cwt ? Anf 81 gain. 44. If, by remitting to Holland, at 34s 6d per 1 fterling, 4-J per cent, be gained ; how goes the exchange, wbei by remittance, I clear 10 per cent. ? - Anf. 36s ^i^i^' 45. Sold goods for 6d guineas ? and by fo doing Loft 17 PROMISCUOUS QUESTIONS. I37 -percent., whereas I ought, in dealing, to have cleared 2© per cent. : Then how much under their jufl: value were they fold? - - - Anf. 28I is 8f|d. 46. If, by felling goods at 2"jd per lb, I gain cent, per- cent., what do I clear per cent, by felling for 9 guineas per cwt ? Anf. 50 per cent. 47. If 20 men can perform a piece of work in 1 2 days, how many will accomplifh another that is thrice as much in one 5th of the time ? - - Anf. 300. 48. A younger brother received 63C0I, which was juft ~ of his eldeft brother's fortune : What was the father worth at his death ? ... Anf. 14400]. 49. A perfon making his will, gave to one child \£ of his eftate, and the reft to another \ and when thefe legacies came to be paid, the one turned out 600I more than the other z What did the teftator die worth r* - Anf 20C0u 50. A father devifed T \- of his eftate to one of his fons, and ytt °f the refiaue to another, apd the furplus to his re- lict for life $ the children's legacies were found to be 257I •3s 4d different : Pray what money did he leave the widow the ufe of? - - Anf. 63 $1 io||d; 51. What number is that, from which if you take \ of -|, and to the remainder add T \ of 3^-, the fum will be 10 ? Anf. io^Vo- j 2. There is a number which if multiplied by ■§■ °f •£ °£ i|, will produce 1 ; What is the fquare of that number ? Anf. 1$$. 53.- A perfon dying left his wife with child, and making Tiis will, ordered that if it mould be ;a fon, \ of his eftate mould belong to him, and the remainder to his mother \ and if a daughter, he appointed the mother -|, and the girl the remainder; but it happened that me was delivered both of a fon and a daughter ; by which (he loft in equity 2400I more than if it had been only a girl : What would have been her dowry had (he had only a fon ? - Anf. 2icol. 54. Three perfons purchafe together a fhip, toward the payment of which A advanced £, and B ^ of the value, and C 200I : How much paid A and B, and what part of the vefTel had C ? Anf. A 90 j£I, B n6 T 4 T l, C Impart. $5. A and B clear by an adventure at fea, 60 guinea*, with which they agree to buy a horfe and a chaife, of which N I38 PROMISCUOUS QUESTIONS. they were to have the ufe, in proportion to the fums adven- tured, which was found to be A 9 to B 8 ^ they cleared 45 percent.: What money did each fend abroad ? Anf. A 74I 2s 4 T 4 T d > and B 65I 17s 7ffd* 56. In an article of trade, A gains 18s 3d, and his ad- venture was 40s more than h's, whofe (hare of profit is but 12s : What are the particulars of their ftock ? Anf. A 5I 163 9|d, and B 3I 16s g}d. 57. Three perfons entered joint trade, to which A con- tributed 240!, and B 210I; they clear 1 20I, of which 30I belongs of right to C : Required that perfon's ftock, and the feveral gains of the other two ? Anf. C's flock 150I, A gained 48I, and B 42J. 5S. A and B in partnerfhip equally divide the gain *, A 9 s money which was 96I I2s, lay for 15 months, and B's for no more than 6 : What was the adventure cf the latter ? Anf. 24 1 1 1 os. 59. Put out 420I to intereft, and in 6J year's time there was found to be due 556I 10s : What was the rate of inte* reft ? - Anf. 5 per cent. 60. A clears 12I in 6 months, B 15I in 5 months, and C, whofe flock was 40I, clears 21 1 in 9 months : What was the whole ftock ? - - - Anf. 1254L 61. A had 12 pipes of wine, which he parted with to B at 4J per cent, profit, who fold them to C for 40I 1 2s advan- tage ' 7 C made them over to D for 605I 10s, and cleared thereby 6 per cent. : How much a gallon did this wine coft A? ,- - ■ - • Anf. 6s 8/ T ^ T ° T d. 62. A of Amftetdam, orders B of London to remit to C of Paris, at 52-§d fterling a crown, and to draw on D of Antwerp for the value at 34^ flem. a 1 fterling •, but as foon as B received the commifiion, the exchange was on Paris at 53d a crown : Pray at what rate of exchange ought B to draw oa D, to execute his orders, and be no lofer ? Anf. 34s 2/ 7 d. 63. A with intention to clear 20 guineas on a bargain with B, rates hops at I5d a lb, which coft him 10-Jd ; B ,apprifed of that, fets down malt, which coft 20s a quarter, at an adequate price : For how much malt did they con- tract ? Anf. 49 qr. 64. A and B venturing equal fum? of money, clear, by ■ PRO-MFSCUOUS QUESTIONS. IJ^? joint trade 180I. By agreement A was to have 8 per cent., becaufe he fpeat time in the execution of the project, and B was to have only 5 : What fum was allotted to A for his trouble! 5 T - Anf. 41I ics 9/-^. 6$. Laid out in a lot of muilin 500I, upon examination: of which, 3 parts in 9 proved damaged, fo that I could make but 5s a yard of the fame ; and by (o doing find I led 50I by it : At what rate per ell am I to part with the undama- ged raufiin,. in order to gain 50I upon the whole ? Anf. us 74*1. 66. A at Paris draws on B in London 1400 crowns, at j;6d fierling a crown, for the value of which B draws again on A at 57CI (lerling a crown, befides reckoning commiflion i percent. ; Did A gain or lofe by this tranfaclion, and what? Anf. He gained Vji% crowns. 67. A, B, and C are in company 3 A pat in his fhare of the ttock for 6 month?, and laid cairn to ~ of the profits; B put in his for 9 months \ C advanced ccol for 8 month.*, and required on the balance -3- cf the gain : Required the flock of the other two adventurers ? Anf. A 185I 3s 8fd, and B 172I 16s 9^-d. 68 A young hare ilaits 40 yards before a grey-hound, and is not perceived by him till (he has been up 40 feconds, fhe feuds away at the rate of 10 miles an hour, and the dog on view,, makes after her at the rate cf 18 : How long will the courfe hold, and what ground will be run over, beginning with the outfetting of the deg ? Anf. 60 i\ fee. and 530 yards run, 69. If I leave Exeter at 8 o'clock on Monday morning for London, and ride at the rate of 3 miles an hour without ntermiffion ; and B fets out from London for Exeter at 4 the fame evening, and rides 4 miles an hour conftantly : Sup- pofing the diftance between the two cities to be 130 miles, whereabout on tRe road mall we meet ? Anf. 694 miles from Exeter, ^o. A refervoir for water has two cocks to fupply it ; by the firit alone it may be filled in 40 minutes, by the fecond tn 50 minutes ; and it hath a difcharging cock, by which it nay, when full, be emptied in 25 minutes : Now, fuppoiing that thefe three cockg ar& all left open, and that the water N z i 140 PROMISCUOUS QUESTIONS. comes in, in -what time, fuppofmg the influx and efHux of the water to be always alike, would the ciftern be filled ? Anf. 3 hrs 20 min, 71. A fets out of London for Lincoln, at the very fame time that B at Lincoln fets forward for London, diftant ic® miles •) after 7 hours they -meet on the road, and it then appeared that A had rode 14 miles an hour more than B : At what rate an hour did each of them travel ? Anf. A 7-?-|, and B 6?} milce. 72. A and B truck •, A has \2\ cwt of Farnham hops, at 2l 16s a cwt, but in barter infills on 3L B has wine worth 5s a gallon, which he raifes in proportion to A's de- mand. On the balance, A received but a hhd of wine : What had he in ready money ? - Anf 20I 12s 6cL 73. A of Amfterdam owes to B of Paris, 3000 guilders of current fpecie, which he is to remit to him, by order, the exchange 9id rlem. de banco a crown, the agio 4 per cent. \ but when this was to be negociated, the exchange was down at pod a crown, and the agio 5 per cent. : What did B get by this turn of affairs ? Anf. 5 liv 12 fol 8^/3- den. 74. If 100 eggs be laid down upon the ground, in a ftraight line, one yard from each other, and the firft of them one yard from a bafke't : What fpacc fhall a man walk over in bringing the eggs one by one to the bafket ? Anf. iroioc yards, or 5 miles J300 yds. A COURSE OF BOOK-KEEPING, ACCORDING TO THE METHOD OF SINGLE ENTRT; With a Description of the Books, and Directions for ufing them;. Very ufeful either for young Book-keepers entering into Bufinefs, or for Matters to teach in their Schools. TT it very neceffary that alraoft every perfon who is in* •*■ tended for bufinefs, fhould learn a courfe of Book-keep- ing of this kind, becaufe it is ufed in almoft every fhop. The Italian method alone is not fufficient \ for it is a con- flant complaint among the merchants, and others, who ufe this method, that their boys, having learnt only the Italian method, when they fir ft come to bufinefs, are almoft as ig* norant in the management of their Books, as if they had ne- ver learnt any method at all. There are fome boys who have not time to learn, or perhaps a capacity to underftand a complete courfe of the Italian method \ there are alfo many intended for fuch kinds of bufinefs, as that the Italian-method would be thrown away upon them : To all fuch then, this method will be very ufeful. And even foppofing a boy were intended for a bufinefs which requires the Italian method a. lone, I would. notwithftanding, have him taught this method firft, if it were only to facilitate his acquifition of the other. This method is fo eafy, that it may alfo be taught in a few week's time to young ladies as well as young gentlemen. The forms of the Books may be fufficiently known by in- fpeflion. — In the Day-Book, every perfon's name is fet down Dr. to the things he receives from you on truft, and Cr. by thofe which you receive from him. In 'the margin of the Day-Book are written the pages where the accounts fiand in the Ledger: Inttead of thefe marginal figures, fomc make only a ftroke or dafh with the pen, to (hew that the account has been polled, that is, entered in the Ledger} but it is better.. to ufe the figures, for they fhew, not only that the account has been ported, but likewife where to find it in the Ledger, without looking* ic the alphabet. In the Day-Book I have N,3 142 BOOK-KEEPING BY $IN<2Lfi ENTRY. fet down only the total amount of all the articles of each day, collected into one fura ; having purpofely omitted the amount or value of each {ingle line of article, every one of which the learner is to compute by way of exercife, and as it were in real trade, and enter in their proper columns in the Day- Book as he copies it out. Then the printed fum to- tals will ftiew him if he has computed the particulars rightly. I have entered in the Day-Book what is received as well as what is delivered, which is abfolutely neceflary in teaching; for the learner ought to make out all his own Ledger from his Day-Book. There are feveral other books, kept by mod merchants^ as the Cafh-Book, the book of Houfe-Expences, the Invoice- Book, &c. DireB'ions for the Learner, Having ruled your Books in the proper form, copy into the Day-Book one month's accounts % 7 then calculate them upon your (late or wafte paper, to find if they be rightly call up, and to exercife you in calculations. Next rule your flate or wafte paper in the form of the Ledger, and upon it poft the accounts that were copied in the Day-Book ■with their dates prefixed \ obferving to fet on the Dr. fide of each perfon's account, thofe accounts to which he is Dr. an the Day-Book, and on the Cr. fide, thofe by which he is Cr. And if any account confift but of one article, you are to exprefs it particularly, with its money, in the columns; but if of feveral, write to or by fundries, placing the fum of the amounts of all the articles in the columns. After the ac- counts are, by correcting if neceflary, placed according to the teacher's mind, tranfcribe them into your Ledger, lea- ving a proper fpace under each perfon's name to receive more accounts. Then under the proper letters in the alphabet, enter thofe names with the pages where they ftand in the Ledger ; and, laftly, write the Ledger pages to the feveral accounts in the Day-Book. Do the fame with the next month's accounts ; and fo on, till the whole be finifhed.— But obferve that you muft not enter any perfon's name down again which has been entered before, till the fpace firft af- figned to it (hall be filled with articles ; and then the ac- count muft be transferred to a new place, as you may obferve is dome with Lady Strawberry's account* DAY-BOOK, 143 When the firfl Ledger, titled A, is filled with accounts, you fnuft, as is done with the following Ledgers, transfer the unbalanced accounts to the fecond Ledger, titled B, and fo on, according to the order of the letters of the alphabet ; and at the end of the old Ledger draw out a balance account, placing your debts on one fide, and your credits on the other. THE DAY-BOOK. January I, i 783. Mr. James Elf or d of Bath, Dr. s d To 15 yds of fine broadcloth at 13 6 — 24 - fuperfine - — • 18 9 George Rob/on , Efq. of 2ori 9 Dr. s d To 12 gal palm fack - at 8 6 — 17 — red port - - —-58 — 9 — claret - — 8.9 Mrs. Mary Majlerman, Dr. s d To i£ lb green tea - at 16 o — 2y — congou - - — 9 6 — o£ done of fugar - — 5 c — A lump of fugar, wt 20-§- lb — o 8 Lady Strawberry , Dr. To 94. yds of fflk - , — 13 - flowered ditto s at 12 — i$ 3 2 *3 16 12 J 7 3i n M4 BOOK-KEEPING BY SINGLE ENTRY. \ : "r-/' ' \ January 20, 1783. — « Sir Jr.nas Moore, Dr. Fo a ream of thick polt paper 27 M . James Wilfon, Schvolmqfter, Dr. s To 6 Hutton's Arithmetic - at 2 — 3 dozcopy books - — 2 — 2 quires of foolfcap - — O 10 . 1 quire thin pod Feb. 5. Mr. Alderman Ableman, Dr. 1 To a ledger ruled — 5c quills - - at o — 3 reams thick poft - — 1 — 6 quires pot — 40 reams blue demy ^ - — o . 2 pen-knives and an ink-ftand — ■ ■■ — * ■ ■■ '- j. a — William Wintw, Efq. Dr. To 20 oz nutmegs — i 54 lb coffee, _ 3JL — cocoa _ * 4 — almonds- . 8 § — raifins s at o -T- 4 — 2 — 1 — o Mr. William Watf on, To 2 gal rum 4 — brandy j_ 3 _ Eng. gin 3 & 15 *5 17 Dr. s at 10 -- 10 — 5 d o 6 o ©AY-BOOK. 145 Sir Jeffery Slingstone, Dr. 02 dwt gr s cl To a Giver punch-! k ^ ^ . , l > wt 21 4 o at c ic — a tankard • 1036 — 62 — a tea pot and lamp 30 5 12 — 7 3 — 6 p!ate8 - 73 1 1 5 — 6 1 — 18 fpcons - 41 o 10 — 6 3 ■ Feb. 27, 1783. — &> Jonas Moore, Cr. By Cafh received of him m full — March 10 — 22 - George Rob/on, Efq. of York, Dr. s d To 27!- gal of (berry - at 6 2 — 22-5- — rhenifti - - — 6 4 — 34 _ LifDOn • —4 10 April 7 &*r Thomas Law/on, Dr. To 7-J- yds of fcarlet cloth — 4 — • fuperflne blue — oj — velvet "— 3° — ■ g°ld * ace s at 21 — 20 — 18 — 10 12 Za Dr. s d To 14 yds blue ribbon - at o 7-^ — 21 «— white - - — o 6 — \l\ — lace - — 3 6 9, pair kid gloves — 2 /ifr. James Wilfon, Schoolmfijler^ Cr. By ca(h received in full 34 12 10 Mr. Roger Retail of NewcaJlleuponTyne^ Dr. s d To 24 *r lb royal green tea at 18 6 — 21^- — imperial - — 24 o 3 Si — beii bobea - — 13 10 J 7i — coffee - — 5 4 — 25 — double refined fugar — 1 i-£ — 9 fugar loaves, wt 13.7 lb — .0 7^ 10 *7 ikfr. Charles Anderfon, To 6 mahogany chairs — 2 elbow ditto • 2 pier glaffes « » Dr. s at 18 — 25 ~ 3 6 83 I . 11 *3 -DAYBOOK. To 25 yds curtain (luff — 12 — tieking — 3 ftone of feathers .— 2 pie* tables July 24, 1783. Mr. Charles Anderfon, Dr at s d 2 2 — * 3 — 2$ .O — JO • si d 28 Captain James Dixon, Dr. To 12 bufh peas — 9 — beans — 17 — malt — 25 lb hops s at 2 ~3 — 4 Auguft 1 -- William Winton, Efq* Dr. To 10 grofs of bottles — 9 — fmall ditto — 2 doz of wine glaffes — 3 decanters s at 22 — 15 ~ 4 — 1 Mr. Alderman Ableman, Cr. By a note upon Dr. James for — cafh in full - 4 12 David John/on, Efq. Cr. By cafh, in part 16 Mr. Charles Anderfon % Cr. By 5 pockets of hops, at 48s O 12 .4 16 18 10 5 5° }5° Book-keeping by single entrt. Mr. Charles Anderson , Dr. To a mahogany bedftead — 2 ftools, at > pocker, tongs,, and fender • 2 other fets of irons Auguft 18, 1783. 5 3 21- ic m c o 5 10 Mr. Conrade Compound y of 'Exeter ', Dr s d To 2i| lb cochineal - at 29 6 -— 6j— -opium - - — 6 4 — • 53-f- — fcammony • ■ — 8 10 26 Mr. John Baker, Dr. s d To 5 grofs of brafs buttons, at 18 o — 2 — white - — 15 o — 7 doz pair o f buckles •— 2 2 a pair — 12 trunk locks , — o 10 each • 6 chamber ditto — 26 Sept. Mrs. March, Dr. To 8 farcenet hoods, at s d A 3 56 J 9 16 iWr. James Wilfon, Scboo/majter, Dr. s d To 6 Hutton's Arithmetic at 2 3 — «■ 1 thouf. pinions «— 3 doz copy books • -— 2 6 — 3 quires of thin pod • «~ I O — Lowtji's Eng. grammar « 2 6 4 10 DAY-BOClC, rjr f— «*— — . September 6, 1783. ■■ . ■ - ■■ ' ■ 2j Ztf^/y Strawberry, Dr. s d To 12J yds latin - at - 1 Or Mr. Nicholas Norton y Cr. By a bank-note for _ ;«Ju - 1 2 — — ■ Z#//)> Strawberry, Dr. s d To u£ velvet - at - 18 o James Wi/fon 9 Scboolmo/ler, Dr. To the Univerfal Penman _ X 6 k Afrf. iH/arf£ ofChefter, Dr. To 17 India fans, at 3* lod , !8 Mrs. Mary Majlerman^ Dr. To cam in full 22- Lady- Strawberry, Cr. By cafli received of the fteward 24- < Mr. Charles Anderfon^ Cr. By cam in full 27 Mrs. March, Dr. s d To 21 yds filver ribbon ^ at 2 2 — 1 if fine lace - jo 6 October 2 Mr. Samuel Edwards^ Dr. To 14 lb flax, at is 4 Afr, 2?. Barber ', Briftol, Stationer, Cr. By 30 reams offoolfcap paper, at 12s 6d * 6 Ztf$* Strawberry, Dr. To 27 J yds holland - at 5s 6d O 2 20 18 4 20 17 i$z BOOK-KEEPING BY SINGLE ENTRX, Q&ober 26, 1783. David Jo hnfan , Efq. Cr. By cafh in full -~— . 10 .— — Mr, Matthew Milton of Norwich, Dr. s d To 4Q ells dowlas - ail 6 — 34 3* diaper hglland - 1 41 -? 8 T 3' Lady Strawberry, Dr. Ta 40 yds of Irifh cloth, at 35 4d *5 Mr.. Henry fojier, Dr. To 2f cwtiron, at 185 94 * _ - 21 » ■ " ■■ " ■' • ■ Mrt. hiary Grey, Cr. 8y 3 ps of Irifh cloth, qt 87 yds, at 2s 2d 23 Mr. John Bahr, Cr. By cafh in part Mrj. March, Dr. To 9 pair kid gloves — 5 doz pair lamb's di-tto j 12 pieces of bobbin d 6 14 10 s d at 2 2, — 1 2 — o 6 25- Ztf*/y Strawberry, Cr. L>y cafh in full 27. George Rob/on, Efq. Cr. By cafh in full 39 37 *J DAY-BOOK. '53 ■ Oft. 30, 1783- -1 I s Mr. Samuel Edwards, Dr. To 12 lb of flax — 14 - - « d at 10 — 9 1 1 2 *4 3 2 7 1 9 Mr. Matthew Milton, Cr. By 30 gal brandy, at 8s 6d — cafh in full 10 ..._. _ -J . 1 r , t Samuel Sim /-/on, Efq. Dr. To 3 fugar loaves, wt 32flb, at 8fd 1 ■ ^ . 1.3 AZr. Ja/^j Eiford, Cr. By a bill for - 5 Captain James Dixon, Cr. By 3 ps of holland, qt 112^ ells, at h t rs 6d 5 Captain James Dixon, Dr. To ca(h in full - - IC Samuel Simpfon, Efq. Dr. To 15-flb of currants, at 4c! zz Mr. Thomas Grey, Dr. To 2 doz knives and forks at — a fet of china - * — 18 china plates — — . 3 dirties — — a. mahogany tea-board s d 2 3 4 6 10 J 03 154 BOOK-KEEPING BY SINGLE ENTRY. 10 20 To I7|lb Malaga raifins — 19! - raifins of the fun — 17 - rice *0 *I — .— — November 26, 1783. - Afr. Thomas Grey, Cr. By 42 ells of holland, at 5s 6d 28- Sir Jeffery SlingJlone> Cr. By cafh in full ... 29 ■ Samuel $impfon y Efq. Dr. x " P e PP e ? — 13 oz cloves s d at o J| — o 6 — © 3i — 1 6 — o 9 Dec. 1 Mr. James Wilfon, Schoolmojier^ Cr. By cafh in full - « - 3 Mr. Alderman Ableman^ Dr. To a pipe of wine 6 William Winton, Efq. Cr. By 30 gi*ll brandy, at 7s 6& — cafh in full ... 8 Mr. Thomas Hunter, Dr. To 3 chaldrons of coals, il 15s 10 Mr. William Watfcn, Ct*: iBy cadi in full __— ^mmmmim 12-— " Peter Thorn [on ofWorceJler, Efq. Dr. To f butts of oi), wt 55 cwt i qr 20 lb grofs, tare 18 lb a cwt, at 24I I OS a ton of 236 gal, ..and 7 -Jib neat to the garlloa +': ' • - • 36 20 72 DAY-BOOKi *ss 5 JO By cafh in full December 13, 1783. * Mr. Henry Fq/ter, Cr. *5_ Sir Thomas Law/on, Cr. By 3 c 2 qr 14 lb of tobacco, at 4! a cwt 18 Mrs. Shields, Dr. To a lamp of fugar, wt 22^\b y at 8|d 20- Samuel Simp/on, Efq. Cr. By cafh in full - - - ri 22 Mifs Fanny Daw/on, Cr. By ca(h in full n Mr* Edward Youngs D/\ To 3 cwt 1 qr cheefe, at 30s 24 « Mr. Roger Retail, Cr. By a bill upon Thomas Williams, Efq. for Mr. Shields, Cr, By cafn in full 29 Mrs. March, Cr. By cafh in full d *3 o^ 50 18 o: 6| 11 10 BOOK-KEEPING BY SINGLE ENtRT, Ledger A, THE ALPHABET. Aider* Ableman 3 Mr. C. Anderfon 6 D Copt, J. Dixon Mifs F„ Dawfon 5 6 G Mrs. M. Grey Mf\ T. Grey 9 10 N Mr. N. Norton P.Tbvmfon t Efq. 11 Mr J. Baker 7 Mr. C. Compound f Mr. R. Barber 8 Balance 1 1 Mr Mr E J. Elford . S. Edwards 1 8 H Mr. T. Hunter 10 Mn H. FoJIer 9 D. John/on, Efq. 4 M Sir T. Law/on qlSir Jonas More 2 M* Majierman 1 Mrs. March 7 Mr. M. Milton 9 O R Gy Robfon y Efq. Mr. R. Retail Mr. £. Young 1 1 L. Strawberry 2 2 8 Sir J. Slings tone 4 Mrs. Shields 5 Si Simpfon, Efq. IO. W Mr. y. Wilfon z W* Win ton, Efq r 3 Mr< W. Wat/on 3 LEDGER A. J .57. o moo j m o ^f o ^o _ O CO | CO j ^ \ co ^ i r>. un { «-<-> ! co ( 1 pq C4 u bo CI ■D > - ^ «4 j • « . ,bO M Q ***"• «- eg en u 3^ w o oq cq pq w2 CO CQ ci ^ CO co CO '? *^ o oo . oo • <% ^ EL G ^ ^ "s *C> <►-> * ^3 v© ^- J J3 - M r m j t«» «T t> o S • ft* «4 t • $ ■ 1 • 1 © 5 CO v> 60 m M % OO • Wi CO aj 4^ *"- c « r^. C B* r^ c C 04 K >H rj8 ? BOOK-KEEPING BY SINGLE ENTRT. -d o o I o *-« t m 1 ! °° 2 I ^ ' »°?!^ 1 co j r^ t m cs | co 6 y CO DQ 03 <5 ^ 3 23 S -5 •2.2 s CS CO OO J CO ,m «« 6 b" > *r> to n OJ cu no -a V. ' 1^ *-» >*» i2 -a G 3 m M o O FH u, h * On e^ vo d o -. >-l 1 "> > m to u G B 1 £ D ns T3 a B u .3 n -n: C3 •**! ^» o o o HHH r— <*- cs M >~l CO OO . * • r- c a* m w ti t"^ CO LEDGER A, ■*S9 >*& AO O 1 VD 1 O 1 o " * £T ° 1 S" I M 1 J^» 1 o 1 CO g 1 PQ ci in +j 'C 3 c o CO 03 ■£ 4 a *o .5* o o ,.HH d *X3 •£ .CO g . ' »o co ro , CO M « J{ H3 *9 C C Q O CO \ £ >> m ^ vO O j ^O ' 1 ° „ i>. o | r- co r^ j m 1 t^ t o-> | o i . >^ >N fa? 03 CO o o W 03 H cc a: aQ 1 DQ CQ M OO o ^o c* \o ' c w M M M M ft: CO to ^ . ••5 30 2 > £ ^-^d I H 3^ "SO " *> >^ >% >, W pq oq CQ CQ £ 5 5$ *0 'ON CO 3 CO oo j; it 1 1 ^0~ 1 H* [jo 0\ i lO^O 1 ~ wo,* | ^ H S m 2q W 1 £> atfa BOOK-KEEPING BY SINGLE ENTRY. -d O i O OS | C\ | O 00 |00 en *£ i O *0 ; ^ | o t^ i ^ - tf- t o *o j CO 1 -n co 1 oo 1 ttJ &* 1 ■J <3 oo ft: 1 ^ o •a r— • ♦J c2 §■ co 3 C . C m ^.S :> — . 3 >» p ••■ , eg * 3 8 fc S.* ^N >> N >-» >* 03 fes oq oq PQ03 . oo u & M o Q m rj o v. »© T3 O I 1 o ^ i • m V> in iA a «j V V 4) J) S3 £m J- V4 M *« . u T3 ^ T3 -O "3 a Q . C .a n c c a 3 3,3 v3 ti <;d^ o • o o o o H H hh^ CO o H r>- ^-co "*' mOn co CO «*> OO >> OO ►> ^ §> M a M S3 ~A < ■*-i Hn LEDGER to PQ ».. no N >% oq cq pq CO ci « 1 ** 1 o — ^1 OVO 1^2 I 00 S3 ,3 £> "*> oo "0 *" 1 o | O l H 1 m rooo ^ | oo f -* 5 *Q T3 *<* ^TJ s 1 i *l • 1 «* CO i t i » »• S.2 CO u% V u U lv W *0 w V Q Q T3 no C oo * c c £v2 o o o o o o p h Kt" KH «-* VO ror^t> 0* cs M M Ci co CO «"> oo ba DO buO 00 il m t*>» ■ 3 r>. 3 *^ cu O < H < M 4* CO 1 64 BOOK-KEEPING BY SINGLE ENTRY. ^3 ! 1 liPj <3- ^ 1 O ^ M * » M <2 OS - CO « H3 vo M n U c* bjO. H nj «M G r« • a! co $ $ ViJ vS r— * O <2 m O w2 u -sT i d u r2 O CTJ w "St rt" u t~» k ►. v. t>-> PQ ^3 DQ «• P3 •i? ^* C\ •^J ci 1 k3 > CO CO M -« j O ~ O M 1 M 1 OO 1 ro o o <3 • $ 0) WD «J *-« CO r- 5 .O CJ a o u 8 £* so so *0 CO £ O ^3 *-* ***• C O o o -a +^ J2 >-« OS ^ o CO OO M CO © 000 CO vQ co <3 o. LEDGER A. 1*5 ^ ■ ^ | 1 2 vo m c* vo | 00 -? H m 1 ON CO X 00 ■ £ £ o C 02 .5 U 00 <3 <5 CO ^j CO o CO oo o VO ~ob~" 1(56 BOOK-KEEPING BT SINGLE ENTRY;, •tf H £ 1 O 1 O v> S?.l 1 *5 - "M 1 ««i * s J3 C^ £ PQ- o u o co oo o J5 i9 na Hint -It | H>N | O 'Z r— r-+ -r-t o o o O O HHH hh *^q o<» " ■ c* t cs ct ci ,£° CO co 00 > o OO • 00 i > o O u u CO o 00 LEDuER A. 167 ^ti VO ovovo 1 - N 1 £1 ^00 O j - S-l "^ 00 on <3- i -» >-, >s PQ 03 oq ~0 *f [ VO 1 1 -■« 1 S* "^-0 ON^^ONr^^-^c^r^i -. r* " «> P § 3 *-U«L-*-« v -'*" , l-*- , *"**-« OOOOOOOOOOO CO CO 00 [ 1 68- I Ledger B. THE ALPHABET. A &der. Ableman i 1 B Mr. J. Baker 3 Mr. R. Barber 3 C Mr. C. Compound 2 D E Mr. J: Elford 1 Mr. S. Edwards 3 F G Mrs. M. Grey 3 Mr, T. Grey 4 H Mr. T. Hunter 4 I K L Sir T. Law/on 1 M N Mr. N. Norton 2 O 1 P CL R Mr. R. Retail 2 S T F.Thompfon, Efq. 4 V W X Y Mr. E, Young 4 1 1 z LEDGER B. ■a c. »V ^J • *5 eq •^3 l* & 1 O 1 s «>*- 1 o 1 ? tf*r- «*> 1 1 S < S3 < Q <£■ i- £ u £ lx p u C5 V bO -5» btO r* 3 bO 1 (1 •*5 no H 4 .o CQ «a Wm Vm «■( <*-> ♦j a cd rt *-» 4-1 4-* a a c 3 , 3 , 3 o w o w. o o u Q u Q u N cj CO o o o E* H H ^ ^ ^t- oa oo oo r>» r* r>» **% ." M K-- BOOK-KEEPING B? SINGLE ENTRTT. 4 6- 5? ■3 3 4 ff ^ £" *. 1- UP> ^5 — £ ^ On i CO CO VO * 4 < 55 M 6 %1 p «l «5 twO 03 H3 -« « *~* i vo si • <2 u2 CO !§ 4-» C8 Ctf 4* 4-* c C3 fl a S3 S3 o O O o u Cf o U u O 1! U u CO p K. N Tf *•• *• f~\ 00 oo ' 00 c^» r>» r* Qti M H **>., v^ > LEDGER B. I?* vo i { OO 1 C\ - 1 M hi ^ 4J ■a ^ A C3 i <* 3 O u p ^ o o u J •^ cfl OS > ' It ^ CQ ^ QQ CO ' rj- s% rj- ^ co OO Hi H «4j * ^ ao v. < **»* *» < s u at u u £ q =1 u 3 nt3 U oo r-4 *8 «5 1 1 +* +■» 03 ♦ « J-» 4-» C a a • =J o v. o L. V. u u q Q Q H w O o H H t*- nr 00 OO * gs r^ j •-« M BOOK-KEEPING BY SINGLE ENTRY. I i I vo Si T3 ( ° "$ « CD 1 - C* £- — 1 * £ 1 * u Q Q a o o o «J o k m a o u o a o be H3 3 O u o etf O rf- *f *f oo oo oo r-» 1-- *^ M M M I 273 1 BOOK-KEEPING BY DOUBLE ENTRY; Or, according to the Italian Method. THIS method is faid to be by Double Entry, becaufc every article is twice entered in the Ledger, viz. on the Dr. fide of one account, and on the Cr. fide of another : and it is called the Italian method, becaufe of its having been invented in Italy. In this method we generally confider three books, viz. the Wafte-Book, the Journal, and the Ledger, of each o£ which I mall give a fhort account. I. Of the Waste-Book. The Wafte-Book contains an inventory of a perfon's ef- fects and debts, with a diftincl: hiftory of all his tranfaclions in a way of trade, narrated in a plain, fimple ftile, and- in order of time as they fucceed one another. The Wafte-Book opens with an inventory of the perfon's effects and debts, which, when he firft fets out in trade, is to be gathered from a furvey of the particulars that make up his real eftate ; but ever after, it is to be collected from the balance of his old books, and carried to the new ones. After this foundation is laid, all tranfaclions in trade are fet down ftmply as they happen, fpecifying their dates, the names, &c. of the perfons dealt with on truft, the condi- tions of bargains, prices of goods, with the fums of money, -or any thing elfe that may be neceffary to make the tranf- aftions eafily underftood \ the manner of doing all which will appear from a view of the following fpecimen of a Wafte-Book. II. Of the Journal. In the Journal, or Day-Book, the tranfactions recorded in the Wafte-Book are prepared to be carried to the Ledger, by having their proper debtors and creditors afcertained and pointed out. It agrees with the Wafte-Book in the form or manner 6t ruling, of dating, and in the order of fuoceffioti of the *o 174 BOOK-KEEPING BY DOUBLE ENTRY. counts according to their dates ; but differs from it by ha- ving the debtors and creditors of all accounts fpecified. On the right-hand margins of each folio or page of tho Journal and Wafte-Book, are ruhd three columns for; pound3, (hillings, and pence, and on the left-hand margins, a column to receive the figures exprtfling the folios or pages where the fame accounts are entered in the fuccceding book, viz. in the Wafte-Book margin are fet the correfponding Journal pages ? and in the Journal margin, the Ledger pages. III. Of the Ledger. In the Ledger the feveral articles of each account arc collected from the different pages of the Journal through which they are Scattered, and difpofed all in one place ap- pointed for them \ the debtor parts of the accounts being placed on one fide of the folio, and the creditor parts of the fame accounts on the other fide of the fame folio, and di- rectly facing the former parts, by which difpofition, all the tranfaftions relating to each account appear at one view, and in one place. From the other Books then, the Ledger differs very con- fiderably : Each folio or page of it is divided in the middle, from top to bottom, into two equal fpaces or fides, the left- hand fide receiving the debtor articles, and the right fide the creditor articles of the accounts ^ being titled, at the top, Dr. and Cr. refpedlively } with the name or title of the account in the middle between, and upon a line with, the faid titles of Dr. and Cr. The right-hand margin of each fide is ruled into three columns for money, 2nd one for the figures exprefTing the folios where the fame articles ftand on the other fide of the folio of fome other accounts 5 and on the left-hand margin is formed a column for the dates of the articles. The Ledger hath belonging to it an Index or Alphabet,, in which the titles of the accounts are entered under their initial letters, with the figures of the pages in which they ftand ) by means of which they are readily found. GENERAL DIRECTIONS. t*J J DIRECTIONS FOR THE LEARNER. Having ruled your Books according to the forms of the following fpecirnens, copy into your Wafte-Book the firft month's tranfactions, as they ftand in the following Wafte- Book, but omitting the left-hand marginal figures, which are to be inferted according to the directions afterwards gi- ven. Then calculate every article to find if they be right, and to makeyourfelf ready at calculations. Thcfe articles are then to be entered, one by one, in the Journal, and according to the Journal form ; and as foon as any article is entered in the Journal, turn to the fame arti- cle in the Wade-Book, and directly againfl it in the margin write the number of the folio where it ftands in the Journal. As to the form of entering in the Journal, having written the date in the fame manner as in the Waile-Book, obferve the two following articles : i. In a fimple poll, viz. in which only one debtor and one creditor are concerned, let the Dr. be exprefsly mentioned, then the Cr., and laftly the fum, all in one line ; below which infert the narrative, or reafon of the entry, in one or more lines, in a full and particular manner, fo that the whole of the tranfaclion may be eafily underftood. 2. In a complex poft, let the fundry Drs. and GrH be ex- preffed in the firft line by the word fun dries, and the reft of the line filled up as in the former cafe ; after which, let the feveral Drs or Crs. be particularly mentioned, each in a line by itfelf, with their refpeclive fums adjoined to them j which are to be added up, and their total carried to the money columns. In this cafe obferve alfo that, in men- tioning the feveral Drs. or Crs., the Crs. take the word To before them, but the Drs. are expreffed limply, without any word prefixed. Farther, »n any entry, to know what to make Dr., and what Cr., obferve the following rules : 1. Things received are Drs., and things delivered are Crs,: conlequently, a thing received is Dr. to the thing given for it. 2. A thing received on truft, is Dr. to the perfon of whom. it is received. Q,2 1J& BOOK-KEEPING BY DOUBLE ENTRT* 3. The perfon to whom a thing is delivered on truft, mm Dr. to the thing delivered. 4. In antecedent and fubfequent cafes, parts that are na- turally the reverfeofone another, are alfo oppofed in re- fpeft of terms. 5. In cafes where perfonal and real Drs, or Crs. are want* ing, the defect muft be fupplied by ficlitlous ones, accord- ing to your judgment. 6. In complex cafes, the feveral Drs. or^ Crs. are to be snade out from the preceding rules jointly taken. The fictitious Drs. and Crs. that are moftlyufed, are, 1. Stocks which is made Cr. lor the effecls in the inventory, and Dr. for the debts in it. 2. Profit and Loft, which is made Dt- for every fum delivered for which nothing is expected as an equivalent, and for loffes upon bargains or goods 3 and Cr. for the contrary fums, viz. for all gains upon goods, &c. 3< Voyage . fuch a place, which is made Dr. for things fent as an adventure, and all expences upon them \ and Cr. for their nett proceeds. And feveral others,, aa may be feen in the following courfe of Books. Having finhned the Journal entries for the month, yoiv then proceed to poft them into the Ledger \ by writing down every article twice, once on the Dr. fide, and once on the- Cr. fide, viz. on the Dr. fide of that account which is Dr. and on the Cr. fide of that which is Cr. ; having firtfc appoint- ed fpaces for thefe accounts by eflimation large enough to- contain all their articles, and titled them according to the defcription of the Ledger, and entered fuch titles in their proper places in the alphabet, immediately on writing them/ in the Ledger. In. entering the Dr. fide article, you write down the name of the Cr , preceded by the word To, and immediately after, in the line, the condition of the article, fuch as price, quantity, &c« Placing the whole : fum of money in the columns with the folio figure where the Cr. ftands, and prefixing to the whole its particular date : And in entering the Cr. fide article, write down the name of the Dr., preceded by the word %, writing the Dr. folio number, and the date, money, &c. as before. Thif done, turn to the article in the Journal, and againft it in the left-hand margin write the Dr. and Cr. folio aumbsr*, GENERAL DIRECTIONS. 1^7 die former above the latter, with a line between, like a vul- gar fraction $ but when there are feveral Drs. in a Journal poll, the Cr folio number need be but once entered, viz. un- der the loweft Dr. number •, and when there are feveral ,rx. f write the Dr. folio number above the upper Cr. number only. Having, in this manner, polled all the journal entries for the month, in the Ledger, write another month's ac- counts in the Waite-Book, from thence to the Journal and thence to the Ledger, as before j after this, go through the next month, and fo on, till the whole be finifhed; tranfpofing from one folio to another, in the Ledger, fuch accounts as happen to be too large to.be contained in the fpace* firii af- figned them, as is done with the Cafih and Profit and Lofs accounts in the following Ledger. Having ported all the accounts into the Ledger, let thenx be balanced thus : Let all accounts have their fums of mo- ney on both fides added up into two totals : but before thefe totals are fet down, if they be unequal, let the fides be made equal ( n e. balanced) by writing down on the lighter lide as much as will make it equal the heavier, which fum is generally charged Dr. to y or Cr. by either Profit and Lofs or Balance, thefe being the two accounts by which moft others ar< balanced : All cafes in which any thing is either gained or loft, being balanced by means of Profit and Lofs ; and all thofe in w T hich goods or any thing remains in your hands, or in which any thing is due to or by you, are clofrd by means or the B alance account, which is drawn out at the end of the Ledger to receive all fuch balances into 7 and the articles clofed by Profit and Lofs being carried to the Profit and L f fs account. By thefe means all the gains and lolTes are collected into one place under the title of Profit and Lofi, and all the effects and debts into another, under the title of Balance ; and therefore by doling or balancing thefe two general accounts by means of the flock account, and carrying the two equalling fums to their proper fides of the flock account, thofe two fides then added up, will be exactly equal to each other ; otherwife the work is fomewhere wrong, and mutt be examined till the error be rectified Some accounts are clofed by me*ns of both Profile ana Lofs^ and Balance /for where any thing remains on hand, it is. firit r?8 BOOK-KEEMNG B7 DOUBLE ENTRT. fet down on the left fide by charging Balance with it at prime coft, or the original value ; after which the whole account rnuft be equalled by Profit and Lofs. The Books are then finifhed, and in beginning a fet of new ones, make out the inventory in the new Wafte-Book, from the balance of the Ledger •, for the inventory confifts of all the articles in the balance account of the old Ledger, Additional Obfervations. Inftead of writing in the direction of the length of the page, after the manner of the Ledger printed in this book, you may eafily vary or alter the form of your Ledger as you pleafe, ruling the columns either acrofs the leaves, or from top to bottom ; and in this cafe, placing the Dr. articles on the left-hand page, and the Cr. articles facing them on the- right-hand page, if the leaves of your book are very narrow^ but if their breadth will admit of it, place both Drs, and Crs* on one leaf, dividing it down the middle for that purpofe. Some teachers think it beft to ufe only a Wafte-Book and Ledger ^ and thofe who think fo, may eafily direft their pupils to poft immediately into the Ledger from the fpeci* men of Wafte-Book here printed, and omit the Journal. And of this opinion is an ingenious preceptor who fent me the fpllowing collection of plain and particular rules or ca- fes, which may be ufed either with or without a Journal. Rules for making Transfers from a Day-Book to a Ledger ■, according to the Italian Method of Double En try > adapted to thefpecimen in p. 184, &c. By Mr. A. C&ocker. CASE I. THE firft transfer which is to be made, is of the flock, &c. which you have in hand, or whatever you owe, at the time you begin your accounts.— You rauft, therefore, make each article which you are poffeffed of, (exprefliqg the prices per yard, cwt, &c, as well as the quantities) deb*. GENERAL RULES. 179 tor to ftock ; the fame of a perfon who owes money ; and then make ftock creditor by fundries for the whole value, or by each article Separately for its value.— Alfo, you muft make ftock Dr. to fundries for what you owe y and each ar- ticle which you owe for (exprefling the prices and quantities, or perfon whom you owe any thing to) creditor by flock. 2. When you buy goods for ready money.— Make the goods (exprefling price and qaantity) Dr. to cafh j and cafh Cr. by the goods, without expreiling the price per yard or quantity. 3. When you buy goods on credit.— Goods (exprefling price and quantity) Dr. to the perfon bought of, and the perfon Cr. by goods. 4. When you buy goods for part money and part credit, —Goods (exprefling price and quantity) Dr to fundries ; and cafh Cr. by goods, for as much as was paid; and the per- fon of whom bought, Cr alfo, for as much as remains due.— Jf part be paid by bill drawn by you, or Tome other perfon, you muft make the perfon, on whom it is drawn, Cr. by bill, exprefling its value, and to whom payable. 5. When you buy goods and pay a bill on another for their value. — Goods (exprefling price and quantity) Dr. to the perfon on whom the bill is drawn ; and the drawee Cr. by bill on him, payable. to the perfon from whom you bought goods. 6* When you buy a fhare of a fhip — The (hip (callingit by its name) Dr. to cafh, (exprefling your fhare ;) and cafh Cr. by fhip, mentioning your fhare and fum paid. 7. When you take goods from another beyond the feas at your own rifk. — Good? (exprefling price and quantity) Dr. to voyage, mentioning from whence \ and the voyage Cr. by goods 8. When you receive goods to fell for another, and pay any charges for freight, &c. — The perfon's goods Dr. to cafh, (exprefling for what-)) and carh Cr. by freight on per- fon's goods. 9. When you fell goods for ready money. — Cafh Dr. to goods, without mentioning price and quantity, and goods (exprefling price and quantity) Cr. by cafh. JO. When you fell goods on credit,— The perfon to tffrX BOOK-KEEPING BY DOUBLE ENTRY. whom fold, Dr to goods, (the price and quantity of which may, or may not be expreffcd j) and goods Cr. by the per- fon (exprefling price and quantity.) ii. When you fell goods, for part money and part credit. ~— Ca(h Dt\ to good3, for the fum received ; and the buyer Dr. to goods for the remainder ; and goods (exprtfiing price and quantity) Cr. by fundries. 12. When you barter one kind of merchanclize for other kinds of merchandize of equal value.— Make goods received Dr. to £Oods delivered, (exprefling price and quantity }) and g^ods delivered Cr. by goods received, exprefling alfo price and quantity. 13. Vh?n you difpofe of goods for part barter, and the remainder caQi. — Goods received Dr. to goods delivered, for the value of goods received, and cafh alfo Dr. for what you receive ; and goods difpofed of Cr. by fundries. 14. When you fell goods, and receive a bill on another perfon for payment. — The perfon on whom the bill is drawn, is made Dr. for a bill received of the drawer, or indorfer, and goods (exprefling price and quantity) Cr. by bill, ex- prefling on w T hom drawn. — Or bills receivable may be made 15 When you fell goods, for part cafh, and the remain- der a bill — Cam Dr. to goods for what is paid in cafh ; and bills receivable Br for the other part; and goods (exprefling price and quantity) Cr. by fundries. 16. When you fend goods to a factor on your own ac- count. — Voyage, or adventure Dr. to goods, exprefled ei- ther by the, woid fundries, or each particular, as you will; and goods ^expreffed particularly, as well a& the price and quantity) Cr. by voyage or adventure : alfo cafh Cr. by char- ges of cuftom, &c. 17. When goods are fent-to a perfon defiring him to fell them for you, or return them.— Sufpence account Dr. to goods; and goods (exprefling price anu quantity) Cr. by iuipence account. 18. When you fell for ready money any goods which were fent to you to fell for another. — Gafh Dr. to the per- ron's goods, and the perfon's goods Cr. by cafh. 19. When you have fold all, and charge the perfon €ommiflion.«— Perfon's goods &r. to profit and lofs for GENERAL RULES. iSf y€urxommiflion, and profit and lofs Cr. by the perfon's goods. 20. When you fend goods to another perfon, beyond the feas, to fell on your account.— Voyage to the place where fent Dr. to fundries} and goods, under each title (expreffing, price and quantity) Cr. by voyage to faid place } alio cailv Cr. by charges of cuiloms, &c. 2i, When you fend a perfon goods, to fell, io partnership with yocrfelf. — The perfon's account in Co. Dr. to goods, for your own (hare,, and perfon himfelf Dr. to goods for his fharc, exprefling in bothy price and -quantity - } and goods (exprefling price and quantity) Cr. by fundries. 22. When you fell goods in psrtnerfhip, — Calh Dr. to goods in company, mentioning the partner's name j and goods (exprefling price and quantity) Cr by cadi. 23. When you fell goods in partnership on ere.*-:':. — Per- fon to whom fold Dr* to goods in company y and goods (ex- prefling price and quantity) Cr. by the perfon to whom fold. If part be paid and part remain due,- the perfon and cafh are- refpeclively Dr. to goods in company, and the goods Cr. by fundries. 24. When, goods in company, which have been fo ? d on credit, are paid for. — Cafh Dr. to the perfon, and the perfon Cr. by cam- 25. When you pay money. — The perfon to whom it is jaid Dr. to cafh, (exprcfnng whether in full or part j) and- cafh Cr. by the perfon. 26. When you pay money, or difcharge a debt by draft, ©r bill on forne other perfon —The petfon to whom the bill is delivered, Dr. for a bill, (exprefTmg on whom drawn, and the fum ;) and the perfon on whom it is drawn, Cr. by bill payable to him to whom it is ol^ivered, exprt fling the fum. 27. When you lend money en bond, or other fecurity* —The peripn to whom it is lent, (exprefUng the fum and fecurity ) Dr. to cam ; and cafh Cr. by the borrower, expreflV \ng the fecurity. 28. When yeu pay fervanfcs' wages, houfe expences, &c 9 — Profit and lofs Dr. to ftrvants' wages, &c. and cam Cr. by fervants' wages, $tc. 29. When you receive money which was due to you.— Cafh Dr. to the perfon who pays - 7 and. lh? perfon Cr.. me^t lioning whether, in pa.it or in full.. 1 81$ BOOK-KEEPING BY DOUBLE ENTRTi 30. When you receive a (hare of the profits of afhip.— Cafe-"' Dr. to fliip ; and (hip Cr. by cafh for the (hare of the profit, 31. When you receive intereft for money lent. — Cafh Dr. (expreffing for what) to profit and lofs y and profit and lofs (exprefiing for what) Cr. by cafh. 32. When you receive cafh for a bill before due.— Cafh Dr. to bills receivable, (expreffing from whom $) and bills receivable Cr. by cafh, expreffing likewife from whom. 33. When you have a legacy left you* — The executor j0r o to profit and lofs ; and profit and lofs Cr. by executor. 34. When you pay charges on a voyage to or from a place.— Voyage Dp. to cafh, (expreffing for what \ and cafh Cr. by voyage. 35. When you receive a freight of a (hip, — Cafh Dr. to* ihip ; anu (hip Cr. by cafh, (mentioning for freight.) 36. When you infure a (hip. — Ship Dr. to cafh for innK ranee ; and caih Cr. by (hip for infurance. 37. When you pay for repairs of a (hip. — Ship Dr, to cafh for repairs ; and cafh Cr. by lhip for repairs. 38. When you pay charges on goods in company. — Goods in company Dr. to cafh ; and cafh Cr: by croods in company. 39. When you pay your partner his (hare of neat pro- ceeds, on goods, in partnerfhip fold ; deduct commiffion and charges, and make perfon Dr. to cafh, for his (hare of neat gain on goods ; and cafh Cr. by perfon for his (hare of neat gain on goods. Alfo, cafh Dr. to goods for the whole, and goods Cr. by cafit. 40. When you receive principal and intereft of a bond, or other fecurity, — Cafn Dr. to perfon paying money, (men- tioning the fecurity;) and perfon Cr. by cafh in full for prin* oipal, and profit and lofs for intereft. 41. When a Dr. fails, and makes a compofition — Cafh Dr. for what you re c eive, and profit and lofs Dr for what you lofe by him •, and the perfon himfelf Cr. by fundries in full. 42. When you draw a bill on a perfon.— Cafh Dr. to drawee, and drawee Cr by cafh 43. When you barter goods of different forts, for differ- ent forts of other goods-— -Perfon with whom you barter, JDr. for fundries j and each fort of goods delivered (expreffing price and quantity) Cr. by perfon •, again, each fort of goods received (expreffing price anc quantity) Dr. to perfon, of whom received y and the perion Cr. by fundries. r €JEN ERAL RULES. 1*83 ,44. When goods are barttred for other goods of equal value. — Goods received (exprefling price and quantity) Dr. to goods delivered (the price of which, may or may not be exprefled) and goods delivered (exprefling price and quan- tity) Cr. by goods received, the price of which may or may not be exprelTed. 45. When returns are made of a fufpence account.— Goods returned (exprefling price and quantity) Dr. to fuf- pence account; and fufpence account Cr. by goods returned,, without exprefling price and quantity. 46. When you receive advice of the fale of goods, fent on & voyage. — The perfon to whom the goods were coni]gned s Dr. to the voyage j and the voyage Cr, by the fame perfon. 47. When the petfon, to whom you made confignment of any goods, makes a remittance in goods.— Voyage from the place Dr. to the peifon on account current; and the per- fon Cr. by voyage from the place, from whence he fendeth the goods \ but if the remittance be made by bills, the per- fon's account current Cr. by bills receivable $ and bills re- ceivable (mentioning on whom drawn) Dr. to the perfon's account current. - 48. When you receive goods confined to you.— Goods, (exprefling price and quantity of each fort) Dr. to voyage, &c. and voyage Cr. by fundries. 49. When you buy a fhip in partnerfhip with another, —Ship (calling it by its name) Dr. for your mare paid 5 and calli Cr. by fhip 50. When a perfon to whom you have delivered goods to fell in partner fhip, has fold them, and made return — Cafh 9 or goods received ^r. to the perfon 5 and the perfon Cr. by cafh or goods. 51. When you fend goods to a perfon according to Ins order.— Perfon to whom fent Dr. to goods, (exprefling ,price and quantity) and goods, (exprefling price and quan- tity) Cr. by the perfon. — If there be fundty forts of goods ordered for, and luch as you have not got in your ware- houfe, but procure them upon commiflion, you may make the perfon's account current Br. to fundries j and each arti* cle Cr. by the perfon's account current. All or moft of ,thefe xv*les are applied in the following ac- counts, **4 BOOK-KEEPING B? DOUBLE ENTItfr. SHE WASTE-BOOK. London, January- j, 1789. 500 225 o c o c } } } An Inventory of the money y goods , and debt \ due to or by me ,<\— . B — I have in ready money 300 yards fuperfine broad "} cloth, at 15s a yard 3 1 1200 yds linen, at 2s 6d 7 a yard - - £ 800 pieces lead, weigh- ~ ing in all 44 ton, at 16I a ton 25 puncheons rum, at 3 81 } a puncheon - 3 12 hhds fugar, contain ing 140 cwt, coft f of the fhip Endeavour William Johnfon owes me, per note due the ift of March James Gibfon, per bond, with interefr, at 5 per cent., from the ift of November laft ijo o o 704 o c 950 o o 3°4 300 6 c o o 200 O 500 o o I owe as follows : To Edward Young on demand —Charles Wilfon, Efq. due the! 1 2th inftant - $ ■William Mercer, per account §33: 120 c 7 4 <=> 65 10 6 27214 ; 2 ) i — WASTE^BOOK, Jan 4, 1789. Bought 4C0 yds fhalloon, at is 3d a yard J 25 I Paid Edward Young in full 12 Bought of Ifaac Onflow 18 hhds Oporto wine, at 9I a hhd *7 Paid to Charles Wilfon, Efq. in full Sold 150 yds linen, at 3s 2d a yard 24 « Bought of Timothy Ciarkfon 12 bags hops, qt 40 cwt 2 qr, at 46s a cwt, payable in 2 months • 3° Bartered 5 puncheons rum, at 40I a pun cheon, for 20 hhds Lifoon wine, at 10I a hhd i*5 d o 120 162 87 2^ 93 200 Feb. 2. Sold Thomas Draper 100 yds of broad cloth, at 18s a yard 6 2 Shipped on board the Diligence, Captain- Tempeft, for Jamaica, the following goods, addreffed to Abel Fa&or, on my account, viz. 800 yds linen, at 2s 6d a yard 100 O 200 — broad cloth, at 15s 150 O C 8 pieces of holland, bought ofl Tho. Draper, at ioi a piece} l&1 ° ° Paid, duty and fees, &c. - 21 4 2 Ditto for infurance of 400I by*l Hazard and Co. at 5 per cent. ^ 2 ° oc R 443 4 J 5 o l$0 BOOK-KEEPING BY DOUBLE ENTRY. 3, Bought of Charles Wilfon, Efq. 1500 yds Scotch linen, at 2s 4c! a yard Paid him part in money - 50 o o Given him a bill on Thomas 7 Draper for . • | 5© o o Reft due in a month ■ 75 o o Feb. 8, 1,789. ~ 12 Sold to Ifaac Onflow 20 puncheons rum, at 44I a puncheon 19 Given William Mercer a bill upon Ifaac Onflow for - - 23 Bought of Thomas Draper 20 pieces hoi- land, at 18I a piece 27 Paid Thomas Draper in full — March 4 -«• Received of William Johnfon in full Lent James Dixon, upon bond at 5 per cent. Sold Charlas Wilfon, Efq. 20 tons lead, at 19! » 14. Sold David Robinfon 54 cwt fugar, at 2I 16s Received in part • - 100 00 Reft due in 2 months " • 51 4 o s d *75 880 5° 360 .472 200 1000 380 *5* U) wasts-eook:., 4 Sold Eugene Arden 23 cwt 2 qr hops, at ?os a cwt, for payment of which he has given me a bill on Timothy Clarkfon, payable at fight March 14, 1789. 22 Bought of George Wood 500 yds broad cloth, at lis 6d, for which have given him a bill on Ifaac Onflow *7 Paid William Mercer in full 3° Said my fervants their quarter's wages, which, together with the expencesof my houfe and pocket, &c. for the laft quar- ter, is in all April 2 Bartered 300 yds broad cloth, at 145 yard, for 10 pieces Indian chintz, of the fame value, at 21I a piece , , — , 5 ; The owners of the fhip Endeavour have fettled the accounts of the faid fhip, and paid me my (hare of neat gain from Mi- chaelmas to Lady-day Received of Ifaac Onflow in full Bartered 15 tons lead, at 18I a ton, foi the following goods of the fame value, viz. 200 lb tea, at 1 2s a lb - 120 o c 12 bales muflin, at 12I 10s a bale 150 o J S d 58 312 *s *5 10 10 i 93 210 no 94 355 *3 10 270 34 i38 BOOK-KEEPING BY DOUBLE ENTRY. (5 1 Bartered 15 hhds Lifbon wine, at ni 10s a hhd, for l cwt cochineal, valued at 120 o c The balance I have received in 1 > 52 10 o 3 3 TI No April 24, 1789. money M ay Bartered with Thomas Young, 250 yds of linen, at 3s a yard And 9 tons lead, at 18I a ton For 120 lb cinnamon, at 8s 4d 7 alb - - S And 1 2 bag 5 ; cotton, qt 34 cwt 1 2 qr, at 5I 10s a cwt 3 37 162 10 o o c 50 o o 120 IJ O Sold 40 cwt fugar, at 58s a cwt, for which received a bill for 50I on Samuel Ward, cue in 30 days 172 199 aud 661 in ca(h S board the Speedwell, 140 162 o o o o Shipped en board the Speedwell, John Gibfon, mailer, by order, and for account of Timoleon Jar.fen, merchant in Leg- horn, the following goods, marked, and numbered as per margin, viz. 200 yds my own broad cloth,! at 14s - - y to ton lead, prefently bought! for ready money, at 16I 4s 3 16 pieces drugget, at 7I 7s, ^ bought of James Horton 3 Paid cuftom and other charges, 1^ till on board • 3 Paid Hazard and Co.. for in-1 Turing 400I on the whole 3 My commillion on ditto, at 2j 1 per cent. 3 117 12 o 12 iq 10 o 11 10 10 170 ; 116 453 11 10 (6) WASTE-BOOK. May 13, 1789. Sent William Lawfon, at Briftol, 1000 yds Scotch linen, at 2s 6d 125 o o And 400 yds ftiailoon, at is 5d 28 6 8 Defiring him to take them at the above prices, or return thetn on my charges *7 Received of James Gtbfon for a half year's intereft of 500I, due the firft inft. 25 Drawn my bill on Timoleon Janfen, in Leghorn, for 1200 piafters, at 5od each, payable to James Johnfon, or order, for value here received June 3 Received of David Robertfon in full Received of Sam. Ward for th« bill on him W.d William Johnfon, 12 pieces holland, at 20I a piece 19 ~ Received 500 yds Scotch linen, returned by William Lawfon, he keeping the other 500 yds and the fhalloon 28 Received from Abel Factor, of Jamaica, fales of 800 yds linen, 200 yds broad cloth, and 8 pieces of holland, by the Diligence, Capt. Tempeft, on my ac- count ; neat proceeds amounting to 794I 13s 4d currency, exchange at 140 per cent. - - - R 3 18$ s Id J J3 12 10 259 Si jo 240 10 567 1?, 4*i *9© BOOK-KEEPING BY DOUBLE ENTRY. 5 8 June 28, 1789. Received from Abel Factor, of Jamaica, invoice of 7 puncheons rum, 8 barrels indigo, and 6 hogfheads fugar, (hipped by him on board the Diligence, on my account and rifk, amounting to 487I 16s iod currency, exchange at 40 per cent J^y 4 My uncle Humphrey Adams is dead, and hath left me a legacy, payable by his executor James Gibfon, the fum ia S Paid houfe and (hop re-nt, and fervants wages, which, together with houfe-keep- ing and other expences till Midfummer, amount in all to *3" Drawn my bill on Abel Fa&or, of Jamaica, payable to Edward Young, or order, of value due by Ditto Young, at 14 days ; Hip Dilligence is arrived fafe with my goods from Jamaica •, freight, duty, and other charges paid here, amount to Sold David Robinfon my 6 hogflieads fugar on the key Received in part - - 70 O o Reft due in 6 months - 70 o o — 25 — Brought into my ware-houfe My 7 puncheons rum, at 23I 161 And 8 barrels indigo, contain- ing 125 lb per barrel, at 2s > 3d per lb * 1 s (7) d 348 300 130 219 83 I4O 273 io| o (8) WASTE-BOOK, July 30, 1789. 8 Received of Edward Young in full for my bill on Abel Factor — Auguft 2-' Received from on board the Dolphin, James Scot, matter, the following goods, to fell for Frederick Van Dyke, merchant in Amiterdam, viz 5 butts currants, and 12 cafk>* railins Paid cuftorn, freight, wharfage, porterage, &c. Received of William Lawfon in part 10 Sold Frederick Van Dyke's 5 butt6 currants qt 84 cwt, at il 1 2s per cwt .14. Sold to James Dixon, tor account of Fre derick Van Dyke, his 12 cafks raiiins, qt 76 cwt, at 2l per cwt *7 Paid ftorage, brokerage, &c. on Van Dyke's goods * My commiiBon on 30 il, at 2| per cent. comes to ■26. p Shipped on board the Dolphin, James Scot, mailer, the following goods, by order of Frederick Van Dyke : My 46 cwt fugar, at 56s 128 16 6 hh-ds tobacco, for which IT ^ have paid - j P^id cuitomand other charges 812 My coramiffion, at 2? per cent. 4810 I 219 19* d 21 12 14 5° *34 152 10 X82 3 1$Z BOOK-KEEPING BY DOUBLE ENTRY. September 2, 1789. 9 Remitted Frederick Van Dyke a bill 10 JO of 866 guilders, drawn by Thomas Young on James Joliffe, merchant in Amfter dam, value paid here, exchange at 35s 6d Received of James Dixon 6 months intereft of ioool ... Shipped on board the Shark, Capt. Blunt, for Hamburgh, the following goods, ad- dreffed to James Conyers, on my account, VIZ;. My 34 cwt 2 qr cotton, at! 3I 10s - - $ Alio my 10 pieces India"! chintz, at 21I - y Paid cuftom and other charges 120 15 o 210 o 24 7 .24. Sold James Dixon 12 hhds Oporto Wine r at iol a hhd, for payment of which he hath given me a bill en William Jones, due at 3 days - * — : 30—: jo Received as my (hare gained by the fhip Endeavour fince Lady-day Oaober 4 XO.Recewed of William Jones, in full of J. Dixon's bill 8 10 Delivered to William Anderfon 12 bales j muflin, at 12I 10s a bale, to fell for our I account^ each one half i 25 35S 120 J8 »3 120 *5° C«o)- WASTE-BOOK, 10 10 Oaober 8, 1789. ~ i he ex pence of my houie, (hop, krvants &c. till Michaelmas w 12 II II II II 11 Received of William Anderfcn, in full for his half (hare of 12 bales muflin *4 Paid William Andedon, for my half ihart 12 hhds tobacco, which he has bought on o our joint account r Bought of Adam Ainfley, for account oi Samuel Edwards and myfelf in company, each a half 14 hhds tobacco, at 5I 10s Due on demand. j6 n Paid the owners of the (hip Swallow, each one half of faid fhip, bought of them in company with Samuel Edwards 21 I William Anderfon having difpofed of our muflin and tobacco, haih paid me my fhare of neat proceeds, as follows, viz. 15 pieces kerfeys, at 61 - 90 o o A bill on George Drake for 50 o o HThe reft in money, viz, - 53 7 4 24 Received of Nicholfon and Company, af- freighter of the fhip Swallow, for one month's freight, advanced by them, on account of her voyage to Liifcon Paid premium of 900I infured on the fhip oKaliow, for her voyage to Lifban, at i\ per cent. • ne 77 920 m 25 *3 193 Ip4 BOOK-KEEPING BY DOUBLE ENTRY* ( n 12 12 12 12 12 12 O&ober 25, 1789. ^aid Hugh Aditris, for repairs of the fLip Swallow, as per his account. :>old o ;r i^hhds tobacco, at 7! 26 Paid carriage and other charges on oar to« bacco 28 Paid Adam Ainfley in full for tobacco ■30- VTy commiflion on the purchafe, fales, and charges on our tobacco, at 2 per cent., is Paid Samuel Edwards, in full of his half (hare of neat proceeds on tobacco Nov ember 2 •Shipped en board the Aftive, Capt Brown, the following goods, in company with Edmund: Eilis and Nicholas Norton, each f, to the addrefs of Peter Thornton of Lifbon, v\r. Furnished by Edm. Ellis, 650 yds braad cloth, at 35s 487 10 o Furnifhed by Nich. Norton, no pieces flannel, at 36* 216 FurnHhed by me, 15 pieces fcerfeys, at 61 a piece 8 pieces holland, at 18I Duty and fees of entry, &c» 1 paid by me - 3 Paid alfo premium of infuring 900I oh faid goods, per Ha zard and Co., at 2 per cent o o 90 144 28 10 o o o o o 3 18 o 14 98 d 4 77 3 7 10 10 11 11 984 i »o WASTE-BOOK. *3 .'3 33 - J 3 >3 *3 i* November 4, 1789. *9S d Nicholas Norton and I have paid Edmund Ellis, on account , of the above cargo, our proportions to make our ihares equal, viz. Paid by Nicholas Norton 112 o c Paid by me - - 47 10 o Received of James Gibfon in full of hi bond with the xntereft due the lit inft. The principal is - 500 o o rhe intereft comes to ■ 12 10 c James Dixon having failed, I have cor pounded his debt at 15s per 1, viz. Receivtd for myfelf • 750 O Received F. V. Dyke - 114 o Allowed for my felf - 2500 Ditto for F. V. Dyke - 38 o 8 Bought of George Emerfon in company, with Edmund Ellis and Nich. Norton, 18 pipes madeira, at 25 1 Received of William JUawfon in full 1A We have paid George Emerfon in full for our madeira as follows, viz. Edmund Ellis -has given him 1 goods to the value cf 3 Nicholas Norton hath counted 1 with him for - 3 I have paid him the reft in money 94 o c 170 o 186 o *S9 J 1 * 10 1132 450 40 16 * 45° l$6 BOOK-KEEPING BY DOUBLE ENTRY. ,{ l£ j November ir, 1789. *4 *4 *4 54 14 H Edmund EUia hath evened our accounts by paying 20 O O 36 O O To Nich. Norton And to me 12 « Sold William Wright 4 pipes madeira ir Co. with Edmund Ellis and Nicholas Norton, at 27 1 '3 Sold James Thompfon 3 pipes of our ma- deira, at 231 Received in part - - 34 o c Reil due in two day* - 50 o o «<• Bartered 11 pipes of cur madeira, at 28! for 14 pipes Canary wine, at 22I 16 5 6 ! Q 108 Received of William Wright in full for madeira Paid James Horton in full for Trmoleon Janfen's druggets Compounded with James Thompfon, who has failed, as folio vs, viz. Compofition received - 30 o c Remainder allowed him • 20 o o *4 .27 Sold Thomas Young my cwt cochineal for « « 84 108 117 50 130 ,,, WASTE-BOOK. 34 H *5 *5 2 5 *j ^5 *5 November 29, 1789. - Sold 2 pipes of our canary, at 25I December 5 Divided between E. Ellis, N. Norton, and myfelf, our remaining 12 pipes of canary, at 22l - Received advice from James Conyers of Hamburgh,* that he hath received and difpofed of my goods, the neat proceeds, as per account of fales, amounting to 676I 15s 6& Flemifh, exchange 34s 6d, makes (terling 14 Sold Timothy Clarkfon 5Q0 yards Scotch linen, at 2s rod 18 James Conyers hath remitted to me in full, exchange at 34s 2d, in bills as follows, viz. One on Charles Cooke for 300 o o One on William Webiter for 96 3 %\ 24. Received of Nicholfon and Co. in full for freight of Ship Swallow's voyage to and from JLilbon Sold Nicholfon and Co. eur Ship Swallow, payable in 3 months 2$ Expences, houfe, fervants, (hop, &c, till Chriltmas, amount to - l 97 1 5 C 264 39 2 70 16 396 25 1000 9 1 ig% BOOK-KEEPING BY DOUBLE ENTRY, JOURNAL. London, January , i, 1789. Sundries Drs. to Stock, £8333 6 p Cafh, for ready money - 5000 o o Broad Cloth, for 300 yds, at") q q 15s a yard 3 Linen, for 1200 yds, at 2s 6d\ ^ a yard - - 3 ■* 2 'Lead, for 800 pieces, \vt44ton8, 1 at 16I a ton - - 3 Rum, for 25 punchepns, at 3 811 Q q a puncheon • 3 Sugar, for 12 hhds, qt 140 cwt 304 6 o Ship Endeavour, for -§• coft 300 00 William Johnfon, per note due 1 the 1 ft March - 3 James Gibfon, per bond bearing intereft at 5 per cent., from }> 500 o o the 1 ft of November 3 3 4 .1 200 o o *} Stock Dr. to Sundries £272 14 6 To J\Jr. Edmund Young on de« 7 mand - 3 To Charles 1 2th 120 o © tries Wilfon, Efq, thel g inftant - J ' ' 4 To William Mercer, per ac-1 6 c 10 6 coun,t • - 3 s \ 8333 272 14 &*f JOURNAL. -Jan. 4, 1789. Shalloon Dr. to Cajh £2$ O O For 400 yards, at is 3d a yard Edward Young Dr. to Cajh £120 o o Paid him in full 12 Oporto Wine Dr. to Ifaac Onflow £162 o o For 18 hhds, at 9I per hhd *7 Charles Wilfon, Efq. Dr. to Cafi £87 4 o Paid him in full Cajh Dr. to Linen £2$ 15 O For 150 yds, at 3s 2d a yard 24. Hops Dr. to Timothy Clarhfon £93 3 o For 12 bags, qt 40 cwt 2 qr, at 46s a cwt, payable in 2 months 3° Lijhon Wine Dr. to Rum £200 o o For 20 hhds^ at iol a hhd, received in barter for 5 puncheons, at 40I a puncheon Feb. 2. . Thomas Draper Dr. to Broad Cloth £90 For 100 yards, at 18s a yard 6 Voyage to 'Jamaica Dr. to Sundries £443 4 * 2 To linen, 800 yards, at 2* 6d JOO o c 2 To broad cloth, 200 yards, at ies 150 O o 7 Tp Tho. Draper, for 8 pieces! holland, at 19I - j" *5 2 3 Focafh, for duty, infurance, &c. 41 4 2 S % 1 s 25 120 162 87 4 23 l 5 93 , 3 200 c 90 | 1 443 4 *9£ d 3C0 BOOK-KEEPING BY DOUBLE ENTRY, Feb. 8, 1789. Scotch Linen Dr. to Sundries £17 5 O O To cafh in part for 1 500 yards, 1 at 23 4d - - i" To Tho* Draper, for a bill on him 5.0 o o To Charles Wilfon, Efq. for the reft 75 o c Ifaac Onflow Dr. to Rum £880 O O For 20 puncheons, at 44} 19. William Mercer Dr. to Ifaac Onflow £50 For a bill given to him for 23' Holland Dr. to Thomas Draper £360 O o For 20 pieces, at 1 81 *7- Thomas Draper Dr. to Cajh £472 o O Paid him in full - , - -- March 4 — Cajh Dr. to William John/on £200 o O Received of him in fall James Dixon Dr. to Cafh £ioco o o Lent him on bond, at 5 per cent. Charles Wilfon, Efq. Dr. to Lead £380 For 20 tons, at 19I a ton 14. Sundries Drs. to Sugar £151 4 O Cafh in part for 54 cwt, at 56s igd O o David Robinfon, reft in 2 months 51 4 o { 3 ) 175 880 53 360 472 200 1000 380 *5' (4) JOURNAL. i ■ March 14, 1789. Timothy Clarkfon Dr. to Hops £58 15 o -j? For 23 cwt 2 qr, at 50s, fold to Eugene Ar den, for which have received a bill on T Clarkfon, due at fight 22 TC Broad Cloth Dr. to Ifaac On/low £312 10 For 500 yds, at 12s 6d, bought of George Wood, for which have given him a bill on Ifaac Onflow 27 William Mercer Dr. to Cajh £i$ 10 6 Paid him in full - 3° Profit and Loft Dr. to Cajh £93 4 10 For fervants' wages, with houfe and pocket expences during the lad quarter April 2 India- Chintz Dr, to Broad Cloth £210 Bartered 300 yd?, at 14s, for ia pieces, at 21I 10 10 2 Cajh Dr. to Ship Endeavour £94 138 For my (hare of the neat gain from Mi chaelmas to Lady-day 9 Cajh Dr, to Ifaac Onjlow £355 10 O Reteived of him in full Sundries Drs. to Lead £270 o o Tea 200 lb, at 12s a lb - 120 o c Muflin, 12 bales at 12I 10s a bale 150 o o Received in barter for 15 tons, at 1 81 a ton S3 201 58 *S 312 *5 10 10 93 21c 94 355 10 o. l 3 270 ao2 BOOK-KEEPING BY DOUBLE ENTRY. (5) i 10 i ~6 ii 2 2 II II II I 11 2 1 J2 .9 April 15, 1789. Timothy Clark/on Dr. to Cajh £34. 8 O Paid him in full - 24 Sundries Drs. to Lijbon Wine £172 10 o Cochineal, 1 cwt valued at 120 o c Ca(h, received in money 52 10 c Received in barter for 15 hhds, at nl 10s a hhd. May 1 Thomas Young Dr. to Sundries £199 I0 ° To linen, 250 yards, at 3$ 37 10 o To lead, 9 tons, at 18I 162 o c Delivered in barter. Sundries Drs. to Thomas Young £170 15 o Cinnamon, 120 lb, at 8s 4d 50 o o Cotton, 34 cwt 2 qr, at 3I 10s 12.0 15 o Received in barter. Sundries Drs. to Sugar £116 o o Ca(h, in part for 40 cwt, at 58$ 66 o o Bills receivable, one on SamueO Ward, for the reft, payable > 50 O o in 30 days - J 8 Timoleon Jan/en's Account Current Dr. to Sundries £453 11 10 ro broad cloth, 200 yds, at "" 14s ro cafh, for 10 tons lead, at 1 6s 4d with charges To James Horton, for 16 ps 1 druggets, at 7I 7s 3 To profit and lofs, for my? coramiflion - 3 140 o o 184 18 7 117 12 o 11 1 3 1 34 172 d 10 199 10 170 *5 116 453 11 10 1 r* JOURNAL. May 13, 1789. Sujpence Account Dr. to Sundries £l$3 6 8 To Scotch linen, 1000 yds, at! 2s 6d - y To fhalloon, 400 yds, at is 5d Sent to William Lawfon, defiring him to take them at the above prices, or re- turn them. — F01 9 8 11 12 12. 125 O © 28 6 8 r 7 . Cajh Dr. to Profit and Lofs £12 10 o :>r half year's intereft of 500I of Jarne: Gibfon - 25 Cajh Dr. to Timoleon Janfdn's Account Current £250 o o Drawn my bill on him for 1200 piafters, at 5od each, payable to James Johnfon, 01 order, value received June 3 Cajh Dr. to David Robin/on £51 4 o Received of him in full Cajh Dr. to Bills Receivable £$0 o o Received of Samuel Ward for the bill on him ... William Johnfon Dr. to Holland £240 For 12 pieces, at 20I a piece *3 Sundries Dr. to Suf pence Account £*S3 6 8 William Lawfon, for 500 yds"l Scotch linen and 400 yds J- 90 16 8 fhalloon, kept - J Scotch linen, 500 yds returned 62 10 o d 153 12 10 250 5i S° 240 153 204 EOOK-KEEPIKG BY DOUBLE ENTRYo Jane 28, 1789. ■■■ ■ 1 1 Abel Fa£Ior 9 my Account Current Dr. Voyage to Jamaica £567 12 a\\ —2 For neat proceeds of 800 yds linen, 200 yds broad cloth, and 8 pieces holland amounting, per fales, to 794I 13s 4d currency, exchange at 140 p^r cent. *3 9 *3 11 Voyage from Jamaica Dr. to Abel Fat~lor % my Account Current £348 9 2. For coft and charges of 7 puncheons rum, 8 barrels indigo, and 6 hogftieads fu^ar amounting, per invoice, to 487I 16s iod currency, exchange at 140 per cent. July 4 James Gib/on Dr. to Profit and Lofs £300 o o For a legacy left me by my uncle Hum- phrey Adams, and payable by ditto Gib fon, his executor 8 Profit and Lofs Dr. to Cafh £130 o o For houfe and (hop expences, and fervants' wages, till Midfummer *3" Edward Young Dr. to Abel Facler my Ac count Current £219 3 2-J- For my bill on him, payable to ditto Young, value due by him, at 14 days 23 Voyage from Jamaica Dri to CaJJj £83 4 8 For freight, duty, and other charges paid here - - 567 12 348 300 130 219 83 ( 2 ) I J* *3 JOURNAL. July 23, 1789. Sundries D*s. to Voyage from Jamaica £140 o o. Cafe in part for my 6 hhas fugor 70 o o David Roblnfon.for the reft at ? ■ • S 6 months. 70 c o 3 x 3 1$ 4 Cafh Dr. to Edward Young £219 3 2j Received in full for my bill en Abel Facloi 11 14 12 £4 *5 -.25 Sundries Drs. to Voyage from Jamaica £273*10 O Rum, 7 puncheom, at 23I 161 o c Indigo, 8 barrels, qt each 1 251b,! each at 2? 3d • J 112 10 o Brought into my ware-houfe. .30-. »- Auguft 2 t- Fred. Van Dyke's Account of Goods Dr. to Cafh £12 14 6 For cuftom, freight, wharfage, porterage, &c. paid here on 5 butts currants, and 12 cafks raifins, lent me here by the Dolphin, James Scott, matter, to fell for him - Cafh Dr. to IV. Lawfon £50 o o Received in part 10 Cafh Dr. to Fred. Van Dyke's Account of Goods £134 8 o For his 5 butts currants, qt 84 cwt, at il J2S » 2©f 140 273 10 219 12 14 SO 134 ac6 BOOK-KEEPING BY DOUBLE ENTRY, 8 Auguft 14, 1789 Uigutti^, 'James Dixon Dr. to Fred. Van Dyke's Ac- count of Goods £152 o o — For his 12 calks raiiins, qt 76 cwt, at 2l 17. £5 *4 1$ ~9 13 3 *4 Fred. Van Dyke's Account of Goods Dr. to Cajh I 2 6 For fiofage, brokerage, &c. paid Fred.Van Dyke's Arcount of Goods Dr. to Profit and Lofs £7 10 6 For my commiflion on 301I, at 2\ per cent. - .26- Fred. Van Dyke's Account Current Dr. to Sundries £182 3 5 To fugar, 46 cwt, at 56s To cafh for 6 hhds tobacco,! with cttftom, &c J To profit and lofs, for my ) conarniflion- - y 128 16 48 18 4 8 ic Fred. September 2. Van Dyke's Account Current Dr. to Cafh £81 6 3 4 — 1'For a bill fent him of 866 guilders, exchange *4 at 35s 6d, drawn by Thomas Young on James Joliffe of Amfterdam 11 9 Cafh Dr. to Profit and Lofs ^25 o O For 6 months intereft of ioool of James Dixon - 152 10 182 81 25 i lo ) JOURNAL, ZOJ XI JO *4 II 14 I 4 II 16 16 10 M if 16 September 20, 1789. Voyage to Hamburgh Dr. to Sundries £355 2 * To cotton, 34 cwt 2 qr, at 1 3I JOS - - J To India Chintz, 10 pie-1 ces, at 21I - 3 To cafh, for eu Horn, &c. 2'4 7 120 13 o o c 24 .B^Y/j" receivable Dr. to Oporto Wine £ 120 For 12 hhds at iol, for payment have taken a bill en William Jones, due in 3 days ■30- Cajh Dr. to Ship Endeavour £58 13 2 For my mare of the gain fince Lady-day October 4 Cafh Dr. to Bills receivable £120 O O Fox J. Dixon's bill on William Jones 8 Sundries Drs. to Mujlin £150 William Anderfon my account*} in Co. foe my half fhare of > 75 o o 12 bales, at 1 2l 1 os J William Anderfon, for his fhare 75 o c 'Profit and Lofs Dr. to Cafh £110 3 4 For houfe expsnees, &c. till -Michaelmas 12 Cajh Dr. to William Anderfon £75 o o Received for his fhare of 12 bales muflin 1 s 355 120 & '3 I2C *J° ■110 75 ..£t>8 BOOK-KEEPING B7 DOUBLE ENTRY. 2* 16 -~ prober 14, 1789. 12 *7 Sh f p Siuallow in Co, with Samuel Edwards D\ to Sundries £920 o O To cam, for my half mare "paid 460 o o To Sam. Edwards, for his half 460 o o J8 II 11 16 14 12 *4 William Anderfon my Account in Co, Dr, to Cajh £72 o o For my (hare of 12 hhds tobacco paid him Tobacco in Co, with Samuel Edwords Dr. to Adam A in /ley £77 O O For 14 hhds, at 5I 10s, bought of him in Co. due on demand - - 16 21 Sundries Drs. to William Anderfon my Ac- count in Co. £193 7 4 Kerfeys, 15 pieces, at 61 90 Bills receivable, 1 on Geo. Drake 50 Cam, received in money, being"} in full for my mare neat pro- > 53 ceeda on muflin and tobacco \ 7 4 24 Cajh Dr, to Ship Swallow in Co, with Sam Edwards {25 o O For 1 month's freight received of Nichol- fon and Co, affreighters of faid (hip on a voyage to Lifbon Ship Swallow in Co, with Sam, Edwards Dr to Cafh £13 10 O For the infurance of 900I paid on her voy- age to Lifbon, at i£ percent. 7* 77 920 *93 25 10 lit) JC5URNAL. — 0<9:ober 25, 1789.— \Ship Swallow in Co. with Samuel Edwards^ I Dr. to Cajb y .£14 64 — Z For the repairs paid to Hugh Adams [Cajh Z)r. to Tobacco in Co. with Sam. Ed J wards £98 o o --For our 14 hhds, at7l 16 *4 Tobacco in Co, with Sam. Edwards Dr. to Cojb £2 7 3 For carriage, &c. paid by me 28 3° Tobacco in ~Co. with Sam. Edwards Dr. to Profit and Lofs £3 10 11 For my commiflion, at 2 per cent. Sam. Edwards Dr. to Cajh £7 10 II Paid him his fhare neat proceeds on our to. bacco Adam Ainjley Dr. to Cajh £77 o O "iZ Paid him in full for tobacco 14 16 9 22 14 18 18 8 *4 November 2 Voyage to Lijbon per the Aclive, Capt. Brown , in Co. with Edm. Ellis and Nich. Norton Dr. to Sundries £984 o o To Edm. Ellis, for 650 yards") Q broad cloth, at 15s - J 4*7 *° To Nich. Norton, for 120 psl * flannel, at 36s . ; J 2l6 ° ° To kerfeys, 15 pieces, at 61 90 o C To holland, 8 pieces, at 18I 144 o o To ca(h, for infurance, duty, ~ &c. '} 46 10 o Addreffed to Peter Thornton. T 1 H I 14 98 "209 d 77 10 ic 11 IX 984 220 BOOK-KEEPING BY DOUBLE ENTRY. R*3 18 *9 £4 4 9 14 8 19 £4 12 12 18 14 J9 14 • j November 4, 1789. £*//#. E/Z/j" Dr. /o Sundries £1 59 10 O . ro Nich. Norton, paid by him 112 o c To cafh, paid by me - 47 10 c Cajh Dr. to Sundries £512 I0 ° To Tames Gibfon, for the 1 J . . , ' J- coo o o principal 3 To profit and lofs for inrtereft 12 10 o Sundries Drs. to James Dixon £1152 O o Ca(h, received for myfelf and 7 ^ F.V.Dyke - j 8 ^°° Profit and lofs for my abate- 7 J- 2C0 O O ment - - 3 F. V. Dyke's account of goods 1 n for his ditto - 3 S Madeira in Co. with Edm. Ellis and Nich, Norton Dr. to Geo. Emerfon £450 O o For 18 pipes, at 25I Cajh Dr. to William Law/on £40 \6 8 Received of him in full 11 94 O Gee. Emerfon Dr. to Sundries £450 00 To Edm. Ellis, -for goods 1 accounted with him $ To Nich . Norton, accounted 7 with him for - y To cafh, paid by me - 168 o Sundries Drs. to Edm. Ellis £56 00 Nich. Norton, paid to him 2000 Cafh, paid to me - • 36 o l 59 10 5" ic II C2 450 40 16 450 56 ©1-c ( M ) JOURNAL. axi *9 *4 20 *9 20 I9 £4 20 November 1 2, 1789. M 20 William Wright Dr. to Madeira in Co. wi;h Edm< Ei/is end N. Norton {108 o o For 4 pipe*, at 2/1 • - - *3 Sundries Drs. to Madeira in Co. with E Ellis and N, Norton £84. o o Cafh, in part, for 3 pipes, at 28I 340 c James Thomfon, reft at 10 days 50 o c *5 Canary in Co. with E. Ellis and iV. Norton Dr. to Madeira in Co. with ditto £308 o c For 14 pipes, at 22], received in barter for 11 pipes, at 28i 16 Cajh Dr. to William Wright £108 O O Received for madeira *9 James Horton Dr. to Cajh £11 7 12 O Paid him in full for T. JanferTs dvuggets 2 3 — Sundries Drs. to James Thompfon £50 o c Cafh, received cempofition 30 o o Madeira in Co. with E. Ellis acd 1 N. Norton, allowed him j; 20 ° c 27. Thomas Young Dr. to Cochineal £130 o o For 1 cwt ... 29 Cajh Dr* to Canary in Co. with E. Ellis and N. Norton {50 o o For 2 pipes, at 2jl T z 101 84 308 108 117 12 5° 130 5° ai2 BOOK-KEEPING BY DOUBLE ENTRY. 38 21 20 21 11 21 Sundries Drs. to Canary in Co. with E. El- lis and N. Norton £264 o O £ Ellis, for 4 pipes taken as his (hare 88 N. Norton, for 4 ditto - - 88 Canary, for rny 4 ditto, at 22I 88 James Conyers, my Account Current Dr. to Voyage to Hamburgh £392 6 8 For neat proceeds of my adventure, 676I ijs 6d flemifti, exchange at 34s 6d 14 *7 21 *7 December 5, 1789. *4 Timothy Clark/on Dr. to Scotch Linen £70 16 8 For 50c yards, at 2s lod 18 Bills receivable Dr. to James Conyers my Ac- count Current £396 3 2-J Remitted rae in full, exchange at 34s 2d, in bills, viz. One on Charles Coke, for 300 O O One on William. Webfter, for 96 3 2* 24 Cajh Dr. to Ship Swallow in Co. with Sam. Edwards £25 o Q For. freight of faid Ship of Nicholfon and Co. ... 26 Nicholfon and Co. Dr. to Ship Swallow in Co with Samuel Edwards £1000 o o For the faid fhip fold them payable in 3 months - - ( 15 ) d 264 392 70 16 J96 25 28 J9 Profit and Lofs Dr. to Cajh £91 2 8 24'For houfe, &c. expences till Chriftmas 1000 9 1 C 213 ] LEDGER, 1789. INDEX oa ALPHABET. Anderfon, Wm.myl , Account in Co. 3 Anderfon, William 16 Airifley, Adam 17 B Broad Cloth Bills Receivable Balance Draper, Thomas 7 Dixon. James S Dyke's Ace. of Goods 15 Dyke 1 s Ace. Current ic Gibfon, James Kerfeys KL Edward:, Samuel Ellis, Edmund Emerfon, George H Hops Holland Ho ft on, James N Norton, Nicholas Nicho!fG7i and Co. *9 oooM m ^ ^ i 1 oo^^oooo ^too OO O tlN on so « (no mvo oo. l i on en ON OO ! ^o r^<-« o ch o ^fn^rro coon d (SCO 't^I^O M o\ cooo COOO oo M- ^" O M M ON »■« ; CO VO vo ,5 J? I c#-> "d- ^f- *^ r^ r^oo io onvo « on^^- cm MM *" 1 rt ._ d -3 5-5 »- c § £ CO Cm w 3 U u o o 1 -a »-5 ^ ^ U2 • O P B g 8 s ^j- 1-* i^.>© 30 i^i^i^o »^>e© od *» >s ooodoooooooooo i I vO O "vo w Tt-S j M f 0*000*000000^00 I CH M M I > M I M M M M , -I j M VO i OS I ,o t> roO O^^dVOMO^oO ^h *^^o t ^ n o ^ o o 0\ 'n °*>vo m ^^i^ts « vo . Ov ^ O dM co d oo oo ' O ^"> O O VO co VO On oo H c* rj- co LEDGER, I589. j O OO u pq -a 1 ° 1 ° 1 ° I ,00 O 1 ° ^r> 1 ^ 1 ° O '16 O ON *0 1 ^3 t-i M - Lead 704 OO O *-< 00 "0 .-1 co On 1 >-t On 1 * On . gn O O g I A O la t^ *T3 O O l-o ~C O c ^ 4 >>» r» OO 2,^ T3 ^- H3 v a c etf tuO CJ WK £1 GO. rt m 02 02 as • 0? l«l OO ■ft T3 h4 X>- CJ O B os c vi a •-« *» d ca C3 *0 4J 14 U3 ■ ^4 to- . O u O O u v3 i-j Oi CO Pi CO Pw O O ^ HH H \~* N 1 H H AS 21 ^r ^> ^ cu O- | j M a So ' **? O **" Ur> C & a° **- rr 4* *j *■» ■ ^ co »g ♦ C9 CO r-- CO .H ° co C SO OO u *o »o 3 « CO . 5=3 O C* ►%« s Vj .2£ »M CO 1*1 3 3 >> 2S * •* G S « c3 «« ►jfacq cottJQ JX, >S *-» >%>-,>> J! X>N^ DQ CQ.CQ 02 a) ca bj 23 QQ CO O cs co •-< t< ss< »o O _ co 'S- ^ < co 5* OS 00 M tO c* I "O 1 2 2 « b SO Tf M u O-VO" VO 1— 1 ,j O OS M VO 1^ co M If c* '4 rf- m ov co Cs CO O CO »o CO M 1 «■ 02 co M O', 1 M OS H OS ,g 10 r^ j * iG «S « ^ O.O O O CO ^ P4 CO Pu CO MH O O Q O O O O H trf H KH . hH *d oo LEDGES, i;8p* O O I O J - P" , 217 O O OIO «. O O U ro , "or; j ^ro'oi rh O O 5r v^ co O 00 ch i CN 1-1 " M 1 On <0 CO t r^ -0 ^1 r— 00 r- IVO CO rj- •7: « cs ! "TT l M t> CM. . s • S M i: £ ^ >% ^Q *Q H3 U 3 J 3 u «*-. a B 8 — C M C t£= J2 ~c5 ^ a M fc c« * V & ^ucq ^ >^ ^ >> >-> oq oq (^ £0 £0 £ S PQ'3Q C % ^ pq co oq & u a 3.3 *r ; M 00 u s > 55 > — iHh •u 1 ! O O 1 I'tt Q O 1 ° « 1 ° O O j 1 «, 1 * ^O " CO CO l r* O 00 00 CO 1 * "o M °° M OS »-*■ CO M CS *2 1 « 1 Cur. 5 Q *S Ji ^ , J % ef | • ^ «^* zl -a * g- *o w 3 fc«F , a> C C8 05 3 %i $ BOCK-KELPIKG EY DOUBLE ENTRY. 13 O O O | O I o o o o i c o o o ON oo .v. L o eg i CI U CS «-rs O mm N3" M •2 " M . . M fvl «• o ^ 1 1 T? o -o tqvo oo ^ r*» 5- irj T3 t t ■ ! l« 01 o . U t C 6 Wine i Merc Cloth full rim c CI 1> u O CO g'« P3 - O >-» t>~, >-» >^ >% >> IS PQ oo I PQ CO O.CQ CO QQ CQ o o O O I o I g I O vo |^0 O Q CO O *-nl ^ ■>• ^o coioo oi > ! "o ^ - o\ i «^> C\ CO Wm w« q c, ^2- i ^<> £ °° a 4> ^ • • C3 a Khd , gain fl CO o . ^'sj ^ CO ^ «>4- o 3 Q ~a.. Dr. flow, t and 9 Ifaac Cafh, [ 4$ .CO u, O o o o o o o o O hH ^H HH H ~C\ r^ 1 ^r 1 ^ CH IH cv * H •s * « i JD \9» I A \ A i M .XEDCEK, *1$9< 11 f t& P cQ rl O j O 00 j 1 O O o w .- «* <* : | js wo ST o o o "" oo -on s r^ » <0 O • M r-l c* «J3 £ ^ *- •» ' cd r~. O • U o £ £ E-pq >^ t>% eo-ao T3 o ° 1 o O OO oo oo ° 1/3 on •f 1 '"JT •ojo so ON ° 2 1 o «— I CO ON ^t- co to ^o cry*"* CO VO M O -OAO *- CO Ob l 2 oo & -a Hi t*> >> >> CQ CO DQ £ 2 c c JJLL I 2 | O « I— On £? vo co "f ^1- H. 1 ^ vo 1 ^ r- vo M 3 ? - CO rs ^^ 1 ^ i •"0 O 5> en f^*"^ La tj O q ^C "tfl »-* e £> .O M CO *M *S c H *-> ed Sp O u n hi ■3«E •a j c cu u u O"* R 6 *£3 C CO igar, du oyage fr due • H & uri O *- jH PH Hf- *£ r^ t- 1 "3" co C* M m rt JO 4 Mar. Aug 1 u 2 22 BOOK-KEEPING BY DOUBLE ENTRY. ^J CO O O O O O M O O O O O O O OO O O o m » o b o oo o h-f O o o ^o o 00 - o o CO CO M O « co CO ^ ^t" ro ON ^*J •-• CO o O* M -rr »-o »-o r}- vo ^f M e* cs CO co co L o ^o^O oo By T. Janfen's Ace. Current By Ca(h, for intereli of 500I By fa. Gibfon for a legacy By F. V. Dyke's Ac. of Good< By F. V. Dyke's Ac. Curreni By Cafh, for intereli By Tobacco in Co. with 1 S. Edwards C By Caih, for intereft of 500I By Broad Cloth, gained By Linen, gained By Lead, gained - By Rum, gained * By Sugar, gained By Ship Endeavour, gained By Shalloon, gained By Oporto Wine, gained By Hops gained 00 N^i- *^\o 0\ M ft C>< CO HJ O O W" O 00 (N | a* cd £5 &<& CU Cu V 5J I) o 00000 1 ■" o 00 00 o 00 cs 0) O £ tc o w Or »-.•■• o o\ CO tV " HJ U <*■» LEDGER, I789 O I O 223 r-4 "-* ► ° CS O fc « 1 M M 5 2? cs ^O ft CS M >-< P-. Jd O r-i 5 $ 2 *-» -co M tr> v, Etf+< s O ^ Vt *- ^ W '•►"■ .O P M •** U-. .*£> CS s tS ra cs 1 is. >-t A t2 1! c a $ . ] Sj =3 O to ►• a rt "O O w 1 c J3 > » CO H >-» >^ ^ >N CQ CO B2 PQ O O r^ cs •» M Q tS d. > O CO ■■■J" '' — : ' ' ■■■». * '■ ■ £ o o 1 1 o o p> o cs i 2 I > O ^ 5 1 75 -a 15 CJ CS !> c T3 «* re *T3 *o .tJ O CO 2 CI PQ M »j hi a, © - fr- J— H . hh cs G\ cn tj- cs u M lZ a» CM CL, Oi 1 » W ■*d O O ° f ° ° o o 2 1 ° ? o o CO CO o oo •-4 .r>. *o M M .o »o o o o *o c* M SO | vo 3 3 i M ! ? -J C* T3 ■^t- u CO PQ PQ s 2 »: ffi To a) > jo CO i P. fa o o B 5 '53 b t V 1* u o H , <* «-»-< V c a •F eg ft CO rtf fl| '£ u u pq £ >s >s >* o^pa pq pq Oy ~o 6 o | of ° I ° 1 o o o O N(M Itt - S° 1 2 ! 6 ? 1 o o o O co 1 m On q i r-. On-co ON » CO ft 1 o *o o cs o o M o * O VO O ON CO |l 3 N 2 a 1 s CO *0 v*D Is < M o © o u "2 - c 2 fl o n c3 hOO £&« gcSO ft LEDGER, I 78 9 . ^25 •SO O | o j O 00 000 1 00 2 i rlSJ'iJ no vo ? ^ *o o en CO £L 1 ON % 1 >% >H >> ^ QQ« 03 CO N PQPQ . § ^ 00 On r- o\ * o ^ X V b>0 > ? s rt c 3 3 O Jus ■ "8 td g ° I OO 00 «5* (A M cs VO no « ^ M I tit , M - . 2*S CO x^ co *» ~ ^ >H ^ ON g ^*- M .*> B "o 2" ] cs s ^2 ... rt R 5 J S ^ « £ • eg S *- * **• 4J C. D O * +J U t* j a V Q *3 & <-. kM .S if in 4J . G CN ^5 ^r t>4 -3 Qk 00 £5 C3 3 cr* X*^ 3 a 3 ' 3' M CO o CO CO o o O H H K H - /-s °° 2 s I s s> 1 • ON n « !>* > ^ V m « o « a ^ £ a ■] s 1 ; 3 u 3 226 BOOK-KEEFJNO BY DOUBLE ENTRY, ^ « « I 1 o o o 1 2 ! o, v> On CO, c* O O oo CO | o -4 M M 1 M- oo 0\ 1 r^ o co oo M t c<* m* *& 2 f VO rj" t^ M CO M co * ! *>-> M CS ^f ! W *o <$ " ■■%* CO I CS OS US M M c< oo Hi *cS g S B- a c r« 3 §.S o JC O O O M t! s-s o o On ° % 80 co ^ c 3 O 1* P- o . q co en C Ur- B c^ K > •i •Si c c o CO CO Ph. a C3 21 t*» >s es t»s >> >^ >» ^J CQ « £> OQ CQ CO OQ t* oo CO CO »-0 1 c* M £ M o ^S .^ c 3 3 *, 'S ■f ^ »— » >— » ^^1 * I 1-4 «N 35 ,-* vo * i ^- 00 co ^i ►-i »-o ! CO rf •^1 H £ •* 3 ** CO *+ ! CO * J-t CO 91 J* o ' — * o .2 ^ o £ £* M ft u O *5 3 r^-n . 2 »*. « "O ON O C a> B-S.S? CtjO M ^ ' • 4 2 - 1 - o w o O o ** H H H CO 1 OO c^ *o *"\ c* ti co u c 3 1 , c ^* 3 3 >— > 1-1. *3 LEDGER, I7S9* £2? o ^t- co o o o o 00 o»y I "& 1 en ^ O 00 \o r^ co O O o O r»* O O O O cs " c* vo VO m rj-oo cs m r*» vo m m r^ ^t* M <3- . O • O ee ,S ,-#4 w j- ^ Q felQ > Oi £ co >* >s x x >> £»,£*,£* pQflQCQflQPQPQPQPQ H (^ CS MM e 5 S I gfw 2 Q Q h < ot> ^ M O ►> fr-. >s >-, >* *-..£ >•> X pQoQcqnoaao s w oq ^£« >^ >^ >n 30 PQ OQ rf «-0 ^OO o* cs cs cs bo &4 It CO O O ci CO ^ M On 00 6 O & — ■ 3 M OOOC^OO"^- o 000000 r^coooo o co o o r^ o poo ^o o 0000 VO 1-4 M 0\0\0 ^r ^00 O ^o *n 00 >-< *o co c* ^ocs ^.^o ON- J-S -2 Sq « * « .2? . OOOO HHHh c O 9 JD no i_2 £ rt C _^ 4i .5 « « la OOOO fe£iife 5 6^' u^ 228 BOOK-KEEPING BY DOUBLE fcNTRT. -O* • o o o r n!c i -te* « OO 1 00 «o 00 o 00 1 CS N 1 * 1 vo i— < oo C4 o» CO t vo 1 °* ! c* ON CO ON* OO «■ ^-co *o ^ & ID ^ — c O I* • o Currant >r his ra o o a o 4-» u £ ° w a; B c« P3 i? o >«> >n ► ^ OC CQ I5.CQ oo cq o .-t bO ►a? o u 3 xO •^ z < Q Q ' vo vo vo ^ o 2 t^ oo vo CO M oo o CO VO I °M vo oo ^ n- o\ oo oo 00 * ^ o c £ -*-» ^ ii £ « VO «« oo -o O |3 H m > «s o ^ re o o ^ CO CO ON CO t42 o : ^ R c c o co PL< .P ° vo ts o LEDGER, I 7 89. 229 rh 1 ^ ! . ° i ? 00 .55 t> r^-n *je« «J M -O .22 w 13 c: <$ . J3 Tt C u *f > **& M u u u (U > ■jf to. CO u k u w <3 p *£ ^ j-. ^ "0 c 9 ^3 CO •So » >> <£ >> cS t>* CO ^CQ CO IH <£' •-0 c* ►h <5 M « « P # O I 23° BOOK-KEEPING BY DOUBLE ENTRY. H3 O 1 o 1 ° o M o I c * «5 ° ° 6 ° ° | o o : 2 i>* O-N U* n U o 1^ OO »— i r^ ' • H N* w rt >> >-, >» OQ CO CQ £ i ^ ^3 . cu CO co oo 'c3 Id ^ "Jf<3 So «S £ ca q. o -d >s >* 00 00 N O v. ON OO OO Q 3 cs c* cs OQ O -a o o o u* m * <>> 3 N •a .5 § is is 'IB t?5 c? ™ U 3 a c o 4-» to Sundries, coft Cafti, for infui Cadi, for repa Sam. Edward Profit and Lo c Q Cafh, paid him gain on Nicholfon and 4> d i o o o o o o o o o H HH h**** t-> H H H \o ^ ^ m CH Ci o o o LEDGER, I789. 231 *tJ O ! O O O j 1 OOOOO 1 ° v> O 1 O O O j 1 OOOOO 1 2 - 0\ OOOOOD 00 «1- 00 ON r^ ^i-^O O cs OO C\^H VO On rzr"~ 00 1 -. - > •£ o CO o* o % X 00 < ^ c _ o •-« Co £ ? s § < ^ ■fej £ s •-•CO M* Z2 O cj HZ J >-» !*> ^ pq PQ QQ O H c o. 5 "" <2 c -%. J3 UOrt so T3 O 3 <-£ *i U ,^ o o » >> >v >S copq cspq pq o T3 i 0000 CO O 0000 2 - ■ ON 00 ON Onoo 00 ^j- M CO 5 OO M | 1 00 PP 0000 2fi ^ BOOK-KEEPIKC BY DOUBLE tNTRT. -d o o o O O I O I I o o 9 I O i SO 0* O cn M M< l-i I'M 00 ^ 00 O QO O 1-1 CO O O O "O O O O O O Q O ( ° 1 ° A 31 * O 1 1 OOOOO. 00 ^ MOO OO 61 CO O O O O O O «-n C* M fi M 1 ° 1 d ^ go 00. On O 00 Cn kq t-00 T3 O O i2 n Co. ining m 1 .s -^ ^ \.*1 to ^% ^ .2 c s -^ "§ *>* q CO CQ > O > 0\ ^o 000 OO OO CS M CS *j- ~ *o O N* M CO CO "O <=* G\ j 0\ OO G\ co c: *- M j W M «-l > > O O O X m BOOK-KEEPING BY DOUBLE ENTRT, On bo P* ! O O I * «A O J »— « 00 VO ON O O O O 2 .O c* 3 k VJ It Xi K > ^ o» i V. a (A 55 tt s >* 1* oa s MC-J • & ^ J5* 00 s* ? fl T3 1 OO VO « • O »— 1 M **» a 0} c* .S 4 &0 c ^ "O V> £ ? •4-> m 1 % 4) £ a « *5 .c/3 L >* 2 v. a* 9 £* CO H HH H vo vo r o o LEDGES, I789. 2J5 OOO OIO m r^ r^ no vo 00 00 00 I^OO Q\ o £ £ 3 ^s o £ gWZ S W * 1 !>> >-. >% C3 «CQ GO * vo O O O O O 00 00 OOO O^mO^OQO M O cs O VO 0^0 u Ocq^ox^OOO 00 M L'n ^^-o\o 00 00 O vo CO e* Jt-*oo 00 Y? ftVO o rj- O O ^fO^ 1 ^ 1 ^^^ ^i'O^O ONO « cn«oo I <* - CON coro m»-* coCSmco ^">|00 joo CO (Y) ^f V rf u^vo VO VO 1^-00 O .•-» m N ff) «-oCO m h< ocooooooooooooo H^-hp^(N^U^fH f--< £-» ^ P- ^ 2$6 BOOK-KEEPING BY DOUBLE ENTRT. T3«0«^000»t h o ovo I t "» — Hi M« o ^f r- o - o o vo a oo c* ^-vo Tf _ VO CS CS M M C* H o H 1^. O « CO •si- rs VO '3 C\vo i^ -f^oo o »-o VO \o x^- ON O - w- C* „ ' o o o o , £ IjJ >.co SU>^ .S -2 o *s HS3 o u 4> 03 15 § Ok pQCQPQGQ OQ 03 CQ 03 co ca >s >% >% CO CQ 03 5 ~v o VO 1 rf ^ * <*> ^ | On vo i ^ O I | Oxgate 1 74880 | 2080 | H '? 1 2* I * | Pound 1. 299520 ! 8320 | 208 | 52 1 84 1 4 1 The length, of a Scots ell is 37 J Englifti inches, EXAMPLES. ft lb oz drs ac r f ell Add 13 12 13 11 Add 12 3 32 19 15 14 15 10 14 2 36 28 14 15 9 8 15 1 18 32 ADDITIONS TO MULTIPLICATION. To multiply by any number of nines. Annex to the mul- tiplicand as many ciphers as there are nines, and fubtract the multiplicand from it. Multiplicand. Multiplier. Produft. 73864 999 7379 01 3<5 5936 ww% <93594 o6 4 / APPENDIX, 23£ COMPOUND MULTIPLICATION. RULE I. When the multiplier is below 12, begin at the loweft place, and carry according to the value of the higher. RULE IT. When the multiplier is a eomoojtte number^ multiply fuc* cefln r ely, by its component parts. Mult. £23 11 9 1 by $6. Prod. £1320 19 2. £3 8 *3 li h Y *32. £>j;c6 1 3* RULE III. When the multiplier is not compoji'e, multiply by the near- eft compofite number, and by the difference ; and add the produ&s when the compoftte number is lefs than the given number, otherwife fubtraft them. Mult. JT 13 16 74- by j 1. Prod.£ 705 7 ioj; £27 19 9^ by 59. £ 1651 8 iig. RULE IV. When the multiplier is a large number, multiply as many times by 10, as there are figures in the multiplier above the unit's place } then multiply the given multiplicand by the unit's figure, the firft produft by the ten's place, the fecond by the hundreds, and fo on -, then add the products. Multiply £3 U 9 \ by 7386 £3 11 9f x 6= - 21 10 9 10 35 17 11 x8= - 287 3 4 10 358 19 2x3= - 1076 17 6 10 3589 11 8 x 7= 25127 1 8 £ 26512 13 3 Mult. £ 4 9 10J by 975. Prod. £ 43^ 8 jf . Mult. Cwt 13 3 17 by J93. Prod. Cwt. Abundance of examples qn thefe rules will be found ia Compound Multiplication, and the firft rule of practice* 1 243 APPENDIX DIVISION. To divide by any number of nines. Under the dividend place it over again, but as many places more to the right as there are nines \ under this, place it a- gain as many more places to the right, and fo on-; then add thefe lines, and from the right of the fum cut off as many figures as there are nines, for the remainder, the reft is the quotient ; but the carriage from the left of the remainder muft be added to the right, and if the remainder be all nines, make it nothing, and add one to the quotient. Divide 739428647 by 999. 739428647 739428. 739 Quotient 740168J814 Remainder 815 Divide 6 ,'39482 by 9999. Quot 654. Rem. 136. Divide 7538324616 by 99999. Qnot. 75384. - Anf. 61 3s 6d. 7*0 divide when the divifor contains afaclion. Multiply both divifor and dividend by the under figure cf the fraction, adding in the upper figure to the product of the divifor, then divide the products. Divide 394I us 7d among 5 men and a boy, and give the lay f of a man's (hare. - Man's (hare, 71I 14s jod. Boy's fhare, 35I 17s $d. Thirteen 3.4th yards cloth coft 81 17s 7*d : What is that the yard ? Anf. 12s nd. MIXT REDUCTION. When money, weights, &c are to be brought from one denomination to another, and one of them does not cont i»n the other, it is called mixt reduction, where both multi- plication and divifion are ufed. RULE. Bring down the given number to a denomination which is- contained in that required, then bring it up to the required denomination. Reduce 739I to guineas. - Anf. 703 gs 17s. In 634 guineas, how many pounds ? Anf. 665I 14s. How many guineas are there in 7382 crowns ? Anf. 1757 gs 13s. In 736 crowns, boiv many dollars at 4s 6d ? Anf. 817 d 35 6d. In 946 dollars, how many quarter guineas ? Anf. 810 qr gs 4s 6d. In 934 half guineas, how many marks at 13s 4d ? Anf. 735 m fs. AFPEKPrX. 243 In 564 marks, how many pounds ? - Anf. 376I. in 964I, how many moidores at 278 ? Anf. 714-1110 28. How many guineas in 853 joannefes at 363 ? Anf. 1462 gs 6s # In $36 hds, how many pun, at 84 gal ? Anf. 402 >un. In 735 drams, apothecaries, how many pennyweights troy? - - . Anf. 18374. In 964 pounds troy, how many pounds avoirdupois ? In 639 pounds avoirdupois, how many pounds troy ? How many ells Englifh are there in 936 yards ? Ifi 738 crowns, how T many marks Scots at J3fd ? A gentleman was robbed of 137 guineas, 326 moidores, 536 piftoles at 17s 9d, and 536 dollars : How much did he lofe in all ? Note. The work may be often abridged, by adding or fubtracting a certain part of the given number. Thus.: To reduce guineas to pounds, add -^5 to moidores, fub- tta6l £ ' y to joannefes, to -J of them acid ■§■ of that half 5 to crowns multiply by 4f j to reduce moidores to guineas, add ■f- ; to pounds add \ and -j- } to reduce pounds to marks, add \ ; to reduce marks to pounds, fubtracl j-. Thus, 360 guineas are 378I 3 280 moidores \ 210 joannefes , or 1522 crowns ? IN PRACTICE, The feventh and eighth rules make but one rule; namely, when the price is (hillings, multiply by half of the millings, and double the right-hand figure for (hillings j the reft are pounds. What coft 357 yards, at 9s > 357 4f 1428 178 1 £ 160 13 Here 1 remains when taking f, which is is, and is to be added to the doubled right-hand figure. What coft 753 yards, at 13s ? •' Anf. 489I 9s. What coft 921 yards, at 27s ? . Anf. 1243I7?. 244 APPENDIX. ADDITIONS TO DECIMAL FRACTIONS, To reduce Jhillings and pence to the decimal of a pound. Half of the (hillings gives the firfi: figure of the decimal 3 the farthings in the given pence and farthings, increafed by one, if they amount to 24 or more, give the fecond and third figures \ but 50 mud be added to them if the (hillings be odd ; and if they amount but to one figure, prefix a ci- pher to it. To complete the decimal, call thefe two figures, or their excefs above 25, 50, or 75, fo many pence, and the farthings in them increafed by 1 for every 24 in them, give other two figures ; and this method is to be continued till the decimal end or repeat. Reduce 6s 4*d to a decimal. - Anf. '318751. Reduce 4s io^d to a decimal. - Anf. '24479l'6\ Reduce 93 7^ to a decimal. Reduce is 2^d to a decimal. Reduce 3s i|d to a decimal. Reduce is to a decimal. Reduce 8|d to a decimal. - Anf. •036458 , 3'. Reduce j^d to a decimal. - Anf. '00625. To "value the decimal of a pound Jler ling. Double the flrft figure for (hillings ; and, if thejecond be live or more, take five from it, and add 1 to the (hillings for it - y then the fecond and third figures are farthings, after throwing away 1 for every 25 in them. Value / C"74 2 - Anf. 14s io£d. •385- 7s 8^d. •666. « 13s 4**. - Anf. 26*8'8'8\ y 2 ?43 APPENDIX;*- Divide 363*j c 740 v by 48-3'. - Anf. fr$ it afterwards rofe to I2\d : What was then gained per cent. ? Anf. io|d : 103,: : I2jd : I20£ the price at which iool •worth was fold \ and the difference between this and iool is the gain or lofs. Sold tobacco at 2s, by which was gained 2 per cent., but it afterwards fell to 22d : What was then gained or loft per cent. ? Anf. loft 64-. Sold (lockings at 3s 9!, by which was loft 3 per cent., they afterwards rofe to 4s : What was then gained or loit per cent ? Anf gained J^V? When one gains a penny en a fhilling, what is that per cent. ? Anf. 8|. In 9738 guilders banco of Holland, how much current money, agio 3 per cent. ? Anf. 100 : 103 : : 9738 : 10030*14 current money. In 8647 current guilders, how much banco, agio 3J per cent, ? Anf* 1034 : 100 : : 8647 : 8354*589 banco* Y 3 250 A-rpENDnr. In 738I fterllag, how much Irilh money, exchange 7J per cent. ? Anf. ico : i- lowing 5 per cent, intereft when the balance is due to the Bank, and 3J per cent, when due to W. Smith, Dr. Batiks Account Current with IV. Smith, Cr. Dec. 31. To Balance £. 436 Apr. 18. To Cafh - 526 Sept. 23. To Cafh- - 498 Dec. 3. To Cafh - 634 Mar. 22. By Cafh £. 394... May 17. By Cafh 673 Nov. 3. By Cafh 428- Dec. 26. By Caih 394-. Dates £• Days. Dr. Produ&s. Cr. Products, Dec. 31. Mar. 22. Dr. Cr. 436 394 8* 436 3488 4pr. i8 v Dr. Dr. 42 526 27 394 84 May 17. Dr. Cr. 568 673 29 5112 1136 Sept. 23. Cr. Dr. 105 498 1 29 94S 1.260 Nov. 3. Dr. Cr. 393 428 41 393 JJ72 Dec. 3, Cr. Dr. 35 634 30 1050 .• Dec. 26. Dr. Cr. 599 394 n 1797 j 198 • Dec. 310 Dr. 205 s 1025 3 6 5 ~8383~7- 7 586859 M5950 14595' M595a- 73000)440909(6! os t>i Rule for extrafiing the Cube Root. Divide the number into periods of three figures, and take the cube root of the left-hand period for the firft figure of the root, and fubtract its cube from the fijft period, and to the remainder bring down another period for a refolvend. Multiply the fquare of the figure fonnd by 3 for a diyifor, by which divide the refolvend, neglecting the tv*o, right-- hand figures : The quotient is to be tried for another figure of the root. To three times the former part of the root annex this trial figure, and multiply the fum by it, and place the product below t,hc divifor but two figures farther to the right, and add thefe two lines together to complete the divifor •, then multiply it by the trial figure, and fubtraft the produft from APPENDIX. 253 e- refolvend, and to the remainder bring down another period for a new refolvend. Below the complete divifor place the fquare of the figure of the root Jait found, and add it and the two lines above it, and the fu.n is a new divifor 5 with, which proceed a* before. What is the cube root of 33396648567-2 ? 333966485672^6938 2,16 jo8 ) 117966 1701 1 2501 X 9 = 1 1 2509 81 t 54574 8 5 14283 6219 •*4345*9*3- = 43°3557 9 ( H539^ 6 7>^ 1440747 166384 144241084x8=: 1153928672 9 tyejlions that require the Square or Cube Root to be tahen* I Required to find two numbers, of which the differ- ence is 4 and their produft 22W Anf. 17 and 13. Add the fquare of f the difference to 221, and the fquare root of the fum is f the Turn of the numbers, to which add $ the difference to get the greater y or fubtratf them to get 2. What two numbers are*hefe of which the product is 48, and their ratio as 4, to 3 ? Anf. 8 and 6. As 4 : 3 ;: 48 : 36; the fquase root of .-which is ths Ter. kiTer. 254 APPENDIX. 3. To find two numbers, of which the fum is 16 an f in pts Length Breadth 3 1 8 4 7 6 4 2 4 3 2 8 4 1 10 1 10 Surface Thicknefs 6 1 4 9 6 6 2 4, 6 6 2 6 Solid content 8 4 6 8 4 6 If the dimenfions be taken in yards or higher denomina- tions, reduce them to feet and find the content, which may afterwards be reduced to higher denominations, by dividing the fu£erficial feet by 9, or the folid feet by 27, &c. In meafuring fquared timber, the breadth and thicknefs are ufually taken in the middle ur the tree, and their pro- duel multiplied by the length gives the content. In meafuring round timber, it is ufual to take the girth in the middle with a line, J of this girth is multiplied by it ft If, and then by the length to get the content. Though the fallacy of thefe methods have been expofed for many year-, they are flill univerfally ufed in practice by • the common meafuiers. In taking the dimenfions of artificers' woike, the length and breadth are only meafured, a certain rate of thicknefs being generally fettled upon as a ftandard ; thus the ftand- ard thicknefs of done walls is 2 feet, and of brick-work i-f- bricks, &c. ; but when the work is not of the ftandard thick- nefs, the content ought to be reduced by adding or fub- tracling a proportional part according as the thicknefs ex- ceeds or fails fhort of the ftandard : Thus 7 the content of a APPENDIX. 257 wall 2-Jfeet thick ought to have | of it added, and that of a wall 1 i foot thick ought to have £ of itfelf fubtradled. In the fame manner, if the wall be 2 bricks thick add -| of the content ; or if it be brick thick fubtra&-|. Though the contents be always found in feet and lower denominations, they are only fmall pieces of work, fach as glazing, fkirt-boards, window-frames, and*hewn-work, that are eftimated by the fquare foot 5 greater works are eftima- ted in higher denominations : Thus, mafon and ilater-work are eftimated by the rood of 36 fquare yards, and fometimes by a fquare of 100 feet; as are alfo roofing and flooring in England, where brick-work is eftimated by the rood of 272 fquare feet : Other works are generally eftimated by the fquare yard. Some parts of a work require much more labour than others ; ana\ this u generally allowed for, in taking the dimeniions : Thus in mafon-work a foot is added to the height of fide- ■Walls for levelling, and the fame for every belt 3 and circu- lar work i* eftimated double meafure, and doors and win- dows, work and half work. For the fame reafon, no de- ductions are made from flooring for the hearths, nor in brick-work for the vacancy at the fire ; and on this account, partly, it is that 9 inches are added to the Hoping fide of a flate-roof for the eaves, and 4 inches for joining a flate-edge to a tile-roof. Cuftom has eftablifhed certajn methods of taking the di- meniions in different cafes : Thus, in meafuring the work- manfhip of a houfe, the girth on the outfide is taken for the length, and no deductions are allowed for doors, windows, &c. •> but when the charge is to be made for both materials land workmanfhip, the length is taken on the outfide and the breadth on the infide, and allowance is made for vacancies. Likcwife, in meafuring hewn-work, the line is bent into all hollows, and over all projections y and in meafuring parti- tions, plafter, and painter-work, the height is taken by- bending over all the mouldings, but the length is taken in a ftraight line ; and the breadth and half breadth of a houfe are often confidered as equal to the two floping fides of a roof, in meafuring carpenter or fiater-work. I How much fhouid be charged for the workmanfhip of a houfe 36 feet long and 24 feet broad within walls ; the fide- Z 2 5 8 APPENDIX. walla-. 2 feet thick and 24 feet high, meafured from the foun- dation 5 the gables 3 feet thick and 8 feet higher than the fide-walls to the bottom of the chimney-ftalks, which are 7 feet broad, 3 deep, and 8 high, the height of all the vents no feet, at 2I per rood for rouble work, reduced to the ftandard of 2 feet, and i4d per foot for carrying up vents I feet Side-walls 42 End-walls 28 i for 3 f. thick 14 T 4 84 Length Height Levelling 7 fide-walls £ Content 168 24 672 33 6 4032 72 4104 feet Breadth of) g gable-ends 3 Ofchimnev- 1 (talks ' $ ' -§• of it for 3 f thick Height • } 35 174 8 Two ga- ble-ends 420 420 J 840 fee Breadth ofl chimney- >• ftalk J Depth of it fofit fori 3 f. thick y Height Two chim-7 ney-ftalks Jf feet Content of the body of the houfe 4104 Two gable-ends - - 840 Two chimney-ftalks - 240 9)5184 36) 57S 16 roods at 2l £32 o tio feet of vents at i4d 6 8 Etfpence of worktnanfhip £38 8 ^ Here the content of ^he gable-ends b found by adding the breadth of the houfe, and the breadth of the chiraney-ftalks, APPENDIX. 259 and multiplying the fum by f the height, from the top of the fide-wall to the top of the chimney-ftalk ; and the content of the chimney-ftaik is found by adding its breadth and thick- nefs, and multiplying the fum by its height. In brick chim* ney-ftalks the girth is multiplied by the height. How much flating in a pavilion roof with a platform, the length at the eaves 50 feet, and the breadth 30 feet ; and the length at. the platform 30 feet, and the breadth 10 feet ; the Hoping fide 14 feet taken the (horteft way from the platform to the eaves, and 17^ feet taken along the hips ? Half the circuit at the eaves Half the circuit at the platform Mean circuit Breadth 1680 4 Hips, each ljf by i\ feet - 105 Eaves, 160 feet by 9 inches - 120 ai 1 yards 6 feet 5° + 3° 30 + 10 feet = 80 = 40 : 120 FINIS. N.'B. Juft Publiihed, (Price 3s. 6d.) A Key to this Arithmetic, containing the Solutions, at large, of all the Queftions in all the Rules of this Book j for the ufe q£ thofe Preceptors aad Others who ufe the Book. CROSS, PRINTER, EDINBURGH \ . \ i M283887 T THE UNIVERSITY OF CALIFORNIA LIBRARY m