f ,v^^ / 'V dt ^ r-f^ DUKE UNIVERSITY LIBRARY Treasure %oom 1' THE FEDERAL ACCOUNTANT: CONTAINING, I. COMMON ARITHMETIC, the Rules and Illustrations. II. EXAMPLES aod ANSWERS with BLANK SPACES, sufficient for their operation by the Scholar. III. To each Rule a SUPPLEMENT, comprehending, 1. QUESTIONS on the oMature of the Rule, its Usk and the nianuer of its Operations. 2. EXERCISES. IV. FEDERAL MONEY, with rules for all the various operations in it. to reduce Federal to Oi.b Lawful, and Old Lawful to FEDERAL MONEY. V. Interest cast in Federal Mo.vey, with Compound Multiplication , Compound />irmo7(,and Practice, -wrought in Old LAwrur and in Federal MoNEv ; the same questions being put in separate columns on the same page in rach kind of money, these two modes of account become contrasted, and the great advan- tage gained by reckoning in Federal Money easily discerned. VI. Demonstrations by Engravings of the reason and nature of the various stops in the eKtraction of the Square and Cube Roots, not to b» found in any other treatise on Arithmetic. Vn. Forms of Notes, Deeds, Bokds and other Instruments of Writirg. THE WHOLE IN A FORM AND METHOD ALTOGETHER* NEW, FOR THE EASE OF THE MASTER AND THE GREATER PROGRESS OF THE SCHOLAR. BY DANIEL ADAMS, M. B. STEREOTYPE EDITION, KETISED AND CORRECTED, WITH ADDITIONS. KEENE, N. H— PRINTED BY JOHN PRENTISS, [proprietor of THE COPY RIGHT.] Sold at his Bookstore, and by the principal Booksellers in the New-England Stat«»- ' and New- York. — IS)9. — «=>*-o— 'Price 10 Dollars per liostn, 1 Dollar siuglt. NcivhaDrpshirc Dialricty ss. I^li IT RKMEMBERED, that on the seventeenth day of July, in tLe thirty L. s. niuth year of the Imlcpendence of the United States of America, Daniel Adams, of Mont Vernon, in said District, hath deposited in tiiis office the Title of a Book, tlje riijht whereof Ire claims as Author, in the following words, to wit : — " The Scholar's .ifitfttnelic. : or Federal Accountant, (.'ontaining I. Common Arithmetic, the Rules and Illustrations. — II. Examples and Answers, with blank spaces sufficient for their operation by Ori Scholar. — III. To each Rule, a Supplement, comprehending, 1. Questions on (lie nature of the rule, its use, and the manner of its operations. — 2. Exercises. — IV. Federal Monej', with rules for all the various operations in it, to reduce Federal to Old Lawful, and Old Lawful to Federal Money. — V. Interest cast in Federal Money with ("om|»ouiid Multiplication, Compound Division and Practice wrought in Old Lawful and ill Federal Money, the same questions being put in separate columns on the same page in each kind of money, by which these two modes of account become contrasted, and the great advantage gained by reckoning in Federal Money easily discerned. — VI. Demon- strations by engravings of"^ the reason and nature of the various steps in (he extraction of the S(juai"e and Cube Roots, not to be found in any other treatise on Arithmetic. — VII. Forms of Notes, Deeds, Bonds, and other instruments of writingr — The whole in a form and method altogether new, for the ease of the Master and the greater i)rogi'es3 of the Scholar.— Bv DANIEL ADAMS, M. B." In conformity to the act of Congress of the United States, entitled '• an act for the encouragement of Learning, by securing the copies of Maps, Charts and Books, to the authors and proprietors therein mentioned, and extending the benefit thereof to the ails of Dfsiguiug, Engraving, Etching, Historical and other prints. G. AV. PRESCOTT, Clerk of tht U. S. Court, JV. H. Disfrict. PREFACE. IT 13 fourteen years siace the first Edition of the Scholar's Arithmetic was offered to the Public. It has now gone through nine editions, and more than Forty Tliousand copies have been circulated. In those places where it has been introduced, it never has, to the best of our knowledge, been superseded by any other work which has come in competition with it. A knowledge of these facts is, perhaps, one of the best recommendations which can be desired of the work. It has now undergone a careful revisal. Some of the rules have been thought to be deficient iii examples ; in tliis revised edition, more than six- ty new examples have been added under the different rules. Some have expressed a desire that answers might be given to the " Miscellaneous Questions,'" at the end of the book ; these have been added accordingly, and the number of these questions increased. But what more particularly claims attention in this revised edition, is the introduction of the rule of Exchange, where the pupil is made acquainted with the different curren- cies of the several states, (that of S. Carolina and Georgia, only excepted,) and how to change these currencies from one to another ; also, to Federal Money, and Federal Money to these several currencies. This has been done more particularly with a view to the accommodation of the State of New-York, and other more southern states, where this work has already acquired a very considerable circulation. Answers are given to many of the questions in different currencies, so that the pupil in N. England, N. York, &:c. will find an answer to the question, each in tlie currency of his own particular state. These comprehend the only additions in the present new edition. Wo have now the testimony of many respectable Teachers to believe;, that this work, where it has been introduced into Schools, has proved a vpry kind assistant towards a more speedy and thorough improvement of IT PKEFACE. Scholars in Numbers, and at the same time, has relieved masters of a hea?y burden of writing out Rules and Questions, under which they have so long labored, to the manifest neglect of other parts of their Schools. To answer the several intentions of this work, it will be necessary that it should be put into the hands of every Arithmetician : the blank after each example is designed for the operation by the scholar, which being (irst wrought upon a slate, or waste paper, he may afterwards transcribe into his book. The Supplements to the Rules in this work are something new ; ex- perience has shown them to be very useful, particularly those " Questions,^' unanswered, at the beginning of each Supplement. These questions the pupil should be made to study and reflect upon, till he can of himself devise , the proper answer. They should be put to him not only once, but again, and again, till the answers shall become as familiar with him as the num- bers in his multiplication Table. The Exercises in each supplement may be omitted the first time going through the book, if thought proper, and taken up afterwards as a kind of review. Tlirough the whole it has been my greatest care to make myself intelli- gible to the scholar ; such rules and remarks as have been compiled from other authors are included in quotations ; the Examples, many of them are extracted ; this I have not hesitated to do, when I found them suited to my purpose. Demonstrations of the reason and nature of the operations in the ex- traction of the Square and Cube Roots have never been attempted in any work of the kind before to my knowledge ; it is a pleasure to find these have proved so higldy satisfactory. Grateful for the patronage this work has already received, it remains only to be observed, that no pains nor exertions shall be spared to merit iti continuance. DANIEL ADAMS. MoiU-Vcrnnu, (A'. //.) December 26th, 1815. RECOMMENDATIONS. JVew-Salm, Sept. 14f/«> 180!. HAVING attentively examined " The Scholar's Ariihmf.Hc," I cheerfully give it as my opinion tliat it is well calculated for the instruction of youth, and that it will abridge muoh of the time now necessary to be spent in the communication and attainment of such Arithmetical kno^vledse as'is proper for the discbarge of business. WARREN FIERCE. Preceptor of A'eic-5a/eni Academy- Crohn Jicadtmy, Sept. 2, 1801. Sir I liave pnniscd with attention " The Scholar's Arithmetic," which you transmitted to me some time since. It is in my opinion, better calculated to lead students in our bchools !uid Academies into a complete knowledge of all that is useful in that branch of lilcratun , ilinn any other work of the kind I have seen. With great sincerity I wi.sh you siiccess in your exertions for the promotion of useful learning ; and I am confident that to be generally approved your work needs only to be generally known. WILLIAM M. RICHARDSON, Preceptor of the Academy. Extract of a Letter from the Hon. JoHi» Wheelock, Lt. D. President of Dartmouth College, to the Author. " The Scholar's Arithmetic is an improvement on former productions of the same nature. Its distinctive order and ?upplement will help the learner in his progress; the part on Fe- deral Money makes it more useful ; and I have no doubt but the whole will be a new fund of profit in our country." September "Jfh, 1807. The Scholar's Aritiimcllc contains most of fne important Rules of the Art, and some- thing, al?o. of (he curious and entertaining kind. Tlie subjects are handled in a simple and concise manner. While the questions arc few, they exiiibit a ccnsideral.-le varietj'. While they are gene- rally easy, .some of them afford .=cope for tlic exercise of ihc Pniiohir's judgnicnt It is a good (juality of the Book, that ithas.=:o much to do with Federal Money. The plan of sliow in^ the reasons of the ojwjrations in the extraction of (i-.e Squard and Cube Roots is good. DAiMEL HARDY, .lus. Preceptor of Chesterfield Academy. Exlracl oj a LeHer fiom thz Rev. Laban Ainswohtti cf -Jtiffrcy, to (he publish^ of (he fuurlh Edition, dated August 3, 1S07. '• The snperioriiy of the Scholar's Arillimetic to any book of (!k^ 1 ind in ray knowledge, ( learly appears from its good rlfect in fhe schools I aimually visit. — Previous lo its intm- " r» i..!ioniil r.ccoisnt of the seveial c|)('r;itions in Arithia'j'iir; whi'oli is the fjst proof of "i^ir having jparned togoed purj'osn ' CONTENTS. Inlroduclioa P^gf 7 >'otation 7 Numeration Poge 8 Explanation of Characters ..... 10 SECTION I. Fundamental Rules of .'irithmelic. Simple Addition 11 do. Siitdrarlion ....... 16 do. MultipliciUion 19 Simple Division . . Compound Addition do. Subtraction 27 35 46 SECTION II. Pules tssenlially necessary for every person to fit and qualify than for the transaetion of business. Rfduction 51 I Method of casfinsr Interest on Fractions 68 DerimnI Fractions 69 Fedf-ral Money 8f) Kxcliarige 84 Tal)lf (o reduce .shiilinj^s and ^ pence to cents and mills ^ * Tables of Exchange 92 Interfst 9.3 Easy mctiiod of casting Interest ... 97 91 99 Notes and Bonds when par- f tial payments at diflerent / times have been made J Compound Interest 104 do. Multiplication .... H)5 do. Division 110 Single fiule of Three HT Double Rule of Three l.W Practice HI SECTION HI. Rales occasionally vstfai lo men in parliculur rmplouncnts of life. Involution 15" Evolution 1.57 Extraction of the Square Root . . . 1§8 Demonstration of the reason and \ nature of the various steps in the operation of extracting the S()UHre Root Extraction of the Cube Root Demonstration of the reason and nature of the various steps in the operation of extracting the Cube Root 1 Sinjile Fellowship 177 Double Fellowship 171> Barter 182 Loss and Gain 185 Duodecimals 188 159 167 168 Examples for measuring wood . . . 189 Boards . .190 Painter's and Joiner's work .... 192 Glazier's work 192 Alligation 193 Medial 193 Alternate 194 Position 198 Single 198 Double 199 Discount 201 Equation of paymeHls 201 Guaging 203 Mechanical Powers 203 The Lever 203 The Axle 204. The Screw 204 2M PEr.BLEMs 1st. To find (he circumference of a circle, the diameter being given - 2d. To find the area of a circle, (he diameter being given 264 3d. To measure the solidity of an irregular body '•- 204 Bliscellaneous Questions --..----, 206 SECTfON IF. Form of Koles. ^c. Notes 211 Bonds 212 Ficceipls 2)."-{ Orders 211 Deeds - - - ., 214 Indenture .--.----- 215 Will 216 THE SCHOLAR'S ARITHMETIC INTRODUCTION. ARITHMETIC is the art or science which treats of numbejF?. It is of two Iciods, theoretical and practical. The Theory of Arithmetic explains the nature and quality of number*?, and demonstrates the reason of practical operations. Considered in this sense, Arithmetic is a Science. Practical Arithmetic shews the method of working by numbers, so as to be most useful and expeditious for business. In this sense Ajrithmetic is an A7-t. DIRECTIONS TO THE SCHOLAR. Deeply impress your mind with a sense of the importance of arithmetical knowledge. The great concerns of life can in no way be conducted without it. Do not, therefore, think any pains too great to be bestowed for so noble an end. Drive far from you idleness and sloth ; they are great enemies to improvement. Remember that youth, like the morning, will soon be past, and that opportunities once neglected, can never be regained. First of all things, there must be implanted in your mind a fixed delight in study ; make it your inclination ; " A desire accomplished is s-xeet to the soul.'''' Be not in a hurry to get through your book too soon. Much instruction may be given in these few words, UNDERSTAND f.vf.ry tiiin(; as you go Ai.oNt;. — Each rule is lirst to be committed to memory ; afterwards, the examples in illustration, and every remark is to be perused with care. There is not a word inserted in this Treatise, but with a design that it should be studied by the Scholar. As much as possible, endeavour to do every thing of your- self ; one thing found out by your own thought and rcllection, will be of more real use to you, tiian 'tzccnti/ (Iiings told yon by an Instructor. Be not overcome b}' little seeming dilTicullies, but rather strive to overcome such by patience and application j so shall 3'our progress be easy and the object of your endeavours sure. Off entering upon this most useful study, the first thing which the Scholar lias to regard, is NOTATION. Notation' is the art of expressing numbers by certain characters or fig- ures : ' of which there are two methods.; 1. The Unman mclltiMt'hy Letters. -. The .Arabic riietlioif, by i'igi-res. The hitter ii' that of ge^fifl u-.e. 8 INTROpUCTIOX. In the Arabic method all numbers are expressed by these ten characters or figures. 1 2 .3 4567890 Unit; or two ; three ; four ; five ; six ; seven ; eight j nine ; cypher o"<^ . [or nothing-. The nine first are called si^niJicaiU, figures, or digits, each of which standing by itself or alone, invariably expresses a particular or certain num- ber ; thus, 1 signilies one, 2 signijjes t-xo, 3 signifies three, and so of the rest, until you come to nine, but for any number more than nin£, it will always require two or more of those figures set together in order to ex- press that number. This will be more particularly taught by NUMERATION. • Nameration teaches how to read or tc-r/fc any sum or number by fio-ures. In setting down numbers for arithmetical operations, especially with be- ginners, it is usual to begin at the ri^U hand, and proceed towards the left. Example. If you wish to write the sum or number 537, begin by setlini^ down the seven, or right hand figure, thus 7, next set down (he three, at the left band of the sevon, thus 37, and lastly the fie, at the left hand o( the three, thus 537, which is the number projiosed to be written. In this sum thus written you are next to obsei"ve that there are three places, meaning the situations of the three different figures, and that each of these places has an appropriated name. The frst place, or that of the right hand ' figure, or the place of the 7, is called vnit^s place; the accond place, or that of the figure standing next to the right hand figure, in this the jdace of the 3, is called ten's place ; the third place, or next towards the left hand, or place of the 5, is called hundred's place ; the next or fourth place, for we may suppose more figures to be connected, is thousand's place ; the next to this tens of thousand's place, and so on to what length we please, there being particular names for each place. Now every ti^ure signifies difTerentlv, ac- cordingly as it may happen to occupy one or the other of these places. The value of the first or right hand figure, or of the figure standing in the place o( units, in any sum or number, is just what the figure expresses stand- ing alone or by itself; but every other figure in the sum or number, or those to the left hand of the first figure, have a dilferent signification from their true or natural meaning ; for the next figure from the right baud towards the left, or that figure in the place of tens, expresses so many times ten, as the same figure signifies units or ones when standing alone, that is, it is te?i times its simple primitive value ; and so on, every removal from the right hand figure, making the figure thus removed ten times the value of the same figure when standing in the place immediately preceding it. Example. Take the sum 3 3 3, made by the same figure three times repeated. The first or right hand figure, or the figure in the place oi' units, has its natural meaning or the same meaning as if standing alone, ar.d signi- fies three units or ones ; but the same figure again towards the left hand in the second place, or place of tens, signifies not three units, but three tens, that is thirty, its valne being increased in a tenfold propoi-tion ; proceeding on still further towards the left hand, the next figure or that in the third place, or place onnmdreds signifies neither three nor thirty, but three hundred, which is ten times the value of that figure, in the jdace immediately preceding it, or that in the place oiLiens. So you might proceed and add the figarc 3, iifty or INTRODUCTION. 9 an hundred times, and every time the figure nas added, it would signify ten times more than it did the last time. A Cypher standing alone is no signitication, yet placed at the right Iland of another figure it increases the value of that figure in the same ten- fold proportion, as if it had been preceded by any other figure. Thus 3, sfanding slone, signifies three ; place a cypher before (30) and it no longer signifies three, but thirty ; and another cypher (300)'%nd it signifies three lionrlrefl. The value of figures in conjunction, and how to read any sum or num- ber agreeably to the foregoing observations, may be fully understood by the following TABLE. ri -: ^ The words at the head of the Table shew ■^ ^ i ^ rt ^^^ signification of the figures against which '^ C ;2 .2 g ^ they stand ; and the figures shew how many ,•, • S -^ 2 -a § of that signification are meant. 'Thns Units §=£^^2 Ha in the first place sio-nify ones, and 6 standinjf £"^°^i3 <^-^OT . against it, shews that six ones or individuals .t«^'c-5!£ •-p^'c-S arc here meant ; tens in the second place c'^oS-iJoc-iofj;?:; . shew that every fitjure in this place means so P -a '« 3 -TD -fi .2 "3 tfi S '^ -/• jn _ J o . J- ■ ^ -J. I uacaocK^acocG.t: many tens, and 3 stanamfr aaramst it, shews cq^U(t_i2jf_,;52^f_U(^E-(ti *hat three tens are here meant, equal to thir- 21G7235 4 21836 iy,what the figure really signifies. Hundreds 3407G214G312in the third pl«ce shew the meaning of fig- 13025037645 ures in this place to be Hundreds, and 8 41393210G4 shews that eight hundreds are meant. In 27021367 5 the same manner the value of each of the re- 46327291 maining figures in the table is known. Har- 12 3 4 6 3 2 ing proceeded through in this way, the sum 2 3 4 5 6 7 of the first line of ligures or those immedi- 8 9 9 8 ately against the words, will be found to be 7 6 5 4 Tv:o Billions, one hundred sixty seven thou- 12 3 sands, izi'o hundred and tluriy-Jisc Millions : 4 5 four Jiundred tzt:c7ity-one thousands ; eight hun- 7 dred and thirty-six. In the like manner may be rerii] all the remaining numbers in the Table. Those words at the head of the Tnble are applicable to any sum or num- ber, and must be committed perfectly to memory so as to be readily appUed on any occasion. For the greater case of reckoning, it is convenient and often practised in public ofiices, and by men of business, to divide any number into periods and half periods, as in the following manner : 5. 3 7 9, G 3 4. 5 2 1, 7 6 8. 5 3 2, 4 6 7 fiO C^ ^ Co ■^ "^ "^ ^ C;=:;=3=;i? S£:5-C-CJ eaa*:Hi2 ^ -G -C) ~Ci --i . o^S^ e r* £ 3 ^ S S -c; 5 e ^- c-i -^ c - ?: .^ ;^ ~ T3 s: „ crs in the I'oregoinj^ Table, the learner may proceed to write the follow- ing nnmbers out in figures. Two hundred and sixty-three. ^ Five thousand one hundred and sixty. One hundred thousand, six hundred and four. Five miHion, eighteen thousand, seven hundred & six. Two million, six hundred and fifty thousand, one hundred and thirty-seven. Seven hundred and ninety-four million, one hun- dred and forty-nine thousand, live hundred and twenty. Three thousand, nine hundred and foity million, four hundred and two thousand, eight hundred and four. Five hundred thirty six thousand, two hundred and seventy two million, one hundred and three thousand and six. ^ ^ Four })ill'.on, six hundred thousand million, seven < hundred thousand, two hundred and ninety- 4 f two. ^* ■ — Explanation of the CJiaracters made use of in this Work. __ ^ The sign of equality ; as 100 cts=l Dol. signifies that 100 centg^ f are equal to 1 dollar. , ^ Saint Gporge's Cross, the sign of addition ; as 2-|- 1=6, that is, "i "^ I addpd to 4 ;)r-j equal to 6. . J — The sign ofi^-ubt.artion ; as 6 — 2=4; thatis,2 takenfrom 61eareK4. ^ Saint Aniircw's Cross, the sign of multiplication ; as 4X G=~i ; ( that is, 4 times 6 r:re equ?l to 24. w i Reveiscd rarentheees, the sign of divismn ; as 3)6(2, that ^^ I is, 6 divided by 3 the qiKti3nt is 2, or G-i- .3=2. - The s^on of proportioi. ; as 2 : 4 t : 8 • 16, that is, as 2 to 4 so is to lti» r X SECTION I. FUNDAMENTAL RULES OF ARITHMETIC. 1 HESE are four, Additiox, Subtraction, Multiplica- tion, and DIVISIO^'; they may be either simple or compotmdj simple, when the numbers are all of one sort or denomina- tion ; compound, when the numbers are of different deno- minations. TuEY are called. Principal or Fundamental RuleSy be- cause all other rules ancl operations in arithmetic are not!iin2; more than various uses and repetitions of these four rules. Thfi object of every aritlnnetiral operation, is, hy certain given qnantilips which arr known, to find otit others which ;ire iinknowti. This cannot be done hut by changes ef fectcd on the given niinibcrs ; and as the only way in whicii numbers can be changed i'- either by increasins; or diminishing^ tlieir (juantitie?, and as there can be no increase or diminution of numbers but by one or the other of the al)ove operations, it consequently lollows, tiiat tlusc full)- rules embrace the whole art of .\rilhmctic. ^ I. SIMPLE ADDITION. SiMFi.E Addition is the putting together of two or more number?, of the same denoiflhiation, so as to make them one whole or total number, Called the sran, or amount. RULE. ilPl^ce the numbers to be added one under another, with units under unit?, lens tinder tens. &.c. and draw n line under the lowest number. 2. Add the right hand column, and if the sum be under ten, write it under the col'imn ; but if it be ten, or any exact number of tens, rxrite a cypher ; and if it he not sn exact number of tens, write the excess above tens at the foot of (lie" rolnmn, and for every tin the sum contstns, carry one to the next column, and add it in the same manner as the former. 3. Proceed in like manner to add the other columns, carrying for the tej^iS of each to the next, and set down the full sum of the last or left hand column. PROOF. Reckon the figures from the top downwards, and if the work be right, this amount will be equal to the first ; — or, what is often practised, " cut " off the upper line of figures and (ind the amount of the rest ; then if the *' amount and upper line when adrjed, be equal to the stim total, the work " is supposed to be right.' 12 SIMPLE ADDITION. EXAMPLES. Sect. I. 1. 1. What will be ^ | E^ il the amount of ... 3 6 1 2 dolls, dollars Avhen added together ? » s oJ -2 2 5 ? "S 8 4 3 dolls. o 6 5 1 dolls, and of 3 tS 3 a 6 1 2 dollars 8 6 4 6 3 1 3 dollars dollars dollars 1 2 3 9 dollars 8 6 9 7 Here are four sums given for addition ; two of them contain V7iits, tens, hundreds, "©^ tJioiisands ; another of them contains units, ^ lens, hundreds; and a fourth contains units onh'. The first step to prepare these sums for the ojieration of addition, is to write them (loun, units under units, tens under tens, ^c. thus ; — Answer, or Amount, Amount of the three lower lines, Proof, 12 3 9 To find the answer or amount of the sums given to be added, begin with the right hand column, and say, 3 and 1 make 4, and 3 are 7, and 2 are b ; xvhich £um (9) being less than ten, set down directly under the column you added. Then proceeding to the next column, say again; 5 and 4 are 9, and 1 is 10 ; beinj; even ten, set down 0, and carry one to the next column, saying 1, v/hich I carry to 6 is 7, and is nothing, but 6 make 13 ; which sum (13) is an excess of 3 over even ten ; therefore set down 3 and carry 1 for the ten to G in the next column, saying 1 to 8 is 9, and 3 are 1'2 ; this being- the last cohimn, set down the whole number (12) placing the 2, or unit figure directly under the column, and carrj'ing the otiier figure, or the 1, forward to the next place on the left hand, or to that of Tens of thousands, and the work is done. It may now be required to know if the woiv. be right. To exhibit the method of proof let the upper line of figures be cut off as «een in the ex- .'imple. Then adding the three lower lines which remain, place the amount (.jG97) imdeF the amount first obtained by the addition of all the sum^-', ob- serving carefully that each figure fail directly under the column which pro- iluced it; then add this last amount to the upper line which you cut ofi'; llius 7 to 2 are 9 ; 9 to 1 are 10 ; carry one to 6 is 7 and 6 are 13 ; 2„jv!vi*'.h 1 carry again to 8 is 9 and 3 are 12, all vv'hirh being set down in their pw'pcv t'liices, and as seen in the example, compare the amount (12309) Jast o!)- itined, wiih the first amount (12.309) and if they agree, as it i^i seen in thif« case they do, then the work is judged to be right. Note. — The reason of carnjing for ten in all simple numbers is evident from v.'hat has been taught in Notation. It is because 10 in an inf-'iioj . oiiunn is just equal in value to 1 in a superior column. As if a man shov.ld te holding in his ri^ht liand half pistareen-, and in his left, dollais. !l you should take 10 half pistareens ivum Iris right lir.nd, and put one dollar.into his left hand, you would not rob the nsan (.f any of his money, because 1 ot those pieces in his left hand is just equal iu vulue to 10 of tiiosa in h;^ y\'>A\'. hand. Sect. I. 1. SIMPLE ADDITION. 13 2 3 4 = = =3 1 6 7 5 2 6 3 7 3 4 7 1 o G 5 4 3 8 2 4 8 5 7 3 2 2 6 3 7 1 2 6 5 1 4 2 1 G 8 3 1 6 4 3 7 1 3 6 2 1 5 2 4 3 6 3 5 4 2 8 6 6 3 9 8 1 7 5 2 1 3 2 4 3 2 2 1 4 3 1 6 'i 7 5 2 6 3 1 9 8 5 1 7 6 4 5 6 3 9 8 7 5 1 3 4 5 6 7 8 9 2 4 6 8 2 3 7 6 9 8 6 5 3 5 1 2 8 7 5 4 1 7 3 8 7 9 T) 2 8 6 7 8 5 4 3 9 6 7 4 9 8 1 2 6 4 7 3 1 0. ] 3 2 1 3 3 6 9 10 4261783 653865076 6402531 983124 3 52643 40750208 7 6 2 3 4 1 6 3 4 1 2 5 7 8 1 7 6 Cf 4 6 8 2 2 9 2 1 1 3 5 3 6 4 o O 3 5 6 8 7 3 6 1 7 3 __ 14 SUPPLEMENT TO ADDITION Sect. I. 1. SUPrLEMENT TO ADDITION. THK altPnlive t-cholar nho has understood, and still carries in his mind, what Iras alrea«ly been tanj^ht him of Addition, will be able to answer his instructor to the following QUESTIONS. L What is simple Addition ? ^ a. How do you place numbers to be added ? 3. Where do you be^in the addition ? 4. What is the answer called ? 5. How is the sum or amount of each column to be set down ? 6. AVhat do you observe in regard to setting down the sum of the last column ? 7. Why do you carry for ten rather than any other number ? 8. How is addition jproved ? NoTi:. Should the learner find any difficulty in giving an answer to the above questions, he is advised to turn back and consult his Rule, with its illustrations. EXERCISES. 1. What i^ the amount of 2801 2. Suppose you lend a neighbour dollars ; 765 dollars ; and of 397 £210 at one time, £,1G at another, rlo]':.:>, when added together ? £17 at another, and £,9 at another. Ans. 3963 dollars. What is fhe sum lent ? Ms. £312. Note. The scholar who looks at greatness in his class, will not be dis- couraged by a little difficulty which may at first occur in stating his ques- Uoii, but will apply himself tlic more closely to his Rule, and to thinking. :hat if pos-^ible he may be able himself to aiiswer what another may be obliged to have taught him by his instructor. o. A tree was broken off by the 4. A ma« being asked his age, \Aw\, 27 feet from the ground ; the said he was 27 years old when he port broken off was 71 feet long ; married, and he had been married vhat was the hei«rht of that tree be- J5 years. What was the man's fore it wa= broken ? ag« ? .'3«f OS frr,^ Sect. I. 1. SUPPLEMENT TO ADDITION. 15 5. Washington was born in the 6. There are two numbers ; tkc year of our Lord 1732 ; he was 67 less number is 3761, the difference years old when he died ; in what between the numbers is 597 ; what year of our Lord did he die ? ^is the greatest number ? Ans. 1109. 'w Ans, 9352. 7. From the Creation to the de- parture of the Israelites from Egypt was 2513 years ; to the siege of Troy, 307 years more ; to the build- ing of Solomon's temple, 180 years ; to the building of Rome, 251 years ; to the expulsion of the kings from Rome, 244 years ; to the destruc- tion of Carthage, 363 years ; to the death of Juhus Cajsar, 102 years ; to the Christian aera, 44 years ; re- quired the time from the Creation to the Christian sera ? Am. 4004. 8. At the late Census, talien A. D. 1810, the number of inhabit- ants in the several jXew-England States was as f^illows ; viz. Maine, 228705; N. Hampshire, 214460; Vermont, 217895 ; MassaGhusctti, 472040 ; Rhode-bland, 76931 ; Coti- ?iecticut, 261942; what was the num- ber of inhabitants at that time in New-England ? Ans^ 1471973. 9. Thefc are five numbers, the first is 2617; the second 893; the third 1702 ; the fourth as much as the three first ; and the fifth twice as much as the third and fourth ; what is the wh©Ie sum ? An^. 24252. 10. A gentleman left his son, 2475 dollars more than his daughters, whose fortune was 25 thousand, 25 hundred, and 25 dollars ; what was the son's portion, and what was the amount of the whole estate ? Ans. Son's portion, 30000. Whole estate. 57525. IQ SIMPLE SUBTRACTION. Sect. I. % i 2. SIMPLE SUBTRACTION. SIMPLE SUBTRACTION is taking a less number from a greater of the same denomination, so as to shew the difierence or remainder ; as 3 taken from 8, there remniins 3. ' The greater number (8) is called the Minuend, the less number (5) the S:illrah4:n(1, and t)>e difference (3) or what is left after Subtraction, the I'ernuitider. RULE. " Place the less number under the greater, units under units, ten? under tens,' and so on. Draw a line bc'ow ; then bcg-in at the right hdm", and gul»lract"each iijjure of the less number from the iigure above it, and piace ti:e remainder directly below. When the tigure in the lawer line exceeds th^ figure ybove it, suppose 10 to be added to the upper ligurc ; but in this case you must add 1 to the under figure in the next column before you iiiibtract iL This is called, borrowlitir len.^'' PROOF. Add the rcmaiiider and subtrahend together, and if the sum of them cor- rerpoiid Willi the minuend, the work is supposed to be right. Jlinuend 6 5 3 The numbers being Jilaced with the larger uppermost, as the rule directs, I begin \vith the Subtrahend 5 2 7 1 unit or right hand Iigure in the subtrahend, and say, ] from 3 lliorc remain 2. which I set Remainder 3 3 G 2 down, and proceeding to tens, or the next figure, 7 from 5 1 cannot, I therefore borrow, or sup- Pronf 6 5 3 pose ten to be added to the upper figure (5) wiiicli make 15, ti.en I say 7 from 15 and there remain 8, which I set down ; ilu-n proceeding to the nest place, I sa}', 1 •which I borrowed to Sis ^, and 3 from and there remain 3 ; this I set down, and in the next place laay 5 liorn and tijere remain 3, which 1 set down and the work is done. PnooF. I add the remainder to the subtrahend, and finding the sum juic efj'iul 1o the minuend, suppose the work to be right. Norn,. The reason of borroxn-ii'^ ten, will appear i^ we consider, that, wh".n iwo numbers arc equally increased by adding the same to both, their diiTi'.ronce will be equal. Thus the dilTerence between 3 and 5 is 2 ; add the niunber 10 to each of these figures (3 auil 5) they become 13 qnd 15, sLiH the xlilTerence is 2. When v»e proceed as above directed, w7*^add or suppose, to be added, 10 to the minw.mh and we likewise add,pi|i^ to the next higiier place of the suhiraJhcnd, which is just equal iu value to 10 of I'iC. lower place. 5 14 6 5 the minuend, 1 2 3 4 2 the subtrahend. Ren?ainder. From Take 1 2 7 G 8 7 6 5 r 3 3 G rUtrT. I. 2. SIMPLE SUBTRACTION. 17 ,Vt -in case of borrowing ten, it is a matter of indifference, as it re- spects ti.i operatioH, whether we suppose ten to be added to the upper figure, and from tht sum subtract the lower figure and set down the dif- ference ; or, as 21^. Pike directs, first subtract the lower figure from 10, and adding the difference to the figure above, set down the sum of this difference and the upper fijrure. The latter method may perhaps be thought more easy, but it is conceived, that it does not lead the understanding of youth so directly into the nature of the operation as the former. 1. From 10236742317 981062 Take 8791284506703281 Rem, 2. From Take Rem. 10236742317981062 8 791284506703281 X From 21468317012101, take 668497067382. Rem. 20899819944719. 4. From 3G47 10825193, take 279403865746. 6. From 168012372458, take 89674807683. 6. From 100610528734, take 99874197867. 7. From 628103570126, take 248976539782. Rem. 85306959447. Rem. 78337564775. Rem. 736330867. Rem. 379127030344. 8. From 10000, take 9999. Rem. 1. 9. From 10000, take 1. Rem. 9999. The distance of time since any remarkable event, may be found by sub- tracting the date thereof from the present year. EXEP.CISFS. 1. How long since tbe Ameri- can Independence, which was declared in 1 776 ? 18 17 present time. 17 7 6 date of Ind. Ans. 4 1 years since. 2. King Charles, the martyr, was beheaded 1648 : how many yeans is it since ? i So, likewise, the distance of time from the occurrence of one thing to that of another, may be found by subtracting the date of the thing first happening, from that of the last. EXAMPLE. 1, How long from the discovery of America by Columbus, 1492, to the com- mencement of the war, 1775, which gain- ed our Independence ? 17 7 6 14 9 2 Ans. 2 8 3 years^ 2. How long from the termination of the war in 1783, which gained our Inde- pendence, to the commencement of the last war between the United States and Great Britain in 1812 ? Ans. 29 ycc.ri >\ 18 SUPPLEMENT TO SUBTRAGTION. Sect. { 8. SUPPLEMENT TO SUBTRACT"-^!.. QUESTIONS. 1. What is Simple Subtraction ? 2. How many numbers mu«.t there be given to perform that operation? ■3. How must the given numbers be placed ? 4. What are thjy called ? 5. When the fi£,'urc in the lower number is greater than that of the up- per number from which it is to be taken, what is to be done ? 6. How does it appear that in subtracting a less number from a greater, the occasional borrowing of ten does not affect the difference between these two numbers ? 7. How is subtraction proved ? EXERCISES. 1. What is the difference between 2. From a piece of cloth that 78360 anJ 5421 ? measured 691 yards, there were sold Ans, 72939. 273 yards ; how many yards should there remain ? Ans. 418. 3. There are two numbers, whose 4. What number is that which diflerence is 375 ; the greater num- taken from 176 leaves 96 ? ber is 862 ; I demand the less ? Ans. 79. Ans. 487. V 5. Suppose a man to have been born in the year 1745, how old was he in 1799? Ans. 54 years. 6. What number is that to which if you add 789 it will become 6350 ? Ans. 6561. in 7. Supposing a man to have been 63 years old in the year 1801 what year wijij^born ? Ans. in the year 1738. 8. At the census in 1800, the number of inhabitante in the New-England States was 12^3011 ; at the late census in 1810, the number was 1471937; s What was the ifcre;u3e of th'* ponuIatioD ia the New-Eng-lantl States in the : 'on years between lOOOand JfJlO? ''-~ '"^'^926 Sect. I. 3. SIMPLE MULTIPLICATION. 19 i 3. SIMPLE MULTIPLICATION. Simple Multiplication teaches, having two numbers given of the same denomination, to find a third which shall contain either of the two given cumbers as niafiy times as the other contains a unit. Thus, 8 multipUed by 5, or 5 timfes 8 is 40 — The given numbers (8 and 5) spoken of together, are called Factors. Spoken of separately, the first or largest number (8) or number to be multiplied, is called the Multiplicand; the less Jjujnber (5) or number to multiply by, is called tho- multiplier, and the amount (40) the Product. Before any progress can be made in this rule, the following Table must be committed perfectly to memory. MULTIPLICATION TABLE. H 2 ~~3 4| 5 6 7 8 9 10 1 11 r: 2| 4 e 8 1 10 12 14 16 18 20 i 22 24 3| 6 9 12|15 18 21 24 27 30 1 33 36 4| 8 12 IG 1 20 24 i 28 32 36 40 44 48 51 10 lo 20 1 25 30 i 35 40 45 50 1 56 60 C| 12 18 24 1 3(» 36 42 48 54 60 1 66 72 7| 14 21 28 1 3r> 42 ! 41) 56 63 70 77 84 M 16 24 32 1 40 43 56 64 72 80 1 88 96 •^1 18 - • 36 , 46 51 i f^2 7'.'. 81 90 j 99 108 10 i 20 30 40 1 5(.i GO ! 70 80 90 100 1 no 120 HI 22 33 44 1 65 C6 1 77 88 99 110 i 121 132 12 j 24 36 18 I 60 72 84 96 108 120 1 132 144 1/ By this table the |>roduct of any two figures will be found in that square which is on a line ynttk thfe one and directly under the other. Thus, 56 the product of 7 and 8, will be found on a Une with 7 and under 8 : so 2 times 2 is 4 ; 3 times 3 is 9, &c. — In this way the table must be learned and remembered. ^ RULE. 1. Flare the numbers as in Subtraction, the larger number uppermost with units under units, o:c. and then draw a line below. 2. Ulien the multiplier does not exceed 12 : begin at the right hand of the tnuUiplicand, and multiply each figure contained in it by the multiplier, set ling down all over even tens and carrying a- in addition^^^. 3. Wlten the multiplier exceeds 12 ; multiply by e»4 by U. ■'-■ ' Sect. I. 3. SIMPLE MULTIPLICATION. fl 3. Multiply 7 6 5 3 2 Multiplicand. 6 5 Multiplier. PROOF. 3 8 2 6 6 10 1 4 5 9 18 12 2X5 1 49744630 Product. Proof. I cast the 9's out of the Multiplicand and set the cemauider (6) at the right hand of the cross ; I do the same with the MuUrplier, and set the remainder (2) at the left hand of the cross ; the^e remainders I multi- ply together, and casting out the 9's fi^m the product (10) the remainder is 1, which 1 set at the top of the cross ; I then cast out the 9's from the Product (49744630) and set the remainder (1) at the bottom of the cross, which as it agrees with the remainder at top, I suppose the work to be right. There is nothing more easy than proving Multiplication by this method •o soon as the scholar shall have giveo it such attention, as to make it a little familiar. Note. Should the Multiplier or Multiplicand, either or both, be less than 9, they are to be taken as the remainders. 4. Multiply 37846 by 235. Pro- 5. Multiply 14356 by 648. Pro- duct, 8893810. duct, 9302688. 6. Multiply 29831 by 952. Pro- 7. Multiply 93966 by 8704. Pro- duct, 28399112. duct, 817793024. 22 SIMPLE MULTIPLICATION. Sect. I. 3. 8. Multiply 98704563 by 75604 Prod. 746245978105? 9. - 3462321- 96484-- 334058570364 10. - A27535-15728 - 8297070480 11. - 2758^7-19725 - 5440687575 12. - 696374-463957 - 323087591918 Contractions and varieties in Multiplication. Any number which may be produced by the multiphcation of two or mpre numbers, is called a composite number. Thui* 15 wuicli arises from the multiplication of 5 and 3, (3 times 5 is 15) is a composite uumber ; and these numbers, 6, and 3, are called component parts. Therefore. 1. If the multiplier be a composite number ; multiply first by one", of the component parts, and that product by the other ; the last product will be the answer sought. EXAMPLES. 1. Multiply 6 7 by 1 5 OPERATION. 6 7 5 one of the component parts. 3 3 6 3 the other component part. 10 5 Product of 67 mult, by 15. 2. Multiply 367 by 48, Product, 17616. OPERATION. Consider first whut two numbers mu- tiplied together will produce 48 ; that is, what arc the component parts oi 48 ? — Answer, G and 8 (6 times is -IC) therer fore multiply 367 first by one of the com- ponent parts, and the product thence arising by the other ; the last product will be the answer sought. 3. Mult. 583 by 66. Prod. 32648. 4. Mult. 1086 by 72. Prod. 78192. OPERATION. * OPERATION. 2. " When there are cyphers on the right hand of either tha multiplicand or *' multiplier, or both, neglect those cyphers ; then i)lace the significant fig- " ures under one another, and multiply by them only ; add them together " as before directed, and place to the right hapd as mcaiy cyphers as there " are in both the factors." Sect. I. 3. SIMPLE MULTIPLICATION. S3 EXAMPLES. I. Multiply 65430 by 5200. OPERATION. 6 5 4 3 5 2 Here in the multiplication of 65430 by 5200, the cyphers are seen neglected, and regard paid only to the significant figures. To the product are annexed 3 cyphers ; equal to the number of cy- phers neglected in the factors. 340236000 i. Mult. 3 6 5 By 7 3 3. Mult. 78000 by 600. Product^ 46800000. Prod. 2 6 6 4 6 3. When there are cyphers between the significant figures of the multiplier, omit the cyphers a^d multiply by the significant figures only, placing the first figure of each product directly under the figure by which you multi- ply, and adding the products together, the sum of them will be the product of the given numbers. EXAMPLES. J. Mult. 154326 by 3007. OPERATION. 15 4 3 2 6 3 7 10 8 2 8 2 4 6 2 9 7 8 464058282 In this example, the cyphers in the mul- tiplier are neglected, and 154326 multi- plied only by 7 and by 3, taking care to place the figure in each product directly under the figure from which it was ob« tained. 2. 3 4 6 7 3 2 10 4 4 14 U SIMPLE MULTIPLICATION. S«ct. I. 3; 48976850 4 3 19592209305500 4. When the Miikiplier is 9, 99, or any number of 9's, annex as many cy- pher's to the Multiplicand, and from the number thus produced, subtract the multiplicand, the remainder will be the product. EXMIPLES. 1. Mult. 6547 by 999 Write down the Multiplicand, place as OPERATION. many cyphers at the right hand a^ there 6 5 4 7 are 9's in the Multiplier for a minuend ; 6 5 4 7 underneath write again the multiplicand for ■ a Subtrahend, subtract, and the remainder 6 6 4 4 6 3 is the product of 6547 multiplied by 999. 2. 3. ^^99 ( ^'■'''^"^^ 640827 "^^^J i PrfidpCt, 695079 4. " $364976 9 9 9 9 i Preduct, 63844375024 Sect. I. 3. SUPPLEMENT TO MULTIPLICATION. 25 SUPPLEMENT TO MULTIPLICATION. \ QUESTIONS. 1. What is Simple Multiplication ? 2. How many numbers are required to perform that operation ? 3. Collectively or together, what are the given numbers called ? 4. Separately, what are they called ? 5. What is the result, or number sought, called ? 6. In what order must the given numbers be placed for multiplication ? 7. How do you proceed when the multiplier is less than 12 ? 8. When the multipHer exceeds 12 what is the method of procedure? 9. What is a composite number ? 10. What is to be understood by the component parts of any number ? 11. How do you proceed when the multiplier is a composite number? 12. When there are cyphers on the right hand of the multiplier^ multi- plicand, either or both, what is to be done ? 13. When there are cyphers between the significant figures of the mul- tiplier, how are they to be treated ? 14. When the multiplier consists of 9's how may the •peration be con* tracted ? 15. How is Multiplication proved ? 16. By what method do you proceed in casting out Lie y's from any number ? 17. How is Multiphcation proved by casting out the 9's ? EXERCISES. 1. What sura of money must Note. The scholar's business in all be divided between 27 men, so questions for Arithmetical operations, is that each may receive 115 wholly with the numbers given; these dollars. .fins. 3105. are never less than two ; they may be more, and these numbers in one way or another, are always to be made use of to find the answer. To these, therefore, he must direct his attention, and carefully consider what is proposed by the question to br known. D 2C SUPPLEMENT TO MULTIPLICATION. Sect. I. 3. 2. An army of 10700 mcD having plundered a city, took so much money, that .when it was shared among them, each man received 46 dollars ; what was tlie sum of money taken ? Ans. 492200. 3. There were 175 men employed to finish a piece of work, for which each man was to receive 13 dollars; what did they all receive ? Ans. 2276. 4. .Suppose a certam town coii- tiiins 145 houses, each house two families, and each family 6 inhabit- ants ; how many would be the inhab- itante of that town ? Jlns. 1740. 5. If a man earn 2 dollars per week, how much will he<»^n in 5 years, there being 62 weeks in a year ? Ans. 620 dolls. 6. How much wheat will 36 men thrash in 37 days, at 5 bushels per day each man ? Ans. 6660 bushels. 7. If the price of wheat be 1 dol- lar per bushel, and 4 bushels of wheat make 1 barrel of flour, what will be the price of 175 barrels of flour ? Ans. 700 dolls. I Sect. I. 4. SIMPLE DIVISION. 27 « 4. SIMPLE DIVISION. SIMPLE DIVISION teaches, having two numbers given of the same denomination, to find how many times one of the given numbers contains the other. Thus, it may be required to know how many times 21 contains 7 ; the answer is 3 times. The larger number (21) or number to be divided, is called the Dividend ; the lesser number (7) or number to divide by, is Cj^led the Divisor; and the answer obtained, (3) the Qttotient. After the operation, should there be any thing left of the Dividend, it is called the Remainder. This part, however, is uncertain ; sometimes there '\s no remainder. When it does happen it will always be less than the di- visor, if the work be right, and of the same name with the dividend. RULE. 1. " Assume as many figures on the left hand of the dividend as contain " the divisor once or ollenor ; find how many times they contain it, and " place the answer as the highest figure of the quotient. .2. ^' Multiply tjie divisor by the figure you have foupd, and place the " product under that part of the dividend from which it was obtained. 3. •' Subtract the product from the figures above it. 4. " Bring down the next figure of the dividend to the remainder and ■' divide the number it makes up as before." When you have brought down a figure to the remainder, if the number it makes up be still less than the divisor, a cypher must be placed in the quotient, and another figure brought down. EXAMPLES. I. Divide 127 by 5. Divisor. Dividend. Quotient. The parts in Division are to stand ")) 1 2 7 (2 6 thus, the dividend in the middle, the 1 divisor on the left hand, the quotient on the right, with a half parenthesis 2 7 separating them from the dividend. 2 5 2 Remainder. Proceed in this operation thus-i-It being evident that the divisor (5) cannot be cotitaiticd in the first figure (1) of the dividend, therefore assume the two first fiii^uros (12) and inquire how often 5 is contained in 12 ; find- ing it to be 2 limes, place 2 in tlie quotient, and multiply the divisor by it, saying 2 times "i is 10, and jdace tlie sum (10) directij' under 12 in the dividend. Subti s't 10 from 12 and to the remainder (2) bring down the next tigure (7) at the right hand, making with the remainder 27. Agam inquire how >nauy times 6 in 27 ; 5 times ; place 5 in t)ie tjuotient, multiply the divisor (6) by the last quotient figure (o) saying 5 times 5 is 26, place the sum (25) under 27, subtract and the v.ork is done. lience it appears that 127 contains 5, 25 times, with a remainder of 2, which was left after the last subtraction. " V This Rule, perhaps at first will appear Intricate to the young student, although it is attended with no dilhcuUy. His liability to errors will chielly arise from the diversity of proceedings. To assist his recollection, let him notice that ^ 1. Find how many times, Lc. rri X rn- • • r ; -• Multiply. 1 he step?: of Divi':i'>n arc Jour < ,, c i » "! ^ ^ •• Subtract. Turing down- 28 SIMPLE DIVISION. S«ct. h s It is sometimes practised to malce a point (.) under the figures in the'j dividend, as they are brought down, in prdcr to prevent mistakes. \ When the divisor is a large num1>er, it cannot always certainly be kfu>wn i how many times it may be taken in the figures wbicn are assumed p;:'tUdj left hand of the dividend till after the first steps in division axe goii^ ever, but the learner must try so many times as his judgment may best dicute, •and after h(j has raultipUed, if the product be greater than the number as* - sumed, or that' number in which the divisor is taken, then it may alwayg be kpowp that the quotient figure is too large ; if after he has muhiplied and subtracted, the remainder be greater than the divisor, then the quotient figure is not large enough, he must then suppose a greater number of time?, and proceed again. This at first may occasion some pezT)lexity, but ' the attfjntive learner after some practice, will generally hit on the right j II umbo r. 2. Let it be required to divide 7012 by 62. OPERATION. Divisor. Dividend. Q,uotient. In this operation it is left for the pchol- » 2) 7 1 2 (I 3 4 ar to trace the steps of procedure without 5 2 having them particularly pointed otit to him by words. 18 1 1 5 6 2 5 2 8 8 4 4 Remainder. PROOF. Division may be proved by multiplication. v RULE. " Multiply the Divisor and Quotient together, and add the remainder, if j " there be any to the product ; if the work be right, the sum will be equal •' to the dividend." Take the last Example. The^Quotieat was 1 3 4 | ..^j^.^j^ ^,^^ ^^^^^^^^^ 2 6 8 6 7 4 4 Remainder added. 7 12 Equal to the dividend. Another and more expeditious way of proving Division is By casting out the 9's. | Cast out the 9's from the Divisor and the Quotient, multiply the resulte, and to the product, add the remainder if any after division ; from the sum of the^e cast out the 9's, also ca.st out the 9's from the Dividend, and if the two last results agree, the work is supposed to be right. I^KCT. I. 4. SIMPLE DIVISION. 59 3. Divide 17364 by 86. OPERATION. PROOF. Divis. Divid. Quot. 9's oiii of (Divis.) 86 Rem. 5 > Multiplied 86)1 7 3 5 4(2 1 {Quot.) 201 Rem. 3 J together. 1 7 2 16 _^,^ 15 4 Remainder 68 added. r..- 8 6 — 9'9 out of 83 Rem. 2 } agreeing 6 8 Rem. 9*8 out of {Divid.) 17334 Rem. 2 J togetWr 4. Divide 153598 by 29. quotient 5296. Rem. 14. 6 Divide 8893810 by 235. Qwr. 37846. 6. Divide 30114 by 63. (Quotient 418. 7. Divide 9302638 by 648. quot. 14356. 30 SIMPLE DIVISION. 8fct. I. 8 Divide 974932 by 365. Quotient 2671. Rem. 17. ii 9. Divide 5221580 by 68705. Ouot. 7t 10. Di>id*» 3228242 dollars equaUy among 563 men; how many dollars must each man receiye ? Jlns. 6734. From a view of the question, it is evident, that the dollars must be divided into as many parts as there ^^ are men to receive them; conse- '^' quently, the number of dollars must be made the dividend, and the num- ber of men the divisor; the quotient will then show how many dollars each man must receive. 1 1. How manv limes does 1030603Gk". contain 3215? .'3ns. 320561 times Sect. I. 4. SIMPLE DIVISION. 31 Contractions and varieties in Division. I. When the divisor does not exceed 12, the operation may be performed without setting down any ligures excepting the quotient, by carrying the computation in the mind. The units which would remain alier subtractins^ the product of the quotient figure and the divisor Irom the ligures assumeci of the dividend, mnst be accounted so many tens, and be «i'jj)posed to stand at the left hand of the next figure in the dividend, then consider again how often the divisor may be had in the sum of them. Proceed in this waj' till ali the figures in the dividend have been divided. This is called short division. EXAMPLES. 1. Divide 732 by 3. OPERATION. 3)732 Here I say, how often 3 in 7, knowing it to be 2 times, I place 2 in the quotient, 2 4 4 Q^uotient. then considering that the quotient figure (2) and the divisor (3) multiplied to- gether would be 6, and that this product (6) subtracted from 7 in the dividend, would leave 1, I then consider tl)i.> remainder (1) as standing at the lefl hand of the next figure (3) of the divi- dend, which together make 13. i now say how many times 3 in 13 — I times, therefore I place 4 in the quotient, which multiplied into the divisor (3) would be 12, and 12 subtracted from 13 would leave 1, which consid- ered as standing at the left Rand of the next or last figure (2) of the divi- dend would make 12 ; again, how many times 3 in 12 — 4 times — I then place 4 in the quotient, which multiplied into the divisor (3) is 12, this product (12) I consider as subtracted from 12, I find there will be no re- mainder, and the work is done. 2. Divide 37426 by 7. OPERATION. 7)37426 ^uot. 5 3 4 6 Rem. 4 Here I say how often 7 in 37 ? 5 times and two remain ; then how often 7 in 24 ? 3 times and 3 re- main ; how often 7 in 32 ? 4 times and 4 remain ; lastly, how often 7 in 46 ? 6 times, 4 remain. Cluot. 77664505790 124215042924 3.6537338604 - 38306588315 - 60702378652 Rem. 2 - - 1 - - 4 - - 7 3. Divide 310658023162 by 4 4. - - - 621075214621 - 5 5. - - - 213524031628 - 6 6. - - - 306452706527 - 8 7. - - - 546321406968 - 9 II. JVhen there are cyphers at the right hand of the divisor, cut them off, also cut off an equal number of figures from the right hand of the dividend and place these figures at the right hand of the remainder. EXAMPLES. 1. Divide 6203946 by 5700. OPFRATION. 57 I 00)62039 | 46(1088 57 . . . 603 456 479 456 2346 Here are two cyj^hers on the riglit hand of the divisor which I cut ofl", also I cut off two figures (4G) from the dividend and to the right hand of the re'mainder after the last divisiou (23) I place the figures cut otT from the dividend (40) which make the whole rr»maindfr ?346. 32 SIMPLE DIVISION. Sect. I. 4 2. Divide 379432 by 6500 q,uot. 68. Rem. 2432. 3. Divide 2r64303721 by 83000. Quot. 33307. Rem. 22721. 4. J0ie7i the divisor is 10, 100, 1000, or 1, ti'j'^A any number of cyphers an- nexed, cut oif as many figures on the right hand of the dividend as there are cypliers in the divisor j the figures vehich remain of the dividend coin- pose tlie quotient ; those cat off, the remainder. EXAMPLES. 1. Divide 1576 by 10. Here Ave have one cypher in the divi- j OPERATION. sor, therefore cut off one figure (6) from ' 1 I 0) 1 5 7 I 6 the dividend; what remains (157) is the quotient, and the figure cut off (6) the remainder. 2. Divide 3217 by 100. Q^uot. 32. Rem. 17. ?y. Divide 76421795 by 1000. Qvot. 76421, Rem. 795 8tcT. I. 4. SUPPLEMENT TO DIVISION. 33 SUPPLEMENT TO DIVISION. QUESTIONS. 1. What is Simple Division ? 2. How many numbers must there be given to perform the operation ? 3. What are the given numbers called ? f 4. How are they to stand for Division ? • '5. How many steps are there in Division ? 6. What is the first ? the second ? the third ? the fourth ? 7. What is the result or answer called ? 8. Is there any other or uncertain part pertaining to Division ? What is it called ? 9. Of what name or kind is the remainder ? 10. What is short Division ? 1 1. When there are cyphers at the right hand of the divisor, what is to be done ? 12. What do you do with figures cut off from the dividend when there are cyphers cut off from the divisor ? 13. When the divisor is 10, 100, or 1 with any number of cyphers annex- ed, how may the operation be contracted ? '14. How many ways may Division be proved ? 15. How is Division proved by Multiplication ? 16. How may Division be proved by casting out the 9's ? EXERCISES. 1. Suppose an estate of 36582 dol- 2. An army of 15000 men having lars to be divided among 13 sons, how plundered a city, and took 2G25000 much would each one receive ? dollars, what was each man's share ' Jlns. 2814 dolls. .Sns. 175 dolU. E 34 SUPPLEMENT TO DIVISION. Sect. I. 4» 3. A certain number of men were 4. If 7412 eggs be packed ia 34 concerned in the payment of ^18950 casks, how many in a cask ? and each man paid 25 dollars, what Ans. 218. was the number of men ? Ans. 768. I 5. A farm of 375 acres is let for 6. A field of 27 acres produces i 125 dollars ; how much does it pay 675 bushels of wheat ; how much is per acre ? Ans. 3 dolls. that per acre ? Ms. 25 bushels. 7. Supposing a man's income to 8. What number must I multiply be 2555 dollars a year ; how much by 13, that the«product may be 871 ? is that per day, there being 365 days ^ns. 67. in a year. , Ans. 7 dolls. Sect. I. 5. COMPOUND ADDITION. S5 J 5. COMPOUND ADDITION. COMPOUND ADDITION is the adding of numbers which consist of articles of different value, as pounds, shillings, pence, and farthings, called different denominations ; the operations are to he regulated h^j the value of the articles, which must be learned from the Tables. RULE FOR COMPOUND A»DITIO?f. 1. Place the numbers so that those of the same denomination may stand directly under each other. 2. Add the first column or denomination together, and carry for that number which it takes of the same denomination to make 1 of the next higher. Proceed in this manner with all the columns, till you come to the last, which must be added, as in Simple Addition. 1. OF MONEY. TABLE. 4 Farthings qr. \ L Penny, marked d. 12 Pence > make one /.Shilling, s. 20 Shillings ) (Pound, £. EXAMPLES. 1. What is the sum of £61. 17.?. 5d. £13. 3s. 8c/. and of £5. 16s. llrf. when added together ? OPERATION. £. s. d. ^t. *o q( Those numbers of ^the same denomination placed 61 17 .p ." ^ under each other, .as the rule direct? 80 18 I begin with the right hand column or that of pence, and having added it, find the sum of the numbers therein contained to be 24 ; now as 12 of this denomination make one of the next higher, or in other words 12 penC'i make one shilling, therefore in this or in the column of pence I must can y for 12 ; I now inquire how often 12 is contained in 24, the sam of the fii-t column or that of pence ; knowing it to be 2 times and nothing over, I set down under the column of pence, and carry 2 to that of shillings, to be added into the second column, saying, 2 I carry to 6 are 8, and 3 are 1 1 , and 7 are 18, and 10 to 18 are 28, and ten again are 38 (for so each figure in ten's place must be reckoned, 1 in that place, being equal in value to 10 units.) Now as 20 shillings make one pound, therefore in the column of shillings, I carry for 20 ; I then inquire how ofi:en 20 in 38 '.' cnce, and 13 remains ; therefore, I set down directly under the column of shilHiigs 18, what 38 contains more than 20, and for the even 20 carry 1 to pounds oi? the last column, which is to be added alter the manner of Simple Addiiion. JVote. — The method of proof for Compound Addition is the same as ta make one ^ Quarter ofahund. weight, qr. 4 Quarters V ) 1 00 weight,or 1 Impounds, ca^ 20 Hundred weight ) \ Ton, T. EXAMPLES. 1. T. crt'f. qr. lb. oz. dr. 186 3 2 25 II 8 4 17 2 3 7 6 9 8 3 7 2 5 2 3 1 16 5 11 2. T. crut. qr. Ih. oz. dr. 8 1 3 2 2 5 1 1 8 I 7 19 3 14 5 6 8 6 2 6 15 3 7 1 6 4 Note. — " 175 Troy ounces are precisely equal to 192 Avoirdupois Ounces, and 175 Troy pounds are equal to 144 Avoirdupois. 1.1b. Troy=5760 grains, and 1 lb. Avoirdupois=7000 grains." SscT. I. 5. COMPOUND ADDITION. 99 4. OF TIME. - 60 Seconds 60 Minutes 24 Hours 7 Days 4 Weeks 13 Months, s. Id.fc Ih. , TABLE. • make one • Minute, marked Hour Day, Week, Month, ♦ Julian Year, m. /i. mo Y, EXAMPLES. 1. F. 1 6 mo. 1 ■w. d. 3 6 h. 23 m. 57 s. 43 28 7 2 5 1 6 28 32 39 6 1 3 1 7 38 1 1 87 4 1 1 4 1 5 1 7 2. F. mo. TO. d. h. m. 9. 89 1 1 3 6 22 45 36 36 1 2 5 6 5 5 44 87 ' 2 1 ' 1 ] 22 33 36 4 3 3 5 8 7 The number of days in each Calendar month maybe remembered by the following verse : Thirty days hath September, April, June and November; February twenty-eight alone ; all the rest have thirty-one. * " The civil Solar year of 3G5 days being short of the true by 5h. 48m. 67s. occasioned the beginning of the year to run forward through the sea- eon nearly one day in four years ; on this account Julius Cassar ordainetl that one day should be added to February every fourth year, by causing the 24th day to be reckoned twice : and because this 24lh day was the sixth (sextilis) before the kalends of March, there were in this year two of these sexliles, which gave the name of Bissextile to this year, which being thus corrected, was, from thence called the Julian year." 40 COMPOUND ADDITION. Sect. I. B. GO Seconds 60 Minutes 30 Degrees 12 Signs, or 360 Degrees .'->. OF MOTION. TJIBLE. f Prime Minute, marked " ' I Degree, ** ' make one < Sign, s. j The whole great circle of the L Zodiac. EXAMPLES. 2 5 1 7 6 1 1 7 4 9 3 5 1 7 1 8 5 6 2 4 1 6 8 2 6 1 8 9 6 5 4 4 3 6 3 3 4 4 5 5 1 2 2 8 6. OF CLOTH IMEASURE. Inches, one fifth in. r Nail, marked na. Nails, or 9 inches Quarter of a yard, qr. Quarters of a yard or 36 inches Yard, yd. Quarters of a yard, or 27 inches Ell Flemish E.FI. Quarters of a yard or 45 inches ^make one • Ell English E. E. Quarters of a yard, or 54 inches Ell French E. Fr. Quarters 1 i nch and 1 fiftl i,or 37 inches and one fifth Ell Scotch E.Sc. Quarters and two thirds , Spanish Var. 1. EXAMPLES. 2. Yds. qrs. n. E. E. qr. n. 6 1 4 3 1 3 1 9 3 2 3 6 1 2 6 6 1 3 7 1 7 2 2 1 5 3 2 1 8 2 I Sect. I. 5. COMPOUND ADDITION. 41 7. OF LONG MEASURE. By Long Measure are measured distances, or any thing where leDgth li considere J without regard to breadth. TABLE. 3 Barlej ' corns, fcar. "] p Inch, nuxrked in. 12 Inches Foot, ft 3 Feet Yard, yd. 5\ Yards, or ICh feet Rod, Perch or Pole, poU 40 Poles 8 Furlongs ■ make one • Furlong, Mile, rmle ( Degree I C-i of a great 69i Statute miles, near/y. rcle. i A great - t the Circle of SCO Degree* Earth. , EXAMPLES 1. Des^. mi. y«r. po/. ft- in. bar. 1 6 8 5 7 7 2 6 1 5 1 1 2 1 2 4 5 3 6 1 8 7 6 1 7 9 3 6 1 7 9 I 4 7 3 3 2 1 2. jDc^. mi. fur. pol. y?. in. 1 3 5 6 5 1 3 8 1 4 9 1 8 1 2 7 1 6 2 2 6 7 1 2 3 I 6 9 2 9 8 5 3 1 4S COMPOUND ADDITION. Sect. I. 6> 8. OF LAND OR SQUARE MEASURE. By Square Measure are measured all things that have length and breadth. TABLE. 144 Inches 'j f Square Foot. > 9 Feet Yard. 30]- Yards, or ) 2721 Feet J 40 Poles • make one - Pole. Rood. 4 Roods 1 f.O rods } or 4840 yards I Acre. 640 Acres J [ Mile. EXMIPLES. Acres. rood. pal. ft. iff. 3 7 6 3 3 6 9 3 1 2 1 5 6 8 1 2 7 5 8 7 6 2 4 7 2 3 5 6 1 2 4 9. OF SOLID MEASURE. By Solid Measure are measured all things that have length, breadth and thickness. TABLE. 1728 Inches 1 f Foot. 1-7 Feet Yard. 40 OI Feet of round timber } ' 50 feet of hewn timber \ > make one ■ Ton or load. 128 Solid feet, i. e. 8 in) le ngth, 4 in breadth and ) Cord of wood. 4 in bei .''^t ) J k. EXAMPLES. 1. t 2. Ton. /(' in. Cord. ft- in. e5 3 7 22 9 3 9 1 1 8 102 1 ?9 2 6 120 7 4 56 1 3 7 Z6 1 7 54 18 72 6 59 Z^7 3 3 G 2 9 86 1 24 Sect. I. 6. COMPOUND ADDITION. 10. OF WINE MEASURE/ By Wine Measure are measured Rura, Brandy, Perry, Cider, Mead, Vinegar and Oil. T.WLE. 1 r 2 Pints pts. 4 Quarts 10 Gallons 18 Gallons 311 Gallons 42 Gallons 63 Gallons 2 Hogsheads 2 Pipes Hhd. 3 9 1 6 3 5 2 9 marked }■ make one { Quart, Gallon, Anchor of Brandy, Runlet, Half Hogshead. Tierce, Hogshead, Pipe or Butt, TUD. EXAMPLES. 1. gal. 5 2 2 7 1 2 3 8 qts. 3 1 2 pts. 1 1 r. hkd. ga/. ^Is. pti 8 6 2 5 3 3 1 3 5 1 3 6 1 1 7 2 9 2 1 gal. anc: rwn. ^hhd. tier. hhd. or B. T. N. B. A Pint Wine Measure is 28| cubic inches. <» 44 COMPOUND ADDITION. Sect. I. 5". »« 11. OF ALE OR BEER MEASURE. TABLE. N. B. A Pint Beer Measure, is 35i cubic inches. 2 Pints "] 4 Quarts Quart, marked Gallon, gcK 8 Gallons Firkin of Ale in London, A.fir. Z\ Gallons Firkin of Ale or Beer. 9 Gallons 2 FirkirH > make one < Firkin of Beer in London, B.^r. Kilderkin, kill. 2 Kilderkins Barrel, bar. 1i Barrels, or 54 gallons ^ Barrels Hogshead of Beer, Puncheon, hhd. pun. 3 Barrels, or 2 hogsheads^ . BuU, bun. EXAMPLES. 1. 2. hhd. gall. qts. B.Jir. gal. 327 48 2 23 6 qfs. 2 2 8 5 13 4 5 2 3 173 24 1 98 T 1 2 7 16 3 6 8 iti^l' 12. OF DRY MEASURE. By Dry Measure are measured all Dry Goods, such as Corn, Wheat, Seed, Fruit, Roots, Salt, Coal, kc. T.WLE. 2 Pints 2 Quarts 2 Pottles 2 Gallons 4 Pecks 2 Bushels 2 Strikes 2 Cooms 4 Quarters 1J Quarters n Quarters 2 Weys ' make one * Quart, marked Pottle, Gallon, Peck, Bushel, qts. pot. gal. pk. hush. Strike, str. Coom, CO. QuaVter, Chaldron, ch. Chaldron in London. ^'Vey, Last, StcT. I- 6>. COMPOUND ADDITION. 45 L, f:j.aji/pz,£5. ]. 2. =: — T 6«s/». pk. qt. ;jf. Ck. lush. pk. qts. 2 7 2 6 I 3 7 1 6 2 5 1 8 8 7 2 6 2 8 3 7 2 I 1 1 8 1 2 1 a 1 1 3 1 7 2 5 3 6 \ _ N. B. A gallon, Dry Measure, contaios 268| cubic inches. JTie follorvins^ are denominations of things cmiuted by the " Tale." 12 Particular things make 1 Dozen, » I „ 12 Dozen 1 Gross, 12 Gross or 144 doz. . 1 great Gross. ALSO, ., 20 Particular things make 1 Score. Denominations of Measures not included, in the Tables. 6 Points make 1 Line, 12 Lines . . Inch, 4 Inches . . Hand, I 3 Hands . . Foot, 6G Feet, or 4 Poles, a Gunter's Chain, o Miles . . League. A Hand is Msed to measure Horses. A Fathom to measure depths. A League iii reckoning distances at Sea. N". B. A G.uintai of Fish weighs 1 cwt. Avoirdupois. 46 COMPOUND SUBTRACTION. Sect. I. «. } 6. COMPOUND SUBTRACTION. COMPOUND SUBTRACTION teaches to find the difference between any two sums of diverse denominations. I RULE FOR COMFOUND SUBTRACTIOX. " Place those numbers under each other, which are of the same denom- " ination, the less being below the greater ; begin with the least deoomina- " tion, and if it exceed the figure over it, borrow as many units as makd; " one of the next greater ; subtract it therefrom ; and to the difference' •' add the upper figure, remembering always to add one to the next supe- •' rior denomination for that which yoi^ borrowed. Proof. — In the same manner as Simple Subtraction.^ • ' ^'* ' 1. .OF MO^EY. ^ ' 1. Supposing a man to h;ive lent j£i85 10s. 7c?. ai^ to have received again of his money, £93 15s. how much remains due ? \ OPERAT10^. £. Lent 1 8 5 Received 9 3 1. s. 1 1 5 d. 7 £. From 3 1 Take 8 5 2. s. 1 6 Due 9 1 1 6 7 ^., Proof 18 5 1 7 3. £. s. d. Lent 6 3 7 1 7 8 The sum of the sever::-!, payments I must first be added toj^ether, and / 1 6 3 2 6 the amount subtracted ^:om the sum Received V 7 8 4 lent, at sundry Sect. II. I. REDUCTION. ' 5^ REDUCTION ASCENDING. RULE. Divide the lowest denomination given by that number wb;eh it takes of the same to make one of the next higher, and so continue to do till you have brought it into tJie denomination which your question requires. EXJiMPLES. 1. In 16971 farthings how many pounds ? OPERATION. Farthings in a penny 4)16971 Zqrs. Pence in a shilling 12)4242 6(£. Reduction descending 5J»d ascend- ing reciprocally prove each otherc Shillings in a pound 2j0)35|3 \3s. £17 Ani.£\l 13s. Gd. Zqrs. 2. In 1765 pence, how many 3. In 38976 ftrthings how many pounds ? Jins, £7 7s. \d. guineas ? Jins. 29 4. In 6950 sixpences, how many 5. In 3792 farthings, how many pounds? .4ns. £173 15.?. crowns? Mt. U. 64 REDUCTION. Sect. II. i. G. In ISneO farthing, how many 7. In 6952 three-pences, how pence, thrce-pences, six-pences and many pistoles at 22x. Ms. 79. dollars ? Ans. 12240 pence, 40BO three-petices, 2040 sii-pences, 170 dollars. REDUCTION ASCENDING Sf DESCENDING. 1. MONEY. J. In 57 moidores, at 365. each, In this question the first step will how many dollars ? be to bring the moidores into shillings: Ans. ^342. lastly bring the shiUings into dollars. 2. In 75 pistoles how many pounds ? Ans. £82 10s. 3. In jG73 how many guineas ? 4. In £,C'3 and o guineas, how Aas. 52 guineas, 4s. many dollars ? Ans. ^233 2*. SrcT. II. 1. REDUCTION. 65 " When it is required to know how many sorts of coin of di^erent values and of equal number are contained in any number of another kind ; reduce the several sorts of coin into the lowest denomination mentioned, and add them together for a divisor ; then reduce the money given into the same denomination for a dividend, and the quotient arising from the division will be the number required." Note. Observe the same direction in weights and measures. 1. In 54 guineas, how many pounds, dollars and shillings of eac^ an equal number ? OPERATION. £1 is 20 shillings 54 guineas 1 dollar is 6 shillings 28 shillings is a guinea 1 shining is 1 shilHug _ 432 Divisor 27 shillings 108 Dividend 1512 shillings. 27)1512(56 of each ; that is, 54 guineas include the value of one pound, 135 one dollar, and one shilling 56 times. 162 162 000 2. In 1 72 moidores how many eagles, dollars and nine-pences, of each the like number? Jlns. 92 of each, and 68 nine-pences over. 3. In 237 guineas how many moidores, pistoles, pounds, and dollars, ot each the like number ? -Ins, 79 of each. 66 REDUCTION. Sect. II. I TROY IVEIGHT. •, 1. In 4/6. boz. and IGpvts. how many grains ? OPERATION. Ih. oz. pK'U 4 6 16 12 oz. in a pound. 63 ounces. 20 pwts. in an ounce. 1076 penny wei miles to barley cornsi OrEIlATION. 16 Mies. 128 Furlongs. 40 5120 Rods. 6i* 25600 2660 28160 Yards. 3 34480 Feci. 12 1013760 Inches. 3 pnoor. 3)3041280 12)1013760 3)84480 111)28160 2560 4{0)512j0 8)1£8 16 MiUs. t Divide by 11 for 5^ and multiply tlie quotient by 2. TJie reason is be- cause 5i reduced to half yards is 1 1-. .^risa-er, 3041280 bar. corns. * To multiply by one half (^) it is only to take half the multiplicand. 2. In 47520 feethow many leagues ? Ans. 3 leagues. Sect. II. 1. REDUCTION. 61 3. How many times does the wheel which is 18 feet 6 inches in circum- ference, turn round in the distance of 150 miles ? Jins. 42810 times, and |80 inches over. 4. How many barley corns will readli round the Globe, it being 360 degrees? ^ms. 47558016Q0. 6t REDUCTION. Sect. II. 1. 6. LAND OR sqUARE MEASURE. 1. In 13 acres, 2 roods, how many poles ? OPERATION. PROOF. Ac. R. 4)0)2 16|0 13 2 4 64 40 Ans. 2160 Poles. 2. In 2862 rode how many acres ? 4)54 13Ac. 2R. Ans. 11 A. 3R. 12P 7. SOLID MEASURE. 1. Ih 1296000 solid inches how many tons of hewn timber ? OPERATION. 6|0 1728)1296000(75|0 12096 8640 8640 00 15 Tons, the Answer. PROOF. 15 50 760 1728 6000 1500 6250 760 1296000 Inches. Sect. II. 1. REDUCTION. 63 2. In 5529600 solid inches, how many cords of wood ? Ans. 25. 3. How many solid inches in a cotd ? -'ins. 22 11 84. 8. DRY MEASURE. 1. In 75 bushels of com how many pints ? OPERATION. 75 4 300 8 2400 2 PROOF. 2)4800 8)2400 4)300 75 Bushels. Ans. 4800 pis. 2. In 9376 quarts how many buehels ? Ans. 293 It would be needless to give examples of Reduction in all the weights and measures. The understanding which the attentive Scholar must ah-eady have acquired of this rule, by the help of the tables, will ever be si^hcient for his purpose. 64 SUPPLEMENT TO REDUCTION Sect. II. 1 Hi SUPPLEMENT TO BUDUCTION. QUESTIONS. 1 . What is Reduction ? '2. Of how many kiads is Reduction ? What are they called ? Wherein do these kinds differ one from the other ? Which of the fundamental rules are employed in their operation ? 3. How is Reduction Descending performed ? 4. l!ow is P».cduction Ascending performed ? b. When it is required to know how many sorts of coin, weights or measures of different values, of each an equal number, are contained in an}' other number of another kind, what is the method of pro- cedure ? EXERCISES. 1. lu 36 guineas, how many 2. How many rings, each weigh- Ci'OWns ? ing bp'wts. Igrs. may be made of Jlns. 163 crotf?j«t^ 9 J. over. oil. ooz. IGvz^ts. 2^rs. of gold? -iv. .'ins. loo. i SrcT. II. I. SUPPLEMENT TO REDUCTION. ' 65 3. How many steps of 2 feet 5 inches each, will it require a man to take, travelling from Leominster to Btston, it being 43 miles ? Ans. 93947. 4. Let 70 doll^rg be distributed among three men in such manner that as often as the first has os. the second shall have 75. and the third 9*. What will each one receive ? Jlns. first gl6 4s. second" ^23 2j, fjiird $30. 5. How many squjare feet in a square mile ? Ms. 27878400. 66 SUPPLEMENT TO REDOCTION. Sect. II. 1. 6. If a vintner be desirous to draw off a pipe of Canary into bottles, con? laining pints, quarts, and 2 quarts, of each au equal number, how many mast he have? v - Ans. 144 of each. 7. There are three fields, one contains 7 acres, another 10 acres, ancl the other 12 acres and 1 rood: how many shares of 76 perches each, are contained in the whole 2 Jins. 61 shares and 44 perches ovev. Sect. II. 1. SUPPLEMENT TO REDUCTION. 67 8. There a/e 106/6. of silver, the property of 3 men; of which A re- ceives 17Z6. lOoz. I9pwts. 19grs. of what remains, B shares loz. Igrs. so Qflen as C shares ISpruts. What are the shares of B and C ? Ans. B's share 53/6. Boz. Spwts. 5grs. Cs share Hlb. 4oz, ISpaft. FRACTIOXS. Sect. LI. i\ ^ 2. FRACTIONS. ■WHEN tlie thing: or things signUietl by figures are whole ones, then (he fii^'jrcs which signify them are called integers or tc7(o/e numbers. But wliun only seme pans of a thing are .sigiiified by ligures, as t%!;o thirds of -.iDY liiiiK.r, ^i-'c iixths, seven fcnths, i5'f. then the figures which signify these I'Ciris of a thing being the expression ai' some quantity less than one, are ctilled FRACTior.s. • Fractions are of two kind;*, Fuigar and Decimal; they are distinguish- ed bv the manner of representing them; they also differ in their modes of operation. VULGAR FRACTIONS. To understand Vulgar Tractions, the learner must suppose an integer (or tiic number 1) divided into a number of equal pa^-ts ; then any number t'f these parts being taken >vould make a fraction-,.', which would l>e re- pre-cuted by two luimbers placed one directly over the other with a thort lino between them thus, -| tuo thirds, ^- three Hflhs, {- seven eic;hths, ^'c. Each of thesfl figures have a difierent name aiid a dill'erent signification. The figure below liie line is called the denominator, and shews into how m.'.iiv parts an integer, or one individual of any thing is divided — tht: figurti ••ibove the line is calh^d the numerator, and shews how many of those parta are signified by the fraction. For iilustiation, suppose a silver plate to be divided into r\>ne equal parts. Xo'Stone or more of these parts make a fraction which will be represented bv liie figure 9 for a denominator placed underneath a short line shewing tho plate to be divided into nine equal parts ; and supposiiig tn-o of those part:* (0 be lakcu for the fraction, then the figure 2 must be placed directly abovi*. the y iind o\'er the line (f) for a numerator, showing that two of those parts are signified by the fraction, or iv^o ninths of the plate. Now let o parts of this plate, which is divided into 9 parts be given to Joh.n, his fraction would be ^ free ninths; let 3 other parts be given to Harry, his fraction would be ij three ?iinths ; there would then be one part of the plate remaining still (5 and 3 are 8) and this fraction would be expressed thus {; one ninth. In this way all vulgar fractions are written ; the denominator or number below the liiie, L-hewing into how many parts an}' thing is divided, and the nunjcrator, or number above the line, shewing how many of those parts itre taken or signified by the fraction. To ascertain whether the learner understands what has novv been taught hill) of iVactions, let us again suppose a dollar to be cut into 13 equal parts ; — let 2 of these parts be given to A ; 4 to B ; and 7 to C. i A's fraction — Required of tiie learner tliat he should write . arc a!.-o expressions of parts of an integer; or are v.: laliie soiacti)injj k-ss than one of any tliins", %vhalover it may be v.hich i^ signilled by ttion!, in decini.ils an inicc;er, or tlie number otic, as 1 foot, ! dollar, 1 year, &ir, is conceiv(ul to be divided iuto^!.'/; equal parts, (ui vuli^arfraciions, an inte- ger may he divided into any number of parts) and each of tliese parts is sub- divided into ten lesser parts, and so on. lo this way the denoirnnator to a decia^.al fraction in all r.-^ges, will be either 10, 100, 1000, or unity (1) H'ilh a niunbcr of cyphers annexed ; and this nunibcc of cyphers will aluaya be equal to the nnmbcr o( places in the numerator. Thus, /^ fj^ lY,/^^ zsc Derimnl Fractions, of whicjj the cyphers in the denoniicator of each arii equid to the number of places in its own numerator. " As the deno:r.inator of a decimal fraction is always 10, 100, 1000, &>'. ' the denominators ncediiot be expressed ; for the numerator only maybe '• made to express the true value ; for this purpose it is only ro;p,ireu to '" write the numerator with a point (,) before it, called a separatrix, at the " kit hand, to distins^uish it from a whole number ; thus, ."j is written ,6 ; Vv'iicn inteirers and deciniais are expressed together in the same sui.:. iliat suih is called a ??j/,rf(/ number ; thus, 25,^3 is a mixed number ; 2n. or all the tigures to the left hand of the separalrix being integers, and ,t ._> or all the fii^ires to the right hand of the same point being decimals. The first tigure on the right hand of the decimal point signifies tenlh parts ; the next, hundredth parts ; the next, thousandth parts, and so on. ,7 seven signifies seven tenth parts. ,(>7 — seven hundredth parts.- ,"7 — two tenth parts and seven hundredth parts ; or twenty seven hundredths. ,.3o7 — three tentli parts, five hundredth parts, and seven thoui- andth parts ; or 357 thousandths. 5,7 — five, and seven tenth parts. 5,007 — five and seven thousandths, i The value of each figure from unity, and the decrease of decimals Iq- ward the right hand may be seen in the following TABLE. '/) 'A -r. ^ G S "^ 'c <" 'o « ^_ ,-. '_ t» T* r* r« t- c =: rr ^ ^ j; -- vj w^ r^ K* v- '*- ^J J— — ■--.ococc.t;:;;cooo~ — — OX ox xo xo .9 87G5 1321,23450739 Cyphers placed to the right band of decimals do not alter their value,-' Placed at the left band Ihey dimini-h thoir vabie in a tenfold proportion. 70 DECIMAL FRACTIONS. ADDITION OF DECIMALS. Sect. II. 2 - > RULE. " I. Place the aumbers whether mixed or pure decimals> under each " other according to the value of their places." " 2. Find theiy sum as in whole numbers, and point off so many places " for decimals as are equal to the greatest number of decimal places in any " of the given numbers." EXAMPLES. 1. What is the amount of 73,612 guineas, 436 guineas, 3,27 guineas, ,8632 of a guinea, and 100,19 guineas when added together. OPERATION. The decimals are arranged from 73,612 the separatrix towards the right 436, hand, and the whole numbers from 3,27 the same point towards the left hand. ,8632 The greatest number of dcciro.al 100,19 places in any of the numbers is four, consequently four liguros in the pro- duct must be pointed ofl'for decimals. Ans. 613,9352 guineas. 2. 345,601 ,3724 63,1 672,313 7,5462 3. Required the sum of 37,82 1-1- 646,36-f8,4-f37,325. Ans. 620,896. 4. What is the sum of three hun- dred twenty-nine and seven tenths ; tliiirty-seven and one hundred and sixty-two thousandths ; and sixteen hundredths, when added together ? Ans. 367,022, &. Add six hundred and five thou- sandths, and four thousandth and thre£ hundredths ? Sum 4600,035. Note. — When the numerator has not so many places as the denominator has cyphers, prefix so many cyphers at the left hand as will make up the defect i so -^f^-^ is written thus, jOOj, kc. Sect. II. 2: DECIMAL FRACTIONS. 71 SUBTRACTION OF DECIMALS. RULE. " Place the numbers according- to their value ; then subtract as io whele numbers, and point off the decimals as in Addition." EXAMPLES. 1. From 716,325 take 81,6i?01. 2. From 119,1384 take 95,91. OPERATION. Kern; 23,?284. From 71(3,325 Take 81,6201 634,7049 3. What is the difference between 4. From 67, take ^92. 207 and 3,115 ? Arts. 283,885. Ron. 66, 08. All the operations in Decimal Fractions are extremely easy ; the onljr liability to error will be in placing the numbers and pointing otT the deci- ipalfs ; and here care will always be security against mistakes. MULTIPLICATION OF DECIMALS. RULE. " 1. Whether they are mixed numbers or pure decimals, place the fac- tors, and multiply them as in whole numbers." " 2. Point off so many figures from the prodact as there are decimal places in both the factors ; and if there be not so many decimal places in the product, supply the defect by prefixing cyphers." EXAMPLES. 1. Multiply ,0261 by ,Q035. In tl?is example, the decimals in the OPERATION. two factors taken together are eight; ,0261 the product falls short of this number ,0035 by four figures, consequently, four ■ cyphers are pretixed to the left ban(!l 1305 of Uie product. 783 ,00009135 Product. 72 DECIMAL FRACTIONS. Sect. II. £. J. Multiply 31,72 by G5,3. Product, 2071,316. OPERATION. 3 1,72 6 5, 3 Multiply 25,238 by 12,17. Product, 307,14646. Multiply ,62 by ,04. Product, ,0248. Multiply 17,6 by ,75. Product, 13,2. DIVISION OF DECIMALS. RULE. " 1. The places of decimal parts in tl;e divisor and quotient counted together must be always equal to those in the dividend, therefore divide as in whole numbers, and from the right hand of the quotient, point ofi' ?o many places for decimals, as the decimal places in the dividend exceed those in the divisor. " 2. If the places of the quotient be not so many as the rule requires, supply the defect by prefixing cyphers to the left Land. " 3. If at any time there be a remainder, or the decimal places in the divisor be more than those in the dividend, cyphers may be annexed to the dividend or to the remainder, and the quotient carried on to any degree of exactness." Divide 2,735 by 51, S OPERATIOX. 51,2)2,735(,0534 + 2,560 1750 1536 2140 2048 92 EXAMPLES. In this example there zrejive decimals in the dividend (counting the two cy- phers which were added to the remain- der of the dividend after the first division) that the decimals in the divisor and quo- tient counted together may equal that number, a cypher is prefixed to the left band of the quotient. Si;cT. II. 2. DECIMAL FRACTIONS. 73 In the division of decimals it is proper to add cyphers so long as there continues to be a remainder, this however is not practised, nor is it neces- sary ; four or live decimals being sufficiently accurate for most calculations. 2. Divide 315G,293 by 25,17. Quotient, 125.34- NoTE. Tlie separatiix is omitted in the ans\rrrs to the exiunples on this pagn to exercise the scholar in placing it according to rule ; to this the In- structor should be jwirticularly atteu- live. 3. Diride 5737 by 13,8. Quotient, 431353-f _ Divide 173948 by ,375. Quotient, 463C61-f 5. Divide 2 by 63,1 Quotient, 037+ G. 'Divide ,012 by ,005. Quotient, 24. 74 DECIMAL FRACTIONS. Sect. II. 2. REDUCTIO.^ OF DECIMALS. CASE 1. TO REDUCE VULGAR FRACTIONS TO DECIMALS. RULE. Annex a cypher to the numerator and divide it by the denominator, an- nexing a cypher contiaually to the remainder- The quotient will be the decimal reqiyred. EXAMPLES. 1. Reduce | to a decimal. 2. Reduce -} to a decimal. OPKRATION. OPERATION. 5)3,0(,C ^Ms. The numerator in these 7)l,0(,1428+^ns. 3 operations is considered as 7 an integer, and always re- ■ ■■ G quires the decimal point to 30 be placed imtnodiately af- 28 ter it, the cyphers annexed occupy the places — ! — * of decimals, the quotient must be pointed off 20 accord iflg to the rule in division. 14 60 56 3. Reduce -! , \, and 4^ to dec:mals. Answers, ,2p ,5. ,76. \ 4. Reduce >,-, ^{,j, and jf^^ to decimals. Ans. ,1923+,02o ,00797^-1 CASE 2. To reduce nwnhers of different denominations, as of Money, Weight and Measure to tJieir decimal values. RULE. " I. Write the given numbers pfjr[)endicularly under each other for dividends, proceeding orderly from the least to the greatest. Sect. II. 2. DECIMAL FRACTIONS. 75 " II. Opposite to each dividend on the left haiuT; place such a number • lor a divisor as will bring it to the next superior denomination and draw ' a line-perpendicularly between them. . " III. Begin with the highest and write fiie quotient of each division, as '• decimal parts on the right band of the dividend next below it, and so on, '• ;ill they are all used, and the last quotient will be the decimal sought." EXAMPLES. 1. Reduce 10s. G|c?. to the fraction of a pound. The given numbers arranged l^r the op- eration, all stand as intogers. I then sup- pose 2 cyphers annexed to the 3(.3,00; which divided by 4, the quotient is 75, which ,628125 Aim. I write against six in the next line, and the sum thus produced (0,76) I divide by 12, placing the quotient, (5625) at the right hand of the 10; lastly, 1 divide by 20 and the quotit^nt (,528125) is the decimal required. 2. Reduce 13s. 6Jrf. to the deci- 3. Reduce I2pri-ts. 14g rs. to the mal of a pound. Aiis. ,6729+ decimal of an ounce. Ans. ,6291, OPERATIOr.. 4 3, 12 6,75 20 10,6625 CASE 3. To find ihe value of any given decimal in the terms of an integer. RULE. Multiply the decimal by that number which it takes of the next less de- nomination to piake one of that denomination in which the decimal is given, and cut oil so many ligures for a remaindor to the right liai.d of the quo- tient, as there are plates in the given decimal. Proceed in the same man- ner wilh (he remainder, and continue to do so through all the parts of the integer, aud the several <3«'nominalions i.tandirg on the lef: hand make the answiT 76 DECIMAL FRACTIONS. Sect. II. 2. EXMIPLE'i. 1. What is the value of .528125 of a pound ? OPERATION'. This question is the fust example ,528125 in the preceding case inverted, hy 2 0' which it will be seen that questions in these two cases may reciprocally prove each other. The given ilecimal being the deci- mal of a pound, and shillings being the next less inferior denomination, because 20 shillings make one pound, I multiply the decimal I>y 20, and cnt- FarthiiiL(s 3,0 ^ ting off from tl:e right' hand of the .i7is. lOs. 6^d. product a number of figures, for a remainder equal to the number qi figures in tlie given decimal, leaves 10 on the left liand which are shillings. i then multiply the remaindet", which is the decimal of a shillirig by 12, and cutting ofif as before, gives 6 on the left hand for pence ; lastly., I multiply this last remainder, or decimal of a pennj' by 4, and find it to be 3 fnrthings, without any remainder. It then appears that 552t>125 of a pound is in va- lue 10s. 6id. 2. What is the value of ,73968 of 3. Vvhat is the value of ,768 of a a pound ? Jlns: 14.?.. 9^d.. pound Troj ? Sliillings 1 0,5 G o 6 1 a Fence G,7 5 4 * II is the last remainder, G80 reduced to its lowest ternw. A fraction is said to be reduced to its lowest terms, when there is no number which will divide both the numerator and denominator without a remainder. — Thus, set to the fraction its proper denominator tVo'V' then divide the nu- merator and the denominator by any number v/hich will divide them both, without a remainder, continue to do .so as long as any number can be found that v/ill divide them in that manner. 4. Q'S _8J>JL S -, —s 1 7 "y I o i-ji 2T' Sect. II. 2. SUPPLEMENT TO FRACTIONS. 7T SUPPLEMENT TO FRACTIONS. QUESTIONS. 1. Wh?ct are fractions? 2. What are iiilogerd or Avhole nuuibcrs ? 3. What arc mixed riuiubcrj .' 4. Of how many kiml^ arc fnictions ? : 3. ifow are Vulgar Fraclioiis written? m 6. , What is signified by the denominator of a fractioa? j 7, What is eignifiud by the numerator ? G. How are Decimal Fractions written ? 0. How do Decimals tlifl'er from Vulgar Fractions ? JO. How can it be ascertained what the denoaiinator to a Decimal Frae- tion is if it be not expressed ? 11. How do cyphers placed at the left hand of a Decimal Fraction affect its value ? 12. How are Decimals distinguished from whole numbers ? 13. In the addition of Decimals what is the rule for pointing off? M. What is the rule of pointing off Decimals in Subtraction ? In Multi- plication ? aiid in Di\ ision ? 15. In what manner is the reduction of a Vulgar Fraction to a decimal performed ? IG. How are numbers of di (To rent denominations, as pounds, shillings, pence, &.c. reduced to their decimal values ? 17. If it be required to find tiie value of any given decimal in the terms of an integer, wliat is the method of procedure ? EXERCISES. 1. What is the sum of 79]- 6i and In Case 1. Ex. 3d, under Ftcd-ic- of -i? when added together. GPEiiATio.v. lion of decimal fractions, the Scholar 79,5 G,25 may notice tliat i, J and ■} reduced ,75 to decimals are, ,25, ,5 and ,7^. When numbers, therelbre, fur ope- 2. From 17 take f orcRATioN. rations in either of the fundamental 17, _ s^S Ptulcs, are incumbered with these 86,60 Ans. 16,25 Rsmainder. fractions I, ■]. {, substitute fjr them sXrPLEMENT TO FRACTIONS. Sect. II. 2- 3. Multiply GQ\ by 5J. OPERATION. 6 8, 2 5 5, 5 3 4 12 6 3 4 12 5 3 7 5, 3 7 5 Product: 4. Divide 26a by ^. OPERATION. 2,5)26,25(10,5 Qtuotieitl. 25 125 125 "ill their equivalent decimal fractions, ^! (hat is, for i ,25 for •} ,6 ibr 3 ,75 then proceed according to the rule^ already given for these respectiva^ | operations in decimal fractions. Many persons arc perplexed by occurrences of a similar nature to the examples above. Hence is seen in some measure the usefulness of frac- tions, parlicuiarh-^ decimal fractious. The only thing necessary to render an}'^ person adroit in these operations is to have riveted in his mind the rules for pointing as taught and explained in their proper places. They are not burthensome ; every scholar should have them perfectly committed. 6. If a pile of wood be 18 feet A cord of wood is 128 solid feet ; long, 1 H wide, and 7-^^ high, how the proportions commonly assigned many cords does it contain ? ^7nr. 12corrf.^. 08 />"?/• 43: are, 8 feet in length, 4 in breadth, and 4 in height. The contents of a load or pile of wood of any dimensions may be found by multiplying the length by the breadth, and this product by the height ; or. by multiplying the length, breadth and height into each other. The last product divided by 128 will shew the number of cords, the remainder, if any, will be so many solid feet. > * The 432 inches is the frsction, .25 of n foot, valnrd acrordhng {n C..Kf,x 3, deduction Decimal Fractions. Sect. II. 2. SUPPLEMENT TO FRACTIONS. 79 6. If a load of wood be 9 feet long, 7. What is the value of ,725 of a 3i feet wide, and 4 feet high, how day? Arts. 11 hrs. 2\min. many square feet does it contain ? Ans. 126 feet, •a.'hich are in,'o feet short of a cord. 8. What is the value of ,0625 of a 9. Reduce SCut. Ogrs. 7/6. Boz. shilling ? Ans. 3 farthings. to the decimal of a ton. Ans. , 1 533482 1-f 10. Reduce 3 farthings to the deci- 11. Reduce j|y to a decimal frac- mal of a shilling? ./??!s. ,0625. tion. Ans. fil2b. €0 FEDERAL MONEY. Ssct. II. 3. ^ tJ. FEDERAL MONEY. Federai. I^foNEY is the coin of the United States, established by Con- gress, A. D. lliiG. Of all coius this; is the most simple, aud the operatious iu it the most easy. Tho denoDiinaticns arc in a den'mal proportion, as exhibited in the fol- Icwiug TABLE. 10 Mills \ r Cent, 10 Cents \ .u r,f Dime, 10 Dimes C "^ i Dolhv, marked thus, ^ 10 Dollars ) ( Eagle. Tlie expression of any sum in Federal Money is sinrply the expression of a mi.itd number in decimal li-actions. A dollar is the Unit Money ; dollars thejol'ore nu'.st occupy the pluCe of units, the less dcnotainalions, as dimes, cents, and mills, are de<:i;ual parts of a dollar, and may be distinguished from dollars in the same way as any other decimals by a comma or separa- trix. AH the figures to the left hand of dollars, or beyond units place are ea;^les. Thus, 17 eagles, 6 dollars, 3 dimes, 4 cents, and 6 mills are written — Of these, four are real coins, and one is imaginary. The real coins are the Eagle, a gold coin ; the Dol- lar and the Dime, silver coins ; and the Cent, a copper coin. The Jlill is only imaginary, there being no piece of money of that denomination. There are half eagles, half dollars, double dimeg, half dimes, and half cents, real coius. These denorainalions, or different pieces of money, being in a tetifold proportion, consequently any sum in Federal Money does of itself exhibit the particular number of each diiTcrent piece of money contained in it. Thus, 175,346 {seventeen eagles. Jive dollars, ikree dimes, four cents, six mills) contain 1753:0 mills, 17534 ^% cents, 1753 -^Vo dimes, 175 V^.W dolls. 17 -,VW'd tagles. Therefore, eagles and dollars reckoned together, ex- press the number of dollars contained ia th«i sum ; the same of dimes and cents ; and thic indeed is the usual way of account, to reckon the wkole sum in dollars, cents, and mills, thus : $175-34 6 The Addition, Subtraction, Mulliplication and Division of Federal Money is perlbrnu'd in all respecis as in Decimal Fractioiis, to which the Scholar is referred for the use ufi-ul*^^ in these oueratious. ■a ^ -a -2 C! ■5 m 3 ^V ^ -?; -o '^ -3 r** 'c,^ 5 S o ^^..'-^g fcT t- .'• ^ irj c 3 f^ O -:!: 2 ^ '^^ w -o.E 5 = ".^v^. k— 1 wJ '^^ ct; 1 7 5, 4 tj 5ect. II. 3. ADDITION OF FEDERAL MONEY. 81 ADDITION OF FEDERAL MOJVEY. 1. Add 16 Eagles ; 3 Eagles, 7 Dollars 5 Cents ; 26 Dollars, 6 Dimes, 4 Cents, 3 Mills ; 75 Cents, 8 Mills, 40 Dollars, 9 Cents together. OPERATION. 2. If I am indebted 59 dollars, 112 ^ . . dollars, 98 cents, 113 dolls. 15 ct3. 15 to ^ J « ?i dollars, 21 dollars, 60 cents, 200 dol- l^ q Q Cj ^ lars, 73 dollars, 35 dollars, 17 cents, ^--^^ 75 dollars, 20 dollars, 40 dollars, 33 16 0, cents and 16 dollars. What is the sum 3 7, 5 which I owe? Am. %n\ 13. 2 6, 6 4 3 , 7 5 8 4 0, 9 4, Or the sums may be all reckoned in dollars, cents and mills, thus, 15 § i^ ^160 37 05 26 64 3 75 8 40 09 g264 54 1 • Accountants generally omit the comma, and distinguish cents from dollars by setting them apart from the dollars. SUBTMCTIO^: OF FEDERAL MO^rEY. 1. From ^863, 17 take $69, 82. 2. From ^681 take ^57,63, OPERATioif. Remainder, ^623,37. 8 6 3, 1 7 6 9, 8 2 Remainder, 7 9 3, 3 6 8e MULTIPLICATION OF FEDERAL MONEY Sect. II. 3. MULTIPLICATION OF FEDERAL MONEY. 1. If flour be glO,25 j>€r barrel, what will 27 barrels cost ? OPERATIO\. 1 0, 2 5 2 7 2 $2 7 6, 7 bAns. 2. Multiply ^76,35 by $37,46. Prodvct, $20GO,O71O. Point off the decimals in the pro- duct according to the rule in Multi- plication of decimals ; if at any time there shall be more than three de- cimal figures, all beyonc^mills or the third place, will be decimal parts of a mill. 3. Multiply $24,675 by g 13,63, Product, $336,320yV-o- DIVISION OF FEDERAL MONEY. 1. If 2728 bushels of wheat cost $2961, how much is it per bushel? OPERATION. Bushels. Dolls. D. 2728)2961(1, 2728 m. 5 Ans.. 23300 21824 14760 13640 1120 2. Divide $3766 equally among 13 men ; what will each man re- ceive ? Ans. $288,923. When the dividend consists ot dollars ottly, if there be a remain- der after division, cyphers must be annexed as in division of deci- mals. Divide $16,75 by 27. Quotmit, 62 cents. Sect. II. 3. SUPPLEMENT TO FEDERAL MONEY. 83 SUPPLEMENT TO FEDERAL MONEY. 1. QUESTIONS. What is Federal Money ? When was its establishment, and by what authority ? 2. What are the denominations in Federal Money ? 3. Which is the Unit Money ? 4. How are dollars distinguished from dimes, cents, and mills ? 5. What places do the different denominations occupy, from the decimal point ? 6. How is the addition of Federal Money perforn»ed ? Subtraction ? Multiplication ? Division ? EXERCISES. 1. A man dies, leaving an estate of ^71600, there are demands against the estate of ^39876,74 ; the residue is to be divided between 7 sons ; what will each one receive ? Ans. ^4331 Zdcts, 2. A man sells 1225 bushels of wheat at $1,33 per bushel, and re- ceives ^93,76 for transportation ; what does he receive in the whole ? w5?M. $1723,01. 3. What will 3 hogsheads of sugar cost, each weighing 3Ca'<. ^qrs. lib. at 16crs. 1 mills per lb. ? Ans. gI99,899. 4. Divide seven tJiousand six dol- lars, one cent and three mills, by five hundred seventy six dollars, thirty four cents and tv.'o mills. Ans. $13,155.. 84 EXCHANGE. Sect. II. 4. ^ 4. EXCHANGE. Exchange is the giving of the bills, money, weight, or measure of one place or country, for the like value in the bills, money, weight or measure of another place or country. Norn 1. The Currencies in the >c\v EnglanJ States, and in Virginia, are the same and \vili be all comprehended under the term A". E. airreJtcr/ ; those of ^ew- York, North Carolina and Ohio, are the same, and will be comprehended under tlie term JV". York Ciirrenni ; thoj^e of N. Jersey, lennsj Ivania, Delaware and Maryland, are the same, and will all bo comprehended under the tcnn Fenn. Currency. Note 1. It will be sufficient perhaps in most cases, that the pupil be re the note above, the right hand figure in the r«- 12)4098|5 mainder (5) being to be reckoned 2 farthings, I set it down in the answer ^d. — the other figures 2|0)34[1 of the remainder (4098) being pence, I divide Jins. £,n Is. 6^d. by 12, in doing which there is a remainder of 6, which are pence ; these I also set down in the answer. The shillings (341) divided by 20, cutting off one figure from the divisor and one from the dividend as is usually practised in reducing shillings to pounds, give £17, and the 1 cut off from the dividend is 1 shilling, which completes the answer. 2. Change gl35 to pounds, &c. 3. Change ^287,74 to pounds Ms. £.50 12s. 6d. Ans. fA^l 18s. Q\d. Sect. II. 4. EXCHANGE. 89 To change the Nexs: -England to the New-Yorh currency ; add one third. To change the JVew-York to the JVe-so-England currency; subtract one fourth. To cfutnge the JVew-England to the Pennsylvania currency ; add one fourtli. To change the Pennsylvania to the Neva-England currency ; subtract one fifth. To change the New-York to the Pennsylvania currency ; subtract one six- teenth. To change the Pennsylvania to the New-York currency? add one fifteenth. SUPPLEMENT TO EXCHANGE QUESTIONS. 1 . What is Exchange ? 2. How do you change N. England 2. How do you change Pcnnsyl- and Virginia currencies to Fed- vania, &,c. currency to Fcde- eral Money ? — New-York cur- ral Money ? rency ? — and wherein consists the difference ? 3. If pounds only are given to be 3. If there are farthings in the changed, how do you proceed ? given sum, how do you proceed ? 4. When there are no shillings, 4. If the given sum be pounds or only one in the given sum, only, how do you proceed ? how do you proceed ? 5. How do you change Federal 5. How do you change Federal Money to N. England currency ? Money to Pennsylvania, &c. N. York currency ? — Wherein currency ? consists the difference ? G. How do you change New-England to New- York currency ? — New York to New-England ? — New-England to Pennsylvania ? — Pennsylvania to New-England ? — New- York to Pennsylvania ? — Pennsylvania to New- York currency ? M 90 EXCHANGE. Sect. II. 4. EXERCISES. 1. In £3G Is. 61,1 N. Enp:. cur. or £48 2.v. Oic/. N. York cur. or £4* 1.1. llr/. Penn. rnr. how many flollars, cents and mills ? 4n5. ^120,257 X. E. cur.— $l20,2oo N. Y. t»r.— gl20, 256 Penn. cur. NoTK. — In making llip oxcliange from one ciinency into another tliere will fi-equently I'p the loss of snme fiartioni of a fnrfhing; far this reason when the exchange is again made into Federal iMoney, there will be the diflerenco of some mills in the answers obtained. 2. Cliange £100 12s. N. E. cur. to N. Y. cur. Penn. cur. and to Federal Money. Ans. £240 IGs. N. Y. cur.— £225 15s. Penn. cur.— $602 F. Money. 3. Chang:e ^150,25 to N. England, N. York, or Penn. cur. accordingly as the pupil may have been instructed in one or the other, or all of these rules. Jns. £45 Is. 6d. N. E. cur.— £60 2s. N. Y. cur.— £56 6s. IQic^.Penn. eur. 4. Let the pupil be required to change the sums in New-York and in Pennsylvania currency in the above answer, to New-England currency ; tl)e same in New-England and in New- York to Pennsylvania currency ; and the same in New-England and Pennsylvania to New- York currency, the answers of which will reciprocally prove each other. 5. Change «>345,625 to N. Eng. or N. York, or Penn. currency. Ans. £103 iSs. 8^d. N. E. CMr.--£l38 5s. N. Y. cur.— £129 12s. 2|d. Penn. currency, 6. Change 75 cents into N. E. or N. Y. or Penn. cvr. Ans. 4s. 6d. N. E. cur. — 6s. N. Y. cur. — 5s. l^d. Penn. cur. 7. Change £45 Is. 6d. N. E. cur. or £60 2s. N. Y. cur. or £56 6s. \0\d. Penn. cur. to Federal Money. An.s. ^150,25. 8. Change 4s. 6d. N. E. cur. or 6s. N. Y. cur. or 5s. "i^d. Penn. cur- rency to Federal Money. Aas. 75 cents. 9. Change £46 10s. 6icZ. considered in either currency to Federal Money. ,. Ans. ^155,09 N. E. cwr.—^l 16,317 N. Y. cwr.— ^124,072 Penn. cur. 10. Change f\61 to N. E. or N. Y. or Penn. currency. Ans. £50 2s. N. E. cur. — £60 16s. N. Y. cvr. — £62 12s. 6d. Penn. cur. 11. Let the pupil be required to change the sums in New- York and Pennsylvania currency, in the above answer, to New-England currency, j fcc. as in the 4tii exercise above. 12. Change 6^r7. to Federal Money. Ans. 9 cents N. E. cur. — 6 cents 7 mills N. Y. cur. — 7 cents 2 mills Penn. currency. 13. Cliangp £263 to Federal Money. Ans. $1376,666 N. E. fwr.— ^657,50 N. Y. ciir.— ^701,333 Penn. cur. Shct. II. 4. EXCHANGE. &t TABLE FOR REDUCING NEW-ENGLAND CURRENCY TO FEDERAL MONEY. ' shill. shiil. shill. shill. shiU. 1 o 3 4 5 Pence. Os. M. Cts. M Cts. M. Cts. M. Cts. M. Cts. M. IG 7 33 3 50 66 7 83 3 1 1 4 18 1 34 7 51 4 68 1 84 7 2 2 8 19 5 36 1 52 8 69 6 86 1 3 4 2 20 9 37 5 54 2 70 9 87 5 4 5 6 22 3 38 9 55 6 72 3 88 9 5 7 23 7 40 3 57 73 7 90 3 6 8 3 25 'll 6 58 3 75 91 6 7 9 7 26 4 43 59 7 76 4 93 8 11 1 27 8 44 4 61 1 77 8 94 4 9 12 5 29 2 45 8 62 5 79 2 95 8 10 13 9 30 6 47 2 63 9 80 6 97 2 11 15 3 32 48 6 65 3 82 98 6 TABLE FOR REDUCING NEW-YORK CURRENCY TO FEDERAL MONEY. shill. shill. sh>il. shill. shill. 1 2 3 4 6 Pence. Cts. M. Os. M. Cts. .If. Cts. J\I. Cts. M. Cts. M. 12 5 25 37 5 50 62 5 1 1 13 5 26 38 6 51 63 5 o 2 1 14 6 27 1 39 6 62 1 64 6 3 3 1 15 6 28 1 40 6 53 1 65 6 4 4 2 10 7 29 2 41 7 54 2 G6 7 5 5 2 17 7 30 2 42 7 55 2 67 7 6 6 2 IG 7 31 2 43 7 56 2 68 7 7 7 2 19 7 32 2 44 7 57 2 69 7 8 8 3 20 8 33 3 45 8 58 3 70 8 9 9 3 21 8 34 3 46 8 59 3 71 8 10 10 5 23 35 5 48 60 5 73 11 11 5 24 3G 5 49 61 5 74 To find by these Tables the Cents and pence under one dollar, look the shillings hand column : then under the former, 9inil found the cci^ts ant' mills sousrht. Mills in any sum of shillings and at top, and the pence in the left on a line with the latter, will be 92 TABLE REDUCING POUNDS, fee. TO DOLLARS, &c. Sect. H. 4. TABLE FOR REDUCING THE CURRENCIES OF THE SEVERAL UNITED STATES TO FEDERAL MONEY. u. r •li 1 2 3 1 2 3 4 5 G 7 8 9 10 11 1 o 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 N. Hamp. 1 N. Jersey, Mass. New-York Pennsylva'a S, Carolina, Rh. Island. and Delaware, and Conn, and N. Carolina. and Georgia. Vir2:iiiia. Maryland. 1). cts. 7n. D. cts. in. D. els. m. D. cts. m. , 3 > 3 , 3 , 4 , '7 , 6 , 6 , 9 , 10 , , 8 , 14 , M , 10 11 , 18 , 28 , 21 22 , 36 , 42 , 31 33 , 54 , 66 , 42 44 , 71 , 69 , 52 56 , 89 , 83 , 62 67 ,107 , 97 , 73 78 ,125 ,111 , 82 89 ,143 ,125 , 94 100 ,161 ,139 ,104 111 ,179 ,153 ,114 122 ,198 ,167 .125 133 ,214 ,333 ,250 267 ,429 ,500 ,375 400 ,643 ,6G6 ,500 533 ,857 ,C?3 ,625 667 1,071 1,000 ,760 800 1,286 1,167 ,875 933 1,500 1,333 1,000 067 1.714 1.500 1,125 200 1,929 1,667 1.250 333 2,143 1,833 1,375 ,467 2,357 2,000 1,600 600 2,671 2,167 1,625 733 2,785 2,333 1,750 867 3,000 2,500 1,875 2 000 3,214 2,667 2,000 2 133 3,428 2,833 2,125 2 267 3,643 3,000 2,250 2 400 3,857 3,167 2,375 2 633 4,071 Sect. II. 4. TABLE REDUCING POUNDS, &c. TO DOLLARS, &c. 93 TABLE FOR REDUCING THE CLtRRENCIES, &c. CONTINUED. New-Hamp. New-York, New- Jersey, S. Carolina, kc. &c. &c. &c. 6cc. £. D. c. m. D. c. m. D. c. in. D. c. m. 1 3,333 2,5 2,666 4,286 2 6,667 6,0 5,333 8,571 3 10,000 7,5 8,000 12,857 4 13,333 10,0 10,667 17,143 5 16,667 12,5 13,333 21,42i) 6 20,000 15,0 16,000 25,714 7 23,333 17,5 18,667 30,000 8 26,667 20,0 21,333 34,286 9 30,000 22,5 21,000 38,671 10 33,333 25,0 26,667 42,867 20 66,667 50,0 53,333 85,714 30 100,000 75,0 80,000 128,671 40 133,333 100,0 106,667 171,429 60 166,667 125,0 133,333 214,286 60 200,000 150, 160,000 257,143 70 233,333 175, 186,667 300,000 80 266,667 200, 213,333 342,857 90 300,000 225, 240,000 o8o, 714 100 333,333 250, 266,667 428,671 1 200 f 300 666,667 500, 533,333 857,143 1000,000 750, 800,000 1285,714 400 1333,333 1000, 1066,667 1714,286 500 1666,667 1250, 1333,333 2142,857 600 2000,000 1500, 1600,000 2571,429 700 2333,333 1750, 1866,667 3000,000 800 2666,667 2000, 2133,333 3428,571 900 3000,000 2250, 2400,000 3857,143 1000 3333,333 2500, 2666,067 4285,714 TABLE FOR REDUCING FEDERAL MONEY TO THE CURRENCIES OF THE SEVERAL UNITED STATES. New-Hamp. New-York, N. .1 ersey. S. Carolina, &c. &c. &c. fcc ,.kc. &c. DOLL. 6.S, DOLL. 8s. DOLL 7s. Gd. POLL. As. Gd. D. cts. £. s. d. q. £. S. d. q. £• s. d.q. £. s. d. q. ,01 3 1 1 2 ,02 1 2 2 1 3 1 ,03 2 1 3 2 3 1 3 ,04 3 3 3 3 2 2 1 ,05 3 2 4 3 4 2 2 3 ,06 4 1 5 3 5 2 3 1 ,07 5 6 3 6 1 4 ,08 5 3 7 3 7 1 4 2 ,09 6 2 8 3 8 6 ,10 7 1 9 2 9 6 2 94 TABLE REDUCING DOLLARS, Sec. TO POUNDS, &c. Sect. H. 4. TABLE FOR REDUCING THE ClKllE.NCIES, kc. CONTINUED. Niew-Hamp. New-York, New-Jersey, S. Carolina,. &c. kc. &ic. &c. &c. Uvll. cts. £. s. (/. f/. £. s. d. q. £. s. d. q. £. s. d.q. \,20 1 2 2 1 7 1 1 6 11 1 ;:30 1 9 2 2 4 3 2 3 1 4 3 ,40 2 4 3 3 2 2 3 1 10 2 ,50 3 4 3 9 2 4 ,60 3 7 1 4 9 2 4 6 2 9 2 ,70 4 2 2 6 7 1 5 3 3 3 1 ,00 4 9 2 6 4 3 3 8 3 ,90 6 4 3 7 2 2 6 9 4 2 2 1 6 8 7 6 4 8 2 12 16 15 9 4 3 13 1 4 1 2 6 14 4 1 4 1 12 1 10 18 8 5 1 10 2 1 17 6 1 3 4 6 1 16 2 8 2 5 1 8 7 2 2 2 16 2 12 6 1 12 8 8 2 8 3 4 3 1 17 4 9 2 14 3 12 3 7 6 2 2 10 3 4 3 15 2 6 8 20 6 8 7 10 4 13 4 30 9 12 116 7 40 12 16 15 9 6 8 60 15 20 18 15 11 13 4 GO 18 24 22 10 14 70 21 28 26 5 16 6 8 80 24 32 30 18 13 4 90 27 36 33 15 21 100 30 40 37 10 23 6 8 200 60 80 75 p 46 13 4 300 90 120 112 10 70 400 120 IGO 160 93 6 8 500 150 200 187 10 116 13 4 600 180 240 225 140 o« 700 210 280 262 10 163 6 8 800 240 320 300 186 13 4 900 270 560 337 10 210 1000 300 400 375 233 6 8 £000 600 800 750 466 13 4 3C00 900 1200 1125 700 40(^0 1200 ICOO 1500 933 6 8 5 3G00 3375 2100 lOtOO 3000 4000 3750 2333 6 8 SCCT. II. 5. SIMPLE INTEREST. ^ 5. SIMPLE INTEREST. INTEREST is the allowance given for the use of money, by the bor- rower to the lender. It is computed at so many dollars for each hundred lent for a year, (/)«r annum) and a like proportion for a g^reater or less time. The highest rate is limited by our laws to 6 per cent, '^ that is 6 dol- lars for a hundred dollars, 6 cents for a hundred cents, ^G lor aX)100,&;c. This is called legal interest, and is always understood when no other rate is mentionGd. There are three things to be noticed in Interest. 1. The PniNciPAL ; or raancy lent. 2. The Rate ; or sum per cent, agreed on. 3. The Amount ; or principal and interest added together. Interest is of two sorts, Simple and Compound. 1. Simple Interest is that which is allowed for the principal only. 2. Compound Interest is that which arises from the interest being added to the principal, and (continuing in the hands of the lender) becomes a part of the principal at the end of each stated time of payment. GENER.iL RULE. 1. For one year, multiply the principal by the rate, from the product cut off the two right hand tigures of the dollars, which will be cents, those to the left hand will be dollars ; or, which is the same thing, remove the separatrix from its natural place two tigures towards the loft hand, then a".\ those figures to the left hand will be dollars, and those" to the right hand will be cents» mills, itnd parts of a mill. Jn the same zcay is calculated the interest on any sum of money in pounds, shil- lings, pence and farthings, with this difference only, that the tzco figures cut. off" to the right hand of pounds, must he reduced to the loxjDest dciiomination, each time cutting off as at first. 2. For tivo or more years, multiply the interest of one year by the num- ber of years. 3. For months, take proportional or aliqiiot parts of the interest for one yestr, that is, for G months, i ; for 4 months, -J- ; for 3 months, -]-, &.c. For days, the proportional or aUquot parts of the interest for one month, allowing 30 days to a month. EXAMPLES. 1. What is the interest of^86,44C for one year, at 6 per cent? OPERATION. Dolls, cts. mills. In the product of the principal mul- 86 44 6 principal. tiplied by the rate is found the answer. G rate. Thus cutting oif the two right hand figures from the dollars leave five on 5J 18 67 6 interest. the left hand which is dollars ; the two figures cut oil (18) arc cents, th« aext figure (6) is mills ; all the figures which may cliaace to be at the right kand of mills, are parts of a mill ; hence we collect the Ans. ^5 IQcts. 6,^^^^;. • III New- York the law allows 7 per ceut. 96 tJlMPLE INTEREST. Sect. II. 6. 2. Whut is the interest of ^3G5 lids. Gmills, for three years 7 months and C da^s ? OPERAflON. 3 6 5, 1 4 principal. 6 rate. •6 months i)2 1 j 9 0, 8 7 6 inlerest for 1 year. J 3 G 5, 7 2 6 2 8 interest for 3 years. 1 month ^) 1 0, 9 5 4 3 8 interest for (J motiths. 6 days 4) 1,82573 interest for 1 month. ,36 514 interest for 6 days. ^7 8, 7 15 3 interest for 3 years, 7 months and 6 days ; that is $78 87^. l^^^mills. Because 7 months are not an even part of a year, take two such num- bers as are even parts, and which added together will make 7 (6 and 1) 6 months are ^ of a year, tlierefore for 6 months, divide the interest of one year by 2 ; again 1 month is ■} of 6 months, therefore for 1 month, divide ihe interest of G months by G. For the days, because 6 days are j of a mouth, or of 30 days, therefore /or G days, divide the interest of 1 month by 5. Lastly add the interest of all the parts of the time together, the sum is the answer. 3. What is the interest of £71 7s. GJcZ. 4. What is the interest of for 1 year at 6 per cent ? lOs. 8d. for 1 year ? Ans. 1$. OPERATION. £. s. d. g. 71 7 6 6 y.3'52 .ins. £4 5j. 7frf. Sect. li. 5. SIMPLE INTEREST. 97 When the rate is at 6 per cent, there is not perliapa a more concise and easy way of casting interest, on any sem ol" money in Dollars, Cents, and Wills, than by the iollowing METHOD. Write down half the greatest even number of months for a multiplier; it there be an odd month it niiist be reckoned 30 days, for which and the given days, if any, seek how many times you can have six in the sura of them, place the ligiire for a decimal at the riglit hand of half the even nuiii- ber of months, already found, by which multiply the principal ; observing in pointing ofl' the, product to remove the»decimal point or separatrix tzio figures from its natural place towards the left hand, that is, point off txro more places for decimals in the product, than there are decimal places in the multiplicand and multiplier counted together ; then all the ligurcs to the left hand of the point will be dollars, and those to the righfhand, dimes, cents and mills, &c. which will be the interest required. Should there be a remainder in taking ohg sixth of the days, rediKe it to a vulgar fraction, for which take aliquot parts of the multiplicand. Thus. If the remainder be 1=|, divide the multiplicand by G If 2=i, by 3 If 3=^, - by 2 If 4=|, by 3 twice. If b=\, and i, by 2 and 3. The quotients which in this way occur, must be added to the product of the principal multiplied by half the months, Lc the sum thus produced will be the interest required. IMien there are days, but a less number than G, so that 6 cannot be contain- ed in them, put a cypher in place of the decimal at the right hand of the months, then proceed in all respects as above directed. Note. In casting interest, each month is reckoned 30 days. EXAMPLES. 1. What is the interest of ^76,54 for 1 year, 7 months and 11 days ? OFERATION. The number of months being 19, the greatest even number is 18, half of which is 9, which I write down ; then seeking how often 6 is con- tained in 41, (the sum of the days in the odd month and given days) 1 find it will be Gtime?. which I set down at the right hand of half th'- even number of months for a decimal, by whi^h together 1 multiply the principal. In laki-y- Ans. 7, 4 1 1 6 2 one sixth of the days (41) there will bev remain ; I set down ti.e 8 and reduce the 2 pence to f*rthings (8) in which I have the di- visor once (1 qr. or ^u'.) and aremain- dei- of ^ of a farthing, which being of small value is neglected. 2. If 48 y^r'U of clolh cost £4 ICs. 4-^ii. vvhut j3 that per y.ird ? Jiij.'£o 2s. IN DOLLARS, CENT*, MILLS. 1 IN ALL CASES. Divide the price by the quantity^ and point off so many places for cents and mills in the product ag there are places of cents, and mills in the dividend. If the quantity he a composite num- ber, that is produced by the multipli- cation of two numbers, the operation may be varied by dividing the price first by one of those numbers, aod this quotient by the other. EXAMPLES. 1. If 5 yards of cloth cost g 12,25, what is that per yard ? D. 5)12, OPERATION. Cts. 25 Jins. 2, 46 There are two decimal places ia the dividend. I therefore point •ff two places for decimals or cents in the quotient. 2. If 48 yards of cloth cost $16,06, what is that per yard ? Ans. $0 33 cents* Sect. II. 7. COMPOUND DIVISION. Ill PO' .t>S, SHI. .," ^i'GS, PENCE, FART ',. 3. If 24^6. of tea cost £2 75. 9|c/. srhat is that per /6. ? Ans. £0 Is. ll|rf. 4. If 36 yards of cloth cost £42 §1. 71c?. what is that per yard ? An$. £1 4s. 2i(/. CASE 2. 1. " Having the price of a hundred weight (1 12/i.) to find the price of Mb. divide the given price by 8, that quotient by 7, and this quotient by 2, and the last quotient will be the price ©f 1/6. required." 2. If the number of hundred u-eight be more than one, first divide the whole price by the number of hun- dreds, then proceed as before. EXAMPLES. 1. \i\cTit. of sugar cost £3 ".s. &d. what is that per lb. ? OPERATION. £. s. d. q. 8)3 7 6 price of \cwt. 7)0 8 5 1 price of Ulh. or ic^L 2) 1 2 2priceof2lb. or-'^czvi. j .'?«s. 7 1 priiy- »/ ! '•■. DOLLARS, CENTS, MILLS. 3. If 24/6. of tea cost $7,97 what Is that per lb. ? Am. $0,233. 4. If 35 yards of cloth cost $141, 103, what is that per yard ? Ans. $4,031. The same may be done in Federal Money. 6. If 1 cwt. of sugar cost $11,25, what is that per lb. '! Ans. iO cents. 112 C031P0UND DIVISION. Sect. il. 7. rOUVDS, SHILLINGS, PE.VCE, FARTII. 2. If 8ca/. of cocoa cost £15 7s. 4d. what is that per lb. ? Jhis. 4d. nOIXARS, CENT6, MILLS. 6. Uilcxct. of cocoa cost g51,223, what i« tl)at per lb. ? Ant. bets. 7m. 3. If 5fr:'<. of sugar cost £15 135. what is that per lb. ? Jlns. lid. CASE S. " When the divisor is such a number as cannot be prodvred by the inultipH' cation of small nuvibers, divide after the manner of Ion? fii>.'i;si(»n, settins: down the work uf di\idinj^ and rc- ducinff." 7. If Scwt. of sugar cost ^52^67 what is that per lb. ? Ans. \5cU. 5m. Sect. II. 7. COMPOUND DIVISION. IIS POUNDS, SHILLINGS, PENCE, FART». EXAMPLES. 1. If 46 yards of cloth cost £53, lOs. 6u'. what is that per yard .' OPERATION. £. s. d. £. s. d. 4G)53 10 6(1 3 3i.3ny. 46 46)130(3 138 12 12 46)150(3 138 12 4 46)48(1 46 2. If 263 bushels of wheat cost £86 Is. lOd. what is that per feushel ? Ans. 6«. e^d. 3. If 670 gallons of wine cost £147 1». lid. what is that per gal- lon ? Ans. 4s. 4\d. DOLLARS, CENTS, MILLS. 8. If 46 yards of cloth cost $119-, 416, what is that per yard ? Anj. ^3,878. 9, If 263 bushels of wheat cost 1^287,973, what is that per bushel ? , Ans. $1,094. 10. If 670 gallons of wine cost §490,32, what is that per gallon ? Ant. $0,73. 114 SUPPLOIENT TO COMPOUND DIVISION. Sect. II. 7. SUPPLEMENT TO C03IP0UND DIVISION. QUESTIONS. What is Compound Division ? When the price of any quantity not exceeding 12, of yards, pounds, &.C. is given in pounds, shillings, pence and farthings, how is the price of one yard found ? When the quantity is such a number as cannot be produced by the multiplication of small numbers, what is the method of procedure ? Having the price of an hundred weight given, in what way is found the price of 1 lb. ? If there be several hundred weight, what are the steps of operating? When the price is given in Federal Money, what is the method of operating ? EXERCISES. rOLNDS, SHILLINGS, PENCE, FARTH. 1. If 10 sheep cost £4 5s, Id. what is the price of each ? Ans.^s. 6irf. 2. If 84 cows cost £253 13j. what is the price of each ? Ans. £3 Os. 4}J. DOLLARS, CENTS, MILLS. Let the Scholar reduce the price of sheep and of the cows to Federal Money, and perform the operations in Dollars, Cents and Mills. Price of 1 sheep $1,426. Price of 1 cow, ^10,065. - Sect. II. 7. SUPPLEMENT TO COMPOUND DIVISION. 115 3. If 121 pieces of cloth measure 2896 yards, 1 qr. 3 na. what does each piece measure ? Ans. 23ijds. 3qr. 3na. 5. If 2cwt. of rice cost £2 11j. $ld' what is that per lb. ? Ms. nd. 4. If 66 tea-$poong weigh 2Ib. lOoz. 14pwt. what is the weight of each ? Jlns. lOpwt. 12^grs. 6. At £2 lis. 6-\d. for 2cwt. of rice, what is that in Federal Money, and what is that per lb. ? Price of lib. Sets. 8m. 7. If 47 bags of indigu wei^h j 8- If 8 liorse? eat 'J«X)bu9hel? and 12cwt. Iqr. 261b. 4oz. what does 1 peck of oats in 1 year, how much each weigh? I will each horse eat per day ? Ansi Iqr. lib. 12oz, I Ans. Ipk. Iqt. Int. Z^illt. 116 SUPPLEMENT TO COMrOL'ND DIVISION'. Sect. II. 7. Divide jC-97 25. 3(^ anaonfi;' 4 men, 6 boys, and give each man 3 times 5o ujuch as one boy ; what will each man share, and each boy ? OPEHATIOX. The men have triple .£. *. d. £. s. d. q. shares, therefore inol- ri)^dl 2 3 (16 10 1 2=1 hoy's share. U\)\y the number o/aion ' ;) by 3, and acid ihe nwmbcr of boys, (G) Ijf a diviaor. tncn. boijs. 4 h a 12 IC ihe number of crjuul shares m t!is :..':c/f.=^ Divisor. 18 117 108 9 20 )1U2(10 18 2 J2 ;27(i Ans. 4D 10 4 2=?1 ;nan's *'iare. PROOF. £49 10 4 2 4 198 1 C Omcr/«sAar«. IC 10 1 2 and C 99 9 boys' share. £237 2 3 added. 9 4 3G 10. Divide £39 12s. ud. among 4 men, 6 women, and 9 boys; give each rjan double to a woraan, each woman double to a boy. £. s. d. 1 15a boy^s share. .Ins. (2 2 10 a ztoman^s share. 4 5 u c jrtcji.'* share. II. 8. SINGLE RULE OF THREE. IT ^ 8. SINGLE HULK OF THREE, THE Single Rule of Three, sometimes called the Rl'le of PROPORTtoN is knouTi by harinij three terms giren to find the fourth. It is o{ tiio kinds, Direct and Indirect, or Imcrse. SLYGLE RULE OF THREE DIRECT. The Singrle Rule of Three Direct teaches, by having three numbers given to find a fourth, which shall bear the same proportion to the third that the second docs to tlie first. It is evident, that the value, weight and measure of any commodity is proporiionate to its quantity, that the amount of work, or consumption is proportionate to the time ; thnt gain, loss and interest, when the time 13 tixed, is proportionate to the capital sura from which it arises ; and that (he elTect produced by any cause is proportioned to the extent of that cause. These are cases in direct proportion, and all others may be known to bs so, when the number sought inrre^scs or diminishes along with the ternj from which it is derived. Therefore, If viore require more, or less require less, the question is always known to belong to llie Rule of Three Direct. Alorc requiring more, is when the third term is greater than the first, and requires the fourth term to be greater than the second. Less requiring less, is when the third term is less than the first and re- quif'cs the fourth term to be less than the second. RULE. ** 1. State the question by making that number which asks the question, "the third term, or putting it in the third place; that which is of the " same name or quality as the demand, the fir^t term, and that which is of " the same name or quality with the answer required, the second term." " 2. Multiply the second and third terms together, divide by the first, " and the t})ioticpt will be the answer to the question, which (as also tl;e " remainder) will be in th(^ same denomination in which 30J left the second " term, and may be brought into any other denomination required." The chief difficult}' that occurs in the Rule of Three, is the right placing of the numbers, or staling of the question ; this being accomplished, there is nothing to do, but to mi.itiply and divide, and the Avork is done. To this end the nature of every question must be considered, and the circumstances on which the proportion depends, observed, and common sense will direct this if the terms of the qucstioo be understooij. The method of proof is by inverlieg the order of the question. »Voie I. If (lie firct and third terms, both or either, he of different de- nominations, both terms muct be reduced to tlie lowest denomination raen- Uoned in either, before stating Ihe que-tion, 2. If the second term consists of difi'erent derjominafions, it must be re- duced to the lowest denomination ; tho. fourth term or nnswer vi'ill t'len be found in tho same denomination, and iau?t he reduced back ?gaiu to the hig!;esl denominaiion pov^ble. 3. After division if there be any rcmaind?r, and the qj?oticnt he not in tlie loivest denomination, it must be reduced to the ne;;t less der.omiiution, dividing ?" beiore. So continue to do till it is bio;;^'iit to the lowest de- nomiPiatipn, or till nothing remains. 4. In every qu<"'sti<)n there is a si^pposilion and a der.'iand ; ihc supposition ii implied in ihe t'.vo lir:-! toruisorihc slalemont, ihc irects — first consider which of the oriven numbers it is that asks the question ; that determined on, put it in the third place, then seek for another number of the same name, or kind, put that in the first place, the second place must now be occtipied by that cumber which is of the same name or kind with the number sought; when these steps are cautiously Ibllowed, the scholar cannot fail to make his statement right. 7. If an ingot of silver weigh 3Goz. 8. A Goldsmith sold a Tankard lOpa.'^. what is it worth at 6s. per for £10 12.?. at the rate of Bs.Ad. ouace ? Ans. £,9 2s. 6d. per ounce, I demand t'.je v/eight of it. Ans.oSoz. lopzet. 9. If the moon move ISdesr. lOmtn. 35sec. in one day ; in what time doQ» it perform one revohit-on ? Ans. ^Idai'^.'^kn-. A.3inin, Sect. II. 8. SINGLE RULE OF THREE DIRECT. 121 10. If a family of 10 persons spend 11. If a fattiliv of 30 persons n bushels of rnalt in a month, how spend 9 bushels of malt in a month, many bushels will serve them when how many bushels will serve a farn- there are 30 in the family ? ily of 10 persons the same time ? .Ins. hisltrh. -'ins. 3 lusheh. 12. If 12 acres 3 roods, produce 78 quarters 3 pecks, Fa/)w much will 35 acres, 1 rood, 20 poles produce ? Am. 21G qrs. Obush' l^peckf. ^'- 122 SINGLE RULE OF THREE DIRECT. Sbct. II. 8 13. If 5 acres, 1 rood produce 26 quarter?, 2 bushels, how many acres will be required to produce 47 quarters, 4 bushels ? Ant. 9 acres, 2 mods. 14. If 365 men consume 75 bar- rels of provisions in 9 months, how much will 500 men consume in the same time ? Ans. 10244- barrels. Note. In the 15th example, in order to embrace the fraction (^f «/ g. barrel) the integers 102 bar- ^rels 'must be multiplied by the de- nominator of the fraction (73) and the numerator, (54) added to the product. After division, the quotient must be divided by the denominator of the fraction, and (his last quotient will be the arrswor, all which may be seen in the example. The Scholar must remember to do the same in all similar cases. 15. If 600 men consume 102^f. barrels of provisions in 9 months, how much will 365 men consume in the same time ? OPERATION. barrels. l02^ _ •• Multiplied by 73 the denominator of the fraction. 306 714 Add 54 the numerat&r. As 500 : 7600 :: 365 7500 182600 2555 5j00)27375|00 73)5476(75 An$. 511 365 366 16. How much will 4 pieces of linen contriinins:, vIt:. 35}, 56. 37-J-, and .■^8 yards c»me to at 79 cents j>er yard ' Ans. j^l 16,13. Sect. II. 8. SINGLE RULE OF THREE DIRECT. 123 17. If I give ^6 for the use of 18. How many tiles of 8 inches ^100 for 12 months, what must I square will lay a floor 20 feet long, give for .'557,82 the same time ? and 16 feet broad ? jlns. $21,469. .Iiu. 720. " 19. If 2lb. of sugar cost 25 cents, 20. If £3 sterling bo equal to £i what will 100/6. of coffee cost, if 8/6. N. England currency, how much of sugar are worth 5/6. of coffee ? N. England currency will be equal Ms. $20. to £1000 sterling ? Ans. £1333 6y. Sd. 21. If I buy 7/6. of sugar for 75 N, B. Suras in Federal Money rents, how much can I buy for 6 are of the s;une denomination when dollars ? Ans. 66/6. the decimal places in each are equal. To reduce satins in Federal MoJicy to the same denomination, annex so many cyphers to that sum which h;is the least number of decimid place?, or places of cents, mills, ^c. ;ts shall make up the deliciency. 124 SINGLE RULE OF THREE DIRECT. Sect. II. 8. 22. If I buy 76 yards of cloth for 23. A man spends g3,25 per till 13,17, what did it coat per Ell week, what is that p^r annum ? English? Aks. ^lyUQU Am. ^iG9,-le>4. 21. if 3 liorses :ind 4 oxen be worth 9 cows, how many cows will 8 horses and 8 oxen be worth : /Ins. I H. 25. Bougl^t a silver cljp, weighing 9oz. Apn-t. Xf^^rt. for £3 2^. Zd. o|.j. what was that ner otmca l Arts. '6i. 'dd. Sec*. II. 8. SINGLE RULE OF THREE DIHECT. 126 26. There is a cistern which has 4 cocks, the first will empty in 10 minutea, the second in 20 minutes, the third in 40 minutes, and the fourth in 80 minutes ; in what time will all four running together empty it? Min. Cist. ( 10 Cist. Min. (6 )20 : 1 : : 60 : )3 yio )],5 (so ( ,75 In 1 hour the 4 cocks would empty - - - - 11,25 Cw Then, Cist. Min. Cist. Min. As 11,25 : 60 : : 1 • 5,33 Ans, 27. A man having a piece of land te plant, hired two men and a boy to plant it, one of the men couM plant it in 12 days, the other in 15 days, and the boy in 27 days ; in how long time would they plant it if they all worked together ? Ans. 5t2'l6 days. i 28. A merchant bought 270 quin- tals of cod lish, for ;f5780 ; freight §37,70 ; duties and other ^charges ^30,60 ; what must he sell it at per quintal to gain ^143 in the whole ? Ans. $3,671. The sum of all the expenses of the tish with the Merchant's gain must be found for the secocd term. 29. If a staff 5/r. 8m. in length cast a shadow of 6 feet ; how higli is that steeple whose shadow measures 153 feet? Ans. U4yeet. ii-^ H ^ tic 116 oiNGLE RULE OF THREE DIRECT. Sect. II. 8. 30. Bought 12 pieces of cloth each 31. Bought 4 pieces of Holland, 10 yards, at g 1,73 per yard, what each containing 24 Ells English, for came they to ? g96 ; how much was that per yard ? Ans. ;J210. Ans. QO cents. 32. Bought 9 chests of tea, each weighing 3C. 2grs. Zllb. at £4 9#. per cwt. what came they to ? Am. J£147 13*. B^d. Sect. II. 8. SINGLE RULE OF THREE DIRECT. 127 33. A bankrupt owes in all 972 dollars, and his money and effects are W ^607,50 ; what will a creditor receive on $11 ,333 ? Ms. $7,083 I 34. Bought 126 gallons of mm for $110, how much water must be added to it to reduce the first cost to 75 cents per gallon?^ Ans. 20|ga/. 36. A owes B £3475, but B com- 36. If a person whose rent is pounds with him for 13s. 4d. on the $145 pays $12,63 of parish taxes, pound ; what must he receive for how much should a person par his debt ? Am. £2316 13j. 4d. whose rent is $378 ? Ans. $32,925. I'ia SINGLE iiULE OF THKEE iN VERSE. Sect. H. 8. [nvcrsc Froportion. IN some questions the number sought becomes less, when the circum- stimccs from which it is do ri veil become greater. Thus wlien the price ot' {roods iiicrciise the qnantily which may be bought ibr a given sum, is smaller When the number of men employed at work is increased, the time in which Ihey may complete it becomes shorter ; and when the activity of «ny cauie is increased, the quantity necessary to produce any given effect is di.^lini!:•hed. These and the lilce cascj belong to the SINGLE RULE OF THREE INVERSE. The Single Rula of Three Inverse teaches by having three numbers eivcn to find a fourth, having- the same proportion to the second, as the fir:«t has to the third. If more require less, or less require more, the question belongs to the Single Rule of Three Inverse. Mirrc rujuirin^ Icsi, is when the third term is greater than the first, and requires the fourth term to be less than the'second. IjC^s rcijuiri/ig more, is when the third term is IccS than the first, and re- quires the fourih term to be (greater than the second. RULE. " State and reduce the terms as in the rule of three direct ; then multi- ply tlic iir-st and second ti.rms together, divide the product by the tliird, and the quoiient will be the ansv.er in the same denomination with tlic &ccfuid term." EXAMPLES. " 1. If -18 men build a wall in iii days, hov/ many men can do the same in 1P2 days t OPKUATIO.V. Men. Days> Aleiu Here the third term is greater than me Aa 48 : 24 : : 192 first, and coaimon sense teaches the fourth 43 term, or answer m men can do the work in 2 1 102 4ay3, certainly 192 men will do it in less 9G time. In this way it may be determined ^ — - if .a queslioii Jaelooii- to the Rule X)f Three i Id2)\l52^r> answer. Inverse. i £. !f a board be 9 inches broad, 3. How many yards of sarcenet, | how much in length will make a "qrs. wide, will line 9 yards of cloth, square fool i '7i}i,s. IC inches. ofS^r*. wide? Ans, 24 yards. _, Sect. II. 8. SINGLE RULE OF THREE INVERSE. 129 4. Lent a friend 292 dollars for 6 months ; some time afterwards he lent me 806 dollars : how long may I keep it to balance the favor ? Jins. 2 months^ 5 days. 5. A garrison had protision for 8 months, at the rate of 15 ounces to each person per day ; how much must be allowed per day in order that the provision may last 9^ months ? ^ Jlns. 12|2. ounces. 6. A garrison of 1200 has pro- visions for 9 months at the rate of 14 ounces per day, how long will the provisions last at the same al- lowance if the garrison be reinforced by 400 men ? Jinx. 6 * months. 7. Flow must the daily allowance be in order that the pro- visions may last 9 months after the garrison is reinforced ? Jlns. 10^ ouncet. 6. How much land at ^2,50 per acre should be given in exchange for 360 acres at $3,75 per acre .' Jlns. 540 urrc. 9. What sum should be put to in- terest to gain as much in 1 month aa ^127 M'ould gain in 12 months ? Ans. $1521. R 150 SINGLE RULE OF THREE INVERSE. Sect. II. 8. 10. If a man perform a journey in 13 days, when the day is li hours loiijr, in how many will he do it when the day is but 10 hours ? dns. 1 3 days. 11. If a piece of land 40 rods in length, and 4 in breadth make an acre, how wide must it be when it is but 25 rods long ? Ans. G^rods. ["J. There wns a certain building tr.ir^i?d in G months by 120 workmen, but the sjunc being demolished, it is it'tiuircd to be built in two months ; i demuid how many men must be (TU'loved about it ? Ans. 480 men 13. How much in length, that ia 3 inches broad, will make a square foot ? Jns. 48 inches. 14. There is a cistern having 1 pipe which will empty it in 10 hours, how many pipes of the same capacity will empty it in 24 minutes ? Ans. 26 pipes. 15, If a field will feed 6 cows 91 da3's, how long will it feed 21 cows 1 Ans. 26 days. 16. If thequiirtern loaf weigh 4i pounds when wiieat is ^2 per bush- el, ;*liat nin?t it weigh when wheat !> ^],riO the bushel? Ans. Gib. 17. How many yards of baize, 3 quarters wide, will line a cloak Avhich has in it 12 yards of camblet, half yard wide ? Ans. 8 yards. Sect. II. 8 SINGLE RULE OF THREE INA'ERSE. 131 GENERAL RULE For stating all qnestimis n^hethcr direct or inverse. 1. Place that number for the third term, nhich signifies the same kind •of thing, with what is sought, and consider whether the number sought will be greater or less. If greater, place the least of the other terms for the first ; but if less, place the greater for the first, and the remaining one for the second term. Multiply the second and third terms together, divide th^ product by the first, and the quotient will be the answer. EXAMPLES. 1 . If 30 horses plough 12 acres, how many will 10 plough in the same time? OPERATIOIVS. //. H. Aq. Here because the thing sought is a number of 30 : 40 : : 12 acres, we place 12, the given number of acres, 12 for the third term ; and because 40 horses will ■ plough more than 12, we make the lesser num- 30)480(10 Ans. ber, 30, the first term, and the greater number, 40, the second term. 2. If 40 horses be maintained for a certain sum on hay at 5 cents per stone, how many will be maintained, on the same sum, when the price of hay rises to 8 cents per stone ? C. C. H. Here, because a number of horses is sought, 8 : 5 : : 40 we make the given number of horses, 40, the 40 third term, and because fewer will be maintaia- ed for the same money, when the price of hay 8)200(25 Ans. is dearer, we make the greater price 8 cent>i, 16 the first term, and the lesser price, 5 rent5, the — second. 40 40 The first of these examples is Direct, the second Inverse. Every question consists of a supposition and a demand. In the first the supposition is, that 30 horses plough 12 acres, and the de- mand horv many 40 nnll plough ? and the first term of the proposition, 30. is found in the supposition in this and every ofher direct question. In the second, the supposition is that 40 horses are maintained on hay at 5 ceiUs per stone, and the demand, how many az// be maintai^ied on hay at 8 cents? and the first term of the proportion, 8, is found in the demanJ, in this and every other inverse qtiestiou. 3. If a quarter of v.'hcat aObrd GO 4. If in 12 months, 100 doUar^i tenpenny loaves, how many eight- gain 6 dollars interest, what will gain penny loaves may bo obtained from the same sum in 5 months? it ? Ans. 75 loaves. Ans. 240 dollars. 132 SUPPLEMENT TO THE SING. R. OF THREE. Sect. H. 8. SUPPLEMENT TO THE SINGLE RULE OF THREE. qUESTlONS. 1. What is the Single Rule of Three ; or the Rule of Proportion ? 'J. How many Ivinds of Proportion are there ? 3. What is it that the Single Rule of Three Direct teaches ? 4. How can it be known that a question belongs to the Single Rule of Three Direct ? 5. What is understood by more reriuiring more and less requiririg less ? (i. How are questions in the Rule of Three stated ? 7. Having stated the question, how is the answer found in direct pro- portion ? 8. What do you observe of the fjrst and third terms concerning the tiiflcrent denominations, sometimes contained in them ? L^.. Whc-n the second term contains different denominations, what is to be done ? 10. How is it known what denomination the quotient is of? 11. If the quotient or answer be found in an inferior denomination, what io to bo done ? 12. W4^en\t]ie terms are given in Federal Money, how is the operation conducted ? 13. How arc the sums in Federal Money reduced to the same denomi- nation ? 14. V»'hen .sny number of barrels, bales, pieces, &c. are given, what is the method of procedure ? \h. What is it that the Single Rule of Three Inverse teaches ? lb'. How are the questions stated in Inverse Proportion ? 1?. What is understood by more requiring less and less requiring more ? IH, How is the answer found in the Rule of Three Inverse ? 1'?. What is the general Rule for stating .ill questions, whether Director Inverfce ? EXERCISES. ■ l! liiy horsff and saddle are worth 13 guine.is, and my horse be worth •: . 'lie- >o ui;\i;it ;.s; the saddjc',. prav what i.s the value of mv horse ? Ant. T'j'dollurs. Sect. II. 8. SUPPLEMENT TO THE SING. R. OF THREE. 133 2. Howmany yards of matting that 3. Suppose 800 soldiers were IS half a yard wide will cover a room placed m a garrison, and their pro- that is 18 feet wide and 30 feet long ? visions were computed sufficient for Ans. ]20 yards. two months ; how many soldiers must depart that the provision mny serve them 6 months ? Ans. 480. 4. I borrowed 185 quarters of corn when the price was 19s. how much must I pay to iademnily the lender when the price is 17s. 4d. 1 Ans. 2021^. 6. Bought45barrelsof beef at ^3,50 per barrel, among which are 1 6 barrels, whereof 4 are worth no more than 3 of the others ; how much must I pay ? _ Ans. ^143,60. 1 134 SUPPLEMENT TO THE SING. R. OF THREE. Sect. U. 8. 6. A and B depart for the same place and travel the same road ; but A goes 5 days before B at the rate of 20 miles per day ; B follows at the rate of 23 miles per day ; in what time and distance will he overtake A ? •fln*. B will overtake A in 20 days, and travel 500 miles. Here two st;«tements will be necessary, one to ascertain the time, and the other to ascer- tain the distance. Method of assessing town or parish taxes. ' 1. An inventory of the value of all the estates, both real and pcr^nal, and the number of polls for which each person is rateable, must be^^^Ji-en in separate columns. Then to know what must be paid on the dollar, make the total value of the inventory the first term ; the tax to be assessed the second ; and 1 dollar the third, and the quotient will shew the value^n the dollar. NoTit. Tliis method is taken from Mr. Pike's Arithmetic, with this difference, that here the money is reduceU to Federal Curreucy. Sect. II. 8. SUPPLEMENT TO THE SING. R. OF THREE. 135 2. Malce a table, by multiplying the value on the dollar by 1, 2, 3, 4, 5, &c. 3. From the Inventory take the real and personal estates of eac^j man, and find them separately, in the table, which will shew you each man's proportional share of the tax for real and personal estates. If any part of the tax be averaged on the polls, before stating to find the value on the dollar, deduct the sum of the average tax from the whole sum to be assessed ; for which average make a separate column as well as for the real and personal estates. EXAMPLES. Suppose the Geaoral Court should grant a tax of 130000 dollars, of which a certain town is to pay ^3250,72 and of which the polls being 624 are to pay 75 cents each ; the town's inventory is G9568 dollai-s ; what vnW it be on the dollar ; and what is A's tax (as by the inventory) whose estate is as follows, viz. real, 856 dollars ; personal, 103 dollars ; and he has 4 polls ? Pol. Cls. Pol. Dolls. 1. As, 1 : ,75 : : 624 : 468 the average part of the tax to be de- diicted from ^3250,72 and there will remain J^2782,72. Dolh. Dolls. Cts. Dolls. Cts. 2. As G9568 : 2782,72 : : 1 : 4 on the dollar. TABLE. Dolls. Dolls, cts. Dolls. Dolls . cts. Dolls. Dolls 1 is 4 20 is 80 200 is 8 2 — 8 30 — 1 20 300— 12 3 — 12 40 — 1 60 400— 16 4 — 16 50 — 2 00 500 — 20 5 — 20 60 — o 40 600 — 24 6 — 24 70 — 2 80 700 — 28 7 — 28 80 — 3 20 800 — 32 8 — 32 90 — 3 60 ' GOO — 36 9 — 36 100 — 4 00 1000— 40 10 •— 40 Now to find what A's rate will be. His real estate being 856 dollars, I find by the Ta ble that 800 dollars is g532 cts. that 50 — — 2 that 6 — — 24 Therefore the tax for his real estate is 34 24 In the like manner I find the tax for his personal estate to be His 4 polls, at 75 cents each, are 3 "I 4 12 pA 36 I Real. \Dolls. Cts. 34 Personal Dolls. Cts 12 Polls. Dolls. Cts. Total. Doils. Cts. 41 36 1 30 DOUBLE RULE OF THREE. Sect. II. 9. ^ 0. DOUBLE ]U LE OF THREE. THE Double Rule of Throe, sometimes called Compound Proportion, li'aclies by havint; five nnmhei-s i;ivpn to find a sixth, which, if the pro- ))ortion be direci, Uiust bear the same pioportion to tlie fourth and fifth as the thiid does to the fi^^t and second. But if tlie proportion be inverse, tlie sixth number must bear the same proportion to the fourth aud fifth, as the first does to the second and third. RULE. 1. " Slate the question, bv placing the three conditional terms in such or- der (ha( that number whicli is the cause of gain, loss, or action, may possess tlie first place ; that which denotes space of time, or distance of place, the second ; and that which is the .1,'ain, Joss, or action, the tliird." 2. •' Place the other two terms, which move the question, under those of the same name." 3. " Then, if the blank place, or term sought, fall under the (bird jtlare, the propoiiion is direct, tlierefore, multiply the thieo last terms together, for a dividend, and the other two for a di\isor ; then the quotient will be the answer." 4. *' But if the blank f:dl under the first or second place, the prcj»orlion is invcr-so, wherefore muUij)ly the first, second and last terms t0G;cther, for a dividend, and the other two, for adiVisor ; the "quotient will be the answer." EXAMPLES. If 100 dollars gain 6 dollars in 12 nionths, what will 400 dollars gain in C months ? Statement of the uuestion, D. M. D. KJO : 12 : : Terms in the supposition, or conditional tenns. 400 : 8 Terms Xi'hich move the nuestion. Of the three conditional terms, it is evident that 100 dollars put at inte- rest, is that one uhich is the cause of gain ; consequently 100 dollars must be the first term ; and because 12 months is the space of time in which the jiain is made, this must be tl)e second term ; and G dollars which is the gain, liie third term. The other two ierziis must then be arranged under those of tlie same name. Now as the blank fills under the third place, therefore, the question is in direct proportion, and the answer is found by multiplying the three last ternw together for a dividend, and the two first for a divisor. Then, 12|00)192l00( Dolls. leArns. 1 200 Divisor. 1 f -£ 00 Dividend. 2. If JOO dollars gain C dolljirs in 12 montLs, \u wh;,t time will 4P0 dol lars gain 16'!' OPERATION 100 : 12 : : 400 8 8 100 12 3200 6 Sect. II. 0. DOUBLE RULE OF THREE. 137 OPERATION. D. M. D. 100 : 12 : : 6 Here the blank falling under the second term, the 400 16 proportion is indirect. 6 12 Therefore multiply the first, second and last terms together for a dividend, and the other two for a divisor. 2400 div. 192 100 10200 dividend. M. Then 24|00)192|00(8 Answer. 192 3. A Farmer sells £04 dolls, worth 4. If 7 men can reap 84 acres of of grain in 5 years, when it is sold wheat in 12 days, how many men ;it GO cents per bushel, ^vhat is it can reap 100 acres in 3 days ? jiL-r bushel wlien lie sells 1000 dolls, worth in 18 years, if he sell the sawe quantity yearly ? Os. Y. D. CO : 5 : : 204 cts. m. IB : : 1000 : ,816 Am. M. D. A. 7 : 12 : :84 M. 6: : 100 • 20 Alls 5. If a family of 9 persons spend 450 dollars in 5 months, Ifow much ^vould be sufficient to maintain them 8 months, if 5 persons more were added to the family ? . Ans. $1120. 138 SUPPLEMENT TO THE DOUB. R. OF THREE. Sect. II. 9. SUPPLEMENT TO THE DOUBLE RULE OF THREE. QUESTIONS. 1. What is the Double Rule of Three ; or Compound Proportion? 2. How are quc?fions to be stated in the Double Rule of Three ? 3. How is it known after the statement of the question, whether the Proportion be Direct or Inverse ? 4. When the Proportion is Direct, how is the answer to be found ? 6. When the Proportion is Inverse, how is the answer to be found ? EXERCISES. 1. If 6 men build a wall 20 feet long, 6 feet high and 4 feet wide in 16 days, in what time will 24 men build one 200 feet long, 8 feet high and G thick? Ms. 80 days. The solid contents in each piece of wall according to the given dimensions, must be found before stating the question. i.'. If 40/w. at Boston make 3G at Amsterdam, and 90/A. at Amsterdam make 116 at Dantzick, how many lb. at Boston are equal to 260/6. at Dantzick ? Ans. 224^/6. N. B. The answer to this question is found by two statements in the Rule of Three Direct. Sect. II. 9. SUPPLEMENT TO THE DOUB. R. OF THREE. 139 3. If the freight of UCtvt. 2qrx. 6/6. 275. miles cost ^27,78 ; how far mny eOCzi't. ^xjr.i be bhipped for C234,78 ? Jns. 480 miles. 4. An usurer put out 75 dollars, 6. If 7 men can make 81 rods of at interest, and at the end of eight wall in 6 days : in what time will 10 months received for principal and men make 160 rods ? interest, 79 dollars ; I demand at Ans. 7i days. what rale per cent he received in- terest ? Ans. 8 per cent. 140 SUPPLEMENT TO THE DOUB. R. OF THREE. Sect. II. 9. 6. If the freight of 9hhds. of sujar, earh wei^^hio*^ I9rw or, u„ mg .1 C«»^ 100 leagues ? ,4„,^ ^^^2 1 1,. 10%d'. Sect. II. 10. PRACTICE. ^ 10. PRACTICE. 141 " Practice is a contraction of the Rule of Three Direct, when the first term happens to be an unit or one ; it has its name from its daily use among' Merchants and Tradesmen, being an easy and concise method of working finest questions which qccur in trade and business." Proof. By the Single Rule of Three, Compound Multiplication, or by varying the parts. Before any advances are made in this rule, the learner must commit to memory the following TABLES. ALIQUOT, OR EVEN PARTS Parts of a shill.' of a £. d. s. 6 is 1 4 3 2 — J. 3 k i H 1 — i — 3. , _L and jj. 4 8 _Jl_ 5d. is the sum o{4d. 4' id. ■ 7c/. G(/. 4' Id. 8rf. is twice id. 9d. is the sum of 6(/. ^ 3d. \0d. Gd.^-Ad. lid. Gd.3d.4'2d. Pts. s. 10 6 5 4 3 2 1 1 1 1 OF MONEY. of a pound. Practice admits of a great va- d. is £,' riety of cases, the multiplicity — ^ of which serves little else than 8 — i that of confounding the mind of — A the ffcholar ; a different method — i will be pursued here, and the 4 — J. whole comprised, in a few cases, G i such as shall be useful and easy 5 — _!_ for the scholar to bear in his 4 3 10 oi _!_ memory, jL The small number of exam- j_ pies under each case will be _i_ made up in the Supplement ; _*_ this will lead the scholar to a _i_ more particular consideration of 1 them. 9S OPERATIONS. POUNDS, SHILLINGS, PENCE, FARTII. DOLLARS, CENTS, MILLS. When the price of the given quan- RULE, tity is j£l. Is. Id. per pound, yard. Multiply the quantity by the price &c. then will the quantity itself be of one pound, yard, &:c. the product the answer at the supposed price. — will be the answer. Therefore, CASE 1. When the ■price of 1 ynrd, pounds ^c. consists of faj-things onh; ; If it be one farthing, take a fourth part of the quantity ; if a half penn}', take a half ; if three firthings, take a half and a fourth of tiie quantity and add them. Thi-? gives the value in pomre. which mu?t bw it'ducod to pounds. 142 PRACTICE. Sect. II. 10. POUNDS, SHILLINGS, PENCE, FARTH. EXAMPLES. 1. What will 362 yards cost at iJ. per yard ? OPERATION. 2)362 n)\Q] pence. 155. Id. Jlns. Here the quantity stands for the price at one penny per yard, but as two farthings are but half one penny, therefore dividing the quantity by 2, gives the price at half a penny per yard, which must be reduced to shil- lings. 2. What will 354A yards cost at ■^d' per yard ? ^ OPERATIOX. H d. q. 4)354 2 12)80 2 7s. Ad, 2 .ins. 3. What will 263 yards cost at Zq. per yard ? Ans. 1 6s. 5^^. DOLLARS, CENTS, MILLS. 1. What will 362 yards cost at 7 nulls per yard ? OPERATION. 3 G 2 quantity. ,0 7 price. ^2, 6 3 4 Ans. Note. The answers in the different kinds of money will not always com- pare, because in the redaction of the price, a small fraction is often lost or sained. 2. What will 3541 yards cost at 3 mills per yard ? operation; 3 5 4 ,5 quantity. ,0 3 price. gl ,0 6 3 5 3. What will 263 yards cost at 1 cent per yard ? Ans. ^2,63. 4. What will 816 yards cost at \q. per yard? Ans. 17s. 4. What will 816 yards cost at 3 mills per yard ? Ans. $2,448. Sect. II. 10. PRACTICE. 143 POUNDS, SHILLIN-GS, PENCE, FARTH. 5. What will 97 yards cost at 3q. per yard ? Ans. 6s. O^d. 6. What will 126 yards cost at id. per yard ? Jins. 5s. 3d. CASE 2. When the price of lib. lyd. ^-c. consists of pence, or of pence and farthings; if it be an even part of a shilling, find the value of the given quantity at Is. per yard, (the quan- tity itself expresses the price at Is. per yard ; if there are quarters, iait write for i 3d. for i 6d. for | 9d.) and divide by that even part which the price is of 1 shilling. If the price be not an aliquot or even part of one shilling, it raust be divided into two or more aliquot parts ; cal- culate for these separately, and add the values ; the answer will be ob- tained in shillings, which must be reducct! to pounds. DOLLARS, CENTS, MILLS. 5. What will 97 yards cost at 1 cent per yard ? Ans. ,97c<*. 6. What will 126 yards cost at 7 mills per yard ? Am. $0,882. 144 PRACTICE. Sect. IT. 10 rov.VDS, SHILLINGS, PENCE, FARTH. DOLLARS, CENTS, WILLS. EXAMPLES. 7. What will ■476 3'aril9 come to at 1 . What will 476 yards cost at 7^. 10 cents 4 mills per yard ? per yard ? OPERATION. OPERATION. 476 S. ,104 Cd. 1 476 Price at Is. per yard. llrf. 1 238 Price at Gd. per yard. 1904 59 Gd. price I !^d. per yard. 4760 2|0)29(7 Gd. price at 7! per yd. ^49,504 Ans. , i l]4 lis. Gd. Ms. FROOF. PROOF. 1. By the Rule of Three. F. £. s. d. Y. A9 47G : 14 17 G : : 1 20 297 12 47G)3570(7(/. 3332 238 4 )952(2^r. 952 2. By Compound Multipli- cation. £. s.' d. 7^ price of 1 yard. 10 6 3 price of 10 yards. 10 3 2 6 price of 100 yards. 4 12 10 price of 400 yards. 2 3 9 price of 70 yards. 3 9 price of G yards. JL'l'f 17 C price (f 4." 6 yards. cts. m. D. cts. m. yds. ,10 4)4 9 50 4(476 4 1 6 7 9 7 2 8 6 2 4 6 2 4 Sect. H. 10. PRACTICE. 14& POUNDS, SHILLINGS, i^ENCE, FARTH. What will 176 yards cost at 9iJ. per yard ? OPERATION. s. [ 6ii. i I 176 value at Is. per yard. I 3.'^ I j 88 value at 6d. per yard. i ii. I of 44 value at 3rf. per yard. 7 4d. val. at -kd. per yd. 2|0)13!9 4c/.— atgiJ. per yd. £6 I9s. 4d. Ans. FllOOF. I i 3. What will 568^ yards cost at 7<^. per yard I Ans. £,16 lis. S^d. DOLLABS, CENTS, MILLS. 8. What will 176 yards cost at 13 cents, 2 mills, per yard ? Ans. g23,232. 9. What will 568^ yards cost at 9 cents 7 mills per yard ? Ans. |55,12. 146 PRACTICE. Skct. II. 10. POUNDS, SHILLINGS, PENCE, FARTH. 4. What will 6862 yards come to at 2|d. per yard ? Jlns. £7 2s. 101(7. DOLLARS, CENTS, MILLS. 10. What will 685^- yards come to at 3 cents 5 mills per yard ? ^«j, $24,001. 5. What will 649} yards cost at iOc/. per yard ? Ans. £21 1$. O^d. 6. What will 6833 yards cost at 8ic/. per yard ? Ans. £23 10s. 03^. 11. What will 6491 yards cost at 13 cents 9 mills per yard 1 Ans. $90,245. 12. What will 683f yards cost at 1 1 cents 7 mills per yard ? Ms. 79,998 Sect. II. 10. PRACTICE 147 POUNDS, SHILLINGS, FENCE, FARTH. CASE 3. If the price of lib. lyd. ^c. be sJiillings and pence and an even part of £,1, Divide the value of the given quantity at £l per yard by that even part, which the price is of i;l. The quotient will be the answer. EXAMPLES. 1. What will 7191 yards cost at Is. 4c?. per yard ? OPERATION. £. s. I Is. 4d. I yij I 719 10 price at £1 [per yd. 143 18 price at 4s. [per yd. illns. 47 19 4d. at \s. 4d. [per yd. Here for the sake of ease in the operation, because 5x3=15, there- fore I divide the price at one pound \H'.r yard by 5, and that quotient by 3 wjiich gives the answer. DOLLARS, CENTS, HILLS. Is. •. What will 648 yards cost at Qd. per yard ? Ans. £54. 3. What will 1671 yards cost at 3s. 4d. per yard ? .^ns.£27 18s. 4d. 13. What will 7191 yards cost at 22 cents, 3 mills per yard ? Ans. $160,448. 14. What will 648 yards cost at 27 cents and 8 mills per yard '? Ans. g' 180, 144. 15. What will 1671 yards cost at :5 cents, 6 mills per yard ? Ans. $93,13. 148 PRACTICE. Sect. II. 10. PoLxns, sniLr.iNGS, rcNcE, faktu. 4. What will 6871 yards coa-t at 6j. per yard ? Jlns. £lll Ms.' 6d. CASE 4. When the price of \yd. 4'C. is shil- linai), ur shillings, pence and farthings, iLid noi an even part of £,1. Multi- ply tlie value of the quanlity at Is. )>€r yard by the number of shillings ; f<»r the pence and tartiiint^s, take !» trts, as in case 2, the resuUs added will i;ive the answer, which must be rediued to pounds. //" (he price be shillings only, and an even nuinber ; multiply by half the price or even number of shillings for one yard, double the unit figure of the product for shiUings, the remain- ing figure will be pounds. Note. When the quantity con- triins a fraction, work for the inte- gers, and for the fraction take pro- portional jfarts of the rate. EXAMPLES. 1. What will 167^ yards cost at 1 7 J. ijd. per yard ? OPEKATION. s. I i^d. I i I 167 17 1169 167 2839.pncc at Ms. per yd. 83 6 — at 6d. per yard. 8 9 price of 3. yard. 2|0)5>93|] 3d. ■Int. £i4ii I l.v 3d. DOLLARS, CENTS, MILLS. 16. What will 687i yards cost at 83 cents, 3 mills per yard ? Ans. ^572,687. 17. What will 167| yards cost at $2,916? ^7j*. j^4 88,43. Sect. il. 10. PRACTICE. 149 POi-NDS, SHILLINGS, PENCE, FARTH. 2. What will 5482 yards cost iit 1 2s. 4id. per yard ? Ans. £3391 IPs. 9d. 3. What vUI G14 yards cost at 16s. per yard ? OPERATION. G14 8 half the price. 4012 double the first figure £491 4s. Ans. \for shill. 4. What will 176 yards cost at 12s. per yard 1 Am. £105 12s. DOLLARS, CENTS, MILLS. 18. What will 5482 yards cost at ^2,063 per yard ? Ans. gl 1309,366 19. What will 614 yards cost »t $2,667 per yard ? Ans. $1637,538. 20. What wiU 176 yards cost ai $2 per yard ? Ai\s. '^2>b-i. 5. What will 36 yards cost at 7s- 6d. per yard? JJns. £13 lOs 21. What will 36 yarda cost ai $1,25 per yard? Ans. %\b. 150 PRACTICE. Sect. II. 10. TOVXDS, SHILLINGS, PENCE, FARTH. CASE 5. jyjien the price of hjd. lib. <$•< . is pounds, shillings and pence ; multiply the quantity by the pounds, and il" the shiUings and pence be an even part of a pound, divide the given quantity by that even part, and add the quotient to the product for the answer ; but if they are not an even part of £,1, take parts of parts and add them together. Or, you may reduce the pound in the price of 1 yard, &c. to shilhngs. and proceed as in the case before. EXAMPLES. 1. What will 59 yards cost, at £6 7s. 6d. per yp.rd ? OPERATrON. £ 6s. is ^ of £1. 59 value of £,1 per yd. 6 DOLLARS, CE.NTS, MILLS 354 — at £6" per yd. 2s. erf.isiof 5. 14 15s. oJ 6s. per yd. 7 7 6d. at 2s. 6d. [per yd. Ans. £376 2s. Gd. at £6 [7s. 9d. 3. What will 163 yards cost, at £2 8s. per yard? ^ns. £391 4j. 22. What will 59 yards cost, at ^21,25 per yard ? OPERATION. D.C. 21,25 69 191 25 1062 5 gl253 75 .^TW. 23. What will 163 yards cost, at 3 per yard? ^ns. $1304. , Sect. II. 10. PRACTICE. 151 ro'JNDS, SHILLINGS, PENCE, FARTH. J. What will 76 yards cost at £3 '2s. Id. per yard ? OPERATION. S. 6d. is i of Is. 78 value at Is. per yd. (j2=shillings in £3 2s. 1 52 value at 2s. per yd. 456 — at 60s. per yd. 1 d. i? i of 6rf. 38 — at 6d. per yard. 6 4d. — at Id. per yd. 2|0)475|6 Ans. £237 16s. 4d. 4. What is the value of 84 yards at £2 1 Is. per yard ? ^7iy.£226 16s. DOLLARS, CENTS, MILLS. 24. What will 76 yards cost at ^10,43 per yard? Ane. $792,68. 25. What is the value of 84 yards at ^9 per yard ? Ans. g766. 152 SLTPLEMBNT TO PRACTICE. Sect. H. 10. SUPPLEMENT TO PRACTICE, QUESTIONS. 1. What is Practice ? 2. Why ia it so called ? 3. When the price of 1 ^'anl, &c. is farthings, how is the value of any given quantity found at the same rate ? 4. When the price consists of pence and farthings, and is an even part of Is. how is the value of any i^iven quantity found? fi. When the price is pence and farthings and not an even part of Is. what is the method of procedure ? 6. When the price consists of shillings, pence and farthings, how is Ihr- value of any given quaniity found ? 7. When the price contains shillings and pence and an even part of Xi, how is the operation to be conducted? 8. When the price consists of shillings only, and an even number, wh^t is the most direct way to find the value of any given quantity ? 9. When the quantity contains fraction;, as \, }, £, &,c. how are they b> be treated ? 10. When the price consists of pounds, and lower denominations, how ii the value of any given quantity found ? 11. When the prices are given in dollars, cents, and mills, how is the value of any given quantity found in Federal Money ? 12. What is the method of proof? 13. How are operations in Federal Money proved ? EXERCISES /JV PRACTICE. In the following exercises the attention of the scholar must be excited first to consider to which of the preceding cases each question is to be re- ferred. That being ascertained, he will proceed in the operation accord- ing to the instruction there given. 1. What will 7453 yards cost at lit/, per yard ? Ans. £31 3s. l^d. Under which of the preceding cases does this question properly belong ? Wiiat must be done with the fraction (| of a yard) in the quantity ? i^ECT. II. 10. SUPPLEMENT TO PRACTICE. 133 \2. What will 964 yards cost at Is. Bd. per yard ? Ans. £80 65, Bd. OFBRATION FROOF. 3. What will 354^ yards cost, at 4. What will 316 yards cost, at ^d. per yard ? Ms. Is. 4^d. ^d. per yard ? Am. \9s. 9d. 5. Whjt will 667i yards cost, at ^id. per yird ? Ans. £3 10s. llirf. 6. What will 9131 yards cost, at Gd. per yard ? Ans. £22 16s. &d. U 154 SUPPLEMENT TO PRACTICE. Sect. II. 10. 7. What will 912i yards cost, at 8. What will 76 yards cost, at i?(/ 9d. per yard ? per yard ? Ans. £34 4s. Ud. Ans. 12s. Qd i 9. What will 845 yards cost, at 8s. 10. What will 91 yards come to per yard 1 at 16s. per yard ? Jlns. £338. Ans. £72 16s. 11. What will 1561 yards come to, 12. What will 96 yari cost at at 6s. Ad. per yard? 10s. \\d. per yard ? Ans. £49 lis. 2d. Ans. £.8 12s. Sect. II. 10. SUPPLEMENT TO PRACTICE. 155 13. What will 671 yards cost, at 14. What will 843 yards cost, at ! 12*. 2d. per yard ? Jlns. £41 Is. 3d. 6». 8d. per yard ? Ans. £281. ' 15. What will 75 yards cost, at 16. What will 59 yards come to, £3 3s. 4d. per yard ? at £6 7s. 6d. per yard ? Ans. £237 10s. Ans. £376 2j. 6d. 17. What m\l 59| yards come to, 18. What will 68 yards cost, at at £3 6s. 8c/. per yard ? £4 6s. per yard ? Ans. £199 3s. 4(/. Ans. £292 8s. ISe SUPPLExAlENT TO PRACTICE. Sect. II. 10. N. B. The following questions are left without any aaswers, that the Scholar may operate and prove each question. 19. What will 1 1 yards of flannel, at 2s. 6d. per yard come to ? OPERATION. PROOF 20. What will 131b. of cotton cost at 3s. 4c?. per lb. ? 21. What will 183 yards of ribbon come to at Sd. per yard ? THE SCHOLARS ARITHMETIC. SECTION III. lULES OCCASIONALLY USEFUL TO MEN IN PARTICULAR CALLINGS AND PURSUITS OF LIFE. ♦ 1. INVOLUTION. Involution, or the raising of powers, is the multiplying of any giren number into itself continually, a certain number of times. The quantities in this way produced, are called powers of the given number. Thus, 4x4= 16 is the second power or square of 4. =4' 4x4x4= 64 is the 3d power, or cube of 4. =4' 4X4X4X4=266 is the 4th power or biquadrate of 4. =4* The given number, (4) is called the first power ; and the small figure, which points out the order of the power, is called the Index or the Ex- ponent. J 2. EVOLUTION. Evolution, or the extraction of roots, is the operation by which we find any root of any given number. The root is a number whose continual inuUiplication into itself pro- duces the power, and is denominated the square, cube, biquadrate, or 2d, 3d, 4th, root, &c. accordinely as it i«, %vheu raised to the 2d, 3d, 4th, &c. power, equal to that power. Thus, 4 is the square root of 16, because 4X4=16. 4 also is the cube root of 64, because 4x4x4=64 ; and 3 is the square root of 9, and 12 is the square root of 144, auJ the cube root of 17-28. because 12xl2xl"=«IT'J8, and no ou. 158 EXTRACTION OF THE SQUARE ROOT. Sect. III. 3, To every number there is a root, although there are numbers, the pr« eise roots of wliich can never be obtained. But by the help of decimals, ue can approximate towards thofe root*, to any necessar} degree of exjictness. Such roots are called Surd Roots, in distinction from those perfectly accu- rate, which are called Rational Roots. 'i'he square root is denoted by this character •%/ placed before the power ; the other roots by the same character, with the index of the root placeil over it. Thus the square root of 16 is expressed V* 1*^5 and the cube root of 27 is a/ 27, &c. When the power is expressed by several numbers, with the sign + or— between them, a line is drawn from the top of the sign over all the parts of it; thus tlie second power of 21 — 5 is \/ 21 — 5, and the 3d power of 66-f8 is V5G-f#, &c. The second, third, fourth and fifth powers of the nine digits maybe seen in the Ibllowing T.WLE. Boots ... - 1 or 1st Powers 1 ^ 3 4 5 1 6 1 7 8 9 Sfiuarcs - - - 1 or i-'d Towers 1 4 i'l 16 25 1 36 1 49 64 bl Cubes - - - - 1 or 8d I'owers «l 27 1 64 125 j 216 ! 343 ■512 729 Bi<]iiudratcs 1 or4tli Powers 1 16 81 25(> 625 1 ]2y6 1 2401 4096 6561 Sursolids - - 1 orotli Powers 1 ■S2 243 1024 3125 1 7776 j 16807 3276S 5yu4a ^ 3. EXTRACTION OF THE SQUARE ROOT, f A To extract the square root of any number, is to find another numbej' which multiplied by or into itself, would produce the given number ; and after the root is fouud, such a multiphcaiion is a proof of the work. RULE. 1. " Distinguish the given number into periods of two figures each, by putting a point over the ])lace of units, another over the jdace of hunchcds, and so on, which points shew the number of figures the root will consist ofi 2. " Find the greatest square number in the first, or left hand period^ place the root of it at(the right hand of the given number, (after the m;in- ner of a quotient in division) for the first figure of the root, and the squiire number, under the period, and subtract it therefrom, and to the remainder i bring down the next period for a dividend. 3. " Place the double of the root, already found, on the left hand of the dividend for a divisor. 4. " Seek how often the divisor is contained in the dividend, (except the right hand ligure) and place the answer in the root for the second tigurc of it, and likewise on the right hand of the divisor ; multijdy the divisor with the fi;.'ure last annexed liy,ilie .figure last })lared in the root, and subtract the j. 1. form. .As if :•- m.in slujtild have 625 yards of carpeting 1 yard v.ide, if lie extract U>e t<^ufire root oi that rm!nl)er (625) lie tvill then b^ive flie IcngUi o( uiic si^k- iA u sijUHi-e room, llie floor uf ^vLicb,62o yards will be just b;iflicieitt lo «;over. To procet-d (hen to the d:^aion?tralion. llx>v!!LE Si. .S(ippO!?tv.g a iiKtn hius G25 yards of carpeting, 1 3'ard wide, what will be Hie lenqlli oi* one side ol" a square room, the floor of whit: h his c;!!|)eting wili cover. The iirst step is to j»oint off (he number into pointed off as the rule directs, we find we have two periods, con- soquently the root will consist of two figures. Tlie greatest square number in the left hand period (G) is 4, of which two is the root, therefore, 2 is the first figure of the root, aj)d as it is certain we have one figure more ' to fini' in the root, wc may for the pres^ent supply the place of tl.at figure by a cypher (2t.') then 20 will express the just value of that pnrt of the root now obtained. But it mui^t be remembered, that a root is the side of a square of equal sides. Let us then form a square. A, Fi^. I. each side of which Bhall be supposed ^0 yards. Now the side a b of thi?) square or either of the sides, shcv.': in i« - ality 400, as might be seen also by filling the pl;;ces to the right hand with f \\- phers, then 1 subtracted from 6 and to the remainder (2) the next period (2 ) beiiiji; brought down, it is plain, the sum G25 has been dimini-hed by the u- di'.ction of sOO, a number equal to tiie ^uperficial contents q'l the square A. IJcrice Fit!. I. exiiibils the exact progress of the operation. By tie operation 400 yards of the carpeting have ])een dispo.sed of, and by tiie iigure is seen the disposition made of them. Now the square A is to be enLrged by the addition of the 225 yards which remaiTi, and this addition must be so made that the Iigure yt the fcume time shall conlir.ue to be a complete and perfect square. If the ad- dition be made to one side only, the figure would lose its square form, it nmst be made to two sides ; for this reason the rule directs, *' place the eouble of the root ahetidy found on the left hand of the dividend fo'r u di- visor." The double of ll.c' vcv)l i- j;ist equ;-d to two j;ides b c and c d o( i'..c square, A, as may be ■^Hksu by whA fidJows. Sect. III. 3. EXTRACTION OF THE SQUARE ROOT. OPERATION continued. 161 I 625(25 4 45)225 225 The double of tbe root is 4, which placed for a divisor in place of tens {fur it must be remem- bered that the next figure m the root is lo be placed before it) is in reality 40. equal (o the sides b c (20) and c d (20) of the square A. Fig. II. /S 20 6 c 6 D 5 25 100 c A C 20 20 20 5 400 100 20 b 6 h =400 yards. =100 — =100 — = 25 — Again, by l]je>ule, " seek how often tTie divisor is contained in the dividend (except the right hand figure) and place the an- swer in the root, for tlic»second figure of it, and on the right hand of the divisor." Now if the sides b c and c d o{ the square A Fig. II. is the length tOAvhich the remainder 226 yds. are to be added, and the divisor (4 tens) is the sum of these two sides, it is then evident that 225 divided by the length of the two sidcs,that is by the divisor(4 tens) will give the breadth of this new addition of the 225 yards to the sides b c and c d o£ the square A. But we arc directed to " ex- cept the right hand figure," and also to '''^ place the quotient figure on tJic right hand of the divisor ;*' the reason of which is that t'.ie addition, C ef and C g h to the sides b c and c d of the square, A, do not leave the figure a completp square, but there is a deficiency D, at the corner. — Therefore in dividi. ;. the right hand figure is excepted, to leave something of the dividend, f^r this deficiency ; and as the deficiency D, is limited by the additions C cf :ind C g h, and as the quotient figure (5) is the width of these additic.)?, consequently equal to one side of the square D ; therefore the quotient figure (5) placed to the right hand of the divisor (4 tens) and multiphed into itself, gives the contents of the square D, and the 4 tcns=to the sum of the sides, be and c d of the addition of Ce/and Cg h, multiplied by the quotient figure (5) the width of tliose additions give the contents C cy'and C g h, which together subtracted from the dividend, and there being no re- mainder, shew that the 225 yards are disposed in these new additions C cf, Cg h, and D, and the figure is seen to be continued a complete square. Consequently, Fig. if. shews the dimensions of a square room, 25 yardi^ on a side, the floor of which G25 jards of carpeting, 1 j-ard wide v/ill be sufTicient to cover. The Proof is seen by adding together the different parts of the figure. Such are the principles on which the operation cf extracting the square root i= ffroLindcd. W Proof 625 yards. 162 EXTRACTION OF THE SQUARE ROOT. Sect. III. 8» 3. What is the square root of 4. What is the square root of 10342656? Ms. 3216. 43264? jJns. 208. 5. Whil is t^ViQaA; roc* of 964,5192360241 ? Ans. 31,05671. V ^acT. III. 3. EXTRACTION OF THE SQUARE ROOT. ' 163 6. What is the square root of 7. What is the square root of ^98001? Ms. 999. 234,09? Ans. 15,3. 8. What is the f^dve root of 1030892198,4001 i Am. 32107,51. 164 tiUPFLEMENT TO THE SqUARE ROOT. Sect. III. 3, SUPPLEMENT TO THE SQUARE ROOT, QUESTIONS. 1. What is to be understood by a root ? A power ? The second, third, and fourth powers ? 2. V.hat is the Index, or Exponent? 3. What is it to extract the Square Root ? A. Why is the given sum pointed into periods of two figures each ? b. Jn the operation, having found the first figure in the root, why do wc subtract the square number, that is, the square of that %ure from the period in which it was talccn ? 6. Why do we double the root for a divisor ? 7. In dividing why do we except the right hand figure of the dividend ? S. Why do we place the quotient figure in the root, and also to the right h;md of the divisor ? # 9. If there be decimals in the given number how must it be pointed ? 10. How is the operation of extracting the Square Root proved ? EXERCISES W THE SQUARE ROOT. 1 . A Clergyman's glebe consists of three fields : the first contains 5 acres, Qr. 12/). the second, 2 acres, 2r. 15p. the third, 1 acre, Ir. 14p. in exchange for wliich the heritors agree to give him a square field equal to all the three. Sought the side of the square ? Ans. o9polfs. 2. A general has an army of 409G men ; how many must he place in rank and file to form them into a square ? Ans. 64 Sect. III. 3. SUPPLEMENT TO THE SQUARE ROOT. I65 3. There is a cirde whose diameter is 4 inches, what is the diameter ©f a circle 4 times as large ? Ans. 8 inches. Note. Square the given diameter, multiply this square by the given proportion, and the square root of the product will be the diameter required. Do the same in all similar cases. If the circle of the required diameter were to be less than the circle of the given diameter, by a certain proportion, then the square of the given diameter must have been divided by that pro- portion. 4. There are two circular ponds in a gentleman's pleasure ground ; the diameter of the less is lOO feet, and the greiter is three times as large. — What is its diameter ? .^ns. 173,2 -f 5. If the diameter of a circle be 12 inches, what will be the diameter of another circle half so large ? Ans. B,4Q-{-inches. X9 SUPPLEMENT TO THE SQUARE ROOT. Sect. III. 3 6. A wall is 36 feet high, and a ditch before it is 27 feet wide ; what is the length of a ladder, that will reach to the top of the wall from the oppo- site side of the ditch ? Jlns. 45 feet. Note. A Figure of three sides, like that formed by the wall, the ditch and the ladder, is called a right an- gled triangle, of which the square of the hypothenuse, or slanting side, {the ladder) is equal to the sum of the squares of the two other sides, that is, the height of the wall and the width of the ditch. 7. A line of 36 yards will exactly reach from the fi) of a fort to the opposite kink of a river, known to be 24 yards broad ; the height of the wall IS required? Ms. 20,^3+ yards. Sect. III. 4. EXTRACTION OF THE CUBE ROOT. 167 8. Glasgow is 44 miles west from Edinbnrgh ; Peebles is exactty south from Edinburgh, and 49 miles in a straight line from Glasgow ; what is the distance between Edinburgh and Peebles ? .ins. 2\y5-{-miUi. ^ 4. EXTRACTION OF THE CUBE ROOT. To extract the Cube Root of any number is to find another number, which multiplied into its square shall produce the given number. RULE. 1. " Separate the given number into periods of three figures each, by piitlinga point over the unit figure, and every third figure beyond the place of units. 2. " Find the greatest cube in the left hand period, and put its root rn tlie quoliont. 3. " Subtract the cube thus found, from the said period, and to the re- mainder brin,^ down the next period, and call this the dividend. 4. " Multiply the square of the quotient by 300, calling it the triple square, and the quotient by 30, calling it the triple quotient, and the sum of these call the divisor. 5. " Seek how often the divisor may be had in the dividend, and place the result in the quotient. G. " Multiply the tiiple square by the last quotient figure, and write tlie product under the dividend ; multiply the square of the last quotient figure by tlie triple quotient, and place this product under the last ; under all, set the cube of the last quotient figure, and call their sum the subtrahend. 7. " Subtract the subtrahend from the dividend, and to the remainder iiring down the next period for a new dividend, with which proceed as be- fore, and so on till the whole be finished. NoTK. "The same rule must be observed for continuing the operation, ^nd pointing for decimals, as in evident from what follows : Side A B=20 \ 2 quotient figure. Side A F=20 ( r,, r 2 > oj the jace a. Superficial coTiients'=^'iOO ) 4 the square of 5. 3 300 The triple square l'zOO=the The triple square 1200=the superficial .'i}rj)prjicial contents of the faces contents of the faces a, c, and b. a, '-, andb. Here the quotient figure 2 is properly, 1'he two sides A B fc A F txco tens, for there is another figure to fol- of the fdcc a, multiplied into low it in the root, and the square of 2,_ each other, give the superfi- standing as units, i.s 4, but its true value l^cial content of c, and as the is 20 (^the side A B) of which the square faces, a, c, and 6, are all equal, is 400, we therefore lose two cyphers, and therefi^rc the content of iUcc these two cyphers are annexed to the fig- 0, multiplied by 3, will give ure 3 — Hence it appears that we square the coatenls of*;, c, and h. the quotient with a view to find the super- ficial content of the face or square a, we multiply the square of the quovient by 3, to find the suparRcial contents of tiie three squares, c, c, and b, and two cyphers are annexed to the 3, because in the square of the quotient tr^o cyphers were lost, the quotient requiring a cypher before it in order to express its true v.«» jc, which would throw the quotient ('!) into the place of tens, whereas now it stands in the place of units. Now wlu n tlie additions are made to the squares a, c, and h, there will f'videiitiy bo a deficiency, along the v/liole length of the sides of the squares, Isctvveen each of the additions, which must be supplied before the figure can hf. a complete cube. These dcncicncics will be 3, as may be seen, It'i^r. ll. nun. X 170 EXTRACTION OF THE CUBE ROOT. Sect. III. 4. Therefore it is, that we are directed, " mvUipIy the quotient by 30, calling it the triple quotient.^ ^ The triple quotient is the sum of the three hues, or sides, agninst which «rc the deGcitiicii;s n n n, all which meet at a point, nigh the centre of the fjgure. This is evident froai what follov.'s. The deficiencies are three in number, they are the ^vhole length of the sides, 2 quotient. the length of each .'ide u 20 feet, oO Therefore 'i(i — 3 Triple quotient 60=^o the length of 3 sides ■where are dejiciencics to be jilled. Triple quotient 60 equal tlw. IcngtJi of 3 sides, fyc. Here as before, the quotient lacks a cypher to the ri^ht hand, to exhibit its true value ; the quotient itsnlf is the length of one of the sides, whore are the deficiencies ; it is midtiplied bj' 3 becaujo there are 3 deficiencies, and a cypher is annexed to the 3 because it has I'cen omitted in the quotient, which gives the same product, as if the true value of the quotient 20, had been multiplied by 3 alone. 1200 the triple square. We now have ^ 60 the triple quotient. The sum of which, 1260 is the divisor, equal the number of square feet contained in all the points of the figure or pile, to which the addition of the 5824 feet is to be matle. OPERATION 138 continued. 1(24 the root. Divis. 1260)o82'l the dividend. "4800" 960 64 A 20 F 1200 triple square. 4 last quotient Jtgure. This Figure in the root (4) shews tl)e depth of the addition on every point where it is to be made to the pile or figure, reproeented, Fig. I. Frc, II. exhibits the additions made to the squares a c h, by which they are covered or raised by a depth of 4 feet. The next step in the operation is to find a subtrahend, which subtrahend is j the number of solid feet contained in all ♦ the additions to the cube, by the last figure 4. Therefore the rule directs, multiply the triple square by the last quotient Jigure. The triple square it must be remem- bered, is the superficial contents of the faces a c and b, which multiplied by 4, the depth now added to these faces, or squares, gives the number of solid feet cniitained in the additions by the last quotient figure 4. 4800 feet, equal the addition trade to the squares, or faces, r. c, h, of Fig. I a depth of 'i feet on each. EXTRACTION OF THE CUBE ROOT. 171 o 60 16 F4n triple quotient, square of the last quotient Then, " Multiply the square of the last quO' tient figure by the triple quotient." This is to fill the deficiencies, n n n, Fig. II. Now these deficiencies are hmited in length by the length of the sides (20) and the triple quotient is the sum of the length of the deficiencies. They arc hmited in width by the last quotient figure (4) the square of which gives the area or super- ficial contents atone end, which multiplied into their length, or the triple quotient, which i« the same thing, gives the contents of tliose ad- ditions 4n4, 4n, 4», Fig. III. 3G0 60 9G0 feet disposed in the deficiencies, between the additions to the squares a c b. Fig. III. exhibits these deficiencies, supplied 47)4, 4«, 4n, and discovers another deficiency u-here these approach together, of a corner wanting to make the figure a complete cube. Lastly, " Cube the last quotient figure.'''' This is done to fill the deficiency, tig. III. left at one corner, in filling up the other deficiencies, n n n. This corner is limited by those defi- ciencies on every side, which were 4 feet in breadth, consequently the square of 4 will be the solid content of the corner which in Fig. IV. e e e is seen filled. Now the sum of these additions make the subtrahend, which subtract from the dividend and the work is done. 16 4 64 feet is the cornj- » « c, n'here the additions n n n, approach together. Figure IF. Shews the pile which 13824 solid blocks of one foot each, would make when laid together. Tlie root (24) shews the length of aside. Fig. I. shews tlie pile which would be i;>rnied by 8000 of Iho.-e blocks, fir?t hud together; Fig. II. F'ig. III. and Fig. T-^ shew the changes which the pile passes through in the addition of the re.Tiaining 5824 blocks or feet. Frnrf. By adding the contents of the first figure, and the udditi^ns ex- ..bibitcd in the other fi.'rares tos-othei'. 17J EXTRACTION OF THE CUBE ROOT. Stci. III. 4. c- Feet. 8000 Contents of Fig. I. 4800 addition to the faces or square a, c, and 6, Fig. U. 960 addition to fill the deficiencies n, n, n, Fig. ///. 64 addition at the corner, c, c, c, FjV. /r". where the additio which fill the deficiencies n, n, n, approach together 4 13824 Number of blocks or solid feet, all which are now disposed in Fig. IV. forming a pile or solid body of timber, 24 feei on a side. Such is the demonstration of the reason ai»d natureof the various steps in the operation of extracting the cube root. Proper views of the figures, and of those steps in the operation illustrated by them, will not generally be acquired without some diligence or attention. Scholars more especially will meet with difficulty. For their assistance, small blocks might be formed of wood in imitation of the Figures, with their parts in diflfcrent pieces. By the help of these. Masters, in most instances, would be able to lead their pupils into the right conceptions of those views, which are here given of the nature of this operation. 3. What is the cube root of 2102457G ? Ans. 276. Sect. III. 4. EXTRACTION OF THE CUBE ROOT. 173 4. Wh?t is the cube root of 253395799552 ? Ms. 6328. 174 EXTRACTION OF THE CUBE ROOT. Sect. III. 4. I 5. What 1*9 the cube root of 84,604519 ? jj^,. 4 3^. 6. What is the cnbe root of 2 ? JIns. 1,25+ Sect. III. 4. SUPPLEMENT TO THE CUBE ROOT. nr, SUPPLEMENT TO THE CUBE ROOT. QUESTIONS. 1. What is a Cube? 2. What is understood by tlie cube root ? •3. What is it to extract the cube root ? 4. In the operation, having found the first figure of the root, why is the cube of it subtracted from the period in which it was taken ? 6. Why is the square of the quotient muUipUed by 300 ? C. Why is the quotient mulliphed by 30 ? 7. Why do we add the triple square and triple quotient together, and tlie sum of them call the divisor ? 8. To find the subtrahend, why do we multiply the triple square by the last quotient figure ? the square of the last quotient figure by the triple quotient ? Why do we cube the quotient figure ? Why do these sums added, make the subtrahend ? 9. How is the operation proved ? EXERCISES IjY THE CUBE ROOT. 1 . If a bullet 6 inches in diameter weigh 321b. what will a bullet of the same metal weigh whose diameter is 3 inches ? Jltis. 41b. Note. " The solid contents of similar fig- ures are in proportion to each other, as the cubes of their similar sides, or diametei-s." 176 SUPPLEMENT TO THE CUBE ROOT. Sect. Ill 4 2. What is the fide*of a cubical mound, equal to one 288 feet long, 216 broad and 48 feet high ? Ans. 144 feet. ". There is a cubical vessel \vhose side is two feet ; I demand the side of a vessel which shall contain three times as much ? Ans. 2 feet ten inches and ^nearly. Note. Cube the given side, multiply it by the given pro- portion, and the cube root of the product will be the siJe souffht. i Sect. III. 5. FELLOWSHIP. 177 * 5. FELLOWSHIP. FELLOWSHIP is a rule by which merchants and others, trading in partnership, compute their particular shares of tUe gain or loss, in propor- tion to their stock and the time of it3 continuance in trade. It is of two kinds, Single and Double. SINGLE FELLOWSHIP, Is when the stocks are employed equal times. ^ RULE. As the whole sum of the stock is to the whole gain or loss, so is each man's particular stock to his particular share of the gain or loss. Pkoof. Add all the shares of the gain or los.^ together ; and if the work be right, the sum will be equal to the whole gain or loss. EXAMPLES. 1. Two merchants, A and B, make a joint stock of 200 dollars ; A puts in 75 dollars, and B 125 dollars ; they trade and gain 50 dollars ; what is each man's share of the gain ? OPERATION. Dolls. Dolls. Dolls, As 200 : 60 : : 75 As 200 : 50 : : 125 75 125 250 250 350 100 D. cts. 50 200)3750(18,75 A's share. , D. cts. 200 200)6250(3 1 ,25 B's share. 600 1750 1600 250 200 1500 1400 600 400 1000 18,75 A's share. 1000 31,25 B's share. 1000 1000 50,00 Proof. 2. Divide the number^60 into 4 such parts, wliich shall be to each other 3, 4, 5, and 6. 60 \ ^ll\ Answer. 120} *'oO Proof. 178 SINGLE FELLOWSHIP. Sect. III. t. 3. A man died leaving 3 sons, to whom he bequeathed his estate in the following manner, viz. to the eldest he gave 184 dollars, to the second 155 dollars, and to the third 96 dollars ; but when his debts were paid, there were but 184 dollars left ; What is each one's proportion of his estate ? Ans. 77,829 ) 65,563 ) Shares*' 40,606 ) 4. A and B cor.panied ; A put in ,^45, ami took f of the gain ; What did B }.ul in ? Ms. £30. Sect. III. 5. DOUBLE FELLOWSHIP. 179 DOUBLE FELLOWSHIP. DOUBLE FELLOWSHIP, or Fellowship with time, is when the stocks of partners are continued uiici|uai times. RULE. Multiply each man's stock by the time it was continued in trade. Then, As the whole sum of the products is to the whole gain or loss, so is each mau's particular product to his particular share of the loss or gain. EXAMPLES. 1. A, B, and C, entered into partnership ; A put in 85 dollars for 8 months ; B put in 60 dollars for 10 months ; and C put in 120 dollars for [\ months ; by misfortune they lost 41 dollars : What must each man sus- tain of the loss ? OPERATION 120 680 A's product. 3 600 B's product. 85 60 8 10 680 600 s 1640 : 41 : : 680 680 680 2720 164|0)2788|0(17 A's loss. 164 1148 1148 0000 As 1640 : 41 : : 360 360 2460 123 164|0)147610 1476 ^9 C'3 loss. 360 360 C's product. 1640 As 1640 : 41 : : 600 600 164jO)2460|0(15 B's loss. 164 820 820 Dolls. 17 A's loss. 15 B's loss. y L'^ 1863. 0000 41 Proof. 180 DOUBLE FELLOWSHIP. Sect. III. 5. 2. A, B, and C, trade together ; A, at first put in 480 dollars for 8 months, then put in 200 dollai-s more and continued the whole in trade 8 months longer, at the end of which he took out his whole stock ; B put in 800 dollars for 9 months, then took out $583,333 and continued the rest in trade 3 months ; C put in ^366,666 for ten months, then put in 250 dol- lars more, and continued the whole in trade 6 months longer. At the end of their partnership they had cleared 1000 dollars ; what is each man'^ share of the gain ? Ans. ^378,827 A's share. 320,452 B's share. 300,721 C's share. SrxT. III. 5. SUPPLEMENT TO FELLOWSHIP. 181 SUPPLEMENT TO FELLOWSHIP. QUESTIONS. 1. What is Fellowship? 2. Of how many kinds is Fellowship ? 3. What is Single Fellowship ? 4. What is the rnle for operating in Single Fellowship? 6. What is Double Fellowship 1 6. What is the rule for operating in Double Fellowship ? 7. How is Fellowship proved ? EXERCISES LV FELLOWSHIP. A, B, and C, hold a pastjire in common, for which they pay £20 per annum. In this pasture, A had 40 oxen for 76 days ; B had 36 oxen for TjO days, and C had 50 oxen for 90 days. I demand what part each of these tenants ought to pay of the £20. £ f. d. q. Ans. 6 10 2 IffAA A's part. 3 17 1 0|fi§ B's part. 9 12 8 2f §j| C's part. 182 BARTER. Sect. III. G * ^ 6. BARTER. BARTER is the exchanging of one commodity for another, and tn;t< hes morrhants so to proportion their quantities, that neither shall rnstaiij lo«6. PiiooF. P>y changing llie order of the question. RULE. 1. When the quantity nf (me commodity is givc;i n:ith its value or ih^ value of its integer, as also the vahiK of the integer of some other commodity to be itat is gained or lost per cent. First, find what the gain or loss is by subtraction, then as the price it cost is to the gain or loss, so is ^100 (or JCIOO) to the gain or loss per cent. EXMIPLES. 1. If I buy candles at 16 cents 7 2. Bought indigo at ^1,20 per lb. mill? per lb. and sell them at 20 cts. and sold the same at 90 cents per lb. per lb. what shall 1 gain per cent or what was lost per cent ? iM hying out 100 dollars ? Ans. ^25. OPERATrON. I sell at ,20 per lb. Bought at ,167 per lb. I gain ,033 per lb. Then as ,167 : ,0 3 3 :: 100 1 D. cts. ,167)3,3 0(19,76 Ans. 1 6 7 J 6 3 15 3 12 7 116 9 10 10 10 2 .3. Bo'ight 37 gallons of Brandy at 4. Bought hats at 4*. a piece, and ^1,10 per gallon, and sold it for sold them again at 4s. 9 J. ; what is ,<,'10; what was gained or lost per the proHt in laying out £1^*0 ' cent? ^ns. ^1,719 loss. .lyrs. £1 8.15s. 186 LOSS AND GAIN. Sect. IU. 7. CASE II. To knozv hpw a commodity must be sold to gain or lose 90 much per cent; As 100 dollars (or £lOO) is to the price ; so is 100 dollars (or £100) with the profit added or the loss subtracted to the gaining or losing price. EXAMPLES. 1. If I bu}' wheat at ^1,25 per bushel, 2. If a barrel of rum cost ijow must I sell it to gain 15 per cent ? 15 dollars, how must it be sol4 to lose 10 per cent? Jlns. $13,50. OPERATION. As 100 1,2 5 : : 115 1 1 5 6 2 5 1 2 5 1 5 — D. cts. m. y30)i 4 3,7 6(1,43 1 Am 1 4 3 7 A 3 7 5 3 7 6 7 6 3. If 1301b. of steel cost £7, how mu^t I sell it per lb. to gain 15^ per cent? Ans. Is. ^d.perlh' Sect. III. 7. SUPPLEMENT TO LOSS AND GAIN. 187 SUPPLEMENT TO LOSS AND GAIN, QUESTIONS. 1. What is Loss an4 Gain ? 2. Having the' price at wliich goods are bought aqd sold, how is the loss or gain estimated ? 3. To knoTv )iow much a commodity must be valued at to gaio or lose so much per cent, what is the method of procedure ? 4. How may questions in Loss and Gain \te proved ? EXERCISES. 1. A draper bought 100 yards of broadcloth for £56. I demand how he must sell it per yard to gain £15 in laying out £100 ? Ans. 12s. lOd. 2^. 2. Bought 30 hogsheads of molasses at ^600 ; paid in duties ^20,66 ; for freight <^40,78 ; for porterage ^6,05, and for insurance, $30,84 j If I eel! it at ^^6 per hogshead, how much shall I gain per cent ? .3«s. g 11,695. 108 DUODECIMALS. Sect. III. fl. $ 8. DUODECIMALS ; OR, CROSS MULTIPLICATION. This rule is particularly useful to Workmen and Artificers in casting up the contents of their work. Dimensions are taken in {e^ei, inchfs and part?. Inches and parts are sometimes called primes (') seconds (") thirds ('") and fourths ("") TAIiLE. By this rule also may be calcula- 12 Fourths make 1 Third. ted the solid contents of bodies, nav- 12 Tldrds - - - 1 Second. in 6 y give the product of IG feet 4 inches, multiplied hy 2 feet 8 inches. '.). Another board i>* 18 feet 9 inches in length, and 2 feet 6 inches wide, how many square feet does it contain ? Am. 46ft. lOin. 6" By Practice. By Duodecimals. 32 ' 9 8 2 8 8 10. There is a stock of 16 boards, 12 feet 8 inches in length, and 13 inches wide ; how many feet of boards does the stock contain ? Ans. 205ft. lOin. By Practice. By Duodecimals. Sect. III. 8. ^t;FPLEMENT TO DUODECIMALS. 191 SUPPLEMENT TO DUODECI3IAL8. QUESTIONS. J. Of what use are Duodecimals ? To whom more specially are they useful ? 2. In what are dimensions taken ? 3. How do you proceed in the multiplication of duodecimals ? 4. For what number do you carry ? 5» What do you observe in regard to setting down the product different from what is common in the multiplication of other numbers ? 6. Of what term is the product which arises from the multiplication of feet by inches ? feet by seconds ? inches by inches 1 inches by sec- onds ? seconds by seconds ? 7. In what way can the operation be varied ? EXERCISES. 1. Multiply 76 feet 3 inches 9 2. What is the product of 371ft. seconds by 84 feet 7 inches 11 sec- Sin. 6 seconds, multiplied by 181A. ends. lin. 9" ? OPERATION. Ans. 67242ft. lOin. 1" 4'" 6"" F. I. " 6 inches is |)76 3 9 84 7 11 76X 4=304 76X 8=608 3X84= 21 9X84= 5 3 m /. 11) 38 1 10 6 " 61) 6 4 3 9 till Si) 4- 21) 3 2 1 10 6 1 7 11 3 1 8 7 6 Prod. 6460 7 18 3 3. How many square feet in a stock of 12 boards, 17ft. 7' long, and 1ft. Sin. wide ? Ans. 298ft. 1 1' 4. Ho^ many cubic feet of wood in a load 6ft. 7' long, 3ft. 6' high and 3ft. 8' wide ? Ans. 82ft. 5' (.'' 4'" 192 SUPPLEMENT TO DUODECIMALS. Sect. IIL 8. The dimensions of wainscotting, paying, plastering and painting are taken in feet and inches, and the contents given in yards. PAINTERS AND JOINERS To find the dimensions of their work, take a line and apply one end of it to any corner of the room, then measure the room, going into every comer with the line, till you come to the place where you first I)egan ; then see how many feet and inches the string contains ; this call the Compass or Round, which multiplied into the height of the room and the product di- vided by 9, the quotient will be the contents in yards. EXAMPLES. 1. If the height of a room painted 2. There is a room wainscotted, be ISft. 4\n. and the compass 84ft. the compass of which is 47ft. 3' and 1 lin. How many square yards does it the height 7ft. 6'. What is the con- contain ? Am. liey. 3ft. 3' 8" tent in square yards ? Ans. 39Y. 3ft. 4' 6" GLAZIER'S WORK BY THE FOOT. To find the dijnetisions of their work, multiply the height of windows by their breadth. EXAMPLE. There is a house with 4 tiers of windows, and 4 windows in a tier; the height of the first tier is 6ft. 8' ; of the second 5ft. 9' ; of the third 4ft. 6' ; and of the fourth 3it. 10'; an' — What will the glazing- come to at 19 cents per foot ? Ans. §53,88. Sect. III. 9. ALLIGATION. 193 J 9. ALLIGATION. ALLIGATION is the method of mixing two or more simples of di/feireAt quaiitiee , so tlrat the composition may b« of a mean or middle quality. It is of two kind3, Medial and Alternate. ALLIGATION MEDIAL, AUigation Medial is when the quantities and prices of several things 3tre g-iven to find the mean price of the mixture compounded of those things. RULE. As the sum of the quantities or whole composition is to their total value, Fo is any part of the composition to its value or mean price. EX.iMPLES. 1. A farmer mingled 19 bushels of wheat at 6s. per bushel, and 40bueh« «1s of rye at 4s. per bushel, and 12 bushels of barley at 3s. per bushel te- c:e(hcr. I demand what a bushel of this mixture is worth ? OPERATION. Bush. s. £,. s. Buih. £. s. Bush. 19 Wieat at, . 6 is 5 14 As 71 : 15 10 : : I 40 Rye — 4 — 8 20 12 Barley — 3 — 1 16 sum of the— 71)310(45. 4d. l^{q. Ajis simples 71 Total value. 15 10 284 ?. A Refiner having 5/6. of silver bullion, 26 o( V.oz. fine, lOlb. of loz. fine, and I5lb. of 12 Goz. fine, would melt all together ; I demand what finaness lib. of this mass shall be ? )312(4i. Ans. 6o2. ISp-X'ts. Sgrs. Jine. 284 28 4 )112(ly. 71 41 A \ 194 ALLIGATION. Sect. IIL 9, ALLIGATION ALTERNATE, Is the method of finding what quantity of any number of simples, whose rates are given will compose a mixture of a given rate, it is therefore, the reverse of Alligation Medial, and may be proved by it. RULE. 1 . Write the prices of the simples, the least uppermost, &c. ia a column under each other. 2. Connect with a continued line the price of each simple or ingredient, which is less than that of the compound, with one or any number of those that are greater than the compound, and each greater rate or price with' one or any number of those that are less. 3. Write the difference between the mean rate or price and that of each of the simples opposite to the rates with which they are connected. 4. Then if only one difference stand against any rate it will be the quantity belonging to that rate, but if there be several, their sum will be the quantity. Note. Questions in this rule admit of as many various answers as there are various ways of connecting the rates of the ingredients together. EXAMPLES. A goldsmith would mix gold of 18 carats fine with some of 16, 19, 22, and 24 carats fine, so that the compound may be 20 carats fine ; what quantity of each must he take ? OPERATION. PROOF. oz. car. fine. 4 of gold at 16^ 16X4=64 18 18X2=36 Mix ^0 car. { 19- 22- f 16 X 4 2 2X1 (^24 ' 4 2 19 \Jins. 19X2=38 3 ..... 22 22X3=66 4 24 J 24X4=96 02 15 ^Q carats fine. 15)300(20 car. /ne. 2. A druggist had several sorts of tea, viz. one sort at 125. per lb. another sort at 1 1*. a third at 9s. and a fourth at 8s. per lb. I demand how much of each sort he must mix together, that the whole quantity may be afforded at 105. per lb. Ih. s.p.lb. '3 at 12 I Aiis.^C 1' 'A 2 Ajis.{^ ^l ^g 3 Ans. 8 s.p.lb. at 12 4 Ans. f. Gal. Gal. Gal. r^^ C 0->28 Gnl. Gal. <■ 20 : 11 -\ of Hater. "'' \ 1,20-^92 As 120 i 75 : : } 92 : 57^ - Brarldy. Sum 1 30 75 given quantity. 2. Suppose I have 4 sorts of currants of Bd. 12c?. IBd. and 22f/. per lb. of wliich I would mix 120lh. and ?n much of each sort as to sell them at 16^. jrer lb. how much of each must I take ? lb. at d. '^"*' < Qd 1 fi / P^^ '^* ". A Grocer ha<»4urrant3 of 4d. Gd. 9d. and ild. per lb. and" he wouM make a mixture of 2401b. so that it might be afforded at 8d. per lb. how much of each sort roust he take ? lb. at d 72 — 4 24 — e 48 f) PG — n ./Ins. «( ■; n/ per lb. SECT. III. 9. SUPPLEMENT TO ALWGATION. 197 SUPPLEMENT TO ALLIGATION. QUESTIONS. 1. What is Alligation? 2. Of Iiow many kinds is Alligatioa ? 3. Wliat is Alligation Medial ? 4. What is the rule for operating ? 5. What is Alligation ALTl;p.^ATC ? 6. When a number of ingredienls of different priced are mixed together. How do we proceed to find the mean price of the compound or mixture ? •7. When one of the ingredients is limited to a certain quantity, v;hat is the nielhod of procedure ? 8. When the whole composition is limited to a certain qr the given time and rat-e is to 100 dol- lars, so is the given sum to its present worth, which subtracted from the given sum leaves the discount. EXAMPLES. 1. What is the discount of 1^321,63 2. What is the present worth of due 4 years hence at per cent? ^42G payable in 4 years and IJ (Jays, discounting- at the rate of .*} o^•l:;RATIo^'. percent? .i/is. ^354,51^. Dolls. 6 interest of 100 doUa. 4 years. [1 year. 24 100 124 amount. Then, as 124 : 100 : : 321,63 321,63 124)32163,00(259,379 321,03 given sum. 259,379 present worth. Mns. G2,25I discount. i r2. EaUATION OF PAYMENTS. EQUATION OF PAYMENTS is the fmding of a time to pay at once, a.cveral debts due at different times, su that neither party shall sustain loss.> RCLE. Sfulfiply each payment by the time at which it is due ; then divide the ?um of the pnxhicLs by the ^■um t.f the payments, and the quolient will be the cd. per cwt. for •:ti2:ir, what is that per lb. ? Ans. Id. 15. How much m length of a board that is 10 inches wide will it require to make a square foot ? Ans. 1 4 j\ feet. 16. How many square feet in aboard 1 foot 3 inches wide, and 14 feet 9 jrKhp= fong ? .Ins. IS/, o 3" 17. How much wood in a load 9 feet long, 31 wide, and 2 feet 9 inches Irigh ? Ans. 86/. 7' 6" 13. At ^1.33 per yard for clotb, what nousti give for 72 A'ards ? Ans. ^95,76. SOS MISCELLANEOUS QUESTIONS. 19. If 2i cwt. of cotton wool cost £11 17s. 6d. what is that per lb ? ^ns. llirf. 20. If 1832^ gallons of wine cost £44 6s, what is that per gallon ? Aiis. 3|i 21. What will o3ilh. of beef cost at 5cts. 5m. per lb. ? Ans. ^2,942. 22. What will 50 bushels of potatoes cost at 21 cents per bushel ? Ans. ^10,50. 23. At $10,76 per cwt. for sugar, what is that per lb. ? Aiu. ^0,096. 24. What will be a man's wages for 6 months, at 43 cents per day, working 5A days per week ? Ans.^6\,49. 25. What must 1 give for pasturing my horse 19 weeks, at 33 cents per week ? Ans. $6,27. 26. Mow many revolutions doe? the moon perform in 144 years, 2 days, 10 hours. One revolution being in 27 days, 7h. 43m. Ans. 1925. 27. What will 7 pieces of cloth containing 27 yards each, come to at 15s. i^ii. per yard ? Ans. £145 ds. lOirf. 28. A man spends 23 dolls. 69 cents, 6 mills, in a year, what is that per day ? . Ans. i?0,064j^ 29. Suppose the Legislature of this State sjould grant a tax of 7 cent^^ mills on a dollar, nliat will a man's tax! be, who is 142 dollars 40 cents on the list ? .^ny. S 10,395. 30. A Bankrupt, uhose effects are 3948 dollars, can pay his creditors but 2G Gts. 5 mills on the dollar. What does he owe ? Ayis. g 13852,63 1. 31. .Suppose a cistern having a pipe that conveys 4 gallons, 2qts. into il in an hour, has another that lets out 2 gallons, Iqt. Ipt. in an hour, if the cistern contains 84 gallons, iu what time will it be filled ? Ans. S9h. 31m. 45\^s. 32. If 80 dollars worth of provisions will serve 20 men 25 days, what number of men will the same provisions serve 10 days ? Ans. 50 men. 33. If 6 men spend 16 dollars 7 cents in 40 days, how long will 135 men be spending 100 dollars ? Ans. 11 days, Ih. 30m. 10s. 34. A bridge built across a river in 6 months, by 45 men, was washed away by the current ; required the number of workmen sufficient to build another of twice as much worth in 4 months ? Ans. 135 men. 35. Four men. A, B, C, &. D, found a purse of money containing 12 dol- lars, they agree that A shall have one third, B one fourth, C one sixth and D one eighth of it, what must each man have according to this agreement? Ans. A's share $4,511^. B's share, |3,4284. C's share $2,285f D's share, |l,714f. 36. A certain usurer lent £90 for 12 months, and received principal and interest £95 8s. I demand at what rate per cent he received interest. Ans. 6 per cent. 37. If a gentleman hare an estate of £1000 per annum, how much may he spend per day to lay up three score guineas at the year's end ? A71S. £2 105. 2d. IHq. 38. What is the length of a road, which being 33 feet wide contains an acre ? Ans. 80 rods in length. 39. Required a number from which if 7 be subtracted, and the remain- der be divided by 8, and the quotient be multiplied by 5, and 4 added to th."" T>ro.'iiict, the square root of the sum extracted, and three fourths of that root cubed, the cube divided by 9, the last quotient may be 24 ? Ans. 103. 40. If a quarter of wheat aft'orJs GO ten jxnny loaves, how many eight peunj' loaves may be obtaijiod from li'i Jins. 75 ei^ht penny loames. . MISCELLANEOUS QUESTIONS. 207 41. If the carriage of 7 cwt. 2qr. for 105 miles he £l 5s. how far loAy 5 cwt. Iqr. be carried I'or the s44- 56. Tf I give HO bushels of potatoes at 21 cents per bushel, and 240/6. of flax, at 15 cents per lb. for G4 bushels of salt, what is the salt per hnshel ? .'Ins. 1^0,025. 57 Whpt is the present ivorth of 4C2 dollars, payable 4 years hence, dis- counting' at the rate of 6 per cent ? Arts. ^088,709. 53. i have owina: to me ns follows : vi/. ^18,73 in 8 months : $46,00 in ."i raontli? ; and 104,84 in P> months ; what is the mean lime for the payment of the whole ? .Ins. 4 inonths 2 days. 59. If I sell 500 {\oA^ at \'jd. a piece, and lo^c £,2 per cent, what do I lose in the whole quantity I Ans. £,2 IBs. 3d. no. If I buy 1000 El!=i Tlemi'!) of iirr^n for £.90, what may I sell it pfr ni in London, to piin £10 in the whf'o '! Ans. 3s. 4f/. 61. Ilovv many wine p;alIons in a cask, whose bunpf diameter measures C7 iuchct. head diamotcr 2] inches, and length 30 inches? ^. .ins. 63 gals. 3g1s. r>?. A ni'iitnrv viTiri^y drovv !ip hi--, uoldif^rs in rank and tile, having the P'unb* r in r.iuk and file erjir.d ; on beinj^ reititbrced with tliree times hi? first number of men, he placed them all in the same form, and then the number in rank and tile was just double what it was at first ; he was again reinforced with three times his first number of men, and after placing the whole in the same form as at first, his number in rank and file was 40 men each ; How miny men had he at first? Anx. 100. C3. Two ships A and B sailed from a certain port at the same time ; A sailed north 8 miles an hour, and B east 6 miles an hour ; What was their distance at the end of one h6ur? Avs. 10 miles. 64. A hare starts 12 rods before a bound ; but is not perceived by him >3ntil she has been up 45 seconds ; she scuds away at the rate of ten mile*; an hour, and the dog on view, makes after at the rate of 16 miles an hour. How lung will the hound be in overtaking the hare, and what distance will hfi run ? Ans. 97^ seconds, he Zinll rvn 22iiGfeef. Go. A fellow said that when he counted his nuts two by two, three by tliree, four by four, five b}' five, and six by six, there was still an odd one ; b-it when he counted them seven by seven they came out even ; How ma- ny had he ? * Ans. 721. 66. There is an island r)0 miles in circumference, and three men start toi^ether to travel the same way about it ; A goes 7 miles per daj', B 8, and C [> ; when will they all come together again, and how flir will they trav- el ? Ans. £>0 days. A 350 tniles. B 400, and C "450. TT. If a weight of 14401b. be placed 1 foot from the prop, at what dis- tance fiom the prop must a power of I60lb. be applied to balance it? Ans. 9 fret. G3. Sound, ui^ir)tcrnTpled, moves about 1142 feet in a second ;.suppos- ir.'^ in a thundor storm, ihe space between the hghtning and thunder be six seconds : at what distance was the explosion ? Ahs. I mxlc, 04roffs, 2 1 /«/'/. ^ MISCELLANEOUS QUESTIONS. 209 CD. A cannon ball at the first discharge, flies about a mile in eight seconds; at this rate, how long would a ball be in passing from the Earth to the Sun, it being, as astronomers well know, 95 173000 miles distant ? Ans. 24 Tjears, 46 days, 7 hours, 33m. iOs. 70. A general disposing his army into a square battalion, found he had 531 over and above ; but increasing each side with one soldier, he wanted ■4i to fill up tb« square ; Of how many men did his army consist ? Ans. 19000. 71. A and B cleared by an adventure at sea 45 guineas, which was £35 per cent upon the money advanced, and with which they agreed to purchase a genteel horse and carriage, whereof they were to have the use in pro- portion to the sums adventured, which was found to be 11 to A, as often as er, and he would pay the value of his proportion. This being agreed to, A produces 5 loaves, and B 3 loaves, which the travellers eat together, and C paid 3 pieces of money as the value of his share, with which the others were satisfied, but quarrelled about the di- viding ('fit. U})on this, the affair was referred to the judge, who decided the' dispute bj' an impartial sentence. Reqtiired his decision ? ^f^'^ / 7. Suppose the 9 digits to be placed in a quadrangular form ; I demand in wliat order they must stand, that any three figures in a right line may make just 16 ? l'. a countryman having a Fox, a Goose, and a peck of corn, in his journey, came to a river, where it so happened that he could carry bat one over ai a time. Now as no two were to be left together that might de- slro} each other ; so lie was at his wits end how to dispose of them ; for s;iys iie, tho' tlie corn can't eat the goose, nor the goose eat the fox ; yet tiitj fox can eat the goa-se, and the goose eat the com. The question is, how he must carry them over that they may not devour each other ? 9. Three jealous husbands with their wives, being ready to pass by night over a river, do find at the water side a boat which can carry but two persons at once, and for want of a waterman they are necessitated to row tiienjselves over the river at several times : The question is, how those six persons shall pass by 2 and 2, so that none of the three wives may be found in the company of one or two men, unless her husband be present ? 10. Two merry companions are to have equal shares of 8 gallons of wine, which are in a vessel containing exactly 8 gallons; now to divide it eqsially between them, they have only two other empty vessels, of which one contains 5. gallons, and the other 3 : The question is, how they shall divide the said wine botweon them by the help of these three vessels, so that tboy may liave four gallons apiece ? SECTION IV. .^j, FQRilS OF NOTES, DEEDS, BONDS, AND OTHER INSTRUMENTS OF J, * WRITING. § 1. OF NOTES. Ab. /. Overdean, Sept. 17, 1802. — For value received I promise to pay to OH- ter Bountiful, or order, sixty three dollars'fifty four cents, ott demand, with interest after three months. William Trusty. Attest, Timothy Testimony. No. 11. ^ Bilfort, Sept. 17, 1802. — For value received, I promise to pay to O. R. or bearer • — dollars cents, three months after date. Peter Pencil. By two Persons, .irian, Sept. 17, 1802. — For value received, we jointly and severally promise to pay to C. D. or order, dollars, cents, on demand with interest. Attest, Constance Mley. Alden Faithful. James Faikfacb. OBSERV.^TIONS. 1 . No note is negotiable unless the words, or order, other\*T«e or bearer j be inserted in it. 2. If the note be written to pay him ^' or order,^^ (No. 1.) then Oliver Bountiful may endorse this note, that is, write his name on the backside and sell it to A, B, C, or whom he pleases. Then A, who buys the note, calls on William Trusty for payment, and if he neglects, or is unable to pay, A may recover it of the endorser. 3. If a note be written to pay him " of bearer,^* (No. 2.) then any per- son who holds the note may sue and recover the same of Peter Pencil. 4. The rate of interest establislied by law, being s/x ^5er cent per annum, it becomes unnecessary in w riting notes to mention the rate of interest •, it »s sufficient to write them for the payment of such a sum, with interest, for it will ^e understood legal interest, which is six per cent. .5. All notes ai-e either payable on demand or at the expiration of a cer- tain term of time agreed upon by t)ie parties and mentioned in the note, as three months, a year, &:c. 6. If a bond or note mentioU no time of payment, it is always on de- mand, whether the words, on dnncnd, be expressed of not. 21S F0R3IS or BONDS. 7. AH notes pa3'abic at a certain time ar« or iaterest as soon a.' they be- come due, thoug;!! in ench notes there be no mention made ol' interest. This rule is founded on the pr;ncij)lo that every man ought to receire Lis money when due, and that the noo payment oth at that time is an inju- ry to him. The law, therefore, to do him justice, allows him interest irom the time the money becomes due, as a compensation for the injury. y. Upon the same principle a note payable en demand, without any men- tion made of intere^st, is on interest aAcr a demand of payment, for upoa demand such notes immediately become due. 9. If a note be given for a speciric article, as rye, payable in one, two, or three months, or in any certain time, and the signer of such note suffers the time to elapse without delivering such ai-licle, the holder of the note will not be obliged to take the article afterwards, but may demand and re- cover the value of it in money. ^ 2. OF BONDS. A BOND WITH A CONDITION FROM ONE TO ANOTHER. KNOW all men by these presents, that I, C. D. of &c. in the county of &c. atn hel.l and iirndy bound to E. F. of &c. in two hundred dollars, to be paid to the said E. F. or his certain attorney, his executors, admiuistratoi-s or jwsigns, to which payment, well and truly to be made, I bind myself, my heirs executors and administrators, firmly by these presents ; Sealed with u.y seal. Dated tlve eleventh day of in the year of our Lord one thou- sand eigiit hundred ar.d two. The Condilion of this obligation is such, that if the above bound C. D. his heirs, executor?, or adaiinistrators, do and shall well and truly pay, or cause to be paid unto the above named E. F. his executors, administrators cr assigns, the fLjll sum of two hundred dollars, with legal interest for the same, on or before the eleventh day of next ensuing the date hereof : Tlien this obligation to be void, or otherwise to remain in full force and virtue. Signed, 4"C. A Condition of a Cowiter Bond, or Bond of Indemnity , where one man becomes bound for another. THE condition of this obligation is such, that whereas the above named A. B. at the special instance and request, and for the only proper debt of {lie above bound C. D. together with the said C. D. is, and by one bond or obli.ofation bearing equal date with the obligation above v/ritten, held and iirmly bound unto E. F. of &.c. in the penal sum of dollars, conditioned for the payment of the sum of, &c. with legal interest for the same, on the day of next ensuing the date of the said in part reci- ted obligation, as in and by the said in part recited bond, with the condition thereunder written may more fully appear: If therefore the said C. D. his heirs, executors, or administrators, do and shall well and truly pay, cr cause to be paid unto the said E. F. his executors, administratox-s, or assigns, the said sum of, iic. with legal interest of the same, on the said day of, &lc. next ensuing the date of the ?aid ia part recited obligation, according to the l.rue intent i'nd meaning, and in full discharge and satisfaction of the said in part re.:iiou bond or oblt;;ution : Then, o:c, Otherwise, &ic. FORMS OF RECEIPTS. 2 IS Note. The principal difference between a note and a bond, is that the latter is an instrument of more solemnity, being given under seal. Also, a note may be controuled bj' a special agreement, different from the note, whereas in case of a bond, no special -agreement can in the least controul wliat appears to have been the intention of liie parties as expressed by the words in the condition oi' the bond. i^ 3. OF RECEIPTS. No, I. Sitgrieves, Sept. IP, 1802. Received from Mr. Durance Adleg^ ten dollars in full of all accounts. Orvand Constance. No. II. Sitgrieves, Sept. 19, 1802. Received of Mr Orvand Constance, five dollars in full of all accounts. Durance Adlev. No. III. Receipt for an cndorscnicjit on a Note. Sitgrieves, Sept. 19, 1802. Received of Mr. Simpson Easily, (by the hand of Titus Trusty) sixteen dollars twenty five cents, which is enilorscd on his note of June 3, 1802. Peter Cheerfi'l. No. IF, A Receipt for money received on account, Sttgrieves, Sept. 19, 1802. Received of Mr. Orand Landike, fifty dol- lars on account. Eldro Slacklev. No, V. Receipt for interest due on a Bond, Received this -day of of Mr. A. B. the sum of five pounds in full of one year's interest of £100 due to me on the day of last on bond from the said A. B. I say received. By roe C. D. OBSERrATIONS. 1. There is a distinction between receipts given in full of nil accounts, and others in full of all demands. The former cut oil accounts nnly ; the latter nit off not onlv all account*, but all obligations an:l ri-?;bt ot action. 2. When any two" persons make a settlement and pass receipts (No. I. and 11.) each receipt must 9]^ec\fy a particular ?nm recpived, less or more it is not iiecessarv that the sum specified in tlic receipt, be the exact sum 'received. £11 FORMS OF ORDERS. ^ 4. OF ORDERS. No. I. Mr. Stephen Burgess, Sir, For value received, pay to A. B. Ten Dollars, and place the same to my acco'ut. Samuel Skinner. Archdale, Sept. P, 1802. No. IL Sir, BostotiySept. 9, 1802v For value received, pay G. R. eighty six cents, and this with his receipt shall be your discharge from me. Nicholas Reubens. To Mr. James Robottom, ^ 5. OF DEEDS. No. L A Warrantee Deed. K.voAv ALL MEN BY THFst: PRESENTS, That I, Peter Careful, of Leomin- ster, in the County of Worcester, and Commonwealth of Massachusetts, gentlecian, for and in consideration of one hundred and fifty dollars, and forty five cents, paid to me by Samuel Pendleton of Ashby, in the County of Midf'lesex, and Commonwealth of Massachusett.?, yeoman, the receipt w!; reof I do hereby acknowledge, do hereby give, grant, sell and convey to i!ie said Pamnel Pendleton, his heirs and assigns, a certain tract and par- cel of hnd, bounded as follows, viz. [Here insert the bounds, together with all the privileges and appurtenanceg thereunto ^efoa^ing.] To have and to hold the same unto the said Samuel Pendleton, his heirs and assit^ps to his and their use and behoof forever. And I do covenant with the said Samuel Pendleton, his heirs and assigns, that I am lawfully seis;ed in see of the premises, that they are free of all incumbrances, and tliat I will warrant and will defend the same to the said Samuel Pendleton. liis heirs and assigns forever, against the lawful claims and demands of all persons. In witness whereof, I hereunto set my hand and seal this day of — in the year of our Lord one thousand eight hundred and two. Signed, sealed and delivered i Peter Careful, O in presence of ) L. R. F. G. No. IL Quitclaim Deed. Kn'ow all men by these presents. That I, A. C. of, &c. in considera- tion of the sum of to be paid by C. D. of&c. the receipt whereof I do hereby acknowledge, have remissed, released, and forever quitclaimed, and do by these presents remit, release, and forever quitclaim unto the said C. D. his heirs and assigns forever {Here insert the premises.') To have and to hold the same, together with all the privileges and appurtenanceg thereunto belonging, to him the said C. D. his heirs and assigns forever. — In n'iinessy ^c. FORMS OF DEEDS. ?16 No. III. A Mortgage Deed. Know all men by these presents, That I Simpson Easley, of in the County of in the State of Blacksmith, in consideration of Dollars Cents, paid by Elvin Fairface of in the county of in the State of Shoemaker, the receipt whereof L do hereby acknowledge, do hereby give, grant, sell and convey unto the said Elvin Fairface, his heirs and assigns, a certain tract and parcel of land, bounded as follows, viz. (^Here insert the bounds, together uiih all the privileges and appurtenances thereunto belonging.') To have and to hold the afore granted premises to the said Elvin Fairface, his heirs and assigns, to his and their use and behoof foiever. And 1 do covenant with the said Elvin Fairface, his heirs and assigns, That I am lawfully seized in fee of the afore granted premises. That they are free of all incumbrances : That I have good right to sell and convey the same to the said Elvin Fair- face. And that I will warrant and defend the same premises to the said Elvin, his heirs and assigns forever, against the lawful claims and demands of all persons. Provided nevertheless, That if I the said Simpson Easley, iny heirs, Executors, or administrators shall well and truly pay to the eaid Elvin Fairface, his heirs, executors, administrators or assig^ns, the full and just sum of dollars cents on or before the day of which will be in the year of our Lord eighteen hundred and with lawful interest for the same until paid, then this deed, as also a cer- tain bond [or note, as the case may be] bearing even date v\ith these pre- sents given by me to the said Fairface, conditioned to pay the same sum and interest at the time aforesaid, shall be void, otherwise to remain in full force and virtue. In witness whereof, I the said Simpson and Abigail my wife, in testimony that she relinquishes all her right to dower or alimon}' in and to the above described premises, hereunto set our hands and seals this day of in the year of our Lord one thousand eight hundred and five. ' Signed, sealed and delivered } Simpson Easley. O in presence of y Abigail Easley. O L. N. V. X. ^ 6. OF AN INDENTURE. A common Indenture to hind an Apprentice. THIS Indenture witnesseth, That A. B. of, kc. hath put and placed, and by these presents doth put and bind out his son C. D. and the said C. D. doth hereby put, place and bind out himself, as an apprentice to K. P. to learn the art, trade, or mystery of The sal 1 C. D. after the manner of an apprentice, to dwell with and serve the said R. P. from the day of the date hereof, until the day of which will be in the vt.ar c^ our Lord one thousand eight hundred and at which time the Sixi't ap- prentice, if he should be living, will be twenty one years of age : During which time or term the said apprentice his s-.nd master well and frtithfu'ly shall serve ; his secrets keep, and his lawful commands every where, r.nd at all times readily obey. He shall do no damage to his said mastiff, nor wilfully suffer any to be done by others ; and if any to his kncnUdg^:^ be intended, he shall give his master seasonable novice thereof. lie sImI! not waste the goods of his said master nor lend them unlavvfiiHy lo any , at cards, dice, or any unlawful gan. >, he shall not play ; fornication he sh<)li not commit, nor matrimony contract during the said term; taverns, ale houses, or places of gaming he shall not hauat or fiequent : From the ser- 216 FORM OF A WILL. vice of his said master lie fhall not absent hi!r.?f!lf ; but in all tiling;, nnd al all times hs shall cariv and beliave himself as a good and faithful aj'prcDtice ought, duri'.ig the whole lime or term aforesaid. And the said K. P. on his part dot'.i hereby promise, covenant and ai^ree to teach and instruct the said appienti':e, or cause him to be taught and in- structed in the art, trade or calliu^^ of a— ———'by the bojt \ray or nicars he can, and also teach and instruct th'^ said apprentice, or cause him to be tanc^ht and instructed to read and wiile, and CA'puer ds fir as the rule of Three, if the said apprentice be capable to learn, pnd shall nell and failh- full}' find and proviJe for the said apprentice, p^ood and sufficient merit, drink, cloathiiig, lodging and other necessaries tit and convenient for sucli an apprentice, during the term aforesaid, and at the exp'iration thereof, shall give unto t!v3 said apprentice, two suits of wcaiing apparel, one suit- able for the Lord's day, and the other for working days. In testimony whereof', the said )',artie8 have liereunto interchangeably" set their hands and seals, this said day of in the year of our Lord one tho'isand eight hundred and (Seal) Sipied, scaled and delivered } (Seal) in presence of y (Seal) ^ 7. OF A WILL. The form of a Will mth a Devise, of a Heal Estate, Lease- koldy Si'C. I.i the r.ame of God, Jlmfn, I, A. B. of, &c. being weak in body, but of sound and perfect mind, and memor\', {or yov may say ifms, considering the tjncertairity of this mortal life, and being of sound, Lc.) blessed be Almightjf God for the same, do make and publish this as my last Will and Testament in a manner and form following (that is to s&y) Firsts I give and bequeath un- to my beloved wife, J. B. the sum of 1 do also give and bequeath un- to my eldest son G. B. the sum of 1 do also give and bequeath un- to my two younger sons J. B. and F. B. the sum of^ apiece. 1 also give and beqneatii to my daughter in law, S. H. H. single woman, the sum of which said several legacies or sums of money, I will and order shall be paid to the said respective legatees within six months after my de- cease. I further give and devise to my said eldest son G. B. his heirs and assigns, All that my messuage or tenement, situate, lying and being in &c. together with all my other freehold estate v, halsoever, to hold to him the said G. B. his heirs and af?signs forever. And I hereby give and bequeath to my said younger song J. B. and F. B. all my leasehold estate of and in all those messuages or tenements, with the appurtenances, situate &.c. equally to be divided between them. Ar.d lastW, as to all the rest, residue and re- mainder of my pen-ional estate, goods and chattels, of what kind and nature foever, I give and bequeath the same to my ssid beloved wife J. B. whom I hereby appoint sole executrix of thi'^ my last Will and Testament ; and hereby revoking all former Wills by me made. In zvitness n-hcr€'cf, 1 hereimto set tny hand and seal, (his - ■ ■ day of in the year of our Lord Signed, denied, paliliphed ftriii tlec'.ared by llie nbovn nnmcri A. B. (Seal) A. B. to Le Lis last A\ill and Tfstament in the presf.'ico of ns, who linvc liereunto subscrilied our napies ?,■< \\iti)e.=si>--, in the presence of lue testator, 11. S. V.-. T. 'J . \\\ APPENDIX. VULGAR FRACTIONS. ULGAR FRACTIONS are parts of an unit, or inte- fi^c: ; and arc represented by two numbers, placed one above the other, with a line drawn between them. The number above the line is called the numerator, and that below the line the denominator. The denominator shews how many parts an integer is divided into, and the numerator shews how many of those parts are meant by the frac- tion. Fractions are either proper, improper, compound, or mixed. A proper fraction is when the numerator is less than the denomina- tor • a'? 1 -5 11 &,c An improper fraction is when the numerator is greater than the de- rominalor ; as f , |, IJ.^ ^c. A compound fraction is the fraction of a fraction, coupled by the word oj; as a of |, ^c. A mixed number or fraction is composed of a whole number and a fraction ; as 7 f , 28 |, &c. Reduction of Vulgar Fractions. J. To reduce a given fraction to its lowest terms ^ RULE. Divide both the numerator and the denominator by some one number ihat will divide them both without a remainder : divide the quotients in the same manner, and so on till no number greater than 1 will divide ihem both, and the last quotients express the fraction iu its lowest terms. 1. Reduce jif to its lowest terms. 4) 3) Thus, 8 f |i=3A=^j._^a the Answer. 2. Reduce ^ff to its lowest terms. ^ns. |. .'^. Reduce ^\^ to its lowest terms. •5«s. \. I. Reduce ^eif- to its lowest terms. iJ>'W- {h D 2 . " 218 VULGAR FRACTIONS. Appenbix Or ; — Find a common measure, thus, Divide the denomiuator by the numerator, and that divisor by the re- mainder, continuing so to do till nothing remains ; the last diviscr is tlie common nieasui-e ; then divide both terms of the fraction b} the com- mon measure, * and the quotients will express the fraction reqiiired. 1. Reduce ||||' to its lowest terms. 1080)1224(1 1080 144)1080(7 1008 Common Measure. 72)144(2 144 Then 72)ifi|(|4 the Ansziuer. 2. Reduce ^|^^ to its lowest terms. Ans. Jj'^. //. To reduce a mixed number to an improper fraction. RULE. Multiply the whole number by the denominator of the fraction, and to the product add the numerator for a new numerator, and place it over the denominator. 1. Reduce 127y\ to an improper fraction. J- Here we multiply the whole nurn- ber, 127 by 17 the denominator of the ooQ fraction, adding in the numerator 4 ; ,aj the sum 2163 is the numerator to the xr.,«,«^„+«- .. fraction soua;ht, and 17 the denomina- JNumerator, 4 . xu * o,«t • *u • f tor, so that 21^3 ig the improper irac- 2 if 3 Ans. ^'°°' equal to 127^^. 2. Reduce 653 ^-^ to an improper fraction. Ans. '-/^». ///. To reduce an improper fraction to its proper terms, or mixed number. RULE. Divide the numerator by the denominator, the quotient will be the t\'hole number, and the remainder, if any, will be the numerator to the given denominator. EXAMPLES. 1. Reduce y to a mixed number. 6)15(2| ^?i0 for f 4x^X8=160 the common denominator. • -' Hence the new equivalent fractions are y'-''^, /j^"^ and {^^ the answer. 2. Reduce a, § of |, 7J, and J^ to a common denominator. .IMS. fa, J, 187-i? T3T2 ' TafS- 3. Reduce -f^, | of 2^, J-^ and | to a common denominator. *'^''*- TiiS.i' TfilTT' TlfSu^ ^"" "rfSSo- VII. To reduce a fraction of one Senomination to the fraction of miother, hut greater, retaining the saute talue. RDLE. Reduce the given fraction to a compound one by companng it with all the denominations between it and that denomination you would reduce it to ; then reduce that compound fraction to a simple one, by Case V. EXAMPLES. 1. Reduce \ of a penny to the fraction of a pound. By comparing it, the compound fraction will be ^ of Jj of jL. Then "iX IX 1 =-J-- of a poii/nd, the Answer. 8X12X20 2. Reduce ^ of a pound Avoirdupois to the fraction of 1 cwt. ' Ans. ylg- Ck'/. . 3. Reduce | of a Pennyweight to the fraction of a pound Troy. Ans. ji- lb. 4. Reduce | of a penny to the fraction of a Guinea. Ans. ^5^ firifiinea. VIII. To reduce the fraction of one denomination to the fraction of another, hut less, retaini/ig the same value. JIULE. Reduce the given fraction to a compound one, as in the preceding case, only observing to invert the parts contained in the integer ; then reduce thi: compound fraction to a simple one, and that to its lowest terms. EXAMPLES. J. Reduce ^0^0 °^^ pound to the fraction of a penny. ■ 7 V 20 V 1 2 ^ Thus ^./^ of V of '/ . Then --— -— — ==J5| o==,i d. T.20 1 1 1920X1X1 '^"' * 2. Reduce ^Jj-^ of a pound to the fraction of a firthing. Ans. ^q. .'3. Reduce frj-^n of a ft Troy to the fraction of a p'Ji't. ._^ Ans. I pwt. 4. Reduce -^ * of a guinea to the fraction of a pound. Ans. a£. * C'omjjared thus ? of ^y of ^'-. Appendix V OT^GaH FRACTIONS. tH IX. To find the value nf a fraction tn the known parts rf the mtcgier. RULE. Multiply the numerator by the piirts in the next inferior denomination, and divide the product by the denuminator ; and it' any thin^; reiaiiin, mul- tiply it by the next inferior denomination and divide as before, and io ou^ as far as necessary ; the several quotients will be the answer. EXA3IPLES. r. What is the value of § of a pound ? 2 " Multiply the numerator of the 20 fraction ('J) by 20, the number of 3\.jQ shillingSiiu a pound ; the product, — ~r J (40) divided by 3, the denominator, .(, gives 13, the number of shillings, 1 and 1 remaining;, being mullijdicd by ^)^^ 12 the number of pence in a shiliins;, 4d. and the product, (12) divided as bc- Ans. 13s. 4d. fore by 3, gives 4, the number of pence, and no remainder. Hence the auswer, 13s. 4t/. 2. What is the value of 5 of a pound Troy ? .2ns. 7 oz. 4 pz^t. 3. What is tlie value of * of a pound Avoirdupois ? Ans. 12 oz. 12} dr. 4. Reduce l of a mile to its proper quantity. Ans. 6 fur. 16 poles. H) ( "io ! l'. To reduce any given (jitantity to the fraction of a greater denomination of thi same kitui. RULE. Reduce the given quantit^f^ to the lowest denomination mentioned for a numerator ; then reduce the integral part to the same denomination for a denominator, which placed under the uumerator before found will express the fraction required. EXAm'LES. 1. Reduce 16s. Sd. to the fraction of a pound. !£. Integral part. 16s. Sd. 20 / 12 20 200 Xumerator. 12 240 Denominator. Ans. |i^=|£' £. Reduce G furlongs and IG poles to the fraction of a mile. .i«s. \ of a mile. 3. Reduce 12 or. 12^ dr. to the fraction of a pound Avoirdupois. Ans. { of a poimtZ 22? VULGAR FRACTIONS. Api'f.ndix. Addition of Vulgar Fractions. RULE. Reduce compound fractions to single ones, mixed numbers to improper fractions, and tractions of different integers to those of the same, and nU of them to a common denominator ; then the sum of tlie nameratdr9 written over the common denominator will be the sum of the fractioB required. EXAMPLES. 1. Add ^, 9 i, and | of i together. First, the mixed number 9}='/ ; the compound fraction | of |==|. Then the fractions are ^, y and | ; which reduce to a common denomi- nator. 3x5x6:= 90 40x7x6=1932 2 X 7 X 5=_^ 7 X 5 X 6=VtV=5vH ^insrcer. 2. Add 1 Jj, 2^«^, 3^^ and llj^ toget-her. Ans. 22. 3. Add i£. -fs. and fd. together. Ans. 2s. Qf%*jd. 4. Add f of 17£. 9f£. and | of i of ^£. together. . t mi ai cu i>r:>-i...o'iu Ans. 16£. 12s. 34rf. Subtraction of Vulgar Fractions RULE. Prepare the fractions as in addition, and ti.e difference of the numera- tors written above the common denominator will give the difference of the fi'actions required. To subtract a fraction froin a n;hole number; from the denominator of tlie fraction subtract the numerator and place the remainder over the de- nominator : then deduct 1 from the integer or whole number. EXAMPLES. 1. From If take ^. 49 X 9=441 5x50=250 50 X 9=:450 com. denom. Therefore ||-i=|||=i|L the Answer. £. From 13 J take | of 15. Ans. 2Jj. 3. From '{£. take | of a shilling. Ans. 14*. 3a'. 4 From 7 weeks take 9^^ days. Am. 5a'. 4J. 7fe.''12^! 5 From o take -,-} . Ans. 9.\-^. Appendix. VULGAR FRACTIONS. 223 Multiplication of Vulgar Fractions. RULE. Reduce compound fractions to simple ones, and mixed numbers to im- proper fractions ; then multiply the numerators together for a new nume- rator, and the denominators for a new denombator. EXAMPLES. 1. Multiply 4i by j. U=%. Then£2ll=-i!L the Answer. 2X8 2. Multiply 1 of 5 by f of a. -^ns. y^. 3. Multiply 48 1 by 13f Ans. GTS-j'^. 4. Multiply /^ by I of f off Ans. ^^ Division of Vulgar Fractions. RULE. Prepare the fractions, as already directed ; then invert the divisor aad proceed as in multiplication. EXAMPLES. 1. Divide A by f. The divisor ^ iuverted will be f . Theni^=4f=f ^"J- .' 7X2 2. Divide 5 by J_. Ms. 7f 3. Divide 9^ by i of 7. Ms. 2if 4. Divide f by 9. Ms. ^. 5. Divide 7 by f Ans. ISf. 6. Divide 52051 by f of 91. Ms. 7U. Rule of Three Direct in Vulgar Fractions RULE. Having stated the question, make the necessary preparations, as in Re- duction of Fractions, and invert the first term ; then proceed as in Multi- Jt plication of Fractions. EXAMPLES 1. If i of a yard cost § of a pound, what will 2 of *^ ell English come to, at the same rate ? 224 VULGAR FRACTiONi:. Appendix. First, reilace the J of a yard to the fraction of an Ell Eiigliih ; thus i ot' E. E. £. E. E. Thien, as ^^ : §• :: |. Invert the first term, and pro. Geed as in Multiplication — Thus!22ll2iP-=i22f=£o. 0. 4X3X5= 60= 2. If -5 yd. cost l£. what will 40^ yds. come to ? Ans. £59 1«. 3(''. 3. If -j^ of a ship cost £51, what are f.r of her worth ? Ans.£\0 18s. Gd. 3}q. 4. At £3| per Cu-t. what will 9| lb. come to ? dns. Qs. 3^d. 5. Iff yd. cost ^ of a £. what will /j- Ell Eoglish cost ? Ans. lis. Id. 2f«3. C. A man owning | of a farm, sells | of his share for £171 ; what is the whole farm valued at ? Ans. £380. Rule of Three Inverse in Vulgar Fractions. EXAMPLES. 1. If 25?s. will pay for the carriage of an Cwt. 1451 nniles, how far ma} 6i^Cwt. be carried for the same money ? Ans. 22Jy miles. 2. If the penny white-loaf weigh 7 oz. when a bushel of wheat cost 5s. 6d. what is the bushel worth when the penny white-loaf weighs 2i oz. f Ans. 15s. \U{. 3. How much shalloon, that i3 j yard wide^ will line 6^ yards of cloth that is Ji yard wide ? .3ns. 11| yards. ^yluw4t^ (TJ-^ ^-^^^' .«3i:-Si"*»wi«sifc»;v*s. ." \ ^,f L «^ n.ryj' .1