.-J V University of California • Berkeley Gift of Miss Helen Pardee T II E SCHOLAR'S ARITHMETIC; O R FEDERAL ACCOUNTANT, CONTAINING, I. Common Arithmetic, fHE Rules and iLLvstRAfioys. n. Examples and Answers ivifn rlank spaces, sufficient for fHEiR OPERATIONS nr riiE SCHOLAR. III. To EACH MULB A SvVPLKMENTy COMPREHENDING, L QUESTIONS ON THE NATURE OF THE RULE, iTS USE, AND THE MANNER OF ITS OPERATIONS. 2. EXERCISES. XV. Federal Money, ^F/^i/ rules for all the various operations in it -^To REDUCE Federal to Old Lawful and Old Lawful To Federal Money. . V. Interest cast in Yehkral Money, /f/t'// Compound Multiplication, Compound Division and Practice, wrought in Old Lawful and IN Federal Money, the same questions being put, in sepa- rate columns, on the same page, in each kind of monet, bi" which these two modes of account become contrasted, and the great advantage gained br reckoning /:/ Federal Money easilt discerned. VI. Demonstrations sr engravings of the reason and nature of the VARIOUS STEPS IN THE EXTRACTION OF THE SQUARE AND CuBE RooTS, NOT To BE FOUND IN ANT OTHER TREATISE ON ARITHMETIC. VII. Forms of Notes, Deeds, Bonds^ and other instruments of writing, THE WHOLE IN A FORM AND METHOD ALTOGETHER NEW, FOR THE EASE OF THE MASTER AND THE GREATER PROGRESS OF THE SCHOLAR. — > ^A y -J'.'r ^ -5:i- ■~r — BY DANIEL ADAMS, M. B. — ^ -;;'? ^ i\k .f -Jii- >>• — FIFTH EDITION. PUBLISHED ACCORDING TO ACT OF COAGRESS. KEENE, N. H.— PRINTED BY JOHN PRENTISS. [Proprietor of the copy-riii^ht] EOR HASTINGS, KTHRIDGE isf BLISS, BOSTON. 180». * 1 Doli. iinglc.J District of Massacbiiseu^ District, to wit : ^^^ TK U L s )| J-Je it remembered, that on the Nintli day of September, irt aL Jl the 26th year of the Independence of the United States of lli§5=3<&l America, DANIEL ADAMS, of the said District haih depos- ited in this Office the Title of a book, the right whereof he chiims as Author, in the following words, to wit : — " The Scholar's Jiui'iiMEfic : or Federal Accountant. Containing, I. Common Arithmetic, the rules and illustrations. — IL Examples and Answers with blank spaces sufficient for their operation by the Scholar — III. To each rule a Supplement, comprehending, 1. Ques- tions on the nnture of the rule, its use, and the manner of its operations — 2. Exercises — IV. Federal Money with rules for all the various operations i^i it, to reduce Federal to Old Lawful and Old Lawful to Federal Money — V. Interest cast in Federal Money, with Compound MuUipiierition, Com- pound Division, and Practice wrought in Old Lawful and in Federal Money, the same questions being put in separate columns on the same page, in eaeh " kind of money, by which these two modes of account become contrasted and the great advantage gained by reckoning in Federal Money easily dis- cerned — V'l. Demonstrations by engravings of the reason and nature of the various steps in the extraction of tlie Square and Cube Roots, not to be found in any other treatise on Arithmetic. — VII. Forms ot Notes, Deeds, Bonds and other instruments in writing — The whole in a form and method altogether new, for the ease of the Master and the greater progress of tlve Scholar. By DANIEL ADAMS, m. b." In conformity to the act of the 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 of such Copies, during the times therein mentioned." '^.QOODAh'E., Clerk of t/ieDistrici A true Cofni of Record. \ of Massachusetts District. ^//c6/,N.GOODALE, Clerk. RECOMMENDATIONS t J- 00000 ^\> 00000 J- * Nciv-Salem, Sept. 14//6, 1801. HAVING attentively examined *' The Scholar's Arithmetic,'^ I cheerfully give it as my opinion, that it is well calculated for the instruction of youth ; and that it will abridge much of the time now necessary to be spent in the communication and attainment of such Arithmetical knowledge, as is proper for the discharge of business, WARREN PIERCE, Preceptor of Nenv- Salem Acacle?ny, Groton Academy, Sept, 2, 1801. Sir, I HAVE perused with attention ** The Scholar'' s Arithmetic^ which you transmitted to me some time since. It is in my opin- ion, better calculated to lead students in our Schools and Acade- mies into a complete knowledge of all that is useful in that branch of literature, than any other work of the kind, I have seen. With great sincerity I w ish you success in your exertions for the promotion of useful learning ; and I am confident, that to be gcn- erallv approved your work needs onlv to be gencrallv known. WILLIAM M, RICHARDSON. Preceptor oj the Academy, EXTRACT Of a letter from the Hon, JOHN IVHEELOCK, L, L. D. President of Dartmouth College, to the Author. ** THE Scholar's Aridimetic is an improvement on former productions of the same nature. Its distinctive order and supple- ment will help tke learner in his progress ; the pait on Federal Money makes it more useful ; and I have no doubt but the whole will be a new fund of profit in our country." September 1th, 1807. THE Scholar's Aridimetic contains most of the important Rules of the Art, and something, also, of the curious and enter- taining kind. iv . RECOMMENDATIONS. The sul)jects are handled In a simple and concise manner. While the Questions are few, they exhibit a considerable va* riety. While they are, generally, easy, some of them afford scope for the exercise of the Scholar's Judgment. It is a good quality of the Book, that it has so much to do with Federal Money. 'i'he plan of showing the reasons of the operations in tlie ex- traction of the Square and Cube Roots is good. DANIEL HARDY, Jun. Preceptor of Chesterfield Academy, Extract of a letter from the Rev. LAB AN AINSWORTH, of Jaffrey, to the publisher of the 4>th Edition, dated Aug, 3, 1807. ** THE Superiority of the Scholar's Arithmetic to any hook of the kind in my knowledge, clearly appears from its good ef- fect in the Schools I annually visit. Previous to its introduc- tion. Arithmetic was learned and performed mechanically ; since, scholars are able to give a rational account of the several opera- tions in Arithmetic, which is the best proof of their having Icarn* ed to good purpose." PREFACE [TOTHE2clEDiriON] DEDICATORY TO SCHOOL MASTERS, iww I urn -; rrr tn — — GEJVTLEMEJ\r, After .expressing my sincere thanks for j^onr kind and very ready acceptance of the first Edition of the Scholar's Arith- Me TIC, permit me "now to offer for your furdier consideration and favor, the Second ELdition, which with its Corrections and Additions, it is hoped, will be found still more deserving of your approbation. The testimony of many respectable Teachers has inspired a confidence to believe, that this work, where it has been introduc- ed into Schools, has proved a kind assistant towards a more speedy and thorough improvement of Scholars in Numbers, and at the same time, has relieved masters of a heavy burden of writ- ing out Kules and Questions, under which they have so long laboured, to the manifest neglect of other parts of their Schools. To answer the several intentions of this work, it will be neces- sary 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 first wrousjht upon a slate, or waste paper, he may afterwards transcribe into his book. 'J'liE Supplement to each Rule inthis work is a novelty. I have often seen books with questions and answers, but in my humble opinion, it is no evidence that the Scholar comprehends the principles of that science which is his study, because that he may be able to repeat verbatim from his book the answer to a question ot\ which his attention has been exercised, two or three hours to commit to memory. Sttidy is of but htde advantage to the human mind without reflection. 'I'o force the Scholar in- to rdlcctions of his own, is the object of those l^iestions unan- swered, at ti^c beginning of each .Supplement. The Exifcises are designed, tests of his judgment. 'J'he Supplements nfcly be omitted the first time going through the book, if thought proper, and rnkcn up aficrwards as a review. Thro' t|ie whole it has been my greatest care to make myself tnteligiblc to the Sciiolar ; such rules and remarks as Iiavc been vi PREFACE DECICATORY. ompiled from other authors are indiuled in quotations ; the E^camples^ 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 t'le operations in the extraction of the Square and Cube Roots have never been attem])ted, in any work of the kind before to my knowledge. It is hoped these will be found satisfactory. 1 HAVE only to add, that any intimation of amcndmejits or de- fects by the candid and experienced of your order, will be thank- ililly received by Gentlemen, Tour most humble^ and most obedient scr^uant, DANIEL ADAMS. Leominster, (Mass.) Oct. 1, 1802. FIFTH EDITION. The Fifth Edition is printed page for page from the Fourth., and the Errors corrected. k m CONTENTS. ■"■■• ''•'"■ ''•'" ■••'■■ ^ "'" "•*'*■ '*'•'" *"'■ Ifntro&uction* NOTATION AND NUMERATION. SECTION I. FUNDAMENTAL RULES OF ARITHMETIC. PACf.. Shnftlc Jddition ----------------- ------:-'-----13 Siibtracfio7i 1-9 Mutti/ilication -- -- 23 ■ — Division --..----------.---------33 Comfiound Addition ---------------------------- 42 tiubtraclion ---------------- -.--53 SECTION II. RULES ESSENTIALLY NECESSAIlY FOR EVERY PERSON TO FIT AND QUAL- IFY THEM FOR THE TRANSACTION OF BUSINESS. Reduction ---------------------------------53 Fractions ------------ — .----_._ .-_. ..,...,y^ Decimal Fractions -- ----.-----.---. -.7 5 Federal Money ------- --- ----- ---;.--_, . „_ . _-87 Table to reduce Shillings and Pence to Cents and Mills --.------94, Tables of exchange --------------------------- -95 Interest ----------.-------. ---------------gs Easy Method of casting interest -------- — ---._------ iqo Method of casting interest on JVotes and Bonds ivhen partial fiayments at different 'imes have been made -------.- .^»«--a ]02 Coinjiound interest - - ----- ---.-...- -..--,.,^ ---•104 Comfiound Multifilicatioii ----.-•■ — ^ ---,..••--- io7 Division - --u------------^-----.-.-- \\2 Single Hide of Three ---«---------- — -..-..... jjg Double Rule of Three -----.------.-----«---.- 133 Practice --• -.,_--.-,-_-_, -.--.,^,,^»..143 SECTION III. RULES OCCASIONALLY USEFUL TO MEN IN PARTICULAR EMPLOYMENTS GF LIFE. Involution 1 59 Evolution , .159 Extraction oj the Sqvare Root . .^. 1 60 Demonstration of the Reason and nature if the various steps in M^^^fl- era/ion of extracting the Square Root 161 Extraction of the Cube Root ...:... 169 Demonstration oJ the reason and nature of the various str/ts in thr of'.- cration of extracting the Cube Root .170 Single Eelloivshi/i , • . . 179 viii CONTENTS. Double F^llovfdhlfi .181 Barter 184 J.o&a and Gain . . . ' ,187 JJuodecimaU^or Cross AlultifilicaUon v 199 Jixam/iles for measuring Wood 19 1 — ■■ ■ ■ " — Boards 192 J^ainter^s and Joiner'' s Work J 94 (i/azier*s Work . 194 Jilligalioii . .. . ; 195 Medial , 195 Alternate 196 I*osition 200 Single 200 Double ' 20 I Discount • 203 iH.(l nation of Payments ........... ^' 203 uaaging- , 205 Mechanical Powers ' . . 205 The Lever ^ 205 The AxLe 206 The 8crenv 206 Problems , 206 1*^. To find the circumference oj a circle the diameter being given . 206 2c/. To find the area of a circle the diameter being given . , . 206 ^d. To measure the solidity of an irregular body ^06 SECTION IV. MISCELLANEOUS ^ESTIONS. SECTION V. fOllMS OF NOTES ^C. ^''^tes . . . , i ... 2 1 I Bfiuds .' 212 Recei/its 213 Order's 214 Deeds .... - 214 Indt^nture ....f2l5 ^y^U 2 i 6 i-XPLANAIiON OF THE CHARACTERS MADE USE OF IN THIS WORK. ._ 5' The sis^n of cquaiiiy ; as 100 c^s.ml i^o/. signifies that 100 cents (are equal to 1 tiolliir. \. Saint Gkouge's Cross, the sign of addition; as 2-f-4=i6, that is "T" ^.2 addtU.40 ^i is'ct|Ua] io 6. — ^The ^s>:^ of subtracton ; as 6 — 2rr-4. ; that is, 2 taken from 6 leaves 4. •^ Saint ANr>uEw's Cross, the sijjii of multiplicaiion ; as 4x6:zi24 ; ^ ^ that is, 4 limes 6 is tqual to 24. ERSED Parenthesis, the sign of division ; as 3)6(2, that is, 6 di- by 3 is equal to 2, or 6-f-3:p:2,. The sii;n of proportion ; iis, 2 ': 4 . : S : 16, that is, As 2 is to 4 so is 8 to 1 . THE SCHOLAR'S ARITHMETIC <:::0«C«DO!C:5:0»000<-s INTRODUCTION. A RITHMETIC is the art or scieiice which treats of numbers. It is of two kinds; theoretical and firactical. TflE TiiEoiiY of Arithmetic explains the nature and quality of nwmbcrsand demonsti-ates 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, Arithmetic is an Art, Directions to the Scholar. Deeply impress your mind with a sense of the importance of arilhmefcical knowledge. The great conccras of life can in no wav 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 improyemcnt. Remember that youth, like th« morning, will soon be past, and that opportu- nities once ncjjlected, can never be regained. First of all things, there must be implanted in y©ur mind a fixed delight is study ; make it your inclination ; *< j1 cienire accemfilished is stveet to the soul.''* Be not in a hurry to get thro* your book too soon. Much instruction may be given in these few words, ww- derstand every thing as you go along. Each rule is first to be com- mitted to memory; afterwards, the examples in illustration, and, every r«- ma^k are to be perused with care. There is not a word inserted in this Trea- tise, but with a design that it should ba studied by the Scholar. As mucU a« is possible, endeavor to do every thing of yours«lf; on^ ^ fN b Example. Take the sum 3 3 3^ made by the same figurfe three time?^ repeated.' The first or right hand figure, or the figure in the place o^ units. has its natural meaning or the same meaning as if standing alone, and signi- fies three units or ones ; but the same figure again towards the left hand in the second place, or place of teiis, signifies not three units, but three ? 6 r ^^^^^ °f figures or those immediately against tlo9 8 '^^^^ords, will be found to be, Two Bi//iom, T fi K A one hundred sixty seven t/tousandsy tivo hiai- o ^^<^(^ «^<^' thirty Jive Alillionfi ;/o2ir hundred twenty one thousands ; eight hundred and thir.^ ty six. In like manner may be read all the remaining numbers in the Table. Those words at the head of the Table are applicable to any sum or nun^- ber, and must l»e committed perfectly to memory so as to be readily applie#< on any occasio.n. ^ For the greater ease in reckoning it is convement and often practised in public oflices and by men of busines, to divide any number into periods an^f half periods, as in the following manner. 5.3 7 9,6 3 4.5 2 1,7 6 8.5 3 2, 4 6 7 ODOeoDCo«o looicoaseo ojajtoorosso ooooo ooooo 5:KR'»Jp^^ ^5 § "§ § '^ r§ i§ § S ^§ § ^ § -^ ^ jS >^"^l| >'^1"^s -SSg^ 12 INTRODUCTION. The first six figures from the right hand are called the unit fieriod^ the next six, the TwzY^/o/i /ifi-zof/, after which the trillion^ quadrillion^ quintilliony periods, Sec. follow in their order. Thus, by the use of ten fig^ure may be reckoned every thing which can be numbered; things, the multitude of which farexceeeds the comprehension ofmnn. " It may nat be amiss to illustrate by a few examples the extetrt of num- " bers, which are frequently named without being attended to. If a person " employed in telling money r«ckon an hundred pieces in a minute, and con- " tinue at work ten hours each day, he will take seventeen days to reckon a " million ; a thousand men would take 45 years to reckon a billion. If we *' suppose the whole earth to be as well peopled as Britain, and to have been « so from the creation, and that the whole race of mankind had constantly " spent their time in telling from a heap consisting of a quadiillion of pieces, <^ they would hai'dly have yet reckoned a thousandth part of that quantity." After having been able to read correctly to his instructor all the laurabers in the foregoing Tablt^ the Learner may proceed to write the following num- bers out in words. 9 8 4 3 7 6 12 7 2 8 4 5 14 9 7 0^ 9 7 8 3 16 5 3 7 2 16 8© SECTION!, FUNDAMENTAL RULES OF ARJTHMETie. X HESE are four, Addition, Subtraction, Multipli- cation, and Division ; they may be cither simple ox conipoimd ; simple, when the numbers are all of one sort or denomination ; "ttompound, when the numbers are of different denominations. They are called, Principal or Fundamental Rules, because that all other rules and operations in arithmetic are nothing more than various uses and re]>etitions of these four rules. The object of every arithmetical operation, is, by certain given quantities which are known, to find out others which are unknown. This cannot be done but by changes effected on the given num- bers ; and as the only way in which numbers can be changed is ei- ther by increasing or by diminishing their quantities, and as there •an be no increase or diminution of numbers but by one or the other of tlie above operations, it consequently follows, that these four rules embrace the whole art of Arithmetic. — <^ -;.:- -::?• •?*> ^;«- ^ •»:;- '.u- -.u- 45* %> — § u .Simple "Siftitiition. :/ Simple Addition is the putting together of two or m»re numbers, o(thc same (lenomination, so as to make them one whole or total number ; as 3 dol« lars, 6 dollars, and 8 dollars added or put together, make 17 dollars. RULE. "Write the numbers to be added one under another, with units under ^* units, tens under tens, and soon. Draw n line under the lower number, then "add the right hwid ciilumn ; and if the sum be under tniy write ii at the " foot of tlie column ; but if it be tetiy or an exact uiimbn- of tens, write a cy- ^^ pher ; and if it be not an exact number of tens, write the excc%9 above tent ^'at the foot of the column ; and for rvery ten the sum contiuni, carry one "to.the next column, and add it in tlje same manner as the farmer. Pro- ♦* cecxlin like manner to add the other columns carrying for the tens of cacK ♦* lo thp next, aud mark down the full sum of the Icfl hand col\imn." 14 SIMPLE ADDITION, Sec. I. 1 PROOF. l^ECKON the figures from the top downwards, and if the work be rit^ht, th amount will be eqiml to the first ; — or, what is often practised, " cut oif th "upper line of figures and find the amount of the rest ; then if the amount <"and upper line when added be ecjual to the sum tQtal, the work is supposed *'io be right. EXAMPLES. i "^ -• d § "^ =.' 5' '^* ^ i t* •- j; -^k »» "^ ?; '^4 s S: ■*>* f!^ k5 ►<>-> I.S t^t >s ^5j ^s: s i, c s> «!: f^ b ^ ti: L-s, b t-^; CN b 1. What will be tl>e amount of S 6 1 2 dollars; 8 4 3 dollars ; 6 5 1 4 he ^1 b dollars, and of 3 dollars, when added together ? Here are four sums given for addidon ; two of them contains wwV^, tens^ Jiundreds, i/wusands ; another of them contains units tens, hundreds ; and a fourth contains unitx only. The first step to prepare these sums for the op- eration of addition, is to write them down, un^ts under units, tens under tens, (^id 60 on, as in the following manner. "s>ilg.5 { 3 6 12 dollars^ -■^J^'- f"".- given sums for acI-^< g 4 7 doHar>. ditioii plapcd ..s tli= r«k directs.-^ ^ ^ , ^^^^^^_ [_ 3 doUars. Answer, or amount, 12 3 9 dollars^ Amount of the thi'ee lower lines, Proof, 12 3 9 To find the answer or amount of the sums given to be added, begin with du right hand column, and say 3 to 1 is 4, and 3 is 7, and 2 is 9 ; which sum i(9) being less than ten, set down directly under the column you added- Then ■$)roceeding to the next column, say again ; 5 to 4 is 9, and 1 is 10, being even ten, set down 0, and carry 1 to the next cohimn, saying 1, which I carry to 6 as 7, and is nothing, but 6 is 13 ; which sum (IS) is an excess of 3 over even tens; therefore, set down 3 and carry 1 for the 10 to 8 in the next column, say- ing 1 to 8 is 9, and .1 is 12 ; this being the last column, set down the whole number, (12) placing the 2, or unit figure directly under the column, and carrying tlie other figure, or the 1, forward to the next^iiaec on the left hand, or to tliat of Tens of J'/iousands^ and the work js done.> It may now be required t© know if the whole be right. To exhibit the meth- od of prooflet the upper line of figures be cut off as seen in the example. Then adding the three lawer lines which remain, place tke amount (8657) under the amount first obtained by the addition otall the sums, observing carefully that each figure fails directly under the coluain which produced it; ^hen m](\ this last amount to the upper liiie which you cut off; thus, 7 lo 2 is 9 ; 9 to 1 is 10 ; care ry 1 to.t3 ib 7 and^G is iJ ; 1 which I cany again to G is 9 ani 3 iu 12, all whick 8 4 3 i 6 5 1 I 3 i 1 2 3 9 8 6 9 7 Sect. I. I SIMPLE ADDITION^ 15 being set down in their proper places, and as seen in the example, compare the amount (12309) last obtained, with the first amount (12309) and if they agree, as h is seen in this case they do, then the work is judged to be right. .Ve7'£. The reason oi carrying for ten in all simple numbers is evident from what has been taught in Notation. It is bccaust; 10 is an inferior column is just equal in Value to I rii a superior column. As if a man should be holding in his right h 4 2 9 6 4 5 3 7 6 5 8 6 7 3 7 3 4 3 2 1 7 S I 16 SIMPLE ADDITION. Sect. I. 1 10 1 7352 S 6 S 2 26871 3098 3 571 65734 567 5' 942 6 83475 1298 3678 32786 12 13 6 3 9 8 7 5 3 4 5 6 7 8 ^ 4 6 8 2 3 7 9 8 6 4 3 5 1 2 8 7 5 4 1 7 3 8 7 9 5 2 6 7 8 5 4 9 6 7 4 9 8 I4t 15 3 5 7 8 5 "S^, 2 6 8 6 7 2 1 3 4 5 3 7 8 6 7 9 8 9 8 3 6 7 7 6 0' Sect. L' 1 SUPPLEMENT to ADDITION. 17 Supplement to ^fjtlition* The attentive Scholar who has understood, and still carries in his mind, what has already been taught him of Ad^tion, will be able to answer his In- structor to the following. QUESTIONS. H 1 . TVha r is Simfile jiddition > 2. Hoxv do you f dace number b to be added ? S. Where do you begin the jlddition ? 4. How is the sum or amount of each column to be set doivn ? 5. What do you observe in regard to setting down the sum of the last column ? 6. Wiir do you carry for 10 rather than any other number ? 7. How is Addition firoved ? 8. Of what use is Addition ? JVoTE I. Should the Learner find any dlffioulty in giving an answer to the above questions, he is sulvised to turn back and consult his Rule, with its Illusti'utions. JVofE. 2, In treating of the Rules of Arithnraetic the Scholar, in all in- stances, i<* not particularly instructed in the use and application of them to the put poses^of life. This is a point, however, to which his thoughts should be called ; tlTerC'fore it is mad^a question here. A consideration of the Rule and of the quest-ions, which it involves, naturally suggest an answer. To cousideraiion, therefore, let the Scholar apply himself. The mind acquires strength by exercise ; instruction ought ever to be plain, but never so full as to preclude a necessity, that the Scholar should in some degree exercise his own thoughts ; it shQuld be given in such a manner as io force him into isome reflections of his own. EXEKCISES. 1. What is the amount of 2801 2. Suppose you lend a neighbor dollars; 765 dollars; and of 397 /"210 at oi>e fime, ^76 at anoihev, dollars, when added together. ^'17 at another, and £9 at another, ^na. ^9Qo dQllars. what is the sum lejJt ? Ans.^212, A^jfE. The Scliolar who looks at greatness in his class wiJl not be discotrr- aged by a little diniculty which may at first occur in stating his (jucslion, but will apply irnnself the more closely to his Rule and to tijinking, that if possi- ble he may be able ot himsull to answer, what anotiier muy l)c obliged to U^c taught him hy his Instructor. 18 SUPPLEMENT to ADDITION. Sect. I. L 3. Wasiiingto?! was born in the year of our Lord 1732 : he was 67 years old when he died : in what year of our Lord did he die ? 4. There are two numbers : the less number is 8761, the difterence between the numbers is 597 : what is the greatest number ? 5. From the creation to the depar- ture of the Israelites from Egypt was 2513 ye^rs ; to the siege of Troy, 307 years more ; to the building of Sol- omon's Temple, 180 years ; to the building of Rome, 251 years ; to the expulsion of the kings from Rorne, 244 years ; to the destruction of Car- thage, 363 years ; to the death of Julius Cajsar, 102 years ; to the (jHiristian sera, 44 years ; required the time frora the Creation to the Chistian xra ? ji?is. 4004 yean. 6. At the late census, taken A. D. 1800, the number of Inhabit- ants in the JVenv- England States was as follows, viz. JVew- Hampshire 183858 ; Massachusetts A22%'^5 -, Maine^ 151719 ; Jihode - Island, 69122 ; Connecticut, 251002 ; Ver- mont, 154461; what was the num- ber of Inhabitants at that lime in JVeiu -England ? Jus. 1233011 Inhabitants, Sect. I. 2. SIMPLE SUBTRACTION. 19 § 2. pimple Subtraction* Simple Subtraction is the taking of a less number from a greater of the same denominaiion, so as to shew the difference or remainder ; as 5 take;n from 8, there remains 3. The greater number (8) is called the Mlnuendfihe less number (5) the Sud- trahendj-dnd the difference (3) or what is left after Subtraction, the Remai7ider, ♦ RULE. "Place the less number under the greater, units under units, tens under "tens, and so on. Draw a line below: then begin at the right hand, and " subtract each figure of the less number from the figure above it, and place "the remainder directly below. When the figure in the lower line exceeds " the figure above it, suppose 10 to be added to the upper figure ; but in thi^ " case you must add 1 to the under figu^*e in the next column before you " subtract it. This is culled borroiinng ten** PROOf. Add the remainder and subtrahend together, and if the sum of them cor- respond with the minuend, \\\£ work is supposed to be right. EXAMPLES. Minuend 8 6 5 3 The numbers being placed with the larger uppermosr, as the rule directs, I begin with the;- Subtrahend 5 2 7 1 unit or right hand figure, in tht subtrahend, and say I from 3 and there remain 2, which I Remainder 3 3 8 2 set down, and proceeding to tens or the next — — — — figure, I say 7 from i 1 cannot, 1 therefore bor- Proof. 8 6 5 3 row or suppose 10 to be added to the upper figure (5) which make 15, then I say, 7 from 15 and there remain 8, which 1 set down ; then proceeding to the next place, I say, 1 which I borrowed to 2 is 3, and 3 fi*om 6 and there remain 3, this I set down, and in the next place I say 5 from 8 and ther^ remain 3, which I set down and the work is done. Proof. I A«d the remainder to the ijubtrahend* o« finding the sum just equal to the minuend, and suppose tlie work to be right. A*©r£. The reason of Ixfrro^ini^ ten^ will appear if we consider, that when two numbers are equally increased by adding the same to both, their difler- cncc will be equal. Thus the difference between 3 and 5 is 2 ; add the num- ber 10 to each of these figures (3 and 5) they become 13 and 15, still the dif- ference is 2. When We proceed as above directed, we add or suppose tQ be added, 10 to !he itimiinu!^ and we likewise add 1 to the next higher place of the subtrahend^ which is just equal in vjilue to 10 of tiie lower place. 2. From 3 278G532146 5 the minuend. Take I 6 7 9 3 6 1 2 3 4 2 the subtrahcud llemaiaidcv 29 SIMPLE SUBTRACTION. Sect. I. 2. 3. From 3 16 dollars, Take 1 7 dollars. Remainder Proof 5. From 1 Take r 2 3 6 8 7 9 1 4. From 7 6 3 5 gOiiieas, Take 2 7 8 3 giriiieas. Remainder Proof 7 4 2 3 2 8 4 5 17 9 8 6 7 1 3 2 Remainder l^YVf]_dl1l_ > T} ^ zTfTTTJTJJ~~ 6. From 3 7 5 1 dollars, take 16 7 4 dollars. Write the less number under the greater, with units under units, Sec. as the rule directs. 7. From 2673105, the minuend ; take 178932, the subtrahend. Thus, 3 7 5 1 16 7 4 ■Rem binder 8. From 10000000. Subtract 9999999 OPERATION. Minuend Subtralicnd The distance of time since any remarkable eveiit, may^be found by sub- tractini^ the date thereof from the present year. EX. How long since the Amer- ican Indepeiultlkne, which %vas declared in 1776. 18 8 present time 17 7 6 date of the Ind. Ans. o 3 years since. So, likewise, the distance of time from' the occurrence of one thing to that of an- other, may be found by subtracting the date of the thing first happening, from that of the last. ex. How long from the discovery of A- nierica by Columbus, 1492, to the com- mencement of the war, 1775, which gained our Independence. 17 7 5 14 9 2 Ans. 3 8 S years. StcT.L.2. SUPPLEMENT TO SUBTRACTION. 21 Supplement to ^ftttaCtiom QUESTIONS. 1. WiiAf is Simfile Subtraction? 2. How many numbers must there be given to jierfcrm that ofieration ? 3. Honv must the given numbers b^ placed ? 4. Wha r are they called ? 5. When the figure in the lower number is greater than that of the upfier mimberfrom which it is to be taken, what is to be done ? 6. I4ojv does it appear, that in sub trcK ting a less number from a greater, the occamnally boiTowing often, does not affect the diff'erence between these two numbers ? 7. How is subtraction proved ? 8". iViiE^y and honu may Subtraction be of usv to a man engaged in the pursuits ofHJe P JEXERCISES. 1. What is the difference be- 2. F.rom a piece of cloth thatmeas- tween 78360 and 5421 ? ured 691 yards, there were scld 278 Ans. 72939 yards ; how many yards should there remain ? ^ns. 413. NoxE. In cVLi^oHorrowifig ten, it is a matter ofindifrerence, as it respects tjis; operaiiou, whether we Mipposo ten to be added to the upper figure, and from the sum subtract the lower fij^ure cmd set down the difference ; or, as IVIr. Pike directH, fust, subtract the lower figure from 10, and adding the difference to tJie fij:;»re rtbovc, set down the sum of tliis difference and the upper lij;ure. The l;itttr nuthoU may, perha4>s, be thought more easy, but it is conceived that it docs not lead the UMdcrsiuiiding of youth so direct- ly ii*io the nature of the oprr^liuu au the former. 22 SUPPLEMENT TO SUBTRACTION. Sect. 1. 2. 5. There are two numbers whose diflercnce is 3 7 5, tire great- er number is 8 6 2 ; I demand the less ? ^ir.a. 487. 4. What number is that, which taken from 1 7 5 leaves 96 ? Am. 79. ^ 5. Th£ capture of Gen. Bur- coYKE and his army happened in the year, 17 7 7, that of Coniwal- lis, in 17 8 1? Ijow many years between these events ? Arts. 4 years. 6. Suppose you should lend a neighbour 2 7 6 5 dollars at a cer- tain time, and he should pay you 973 at another ; how much would remain due ? Ans. 1792 dollars. 7. Supposing a man to have been born in the year 1745, how old was he in 1799 ? Am. 54 years. 8. What nnmbcr is that, to which if you add 789 it will become 6S30 I Ans, 5561. §. ^Fppo8E a man to have been 63 year§ old in the year 1801 ^ in wiiul yuar was he born ? • Ana. ' In ihc y car 1738. 10. King Charles, the Martjfr >yas beheaded 1684 ; how many years since I Sect. 1. 3. SIMPLE MULTIPLICATION. 23 § 3, ..^tmple Jl^ultiplication. Simple Mulxiplication ttaches, having two numbers given of the same denomiuation, to find a third which shall contain ciditM- of the two givea numbers as many times as the other contains a unit. — Thus, 8 multiplied by 5, or 5 times 8 is 40. — The given numbers (8 and 5) spoktn of together are called Factors. Spoken of separately, the fi^-st or largest number (8) or num- ber to be multiplied, is called the MnltifUicand ; the less number, (5) or num- ber to multiply by, is called the MuUiplier ; and the amount, (40) the firoduci. This operation is nothing else than the addition of the same number sev- eral times repealttd. If we mark 8, five times underneath each oth- 8 er, and add i.hem, the sum is 40, equal to the pr v- 'net of ^ and 8 8 multiplied together. But as this kind of addition is of frequent 8 and extensive use, in order to shorten the operation we mark down 8 the number only- once, and conceive it to be repeated as often as 8 there are units in the Multiplier. — Before any progress car. be made in this rule, the following 40 Table must be committed perfectly to memory. MULTIPLICATION TABLE. ll 2| 3| 4| 5\ 6 7 8| 9 1 10 1 iH '"' 1 2 4| 6| 8 1 10 1 12 1 14 1 16 1 18 1 20 1 22 1 24 3| 6| 9| 12 15 1 18 211 24 27 1 30 1 33 1 S6 4| 8| 12 1 16 1 20 1 24 1 28 1 32 1 36 1 40 1 44 48 5| 10 1 15 1 20 1 25 1 30 1 35 40 1 45 1 50 1 55 1 60 6| 12 1 IM 24 1 30 1 36 ! 42 48 1 54 1 60 1 66 1 72 7! U\ 21 28 1 35 42 48 49 56 1 63 1 70 1 77 1 84 s| 16 1 18 : 24 32 1 40 56 64 72 1 •80 1 88 1 96 9 ; 10 27; 36 J 45 54 : 6S 72 1 81 i 90 ; 99 ; 108 ; 20 30 ; 40 ; 50 60 •70 ; «0 ; 90 : 100 : no: 120 11; 22 I 33 ; 44 I 55 66 i 77 ; 88 : 99 \ no i 12L i 132 i 132 144 12 124 ; 36 ; 48 i 60 : 72 ;84 i96: 108 : 12Q ; By this Table the product of any two figures ntiII be found in t IjuI sijuarc which is on a line with the one and directly unden* the olhcr. Thus, Su the product of 7 and 8, will be found on a line with 7 «nd tinder 8 : so ,2 tj|i»c.s ':■ is 4 ; 3 times 3 is 9, kc, in this Way the Table n\ust be k-jrncd aivd remcm be red. 24 SIMPLE MULTIPLICATION. Sect. L 3, RULE. 1. Place the numbers as in Subtraction, the larger number uppermost with units under units, &c. then draw a line belqvv. 2. M^^HEN the Mul'iiiicr does not exceed 12 — begin at the right Hand of the multiplicand, and multiply each figure contained in it by the multiplier, set- tiiig down all over even terifs and carrying as in addition. 3. When the multifilirr exceeds 1 2 — muiti ply by each fi.^urc separatel v, first by the w«i7.s- of the multiplier, as diitcted above, then by the ^e/75, and the other figures in their order, remembering always, to place the first fi;-^ure of each product directly un.der the figure ^y whicii you multiply ; having- gone through in this manner vrivh each figure in the JVIuUpSier, addthcir several products together, and the sum of them will be the product required. » , EXAMPLES. 1. Multiply 5 2 9 1 by 3. OPERATION. 5 2 9 1 Multiplicand. 3 Multiplier 8 7 3 Product. 2. Multiply 5 6 2 by 12. OPERATION. 3 6 2 1 2 2 2 4 The numbers being placed as seen Vnder the operation, say — 3 lines 1 is 3, which set down directly un- ci'er the multiplier ; then 3 times 9 is 27. set down 7 and carrf 2 : again 3 times 3 ib 6, and 2 I carry is 8, set down 8 ; then lastly, 3 tim.es 5 is 4 5, which set down and the work is don». The numbers being properly plactd, proceed thus — 12 times 2 is 24, set down 4 and carry 2 ; — 12 times is nothing, but 2 I carried is 2, which set dovvn';^^ — -then 12 limes 6 is 72, set down 2 and carry 7 ; lasdy 12 limes 3 is 36, and 7 I number, 5. What is the product of 4175 multiplied by 37 ^ "4175 Multiplicand. 3 7 Multiplier. Place the Factors thus, 2 9 2 2 5 Prod«f.t by the units (7) of the multiplier. 2 5 2 5 Product by the if72s (3) 15 4 4 7 5 Product or answer. \^ this example, as tlve Mu 'dpUer exceeds 1 2, the«pfore, you must multi- ply hv each figure, separately. -First, by the units (7) just in the manner of the olhtr examples. Secondly, by the tens (3) in the same way, excepting c.niv, that the first figure of the product in the muhiplication by 3, must be f)h.ced under the 3, that is, under the figure by which you multiply. Lastly, ad'J these two products together, the sum of them is the answer. PROOF. 51uLTirucATio» ir.av be proved by Division, but a method more concrs* y htm Sect. I. 3. SIMPLE MULTIPLICATION. 25 and easy, often practiced by accountants, and which I shall recommend, is called Casting out the 9'^. Casting out the ^'s from any sum or number, is the exhausting of that number by the figure 9, till there is nothing left of it but a remainder, or ex- cess over even nines which remainder or excess is the thing sought. How to cast out the 9V. Whatever method may be adopted, this in effect, is nothing else than di- viding the number by 9, The operation, however, would be tedious as nat- urally practised by division ; besides, as yet, we do not suppose the learner acquainttd with it. A shorter and more successful way is tlie following METPIOD. Beginning at the right hand of the number, add (he figures, and when the sum exceeds 9, drop the sum and begin anew by adding, first, th€ figures, which would express it. Pass by the nines, and when the sum conies out exactly 9, ngglect it ; what remains after the last addition will be the remain- der sought, t EXAMPLES. If it be required to cast the 9's out of 576394, proceed thus ; — 5 to 7 is 12, which sum (t%t}elve) as it exceeds 9 you must drop, and begin anew, first add the figures (12) which would express tvjelx*e^ saying 1 to 2 is Sand (proceeding with the other figures, which remain to be added) 6 is 9, being •j- This Method of Casting out tlie 9'« succeeds on a PRINCIPLE. That every §figure, in rising ^^^^''l """'"'^T '^"'" '7*^ '''• '^r t^""'^ '-^ . Jinits js the exfircssion qj an individual or from the place of units to that ^„^^ ^^ ^j^^ ^^^^^^ ^^ ^^„^ ^j^,^ .^ -^ ^j^^ ^^_ of tens, takes to itself the addi- pression of ten individuals or ones ; there" lion of 9 times its value. The fore taking 1 (one) its sigvijicaucn in units same from tens to hundreds, See. P'^<^ce,from 10 (ten) iis sigwfcaiion in ttnt jilace^ leaves 9, the increase 0/ \y or 9 timci Vsvaluc^ in rising from the place of units to that of tens. * A; removed from unit% filace by a cufiher is 40, ivhlch divided by 9 leaves 4 (^4 times 9 is '^6. J X 6 removrd by a cypher is 60, tvlnch di- vided by 9 leaves a remainder of 6 ; cr 600 divided by 9 stilt the remainder is 6, the re- mainder always begins the same figure what' ever may be the place of its removal if divided by 9. tt Thus, 5683 divided by 9, the remainder is 4 ; let the figures ivhich express the num- ber 5G83 be added together — 5 to 6 is 1 \,and 8 is \9and 3 is 22, ivhich number (2"!) divi- ded by 9 leaves a remainder o/"4, tlie same as ivhen the number 5683 was divided by 9. Thls-e properties of the figure 9 belong to none other of the Digits, cx- ciptingto the figure 3, and this figure (3) possesses them inconsequence '•:)>- (.'1 beinr an even part of 9. Consequently, if any figure for instance *4 be removed from units place and divided by 9, it will leave a remainder of 4 ; the Mune of any other X figure, re- moved and divided by 9, it will leave a remainder of itself, aftd that only. Theuefore, itanyttnum- l)..i- be divided by 9 ; or, tlie fig- ures which express that number be added together and the sum of them divided by 9, the re- mainder will be equal. 26 SIMPLE MULTIPLICATION. Sect. I. .3. eTactly nine^ neglect it, and begin again ; S to 9 twelve ; ngain, drop the num (iiodve) and add the figures (12) which would express it, 1 to 2 is 3 and 4 is 7, which sum (7) is the remainder after the last addition, or the thinp- soLiijht, and is the remainder that would be left after dividing ihe sum 57 6394 by b. To Proi^e Mulnplication. Cast the 9's out of the MuUi/itkand by the foregoing method, and mark down the remainder ; cast the 9's out of the Mii/d/iiier, mark the remainder, then multiply tht- remainder first obtained by this last remainder, and cast the 9's out of the //ror/MCif ; also, cast the 9^ s out o^ iht answer ov product of the Multiplicand and Muliipiicr, then if these two last remainders correspond, j^ the work is Mipposed to be right. EXAMPLES. Let 7 6 6 3 2 be multiplied by 6i. Cast out the 9's from 7 6 5 5 2 Remainder 5? Remainders multiplied . from 6 5 Ijtemainder 23 together. 3 8 2 6 5 10 9's from 10 Rem, 1 f Corresponding 4 5 9 18 12 ^ with each oth- er. 9*soutof4 9 7 4 4 6 3 Remainder 1 There is nothiiK^ more easy than pi'oviiig Multiplication by this method, so soon as the Scholar bhall have given it sucii attention, as to make it a lil- xW familiar. Note. Should the Multiplier or Muliiplicand, either or both, be less than 9, they are to be taken us the remainders. The examples which follow are to bt wrought and proved according to the illustrations already given. 4. Multiply 6 ' 2 3 7 5 proof. By 8 4 5 2 3 9 5 ProducL 5. Mult. 3 7 8 4 6> Bv 2 3 5 WVct/z^c-:, S893910. Sect. I. 3. SIMPLE MULTIPLICATION. 27 6. What is the product of 14356 multiplied by 648 ? Jus. 9302688. 7. What i« the product of .939 £6 multiplied by sroi ? Jyis. S 1 7793024, Multiply 34623217' , , •• ,,>^„-m«,^. iiy 9 6 4 8 4 J ^'''^^'Jy SS4058579364, 28 SIMPLE MULTIPLICATION. Sect. I. 3. Contractions and Varieties in Multiplication, Any number which may be produced by the multipliGation of two or more numtiCrs, is called a composite number. Thus 15 which arises from the mul- tiplicf.tiun of 5 and 3(3 times 5 is 15) is a composite number; and these luimb^rs, 5 and 3 yre called Comfionent parts. Therefore, 1 . If the Midciplier be a Composite number — Multiply first by one of the component purls, and that product by the other ; the last product will be thi^ iwer sought. 1. Multiply 6 7 by 15. OPERATION. 6 7 EXAMPLES, 5 one of the component parts. 3 3 5 3 the other component part. 10 5 Product of 6 7 multiplied by 15, 2. Multiply 367 by 48 Product, 17616. OPERATION. Consider first, what two numbers multi- plied together will produce 48 ; that is, what ara the component parts of 48 ? Answer 6 and 8(6 time? 8 is 48 ) therefore, multiply 367 first by one of the component parts, and the product thence arising by the other ; the last product will be the answer sought. S. Mult. 583 by 56. Prorf, 32648. 4. Mult. 1086 by 72. Procf. 7819 :i OPERATI-ON, OPERATION. 2, " IVken there are cyphers on the right hand of either the Multiplicana or '^ J\^Uiplier, or 6o//^, neglect these cyphers ; then place the significant fig,- *' ures under one another, and multiply by them only ; add them together *' as before directed, and place to the right hand as many cyphers as there aje *' in both the factors." Sect. 1. 3. SIMPLE MULTIPLICATION. 29 EXAMPLES. Muliiply 65430 by 9200, OPERATION. 6 5 4 S 5200 Mere in the multiplication of 65430 by 5200, th-e cyphers arc seen neglected, and regard paid only to the significant fig- ures. To the product are prefixed 3 cy» pliers equal to the number of cyphers ne£?lccted in the factors. 3 4 3 6 2. Mult. 3 By 6 5 7 3 3. Mult. 78000 by 600. Frodiict, 46800000, Frod, 26645000 3. IVhen there are cyfihers between the significant fgures of the ATulti filter g emit the cyphers, and multiply by the significant figures only, placing the first figure of each product directly under the figure by whicli you multiply, and adding the products together, the sura of them will be the product of the given numbers, EXAMPLES. 1. Mult. 154326 by 3007, OPERATION. 15 4 3 2 6 3 a 7 10 8 2 8 2 4 6 2 9-78 464058283 In this example the cyphers in the multiplier are neglected, and 154326 multiplied only by 7 and by 3, taking care to place the figUKe in euch product directly under the figure frQra which it was obtained. 10 4 4 I 30 SIMPLE MULTIPLICATION. Sec. I. 3J 3. 48976850 4 3 195922O93@55O0 4. WhrH the Midlifilier is 9, 99, er any number of9's ; annex as many cy- phers to the MullipJicand and from the number thus produced subtract the multiplicand, the reri^ainder will be the product. EXAMPLES. I. Mult. 6547 by 999. OPERATION. 6 5 4 7 Write down the Multiplicand, place as many 6 5 4 7 cyphers to the rii^ht hand as there are 9's in the multiplier for a wmw^72c/, underneath write agsin 6 5 4 4 5 3 the mulliplicand for a subtrahend^ subtract, and the remainder is the product of 6547 multiplied by 999. 2 3. 6473 \ Pr&ducl, 640827 ' 7021 99S 99 Prod. 695079 4. 8 4 9 7 6 7 Proc?. 5 3844S75C24 9 9 9 9^ Sect, J. 3. SUPPLEMENT to MULTIPLICATION. 31 Supplement to JJlMtipCattOn. QUESTIONS. 1 IV/iat is Simple MuUiJilicadon ? 2. //ocy many numbers arc required to fierforin that oficralion ? 3. CoLLKctiYELT^ OY together^ ivhat are the givai numbers called ? 4. SEPARAfEir vjhat are thcij called? 5. What is the result^ or number sought, called ? 6. In \:hat order must t/ie given numbers be placed Jar multi/ilicaticn? 7. ffo:rdo ijou firoceed when the Multi/ilier is less than 12 ? 8. When the multiplier exceeds 12, rohat is the method of procedure ? 9. WnAf is a composite number ? 10. What is to be understood by the Component parts of any number ? 3 I. How do you proceed -when the Multiplier is a Composite Jiumbcr ? 12. When there are cyphers on the right hand of the multiplier^ mulliplicandy 13. Whet^ there are cyphers between the significant figures of the multiplier* how are they to be treated? . 14. When the multiplier consists oj 9*s how may the operation be contracted ? 15. Ho IV is multiplication proved ? 15. By what method do you proceed in casting eut the 9*8 from any number ? 1 7. HoJV is multiplication firoved by casting out the 9'« ? 18. Of%vhat use is Mulli/dication ? EXERCISES. 1. What sum of money must be divided between 27 men so that each may receive 115 dol- lars? vc/;j*. 3105. J^ofE. The Sc!ioIar*s business, in all questions for Aiiihmciical operations, is "whol'y Avith the numbeis j^iven ; these arc never less than two ; tlicy may be mort ; and these numbers, in ont way or aviilher^ are always to be made use of lo find'. Subtract. [Ji. BrinjfV.own. v\ 34 SIMPLE DIVISION. Sect. I. 4. It is sometimes practised to make a point ( . ) under the figures in the Di- Tidend as they are brought down, in order to preveiit mistakes. When the divisor is a large number it cannot always certainly be known liow many times it may be taken in the figures which are assumed on the left Jiand of the dividend till after the first steps in division are gone over, but the learner must try no many times as his judgment may best dictate, and after he has multiplied, if the product be greater than the number assumed or that number in which the divisor is taken, then it may always be known that the quotient figure is too large ; if after he has multiplied and subtracted, the re- mainder be greater than the divisor, thea the quotient figure is not large e- nough ; he must then suppose a greater number of times, and proceed again. This at first may occasion some perplexity, but the attentive learner after some practice will generally hit on the right number. 2. Let it be required to divide 7012 by 52. OPERATION. In this operation it is left for tlie Schola-r to trace the steps of procedure without hav- ing tliem particularly pointed out to him \jy words. Ml visor. 5 2 Dlvideyid. ) 7 1 2 5 2 1 8 1 1 5 6 Quotient. (13 4 2 5 2 2 8 4 4 Retjiainder. TROOr. Division may be proved by Mul tiplication. RULE. « Multiply the Divisor and Quotient topjether, and add the remainder, if «« there be any, to the product ; if the work be right, the sum will be equal *« to the Dividend." Take the last example. The Quotient was 134^^^^^j j^^^^^^ j^^^^ 1 he Divisor 52 > ^ ^ '^ 268 670 44 Remainder added. 7012 Equal to the Dividend. Another and more expeditious way of proving Division it By casting out the 9V W Cast out the 9's from the Divisor and the Quotient, multiply the results* and to the product, add the remainder if any :\fter division ; from the sum of these cast out the 9's, also cast out the 9's from the Dividend, and if the two last results agree the work is right Sect. I. 4- SIMPLE DIVISION, 3. Divide 1/354 by 86. OPERATION. PROOF. Divisor. Dividend, Qjwtient. 9^^ onio^ fDivls.J 86 Bern. 5 7 Multiplied lo- se)! 7 3 5 4(2 1 (Quot.) 201 Rem. 3 5 gether. 1 7 2 • • 1 5 4 8 6 6 8 Rem. A. Divide 153598 by 29, OPERATION. 2 9) 15359S ( 15 Remainder 68 added. 9 make one ^ Ounce oz. 12 Ounces J / Pound lb. EXAMPLES. 1 lb. oz. pwts. grs. 7 10 13 4 Because 24 grains make a 3 9 7 16 pennyweight, you carry one to 2 8 © 5 the pennyweight column for ev- -^ 3 6 2 ery 24 in the sum of the column f/\/y'^ t t " SC" "Pj of grains : because 20 pcr^y- lOy II 7 lj7 weights make 1 ounce, you carry — 4> ^ for 20 in pennyweights, and be- cause 12 ounces make 1 pound ■ — — you carry tor 12 in the ounces. This is called carrying accord- ing to the value of th« higher place. 2. 3. lb.' lb. oz. pwt. 1 6 1 7 1 9 6 5 6 2 8 1 4 3 7 19 6 s 6 3^ f 7 oz. 7 pwts. 1 4 2 S 2 6 1 1 1 3 5 1 1 2 7 53 ^.. / U / y^ /y/^ S [y 35 S J Note. The fineness of gold is tried by fire and is reckoned in ccra^*, by which is understood the 24th part of any quantity ; if it lose nothin«g in the trial, it is said to be 24 carats fine ; if it lose 2 carats, it is then 22 carats fiiie, which is the standard for gold. Silver which abides the fire without loss is said to be 12 ounces fine. The standard for siver coin is 1 1 oz. 2 pwts. of fine silver, and IS pwts. of eopper melted together. Sect. 1. 5. COMPOUND ADDITION. 45 3. OF AVOIRDUPOIS WEIGHT. By Avoirdupois weight are weighed all things of a coarse and drossy nature, as tea, sugar, bread, flour, tallow, hay, leather, and all kind of metals, except gold and silver. TABLE. 16 Drams dr, "^ f Ounce, marked, oz, 16 Ounces | | Pound, — lb. 2g Pounds J>make one<^ Quarter of a hundred weight, yr. 4 Quarters | | 100 weight, or 1 12 pound, civt, 20 Hundred weight J L'^'on, — — T, EXAMPLES. I T. civt. qr. lb. tz. dr. 186 3 225 11 8 417023 7 6 9837 2 5 2 3 1«16 5 II toz /3 77? Id 7^ 2. T. cwt. yr. lb. ez. «^. 8 1 3 2 2 5 1 I 8 7 1 9 3 1 4 5 ^ 8 6 2 6 1 5 3 7 1 6 i 9 9 (? /t 3 Itf (1^ / 9 r ? 10 /i y 8i d rL s /s 8 / Note. " 175 Troy Ounces are precisely equal to 192 Avoirdupois Ounces, .1 to U4 Avoirdupois. 1 lb Troy=:5760 y;rain», and 175 Troy pounds are equa Mud I lb Avoirdupoisiz:7000 grains /' 46 COMPOUND ADDITION. Sect. 1. 5 4. OF TIME; TABLE. • 60 Seconds 9. •" f Minute, marked^ m. 60 Minute* 1 Hour, h. 24 7 Hours Days i>makeone^D^Jj^^ d. IV. 4 Weeks 1 1 Month, mo. 13 Months kf. if U.J 1^* Julian year. Y. PVAMPLES. r. mo. «>. u h. m. ». 1 6 1 3 6 2 3 5 7 4 3 2^8 7 2 5 1 6 2 8 3 2 3 9 6 1 3 1 7 3 8 1 1 8 7 4 1 1 4 1 5 1 7 i/i 3 u /^ '/y^ ISS iS- ^ 2^ ^ /7X :3 ^ 4^ ^ /^ ^.3 F. 77219. W. make onemake one o E. E, 1 9 5 6 7 6 3 1 8 /^.f 2. qr. 3 1 2 2 /{^S- \JS J ^ o ^v3 COMPOUND ADDITION. Sect. 1. 5. 7. OF LONG MEASURE. Bv Lonj^ Treasure are raeasured distances, or any thing where length is con^iderfcd Avithout regard to breadth. TABLE. 3 Barley corns bar. "^ rinch marked m. I'i Inches Foot, /^ 3 Feet Yard. VcT. 5-^ Y^rds, or 16.V feet Rod, Perch, or Pole,/zo/. 40 Poles ^make oner. 8 Furlongs Mile, C Degree of a Thile 691 Statute miles nearly I great Circle ^ A great Circle deif. 30O Degree* -< L ^ of the Earth. EXAMPLES • Drg: ml. >r. 1- ft. in. bar 16 8 5 7 7 2 6 15 11 2 12 4 5 3 6 1 8 7 6 1 7 9 3 6 1 7 9 1 4 7 3 3 2 1 ^;/ /^ ^ 13 3 6 1 ^^(^ l/i z. 16 3i 6 Z d77 /6 Z. /5 / 2. Brg. wf. fur. fioL Z'^. in. 1 3 5 6 5 1 3 8 1 4 9 1 8 1 2 7 1 6 Q 2 6 7 1 2 3 1 6 9 2 9 8 5 3 1 J^^ 2St ^ i^3 3 4^ ■ j^r 3S> r ^ J/t .3 J>"^ '^"rt L - Z3 ~~^3 ?■ Sect. I. S. COMPOUND ADDITION. 8. OF LAND OR SQUARE MEASURE. By Square Measure are measured all things r.lvat have length and breadth. TABLE. 144 Inches Lmake ! fSqu are foot, 9 Feet -Yard, 30\ yards, or") 272-L Feet 3 40 Poles one or 4840 yards 3 /^ _ -Acre, 640 Acres J L — -Mile. EXAMPLES. Jcres. rood. poL Jt. i?i. 3 7 6 3 3 6 9 3 1 2 I 5 6 8 1 2 7 5 8 7 6 2 4 7 2 3 5 6 I 2 4 7//J /^' lid ;/ 816 ix in ^'^ //9S IS 213 // 9. OF SOLID MEASURE. By Solid Measure are measured all things that have length, breadth, and thickness. TABLE. 1728 inches f fFoot. 27 Feet | | Yard. 40 Feet of round tinxjjer, or > J make one J r^ r r.^,\ 50 feet of hewn limber S \ ? ^"" °' ^''^^' 128 Solid feet, i.e. Sin leuirth? \ r- ^ e ^xt \ 4 in breadth, & 4 in heighi \ J [^^^^ ^^ ^^°°^- EXAMPLES, 1. 2. Ton. y?. z«. Con^ ft. in. 65 37 229 39 118 1021 19 2 6 1207 3 SO 437 36 17 54 18 72 65 9 57 38 629 86 124 W Jd~~JW6 Tt 7/ SL^ W lu; SI sb~rito JTl To VJU Tf TT 'TT'^ G so COMPOUND ADDITION, Sicr. I. 5. 10. OF WINE MEASURE By Wine measure are measured Rum, Brandy, Perry, Cyder, Mead, Vinegar and Oil. TABLE. 2 Pints fits. r 1 Quart, marked qts. 4 Quarts Gallon, gal' 10 Galk)ns Anchor of Brandy, arte. 18 Gallons ■»- Runlet, ru?i. ,1^ Gallons < make one<| Haifa 1 tiogshead Ihhd. 42 Gallons Tierce, tier. 63 Gallons Hogshead, hhd. 2 Hogsheads Pipe or Butt, P. otB. 2 Pipes J Ltun, T, EXAMPLE 1. Hhd. gal. y^*. fits. 3 9 5 2 3 I 1 6 2 7 1 3 5 1 2 1 2 9 5 8 2 nil ^30 8t \U 3 1 T. hhd. ^ S 2 S 5 1 1 7 2 3 2 ^ 5 gal. qts. fit*. 5 8 3 1 3 Q 1 2 9 2 1 12 1 //^ / II 77 IS / 114 I II N. B. A PINT ^vine meaaui'c, is 28J cubic inches. Sect, 1. 5. COMPOUND ADDITION. 51 "2 Pints 4 Quarts 8 Gallons 8|- Gallons 9 Gallons 2 Firkins 2 Kilderkins li Barrel, or 54 Gallons 2 Barrels ;3 Barrels,or 2 hogshead^ Jl. OF ALE OR BEER MEASURE. TABLE. Quart, marked 9?*- Gallon, gal- Firkin of Ale in London, A. fir. Firkin of Ale or Beer, •i makeonc .^D ADDITION. EXAMPLES. 2. Sj ECT. 1. 5. Bufi, ilk. qta. ptB. Ch. tus: /2^. 9?*. 2 7 2 6 1 3 7 1 6 2 5 1 8 3 7 2 6 2 8 3 7 2 1 1 1 8 1 5 1 19 1 3 t\> ^7 ^ ^ /i? / 3. 1 17 2 5 irr i^ '} i N. B. A GALLON, Dry Measure, contains 268 ^ cubic inches. The folloxmn^ are denominations of things counted by //^ Table. * 12 Particular things raake 1 Dozer, 12 Dozen 1 Gross, 12 Gross or 144 dozen great Gross. ALSO 20 Particular things make one Score. Denominations of measure not included in the Tables. 6 Points make I Line, 12 Lines — Inch, 4 Inches — Hand, 3 Hands — Foot, 66 I'\;et, or 4 Pole^a Gunter's Chain, 3 Miles League. A Hand is used tp measure Horses A Fathom, to measure depth*.— A League, in reckoning distances at Sea. N. B. A Quintal of Fish weighs I Cwt. Avoirdupois. Sect. I. 6. COMPOUND SUBTRACTION. ^ § 4. Compound ^Subtraction, Compound Subtraction teaches to fuid the difference between any tw« sums of diverse denominations. RULE FOR COMPOUND SUBTRACTION. "Place those numbers under each other, which arc of the same denomi- <'-.nation, the less bc'un^ below thegreatei'; begin vith the least dcnomina- " nation, and if il exceed tlie figure over it, borrow as many units as make one « of the next greater ; subtract it therefrom ; and to the difference add the " upper figure, remembering, always to add one to the next superior denomi- " nation, for that which you borrowed. Proof. In the same manner as simple Subtraction. 1. OF MONEY. 1. Supposing a man to have Ient;6*l85 10s. 7d. and to have rccerved again of his money, ^93 1 5s. how much reraaisns due ? I.ent Received OPERATION. 1. 18 5 1 0, 9 3 15 d. 7 From Take 2. 5 1 8 5 9. 1 5 Due 9 1 1 5 7 Zf7[ 00 Proof 1 8 5 1 % 7 3. Lent 6 3 7 1 7 8 f 1 6 3 Received | 7 8 at C'195 13 U how much is there yet due ? ^ , 4ns. £i7^ 12 7. e^r 21^ // i_ ^^ >t-cl.i ^78 /Z 7 2. OF TROY WEIGHt. From 7 6 Take 3 1. or. 8 9 inn. 1 6 I 7 5T. 1 3 6 3. lb. OT. 7 3 2 8 fiwt. 5 9 Remain* ^ 10 /^ -I'r " 4 6 /^ Proof f ^ S /6 !■& r'±. r 5. OF AVOIRDUPOIS WEIGHT. ■f- 1. o /^. cz. dr. T. cwt. gr. lb. 9 1 5 5 6 1 1 1 1 5 • « 7 1 5 I 1 dr. 3 8 4 8/^ S S 3 Z^ /<) II 9 /r r Y. 7)19. S 9^ . 6 . I 6* ^ 9 1 4. OF TIME. 1. d. h. 6 2 2 18 • 4 4 5 45 ^5 9 A 5 Z v fZ ? X ^ / ^^ ^^ SfiCT. I. 6. COMPOUND SUBTRACTION. 55 5. OF MOTION. I. 2. • 1 6 8 2 r 3 4 ft S 3 2 3 • 6 8 3 9 5 1 5 r 7 53 ■""27 ■ ' 35 2 ^^' 6" (5* --JT OF CLOTH MEASURE. 1. 2. ^rf». yn w. -E.£. qr. Hi 2 7 12 2 6 2 1 16 13 17 3 2 7. OF LONG MEASURE. ^^S' mi. fur. p. yds. ft, in. bar, ^6 13 526 2 1 8 1 ^ '^ 15 2 27 1 2 9 2 Ti A3 J 26 Z~7 ^/ f . OF LAND OR SQUARE MEASURE. 1. 2. A. R. fiol. fiol. fi. in 17 117 is 16 11 16 1 16 10 201 130 / I J__l6±_ Z£ 5^ COMPOUND SUBTRACTION. Sect. 1 6. 9. OF SOLID MEASURE. 1. 2. Tons. ft. in. Cords. ft. in. 45 2' 9 186 68 23 810 1934 1237 6 127 1529 50ZZ1ZZ IlZZMIUIl 10. OF WINE MEASURE, hhd. 6 6 1 7 I Tl < .V sal. gfs. Uhd. 3 1 2 7 5 3 3 3 2 4 1 gal. 1 6 4 3 ^^ "gg 3 ■ 7g 3 3^ II. OF ALE AND BEER MEASURE. 1. 2. Hhd. sal. §r/*. Butt. hhd. .^a/. 8 9 1 9 2 6 S 1 1 6 3 7 25 S 29 1 1^" S} ^r 3 34 I 12. OF DRY MEASURE. I. 2. ^"- y^-^-. ?^9. CkaL bu. Jik. 6 1 1 2 1 7 I I 3 1 5 14 7 6 2 3 2 ^S S 6 • __.f7^ '17 3 THE SCHOLAR^S ARITHMETIC OBSERVATIONS. HE Scholar has now surveyed the ground work of Arithme- tic. It has before been intimated, that the only way in which numbers can be affected is by the operations of Addition, Sub- traction, Multiplication and Division, These rules hav^ now been taught him, and the exercises in a supplement to each, suggest their use and application to the purposes and concerns of life. Further, the thing needful, and that Which distinguishes the A- rithmetician, is to know how to proceed by application of rZ^d-i-^? four rules to the solution of any arithmetical question. To af- ford the scholar this knowledge is the object of all succeeding rules. ^ -::::< -;:?■ — — SECTION II. Rules essentially necessary for cucry person to fit and qualify them for the transaction of business, Tkese arc nine : reduction, fractions,* federal money, interest^ COMPOUND MULTIPLICATION, compound DIVISION, SINGLE RULE OF THREE) DOUDLE RULE OF THREE and PRACTICE. A THORouoH knowledge of hiese rules is sufficient for every ordinary oc- curntnce in life. Short of this a person in any kind of business, will be liable to repeated embarrassments. It is the extreme usefulness of these rules which commends them to the altenUQU of every Scholar. * Fr ACTIOS s are tairn u/i here no further than is necessary to shew their tiff" nification^aihd toilluutratc the /irinci/ilci of FMOMJiAi JMoNgr, H SS REDUCTION. Saci. II. § 1. i^^tjuction- ^j^-v::-4::-v^< " RiJDt'CTioN teaches to bring or exchange numbers of one denommalion t' to otheis of different denominations, retaining the same value." // is of two kinds, I . When high denominations are to be brought into lower, as pounds into shillings, pence, and fartJiings ; it is then caiicd reductio-n descending-, and is ptrformed by MulUjilication. II. IVhen lower deriominations are to be brought into higher, as farthings into pence, or into pence, • shillings and pounds ; it is then called reduc- tion ASCENDING, and is performed by Division. Reduction Descending, RULE. Multiply tlie highest denomination by that number which it takes of th» next It SS to make one of tirat greater ; so continue to do, till you have brought it as low as your question requires. Pkoof. *' Change the order of the question, and divide yonj? last product by the labt multiplier, and so on.'*i E2?AMPLES. 1. 1^ £\7 13s. 6rf. "qrs. how many farthings ? orERATTON. £. s. d. g.rs. In this example, the highest, denonl - 17 13 6 3 tnalion is pounds, the next less, is shil- 2 Shillings inpoimd. Hngs, and because 20 shillings make — one pound, therefore, I multiply ^'17 by 3 5 3 Shillings In £\7 13s. 20, increasing the product by the addi- l 2 Faice in a shilling, tion of the given shillings, (13) which it must be remembered, must always be 4242 Fence ill jQ\7 ISe^. 6c?. done in like cases ; then, because 13 4 Farthings znaj2enny. pence make one shilling, J multiply the .^ shi-Iings, (353) by 12, adding in the §^iv- ./T71SA & 9 7 1 Farthings en pence (6f/.) lastly because 4 farthings make one penny,! multiply the pence (4242) by 4 and add in the given farthings (Syr^.) I then find, that, in 17 '35. ed. b(jrs. there are 16971 fratbings. '^■ PROOF. 4) 1 6 9 7 1 To prove the above question change the ordes^ — — of it, and it will stand thus j in 16971 farthings, 12)4 2 4 2 3_:^rs. hov/ many pounds ? Divide the last product by the last multiplier, 2jO) 3 DiS 6d. the remainder will be fart.'iings. Proceed in tlus — v/ay till iiii the steps oC the operation have been £1 7 IZs. retraced back ; the last qiftotient with the remain- ders will be proof of the accuracy of tlie opera- tion if the j? agree ^\ilh tlae sum given in the ques- lioG. I Sect. If. L REDUCTION., 69 2. l^ £7 \A5.6d. I^r. how marif .3. In ^7 6s. Ad, how many pence ? Earthings -f* J?}». 7417 ^rs, ^ Ans, \7o^d. 7./^.6// ■ 7.6.4 2<'o 4. In 29 Guineas, at 28s- how many 5. In ^173 15?. how many sixpen- farthln^^s ^««. 3§9 76 ^s. ces ? ^^jy. 69oO. Y. r.>r 671 eai>Ies, at 10 dollars, each, how mmy shillings, threepences, pence 6. In 12 crowns, at Cf7^ how and favlliiii^-s ? Jwi. A02Zd ff-.iU. liny pence and lunliinj^s ? 1G1040 three fienccsy 4^^120 /. i Ann. 94J3 d. 37^2 (jn- 193?4ill> fjrs^ 6 / 6 "TWa — ^ 60 REDUCTION. Sect. II. .1 Reduction Ascending. RULE Divide the lowest denomination given by that number which it takes of the same to make one of the next higher, and so continue to do, till you have brought it into that denomination which your question requires. EXAMPLES I. In 16971 farthings how many pounds ? OPERATION. Farthings in a ^lenny 4)16971 Pence in a s/iiiHng 12)4242 Sqrs. JShiilings in a/iound 2/0)35)3 6d. j^ns. 17 136(. 6d. Sgrs, ■■%: Reduction descdfifiing and as- cending reciprocstSBbrove each other. ^ 2. In 1765 pence, how many pounds ? jins. £7 7s. \d. 4. In 38976 farthings, how many guineas ? Ans. 29. t9 ^j^- 4. In 6950 sixpences, how many pounds ? Ans.£\7o. I5s 5. In 3792 farthings, how many crowns Ans, 12. I S8 Sect. II. 1, REDUCTION. 61 6. In 48960 farthings, how many pence, 7. In 6952 three pences, how three-pences, six pences, and dollars. many pistoles at 22s. each ? jins» l'2'24=0 pence, 4080 i/irec'/icnces, jins.79, 2040 six'pences. 170 chllars^ 6) 1020 ^ ^'V^' lyo . /9 8 Reduction Ascending and Descending. 1. BIONEY. 1. In 57 nioidores, at 36s. each, how many dollars ? Ans. 342 dollar: 3 3^Z In this question, the first s;tep will be to bring the moidores inco shillings I lastly bring the shillings into dollars. 2. In 75 pistoles ho\v many pounds ? Ant. £82 lOs. 3. In ^73 how many guineas. uiJis. 52 ^tiinea/if 4*. Z0 /40 V 60 56 >/^//. 4. In ^63 and 5 guineas how many dollars ? Am. 233 dollart. 2.5. — 1^' ^L 63 ~nnT?r ^ 62 REDUCTION. Si:ct. II. 1, " JVhen it is required to hioiv hoto many sorts of coin of different values^ and of equal number^ are contained in amj number of another kind / reduce the seVf cral sorts of coin into the lowest denomination mentioned, and add them to- gether for a divisor ; then reduce the money given, into the same denomintv- tion for a dividend, and the quotient arisiinj from the division will be the number required." - 172 moidores, how many eagles, dollars and nine-pences, of each the like Ttumber ? Jns, 92 of each ^ and 68 nine-^iences over, 56^. /0S2. rzo9 • Ecr. IL 1. REDUCTION. TROY WEIGHT, 1. In Aid. 5oz, \6Jiwis. how many grains ? lb. 4 12 OPERATION. oz. /iwts. 5 16 oz. in a fiound^ 53 20 Ounces, fiivts. in an ounce. 1076 24 Penny weights, grs. in one fiivt. Fr 00/, 4364 2152 24)25824 20)1075 (xrainsy the answef^ 16 /iwts. 12)53 5oz.. 4 lb. C3 %. Ik \Olb. of silver, how many spoons, each weighing 5 az* \0 fit^Si Amt'%1 s/ioontt) and 9Q fiwJs, over, /.^ »t If) 2.0 120 I I Out: / 20 • ^ i k>) 14 olo , .. A 64 REDUCTION. Sect. IL 1 5. In 45681 grains of silver how many pounds? OPERATION. 20 12 34)45.68T(1903('95(7/^>, Jnsiver, 7}h . l\oz, Spivts. 9grs, 24 180 84 12 216 103 11 oz. 95 21S 100 20 081 003 /iivts» 1903 72 24 09 grs, 45681 Proof. 4. In 4560 grains of silver, how many tea-spoons, each one ounce ? Jns. 9^tea-s/2007i8. 2J 6 3. AVOIRDUPOIS WEIGHT. Civt. qr. lb. oz. I.In er 1 13 11 how many drams ? 269 28 2165 538 7545 16 452*1 7545 120731 16 724386 120731 1931S96 PllOOF. 16)1931696 16)120731 11 02. 28)7546 4)269 13 lb. 1 qr. «7 / Civt. Sect. II. 1. REDUCTION. 65 2. In 14048 oz. how many hundred weight ? Ana. 7C. Sqrs. 10/6. 2ff ifri'^ O^i. 9*- ^ J \ J) /I ' ' 'a/j <;a> 3. In 470 boxes of Sugar, each 26/6. how manyCwt. ? Ans. 109 C. oqra. \2lb 2810' fs Ton 8U m. 4. In 17Cwt. Iqr. 6lb.of Sugar, how many parcels, each \7 lb.? t/J.6 . 4l 275 _L2L €<9 66 i REDUCTION. Sect. 4 TIME. 1. In 121812 seconds, how many hours! OPERATION. pRooy. 6)0)121812 6lO)203[0 12 sec. H. m. s. 33 50 12 60 Ans. 33A. SOm. 12s. 2030 60 \ 121812 2. Supposing a man to be 21 years old, how many seconds has he lived, allowing 365 days, 6 hours to a year. Ans. 66^2709600 seconds. 750 d 7S6 60 sTsTtToc tTzToJJc7 c^47^ - 5. How many minutes from the commencement of the war between A- merica and England, April 19,1775,10 the settlement of a general peace whichtookplace, Jan. 20, 1783 ? Ans. ^^7^ \^0 minutes. ^ _I&1. ^rti^^ T¥J6 wv. ^0 7 9 1' 6 a Sect. II. 1. REDUCTION. 4. In 413280 nJnutcs liow many weeks? Jins. 41. ^8 IG8 /6(9 67 5. LONG MEASURE. 1. Reducji 16 miles to barley-c9rns. OPERATION. 1,6 Miles. 8 128 Furlongs. 40 5120 R9d8. 2560Cr 2560 28160 Yards. 3 1>ROOF. 3)3041280 12)1013760 3)844SX) t 11)28160 2560 2 4|0)512[O 8)128 84480 Feet 12 1013760 Inches. 3 16 Miles. t Divide by 1 1 for 5^ and multipljr the quotient by 2. The reason is be- cause 5 } reduced to half yards is 11. Ans. 2Q^\2tQ Barley Corns. *To multiply by one half{^) it is only to take half the Dividend. 2. In 47530 feet how many Leagues ? Jins. 3 Leagues. Z 4i 5 68 REDUCTION. Sect. II. 1. .1. How many times does the Wheel, which is 18 feet 6 inches In circuTn- ference, turn round in the distance of 150 miles? 'U- /*^ .^^5. 42810 ;/mf5, anci 180 znc/zM f9rv-r„ —^-4^ / /20C 1» // /_%_ /^ -- ,/ S (9 (9 * * * ^ f 900 2^0 4. How many barley-corns will reach round the Globe it being 360 dc: grees i «fns. 4755801600 .i^^ t G9-^ 2.1 BO I so 8 ZCo I 60 2/^05 rZ 60 3 /2. 3 1/TsTsoi boo u4ii. Sect. II. 1. REDUCTION. 6. LAND OR SQUARE MEASURE. ^. In 13 acres, 2 roods'^iow many poles ? ^ OPERATION. Jc. r. 13 2 S4> 40 PROOF. 4'0)2l6iO 4)54 13 ^c. 2 roodi. Ans. 2 1 60 Poles 5. In 2852 rods how many acres ? Ans, \7 A, 3 R. \2 P., 9. SOLID MEASURE. 1. In 1296000 solid inches, how many tons of hewn timber ? oreration; 5,0 1728)1296000(7510 12096 the Answer rRoor- 15 50 750 1728 6000 1500 5250 750 8640 8640 00 1396000 I/u-/ic*. 70 REDUCTION. 2. Ix 552960G solid inches, how many cords of wood ? Sect. II: Jns, 25 3^M oo 8. DRY MEASURE. 1 In 75 bushele of corn how many pints ? OPERATION. 4 300 8 2400 2 PROOF. 2)4800 8)2400 4)300 75 bushels. ^ns. 4800 pints 2. In 9376 quarts how many bushels? jins. 293. zss It would be needless to ^ive examples of Reduction in all the weights and measures. The understanding, which the attentive Scholar must already have acquired of this rule, by help of the Tables, will e-ver be sufficient for his pur- pose. Sect. I. II. SUPPLEMENT to REDUCTION. Supplement to IHctlUCtiOn* mm' -;> -r ■!•> -r v.c- -^ >^ -^ -J'.c- ^ .-J. wT 4;;- — QUESTIONS. 1. WHAf is Reducticn. 2. O/- Aow 7nany kinds is Reduction ? nvhat are they called ? toherein do these kinds differ one from the other ? Which of these fundamental r\des are em* ployed in their operations ? 0. How is Reduction Descending performed ? A. Hoiv is Reduction Ascending perjormed? 5. When it is required to know /tow many sorts of coin ^ nveights or measures of different values^ of each an equal number ^ are contained in any other number of another kindy wMt is the method of procedure ? EXERCISES. 1. In 36 guineas how many cEowns ? Jns, 153 croions, and 9d. over. IOCS' ~TiT 2Z/6 r 12 SUPPLEMENT to REDUCTION. Sect. II. 1. 2. How many steps of 2 feet 5 inches each, will it require a man to take, going from Leominster to Boston, it being 43 miles. i/ins. 93948 ste/is ; JCI/**MJ Second 23 dolls. 2/ Third 30 dolU' ^% 6)1 , , , /^ 6/700 Sect. II. 1. SUPPLEMENT to REDUCTION. 73 4. If a vinter be desirous to draw off a Pipe fif Canary into bottles contain- ing pints, quarts, and 2 qiiurts, of each an equal numbtr, how many must he have ? ^ns. 144 gJ each. T )I0 08 , 5. There are three fields ; one contains 7 acres, another 10 acres and the other 12 acres and 1 rood ; how many shares of 76 perches each) are contained in the whole ? Ans. 61 shares and 44 Jierc/ies ever. t 1^ 7 f^ '. /z-/ £9-/ 4^ / /^/^ o-^^-rh ^^ F6) 46 80 [6 J. ^^w^'^^. l%0 r^ f / 74 SUPPLEMENT TO REDUCTION. Sect. II. 1. 6. There are 1061b. of silver, the property of 3 men ; of which A receives \7lb. XOnz. \9/iiv(fi. I9grs. of what remains, B shares lor. 7grs. so often as C shares \3fiwts. W^at are the shares of B. and C ? Ansnver, B*s share 5ilb. ioz. 5/iwts, S^rs, C*s share 24lb. 4oz AB/i^ts, /G6 . 00. 00. 00 , . I />" . / O ' 19 . /? ^ <^X^/^ ii^ to /y ^^^^ 7i*rt5K'*^^ /to I f f.^. Sect. II. 2. FR ACTONS. 75 § 2. fraction,^* — !■■ "5:> -r -,::- ^ -i'.j- -r j^c -/■ -;.s- ^ -v.* ->~ -/.;- •■ i When the thing or things signified by figures are ivhoh ones, then the fig- ures which signify them are called Integers qv ivhole numbers. But ui.eii only some parts of a thing are signified by figures, as tivo thirds of any thing, Jive sixths, seven tenths^ Ij'c. then the figures which signify these f.arts of a thing being the expression of some quantity less than owe, are called Frac- tions. Fractions are of two kinds, Vulgar and Dedmal ; ihey are distinguished by the manner of representing them ; they also differ in their modes of ©pera- tion. \ VULGAR FRACTIONS. To understand Vulgar Fractions, the learner must suppose an integer (or the number 1) divided into a number of equal parts ; then any number of these parts being taken, would make a fraction, which would be represented by two numbers placed one directly over the other, with a short line between them, thus -| two thirds, | three fifths, ^ seven eights, ^c. Each of these figures have a different name and a different signification. The figure below the line is called the Denominator and shews into how many parts an integer, or one individual of any thing is divided— the figure above the line is called the nwnlerator and shews how many of those parts are sig- nified by the fraction. For illustration, suppose a silver plate to be divided into vine equal fiarts. Now one or more ot these parts make a fraclien, which will be represented by the figure 9 for a denominator placed underneath a short line shewing the plate to be divided into nine equal fiarts y and supposing I'-iUo of those parts to be taken for the fraction, then the figure 2 must be placed directly above the 9 and over the line (-Dfor a Numerator, shewing thai ivv© of those parts are sig- nified by the fraction, or t%vo ninths of the plate. Now let 5 parts of this plate, ^hich is divided into 9 parts, be given to John his fraction would be ^five ninths ; let 3 other parts be given to Harry, his fraction would be -i three ninths; there would then be one part of the plate remaining still (5 and 3 arc 8) and this fraction would be expressed thus -i one ninth. In tliis way all vulgar fractions are written ; the Denominator, ornumber below the line shewing into how many parts any thing is divided, and the nuw merator, or number above the line, shewing how many of those parts are ta- ken, or signified Uy the fraction. To ascertain whether the Learner understands what has now been taught kim of fractions, let uf again suppose a dollar to be cut into 13 equal parts; — let 2 of those ])aris be given to A ; 4 to B ; and 7 to C. f A's fraction . -^ Required of the Learner that he should wiite<^ B's fraction . J^ LC's fraction . -f^ It is from Division only that fractions arise in Arithmetical operations : the remainder after division is a portion of the Dividend undivided; and is always the Numerator to a fraction of which the Divisor is the Denomina- tor. The Quotient is so many integers. The Arithmetic of Vulgarl'ractions is tedious and even intricate to beginners. Besides, tlicy are not of necessary use. We shall not, therefore, enter into any further coniiideration of lliem here. This difficulty arises chitfiy from the varie- ty of denominators ; for when numbers are divided into diffvrent kii-ds, or parts 76 DECIMAL FRACTIONS. Sect. II. 2. they cannot be easily compared. This consideration gave rise to the inven- tion of DECIMAL FRACTIONS. Decimal Fractions are also expressions of parts of an integer ; or, are in value something less than one of any thing, whatever it may be, which is sig- nified by them. In decimals, an integer, or the number one, as 1 foot, 1 dollar, 1 year, Sec. is conceived to be divided into ten equal parts, (in vulgar fraaions an integer may be divided into any number of parts) and each of these parts is subdivided into ten lesser parts, and so c%. In this way, the denominator to a decimal fraction in all cases, will be either 10, 100, 1000, or unity (1) with a number of cyphers annexed ; and this number of cyphers will always be equal to the number of places in the numer%M^ Thus, -j^^ -^y^ ^^^^^ are Z^eczma/ i'Vac- /fo?2.T ,OZZ 5. Add six hundred and five ihaii- sandths,and four thousandrfc* and three hundredths \ Suniy 4600,035 ' '2/1 do; OS S'' Note. When the numerator has not so many places as the dononiin.^- tor has cyphers, prefix so many cyphers at the left l.anil ns nvIII uvxkv va tlie defect ; so y^',^ is written thus, ,005, &c. 78 DECIMAL FRACTIONS. Sect. II. 2. SUBTRACTION OF DECIMALS. RULE. « Place the numbers according to their va'ue ; then subtract as in whole numbers, and point oif the decimals as in addition.'* EXAMPLES. 1. From 716,325 take 81,6201. OPERATION. i^ro7« 716,325 ^ Take 8 1,6201 Fern. 634,7049 2. From 119,1384 take 95,91. Bern. 33,2284. 3. What is the difference between 2S7 and 3,1 15 ? Answer^ 283,885 28d;S8S 4. From 67 <.ake ,92 Bern. 66,08 b& f 8 All the operations in Decimal Fractions are extremely easy ; t^he only li- abiHly to error will be in placing the numbers and pointing ofFthe decimals ; and here care will always be security against mistakes. MULTIPLICATION OF DECIMALS. RULE. "Whether they be mixed numbers, or pure decimals, place the factors and multiply them as iu whole numbers." " 2. Point off so many figures from the product as there are decimal pla- ces in both the factors ; and if there be not so many decimal places in the product, supply the detect by prefixing cyphers.'* EXAMPLES. 1. Multiply, ,0261 by ,0035 OPERATION. ,0261 ,0035 1 305 In this example, the decimals in the two factors taken together are tij^ht ; the product falls short of tliis number by four figures, consequently, four cyphers are prefixed to the left hand of the pro- duct. 5O00OL)135//rou'i/r;. Sect. II. 2. DECIMAL FRACTIONS. 79 2. Multiply 31,72 by 65,3 Product y 2071,316 operation; S 1, r 2 6 5,3 9^1 s IS860 1671,5 I 6 4>. Multiply ,62 by ,04, Product ,024^ ,6% .[0%^S 5. Multiply 25,238 by 12, ir J'roduct, S07,li6i6 2^-13 8 S. Multfply 17,6 by ,75 Producty 13,3 80 Id'} Z /3,ZO0 DIVISION OF DECIMALS- RULE. *' 1 . The places of decimal parts in the divisor and quotient counted togeth- er must be always equal to those in the dividend, therefore divide as in whole numbers, and from the rit^ht hand of the quotient, point off so many places lor 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, sup- ply the dtfcct by prefixing cyphers to the left hand. "3. If at any time there be a remainder, or the decimal places in the di- visor be more than those in the dividend, cyphers may be annexed to the div- idend, or to the remainder, and the quotient carried on to any degree of ex- actness." Divide 2,735 by 51,2 OPERATION. 51,2)2, 735(,0534-f 2 560 1750 1536 2 140 2048 EXAMPLES. In this example, there are ^five decimal* in the dividend (counting lUe two cyphers which were added to the remainder of the dividend after the first division) t!»at the decimals in the divisor and quotient counted together may c- qual that number, a cypher is prefixed to the left hand of the quotient. 92 80 DECIMAL FRACTIONS. Sect. II. 2, Ik division of decimals it is proper to add cyphers so lorg as there contin- ues to be a remainder, this however is not practised nor is it necessary ; four, or five decimals being sufficiently accurate for most calculations. 2. Divide 3156,293 by 25,17. Quotient, 1253-f. The Scholar is requested to point the following example as the rule di*^ rects. 4. Divide, 173948 by ,375. ^wo^^nr, 463861-j- nsc MOO -^500 ^00 3. Divide 5737 by 13,3 Quotient ^ 4313534- ~m Divide 2 by 53,1 Quotient , Qo7-\- mr Sect. II. 2. DECIMAL FRACTIONS. 81 REDUCTION OF DECIMALS. CASE 1. To Reduce Vulgar Fractions to Decimals. RULE. Annex a cypher to the numerator and divide it by the denominator, annex- ing a cypher continually to the remainder. The quotient will be the decimal required. EXAMPLES. 1. Reduce 4 to a decimal. 2. Reduce | to a decimal. OPERATION. OPERATION. 5)3,OC,6 ./fniw^r. The numerator in these 7)I,0(,1428+^nff, 3 Operations is considered as 7 ■ an integer, and always re- — quires the decimal point 30 to be placed immediately 28 afte-r it, the cyphers annexed occupy the places of — decimals, the quotient must be pointed off ac- 20 cording to the rule in Division. 14 60 56 5. Reduce i, -J, and ^ to decimals. Antwer9) ,25. ,5. ,7S, ^;a^(.2^ 2)1^^ . m{u zo "^^ -^ M- %o^ TT ov 4. Reduce ^^j, ^^y andTiW to decimals. An^ 1923+j025 ,007974. ff2.9)y*ooa{,ocr9't^ %^o ^^m ) C^ 70 ^. tC I6J % CASE, 2. To reduce numbers of different denominations^ 0ts of money ^ weig/jt and measure^ to their decimal lvalues, RULE. " 1 . Write the given numbers perpendicularly urtdereach Other for divi* " dcnsls, proceeding orderly IVom the least to the greatest. L !2 DECIMAL FRACTIONS. Sect. II. 2 " II. Opposite to each dividend, on the left hand, place such a number for <' a divisor as will bring it to the next superior denomination, and draw a line "perpendicularly beiweenthem. "Ill, Begin with the highest, and write the quotient of each division, as " decimal parts, on the right hand of the dividend next below it, and so on, " tin they are all used, and the last quotient will be the decimal sought." EXAMPLES. 1. Reduce 10*. 6|f/. to the fraction of a pound. OPERATION. 3, 6,75 20 I 10,5625 ,528125 ^ns. The given numbers arranged for the op- eration, all stand as integers. I then sup- pose 2 cyphers annexed to the 3 (3,00) which divided by 4, the quotient is 75^ which I write against 6 in the next line and the sum tiius produced (6,75) I divide by 12, placing the quotient, (5625) at the right hand of the 10 : lastly, I divide by 20 and the quotient, (,528125) is the decimal required. 2. Reduce mal of a pound. Am. ,6r29-f- 3^. 5|d. to the deci- 3. Reduce 12pwts. 14grs. to the de- cimal of an ounce. Ans, ,629 1. 20 \MsS3 6 719+ 2^ 61?/+ CASE 5. To find the value ofanyghen decimal in the term^ of an integer • RULE. IMuLTipLY the decimal by that number, which it takes of the next less de- nomination to make one of that deitoiunraion in which the decimal is given, and cut off so many figures for a remainder to the ri,u,ht hand ©f the quotient, as there arc places in the given decimal. Proceed in the same manner with the remainder, and continue to do so thro* all the parts of the integer, and the several denominations standing on the left hand make the answer. Sect. II. 2. DECIMAL FRACTIONS. 83 EXAMPLES. J. WhatIs the value of ,528125 of a pound ? This question is the first example OPERATION. in the preceding case inverted, by ,528125 which it will be seen, that questions 2 in these two cases may reciprocally prove each other. The given decimal being the deci- mal of a peund, and shillings being the next less inferior denomination, because20 shillings make one pound, I multiply the decimal by 20, and cutting off from the right hand of the Farthings 5,0 product a number of figures, for a re- ^Tzs. 10s. 6|d. mainder, equal to the number of fig- ures in the given decimal, leaves 10 on the left hand which are shillings. I then multiply the remainder which is the decimal of a shilling by 12, and cutting off as before, gives 6 on the left hand of pence; lastly, I multiply this last remainder, or decimal of a penny by 4 and find it to be 3 farthings, without ' any remainder. It then appears that ,528125 of a pound is in value 10s. 6|d. 2. What is the value of ,73968 of 3. What is the value of ,7,68 of a ^ pound? jins.jQ\4, 9lci. -poundTroy I Jns. 9 ozAfiwt. 7 ^"grs. ,75968 ■ .«! Shillings^ 1 ©,5 6 2 5 1 2 PencCi 6,7 5 4 /^. 79J60 ^'^^io ; £ i%So il 18 the iast remainder, ^80 reduced to it» hweat ferm». J fraction is said t9 be reduced to its lowest terms, vjhen there is no number which will divide both tht numerator and denominator without a remainder. TiniSy set f the fraction Its proficr denominator -f^o^, then divide the numerator and the denominator by any number which will divide them both without a remainder, continue to do S9 ts long as any number can be found that will divide them in tfiaC manner. PusS)^A\^^l-li S4/ SUrPLEMENT TO FRACTIONS. Sect. II. 2. Supplement to ^ptaCtlOltJ^t ^- ,M» 'f "• -JU- ® C- v» vU'^r ■! ■ ! QUESTIONS. 1 . JF/f y< y are fractions ? 2. WiiAf are integers^ or ivhole numbers ? 3. Wha'T are mixed numbers 7 4. Cf hcv) many kinds are fractions ? 5. How are Vulgar Fractions written ? 6. WHAris sig7iified by the denoTninat or of a fraction? 7. WnAr is signified by the numerator ? 8. //b«^ ore Decimal Fractions written ? 9. How do Decimals differ from Vulgar Fractions? 10. How can it be ascertained, what the denominator te a Decimal Fraction i^t if it be not exfir eased? 1 1 . How do cyfihers filaced at the left hand of a Decimal Fraction affect its value ? 12. How are Decimals distinguished from whole numbers ? 13. /i^ the addition of Decimals' what is the rulefsr fiointing off? 14. What is the rule for fiointing off Decimals in Subtraction ? In Multiplica- tion ? and in Division ? \S. In what manner is the reduction of a vulgar Fraction to a decimal fierf or me d? 16. How are numbers of different denominations as Jiounds, shillings, pence, life, reduced to their decimal values ? 17. If it be required to find the value of any given decimal in the terms of an integer what is the method of procedure ? EXERCISES, 1. What is the sum of 79-} 6iand In Case 1. Ex. 3c?, under Re- of J when added together ? Operation. duction of decimal fractions the 79,5 6,25 Scholar may notice, that i, I and ,75 — ~- I reduced to decimals are, ,25 ,5 86,50 Jins. and ,75. "When numbers, there- 2. FnoM 17 take I operation. fore, for operations in either of ,75 Ihe fundamental Rules, are in* 16,25 Remainder. cumbered with these fractions J^ Sect. II. 2. SUPPLEMENT to FRACTIONS. S5 5. Multiply 685^ by 5 | OPERATION. 6 8,2 5 5,5 3 4 12 5 1 2 5 i. Divide 26|-by 5^ OPXRATION. 2,5)26,25(10,5 Quotient, 25 h h substitute for them their equivalent decimal fractions, that is, for a ,25 for g- ,5 for 1 ,75 then proceed according to the rules already given for these respective operations in decimal fractions. Many persons are perplexed by occurrences of a Fimilar nature to the ex« fimples above. Hence it is seen in some measure the usefulness of Fractions, particularly decimal fractions. The only thini^ necessary to render any per- son 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 bur- thensome j every scholar should have them perfectly committed. 5. If a pile of wood be 18 feet long, ll| wide, and 7^ high, how many cords does it contain ? jins. \2 cords 68 /eel* 432 inc/^cf^ IS 9 IS to 7,0 f03SO l^^9 6^ A CORD of wood is 128 solid feet; the proporlionscommonly as- signed arc, 8 feet in length, 4 in breadth, and 4 in height. The contents of a load or pile of wood of any din.ensions may be found by multiplying the length by the breadth and this product by the height ; ow, by multiplying the length, breadth, and height into each other. The last product di- vided by 128 will shew the number of cords, the remainder, if any, will J)e so many solid feet. XO 00 ^0 J3Y trt^/tJ*i. The 432 inc/ica in thcfvactiony ,25 of a foot valued accordirff to Case 3, lie due. Dec. fractions. 36 SUPPLEMENT TO FRACTIONS. Sect. II. 6. If a load of wood be 9 feet long 7. What is the value ,725 S^ feet wide, and 4 feet high, how nia- day ? ny square feet does it contain ) ^ns. 17 hours 24 minutes, f4ns. 126 feet, which is 2 feet short of a cord. i± 4'y 1^ 5 IS JL , I 16jO :J/^. 17,4^0 dO V,000 «. What is the value of ,0625 of % shilling ? Ans, 3 farthings. 9. RjiDucE ^Cwt. Qqrs. 7lh. ^oz. ta the decimal of a Ton. 4ns. ,l5334821-{. lb 8, iH).2bn^7f\ 10. 10. Reduce 3 farthing;sto the deci- 1 1. Reduce ^^ to a decimal fracn Kial of a shilling. *^ws. ,0625 tion. Jns. jQ\25. /2L 3. ISLOO 2.^00 Sect. II. 3. FEDERAL MONEY, 87 § 3. ftnttai Mmtp, -^mmm 'Ai -:::- -:'.5- >|c ^> •»/.;- -i:?- <^ — Federal Money is the coin of the United States, established by Congrcss> A. D. 1786. Of all coins this is the most simple, and the operalions in it, the most easy. The denominations are in a decimal firofiortkn^ as exhibited in the following TABLE. 10 Mills ) C Cent, 10 Cents C 1. ) Dime, 10 Dimes C "laKeone \ DoJ,^,.^ marked thus, % 10 Dollars ) C Eagle, The expression of any sum in Federal Money is simply the expression of a mixfd number in decimal fractions. A dollar is the Unit money ; dollars therefore must occupy the place of units, the less denominations, as dimes, cents, and mills, are decimal parts of a dollar, and may be distinguished from dollars in the same way as any other decimals by a comma or separatrix. A'l4 tlie figures to the left hand of dollars, or beyond Units place arc eagles. Thus, 17 Eagles, 5 dollars, 3 dimes, 4 cents, and 6 mills are written — V. V. ^ ^ -^ <; Or these, four are real coins, and one is imagin- a -c -^i S R "Q "2 ary. ^1 K C '^ -: ^ -^ § ^"^' ^^^^ ^^^"^ ^^® ^^^ Eagle, a gold coin ; the ^ ^ '^ § o Dollar and the Dime, silver coins ; and the Cent, a tSs^^ li^ g copper com. The mill is only imaginary, there bc- "< o o C ;:7 i"g no piece of money of that denomination ," •- ••> f ° There are half-eagles, l;alf-dollars, double-dimes, ^ ."° S ^^ ^ half-dimes, and half cents, real coins. '^ o ^ - - 17 5, 3 4 6 These denominations, or different pieces of money, being in a tenfold J»roportion, consequently, any sum in Federal Money does of itself exhibit tiie particular number of cacli diffcrci»t piece of money contained in it. Thus' 175,346 (neventfen cuglea.Jive dollars, three dimes, four cents, six miUs) con- tain 175346 mills, 17534 -^6, cents, 1 75 3^^^^ dimes, 1 7 5-,^/^, dollars, 17-,%3,*«, eagles. 1 herelore, eagles and dollars reckoned together, express the num- ber of dollars contained in the sum ; tke same of dimes and cents ; and thia indeed IS the usual way of account, to reckon tfee whoio sum in dollars, cents, and mills thus, ' S 175 34 G _ Wl Addition, Subtraction, Mwltiplication and Division of (Federal Money Us performed in all respects as in Decimal Fraciions, to which the Scholar it f referred lor the use of rules in these operations. 83 ADDITION or FEDERAL MONEY. Sect. II. 3. ADDITION OF FEDERAL MONEY. 1. Add 16 Eagles ; 3 Eap^les, 7 Dollars, 5 Cents; 26 Dollars, 6 Dimes, 4 G;iius, 3 Mills ; 75 Cents, 8 Mills j 40 Dollars, 9 Cents together. Operation. Or, the sums may be reckoned in dol- lars, ce;its, and mills, thus, Dime Cts. 1 /^-^^ 16 0, 3 7, 5 2 6, 6 4 3 * 7 5 8 4 0, 9 ^26 4, 5 4 1 1 a ^ ^ 160 37 05 26 64 3 75 8 40 C9 S264 54 1 2. If I am indebted 59 dollars, 1 12 dollars,98 cts. 113 dolls. 15 cts, 15 dolls. 21 dolls. 50 cts. 200 dolls. 73dolls. 35 dolls. 17 cts. 75 dolls. 20 dolls. 40 dolls, 33 cts. and IG dolls. What is the sum v/hich I owe ? jins. 1^781 13 / / ^ 9^S^ AticouNTANTs generally omit the / / 3 / "T" comma and distinguish cents from , j\. dollars by setting them apart iVora L'l iW} the dollars. ZOO J6 T78I t2> u^i^. Segt. II. 3. SUBTRACTION of FEDERAL MONEY. 89 SUBTRACTIOIJ OF FEDERAL MONEY. 1. From Dolls. 863,17 take, Dolls. 69,82 OPERATION. 8 6 3,1 7 6 9,8 2 Remainder J 7 9 3,3 5 2. From Dolls. 681 take, Dolls. 57,63 Remainder ^ DolL 62 3,37 6Sh 00 MULTIPLTCATION OF FEDERAL MONEY. 1. If Flour be Dolls. 10,25 //cr.^ 157 dollars, 98 cents, and o^niltn. l§s "■^ s '^ •^ <3 fco •s ^.s ^ "v3 O^ ^ o i^ -^^ ^ ^ «^ s ~ •? ^ a ? ^ H e< . « t^ C3 ii "5 :S S S O V - •- ;^ fis ^^ < ^ ^ <4j »§ '■== -J >. ^ ?^ ^ v^ /^A.^^ r'.-\>-> DrviDE by 3) 4 : r 3 9 5 Do! 1 5 7, 9 8 Sect. II. 3. REDUCTION of FEDERAL MONEY. 91 Jf fiounds only are given to be reduced^ a cypher must be annexed and the number divided by 3 : the quotient will be dollars. If there be a remainder annex more cyphers, and divide, the quotient will be cents and mills. TVhen there are no shillings^ or only 1 shilling in the given mim^ so there be no even number^ w\\\G -A cypher \x\ place or half the even nurpiber of shillings, then -proc&ed with the pence and farthings as in other cases* If it he required to reduce fiounds^ shillings^ fience^ iD'c. to Dollars and cents only, the cypher must not be annexed ; in this case two figures only must be cut oif from the quotient. A LITTLE practice will make these operations extremely easy. 2. lN;C"r63 how many dojiars, cents and mills ? Am. 2543 Dolls. 33 cts, 3 m. 2) )r62)OO00 3. In £\7 \s. &\d. hew many dol- lars, cents and mills ? Ans. Dolls. 5-6 92 3. 3) iro rr 4- In ^109 S*. 8cf. how many (dollars, and cents ? Jns. Dolls. 363,944.4- ^6 5,9 ^7? -i- 5. In ;C86 6*. 5^r/ how many dol- lars, c€»its and mills ? Ans. Dolls. 287,74. 2<9 z r4~ CASE 2. To reduce Dollars, Cents ^ and Mills, to Pounds, SbilUngs, Pence, and Farthings, RULE. ..'^rv^^ !^'^ ?'"'" '""" '^>^ ^' "'"' °" ^'^^ ^«"^ '-'^'^ ''--^"^^ ^'^urcs, which will be decimals of a pound, the left hand figures will be the pounds. To f.nd the value of the deiimals, douhle the first figure for shillings, «nd if the figure- in «^^y;^"^,P "ce be 5, add another shilling, then call ihe figures in.thc sec - ond and hird places, af^er deducting the 5 in the second place, so many far- things, abaiu^ 1 when they arc above 12, and 2, when Ihcy arc above 5G. 92 REDUCTION OF FEDERAL MONEY. Se<:t. If. 3. JEXAMPLIIS. 1. Reduce 255 dollars 40 cents, 6 mills, to pounds, shillings pence andj farthings. OPERATIOK. 255406 •■•' — -^ 76]6216 r^C/'e 12.S. 5(f. Jinsiver. In this example having mnlti plied t1i€ given sum by 3 and cut off the four right band figures of the product, I double the first figure (6) for shillings, the figures in the second and third places (21) abating 1 for being over 12,(20) I consider as farthings, equal to 5c?. In Dolts. 255,40^ therefore, are ;C'^6 Igs 5d. Here 8 in the fourth place of decimals (tt^Ioo of a pound) beinj^pf infe-* rior value is not reckoned. The loss in this place is always less than one tar- thing. If there be no mills in tike given «Mm, multiply as before and cut off Sj^gure* only. If there be neither cents nor milU that is, if the given sum be cfo//flr«, multiply by 3 and cut off one figure only. 2. In Dolls. 392,75 how many pounds 3. In Dolls. 39,635 how msiny shillings, pence and farthings ? pounds, shillings pence, Uc. ? 4n8, ^117 'l6«. 6rf. 4n8. f; \\ 17« 9\d, £u.t7.9i 4. Reduce 134 dollars, 65 cenisto pounds, shillings, pence, and farthings. Ms, £40 7s. \Old, WSJT 5. Redug* Dolls. 684 to pounds and shillings, Ans. >^205 4«» G8^ 3 Sect. II. J. SUPPLEMENT to FEDERAL MONEY. 93 Supplement to 5f0t!trai ^Ofitp. QUESTIONS. ). WhaTis F^DRRAJL MoNEr ^ When was its establishment, and by v^hat authcriiy ? 3. IVfiAtare the denominations in Federal Money ? 3. Which is the Unit Money? 4. How are dollars distinguiahedfrom dimes, cents and jniils ? 5. Wha rfilaces do the. different denominations occufiy,from the decimal t^oints^ 6. Ho IV is the Addition of Federal Money pei'formed? Subtraction? Mulf tifilication ? Division ? 7. By what method are Pmmds, Shillings, Pence and Farthings reduced to Federal Money 7 S. Hqw ctre Dollars^ JDimes, Cents Qnd Mills, reduced to Founds^ Shillings Fence and Farthings ? FXERCISES. 1. A WAN dies leaving an estate of 2. A man sells 1225 bushelfcof 71600 Dollars, there are demands a- "wheat at Dol. 1,33 per busl-wel, and gainst the estate of Dol. 39876,74 ; the receives Dol. 93, 76 for transporta- residue is to be divided between 7 Bons J tion ; whatdoeshe receiveiulhc ^vhat mil each one receive ? -whole ? Jn9. 4531 Volls. 89 c(s, 4n8, Dolls, 1723,01. i 9 8r6.r// ^, A 5 5 ^ T H I 715.16 ^{^ ^^3) .89.5 .4^. /fi^^ T 71^.0) 'M. 94 SUPPLEMENT to FEDERAL MONEY. Sect. IL 3, 3. Reduce yC "75 Is. e^rf. to Dol- 4. In /7 13*. 8cL how many dollars I^is and cents, Jus. Dolls. 1250,2 56 cents and juills ? j^ns. Bolls. 25 61 3}161A ZC,6/ 5. Reduce Dolls. 781,27 to pounds 6. Reduce Dolls. 98,763 to pounds phillings, pence, and farthings* shillings, pence, and farthings. Ms. £23i 7s. r^d, Jns.£ 29 12s. 7d. TABLE For reducing Shillings and Pence to Cents and Mills, Shill. Shiil. Shill. Shill. Shili. 12 3 4 5 Pence 1 Cts.Mills I 4 Cts.Mills 16 7 18 1 Cts.Mills 33 3 34 7 Cts.Mills 50 51 4 Cts.Mills 1 66 7 68 1 Cts.MilU 83 -3 84 7 2 2 8 19 5 36 1 52 8 69 5 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 41 1 58 3 75 91 6 4 9 7 26.## 43 59 7 76 4 93 8 11 k ^7 'Hi ►44 4 61 1 77 8 94 4 9 10 . 12 13 5 9 2^9-^2 . 30 . 6 45^ 47 8 2 6^ 63 5 9 79 80 2 6 95 8 97 2 11 15 3 32 "' 48 6 65 5 82 igS 6 To find by tliis Table the Cents and Mills in any sum of Shillings and Pence under one dolhr, look the Shillings at top, and the Pence in the left hand column, then under the foriner and on a line wilh the latter, will be found the Cents and Mills sougist* Sect. II. ^. Table for reducing poundsy ^c, to dollars y ^c» 95 TABLE. For rsducing the Currencies of the sender al United States to Fed* eral Money, N. Hamp. N. Jersey. Mass. N. York. Pennsylvania S. Carolina. Rh. Island, and Delaware, and Conn, and N. Carolina. and Georgia. Virginia. Maryland. D.cis.m. D.cts.iii. D.cts.in. 1 D.cu.m. :f 1 , 3 , 3 , 3 , 4 1 2 , 7 , 5 , 6 , 9 , 3 , IQ , 8 , 8 , 14 1 , 14 , 10 , il , 18 2 , 28 , 21 , 22 , 36 3 , 42 , 31 , 33 , 54 4 , 5 6 , 42 , 44 , 7 1 5 , 6 9 , 52 , 56 , 89 .• 5 , 83 , 62 , ^7 ,107 , 97 , 73 , 78 ,125 8 ,111 , 8 3 , 89 ,143 9 ,12 5 , 94 ,100 ,161 10 ,13 9 ,104 ,111 ,179 II ,153 ,114 ,122 ,196 = 1 ,167 ,125 ,133 ,214 2 ,3 3 3 ,250 ,267 ,429 3 ,5 ,375 ,400 ,643 4 ,6 6 6 ,500 ,533 ,857 5 ,833 ,625 ,66 7 1,171 6 1,0 ,750 ,800 1,286 7 1,167 ,875 ,9 33 1,500 ' 8 1,3 3 3 1,000 1,067 1,714 9 1,500 1,125 1,200 J, 929 10 1,6 6 7 1,250 1,333 2,143 H 1,833 1,37 5 1,467 2,6 5 7 12 2,0 l,5oO 1,600 2,5 71 13 2,16 7 1,62 5 1, 733 2,786 14 2,3 3 3 1,750 1.8(57 3,oog ■ 15 2,500 1,875 2,000 3,214 16 2,6 6 7 2, 2, l33 3,424 17 2,8 3 3 2, 125 2,267 o^e'i^ 18 3,000 2. 2 5 2,400 3,857 19 / 3,16 7 2,3 75 2.53 3 4,071 96 Table far reducing Pounds ^ ^&. to Dolhirs, Wc. Sect. II. TABLE For reduc'wg the Currencies^ ^c. continued. 1 2 4 S 6 7 8 9 10 20 30 40 50 60 ro 80 90 100 200 3D0 400 500 6D0 7-00 8100 900 1.000 New-Hamp. New-York, New- Jersey, South-Caroli. S:c.&c. &c. 8CC.&C. 8cc. D.c.m. D.c.m. D.c.m. D.c.m. :' 3,333 2.5 2,666 4,286 1 6,667 5,0 5,333 8,571 10,000 7^5^ 8,000 12,857 ' 13,333 10,0 10,667 17,143 10,667 12,5 13,333 21,429 20,000 15,0 16^000 25^714 33,333 17,5 18,667 30,000 26,667 2(^,0 21.,333 34,286 30,000 22,5 24,000 38,571 3^,^33 ' 2«^,0 26,^67 42,8 57 6&,&67 50,0 53.333 85,714 100,000 75,0 80,000 128,571 133,333 100,0 106,667 171,429 166,667 125,0 133,333 214,286 200,000 150, 1 60,000 257,143 233,333 175, 186)667 300,000 256,667 , 200, 2131333 342,85 7 300,000 ' 225, 240,000 385,714 333,333 250, 266,667 428,571 666,667 500, 533,333 857,143 lOOO.OOO 750, 800,000^ 1285,714 1333,333 1000, 1056,667 1714,286 1666,567 1250, 1333,333 2142,857 2000,000 1500, 1600,000 25 7r,429 3333,335 1750, 1865,667 3000^000 2656, 6£6 2000, 2133,333 3428,571 3000,000 2250, 24.00,000 385 7,143 3333,333 25.00, 1 2666,667 4285,714 ^or reducing Federal I New-Hamp. | I £cc. ^c> I I Dol.6/; I TABLE Money to the currencies of the several Ujiiied States, New- York, j Nevy-Jersey, [' South-CaroU. &c. I 8cc. &c. Sec. Dol. Bf. I Pol. 7f6. Dol. 4/8. D. Ct3. 1 £. s.d.q; 1 £. s.d.q^ ! ^.s.d.q. ^. s.d;q. ,01 o 1. 2 ,02 1 2 2 1 ,03^ 2 1 3 2 3 1 3 ,04- 3. S 3 S 2 3 1 ,05 3 2 4 3 4r 2 2 3 ,06 : * 1 5 3 5. 2 3 1 ,0.7- 5,0 C 3 6 1 4 vOS 5 3 7 3 7 1 4 2 ,0.9' 6 2 8: 3 8 5-0 ,10 7 1 9 2 9 5 2 Sect. II. 5. Table for reducing Dollars, ^c, to Pounds, ^c. 97 TABLE For reducing the currencies, ^c. continued. New ■Hamp New -York, New- Jersey, South Cai olinai 8cc . &c. &c. Sec. &c. £cc 3. d. q. Dolls, cts £ s.d.q. £- s. d. q. £ . s. d. q. £' ,20 1 2 2 1 7 1 1 60 11 1 .30 1 9 2 2 4 3 2 3 I 4 S ,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 5 7 1 5 3 3 3 1 ,80 4 9 2 6 4 3 '6 3 8 3 ,50 5 4 3 7 2 2 6 9 4 2 2 1, 6 8 7 6 4 8 2, 12 16 15 9 4 3, 18 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 d ^ 15 2 6 8 20 6 8 7 10 4 13 4 oO 9 12 11 5 7 40 12 16 15 9 6 8 50 15 20 18 15 11 13 4 60 18 24 22 10 14 70 21 28 26 5 16 6 S 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 46 13 4 300 90 120 112 10 70 6 400 120 160 150 93 6 8 500 150 200 187 10 116 13 4 600 180 240 225 140 700 210 280 262 10 1C3 6 8 800 240 320 300 186 13 4 900 270 360 337 10 210 1000 300 400 575 333 6 8 2000 600 800 750 466 13 4 r,ooo 900 1200 1 125 700 4000 1200 1600 1500 9 33 6 8 5000 1500 2000 1875 1 166 1 :1 A. 6000 18(A) 2i00 2250 Uo.» 7000 2 100 2800 2625 165:1 8(K)0 2. 100 3200 5000 l8t*M"i yooo 2700 5600 .« . — ,. ' 1 ( .( : 10000 3000 4000 N 98 SIMPLE INTEREST. Sect. II. 4. § 4.. mmm. Interest is the allowance given {or the use of money, by the Borrower to the Lender. It is computed at so many dollars for each hundred lent for a year, ffiey- annum) and alike proportion for a greater or less time. The high- est rate is limited by our laws to 6 per. cent, that is 6 dollars for a hundred dollars, 6 cents for a hundred cents, ^6 lor a ;ClOO, Sec. This is called le^aL interest^ and is always understood when no other rate is mentioned. There are three things to be noticed in Interest. 1. The Principal ; or, money lent. 2. The Rate ; or, sum/ztr cent^ agreed on. 3. The Amount ; or, Principal and Interest added together. Interest is of two sorts, simfile and comfiound. 1. Simple Interest is that which is allowed for the principle only. 2. Compound Interest is that which arises from the interest being added to the principal and (coniinuing in the hands of the lender) becomes apart of the principal, at the end of each stated time of payment. GENERAL RULE. 1. For one year., multiply the principal for the rate, from the product cut off the two rii^ht hand figures of the dollars, which will be cents, those to the left hand will be dollars ; or, which is the same thing, remove the nefiaratrix^ from its natural place two fij^ures towards the left hand, then all those fig- ures to the left hand will be dollars, and those to the light hand will be cents, luills, and parts of a mill. In the same way is calculated the interest on any sum of money in fioxinds.^ shil- lings., pence and farthings^ ivith this difference only., that the two figures cut off to the right hand of pounds., must be reduced to the Ictuest derio-ndnatiovy each time cutting off as at first. 2. For t%vo or more years, multiply the interest of one year by the number of years. 3. For months, take proportional or aliquot parts of the interest for 1 year that is, for 6 months, -^- ; for 4 months, | ; for 3 months, i, Sec. 4. For days, the proportional or aliquot parts of tiie interest for 1 month, allowing 30 days to a month. EXAMPLES. 1. What is the interest o^ Bolls. 86,446 for one year, at 6 per cent ? OPERATION. JDolls. cts. m. Ix the product of the principal mulli- 86, 44 6 principal. plied by the rate is found the answer. 6 rate. Thus, cutting off the two right hand figures from the dollars leaves 5 on the 5/18,67 6 Interest. left hand which is dollars ; the two fig - • _ _ ures cut off (18) are cents, the next Cg- nre (6) is mills, all the figures which may chance to be at the right hand of mills, are parts of a iiiill, hence we collcct'the Jno-vcr. 5 /Jo//?. \^'cts. 6 -{^m. Sect. II. 4. SIMPLE INTEREST. 99 2. What is tlte intefeSt of Dolls. 365 14c/«. 6m///5, for three years, 7 months and 6 duys ? OPERATION. 3 6 5, 1 4 6 firincifial. 6 rate. 6 Months ^)2 1[ 9 0, 8 7 6 interest for one year. 3 6 5, 7 2 6 2 8 interest /•r Z years. 1 Month |)1 0, 9 5 4 3 8 interest for 6 months. 6 Days -5) I, 8 2 5 7 3 interest for I month. ,36514 interest J or 6 days. 7 8, 8 7 15 3 interest for 3 years, 7 months, and 6 days ; that is 78 dolls. BTcts. l-f-^\m. Because 7 months is not an even part of a year, take two such numbers as are even parts and which added together will make 7 (6 and 1,) 6 months is -| of a year, therefore, for 6 months, divide the interest of one year by 2 ; «gain, 1 month is | of 6 months, therefore for 1 month^ divide the interest of 6 months by 6. For the days, because 6 days is | of a month, or of 30 days, lhcrefore,ycr 6 days, divide the interest of 1 month by 5. Lastly add the in- terest of all the parts ©f the time together, the swm is the answer. 3. What is the interest of ^71 7s. 6^d. 4. What is the interest of 16s. for year, at 6 per cent OPERATION. £. s. d. q. 71 7 6 2 6 fid. for 1 year ? Ans. s 6 S.0 ^4 28 5 20 3 ^/,oo «. 5165 12 d. 7(83 4 q. 3)32 Ans.^ 4 5*. 1\d. 100^ SIMPLE INTEREST. Sect. II. 4. When the rate is 6 per cent, there is notj perhafis, a more concise and easy way of'casting interest^ on any su7n of money in Dollars^ CentSy and Millsj than by ihe following' METHOD, Write down half the greatest even number of months for a multiplier ; if there be an odd month it mast be reckoned 20 days, for which and the given days, if any, seek how many times you can have 6 in the sum of them, place the figure for a decimal at the right hand of half the even number of months, already found, by which multiply the principal ; observint^ in pointing off the product, 10 remove the decimal point or separatrix tn^o figures from its natur- al place towards the left hand, that is point off t'lvo more places for decimals in the product, than there are decimal places in the mwltipiicand and multiplier counted together; then all tlie figures to the left hand of the point, will be dollars, and those to the right hand, dimes, cents, mills. Sec. which will be the interest required. Should there be a remainder in taking one sixth of the days, reduce it to a vulgar fraction, for which take aliquot parts otthe multiplicand. Thus, If the remainder be Izz:',) divide the multiplicand by 6 If 2=ii by 3 If 3=i by 2 If 4=1 by 3 twice; If 5 :^J and I by 2 and 3 The quotients which in this way occur, must be added to the product of the principal niuliip'icd by half the months, Sec. the sum thus produced, will Ije the interest required. When there are days, but a less number than 6, so that 6 cannot be contained 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. Jn .casting interest, each month is reckoned 30 days. EXAMPLES. 1. What is the interest of Dolls.76,5A for 1 year, 7 months, and 1 1 days ? The number of months being 19, the greatest even number is 18, half of which is 9, which I write down ; then seeking hovr often 6 is contained in 41, (the sum of the days in tUe odd month and the given days) I find it will be 6 times, which I also set down ut the right hand of half the even number of months Ti-r a decimal, by which Ans. r, 4 1 1 6 2 together I multiply the principal. In taking one sixth of the days (41) there will be a re- mainder of 5z=j and ;^Vh^t Avouldbe due July 11, 1801 ? Jns. Dolls. 215,103. s-zo 3 ^8 iZ0398 y ^ ^5^. dJ6y ^— A/dPd?^. ^^// A /<£> o y, /jj. u^/f]t. u.i>K 104 COMPOUND INTEREST. Sect. II. 4. COMPOUND INTEREST, Is calculated by adding the interest to the principal at the end of each year and inakini; the amount the principal for the succeeding year ; then the given principal subtracted from the last amount the remainder will be the compound interest. A concise and easy Method of casting Compound Interest ^ at 6 per cent, on any sum in Federal Money, RULE. Multiply the given sum, if ror 2 years^ by 112,36 For 7 years, by 150,3630 3 years — 119,1016 8 years — 159,3848 ■ 4 years — 126,2476 9 years — 168,9478 5 years — 133,8225 10 years — 179,0847 6 years — 141,8519 U years — 189,8298 Note. 1. Three of the first highest decimals, in the above numbers, will be sufficiently accurate for most operations ; the product, remember- ing to move the separatrix^wo figures from its natural place towards the Jefi hand, will then shew the amount of principal and compound interest for the given number of years. Subtract the principal from the amoutit and it will shew the compound interest. 2. When there are months and days ; first, find the amount of principal and compound interest for the years, agreeable to the foregoing method, then, for the months and days cast tlie simple interest on the amount thus found ; this added to the amount will give the answer. 3. Any sum of money at Compound Interest, will double itself in 11 years, 10 months, and 22 days. i EXAMPLES. 1. What is the compound in- 2. What is the amount of 1^236 at lerest of $56 75 for 11 years? compound interest, for 4 years, 7 months an4 6 days OPERATION. OPERATIOHr, 5 6, 75 ^ 12 6, 2476 18 9, 829 236 51075 7574856 11350 3787428 45400 2 524952 6 10 7 5 5400 ^29 7, 944336 Jmmint 6 7 5 3, 6 [for 4 years. glO 7, 7279575 Jjnount. 17 8 7 6 6 4 5 6,7 5 Principal subtracted. 8 9 3 8 3 2 {mo. 6 daijs. ^5 Oj 9 7 CompQund interest. ^10, 725984 Interest for 7 2 9 7, 9 4 4 Amount for 4 years S 3 8, 6 6 9 Jnsiver. [added. Sect. II. 4. SUPPLEMENT lo S. INTEREST. 105 Supplement to ^^. ^nterc^t. QUESTIONS. 1. IVHAfh interest ? 2. JVhjT is understood bxj 6 P£/? cENr ? 3 J*£ff cEN-r? 8 P£fl cr^r, fJJ'c. 3. WHAffier cent.fier annum is alloived by Law to the Lender for the use cf his Money ? 4. WHAfis understood by the Principal ? the RAfE ? the AMOUNf ? 5. Of how many kinds isinteres/ ? in what does the difference consist ? 6. Hoif/- is simfile interest calculated Jor one year in Federal Money ? 7 . For more years than one^ how is the interest found ? 8. Wheh there are months and days what is the method of fir oce dure ? 9. What other MEtHOD is there oj casting interest on sums in Federal Money ? 10. When the days are a less number than 6, so that 6 cannot be contained in themj what is to be done ? 1 1. How is simfiU interest cavt in pounds, shillings, /lence, and farthings ? 12. When fiartial fiayments are made, at different times, how is the interest caU ciliated ? EXERCISES. 1. What is tlie interest of 91 eZ^o//*. 2. What is the interest of 93 Dolls, 72 cts. for 1 year and 4 month's ? 17 ct3. \ 1 clays ? Ans. DoU».7j^o^7. Ans. 17 cts. 93 / r . 3. What is the interest of Dolls. 5,19, 4. What is the interest of Dolh. for 7 months ? 1,07 for 3 years 6 months, and IS jitis. IS cts. \m. days ? jins 22 cts. 7m. l_ I S,I6 O l,OT 2/^ 12/3 ^r IQO i SUPPLEMENT to S. PNTEREST. Sect. IL 4. 5. Whatis the ihter&stof^ 41 lis. 6. What is the interest of Dolls. 3|d. for a year and 2 raonths ? 273,51 at 7 fier cent for 1 year and- Ans.^l 18 2i 10 days? jim. JDolls,. 19^677. %r:>. r/ ^/. II. 3^ 2J90./8.I / to 0% IS/IS Z/t6 /£^l£>6 2. r3 57 t. ScpfoSiKG a note of i)o&. 3 1 /,92 dated July S', lYsr, on which \*ere the following payments, Sept. 1 3, 1799, BolU. 208,04. March 10,1800 Ddlu 76 ; whatwas the sum due Jan. 1, 1801 \ Jina. Dolla. 85,991, 62cr. 11. 5. COMPOUND MULTIPLICATION, ao7 § 5, Compound Mttlti0mtmh Co^irouND Multiplication is when the multiplicand consists of sever- al denominations. It is paniculufly Aisefui.in finding the value of Goods. The different denominatrions in wh^t was formerly called Lnnv/ul Money, •render this rule with some others in Arithmetic, as Comfiound Division and /-'rac/'/cney is our J^ational Currency ; the Schol- lar might encompass the most uscfal rules of Arithmetic in half the time ; the value of commodities bought and sold, might be cast with half the troub- le, and with much les« liability to errors, were all the calculations in money universally made in Dollars^ Cenr.Sy.and Mills. But this, to be practised must '^bc taugiit ; it must be taught in our Schools, and so long as the prices of goods, and almost every ma.V's accounts are in Pounds^ Shillings^ Penccy a?id Far- t/iingSf this mode of reckoning must;not be left untaught. To comprise the greater usefulness, and also to shew the great advanta^ which is gained by reckoning in Federal Money, I have contrasted the two ^odesof account, and in separate columns, on the same page, have put the same questions in Old JLaitful and ^n Federfll Money. litnes 1 0yd. 70 J 5 O firice (f 60 yds. 2, limes 1 yr/i ^ 19 -9 firice of 3 yds. Jins.n^ 6 5firiccof56^yds. 2. What will 328 yards of cloth j cost at lOs. 6^-d. yer yard ? ►er yard ? OPERATION. Yards. 5 6 3 ^ 4, 4 3 i 16 8 9 $ 2 5 2 "5252 jg 2 4 9 4, 9 ^n#. 5. What will S28 yards ofdot^ cost at S 1)757 per yard ? 4ns. % 576,295 3. WhiU will G24 yards of cloth cout at 12s. ^d. per yard ? J7is, £295 4s. 10. What will 6^4 yards of clotk cost at $2,111 per yard ? Jns. S 1317,264, Stct.ItS. SUPPLEMENT TO C. MULTIPLICATION. iU, Supplement to CompOUttb JiBultipltCatlOn^ QUESTIONS. 1. What ia dqinjiound Multiplication ? 2. What is its use ? 3-. jIre ofieratiorii nioiteaiy in Old LawPvI^ 6r in FkotRAL MdNsr. 4. WHAtistheruleforcomfioundMuliililication? 5 . TVhen the quantity that is the Multifiliery excet ds 1 2 and is within the Mul* ti/ilicers, and this quotient by the other. 6il. EXA^TPLES. . If 5 yards of cloth cost ^'3 what is that per pard ? 13s. OPERATION. 5)3 13 & firice of 5 yards. U Q^ firice of I yard. FiNDiTCG I cannot have the divisor (5) in the first d*:nominaiion (^.3) I reduce it to shillinc^s, (GO) and add in the 13 shilliiis^s, which make 73 shil- lings, in wiiich the divisor (5) is con- tained 14 limes, and 3 remain ; I set down the 14 Sc the remainder )3 shil- linijs) reduced to pence (36) and the €d. added make 42 pence in which the divisor is contained 8 times and iwo remain ; I set down the 8, and re- duce the two pence to farthings (8) in which I have the divisor ance{\qr. or |f/.) and a remainder of -5 of a far- vhing, which being; of small value is iitgkcted. 2. If 48 yards of cloth cost £4 I65. Dollars, Cents, Mills. IN ALL CASES. Divide the price by the quantity, and point off so many places tbr cents and mills in the product as there are places of cents and mills in the divi- dend. If the quantity be a com/iosite num- bery that is produced by the multi- plication of two numbers, the opera- tion may be varied by dividing the price first by one of those two num- bers and this quotient by the other. EXAMPLES. 1. If 5 yards of cloth cost Dolls. 12, 25, what is that per yard ? n. 5)1 OPERATION. Cts. 2,2 5 There are twt> decimal places in .^ns. 2, 4 5 the di\idend. I, therefore, jxjint oit two places for decimals, or cents in the quotient. 2. If 48 yards of cloth cost Dotls, 16,06 what is that per yard ? .4ns, Dolls. Oyoo, Aid. what is that per pavd ? Sect. II. 6. COMPOUND DIVISION. 115 Pounds, ShiU.Peiicey Far things y Dollars^ Cents ^ Mills. 3. If 24lb. of tea cost £.2 7s, 9^d. 3. If 24 lb. of tea cost Dolls, 7,97 what is that per lb. ? what is that per lb. ? Jas./C.O Is. lUd. Ans. Dolla, 0,2Z2 4. If 35 yards of cloth cost £A2 is. 7\d, what is that per yard ? CASE 2. 1. " Having the firice of an hund- red iveight ( 1 1 '214).) tojind the firicd of I lb. divide the given price by 8, that quotient by 7, and this quotient by 2, and the last quotient will be the price of 1 lb. required." 2. If the number of hundredweight be vvQre than one, first divide the whole price by the number of hundreds, then proceed as before. EXAMPLES. 1. If 1 cwt. of sugar cost £S 7*. 6d. what is that per lb. ? OPERATION. £. s. d. q. 8)3 7 6 firice 1 av)t. 7)0 8 5 1 firice of \ Alb. or I civt, 2) 12 1 firice of 'Xlb. or ^^ctut. Ann. 7 1 firiee of Mb, 4. If 35 yards of cloth cost JDolU 141,103 what is that per yard ? Ana. Dolls. 4,031 The same may be done ia Federal Money. 5. Ir 1 cwt. of sugar cost Dolls 1 1, %5 cts. what is that per lb. ? Ana, 10 cents. 114 COMPOUND DIVISION. Sect. II. 6, Pounds^ ShilL Pence ^ Farthings, 2. If 8 cwt. of cocoa cost jCl5 *ts. 4d. what is that per lb. ? Jns, 4id. 3. I? 3 c-wfr. of sugar cost ^15 13*. Vhat is that per lb ? Jns. 1 Id, Dollars, Cents, Mills. 6. If 8 cwt. of cocoa cost jg5 1,223 what is that per lb ? ^ns, 5 cents, 7 mills. 7. If 5 cwt. of sugar cost ^52,167 what is that per lb ? jfns. 15 cents, 5 mills. CASE. 3. 5< WffSN the divisor is such a number <25 cannot be produced by the multi/ili- cation of small numbers, divide after the manner of long division, setting down the work of dividing and re- ?H»eing." Sect. II. 6. COMPOUND DIVISION. 115 Founds ^ShilL Pence y Farthings. EXAMPLES. 1. If 46 yards of cloth cost aC.53 10*. 6d. what is that per yard ? Z ^Z^Ana, OPERATIOJ^r. /;. s. d. £, s. d. 46)53 10 6(1 46 7 20 46)150(5 138 12 12 46)150(3 158 12 4 46)4i(l 46 2. If 263 bushel of wheat cost C147 !<• 1 Id. what ii that per ^^«lloi» ? ^//A'. 4.S. 4j.«/. Doilans, Cents, Mills. 8. If 46 yards of cloth cost gl^d ,4 1 6 what is that per yard ? Am. g3,878. 9. If 263 bushels of wheat coBt g28f j973, what is that per bushel ? An8, 21,093 10. If 670 gallons of wine cott S490,32 ; what is that per gallon ? Arnt, ^0,7 3. 116 SvPFLEMENT 10 COMPOUND DIVISION. Sect.1I. 6. Supplement ToCompOUnD^Ditjij^lOn. — — "iWli^ -:;=• ■i'.i -Jl'c ^ •*'/:• •5!?> "Jlfr <■■*■—— QUESTIONS. 1. IVHAf is Comfiound Division ? 2. When the firice of any quantity^ Hot exceeding 12, ofyards^ Jiounds^ Isfc. ia given in fioundt, shillings ^fience Isf Jarthings^ honv ia the firice of one yardfuund ? 3. When the quantity is such a number as cannot be firoduced by the multi' plication of small numbers^ what is the method of procedure ? 4. Having the firice of an hundred weight given^ in what way is found the price of Mb.? 5. If there be several hundred weighty what are the stefis of ofierating ? 6. When the price is given in Federal Money ^ what is the method of operating ? EXERCISES. Founds, ShilL Pence, Far things. 1. Ir 10 sheep costyC4 5«. 7rf. what is the price of«ach ? Jns. 6/61. Dollars, Cents, Mills. Lfet the scholar reduce the price of the sheep and of the cows to Federal Money, and perform the operations in Dolls. Cents and Mills. Price of \ sheep, %\,A,%6. 2. If 84 cows cost £252 13s. what js the price of each ? jins.^2 Aid. Price cf\ a>ty,S 10,06- Sect. II. 6. Supplement to COMPOUND DIVISION; 11 3. If 121 pieces of cloth measure 2896 yards, \qr. 3na. what does each piece measure ? j^ns 23 yardsy 3qr, 2na. 4. If 66 tea-spoons weigh 215, \Ooz, 14/jwr. what is the weight of each ? yins. lOftwt. ISII^-r*. 2 cwt. of rice cost x; 2 11*. ? Am.^ld, 4 5 Ir «. w.. (>\d. what is that per lb 6. AT)C,.2 11*. 6|J for 2 cwt. of rice, what is that in Federal Money, and what is that per lb ? Price of I Id, 3 cents, 8 mill*. 7. If 47 bags of Indigo weigh 12 cty/r. ](/r. 26lb. A,oz. what does each weigh ? ^«*. Iryr. 1/*, 12or. 8. If 8 horses eat 900 bushels, and 1 peck of oats in 1 year, how much will each horse eat per day ? Ans. 1 ficck^ \qt. \/it. 2^ilis. 118 Supplement to COMPOUND DIVISI0N. Sect. II. 6. 9. Divide x;297 25, 3d. among 4 men, 6 boys, and give each man 3 times 80 much as one boy ; what will each man share, and each boy ? OPERATION. The men have triple £. s. d. £ . s. d. q. shares, therefore, multi- 18)297 2 3(16 10 1 2^zl boi/s share. ply the number of men (4) by 3, and add the number of boys (6) for a divisor. men. boys, 4 and 6 3 12 6 18 117 108 9 20 .4ns. 49 10 4 2zzi\ man* s share. Proof. £A9 10 4 2 4 )182(10 18 1 3 the number of equal 1 2 shares i?i the ivhole — » rrDivisor. )27(1 18 198 1 6 © men*s share. 16 10 1 2 and 6 99 9 boy^a share. £.197 2 3 added. 9 4 )36(2 36 10. DiviiiEjC-39 12s. 5(s;. among 4 men, 6 women, and 9 boys ; giv« c«ch man double to a wx)man, each woman double to a boy. X . £. s. d. a boy*s shi share. C \ 1 5 a boy's share. Answer. -? 2 2 \0 a woman's shai (^4 5 8 a inan*s share Sec. II. 7. SINGLE RULE of THREE. 119 The Single Rule of Three, sometimes called the Rule o/Proportiox, is known by havinj:^ ihrtc terms given to find the fourth . It is of TWO kinds, Direct and Indirect^ or Inverse. — Single Rule of Three Direct. The Single Rule of Three Direct teaches, by having \hree numbers given to find a fourth, which shall bear the ianae proportion to the third that the se- cond does to the first..— It is evident that the value, weight, and measure of any commodity is pro- portionate to its quantity, that the amount of work, or consumption is propor- tionate to the time ; that gain, loss, and interest when the time is fixed, is proportionate to the capital sum from which it arises ; and that the effect pro- duced by any cause is proportionate to the extent of that cause. These are cases in direct proportion, and all others may be known to be so, when tha number sought increases or diminishes along with the term from ,which it is derived. Therefore, i If 7nore require more^ or less require less^ the question is always known to 'belong to the Rule of Three Direct.; ' More 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- quires 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 whicli is of the same *' name or quality as the demand, the first term, and that, which is of ihe 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 quotient will be the answer to the question, which (as also the re- " mainder) will be in the same denomination in which you left the second " term, and may be brought into any other denomination required.*' - The chief difficulty that occurs in the Rule of Three, \^ the right placing* of the numbers, or stating of the question ; this being accomplished there i*} nothing to do, but to multiply and divide, and the work is done. To this end the nature of every question must be considered, and the cir- cumstances on wiiir;h the proportion depends, observed, and common cense will direct this if t lie terms of the question be understood. The method of proof is by inverting the order of tiic question. Mjie. I. Ik the first and third tenns, bcth or either', he of difTirent de- noniinalions, both terms must be reduced to the lowest denomination men- tioned in either, before stating the question. 2. If the second term consist of difl'erent denominations, It must be redu- ced to the lowest Ucnominaiion ; the fourth term, or answtr will then be found in the same denomination, and must be reduced back again to the highest de- nomination possible. 3. After division if there he any remainder, and the quotient be not in the lowest denomination, it must be reduceil to the tiext less denomination, divi- ding aft bcfor.c. So continue to do, till it is brought to the lowest denomina- tion, or till-nothing remains. 4. In every question there is a supposition and a demand ; the svTpposition is implied in the two first terms of the statement, ilie demand itt the third. 120 SINGLE RULE of THREE DIRECT. Sec. II 1 5. When any of the terms are given in Federal Money the operation is con- ducted in all respects as in simple numbers, observing only to place the point, or separatrix between dollars and cents, and to point off the results according to what has been taught already in Decimal Fractions^ Federal Money ^ and further illustrated in Compound Division. 6. When any number of barrels, bales, or other packages, or pieces are given, if they be of equal contents, find the contents of one barrel or price, S;c. in the lowest denomination mentioned, which multiply by the number of pie- ces. Sec. the product will be the contents of the whole.— If the pieces, Sec. be of unequal contents find the content of each, add these together, and the sum of them will be the whole quantity. 7. The term which asks the question, or that which implies the demand, is generally know by some of these words going before it ; How much ? How many ? How long ? What cost ? What will ? Sec. EXAMPLES. 1. If 9/^5 of tobacco cost 6*, what will 25^6* cost ? ♦ OPERATIOU. Ibfi. 25 : to the answer. lbs As 9 6 25 30 12 8, d. 9)150) 16 8 answer, 9 60 54 6 12 Here %Slbs which asks the question, (what will V-Slbs. ^c.J is made the third term, by being put in the third place ; 9lbs. being of the same name, the first term, and 6s. of the same name with the term sought, the second term. I MuTiPLY the second and third terms together and divide by the first. The remainder (6) I reduce to pence, and divide as before. The quotients njake the answer, 16/8. 9)72(8 72 00 By inverting the order ot the question it will stand thus, 2 If 6s. buy 9lbs, of tobacco, wliat will 1668 buy ? 6 12 16 12 72 pence 200 pence. pence, lbs. pence. Js,72 : 9 : : 200 200 Here the term which asks the question (16s8) is of different de- nominations; it must, therefore, be reduced to the lowest denomina- tion mentioned (pence) as must also the other term of the same name, consequently, to be the first term. 72)1800)25 Ibs.antwar. 144 160 160 Sect. IL 7. SINGLE RULE of THREE DIRECT. 121 Jgain — By inverting the order of the question. 3. If 16/8 (=200 fience) buy 25lbs. of Tobacco, how much will 6s(=72 fience) buy ? OPERATION. d. lbs. d. As 200 : 25 : : 72 72 50 175 These three questions arc only the — — first varied ; they shew how any ques- 2100)18100(9/^5. Ans. tion in this Rule, may be inverted. 18 4. If \oz. of Silver cost 6/9 what will be the price of a silver cup that weighs 9oz. Afiiut. \6grs, oz. s. d. oz. tiwt. grs. \ 6 9 9 4 16 20 12 20 M As each of the 20fiivt. %\pence. I84jiwt. terms contains dif- 24 * 24 ferent denomina- -„ tions, they must 80 752 all be reduced to 40 368 the lowest denom- ination mentioned. 480 g-w. 4432 ^r*. grsr d. grs. As 480 : 81 : : 4432 4432 162 243 324 324 d. q. 480)358992(747 3 J Amivcr, which must be reduced to 3360 the highest denomination ; thus, 1 2)7 4 7 3|y. 2299 1920 2|0)6|2 M. ^'3 Is 3d. S^gr. Ans. 3792 3360 432 4 ) 1 728(11 1440 288 Q 122 SINGLE RULE or THREE DIRECT. Sect. II. 7. 5. If 6 horses eat 21 bushels of oats in 3 weeks, how many bushels will 20 horses eat in the same litne ? ^ns. 70 biidheii. The same quefitio7i inverted. 6. If 20 horses eat 70 bushels of oats in 3 weeks, how many bushels will 6 ? Jns. 21 bushels. The statement of every question re- q\iires thought and consideration ; — here are Jour numbers given in the question ; to know which three are to be employed in the statement there can be no difficulty if the Scholar proceed , deliberately and as his rule directs — first, consider which of the given num- bers it is, that asks the question ; that determined on, put it in the third place, then seek for another number of the same name, o?M^ind, put that in the first place, the second place must now be occupied by that number which is of the same name or kind with the number sought ; when these steps are cautiously followed, the Scholar cannot fail to make his statement right. 7. If an Ingot of silver weigh 36oz. 8. A Goldsmith sold a Tankard lOpwt. what is it wortr, at 5.9. per ounce ? for C^O 12.9. at the rate of 5s. 4d. * Jns./19 25. 6d. per ounce, I demand the weight of it. Jns. 3 9 or. ISJiwt. t^6 ^^ it: 2«.-tf^|_j,^ 6T6 310 2)Z0 Sect. II. 7. SINGLE RULE of THREE DIRECT. 123 9. Tf a family oF 10 persons sper.d | 10. If a family of 30 persons spend 9 3 busiielsof malt in a month, how | bushels of malt in a month, how many many biisliels will serve ihem when | bushels will serve a family of 10 per- there are 30 in the family \ sons, the same time ? Jtis, 9 bushels, -/ins. 3 bu^iek. U. If 12 acres, 3 roods produce 78 quarters, 3 pecks, how much will 35 acres, 1 rood, 20 poles, produce ? Ans, 216 quarters^ 5 bushels, 1^ fieck, /Z."^: T8.^y: 3T.J .to ^ z ^ s/ inr /^/ 1_ ■ ,4 /90^ ' ZI6—S--li c '>'-■>. I (S?>6 1:24 SINGLE RULE of THREE DIRECT. Sect. IL 12. If 5 acres 1 rood, produce 26 quarters 2 bushels, how many acres will be required to produce 47 quarters 4 bushels ? »^ns. 9 acres, 2 roods. 2.6.2: S.l::^/.^ ±10 1/ vIoJJTio 65 169 5&0 1680 63 As 13. If 565 men consume 75 barrels of provision in 9 months, liow much wili 500 men consume in the same time ? jiri^s, 102^| barrels* T50 5- ; zro ^ ^^ 14. If 500 men consume lO^ff barrels of provisions, in 9 months, how much will 365 men consume in the same time ? OPERATION. bar r eh* 102f| Multifily by 73 the denominator — — of the fraction* 306 714 jidd 54 the numerator* ^s 500 Note. In the 14th example, in order to em- brace the fraction (4* of a barrel) the inte- l^ers, 102 barrels must be multiplied by the clenominutor of the fraction, (73) and the luimeravor, (54) added to the product. After division, the quotient must be di- vided by the denominator of the fraction, and this last quotient will be the answer, all which may be seen in the example. The Scholar must remember to do the same in all similar cases. 7500 : : 365 7500 182500 2555 5]00)27375!00 :)5475(r5 Jns. 511 165 Sect. II. 7. SINGLE RULE of THREE DIRECT. 125 1-5. If I give 6 Dolls, for the use of 100 Dolls, for 12 TTJonths, what must I give for Dolls. 337,^2 tVie same length of lime ? OPERATION. Here in the tliii'd term I had two decimal places, (82) or places of cents multiplied by the second terni (6j I point off two places for cents (,92) in the product, which divided by 100, I point off three decimal pla- ces in the quotient equal to the num- ber of decimal places in ^e dividend (^,92 cents and annexed to the remain- der) ihere being no decimal^ in the divisor. As X). IOC D. :6 D Cts. : : 357,82 6 100' 2146,92{21,469A 200 146 100 469 400 692 600 920 900 Jus 20 16. How much land at ^2,50 per acre should be given in The Scholar i» exchange for 360 acres, at S^75 per acre ? | desired to invert and Ans* 540 acres, \ firove the question. %Soi 360:: 57 r 2?60 ZXSOO /oo /oo ^ z?ooo io8o 3;^^ 17^^06177360. ^v^^, O 17. If I buy 7lb. of sugar for 75 cents, flow much can I buy for 6 dollars ? Jns, 56/6. rs- r: 600 600 , ys)4zooiS6 -^. N. B. Sums in Federal Money are of the same denomination when the decimal places in each are equal. To reduce sums in federal money to the same denomination^ annex so many cyphers to that sum which has the least number of decimal places, or places of cents, mills &c. as shall make up the dcTi- r.icnry. 126 SINGLE RULE OF THREE DIRECT. Sect. IL 7$ 18. If I buy 76 yards of cloth for | 19. A man spends g3,25 per §113,17 what did it cost per Ell Ek- | Week, what is that per annum ? glish ? ^n.f. Sl,861. .4ns. gl69,4G4. f/^ 50^ lOG 20. Bought a silver cup weighing 9oz.AJi'ivt, \&grs for iC3 3*. 3fl?. 3|y, what was that per ounce ? Ans. 6*. 9c/. JlfT "JW ^0 11966I40 ISXOO /3Z66 Sect. II. 7. SINGLE RULE of THREE DIRECT. 127 2 1 . There is a Cistern, which has 4 cocks ; the first will empty in 10 min- utes ; the second, in 20 minutes ; the third in 40 minutes ; and the fourth in 80 minutes ; in what lime will all four running together empty it ? Min. , Cisi. Min, 1 : : 60 : flO J 20 •^ 40 1^80 In 1 hour the 4 cocks would empty Then, Cist. Min. Cist As 11,25 : 60 / Cist. r6 S 1,5 L .75 22. A MAN having a piece of land to plant, lured two men and a boy to plant it, one of the men could plant it in 12 days, the other in 15 days, and the boy in 27 days ; in how long time whould they plant it if tliey all worked together ? Ans, 5,346 days ' 60 11,25 Cist. 6d Min. : 1 : 5,33 Jns. JJ,K) 6000(6%'^ 3 -^-^' // >%x^ //,2^ 60000 \^)3^6»^ 58fOO flfto fTsrr 23, A MERCHANT bought 270 quin- tals- of cod fish, for g780 ; freight ^37, 70 ; duties and other charges §30,60 ; what must he sell it at, per quintal to gain §143 in the whole ? ] ylns. §3,671 The sum of all/ he exficncea ofthcjish vjilh the Merchant's gaiJi must be found for the second term. 24. If a staffs//. 8m. in length, cast a shadow of 6 feet ; how high is that steeple whose shadow measures 153 feet I zro no nj Ans 1836 68 \ Ail feet /8S6 4 128 SINGLE RULE of THREE DIRECT. Sect. II. 7. 25. Bought 12 pieces of cloth each | 25. Bought 4 pieces of holland 10 yards at ^1,75 per yard, whai came | each containing 24 EUs-EnglishjCor they to ^ jin». ^2 10 | $9^ ; how much was that per yardi ? ^ns, 80 CenU. f6 o 27. BotJG'HT 9 Chests of tea, each weighing SC S/rs'. 21/5. atyS'4 9s.f$fr cv>i, what came they to ? jlns. ^147 13x. 8M. C..,,. ti> ^ 7f /B?> %o 9 zoos 601 ~ 11%. ^'if Sect. II. 7. SINGLE RULE of THREE DIRECT. 129 28. a: but S607, 28. A Bankrupt owes in all 972 dollars, and his money and effects are S607,50 : what will a creditor receiva on %\ 1,333 ? Mi. S7,0«S. 6ozro dOt"f Xf/b 1^1 \ S9. A owes B iG347.'^,but B com- 50. If a person whose rcHt is SM5, pounds with him for 13*. 4^/. on the pays ^12, 63 of parish taxes, how pound ; what must he receive for his much should a person pay whose rent debt? ^;i.».>C2316 13«. W. ./ is ^378. .i/»s. §32,925. ^S'O ~TiY27 UhJTzJ^iWoJoX^-^6 00 ~TSW 00(1 R I 130 SINGLE RULE of THREE INVERSE. Sect. IL 7; INVERSE PROPORTION. In some questions the number sought becomes less, when the circum- stances from which it is derived become greater. Thus, when the price of goods increases the quantity whieh may be bought for a given sum is small- er. When the number of men employed at work is increased, the time in which they may complete it becomes shorter ; and, when the activity of any cause is increased, the quantity necessary to produce any given effect is di- minished. These and the like cases belong to the SINGLE RULE OF THREE INVERSE. The Single Rule of Three Inverse teaches, by having three numbers giv- en to find a fourth, having the same proportion to the second, as the first has to the third. If more require less, or less require more, the question belongs ta the Sin- gle rule of Three Inverse. More requiring less, is when ihQ third term is greater than the first, and requires the fourth term to be less than the second. Less requiring more, is when the third term is less than the first, and re- quires the fourtk term to be greater than the second. RULE. " State and reduce the terms as in the rule of three direct ; then, multiply the first and second terms together, divide the product by the third, and the quotient will be the answer in the same denomination with the second term.'* EXAMPLES. 1. If 48 men cMn build u wall in 24 days, how many men can do the same in 192 days ? OrERATION. Men. Days, Men. * Here the third term is greater than the As 48 : 24 : : 192 first, and common sense teaches the fourth 48 term, or answer must be less that the second, for if 48 men can do the work in 24 days, certainly 192 men will do it in less time. In this way it may be determined if a question belong to the Rule of Three Inverse, 192)1 152(6 ./fn*. 1152 2. If a board bq 9 inches broad. 3. How many yards of sarcenet, Sjr*. how much in length will make a wide, will line 9 yards of cloth of Sf/r*, square foot ? wide ? yins. 24 yards. InU. Inh. InB. Inh. Js 12 : 12 : ; 9 : 16 Antt. Sect. II. 7. SINGLE RULE of THREE INVERSE. 131 4. Lenta friend 292 dollars for 6 5. A garrison had provisions for 8 months; sometime afterwards, he months at the rate of 15 ounces to lent me 806 dollars ; how long may each person per day ; how much must I keep it to balance the favor ? be allowed per day in order that the Jm. 2 months S days, provisions may last 9|^ months ? Uns'. 12*1 ounces 6. A garrison of 1200 has provis- 7. ' How must the daily ions for 9 months at the rate of 14 allowance be in order that the pro- ounces per day, how long will the visions may last 9 months after the ])rovisions last at the same allowance garrison is reinforced ? if the garrison be reinforced by 400 ^ns. 10^- ounces. men ? Jns. 6| Months. 132 SINGLE RULE of THREE INVERSE. Sect, II. 7. 8 If a man perform a purney in 9. If a piece of land, 40 rods in I5days,when the day is 12hours long, length, and 4 in breadth make an in how many will he do it when the acre, how wide must it be, when it day is but 10 hours ? ^«a. 18 days, is but 25 rods l®ng ? jins. 6| rods. 10, There was a certain building raised in 8 months by 120 workmen, but the same being demolished it is required to be rebuilt in 2 months : I demand how many men must be em- ployed about it ? Jns. 480 men. \\. How much in length, that is 3 inches broad, .will make a square foot? jUns. ASinches. 1 2. There is a cistern, having 1 pipe 13. If a field will feed 6 cows which will empty it in 10 hours; how 91 days, how long will it feed 21 many pipes of the same capacity will cows I Ans 26 days. empcy it in 24 minutes ? Ana. 2 5 pipes. Sect. II. 7. SINGLE RULE of THREE INVERSE. 133 GENERAL RULE For stating all questions ivhether direct or inverse. 1. Place that number for the third term, which 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 se- cond term. 2. Multiply the second and third terms together, divide the product by the first, and the quotient will be the answer. EXAMPLES. 1. Jf 30 horses plough 12 acres, how many will forty plough in the same time ? OPERATION. H. H. Ac. 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)48O( 1 6 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 ot horses, 40, the 40 third terra, and because fe^ver will be main- tainedfor the same money, when the price of 8)200(25 Ansiver. hay is dearer, we make tlie greater price, 8 16 cents, the first term, and the lesser price, ^ — — cents the second. 40 40 The first of these examples is direct, the second inver&e. Every question consists of a supposition and a demand. In the first the supposition is, that 30 horses fUongh 1 2 acres, and the demand hoiv many 40 willfilough ? and the first term of the proportion, 30, is found in the supposition, in this and every otiier direct question. In the second, the supposition is that 40 horses are waintained on hay at 5 cents per stone, and the demand, hoiv many will be jnaintaincd on hay at 8 Cktits ? and the first term of the proportion, 8, is found in tlie demand, in this and •very other inverse question. 3. If a quarter of wheat afford 60 4. If in 12 months, 100 dollars gain tcnpenny loaves, how many eight pen- 6 dollars interest, what will gain the ny loavea may be obtained from it ? same sum in 5 months ? Ar.s. 75 luavcff. Afi.sr.'rr, MO dollars. 134 Supplement to the SING. R. of THREE. Sect. IL 7: I Supplement to the ^illgtc ^Hufe Of <^\jttL mm Y " QUESTIONS. 1 1. WHAris the Single Rule of Three ; or^ the Rule of Proportion ? 2. How many kinds of Profiortion are there ? 3. WHAfis it, that the Single Rule of Three Direct teaches ? 4. How can it be knovfn, that a question belongs to the Single Rule oj Three Direct ? 5. WHAfis understood by more requiring more, and less requiring less ? 6. JFTbware questions in the Rule of Three stated ? 7. Ha vjng stated the question, how is the answer found in direct Profiortion ? 8. JVhat' da you observe of the frst and third terms concerning the different denominations, sometimes contained in them ,? 9. When the second term contains dijfferent denominations %vhat is to be done ? \0. How is it known what denomination the quotient is oJ ? 1 1. If the quotient, or answer, be found in an inferior denomination, what is t9 be done ? 12. When the terms are given in Federal Money, how is the ofieration conducted? 13. How are sums in Federal Money reduced to the same denomination ? 14. When any number of barrels, bales, or pieces, Ijfc. are given, what is the method of procedure ^ 15. WHAfis it that the Single Rule of Three Inverse teaches ? 16. How are questions stated in Inverse Proportion ? 17. What is understood by ^ore rej^uirinc less Isj" less req^uirjnq more ? 18. How is the answer found in the Rule of Three Inverse ? 1 9 . Wha f is the general Rule for stating all questions whether Direct or Inverse 2 EXERCISES. 1. If my horse and saddle arc worth 18 guineas and my horse be worth liix times as much as my sriddle, pray, wjiat is the value of my horse ? Jnswer 7Z dollars. Sect. II. 7. Supplement to the SING. R. or THREE. 155 2. How many yards of mattin, that 3. Suppose 800 soldiers were placed is half a yard wide, will cover a room in a garrison, and their provtsions Wiat is 18 feet wide, and 30 feet were computed sufficent for 2 months; long? Ann. \'iO yards. how many soldiers must depart, that the provisions may serve them 5 mouths ? Ana. 480. 4. I borrowed 185 quarters of corn when the price was 19«.how much^ most I repay, to indemnify the lender, when the price is 17*- 4rf. Am, 202|J. 136 Supplement to the SING. R. of THREE. Sect. II. 7. 6. A and B depart from the same place and travel the same road ; but A. goes 5 days before B at the rate of 20 miles per day ; B follow at the rate of 2i miles pei* day : in what time and distance will he overtake A ? J?is. B will overtake Am 20 days, and travel 500 miles. Here two statements will be necessary ; one to ascertain the lime and another to ascertain th« distance. METHOD Of assessing towji or parish taxes. 1. An invcjitory of the value of all the estates, both real and personal, and the nuniber of polls, for which eacli person is rateable, must be taken in sepa- rate tuiumns. Then to know v\h::it must be paid on the dollar, make the total value of the inventory the first term ; the lax to be assessed, the second ; and 1 dollar, the third, and the quotient will shew the value on the dollar. JVora. lids method is taken from Mr. Pikf.''s Arithmetic^ "Milk tins differ-- encCf that here the money is reduced to Federal Currency, SicT. II. 7. Supplement to the SING. R. of THREE. 137 2. Make a table, by muUiplying the value on the dollar by 1,2, 3, 4, 5, Sec. S. From the Inventory take the real and personal estates of each man, and find thsm separately, in the table, Avhich will shew you each man's pro- J>ortional share of the tax for real and personal estates. If any part of the tax be averajjed 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 miake a separate column as well as for the real and personal estates. EXA.MPLES. Suppose the General Court should grant a tax of 1 50,000 dollars, of which a certain town is to pay Dolls. 3250,72 and of which the polls being 624 are to pay 75 cents, each ; — the town's inventory is 69568 dollars ; what will 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. Ctfi. Pol. Dolls. 1. As 1 : ,75 : : 624 : 468 the average part of the tax to be deducted from ^3250,72 and there will remain ^2782,72 Dolls. Dolls. Cis. Dolls. Cts, 2. As 69568 : 2782, 72 : : 1 : 4 on the dollar. TABLE. Dolls. Dolls, cts. Dolls. Dolls. cts. Dolls. Dolls 1 is 4 20 is 80 300 is 8 2 — 8 30 , 1 20 300 — 12 S — 12 40 — « 1 60 400 — 16 4 — 16 50 — 2 00 500 — 20 5 — 20 60 — 2 40 600 -^ 24 6 — 24 70 — 2 80 700 — 28 7 — 28 80 — 3 20 800 ■— 52 8 — 32 90 — 3 60 900 — 36 9 — 56 100 — 4 00 1000 — 40 10 — 40 Now lo find what a's rate will be. His real estate being 856 dollars I find by the Ta- ble that 800 dollars is g32 cts. . that 50 — — 2 that 6 — — 24 Therefore the tax for his t*eal estate is 34 24 In like manner I 'find the tax for his personal estate to be Hi* 4 polls, at 75 cents each, are 3 4 12 Real Dolts. Cm. Personal. Dolls. Cts. Polls. Dolls. Cts. Total. Dol/s. Cts. 34 24 I 4 12 I S 36 133 DOUBLE RULE OF THREE. Sect. IL 8. § 8, «©ouMe0u!eof€!jm. WiM •'.' '-.r -.,.- ^ v.c -vK- -i.? ■■^' The Double Rule of Three, sometimes called Compound Proportion", teaches, by having five numbers given to find a sixth, which, if the proportion be direct, must bear the same proportion to the fourth and fifth as the thiid does to the first and second. But if the proportion be inverse, the sixth num- ber must bear the same proportion to the fourth and filth, as the first does to the st-xond and third. RULE. 1. State the question, by placing the three conditional terms in such order, that that number which is the cause of gain, loss, or action, may pos- sess the first place ; that vvliich denotes space of time, or distance of place, the second ; and that which is the gain, loss, or action, the third." 2. " Place the other two terms, which move the question, under those of the same namu." " 3. Then, if tlie blank place, or term sought, fall under the third place, the proportion is direct, therefore, muliiply the three last terms together, for a dividend, and the other two for a divisor ; then the quotient will be the answer " 4. But if the blank fall under the first or second place, the proportion is inverse, wherefore, multiply the first, second, and last terms together, for a dividend, and the other two, for a divisor ; the quotient will be the answer.** EXAMPLES. 1. If 100 dollars gain 6 dollars, in J 2 ninths, what will 400 dollars gain in a months ? Statement of the questio7t. ■■ D. JSL D. 100 : 12 : : 6 Terms in the ftupfiosition, or conditional terms* 400 : 8 Terms which 77iove the quesdon. Of the three conditional terms, it is evident, that 100 dolfars p«t at inter- est is that omt, which 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 gain is made, this must be the second term ; and 6 dollars which is the gain, the third term. The other two terms must then be arranged under those of tlje same name. Now as the blank falls under the third place, therefore, the question is in direct proportion, and the answer is found by multiplying the three last terms together for a dividend and the two first for a divisor. Then, 12l00)192l0O( OPERATION. 100 ': 12 : : 6 400 8 8 0(3 3200 12 6 Dolls. 16 Ansnver. JOO Div. 19200 Dividend. 2, If 100 dollars gain 6 dollars in 12 months, in what time will 400 dol" hrsgain 16 ? Sect. II. 8, DOUBLE RULE of THREE. 139 OPERATION. D. M. D. 100 ; 12 ; . 6 Here the blank falling; under the second 400 16 term, the proportion is indirect. 6 12 Therefore raultiply the first, second and . . last terms together for a dividend, and the 2400 divis. 192 other two for a divisor. 100 19200 divided. M, Then. 24)00) 192|00( 8 Jns. 192 3. A FARMER sells 204 doHars worth of grain, in 5 years, when it is sold at 60 cents per bushel ; what is it per bushel when he sells 1000 dol- lars worth, in 18 years, if he sell the same quantity yearly ? Cu. Y. D. 60 : 5 : : 204 cU.m. 18 : : 1000 : ^^\^ Ans. 4. If 7 men can reap 84 acres of wheat in 12 days ; how many men can reap 100 acres in 5 days I M. 7 : D. 12 : 5 : J. 84 M. 100 20 .^ns. 140 Supplement TO THE DO. R. OF THREE. Sect. 11.3. Supplement to the vDOUMe MlXlt Of Cj^tCet ■ M B - - -; - "I" - -' - - IT QUESTIONS. 1. WHAfis the Double Rule of Three ; or Compound Propori'ion ? 2. How are (juestiona to be stated in the Double Rule of Three ? 3. How is it knoivn^ after the statement of the question^ whether the firofior-T tion be direct or inverse ? 4. When the firofiortion is Direct ^ how is the answer to be found ? 5. When the proliortion is Inverse j how is the answer to bejound ? EXERCISES. 1. If 6 men build a wall 20 feet long, 6 feet hig-h, and 4 feet wide in 16 days, in what time will 24 men build one 200 feet long, 8 feet high and^S thick? The solid contents in each piece of wall, according to the giv- en dimensions, must |)e found before stat^ ing tl>c question. Sect. II. 8. Supplement to the DO. K. of THREE. Ul 2. If the frr ight of \2C'wt. ^qra, 6/6. 275 miles, cost §27,78: 1k>\v fap piay eoCw/. Syr*, be shipped for §234,78 I 4ns, 480 wtV?^, S. An usurer put out 75 dollars, at I 4. If 7 men can make 84 rods ©f Interest ; and at the end of 8 months j wall in 6 days ; in what time wili received for principal and interest, 79 I IQ men make 150 rods ? dollars ; I demand at what rate per | jins. 5^5 days, f ent he received interest ? I Am. ^/icr cent. X42 Supplement TO THE DO. R. OF THREE. Sect. II. 8- 5. If the freight of 9hhd. of sugar, each weighing \2Civt. 20 leagues, cost _ comprised, in a few cases, such as 4 — y^ shall be useful and easy for th« 3 — j_ Scholar to bear in his memory. — 2^ The small number of examples — 2T "nder each case will be made up 8 — 3^'^ in the Supplement ; this will lead 5 — 4?g- the Scholar to a more particular 2\ — ^'g- consideration of them. 10 6 5 4 n o 2 L 1 1 1 OPERATIONS. Pounds, SbilL Pence, Farthings. Dollars, Cents, Mills. When the price of the given quan- | RULE. tity is 1/;. 1*. l(i per pound,yard. Sec. I Multiply the quantity by the then will the quantity itselfbe the an- I price of 1 pound, yard, 8cc. the pro- swer at the supposed price. There- | duct will be the answer, fore, CASE 1, WifES the firlce ofUjd. lb. l:fc. con^ 9iat.i of farthrngft only; If it be one farthing, take a fourth of the quantity; if a half penny, Xxxkc a half ; if three farthings take a half and a fourth of the quantity and add them. This gives the value in pence, which must be reduced to pounds. 144 PRACTICE. Sect. II. 0. Founds t SbilL Pence, Farthings, EXAMPLES. K What will 362 yards cost, at Jrf per yard ? OPERATION, 2)362 \'2)\%\ pence. \5s. Id. jins. Here the quantity stands for the price at one penny per yard, but as two farth.lngs, are but half one penny, therefore dividing; the quantity by 2 gives the price at half a penny per yard, which must be reducetl to shil- lings; 2. What will 554^ yards cost, at \d, per yard ? OPERATION. d. q. 4)354 2 12)88 2 laAds 2 Am. 3. What will 203 yards cost at 3;/, per yard I Ans. 16*. 5^-^. 4. What will 816 yards cost at .q. per yard? * Am. 17«. t)tllars. Cents, Mills. 1. What will 362 yards cost at / mills per yard ? OPERATION. 3 6 2 quantitij. ,0 7 iirice. g2, 5 3 4 AmiDtr. JYorE. The answers in the differ- ent kinds of money will not always compare, because in the reduction of the price, i\ small fraction is often lost or gained. 2. What will 3 54 J yards cost, at 3 mills per yard ? OPERATION. 3 5 4,5 quantity. ,0 3 firice. ^10 )6 3 5 Ansiver. 3. What will 263 yards cost, at i cent per yard. Ans, % 2,63, 4. What will 816 yards cost at S mills per yard ? Ans. % 2,448, S£CT. II. .9. PRACTICE. 145 Pounds-^ ShiL Pence^ Farthings, 5. What will 97 yards cost at 3y. per yard ? Ans» 68. 04, 4 6. What will 126 yards cost at Id. per yard ? Ans. 5s. 3d. CASE 2. Wmen the firice of Mb. I yard^ Isfc. consists of pence, or oj pence and far- things i if it be an even part of a shil- ling, find the value of the given quan- tity at \s per yard, (ihe quantity it- self expresses the price at \s.per yard; if there are quarters, Is^c. tyrite for ^ od.for \ &d.for i 9d.J and divide by that even part, wliich the price is of 1 shilling. If tlie price be not an al- iquot or even part of 1 shilling, it must be divided into two or more al- iquot parts ; calculate for these sep- arately, and add the values ; the an- swer will be obtained in shillings, ■which must be reduced to pounds. T Dollars, Cents^ Mills, 5. What will 97 yards cost, at 1 cent per yard? Jins. ,97 cents. 6. What will 126 yards cost at 7 mills per yard ? Jns. gO,882. 146 PRACTICE, Pounds, Sbili, Pence, Farihings EXAMPLES. 1. What will 476 yards cost, at 7ld per yard ? OPERATION. fl. ^^,\ I f I ^'\^ Price at \s. /ler i/ard. ^'28 firzce at 6d./ieryard. 59 6d./iriceac7{d./ieryd. Irfi 2(o)29|7 Gd./irice at 7\d. fier yd. ^ 14 17*. 6d. jinswer. PROOF. 1. By the Rule of three. Y. £. s. d. Y. As 476 ; 14 17 6 :: I 20 297 12 476)3570(7^. 238 4 )952(2-/. 952 % By Compound Multiplication, £. s. d. 7^ price of 1 yard. 10 6 3 firice 10 yards, 10 2 6 price ef 100 yards. 4 12 10 price, of 4.00 yards. 2 3 9 price of 70 yards. 3 9 price of^ yards. r.\i> 17 €> price of 4.7 6 yards. Sect. II. £), Dollars, Cents, Mills, 7. What win 476 yards cdme to at 10 cents 4 mills per yard ? OPERATION. 476 ,104 1904 4760 ^49,504.^^5. PROOT cts^ fn. D. ct8. m. yds. ,1 4)4 9 5 4(476 4 1 6 7 9 7 2 8 6 2 4 6 2 4 Sect. II. 9. PRACTICE. U7 Founds, ShilL Pence, Farthings^ 2. What will 17G yards cost, at 9~_d. per yard ? ed. 5d. id. OPERATION. S. ■J I 176 value at \s. fier yd. ^ ! 88 value at ed.fier yd. -|of 44 value at 2d, per yd. 7 4a?. value at \d.iier yd. 2I0>13I9 ^d.—at'^\d. per yd. ^6 \9s.Ad. Jins. PR90J?. 3. What will 5681 yards cost at 7d. per yard? Ans.€\& \\s. s^d. Dollars, Cents, Mills. 8. What will 176 yards cost, at IS cents, 2 mills per yard ? Jns, g23,232. 4. What will 568^ yards cost at 9 cents, 7 mills per yard ? -^^w. g 55,12. 148 PRACTICE. Sect. 11.9.^ Pounds, SbilL Pence, Farthings. 4. What will 685| yards come to, at '2>\d. per yard ? Jns, £7 2s. lO^rf, Dollars, Cents, Mills, n 10. What \7ill 685 2; yards come to, at 3 cents, 5 mills per yard ? 4ns, 3S24,OOU 5, What will 649 * yards cost, at lOd. per yard ? Ms. £27 U. Ojrf. 11. What will 6491 yards cost, at 13 cents, 9 mills per yard. Jins. S9Q,245, 6. What will 6831 yards cost at Slf/. per yard ? Jins. jC23 10*. 0|f/. 12. What will 683-1 yards cost, at Sect. II. 9. PRACTICE. 149 Pounds.ShilL Bence^ Farthings, CASE 3. If the price of Mb. 1 yard^ ifc. be shillings and pence^ and an even part of 1 f^. Divide the value of the given quantity at \{^ per yard by that ex'en part, which the price is of ^Cl. The quotient will be the answer. EXAMPLES. l.What will 7 19 J yards cost at 1*. 4.i/. per yard ? OPERATION. ^. Si I 1/4 I -I5. I 719 \0 price atU per yd. 143 1 Q price at As. per yd. Dollars, Cents, Mills. Jns. 47 19 4d. at \f4 per yd. Herk for the sake of ease in the operation, because 5X3:=: 15, there-? fore I divide the price at one pound per yd. by 5, and that quotient by 3, which gives the answer. 2. What will 648 yards co»t, at 1/8 per yard ? jins. /C54. 13. What will 7 19 J yards cost, at 22 cents, 3 mills per yard ? jins. g 160,448. 14. What will ;^S yards cost, at 27 cents, 8 mills per yard ? Am, g 180 j 144. ISO PRACTICE. Sect. II. f'ourtdsy SbilL Pence ^ Farthings^ 3, Whativiil 687| yards cost, at 5». per yard? Ans.fAY \7s. 6d CASE. 4. Whek thefiriceof 1 yard, i^c. is shillings f or shillings pence b farthings and not an even part of \C. Multiply the value of the qnaniity at \s. per yard by the number of shillings ; for the pence and farthings take parts, as in Case 2. the results added will give the answer, which must be reduced to pounds. If the price be shillings onlt/y and an even number ; multiply by half the price or even number of shillings for one yard, double the unit fig- ure of the product for shillings, the remaining 6gures will be pounds. Note. When the quantity contains a fraction, work for the integers, and for the fraction take proportional parts of the rate. EXAMPLES. l.What will 167^ yards cost at 17*. 6rf. per yard ? OPERATIOBT. *- I 167 17 1169 167 2839 price at 1 7s. per yd. 83 6 — at ed.per yd, 8 9 price of \ yd. I ^d- 210)293(1 Zd. Jn». xn46 11«. Zd. Dollars f Cents, Mills. 15. What will 687^ yards cost, a|: 83 cents, 3 mills per yard ? Ans, 8^72,687. 16. What will U7f yards cost, at g2j916? ^n«. 488,43 Sect. II. 9. PRACTICE. 1^1 Pounds, ShilLPetice, Farthings, 2. What will 5482 yards cost, at \2s. 4^f/per yard ? jins, £ 3391 19*. 9d. What will 614 yards cost, at 16s. per yard. OPERATION, 614 8 /laff the firice. 4912 double the Jirst figure ^491 4s. ^ns. [for shill. 4. What will 176 yards cost, at i25.peryard? Jns,£lQ5 I2s. 5. What will 36 yards cost, at 7#, 6^. per yard. ^«*. X;i3 10*. Dollars, Cents, Mills. 17. What will 5482 yards cost, at S2,063 per yard ? 18. What will 614 yards cost, at S 2,667 per yard I Ans. g 1637,538 \'^. What will 176 yards cost, at 2 dollars per yard I Akm, gSoSl 20 What will 36 yanlscost,at S 1,25 per yard? ^««,S45, \ 152 fRACTJCE. SUCT. II. 9. Founds f SbilL Pctice ^Farthings, CASE 5i When the price of 1 yard^ 1 Ih. IsfC. is fioUhds^ shillings^ and pence ; Mul- tiply the quantity by the poimds and if the shillings and pence be an even part of a pound, divide the given quanliiy by that even part ^ and add the quotient to the product for the answer ; but if they are not an even part of \{^. take parts of parts and add them together. Or, you may reduce the pounds in the price of 1 yard, 8cc. to shilling's and proceed as in the Case before. EXAMPLES. 1. What will 59 yards cost at iC6 78^6d. per yard ? 55. is ^ of ^1 2/6 is i^ of 5.9. OPERATION. 59 valhe at £\ per yd. 6 354 — at £6 per yard, 14 15.9. at 5s. per yd, 7 7 ed.at2/eper yd. Jins. £076 2 &'at £ 7s, 9d. 2. What will 16 3 yards cost, at £2 8s. per yard ? Jns. /?39i 4s. Dollars, Cents^ Mills. 21. W^hat will 59 yards cost at g21,25 per yard ? OPERATION. D. C. 21,25 59 191 25 1062 5 gl253 75 Jins. 22. What will 163 yards cost at 8 dollars per yard ? Ans, g 13,04 Sect. II. 9. PRACTICE, Pounds y SLilL Pence ^Farthings, 3. What will 76 yards cost, at aCS 2s. Id. per yard ? OPERATION. ». $d. is \ of ,U« 76 value at Is. per yd. ,^ ^ " fizzzshiUs. in £3 Sa. 152 value at "is.fier yd. 456 — At 608. tier yd. \d. is -J- of Ct/. 38 — ar ed.fier yd. 6 4c/. — at Id. /ler yd. 2|0)475/6 4d. jins.£237 16*. 4\ hovi is the operation to he conducted ? 8. When. the price consists of shillings only, and an even number, what is the most direct way to find the value oj any given quantity ? 9. When the quantity contains fractions, as ^, |) ^^, Ufc. how are they to be treated ? 10. When the price consists of pounds, and lower denominations, how is the value of any given quantity found ? 11. When the prices are given in Dollars, Cents and Mills, how is the value tfany given quantity J ound in federal money ? 12. WHAf is the method of proof ? {%„ How are the operations in Federal Money proved ? EXERCISES IJV PRACTICE. . In the followirig exercises, the attention of the scholar must be excited first to consider t6 which ot" the preceeding cases each question is to be referred.. Thai heini; ascertained, he will proceed in il^ operation according to the in- struction there given. 1. What will 745-^ ysfrds cost at 1 \d. per yard ? Jins. x;34 3*. 7\d. tJNDER which of the ' proceeding cases docs this question properly * belong ? ^ What must be done, with the fraction (^ of a yard) in the quantity ? H ^* # #■ Sect. II. 9. SUPPLEMENT to PRACTICE. 155 i 2. What will 964 yards cQSt,at Is. 9d, per yard ? ^na. £^0 6s. 8d. OPERATION. PROOF. f-» ^ # % 3, What will 354^ yards cost, at ^d. per yard ? ^ns, 7.9. A,^d , 4. What will 316 yards cost, at \d. per yard? Aris^ 19*. 9rf. -■■4 5. What will 567^ ywds GOSt> at IJflf. per yard ? Jns.cz 10*. \\\d. 6. What will 913J yardacosc|Ntt Crf. per yard ? 'r^ ATt9. £22 16#. W. 156 SUPPLEMENT to PRACTICE. Sect. II. §. 7. What will 912| yards cost, at 9cl. per yurcl i Ma. £34, 4». 4^d. 8. What will 76 yards cost, at 2d. per yard ? y(f««. 125. 8i/. r I* ' # ^ « 9. What will 845 yards co&t, at 8*. per yard ? If). What will 9 1 ya»ds come lo at 1 6«. per yard ? Jns, £72 IGs. 11. What will 156^ ya^ds eotoc to, at 6«. 4:d. per yard. jins. £4t9 lU, St/. 12. What will 96 yards cost at 10s. U«/. per yard ? ^«*. jC;48 12«. m ^^ ' Sect. II. 9. SUPPLEMENT to PRACTICE. 157 13. What will 67 1 yards cost, at 12&. 2d. per yard ? ^w*. ^41 1*. 2d. 15. What will 75 yards cost, at £3 3«. 4^/. per yard ? Jns, £2o7 10«, 14. What will 843 yards cost, at €«. 8J, per yard ? Jns. £2&i. &^ 16. What will 59 yards come to, «it jp6 7s. 6d. per yard ? .4ns. £376 2s. 6d. y I 17. What willj9^- yards come I to, *t/C3 ««. 8d. per yard? • Jns. /;199 3«. 4d. 18. What will 68 yards cost, at /C 4 6*. per yard ? Jm. 292 S<. V 158 SUPPLEMENT to PRACTICE. Sect, II. 9? N. B. The following questions are left "without any answers, Chat the Sciiolar may operate and prove each question. 19. What will 11, yacds of flannel, at %s. ^d. per yard, come to ? * OPERATION'. «fc PROOF. 12. What will IS lb of cotton cost, at S*. 4c?. per Ife 2 } , Wk AT will 1 &S yards of ribbon come to, at &d. per yard I -^.^ f l'*1k •>• ^^ -■ ' 1 -f^ * *» % THE ^ SCHOLAR'S ARITHMETIC. ► -r -;::- v15- >r >^ -r *> -j'.:- a* « SECTION III. I Rules occasionally useful to men in particular callin^.^ and pur- suits of UJe. § 1. 1'nUoluttan. Tnvolutjon, or the raising of powers is the multiplying of any given num- ber intoitself continually, a certain number of times. The quantities m this way produced, are called powers of the given number. Thus, 4x4=: 16 is the 2d. power, or square of 4. r-4'* 4X4x4zi64 is the 3d. power, or cube of 4, zr4^ 4X4X4X4::=256 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 povyer, is called the Index or the Exfioncnt. ■\ — g>>i The root is a number whose continual n\ultiplication into itself prodoces the power, and is denominated the square, cube, biquadrate, or 2d, 3d, 4th, root, &cc. accordingly as it is, when raised to the 2d, 3d, 4lk, Sec. power, equal to that power Thus, 4 is the square foot of 16, bccatise 4X'1:=:16 4 also is the cube root of 64, because 4x4X'^=:64 ; and 3 is the square root of 9, and 12 is the square root of 144, and the kubc root of 1728, bccaud« 12x]2xl2:::;i72y, and soon. 160 EXTRACTION of the SQUARg ROOT. Sect. III. 3 To every number there is a foot, although there are numbers, the precis* roots of which can never be ebtained. But, by the help of decimals, we can approximate towards those roots, to any necessary degree of exactness. Such roots are called Surd Roors^ in distinction from those, perfectly accurate, which are called Rational Roots. ^1 The square root is denoted by this character ij placed before the power ;f the other roots by the same character, with the index of the root placed over it. Thus, the square root of 16 is expressed ^ 16, and the cube root of 27 3 is y/ 27, kc. \y;hen the power is expressed by several numbers with the sii^n -|- or— i between them, a line is drawn from the top of the sign over all the parts of it; tlius, the second power of 21 — 5 is v'^l — 5, and the 3d. power of 5 6-{-8 3 is ^56 + 8, Sec. Ihe second, third, fourth, and fiftli powers of the nine digits may be seen in the following TABLE. Hoots, - or 1st. Powers. * 1 2 4 8 16 Squares, or 2d. Powers. Cubes, - or od. Powers. Bi quad rates or 4th. Powers. • Sursolids, |or 5th. Powers. 32 3| 4 5 6 7 49 8 9 9 16 25 36 64 8.| 27J 64 125 216 343 2401 512 729 ! 81 256 625 1296 4096 6561 243]l024 3125 777^ 16807 32768 59049 1 1 § 3. (ffitrattion of tijt Ji^uare Boot. To extrnrt the square root of any number, i.s to find anothertiumber which mulriplifd by, or into itself, will produce the given number; and after the root is found, such a multiplication is a proof of the work. RULE. 1. *' Distinguish the given number into periods of two- fic^ures each, by pulling a point over the place of units, another over the place of hundreds, and so on. which points shew the number of figures the root will consist of. 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 ihe manner of a quouenl in division) for the first figure of the root, and the, square number, under the period, and subtract it therefrom, and to the remainder bring down the next period for a dividend. 3. " PLy* cE 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 (excpt the riglit hand figure} and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor ; multiply the divisor with the fig- ure last annexed by the figure last placed in the rc>ot, and subtract the product from the dividend ; To the remainder join tlie next period for a new dividend. Sect. III. 3. EXTRACTION or the SQUARE ROOT. 161 .5. ^' Double the fiojures already found in the root, for a new divisor^ (or bringdown your last divisor for a new one, doubling the rii;ht hand figure o.f it) and from these, find the next figure in the root as lust directed, and con- tinue the operation in the same manner, till you have brought down all ilie periods. . . . " Note 1. If, 'when the given power is pointed off as the power ijequires, the left hand period should be deficient, it must nevertheless stand as.thp first period. . ' " Note 2. If there be decimals in the given number, it must be pointe*! both ways from the place of units : If, when there are integers, the first pe- riod in the decimals be deficient, it may be completed by annexing so many cyphers as the. power require* : And the root must be made to con- sist of so many whole numbers and decimals as there are periods belonging to each ; and when the periods belonging to the given number are exhausted, the operauon may be continued at pleasure by annexing cyphers.'* EXAMPLES. - I. What is the square root of 729 ? OPERATION. 720(27 the root. 4 The given number being dislinguished into periods, I seek ilie greatest square number ia the left hand period (7) which is 4, of which the root (2) being placed to the right hand of the given number, after the manner of a quotient, and the square number (4) subtract- ed from' the period (7) to the remainder (3) I bring down the next period (29) making for a dividend, 329. Then the double of the root (4) being placed to the left hand for a divisor, I say how often 4 in 32 ? (excepting 9 ///(? right handJiguTeJ the answer is 7, which t place in the root for the second figure of it, and 729 also to the right hand of the divisor ; then multiplying the divisor thus increased by the figure (7) last obtained in the root, I place the product underneath the dividend, and subtract it therefrom', and the work is done. DEMONSTRATION Of the reason and nature of the ^various steps in the extraction oj the Sq^UARE Root. The superficial content of any thing, that is, the rnunaber of square feet, yards, or inches. Sec. contained in the siirlace of a thing, as of a tabic or floor, a picture, a field, ecc is found by mulliplying the length into the breadth. If the length and breadth be equal, it is a square, then the measure of one of the sides as of a room, is the root, of which the superficial content in the floor oF that room, is the second power. So that having the superficial contents of thefloor of a square room, if we extract the square r»ot, wc shall have the length of one side of that room. On the other hand, having the length of one bide of a square room, if we multiply that number into itself, tha\ in to I'aise it to the second power, we shall then have the superficial contents of the floor of that room. The extraction of tlie square root, therefore has this opperation on num- "S, to arrange the nuJiibcr oj'vjhich ive txtruct the root into a ftqtKirc form. As w I 16^ EXTRACTION or ihe SQUARE ROOT. Sect. III. 3: if a tnaTi should have 625 yards of cnrpetirig;, 1 yard wide, if he extract the square root ol that number (625) he- will then have the len^nh of one side of a square room, tiie floor of wh-ch, 62 'i yards, will be just sufficient to cover. to proceed then to the demonstration. Example 2. Strppos-rxG a n25(30 TSE nurn'bef berng pointed off as the rule 4 directs, we fi-nd we have two periods- ; conse- ' quently; the root will consist of two figures. 225 The greatest Sqilare number m the left hand periiud (6)15 4, of which two is the r»ot ; there- "^^Tcr. T. fore, 2 is the first figure of the root, and as it is - certain we have one figure more to find in the root, we may for the presenc supply the place of that figure by a cyphei', (20) then 20 will ex- press the just valae of that part of the root now obtained. But it mulacecl for ■a divisor in place of tens (for it inust he re member ed^ that the next Jijure in the root it to be placed before it J is in reality 40, equal Xa the ,§ides b c (20) and c d (^0) of the square A. OQO '0 A 20 400 C. 30 5 100 20 T 5 h The square A Fio. n. AiiAii^, hy the mle, " Seek y 5 ^ow often the divisor is contain^ ~^- ~ 20 5 I f^' ^*« ^^^ dividend feafe/it the Q 5 D 5 J5 right hand f^nire J and /dace the ^ ' I QQ ~25~^i ananver in the root^fctr ihe sec^ ond figure of its and dpi Hie right Jhand 9f the dix'jsQr'* ^w If t'he sides b c and c d of the square A, Fig. II. is the lenj^th to "Which the remain- ing 225 yards are to be added, and the divisor ^4 /ew.^J is the &\\m of these two sides, it is then evjdcm, that 225 divided by the leagth of the two sides, 'that is by tlve idivisor (A tens} will give .the breadth of this new addition of the .225 yards^ to the sideA 6 c aod.c^ of the «quarc, A. But we are directed to " except the rig%t /land figure ^'^ and also to ^'' place the quotient — — — figure on the right hand of the divisor ;" the froofQ35 ijd9 reason of which is that t\he additions, C>y^ and C g h to the sides he and cd of the square, A, do not leave the figure a com- plete square, but there is a deficiency, D, at the corner. Therefore, in di- . viding, the right hand figure is excepted, to leave -something of the dividend, for this ileliciency ; and as the deficiency, D, is limited by the additions C e f and C .5- /i, and as the quotient figure (5) is the width of these additions, con- sequently equal to one side of Ihe square, D ; therefore, the quotient figure (5) placed to the right hand of the divisor {^tensj and multiplied into itself, gives the contents of the square, 1), and the 4 fcw.^rrito the sum of the sides, be and cd of the addition Cef and Cgh^ multiplied by the quotient figure, (5) the width of those additions, give the contents C r/and C.^ A, which together sub traded from the dividend, and there being no remainder, shew that the 225 yards are di<*posed in the new additions Cf /", C^A, and D, and the figure is seen to be continued a complete square. Consequently, ^f II. shews the dimensiors of u square rcom, C5 yard^ on a side, the floor of which, tJ25 yards of carpeting, 1 yard wide will be suffi- cient to cover. The proof is seen by adding together the different parts of the figtirr. Such are the principles, on which ihc operation of extracting the square root is grounded. —400 yds. — 100— P^/i— 100 — D — 25— 1^4 E XTRACTION of the SQUARE ROOT. Sect. III. 3. 3. What is the square root of 4. What is the square root of 1O3426'50 ? Jns. 3216 43264? ^/i*. ? 208. 5. What is the square root of 964,5192360241 ? ^/z^,- 51,05671 Sect. III. 3. EXTRACTION of the SQUARE ROOT. 169 6 Wh 99800 iHkT is the square root of 7. "What is the square rpot of 1 i" ^72*. 999. ^y:T^S.4,jQ9V" jinn, 15^3 Zi Wlvatis the square root of ipqq892198,40ai.? . yins. SSlQT^l 166 SUPPLEMENTto t«e SQUARE ROOT. Sect. III. 3. Supplement to the ^quatC ClOOt .. II—. -:> ^ iit jfc -irr iVe •»> — i | ii . QUESTIONS. 1. JVxjristo he understQod by a root ? A Jio^er ? The second) thifd^ and fourth fioivers P 2. WnAfia the IndeXy or Exfionent ? 5. If- HAT is it to extract the Square Root ? 4f. Why is the given sum peinted off into fieriods of tv)0 Jigures each ? 5. Ik the operation^ having found the Jirst figure in the rooty tvhy do we sub- tract the square number^ that is y the tqucire of that Jigurey from the peri- od in ivhich it ty«* taken ? 6. Why do "JDe double the root of a divisor ? 7. Ivi dividing why do mxe except the right hand figure of the dividend ? fjt. Why do ive place tfje quotient fgure in -the root and also to the right hand of the divisor ^ 9. Ip there he decimals in the given numbery hoxv mu^t it be pointed ? JO. Noiv is the operation of extracting t^e Square Root proved ? EXERCISES IN THE SQVARJ^ ROOT. i. A Clergyman's glebe consists of three fields; the first contains 5 Jcr. 2 r \2 p. the second, 2 ac. 2 r 15 p. the third 1 ac, I r. 14 fi. in ex* change for which the heritors agree to give hini a square field equal to all tl>c ihree. Sought the side of the ^uare I Ana. 39 poles. 2. A OENERAL nas an arm)' of 4(596 men ; how many must he place in rank ar^d file to form them into a square ? Jnsiver 64. Sect. III. 3. SUPPLEMENT to the SQUARE ROOT. 167 3. There is a circle whose diameter is 4 inches, what is the diameter of a circle 4 lime* as large ? -^«*. 8 inches. ^ Note. Square the given diameter, multipiy this square by the given proportion, and ihc square root of the product will be the diameter requ'ired. Do the same iR all similar cases. If the circi»; of the required diameter were toi be less than the circle of the given diameter, by a certain proportion, then the square of the giv- en diameter must have been divided by that prop«rtioa^ 4. There are two circular ponds in a gentleitian's pleasure gtound : the diameter of the less is 100 feet, and the greater is three times as large. What is iis diameter. ^mivcry 173, 2-^ 5. If the. diameter of a circle be 12 inches, what will be the diameter of another circle, half so large i Jint. 8, 48 -frnc>if#. 168 SUPPLEMENT to the SQUARE ROOT. Sect. 111,3. 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 sides of the ditch ? Answer 4:5 feet. Note. A figure of three sides, like that form- ed by the wall, the ditch and the ladder is called a right angle triangle^ of which, the square of the hypotenuse, or slanting side, (the ladder) is equal to the sum of the squiAres of the two other sides, that is, the heighth of the wall and the width of the ditch. 7. A LINE of o& yards will exactly reach from the top of a Fort to the op- posite bank of a river, known to be 24 yards broad ; the height of the wall is i-equired ? Ansnver 26^%o-\-yards. Sect. III. 4. EXTRACTION of the CUBE ROOT. 169 8. Glasgow is 44 miles west from Edinburgh : Pceble?^ is exactly soutli from Edinburgh, and 49 miles in a straight liae from Glascow ; whatis the distance between Edinburgh and Peebles ? AiiHioer,, 2\^o-\-miles. § 4, (iHjetraction of t&c Culje 0oot, 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- putting a 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 in the quotient. 3. " Subtract the cube thus found, from the said period, and to the re- mainder bring down the next period, and call this the dividend. 4. " MuLTU'LY 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 qualient. 6. " Mui.Tin.Y the tripple square by the last quotient figure and write the product under the dividend ; muliiply the square of the last (juolicnt fig- ure i)y the triple quotient, and place this product under the last ; under ulU sot the cube of the list qiioiioiil fK^ure, and c.ill tijcir sum the subtrahend. 7. " SuimiAcr the subtrahend from the dividend, and to the remainder bring down tiie neKt period for a new dividend, with which proceed as be- fore, and so on till the whole be rmii.hed. Note. The same rule must be observed for continuing the operalion> and pointing for decimals, as in the square root." X 170 EXTRACTION ox the CUBE RQOT. Sect. III. 4. Wh^t is the cube root of 373248 ? Oi'ERATIO:?. Divisor ; 49 10)30248 373248(72 the root. 343 7^'7y^^00i=:\^700,the tri/Ue square, 7y.oQ z=. 2 1 G r//e trijile qxiQtiau, 14910 //;c 29400 S40 14700x2—29400 ^ 2X2X210=: 840 2X2X2 = 8 J0248 r 30248 the sukrahend. 00000 DEMONSTHATION OJ theRecffif)!). and Kature of Ike the various stefis in the ojieratioii of extracting the CUBE ROOT. Any solid body \\z.\i\-\^ Ri:x: equal sides^ and each of these sides an exact (square is a Cube, and the measure ia len?:lh of one of its sides is the root of that cube. For if ihe measure in teet of any one side of such a body be multipli- ed three time« into itself, that is, raised to the tiiiid power, the product wiU be the number of solid feet the whole body contains. AxD on the other hand, if the cube root of any number of feet be extracted this root will be the length of one §ide of a cubic body, the whole contents oi which will be equal to such a number of feet. Slpposincv a man has 13824 feet of timber, in distinct and separate blocks ef one foot each ; he wishes to know how large a solid body they will make when laid together, or what will be the length of one of the sides of that cu- bic body ? To know thi?, all that is necessary is to extract the cube root of f hat num- ber, in doing whicii I propose to illustrate the operation. OPEllATlOK. In this number, pointed off as the rule 15824(30 directs, there are two periods, of course ^ there will be two figures in the root. The {greatest cube in the right hand pe- riod, (13) is S, of which 2 is the root, there- fore, 2 placed in tlie quotient is the first fig- ure of the root, and as it is certain we have one figure more to find in the root, we may lor the present supply the place of that one figure by a cypher (20) then 20 will express I the true value ©f that part of the root now ^; obtained. But it must be remembered, that V ./ %^^^ cube root is the length of one of the sides I {^ . _ _._:^i^ of the cubic body, whose length, breadth, and A 20 F tliickness are equal. Let us then form a 2C/ cube. Fig. I. each side of which shall be sup- posed 20 feet; now the side A. B. of this 400 cube, or either of the sides, shews the root, 20 (20) which we have obtained. Sl'l-i FiG. r. c ' n k 5000 feetzzzthe Qolid contents of the CuBEt Bect. III. 4. EXTRzlCTION o? fnE CUBE ROOT. 171 The Rule next directs, subtract the cuhe^ thus founds from the stiicl period and to the remainder bring doion the next fieriod^ ijfc. Now this cube (8) is the solid contents o( the figure wc have in representation. Made evident thus — Each side of this figure is 20, whith being raised to the 5d ])ower, iliat is, the Itngth, breadth and thickness being multiplied into each other, irives the solid contents of that figurerrSOOO feet. And the cube of the root, (2) which we have obtained is 8, which placed under the period from which it was tak- en as it falls in the place o^ tHonsands^ is 8000, equal to the solid contents of the cube A B C D E f, which being subtracted from the given number of feet, leaves 5824 feet. Hence J'^ig. L exhibits the eiiact progress of the operation. By the opera- tion 8000 feet of the timber are disposed of, and the figure shews tlie disposi- tion made of them, into a square solid pile Which measures 20 feet on every side. Now this figure or pile is to be enlarged by the addition of the 5 824 feet, which remains ; and this addition must be so made^ that the figure or pile, shall continue to be a complete cube; that is have the measure of all its skies equal. To do thi^ the addition must be made equally to the three different squares, or faces a, c and d. ThE next step, in the operation is, to find a divisor ; and the proper di- visor will be, the number of square feet contained in all the points of the figure, to which the addition of the 5 824 feet ia to be made. Hence we are directed " multiply tke sgtiare of the quotient by 300," tlie object of which is, tt* find the superficial contents of the three faces a, c, ^>, to wl»ich the addition is now to be made. And that the square of the quotiuiit, multiplied by 300 gives the superficial contents of the fscesjn, c, b^ h evi- dent from what follows. Side A B:zi20^ 2 quolieiit f^nra Side AF = 20 t' ,. ^ 2 f>o/ the facey a. Superficial content— z40o} 4 iiic sc^vare rf 2 00 The triple square 1 'lOOz^the sufier- The triple sqiiarr rf \ 2Q0-=ihe super- ficial contents oj the faces^ a, c, and 3, fcial contents (fthefarca «, c, andb. The two sides A B and A F of the Here the quotient figure 2, is prop- face, a, multiplied into each other, crly, fwo /f?.'^ for there is another fig- give the superficial content of c, and ure to follow it in the root, and the as the faces, a, c, and 6, ar i all equal, square of 2, standing as units^ is 4, but therefore, the content of the face, a its true value is 20 (r//r ddc A B) of multiplied by 3,will give the contents which the square is 400, w^hercfore ofa, c, and^. lose two cyphers, and ihero two c0l)lp* QUESTIONS. 1. WRAf is Felloivshi/i .? 2. Of Aow mamj kinds is Fellozvshifi f 3. WuAr is single Fellowshifi ? 4. WHA'Tia the rule for operating in single Felloiushifi ? 5. Whav is double Fellotuahifi ? 6; WHAtis the rule for operating in double Fellcwshi/i ? 7. How is Felloiuship proved ? EXERCISES lA' FELLOWSHIP. A, B, and C, hold a pasture in common for which they pzy£20fier amrurrt^ In this pasture, A had 40 oxen for 76 days ; B had 36 oxen for 50 days, and C had 50 oxen for 90 days. I demand what part each ot these tenants ought to pay for the jC20 ? £. s. d. gr. - Ans. 6 10 2 l||t§A*spart. 3 J 7 I Oi^no B's part. 9 12 8 2|ffJ CSparl, 1(34 BARTER. Sect. III. § 6. barter. iO-^o^o-^o y -.'.'f^o^o-^O'^o* Barter is the exchanging bf one toinmodity for another, and teaches merchants so to proportion their quantities, that neither shall sustain loss. Proof. By changing the order of the question. RULE. 1 . When the quantiiy of ine ccmmodity is ^iven^ with its valucy or the value of its integer^ as also the value of the integer of some other commodity to be ex- changed for ity to find the quantity of this commodity : Find the value of the commodity of which the quantity is given, then find how much of the oth- er commodity at the rate proposed, rhay be had for that sum. 2. /> the quantities of both commodities be given, and it should be required to find honv much of some other commodity , or honv much money should be given, for ihe inequality of their "valines : Find the sepjtrate value of the two given com-* inodities, subtract the less from the greater, and the remainder will be i\\b balance, or value of the other commodity. < 3. If one com7nodity is rated above the ready money fir ice, to find the barter* i^S firice of the other : Say, as the ready money price of the one is to the bar* tering.price, so is that of the other to its bartering price. EXAMPLES. 1. How milch coffee, at 25 cents per 2. I have 760 gallons of mo- lb. can I have for 56 lb. of tea at 43 lasses, at 37 cents, 5 mills, per cents per lb. gallon, which I would exchange OPERATION. for 66 Civt. 2qr. of cheese, at 4 5 6 lb. of tea. dollars fier Civt. Must I pay of ,4 3 /ler lb. receive money and how much ? jins. must receive 19 dolls^ 1 6 8 2 2 4 ^Ib. or. 5)2 4,0 8(96 5^\ ans-iv 2 2 5 1 5 8 15 8 1 6 2 5) 1 2 8(5 1 2 5 Sect. III. 6. BARTER. ISB 3. A and B» barter ; A has 150 bushels of wheat at 5*. 9d. per bushel, for which B gives 65 bushels of barley, worth 2s, lOd. per bushel, and the bal- ance in oats *t 2s. Jrf. per bushel ; what quantity of oats must A receive fr«m B ? Amwer, 325^| bushels. 4. A HAS liimen cloth worth 20rf. an Ell, ready itioney ; but in barter he will have two shillings ; B has broadcloth worth 14«. 6^. per yard ready Taoney j at what price ought the broad cloth to be rated in barter i Amnvevy 17«. 4ci, Sgr. ^^ t^er yard. 186 SUPPLEMENT to BARTER. Sect. III. 6. Supplement to ^attCl% QUESTIONS. 1. WtiA \pt. and \ very nearly. Sect. III. 7. LOSS AND GAIN. 187 § 7. nm anti (Sain *< Loss and Gain is a rule wliich enables merchants to estimate their profit or loss, in buying and selling goods ; also, lo raise or fall the pidce of them, so as lo gain, or lose so much per cent/* CASE I. 7o k?iow what is gained or lost fier cent. First, find what the gain or loss i» by subtraction : then, as the price it cost is to the gainer loss, so is 100 ■dollars (or C l&O) to the gain or loss, per cent. EXAMPLES. 1. If I buy candles at 16 cents, 7 2. Bought indigo, at g 1,20 per lb. mills per lb. and sell them at 20 and sold the same at 90 cents per cents per lb. what shall I gain fier lb. what was lost/zer cent ? cent, or in laying out 100 dollars ? Answer, 25 dollar 9. OPERATION. I sell at j20 per lb. bought at ,167 per lb. I gain ,033 per lb. Then, as, 167 : ,0 3 3 1 100 ■D. cis. ,J67)3, 3 0(19,76 Ans. 1 6 7 16 3 15 3 12 7 116 9 1 2 3. Bought 37 gallons of Brandy, 4. Bought hats at 4«. apiece, and at S 1>10 per gallon, and sold it for sold them again at 4*9 ; what is the 840 : what was gained or lost / a^d fourths {""). TABLE. By this rule also may be calculated the 12 Fourths make \ Third. solid contents of bodiesjharing the meas- J2 lliirdfi — 1 Second. ures of their different sides, and is very 12 Seconds — 1 /«c/t,or/*rfm how many square feet does it contain ? Jns. 46//. 10m. 6'\ B-y J^raccice, By Duodecimals. OPERATION. Ft. in. 6 inches 18 \ 16 4 2 8 32 8 % \ 3 2 2 8 10. Thcbe is a stock of 15 boards, 12 feet 8 inches in length, and 13 inch« es wide j how many feel of boards does thtc stock contain ? Ans. I^Sfeet^ \0 inches. By Practice. By Duodecimals, Sect. III. 8. SUPPLEMENT to DUODECIMAtS. 159 Supplement to ©UOtltCintaK,^* ^mm -Jl'r •»;> -:'.> H^i •:'.'? 4:v iV-r «■»- QUESTIONS. 1 . Of 'What use are Duodecimals ? To whom more esfiecially are they usejnl « 2. 'In what are dimensions taken ? 3. How do you fiTOceed in the multiplication oj" duodecimals ? 4. For what number do you carry ? 5. WHAfdo you observe in regard Xo setting down the fir oduct different Jrorri what is common in the multifilication of other numbers ? 6. Of what term is the firoduct which arises from the multifilication of Jfet by inches ? Feet by seconds ? Inches by inches ? Inches by seconds ? Sec onds by seconds ? 7. In what way can the operation be varied ? EXERCISES. 1 Multiply 76 feet 3 inches 9 sec- 2. What is the product of 371 onds, by 84 feet 7 inches 11 seconds. feet 2 inches 6 seconds, multiplied by 181/. Un. 9", ^n«. 67242/. lOin. V 4'" 6"" OPERATION. F. /. " 6 inches is ^)76 3 9 84 7 11 76X 4rr 304 76X 8— 608 3X84— 21 9X84— .5 I-H) ^ 3 '" 1 10 6 31)anrf 21) 3 4 3 9' nft 2 1 10 6 1 7 11 3 1 8 7 6 Vrod. 6460 7 1 8 3 3.' How many square feet in a stock of 12 boards, 17/ 7' long, and \f. Sin, wide? ^ns, 298/ U'. A a many cubic feet of \vood 6/ 7' long, 3/. 5' high, 4. How in a load ^ and y. 8' wide ? Jns. 83/. i' »" 4'" 1^4 SUPPLEMENT to DUODECIMALS. Sect. IIL 8. The Dimensions of Waincoating, Paving, Plastering^, and Painting arc taktn in Feet and Inches, and the content given in Yards. PAIN'2'ERS AND yoiNERS. To find the Dimensions of their work., take a line and apply one end of it td any corner of the room, then measure the room goint^ into every corner with the line, till you come to the place where you first began ; then see how many feet and inches the string contains ; this call the Com/iass or Rounds which multiplied into the height of the room, and the Product di- vided by 9, the Quotient will be the content in yards. EXAJtlPLES. 1. If the height of a room painted 2. There is ^ room wainscotted be 12/. 4m. and the compass 84/. the compass of which is 4 T/. 3' and 1 \in. How many square yards does the height 7f. 6'. What i^ the content it contain I Ansi 1 16 F. 3/ 3' 8" in square yards ? Ans. 39 Y, 3/ 4' 6".. GLAIZERS WORZ BT THE ¥001', To find the dimensions of their ivorky multiply the height of windows by their breadth. EXAMPLES. There is a house with 4 tiers of windows, and 4 windows in a tier ; the height of the first tier is 6/ 8' ; of the second, 5/, 9' ; of the third, 4/ 6' j and of the fourth, 3/ 10' ; and the breadth of each is 3/. 5' ; What will the glazing come to, at 19 cents per foQt ? Ans, ^53,88. Sect. III. 9. ALLIGATION. 195 § 9. mum^m. wm^ ^ -r .;> ^ -::5- ^ < Alligation is the method of mixing two or rr.ore simples of different qualities, so that the composition may be of a mean or middle quality. It is •f two kinds, Medial and Alternate. ALLIGATION MEDIAL. Alligation Medial is when the quantities and prices of several things are given, to find the mean price o»f the mixture compounded of those things. JRULE. As the sum of the quantities or whole composition is to their total value, so is any part of the composition to its valuie or mean price. 9 EXAMPLES. 1. A Farmeu mingled 19 bushels of wheat at &s. per bushel, and 40 bush- sels of Rye, at 4s. per bushel, and 12 bushels of barley, at 3«, per bushel to- gethe r I demand what a bushel of this mixture is worth ? ©PERATiaN. Bush. s. £• s. Bush. £. s. Bush, 19 Wheat, at 6 is 5 14 As 71 : 15 10 :: 1 40 Rye, —4 — 8 20 12 Barley,— 3-^1 16 Sum of the aimfiles 7 1 Total value 15 10 284 26 2. A Refiner having 5lb. of silver bullion, 12 ofSoz. fine, 10/A. of 7oz. fine and 15/6. of 6oz. fine, would melt all together ? I demand what )3 12(4ht. S'b. of each sort. Note. Thesf. seven x\vvswers ariae from a 5 inany different Ways of Hn kin i the liaiei cf the. Simples to:^'.'ther. SECT.IIL9. ALLIGATION. 197 CA.SE 2. When the raten of all the ingredients, the quantity of but one of them, and the mean rate of the ivhole mixture are gra en to find the several quantities of th: r>:st, in firafiortion to the given quantity i take the diftcrence between cm: I), r^ -ice and the mean rate as before. Then say, As the difTerence of that simple whose quantity is given, Is to the given quantity, So is the rest of the differences severally j To the several quantities required. EXAMPLES, 1. How much wine, at 80 cents* at 88, and 92 cents per gallon must be mixed with four gallons of wine at 75 cents per gallon, so that the mixture may be worth 86 cents per gallon ? OPERATION. !75 ^- ^ 6+ 2z=z 8 s(a?ids against the given quantity. 80-,-f-^l 24-6z=8 88- + 92--'- -\-J.-j\ 6-f 11—17 1+ 6=17 gal. cts. 4 at 80 U7 As8 : 4 : :-( 17 : s\ -^ ^8/ier. gal. The answer, al -- 92 2. A MAN being determined to mix 10 bushels of wheat at 45. per bushel, with rye at 3s. with barley at 2«. afid with oats at Is. per bushel y I demand how much rye, barley, and oats must be mixed with the 10 bushels of wheat, that the whole may be sold at 28c^. per bushel. B. fi. CB. CB. . J 2 2 of Bue ^ . J 40 0/ Bye „ . ) 8 o/ Bye ^ '^ of Barley \ S^ of Barley / 10 <2/ Barley 2 of Oats {_20 of Oats ( 14 o/ Oats C B. 11. C B. . A J ^^ ^f Rye .J 3 12 2 of Rye >. ^^„ ^ 2 of Rye . CASE 3. IVhen the rates of the severalingredients, the quantity to be comfiounded, and tjte mean rate of the nvhole mixture are given to find how much of each sort ivili make ufi the quantity ; find the differences between the mean rate, Sec. as in case 1. Then, As the sum of the quantities, or differences, Is to tlie given quantity or whole composition ; So is the difference of each rate, To the required quantity of each rate. EXAMPLES. \. How many gallons of water, of no value, must be mixed with brandy, a.t one dolUr twenty cents per gallon so as to fill a vessel of 75 gallons, that may be afforded at 92 cents per gallon ? OPERATION. Gal. Gal. Gal. 5 0-^ 28 GaL Gal. C 28 : \7^qf Water. ^^^1,20--' 92 As 120 : 75 : : ^92 : 57^ cf Brandy. Sum 120 75 given quantity. 2. Suppose I have 4 sorts of currants of 8^. \2d. \Bd. and 32rf. per lb. of v;liich 1 would mix 1201b. and so much of each sort as to sell them at \6d, per lb. how much of each mu»t I take ? f/d. ai I 36 -.1 Jns.-{ 12 ~ 12 )>fierlb, I 24 -— 18 I ,L48 — 22J o. A GitocER has currants of 4rf. 6c?. 9d. and 1 \d. per lb. and he would make a mixture of 240lb. so that it might be afforded at 8rf. per lb. how much of each sort must he take ? f/6. at d.-^ |72- 4| jins.-^ 24 — 6 )>fier lb. 48- 9\ * ^96 — llj Sect. III. ^. SUPPLEMENT to ALLIGATION. 199 Supplement to Alligations QUESTIONS. 1. WHAf is Alligation? S. Of haw many Idnda is jilligation ? 3. WHAf is Alligation Medial ? 4. JVHAris the rule/or o/ierating ? 5. What" is Alligation ALfER^AtE ^ 6. JVhes a number of ingredients of different firiees are mixed together^ hotO' do we firoceed to find the mean firiee of the comfiound or mixture ? 7. Wheii one of the ingredients is limited to a certain quantity •, tvhat is tfic method of procedure ? 8. Wheh the whole comfiosition is limited to a certain quantity, how do you proceed ? 9. How is Alligation fir oved ? EXERCISES i I. A Grocer would mix three \ 2. A Goldsmith has several sorts of ijold ; some of 24 caiatsfine, some of 22, and some of 18 carats fine, and he would have compounded of these sorts the qtianiity of 60 oz. of 20 ca- rats fine ; I demand how niach of e^ch sort he must have ? Ans, 12oz. 24 carats fine, \2 at 22 ca* rats fine, and S6 at 18 carats fine. sorts of sugar together ; one sort at 1 Orf. per lb. another at 7d. and another at 6d. how much of each sort must he take that the mixture may be sold for 8f/. per lb ? Ans. 311). at lOd, 2 at Id. and 2 at dd. 200 POSITION. Sect. III. 10. § 10. po.^tion. Position h a rule whiclj, by false or supposed numbers, taken at pleasure^ 000{Z\0 Answer, errors. 3 The divisor is the sum of the errors 90-|-and 10 — 2. There is a fish whose Itend is 10 feet long ; his tail as long as his head and hall' the length of his bodvi and his body as long as his head and tail ; what if the whole length of the fish ? Ans. UO/rrt. B b 202 DOUBLE POSITION. Sect. III. 10, 3. A CERTAiK man having driven his Swine to raarkef, viz. Hogs, Sovvs^ and Pigs, received for them all 50/. being paid for every liog I8s. for every sow \6s. for every pig 2s. there were as many hogs as sows, and for every sow there were three pigs ; 1 demand how many there were of each sort ? Jns, 35 fyogSj 25 sovfSi and 75 pigs. A. A and B laid oiit equal sums of money in trade ; A gained a sum equal to -|- of his'stock, and B lost 125 dollars, then A's money was double that of B's ; What did each one lay out ? Jns. 600 dollars. 5. A and B have the same income ; A saves | of his ; but B, by spendin* 30 ciollars per annum more than A, at the end of 8 years finds himself 49 dollars in debt ; what is their income, and what does each spend per annum ? Ans, their i?icome is 200 do'ls, }ier arm, A spends 1 75 dollars^ i^ B 205 fier aim. Sect. III. II. DISCOUNT. S03 Discount is an allowance made for the payment of any sum of money be- fore it becomes due^ and is the difference between that sum, due some time hence, and its present worth. THE/zrr*(?n/ wor^A of any sum, or debt due sonae time hence, is such a surn, as, if put to interest, would in that time and at tlie rate {ler cent, for which the discount is to be made, amount to the sum or debt, then due." RULE. As the amount of 100 dollars, for the given lime and rate is to 100 dollars, so is the given sum to its present worth, which suUractcd from the given $um, leaves the discount. EXAMPLES. 1. What is the discount of 2. What is the present worth of E321,63due4 years hence, at 6 426 dollars payable in 4 years and per ce?ii ? 12 days, discounting at the rate of 5 opERATiiON. fier cent. Dolls. Ms. Dolls. 25i,5l5, 6 interest of \ 00 cloUs, 1 ye^r. 4 years. 24 100 l?4 amount. Thea, As 124 : 100 : : 321,63 321,«3 124)32163,00(259,379 321,63 given sum. 259,379 present ivorth. Ans*, 62,251 discount. § 12, (ZBtiuation of ^apmcnt,^. ij^j'.r^ -.> y ^ -/■ ■::» j-^^^ < Equation of pnymcnts is the finding of a time to pay at oHce, several debts due at dilTcrcut linRs so that neither parly shall sustain loss. RULE. Multiply each payment by the lime at which it is due ; then divide ihd sum of the products by the sum of the payments, aud the quotient will be the equated lime. 204 EQUATION or PAYMENTS. Sect. III. 12. EXAMPLES. 1 . A owes B 1 36 dollars to be paid in 10 months ; 69 dollars to be paid in 7 months ; and 260 to be paid in 4 months *- what is the equated time lor the payment of the whole ? OPERATION. 136/ lOmlSeo 96X 7= 672 260X 4r=1040 492 3072 492)30 7 (6 months 295 120 30 49 2^3600(7 days. 3444 156 be paid I in 10 in 2. I owe S6!,125 to 3 months, -^ in 5 mor months, and the remainder in 14 months ; at what time ought the whole to be paid ? jins. 61 months. 3. A MERcHAN-r ha* owing to him 300/. to be paid as follows, 50/. at 2 months, 100/. at 5 months ; and the rest at 8 months : and it is agreed to inake one payment of the whole; I demand when that time must be I jins, 6 months. 4 A MERCH A If 7' owes mc 900 dol- lars to be paid in 96 days, 130 dollars in 12© days, 500 dollars in 80 days, 1267 dollars m 27 days ; what is the mean lime for the payment of the whole ? Jns. 63 days very nearly. Sect. III. 13. GUAGING. 205 §13. (©ua0in0. mmmm^ooOiVTOOOy^^^ Gu AGING is takinc^ the dimensions of a cask in inches to find its content in gallons by the loUowing METHOD. 1. Add two thirds of the difference between the head and bung diameters to the head diameter for the mean diameter ; but if the stares be but little curving from the head to the bung, add only six twnths of this difference. 2. Square the mean diameter, which muliiplkd by the length of the cask and the product divided by 294, for wine, or by 359 for ale, the quotient will be the answer in gallons. EXAMPLE. 1. How many ale or beer gallons will a cask hold, whose bung diameter is 31 inches, head diameter 35 inches, and whose lengtli is 36 inches ? operation; 31 Bung di,am 25 Bead diam. 25 Head diam. 4 Tnvo thirds diff. 6 Difference. 29. Mean diam. 29 261 58 841 Square of mean diami 36 Length. 5046 2523 359)30276(84 galls. 1 \\^ql3. Note. 1. In taking the length of the cask, an allowance must be made for the thick- ness for both heads of 1 inch, 1^ inch, or 2 inches according to the fize of the cask. NoTK. 2. The head di- ameter must be taken close to the chimes, and for small casks, add 3 tenths of an inch ; for casks of 40 or 50 gallons, 4 tenths, and for larger casks, 5 or 6 tenths, and the sum will be very nearly the head diameter within. § u. .H^ccfianicd J?otocr.^. 1. Of the Lever. To find nvhal weight man be raised or balanced by ^ny given /lower, Say, as tlie distance between the body to be raised or balanced., and the fulcrum or prc/u is to tiie distance between the prop and the point where the power is ai)pUcd ; so rs the power to the weight which it will balance or raise. 206 MECHANICAL POWERS. Sect. IIL 14 EXAMPLE. If a man weighing 150/6. rest on the end of a lever 12 feet long ; what weiglit will he balance on the othpr end, supposing the prop ij foot from the weight ? 12 feet the Lever. 1,5 distance of the no eight from the fulcrum. 10,5 distance irom the Fulcrum to the man. Therefore, Ifeet. Feet. lb. lb: M 1,5 : 10,5 : : 150 : 1050 Jns. 2. Of the Wheel and Axle. As the diameter of the axle is to the diameter of the wheel : so is the power applied to the wheel, to the weight suspended by the axle. EXAMPLES. \. A MECHANIC wishes to make a windlass in such a manner, as that Mb, applied to the wheel, should be equal to 12 suspended on ll»e axle ; now, supposing the axle 4 inches diameter, required the diameter of the wheel ? lb. in. lb. in. As 1 : 4 : : 12 : 48 jins. or diameter of the wheel. 2. Suppose the diameter of the axle 6 inches and that of the wheel 6Q incites ; what power at the wheel will balance \Olb. at the axle ? Ans. 1 lb. 3. Oi the Screw. The power is to the weight to be raised as the distance between two threads ot the screw is to the circuniference of a circle described by the power ap- plied ai the end of the lerer. NoTK. 1. To find drcuinfer^nce of the circle described by the end of t he lever., multiply the double of the lever by 3,14159 the product will be the cirr cumference. Note* 2. It is usual to abate ^ of the eifect of the machine for friction. EXAMPLES. There is a screw v,?hose threads are an inch asunder ; the lever by which it is turned is 36 inrheslong, and the weight to be raised a ton, or 22401b. What power or force must be applitd to the end of the lever suflkient to turn the screw, that is, to raise the weight ? The lever 36X2— 72-}-3, 14159— 226,1944-r/t<; circumference, circiimf. in lb. lb. Then,a« 226,194 : 1 : : 2240 . 9,903 Problems. 1. The diameter of a circle being given to find the circumference^ multiply the diameter by 3,14159 ; the product will be the circumference. 2. To find the area nf a circle ^ the diameter being given ; niulliply the square of the diajaieter l)y ,78539-8 ; ttic product is the area. 3. To measure the solidity of any irregular body nvhose dimensions cannctbe taken ; put the body into some regular vessel and fill it with water, then tak- ing out the body, measure the fall of water in the vessel ; if the vessel be fjquare, multiply the side by itself, and the product by the fall ^f water which gives the solid content cf the irregular body. SECTION IV. Mil Tiri'ii : nr irr --nt— Miscellaneous ^icsdons. In tliis Section there is nothing new to be proposed to the Scholar. Enough of Arithmetic has been taught him for all ordinary occurrences in life. It only remains to lead him into some reflections on the foregoing rules. For this purpose the following questions are subjoined. They are Irft without ansvjcrs^ that the Scholar's only resource of knowledge for \Torking them should be in his own mind. Masters having wroughiout these questions at a leisure hour, may transcribe them with their answers inio a manuscript for their private use, to which on any Occasion, without trouble or hindrance, they may readily advert to satisfy the enquiries of their pupils. i. The Northern Lights wtre first observed in London in 1360 ; how many years since ? 2. What number. Muliplied by 43 produces 88159 ? 3, If a cannon may be discharged twice with 6/^. of powder, how many limes will 7C. Sgrs, \7lb. discharge the «arae piece ? 4 Reduce 14 guineas and £75 13s. 6|f/ to Federal Money. 5. What is the interest of §79,49 one year, and Rve months ? 6. A OWED B S3 17,19, for which he gave bis note, on interest, bearing 4ate July 12tb, 1797. .On the back of the note are these several endorsements, vir. Oct. 17, 1797, Received in cash, g6l,10. March 20th, 1798, Received 17 Cwr. of Beef, at S 4,33 per cwt. Jan. 1st, 1800, Received in cash, 84 dollars. What was there due from A to B, of principal and interest, Sept. 18th, 1801 ? 7. What cost 13^ yards of flannel at US^ per yard ? 8. What must 1 give for 5 Cwr. 'igrs. 13/6. of cheese, at 7 cents per lb f 9. What will 35 yards of broadcloth cost at 236G yer yard ? 10. What will be the cost of a line of Veal,wcighing \6\lb.-A\. 2jf/.per/6.? 11. What will Q7\lb. of tallow cost, at 9\d. per lb ? 12. What will 196 yards of tape cost, at 3 farthings per yard ? 13. What will 56 bushels of oats cost, at 2*-. 3{d. per bushel ? J 4. At >C3 7s. 6d. fier cwt. for sugar, what is that per lb. 15. How much in length of a board that is 10 inches wide will it require to make a square foot ? 16. How many square feet in a board 1 foot, 3 inches wide, and 14 feet, 9 inches lojig ? 17. How much wood in a load 9 feet lofig, 3^ feet wide, and 2 feet 9 inchcB high ? 18. At SU33 per yard for cloth, what must I give for 72 yards ? 19. If 21 civt. of cotton wool cost C\ I 17*. 6c/. what is that per lb ? 20. If 1832^ gallons of wine cost /;44 6*. what is that per gallon ? 21. What wili 53-i/. of beef cost at 5 cents 5 n»ills per/ 35. Four men, A, U, C, and D, found a purse of money containing 12 dol- lars, they agree tliat A shall have one third, B one fourth, C one sixth, and D one eighth of it ; what must each one have according to this agreement ? 36. A certain usurer lent 90/. for 12 months, and received principal and interest 95/. 8*. I demand at what rate per cent, he received interest? S7. Tf a gentlemen have an e&tate of 1000/. per ann. how much may he spend per day to lay up tiiree score guineas at the year's end ? 38. What is the length of a road, which being 33 feet wide contarins an acre ? 39. Required a number from which if 7 be swbtracted and the remain- der f>e divided by 8, and the quotient be mullipiitd by 5, and 4 edded lo the product, 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 ? 40. If a quarter of wheat afford 60 ten penny loaves, how many eight penny loaves may be obtained from it ? 41. If the carriage of 7 cwt. 2qr. for 105 miles be 1/. 5s, how far may 5 cwt. 1 qp. be carried for the same money ? 42. If 50 men coi.>sume 15 bushels of yrain in 40 days, how much will 30 men consume in 60 days ? 43. On the same supposition, how long will 50 bushels maintain 64 men ? 44. A gentleman having 50.9. to pay among his laborers for a days work, would give to every boy 6d. to every woman 8d. and to every man 16d. the the numberof boys, women and men, was the same, I demand the number of each ? 45. A gentleman had 71. \7s. 6:1. to pay among his laborers ; to every boy he gave t>d. to every w'oman 8ij sheep to market, says one to the other, give me one of yours and I shall have au many as yovi ; tiie otiier says, give me on« of yours and I shall have as many again i-.s you. How many hud each ? 3. As I was goin^to St. IveSi I flnet seven wives, Every wife had seven sacks. Every sack had seven cuts, Every cat had seven kits, Kits, cats, sacks, and vvives, How many vvere going to St. Ives ? 4. The account of a certain School ir, as follows,- \(iz. y^^ of the boys learn . geometry,! learn grammar, J'j learn arithmetic, ^^^ learn to write and 9 learn to read ; 1 demand tke number of each ? 5. A MAN driving his geese to market^ was met by another, who said, Good- - morrow, master, with your hundred geese; snys he 1 have not an hundred, but if I had half as many as I now have, and two geese and a half beside the number I now have already, I should have an hundred : How many had he .•• , 6. Three travellers met at a caravansary, or iim, in Persia ; and two of them brought their provisions along with them, accoj^g to the custom of the country } but the third not having provided aiijfpropostd to the others that they should eat together, and he would pay tl«?alue of his proportion. This being agreed to, A produces 5 loaves, and B vSlbaves, which the travel- lers eat together, and C ptiid 8 pieces of money as the vahfe of his share, with ' which the others were aaiisfred, but quarreled about the dividing of it. Up- on this the affair was referred to the judge who decided the dispute by an impartial sentence. Required his decision ? 7. Suppose the 9 Pigits to be placed in a quadrangular form : I demand- in what order they must standi that any three figures in a right line, may make just 15. 8. A Countryman having a FoK, a Goose, and a peck of Corn, in his jour- ney came to a river, where it so happened that he could carry but one over at a time. Now as no two were to be left together that might destroy each, other; so he was at his M'its end how to dispose of them : for, Suys he, the corn can't eat the goose, nor the goose eat the fox, yet the fox can eat the goose, and the goose eat the corn. The Question is how he must carry them over, that the),' miglit nut devour each other- 9. Three jealous husbands with their wives, being ready to pass by night over a river, do fmd at water side a boat which can carry but two persons at, once, and for want of a waterman they arc necessitated to row themselves over the river at several times : The Question is, how those 6 persons shall pass by 2 and 2, so that none of the three wives may be fouiicl in the company of one or two men, unless her husband be present ? 10. Two merry companioris are to have equal shares of 8 gallons of wine, which are in a vessel containing exactly 8 s^allons ; now to divide it equally betweew them they have only tv/o oth.er empty vessels, of wldch one contains 5 gallons, and the other 3 ; Thequesiion is, how they shall divide the said wine between them b/thc help of these three vessels, so that, they may hijive 4 gallons fi-piece ? ^ /" SECTION III. Forms oj Notes J Deeds, Bomls, and other Instruments of IFriting^ § 1. Of Notes. No. I. , Overdcan^ Scfit. 17, 1 802. For value received, T promise to pay to Oliver Bounliful, ©r order, Sixty -three dollars fifty -four cents on demand, with in- terest after three months, William Triisty, Attest, Timothij Testimony. NO. II. Bilforli Se/it. 17, 18©2. For value received, I promise to pay to f). R. 04j bearer > . .dollars cents, three months after date. Feter Pencil. No. III. V Br riVO PERSONS. Arian, Sefit. 17, 1502. For value, received we jointly and severally prom* ise to pay to C. D. o*r order— dollars cents on demand, with interest. Attest, Mden Faithfuls Constance jidley^ James Fairface. ' OBSERVATIONS. 1. No note is negotiable unless the words, or order ^ otherwise, or bearer^ be inserted in it, 2. If the note be written to pay him '• or order,** {No. I.) then Oliver Bonn' tiful may endorse this note, tliatis, write his name on the back side, and sell it to A, B, C, or whom he pleases. Then A, who buys the note, calls on WU^ Ham Trusty for payment, and if he neglects or is unable to pay, A may re- cover it of the Endorser. 3. If a note be wiitten, to pay him " or bearer,** (No. II.) then any persoa who holds the note may sue and recover the same of Peter Pencil. 4. The rate of interest established by law being sijc fier cent.fier annum, it becomes unnecessary, in writing notes to mention the rate of interest ; it is suffiicient to write them for the payment of such a sum, with interest, for it will be underBlood, legal interest, which is .six per cent. 5. All notes are either payable on demand, or at the expiration of a cer- tain term, of time agreed upon by the parties and mentioned in ths note, ai three months, or a year, £cc. 6. If a bond or note mention no time of payment, it is ulwaya oademand^ whether the words *' on demand" be expressed or not. 212 FORMS OF BONDS. 7. All notes payable at a certain time are on interest as soon as they be- come clue, ihough in such notes there be no mention made of interest. This rule is founded on the principle, that every man ought to receive his money when due, and that the nonpayment of it at that time is an injury to him- The law, therefore, lo do him justice, allows him interest from the time the money becomes due, as a compensation for the injury. 8. Upon t'le same principle a note payable on demand, witkout any men- tion made of interest is on interest after a demand of paymeJit, for upon de- mand, such notes immediately become due. 9. If a note be given for a specifie article, ns rye payable in one, two or tliree months, or in any certaiis time, and the signer of such note suffers the time to elapse without delivering such ariicle, the holder of the note will not be oblij^ed to take the article afterwards, but may demand and recover the value of ii in money. § 2. Of Bonds. A Bond, iviib a condition J rom one to another. Know all pnen by these presents, that I C. D. of, Sec. in the county of Sec. am held and firmly bound to E F. of, Sec. in two hundred dollars to be paid lo the said E. V ■ or his certain attorney, his executors, administrators or as- sip;ns ; to which payment, well and truly to be made, I bind myself, my heirs, executors, and administrators, firmly by these presents ; Sealed with my seal. Dated the eleventh day of in the year of our Lord one thous- and eight hundred and two. The condition of this oblig-ntion is such, That if the above bound C. D. his heirs, executors or administrators, do and shall well and truly pay or cause to be paid, unto the above named E. F. his executors, administrators or as- signs, the full sumof two hundred dollars, with legal interest for the same, on or before the eleventh day of--; next ensuing the date hereof: Then this obligation to be void, or otherwise to remain in full force and virtue* A Condition of a Counter 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 the above bound C. D. together with the said C. D. is, in and by one bond or ob- ligation bearing equal date with the obligation above written, held and firmly bound unto E. F. of Sec. in the penal sum of dollars, condi- tioned for the payment of the sum of, Sec. with legal interest for the same on the r-day of next ensuing the date of the said in part recited obligation, as in and by the said in part recited bond, with the condition thereunder writ- ten may more fully appear : If therefore the s«id C. D. his heirs, executors or administrators, do and shall well and truly pay or cause to be paid tinto the said E. F. his executors, administrators or assigns, the said sum of Sec. with legal ii-terest for the same, on the said day of, Sec. next ensuing the dale of the said in part recited obligation, according to tiie true intent and meaning, and in full discharge and satisfaction of the snid in part recited bor.d or oblig'iaion : Then, Sec. Otherwise, Sec- FORMS OF RFXEIPTS, 2ia A^oTe. The principal difference between a note and a bond is that the lat^ ter is an instrument of more solemnity, being given under seal. Also, a note may be controuled by a special ag-ceement, different from the note, where- as, in case of a bond, no special agreement can in the least controul what appears to have been the intention of the parties as expressed by the word* in the condition of the bond. §. 3. Of Receipts, No. I. Sifgrieves, Se/tt. 19, 1802. Received from Mr. Durance Adley, tea dollars in full of all accounts. Orvand Constance^ No. II. Sitgrieves'i Scfit. 19, 1802. Received from Mr. Orvand Constance, five dol- lars in full of all accounts. Durance Adley ^ No. III. A receipt for an endorsement on a Note Siigrieves, Sefit. 19, 1802. Received from Mr. Simpson Easily, (by the Iiand of Mr. Titus Trusty), sixteen dollars twenty five cents, which is en- dorsed on his note of June 3, 1802. Feter Cheerful. No. IV. A receipt for money received on Account. Stf^^rxeves, Sept, 19, 'l 802. i^cceived ©f Mr. Grand Landlike, fifty dollars on account. ^ Eldro Slackley. No. V. A Receipt for interest due on a Bond. Received tliis day of of INIr. A. B. the sum of fire pounds, in full of one year's interest of 100/. due to me on the day of last, Qn bond from the said A. B. 1 say received- By me C. D. Observations. , 1 . Therk is a distinction between receipts given in full of a// accounts^ and otiicrs in full oi all devuinds. The former cut off accounts only ; the latter cut of not only accounts, but all obligations and riglit of action. 2. Whkm any two persons make a settlement and pass receipts (No. Land No. II ) each receipt must specify a particular sum, rectivcd, less or more. It is not necessary that the sum specified in the receipt, be the exact sum re- ceived. $14 FORMS OF ORDERS. § 4. Of Orders. No. I. Mr. Stefihen Burgess. SIR, For value received, pay to A. B. Ten Dollars, and place the same to my account. Samuel Skmner. Archdale, Sept. 9th, 1802. No. II. Boston, Se/it.9t/i, 1802. SIR, For value received, pay G. R. eighty sij^ cents, and this, with his receipt, shall be your discharge from rne. Nicholas Reuhem. To Mr. James Robottom, § 5. Of Deeds, Mo. I. A Warrantee Deed. Know all mkn by these presents, That I Peter Careful of Leominster, in the county of Worcester and Commonwealth of Massachusetts, gentleman, for and In consideration of one hundred fifty dollars, and forty-five cents paid to me by Samuel Pendleton of Ashby, in the county of Middlesex, and Com- monwealth of Massachusetts, yeoman, the receipt whereof I do hereby ac- knowledge, do hereby give, grant, sell, and convey to the said Samuel Pen- dleton, his heira and assigns, a certain tract and parcel of land, bounded as follows, viz. l^Here insert the boundfi, tog^ether ivitb a,ll the privileg^es cind afifiurtenences there- unto belonging.'l To have and to hold the same unto the said Samuel Pendleton his heirs and assigns to his and their use andfoehoof forever. And I do covenant with the said Peter Pendleton his heirs and assigns, That I am lawfully seized in fee of the premises, that they are free of all incumbrances, and that I will war- rant and defend the sitme to the said Peter Pendleton his heirs and assigns forever, against the lawful claims and demands of all persons. In witness wh-sreof I have hereunto set my hand and seal, this day of in the year of our Lord OnQ thousand eight hundred and two. Signed, sealed and de- > ^^^^^ hveredinfiresenccef y j \^ L. R. F. G. No. IL ^litdaim Deed, Know all meic by these presents, That I A. B. of, Sec. in considera- tion of the sum ot to be paid by C. D. of. Sec. the receipt whereof I do hereby acknowledge, have remissed, released, and forever quit-claimed, and do by these presents remiss and release, and forever quit-claim unto the said 'C D. his heirs and assigns forever. [Here insert the firemises."] To have and to hold the same, togethen with all the privileges and appertenances thereunto belongingjto him the said CD. his heirs and assigns forever. — In witness, life. FORM OF DEEDS, 215 No. III. A Mortgage Deed, Know all men by these presents, That I Simpson Easly, of in the county of in the state of > Blacksmith, in consideration of Dollars Cents paid by EWin Fairface of in the conn- • ty of • in the state of— —Shoemaker, the Receipt whereof I do hereby acknowledge, do hereby give, grant, sell and convey unto the said El- vin P'airfiice, his heirs and assigns, a certain tract and parcel of land, bounded as follows, viz. [^Here insert the bounds together with ail the privileges and afi- Jmrtenances thereunto belonging.'] To have and to hold the afore-granted Prem- ises to the said Elvin Fairface, his heirs and assigns, to his and their use and • behoof forever. And I do ctivenant 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 Fairface. And that I will warrant and de- fend the same premises to the said Elvin, his heirs and assigns forever, against the lawful claims and demands of all persons. Frovided nevertheless^ That if I the said Simpson Easly my heirs, executors, or administrators, shall well and truly pay to the said Elvin Fairface, his heirs, executors, administrators or assigns, the full and just sum o f dollars cents on or before the — — day of ^which will be in the year of our Lord Eighteen Hundred and — — - with lawful interest tor the same until paid, then this deed, as also a certain bond [or note as the case may 6e] bearing date with these presents 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 wit- Rpss whereof, I the said Simpson and Abigail my wife, in testimcney that she Telinquishes all her right to dower or alimony in and to the above described. preiTiises, have hereunto set our hands and seals this — —sna^i — day of — in the year of our Lord one thoiuand eigJu hundred and five. Signed, sealed^ and de-'^ Simfison Easly . Q livercdinfirescjice of. 3 ^bigaii £a»ly, q. L. N. V. X. § 6. Of an Indenture. J common Indenture to bmd an Apprentice, This Indenture Witnesneth. That A. B. of, &c. 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 R. P. to learn the art, trade or mistery of The said C. D. after the manner of an appren- tice, to dwell with and serve the said R. P. from the day of the date here- of, until the day of which will be in the year of our Lord one thousand. eight hundred and at which lime the said apprentice if he should be liv- ing, ^vill bo twenty-one years of age : During which time or term the said apprentice his said master well and faithfully shall serve ; his secrets keep, and his lawful coiiimands every where at all times readily obey , he shall do no damage to his said master, nor wilfully sufler any to be doneb>^oth- ers ; and if any to his knowledgo be intended, he shall give his master season- able notice thereof. He shall not waste the goods of his said muster nor lend them unlawfully to any ; at cards, dice, or any other unlawful game he shall not play ; fornication he shall not commit, nor matrimony contract dur- ing the said term ; taverns, ale-houses, or places of gaming he shall not haunt or frequent : From the service of his said master, he shall not abjent him self; butia all things and at all limes he shall carry and behave himself as n good and faithful apprenlice ought, during the whole lituc or term aforesaid § ^IG iFORiM OF A WILL* And the said R.P. on l.is part, cloth hereby promise^ covenant and agree to teach and instruct the said apprentice, or cause him to be taught and instruct- ed, in the art> trade or calling of a ^ ^by the best way or means he can, and also to teach and instruct the said apprentice, or cause him to be taught and instructed, to read and write, and cypher as far as the Rule of 'J^hret?, if the said apprentice be capable to learn, and shall w'ell and faithfully find and provide for the said apprentice, good and sufficient meat, drink, clotiiing^ lodging and other necessaries fit and convenient for such an apprentice, dur- ing the term aforesaid, and at the expiration thereof, shall give unto the said apprentice, two suits of wearing apparel, one suitable for the Lord's day, and the other lor working days. In testimony whereof, the said parties have hereunto interchangeably set their hands and seals, the -day of in the year of our Lord one thou- sand eight hundred and ^ (Seal) Signed., sealed^ and de- > (Seal) iivercd in firesence o/^ ,i (Seal) 7. Of a WilL The form of a P^ill, ivitb a dcoisc oj a Real Estate, Leasehold, CsV. In the name oJ GOD Amen^ I A, B, of, Scc.bting weak in body, but of sound and perfect mind and memory {ovyountay say thus^ considering the uncer- tainty of this mortal life, and being of sound, Sec.) blessed be Almighty God for the same, do make and publish this my last Will and Testament, in manner and form following (that is to say) First, I give and bequeath unto my beloved wife J. B. the sum of -I do also give and bequeath unto ray eldest son, G. B. the sum of 1 do also give and bequeath uwto my two younger sons, J. B. and F, B. the sum of apiece. I also give and bequeath to my daughter-ini^law, S. H. single woman, the sum of- which said several legacies or sums of money I will and order shall be paid to the said respec- tive legatees, within six months after my decease.— I further give and de- vise to my said eldest son G.B. his heirs and assigns, ^/Mhat my messuage or tenement, situate, lying, and being in, 8cc. together with all my other free- hold estate whatsoever, to hold to him the said G. B. his heirs and assigns forever. And I hereby give and bequeath to my said younger sons J. B. and F. B. all my leasehold estate of and in all those messuages or tenements, with the appurtenances, situate. Sec, equally to be divided between them. And las' ly, as to all the rest, residue and remainder of my personal estate, goods ajia chattels, of what kind and nature soever, I give and bequeath the same to my said beloved, wife J. B. whom I hereby appoint sole executrix of this my last will and testament ; and hereby revoking all former wills by me made. In nvilness nvherenf I have hereunto set my hand and seal, the day o f in the year of our Lord. -— Signed, scaled, published and declared by the above A. B. (Seal) named A. B. to be his last will and testament, in the pre^ence of us, who have hereunto subscribed our namcsi as witnesses, in the presence of the tes- tator. ^^' s. W. T. T, W. FINIS. i^ a- r h^ •