University of California Berkeley 
 
THE 
 
 AMERICAN YOUTH: 
 
 BEING 
 
 A NEW AND COiMPLETE COURSE 
 
 O F 
 
 Introductory Mathematics : 
 
 DESIGNED FOR THE USE OF 
 
 PRIVATE STUDENTS. 
 
 B Y 
 
 CONSIDER and JOHN STERRY. 
 
 VOL. I. 
 
 'where the mind 
 
 In endlefs growth and infinite afcent, 
 Rifes fromjiate tofeate^ and world to world. 
 
 THOMSON, 
 
 PRINTED AT PROVIDENCE, 
 
 BY BENNETT WHEELER, FOR THE AITTHORS, 
 
PREFACE. 
 
 TIME, ever big with wonders to be 
 unfolded to the human mind, has ujh- 
 ered in y through a feries of the mofl important events, 
 the rifing Empire of America \ who hath eftablifhed 
 her own Independence^ and the flame of her liberty has 
 Jpread itjelf to the remo&fl parts of the earth j the cf- 
 fett of which great example has not yet Jpent its force, 
 but mufl continue to operate throughout ages, and form 
 a grand ingredient in the affive fermentation, and in 
 the hiftory of nations* 
 
 But the great objett of true national dignity and 
 grandeur ', conjifts in the cultivation of the human mind, 
 whereby the natural fav age barbarity^ rudenefs and im 
 becility of human nature are eradicated, and thofe prin 
 ciples of knowledge and virtue engrafted in the foul, 
 which are the foundation of that knowledge and pre 
 eminence of merit, which isthenobleft of all diftinftions. 
 
 As foon as we begin to exift, that Javage and im 
 becile fpirit takes root in the Joul, and grows as the 
 mind enlarges, tilltheje^ds of knowledge by cultivation 
 do take effectual root, and then like the tender bud 
 it will burfl its native bonds, expand and flgurijh in 
 its own beauty ; Ths veil will then disappear, and an 
 
 infinite 
 
. 
 infinite diverfity offcenes, both plea 
 
 iv ill open themjelves to our view. But in order to f re- 
 fare the mind for thefe f leafing and enlarging views, 
 we muft early employ ourf elves in the Jiudy offomething 
 which is noble and important, whereby our minds may 
 be cultivated aud brought to maturity. cc A jufl and 
 perfect acquaintance with thefimple elements offcience, 
 is a neceffary Jhp towards our future progrefs and ad 
 vancement j and this, ajfiftedby laborious mvejiigation, 
 and habitual enquiry will conjlantly lead to eminence 
 and perfection " 
 
 <f But as the various modes, fituations and circumftan- 
 * "ces of life are various, fo accident, habit and education, 
 have each their predominate influence, and give to every 
 mind its particular bias" It is, therefore, for this 
 re of on, we particularly admire thofe things which are 
 the moft compatible with our genius and purfuits in life* 
 
 " Riches and honours are the gifts of fortune, cafu- 
 ally beftowed, or hereditarily received, and are fre 
 quently abufed by their foffeffbrs -, but the Juperiority of 
 ivifdom and knowledge, is a pre-eminence of merit, thai 
 originates with the man, and is the nobleft of all dif- 
 tintJions." 
 
 Since, therefore, the cultivation of knowledge is a 
 thing of the laft importance, too.many attempts cannot 
 be made to render it univerfal, andjince youth is the 
 time therefor, we have therefore, " only to point out to 
 them feme valuable acquifition, and the means of ob 
 taining it. be active principles are immediately put 
 in motion, and the certainty of the conqueft is enfured 
 from a determination to conquer." But of all thejdences 
 cultivated by mankind^ none are more ujeful than the 
 
 Mathematics, 
 
Mathematics, to call forth a Spirit of enterprife and 
 enquiry, The unbounded variety of their application^ 
 which is of univerfal utility to mankind, fir ft prompts 
 our curicfity to have in pojfeffion a treasure offuch in- 
 6/limable value. By their elegant and fub lime manner 
 of reafoning, our minds are enlightened and our under- 
 ftanding enlarged^ and thereby we acquire a kabit of 
 reafoning, an elevation of thought, that determines the 
 mind and fixes it for every other purfuit -, and none but 
 thofe who either from fordid views, or a grofs ignorance 
 of what they difpife, will ever think their time mijjpent, 
 or their labours ufelefs in the purfuit of that, which is 
 the guide of our youth, and the perfection of our reafon* 
 
 of 'the prefent performance, is Arithmetic 
 and Algebra , the foundation of all our mathematical 
 enquiries. 
 
 Although a great number of books has been publijh- 
 ed on tkefubjeff of Mathematics, yet few of them are 
 adapted to the capacity of young and tender minds. 
 Where is that fimpli city, plainnefs and brevity, which 
 is e.bjohitely necejj'ary for the young and unajfijled be 
 ginner ? That clofe and refined reafoning with which thofe 
 Authors^ writings are replete, renders them unfit for 
 learners in general, and entirely ufelejs to thofe unajjift- 
 ed by a 'Tutor : They have ccnfulted more the elegance 
 of their diffion, and refined demonft rations, than the 
 method of conveying their knowledge to their readers. 
 Others again, in attempting to render their JubjetJs at 
 tainable to the weakeft minds, have been fo prolix and 
 voluminous, as even to difcourage a learner at the fight 
 of their works : Thus, we. fee that writers in general, 
 aim at the extremes, while the true and proper medium 
 is for the moft fart gmitsd* Propriety therefore, and 
 
 compatibility 
 
compatibility ought always to be the grand text, 
 fimpluity joined with brevity leads the chain of argument. 
 
 In all countries, where thefciences are cultivated^ 
 local intere/is have been particularly confidered, which 
 muft therefore exclude thofe who neglect the cultivation 
 if the Arts and Sciences, from many advantages of their 
 works. 
 
 'Taking into ccnfideration the works of thofe who have 
 gone before us on this fubjefl, the utility of an alteration 
 appeared manifeft, while reafon and convenience urged 
 the pr -amicability thereof. 
 
 In the profecution of this plan, we have in the frft 
 Book of the prefent Volume, explained the rudiments 
 end application of numbers ; beginning with the proper 
 ties of an unit, we have led the learner by eafy and na* 
 tural gradations to the moft remote analogies of the 
 Jcience. In all the calculations relating to money, we 
 have made life of the Federal Money, or Money of the 
 United .States, which is not only much more ccncife than 
 the prefent praflice by pounds, /hillings, &c. but it is 
 equally eft im able for its fimpli city and brevity. The 
 denominations of this money being in a decimal ratio, 
 are therefore above all other numbers, the mvft natural 
 and eafy to be manage } d, and which muft confequently give 
 it a preference to any ofher method whatever. 
 
 tfhe futyeft of the fecond Book is Algebra, or the 
 Analytic art ; which above all others is the moft ex- 
 tenfive and fublime. It was by this, with the conji- 
 deration of motion, that one did in fome meafure do ho 
 nour to hnman nature itjelf, by his almojl divine in 
 vention* ; whichfucceeding ages will view with pleaf- 
 ing admiration. Algebra 
 
 * Alluding to Sir Ifaac Newton's invention of Fluxions, 
 
C vii ) 
 
 l ' Algebra Is a general method of dif cover ing truth in 
 all. cafes where proper data can be ejtabiijked, with tbs 
 greateft expedition, elegance and eafe. 
 
 In delivering the rudiments of thisjcience, we have 
 particularly confulted the eaje and accomodation of the 
 learner y by confining every thing within thejphere of the 
 ingenious Student, and therefore, exploding thofe tedious 
 end complicated explanations, which are commonly to be 
 found in authors on this JubjeR. 'The leading queftions 
 are Jhort andjimple, and the method of arguing brief 
 and confpicuous ; which particular, although of the I aft 
 importance to facilitate the progrefs of learners, is too 
 much neglected by mofl writers, and tonfequently, deter 
 many from becoming acquainted with this inter ejling and 
 important acquirement. Great attention has been paid 
 to render the doftrine of irrational quantities plain and 
 intelligible, particularly the method of expanding quan 
 tities into infinite Jeries, and noting their powers and 
 roots, which is a matter of the lafl importance in the 
 higher branches of the Mathematics. And finally^ 
 through the whole of the following Jheets, Simplicity and 
 brevity has been our general aim, and at the fame time 
 to explode all foreign and provincial cuftoms, and adapt 
 the whole to the practice and convenience of the United 
 States. 
 
 fhusfar, for the fatisfacJion of the learner, we have 
 explained the economy of the prefent performance, we 
 /hail now fubmit it to the candid public, and from the 
 pajns we have taken to render thefubjett ufeful to learn 
 ers in general, are not without hopes of its meeting with 
 their approbation. 
 
 THE AUTHORS. ^ 
 
 Prefton (Connefticut) July., 1790. 
 
RECOMMENDATIONS. 
 
 of a letter from Mr. NATHAN DABOLL, 
 Teacher of Mathematics and Aftronomy y in the Aca 
 demic School in Plainfisld> to the Authors -, da fed 
 March i, 1787. 
 
 GENTLEMEN, 
 
 cc T H A V E perufed the firft Volume of your new 
 A courfe of introdu6tory Mathematics, entitled 
 THE AMERICAN YOUTH ; and it appears to me a 
 work well executed, and compatible with its defign. 
 You have given your rules and examples in a con- 
 cife, plain and familiar manner, and confequently 
 well-adapted your matter to the capacities of learn 
 ers : I therefore efteem it a very valuable perform 
 ance, and wilh you fuccefs in its publication, and 
 that it may meet with an encouragement from the 
 public equal to its merit." 
 
 B Frcm 
 
RECOMMENDATIONS. 
 
 From Mr. JARED MANSFIELD, to the Authors \ dated 
 
 ) December, 1787. 
 
 GENTLEMEN, 
 
 *\TO U R Treatife of Arithmetic and Algebra, 
 A I have perufed with care and attention, and 
 have the pleafure of afluring you I think it a work 
 of ingenuity and merit. My reading of mathemati 
 cal books hath been extensive -, yet I know of no 
 writer who hath treated thefe fubjeds in a more 
 fcientific and comprehenfive manner, and at the fame 
 time accommodated his matter fo well to the capa 
 cities of learners, as I find to be done in your work. 
 If you publifh it (which I hope you will not fail to 
 do) I have no doubt it will be received into our 
 Schools and Seminaries, as it is high time that Ward, 
 Hammond, and other inferior treatifes now in com 
 mon ufe, were exploded. For my own part, as a 
 lover of Mathematics, I wifh you all pofiible fuccefs* 
 and that you may be encouraged to proceed, and 
 write on the higher and more fublime branches of 
 the Mathematics j and that a fpirit of emulation may 
 be excited among the Youth of America, to excel in 
 thefe ufeful and exalted, but hitherto much-neg- 
 lefted purfuits." 
 
 > i ^MiiNM^<aMBEPHBgP|PH)p|M^ 
 
 From 
 
RECOMMENDATIONS. 
 
 From Col. SAMUEL MOTT to tbe Authors, dated 
 P reft on, April 28, 1788, 
 
 GENTLEMEN, 
 
 iC ^rr OUR Manufcript Treatife on Arithmetic 
 JL and Algebra, entitled THE AMERICAN YOUTH, 
 has been put into my hands. I have paid particular 
 attention in its perufal. I have heretofore been con- 
 fiderabiy engaged in the reading and ftudy of au 
 thors upon the various branches of Mathematics, 
 though of late I have been more diverted from that 
 purfuit. It has however given me great pleafure 
 and fatisfaction to obferve the ingenuity, concifenefs 
 and perfpicuity which appears in your work, noU 
 withftanding the extenfive and finifhed refearches de^ 
 monftrated in all your rules and examples ; yet it 
 appears to me exceedingly well accommodated to 
 the capacity of a learner, and your method through 
 the whole more eafy than any I have before feen. If 
 you fhould publifh your book (which my high efteem 
 fpr mathematical fcience, and fincere regard for the 
 progrefs of literature among the youth of our coun 
 try, induces me earneftly to wifli you may) juftice 
 obliges me to fay, that I am clearly of opinion it will 
 be found moreufeful among ftudents than any other 
 author now extant upon the fubjeft. I fincerely wifh 
 
RECOMMENDATIONS. 
 
 you fuccefs, and that you may meet with every en 
 couragement which the merit offo important a work 
 deferves." 
 
 From the Rev. JOSEPH HUNTINGTON, D. D. one of 
 
 the Truftees of Dartmouth College y &c. to the Hon. 
 JOHN DOUGLAS, Efq. dated Coventry , May 23, 
 1788. 
 
 cc T H A V E with much pleafure perufed the ma- 
 A thematical competition in which the two Mefirs. 
 STERRY'S are united, and really think it worthy of 
 publication and encouragement : The fcience of 
 Arithmetic and Algebra has hitherto been extended 
 nearly to its bounds, but I efteem this work an ex 
 cellent piece for the iludy of youth, to lead them to 
 the knowledge of this ufeful fcience, fin.ce it is more 
 eafy and intelligible to tender capacities than any 
 \vork of the kind preceding, and this,more efpecially 
 in the moft abftrufe part of the whole fcience, *. e. 
 Algebra. I could wifh that you, Sir, and many o- 
 ther gentlemen, eminent for their friendfhip to the 
 liberal fciences, m'ight pay attention to the work I 
 have alluded to," 
 
TABLE OF CONTENTS. 
 
 PART I. 
 
 Chapter Page 
 
 I. Of Definitions and Illuftrations, 17 
 
 II. Of Notation or Numeration, 20 
 
 III. Of Addition of fimple whole Numbers, 22 
 
 IV. Of Subtraction of fimple whole Numbers, 27 
 V. Of Simpk Multiplication, 30 
 
 VI. Of Divifion of Simple Numbers, 36 
 
 VII. Of Addition of Compound Quantities 48 
 
 VIII. Of Subtraction of Compounds, 60 
 
 IX> Of Multiplication, &;c. of Compounds, 63 
 
 X. Of Reduction, 6S 
 
 PART II. 
 
 I. Of Definitions and Illuftrations, J6 
 
 II. Of Redudion of Vulgar Fra&ions, 78 
 
 III. Of Addition, &c. of Vulgar Fractions, 95 
 
 P "A R T III. 
 
 I. Of Definitions and Illuftrations, 100 
 
 II. Of Addition, &c. of Decimal Fractions, 102 
 
 III. Of Redu&ion of Decimals, 114 
 
A Supplement to P A R T III. 
 
 Chap. Page 
 
 I. Of Definitions and Illuftrations, 119 
 
 II. Of Reduction of circulating ; Decimals, 121 
 
 III. Of Addition, &c. of circulating Decimals, 127 
 
 A Supplement to PART I. 
 
 I. Of Proportion, or Analogy, 132 
 
 II. Of Disjunct Proportion, 149 
 
 III. Of Simple Intereft, 162 
 
 IV. Of Compound Intereftj 179 
 V. Of Rebate, or Difcount, 183 
 
 VI. Of Equation of Pay ments, 187 
 
 VII. Of Barter, 189 
 
 Vill. Of Loft and Gain, 191 
 
 IX. OfFellowfhip, 193 
 
 X. Of Compound Proportion, 200 
 
 XL Of Conjoined Proportion, 204 
 
 XII. Of Allegation, 206 
 
 XIII. Of Pofnion, or the Gueffing Rule, 214 
 
 XIV. Concerning Permutation & Combination, 218 
 XV. Of Involution, 228 
 
 XVI. Of Evolution., 229 
 
 BOOK II. 
 
 I. Of Definitions and Illuftrations 3 241 
 
 II. Of Addition of whole Quantities, 245 
 
 III. Of Subtraction of whole Quantities, 248 
 
 IV. Of Multiplication, 249 
 
 V. OfDivifion, 252 
 VI. Of In volution of whole Quantities, 257 
 
 VIL 
 
Chap. f Page 
 
 VII. Of Multiplication and Divifion of Powers, 265 
 
 VIII. Of Evolution of whole Quantities, 268 
 
 IX. Of Algebraic Fradtions, 273 
 
 X. Concerning Surd Quantities, 285 
 
 XI. Of infinite Series, 296 
 
 XII. Of Proportion, 306 
 
 XIII. Of fimple Equations, 315 
 
 XIV. -Of Extermination of unknownQuantities, 321 
 XV. Solution of a Variety of Queftions, 330 
 
 XVI. Of Quadratic Equations, 341 
 
 XVII. Solution of Queftions with Quadratics, 348 
 
 XVIII. Of the Generis of Equations, 355 
 
 XIX. Of the Transformation of Equations, &c. 362 
 XX. Of theRefolution of Equations byDivifors, 368 
 
 XXI. Of finding the Roots of numeral Equa 
 
 tions, by the Method of Approximation, 373 
 
 XXII, Concerning unlimited Problems, 377 
 
ALTHOUGH the Authors ex 
 amined the Proof-Sheets^ yet the follow 
 ing efcaped their Notice. 
 
 ERRATA. 
 
 PAGE 28, laft line, dele See the Example. P. 
 34, 1> 12. read 69530000. 1. 14, r. 720800. P. 35, 
 1. 1 8, for 3, r. 4. P. 55, 1. 4, f. content. 1. 24, for 
 3 qr. 3 na. r. i qr. 3 oa. 1. 26, for 3qr. 2 na. r. i. 
 qr. 2 na. P. 67, 1. 4, r. 105 dol. P. 74, 1. 9, r. 
 5638^8. 1. 19, r. 15480 yards. P. 77, L 20, for in, 
 r. is. P. 80, 1. 13, for 13, r. 15. P. 85, 1. i, for 
 2o%- -;- 20, r. 20-0 24. P. ioo, 1. 15, for .5, r. 
 5. P. 1 06, L 20, r. preceding. P. 107, 1. 27, r. 
 8rr. P. 113, 1. 12, r. 31.415* &c - p - i3 2 >l-25, 
 r. numbers. P. 157, 1. 13, r. operation. P. 169, 
 I. 13, r. 49cts. P. 193, I. 7, r. 51 dol. 72^4 cts. 
 1. 15, r. fcllowfhip. P. 322, 1. 5, r. 6X2X6. P. 
 245, 1. 16, for -j-20, r. +2#. P. 246, for ax> r. az. 
 P. 249, for \/aw yb y r. ^aw yb. P. 266, 1. 
 
 i6,r. i+j^-5-jf-f.yl*. P. 288, for a*, r. a*. P. 
 353, 1. 1 6, for laft, r.xvi. P. 357,!. 19, r. v = 
 c - I J - 379? 1- 9> ^ad axiom 8. J. 10, r. axiom 8, 
 1. 1 8, r. axiom 8. P. 380, 1. i, r. ax. 7. 1. 2, r. 
 ax. 9. 1. 3, r. ax. 8. 1. 5, r. ax. 8. 
 
BOOK I. 
 
 OF ARITHMETIC. 
 
 !>OO<>^^ 
 
 PART I. 
 
 4RITHMETIC of WHOLE NUMBERS. 
 
 CHAP. L 
 Of DEFINITIONS and ILLUSTRATIONS. 
 
 ARITHMETIC confift&of three parts'; twa 
 of which are natural, and the third artificial* 
 The firft part of natural Arithmetic,- is wherein an 
 unit or integer reprefents one whole quantity, of any 
 kind or fpecies ; and is therefore ftiled Arithmetic 
 of whole numbers. The fecond part of natural A- 
 rithmetic, is wherein an unit is confidered as broken 
 or divided into parts, either even or uneven, which 
 are considered either as pure parts of an unit, or as 
 parts mixed with an unit -, and is ufually ftiled the 
 doctrine of vulgar fractions . The third parr, or ar 
 tificial Arithmetic, is an- eafy and elegant method of 
 managing fractional, or broken quantities ; the oper 
 ations are nearly fimilar to thofe of whole numbers* 
 This part is of general ufe in the various branches of 
 the Mathematics. 
 
 C THE 
 
( 18 ) 
 
 THE operations of common Arithmetic in all its 
 parts, are performed by the various ordering and 
 difpofmg of ten Arabic characters, or numeral fi 
 gures ; which are thefe following, viz. 
 one two three four five fix feven eight nine cypher 
 1234567 8. 9 
 
 AN unit (by Euclid) is that by which every thing 
 that is, is one ; and number is compofed of a multi 
 tude of units* 
 
 NINE of the aforefaid figures, are compofed of 
 units ; each character reprefenting fo many units put 
 together in one fum, as was intended they fhould de 
 note ; nine of thofe units, being the greateft number 
 which is thought bed for any one character to re- 
 prefent ; the lail of the before-mentioned characters, 
 is a cypher, or as fome call it a nothing ; for of itfelf 
 it is nothing ; becaufe, if ever fo many cyphers be 
 added to, or fubtracted from an unit or number, 
 they will neither increafe nor diminifh its value : con- 
 fequently a cypher of itfelf is no aflignable quantity ; 
 but cyphers annexed or prefixed to an unit or num 
 ber, will increafe, or diminifh that unit or number 
 in a tenfold proportion. 
 
 THAT the learner may underftand the following 
 fheets, it is abfolutely necefifary for him to be well 
 acquainted with the following Algebraic figns. 
 SIGNS & NAMSS. SIGNIFICATIONS, 
 
 fis the fign of Addition : as 4+6, 
 
 + Plus, or more,] w ,f ich d <- es . that 6 is to be add- 
 M ed to 4, and is read thus, 
 (.4 more 6. 
 is the fign of Subtraction: as 
 
 4 ~~ 2 > which nifies that 2 is to 
 
 X into, 
 
 Minus o 
 us,o 
 
 .thus, 4 lefs 2. 
 
X into, or with, 
 
 by, 
 
 ~ equal, 
 
 fo is, 
 
 is the fign of Multiplication: thus 
 4X3 denotes, that 3 is to be mul 
 tiplied into 4 ; and is read thus, 
 4 into, or with, 3. 
 is the fign of Divifion : thus 
 6-:-3, is 6 divided by 3, or f, fig- 
 nifies the fame thing ; and is read 
 thus, 6 by 3. 
 
 is the fign of Equality : and 
 whenever this fign is placed be 
 tween any two quantities, it de 
 notes that thofe quantities are 
 equal : thus 99 ; that is, 9 e- 
 quals 9 ; alfo 6+410, is read 
 
 C 6 more 4 equals 10. 
 
 pis the fign of Proportionality ; 
 and is always placed between the 
 iecond and third numbers that 
 are in proportion : thus 
 
 ii 
 
 p is alfo a fign of Proportion, and is 
 
 I placed between the firft and fe- 
 : to, \ cond, third and fourth numbers 
 
 i in proportion : thus a:4::j:6; 
 
 is read thus, 2 to 4 fo is 3 to 6. 
 A +6X2 ("denotes the fum of 4 & 6 mul- 
 
 Jtiplied with 2. 
 fr is the fign of continued Proportion. 
 
 THE whole doftrine of Number is founded on 
 the five following general rules, to wit, Notation, 
 Addition, Subtraction, Multiplication and Divifion, 
 
 CHAP. IL 
 
CHAR IL 
 
 O/ NOVATION or NUMERATION. 
 
 TV T OTATION or Numeration teaches us, 
 J[\| how to exprefs the value of figures ; and con- 
 iequently to note or write do\jn any propofed num 
 'l}er, according to itsjuft value ; in the operation of 
 which, two things muft be obferved, viz, the order 
 of writing down figures, and the method of valuing 
 each in its proper place, as in the following Table ; 
 
 :ON TABLE. 
 
 nds of Millions 
 
 (A 
 
 a, 
 
 
 
 1 
 
 o 
 
 </ 
 
 <Si 
 
 
 
 to 
 
 C 
 
 
 
 
 
 *"^ 
 
 f% 
 
 CO 
 
 /" 
 
 n 
 
 
 
 n 
 
 
 
 
 
 NUMERAT 
 
 Hundreds of Thouf 
 
 Tens of Thoufand 
 
 Thoufands of Millie 
 
 Hundreds of Millio 
 
 en 
 
 G 
 
 .2 
 
 i 
 
 
 
 s 
 
 H 
 
 Millions 
 
 Hundreds of Thouf 
 
 Tens of Thoufands 
 
 Thoufands 
 
 Hundreds 
 
 C/3 
 
 S [5 
 
 
 3 
 
 2 
 
 i 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 I 
 
 HERE, 
 
HERE the order of reckoning begins on the right 
 hand, to wit, at unity, and foon as the table directs. 
 But to make the underftanding of this table plain, 
 it is required to exprefs the value of the numeral fi 
 gures 321. Firft, beginning at the firft figure on the 
 right hand, viz. at i, which ftands in the units* place, 
 where it reprefents its own fimple value, which is an 
 unit, or i ; the next tc^be confidered is the figure 2, 
 which ftands in the tens 1 place, reprefenting lo many 
 tens, as the figure 2 is compofed of units, which are 
 two ; fo that the figure 2 Handing in the place of 
 tens reprefents 2 tens, or 20 ; the next figure, 3, 
 {lands in the hundreds' place,, and fignifies as many 
 hundreds as the laid figure hath units, viz. 3, that is, 
 three hundred: now, if the whole value of thejigures 
 321 beexpreiTed, the expreflion will be three hundred 
 twenty-one. Altho the figure 3, is in the laft place 
 on the right, or the firft on the left, yet when 
 we come to read or exprefs them, we begin with the 
 figure 3 ; becaufe the method of reading figures is 
 the fame as that of words. Hence the firft fi 
 gure in numbering, is the firft figure on the right 
 hand ; but in reading or exprefiing the value of 
 numbers, the firft figure in the exprellion is the 
 firft figure on the left hand. Again, let it be re 
 quired to read or exprefs 7645. Here as before, 
 the firft figure of the propofed number, to wit. 5, 
 ftands in the units' place, and is 5 units, or five, the 
 fecond figure which is 4, is in the tens' place, and is 
 four tens or 40, the third figure which is 6, in the 
 hundreds' place, is called hundreds, and the fourth 
 figure, which ftands in the thoufands' place, is for 
 the fame reafori called thoufands ; and the expreflion 
 for the whole value, beginning as before, is feven 
 thoufand fix hundred forty-five. 
 
 IF 
 
IF what has been laid concerning notation and va 
 luation of figures, be thoroughly confidered, toge 
 ther with the following examples and their anfwers, 
 the whole bufmefs of Numeration will appear plain 
 to the meaneft capacity* 
 
 EXAMPLES. 
 
 What is the value of 56434 ? 
 Anfwer, Fifty -Jix thoufandfour hundred thirty -four. 
 What is the value of 7843217 ? 
 Anf. Seven million eight hundred forty-three thou- 
 Jand two hundred ftvente en. 
 
 What is the value of 640036 ? 
 
 Anf. Six hundred forty thoujand thirty -fix. 
 
 What is 891000002 ? 
 
 Anf. Eight hundred ninety -me million two. 
 
 CHAP. III. 
 
 Of ADDITION of SIMPLE WHOLE NUM 
 BERS'. 
 
 ADDITION is the collecting or putting to 
 gether feveral quantities or numbers into one 
 fum, fo that their total amount may be known : and 
 in order to perform the operations of this rule, two 
 things muft be carefully obferved, which are, Firft, 
 the right placing or fetting each figure in its proper 
 place j that is, units muft Hand under units, tensun- 
 Uer tens, hundreds under hundreds, andfoon, fetting 
 each denomination under that of the fame value: 
 thus 246 + 25 + 163, being fee asdireded, will ftand 
 
 thus, 
 
f2 4 6 
 
 , < 25 
 (163 
 
 thus, < 25 
 63 
 
 THE fecond thing to be ohferved, is the right col 
 lecting or adding together each perpendicular row 
 of figures, placed as before directed , which is per 
 formed as in the following example,, being the fame 
 as made ufe of above, viz. 246 + 25-1-163 : 
 
 f 
 < 
 
 (. 
 
 or thus, <{ 25 
 163 
 
 Then ftriking a line beneath the figures, as in the 
 example ; begin on the right hand at the units' place, 
 adding together all thofe figures which (land in the 
 units' place, and if their fum be under ten, fet it down 
 underneath in the units 1 place , but if their fum ex 
 ceed ten, fet down the furplus, carrying one to the 
 next place, viz. the tens* place : or, more generally, 
 as many tens as the fum of thofe units amounts to, 
 you muft carry to the next place of figures, to wit, 
 the tens' place, adding them up with all the figures 
 that (land in that perpendicular line -, and fo on for 
 the reft ; remembering to carry one for every ten of 
 your aggregate : the whole of which will be illuf- 
 trated in the following 
 
 EXAMPLE. ; 
 
 Find the fum of the following numbers, viz* 
 392+466 + 256. 
 
 THOSE numbers being placed as the rule directs, 
 will (land 
 
 thus,! 466 
 
 se 
 
 1 1 i4=fum required. Then 
 
Then begin with the bottom figure, in the units* 
 place 5 laying 6 and 6 is 12, and 2 is 14; fetting 
 dov/n 4, carry i to the next, or place of tens, faying 
 5 and i that I carry make 6, and 6 is 12, and 9 is 
 21 j here becaufe the aggregate or fum total is 21 
 units (or becaufe it (lands in the tens' place) 2 tens 
 and one unit ; therefore fet down i and carry 2 to the 
 next place, faying 2 and 2 that I carry make 4, and 
 4 is 8, and 3 is 1 1 j which being the ium of the laft 
 place of figures in the example, fee down the whole. 
 [See the work ac the bottom of the preceding page.] 
 THE reafon of fetting down the furplus, or odd 
 figures, and carrying for the tens, as in the laft and 
 all other examples in addition of fimple quantities, 
 is to Ihorten the work under coniideration j and to 
 faye the trouble of ufmg fuperfluous figures. To 
 exemplify which, let us make ufe of the foregoing 
 example, to wit, 3.92+4664*256, which muft be 
 placed 
 
 thus, 
 
 3 
 
 4 
 
 2 
 
 9 
 
 I 
 
 2 
 
 6 
 6 
 
 
 2 
 
 9 
 
 i 
 
 
 
 
 4 
 
 
 
 the fum of the row of units 
 thejum of the row of tens 
 
 the fum cf the row of hundreds 
 
 i 
 
 i 
 
 4 j the J urn of the whole ; 
 
 then adding up each fingle row, frt down its fum in 
 its proper place, in the fame manner as if there were 
 but one fingle row ; fupplying the vacant places on 
 the right hand with cyphers. Hence the refult of 
 this operation is the fame as in the former method of 
 carrying for the tens j and hence alfo it appears, trlat, 
 adding the cyphers, makes no alteration in the value 
 of the fum of 'the other figures. 
 
 THE 
 
THE manner of proving your work., flows as a na 
 tural confequent, from the following felf-evidenc 
 proportion, on which the truth of the rule depends, 
 viz. that every whole is equal to all its parts taken 
 together. Wherefore if you divide, or feperate the 
 given numbers into two, or more parcels, according 
 to your propofition ; and by adding together each. 
 part fo feperated, if the fum of all thofe pans added 
 together, is equal to the fum total of all the given 
 numbers, found before Operation, your work is right. 
 THIS method will appear plain by the following 
 example. Suppofe it were required to add together 
 the following numbers, viz. 3489 + 6725 + 2324+ 
 6744 ; which according to the rule of Notation muft 
 ftand thus, 34^9 
 
 6725 
 
 2324 
 
 6744 
 
 Firftpartj 
 
 19282 fum before federation. 
 
 Second C 2324 
 part [6744 
 
 firft fart. 
 
 z:/#tf2 of 
 fecond fart. 
 
 The fum oftbefrfi and fecond parts | 
 
 Sum of all the farts 19282 
 
 which agrees with the fum total before feperation ; 
 therefore the work is right. But themoft ufual me 
 thods of proving Addition, is either by beginning at 
 the top, and reckoning downwards ; which fum, if 
 equal to that found by cafting upwards, the work is 
 right. Or, firft add together all the propofed num- 
 D bers 
 
( 26 ) 
 
 hers into one fum ; then feperate the upper number 
 from the reft, by a line, and add together the re 
 maining numbers beneath; placing their fum under 
 the former, or fum total before feperation; which be 
 ing done, add the fum lad found to the upper line in 
 your example; which fum, if equal to the fum total 
 or firft addition, the work is right : this is the fame 
 in effect, as the firft method of proof, though a little 
 different in mode, as will appear by the following 
 example. 
 
 34673 
 
 24532 
 12760 
 53865 
 21671 
 
 1 47 506 fum of the whole 
 
 1 1 2$ 2% fum of all but the upper line 
 
 147506=34678 + 1 i<i%i%~fum oftbe wMe : 
 
 therefore the work is right. 
 
 TAKE the following examples, without their an- 
 
 fwers, for practice. 
 
 a 6538764 
 
 3457643 460039 372 875623 
 4567012 914321 42734 43521 
 2354123 675422 8173456 6300 
 1678432 342310 37240 579 
 , 421 $4 
 
 CHAR 
 
CHAP. IV. 
 
 Of SUBTRACTION of SIMPLE WHOLE NUM 
 BERS. 
 
 SUBTRACTION is the taking one number 
 out of another, whereby the remainder, differ 
 ence, or excefs may be known : thus 3 taken out or 
 from 5, leaves 2, which is the difference between 3 
 and 5 ; and is alfo the excefs of 5 above 3. 
 
 HENCE it follows, that the number from which 
 fubtraftion is to be made, muft be equal to, or 
 greater than the fubtrahend, or number to be fub- 
 tra&ed ; and alfo, that Subtraction isthereverfe of 
 Addition ; for Subtraction is the taking of one num 
 ber from another, but Addition is the collecting or 
 putting them together. 
 
 HERE the Notation is the fame as in Addition, 
 to wit, thofe numbers which are of like value, muft 
 ftand directly beneath each other ; that is, units muft 
 ftand under units, tens under tens, &c. After having 
 thus placed your numbers, the lefs beneath the great 
 er, you may proceed to fubtract them apart, by ob- 
 ferving the following 
 
 RULE. 
 
 BEGIN with the firft figureon the right-hand, which 
 (lands in the units' place, and fubtract the lower fi 
 gure from that which (lands directly over it, of the 
 fame value ; fetting down the remainder (if any) be 
 neath in the units' place : If the figure in your fub 
 trahend be equal to the figure which (lands directly 
 over it, you muft fet a cypher for the remainder ; 
 but if the lower, or figure in your fubtrahend, con 
 tains more units than your upper figure, you muft 
 add x.o to the upper figure, or fuppofe it to be fo add 
 ed 
 
ed in your mind ; then fubtrac~b your lower figure 
 from your upper fo increafed, letting down the re 
 mainder or difference in its proper place ; then pro 
 ceed to your next place of figures -, now it is fuppof- 
 cd that the 10 you before added was borrowed from 
 your next fuperior place of figures, where you muft 
 pay what you before borrowed, which is performed 
 as the uiual method is, by calling the lower figure, 
 {landing in that place, one more than it really is ; 
 then fubtrafting it fo augmented, from your upper 
 figure, or figure ftanding direftly over it, fet down 
 the difference as before dire&ec}; and foon, from one 
 place of figures to another, until the whole be complet 
 ed j the whole of which, is illuftrated in the following 
 
 EXAMPLES. 
 
 SUPPOSE, that" from 4567, you were to fubtraft 
 3692 ; which numbers, being placed according to 
 the rule, will (land 
 
 HERTS begin with the 2, faying 2 from 7 and there 
 remains 5, fetting it down as directed -, then proceed 
 to your next place of figures, faying 9 from 6 I can 
 not, becaufe my lower figure, to wit, 9, contains 
 more units than my upper, or figure from which I 
 would fubtract; therefore I fuppofe 10 to be added to 
 the upper figure which makes 16 j then faying 9 from 
 1 6 and there remains 7 3 then proceed to the next 
 place, where you muft pay what you have borrowed, 
 by faying 6 and i that I borrowed make 7 ; then 7 
 from 5 I cannot, but 7 from 54-1015, and there 
 remains 8 ; then to the next place, faying 3 and i that 
 I borrowed make 4, 4 from 5 and there remains 
 i > now there being no more places of figures, fet 
 down the i and the work is done. (Seethe eisamplo?) 
 
 THE ' 
 
( 29 ) 
 
 THE truth of fubtraction is founded on the fame 
 felf-evident proportion, or axiom, as that of Addition) 
 viz. the whole is equal to ajl its parts taken together. 
 From which propofition is deduced the following 
 method of proving your work, to wit, by adding the 
 fubtrahend, or number to be fubtradted, to the re 
 mainder : for the number from which fubtraclion 
 is made, is here confidered as the whole, and the fub 
 trahend, as a part of that whole ; coniequently if that 
 part be taken from the whole, the remainder will be 
 the other part ; therefore if both parts when added 
 together, be equal to the whole, the work is right. 
 
 HENCE it is manifeft that fubtraction may be 
 proved by fubtraction ; for if from 
 
 67834 the whole, 
 is taken 53723 a part of that whole, 
 
 there will remain, 14111 the other part ; 
 and if from 67834 the whole, there is taken 
 the laft part 14111 
 
 there will remain 53723 the firil part, or fubtrahend: 
 confequently, &c. 
 
 AGAIN, if from 17942 the whole, 
 
 is taken 13724 a part of that whole $ 
 
 there will remain 14218 the other part, 
 
 27g42~fum of the fubtrahend 
 and remainder the whole. 
 
 TAKE the following examples for practice. 
 From 37654 394076 2876955 7 6 54i09 
 take 28765 123468 423610 347472 
 
 Rem.' 
 
 CHAP. V. 
 
M 
 
 ( 3Q ) 
 
 CHAP. V. 
 
 Of SIMPLE MULTIPLICATION. 
 
 ULT I PLICATION is a rule by which 
 a given number may be increafed any num 
 ber of times propofed. 
 
 THERE are three requifites in Multiplication : firft, 
 the multiplicand, or number to be multiplied : fec- 
 oncl, the multiplier, which denotes how many times 
 the multiplicand is to be taken ; for by Euclid^ as 
 many units as there are in the multiplier, fo many 
 times is the multiplicand to be added to itfelf : third, 
 the producl, or multiplicand increafed fo many times 
 as there are units in the multiplier. 
 
 SUPPOSE for example, that 7 be increafed 4 times ; 
 that is, to multiply 7 into or with 4 ; thefe numbers 
 muft be placed as in Addition, 
 , T 7 multifile and 
 JS > 1 4 multiplier 
 
 28 produft. 
 
 Now that 4 times 7 make 28, will appear evident 
 by fetting down the multiplicand 4 times, and adding 
 up the whole, as in this, f" 7 
 
 7 
 7 
 7 
 
 i%~fum or fro du ft. 
 
 HENCE it is plain, that multiplication is a con- 
 cife method of Addition. 
 
 BUT before you proceed any further on the fub- 
 jeft of multiplication, you muft learn the following 
 
 Table : 
 
 MUL- 
 
MULTIPLICATION TABLE. 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 H 
 
 16 
 
 18 
 
 20 
 
 22 
 
 2 4 
 
 3 
 
 6 
 
 9 
 
 12 
 
 '5 
 
 18 
 
 21 
 
 24 
 
 27 
 
 3 
 
 33 
 
 3 6 
 
 4 
 
 8 
 
 12 
 
 16 
 
 20 
 
 24 
 
 28 
 
 3 2 
 
 36 
 
 4 
 
 44 
 
 48 
 
 5 
 
 10 
 
 IS 
 
 20 
 
 25 
 
 3 
 
 35 
 
 40 
 
 45 
 
 5 
 
 55 
 
 60 
 
 6 
 
 12 
 
 18 
 
 24 
 
 3 
 
 36 
 
 42 
 
 48 
 
 54 
 
 60 
 
 66 
 
 7 2 
 
 7 
 
 H 
 
 21 
 
 28 
 
 35 
 
 42 
 
 49 
 
 56 
 
 63 
 
 70 
 
 77 
 
 84 
 
 8 
 
 16 
 
 24 
 
 32 
 
 40 
 
 48 
 
 56 
 
 64 
 
 72 
 
 80 
 
 88 
 
 96 
 
 9 
 
 18 
 
 27 
 
 36 
 
 45 
 
 54 
 
 63 
 
 72 
 
 81 
 
 90 
 
 99 
 
 1 08 
 
 10 
 
 20 
 
 3 
 
 40 
 
 5 
 
 60 
 
 70 
 
 80 
 
 90 
 
 IOO 
 
 IIO 
 
 120 
 
 ii 
 
 22 
 
 33 
 
 44 
 
 55 
 
 66 
 
 77 
 
 88 
 
 99 
 
 I IO 
 
 121 
 
 I 3 2 
 
 12 
 
 24 
 
 36 
 
 48 
 
 60 
 
 72 
 
 84 
 
 96 
 
 1 08 
 
 120 
 
 132 
 
 144 
 
 FOR an explanation of the foregoing Table, fup- 
 pofe that it were required to find the product of 3X4. 
 Firft, look in the left hand column for 3, and right 
 oppofite with it in the column under 4 at the top, is 
 12, the product of 3X4. 
 
 AGAIN, to find the product of 9X12. Look for 
 9 in the left hand column as before, and right oppo- 
 fite to it, under 12 in the upper column, is 108, the 
 product required ; and the like is to be underftood 
 of all the reft. 
 
 HAVING given you this fhort, but comprehenfive 
 idea of the foregoing Table, we lhall now proceed ta 
 
 examples, 
 
( __> 
 
 examples, with this caution^ to wit, that in multiply 
 ing, care muft be taken, that the product of the firil 
 figures, ftand directly under its multiplier ; alfo re 
 membering to carry i for every 10 of the product. 
 
 EXAMPLES. 
 
 IT is required to multiply 120X945 which placed 
 as before directed will ftand thus, 
 
 1 20 multiplicand 
 94 multiplier 
 
 --. 
 
 HERE you begin with that figure of your multi 
 plier, which (lands in the units* place, viz. 4, faying 
 4 times o is o, which fet down directly under the 
 figure you are multiplying with ; then fay 4 times 2 
 is 8, which fet under the 9; then 4 times i is 4, which 
 alfo place as in the example ; and the product of the 
 multiplicand with the firft figure of your multiplier, 
 is 480 : then begin with the next figure of your mul 
 tiplier, faying 9 times o is o, which place under your 
 multiplying figure, then fay 9 times 2 is 185 here fee 
 down 8 and carry i to the next place, faying 9 times 
 i is 9, and i that I carry makes 10; now this being 
 the product of the laft place of figures, fet down the 
 whole, and the product of the multiplicand, with 
 the fecond figure of your multiplier is 1080, or more 
 properly 10800 : then adding up both products, 
 their fum is 11280, the product required. (See the 
 example above.) 
 
 IT is required to multiply 2439X421 - 9 thefe num 
 bers placed as directed will ftand 
 
 thus, 
 
( 33 ) 
 
 thus, -j 4 i faffors 
 
 0/2439X1 
 / 2439X400 
 0/2439X421 
 
 THE annexing of cyphers, as in the lad example, 
 is to fupply the vacant places ; and to fhew the fev- 
 eral products are fncreafed in a tenfold proportion, 
 with regard to the places in which your multiplying 
 figures (land. Thus the product of the multiplicand 
 with the fecond figure of your multiplier, is not the 
 product of 2439X2, but the produ6l of 2439X2 tens 
 or 20 ; which product is 10 times more than it would 
 have been, had the multiplying figure (2) flood in 
 the units' place -, fo alfo the annexing of two cyphers, 
 as in the product of the multiplicand with the third 
 figure of the multiplier, to wit, 4, is becaufe that 
 figure Hands in the hundreds' place; and therefore 
 the product is not 2439X4* but really the product 
 of 2439X400 j yet thofe cyphers may be omitted, 
 by obferving the direction in the beginning of this 
 chapter, viz. that the firll figure of the feveral pro 
 ducts (land directly beneath its correfponding figure 
 f the multiplier. 
 
 Find the product of 24354X 32001 
 
 24354 
 48708 
 73062 
 
 779352354= 24354X32001 ~^mfo# required, 
 E " HERE 
 
( J4 ) 
 
 HERE you may obferve that we pafs the cyphers, 
 taking care only to place the next figure according 
 to the foregoing directions. 
 
 WHEN there are cyphers on the right-hand of the 
 multiplicand, or multiplier, or to both, you may mul 
 tiply the figures as before, neglecting the cyphers, 
 until you have found the product of the digets only ; 
 to which annex. fo many cyphers as there are in both 
 faftors : as in thefe, 
 
 848 6953 000 fi c 347 6 5X 200 
 636 
 
 720820^:21200X34 
 
 MB 
 
 24000000? , ' 
 
 24000000 \ J 
 
 96 
 
 48 
 
 57600000000000011 24000000 X 24000000 
 
 IF it be required to multiply any number with 
 10,100,1060, &c. you need only annex to your mul 
 tiplicand fo many cyphers as are in the multiplier, 
 and the work is done ; as in the following, 
 
 20X100-2000 
 300X1000=300000 
 26460X1000012:264600000 
 
 HERE it may perhaps be ufeful, to acquaint the 
 learner of the method of performing Multiplication 
 by Addition - 9 which in fome cafes will be found ufe 
 ful : 
 
ful : the method is as follows : firft, fet down the 9 
 digets, or numeral figures, in a fmall column made for 
 that purpofe -, then againft i, place the multiplicand, 
 againft 2, double the multiplicand, againft 3, three 
 times the multiplicand, and fo on to the laft. 
 
 Find the product of 2439X421 by Addition, 
 
 do. 
 do. 
 do. 
 do. 
 do. 
 do. 
 do. 
 do. 
 
 
 i 
 
 2439irtf/fl 
 
 'I tip I 
 
 
 2 
 
 3 
 
 4878=2 times 
 7317:1:3 do. 
 
 
 4 
 
 975 6 =4 
 
 do. 
 
 
 5 
 
 12195=5 
 
 do. 
 
 
 6 
 
 14634=6 
 
 do. 
 
 
 7 
 
 17073=7 
 
 do. 
 
 
 8 
 
 19512=8 
 
 do. 
 
 
 9 
 
 21951=9 
 
 do. 
 
 r i is 2439 
 
 . 
 
 againft < 2 4878 
 
 
 Sum 
 
 z:pro. req. 
 
 HERE it is evident, that the foregoing table will 
 ferve let the multiplier be any number whatever ; 
 for fuppofe it were required to find the product of 
 2439>< 6 734- 
 
 OPERATION. 
 
 againft 
 
 9756=2439X4 
 
 73^7 2439X30 
 
 17073 2439X700 
 
 14634 1^2439X6000 
 
 Sum 
 
 EXAM- 
 
EXAMPLES. 
 
 691861X26=17988386 
 
 346732X6523:226069264 
 7 9 1 37 5X3000011237041 250000 
 129186X981:12660228 
 7600 1 > 1 302:1:98953302 
 3581X2007=17187067 
 
 THE proof of Multiplication, is bed done by 
 Divifion. 
 
 CHAP. VI. 
 
 Of DIVISION of SIMPLE NUMBERS. 
 
 IVISION is a fpeedy method of fubtrafting 
 one number from another ; to know how many 
 times one number is contained in another ; and alfo 
 \vhat remains. 
 
 THERE are three requifites in Divifion; thedivifor; 
 the dividend, and the quotient; which fhews how 
 many times the divifor is contained in the dividend. 
 
 WHEN any number meafures another, the number 
 fo meafured, is faid to be a multiple of the other : 
 thus, 21 is meafured by 7, for 7 is contained juil 3 
 times in 21 ; confequently 21 is a multiple of 7. 
 
 ONE number is faid to meafure another, by a third 
 number, when it- either multiplies, or is multiplied 
 by the meafnring number, produces the number 
 meafured. (See Euc'/id's 7th book, def. 23.) 
 
 HENCE it follows, that in Divifion the quotient 
 rnuft be fuch a number, which if multiplied with the 
 divifor, will produce the dividend ; confequently 
 
 Divifion 
 
( 37 ) 
 
 Divifion is the reverfe of Multiplication ; and there 
 fore operations in Divifion, muft be performed direct 
 ly reverfe of thofe in Multiplication -, that is, the di- 
 vifor muft be placed firft ; then make a flroke on the 
 right-hancfcof it, and fet down your dividend, on the 
 right-hand of which, make another ftroke, to feper- 
 ate the dividend from the quotient ; then begin on 
 the left-hand, and decreaie the dividend by a re 
 peated fubtra&ion of the products of the divifor and 
 each quotient figure, as they become known. 
 
 EXAMPLES, 
 
 REQUIRED to divide 344 by 4; the operation of 
 which will Hand in the following order, 
 
 dividend 
 divifor 4^344(86 quotient 
 
 24 
 
 24 
 
 oo 
 
 THE explanation of the above is as follows : firft 
 enquire how many times your divifor, which confifts 
 of i figure, is contained in the firft figure of your divi 
 dend, which is o times ; becaufe your divifor (4) is 
 greater, than the firfl figure of your dividend (3), as 
 appears by inflection ; and therefore cannot meafure 
 it ; for a greater number to meafure a lefs is abfurd ; 
 therefore you muft increafe the value of the firft 
 figure of the dividend, by taking the annexed figure 
 (4) into the expreflion ; which will then be 34 (for 
 the reafons before given) ; then enquire how many 
 times your divifor is contained in thofe two figures 
 
 of 
 
( 38 ) 
 
 of the dividend, to wit, 34 ; which is 8 times, for 8 
 times 4 is 32, and 32 being the greatefl multiple of 
 the divifor that can be made under 34 -, confequent- 
 ly 8 mult be the firft figure of the quotient, which 
 place as in the example - y then multiplying the quo 
 tient figure (8) with your divifor, as in Multiplica 
 tion, fubtract their product from thofe two figures of 
 the dividend, by which the faid quotient figure was 
 obtained ; and to the remainder (2) annex the next 
 figure of your dividend (4), and the remainder fo in- 
 creafed becomes 24 ; then enquire how many times 
 4 is contained in 24, which is 6 times > therefore 
 place 6 in the quotient, and multiply it with your 
 divifor, fubtracting their product as before, and the 
 work is done. (See the example page 37.) 
 
 Now the quotient obtained in the example is 86 - } 
 and there being no remainder, fhews that 4 is con 
 tained in 344, juft 86 times. 
 
 THE greateft difficulty in divifion, is when your 
 divifor confifls of many places of figures, and does 
 not exactly meafure the figures of the dividend with 
 which you compare it : therefore to find the right 
 quotient figure, may be done by confidering that 
 the product of the quotient figure with your divifor, 
 rouft never be greater than that part of the dividend, 
 with which you compare it ; nor yet fo fmall, that 
 the number remaining after fubtracting the product 
 of the quotient figure and divifor from the aforefaid 
 part of the dividend, fhall be greater than the divifor. 
 Therefore by fuppofing a figure for the quotient, and 
 multiplying it with a figure or two on the left-hand 
 of your divifor, you may eafily determine the right quo 
 tient figure; which may be obtained by fuch mental 
 operations, on the fecond or third trial, at fartheft. 
 
 BY thoroughly obferving the foregoing directions, 
 you may proceed to the performance of the following 
 
 examples 5 
 
( 39 ) 
 
 examples ; wherein we fhall prove thofe operations, 
 performed in the laft chapter ; in order to which, we 
 {hall begin with the fecond example; taking the pro- 
 duel: of the factors for a dividend, and the multiplier 
 for a divifor ; and proceed as before. (See the oper 
 ation annexed.) 
 
 dividend 
 
 divifor 421 \ 10268 19(2439 quotient 
 ) 84.2- 
 
 3789 
 3789 - 
 
 ,.oo 
 
 Note, // will be~l>:ji to point the figures of the divi 
 dend, as they are annexed to the fever al remain 
 ders > without which you may annex a wrong one. 
 
 HERE you may fee the quotient is the fame as the 
 multiplicand of the example before quoted ; which 
 proves that the product of 2439X42.1:1:1026819. 
 
 Required to divide 779352354 by 32001. 
 
 OPER- 
 
( jt-o ) 
 
 OPERATION. 
 
 32001^779352354(24354:1:77935235432001 
 / 64002 
 
 139332 
 128004 
 
 113283 
 96003 
 
 172805 
 160005 
 
 12800.4 
 128004 
 
 ..... o 
 
 Again, divide 1798836 by 26. 
 OPERATION'. 
 
 26^1798836(69186 179883626 
 Ji 56 
 
 238 
 
 48 
 
 223 
 208 
 
 ~6 
 i$6 
 
 - *- Once 
 
( 41 ) 
 
 Once more, divide 12660228 by 98, 
 
 OPERATION. 
 98 \ 1 2660228(1 291 %6 quotient required. 
 
 286 
 196 
 
 900 
 882 
 
 182 
 9* 
 
 842 
 784 
 
 588 
 588 
 
 IF there be cyphers annexed to the divifor and 
 dividend, expunge an equal number in both faftors : 
 as in the following example. 
 
 Divide 694000 by 2000. 
 
 OPERATION. 
 2(000^694(000(347^:694000-1-2000 
 
 14 
 14 
 
 F IT 
 
( 42 ) 
 
 IT will fometimes happen in Divifion, that the 
 remainder, when augmented by annexing the next 
 figure of the dividend, is lefs than the divifor, and 
 confequently cannot be meafured by it; in which 
 cafe, place o in the quotient, and annex the next 
 figure of the dividend to the former number ; but 
 if this number be ftill lefs than the divifor, place o 
 in the quotient and annex another figure of the div 
 idend ; and fo on, in like manner till the faid num 
 ber be fo in-creafed, that it may be meafured by the 
 divifor. (See this illuftrated in the following.) 
 
 Divide 98953302 by 1302. 
 
 OPERATION. 
 
 1302^98953302(760011198953302-7-1302 
 ' 
 
 ...o 
 
 THE proof of the remaining examples in Multi 
 plication, are left to the fagacity of the learner. 
 
 It is required to divide 32176432 by 3476. 
 
 OPER- 
 
( 43 ) 
 
 OPERATION. 
 
 3476^32176432(9256 
 ^31284 
 
 8924 
 6952 
 
 19713 
 17380 
 
 23432 
 20856 
 
 2576 remainder. 
 
 HERE follows fome examples and their anfwers 
 without their work. 
 
 What is the quotient of 23884044718-7-45007 ? 
 Anfwer. 530674. 
 
 What is the quotient of 34500000-7-100000 ? 
 Anfwer. 345. 
 
 What is the quotient of 24457 2OOO-T-J56 ? 
 Anfwer. 687000. 
 
 What is the quotient of 1332250-7-365 ? 
 Anfwer, 3650. 
 
 THAT DivifiooJs a fpeedy method of fubtra&ion, 
 as before hinted, may ^be thus proved. Suppofe 18 
 were to be divided by 6 : firft fubtraft the divifor 
 from the dividend, and the divifor again from that 
 remainder, and fo on till nothing remains. (See the 
 operation in the next page.) 
 
 OPER- 
 
( 44 ) 
 
 OPERATION. 
 
 1 8 dividend 
 
 6 divifor 
 
 12 remainder 
 
 6 divifor 
 
 6 remainder 
 
 6 divifor 
 
 HENCE it is manifeft, that the divifor is contained 
 in the dividend, jufl 3 times -, that is, 3 times 6:1:18 : 
 confequently, &c. ^. E. D. 
 
 THE next thing to be confidered, is the proof of 
 your work, i. e. whether the quotient found is a true 
 one. The method is directly reverie of that ufed 
 for the proof of Multiplication ; for, as the truth of 
 Multiplication is known by Divifion, fo that of Di- 
 vilion is known by Multiplication -, that is, by mul 
 tiplying the quotient with the divifor, which product 
 inuft be equal to the dividend - 3 therefore multiply 
 the quotient with the divifor, and to their product 
 add what remains after divifion; which aggregate 
 will be equal to the dividend, if the work is right. 
 
 THERE is another method of proving Divifion ; 
 which is much fhorter than the former, and is no 
 more than adding together the products of the fever- 
 al quotient figures with the divifor, as they (land 
 in your operation ; which aggregate, together with 
 the remainder, will be equal to the dividend. (See 
 the following example.) 
 
 Required to divide 8765452 by 3463. 
 
 OPER- 
 
( 45 ) 
 
 OPERATION. 
 
 3463)8765452(2531 
 
 + 6926- ' 3463X2000 
 
 18394 
 + I73IS 1=3463X500 
 
 10795 
 + 10389 =3463X30 
 
 4062 
 + 3463 =3463X1 
 
 + 599 remainder 
 
 8765452= dividend. 
 
 0,6926000+1731500+103890+3463 + 599:= 
 8765452. Therefore, &c. 
 
 A SUPPLEMENT to CHAPTER VI. 
 
 NOTWITHSTANDING what hath been 
 faid on this fubject, refpecting the divifion of 
 fimple quantities, is univerfally true ; yet there is 
 another method of dividing quantities, which is very 
 ready in practice ; and is therefore called Short Divi 
 fion : this method is performed by the following Rules. 
 
 RULE I. 
 
 ARRANGE the factors as before in Divifion ; then 
 by comparing the divifor with the dividend, you 
 
 will 
 
( 46 ) 
 
 will obtain a quotient figure., which muft be let in 
 its proper place, under that part of the dividend by 
 which your divifor was compared ; valuing faid fig 
 ure as though there were no other ; alfo obtain the 
 difference (if any) of the product of the divifor and 
 quotient figure, and the aforefaid part of the divi 
 dend j prefixing that difference in your mind to the 
 next figure of your dividend ; which forms an expref- 
 fion for obtaining the next quotient figure, which 
 muft be fet directly under that figure, to which the 
 difference was prefixed ; and fo on till the whole 
 be completed. 
 
 EXAMPLES. 
 Divide 46782 by 3. 
 
 THOSE numbers being placed as directed will 
 ftand thus, 
 
 3)46782 
 
 15594=46782-3 
 
 Again, divide 68432 by 4 : 
 thus, 4^)68432 
 
 17 iQ%~quotient required. 
 
 Note i. If there be a remainder after the lafl quo 
 tient figure is found, Jet it at a little diflance on the 
 right -hand of your quotient, making a dot with your 
 pen, denoting the federation -, as in the following. 
 
 Divide 13764 by 5 : thus, 5 \ 23764 
 
 / '\ 
 
( J-7 ) 
 
 Again, find the quotient of 732 r 56: 
 thus, 6x73215 
 
 1 2 20 2 . 
 
 Alfo, divide 43206 by 8 : 
 
 thus, 8 \432io6 
 
 54013 . zrem, 
 
 Note 2. If your divifor be 10, federate the fir ft fig 
 ure on the right-hand of your dividend for a re 
 mainder, and the work is done. 
 
 thus, 10^76435(2^^. 
 
 Find the quotient of 645384-^-12: 
 thus, 12^645384 
 
 RULE II. 
 
 1. RESOLVE your divifor into feveral parts fuch, 
 that their continued product fhall be equal to the 
 given divifor. 
 
 2. SUBSTITUTE thofe parts fucceflively as divifors, 
 in the following manner, viz. divide the given divi 
 dend by one of thofe parts, now called divifors, and 
 the refulting quotient by another of thofe divifors, 
 and fo on 5 the laft quotient arifing by fuch divifors, 
 will be the quotient required. 
 
 EXAMPLE. 
 
 Divide 2904 by 24, 
 
 YOUR 
 
YOUR ditflfor refolved into parts as above directed 
 will be, either 8 and 3, 6 and 4, or 1 2 and 2 -, for 
 8X3=24, 6X424) or 12X2=245 therefore let the 
 parts be 6 and 4 ; then 29046^484, and 484-7-4 
 ~ 1 2i~quotient required ; and if the others be tryed 
 they will equally fucceed. 
 
 CHAP. VIL 
 
 ADDITION of COMPOUND QUANTITIES or 
 NUMBERS. 
 
 ADDITION of compound quantities, is the add 
 ing together numbers of different denomina 
 tions, fo that their aggregate, or total amount may 
 be known. The operations are performed by the 
 following general 
 
 R UL E. 
 
 1. WRITE down the feveral denominations fo, 
 that all thofe of the fame name may ftand directly 
 under each other. 
 
 2. BEGIN on the right-hand, at the leaft of the 
 given denominations, adding together the whole of 
 that denomination, as in Simple Addition ; then di 
 vide this fum by fuch a number, as it takes parts to 
 make one of the next greater denomination, placing 
 the remainder (if any) under its own denomination, 
 and carrying the quotient to the faid next greater 
 denomination, add them up with the whole of that 
 denomination, theri divide, as before; and fo on, 
 from one denomination to another, until the whole 
 be completed. 
 
 SECT. 
 
( 49 ) 
 
 SECT. I. 
 
 ADDITION of TROT WEIGHT. 
 
 TROY WEIGHT is that by which gold, filver, 
 jewels, medical compofitions, and all liquors are 
 weighed. It is divided into four denominations, 
 to wit, ft. pounds, oz. ounces, dwt. pennyweights 
 and gr. grains, according to the following 
 
 TABLE. 
 
 .24Odfw/.:ni 2 cz.ni ft . 
 
 EXAMPLES. 
 
 Find the fum of the following, i/j-tb. i loz. i6dwt. 
 I 3^ r - + i9l6. iooz. IT dwt. ijgr. + ijfi). noz. 
 iidwt. 22gr. 
 
 THESE numbers being placed, according as the 
 general rule directs, will Hand 
 
 ft. oz. dwt. gr. 
 fi4 ii 16 13 
 thusX 19 10 17 17 
 
 1 II 12 2 
 
 52 10 7 4^r jto required. 
 
 _ i 
 
 THEN begin at the lead denomination, to wit, grains, 
 and adding together all that denomination, we find 
 the fum to be 52 : now becaufe 24 grains make one 
 pennyweight, divide 52 by 24, and the quotient will 
 be 2, leaving a remainder of 4, which write under 
 grains, and carry the quotient 2, to the next place, 
 and adding it up with that denomination, we find the 
 fum to be 47, which divide by 20 (becaufe 20 
 pennyweights make one ounce) and the quotient will 
 
 G be 
 
be 2j leaving a remainder 7, which write in its prop 
 er place, and carry the quotient 2., to the next place ; 
 this being added up with the denomination, we find 
 the Ium to be 34, which divided by 12 quotes 2, and 
 10 remaining; write this under its own denomina 
 tion, and cany the quotient 2 to the next place, which 
 added up with that denomination, we find the furn 
 to be 525 and becaufe this is the lad denomination, 
 write the whole, and the work is done. Hence we 
 find the fum total to be 5216. 10 oz. *]dwt. and 4gr. 
 as was required. (See the example, page 49,) 
 Jb. oz : . dwt* gr. IJj. oz. dwt. gr. 
 
 37 I0 J 7 19 47 IX *9 2 4 
 
 12 7 12 17 27 8 17 20 
 
 17 10 17 12 19 7 12 17 
 
 18 9 19 23 10 5 15 -17 
 
 SEC T, JL 
 
 ADDITION of MONET. 
 
 *THIS is to find the aggregate, or ium total of fcv- 
 tral (urns of money. 
 
 EVERY nation of the world has a particular method 
 of reckoning their money. Great-Britain makes ufc 
 of pounds, fhillings, pence and farthings 3 and the 
 United States followed the fame method, until the 
 prefent fyflem of government was eflablillied ; by 
 which it is enadlcd, that all the monies of every na 
 tion or kingdom, fhall be reckoned or eftimated in 
 America, in dollars and cents: fo that thefe two 
 fpecies of money are to be made the ftandard money 
 of the United States. 
 
 Hitis thai 100 c$nt$ m.akt om dollar. 
 
 -RX/W 
 
( 5* ) 
 
 EXAMPLES. 
 
 Find the Him of ij^dol. I'-jcts. -\-\y-doL \gcts.-\~ 
 37$^0/. yicts. -\-2j$dol. yicts. Thefe being plage4 
 according to the general rule, will ftand 
 
 cts. 
 
 1023 20 j "urn required. 
 
 Note. Since 100 ^tf/j make one dollar, we mufl di 
 vide thejum of the cents ly 100 ; but to divide by 
 100 is no more than tojeperate the two right-hand 
 figures of the dividend for a remainder, the refl 
 are the quotient, ^herefore^ after you have add 
 ed up the laft flace of figures in the cents' place, 
 proceed to the dollars' place as though the whole 
 was but one denomination. 
 
 Find the fum of i zjdol. v . , 
 $37 do!, igcts.+ izzdol. yicts. -\-ii7 dot. yocts* 
 
 do I. cts. 
 
 127 19 
 
 278 19 
 
 137 19 
 
 122 92 
 
 1 27 90. 
 
 793 39=/^m required. 
 
dol. cts. dol. cts. doL cts. 
 
 127 17 3787 19 2784 19 
 
 172 57 3729 72 1234 27 
 
 189 68 4229 91 3456 78 
 
 Total ZZZZH HHZZI ZZZZZ 
 
 HAVING thus explained the principles, and given 
 a general rule for the Addition of all compounds in 
 whole numbers ; we lhall leave the reft to the faga- 
 city of the learner, who with the affiftahce of the fol 
 lowing tables and examples, will be able to manage 
 any fuch compounds as have relation therewith. 
 
 SECT. III. 
 
 Of AVOIRDUPOIS WEIGHT. 
 
 By Avoirdupois Weight are weighed, flefh, but 
 ter, cheefe, fait ; alfo all coarfe and drofTy commodi 
 ties i as grocery wares; likewife pitch, tar, rofin, 
 wax, iron, flee], copper, brafs, tin, lead, hemp, flax, 
 tobacco, &c. 
 
 THE characters in Avoirdupois Weight zxtdr. oz. 
 Ib. qr. C. c f. that is drachm, ounce, pound, quarter, 
 hundred, tun. 
 
 TABLE. 
 
 16 dr.-=.\ oz. 256^. i60%.=i/. 7168^.= 
 ir. 2867 2 dr.-=^\ 79202;.=! 
 
 57344O ^.=35840 6>2;.^z 
 
 EXAMPLES. 
 
 ?. C. qr. Ib. oz. dr. T. C. qr. Ib. oz. dr. 
 
 346 12 2 16 10 14 576 19 i 16 12 13 
 
 67 16 3 22 8 10 867 4 o 24 14 13 
 
 4"6 10 3 15 12 15 453 6 3 27 3 4 
 
 SECT. 
 
( 53 ) 
 
 SEC T. IV. 
 
 Of APOTHECARIES HEIGHT. 
 
 THE Apothecaries pound and ounce is the fame 
 as the pound and ounce Troy, but differently di 
 vided, as in the following 
 
 TABLE. 
 
 20 jr. 13. 6oT.=39.=i3. 48037. = 249. 
 
 APOTHECARIES make ufe of thefe weights in the 
 compofition or mixture of their medicines, but fell 
 their drugs by Avoirdupois Weight. 
 
 EXAMPLES. 
 
 K- I- 5- 3- Z r - K>- I- 3- 9- Z r - 
 124 10 4 2 14 266 9 5 i 15 
 
 64 8 6 i 16 76 10 4 2 14 
 
 30 ii 7 o 17 96 ii 6 2 10 
 
 50 9 3 i 12 10 7 i i 
 
 SECT. V. 
 
 BY Long Meafure, is eftimaied length, where no 
 regard is had to breadth : or in other words, it meaf- 
 ures the diilance of one thing from another : and the 
 ufual method of dividing and fub-dividmg of length, 
 is into degrees, leagues, mile's, furlongs, poles, yards, 
 feet, inches, and barley-corns, as in the following 
 
 TABLE L 
 
 23760^. 
 190080^. 
 
( 54 ) 
 
 ^63-360 fa.^$2%of.=i76oyd.^32op.==&fur.==: i m. 
 
 570240^.=! 
 
 TABLE II 
 
 zifur. 1 90080^. 
 =63360 /#.=528o/.=i 7 60^.= i 60 t^.=i3/ar,=: i ^. 
 570240 ^.=1 90080 f.=i 5 840/.==528ojK^.==48or, 
 =24/r.=2 w.= i /^. 1 1 404800 ^f. =38oi6oo///.= 
 3 1 68oo/.= 105600^=^9600 -cb.=.4.%ofur.= 60 ;.== 
 
 THE length of a degree as laid down in table ad. 
 is not to be underftood as the true one, but the length 
 of a degree as commonly received and pra6lifed ; for 
 the length of the greatefl degree is yo^Vrniles, and 
 the leaft 67^. miles nearly; a mean degree is there-' 
 fore 6 8-^ miles. 
 
 EXAMPLES. 
 deg. h. m.fur. cb. yd. f. in, be* 
 
 I2O 14 2 6 14 5 2 1O I 
 
 87 12 o 7 12 3 i 50 
 90 19 i 5 18 2 2 42 
 
 332 15 i 7 12 8 i 10 2 
 
 19 a o 14 9 2 9 
 
 i 6 13 5 o 
 
 4 10 4 
 
 9 
 
 S E C TV 
 
( 55 ) 
 
 SECT. VI. 
 
 Of LAND MEASURE. 
 
 THE ufe of this meafure, is to find the area or fu- 
 perficial content of any piece of land in acres, and 
 pares of an acre - 3 which parts are as in the following 
 
 TABLE. 
 
 I . .fy.yd.iofq. cb.=ifq. qr. 43560 
 
 f- ~ ,c/. fl&.=4/j. qr.^ifq. acre* 
 
 EXAMPLES. 
 
 :b. yd. /. ac. qr. cb. yd. f. 
 
 '6 104 - 8 9.2 i 7 loo 7 
 
 j 7 1 1 1 7 27 3 7 98 8 
 
 24 90 7 39 o 7 117 7 
 
 SECT. VII, 
 
 Of CLOTH MEASURE. 
 
 THE divifions of Cloth Meafure are as in the 
 foyowing 
 
 TABLE. 
 
 Ailb, j^r.s=i^// 
 
 EXAMPLES. 
 
 yd. qr. . rt. <// F/. qr. na. 
 
 226 3 2 3-733 
 
 74 3 o 39 2 i ' 
 
 362 2 3 500 3 2 
 
$11 En%. 
 
 qr. 
 
 na. 
 
 ell Fr. 
 
 jr. 
 
 *, 
 
 3^7 
 
 4 
 
 3 
 
 529 
 
 5 
 
 3 
 
 90 
 
 3 
 
 2 
 
 468 
 
 2 
 
 2 
 
 264 
 
 2 
 
 I 
 
 436 
 
 4 
 
 3 
 
 354 
 
 3 
 
 3 
 
 43 
 
 3 
 
 i 
 
 
 SECT. VIII. 
 
 Of DRT MEASURE. 
 
 DRY MEASURE is fo called becaufe it meafurcs all 
 fuch dry commodities as corn, .wheat, rye, oats, bar- 
 ley, peas, beans, and all kinds of grafs-feed -, alfo all 
 kinds of roots and fruits. 
 
 TnEftandard of thismeafnre, is abufhcl of a cylin 
 drical form, of the following dimenfions, viz. 1 8~ in 
 ches in diameter, and 8 inches in altitude $ confe- 
 quently a veflel of fuch form and dimenfions will 
 contain 2150^^ cubic inches, which is the content 
 of the Wincheiter bufhel: Therefore the quart Dry 
 Meafure, contains 67-^5- cubic inches nearly ; and the 
 divifions are as in the following 
 
 TABLE. 
 
 67 . 2 cu. tn.^iqrt. 268 . 8 cu. in. =4 qrt.^i gal. 
 537 .6 cu, in.i=& qrt.~zgaL-=.ipc. 2150.42^. in. 
 zzz^iqrt.Kgal.-^^fc^i bujh. 
 
 EXAMPLES, 
 lujh. pc. gal. qrt. bujh. pc. gal. qrt. bu/h. pc. gaLqrt. 
 
 57 3 i 3 37 3 * l 2312 
 24 002 19000 it 
 
 47 210 33 203 2313 
 
 SECT. 
 
( 57 ) 
 
 SECT. IX. 
 
 Of LIQUID MEASURES. 
 
 IN Liquid Meafures, the gallon is made the (land- 
 ard, and from thence are deduced the other denomi 
 nations made ufe of in fuch meafures. The wine gal 
 lon is fuppofed to contain 231 cubic inches, confe- 
 quently the quart muft contain 57^ cubic inches j 
 from thence is deduced the following 
 
 TABLE of WINE MEASURE. 
 
 $7^eu.in.~iqrt. 231 r#. in.-zz^qrt.^i gal. 9702 
 cu. /#.=i68 qrt.=z4.2gal.-=itr. 14553 cu.in.-=.2$2qrt. 
 =63#/~i ^ tr.=i bhd. 19404 cu. /. 336^77 .=84 
 g0/.=:2 tr.~i^ bbd.i pun. 29106 cu. ^.=504 qrt. 
 ==126 gal.^z /r.=2 bbd.~i* i -fun.'=.i bt. 58212 cu. 
 
 i tun. 
 
 EXAMPLES. 
 
 tun bbd. gal. qrt. tun. bbd. gal. qrt. 
 
 237 2 62 3 279 2 57 2 
 
 234 i 27 o 273 o 39 o 
 
 72 2 25 3 99 2 47 3 
 
 34 o 59 o 93 i 24 2 
 
 Of ALE or BEER MEASURE. 
 
 THE gallon of Ale or Beer Meafure contains 282 
 cubic inches, as in the following 
 
 TABLE. 
 
 *1&^cu.in. iqrt. 282 *. f#.=4. qrt. i gal. 2397 
 cu. *.=34 qrt.-=$)Lgal.-=>\Jir. 4794 #. /'#.=-68 ^r/. 
 =17 tf/.=2/r.=i A//. 9588 c^. ;/;.=:i36 ^.=34 
 
 H gal. 
 
.=i ^r. 14382 r^. 
 kil.i bar.~i bbd. 
 
 EXAMPLES. 
 
 bbd. kit. fir. gal. qrt. bhd. kit. fir. gal. qrt. 
 
 79 2 i 7 2 73 2 i 6 3 
 
 64 3 o 5 ; j o 97 i i 7 2 
 
 49 i 162 37 2120 
 
 SECT. X. 
 
 O/ the MEASURE of flME. 
 
 IN the divifion of Time, a year is made thefland- 
 ard or integer, which is determined by the revolu 
 tion of fome celeftial v body in its orbit ; which body 
 is either the fun or moon. The time meafured by 
 the fun's revolution in the "ecliptic (or imaginary 
 circle in the heavens, fo called by aftronomers) from 
 any equinox or foltice to the fame again, is 365 days, 
 5 hours, 48 minutes, 57 feconds, and is called the 
 
 iblar or tropical year. Although the folar year 
 
 before mentioned, is the only proper or natural year, 
 yet the civil or Julian year is the one which the dif 
 ferent nations of the world make ufe of in the re 
 gulation of civil affairs. 
 
 THE civil folar year contains 365 days, 6 hours ; 
 but in common mathematical computations, the odd 
 hours are generally negledled, and the year taken 
 only for 365 days; from which, the divifions in the 
 following TABLE are made, wherein a fecond is con- 
 fidered (as it really is) the lead part of time that can 
 be truly meafured by any mechanical engine. 
 
 6o*.-=i / . 36oo // .=6o / .=i 
 ?4&.=:i d. 3i536ooo // .=^ 
 ^i year. EXAM- 
 
(59 ) 
 
 EXAMPLES. 
 
 y. d. h. l " y. d. h. ' " 
 
 167 272 14 42 29 173 192 10 17 29 
 
 234 i?3 22 58 59 346 364 23 59 59 
 
 39 290 19 19 19 199 170 19 17 16 
 
 43 222 22 22 22 344 19 IO 34 46 
 
 99 99 20 57 21 79 38 23 43 43 
 
 SECT. XL 
 
 Of CIRCULAR MOTION. 
 
 WHAT is here meant by Circular Motion, is that of 
 the heavenly bodies in their orbits ; which are reck 
 oned in figns, degrees, minutes, and feconds, as in 
 the following 
 
 TABLE. 
 
 / =6o / =io. io8ooo // .=i8oo / =30 d 
 =16". 1 296ooo"=2 1 6oo / =36o=i 2 S* great circle 
 of the ecliptic * 
 
 EXAMPLES. 
 S. * : S. ' ^ 
 
 10 12 30 10 II 13 13 IJ 
 
 9 ii 47 47 8 17 23 43 
 
 4 37 4 7 29 44 27 
 
 7 24 42 36 6 19 38 59 
 
 Note. In the Addition of Circular Motion* when- 
 the Jum of the Jigns exceed 1 2, or any inultifle of 
 if, writs fuch excefs in the place of Jigns , rejecting 
 tht reft. 
 
 Note. 
 
Note. In order to prevent a mijconftruftion of the 
 abbreviations , in the nine preceding TABLES, ive 
 have Jubjoined the following explanation, viz. gr. 
 Jiands for grains. ^Jcruples. % drachms, g 
 ounces, fl3 pounds. be. barley-corns, in. inches. 
 f. feet. yd. yards, ch. chains, p. poles, fur. 
 furlongs, m. miles, le. leagues, deg. degrees. 
 fq.Jquare. qr. quarters, ac. acres. na. nails. 
 Flern. Flemijh. Eng. Englijh. Fr. French. cu. 
 cubic. qrt. quarts, gal. gallons, pc. pecks, bufh. 
 bufhels. tr. tierces, hhd. hog/heads, pun. punch 
 eons, bt. butts. fir. firkins, kil. kilderkins, bar. 
 barrels. "feconds. * minutes, h. hours, d.days. 
 y. years. degrees. S. Signs. 
 
 CHAP. VIII. 
 
 SUBTRACTION of COMPOUNDS. ' 
 
 SUBTRACTION of Compounds is the taking 
 o;ie number from another: and is performed by 
 the following general 
 
 R ULE. 
 
 1. RANGE the given denominations according to 
 the dire&ions in the laft chapter. 
 
 2. BEGINT at the fame place as in Addition, to wit, 
 at theleaftof the given denominations, fubtra&ing the 
 lower number from the upper, as in Simple Subtrac 
 tion, writing the difference under its own name ; but 
 if the number in the fubtrahend or under number, be 
 greater than that which (lands direftly over it (as it 
 prten happens) you mult add to your upper number, 
 fo many units of that denomination as are equal to 
 
 one 
 
one of the next greater ; from which perform thein- 
 tended fubtmetion, writing the difference as before. 
 Then proceed to the next place, where you muft pay 
 what you before borrowed of this denomination, by 
 adding one to the fubtrahend, and then perform fub- 
 tradtion as before > and fo on to the laft place, where 
 the fubtraclion is performed as in fimple quantities. 
 
 EXAMPLES. 
 
 From 37 ft 1002. 17 dwt. 2Or. take 27 ft noz. 
 19 dwt. ijgr. 
 
 Thefe numbers being placed according to the rule, 
 will ftand 
 
 ft cz. dwt. gr. 
 37 io 17 20 
 27 ii 19 17 
 
 9 io 1 8 3 diff. required. 
 
 HERE beginning at the leaft denomination, to wit, 
 at- grains, fubtract 17 from 20, and there remains 
 3, which write under its own namej then pro 
 ceed to the next denomination - 9 but here the under 
 number is the greateft, and therefore cannot be taken 
 from the upper ; wherefore add ao to the upper num* 
 ber (becaufe 20 pennyweights make one ounce) and 
 the fum is 37, from which take 19, and their remains 
 1 8 ; or take 19 from 20, and then add 17, and the 
 fum will be 1 8, as before. Then proceed to the next 
 place ; and here again, the under number is the great- 
 eft, therefore add I to 1 1 for what you before borrowed, 
 and the fum will be 12, which taken from 22, leaves 
 io, which write in its proper place, and proceed to 
 the laft denomination, where paying what you before 
 borrowed, perform the fubtraclion as in whole num 
 bers, 
 
bers, and the remainder will be 9. Hence we find the 
 whole difference to be 9 pounds, 10 ounces, 18 pen 
 nyweights, and 3 grains. 
 
 ft $z. dwt. gr. doL cts. 
 From 27 10 13 17 37 19 
 
 Take 22 8 19 19 21 18 
 
 Rem. 5 i 13 22 16 i 51 17 
 
 As the foregoing rule is general, the learner by du 
 ly obferving the application of it, to the above exam 
 ples, may very readily perform the following ones 
 without any further direction. 
 
 T. C. qr. It. cz. dr. 
 From 324 19 3 17 2 15 
 Take 233 17 2 20 13 14 
 
 yd. qr. na 
 27 3 2 
 204 i 3 
 
 Rem. 
 
 ellFlem.qr.na. ell Bpg. qr. na. e 
 From 5213 42 4 i 
 Take 35 2 i 36 2 3 
 
 HFr. qr. na, 
 53 3 3 
 49 5 o 
 
 Rem. 
 
 T. bhd. gal. qrt. bhd. kil. 
 From 37 3 36 2 33 * 
 Take 23 i 37 3 27 i 
 
 Jlr. gal qrt. 
 * 7 3 
 043 
 
 Hem. 
 
 y. d. b. ' ' ft 
 From 434 320 17 24 42 47 
 Take 329 370 19 47 29 45 
 
 I 2 3 gr. 
 
 10 7 2 14 
 
 8 5 i 17 
 
 Rem. ~ ' ; 
 
 THE 
 
_ 
 
 THE method of proving your work, is the fame as 
 that of Simple Subtraction. 
 
 CHAP. IX. 
 
 MULTIPLICATION and DIVISION of COM 
 POUNDS. 
 
 SECT. I. 
 
 Of MULTIPLICATION. 
 
 MULTIPLICATION of Compound Numbers 
 is the multiplying any fum compofed of divers 
 denominations, with a fimple multiplier, according 
 to the following 
 
 RULE. 
 
 BEGIN the operation as in all other compounds, 
 anultiplying that denomination with your multiplier, 
 as in Simple Multiplication ; then divide this pro- 
 duel by as many units as make one of the next great 
 er denomination, writing the remainder as in Addi 
 tion ; then note the quotient, and proceed to the 
 next place, and multiply that denomination with your 
 multiplier, to which add the aforefaid quotient ; then 
 divide this produd as before, and fo on, till you have 
 multiplied your multiplier with every denomination 
 in your multiplicand j and the refult will be the pro- 
 duel: required. 
 
 EXAMPLES. 
 
 Multiply 120 16 10 02. i$dwt. i J gr. with 4, 
 
 OPER- 
 
OPERATION. 
 
 fe 02. dwt. gr. 
 1 20 10 13 17 multiplicand 
 4 multiplier 
 
 483 6 14 20 froduft required. 
 
 HERE we begin with 4X17=^68 ; then 68-^-24= 
 2, and 20 remaining, which write in its proper place ; 
 then 4X13=52, to which add 2, the quotient juft 
 found, and thefum will be 54; then 54-7-202, and 
 14 remaining, which write in its proper place ; then 
 4X 10=40, to which add the laft quotient 2, and the 
 fumis42; now42-r-i2=rr3, and 6 remaining, which 
 write in its proper place* Laftly, 4X120=480, to 
 which add 3, the laft found quotient, and the fum is 
 483. Hence we find the whole product to be 483 
 pounds, 6 ounces, 14 pennyweights, and 20 grains. 
 
 Multiply \<r]doL ijcfs^ with 6. 
 
 OPERATION. ** 
 
 doL cts. 
 
 127 17 
 
 6 
 
 $: 
 
 10 
 
 o ' 
 
 13 42 
 
 IO 
 
 4 
 
 763 2 froduft. 
 
 yd. qr. na. 
 423 
 6 
 
 5 24 48 40 frod. 28 o 
 
 ellFlem. 
 
ellFL qr. na. ellEng. qr. na, ell Ft. qr, tfa. 
 17 2 i 10 4 2 13 5 3 
 
 7 12 8 
 
 124 o 3 130 40 in 4 Qprod, 
 
 deg. le* tn.fur. p. f. in. be. 
 
 12 10 2 5 10 10 I 2 
 
 4 
 50 3 i 5 2 7 6 zprcduff* 
 
 Note. You may refolve your multiplier into feveral 
 parts, as in Short Divifeon, and if thofe parts when 
 multiplied together, do not exaftly make the gi*uen 
 multiplier y add as many times the multiplicand to 
 the produft, as the product of the f aid parts fall 
 Jhort of the given multiplier ; as in thefe : 
 
 Find the produdt of 127 doL ipc/j-.XiS- 
 HERE the parts of the multiplier are 3 and j. 
 Therefore, | f/ 7 ' 
 
 3 
 
 33 1 57 
 
 ; doi. 
 
 1907 85 (becaufe 3XS=5)= 
 
 Required the pro dud of iy] doL 87 / 
 
 I Lee 
 
( 66 ) 
 
 et the parts be 3 and 7. Therefore, 
 
 dol. cts. 
 197 87 
 3 
 
 593 6 1 
 7 
 doL cfs. 
 
 4155 27=197 87X21. 
 add 2 times iyjjol. Sjcts. or 395 74 
 
 4551 
 
 What is the product of 228) 6 oz. isdwt. iigr. 
 
 X32 ? 
 Anfwcr. 72i]b 4oz . 16 dwt. 
 
 What is the product of i^yd. 3 qr. ina.y(^? 
 Anjw er. 666 yd. 
 
 SECT. II. 
 
 DIVISION of COMPOUNDS. 
 
 DIVISION being directly the reverfe of Multipli 
 cation, needs no other explanation than the follow 
 ing examples ; only obferve, that when any denomi 
 nation is not exactly meafured by the divifor, the re 
 mainder mufl be reduced to the next inferiour deno 
 mination, and added to its then perform thedivifion. 
 
 EXAMPLES. 
 
 lb oz. dwt.gr. dol. cts. dol. cts. 
 
 2V?75 ii 13 14 4)347 12 7)784 49 
 
 187 ii 1 6 1.9 86 78 112 *] quo. 
 
 k. 
 
( 67 ) 
 
 o le. m.fur. cb. yd. f. in. 
 4U7 14 2 6 12 5 i 7 
 
 . -4-6527 ^/. 5o<rAr. and 527^7. 
 
 Likewife, loidol. 50 //.-:- 5=20 </07. 30 cts. (be 
 caufe 6X5X5=150^=3165-7-150 
 
 Miscellaneous ^uejtions for the Learn 
 er s Pratt ice. 
 
 SI R Ifaac Newton was born in the year 1642, and 
 died in 1726 : What was his age whence died r 
 
 There are two numbers, the greater 96, and the 
 lefs 45 : What is their funi and difference ? 
 
 To find a number fuch, that 426 taken from it, 
 will leave 127 remainder. 
 
 A certain number of merchants in trade, gained 
 19140 dollars, which being equally divided, a fliare 
 was found to be 4785 dollars : How many merchants 
 were there in that trade ? 
 
 What is the quotient of 3276 divided by 3, and by9 f 
 
 What number is the divifor of 1530320, when the 
 quotient is 470 ? 
 
 What is the coft of 51 yards of broadcloth, at 
 locts. per yard ? 
 
 CHAP, 
 

 CHAP. X. 
 
 REDUCTION.. 
 
 BYRedu&ion, numbers compoled of different der 
 nominations are brought into one, by unfolding 
 the feveral denomination's by the parts that compofc 
 them. Or, from any number of homologous parts, 
 to difcover the number of certain heterogeneous, or 
 unlike denominations. The former is called Reduc 
 tion by Multiplication, and the latter Reduction by 
 pivifion.' - Reduction by Multiplication has the 
 following general 
 
 RULE. 
 
 % 
 
 BEGIN at the greateft denomination mentioned, 
 multiplying it with as many units as one of this de 
 nomination contains units of the next inferiour de- 
 denomination ; and to the product add the numbers 
 in the lefs denomination ; then multiply this fum as 
 before, add as above, and fo on (multiplying with as 
 many units as it takes thofe of the next lefs denomi 
 nation to make one of the prefent), until you have 
 reduced the given parts to the denomination required. 
 
 EXAMPLES. 
 
 Required the number of cents equal to IGCO dollar, 
 OPERATION. 
 
 JOOO 
 
 of cent 3 in a dollar. 
 iooooo~tumfar of cents required. 
 
 Reduce 
 
Jlediice 1057 dol. 90 as. into ctfnts 
 OPERATION, 
 
 cts. 
 90 
 
 ICO 
 
 ^r.umber *f cents required. 
 
 BUT to reduce the monies of foreign nations, to 
 thofe of the United States, confult the following 
 
 TABLE. 
 
 dol. cts. 
 
 Pound Sterling of Great-Britain^^. 44 
 Livre Tcitrnois of Fran'ce 1 8-J 
 
 Guilder of the United Netherlands 39 
 Mark Banco of Hamburgh 33-^- 
 
 Rix Dollar of Denmark I 
 
 Rix Dollar of Sweden i 
 Real Plate of Spain 10 
 
 Milree of Portugal I 24 
 
 Pound Sterling of Ireland 4 10 
 
 Tale of China I 4? 
 
 Pagoda of India I 94.^ 
 ##/><? 0/ Bengal 5 54. 
 
 Mexican Dollar j 
 
 Crown of France I i j 
 
 Crewn of England I 1 1 
 
 M6te. The go Id coins of France, England) Spain, and 
 Portugal^ are valued at 89 ^#/,f per pennyweight. 
 
 In 
 
( 7Q ) 
 
 In 127 pounds fterling of Great-Britain, how 
 many cents ? 
 
 Here multiply the pounds with 444. 
 127 
 444 
 
 508 
 508 
 508 
 
 56388 -the anfuser. 
 
 In 274 livtes tournois of France, how many cents? 
 
 Multiply with 18, and add half the multiplicand 
 to that produ6h 
 
 274 
 18 
 
 2192 
 
 274 
 
 4932 
 *37 
 
 5069 the an fiver. 
 In 540 marks banco of Hamburg : how many cents ?' 
 
 Multiply 
 
Multiply with 33, and add on$ third of the oiulti 
 plicand to that produft. 
 
 540 
 
 33 
 
 In 424 rupees of Bengal : how many cents ? 
 
 Multiply with 55, and proceed as in the livres 
 tournois of France. 
 
 424 
 55 
 
 2I2O 
 2 1 2O 
 
 23320 
 112 
 
 23532 the anfwer. 
 
 Note. In reducing the following /peeies of money U 
 cent s ^ take the following methods. 
 
 For the Guilders of the United Netherlands > multiply 
 
 with 39 
 
 Real Plate of Spain 10 
 
 Milree of Portugal t 1 24 
 
 Pound Sterling of Ireland '410 
 
fale of China - 148 
 
 Pagoda of Indit 1 94 
 
 Crown of Francs in 
 
 Crown of England in 
 
 In 127 Jbj how many ounces, pennyweights and 
 grains ? 
 
 127 
 i i-^Jiumber of ounces in i found 
 
 of ounces in 127 founds 
 of fenny weights in i ounce 
 
 of fennyw eights in \ 27 founds 
 of grains in i fenny weight 
 
 121920 
 60960 
 
 of grains in 127 founds. 
 
 16. oz. dwt, gr. 
 In 1 2 8 12 4 how many grains ? 
 
 12 
 
 2O 
 
 3052=152x204-12 
 24 
 
 I22I2 
 6104 
 
 73252=3052x24 +4~nuwfor of grains req. 
 
 In 
 
( 73 ) 
 
 In 333 milrees of Portugal : how many cents ? 
 Anfwer. 41292. 
 
 In 555 tales of China : how many cents ? 
 Anfwer. 82140. 
 
 REDUCflON by DIVISION. 
 
 THIS method is dire&ly reverfe of the former \ for 
 where we before multiplied, here we muft divide 
 with the fame number - 9 and therefore admits of the 
 following 
 
 RULE. 
 
 DIVIDE the numbers in each denomination, by the 
 number of units that make one of the next fuperiour 
 denomination ; and the quotients refulting, will be the 
 numbers in the feveral denominations required, 
 
 EXAMPLES. 
 
 In 57200 cents : how many dollars ? 
 1(00^572(00 
 
 Therefore 572 dollars is the anfwer. 
 
 In 73252 grains Avoirdupois : how many penny 
 weights, ounces, and pounds ? 
 
 24^73252 203052 
 
 ^ 
 
 12 152 .. I2rem. 
 125 
 1 20 12.8 rem. 
 
 48 * 
 
 K Therefore 
 
( 74 ) 
 
 Therefore in 73252 grains, there are 3052 penny 
 weights, 152 ounces, or 12 pounds. 
 
 Note, tfhejeveral remainders are of the Jams n'ame 
 of their dividends. 
 
 In 41292 cents: how many milrees of Portugal? 
 41292-7-12411:333, the anjwer. 
 
 In 82140 cents : how many tales of China ? 
 Anjwer. 555. 
 
 In 5^388 cents: how many pounds fterling of 
 England ? 
 
 Anjwer. 127. 
 
 Note. In reducing cents into livres tournois of France, 
 you muft multiply with 2, and divide that pvodufl 
 
 by 37 < The mark banco of Hamburg, multiply 
 
 with 3, and divide that fro du ft by ioo.> - The 
 rupee of Bengal, multiply with 2, and divide bym. 
 
 In 752 nails : how many yards ? 
 Anjw er. 47 yards , 
 
 In 15840 barley cornc : how many miles ? 
 Anfwer. 3 miles. 
 
 In 469 gallons : how many hogfheads ? 
 Anfwer. ^hbd. 3% gal. 
 
 Mifcellaneous 
 
 THE comet of 1680, at its greateft diftance from 
 the fun, was 11184768000 miles: now fup- 
 pole a body had been projected from the fun, with a 
 degree of fwiftnefs equal to that of a cannon ball, 
 
 which 
 
( 75 ) 
 
 which is at the rate of 480 miles per hour : in what 
 time would this body reach the aforefaid comet ; al 
 lowing the year to confift of 365 days ? 
 Anjwer. 2660 years. 
 
 How many times will a fhip of 97 feet 6 inches 
 long, fail her length, in. the diftance of 1 2800 leagues 
 and 10 yards; 
 
 Anfwer. 2079408. 
 
 A MERCHANT bought 4 tuns, 15 hundreds, and 24 
 pounds of fugar, and ordered it to be put up into 
 parcels of 24 pounds, of 20, of 16, of 1 2, of 8, of 4, 
 of 2, and of each a like number. How many parcels 
 will be made of the fugar ? 
 
 Anfwer. 1 24. 
 
 A GENTLEMAN had 15 dollars to pay among his 
 labourers to every boy he gave 10 cents to every 
 woman 20 cents, and to every man 45 cents : the 
 number of men, women and boys was the fame, I 
 demand the number of each fort ? 
 
 Anfwer. 20. 
 
 THERE are five tooth wheels placed in fuch order, 
 that their teeth play direclly into each other : the firft 
 wheel contains 500 teeth thefecond75o the third 
 1500 the fourth 2000, and the fifth 3000: how 
 m^ny times will the fifth wheel turn in 100 turns 
 of the firft ? 
 
 Anfwsr. 600, 
 
 THE velocity of light being at the rate of 10000000 
 miles per minute, takes up 6 years, 32 days, 5 hours, 
 and 20 minutes in coming from the neareft fixed ftar 
 to the earth : what is the diftance of that ftar ? 
 
 Anfwsr* 32000000000000. 
 
 PART, 
 
PART JL 
 
 CONTAINING tHE DOCTRJNE OF 
 VULGAR FRACTIONS, 
 
 CHAP. I. 
 
 DEFINITIONS and ILLUSTRATIONS. 
 
 A FRACTION is a broken quantity, or the 
 parts of an unit, which are exprefTed like quan 
 tities in divifion ; to wit, by writing two quantities, 
 one above and the other below a fmall line $ 
 
 thus \l mmerator prl'X?,,! 
 
 3 \ 4 denominator or divifor $4 4 
 which rs three times the quotient of unity divided by 
 4: therefore in all Vulgar Fractions, unity is divid 
 ed into fuch parts, as are exprefied by the denomina 
 tor ; that is, the denominator exprefles what kind of 
 parts the unit is divided into, and the numerator 
 the number of thofe parts. 
 
 HENCE it follows, that all Vulgar Fra&ions what- 
 foever, reprefent tfye quotients of quantities, which 
 are to unity, as the numerator to the denominator -, 
 thus, if the fraction be ^, it will be 4 : i : : 3 : 4 , and 
 fo on for others. 
 
 ALL Vulgar Fractions whatfoever, fall under the 
 five following forms, viz. proper, improper, fingle, 
 compounded, and mixed, 
 
 A 
 
( 77 ) 
 
 A PROPER fraction, is when the numerator is lefs 
 then the denominator : thus -J-, -, and X 7 T > are proper 
 factions. 
 
 AN improper fraction, is when the numerator is 
 greater than the denominator : thus , -f , and -^, are 
 improper fractions. 
 
 A SINGLE fradtion, is a (imple exprefilon for the 
 parts of an unit : thus 4-, -'-> and-*-, are fmgle fractions. 
 
 A COMPOUND fraction, is a fraction of a fraction : 
 thus, ~ of -^ and.y of ^ of 4? are compound fractions. 
 
 WHSN whole numbers are joined or connected with 
 fractions, they .are fometimes called mixed numbers ; 
 as ioi, and 15 --. 
 
 A MIXED fraction, is when either or both the nume 
 rator and denominator, is a mixed number : 
 r j 2! 17 T 
 
 thus, | i and -~7j, are. mixed fractions. 
 
 ANY whole number may be expreffed in the form 
 of a Vulgar Fraction, by writing unity, or I under it: 
 
 thus, i2Oz= - and Cj2=- &c. 
 i i 
 
 THE commpn meafure of two numbers, j^any 
 number that will meafure both without a remainder : 
 thus, 3 is the common meafure of 9 and 12 ; becaufe 
 it meafures 9 by 3, and 1 2 by 4. 
 
 THE greateft common meafure of two numbers, is 
 the greateft number that will meafure both without 
 a remainder : thus, 7 is the greateft common meafure 
 of 21 and 49 ; becaufe no number greater than 7 can 
 meafure 21 and 49, without a remainder. 
 
 ANY number that can be meafured by feveral other 
 numbers, the number meafured, is called their com 
 mon multiple : thus, 24 is a common multiple of 4 
 and 6, for 2X1224, 4X61124, and 6X424 : the 
 leaft number that can be meafured in this manner, is 
 
 called 
 
( 78 ) 
 
 callecl the lead common multiple: thus, 12 is the 
 kail common multiple of 4 and 6 ; becaufe no num 
 ber lefsthan 12, can be divided by 4 and 6, with 
 out a remainder. 
 
 A PRIME number is that, which is meafured only 
 by unity: as 5, 7, n, 19, &c. 
 
 NUMBERS prime to each other are fuch, as no num 
 ber except unity will meafure both without a remain 
 der : thus, 9 and 4 are numbers prime to each other ; 
 for although 2 will meafure 4 without a remainder, 
 yet it cannot divide 9 without a remainder: 3 may 
 meafure 9, but it cannot meafure 4 : therefore, &c. 
 
 A COMPOSED number is that, which fome certain 
 number meafures : thus, 6, 8 and 12, are compofed 
 numbers; for jx 21 ^^, 4X28, and 2x6=12. 
 
 CHAP. II. 
 
 REDUCTION of VULGAR FRACTIONS. 
 
 REDUCTION of Vulgar Fractions, is the chang 
 ing of one fraction into another of equivalent 
 value ; and thereby fitting them for the purpofe of 
 Addition, Subtradlion, &;c. 
 
 THE whole bufinefs of Reduction, is comprifed in 
 the following Problems. 
 
 PROBLEM I. 
 
 To find the lea/} common multiple of fever al numbers. 
 
 RULE. 
 
 1. RANGE the numbers in a direct line. 
 
 2. FIND what number will divide two or more of 
 them without a remainder; by \^iich divide them, 
 
 and 
 
( 79 ) 
 
 and fet their quotients together with the undivided 
 numbers, in a line beneath. 
 
 3. DIVIDE this line in the fame manner as the firfl $ 
 and fo on, from line to line, until no number, except 
 unity will divide two of them without a remainder ; 
 then the continued product of all the divifors, and 
 the lad quotients, will be the Jeafl common multi 
 ple required. 
 
 EXAMPLES. 
 Find the leail common multiple of 4, 8, and 12, 
 
 OPERATION. 
 4U 8 12 
 
 WHENCE, 4X*X 2 Xj 24, the leafl common 
 multiple required, ij 
 
 Find the leafl common multiple of 2, 4, 6, 7 and 20. 
 OPERATION. 
 
 2 4,6 7 20 
 i ' 2 3 7 10 
 
 1 i 375 
 WHENCE, 2x2x3x7X5=420, the leail common 
 multiple required. 
 
 PROBLEM II. 
 
 fo find the greatefl common meafure of two or more 
 
 quantities* 
 
 R U L E. 
 
 i. FIND the greatefl common meafure- of any two 
 of the quantities, by dividing the greater by the lefs, 
 and the divifor by the remainder ; and fo on, divid 
 ing the laft divifor, by the 1 aft remainder, till noth- 
 
( So ) 
 
 ing remains , and the laft divifor made ufe of, will be 
 the greateft common meafure of thefe two quantities. 
 2. FIND the greateft common meafure of any one 
 of the other quantities, and the common meafure laft 
 found ; and fo on, from one number to another, thro' 
 the whole j and the laft common meafure thus found, 
 will be the greateft common meafure required. 
 
 EXAMPLES. 
 Find the greateft common meafure of 12 and 
 
 OPERATION. 
 
 3 
 
 J 12 
 
 HENCE, 3 is the greateft common meafure required* 
 Find the greateft common meafure of 12, 18, 26, 36. 
 
 OPERATION. 
 
 F : rft find the greateft common meafure of 12 
 and i 8. 
 
 thus, 
 
 HENCE, the greateft common meafure of 12 and 
 18 is 6. 
 
 Again, find the greateft common meafure of 6 
 and 26, 
 
 thus 
 
6(4 
 
 Therefore the greateft common meafure is 2. 
 
 Laftly, find the greateft common meafure of 2 
 and 36 : 
 
 Confequently the greateft common meafure of 1 2, 
 1 8, 26, and 36, is 2; which was to be done. 
 
 PROBLEM III. 
 
 'fo abbreviate, or reduce a Vulgar Frattion to its leajt 
 or moftjimple terms, 
 
 RULE. 
 
 4f 
 
 FIND the greateft common meafure of the mime-* 
 rator and denominator, by the laft problem -, then di 
 vide them by their greateft common meafure, and the 
 refult will be the terms of the fraction required. Or, 
 
 DIVIDE both the numerator and denominator of 
 the given fraction, by fuch a number, as will divide 
 them without a remainder, and the refulting fraction 
 in the fame manner 5 and fo on, till no number ex 
 cept unity, will divide both without a remainder -, 
 and you will have the fraction required. 
 
 EXAMPLES. 
 
 Reduce -^ to its moft fimpla terms, 
 
 L THE 
 
( 82 _) 
 
 THE grcateft common meafure of 64 and 384, is 
 $4. Therefore 64-7-641, and 384-7-646 j con- 
 
 fequently -TT- g> the fraction required. 
 
 Or, Jli=l , and -1* -i; /**/*/ */ fo/w. 
 
 '384-8 4B 4&->8 6' 
 
 Find the value of ~, in its moft fimplc terms, 
 
 Thus. - ^"^iz, the fraflion required. 
 45-^5 9 
 
 Reduce 121, to its moft fimple terms. /Inf. - 
 480 5 
 
 PROBLEM IV. 
 
 ff"0 wn/^ 1 a mixed number, in the form of a Vulgar 
 Fraftion. 
 
 RULE. 
 
 MULTIPLY the whole number with the denomi 
 nator of the fraction, and to the product add its nu 
 merator ; then under this, write the faid denomina 
 tor j and you will have the fraction required, 
 
 EXAMPLES. 
 
 Write 4J, in the form of a Vulgar Fraction. 
 Thus, 4X28, and 8 + 1=9 /^ numerator-, 
 
 Whence ~ is the fraction required. 
 
 6 _ 126 , n i 20 40X100+20 
 
 a T -g. . i zz" > ana 40. . .. 
 
 10 100 100 
 
 100 2C 2e 20 
 
 PROB- 
 
PROBLEM V. 
 
 fo find the 'value of an improper fr a ftion. 
 
 RULE. 
 
 DIVIDE the numerator of the given fradtion by 
 the denominator 5 and the quotient will be the 
 value fought. 
 
 EXAMPLES. 
 Find the value of 2L. 
 
 12 
 
 20 417 
 
 -M 00=40 - j =20~, 
 j;oo ao 
 
 PROBLEM VI. 
 
 Ti write a whole number in the form of & Vulgar 
 Fraction) whoje denominator is given. 
 
 RULE. 
 
 MULTIPLY the whole number with the given de- 
 aominator j and under this produft write the faid de 
 nominator; and you will have the fraction required. 
 
 EXAMPLES. 
 
 Reduce 40 to its equivalent Vulgar Fraftion^ 
 whofe denominator is 10. 
 
 Thus, 
 Whence, 42 is thefraRion required* 
 
 Change 304 into its equivalent Vulgar Fraftion, 
 having 5 for its denominator, 
 
 Thus, 
 
Thus, = the fraction required. 
 
 Change 3476 into its equvialent Vulgar Fraction, 
 having 12 for its denominator. 
 
 Thus, n the f ration required. 
 
 PROBLEM VII. 
 
 f o alter or change a Vulgar Fraction into another of 
 equivalent value -, wbofe denominator is given. 
 
 RULE. 
 
 MULTIPLY the given numerator with the prbpofed 
 denominator ; the product divided by the denomi 
 nator of the given fradtion, will give a new numera 
 tor ; under which write the propofed denominator ; 
 and you will have the fraction required. 
 
 EXAMPLES. 
 
 Change into its equivalent Vulgar Fraction, 
 v/hofe denominator is 20. 
 
 Thus, - - =10 the new numerator. 
 
 Therefore, is the fraction required. 
 
 Change ~* into its equivalent Vulgar Fraction, 
 having 40 for its denominator. 
 
 Thus, "* ^-izjo : therefore -~ is the fraftion req. 
 
 
 Change ~ into its equivalent Vulgar Fraction, 
 
 whofe denominator is 24. 
 
 Thus, 
 
req. 
 
 20 2.0 
 
 PROBLEM VIII. 
 
 fo change a Vulgar Fraction into another of equiva- 
 lent value, whofe numerator is given. 
 
 RULE. 
 
 MULTIPLY the given denominator with the pro- 
 pofed numerator -, and the produft divided by the 
 numerator of the given fraftion, will give a new deno 
 minator 5 over which write the propofed numerator ; 
 and you will have the fra&ion required. 
 
 EXAMPLES. 
 
 Change into its equivalent Vulgar Fraction, 
 whofe numerator is 20. 
 Thus, ^ 40 : therefore, , is the fraftion req. 
 
 7 ... .$ 
 
 Change into its equivalent Vulgar Fra&ion, 
 
 whofe numerator is 8. 
 
 Thus, =IO | . therefore, p w thefraffion req. 
 
 1 IO T 
 
 Change - into its equivalent Vulgar Fraftion, 
 whofe numerator is 37. Anf. 2~~ 
 
 PROBLEM IX. 
 
 To reduce a mixed fraftion to fimfle terms. 
 
 RULE. 
 
 ^ i. REDUCE the numerator and denominator of the 
 given fraftion to improper fradions. 
 
 i, MULTIPLY 
 
( 86 ) 
 
 2. MULTIPLY the numerator of the denominator, 
 into the denominator of the numerator, for a new de 
 nominate* j and multiply the numerator of the nu* 
 merator, into the denominator of the denominator, 
 for a new numerator ; and you will have the terms of 
 the fraction required. 
 
 EXAMPLES. 
 
 4. 
 
 Reduce 2 to fimple terms. 
 
 ?T 
 A. * * ? 
 
 Firft, If (by reducing to impr. fract.) JL iz 
 
 fi 
 
 ~-$Q the f raft ion required, 
 
 OQ 
 
 O 1 
 
 Reduce ~J to (imple terms. 
 10 
 
 Thus,!!-^-^!^!?. andi|i=:^r:^. 
 10 10 2X10 20 16 16 48 
 
 10 
 
 _20O^ 3OO __3OO _ 
 
 PROBLEM ,X. 
 
 3*o reduce a compound fraffion tq a Jimfle one of c~ 
 
 qua! value* 
 
 RULE. 
 
 i. REDUCE all fuch parts of the given fraction as 
 are whole numbers, mixed numbers, and mixed frac 
 tions; according to the foregoing rules j that is, whole 
 and mixed numbers muft be reduced to improper 
 fractions^ and mixed fractions to fimple terms. 
 
 a. Mui,* 
 
2. MULTIPLY all the numerators continually to 
 gether, for a new numerator, and all the denomina 
 tors continually together, for a new denominator ; 
 and the former produd written above the latter* will 
 give the fra&ion required. 
 
 Note. Any number that is found among the nume 
 rators and denominators, may beftruck out cfboth, 
 
 EXAMPLES. 
 
 Reduce 5 of lof-^, to a fimple fraftion. 
 p 3 4.6 
 
 Fhus, 2X3X *=(by ftriking out the 3)^!= ~ the 
 
 3X4X6 J ' 4 X6 24 
 
 raftion required. 
 
 O *"7 
 
 Reduce of /- , to a fimple fraftion-* 
 4 9r 
 
 ~ ; 
 1 9 
 
 f-X 19=76 /^<? ^w denominator : therefore 
 
 ,L. ~ ; then 3 x 1 4=4^ the new numerator, and 
 9r 1 9 
 
 76 
 raftion required. 
 
 ^flofS^Ioffoflof^i^. 
 264*- 269 i 108 
 
 toff of -E=! of 5 of l! = l^><ii- = JL. 
 3 5 V 3 5 88 3X5X^8 1320 
 
 PROBLEM XL 
 
 f<? r^r^ feveral fractions of different denominators, 
 to equivalent fraffionsybaving a common denominator. 
 
 RULE. 
 i, REDUCE all fraftions to fimple tqrms. 
 
( 38 ) 
 
 2. MULTIPLY each numerator into all the deno 
 minators except its own, for new numerators. 
 
 3. MULTIPLY all the denominators continually to- 1 
 gether, for a new and common denominator, and this i 
 written under the feveral new numerators, will give 1 
 the fractions required. 
 
 EXAMPLES. 
 
 Reduce -* -, and ^, to their equivalent fractions, 
 having a common denominator. 
 
 f I X4X6=24 the new numerator for \. 
 
 Firft, < 2x3X^=36 the new numerator for |. 
 
 (. 5X2X4=40 the new numerator for 4. 
 
 Then 2x4x6=48 the new and common denominator, 
 
 Hence -A ~, and ~, are the fractions required. 
 40 4^ 4* 
 
 ~, -, and -, reduced to a common denominator-^ ^ \ 
 9 7X4X9* 
 
 1X7X4^144 and 
 
 ^^'' 
 
 7X4X9 7X4X9^252^252' 252" 
 
 - and - of ~of ~, reduced to a common denominators* 
 3 3 5 7r 
 
 22, and 3X4Q8 = .i3Q > 
 
 224. 
 3X13^ 3X1320 3960 3960 
 
 3? r) J, and - of 4, reduted to a common denominators 
 4 3 , 
 
 720 189 , 448 
 ' , _|, and S-, 
 
 336 336 336 
 
 PROB- 
 
PROBLEM XII. 
 
 To reduce feveral fractions of different denominators, to 
 gthers of equivalent value, having the leaft foj/ible 
 common denominator* 
 
 R U L E. 
 
 1. REDUCE all the fractions to fimple terms. 
 
 2. FIND the leaft common multiple of all the de 
 nominators j and you will have the leaft common de 
 nominator required. 
 
 3. DIVIDE the denominator thus found by the de 
 nominator of each fraction, and multiply the quo 
 tient with its numerator, and you will have new nu 
 merators, under which write the common denomina 
 tor ; and you will have the fractions required. 
 
 EXAMPLES. 
 
 Reduce ~, -, and - to equivalent fractions, that 
 84 2 
 
 flull have the leaft poflible common denominator. 
 Firft, the leaft common multiple of 8, 4, and 2, is 3 ,: 
 Then, 8 8 x i=J> the new numerator for ^ 
 And, 8-f-4X3 == ^ *be new numerator for 
 Alfo, 8-7-2X I:== 4> the new numerator for ~. 
 Hence the fractions required are 4* TJ an <i ? 
 
 Reduce i, -, ~, and ~, to equivalent fractions, 
 
 I 4 5 " 
 
 having the leaft poflible common denominator. 
 
 . Firft, the leaft common multiple of the 3, 4, 5, 
 and 6, is 60. 
 Then, 60-^3X1=20, the new numerator for \- 
 
 >nd, 6o-r-4Xj :: =4S, tie new numerator for ~ 
 
 M Alfo. 
 
( 9 ) 
 
 A!ib,~6o-f- 5 X4~4-S tie new numerator for -J 
 Laftly, 6o-~6x5~50 the new numerator for 
 
 Hence, ~, , -, and , <w the fraftions red. 
 
 OO DO OO r x 6O 
 
 P-RQBLEM XIII. 
 
 fo change the fraction of one denomination to thefraftion 
 of a greater one> retaining its fame value. 
 
 RULE. 
 
 CHANGE the given fraction into a compound one, 
 by writing its value in all the intermediate denomi 
 nations up to the one wherein the value of the frac 
 tion is to be ex;)refied ; and the value of this com 
 pound fraftion, will be the fraction required, 
 
 EXAMPLES. 
 Change - of a nail, to the fraction of an ell Eng, 
 
 Firft, . of a nail~=.~ of a quarter, and -rr:- of an ell : 
 3 3 4 5r 
 
 Therefore, - of a nail^z- of '- of ---> the fraction req. 
 3 4 5* oo 
 
 1 pennyweights, reduced to the fraction of a pound 
 =r of = . 3 grains, reduced to fraction of an 
 
 2fr IX 240 
 
 ounce=: of = |. of a cent, reduced to the 
 24 ao 480 3 
 
 fraction of am ilree of Portugal- of- - = - ; 10 
 
 3 124 j/a 
 
 cents, reduced to the fraction of a pound flerling of 
 
 Ireland^:- = J - 4 ~ of a cent, reduced to the frac- 
 410 41 8 
 
 tion 
 
*7 T *7 
 
 rion of a dollar-^ of =o^-. i drachm Avoir- 
 
 o i oo 8 oo 
 
 i r i ~ x f r f 
 
 is=--i of -7 of of of a tun. 
 
 10 10 112 2O 
 
 PROBLEM XIV. 
 
 2V change the fraRion of one denomination to the frac 
 tion of a lefs one, retaining itsjame 'value. 
 
 RULE. 
 
 MULTIPLY the numerator of the given fra&ion 
 into all the intermediate denominations down to the 
 one wherein the value of the given fradtion is to be 
 exprefled, and under this product, write the given de 
 nominator, and you will have the fra&ion required. 
 
 EXAMPLES. 
 
 Reduce of an ell Eng. to the fraction of a nail. 
 70 
 
 Thus, 1X5X4=20 the numerator 
 Therefore, =~/V thefrattion required. 
 
 Reduce ^- of a lb Troy to the fraft. of a grain. 
 
 Thus, ::= . f btffafKm 
 
 II2O 1120 
 
 of a pound Troy, reduced to the fraction of a 
 
 . i aXiaXao 480 8 c , 
 pennyweight -- - ; - of an hun- 
 1240 1240 17920 
 
 dred weight, reduced to the fra&ion of an ounce= 
 
 8XH2- 
 
J_ of a milree of Portugal, 
 
 17910 J7920 372 
 
 reduced to the fra&ion of a cent= 1^ =-. 
 
 $7* 57*. 5 
 
 PROBLEM XV. 
 
 To find the value of a Vulgar Fr a ft ion in known farts 
 of the integer. 
 
 RULE. 
 
 MULTIPLY the numerator of the given fra&ion 
 with the parts in the next inferiour denomination, and 
 divide the produA by the denominator $ then if there 
 ^e any- remainder, multiply it with the parts in the 
 next inferiour denomination, and divide by the former 
 divifor, and fo on, and the feveral quotients refulting 
 v/ill exhibit the value fought, 
 
 EXAMPLES. 
 
 Find the value of of an ounce Trov, 
 24 
 
 OPERATION, 
 
 5 
 
 
 
 24 \ 100(4 pennyweights* 
 ' 96 
 
 4 
 24 
 
 Therefore* 
 
( 93 ) 
 
 * 
 
 Therefore, -* of an ounce=4 dwt. 4j?r. /J 
 M 
 
 fought. 
 
 Find tfic value of - of an ounce Troy. 
 
 OPERATION. 
 5 
 
 20 
 
 7\ioo( 
 / 14 irtm. 
 
 24 
 
 7)48( 
 f 6 6rtm. 
 
 Therefore i^dwt. 6 6 r gr. is the valug fought. 
 
 Find the value of - of an hundred weight, 
 7 
 
 OPERATION. 
 6 
 4 
 
 3 
 28 
 
 Therefore 3qr. ia/^. is tbe value fought. 
 
 Find the value of of a pound fieri, of Ireland 
 4< 
 
 Thus, 
 
( 94 ) 
 
 us> - jo cts. the valuffougbf. 
 4* 
 
 Find the value of of a pagoda of India. 
 Thus, a I94 ^=4^. /fo value fought. 
 
 PROBLEM XVL 
 
 21? reduce the known farts of an integer to their equiva- 
 lent Vulgar Fraction. 
 
 RULE. 
 
 1. REDUCE the given parts to the lead denomina 
 tion mentioned. 
 
 2. REDUCE the integer to the fame denomination ; 
 and the latter written beneath the former, will be the 
 fra&ion required, 
 
 EXAMPLES. 
 Reduce 3 dwt. *]gr. to the fra&ion of a pound. > 
 
 OPERATION. 
 dwt. gr. cz. 
 
 37 12 
 
 24 20 
 
 79 240 
 
 24 
 
 960 
 
 480 
 
 5760 
 Therefore, -^- is thtfraftion required. 
 
 Reduce 
 
C 95 ) 
 
 Reduce 10 cts. to the fra&ion of a potand fterling 
 of Ireland. 
 
 Thus, is tfa fraction required. 
 
 n. reduced to the fraction of a foot T-f^JZ. . 
 
 12 10 
 
 4- p. reduced to the fra<ft. of an acre=~ : rr-- r nr- . 
 
 1 60 480 30 
 
 CHAP. III. 
 
 ADDITION, SUBTRACTION, MUL 
 TIPLICATION, A*D DIVISION OP 
 VULGAR FRACTIONS. 
 
 SECT. L 
 
 Of ADDITION of VULGAR FRACTIONS. 
 
 RULE. 
 
 EPUCE all the fractions to a common de- 
 nominator, by the rule to problem xi of the 
 .aft chapter : thofe- of different denominations to the 
 Tame, by the rules to problem xm or xiv. 
 
 2. ADD all the numerators together for a new nu 
 merator, under which write the common denomina- 
 :or ; and you will have a fraction equal to the fum 
 *e quired. 
 
 EXAMPLES, 
 
 the fum of- | }-- . 
 
 ZT|T5| 
 
 Thus, 
 
( 96 ) 
 
 Thus, -+--)-- =(by reducing to a common derio- 
 
 mirutor) i?+ + = 
 
 24 24 24 24 
 
 2 * I 
 
 Required the fum of 2-H ? +* of 4. 
 
 4 o 
 
 ,24"^M-^ of 4= j+^+|= 
 720 189 
 
 336 36.336~ 336 
 
 Find the fum of - of a grain -^ of an ounce. 
 
 7 7 I I ? 
 
 Firft, -0/<z grain- of of n-* ^/^ ounce , 
 * 4 4 24 20 1920 y 
 
 /ir/; /<? /^^ becomes - 4-*^: -2 - tbt ftim req. 
 1920 ' 7 13440 y 
 
 SECT. II. 
 Of SUBTRACTION of VULGAR FRACTIONS. 
 
 RULE. 
 
 1. PREPARE the fra&ions as in Addition. 
 
 2. SUBTRACT the numerator of one fraction from 
 the numerator of the other, and the refult placed a- 
 bove the common denominator will be the differ 
 ence required. 
 
 EXAMPLES. 
 
 From -take ~. 
 3 4 
 
 Thus, ~~-=s.~-^ ~iH^ J. fa difference re^ f 
 3 4 ia 12 12 ia 
 
 From 
 
4 l r l 
 
 From - take -of -. 
 
 5 3 3 
 
 ~, 4 i c l 4 i 3 6 5_36~5 31 
 
 THUS, -_ - Of r -- -=- -~ - : = - ,* 
 
 difference required. 
 
 3 i 2__io 6 _io8o i8__io8o 18 
 ~ 9 "~"~""" "~"" 
 
 _io62 y 5 51 _44O 204 236 
 
 From - of an ounce take - of a grain 
 
 7 7 
 
 Firft, - of a grain=. of and ounce : Therefore, 
 
 5 3 95 8 9 j.jr 
 
 -. = . ts jfo difference required. 
 
 7 1920 13440 
 
 SECT. III. 
 Of MULTIPLICATION of VULGAR FRACTIONS, 
 
 RULE. 
 
 1. REDUCE all whole and mixed numbers to im 
 proper fractions, mixed fractions to fimple terms, and 
 fractions of different denominations to the fame. 
 
 2. MULTIPLY all the numerators together for ft 
 new numerator, and all the denominators together 
 for a new denominator; and you will have the terms 
 
 "the fra&ion required. 
 
 EXAMPLES. 
 
 I o 
 
 Required the product of ~X ~- 
 
 \5 T" 
 
 3 
 
 'hus, TTT^^ thefroduft required. 
 
 N 64X 
 
( 98 ) 
 
 4 ^4- v 12 IO I2XIO 12O 
 
 ~ X~ (by reduction) X = r^ = - 
 34.4 ' 10 12 10X12 120 
 
 J ~- 
 
 ; 4 "^3 
 
 SECT. IV. 
 
 Of DIVISION of VULGAR FRACTIONS. 
 
 RULE. 
 
 PREPARE the numbers as in Addition, then multi 
 ply the numerator of the divifor into the denomina 
 tor of the dividend, and the numerator of the divi 
 dend into the denominator of the <3ivifor 5 then the 
 latter written above the former, will give the quo 
 tient required. Or, 
 
 INVERT the divifor, that is, write the denomina 
 tor in the place of the numerator, and the numera 
 tor in the place of the denominator ; then proceed as 
 in Multiplication, and the refult will give the quo 
 tient required. 
 
 EXAMPLES. 
 
 Required the quotient of 2 -r-, 
 Thus, 1X4 4-thenumeraton and i writhe denm. 
 
 4 
 Therefore, -=2 is the quotient required. 
 
 414 
 Or, - X""- the fame as before. 
 
 i 22 
 
 2 8 _27 2_2-X2_54 4 4_J>8 4 
 3~ : 27""" 8 X 3~ SX3 ""2 4 ; 7 ' is" 4 X 7" 
 
 28X4 
 
99) 
 
 =II^=4 5 lM^=(by reduction)-* -J = 
 28 8 i '2 2 
 
 4X7 28 i 24 
 
 4X2 8 i i m i f i . i__3Xi-3 
 ' 4 2*3 '-ST'iXS-S- 
 
 Mifcellaneous >ueftions. 
 
 AM AN at hazard won the firft throw 2| dol 
 lars the fecond throw he won as much as he 
 then had in his pocket the third throw he won 4 
 dollars, and the fourth throw he won double of all 
 that he then had, at which time he found that he hacj 
 in all 45 dollars. How many had he at firft. 
 Anfwer. 3 dollars. 
 
 THERE is a certain club, whereof^- are merchants, 
 4- mathematicians, ~ mechanics, and 13 phyficians. 
 How many were there in the whole ? 
 
 Anfwer. 60. 
 
 REQUIRED the difference between three times thir 
 ty-three and a third j and three times three and thir 
 ty and a third. 
 
 Anfwer. 6oJ. 
 
 A MAN who was driving fome flieep to market, 
 was met by another who demanded the number of 
 Iheep in his drove : the drover to evade a direcl: anf- 
 wer replies, that if I had as many more, and half as 
 many more, and 12! fheep, I fhould have 100. 
 What number had he ? 
 
 35- 
 
 PART 
 
 Anfwer. 
 
 \ 
 
PART III. 
 
 CONTAINING THE DOCTRINE OF 
 DECIMAL FRACTIONS. 
 
 C H A P. I. 
 
 
 
 DEFINITIONS and ILLUSTRATIONS. 
 
 AD E C I M A L Fraction is formed from a 
 proper Vulgar Fraction, by dividing the nu 
 merator with cyphers annexed to it, by the denomi 
 nator ; that is, the equivalent Decimal of any Vulgar 
 Fraction is found by multiplying the numerator with 
 10, 100, or 1000, &c, till it be fo increafed, that it 
 rnay be exactly meafured by its denominator j and 
 this quotient will be the decimal required : 
 
 Thus, ~ X i oo ^21122 122--; 25 j and Xio ~ 
 4 44 2 
 
 22 4 44 
 
 which quotients are exprefTed by writing them with 
 a point on the left-hand : Thus, ^=: .25, i = -5> and 
 J = -75 i which are reflectively equal to T Voj ~, and 
 .jVo-s but thefe denominators are always omitted, and 
 the numerators written as above, where the point dif- 
 tinguifhes them from whole numbers : Thus, 2.3=; 
 &c, 
 
 HENCE 
 
HENCE it appears that every Decimal Fraftion, is 
 equal to a Vulgar one, whofe numerator is the deci 
 mal, and the denominator unity, with as many cy 
 phers annexed to it as there are places of figures in 
 the numerator : Thus, .1, .44, and .127, are refpec- 
 tively equal to T V, -rVo* and -r^J-J-- 
 
 THEREFORE it follows, that in decimals, unity is 
 divided in 10, 100, or 1000, &c. equal parts; and the 
 given decimal reprefents the number of thofe parts : 
 Thus, . i = _~ reprefents one tenth part of an unit, 
 .44 reprefents forty-four hundred parts of an unit, 
 &c. Therefore, in decimals, cyphers annexed neither 
 increafe nor diminilh their value ; but cyphers pre 
 fixed, diminifh their value in a ten fold proportions 
 Thus, .440 x \V6=(by ne nature of Divifion) T t *j 
 rz.44 ; but .04 T-O-O~TO- f (Ar) -4> an ^ f n for 
 any other decimal. 
 
 WHENCE it follows, that the farther any diget or 
 numeral figure (lands from the units' place, or deci 
 mal point towards the right-hand, the lefs will be its 
 value, to wit, in a tenfold proportion. Thus in the 
 decimal .mi, the figure next to the decimal point 
 is ^-V> tne fecond is -y-^, the third -j-oVerj and the 
 
 T _.L^_, which is plainly a feries of numbers in ge 
 ometrical proportion, decreafing by the common 
 divifor 10. Again ,oi23zn: T 4-c-f-T-5 1 oo+T^4-^-i an< ^ 
 the like to be underftood of all others. 
 
 HENCE, the notation of decimals, or the valuation 
 of the feveral places from unity downwards, is the 
 fame among themfelves as that of integers or whole 
 numbers *, therefore every figure is to be valued accord 
 ing to the diftance it Hands from unity downwards. 
 
 
 CHAP, 
 
CHAP. II. 
 
 ADDITION, SUBTRACTION, MUL 
 TIPLICATION, AND DIVISION OF 
 DECIMAL FRACTIONS. 
 
 SECT. I. 
 
 Of ADDITION of DECIMALS^ 
 RULE. 
 
 i. TTT 7 R I T E the given decimals in fuch order, 
 VV that thofe places of equal diftance from 
 unity or the decimal point, may Hand directly un 
 der each other. 
 
 a. Find their fum as in whole number?, then dif- 
 tinguifh with a point as many places of figures on 
 the right-hand, as are equal to the greateft number 
 found in any given decimal $ and you will have the 
 fum required. 
 
 EXAMPLES. 
 
 Find the fum of .176 + .1264+. 34+^94 
 
 Thefe numbers being placed according to the rule 
 will Hand 
 
 .i 7 6 
 
 4 
 
 Find the fum of 34. 123+6437, 27+34? 
 
 ^347^34+347^34-1. 
 
 thus, 
 
r 34-123 
 
 thus 6437 ' 27 
 thUS ' 347-2 
 
 347634.1 
 
 354454.040634=1/^1 required. 
 
 Required the fum of 25.1244-12,247 + 24.3485 
 25 124 
 
 457 8 -74 
 
 4992.5595 /#># required. 
 
 SECT. II. 
 
 Of SUBTRACTION -of DECIMALS. 
 
 RULE. 
 
 WRITE down the numbers as in Addition, then 
 fubtraft the lefs from the greater as in whole num- 
 bersj remembering to point off in the remainder as in 
 Addition j and you will have the difference fought. 
 
 EXAMPLES. 
 Required the difference between 12.19, and 8.9 
 
 3 . 2 9 r: dffiren ce required. 
 
 Required 
 
104 ) 
 
 Required the difference between 342.364, and 
 -199.2437 
 
 thus 1342.364 
 
 us >{ 299.243? 
 
 43. i v&yssjffirince required* 
 
 1 999-9999 8: =473- 002 42 > 
 
 2479-377793o.oooo45=i549.377 
 9999.88888888.99991 1 10.8889 
 
 SECT. III. 
 
 Of MULTIPLICATION of DECIMALS. 
 RULE. 
 
 WRITE the numbers and multiply them as in com 
 mon Multiplication ; then diftinguiih with a point 
 as many places of decimals in the product, as are 
 equal to the number in both factors $ and you will 
 have the product required. 
 
 ftfote. If the number of places in theproduff, are kjs 
 than the number of decimal places in both faftors,, 
 you muftfiipply the deficiency by prefixing cyphers. 
 
 THAT the number of decimal places in the prod 
 uct, ought to be equal to the number in both factors, 
 may be thus demonftrated. 
 
 SUPPOSE .34 were to be multiplied with .27 j the 
 product of theie two numbers by common Multipli 
 cation is 918 ; but .34== T 3 5 * S . and .27= T Vg- ; there 
 fore, .34 X .27 = & X -rVo = -1-1-44-5-= (by thena- 
 ture of decimal notation) . 091 8, confiding of as maray 
 places of figures as there were in both factors j and 
 the fame will hold true in any others. ^. E. D. 
 
 E XAM- 
 
EXAMPLES. 
 
 Required the produdt of 2.438 ^ ,005. 
 OPERATION. 
 
 2.438 
 .005 
 
 .o 1 2 1 90 ^=.produft required. 
 Required the produft of 34.38X24,7 
 OPERATION. 
 
 34.38 
 24.7 
 
 24066 
 6876 
 
 required. 
 
 Required the product 
 
 3. 8 402 z: produfi required, 
 Required the produft of 2.7122X3,2121 
 
 64242 
 64242 
 32121 
 
 2*4847 
 64242 
 
IN the multiplication of decimals, where the factors 
 confift of a great number of decimal places, the oper 
 ation becomes very prolix, and befides, a great part 
 of it is entirely ufelefs, fmce that four or five places 
 of decimals in the product, is fufficient for common 
 purpofes. Therefore to abridge the work by obtain 
 ing the product true to any defigned number of 
 places of decimals, you muft obferve the following 
 
 RULE. 
 
 1 . WRITE the multiplier inverted, fo that the units' 
 place mayftand under that figure of the multiplicand, 
 to whofe place the product is to be found true. 
 
 2. IN multiplying with the feveral figures of the 
 multiplier, you muft reject all the figures of the mul 
 tiplicand, that are to the right-hand of the figure 
 you are multiplying with ; placing the firft figure of 
 the feveral products directly under each other, in- 
 creaied by adding I from 5 to 15, 2 from 15 to 25, 
 &c. of the product of the multiplying figure with 
 the proceeding figure of the multiplicand, when you 
 begin to multiply 5 and the fum of all the products 
 will be the product required. 
 
 EXAMPLES. 
 
 Required the product of 3.2121x2.712, to three 
 places of decimals. 
 
 3.2121 
 
0/3.212X2 . 
 
 0/3.21X7, increafed by adding i for 
 0/3.2x1 [the f rod. 0/7x2 
 
 '/3X2 
 
 required* 
 
 Required the product f 3.24211X2.34634, to 
 four places of decimals. 
 
 3.24211 
 436432 
 
 64842=3.2421X2 
 
 9726=3.242X3. 
 
 1 297 =3. 24x4* increafed by adding i far 4X2 
 
 194=3. 2X 6> increafed by adding 2 for 6X4 
 
 ~3X3> increafed by adding i for 3x2 
 
 required. 
 
 Required the produd of 2,13214X2.21 134, to 
 five places of decimals. 
 
 2.13214 
 431122 
 
 426428=2.13114X2 
 
 42643=2.1321X2, increajedby adding ifor 2X4 
 21.32=2.132X1 
 213=2.13x1 
 
 64=2. 1X3, increafed by adding i for 3X3 
 2=2X4 
 
 required. 
 
 Required 
 
Required the product of 27.i7X*9i4> in inte 
 gers only. 
 27.17 
 4191 
 
 27 2 = 27 . 1 X i > increafed by adding I for I x 7 
 244 27 X9> increafed by adding i /0r 9X1 
 increa/ed by adding i /<?r 1X7 
 increa/ed by adding I /0r 4X2 
 
 required. 
 
 SECT. IV. 
 
 Of DIVISION of DECIMALS. 
 
 IN divificn of decimals, it may at firft appear 
 difficult to determine the number of decimal places 
 the quotient muft coniift of ; but this difficulty will 
 vanifh, when we confider that the quotient mufr be 
 fuch a number that when multiplied with the divi- 
 for will produce the .dividend; therefore it follows, 
 that the number of decimal places in the divifor and 
 quotient taken together, muft be equal to the num 
 ber in the dividend, by the nature of Multiplication ; 
 confequently the difference between thofe in the di 
 vifor and dividend, muft be equal to the number in 
 the quotient j which affords the following 
 
 ' R U L E. 
 
 RANGE the numbers and divide them as in com 
 mon Divifion, then point off as many places of de 
 cimals in the quotient, as are equal to the difference. 
 between thofe in the divifor and dividend ; and you 
 will have the quotient required. 
 
 Note 
 
Note i. If there are not fo many places of figures 
 in the quotient, as are equal to the difference be 
 tween thofe in the divifor and dividend, you muft 
 Jupply the defett by prefixing cyphers. 
 
 2. If the places of figures in the dividend, are lefs 
 in number than thofe in the divifor, you muft an 
 nex cyphers to the dividend. 
 
 EXAMPLES. 
 Required the quotient of 849.186 divided by 24.7 
 
 OPERATION. 
 
 24.7 ^849.1 86(34.38 -^quotient required. 
 '741 
 
 1081 
 
 988 
 
 MM^BM 
 
 938 
 
 741 
 
 1976 
 1976 
 
 Note. If the dhifor be 10, or 100, &V. the" quotient 
 may be found by removing the decimal point in the 
 dividend , as many places towards the left-hand 
 as there are cyphers in the divijor : thus, the quo 
 tient of 1000)2737,45 is 2.73745 and .0234-7- 
 
 IOO.000234. 
 
 Required the quotient of .012190-7-2.438 
 
 OPER- 
 
( no ) 
 
 OPERATION. 
 
 2.438 \. 012190(5 
 ) 12190 
 
 VJ 
 
 HERE, the quotient found by divifion is 5 ; but 
 the difference between the decimal places in the di- 
 vifor and dividend are three j therefore .005 is the 
 quotient required. 
 
 Required the quotient of 2-~42. 
 OPERATION. 
 
 42X200000(^04761 &c. quotient required. 
 A68 
 
 320 
 294 
 
 260 
 
 252 
 
 80 
 42 
 
 38 &c. 
 
 Required the quotient of 1 65, 6995001 2964- 
 
 OPER- 
 
( I" ) 
 
 OPERATION. 
 
 52.7438 ^165.6995001 296(3, 141 sw 
 J 1582314 
 
 746810 
 
 2193720 
 2109752 
 
 839681 
 
 3122432 
 2637190 
 
 4852429 
 4746942 
 
 1054876 
 1054876 
 
 o 
 
 HERE, as in Multiplication, the work may be 
 greatly contra&ed, by finding the quotient true to 
 any determinate number of decimal places : The 
 method is as follows. 
 
 RULE. 
 
 i. RANGE the numbers -as in common Divifion. 
 
 a. TAKE the figures of the given divifor, to as 
 many places of decimals as you intend the quotient 
 fhall confift of, for your firft divifor, and find a quo 
 tient figure by comparing this divifor as in common 
 
 Divifion ; 
 
Divifion ; then fubtraft its product with the divifor, 
 from the dividend as ufual, calling the remainder a 
 new dividend. 
 
 3. REJECT the right hand figure of your former 
 divnfor, and call the refult a new divifor ; then find 
 a quotient figure by comparing the new divifor and 
 dividend together, and place it in the former quo 
 tient, fub trading as before j and fo on, making each 
 remainder a new dividend, and rejecting the right- 
 hand figure of the laft divifor for a new one j alfo 
 remembering to add for the figures rejected as in 
 Multiplication. 
 
 Note i . If there are notfo many places of decimals 
 in the divifor, as you intend there foall be in the 
 quotientljupply the defeft by annexing cyphers. 
 
 2. Tou may determine hew many places of whole 
 numbers there will be in the quotient, by confider* 
 ing that the fir ft figure of the quotient y is always 
 of the fame denomination of that figure of the divi* 
 dendy which ft ands direftly over the units' place of 
 the produft of the fir ft quotient figure and divifor. 
 
 EXAMPLES. 
 
 Required the quotient of 10.1 934-7-4. 2, to three 
 places of decimals. 
 
 QPE& 
 
( "3 ) 
 
 required, 
 
 J 8400 
 
 420)1793 
 1680 
 
 42)113 
 84 
 
 4)29 
 28 
 
 i 
 
 Required the quotient of 165.6995001296-7- 
 52.7438, to five places of decimals. 
 
 52.74380 ^165.6995001 296(3. 141 592= quotient req. 
 J 15823140 
 
 52.7438)746810 
 5274JS 
 
 52.743)219372 
 
 2i0975 = 52.743X4> encreafed by add- 
 [* n S 3 f or 4X 8 
 
 52.74)8397 
 5274 
 
 52.7)3123 
 
 2637=5 27 X5> encrtafed by adding 
 
 [*f or 5X4 
 
 52)486 
 
 47352X9, increajed by adding 5 
 
 IO 
 
 3 P Required 
 
( "4 ) 
 
 Required the quotient of 780.516-7-24.3, in inte 
 gers only. 
 
 OPERATION. 
 24\78o.5i6(32n:<7#0//V;tf required. 
 
 ) 73=24X3, increafed by adding 1/^3X3 
 
 5 = 2X2, increafed by adding i for 2X4 
 o 
 
 CHAP. III. 
 
 Of REDUCTION of DECIMALS. 
 
 PROBLEM I. 
 
 To reduce a Vulgar Fraction to its equivalent de 
 cimal. 
 
 RULE., 
 
 ANNEX cyphers to the numerator, and divide by 
 the denominator till nothing remains, and the 
 quotient will be the decimal required. 
 
 EXAMPLES. 
 
 3 
 
 Reduce to its equivalent decimal. 
 20 
 
 Thus, 30 \ 3 .oo( . i 5;=f fo decimal required. 
 
 ) 20 
 
 
 100 
 IOO 
 
 o Reduce 
 
( "5 ) 
 
 i3 
 Reduce to its equivalent decimal. 
 
 20 
 Thus, 20 \ i8.o(.9=/&<? decimal require/, 
 
 I ,Q~ 
 
 ) 180 
 o 
 
 6 
 
 Reduce to its equivalent decimal. 
 
 *5 
 
 Thus, I5\6.o(.4i=^ decimal required. 
 ) 60 
 
 ^ 8 
 Required the equivalent decimal of . 
 
 9 
 
 OPERATION. 
 
 9^8.oooo(.8888 &V. ad infnitum. 
 
 80 
 72 
 
 80 
 72 
 
 8 
 
 HERE, we have what is called a circulating de 
 cimal for the quotient, that is, a continual repetition 
 of the fame figure without any poflibility of ever 
 coming to an end, as is evident from the example, 
 Therefore it follows, that the equivalent decimal of 
 f can never be found in finite ttfrms ; but may be 
 obtained to any degree of exa&nefs you pleafe. 
 
 Note. 
 
( Ilfi ) 
 
 Note. When a vulgar fraftion is annexed to a 
 number of cents, reduce thefraftion to its equiv. 
 lent decimal, and annex it to the tents, and t 
 whole will become a decimal: Thus, tf^cen 
 
 =3775 
 
 PROBLEM II. 
 
 ?o reduce numbers of different denominations to their 
 equivalent decimal. 
 
 R U L Jb?. 
 
 REDUCE the given numbers to their equivalent 
 vulgar fraction, by problem xvi of vulgar fractions, 
 then proceed as in the laft problem. 
 
 EXAMPLES. 
 
 REDUCE 3 qr. 2 na. to their equivalent decimal of 
 a yard. 
 
 Firft, 3 qr. 2 #0.r=~of a yard ; 
 
 Then i6\i4.ooo(.875=c:/fe decimal required. 
 
 120 
 112 
 
 80 
 
 80 
 
 O 
 
 4 b. 39 : ' 10 " 3 reduced to the decimal of a 
 187615 fcfr. 
 
 8 S. reduced to the decimal of the eclipticir.666 
 . ad infinitum. 
 
 o-iV in. reduced to the decimal of a foot^.9 
 
( "7 ) 
 
 p. reduced to the decimal of an acre = .o 333 
 
 Ad infinitum. 
 
 PROBLEM III. 
 
 *To find the value of a decimal in known parts of the 
 integer. 
 
 RULE. 
 
 1. MULTIPLY the given decimal with the parts 
 in the next inferiour denomination, and point off as 
 in common multiplication of decimals ; and the 
 whole numbers will be the value of the given de 
 cimal in that denomination. 
 
 2. MULTIPLY the remaining decimal with the 
 parts in the next mferiour denomination, and point 
 off as before, and fo on, thro all the inferiour deno 
 mination, if need be i and you will have the value 
 fought. 
 
 EXAMPLES. 
 Find the value of .875 of a yard. 
 
 OPERATION. 
 .875 
 4 
 
 3.500 
 
 4 
 
 .. , qr. na. 
 
 a.ooo tbmfore, .875:1:3 2, the value fought. 
 Find the value of .426 of a pound troy. 
 
 OPER- 
 
.426 
 
 12 
 
 iyg ) 
 OPERATION. 
 
 960 
 480 
 
 ' 16 oz. dwt. gt\ 
 
 S-1&Q fare/ore .4*6 $ 2 5.76 
 
 Find the value of .75 ofapound fterling of Great- 
 Britain. 
 
 OPERATION. 
 
 75 
 
 444 
 
 300 
 300 
 300 
 
 . cts. do I. cts. 
 
 333-00 therefore .75-333-3 33- 
 
 Find the value of .37752 of a pound fterling of 
 Great-Britain. 
 
 cts. dol. cts. 
 Thus .3775^X444-167.61888^:1 6761888. 
 
 Note. There never can be more than two places of 
 cents, and where there are other figures annexed, 
 they are the farts of another cent : thus, in the 
 lafl example, the 6761 888m. is 67 cents, and 
 .61888 of mother. A 
 

 
 
 A SUPPLEMENT to PART III, 
 
 ONTAINING THE DOCTRINE OF 
 
 CIRCULATING DECIMALS. 
 
 CHAP. I. 
 
 DEFINITIONS and ILLUSTRATIONS. 
 
 \ CIRCULATING decimal is generated or 
 produced from a vulgar fraction, vvhofe nu 
 merator and denominator are incommenfurable to 
 ?ach other -, and therefore if the numerator with cy 
 phers annexed, be divided by the denominator, there 
 will always be a remainder, or the quotient will run 
 on fempiternally j confequently the true and ade 
 quate decimal of every iuch vulgar fraction, muft 
 confiftofan infinite number of decimal places, which 
 is therefore not aflignable in finite terms, and confe- 
 tjuently the true and complete decimal impofTible. 
 NOTWITHSTANDING the equivalent decimal of 
 
 very vulgar fraction of the kind above defcribed, 
 if actually completed, would then confift of an in 
 finite number of decimal places ; yet from a few of 
 the firft, we obtain fome certain law by which the 
 figures ever after circulate or return again ; and it 
 is for this reafon they are called circulating decimals : 
 the circulating figures are called repetends, of which 
 there are fowr kinds, viz. fmgle, compound, mixed- 
 fi^ngle, and mixed compound. 
 
 A SINGLE repetend is a continual repetition of tl^e 
 fame figure : Thus .666 &c. and .2222 &V. are fm 
 gle repetends, which are rxprcfTed by writing the re- 
 
 pelting 
 
 
56 &c, 
 
 peating figure with a point over it : thus, for .666 
 
 
 
 write .6 for .2222 &V. we write .2; andfo on foi 
 others. 
 
 A COMPOUND repetend is when the fame figures 
 circulate or return alternately : thus .9595 '&c. and 
 .321321 &c. are compound repetends, which areex- 
 prefTed by writing the combination of figures that 
 circulate or return together, with a point over the 
 firft and lad figure : thus, inftead of -959J &c. we 
 
 write .95 for .321321 &c. we write .321 ; and fo 
 on for others. 
 
 A MIXED fingle repetend is when one or more fi 
 gures occur before the repeating ones : .thus .172444 
 &c, and .1942777 &V. are mixed fingle repetends. 
 
 A MIXED compound repetend is when fevcral fi 
 gures (land before thofe that circulate alternately: 
 thus .1724747 &c. and .41972972 &c. are mixed 
 compound repetends, 
 
 THOSE combinations of figures, which circulate 
 or return together, are called circu ! ates, of which 
 there are three kinds, viz. fimilar, diflimilar, fimijar 
 and conterminous. 
 
 SIMILAR circulates are thofe that confift of the 
 fame number of repeating figures, beginning either 
 
 before or after the decimal point: thus 42.7 and 9.19 
 are fimilar circulates. 
 
 DISSIMILAR circulates are thofe that confift of an 
 unequal number of repeating figures, beginning at 
 
 different places : thus 1.77 and 217.4 are diflimilar 
 circulates* 
 
 SIMILAR 
 
SIMILAR and conterminous circulates, t arc thole 
 which confift of an equal number of repeating fi 
 gures, beginning and ending together : thus, 27.47 
 and 4.73 are fimilar and conterminous circulates. 
 
 CHAP. II. 
 
 Of REDUCTION of CIRCULATING DECI 
 MALS. 
 
 PROBLEM I. 
 
 To reduce a fingle repetend to its equivalent Vulgar 
 Fraffion. 
 
 RULE. , 
 
 UNDER the given repetend, with as many cy 
 phers annexed to it, as there are places of whole 
 (numbers, write as many 9^ as there are places of 
 figures in the repetend j and you will have the Vul 
 gar Fraction required. 
 
 THE reafon of this rule will appear obvious, when 
 
 weconfider, that .91 ; for fz=. in &V. .1 j con- 
 
 fequently ,ix9~iX9 ; that is, .9 f-: i ; whence 
 it follows, that each figure of the repetend is equal to 
 
 that figure divided by 9 : thus .3 =4 ,5=: j.,&fr. 
 EXAMPLES, 
 
 Required the leaftVulgarFra&ion equivalent to .72 
 
 Thus, 
 
Thus, ^-zz^^f raft ion required. 
 
 21300 ^ 64^2^000 
 
 21.3^1 g 643.25=-2-2 1.7421= 
 
 999 . 99999 
 
 174.210 - 1270002000 
 
 - -. 127.0002= ; . 
 
 99999 9999999 
 
 PROBLEM II. 
 
 fo reduce a mixed compound repetend to its equivalent 
 Vulgar Fraction. 
 
 RULE. 
 
 WRITE down as many 9'^ as there are places of 
 figures in the repetend, to which annex as many cy 
 phers as are equal to the number of occurring places 
 of figures in the finite part, (i. e. the figures occurring 
 before the alternate circulates) for a denominator ; 
 then multiply the 9'^ in the denominator, with the 
 finite part, to which product, add the infinite or cir 
 culating part for a numerator j and you will have ; 
 the fraction required. 
 
 Note. When the circulate begins any where in the 
 integral fart, omit the cyphers in the denominator, 
 and annex as many to the numerator as there are 
 places of whole numbers included in the circulate. 
 
 TnEreafon of this rule will appear plain from the 
 following. Suppofe the decimal whofe equivalent 
 
 VulgarFraftion is required, to be .53: Conceive it 
 to be divided into finite and infinite parts -, that is, 
 conceive it to be made of the finite part .5 and the in 
 finite or circulating part .03 j then .53 = .54-.O3j but 
 .3:r~i confequently .ojZZ-rV of -J-= 7 V wherefore 
 
 -53 
 
53= =+ 
 
 ( i*3 ) 
 
 
 
 -i^> which is the 
 90 
 
 fame as the rule. 
 
 EXAMPLES. 
 
 Required the Vulgar Fraction equivalent to .4739 
 Firft, 9990 denominator. 
 
 Then 999X4- :=: 3996prodi{ff of the cfs in the de 
 nominator and finite part -, and 3996+73947351= 
 numerator. 
 
 Wherefore -f-J-14 is the fraction required. 
 
 Required the equivalent Vulgar Fradtion of 5.27 : 
 
 Thus, 52X9+7 -r 900=468+7-*- 900=^54 tbt 
 fraction required. 
 
 Required the equivalent Vulgar Fraction of 42.3 : 
 
 Thus, 990X4+230-7-99 4 f. the fraftion re 
 quired. 
 
 Required the equivalent Vulgar Fraction of 321.7 : 
 
 Thus, "999x3 + 217-7-999 = 3214-7-999 j then 
 "* ^frdSlon required* 
 
 14-00 
 
 PROBLEM III. 
 
 To determine whether the decimal equivalent to any 
 Vulgar Fraction be finite, or infinite -, and if infinite y to 
 find the number of places of figures that ccnftitute the 
 circulate. 
 
 RULE. 
 
 1. REDUCE the given fraction to its lead terms. 
 
 2. DIVIDE the denominator of the refulting frac 
 
 tion 
 
( 124 ) 
 
 tion by i, 5 or 10, as often as you can without a re 
 mainder, making the refult a divifor, and 999 &V. 
 a dividend, divide till nothing remains, then will the 
 circulate confift of as many places of figures as you j 
 ufed places of ffs. 
 
 Note, i . The circulate will begin, after as many 
 places of figures as you made divifions of the de 
 nominator. 
 
 2 t In dividing the denominator as above, if the quo 
 tient become equal to unity, then the decimal is 
 finite, confining of as many places of figures as you 
 made divifions of the denominator. 
 
 THE principles on which this rule is inveftigated, 
 may be fhewn in the following manner. 
 
 Firft, let it be premifed, that if unity with cy 
 phers annexed, be divided by any prime number, 
 except 2, or j, the figures in the quotient will begin 
 to repeat when the remainder becomes unity ; con- 
 fequently 999 &c. divided by any prime lumber, 
 except 2, or 5, will leave no remainder. 
 
 Now if the places of figures in the circulate are 
 any number, when the dividend is unity, they will 
 remain the fame, let the dividend be any other num 
 ber whatever ; for it is plain, that if the decimal be 
 multiplied with any number, every circulate will be 
 equally multiplied, and what one is increafed will be 
 carried to another, and fo on through the whole ; 
 confequently, the places of figures will remain the 
 fame : But to multiply the decimal or quotient with 
 any number, is the fame thing, as to divide the divi- 
 ibr by the fame number before divifion is made; 
 whence^ &c. 
 
 EXAMPLES.. 
 
EXAMPLES, 
 
 Required to know, whether the equivalent deci 
 mal, of-'-rr is infinite or finite, and if infinite, how 
 many places of figures there will be in the circulate. 
 
 Firft, -i-f4 reduced to its lead terms ~ ; then 
 999999-7-7=142857, and therefore the decimal 
 is infinite, whofe circulate confifts of 6 places of fi 
 gures, beginning at the tenth's place. 
 
 Required to know whether the equivalent dicimal 
 of -i-V-sSr is infinite, or finite ; and if infinite, how 
 many places of figures there will be in the circulate. 
 
 Firft, -m^=(by reducing to its lead terms) T y ; 
 then, i6-r-2n 8, 8 -r-2 4, 422, and 2 2=1 : 
 Consequently the decimal is finite, confifting of 4 
 places of figures. 
 
 -Required to know whether the equivalent decimal 
 of l|t is infinite, or finite ; and if infinite, to know 
 how many places of figures there will be in the cir 
 culate. 
 
 Firft, |44 (by reducing to its leaft terms) 44 ; 
 then 70-7-10=7, and 999999-r-7 ::::::: i42857 : Con- 
 fequently the dicimal is infinite, and the circulate 
 confifts of 6 places of figures, beginning at the hun 
 dredth^ place. 
 
 PROBLEM IV. 
 
 To make diffimilar circulates > fimilar and conterminous. 
 RULE. 
 
 i. Find the leaft poffible common multiple of the 
 feveral numbers exprefRng the number of places of 
 figuies in the given circulates, 
 
 2. 
 
2. Change the given circulates into others.confift- 
 ing each of as many places of figures as the leaft 
 common multiple found as above^ and the work will 
 be done. 
 
 EXAMPLES. 
 
 . < * 
 
 Make .727, .179, .12 and .19 fimilar and con 
 terminous. 
 
 Firft the leaft common multiple of 3, 3, 2 and 2, 
 IE 6. 
 
 Diflimilar. Similar and conterminous, 
 
 f -7 27 ~-7 277 27 
 Then, j-?79=.i79i79 
 
 j.12 =.121212 
 1.19 =.191919 
 
 Make 24.3, .4762," 32, ,6 and .576 fimilar and 
 conterminous. 
 
 Diffimilar. Similar and conterminous, 
 
 ^24.3=24.333333333333 
 Thus \ '^7 62=. 47 6 247 6 '2-47 6*2 
 9 j 32.6=32.666666666666 
 1.576^.576576576576" 
 
 C PI A P. 
 
CHAP. III. 
 
 ADDITION, SUBTRACTION, MUL 
 TIPLICATION AND DIVISION OF 
 CIRCULATING DECIMALS. 
 
 SECT. I. 
 
 Of ADDITION of CIRCULATING 
 DECIMALS. 
 
 R ULE. 
 
 MAKE the given circulates fimilar and conter 
 minous, by problem iv, of the lail chapter; 
 then add them together as in common Addition, and 
 becaufe each figure of the circulate is equal to that 
 figure divided by 9, you mufl divide the fum of the 
 circulates, by as many places of $'s as there are pla 
 ces of figures- in the circulate, and writing the re 
 mainder (if any) directly beneath the-figures of the 
 circulate, carry the above quotient to the next place ; 
 then proceed as in common decimals, and you will 
 have the fum required. 
 
 Note. When the remainder confifts of a hjs number 
 of places than the circulate, you miift fitpfly !bt 
 defeffi ly prefixing cyphers. 
 
 EXAMPLF3, 
 
( 128 ) 
 
 EXAMPLES. 
 
 Required the fum of 3.3+4.271+3.725 : 
 
 Similar and conterminous. 
 
 !3-3 .=3-333 
 4.271=4.271 
 3-725=3.725 
 
 ii.330=r///ff* required. 
 
 Required the fum of 24,327425 + 37.274+27.35+ 
 
 34-27 : 
 
 Diffimilar. Similar and conterminous, 
 
 ^24.327425 24.327425425 
 
 Thus, I 37-274 = 37-274444444 
 127.35* =27.353535353 
 134.27 =34.277777777 
 
 * 
 
 1 23. 233 1 8 300 1 ~y#w req, 
 
 SECT. II. 
 
 Of SUBTRACTION of CIRCULATING DECI 
 MALS. 
 
 RULE. 
 
 PREPARE the given numbers, as in Addition, and 
 then fub tract them as in common SubtradHon, only 
 with this difference, viz. when the circulate to be fub- 
 tra&ed, is greater than the one from which Subtrac 
 tion is to be made, you muft make the right-hand 
 
 figure 
 
figure of the difference lefs by unity, than as found by 
 common Subtraction. The reafon of this rule will 
 appear plain from the following. 
 
 SUPPOSE 1.81 were to be taken from 2.72; the 
 difference by common Subtraction would be .91 - t but 
 2.72= Vs? and i.8i~VV% then 2.72^1.81:=: 
 VT O ==?== .90; whence, 
 
 EXAMPLES. 
 
 Required the difference between 6.4729 and 3-49': 
 Diflimilar. Similar and conterminous. 
 29-6.4729729 
 
 ^ |6. 4 7 29:1:6., 
 '' U'49 =3- 
 
 Thus 
 
 ,4949494 
 
 
 
 2 , 97 8 0234^ difference required. 
 
 Required the^ difference between 4. 37 5 2 and 1.1210 : 
 Diffimilar. Similar and conterminous* 
 
 Th 
 
 US> I.iaiO=:I.I2lOl2IO 
 
 3,25424041= difference required. 
 
 SECT. III. 
 
 Of MULTIPLICATION of CIRCULATING DE~ 
 CIMALS. 
 
 RULE. 
 
 INSTEAD of the given circulates, write their equi 
 valent Vulgar Fraftions, and find their product as 
 
 R ufual ; 
 
ufual ; then this produft thrown into a decimal, will 
 give the product required. 
 
 EXAMPLES. 
 
 Required the product of 3.2X-7 
 
 Firft, .32 -Jl- and .7 > wherefore .jix.? 
 which thrown into a decimal is, 
 
 ra/# required. 
 Required the product of 1.8x2.7 : 
 
 Thus, i.^2.7-V 7 XV 5 =\V-5'S 
 the pro duff required. 
 
 Required the product of. 20X^36 : 
 
 Thus, .2o X .36~ r ^X^4=^4 
 produft required. 
 
 SECT. IV. 
 
 Of DIVISION of CIRCULATING DE 
 CIMALS. 
 
 R ULE. 
 
 CHANGE the given decimals into their equivalent 
 Vulgar Fractions, and find their quotient as ulual ; 
 then this quotient thrown into a decimal, will give 
 the quotient required. 
 
 EXAMPLES. 
 
 
 
 Required the quotient of .26 divided by .3 : 
 
 3-~ ' 
 
 Wherefore, 
 
Wherefore, .26^.3=14-H- = T74=.8 tie quotient 
 required. 
 
 Required the quotient of .9-7-.! 08 : 
 
 Thus, .9^08:^-^=^ AV=VT =9.25 
 ^quotient required. 
 
 Required the quotient of 2. 9-4-, 27* : 
 
 Thus, 2.9~.27:=y^^:rY'== n the quotient 
 required. 
 
13* 
 
 A SUPPLEMENT to PART I, 
 
 CONTAINING THE DOCTRINE AND 
 APPLICATION OF RATIOS, OR 
 PROPORTION, EXTRACTION OF 
 ROOTS, C5>, 
 
 CHAP, L 
 
 Of PROPORTION or ANALOGS 
 
 PROPORTION is a degree of likenefs which 
 quantities bear to each other, by a fimilitude 
 of ratios. 
 
 RATIO is the mutual refpect of two quantities of 
 the fame kind ; but they form no Analogy, becaufe 
 there can be no fimilitude of ratios between two 
 quantities, and therefore Analogy confifts of three 
 quantities at leaft, whereof the iecond fupplies the 
 placeof two : Thus the refpect of 2 to 6 3 being com 
 pared with 1 8, it will be, 2'.6::6:i8. 
 
 SECT, I. 
 Of CONTINUED PROPORTION 
 
 O R 
 ARITHMETICAL PROGRESSION, 
 
 WHEN quantities increafe or decreafe by an equal 
 difference, thofe quantities are in Arithmetical Pro 
 portion continued :* Thus, the number i, 2, 3, &c, 
 are a feries of quantities in Arithmetical Proportion 
 
 continued., 
 
( 133- X 
 
 continued, increafing by unity, or i, which is called 
 the common difference of the feries. 
 
 ALSO, the numbers 2, 4, 6, 8, are numbers in 
 Arithmetical Progreflion, whofe common difference 
 is 2 ; but the numbers 9, 7, 5, 3, i, are a feries of 
 quantities in Arithmetical ProgrefTion, decreafing by 
 the common difference, 2. 
 
 LEMMA I. 
 
 If three numbers are in Arithmetical Progreffion, the 
 fum of the two extreme numbers will be double the mean 
 
 or middle number. 
 
 THUS, let i, 3, 5, be the numbers in progreffion ; 
 
 Then, 1+5, the fum of the two extremes ~ 3-^3 
 the double of the mean. Again, in the numbers 
 14, ic, 6, the fum of the two extremes are i4-f 6~ 
 20, and the double of the mean 10+1020 ; and 
 the like will hold in any other numbers. 
 
 LEMMA II. 
 
 If four numbers are in Arithmetical ProgreJ/ion, the 
 Jam of the two extremes will be equal to the fum of tbc 
 two means. 
 
 LET the number be 4,7, 10, 13 ; then4~f-i3~i7j, 
 the fum of the two extremes, and 7~|-io:i:i7, the 
 fum of the two means : Again, in the numbers 16, 
 13, 10,7 i 16+7 13+10. 
 
 AND iince in four numbers as above, the fum of 
 the two extremes, is equal to the fum of the two 
 means, we have no reafon to doubt of the like, let 
 the terms be any number whatever : Whence it fol 
 lows, that in any Arithmetical feries, of any afllgna- 
 ble number of terms whatever, the fum of any two 
 terms equidiftant; from the mean, will be equal to 
 
 the 
 
the fum of any other two terms, equidiftant from the 
 mean j as in thefe, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 ; 
 where 2-f-2O=4-f 18^:6^16=: 8+14=10+1 2 : 
 Therefore, &V. 
 
 LEMMA III. 
 
 /# any Jeries of numbers in Arithmetical Progreffion^ 
 thejeveral terms are formed or made up by the addi 
 tion of the common difference to the frft term, fo often 
 repeated, as there are number of terms to the feveral 
 places, except the firft. 
 
 LET the feries be, i, 4, 7, 10, 13, 16, 19, 22, fc?r* 
 wherein the common difference is 3. 
 
 Now 1+34 the Jecond term, i+jH-3~7> the 
 third term \ i-r-3+3*f-3 10, the fourth term \ 
 
 J *3-Klf 3-f3=i3 **' fifth term i and 1+3X7 
 r:22, the %th term, &c. Confequently the differ 
 ence of the two extremes, is equal to the common 
 difference multiplied with the number of terms lefs 
 i : Thus in the above feries, the common difference 
 
 is 3, and number of terms 8 ; therefore 8 IX37 
 X 32 1 rz difference of the two extremes. 
 
 PROBLEM I. 
 
 find thejum of a Jeries of numbers in Arithmeti 
 cal Progrejfion. 
 
 THERE are feveral ways of deducing a rule for the 
 folution of this problem, but perhaps none more fim- 
 ple and natural than the following. 
 
 LET the feries whofe fum is required, be 2+4+6 
 -fS-fio-f 12. 
 
 Or, 
 
Or, 
 
 22222 
 
 + + + + 
 
 2222 
 
 + + + 
 
 222 
 
 4- + 
 
 2 2 
 
 which is the fame as the former, though differently 
 exprefTed : Now under the given feries place die 
 fame inverted and add up the whole. 
 
 Thus, 
 
 ~ HI 
 
 10 
 
 Co 
 
 t. 
 
 o^ 
 
 ? 
 
 1 
 
 i 
 
 P 
 
 i 
 
 1 
 
 i 
 
 2 -f 
 
 2 -f 
 
 ' ^ + 
 
 2 -f 
 
 2 - 
 
 L 2 
 
 2 
 
 4- 
 
 + 
 
 + 
 
 + 
 
 4- 
 
 4- 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 4- 
 
 4- 
 
 4- 
 
 4- 
 
 "T" 
 
 4" 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 4 
 
 + 
 
 4- 
 
 ~f- 
 
 f 
 
 + 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 -f- 
 
 f 
 
 ~f- 
 
 + 
 
 + 
 
 "f 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 o 
 
 2 
 
 4- 
 
 f 
 
 -f- 
 
 f 
 
 + 
 
 -f- 
 
 2 
 
 12 
 
 2 
 
 2 
 
 2 
 
 - 
 
 2 
 
 14+ 14+ 14 +14 -f 14 HhJ4=^*- 
 
 By this means the terms of the feries are reduced 
 to an equality, to wit, equal to the fum of the firll 
 and laft term ; but the fum above found, is evident 
 ly double the fum of the propofed feries : When a; 
 
 it 
 
C 
 
 it folloxvs, that the fum of an Arithmetical feries, is 
 equal to half the product of the firft and laft term, 
 with the number of terms; wherefore if the firft term, 
 jail term, and number of terms of an Arithmetical 
 ProgrefTion be given, the fum of the feries may be 
 found by the following 
 
 RULE, 
 
 MULTIPLY the fum of the firft and laft terms, or 
 two extremes, with the number of terms, and half of 
 that product will be the fum required. 
 
 EXAMPLES. 
 
 Let the firft term of a feries of numbers in Arith 
 metical Progrcilion,:~i, Lift termizjy, and num 
 ber of terms 19 - 3 required the fum of the feries. 
 
 OPERATION. 
 
 Firft, i ^37=3 8 =:// of the frfl and laft terms : 
 Then 3&X i9-~2~ 722-t-2~j6i the Jum required. 
 
 A MAN bought 20 yards of broad-cloth -, for the 
 firft yard he gave 2 doL and for the laft 80 dol. 
 what did the whole coft ? 
 
 The fum of the two extremes, is 2-/-8o, then 
 
 2 -f-S ox 20 -7-2=1:820 dol. the anjwer. 
 
 A MAN travelled 12 days, the firft day 4 miles, and 
 the laft day 40 miles j what was the diftance travelled 
 in the 1 2 days ? Anfwer. 264 miles. 
 
 PROBLEM IL 
 
 To find the common difference of an Arithmetical Je- 
 ries y when the two extremes and number of terms are 
 
 given. . A 
 
A RULE for the folution of this problem, is eafily 
 deduced from the inference to Lemma in 5 for fince 
 the difference of the two extremes, is equal to the 
 common difference multiplied with the number of 
 terms lefs i, it follows, that if that difference, be 
 divided by the number of terms le& i, the quotient 
 muft be the common difference of the feries ; whence 
 the following rule is evident. 
 
 RULE. 
 
 DIVIDE the difference of the two extremes by the 
 number of terms lefs i, and the quotient will be the 
 common difference required. 
 
 EXAMPLES. 
 
 In an Arithmetical feries, there is given the fir ft 
 term 3, lad term 60, and number of terms i^: 
 Required the common difference. 
 
 OPERATION. 
 
 The difference of the two extremes, is 60 3; 
 ^_I--^ common difference 
 
 201 
 
 required. 
 
 Four men differing in their ages by an equal in 
 terval : The age of the firft, is 1 9 years, *nd the 
 fourth 40 : What are their feveral ages ? 
 
 OPERATION. 
 
 Firft, find the common difference of their ages : 
 Thus, 40 1 9^-4^7=2; i -4-3=7 ?"*rs ; therefore 
 
 S i 
 
< 138 ) 
 
 z6 years, the age ofthefecond, and 26-1-7=33 
 years, the age >cf ^the third-, laftly, 33-j~7rr4O years, 
 the age of tbe fourth, as given above. 
 
 A man owes a certain debt, to be difcharged at 
 8 feveral payments ; all of which are to be made in 
 Arithmetical Progreffion, the firft payment to be 4 
 dol. and the Jail 32 dol. Query, the whole debt and 
 each payment. 
 
 OPERATION. 
 
 w bole debt, and 324-7- 
 
 1 1=4 dol. the common difference -, wherefore 4+4 
 ^=8 dol. the Jecond payment, and 8-j-4~i2 dol. the 
 third payment > alfo, 12+4:1116 dol. ; for the fourth-, 
 moreover 1 6 +4:1:20 dol. for the $th, in like man 
 ner 20 -f 424 dol.for the 6th, and 24-j-4=28 dol. for 
 the jtb -, laftly 284-432 dol. for the laft payment 
 as before. 
 
 PROBLEM III. 
 
 To find the number of terms of an Arithmetical Jeries, 
 when thefirjl term, laft term and common difference are 
 given. 
 
 FROM the laft rule, it is eafy to conceive haw a 
 rule for the folution of this problem may be obtain 
 ed i for fince the difference of the two extremes, 
 divided by the number of terms lefs i, gives the 
 common difference ; it follcfcsj that the difference of 
 
 .V 
 
 the two extremes, divided by^the common difference, 
 muft quote the number of terms lefs r . 
 Whence is deduced the following 
 
 RULE. 
 
RULE. 
 
 DIVIDE the difference of the two extremes, by the 
 common difference, the quotient increafed by unity 
 or i, will be the number of terms. 
 
 EXAMPLES. 
 
 Given the firft term of an Arithmetical ferieszrs, 
 laft termm67, and common difference 3, to find the 
 number of terms. 
 
 OPERATION. 
 
 +I= Ji+ 1=55+1-56 the nmhr of 
 
 3 . . 3 
 terms required. 
 
 A man bought a quantity of broad-cloth ; for the 
 firft yard he gave 6 dol. for the fecond, 10 dol. and 
 fo on, in Arithmetical Progreffion, to the laft yard, 
 for which he gave 246 dol. j what was the quantity 
 of cloth bought ? 
 
 _ OPERATION. 
 
 0466 = 240 + I= 6i, the numlerof yards bought. 
 
 4 ~4 
 
 A man travels from Bofton, to a certain place, in 
 the following manner, viz. the firft day 10 miles ; the 
 fecond day 15 miles, and fo on, till a day's journey 
 is cc miles : In how many days will he perform the 
 whole journey ; alfo, how many miles is the place 
 he goes to, diftant from Bofton ? 
 
 Anfwer. Pie will perform the whole in eleven days, 
 The place diftant from Bofton, 330 miles. 
 
 SECT. 
 
SECT. II. 
 
 Of CONTINUED PROPORTION 
 GEOMETRICAL, 
 
 Or 
 GEOMETRICAL PROGRESSION, 
 
 GEOMETRICAL Progreifion continued, differs frorr 
 Arithmetical Progreflion in this -, in Arithmetica 
 Progreifion, each following term of the feries is form 
 ed or made up by the Addition or Subtraction o 
 the common difference, (as we have before fhewft) 
 Whereas in Geometrical Progreffion, each fucceffive 
 term of the feries, is produced by the Multiplication 
 or Divifion of the preceeding term, with a common 
 multiplier or divifor : Or in other words, Arithme 
 tical Progreflion, is the effect of a conftant Addition 
 or Subtraction -, but Geometrical Progreffion, of a 
 conftant Multiplication or Divifion. 
 
 THUS, 2, 4, 8, 1 6, 32, 64, 128, &V. are a feries 
 of numbers in Geometrical Proportion continued ; 
 whofe refpective terms are compofed by the Multi 
 plication of the Ratio or common multiplier, (2): 
 thus, 2X2:1=4, the Jecond term, 4X2^18, the third 
 term ; 8X2:1:16, the fourth term) and fo on. 
 
 ALSO, 1 6, 8, 4, 2, are a feries of numbers in Geo 
 metrical Proportion, continually decreafmg by the 
 divifion of the Ratio, or common divifor, (2) : Thus, 
 -^i^S, the Jecond term> 4^:4, the third term, ^2, the 
 fourth term, and T 1 * ifajtjftt term. 
 
 LEMMA I. 
 
 Jf three numbers are in Geometrical Progtefficn, the 
 froduSt cf the two extremes, will le equal to the pro- 
 duff of the mean with iff elf. 
 
( HI ) 
 
 LET the numbers be 2, 8, 32 ; where 2X3264, and 
 8X8=64 5 confequently 2X32=8X8. 
 
 LEMMA II. 
 
 In any Geometrical Proportion fonftfting of four 
 terms, the produfl of the two extremes, is equal to the 
 pro duff of the two means. 
 
 IF the numbers are, 2, 8, 32, 128, it will be 
 2x128:118x32 j therefore 2: 8 1:32 1128. 
 
 CONSEQUENTLY, if the product of any two num 
 bers, be equal to the produd: of any other two num 
 bers, thofe four numbers are proportional. 
 
 HENCE it may be eafily underftpod, that if any 
 number of terms are in -ff the product of the two ex 
 tremes, will be equal to the product of any other 
 two terms, equidiftant from thofe extremes. 
 
 LET the feries be 3, 6, 12, 24, 48, 96 ; where 
 3X96=6X48-12X24. 
 
 WHEN numbers are compared together, in order 
 to difcover their relation to each other, the number 
 compared is writen firft, and called the antecedent, 
 and the number by which you compare the other, 
 being written next, is called the confequent : Thus 
 if you would compare 2 with 4, the numbers muft 
 be wrote thus, 2, 4 ; where 2 is the antecedent, and 
 4 the confequent : Again in theie, 3 : 6 :: 6: 12 ; 
 where 3 is antecedent, and 6 its confequent ; alfo, 
 6 the middle term, is an antecedent to 12, its confe 
 quent. Therefore in every feries of numbers in 
 Geometrical Proportion continued, all the terms ex 
 cept the laft, are antecedents, and all except the firft 
 are confequtnrs. 
 
 THUS in (fee feries 3, 9, 27, 81, 243, 729, the 
 numbers 3, 9, 27, 81, 243, arc all antecedents, and 
 
9, 27, 8 1, 243, 729, 
 
 i all confequents ; therefore 
 8 1 :: 81:243:: 243:729. 
 THE Ratio is had by dividing any confequent by 
 
 3 19 1:9 127:: 27 
 
 its antecedent^ 
 
 LEMMA III. 
 
 If any numbers are proportional) it will be, as any 
 cne cf the antecedents, is to its confequent jfo is the fum 
 of all the antecedents, to to the fum of all the confe- 
 qusnts, (! y id. Euclid's fifth book, Proportion 12.) 
 
 LET the numbers be thefe, 4, 8, 16, 32, 64, then 
 4-32 : 84-164-32 + 64, that is, 
 for 4Xi20iz8X6o j therefore, 
 
 8 :: 44-84- ic 
 8 :: 60 : 1 20 j 
 
 PROBLEM I. 
 
 To find tl}3jum of any Geometrical Jeries increofing. 
 
 SUPPOSE the fum of the following feries, i, 4, 16, 
 64, 256, is required : Multiply this feries with the 
 Ratio, which is 4, and the' product will be a new fe 
 ries, 4, 1 6, 64, 256, 1024 : Now it is plain, that 
 the fum of the produced feries, is as many times the 
 fujQDL of the former, as the Ratio hath units ; or the 
 produced feries, is to the propofed, as the Ratio to 
 unity, or i : Subtract the firft feries from the fccond, 
 
 4> l6>64 > 2 5 6 > I02 4 
 1,4, 1 6, 64, 256. 
 
 Thus 
 
 
 
 i, ' 4-1024, or, 1024 i, 
 
 which is evidently equal to the fum of the firfc ftries 
 multiplied with the Ratio, lefs i, by what has been 
 faid ; confequently the fame divided by the Ratio, 
 lefs i, muft give die fum of the propofed feries; that 
 
 is; 
 
THEREFORE, when the firft term, lad term, and 
 Ratio of a Geometrical feries are given, we may find 
 the fum of all the terms by the following 
 
 RULE. 
 
 MULTIPLY the lad term with the Ratio, from 
 which produftj fubtradfc the firft term, divide the re 
 mainder by the Ratio lefs i, and the quotient refuh- 
 ingwill be the fum cf the fen cs. 
 
 MR. WARD, in his introdu&ion to the Mathe 
 matics, page 78, has given an analytical invefiiga- 
 tion of a rule for rinding the fum of any feries infr 
 increafing ; which is afcer the manner following. 
 
 LET a Geometrical feries be given^ fuppofe the 
 following, 2, 4, 8, 16, 32, 64. 
 
 Put x~^Jum of the feries : 
 
 Then, x* 64. fum of all tbe antecedents : 
 
 And x*i^Jum of all the confequents : 
 
 Therefore, 2:4:: #64 :x 2 ; per Lemma 
 in. . 
 
 Cohfeqticntly, x 2X2itf 64X4* 
 
 That is, 2* 4n4.v 256 : 
 
 Then, 4^- 2^^256^ 4 : 
 
 Therefore, (by divifion) ix #1^1128 2 : 
 
 Whence, Arni28 2-r-2 i, wbich affords thefamf 
 rule as that above. 
 
 Or finding tbe value ofx in tbe equation 4* 2#:n 
 
 256 4, to wit) ,VH256 4~-4 i which admits of 
 &e following 
 
 R U L E, 
 
( 144- ) 
 
 RULE. 
 
 FROM the produ6b of the fecond and laft terms, 
 fubtracfc the fquare of the firft, divide the remainder 
 by the fecond term lefs the firft -, and the quotient 
 will be the fum of the feries. 
 
 EXAMPLES. 
 
 In a Geometrical feries, there is given, the firft 
 termzz3, laft term=r243, and Ratio 3 j to find the 
 fum of the feries, per Rule firft. 
 
 OPERATION. 
 
 . 
 
 Firft, 243 X 3 7 2 9 ^'prcduft of the laft term with 
 the Ratio-, then 729 3-7-31^726-7-2 = 363 the 
 fum required. 
 
 A man bought a quantity of cloth ; for the firft 
 yard he gave 2 dol. for the fecond 4 ; and fo on, in 
 continued proportion Geometrical to the laft yard, 
 for which he gave 256 dol. what did the whole coft ? 
 
 Here, is given the firft, fecond, and laft terms, to 
 find the fum of the feries, per Rule fecond. 
 
 256x4 4 1024 4~ ioio~produft of the fecond 
 and laft terms, lefs the fquare of the fir ft \ then 
 
 difference divi- 
 
 4 2 2 
 
 Jed by the fecond term lefs thejirft~fum that the whole 
 cloth coft. 
 
 BUT in finding the fum of the feries by the fore 
 going rules, it is neceijjiry to have the laft term giv^- 
 en : therefore the next thing in order, is, to Ihew 
 how the laft term of the feries, when it is not given 
 in the queftion, may be obtained. 
 
 PROBLEM 
 
( '45 ) 
 
 PROBLEM II. 
 
 The fir ft term > Ratio > and number of terms of a Geo 
 metrical feries being given, to find the laft term. 
 
 I. WHEN the firft term and Ratio are alike. 
 
 RULE I. 
 
 1. WRITE down an Arithmetical feries of a con 
 venient number of terms, whofe firit term, and com 
 mon difference is unity or i. 
 
 2. WRITE a few of the leading terms of the Geo 
 metrical feries, under the firft terms of the Arithme 
 tical one. 
 
 Thus I T> 2 > 3 > 4 > 5 ' Indices, or exponents. 
 I 2, 4,' 8, 1 6, 32, Geometrical feries. 
 
 3. ADD together any two of the indices, and mul 
 tiply the terms in the Geometrical feries, which be 
 long to thofe indices, together, and their produ-ft will 
 be that term of the Geometrical feries, which the 
 fum of thofe two correfponding indices point out. 
 
 4. CONTINUE the addition of the indices, and mul 
 tiply their correfponding terms, of the Geometrical 
 feries, refpectively as before, until the fum of the in 
 dices is equal to the number of terms, the product 
 anfwering thereunto, will be the lad term required. 
 
 II. WHEN the firft term is either greater or lefs 
 than the Ratio, (unity excepted.) 
 
 RULE II. 
 
 i, WRITE down an Arithmetical feries, beginning 
 with a cypher, the common difference, the fame as 
 in the laft rule, 
 
 T , 
 
2. PLACE the leading terms of the Geometrical 
 feries, under the Arithmetical, fd that the cypher 
 may ftand over the firft term of the Geometrical fe- 
 ries ; then add the indices, and multiply their corref- 
 ponding terms as before. 
 
 3. DIVIDE that product by the firft term, and the 
 quotient will be that term of the feries, which is de 
 nominated by the fum of thofc indices : The reft 
 the fame as before. 
 
 III. WHEN the firft term is unity or i. 
 
 R U L E HI. 
 
 W V RTTE down the terms, and place their indices as 
 in the lad rule ; then add the indices, and multiply 
 the terms which they denominate, together, till the 
 fum of the indices is one lefs than the number of 
 terms, and the refult will be the iail term, as requir 
 ed. 
 
 AN example in each of the foregoing rules, will 
 make their application eafy. 
 
 In a Geometrical feries, there is given, the firft 
 term=2, Ratio 2, and number of terms 12, to find 
 the laft term, per rule i . 
 
 OPERATION. 
 
 Thus, I x > 2 > 3> 4> 5> 6> Indices. 
 I 2> 4, 8, 16,32,64, Tf. 
 
 Here, 4+2=6, the index ofthefixtb term ; confe- 
 quently 4x16=64, the fix th term. Again y 6+612, 
 tffltf 7 64X64=4096^/^4^ term* as required. 
 
 Suppofe the firft term of a feries in ~, is 3, Ratio 
 2, and number of terms 15 i required the laft term, 
 per rule 2. 
 
 OPERATION. 
 
( 147 ) 
 OPERATION. 
 
 irft I 3 l > 2) 3> 4> S> Indices. 
 > I 3> 6, 12, 24, 4 8, 9^> #v' 
 
 Firft 
 
 24X96=23045 therefore, 
 3 l6% eighth term. Again y 34-4=7, and 
 ; therefore, 
 
 term. Z^/y, 7 + 8 1 5 ; whence^ - 4 ./. ------- 98304 
 
 O 
 
 : i $tb y and laft term which 'was to be done. 
 
 Given firft term' i, Ratio 4, and number of terms 
 II, to find the laft term, per rule 3. 
 
 )0 OPERATION. 
 
 TVm I > x > 2 3> 4> Indices. 
 
 hUS> 1 i, 4, 16, 6 4 , 256, *. 
 Then, 4-}-3 + 3=io~^w^r ^ /^rwj lefs one 
 index to the nth term ; therefore, 256X^4X64 
 1048576^111^ term as was required. 
 
 Mifcellaneous >ueftions. 
 
 A MAN hired himfelf to a farmer, for 28 weeks 
 upon thefe confiderations ; that for the fir ft week to 
 have i ct. ; for the fecond 2 cts. ; and the third 
 4 tfj. ; and fo on, in 4f '- What did his 28 weeks 
 wages amount to ? 
 
 The laft term by the foregoing rules, is, 1 342 17728, 
 which multiplied with the Ratio (2) produces 
 
 268435456; therefore, - 43545 T 268435455 
 cts. "2684354 dol. $$cts. the anfwer. A 
 
AnJ. { 
 
 ( 14* ) 
 
 A MAN bought 20 yards of velvet, at the follow 
 ing prices, viz. for the firft yard he gave i.cts. -, for 
 the fecond, 4 cfs. j for the third, 8 cts. and fo on, in 
 Geometrical Proportion : How much did the 
 whole coft ? 
 
 Anfwer. 20971 del. 50 cts. 
 
 A MERCHANT fold 24 yards of lace 5 the firft yard 
 for 3 pins, the fecond for 9, the third for 27 ; and fo 
 on, in triple Proportion Geometrical : Now fuppofe 
 he afterwards fold his pins 1 10 for a cent : What did 
 his lace amount to, and what was his gain in the 
 whole, when he gave 50 cts. per yard for his lace ? 
 
 Lace corns to y 4236443047 dot. 20 cts. 
 Gain in ths whole, 4236443035 dol. 20 cts. 
 
 A THRESHER agreed with a farmer to work fer 
 him 25 days, for no other confideration than 2 barley 
 corns for the firft day 8 ; for the fecond 32 ; for the 
 third - 3 and fo on, in quadruple proportion Geometri 
 cal : How much did his wages amount to, allowing 
 7680 barley-corns to make one pint, and the barley 
 to be fold for 25 cts. per bufhel ? 
 
 Anfwsr. 381774870 dol. 75 ~cts. 
 
 SUPPOSE a wheat-corn had been fowed at the crea 
 tion, and continued to increafe in a ten -fold propor 
 tion every year, down to the prefent time 5 now al 
 lowing 5003 years for the elapfe of time: What 
 would be the number of wheat-corns produced ? 
 
 Here the firft term being i, the Ratio 10, and 
 the number of terms 5000, it is therefore plain, that 
 the laft term will be i, haying as many cyphers an 
 nexed, as there are number of terms, ; le-fs one -, con- 
 fequently its value is 1(4999)0^, where the numeral 
 figures included in the parenthefis, exprefs the 
 number of cyphers annexed to the i ; Next to find 
 the fum of the feries. Firft 3 
 
 
( '49 ) 
 
 Firjl, 1(4999)0^X101(5000) tfs y tben 1(5000) 
 oV i =(5000) 9'j ~ /<? number cf y's therefore 
 
 5 OOQ 9 >J , , 
 
 ' ^:i 1 1 ii in i in 1 1 iniii ejv. to cooo 
 
 10 I 
 
 f laces of figures number of wheat-corns produced j 
 which number far exceeds all human imagination ; 
 for the whole fpace occupied by our folar fyitem, 
 which is at lead twenty thoufand million of miles in 
 diameter, is by'mtich too fmall, to contain the afore- 
 faid quantity of wheat: Nay, fuch a quantity would 
 take up more fpace, than is contained in the whole 
 heavens on this fide the fixed ftars. Hence we may 
 learn the great power of progreflive numbers, and that 
 fmall portion of fpace, neceffary to exprefs a number 
 by the help of numeral figures contrived for that 
 purpofe, which Ib far exceeds all our imagination*. 
 
 CHAP. II. 
 
 DISJUNCT PROPORTION, 
 
 OR 
 fb* RULE of THREE. 
 
 WH E N of four numbers, the firft has tha 
 fame Ratio to the fecond, as the third has to 
 the fourth : .Or when the fecond is the fame multiple 
 or quotient of the firft, as the fourth is of the third 5 
 then are thofe numbers faid to be in Disjunct Pro 
 portion. 
 
 IF four numbers are proportional diree~lly, as the 
 firft to the fecond j fo is the third to the fourth ; then 
 will they alfo be proportional ; Inverfely, Alternate 
 ly, 
 
ly, Compoundedly, Dividedly, and Mixtly. (Vid 
 Book ii. Cbaf. xrr.) 
 
 SECT. I. 
 
 DIRECT PROPORTION, 
 
 OR 
 The RULE of THREE DIRECT. 
 
 THIS is fomctimes called the golden rule, from the 
 great benefit people in all kinds of bufmefs receive 
 from it, as well the farmer and mechanic as the mer 
 chant, &c. It confifts of four numbers, which are 
 proportional, as the firft to the fecond -, fo is the third 
 to the fourth, as above : The two firft are a fuppofi- 
 tion, the third a demand, and the fourth the anfwer. 
 The two fuppofitions and the demand are always giv 
 en, and the fourth required. 
 
 Let the four numbers be, a, , c, d. Then a \ b :: 
 c \ dy dire&ly ; therefore, ay^dby^c, or ad~bc, per 
 Lemma ir, of the laft Scclion. 
 
 Whence by the nature of divifion bca^zd, that is, 
 if the product of the fecond and third terms, be di 
 vided by the firft, the quotient will be the fourth. Or 
 iince the Ratio of the firft to the fecond, is the fame 
 ss that of the third to the fourth ; it follows, that 
 'b~ay,c~d, that is, if the fecond term be divided by 
 the iirft, and that quotient multiplied into the third, 
 it will produce the fourth. 
 
 Now, in order to prepare your numbers for obtain 
 ing a fourth proportional, according to the forego^- 
 ing rules, you inuft obferve the following 
 
 RULE. 
 
R U L E. 
 
 WRITE that number which is of the fame name 
 with the number fought, in the middle place, and 
 the other two fo, that the exprefllon may read accord 
 ing to the nature of the queftion. 
 
 Let the following conditions be exprefied in num 
 bers. 
 
 What is the coft of 24lb. of cheefe, when the 
 price of 3lb. is 20 cfs. ? 
 
 Here the middle number muft be coft, becaufe 
 the fourth, or number required, is always of the 
 fame name and denomination of the fecond, by the 
 nature of the proportion : Hence the above condi 
 tions in numbers, is, 
 
 Thus, jib. zocts. 24bl. ; that is, if 3 pounds coft 
 so cts. what will 24 pounds coft ? Then to find a 
 fourth number, proceed as before directed. 
 
 Note. If the firft and third numbers are not &f the 
 Jame name, they muft be made Jo by the rules of 
 reduction : Alfo, if any of the numbers are com 
 pounds, they mujl be reduced to the leafl dejivmin- 
 ation mentioned. 
 
 EXAMPLES. 
 
 If 4lb. of cheefe coft 32 cts. ; what will 32olb coft 
 ic fame rate ? 
 
 OPERATION 
 
( '5* > 
 
 OPERATION. 
 
 Thefe numbers being placed according to the 
 
 \b, cts. Ib. 
 rule, will ftand thus, 4 : 32 :: 320 
 
 3* 
 
 640 
 960 
 
 4) ,10240 
 
 i (oo) 25(60=125 dol. 60 
 [cts. the anjwer. 
 
 Or, 32-7-4^:8 ; therefore) 320X8 2560 ^. 25 
 Jol* 60 els. the fame as before. 
 
 What will 6 yards of holland coft, when the price 
 of 40 yards, is 24 dol. 40 cts. ? 
 
 OPERATION. 
 
 yd. dol. cts. yd. 
 As 40 : 24 40 :: 6 ftated. 
 
 ^heny 24.4O-4-4On=.6i, and 6 X. 6 1=366 cts.^ 
 3 dol. 66 cts. the anfwer. 
 
 Find the value of loolb. of flax, when the price of 
 lib. is 12 cts ? 
 
 OPERATION; 
 
 Ib. cts. Ib. 
 As i : 12 :: 106 
 
 iz 1 2 dol. the anjwcr. 
 
 What 
 
( 153 ) 
 
 What is the coft of4olb. of cheefe, when the 
 price of jlb.^is 15 cts. 
 
 OPERATION. 
 
 Firft, 15-7-3 5, the ratio of the firft term to the 
 \fecond. 
 
 Then, 40X5 200 cts.~i dol. the anfwer. 
 
 What is the coft of Sylb. of tobacco, at 84 cts. per 
 Ib. ? 
 
 OPERATION. 
 
 Ib. cts. cts. Ib. 
 As i : 84- =8.5 :: 87 
 
 435 
 696 
 
 739-5=739-1- ^-= 7 ^ 
 [39i cts - the anfwer. 
 
 A goldfmith fold a tankard for 29 ^/. 97 r/j. at 
 the rate of i */0/. n cts. per oz. : What was the 
 weight of it ? 
 
 Anfwer. ' 27 oz. 
 
 A man bought Iheep at i <&/. 1 1 r/j-. per head, to 
 the amount of 5 1 doL 6 cts. : How many Iheep did 
 he buy ? Anfwer. 46, 
 
 SECT. II. 
 
 RECIPROCAL, or INSERTED PROPORTION* 
 
 OR 
 *ba RULE of THREE INDIRECT. 
 
 THIS kind of proportion, is the reverfe of the 
 former, as to the performance j for the greater the 
 
 U third 
 

 third term is, in refpect of the firft, the lefs will b( 
 the fourth, in refpect of the iecond ; whereas in di- 
 red proportion, the greater or lefs' the third term is.) 
 in rcfpect of the firft, the greater or lefs will be the 
 fourth term, in refpect of the fecond ; but to illuf- 
 trate the former. If two men can produce a certain: 
 effect in 12 days : In how many days would 6 mer 
 produce the fame ? Here it is manifeft, that 6 mer 
 would produce the effect in lefs time than 2 ; anc 
 therefore the greater the third term is, the lefs will 
 be the fourth. Again, if lomen can produce a cer 
 tain effect in 6 days : In how many days would 4 
 men do the fame ? Here it is evident, that 10 men 
 would produce the effect in lefs time than 4 men \ 
 and therefore the lefs the third term is, the greater 
 will be the fourth : Confequently, more requires lefs^ 
 and lefs requires more, in indirect proportion. 
 
 HERE the fame rule is to be obferved, in Hating 
 your queftion, as in the former proportion, and did 
 refults in refpect of names and denominations are the| 
 fame alfo : Then to find a fourth proportional, pro-! 
 ceed with the following rules. 
 
 RULE I. 
 
 MULTIPLY the firft and fecond numbers together, 
 and divide that product by the third ; the quotient 
 reiulting will be the fourth proportional required. 
 
 RULE II. 
 
 DIVIDE the fecond number by the third, and that 
 
 2uotient ^multiplied into the firft, will produce the 
 ajiirth, 
 
 RULE 
 
( '55 ) 
 
 RULE III. 
 
 DIVIDE the third term by the firft, and the fecond 
 :rm by this quotient 5 and the refulting quotient 
 iil be the fourth number. 
 
 EXAMPLES. 
 
 If 5 men can perform a certain piece of work in 
 days : How long will four men be in doing the 
 
 ame 
 
 OPERATION. 
 
 Men. D. Men. 
 S 8 4 
 
 5_ 
 
 4) 40 D. 
 
 Or, 
 
 4_4Q 
 
 8+r~~ 
 
 mo days as before. 
 
 If 2,0 bufliels of grain, at 50 o?#/.r per bulhel, will 
 pay a debt : How many bufhels at 60 cents ^r bufh- 
 il will pay the fame ? 
 
 OPERATION. 
 
OPERATION, 
 cts. Bujb. cts. 
 
 50 20 60 
 20 
 
 6(o)lOO(o 
 164 
 
 Anfwer. 164 bujhds. 
 
 If 2 yards of cloth, i yard and 3 quarters wide, 
 is fufficient to make a coat ; how many yards of i 
 yard wide, will make the fame ? 
 
 OPERATION. 
 
 ^ 
 
 3^ 3~ yards the anfwer. 
 
 A man being defirous to draw off a cafk of bran 
 dy into bottles, finds that if he makes life of three 
 quart bottles, it will require 60 : How many five- 
 pint bottles will it require, to draw off the aforefaid 
 cafk of brandy. Anfwer. 72 bottles. 
 
 A man bought a piece of cloth 9 quarters wide, 
 and ii quarters long : How many yards of 3 quar 
 ters cloth will line it ? Anfwer. 8.1 yards. 
 
 If 3'-y ai *ds of yard-wide cloth will make a coat : 
 How many yards of 7 quarters cloth, will make the 
 ? Anfwer. 2 yards. 
 
 SECT. 
 
( 157 ) 
 
 S E C T. III. 
 COMPOUNDED RATIO. 
 
 COMPOUNDED Ratio is when the antecedent and 
 confequent taken together, is compared to the confe- 
 quentitlelf : thus, a : I :: c : ^, directly, therefore .by 
 cornpofition ; as a-\-b : b :: c-\-d : d. 
 
 Note. The fame Rule is to be oljerved here, as in 
 direft proportion. 
 
 EXAMPLES. 
 
 If A can produce a certain effe6t in 5 days, B can 
 do the fame in 7 days ; fet them both about it toge 
 ther, in what time will it be finilhed ? 
 
 ORERATION. 
 
 12)35(2 days. 
 24 
 
 22 bourf. AnJ. 2 days lib. 
 If A in in 5 hours, can make 1000 nails, B in B 
 hours, can make 2000 : In what time would they 
 ' jointly make 50000 nails ? 
 
 Here 
 
Here you muft firft find in what time each perfon 
 would make 50000 nails, and then proceed as in the 
 lait example. 
 
 OPERATION. 
 
 n. h. n. 
 
 As icoo : 5 :: 50000 : 50000X5-^-1000 =z 250 
 boars, the time it would take A to make 50000 nails. 
 n. h. n. _ 
 
 As 2000 : 8 :: 5000 : 50000X^-7-2000^:200 
 hours, the time it would take B to make 50000 nails. 
 
 e^ as 2504-200 : 200 : : 250 : 200 x 250- 
 450 1 1 r~ hour s y the time it would take them jointly 
 to make 50000 nails, as was required. 
 
 Note. From this operation, we have the following 
 general theorem forjolving all queftions ofaftmil- 
 ar nature, let the ferfons or agents employed, be 
 any numler whatever, 
 
 THEOREM. 
 
 MULTIPLY the joint effect with the time each one 
 would produce his particular effect, and divide the 
 product by the faid particular effect ; then multiply 
 all the refuhing quotients together for a dividend, 
 and make the fum of them a divifor -, then divide, 
 and the refulting quotient will be the time required. 
 
 SECT, 
 
( 159 ) 
 
 S EC T. IV. 
 
 DIVIDED RATIO. 
 
 DIVIDED Ratio is when the excefs wherein the an 
 tecedent exceeds the confequent, is compared with 
 the confequent : fbus, a\b\\ c\ d> direftiy , therefore 
 by divifien as a b : b : : c d : d. 
 
 EXAMPLES. 
 
 If A can do a piece of work in 8 days, A and B 
 can do it in 5 days : In what time can B do the fame 
 work ? 
 
 OPERATION. 
 
 u. h. 
 
 As* 5^3 -.5:: 8: 5X8^3-40-3- '3 8 > ** 
 time required. 
 
 Two flitps, one in chafe of the other, the head- 
 moft (hip is 48 miles diftant from the other, and fails 
 at the rate of 4 miles per hour, and the fternmoft 
 Ihip at the race of 7 miles per hour : How long be 
 fore the fternmoit fhip will overtake the o:her ? 
 
 OPERATION. 
 
 As 7 4 IT 3 * i : : 48 : 48X1-7-316 bcurs, tbe 
 time required. 
 
 A hare is is 50 leaps before a grey -hound, and 
 takes 4 leaps to the grey-hound's three ; but 2 of 
 the grey-hound's leaps are as much as three of the 
 hare's : How many leaps rnuft the ^r?.y-hound take 
 to catch the hare ? 
 
 Her you mufl firft find how many kafs of ths bart, 
 anfivers to three of the grey-hound' V : Tbu^ 2:3^ 3 
 
?hen, as 4.5 4~-5 :3 : *-5 : 3X5o-~-.5 300 
 /fo anjwer. 
 
 The hour and minute-hand of a clock are exactly 
 together at 12 o'clock j when are they next together I 
 
 Here the proportion of the vehfities of the hour and 
 minute-hand^ is as i to 12. Thereforeyii iHn : i 
 
 :: 12 t i2Xi-S-u~ih. 5-/-A the anjwer. 
 
 If A, B and C, can' produce a certain effect in 12 
 days, A can do it in 30 days and C in 50 days, in what 
 time will B do the fame work ? 
 
 Fir ft find the time in which A and C, would produce 
 the effeff jointly, by Ratio of compofition. Thus, 
 
 30-1-50 : 50 :: 30 : 50X30-^70 =21! days, 
 
 as 21^ 12^:91 : 12:: 21 - T : i$^~ y the time requir 
 
 ed. 
 
 There is an ifland 100 miles in circumference, 
 and two footmen, A and B, fet out together, to trav 
 el the fame way round it, A travels 15 miles per day, 
 and B 17 miles: When will they come together 
 again ? 
 
 Firfty find how many miles B muft travel to over 
 take A y after their departure : T'btts, as 17 15:1:2: 17 
 :: 100 : 850, the number of miles B muft travel y 
 which is 50 days journey ; therefore they will be together 
 Ggain 50 days after their departure. 
 
 There is three pendulums of unequal lengths -, the 
 firft of which vibrates once in 12 feconds, the fecond 
 in 1 8 feconds, and the third in 24 feconds : Now 
 fuppofing- them all to move from a line of conjunc 
 tion, at the fame moment of time : When will they 
 come into the fame fituation again, and move on 
 together ? Firft 
 
Firft y find the time when the two firft pendulums 
 W9ve on together y as in the loft example : Thus, 1 8 - 
 12: 1 8 :: i : i% ^1+6^ $ytbfmmber of vibrations $ 
 the firft y which is performed in 36 Jeconds~i vibra~ 
 tions of thefecond. ThereforCy after th? firft has vi 
 brated 3 timeSy and the fecond 2, they will move on to 
 gether again. 
 
 In the next place y we muft examine into the fituatim 
 of the third pendulum , at the conjunction of the two firft. 
 In 36 fecondsy there is 1.5 vibration of the third pen~ 
 dulumy which is therefore y .5 of a vibration , diftant 
 from the conjunction of the other two ; whereforey .5:1 
 :: 3 : 6, the number of vibrations of the fir ft y at which 
 time, they all come into a line of conjunct ion y and move on 
 together. Conjequentlyy when the firft has made 6 vi- 
 brationSy the Jecond will have performed 4, and the 
 third three^zi^y.3=?]ijecondsy the time required. 
 
 If A can do a piece of work in 20 days ; A and 
 B in 13 days ; A and C in 1 1 days ; and B and C 
 in LO days : How many days will it take each perfon 
 i to perform the fame work ? 
 
 OPERATION. 
 
 As 20- 13 : 13 :: 20 : 374- the time that B would 
 1 do it. 
 
 As 20 ii : ii :: 20 : 244 the time that C would 
 do it, 
 
 C HA 
 
CHAP. III. 
 
 SIMPLE INTEREST. 
 
 SIMPLE intereft is a premium of a certain fum 
 paid for the loan of money borrowed for a par 
 ticular term of time, at any rate per cent or hundred, 
 as the borrower and lender fh all agree. 
 
 THUS, if 100 dollars be lent at 6 per cent per an 
 num, the premium for i year will be 6 dollars, for 
 a years 12 dollars, for 3 years 18 dollars; and fo 
 on. 
 
 THE fum lent is called the principal, and the pre 
 mium per 100, the Ratio or rate per cent -, and the 
 amount is the principal and intereft added together. 
 
 ALL the varieties of fimple intereft, are comprifed 
 in the following cafes. 
 
 C AS E I. 
 
 When the fum lent, Is for any number of years > and 
 the rate per cent, any number of dollars. 
 
 RULE. 
 
 MULTIPLY the principal with the number of years, 
 and that product with" the Ratio, and divide by 1005 
 the quotient refulting, will be the intereft required. 
 
 EXAMPLES. 
 
 Required the intereft of 700 dollars, for 4 years, 
 at 6 per cent per annum ? 
 
 OPERATION. 
 
OPERATION. 
 
 700 
 4 
 
 2800 
 6 
 
 1(00)168)00 
 'nfwer. 168 dollars y the inter eft required. 
 
 Required the intereft of 3520 dollars, for 7 years, 
 at 6 per cent per annum. 
 
 OPERATION. 
 
 1(00)1478(40=1478 dol. 4octs. the 
 
 \anjwer. 
 
 What is the intereft of 57821 dollars, for 5 years, 
 
 at 5 per cent per annum ? AnJ. 2891 dol. 5 cts. 
 
 What is the intereft of 5972 dollars, for 12 years, 
 
 at 3 per cent per annum ? AnJ. 716 dol. 64 cts. 
 
 CASE II. 
 
 " When thefum is lent for years and months 5 the Ra 
 tio the fame as before. 
 
 RULE. 
 
 REDUCE the number of months into the decimal of 
 a year, then multiply the principal with the time, 
 
 and 
 
and that product with the Ratio, then divide by 100 
 and you will have the intereft required. 
 
 . Or >. 
 
 MULTIPLY the principal with the number of years, 
 
 and take parts of the principal for the reft part of 
 the time, a^.d add them to the reft $ then proceed as 
 before directed. 
 
 Required the intereft of 735 dollars, for 5 years/ 
 4 months, at 5 per cent per annum. 
 
 OPERATION. 
 
 4 monthszr-J- of a year, 3)735 
 
 5 
 
 3675 
 
 245=735-^3 
 
 3920 
 
 5 ratio. 
 
 1(00)196)00=196 dollars, the 
 [intereft required. 
 
 Required the intereft of 52374 dollars, for 7 years 
 8 months, at 6 per cent per annum. 
 
 OPERATION. 
 
OPERATION. 
 
 *? of a year, 3)52374 
 7 
 
 366618 
 
 i745%- *f 5*374 
 17458 
 
 401534 
 
 i(oo)-249 a )4 24092 4, the in- 
 \tweft required. 
 
 What is the intereft of 32104 dollars, for 4 years, 
 3. months, at 5 per cent per annum ? 
 AnJ. 6827 dol. 10 cts. 
 
 CASE III. 
 
 When the Rath is dollars and farts of a dollar* 
 the reft the fame as * before. 
 
 RULE. 
 
 1. REDUCE the number of months into the decimal 
 of a year, and multiply the principal with the whole 
 
 time. 
 
 2. REDUCE the fractional parts of the Ratio into 
 
 the decimal of a dollar. 
 
 3. MULTIPLY the former refult with the latter, 
 and divide by 100, and you will have the intereft re 
 quired : Or, 
 
 MULTIPLY the principal with the number of years, 
 and take parts of the principal for the reft part of the 
 time, and add them to the former product > then 
 
 multiply 
 
( 166 ) 
 
 multiply this product with the dollar's part of the 
 rate, and take parts of the multiplicand for the reft 
 part of the rate, and add them to the latter product j 
 then divide them by 100, and you will have the in- 
 tereft required. 
 
 EXAMPLES. 
 
 Required the intereft of 700 dollars, for 3 years 
 6 months, at 6~ per cent per annum. 
 
 OPERATION. 
 
 700 
 
 3500 
 
 2IOO 
 245O.O 
 
 6,$ ~ 
 
 122500 
 147000 
 
 doL cts. 
 
 1(00)159)25.00159 25, the anfwer* 
 Or. 6 months^ a year 2)700 
 
 3 
 
 2IOO 
 
 for tbt -per cent 2)2450 
 
 6 
 
 14700 
 1225 
 
 doL cts. 
 
 I () I 59( 2 5= I 59 25 as 
 /"'* Required 
 
i7 
 
 Required the intereft of 3520 dollars 17 cents, for 
 2 years 6 months, at 5^ per cent. 
 
 OPERATION. 
 2)3520.17 
 
 704034 
 176003.5-3520.17-7-2 
 
 4)8800.375 
 5 
 
 44001875 
 2200.093 
 
 doL cts. 
 
 1(00)462(01.968462 i. 968 tie 
 
 \anf. 
 CASE IV. 
 
 When fhefum is lent for any number of weeks. 
 
 RULE. 
 
 REDUCE the number of weeks into the decimal of 
 a year, and proceed as in the laft cafe. 
 
 Or, 
 
 FIND the intereft of the given fum, according to 
 the foregoing rules for one year $ then fay, as 52, the 
 number of weeks in a year, is to the intereft thus 
 found ; fo is the given number of weeks, to the in 
 tereft required. 
 
 EXAMPLES. 
 
 Required the intereft of 720 dollars, for 10 weeks, 
 
 at 5-i per cent per annum, 
 
 OPERATION. 
 
OPERATION. 
 
 720 
 
 .19211: time nearly. 
 
 138.240 
 
 691200 
 691200 
 
 dol. cts. 
 
 7,60.3200:1:7 60/32 the anfwer* 
 
 ix). dol. cts. w. 
 Or, 720 As 52:39 60 :: 10 
 
 * 5-5 
 
 10 
 
 3600 
 3600 
 
 52\396-(7-6*~7 Jol. 61 fts. 
 
 J 20 
 
 312 
 
 80 
 
 28 
 
 Note. Tfareafw why the two methods of operation a- 
 lo r oe>do not bring cut tbefameanjwer, is lecauje the 
 decimal of 10 weeks fan never be exaftly found ; 
 yet the err GUT artfing from any Juch computation, 
 will be inconfidcrabh. 
 
 Required 
 
I. LW 7 / 
 
 dol. cts. 
 
 Required the intereft of 5 27 2, for 13 weeks, at 
 ;i per cent per annum. 
 
 659000 
 659000 
 
 i (00)7 2(49*000=7* dol. 46 cts. the anf. 
 
 C A S E V. 
 
 When tbefum is lent for any number of days. 
 RULE. 
 
 REDUCE the days into the decimal of a year, and 
 
 proceed as in the laft cafe. 
 
 Or, 
 
 T MULTIPLY the given fum with the number of 
 dap, S US P rodua g with theRatio a dmdena 
 
 V MULTIPLY 365, the number of days m a year, 
 with ,00 for a divifor ; then divide, and the quotient 
 will be the intereft required. 
 
 As 365 days, is to the intereft of die principal for 
 gjjff 
 
EXAMPLES. 
 
 Required the intereft of 300 dollars, for 219 days 
 at 6 per cent per annum. 
 
 OPERATION. 
 
 Or, 219 365 
 
 3 . 300 100 
 
 65700 36500 
 180.0 6 
 
 6=ratio y . doL cts. 
 
 d. 365(00)3942(00) 10 80 aslefin 
 
 1(00)10)80,0=1080^.^.365 
 
 292.0 
 2920 
 
 o 
 
 Required the intereft of 1000 dollars, for 35 days 
 at 6 per cent per annum. 
 
 OPERATION. 
 d. dol. d. 
 
 1000 As 365: 60:: 35 
 
 60 
 
 6o - 365x2100 (5 del. 75 cts. 
 
 71825 
 
 275.0 
 
 15150 
 
 1825 
 dot. cts. 
 
 125 
 
 CASE 
 
CASE VI. 
 
 Wbtn the principal, Rath, and intereft are given t* 
 
 id tke time. 
 
 RULE. 
 
 i. FIND the intereft of the principal for one year, 
 
 it the eiven rate. 
 2 SAY as the intereft thus found, is to one year ; 
 
 b is the given intereft, to the time required. 
 
 EXAMPLES. 
 
 Required the time in which 500 dollars will gain 
 150 dollars, at 6 per cent per annum. 
 OPERATION. 
 
 dol. y. JI- 
 As 30:1:: 15 
 6 * 
 
 Find in what time 700 dollars will gain 159 o. 
 cts. at 6iper cent per annum. 
 
 OPERATION. 
 
 700 del. cts. y. dot. cts 
 
 A 45 5 o:i: 
 
f( 172 ) 
 
 **- dol. cts. 
 
 Required the time in which 283 334. will amount 
 to 370 dol. 50 cts. at 6 per cent per annum. 
 
 OPERATION. 
 
 dol. cts. dol. y. dol. cts. 
 
 33r As 17 i 87 i6 
 
 6 i 
 
 17.0000 
 
 17X3=51)261. 50(5,12^*, the 
 2 5S {anfwer* 
 
 140 
 1 02 
 
 CASE VII. 
 
 When the Ratio, time, and amount are riven to 
 find the principal. 
 
 RULE. 
 
 As the amount of 100 dollars, a* the rate per 
 :cnt and time given, is to 100 dollars ; fo is the giv 
 en amount, to the principal required. 
 
 EXAMPLES. 
 
 9 Required the principal that will amount to 7766 
 dot. 40 cts. in 7 years, at 6 per cent per annum. 
 
 OPERATION, 
 
( 173 ) 
 
 OPERATION. 
 
 TOO 
 
 6 
 
 6.00 
 7 
 
 42.00 
 doL dol. cts. dol. cts. 
 
 As 100+421:142 : 100 :: 3766 40 : 2793 23 \\\ 
 
 the anfwer. 
 
 Required the principal that will amount to 861 
 dollars in 4 years, at 6 per cent per annum. 
 
 OPERATION. 
 
 100 100+24^124: 100 :: 868 
 
 4 ioo 
 
 400 124)86800(700 
 
 6 868 
 
 24.00 ooo 
 
 The r efcr e 700 doL is the principal required. 
 
 Requkred the principal that will amount to 270 
 dollars, in 2 years at 6 per cent per annum. 
 
 OPERATION. 
 
( '74 ) 
 
 OPERATION. 
 100 As 100+12=: 1 12 
 
 2 
 
 200 
 
 6 
 
 12.00 
 
 16 
 
 ) 241 doL 7 T |4 cts. is the principal re 
 quired. 
 
 Admit I have a legacy of 196 dol. 66-* cts. to pay, 
 but is not due till the end of 3 years, and the lega 
 tee being in want of money, defires I would lend him 
 fome : What fum muft he have to amount to his 
 legacy in 3 years, at 6 per cent per annum ? 
 
 Anfwtr. 1 66 dol. 66|- cts. 
 
 CASE VIII. 
 
 When the principal, amount > and time are given to 
 find the Ratio. 
 
 R ULE. 
 
 i. SUBTRACT the principal from the amount, and 
 the remainder is the intereft. 
 
 2. 
 
( 175 ) 
 
 2. SAY as the given principal, is to its intereft ; fo 
 is 100 dollars, to the intereft of 100 dollars for the 
 given time. 
 
 3. DIVIDE the intereft of i oo dollars thus found 
 by the given time, and the quotient will be the ra 
 tio required. 
 
 EXAMPLES. 
 
 Required the rate per cent per annum fuch, that 
 1240 dollars may amount to 1400 in 3 years. 
 
 OPERATION. 
 
 1400 1240: 200 :: ico 
 i 200 100 
 
 200 1 24)0} 2ooo(o( 1 6. 1 2 
 
 /I24 
 
 760 
 
 744 
 
 16.0 
 
 124 
 
 360 
 
 248 
 
 dol. cts. 112 
 5.37 the Ratio required. 
 
 Required 
 
Required the rate per cent per annum, that 100 
 dollars in 7 years will amount to 135 dollars. 
 
 OPERATION. 
 
 135 As 100 : 35 :: 100 
 100 100 
 
 3$=intcreft, 1)00(00(35 
 
 7 (35 </<?/. 
 
 $~ ratio req. 
 
 At what Ratio will 3333 dollars 33 J- cents amount 
 to 4000 dollars in 20 years. Anfwer* 6 dot. 
 
 CASE IX. 
 COMMISSION or PROVISION. 
 
 THIS is a premium allowed to faftors for buying 
 or felling goods, wares, or merchandize, at fo much 
 per cent, without any regard to time -, which rate is 
 governed according to the cuftoms of particular 
 places. 
 
 THE method of proceeding, is the fame as in cafe 
 in, except no regard is had to time. 
 
 EXAMPLES. 
 
 If I buy goods for my correfpondent in Philadel 
 phia, to the value of 4000 dollars : What may I 
 demand for my commifTion, at 44- per cent ? 
 
 OPERATION. 
 
OPERATION, 
 
 4000 
 
 . thsanfwer. 
 
 Required the commiflion for felling 5720 dollars 
 worth of goods, at 2~ per cenr. 
 
 OPERATION. 
 
 143*000=143 doL the anf. 
 
 My correfpOhdent fends me word, that he has dif- 
 burfed goods on my account, to the value of 13333 
 dollars 334. cents : What is his commiflion at ai per 
 cent ? Anfwer. 333 doL 33rCts* 
 
 CASE X. 
 
 BROKERAGE. 
 
 BROKERAGE is an allowance of fo much per cent, 
 made to perfoas called brokers, for finding cuftom- 
 ers a and felling to them goods, wares, &c, -which be 
 long to qther men^ 
 
 Z 
 
RULE. 
 
 FIND the intereft of the given fum, at one peri 
 cent i or which is the fame thing ; divide the given 
 fum by 100, and take parts of the quotient, agree 
 ing with the rate per cent. 
 
 Or, 
 
 REDUCE the rate per cent to a decimal, and mul 
 tiply it with the given fum -, then divide by 100, and 
 the quotient will be the anfwer- 
 
 EXAMPLES. 
 
 Required the Brokerage of 1000 dollars, at 25 cents 
 per cent. 
 
 OPERATION. 
 
 1(00)10(00 
 
 25 cents ^ of a dollar, therefore, 10-7-4=2 dollars 
 50 cents = Brokerage of 1000 dollars divided by 4:= 
 .Brokerage required. 
 
 Or, 
 
 1000 
 
 2.50002 dol. 50 cts. as be- 
 
 [fore. 
 
 Required the Brokerage of 324 dollars 40 cents, 
 4 of a dollar per cent. 
 
 OPERATION. 
 
( '79 ) 
 
 OPERATION. 
 
 324.40 
 
 . 20=} of a dollar > 
 
 1(00)64.8800^:64.88 cts. t he Brokerage 
 'jquired. 
 
 What is the Brokerage of 15600 dollars, at 77 
 :ents per cent ? AnJ. 120 doL 12 cts. 
 
 CHAP. IV. 
 
 COMPOUND INTEREST. 
 
 GO M P O U N p. Intereft arifes from the com 
 putation of the intereft of any principal added 
 ,:o its intereft, when the payment Jhould be made ; 
 vhich forms a new principal at every time when 
 I Repayments become due; and is for this realbn, 
 bmetimes called intereft upon intereft. 
 
 THUS, if 100 dollars be put to intereft at 6 dollars 
 )er cent per annum ; at the end pf the firft year, the 
 ntereft will be 6 dollars as in fimple intereft, which 
 f added to its principal will be 106 dollars, for a new 
 )rincipal the fecond year, which principal at the 
 :nd of the fecond year, will amount to 112 dol- 
 ars 36 cents ; which is 36 cents more than if 100 
 lollars had been put out at fimple intereft only. 
 
 THE Compound Intereft of any fum may be found 
 )y the following 
 
 RULE. 
 
 i. FIND the intereft of thepropofed fum for the 
 irftyearatthe given rate per cent, as in fimple in- 
 srcft, 2. 
 
2. ADD this intereft to its principal, which amount 
 makes the principal for the fecond year. 
 
 3. FIND the intereft of the fecorid year's principal,, 
 in the fame manner as \ou did the firft, and add it to! 
 its principal, for the third year's principal* which 
 mull be computed as before ; and fo on, for the time 
 required . 
 
 4. SUBTRACT the given principal from tfye laft 
 amount, and the remainder will be the Compound' 
 Intereft required. Or, 
 
 !FIND the amount of one dollar for one year, at 
 the given rate per cent, and multiply it continually 
 with the principal, as many times as the given num 
 ber of years, and the refill ting product will be the 
 amount ; from which fubtradfc the principal, and the 
 remainder will be the Compound Intereft. 
 
 EXAMPLES. 
 
 "Required the Compound Intereft of 100 dollars, 
 for 3 years, at 6 per cent per annum. 
 
 OPERATION 
 
 tOO IOO 
 
 6 6 
 
 6.00 106 
 
 6 
 
 6.36 6.7416 
 
 , 119 dpi, i o. 1 6 cts. i oo doL 19 d*l. i o. 1 6 
 
 intereft require-d. 
 
 Or, 
 
 106 
 6.36 
 
 112.36 
 6.7416 
 
 112.36 
 
 6 
 
 119.1016 
 
: 106 :: i : i.o6f iooxi.o6Xi.o6X 
 the amount of i dollar for i< t;.<p.6'^: 119.1016=11 9 
 year> at 6 per cent. [dol. 10.16 cts. 
 
 fhen, 119 dol. 10.16 cts. IQQ doL iydol. 10.16 
 cts. the fame as before. 
 
 THE following is a Table of the amount of i dollar, 
 from i to 30 years ; for the more ready computing 
 Compound Intereft.at 6 per cent per annum. 
 
 J? 
 
 & 
 "t 
 
 The amount of j 
 dol. at 6 pgr cent, 
 
 *3c . comp . inter eft . 
 
 I s 
 
 y ^ amount of I 
 </o/. at 6r per cent, 
 &c . comp . inter eft . 
 
 ? 
 
 I he amount of \ 
 dol. at 6 per cent, 
 
 &c.ccmp.intereft. 
 
 I 
 
 .06 
 
 II 
 
 1.898298558 
 
 21 
 
 3-3995 6 36oO 
 
 2 
 
 .1236 
 
 12 
 
 2.012196471 
 
 22 
 
 3-603537416 
 
 3 
 
 .I9IOl6 
 
 *3 
 
 2.132928260 
 
 23 
 
 3.819749661 
 
 4 
 
 .26247696 
 
 H 
 
 2.260903955 
 
 24 
 
 4.048934641 
 
 5 
 
 .338225577 
 
 *5 
 
 2.396558193 
 
 2 5 
 
 4.291370719 
 
 6 
 
 .4185191 12 
 
 16 
 
 2.540351684 
 
 26 
 
 4.549382962 
 
 7 
 
 .503630259 
 
 *1 
 
 2.692772785 
 
 27 
 
 4.822345940 
 
 8 
 
 .593848074 
 
 18 
 
 2-8.54339 I 5 2 
 
 28 
 
 5.111686697 
 
 9 
 
 .689478959 
 
 T 9 
 
 3.025599502 
 
 29 
 
 5.418387899 
 
 10 
 
 .790847696 
 
 20 
 
 3.207135472 
 
 jo 
 
 5-74349 I ?29 
 
 ?v the above table, the amount of any fum may 
 be computed from i to 30 years, by only multiply 
 ing the principal with the numbers {landing againft 
 the number of years in the table, and the product 
 will be the amount required. 
 
 EXAMPLES. 
 
 Required the amount of 127 dollars, for 7 years, at 
 6 per cent per ^nnum. 
 
 OPERATION, 
 
OPERATION. 
 
 dgainft 7 in the table is 1.503630 
 
 127 
 
 10525410 
 
 3007260 
 
 1503630 
 
 ~I90 96.1 
 
 , Required the Compound Intereft of 555 dollars, 
 for 30 years, at 6 per cent per annum, 
 
 OPERATION. 
 
 dgainft 30 ^ /^ tails is 5.743491 
 
 ,555 
 
 28717455 
 28717455 
 
 28717455 
 
 3187.63750511: amount. 
 
 <?ben, 3187 dol. 63,7505 f/j. 555 doL 2632 
 doL 63.7505^.?. the intereft required. 
 
 CHAP. 
 
C H A P. V. 
 
 REBATE or DISCOUNT. 
 
 REBATE or Difcount is when any fum of mo 
 ney is due at a certain time to come, and the 
 debtor is ready to make prefent payment, provided 
 he can have allowance made him at a certain rate per 
 cent per annum, which allowance is called the Rebate 
 or Difcount, and the prefent payment, a fum of 
 money, which ifputtointereft, would amount to the 
 given fum, at the rate per cent and time given. 
 THE Rebate of any fum is found by the following 
 
 RULE. 
 
 As the amount of 100 dollars, at the rate per cent 
 and time given, is to the intereft of 100 dollars, at the 
 fame rate and time ; fo is the given fum, to the Re 
 bate : And from the given fum fubtrad the Rebate, 
 and the remainder will be the prefent payment. 
 
 EXAMPLES. 
 
 A, hath 100 dollars due to him, to be paid at the 
 end of 2 years j but his debtor agrees to make pre 
 fent payment, provided A will make a Rebate at 6 
 dollars per cent per annum ; Required the Rebate. 
 
 OPERATION'. 
 
OPERATION. 
 
 100 
 6= ratio, 
 
 600 
 
 12,00 
 
 2 112 
 
 12 
 
 IJ2)l200(l0.7l 
 
 112 [rebate 
 
 80.0 
 
 784 
 
 1 60 
 
 XI2 
 
 Required the Rebate of 720 dollars, for 2.1 years, 
 at 6 per cent per annum. 
 
 OPERATION. 
 
OPERATION. 
 ^100+15=115 : 15: 1720 
 
 '5 
 
 3600 
 720 
 
 115)10800(93.91 = re- 
 1035 \iate req. 
 
 450 
 
 15.00 345 
 
 105.0 
 I03S 
 
 150 
 "5 
 
 35 
 
 Find what fum ought to be paid down for a debt 
 of 1000 dollars due 3^ years hence, difcQUHting at 5 
 per cent per annum. 
 
 PERSON* 
 
 xoo 
 5 
 
 2)500 
 3 
 
 1500 
 250 
 
 17.50 
 
 Aa 
 
'( J86 ) 
 
 100+17 .50=117. 50 : 17.50 :: rooo 
 
 IOOO 
 
 117,50)1750000(148.93 
 
 11750 
 57500 
 
 47000 
 
 105000 
 94000 
 
 I IOOO.O 
 
 105750 
 
 42500 
 35 2 5 
 
 7250 
 1000 148.93=851 doL 7 cts. the anjwer. 
 
 Suppofe I have a legacy due to me of 4000 dollars, 
 whereof 800 dollars is to be p'aid in 8 months, and. 
 the reft at the end of 16 months : How much ought 
 I to receive for prefent payment, allowing 6 per cent, 
 &c. difcount ? Anjwer. 3732 doL 21 cts. 
 
 A owes B 15000 dollars, one half of which is to be 
 paid in 4 months, and the reft at the end of 8 months : 
 What ought B to receive Fn p!fent payment, allow 
 ing 6 per cent difcount ? <dnf* 14564 doL 58 cts* 
 
 CHAP 
 
( i*7. ) 
 
 C H A P. VI. 
 
 of P ATMENTS 
 <?be COMMON WAT. 
 
 EQUATION of payments, is when feveral fums of 
 money are due at different times, to find a certain 
 time when the whole may be paid without lofs to 
 either party 
 
 RULE. 
 
 MULTIPLY each payment with its refpedtive time, 
 anddiyi.de the. fum of the products by the fum of the 
 payments 5 the quotient refill ting,' will be the time 
 required. 
 
 THIS rule will give the equated time near .enough 
 for common practice in matters of this nature ; but 
 not accurately true, becaufe the rule is founded on a 
 fuppofition that the fum of the interefts of the debts 
 due before the equated time, computed from the 
 times they become due to that time, is equal to the 
 fum of the intereft of the debts, payable after the 
 equated time, computed from that time to their re- 
 fpective terms of payment ; that is, the gain made by 
 the debtor's keeping- thofe debts which become due 
 before the equated time, until that time, is equal to 
 the lofs fuftained by paying thofe debts at the equated 
 time, which are not due till afterwards ; but it ismani- 
 feft, that the gain made by keeping a debt any time 
 after it is due, is equal to the intereft of that debt 
 for that time; but the lofs fuftained by paying a debt 
 any time before it becomes due, is plainly no more 
 than the rebate of the debt for that time -, and fmce 
 the rebate is always lefs than the intereft of the fame 
 
 fum, 
 
fum, it follows that the fupppfition is not true, andr 
 consequently the rule falfe. 
 
 EXAMPLES in equation of payments the common 
 way. 
 
 A owes B 100 dollars, whereof 50 dollars is to be 
 paid at the end of 4 months and the reft at the end 
 of 8 months : Required the time when the whole 
 may be paid without lofs to either party. 
 
 OPERATION. 
 
 Fir ft y 50X4=200 thejirft payment with its time ; 
 
 Secondly y 50X8=1400 thejecond payment with its 
 time : 
 
 fberiy 400 4- 200600 thefum of the produfts. 
 
 Andy 504-50=1100 thefum of the payments : 
 
 Confequently, 6oo-i-ioo=6 months, the time re 
 quired. 
 
 W owes X 865 dollars, whereof 50 dollars is to 
 be paid prefent, 195 dollars to be paid in 8 months, 
 and the reft at the end of 12 months : Required the 
 equated time to pay the whole . 
 
 OPERATION. 
 
 50X150 the fir ft payment with its time :. 
 195X81560 the fecond payment with its time : 
 62oX 1211:7440 the laft payment with its time : 
 And 5O-(- 1 560 +7440^:9050 the Jum of the pro- 
 
 dutts. 
 
 Confequentfyy ?|4T^ 10 won$s 13 ll\days the tim? 
 
 required. 
 
 P owes a debt to be paid at 5 feveral payments, 
 in the following manner, to wit, 4 i n 4 months, ^ at 
 8 months, 4 at 12 months, | at 16 months, and 4 at 
 
 20 
 
20 months : Required the equated time to pay the. 
 whole. 
 
 OPERATION. 
 
 Suppofe the debt =25 dollars, one ffth of which 
 is $ dollars, then 5 X4+ 5X^+5X1 2+5 Xi6 + 5X 
 10=20+40+60+80+10011:300. Therefore, 3 |4z:- 
 12 months, the time required. 
 
 IN the iblution of the above queftion, we madeufe 
 of 25 dollars to reprefent the whole debt ^ but any 
 other number would have equally fucceeded, as may 
 be thus analytically demonftrated. 
 
 Let x~anyfum whatever, to be. paid in manner as 
 above. 
 
 jut * i v* v v* 
 
 xis~> and 7X4+7X8 -f-^X 12+ T X 16 
 
 , , , - ^ 
 
 4- 4- - + - = =fumof 
 
 tleprodufts of the fever al payments with their refpeft- 
 
 ive times : Therefore, -- r^= - ^-= i imonths 
 
 5 5* * 
 
 tbefame as before. <3 E D 
 
 CHAP. VII, 
 
 B 4 R T E R. 
 
 BARTER is the exchanging one commodity 
 for another in fuch a manner, that the parties 
 bartering, may neither of them fuftain lofs. Thus, 
 fuppofe A hath 5olb. of ginger, at 30 cents perlb. 
 and would Barter with B for pepper at 70 cents per 
 
 Ib. 
 
Ib. What quantity of pepper mud B give A for his 
 50 Ib. .of ginger ? 
 
 IN the Iblucion of this queftion, and all others of 
 the like nature, you muft firft find the value of the 
 given quantity at the given price, and then find how 
 much of the quantity fought at its price, will amount 
 to the value of the given quantity, and the refult will 
 be the a-nfwer to the queftion, 
 
 Thus, in the above queftion, the quantity given, is 
 5clb. of ginger, at 30 cents per Ib. and the quantity 
 fought is pepper, at 70 cents per Ib. Therefore, as 
 70 cts. : lib. :: 15 dol. (the price of the ginger) : 
 214 Ib. the quantity of pepper required. Oonfe- 
 quently in Barter, the method of operation is the 
 fame as i:i the rule of three direcl. 
 
 EXAMPLES. 
 
 Required the quantity of flax, at cents per Ib. 
 thatmuft.be given in Barter, for 12 Ib. of indigo, at 
 2 dol. 50 cts. per Ib. 
 
 OPERATION. 
 
 .Ib. doL cts. Ib. dol, 
 Firft, as i : 2 50 :; 12 : 30, the value of the indi* 
 
 go 
 ers. Ib. dol. Ib. 
 
 ?hen> 8 : i :: 30 ". 375, the anfwer. 
 
 A hath rum at 70 cents per gallon ready money, 
 but in Barter he muft have So cents ; B hath raifins 
 at 1 2 cents per Ib. ready money: How many Ib. of 
 raiiins mufl A have for 60 gallons of rum. 
 
 Here you mufl fir ft find what B's raifins ought to le 
 fer Ib. in Barter , which muft be as much more in fro- 
 fortion > as A y s price in ready money > is to his price in 
 
 Barter i 
 
( '9* ) 
 
 Barter ; which to obtain, Jay as 70 cis. '. 80 ct?. :: ia 
 c/j. : 13.71 cts. ~ price of B's raijins $er Ib. in Barter ; 
 /## proceed as before diretled> and the quantity cfrai- 
 fins that B muft. give A will be found=3$o.iM. 
 
 How much wheat at 91 J- cts. per bufhei, mud be 
 given for 8 ewt. of fugar at 8^- cts. per Ib. ? 
 
 Avfwer. 8i~ bujhels. 
 
 A hath rum at 70 cents per gallon ready money, 
 bin in Barter he mud have 84 cents ; B hath corn at 
 50 cents per bufhei ready money : How muchmuit 
 B have per bufhei in Barter for his corn \ alfo, how 
 many bufhei of corn B muft give A for a hogfhead 
 of rum containing 120 gallons ? 
 
 Anjwer. M^n^ft have 57^- cts. per bitfoel in Barter, 
 and muft give A 168 kujbel of corn for the 12.0 gallons 
 of rum. 
 
 D hath 12 cwt. of fugar, which he will fell to H 
 for 8 dollars 33-]- cents per cwt. ready money, but in 
 Barter he muft have 8-i cents per Ib. H hath a 
 horfe which he would fell for 90 dollars ready mon 
 ey, but in Barter he muft have 20 per cent advance: 
 They Barter, D takes the horfe, and H the fugar : 
 Query which is in debt, and how much ? 
 
 Anjwer. H is in debt 3 doL-yj^ cts. ready money. 
 
 CHAP. VIII. 
 
 LOSS and GAIN. 
 
 LOSS and gain is a rule by which merchants' 
 are inftrufted how to raife or fall in the prices 
 of their goods, fo as to gain or loofe fo much per Ib. 
 bag, or barrel, &c. 
 
 The 
 
THE operations are performed by the rule of three 
 direct. 
 
 EXAMPLES. 
 
 Suppofe I buy cheefe at 6 dollars per loolb. and 
 fell it again at 8 cents per Ib. What do I gain in buy 
 ing and felling 6oolb. ? 
 
 Here you muft firft find what 6oolb. comes to, 
 at 6 dollars per loolb. and 6oolb. at 8 cents per Ib. 
 then fubtra6t one fum from the other, and the refult 
 will be the anfwer. 
 
 OPERATION. 
 
 Firft, 6X636 doL the value of^6volb. at 6 doL 
 per ioolb. 
 
 Then, 600X8 cts.~ 48 doL the fries of 6oolb. at 
 8 cts. per Ib. 
 
 And, 4$ 3611 dol. the anfwer. 
 
 When butter coft 7 dollars per firkin of 561b. To 
 find how it muft be fold per Ib. to gain 25 per cent. 
 
 OPERATION. 
 
 <* 
 
 As $6lb. : 7 dol. :: lib. : iz~cts. the price that tfa 
 butter coft per Ib. 
 
 Asioolb.: ii^tts. :: 1004-25=125 : 15.625 cts. 
 the anfwer* 
 
 When tea coft 75 cts. per Ib. To find how it muft 
 be fold per Ib. to gain 25 per cent. 
 
 OPERATION. 
 
 As 100 :75 :: 100+25=125: 93.75 cts. the an- 
 
 Cwer* 
 
 At 
 
C *93 ) 
 
 At I2~ cts. profit in a dollar : How much per cent? 
 
 As i dbL : 12.5 cts. :: 100 dol. ; 12- per cent the 
 'anfwer. 
 
 Bought rum at 50 cents per gallon, and paid i'm- 
 poft, at $ cents per gallon, and afterwards fold it at 
 53 cents per gallon : What do I loofe in laying out 
 00 dollars. Anfwer. 86 dol. 21 cts 
 
 If I buy tallow at ii~ cents per Ib. and give 2%. 
 cents per Ib. to a chandler to make it into candles, 
 and i4oz. of tallow make a dozen of candles, which 
 I fell at 1944 cents per dozen : What do I gain in 
 buying and felling iSolb. of tallow. 
 
 Arifwer. iQ.dcl.$Qcts. 
 
 CHAP. IX. 
 
 FFLLOWSHIP, 
 
 FELLOWSHIP is a rule, when feveral per- 
 fons as merchants, &c. trade in company with 
 a joint flock, to afcertain each man's proportional 
 part of the gain or lofs, which arifes>from the em 
 ployment of the joint (lock, according to the quan 
 tity of goods, Hum of money, &c. each man puts in 
 to the faid ftock j which admits of a two-fold con- 
 fideration. 
 
 SECT, L 
 FELLOWSHIP SING LE. 
 
 SINGLE Fellowfriip is when all the feveral (locks 
 are employed in the common (lock, an equal term 
 of time. Therefore, fince the times of the feveral 
 
 B b flocks 
 
( 1.94 ) 
 
 flocks employed in the joint flock, are all equal ; it 
 follows, that each partner's Ihare of the gain or lofs, 
 is as his Ihare of that Hock : Wherefore it is mani- 
 fed ; if I put in-^ of the whole Hock, I ought to 
 have -^ of the whole gain, or fuffer ^ of the whole 
 lofs : Hence ws have the following 
 
 RULE. 
 
 MULTIPLY each partner's part of the joint flock, 
 with the vdiole gain or lofs, and divide the feveral 
 products by the whole flock, and the quotients re- 
 fulting will be the anfwer to the queftion. Or, as the 
 whole flock is to the whole gain or lofs j fo is each 
 man's particular part of that flock, to his particular 
 part of the gain or lofs* 
 
 EXAMPLES. 
 
 Two partners, A and B, conftitute a joint, flock of 
 300 dollars, whereof A put in 200 dollars, and B 
 100 dollars, and they trade and gain 150 dollars : 
 Required each man's part of the gain. 
 
 OPERATION. 
 
 150 
 100 
 
 3)00)300)00 3)oo) 1 50)00 
 
 i oo A's gain. 50= B*s gain. 
 
 Or, 
 
 As 300 : 150 :: 200 : 150X200^300=100 doL 
 A's fart of the gain. 
 
 As 300: 150:: ioo : 1 50 xi 00^-30050 dol. B's 
 fart of the gain : 
 
 Or, 
 
C *9$ ) 
 
 Of, 150300:3.5. the ratio of the fir ft term to the 
 Jecond : 
 
 Therefore, iQQX.$~ioo A's part, and ioox5 = 
 50 B's part as before. (Vid. Chap, u.) 
 
 Three merchants, A, B, andC, make a joint ftock 
 of 2000 dollars, whereof A put in 1500 dollars, B 800 
 dollars, and G 700 dollars \ and by trading gain 
 400 dollars : Required each man's part of the gain ? 
 
 OPERATION. 
 
 Fir ft, 40O-r aooo.a the ratio of the fir ft term to 
 thefetond. 
 
 f 500X.2IOQ dol A's "I 
 'therefore^ < 8oox-2 160 B's > gain. 
 (. yooX. 2-140 - C's] ' 
 
 Four merchants enter into partnerfhip, and confti- 
 tute a joint (lock of 60000 dollars, whereof A put in 
 15000 dollars 24 cents, B 20000 dollars 76 cents, 
 C 21000 dollars, and D 3999 dollars, and in trade 
 they gain 24000 dollars : Required each partner's 
 (hare of the gain ? 
 
 OPERATION. 
 
 Firft, 24000-7-60000^.4 the ratio of gain : There 
 fore, 1 5000. 24X. 4=6000 dol. 9.6 cts. A y s part of 
 the gain -, and 20000.7 6 X'4 8000 dol. 30.4 cts. B's 
 part of the gain ; alfo y 2 ioooX -4=8400 dol. C's 
 part ; laftly, 3999X4=i599 del. 60 cts. D's part. 
 
 Six farmers, A, B, C, D, E, and F, hired a farm 
 for 300 dollars ; A paid 20 dollars, B 30, C 40, D 
 60, 80, and F 70 dollars ; and they gained 60 
 dollars : What is each man's part of the gain ? 
 
 Answer. 
 
c *?g ) 
 
 4 */ 5V 6, C'jf ., P> xaE'j 56, 
 &nt?F's 14 <?/. 
 
 SECT- JI> 
 CO MPOUND FELL O WSHIP. 
 
 TnEonly difference between Fcllowfhip fingle and 
 compound, is, that in the latter regard muft be had 
 to the time each partner's fjtock continues in com 
 pany 5 whereas in fingle Fellowfhip the times of con 
 tinuance are all fuppofed equal* and when the times 
 are equal, the {hares of gain or lofs, are as their 
 flocks, as we have before fnewn : Therefore when 
 the flocks, are equal, the fhares muft be as the times. 
 Confequently, when neither the ftocks nor times are 
 equal, the fhares muft be as their products \ which 
 affords the following 
 
 RULE. 
 
 1. MULTIPLY each man's flock with the time it 
 is employed, and find the fum of all the products. 
 
 2. As the fum of the products thus found, is to 
 the whole gain or lofs ; fo is the product of each 
 inan's flock with its time^ to its proportional pajt 
 of the gain 'or lofs. 
 
 Or, 
 
 FIND the ratio between the two rft terms, and 
 proceed as in the laft rule, 
 
 EXAMPLES. 
 
 Two men, A and B, made a joint flock of 600 
 dollars, whereof A put in 200 dollars for 2 months', 
 B put in 400 dollars for 4 months -, at the expir 
 ation 
 
ation of which, they find they have loft 200 dollars* 
 Required each man's part of the lofs ?< 
 
 OPERATION. 
 
 Firft, 200x2400:3 A's ftock with its time : 
 Andy 400X4=1600 B's ftock with its time : 
 Then, 4004-1600^:2000 the fum of the prcdufts of 
 each man's ftocky with its time : 'Therefore) as 2000 : 
 
 200 :: 400 : 2oox400-J-2:ooo~4O del. A's fart of the 
 
 lofs; and as 2000 : 200 :: 1600 : 200X1600-7-2000 
 160 dol. B 'j fart of the lofs. 
 
 Or, 2oo-r-2ooo ~ .1 the ratio cf lofs 9 then, 
 40oX.i~4O A' s party and i6ooXi~i6o B's part, 
 the fame as before. 
 
 Three merchants made a joint flock of 8000 dol 
 lars in the following manner^ viz. A put in 1200 dol 
 lars for 3 years, B 2000 dollars for 7 years, and C 
 4800 dollars for 8 years > and at the end thereof, 
 they find they have gained 6720 dollars: Required 
 e&cfx man's part of the gain ? 
 
 OPERATION. 
 
 Firft, i2OoX3~36oozi^'j ftock with its time :. 
 And, 2000X7 ~i4000rn$\f j#0f& with its time", 
 AlfOy 4800X838400 Cs ftock with its time : 
 Then, 3600+ 14000 + 3840056000 the fum cf the 
 
 yrodufts : 
 
 Andy 67 20 -r- 5600011:. 12 the ratio of gain : 
 Therefore, 3600 X- 12:1:432 dol. A* s part of the 
 
 gain-, and I4ooox-I2=ri68o doL B's^party Alfa^ 
 
 3S400X-12 4608 doL Csfart. 
 
 Two- 
 
Two merchants, A and B, made a joint ftock ; A 
 put in at firft, 300 dollars for 7 months, and 4 
 months after put in 500 dollars more : B put in at 
 firft, 700 dollars, and 3 months after put in 200 dol 
 lars more. Now at the end of 7 months, they make 
 a fettlement of their accounts, and find they have 
 gained 1860 dollars : Required each man's part of 
 the gain, according to his (lock and time ? 
 
 F*r ft * 300x41200 the produft of A' s fir ft flock 
 
 with its time.) and 3004-500X3800x3=2400 the 
 product of As increafed ftock, with the remainder of the 
 time : Therefore, 12004-24003600 the produtt of 
 A y s flock with the whole time, according to the queftion. 
 
 Secondly, 700X3=2100 the produft ofB's firft fleck 
 
 with its time, and 7004-200X4900X43600 the 
 frodyfaof B's augmented ftock y with the remainder of 
 the time : therefore, 21004-3600=5700 the product 
 cf B's whole ftock, with the whole time, and 36004- 
 5700^9300 thejum of the produtls. 
 
 Hence ^ 186003001^.2 the ratio of gain : there 
 fore, 36ooX.2i=72o dol^A's part of the gain, and 
 
 of the gain. 
 
 Four merchants, A, B, C and D, enter into part,- 
 nerfhip for 12 months: A put into the common 
 ftock at firft, 300 dollars, B 400, C 500, and D 800 
 dollars, and at the end of four months, A took out 
 200 dollars, and 3 months after that, he put in 100 
 dollars more -, Bat the end of 2 months took out 
 200 dollars, and 2 months after that, put in 200 dol 
 lars more : C at the end of 6. months, took out 300 
 dollars, and two months after that, put in 200 dol 
 lars more : D at the end of 8 months, took out 400 
 dollars, and 2 months after that, put in 200 dollars 
 
 more : 
 
*( 1 99 ) 
 
 more : Now at the end of 12 months; they find they 
 have gained 406 dollars : Required each man's part 
 of the gain ? 
 
 OPERATION. 
 Firfl, 300X4=1200 the produft of A'sjirft flock 
 
 with its time, and 300 200X3 100X3 300 the 
 frodutt of A's remaining fleck for 3 months after the 
 
 taking out of the 200 doL Again, ioo-j-iooX5 = 
 200X5=1000 /<? produft of A's /lock with the re 
 mainder of the time according to the queflion ; then, 
 
 1200+300+1000=2500 the produft of A's flock for 
 the whole time. 
 
 Secondly, to obtain the produfit of B's flock with its 
 time, proceed as before : Thus 400X2 = 800 ; then, 
 
 400200X2=200X231400; and 200 + 200X8 
 
 400X8 = 3200. Hence, 8004-400 + 3200=4400 the 
 
 froduff of E's flock with its time. 
 
 Thirdly, 500X6=30005 then 500 300X2200 
 
 and 2004-200x4=400X4=1600 ; 
 wherefore 3000+4004- 1600=5000 the $ rodutt of Cs 
 ftock with its time. 
 
 Fourthly, 800X8=6400 ; then 800 400X2:= 
 
 400x^=800; and 400+200X2=1600X21200; 
 therefore 6400 + 8004-1200=8400 the frodutl of D's 
 Jlock with its time. 
 
 Confequently, 
 
( ado ) 
 
 -Confiquentl?, 2500 +4400 + 5000+ 8400 ~ 2030*0 
 the fum $f all the prcrdufis according to the queftion. 
 
 Therefore, 406-7-203002^.02 the ratio of gain y and 
 250oX-02rr5o doL A' s part of the gain \ aljo, 440OX 
 .O2rr88 doL B'f part , likewifa 5000X^02100 
 4ol. C'spart ; laflly, 8400X^02=168 dol. D'spart. 
 
 CHAP X. 
 
 CO MP UND PROPORTION. 
 
 COMPOUND Proportion, is ufed in the 
 folution of queftions that require feveral oper 
 ations in fimple proportion, \vtiether diredt or reci 
 procal. 
 
 FOR inftance : Suppoie a footman performs a 
 journey of 240 miles in 8 days, when the days arc 
 1 6 hours long : In what time would he perform a 
 journey of 540 miles, when the days are but 12 hours 
 long. This queftion refolved by fimple proportion 
 is thusj 
 
 m. d. m. 54oX8_ 
 /& 240 : 8 : : 540 : 24Q 18 days. 
 
 THAT is it would require 18 days to perform a 
 journey of 540 miles, when the days are 18 hours 
 long i but it is required to know how many days it 
 will take to perform the laid journey of 540 miles 
 when the days are but 1 2 hours long 5 which is thus : 
 
 As i6. 1540X8 -7-240 ( 1 8d.):: 12: 540X8X12-7- 
 
 240 x 1 2^:24 day s 1 1)' inverfe proportion, 
 
 Now 
 
Now from the laft analogy, is deduced the fol 
 lowing rule, for ftating and working all queftions in 
 compound proportion, at one operation. 
 
 RULE. 
 
 i. PLACE that term which is of the fame name of 
 the term lought, fo that it may.ftand in the middle 
 place : d. 
 
 Thus, < 
 
 2* WRITE the remaining terms of fuppofition, one 
 above the other in the firft places, and the terms of 
 demand- in like manner in the third places, fo that 
 the firft and third terms in each row, may be of the 
 fame name and denomination : 
 
 m. d. m. 
 r 240 : 8 : : 540 
 Thus, 4 h. h, 
 
 L 16: ::i2 
 
 3. HAVING thus ftated your queftion, find your 
 divifOr by comparing the terms in each row : Thus 
 if the firft term gives thefecond, does the third term 
 require more or lefs ? If more, diftinguifli the lefs 
 extreme with a point over it > but if the third term 
 require lefs, point the greater extreme : 
 
 . m. d. m. 
 f 240 : 8 : : 540 
 Thus, \ h. .h. 
 
 I 16: :: 12 
 
 4. MULTIPLY together the terms which are point 
 ed for a divifor, and the remaining terms .for a divi 
 dend, and the quotient refulting will be the anfwcr : 
 
 Thus, 540X8X16-- 240X1 224 days as before, 
 C c EXAMPLES, 
 
( 202 ) 
 
 EXAMPLES. 
 
 If 12 bufhels of corn are fufficient for a family of 
 9 perfons 12 months : How many bufhels will be 
 fufficient for a family of 16 perfons, 20 months ? 
 
 OPERATION. 
 
 Here lujhels are fought > therefore the queftion ftated 
 willjland 
 
 .per. b. per. 
 r 9 : 12:: 1.6 
 Thus, < . m. m. 
 
 (. 12 : ::20 
 
 f hen Jay, if 9 perfons eat 1 2 lujhels in 1 2 months, 
 1 6 perfons will edt more 5 therefore point the lejs ex* 
 ireme, which is 9. Again > Jay y if 12 months require 
 1 2 bu/hels for 9 perfons, 20 months will require more \ 
 therefore point the lefs extreme, which is 12. 
 
 Therefore, 12X20X16-- 12X93840-7- 108=35!. 
 lujhels , the quantity of for n required. 
 
 Note. If the fame quantity is found loth in the divifor 
 and dividend, it may be expunged from loth : 
 
 /.> the above exprefflon, 1 2X 20 x * 6-f- 1 2X9, /fij 
 1 2 wtfj be flruck out of the divifor and dividend ; thus, 
 
 2oX 1 6 -^9^:3 5-1 the fame as lefore. 
 
 If 15 dollars be the hire of 8 men 5 days : What 
 time will 40 dollars hire 20 men ? 
 
 OEERATION. 
 
( 2 3 ) 
 
 OPERATION. 
 
 .do 1. d. dot. 
 IS : 51:40 
 m. m m. 
 
 8 : ::20 
 
 15X20=1600^300=51^^, 
 the time required. 
 
 If 200 dollars in 2 years, gain 15 dollars : What 
 will 150 dollars gain in half a year ? 
 
 thus, 15x150X26-7-200X104= i^doL the 
 anfwer. 
 
 If i50olb. of bread ferve4Oo men 14 days : How 
 many pounds of bread will ferve 140 men 9 days ? 
 
 Thus, 1 500x140X9-7- 400 xi4==337# %oz. the 
 anfwer. 
 
 If 12 Clerks will write 72 fheets of paper in 3 
 
 *days : How many Clerks will write 140 Iheets in 8 
 
 days ? 
 
 Anjwer* ^XjXHO-r-y^X 8 8 ^ Clerks. 
 If 5000 bricks are fufficient to make a wall 4 feet 
 high and 5 feet long : How many bricks of the 
 fame fize will make 7 feet of wall 2 feet high ? 
 
 Anfwer. 3500, 
 
 CHAP, 
 
204 
 
 CHAP. XL 
 
 CONJOINED PROPORTION. 
 
 CONJOINED Proportion is when, in a rank of 
 numbers, the firfl term is compared with the fecond, 
 and the fecond term being increafed or diminiihed, 
 is compared with the third, and fo on $ from thence 
 to determine the equality of any of the terms : Thus, 
 ifj^~4^, and 8~i2r, then will 3a=6c -, becaufe, 
 as 4# : 30 : : 8 : 60 izc, or ^a^Jbc as before. A- 
 gain, if 240 3 2#, 48^^:30^ and ioc$d, then will 
 becauie, as 32^ : 24^ : 
 
 =36^1^ yx, and 
 
 a4=20<r. Again as jor : 24^X48^-7-32^ :: 
 
 =i2^~9, or 24^ 
 Hence from the foregoing analogy we have the fol 
 
 lowing 
 
 RULE. 
 
 1. BEGIN with that term whofe equality with any 
 Other term is required, which call A and write out 
 all the terms up to the one B, by which the afore- 
 faid term is to be compared. 
 
 2. MULTIPLY all the alternate numbers together, 
 beginning with the firfl, for a dividend, and all the 
 remaining ones together for a divifbr. 
 
 3. Divide, and the quotient will be the anfwer, 
 
 EXAMPLES. 
 
EXAMPLES. 
 
 If G in 48 days can produce a certain effect, which 
 will require H 64 days to perform j H can pro 
 duce an effect in 80 days, which will take JL 50 days 
 to perform : Which is the mod profitable to hire, 
 G or L, and what is the difference ? 
 
 OPERATION. 
 
 Fir/}, 48, 64, 80, are the numbers written out ac 
 cording to the rule : 
 
 <Tben, 48x80--- 64=60 50 days' of L, that is, 60 
 days of G are equal to 50 days of L ; and therefore if 
 is the moft profitable to hire L, to wit, in the proportion 
 of 60 to 50, or as 6 to 5. 
 
 IfD in 24 days can do as much as E can in 32 
 days, E can do as much in 48 days, as F can in 30 
 days, and F can do as much in 10 days, as G can in 
 9 days : Which is the moil profitable to hire, D, F, 
 orG ? 
 
 OPERATION. 
 Fir ft, find which is the moft profitable to hire, D orF: 
 
 Thus, 24X48-4-32=36=30 days of F, that is, 36 
 days cfD are equal to 30 days of F -, and therefore F 
 is more profitable to hire than D, 
 
 Again, 24x48X10-7-32X30=12:1:9 days of G ; 
 that is, 1 2 days of D are equal to 9 days of G > and 
 therefore G is more profitable to hire than D ; andfince 
 F is more profitable to hire than D y and G more profit 
 able than F j if fellows, that G is the moft profitable to 
 hire of the three. CH A P. 
 
( 206 ) 
 
 CHAP. XII. 
 
 ALLEGATION. 
 
 BY Allegation we are taught how to mix quan 
 tities of different quality, fo that any quantity 
 colleftively taken, may be of a mean or middle 
 quality ; that is, it fliews us the value of any part of 
 a competition, made of things all of a different 
 quality. 
 
 WE fhall confider Allegation, under the two fol 
 lowing general heads, viz. Allegation Medial, and 
 Allegation Alternate. 
 
 SECT. I. 
 
 ALLEGATION MEDIAL. 
 
 THIS is when any number of things are given, and 
 the price of each : To find the price of any quantity 
 of a mixture compounded of the whole. 
 
 RULE. 
 
 1. MULTIPLY each quantity with its price, and 
 find the fum of all the proslufts. 
 
 2. DIVIDE the fum of the produces by the fum of 
 all the quantities, and the quotient refulting will be 
 the mean price required. 
 
 EXAMPLES. 
 
 A man is minded to mix 20 buftiels of wheat,at 100 
 cents per bufhel, with 10 bufhels of rye, at 50 cents 
 per bufhel : Required the price of a bufhel of this 
 mixture. OPERATION. 
 
( 207 ) 
 
 OPERATION. 
 
 Firfl, 20X100=2000 cts. price of all the 
 wheat, and 10X50500 cts. frice of the rye then 
 2000+5002500 thefum of the produces, and 20+ 
 1030 thefum of the quantities : Therefore, 2500-- 
 3083]- cts. the price tf a lujhel, as was required. 
 
 A man would mix 27 bufhels of wheat, at 75 cents 
 per bufhel, with 40 bufhels of rye, at 60 cents per 
 bufhel, and 24 bufhels of oats,at 24 cents per bufhel : 
 Required the price of a bufhel of this mixture. 
 
 OPERATION. 
 9 
 
 Firft) 27X75 z:: i885 cts. price of the wheat, and 
 40X602400 cts. price of the rye, alfo, 24X24 
 576 cts. the price of the oats ; then 1 88 5-!- 2400-!- 
 5764861 thefum of the products, and 27+40 + 24 
 91 thefum of the quantities. 
 
 Whence 4861 -cts.-r-yi^z price of a bujhel, as was 
 required. 
 
 A mahfter would mix 70 gallons of one fort of beer, 
 worth 12 cents per gallon, with 20 gallons of another 
 fort, worth 24 cents per gallon, and 20 gallons of a 
 third fort, worth 22 cents per gallon : How may this 
 mixture be fold per gallon without gain or lofs ? 
 
 Anfwer. 16 cts. 
 
 Required what a gallon of the following mixture 
 is worth, viz. 60 gallons of malaga, at .5 dollars per 
 gallon, 40 gallons at .7 dollars per gallon, and 12 
 gallons at .3 dollars per gallon. 
 
 Anfwer. .55 dol. 
 
 A Goldlinith melts iSffi. of gold bullion, of 12 
 carats fine, with iofl>. of 16 carats fine, and 20 ]fo. of 
 
 10 
 
( 203 ) 
 
 10 carats fine : How many carats fine is a pound of 
 this mixture. Anfwer. 1 2 carats. 
 
 Note. Goldjmiths fuppoje every quantity of gold to 
 confifl of 24 farts, which they call carats ; but 
 gold is generally mixed with fome other me tats, fitch 
 as copper^ Irafs, &c. which is called alloy ^ and 
 the quality of the gold is eftimated according to 
 the quantity of alloy in it : Thus if 20 carats of 
 pure gold, and 4 of alloy are mixed together, the 
 gold is called 20 car at j fine. 
 
 SECT. II. 
 
 ALLEGATION ALTERNATE. 
 ALLEGATION Alternate confifts of 3 cafes. 
 CASE I. 
 
 When the prices of the federal quantities to be mixed 
 are given y to find what number ofcachjort muft be ta 
 ken, to ccmpofe a mixture whcfe mean price Jhall be as 
 given in the jueftion* 
 
 RULE. 
 
 1. WRITE all the particular rate* or prices di- 
 reftly under each other, and the mean price on the 
 left hand. 
 
 r i 
 
 Thus, mean price, 4 < particular prices. 
 
 I 5 
 
 2. COUPLE or connect the particular prices with 
 lines, fo that one or more of thole greater than the 
 mean price, may be coupled with one or more of 
 thofe lefs. Thus, 
 
Thus, 4 
 
 Or thus > 4 
 
 5- L 5 
 
 WRITE the difference between the mean price 
 md every particular price, direclly againft the one 
 nth which it is coupled. 
 
 fi n i fi 
 
 I A \ <j ^ . I ^ 
 
 hus, 4 
 
 4. THE difference (landing againft each particular 
 price, is the quantity that mutt be taken of that kind 5 
 and where two or more differences are found Handing 
 againft any one particular price, their fum is the 
 
 uantity. 
 
 Amaltfterhas the following forts of beer,viz. at 12 
 cents, 22 cents, and 24 cents per gallon : Required 
 the quantity of each fort that muft be taken to make 
 a composition worth 20 cents per gallon. 
 
 20 
 
 y 
 
 L 2 
 
 OPERATION. 
 2+4=6 
 
 22 I 
 
 24 
 
 8 
 
 Therefore, there muft be taken 6 gallons at 1 2 cts. 8 
 gallons at 22 f/j. tfdf 8 gallons at 24 i:/j. w/^/V^ w^ 
 be proved by Allegation Medial. 
 
 To find how much wheat at 100 cents per bufliel, 
 rye at 75 cents, corn at 40 cents, and oats at 30 cent* 
 per bufhel, may be mixed together, fo that the mix 
 ture may be fold for 50 cents per bufhel, without gain 
 or lofs. 
 
 D d OPERATION* 
 
( 210 ) 
 
 OPERATION. 
 
 anjwtr. 
 
 A merchant has coffee worth 12, 15, 16, and ic 
 cents per Ib. and would make a mixture worth 14 
 cents per Ib. What quantity of each fort muft be 
 taken ? 
 
 OPERATION. 
 
 Ib. cts. Ib. cts. 
 
 fi2 ^\ i at 12 fi6 \2 at 16 
 
 I IS J a . 15 Ii 5 1 4 15 
 
 14 1 i6-n 4 16 ( ^, 12 J 2 12 
 
 LTO_J 2 10 LIO J i 10 
 
 Proceeding in this manner, by varying the order fif link 
 ing the particulars, you will dif cover five more anfwen 
 to this queftion, in whole numbers. 
 
 How thefe kind ofqueftions can admit of various 
 anfwers, is eafy to conceive - 3 for if any two of the 
 particular prices make a balance by their increment 
 and decrement, in refpecl of the mean price, ther 
 will any multiple or quotient of the fame, make i 
 balance alfo : Therefore all numbers which are ir 
 the fame proportion, equally anfwer the queftion 
 Confequently, there are fome queftions which wil 
 admit of an infinite variety of anfwers : Hence it is 
 that thefe queftions are fometimes called indetermin 
 ate or unlimited problems -, yet by an analytical pro 
 
 cefs 
 
:efs, we can difcover all the poffible anfwers in whole 
 inumbers, when thofe anfwers are limited to finite 
 terms. ( Fid. Book n, Chap. xxm.J 
 
 CASE II. 
 
 When tie quantity of one of the particulars is limit '- 
 d or given, thence to proportion all the others in the 
 compofttion by it. 
 
 RULE. 
 
 1. OBTAIN the difference between the mean price 
 and every particular price, as in the lad rule. 
 
 2. As the difference found againft thefimplewhofe 
 quantity is givep, is to the quantity itfelf 5 fo is each 
 difference, to its refpeftiye quantity of the compofi- 
 tion. 
 
 EXAMPLES. 
 
 A farmer would mix 12 bufhels of wheat at 72 
 cents per bufhel, with rye at 48 cents, corn at 36 
 cents, and barley at 30 cents per bufhel, fo that the 
 whole compofition may be fold for 38 cents per 
 bufhel : Required the quantity of each fort that mud 
 be taken. 
 
 OPERATION. 
 
 36- 
 
 jo- 
 
 8 
 
 2 
 IO 
 
 34 
 
 ^ as 8 : 1 2 : : 2 : 3, the quantity of rye, and> as 
 8 : 12 :: 10 : 15, the quantity of corn j alfo, as 8 : 12 
 ; : 34 Si> the quantity of barley,. 
 
 To 
 
To find how many gallons of frontenaic at 81 < 
 cents, cFaret at 60 cents, and port at 51 cents per 1 
 gallon, mull be mixed with 42 gallons of madeini 
 at 90 cents per gallon, fo that the whole compofition | 
 may be fold for 72 cents per gallon, without profit! 
 or lofs. 
 
 21 
 12 
 
 9 
 
 18 
 
 15* 
 
 f ben, 42212 ; therefore, 12X2 24, the 
 quantity of the claret, and 9x2:1:18, the quantity of 
 the frwtinaic >, alfo, 18X2=36, the quantity of fort. 
 
 A tobacconift would mix 6 Ib. of tobacco worth 
 6 cents per Ib. with another fort at M cents, and a 
 third fort at 12 cents : What quantity muft be taken 
 of each fort, to make a mixture worth 10 cents per 
 Ib ? Anfwer, 8 Ib. ofeachjort. 
 
 CASE III. 
 
 When the whole compofition is equal to a given quan 
 tity ; that is, when thefum of all the quantities which 
 make up the compofition, colletJively taken, amount to 
 the given quantity : 'To find the feveral quantities them- 
 fefaes. 
 
 RULE. 
 
 1. LINK or couple the feveral particulars, and find 
 their differences, as in thelaft cafe. 
 
 2. As the fum of the differences, is to the fum of 
 the whole compofition or given quantity ; fo is each 
 difference, to its refjpeftive quantity of the compofi 
 tion. 
 
 f 
 
 EXAMPLES. 
 
EXAMPLES. 
 
 A grocer having fugars at 4 cents, 8 cents, and 12 
 cents per Ib. would make a compofition of 240 Ib, 
 worth 10 cents per Ib. Required the quantity of each 
 fort that muft be taken. 
 
 OPERATION. 
 
 Firft, 10 \ 8 <| I 2 
 
 I 12 LJ 64-2=8 
 
 12 fum efthe differences. 
 
 Then y as 12 : 240 : : t : 40, and y as 1 2:240 : : 2 : 40 ; 
 alfo y as 12 : 240 : : 8 : 160. Therefore, there muft be 
 taken> 40 Ib. at 4 cts. 40 Ib. at 8 cts. and 160 Ib. at 
 12 cts. 
 
 A merchant would mix brandy of the following 
 prices, viz. at 60 cents, 72 cents, and 84 cents per 
 gallon, together with water at o cents per gallon, fo 
 that a compofition of 846 gallons, may be fold for 
 48 cents per gallon, without gain or lofs : Required 
 the quantity of each fort that muft be taken. 
 
 OPERATION. 
 
 Firft, 48 
 
 Then, <w. 216; 846:: 
 
 4 8 
 48 
 48 
 24-^X1+36=72 
 
 216 fum of the differences. 
 '48 : 188 at jicts. 
 
 $; 1 88 at 60 cts. 
 48 : 1 88 at 84 cts. 
 72 : 282 of water. In 
 
( 214 ) 
 
 IN this cafe might beftarted,a variety of very curious 
 queftions about the fpecific gravities of metals 3 but, 
 as they would require the knowledge of fom.e things 
 which are not treated of in this volume, we defift. 
 
 CHAP. XIII. 
 
 Of POSITION, or the GUESSING 
 RUL E. 
 
 POSITION is a method of folving queftions, 
 by fuppofmg numbers, and then adding them, 
 fubtrading, multiplying, &c. according as the refult 
 or number given in the queftion is produced by ad 
 dition, fubtra6lion, multiplication, &c, of the num 
 ber required. 
 POSITION is di ft ingui fried into two kinds, fingle 
 
 and double. 
 
 
 
 SEC T. L 
 
 Of SINGLE POSITION. 
 
 SINGLE Pofition is when one quantity is re 
 quired, the properties of which are given in the quef 
 tion. 
 
 RULE. 
 
 SUPPOSE a number for the quantity required, and 
 multiply or divide it, &c. according as the quantity 
 required was multiplied, divided, &c. then -, as the 
 refult of the fuppofition, is to the fuppofition, fo is 
 the refult given, in the qfieftion, to the number re 
 quired. 
 
 EXAMPLES. 
 
EXAMPLES. 
 
 To find fuch a number, that being divided by 2, 
 4, and 8, refpeftively, the fum of the quotients fhall 
 be 7. 
 
 OPERATION. f 
 
 Suppofe the number to be 24, then, V + V + V 
 
 Whence, 21 : 24 : : 7 : 24X7 + 21 8, /* number 
 required. 
 
 For, 4-+t+T-4+2+i=7; therefore, &c. 
 
 A man having a certain fum of money, {aid one 
 half, one third, and one fourth of it being added to 
 gether, made 13 dollars : What fum had he ? 
 
 Suppofe he had 36 do 1. then ^ -f 3 T 6 + \ 6 = 1 8 + 1 2 
 + 939, which ought to be 13, by the queftion, 
 
 therefore, 39 : 36 : : 13 : 12, the anfwer. , 
 
 Three men found a purfe of dollars, difputed how 
 it fhould be divided between them. A faid he would 
 have one third ; B faid he would have one third anJ 
 one quarter -, well fays C, I fhall have but 2 dollars 
 left for my part : How many dollars were there in 
 the purfe, and how many did each one take ? 
 
 Supfofe the purfe contained 1 1 dollars : 
 
 Then, V+T +V Z =4+4+3 = ii - 
 And, 1 2 i ii^ri, which ought to be 2. 
 Wherefore, i : 2 : : 12 : 24, the number of dollars in 
 the purfe ; whence, ^zrS, the number of dollars that 
 A took; and ^-\-^=^-\-6^\^,the number that Btcek. 
 
 Delivered to a banker, a certain fum of money, to 
 receive interefl for the fame, at the annual rate of 
 6 dollars per cent ; at the end of 7 years, received 
 
 for 
 
for intcreft and principal, 2495 dollars 27^ cents : 
 What was the fum lent ? 
 
 Anfwer. 1736 doL 1 1 ~ cts. 
 
 SECT. II. 
 
 Of DOUBLE POSITION. 
 
 DOUBLE Pofition is when there are feveral un 
 known numbers in the queftion, analogous to each 
 other ; fo that when one or more are found, the reft 
 may be had, either by addition, fubtraction, or mul 
 tiplication, &c. according as the queftion requires. 
 
 RULE. 
 
 1. ASSUME two convenient numbers, and work 
 with them as the queftion directs, finding their re- 
 fults. 
 
 2. FIND the difference between thefe refults and 
 the refult given in the queftion, and call thofe differ 
 ences errors, which place under their refpective fup- 
 
 pofitions. ^r C x, y,fup$ofttions. 
 
 ' I *> &> errors. 
 
 3. MULTIPLY the firft error with the fecond fup- 
 pofition j and the fecond error with the firft fuppofi- 
 tion. Thusy a X J, find b X # 
 
 4. IF the errors are alike, that is, both too great, 
 or both too fmall, or more properly, the numbers 
 from whence they were deduced, are both either 
 greater or lefs than the true ones, you muft divide the 
 difference of the products, by the difference of the 
 
 errors, that is, aXy X#-r-tf ^ ; but if the errors 
 are unlike, that is, one too great and the other too 
 fmall, divide the fum of the products by the fum of 
 
 the 
 
the errors : Thus, ^X^+^x^-H and the quo 
 tient in either cafe, will be the number fought. 
 
 EXAMPLES. 
 
 A, B, and C, difcourfing of their money : Says B, 
 I have 6 dollars more than A : Says C, I have 7 
 dollars more than B : Well fays A, the fum of all our 
 money is 100 dollars : How much had each one ? 
 
 Suppofe A had 20 dol. then B muft have 20+626 
 dol. and C 26 + 7=33. <&/. but 20+26+33=279, 
 which Jhould be 100 by the queftion. 
 
 Therefore, 100 79=121, the fir fl error, t oof mall. 
 
 Again, fuppofe A had 24 dol. then B muft have 24+6 
 =30, <WC 30+737, but 24+30+37=91, which 
 Jhould be 100. Therefore, 100 919, thefecond error, 
 t oof mall. 
 
 Whence, *^*i^$&f=frMti& of the fccondfuppo* 
 Jltion andfirft error ; 
 
 And, 2oX9~i%Q~produftofthefrftfuffofttion and 
 fecond error ; 
 
 Wherefore, 504 1 80-7-21 91^27 dol. A's money ; 
 
 Then, 27 + 6=33 dol.B's money, ^^33 + 740 
 C'j money. 
 
 A man having been to market with hogs, pigs and 
 geefe ; received for them all 190 dollars, for every 
 hog he received 4 dollars, for every pig 75 cents, 
 |and for every goofe 25 cents ; there were for every 
 ipig two hogs and three geefe : What was the num 
 ber of each fort ? 
 
 Suppofe he had 1 2 pigs, then he muft have 24 hogs , 
 
 ** 30 geefe'yby the queftion; and 12 pigs at 7 5 cts. each, 
 
 is 9 doL i\hogs at 4 dol. each, is 96 ^<?/. and 36 */<? ^ 
 
 .5 cts, each, is 9 *&/. but 9 + 96+9^=114* which Jhould 
 
 E e ^<? 
 
b 1 1 90 : Therefore, \ 90 1141^76, thefirft err or, too fmalL 
 Again yfuffoje he hadi 6 pigsjhen he mufl have 3 2 hogs> 
 andafi geefe ; and \6pigs at 75 fAr. is 12 *&/. 32 0.r 
 4 */0/. /j 128 doL and 48 ^^ tf/ 25 r/j. /j 12 ^/. but 
 12+128 + 121^:152 which Jbould be 190. Therefore, 
 190152:1138, thejeconderrvr, toofmalL 
 
 Whence we have i6x76 i2X38-r7 6 3 8 ?6o 
 ^.38^:20, /^ number of figs y and 20X^1140, //?<? 
 number of bogs -\ ^,20X3=60, the number of geefe. 
 
 CHAP. XIV. 
 
 CONCERNING PERMUTATION 
 
 AND 
 
 COMBINATION. 
 
 S E C T. I. 
 O/ PERMUTATION. 
 
 PERMUTATION is the changing or varying the 
 order of things ; and is when any number of quan 
 tities are given; to find how many ways it is poflible 
 to range them, fo that no two parcels fhall have the 
 fame quantities (landing in the fame place, with re- 
 fpe6l to each other. 
 
 PROBLEM I. 
 
 To find all the variations or changes that can be made 
 cf any number of things, all different one from ano 
 ther. 
 
 FIRST it is evident, that any one thingis capable of 
 one pofitiononly,, and therefore cannot poflibly have 
 any change or variation -, but any two quantities ; as 
 
 a 
 
C 
 
 a and b, are capable of change or variatipn ; as a, b, 
 and b a y that is, the number of variations is IX 2 * 
 Again, if there be 3 quantities ; as #, b, c, their vari 
 ations are a b c, a c b y b a c y b c a> c a b y c b a ; for 
 taking only the two firft, a and b> the number of 
 their variations is IX 2 > therefore taking in c> the 
 number of changes is 1x2X3=6; and fo on for 
 any number of quantities. Hence we have the fol 
 lowing 
 
 RULE. 
 
 MULTIPLY together the natural feries of numbers, 
 *> 2, 3, 4, &c. continually, till your multiplier is 
 equal to the number of things propofed, and the laft 
 product will be the number of variations required. 
 
 EXAMPLES. 
 
 In how many different pofitions may a company of 
 8 perfons ftand ? 
 
 Anfwer. 1X2X3X4X5X^X7X8-40320 pofi 
 tions. 
 
 How many changes may be rung with 12 bells ? 
 Anfwer. 1X2x3X4X5X6X7x8X9X10X11 
 
 X 12479001600, the number of changes required. 
 
 PROBLEM II. 
 
 To find all the poffible alternations or changes that can 
 be made of ar.y given number of different quantities, by 
 faking any given number of them at a time. 
 
 THE manner in which this problem is folved, is di~ 
 reftly thereverfe of the laft ; for it is manifefl, that 
 let the number of quantities be ever fo many, and we 
 take one of them at a time,the number of alternations 
 be equal to the number of quantities, Therefore 
 
 it 
 
C o ) 
 
 it follows, that the operation muft begin at the num 
 ber of things propofed, and then decreafe by unity, 
 till the number of multiplications are one lefs than 
 the number of things propofed. Hence we get the 
 following 
 
 RULE. 
 
 MULTIPLY continually together, the terms of the 
 feries, beginning at the number of things propofed ; 
 and decceafing by unity or i, until the number of 
 multiplications, are one lefs than the number of things 
 to be taken at a time, and the laft product will be the 
 number of alternations required. 
 
 EXAMPLES. 
 
 How many different pofitions may a company of 
 9 men be placed in, taking 3 at a time ? 
 
 Here the number of multiplications muft be 2, and 
 the feries 9, 8, 7, 6, &c. Therefore, 9X8X7 504, 
 the number of pofitions required. 
 
 How many alternations will the letters a b b admit 
 of, taking 2 at a time ? 
 
 Anfwer. 3X26, the number of alternations requir 
 ed, and the letters willftand thus, a h, h a, ab> b a, 
 bb>bb. 
 
 How many alternations or changes can be made 
 v/ith the letters a b c d, taken 3 at a time ? 
 
 Anfwer. 4X3X224, the num ber of alternations 
 required \ and the letters willftand 
 
 {abc y acb>baC)bca y cal>) cba~ alter, of ale 
 lacdyadCiCadiCda^dac^dcado.ofacd 
 ' bcd> bdc, cbd> cdb> deb, dbc^^do. of bed 
 
 ^=.do. of dab. 
 How 
 
C 
 
 How many alternations or changes can be made with 
 the letters of the word Algebra, taking 4 at a time ? 
 Anfwer. 7 X 6 x 5 X 4=840* 
 
 PROBLEM III. 
 
 find all the alternations or changes that can be 
 made of any given number of quantities, which confjfl 
 of feveral ofonejort, and feveral of another. 
 
 RULE. 
 
 1. FIND the produft of the feries, 1X2X3X4, 
 &c. to the number of things to be changed, which 
 call your dividend. 
 
 2. FIND all the alternations that can be made of 
 each of thofe things which are of the fame fort, by 
 problem i, and multiply them continually together 
 for your divifor. 
 
 3. DIVIDE, and the quotient refulting will be the 
 anfwer. 
 
 EXAMPLES. 
 
 Find all the variations that can be made of the fol 
 lowing letters, a a b c c c. 
 
 OPERATION. 
 
 ^ 1X2X3x4X5x6 TIG number of va 
 
 riations that can be made of 6 different things, and 
 i X 2 z: 2, tbe variations of the a's ; al/o, i X 2 X3=6 
 the variations of tbe c's. 
 
 Whence, 720-1-6x2.^60, the number of variations 
 required. 
 
 Find all the different numbers that can be made 
 cf the following numeral figures, 1 1 1 22777. 
 
 OPERATION, 
 
( 222, ) 
 
 OPERATION. 
 
 Firft, i^ f iy=&~ r variatiQns of the iV, and 1X2 
 xzi^vtpations of the z's > alfo^ ix^Xj 6 van- 
 it ons of the 7*J. 
 
 Whence, 1X2X3X4X5x6X7X8-5-6X2X7 
 =40320-7-72:1:560, theanfwer. 
 
 SECT. II. 
 O/ COMBINATION. 
 
 COMBINATION of quantities, is, when any number 
 of things are given, to find all the different forms in 
 which thofe quantities can be poflibly ordered, and 
 from thence, all the different combinations in thofe 
 forms, without any regard to the order in which the 
 feveral quantities ftand in thofe combinations. That 
 is, by combination we determine how many ways it 
 is pofiible to combine any number of things, fo that 
 no two combinations fhall have the fame things in 
 both. Combinations of the fame form, are thofe that 
 have a like number of quantities which repeat in the 
 fame manner in both : Thus, a a c d y and yy x z, are 
 of the fame form> but aaa be, and s mn ry> are of 
 different forms. 
 
 PROBLEM I. 
 
 fofind all the different combinations that can be made 
 of any number of quantities all different one from an- 
 ctber, by taking any number of them at a time. 
 t THE rule for the folution of this problem, is eafi- 
 ly deduced from the rule to Problem n, of permu 
 tation, For it is plain, that the number of combina 
 tions 
 
( 223 ) 
 
 tions multiplied with the changes in the number of 
 things taken at a time, gives the number of alterna 
 tions in the whole. Therefore it follows, that the 
 number of alternations in the whole, divided by the 
 changes in a number of things equal to thpfe taken 
 at a time, gives the number of all the different com 
 binations. Hence we have the following 
 
 RULE. 
 
 1. FIND all the alternations or changes of the giv 
 en quantities, taken as many at a time, as are equal 
 to the number of things to be combined at a time 5 
 and call the refult your dividend. 
 
 2. FrND all the changes in as many quantities, as 
 are equal to thofejto be taken at a time -, and call the 
 refult your divifot. 
 
 3. DIVIDE, and the refulting quotient will be the 
 number of combinations required. 
 
 EXAMPLES. 
 
 
 
 Find all the different combinations that can be 
 made with the following numeral figures, i, 2, 3, 4, 
 5, 6, taken 2 at a time. 
 
 Here the number of given quantities are 6 ; and 
 the number to be taken at a time are 2 ; therefore, 
 6X5=30^=dividend ; and i X 22 divifor. 
 
 Whence 30--- 2=115, the number of combinations 
 required - t and the figures will (land as follows : 
 
 12, 13, 14, 15, 16 
 
 23, 24, 25, 26 
 
 34, 35> 3-6 
 4$>4<$ 
 
 5*- 
 
 FIND 
 
( 224 ) 
 
 FIND all the* different combinations that can be 
 made, with the following letters, a b c d b> taken 3 
 at a time. 
 
 Here the number of quantities are 5, and the 
 number to be taken at a time are 3 ; therefore, 5x 
 4X3~6o dividend ; and 1X2x3 6~divifor. 
 
 Whence, 6o-r-6mo, the number of combinations 
 required : and the letters will ftand as follows : 
 
 a b C) a b d, b b b, a c d 
 
 a c b, a d b, bed 
 
 b c b y bah 
 
 a b 
 
 How many different combinations may be made 
 with the following numeral figures, i, 2, 3, 4, 5, 6, 
 7 1 8, 9, taken 5 at a time ? 
 
 Anjwer. 1*26 combinations. 
 
 PROBLEM II. 
 
 find the number of different combinations tbat 
 may be made from any number of Jets y by taking ons 
 out of each fet and combining tbem together ; the things 
 in every fet being all different one from another. 
 
 RULE. 
 
 MULTIPLY the number of things in each fet con 
 tinually together, and the product refulting, will be 
 the number of combinations required. 
 
 EXAMPLES. 
 
 How many different combinations of two letters, 
 may be made of thefe two fets an w and s x y ? 
 
 Here 
 
Here the number of things in each fet are 3 r 
 
 Therefore, 3X3^9, the number of combinations 
 required. 
 
 The method of making the combinations, may be 
 {hewn in the following manner. 
 
 Write down the two fets one beneath the other, 
 and join thofe letters that are to be combined, with 
 iftraight line, 
 
 ~ a n iv 
 
 i i i 
 
 s x y 
 
 Then drawing lines from s to a, from x to , and 
 from y to w y you will have three of the required 
 combinations, to wit, s a, x n y and y w< 
 
 Again, let the fets be placed as before : 
 
 I 
 
 y 
 
 Then joining s and w y x and a, andj to n y we 
 get s w y x a and^ n. Once more, place the fets as 
 ibove . 
 
 s x' y 
 Then joining s and n y x and tu, and y to tf, v;t 
 get s n y x w y and y a. 
 Hence, all the combinations are as follows., 
 
 x a y x n, x w, 
 
 F f Suppofc 
 
( 226 ) 
 
 Suppofe there are three flocks of flieep ; in one of 
 which there is 10, and in the other two, 20 each : 
 To find how many ways it is poflible to choofe 3 
 fheep, one out of each flock. 
 
 Thus, 10X20X20=4000, the anfwer. 
 
 PROBLEM III. 
 
 7 find the number of forms in which any given 
 number of quantities may be combined, by taking any 
 number at a time ; wherein there arejeveral of one fort, 
 and fever al of another. 
 
 RULE. 
 
 1. WRITE the quantities according to the order 
 of the letters. Ttttfc a y a, b y c y d. 
 
 2. JOIN the firft letter to the fecond> third, fourth, 
 &c. to the lad -, and the fecond letter to the third, 
 fourth, &c. to the laft ; alfo, the third letter to the 
 fourth, fifth, &c. to the laft : Proceeding in like 
 manner through the whole, taking care to reject all 
 combinations that have before accrued a and you 
 will have the combinations of all the twos. 
 
 3. JOIN the firft letter to every one of the twos, 
 and the fecond, third, fourth, &c. in like manner 
 to the laft ; and you will have the combinations of 
 all the threes. 
 
 Thus, a a a, a a b y a a c y aa d, ab e y a b d y a c d y 
 b a a, a b , b a c y b a d> bb c> b b d y bed, 
 
 c a a y c c a y c c b y c c d y 
 
 da a y dd a y _ d d c y 
 
 And proceed in this manner, till the number of 
 things in the combination, are equal to the number 
 to be taken at a time. 
 
 Note. All thoje combinations 'which contain more 
 things of the fame fort , than are given of the like 
 kind -in the queftion, muft be rejected. EXAM* 
 
(227 ) 
 
 EXAMPLES. 
 
 Find all the different forms of combination, that 
 can be made of the letters a a b b c c } taken 4 at a 
 time, 
 
 OPERATION. 
 
 a ay a by a c> b by b c y c c~ combinations of the twos. 
 a a by a a Cy b b ay b a c } bb c y b c Cy ac Cy~ combi 
 nations of the threes. 
 
 a abby a ab Cy b b c a y c c a by aa c (> b b c Cy iz 
 combinations of the fours. 
 
 Whence y aabby b b c c y a a c c y and a a c by b b a c , 
 c c a by are the two forms required. 
 
 %ind all the different forms of combination that 
 can be made of the following figures, 22334455, ta 
 ken 3 at a time. 
 
 OPERATION, 
 
 Thusy 22, 13, 24, 25, 33,34, 35> 44>45>5S- 
 combinations of the twos. 
 223, 224, 225, 234, 235, 245, 233, 334, 335, 
 
 345> 244, 344* 44S> 2 55> 355> 455 combinations 
 of the threes. 
 
 Whencey 223, 224, 225, 233, 433, 533, 244, 
 344, 544, 255, 355, 455> ^234, 235, 245, 345, 
 are the forms required. 
 
 THUS far, concerning Permutation and Combin 
 ation. 
 
( 223 ) 
 
 CHAP. XV. 
 
 Of INVOLUTION. 
 
 WHEN any number is multiplied into itfelf, 
 and that product multiplied with the fame 
 number; and fo on, it is what is called Invoiution, 
 and the feveral produ&s refujting, are called the 
 powers of the multiplying quantity, or root. Thus* 
 
 3X37 3X3X3> 3X3X3X3, &c. are the powers of 
 
 3. Aad generally, a^a, ^X^X^, and 
 
 &c. are the powers of a ; whofe height is denomina 
 ted by the number of multiplications more one. 
 HENCE, the 2d power of ip, is loXiomoo 
 the 3d 
 
 the 4th - ioXioXioXio=iQQOO. 
 Therefore it follows, that the powers of any quan 
 tity, are aMeries of numbers in Geometrical Propor 
 tion continued, whofe* firft term and ratio is the fame, 
 to wit, the root of the power : Confequently the 
 height of the power at any particular term, will be 
 pcprelfed by the exponent of that term : AS in 
 
 12 3 4 &c> Expon. 
 
 ?** 10, 10X10, 1,0x10X10, lo 
 
 HERE it is evident, that the index, or exponent 
 of each term of the Geometrical feries, is equal tc 
 the number of multiplications of the firit term witl 
 itfelf, to that place, more one, and is therefor^ call 
 ed the index, or exponent of the 
 
 Thus ---- 
 
 1 5x5x5X5X5^:3125-5^^^^/5: 
 
 on for others. 
 
 WHENCE 
 
WHENCE it follows, that to raife any number to 
 any given power, is no more thart to multiply the 
 given number into itfclf, fo often as there arc units 
 in the index of the power i. 
 
 EXAMPLES. 
 
 Required the 5th power of 9. 
 OPERATION. 
 
 9 
 9 
 
 %i=.id power of 9 
 9 
 
 3d power cf 9 
 9 
 
 656 1 =4/ fewer of 9 
 9 
 
 $tb power of y, as requlr. 
 
 Required the 7th power of 8. 
 Thus, 8X8X8X8x8X8X8 = 2097152 = 7/4 
 power 0/8. 
 
 E 
 
 CHAP. XVI. 
 
 Of EVOLUTION. 
 
 VOLUTION is the converfe of Involu 
 tion j and is when any power is given, to find 
 
 the 
 
the number from whence fuch power was produced, 
 which number (as we before faid) is called the root 
 of the power -, and the bufinefs of finding it, is called 
 extraction of roots. 
 
 ALL powers whatever, are produced by the contin 
 ual multiplication of their roots into themfelves, as 
 is evident from what has been faid ; yet there are 
 many powers which have no finite root, that is, whofe 
 true and adequate root cannot be expreffed in finite 
 terms ; but by approximation may be determined to 
 any affigned degree of exaclnefs. 
 
 THESE powers are called furds, or irrational 
 powers. > 
 
 PROBLEM I. 
 
 2*o extra ft the root of tbejquare orjecond -power of 
 any number. 
 
 RULE. 
 
 1. PREPARE the given nurpber for extraction, i. e, 
 diftinguifh it into periods of two figures each, by be 
 ginning at the unit's place and placing a point over 
 the firft, third, fifth, &c. figures of the given num 
 ber, and if there are decimals, point them in the fame 
 manner, from unity towards the right hand. 
 
 2. FIND a number by the help of a table of pow 
 ers, whofe fquare is equal to, or lefs than the firft pe 
 riod on the left hand, and this number will be the 
 firft figure of the root, which place in the form of a 
 quotient - 3 then fubftracV its fquare from the afore - 
 faid period ; and to the remainder annex the next 
 period for a dividend. 
 
 3. DOUBLE the firft figure of the root for a divifor. 
 
 4. FIND fuch a quotient figure, that when annex 
 
 ed 
 
ed to the divifor and the refult multiplied with the 
 fame number, the produft will be equal to, or lefs 
 than the dividend j and this will be the fecond figure 
 of the root. 
 
 5. To the remainder annex the third period for a 
 new dividend^ and add the figure in the root lad: 
 found to your former divifor for a new one. 
 
 6. FIND the third figure of the root as you found 
 the fecond ; and fo on, till all be done. 
 
 Note i , - If there is a remainder after all the periods 
 are annexed, the given number is a furd, and you. 
 muft approximate to the root, by annexing cyphers 
 two at a time, to the remainder, 
 
 2. If the given number confifts of integers and 
 decimals, you muft ponit off as many places in the 
 root, as there were periods of decimals, in the given 
 number. 
 
 EXAMPLES. 
 
 Required the fquare root of 58081. 
 
 OPERATION. 
 
 ' . 
 
 58081(241 
 
 4 
 \fl divifor 4.4) i%o 
 
 4 i?^ 
 
 id divifor-=4& i ) 481 
 481 
 
 therefore, 241 is the root required, as may be proved 
 ly involution : Thus, 241X241 58081, which is the 
 fame as the given number : Whence > &c. 
 
 Required 
 
Required the fquare root of 1000. 
 OPERATION. 
 
 1000(31.622 fcfc.:=: root required, 
 9 
 
 61)100 
 i 61 
 
 626)39.00 
 6 37S 6 
 
 6322)14400 
 
 V 2 12644 
 
 63242)175600 
 
 2 126484 
 
 fifc. - 
 
 49116 &V. 
 
 Required the fquare root of 105462.5625 : 
 
 OPERATION. 
 
 
( 233 ) 
 
 OPERATION', 
 
 105462.5625(324.75 rev! 
 9 
 
 62)154 
 a 124 
 
 644)3062 
 4 2576 
 
 6487)48656 
 7 45409 
 
 64945)324725 
 324725 
 
 o 
 P R O B L E M II. 
 
 !T0 extract thefquare root of a Vulgar Fraction* 
 RULE. 
 
 EXTRACT the root of the numerator, for the nume* 
 rator of the root ; and the root of the denominator, 
 for the denominator of the root. 
 
 EXAMPLE. 
 Required the fquare root of T |4- 
 
 OPERATION, 
 
( 234 ) 
 
 OPERATION. 
 
 225(15:1: numerator of the root. 
 i 
 
 1024(32 denominator of the root, 
 "5 1_ 
 
 62) 124 
 
 124 
 
 , y~ is the root required, 
 PROBLEM III. 
 
 find the root of the third power or cube, by af~ 
 proximation. 
 
 RULE. 
 
 1. DISTINGUISH the given number into periods 
 of three figures each, by beginning at the unit's 
 place, and placing a point over the firft, fourth, fe- 
 venth, figures, &c. and if there are decimals, point 
 them from the unit's place towards the right hand, 
 in the fame manner. 
 
 2. FIND the root of the firft period on the left 
 hand, by the help of the table of powers, and annex 
 to it, as many cyphers as there are remaining periods, 
 then involve this number to the fame power as the 
 given number, and call the refult the fuppofed cube ; 
 then : As twice the fuppofed cube + the given cube j 
 is to twice the given cube -f the fuppofed cube $ fo 
 is the root of -the fuppofed cube ; to the root requir 
 ed, nearly. 
 
 .3. IF a greater degree of exa&nefs is required, in 
 volve the root already found, to the third power, and 
 
 call 
 
call the refult the fuppofed cube, with which pro 
 ceed as as before, and fo on, to any degree of exa<5b- 
 nefs. 
 
 Note. When the root is finite, you mayfometimes 
 Jave the trouble of repeating an operation, by in- 
 creafing the right hand figure of the root found, by 
 unity. 
 
 EXAMPLES. 
 
 Find the cube root of 1367631. 
 OPERATION. 
 
 Firft, 1 3 67 63 1 is the given number prepared for ex- 
 traftion, the root of whofe firft period (i) is i ; then 
 iooX iooX tQQ=iOGOGGQ~&04 cube ; and, 
 
 ^1000000X2+1367631 : 1367631X2+1000000:: 
 100, /'. e. 3367631 : 3735262 :: 100 
 
 100 
 
 3676310 iu~ root requtr, 
 3367631 
 
 3086790 
 Required the cube root of 729001101. 
 
 Firft, 729001101 is the given number pointed, and 
 the root of the fir ft period (7 29) 9 ; therefore 900 x 
 900X 9007 -lyoQcvooJuppofedtubei then, 
 
 as 729000000X2+729001101 ; 729001101X2 + 
 72900000 o :; 900, 
 
 That 
 
<?bat is, 2^87001101 : 2187002202:: 900 
 
 900 
 
 2187001101)1968301981800(900,0004 
 19683009909 \rbotnearLy. 
 
 9909000000. 
 
 THE cube root of a Vulgar Fraction, is found by 
 cxtra&ing the root of the numerator and denomina-r 
 tor. 
 
 PROBLEM IV, 
 
 '/a sxtraft the roots of powers in general. 
 RULE. 
 
 1. LET the index of the power whofe root is tp 
 be extracted, be denoted by #. 
 
 2. POINT the given 'number into periods of as 
 many figures each, as there are units in #, begin 
 ning at the unit's place 3 and if there are integers and 
 decimals together, let them be pointed both ways 
 
 from unity. 
 
 3. FIND the root of the firft period, by the help 
 of the table of powers, and this will be the firft fi 
 gure of the root. 
 
 4. SUBTRACT the n power of the firft figure of the 
 root, from the firft period, and to the remainder an 
 nex the firft figure of the next period^ which refuH 
 call your dividend. 
 
 5 INVOLVE the root now found to the i power, 
 and multiply the refill t with n for your divifor. 
 
 6. DIVIDE, and the quotient will be the fecond 
 figure of the root. 
 
 7. INVOLVE all the root now found to the n power, 
 Ind fubtract it ( always ) from as many periods, as 
 
 you 
 
you have found figures of the root : But if the num 
 ber to be fubtraded, is greater than the aforefaid pe 
 riods, the laft figure of the root is too great, which 
 mult therefore be diminilhed, fo that the n power of 
 the root now found, may be taken from the aforefaid 
 periods. 
 
 8. To the remainder annex the firft figure of the 
 next period for a new dividend, then find a new di- 
 vifgr as before -, and fo on, till the whok be done. 
 
 EXAMPLES. 
 
 Required the cube root of 61209.566621 : 
 OPERATION. 
 
 Here n 3> therefore the given number fainted is 
 
 61209.566621, and the neareft root of the fir ft -period 
 (6 1 ) is 3, which is thefirft figure of thereof, the n pow 
 er of which is 3X3X3=27 -, and 61 27 34, which 
 having the fir ft figure of the next period annexed to if, 
 
 becomes 342:1: fiv 'ft dividend \ tf#*/ 3X3X3 27 =r/foy/ 
 divifor : Whence, 27 )342(9~ fecond figure of the root> 
 and the whole of the root now found is 39 j therefore, 
 39X39X39=59319= n power of 39, which being 
 Jub traced from the two firft periods, leaves 1890, and 
 
 \*y*$=.Jecond dividend; aljo, 39X39X3 A-S^S 
 Jecond divifor > whence, 4563)18905(4 third figure 
 
 oftheroot. dgain, 394X394X394 z:: ^ i \6-2$%4,whicb 
 Jubtrafted from the three firft periods, leaves 46582, 
 
 then, 465826=: third dividend, 'and 394X394X3 
 465708^=^ third divifor - y whence, 465708)465826(1 
 fourth and I aft figure of the root, and becaufe there 
 are two*periods of decimals in the given number, the 
 root required is 39.41 ; for 39.41X39.41X39.41 = 
 61209.566621^ the number whofe root was required: 
 &c. Required 
 
C 
 
 Required the 6th root of 148035889. 
 OPERATION. 
 
 Firft, txtraft thefquare root, and then the cube roof 
 cf that rejnlt will give the root required : 
 
 Thus, 148035889(1216? 
 i 
 
 22) 4 8 
 
 2 44 
 
 241) 403 
 
 I 241 
 
 2426) 16258 
 6 I455 6 
 
 -4327)170289 
 170289 
 
 w 
 
 Again, 1 2 1 67 ( 23= root required, 
 
 2X^X28 
 
 23X23X23=12167 
 
 o 
 The fame at one operation : 
 
 Thus, 148035889(23 
 2X2X2X2X2X2-64 
 
 2X2X2X2X2X3=96)840 
 23X23X23X23X23X23 = 148035889 
 
 ' IN 
 
( 239 ) 
 
 IN extradling the roots of heigher powers, it will 
 be beft to extraft fquare root out of fquare root fuc- 
 ceilively, as often as the index of the given power is 
 divifible by 2 : Thus, in the i6th power, the index 
 (i6)is divifible by 2, four times - 3 for i6~-2n:8, 8-f- 
 24, 4-~-2~2 > and 2-7-21 : Whence it follows, 
 that the root of the i6th power may be obtained by 
 four feveral extraflions of the fquare root 5 and the 
 like may be fhcwn of all the even powers. 
 
 THE END OF BOOK FIRST. 
 
BOOK II. 
 
 OF ALGEBRA, 
 
 *&*Oto&^i*^^ 
 
 CHAP, 1. 
 
 6f DEFINITIONS 
 
 AND 
 
 ALGEBRA, one of the mod important 
 branches of mathematical fcierice,is a method of 
 computation by figns and fymboh, which have been 
 invented and found ufeful for that purpofe. Its 
 , invention is of the higheft antiquity, and has juft- 
 | ly challanged the praife and admiration of the learn 
 ed in all ages; Arithmetic is indeed ufeful, and is 
 not to be the lefs valued, becaufe it is allowed to be 
 the moil clear and evident of the fciences j yet it is 
 confined in its object,- and partial in its application, 
 Geometry for clearnefs of principles^ and elegance 
 of demftnftration, no lefs deferves, than commands 
 our efteem; but the many beautiful theories, that 
 srife from the application of Algebra and Geometry 
 to each other, fully evince the excclUncf aind etften* 
 
 Hh fiveitefs 
 
fivenefs of the former. The doctrine of Fluxions, 
 Which is eftcemed the fublimity of human fcience^ de 
 pends on the noble fcience of Algebra for its exift- 
 ance and application. In a word, Algebra is juftly 
 efteemed the key to all our mathematical inquiries. 
 
 IN Algebra, like quantities are thofe which have 
 the fame letters : Thus, ax and ax are like quantities; 
 but ax and dx are unlike quantities. 
 
 GIVEN or abiblute numbers, are thofe whofe val 
 ues are known : Thus, 6, 7, &c. are given numbers, 
 becaufe their refpeftive values are known ; but the 
 quantities x, y 3 &c, are not given quantities, becaufe 
 their values are not known, and are therefore called 
 unknown quantities. 
 
 SIMPLE quantities are fuch as have but one term : 
 Thus, , axby and xyz, are fimple whole quantities, 
 
 and~r- and 7- are fimple fractional quantities. 
 
 COMPOUND quantities are fuch as confift of feveral 
 terms connected by the figns-f and : Thus, a-\- 
 Jt-^c </and ax xy are compound whole quantities, 
 
 . a c dx a-\-b . c -. . 
 
 and 7+-. r- i. 7 are compound fractional quan- 
 b d b c d 
 
 tities. Compound quantities ^have fometimes aline 
 
 drawn over them ; as a^b-^c d. 
 
 CO-EFFICIENTS are numbers prefixed to quantities, 
 denoting how many times the quantity to which they 
 are prefixed, ought to be taken : Thus, 3 a denotes 
 that the quantity a is to be taken 3 times $ alfo, na 
 ihews that the quantity a is to be taken as many times 
 as there are units in n : Therfore, co-efficients mul 
 tiply the quantities to which they are prefixed ; and 
 quantities which have no co-efficient prefixed to them, 
 are always underftood to have an unit for their co 
 efficient : Thus, a is i a, x \ #, &c. 
 
 A 
 
A POSITIVE, or an affirmative quantity, is a quan 
 tity having the fign -j- before it ; as -\-a : Alfo, all 
 quantities that have no figns fet before them, as the 
 leading quantity generally hath none, are underftood 
 to have the fign 4-, and are therefore called pofitive 
 quantities. 
 
 WHEN quantities have the fign before them, 
 they are called negative quantities : As a, x ; 
 and when any quantity is to be diftinguilhed, as a 
 quantity to be fubtra&ed, the fign muft be placed 
 immediately before it. 
 
 QUANTITIES are faidto have like figns, when t^iey 
 are all + or all -, 
 
 UNLIKE figns is when the figns are + and . 
 
 A QUANTITY confiding of two terms, as, 
 
 is called a binominal $ a+b+c> a trinominal 
 
 y a quadrinominal, &c. 
 A RESIDUAL quantity, is the difference of two 
 
 quntities, Thus, a , is a refidual quantity. 
 
 THE letters made ufe of to reprefent the unknown 
 quantities, are thofe of the laft part of the alphabet^ 
 and the letters of the firft part, reprefent thofe that 
 are known. 
 
 THE principal figns by which quantities are man 
 aged in Algebra, are the following, in addition to 
 thofe made ufe of in the firft book of this treatife, 
 
 and Explanations. 
 
 is the fign of the fquare root. 
 - - of the cube root. 
 . of the n root. 
 
 . of more or lefs. 
 
I 
 
 x or * T denotes the fquare ropt of x* 
 x or # T the cube root of AT. 
 
 or <*+) T the fquare root Qfa+ 
 or ^4" the root of 
 
 * the reciprocal of #. 
 
 
 the reciprocal of-. 
 y 
 
 <i^ the (urn or difference of ^ and t 
 
 AXIOM S. 
 
 1. IF to thofe quantities that are equal, there be 
 gdcjed the fame quantity, their fum will be equal. 
 
 2. IF from thofe quantities that arc equal, thertf 
 be taken the fame quantity, the remainders will be 
 equal. 
 
 3. IF thofe quantities which are equal, be multi 
 plied with the ftme quantity, their products will be 
 equal. 
 
 4. .IF thofe quantities that are equal, be divided 
 by the &me quantity, .the quotients will alfo be 
 equal. 
 
 5. Two quantities refpeftiyely ecjual to a third,, 
 are equal to each other. 
 
 6. EQUAL powers, or roots of equal quantities^ 
 are equal to each other. 
 
 7. IF to any whole number, there be added any 
 other whole number, the fum will be a whole num- 
 ker.' 
 
 8. IF from any whole number, there be taken any 
 other whole number, what remains will alfo be, a 
 whole number. 
 
 9- 
 
( 345 ) 
 
 g, IF any whole number be multiplied with any 
 p$her whole number, the pro4uft will alfo be a whole 
 number. 
 
 CHAP. II. 
 
 ADDITION of WHQLE 
 A D D I T I O N cpnfifts of three cafes. 
 
 CASE i. 
 
 Wbcn the quantities arc alike, and have 'li 
 RULE. 
 
 ADD the co-efficients together, and to their furn 
 annex the common quantity, prefixing the common 
 Jign, 
 
 EXAMPLES. 
 
 lab 6^ - 2xy zx 2 a 
 tab x iQx 6x ^ 
 
 ixy ax a 
 -^ xy x a 
 
 IT ak >8# i%xy 
 
_ 
 
 av 3 xy' i + iaz b 4^;* 6 0+32^ 
 
 t^ 
 
 tf<tf #jy*-f3 <*z 3 ^ ^ w * 2 *+ * *8 d 
 
 t^ 
 
 av 6 #>*- #- W T - #-0 d 
 
 1 3 av -i o^y * + 1 4^% 5< ^i i w * 9# + 41 
 
 CASE II. 
 
 ^Z>^ /Ad- quantities are alike, but have unli 
 RULE, 
 
 1. ADD all the affirmative quantities into one fum 
 by the Jaft rule, and the negative into another. 
 
 2. SUBTRACT their co-efficients, the lefs from the 
 greater, and to their difference, prefix the fign of the 
 greater, annexing the common quantity. 
 
 THE reafon of the foregoing rule will appear evi 
 dent, if you put a = debt due to B, and a the 
 want of a debt, or a debt due from B j then the bal 
 ance is evidently equal o, or 4- & ^ o : Whence,. 
 
 EXAMPLES, 
 
 3 * + 6 J> 
 S*+2j 
 io ^ 
 
 io ay ~ yo <?/<? negatve. 10 
 {- 5 *? ^-Jum of the affirmative. + 7 ^ 4- 
 
 ^ 5 #y ~Jum required, 
 
4 </* a // **4-jE 3 * 4-4 ^ w* 
 
 8 </*-- 
 
 CASE III. 
 
 ^T^/r the quantities are unlike, and have unlike Jlgns. 
 RULE. 
 
 WRITE the quantities one after another with their 
 proper figns, and they will be the fum required. 
 
 Note. If there be like quantities given, you muft 
 colleft them by the preceding rules. 
 
 EXAMPLES. 
 
 7 V +99? 
 4^4.7 c 
 
 
 8 d < 7 <y 
 
 3 a 4-4 1 2 c 2 d ^=.Jn 
 
 4 
 
 I X 
 
 ^-^^jy" 57 - 2 /fr +2J* 5 " c& % 4-96 3 \/^*~ 
 
 * rf*~- s -/a* 
 
 CHAR 
 
CHAP, III. 
 
 SUBTRACTION of WHOLE 
 TITIES. 
 
 A 
 
 LGEBRAIC Subtra&ion is performed by 
 the following general 
 
 R U L E. 
 
 CHANGE the flgns of the quantities in the fubtri- 
 hend (or fuppofe them in your mind to be changed) 
 then add the quantities with their figns changed, to 
 the number from which fubtra&ion is to be made, 
 by the rules of the Jaft chapter and their fum will 
 be the remainder required. 
 
 THE reafon of this rule will appear obvious, when 
 we confider that fubtraftion is the revrfe of addition j 
 and therefore} to fubtraffc an affirmative or negative 
 quantity, is the fame thing as to add its oppofite 
 kind : Whence, if a is to be taken from + tf, the 
 difference will be + 2 a> for if the remainder 2 a be 
 added to the fubtrahend *z, their fum will be 3: a 
 n the number from which fubtra&ion was made i 
 Whence, &c 
 
 EXAMPLES. 
 
 From 44 4lu^~%b* 
 
 3 a ibu 2^ a 4y+ 
 
 Remains a y,bu~~> b* 
 
 34 V 
 
( 249 ) 
 
 Ic 
 
 Ir any doubt arife, refpeding the truth of the 
 peration, add the remainder to the fubtrahend, 
 rhich fum muft be equal to the other number. 
 
 CHAR IV. 
 
 Of MULTIPLICATION. 
 
 LGEBRAIC Multiplication confifts of 
 three cafes. 
 
 CASE I. 
 
 When loth the faftors are fm fie quantities. 
 RULE, 
 
 MULTIPLY the co-efficients together, and to their 
 roduct annex all the letters in both faffcors, as in a 
 r ord ; this exprefTion being wrote with its proper 
 gn, will give the product required. 
 
 Note. Like figns give +, and unlike Jigns for 
 the produft, 
 
 I i EXAMPLES* 
 
( 25 ) 
 
 EXAMPLES. 
 
 4. 30 3 21 j^ 3 
 
 60 2 2 
 
 1 8 tftffo 4-1 yyy -f-6 tfzejyjp produff. 
 
 CASE II. ^ 
 
 # of the factors is a compound quantity. 
 
 R U L E. 
 
 1. WRITE the compound quantity for the multi 
 plicand, and the fimple quantity for the multiplier. 
 
 2. OBTAIN the product of the multiplier with 
 every particular term of the multiplicand, by the lad 
 rule, and place the terms of the product one after 
 another, with their proper figns, found as in the laft 
 rule, and you will have the product required. 
 
 EXAMPLES. 
 
 a 4- b 3 ab -J- cd 2 aa + 2 ab -{- It ' 
 
 a d ia 
 
 au 4 cv +34 2.7 ddd aaa 
 
 3 auy^ii cvy io2y 8 1 dddw 3 aaaw. 
 CASE III. 
 
 When both tbefySiors arc compound quantities. 
 
 RULE. 
 
R ULE. 
 
 MULTIPLY every particular term of the multiplier, 
 with all the feveral terms of the multiplicand, as in 
 the laft rule, the feveral products collected into one 
 fum by the rules of addition, will give the who!? 
 product required. 
 
 + 
 
 + 
 
 y 
 
 y 
 
 EXAMPLES, 
 a b 
 
 V 2Z 
 V+ 22 
 
 vv -f- vy 
 
 +yy 
 
 aa- ab 
 
 ab 
 
 OT 2 -VZ 
 
 bb -f 2 vz 
 
 vv -\-ivy +yy 
 
 yy -f xx 
 
 yy xx 
 
 yyyy 
 
 aa 
 
 bb 
 
 2X I 
 
 yyxx xxxx 
 
 2 xx 8 x 
 
 *- 2 xy 
 
 yyyy 
 
 xxxx 
 
 4 xxy 
 
 < 2 #j 9 A: 4- 4 
 
 HAT -f X or -r- x + gives , an'd^ X 
 gives -j- for the product, is demonftrable feveral 
 ways, but none more fimple than the following. Sup- 
 pofe a ~ b j then a b zzo : Now it is plain, that if 
 this exprefllon be multiplied with any number what 
 ever, the product will be ~o : Therefore, fuppofe 
 2 b = o, is to be multiplied with -f- n > now ' lt is 
 manifeft, the firfl term of the product a X n will be 
 poiitive j or -f- ##> becaufe bo;h the fadors are pofi- 
 
( 252 ) 
 
 tive y confequently the other term of the product 
 X b muft be negative, or nb ; for both terms 
 of the prod uft taken together,muft deftroy each other, 
 and their amount =o ^Jthat is, na nb zi o : Gon- 
 fequently ~j- X > or X + gives for the prod- 
 
 vft. 
 
 AGAIN, fuppofe a b o, be multiplied with 
 n ; the firft term of the product n x a will, be 
 negative, or na> by what has been proved : Con 
 fequently, the other term # X b will be pofitive, 
 or -j- nb 3 for both terms taken together mufl n o ; 
 thus, na -f- nb m o : Cosfequently, X gives 
 + for the produft. ^. . D. 
 
 CHAP. V. 
 
 of p iris IQ ft. 
 
 DIVISION being the converfe of multipli 
 cation ; it follows, that the quotient muft be 
 iiich a quantity, that if multiplied with the divifor, 
 will produce the dividend > confequently, like figns 
 in divifion give -f, and unlike figns -for the quo 
 tient. 
 
 CASE I. 
 
 Wktrn the dimjarls aftmfle quantify, 
 R U L E. 
 
 T. WRITE down the quantities, in form of a vul 
 gar fradtion, having the divifor for the denominator. 
 
 2. EXPUNGE all thole quantities in the dividend 
 and divifor, that are alike 5 and divide the co-effi 
 cients 
 
dents of the quantities by any number that will di 
 vide them without a remainder ; the refult will be 
 the quotient fought. 
 
 EXAMPLES. 
 
 ^u the quotient > ==i2j-2 S -=* 
 
 2 a 22 a 
 
 ii adz 8 Jcz 
 
 - 42 
 
 IF you divide any quantity by itfelf, the quotient 
 will be unity or i : Thus, -mi jfor if the quotient be 
 
 multiplied with the divifor, the produd will be the 
 dividend ; thus, x x i ~ x - Confequently, if any 
 term of the dividend be like that of your divifor, 
 the quotient of that term will be i : As in 
 
 "U 
 
 3 
 CASE II. 
 
 When the divifor and dividend are both torn found 
 quantities. 
 
 RULE. 
 
 1. RANGE the quantities in the divifor and divi 
 dend, according to the order of the letters. 
 
 2. FIND how often the firft term of the divifor is 
 contained in the firft term of the dividend, and place 
 the refult in the quotient. 3. 
 
3. MULTIPLY the quotient term thus found, with 
 the whole divifor, fubtrad the product from the 
 dividend, and to the remainder bring down the next 
 term o&the dividend ; which forms a new dividend* 
 
 4. DIVIDE the firft term of your new dividend, 
 by the firft term of your divifor, as before ; and fo 
 on, until nothing remains, as in common Arithmetic, 
 and you will have the quotient required. 
 
 EXAMPLES. 
 
 Suppofe it is required to divide i yyy + %yy -f-8^ 
 \)j yy + iy > which being ranged as directed in the 
 rule, the operation will ftand 
 
 Thus, yy + zy)iyyy + *yy + *y(*y +4 
 
 lyyy +4 yy 
 
 Here thefirft term of the dividend, which is zyyy, 
 being divided by tbejirfl term of the divifor yy, the quo 
 tient is iy\ which being placed in the quotient as in 
 vulgar Arithmetic, and multiplied with all the terms of 
 the divifor, the produft is iyyy +4jv>', whichjubtraft- 
 cd from the dividend, the remainder is 4jvy, to which 
 annex the next term of the dividend %y, the new divi 
 dend becomes ^yy -fr-8j>, and dividing q.yy by yy, the 
 quotient is 4 ; which being annexed to the quotient term 
 before found, and multiplied with every term of the di- 
 vipr, produces ^yy-\-^y^ which fubtrafted from the 
 lafl dividend, the remainder is nothing ; and having 
 brought down all the terms of the propofed dividend , 
 the work is done ; therefore, iy -{-4 is the true quo 
 tient, for iy -f 4 X-^y + 2j> ~iyyy +8jjy 4-8 J T 
 the given dividend. Divide' 
 
Divide 6 avv $av -a^ 4-2 ^ +2v i by 
 v i. 
 
 OPERATION. 
 
 f\ sificr 
 
 * * ~4vy 
 
 1> * I 
 
 2V I 
 
 Divide vvv yyy by v j^. 
 OPERATION. 
 
 v y) vvv >yyy(vv+vy+yy 
 
 * + vyy yyy 
 -f- vvy vyy 
 
 yyy 
 
 vyy yyy 
 
 Divide i by i*v 
 
 OPERATION, 
 
256 
 
 OPERATION. 
 
 IN this example, the divifor cannot exactly be found 
 in the dividend, without a remainder > and you have 
 what is called an infinite (tries for the quotient ; that 
 is, if the divifion could be carried on ad iufinitum, 
 you would have a feries of terms for the quotient, 
 that would come infinitely near to an equality with 
 the true quotient, and therefore might be confidered 
 as fuch ; for when ratios from that of equality, are 
 but indefinitely little, or lefs than can be afiigned, 
 they may be confidered as equal -, but as it is impof- 
 fible to carry on the divifion ad infini titty, or take in 
 a fufficient number of terms to exprefs the true quo 
 tient : Therefore, in general you need only take a 
 few of the leading terms for the quotient, which will 
 be fufficiently near for mod purpofes. : But more of 
 this in its proper place, fince the knowledge of Alge 
 braic fractions, is in moft cafes, abfolutely necefTary, 
 in order to obtain an infinite feries by divifion. 
 
 CASE III. 
 
 When tie quantities in the dfoifor cannot be found in 
 the dividend. 
 
 RULE, 
 
, RULE. 
 
 PLACE the dividend above, and the divifor belovy 
 a fmall line,, in form of a vulgar fraction j and the 
 expreffion will be the quotient required. 
 
 EXAMPLES. 
 
 The quotient of a divided by b, is -, 
 The quotient 0/21 Ix -~ d 7 
 
 The quotient cf%ac + dc-~zx + 
 
 ^ 8 <ic + dc 
 
 zx -|- ab 
 
 C H A P. VI. 
 
 INVOLUTION of WHOLE g>JJdN- 
 
 INVOLUTION is the raifing of powers 
 from quantities called roots, and differs from 
 multiplication in this, viz. that in involution the 
 multiplier is conftant, or the fame j therefore when 
 any quantity is drawn into itfelf, and afterwards into 
 that product, and fo on, the mode of operation is 
 called involution, and the number produced, the 
 power, whofe height is ufually denominated by plac 
 ing numeral figures over the right hand of the root> 
 or quantity to be involved, and are called indices or 
 exponents of the powers which they denominate : 
 .Thus, a*~aa denominates the fquare of a, a*~ 
 
 K k aaa 
 
the cube of#, a*- the fourth power of a\ and 
 generally, a n the n power of a. 
 
 INVOLUTION of firnple quantities is performed by 
 the following 
 
 RULE. 
 
 MULTIPLY the index or exponent of the given 
 quantity or root, with the exponent which denomin 
 ates the power required, making the produft the 
 exponent of the power fought. 
 
 Note. If the quantities to be involved, have co-effi 
 cients, the co- efficients muft be involved as in vul~ 
 gar Arithmetic^ to thejame height as the index of 
 the fower required denotes. 
 
 EXAMPLES. 
 
 Jquare of n = a ~ = a* j the cube of a=^ 
 
 -a*i the cube 
 > tic tfb fower of 
 :z: 256 x l *y* 5 
 the n power of x~ x l ^ n =x> 1 . 
 
 IF the quantity propofed to be involved is pofitive, 
 all its powers will be pofitive: Alfo, if the quantity 
 propofed be negative, all its powers whofe exponents 
 are even numbers, will likewife be pofitive ; becaufe 
 any even number of multiplications of a negative 
 quantity, gives a pofitive one for the product, fince 
 X gi^s -f-5 confequently X X X = 
 -f X+ f r tne produdl ; therefore, that power of the 
 negative quantity, only is negative, when its expo 
 
 nent 
 
nent is an odd number : As may be feen in the fol 
 lowing form, 
 
 a the root, * 
 
 a the roof 
 
 a 3 ir cube 
 
 a the root 
 
 a* tf power 
 a the root 
 
 I a*-s=z$tb power. 
 
 INVOLUTION of compound quantities, is perform 
 ed by the following 
 
 RULE. 
 
 MULTIPLY the root into itfelf, and then into that 
 product, and fo on, until the number of multiplica 
 tions are one lefs than the exponent of the power re 
 quired ; the refult will be the power fought, 
 
 EXAMPLES, 
 
 L,et the binomial <?-f-be involved to the 5th 
 power- 
 
 OPERATION, 
 
( 260 ) 
 
 , _ -| - - 
 
 OPERATION. 
 
 a+b the root 
 a + b 
 
 aa-\-ab 
 
 aa ~f- lab -f- bb 
 
 aaa~\-i.aab-\~abb 
 -\-aab \-<2. 
 
 aaab + ^aabb-\-^abbb -\- bbbb 
 
 aaaa -\- ^.aaab + 6aabb + ^abbb + bbbbit^th fower 
 
 aaabb + ^aabbb + ^^^^^ 
 4- aaaab-\-^aaabb -\> 6aabbb + A^abbbb -f- bbbbb 
 
 Involve 
 
Involve a b to the 3d power. 
 OPERATION. 
 
 z lab +b*~ id power 
 a b 
 
 * & 3 ~ 3d power. 
 
 IT is to be obferved in the foregoing examples. 
 
 1 . THAT all the terms in the feveral powers, railed 
 from the binomial a-\-b y are affirmative. 
 
 2. THE terms in the feveral powers raifed from the 
 refidual a b y have the figns -f- and , alternate 
 ly ; the firfl term being a pure power of a, is confe- 
 quently affirmative ; the fecond term hath a nega 
 tive fign, and fo on, alternately j but b is no where 
 found negative, only where its exponent is an odd 
 number; as in a 2 ^a^b+^ab* b* ; where the 
 fecond and fourth terms are negative, becaufe the ex 
 ponent of b in thoje terms, is an odd number. 
 
 3. THAT the firft term of any power, either of the 
 binomial or refidual, hath the exponent of the pow 
 er : That is, the index of the firft term, is equal to 
 the index of the power ; bui in the reft of the terms 
 following, the exponents of the leading quantity, de- 
 creafe in arithmetical progreflion, unity or i, being 
 the common difference ; fo that the quantity a is 
 
 never 
 
( 262 ) 
 
 never found in the Lift term j but the exponents of 
 b y on the contrary, increafe in the fame progreflion 
 that the exponents of a decreafe ; that is, the quan 
 tity b> is not to be found in the firfttermj but in the 
 fecond term, its exponent is unity or i 5 in the third 
 term 2, and fo on in the faid arithmetical progref- 
 fion, to the laft term, where its exponent is equal to 
 the exponent of the power. 
 
 4. That the number of terms in any power, is one 
 more than the number which denominates that; 
 power. 
 
 HENCE from the foregoing obfervations it follows. 
 
 i. THAT the fum of the exponents of both quan 
 tities in any term, are equal to the exponent of the 
 power in which thofe terms belong : Thus, the 6th 
 power of a +b zr a 6 -\-6a*b + i$a*b* + 2oa*l> 3 
 * i$a*&*-\-6at> 5 +b 6 y where you will pleafe to ob- 
 ferve, that the fum of the exponents of a and b y in 
 any term, are equal to the exponent of the power : 
 Thus in the third term, the exponents of a and b y 
 are 4 and 2, whofe fum 6 ^exponent of the pow 
 er. 
 
 2. THE method of writing without a continual in^ 
 volution, the terms in any power of a binomial, or 
 refidual quantity, without their co-efficients : Thus 
 the terms of the 4th power of x +y without their 
 co-efficients, will ftand thus : x^+x^y+x^y* *%-xy* 
 and the terms of the 4th power of # y~x* 
 ,?* xy 3 -f-j*. 
 
 IN order to find the co-efficients ofthefeveral 
 terms, it is necefTary to have the co-efficient of one 
 of the terms given : And becaufe the firft term or 
 leading quantity is a pure power, having its index 
 equal to the index of the given power ; its co-effi 
 cient is therefore unity or i : Confequently, you 
 
 have 
 
have the co-efficient of the firft term given ; thence 
 to find the co-efficients of the reft of the terms by 
 the following 
 
 RULE. 
 
 DIVIDE the co-efficient of the preceding term, by 
 the exponent of y in the given term ; the quotient 
 multiplied with the exponent of x, in the fame term, 
 iiicrcafed by x, will give the co-efficient required. 
 Or, 
 
 MULTIPLY the co-efficient of any term, with the 
 exponent of the leading quantity, in the fame term ; 
 the product divided by the number of terms to that 
 place, will give the co-efficient of the next fubfe- 
 quent term. 
 
 EXAMPLES. 
 
 Given # 4 +# 3 ^4-#^ 7 +*y 3 +.7 4 j to find the co-ef 
 ficients of the feveral terms. 
 
 Firft, the co-efficient of # 4 is i ; thence to find the 
 co-efficient of# 3 j : And becaufe the exponent ofy 
 
 in the given term, is unity or i ; then per rule, I 
 = iX4=4> the co-efficient required : Again, 
 
 i=rlX3n =6, the co-efficientof the third 
 2 22 
 
 term ; and -Xi+ izr-X 2 ~4> the co-efficient 
 
 3 33 
 
 of the fourth term ; but the next term hath the ex 
 ponent of the power,being the lad term of the 4th pow 
 er of A'-f-jy, and confequently, its co-efficient an unit 
 or i . Therefore,the co-efficients of the feveral terms 
 of the 4th power of x+y t are i, 4, 6, 4, i. 
 
 HJSNCIZ 
 
 
HENCE you may obferve, that the co-efficients of 
 the feveral terms increafe, until the exponents of x and 
 y become equal to each other, and then decreafe in 
 the fame order in which they increafed. And geni 
 ally, the co- efficients of the terms increafe, until the 
 exponents of the two quantities become equal in one 
 term, if the exponent of the power is an, even num 
 ber ; arid when the exponent is odd, two of the terms 
 will have equal co-efficients, and then clecreafe in 
 the fame order. Therefore, in finding the co-effi 
 cients, you need only obtain the co-efficients, until 
 they decreafe ; the reft of the terms having the fame 
 co-efficients decreafmg. 
 
 THE n power of ^ + iif+^""V-f-X 
 
 -z^-v+ix^x^*'-^' &c . to- 
 
 2 23 
 
 i, terms. 
 
 Let a -}- b 4- c be involved to the fecond power, 
 OPERATION. 
 
 1 -f ab 4- ac 
 +l* 
 + ca 
 
 a* -J- iab 4- iac-{- b* -fa bc+ c*^ id power. 
 
CHAP. VII. 
 
 Of MULTIPLICATION and DIVISION 
 of POWERS of the fame ROOT. 
 
 M 
 
 ULTI PLICATION of powers of the 
 fame root, is performed by the following 
 
 RULE. 
 
 ADD the exponents of the powers together, and 
 make their fum the exponent of the product. 
 
 EXAMPLES. 
 
 =: 6 s rz 
 
 7776 ; 6 # 3 X4 # 4 =: 
 
 a 6 ~ a 10 ; alfoy 0'X ^ 4 = ^ 3 i in like man 
 
 5 0m/ 
 
 unmerfally> 
 
 DIVISION of powers that have the fame root, is 
 effected by the following 
 
 RULE. 
 
 wfl 
 
 FROM the exponent of the dividend, fubtradt the 
 exponent of the divifor, and the remainder will be 
 the exponent of the quotient. 
 
 LI EXAMPLES. 
 
( a66 ) 
 
 EXAMPLES. 
 
 4 .2_<2 5 3 Z 
 
 5 
 
 HENCE it follows, that in divifion of powers which 
 have the fame root, if you divide a lefs power by a 
 greater, the exponent of the quotient will be nega 
 tive ; for we have fbewn, that to divide any power 
 of a by #, is to fubtract one from the exponent of the 
 
 power of a: Thus, -z:^ 1 ; therefore, -~a 
 a a 
 
 ~a 5 but ~i by the nature of divifion j confe- 
 a 
 
 quently, # i by equality j and therefore, Izif- 
 
 a a 
 
 o i i^ j i # o 2 2 ^ , 
 
 a*~~a T ~ 
 
 ib on for any power of ~; Likcwife, ^ I a z= 
 x-\-y\ " ~^Hhjl = (becaufe, ' 1 3 ) i '; 
 
 confequently, . l ^^!J2l .y^,y|"" . therefore. 
 
 ' 
 
J ; And generally, ' 
 ~x+j\~~ n . Therefore, 0, 4 I > a 2 , ^ 3^ an j^ 
 
 * a* a* 
 
 and of which they are pofitive pow- 
 
 | x+y 
 crs. 
 
 HENCE the propriety of ufmg negative exponents. 
 
 THE multiplication, and divifion of powers which 
 have the fame root, having negative exponents, is 
 performed by the fame rule as thofe powers which 
 have affirmative ones ; that is, add the exponents of 
 the factors in multiplication, and in divifion fubtracT: 
 them. 
 
 EXAMPLES. 
 
 *>O iHi*/t MMW O wmvmm A "'^O 
 
 A multiplied with a ^ r= a * = a ; 
 
 a~ 3 X a~~ l = a~~ l ~ 3 =i a '~'* > a~* X a* ~ 
 
 *\/a X 
 
 -r-0 ^~ (by the nature of fubtra&ion) 
 ==tf -3 ==I ^ 3 . and ,f-3 -f^- 6 zz^-3 
 
 5 but by the nature of multiplication and 
 divifion, a~~^~d~' ~a ^-r-^"*"" 1 ^ X # ^^ 
 
 = a 3 -, likewife, 
 
( 263 ) 
 
 CHAP, VIII. 
 
 EVOLUTION of WHOLE 
 
 TIES. 
 
 EV O L U T I O N is the unfolding of powers 
 produced by involution 3 thereby difcovcriog 
 -the roots with which they are compofed, and is there 
 fore the reverfe of involution, 
 
 THE rule for evolution of powers, whofe roots are 
 fimple quantities, flows from this confideration ; that 
 to involve any fimple quantity to any power, is to 
 multiply the exponent of the quantity, with the expo 
 nent of the power ; making the product the exponent 
 of the required power; consequently, if the expo 
 nent of the power, be divided by the index which 
 denominates the root required, the quotient will be 
 the exponent of the root. Therefore, when the ex 
 ponent of the power whofe root is required, is not 
 a multiple of the number which denominates the 
 kind of root required -, it follows, that the root will 
 be exprefled by a fractional exponent : Thus, the 
 
 s . 
 
 fquare root of a 5 a*, and the cube root of ^ 4 =^ T . 
 Whence, we have the following rule for evolution of 
 fimple quantities. 
 
 RULE, 
 
 EXTRACT the root of the co-efficient, as in vul 
 gar arithmetic, and divide the exponent of the power, 
 
fay the index of the root rquired \ making the root 
 of the co-efficient, the co-efficient of the root. 
 
 EXAMPLES. 
 
 cube root of a 9 a r ~ a* : The fquare root of 
 i2a*~ 2 a* : <?be cube root of 64.x 9 a 3 3 ^64 
 4* 3 a : fhe tfb root of 256 a*b** =1*^/256 
 
 4. I 
 
 X <? T T ~ 4 # 3 : ^be cube root of 27 # 3 zi 3/2 J . 
 But the fquare root of a negative quantity ; as #% 
 cannot be afTigned, becaufe no even number of mul 
 tiplications, either of a pofitive or negative quantity, 
 can give a negative one for the produd, as was fully- 
 explained in chapter vi 5 therefore, the fquare root 
 of #* is an imaginary quanritiy : And fince the 
 iquare of any negative or pofitive quantity, is always 
 pofkivej it follows, that the fqjare root of x* may 
 be-M, or x. Therefore, when the number which 
 denominates the root to be extracted, is odd, thefign 
 of the root will the fame as the fign of the power ; 
 and when the number which denominates the root, 
 is even, the fign of the root may be either -j- or : 
 Thus, the cube root of 2ja t5 b 9 ~ 3 a s b 3 , 
 and the 4th root of 16 *#* 2 a*x or 2 a* ' x ; the 
 
 the ;; power of .v w zr^-; 
 
 EVOLUTION of compound quantities, requires a 
 different method of proceeding from that of fimple 
 ones. 
 
 To extract the fquare root of a compound quan 
 tity we have the following 
 
 RULE. 
 
RULE. 
 
 i. RANGE the quantities according to the order 
 of the letters, fo that the firft term fhall have the 
 index of the power. 
 
 . 2. FIND the root of the firft term, as in evolution 
 of fimple quantities, and place it in t}he quotient* 
 
 3. SUBTRACT the fquare of the root thus found, 
 from the firft term of the power propofed, and to the 
 remainder bring down the reft of the terms for a di 
 vidend. 
 
 4. DIVIDE the firft term of the dividend, by dou 
 ble the root, and write the refult in the quotient, 
 for the fecond term of the root, 
 
 5. ADD the laft term of the quotient to your divi- 
 for, and multiply their fum with the faid quotient 
 term, fubtracling the product from the dividend ; 
 and fo on, to obtain the next term of the root, by 
 the help of jthofe already found, in the fame manner 
 as the fecond term was obtained by the help of the 
 firft. 
 
 EXAMPLE. 
 I Extract the fquare root of a % + 2 ay + y ' ~f- 2 za 
 
 -f 2J2 4- Z\ 
 
 The fquare roof of the fir fl term viz. # % is a y which 
 being placed In the quotient, is the firft term of the roof, 
 (fee the operation annexed) which fquared and fubtraffed 
 from the firft term of the propofed power, leaves no re 
 mainder ; the reft of the terms being brought down for 
 a dividend, the firft term, viz. i ay divided by la 
 (the double of the root) gives y for the fecond term of 
 the root ; which with the divifor, being multiplied with 
 y> and the produtt fubtrafted from the firft terms of the 
 
 dividend, 
 
dividend, the remainder is nothing; the remaining 
 terms being brought down as before and divided by the 
 double oftbe twofrjt terms of the root, gives zfcr tlse 
 third term of the root, which added to the divifor and 
 multiplied with z, the frodutt fubtrafted as before, 
 leaves no remainder : Therefore, the root fought, is 
 
 a 
 
 OPERATION. 
 
 And univerfallyy to extraEl any root. 
 
 R ULE: 
 
 1. RANGE the terms of the given power, as in the 
 laft rule. 
 
 2. EXTRACT the root of the firft term as before, 
 and place it in the quotient for the firft term of 
 the root. 
 
 3. SUBTRACT the power of the root thus found, 
 and to the remainder bring down the next term for 
 a dividend. 
 
 4. INVOLVE the root to a dirnenfion lower by unity 
 than the number which denominates the root requir 
 ed, and multiply the refult with the index of the 
 
 root 
 
root to be extracted, which product call your divi- 
 for. 
 
 5. FIND how often the divifor is contained in the 
 dividend, and write the refult in the quotient for the 
 fecond term of the root. 
 
 6. INVOLVE the whole of the root thus found, tc 
 the dimenfion of the given power, and fubtra<5t the 
 refult from the given power \ and call the remainder 
 a new dividend, 
 
 7. INVOLVE the whole of the root in the fame man 
 ner as you did the firfl term, and multiply the refult 
 as before for a new divifor. 
 
 8. DIVIDE as before, and the refult will be the 
 third term of the root; and fo on, till the whole be 
 finifhed. 
 
 EXAMPLES. 
 
 Required the fquare rootofi6jp 6 
 
 +96^+64. 
 
 OPERATION. 
 
 - 8 is the roof required. 
 
 Required 
 
( 273 ) 
 
 Required the cube root of 8 3 -J- 
 . 
 
 OPERATION. 
 
 # * * * 
 
 y ia+ b, is the root required. 
 
 CHAP. IX. 
 
 Of ALGEBRAIC FRACTIONS or 
 BROKEN QUANTITIES. 
 
 ALGEBRAIC fra&ions are formed by the 
 divifion of quantities incommenfurable to each 
 other : Thus, if x is to be divided byjy, it will be 
 
 X 
 
 (by cafe in, of algebraic divifion) -, which is an 
 
 y 
 
 algebraic fra&ionj wherein x is the numerator and_y 
 the denominator. When fractions are connected 
 
 v /*v -jf j ^ 
 
 with undivided quantities , as a -+--. and a + . 
 
 y a+b 
 
 they are called mixed quantities j alfo, if the denom 
 inator is lefs than the numerator, the fraction is 
 called improper. 
 
 THE various operations, neceflary in managing 
 algebraic fractions, arc comprifed in the following 
 problems. 
 
 Mm PROB. 
 
PROBLEM I. 
 
 3"0 reduce a mixed quantity to an improper fraRion 
 of equal value. 
 
 RULE. 
 
 MULTIPLY the denominator of the fraction with 
 the integral part, to which product add the numera 
 tor, and under their fum, fubfcribe the denominator, 
 for the fraction required. 
 
 EXAMPLES. 
 
 tf^li-rr- 
 
 ^ 2 
 
 : iv 
 
 a 2 ^ 2 
 
 PROBLEM II. 
 
 ^i? reduce an improper f ration to a whole or mixed 
 quantify. 
 
 RULE. 
 
 DIVIDE the numerator by the denominator for the 
 integral part, and write the denominator under the 
 remainder for the fractional part ; and you will have 
 the number required, 
 
 EXAMPLES. 
 
( 27S ) 
 
 EXAMPLES. 
 
 a b 
 PROBLEM III. 
 
 TV reduce fractions of different denomination s> to 
 fractions of the fame value , that Jhall have a common 
 denominator* 
 
 RULE. 
 
 1. REDUCE all mixed quantities to improper frac 
 tions. 
 
 2, MULTIPLY every numerator feparately taken, 
 into all the denominators except its own, for the 
 feveral numerators, and all the denominators toge 
 ther for the common denominator, which being 
 wrote under the feveral numerators, will givti, the 
 fractions required. 
 
 EXAMPLES. 
 
 Reduce 1 and -L to fractions of the fame value, 
 
 2 4 
 
 .having a common denominator. Firft, ArX4^4-v 
 and jX2~2j for the numerators : Then, 2X4=8, 
 
 the common denominator. Therefore, il and 22 
 
 o b 
 
 are the fractions required. 
 
 Reduce -, 5, and - to equivalent fractions, 
 y v c 
 
 having a common denominator, 
 
s=w*'1 
 ~cyz V 
 ~ayv J 
 
 numerators. 
 
 izrvyz: common denominator. 
 Therefore, fZl, 2iand 22 are the fractions re- 
 
 quired; which are rjefpeftively equal to -, -, ?. 
 
 for - = (by the nature of divifion)- $ and the like 
 for the reft. Whence, &c. 
 
 Reduce, -' , , and 2! to a common de 
 nominator, retaining their refpective values. 
 
 "lav iv* 1 
 
 ;=2i> 3 b |> = numerators. 
 
 zv X 2 X ^^14^* common denominator. 
 
 ,_,, .. 2^ < y~~~*2'y a 2*y 3 ^ i A^^V i 
 
 i^ierefore, , . , r , and 2-^, are the 
 
 fra<5lions required. 
 
 _, H, and reduced to a common denomin- 
 x ba ax 
 
 x*a*x cax* , b*acx 
 
 ator, are - '- - , . and 7-- 
 
 PROBLEM IV. 
 
 ^ greateft common meajure of algebraic 
 fractions. 
 
 RULE. 
 
R U L E. 
 
 1. RANGE the quantities as in divifion. 
 
 2. DIVIDE the greater quantity by the lefs, and 
 the laft divifor by the laft remainder, until nothing 
 remains; taking care to expunge thofe quantities 
 that are common to each divifor ; and the laft divifor 
 will be the greateft common meafure required. 
 
 EXAMPLES. 
 
 ~ 
 
 Find the greateft common meafure of 
 OPERATION. 
 
 Therefore, v a, is the greateft common meafure 
 required. 
 
 Z 7fc 
 
 Find the greateft common meafure of- 
 
 * 
 
 OPERATION. 
 
OPERATION, 
 a* at> + l>*) a* ~ !>* (i 
 
 O, ( by cafting out ib) a b} a* iab+ &* 
 
 a 1 ab 
 
 ^Therefore, ab, is the great eft common meafure r^ 
 
 PROBLEM V. 
 
 To reduce fractions to their leafl terms* 
 RULE. 
 
 1. FIND their greateft common meafure by the 
 laft problem. 
 
 2. DIVIDE both terms of the propofed fra6lion by 
 their greateft common meafure, and the quotients 
 will be the refpedivc terms of the fraftion, reduced 
 to its leaft terms. 
 
 EXAMPLES. 
 
EXAMPLES. 
 
 Reduce * a + a to its leaft terms. 
 
 xy* + y*a 
 
 Firft, xa + a*) xy* -\-y*a 
 Or, x+a)xy*+y*a(y* 
 
 * # 
 
 *Then y x + a) xa -{ a % (^= numerator. 
 xa + a* 
 
 a) xy* + y^a (j>* rr denotftinator. 
 
 Therefore, is tbe propofed fraftion in its haft 
 
 terms. 
 
 Reduce -Jlfl to its leaft terms. 
 y s x 2 y 3 
 
 Firft) the great eft common meafure is y* x* : 
 
 then, 
 
 PROBLEM VI. 
 
 jTi? add algebraic fractions* 
 RULE. 
 
 i. PREPARE the given fra&ions by redu<5lion i 
 that is, mixed quantities muft be reduced to improp 
 
cr fractions, and all fra&ions to a common denom 
 inator. 
 
 2. ADD all the numerators together, under which 
 write the common denominator; and you will have 
 the fum required. 
 
 FOR, put -z=<z, and -. b $ then will v ~ya and 
 
 y y 
 
 z~yb by the nature of divifion; confequently ya + 
 jzz v-\-z, and therefore by divifion a+b zz 
 
 But, a + b^+*-> confequently, 2+; = !2 ; 
 
 jr ^ y y y 
 
 which is the fame as the rule. 
 
 EXAMPLES. 
 
 Given -, ^ and 1^ to find their fum. 
 66 6 
 
 u -\-u -j-4 z ^ 2. u + 4 2 tf^/ i!_r: /^^ requir. 
 
 6 
 
 Having ~, and^ given to find their fum. 
 2 y u 
 
 Firft, uxyX u ^ a *Jj and 
 
 -f- 6 j -- 2 #j z=^fum required. 
 
 1^+ ^ -K 2 * 4 *-! S^ 
 "~ ' 
 
 - 
 3 "~2^ ' 3 6 a 
 
 PROBLEM VII. 
 
 Tojubtraff one /ration from another. 
 
 RULE. 
 i, PREPARE the quantities as in the.iaft problem 
 
 2. 
 

 2. SUBTRACT the numerator of the fubtrahend 
 from the numerator of the other fra&ion, and write 
 the common denominator under their difference ; 
 and you will have the fra&ion required. 
 
 1) CL 
 
 FOR put -= m and - n $ then vym and a zr 
 
 y y 
 
 yn ; alfo, yn ym zzza v by equality ; and di 
 viding the whole by j, it will ben m *""??., but 
 
 y 
 
 the difference of m and #, is manifcftly equal to the 
 difference of 2 and -5 confequently, ! - = .lir, 
 
 jr 7 y y y 
 
 Hence, &c. 
 
 From f take. Firjt, aXal=a*l>, and c^b 
 b ab 
 
 ~cbx> alfo> bKab<=,ab\ ^Therefore, 
 
 ae* 
 
 *re the fractions reduced-, and - = difference 
 
 c*x % c*-}-^ 1 * 
 
 required. From - take '-, and it 
 
 7 " 
 
 b 
 
 ;. The f rations reduced arc 
 4 8 
 
 /on?, 4^4-6^,,^;,,, ^ r ^^ required, by the na~ 
 
 O 9 
 
 ture offubtratliw. 
 
 Nn PROB, 
 
( 282 ) V 
 
 PROBLEM VIII. 
 
 fo multiply fractional quantities together. 
 RULE. 
 
 MULTIPLY ttie numerators together for the nume 
 rator of the product, and the denominators together 
 for the denominator of the produd: 5 and you will 
 have the product required. 
 
 fTj + j* 
 
 FOR put-zr m and -~ n\ then v zi zm and a ~ 
 z b 
 
 In j alfo, bnx% m ~ a X v > that is, bznm zz ai) y and 
 
 dividing \yftz, nm ~ 5 but m X n ^: X-J 
 bz z b 
 
 fequently, x-= : Therefore, 
 z b bz 
 
 EXAMPLES. 
 
 - 
 
 3 4 3 4 3 12 4 
 
 or, ^y. 
 
 PROBLEM IX. 
 
 To divide one fraflion by another, 
 RULE. 
 
 MULTIPLY the denominator of the divifor,. with 
 the numerator of the dividend, for the numerator of 
 
 the 
 
, w - ( 383 ) 
 
 the required quotient, and the numerator of the di- 
 vifor, with the denominator of the dividend, for the 
 denominator of the quotient. Or, 
 
 INVERT the terms of the divifor, and proceed as 
 in multiplication. 
 
 For put ~m and - # > then x~ym and z 
 y d 
 
 "zzdn. Multiply z=dn by y> and it will 
 
 in like manner, dx^dym-, therefore, L^-L.j 
 
 aym dx 
 
 but 3 ? z: fby divifion) ~, and therefore by refti- 
 ydm m 
 
 tution ?-4-l=^ : Cpnfequently, &c. 
 d y dx 
 
 EXAMPLES, 
 
 a c a^d ad ~ a c d^a ad , c 
 T ^----9- > ~~ z . Or, 7 ~ :} z:-X I ^: as before j 
 b d cXd cb b d c b cb 
 
 au . a-^-u __a u X v^vauv __ a u . rp, 
 
 fore, in divifion of fractions that have the fame de 
 nominator, caft off the denominators, and divide the 
 numerator of the dividend, by the numerator of the 
 divifor, for the quotient. 
 
 Thus, l ^Slxfe= 
 
 3 
 
 ay ^6 a 
 
 PROBLEM X, 
 
 fo find the fowtrs of fractional quantities. 
 
 RULE, 
 
RULE. 
 
 i. PREPARE the given fradHon, if need be, by the 
 rules of reduction. 
 
 a. INVOLVE the numerator to the height of the 
 power propofed, as in involution of whole quantities, 
 for the numerator of the power required. 
 
 3. INVOLVE the denominator in like manner, for 
 the denominator of the aforefaid power. 
 
 EXAMPLES. 
 
 Find 
 
 X 
 
 z=z power required. 
 
 a*, f^ 
 
 uhe 4tb power of =. 
 zy 
 
 PROBLEM XL 
 
 To find the roots of fractional quantities, 
 RULE. 
 
 1. EXTRACT th root of the numerator, by the 
 rules for extradting the roots of whole quantities, 
 for the numerator of the root required. 
 
 2. EATRACT the root of the denominator in like 
 manner, for the denominator of the required root. 
 
 ' EXAMPLES. 
 
a 6 
 Find the fquare root of . 
 
 * 
 
 0% /or /* numerator of the root, 
 find x ' '' 1 ~x* for the denominator of the root-, there 
 fore, is the root require^. The cube root of .. a .. 
 
 X* Z 3 y* 
 
 -JL 4/flfl-f*! 
 
 ""zp" 2V 6 ~~~z<; 3 ' 
 
 The fquare root of **~ 4 * + 4 =flZ*. 
 J*+6j + 9 jK + 3 
 
 But if th? propofed quantity hath not a true root 
 of the kind required, it muft be diflinguifhed by the 
 
 fign of the root: Thus, the fquare root of? ~~ X 
 
 * 
 
 or 
 
 a* 
 
 CHAP. X, 
 
 CONCERNING SURDS or IRRA- 
 TIO NAL 
 
 IF the whole doftrine of furds, with every thing 
 therein, which might be of ufe, were to be ex 
 plained according to the methods ufed by fome 
 writers on the fubje6b,it would become very complex, 
 and by far the moil intricate and dfficult part of all 
 Algebra 3 and neceflarily fwell this volume beyond 
 
 its 
 

 its defigned limit : And befides, there are many 
 things in the explanation and management of furd 
 quantities, as was taught by many writers on Alge 
 bra, which were then thought neceffary, are now at 
 moft, confidered as ufeful. We fhall therefore, 
 endevour on the one hand-, to avoid all fuch tedious 
 redu&ions, and complicated explanations, as would 
 ferve rather to puzzle, than inftruct the learner : 
 And on the other hand, not to omit any thing which 
 is neceffary, either in the explanation or management 
 of fuch furds as generally arife in algebraic operations. 
 A SURD quantity is that which has no exact root : 
 Thus, the fquare root of 5 cannot exactly be found 
 
 in fiinite terms, but is exprefied by 5*, or \/5 $ the 
 
 JL .-. 
 
 cube root of a by a 3 , or ay ~ 3 \/a : The recipro 
 cal of the fquare root of a-+y, or i divided by the 
 
 fquare root of a +y, is exprefied by 
 
 THEREFORE, the roots or irrational or furd quan 
 tities, may be confidered as powers having fractional 
 exponents; that is, the index fhewing the height of 
 the power, is here placed as the numerator of a frac 
 tion, whofe denominator is the radical fign. 
 
 SECT. I. 
 
 Of REDUCTION of SURD QfTAN- 
 <T IVIES, 
 
 REDUCTION of furds has the following problems. 
 
 P R O B s 
 

 PROBLEM I. 
 
 fo reduce a rational quantity to the form of an irra 
 tional, or fur d quantity* 
 
 RULE, 
 
 INVOLVE the rational quantity to the height of the 
 propofed radical fign, or index fhewing the root to 
 be extradled ; the power diftinguifhed by the radical 
 fign, will be the form required. 
 
 EXAMPLES. 
 
 0, reduced to the form of the 3 i/x, is 3 \/a l *3 rz 
 3 y'tf 3 ; 6 reduced to tie form of the fquare root of 2, 
 
 lX2 
 
 Reduce 2. to the form of a cube root. ^ 
 4 4) 
 
 zr~. 3 = 3 y^^=fsrm required ; u+y reduced to the 
 
 form of a fourth root, is * \/ f u +y 1 ^ ' = 4 \/ 'u+y 
 
 . Alfo y u~\/ii* 
 
 5 
 
 PROBLEM II. 
 
 fo reduce Jurds of different radical ftgns to the fame* 
 RULE. 
 
 REDUCE the indices of the furds to a common de 
 nominator, and the Curds will have the fame radical 
 fign as required, EXAMPLES, 
 

 ( 288 ) 
 
 EXAMPLES. 
 The 3 \/ of a, and the \/ ofb, reduced to the fame 
 
 1X2 1X3 2- 3 
 
 I t 
 
 and-*\/b*. 2 T and y* reduced to the fame 
 1X3 2X4 s 
 
 and iz 
 
 3X2 1X2 
 
 3 x 
 
 reduced to a common radical 
 
 Jign> are ~ 
 
 i/^l 1 : Alfo, y~ ' and y~^~ 
 PROBLEM III. 
 
 To reduce fur ds to their moftjimpk terms. 
 R ULE. 
 
 x. DIVIDE the quantity under the radical fign, by 
 fuch a rational divifor, as will quote the greateft ra 
 tional power contained in the propofed furd without 
 a remainder. 
 
 2. EXTRACT the root of the rational power, and 
 place it before the furd, with the fign of multiplica 
 tion, and the propofed furd will be in its mod fimplc $ 
 terms. 
 
 EXAMPLES. 
 
 Reduce -/J? to its mod fmnple terms. 
 
 tiers 
 
( 289 ) 
 
 ~ 16 the greateft rational power contained in 
 " 
 ; therefore>the /32 = 
 
 4f 
 3 
 
 SECT. II. . 
 
 O/ ADDITION of SURD $JJAN- 
 ffTIES. 
 
 ADDITION of furd or irrational quantities^ confifts 
 of the following cafes. 
 
 CASE L 
 
 When the propofedjurds are of the fame irrational 
 quantity (or can be made Jo by reduftitn) and the ra~ 
 dlcal fign thejame in all. 
 
 RULE. 
 
 ADD the rational to the rational, and to their funa 
 annex the irrational part with its radical fign. 
 
 EXAMPLES. 
 3/20+6/20 z: 3+6X ^209^20 ; \/ Z^K-SC, 
 
 '1 
 
 O o V"*- 
 
CASE II. 
 
 the irrational orjurd quantity, and the radi~ 
 calftgn are not tbefame in all. 
 
 RULE. 
 
 CONNECT the furds with their proper figns + or 
 * -, and you will have the furn required. 
 
 Note. Iftbejum confifls of two terms, it is called 
 a binomial^ or refidualjurd> as tbejign is -f- or , 
 
 EXAMPLES. 
 
 +jv/9X3 = a 3 v/2 + 3 v/j: 
 j^-jv^Xjlr , 2 y 9X3\ 
 "^ 3 36X1! 
 
 added to \Sxyy*i/ax \/xyy*. 
 SECT. III. 
 
 Of SUBTRACTION of SURD QUAN 
 TITIES. 
 
 CASE 
 
CASE L 
 
 all. 
 
 Wloen the radical fign and, quantity are the fame in 
 
 RULE. 
 
 FIND the difference of the rational parts, to which 
 annex the common irrational or furd quantity,, with 
 the fign of multiplication. 
 
 EXAMPLES. 
 
 80-45=: y/ 16 X 5 
 = 4 
 
 - 4X5 -~3 X 5 
 
 3 X 5 ft 
 
 4x5 
 
 _i2^ IOJK 
 
 16x5 
 
 C A S E IL 
 
 When the irrational farts are not the fame in all. 
 RULE. 
 
 CHANGE the fign of the quantity to be fubtraftcd, 
 the%cpreflioa connected, is the difference required. 
 
 EXAMPLES, 
 
EXAMPLES. 
 
 Vf* Jubtratied from 80% =r %/ 1 6 X 5 
 
 4V/5 
 
 6 ^ 16* ; 3 4 v/g 2 fubtrattedfrom + 
 
 SECT. IV. 
 
 Of MULTIPLICATION of SURD 
 . QUANTITIES. 
 
 SURDS being confidercd as powers having fra&ion- 
 al exponents , it therefore follows, that to multiply 
 one furd with another, is to add their fractional expo 
 nents together, making the denominator of their 
 fum the radical fign, and the numerator the index of 
 the root. 
 
 HENCZ is deduced the following rule for multi 
 plication of furds. 
 
 RULE. 
 
 1. REDUCE the indices of the furds to a common 
 denominator. 
 
 2. ANNEX the product of the furds, to the prod- 
 udfc of the rational parts with the fign of multiplica 
 tion i and it will give the product required. 
 
 EXAMPLES. 
 
( 293 ) 
 
 EXAMPLES. 
 
 =4 6 N/32i z* +.XT Xz' +.T l f = 
 
 SECT. V. 
 
 O/ DIVISION of SURD 
 
 RULE. 
 
 1. REDUCE the furds to the fame in^ex. 
 
 2. DIVIDE the rational by the rational, and to the 
 quotient annex the quotient of the furd quantities ^ 
 and it will be the quotient required. 
 
 Note. If the quantity is thejame in both faftoKS, 
 they are divided byJubtraEling their exponents. 
 
 EXAMPLES, 
 
 f* .. - - 2, 
 
 Pp 
 
294 
 
 * 
 
 i _i_ m n 
 
 fin ~r d 1ZL ^ mit 
 
 SECT. VI. 
 
 Of INyOLUflON- of SURD .QUAN 
 TITIES. 
 
 THE powers of furds are found by the following 
 RULE. 
 
 INVOLVE the rational part, as in involution of num 
 bers ; and to the refult annex the power of the furd, 
 found by multiplying its exponent with the expo 
 nent of the power required. 
 
 EXAMPLES. 
 
 of 
 'v/36. The cube of v/3=3 7 ^^if 3^ n \/3 
 
 The fquare o 
 
 ir4 X A; T zz 4 3 \/ ^ + . -7*^ cube cf 3 >/^ 'bx 
 
 ^~bx\ 3 3~ax bx \ Therefore, when the in 
 dex of tha.power required, is equal to, or a multiple 
 of the exponent of the root -, the power of the furd be 
 comes 
 
( 295 ) 
 
 comes rational. The cube of a x\ T ^~^]"^ 
 
 = a x\ T a x] 2 . The n power of fZ = 
 
 IF the propofed furd is a binomial or refidual one, 
 involve it as in chapter vn. 
 
 Thus, thejquare of\/6 -f 2v/*'~ 
 
 * &j //??<? operation annexed.) 
 
 OPERATION. 
 
 \S6+ 
 \/6+ 
 
 4^ 
 
 S "E I 11 "T. 1 VII. 
 Of EVOLUTION of SURDS. 
 
 THE powers of furds are found by multiplying 
 their exponents with the index or exponent of the 
 power to which they are to be involved ; as we have 
 Ihewn ; confequently, ifthofe exponents be divided 
 by the index of the root to be extracted, the quotient 
 will be the exponent of the root 3 which gives the 
 following 
 
 RULE. 
 
 EXTRACT the root of the rational part, as in com 
 mon extraction of roots ; and annex the root of the 
 furd, found by dividing the index of the furd, by 
 the index of the root required, 
 
 EXAMPLES, 
 
EXAMPLES. 
 
 - 
 
 root oj \/a~ a* ' J -s= a 6 
 
 cule root of ^3 = V 
 
 T 6 v/3- 
 
 Jquare root of 3 ^a 3 
 
 foot of x r ~.^ *iz7 x a . If the pro- 
 
 V/ ^v 
 
 pofed furds are binomial, refidua! 5 or trinomial, &:c. 
 find their roots as in Chap. vnr. 
 
 'The Jquare root ofx*-\~6x* \/y + $y = ^ 
 
 20 + i\ x -~ x 20 -f 2 vtf 4- 
 
 CHAP. XL 
 
 Of INFINITE SERIES. 
 
 AN infinite feries, is formed from a fraction 
 whofe denominator is a compound quantity, 
 by dividing the numerator by the denominator ; or 
 the extra&ing the root of a furd quantity, which if 
 continued in either cafe, would run on fempiternal- 
 ly ; that is, the number of terms in the feries would 
 be infinite ; but by obtaining a few of the firft terms 
 of the feries, you will eafily perceive, what law the 
 feries obferve in their progreflion ; by which means 
 you may continue the feries by notation as far as you 
 pleafe, without an aftual performance of the whole 
 operation at large. P R O B, 
 
PROBLEM I. 
 
 fo find an infinite furies by dhifion ; that is y to 
 Tfrow a compound fractional exprejfion into fuch afe- 
 ries, whofe fum, if the number of terms were continued 
 adinfinilum, would be equal to the given frattional ex* 
 preffion. 
 
 RULE. 
 
 DIVIDE the numerator by tkc denominator until 
 you have 3, 4, 5, or more terms in the quotient. 
 
 EXAMPLES. 
 Throw i into an infinite feries. 
 
 OPERATION. 
 
OPERATION. 
 
 I V e U' i V 
 
 V 
 
 o - 
 
 y 
 
 V *V" 
 
 z V 
 
 v 
 
 * _ _ 
 
 HERE the law of the progre/lian which the feries 
 obferve, is plain ; for each fucceeding term is pro 
 duced, by multiplying the preceding one with 
 
 ~ : Thus, the firft term of the feries is 1, which be- 
 y y 
 
 ing multiplied with -, gives for the fe- 
 
 cond 
 
*99 
 
 cond term, and X 2 = .!L.= third term i 
 
 y* y y\' 
 
 IT ^* v *Z> 1> 3 7 , - T , . ,. 
 
 alio, X ~ the 4th term, which mulnpli- 
 
 -V 3 y y 4- 
 
 ed with ~~ will give the 5th term; and fo on, 
 multiplying the preceding term by the common ratio 
 ~, you may find any number of terms at pleafure. 
 
 y ' 
 
 BUT in oriler to have a converging feries, or a 
 feries wherein the terms continually decreafe, the 
 greateft term of the divifor muft fland firfl in the or 
 der of arrangement; for fuppofe in the above exam 
 ple, that y is very great in refpecl: of v ; hen will 
 
 -. be very great in refpecl: of ; fo that in this fup- 
 
 pofition, the terms being multiplied with the powers 
 of'y, and divided by thofe of y\ it follows, that 
 each fucceeding term is very little in refpecl: of the 
 preceding one, and confequcntly the feries, a con 
 verging feries. Again, puti; for the firft term of the 
 divifor (the fuppofition the fame as before ) and the 
 
 i v y 3 " 
 feries will be -- Z,-f^_ &c. and fince y is very 
 
 - v v z v 3 
 
 great in refpecl: of v -, it follows, that ~ is very lit- 
 
 i) 
 
 tie in refpecl of , and very little in refpeft of 
 
 v* v 2 - 
 
 y* 
 
 ^. ; confequently, the feries is a diverging one ; 
 
 that is, a feries whofe terms continually increafe, 
 and therefore, the farther you proceed in them, the 
 farther you will be from the truth. Hence, &c. 
 
AND fince it is impoflible to afTign an infinite 
 number ; it follows, that the number of terms ex- 
 prefling the true value of fuch a feries, is not aflign- 
 able; yet the taking of a few of the firft terms will 
 be fufficient for any practical purpofe. 
 
 Throw > a into an infinite feries. 
 v d 
 
 OPERATION. 
 
 
 v 
 
 , W_^v 
 
 V V* 
 
 
 V 3 
 
 HERE, each preceding term, after the firft, is 
 
 multiplied with -, and theprodudb is the next term 
 
 v 
 
 following; therefore, the law of the progreltion is 
 manifeft. 
 
 Throw - into an infinite feries. 
 
 OPERATION. 
 
OPERATION. 
 
 b* 
 
 HERE thelaw of the, continuation is the preceding 
 terms multiplied with" fr*. 
 
 PROBLEM II. 
 
 fo cxtraft the root of a compound Jurd in an infinite 
 Jeri&s ; that /V , to throw a compound JUT d quantity into 
 * converging feries, whofefum, if the terms were infi~ 
 nitely continued would be equal to the root required. 
 
 RULE. 
 
 EXTRACT the root of the quantity, as in common 
 algebraic extraction ; the operation continued as far 
 as is thought neceflary, will give the feries required, 
 
 V 
 
 EXAMPLES. 
 
 Throw \/a* +y i into an infinite feries. 
 
 OPERATION. 
 
OPERATION. 
 +,. (,+_ 
 
 4* 1 
 
 1 21.4- 
 
 * r 
 
 Tbat is, T^+lc^a + -- -f +&e, 
 
 Find 
 
Find the value of i # 2 ] r in an infinite ferics, 
 OPERATION. 
 
 -- 
 
 2 8 16 
 
 X 
 
 4 
 
 4 
 
 A* V 6 V * 
 
 :L 4- IL. 4*2- 
 4 8 r 6 4 
 
 16 8 64 
 
 il-4- ^* 4- * ' 4- >v ' * 
 " 8"" I6 1 " 64 256 
 
 PROBLEM III. 
 
 To reduce any fur d or fractional quantity into an in 
 finite feries, by the celebrated Binomial Theorem, invent 
 ed by that Prince of Mathematicians, the illuftrious 
 Sir ISAAC NEWTON, which is as follows. 
 
 Binomial 
 
( 3Q4 ) 
 
 Binomial theorem* 
 
 Wherein it is to be obferved, that P + PQJs the 
 quantity whofe power is to be thrown into an infi 
 nite feries ; P reprefents the firft term of the propof- 
 ed quantity 5 Qjthe other terms divided by the firft -, 
 
 the index of the power, whether it be affimative or 
 n 
 
 negative : And A firfl term of the feries *, Bthc 
 iecond j C the third ; D the fourth ; E the fifth ; 
 F the fixth, &c. that is, the feyeral terms of the fe 
 
 ries, are A=P7> B =2 AQ, C =^Z^ BO . D 
 
 n 2ft 
 
 EXAMPLES. 
 
 ir 
 
 Reduce a*+x*\* into an infinite feries. 
 
 Here <?*=:?, ~ ~ Cj, m~\>andn~i : 
 
 Mere/ore, A = Pr = a> B n-AQ=: , C 
 
 ~ r + ^~, fSe. isthejeritsrc 
 
 quired* 
 
 Expand 
 
( 3Q5 ) 
 
 T I-'" I 
 
 Expand -77- = i+a| into an infinite fcries. 
 
 T ; therefore, m~~-i, 
 
 f m \ 
 i A~^pT J 
 
 Find the value of ; in an infinite feries. 
 a+y 
 
 Here ~- -^X^+jF * '> IWerefore,-'? =.a, 
 
 y T * 
 
 -, i>anJn=:i : Wen, A a *, or j 
 
 _ ! i y y* y* 
 That is, vX^+j] = ^ X - ^ + JT 2? 
 
 w v vy vy* vy* 
 
 PROBLEM IV. 
 
 Tofind tbefum of an infinite feries, geometrically 
 decrcafin?. 
 
 R U L E. 
 
 DIVIDE the fquare of the firft term by the differ 
 ence between the firft and fecond, and the quotient 
 will be the fum required, THUS, 
 
( 306 ) 
 THUS, the fum of the infinite ierics ^ *a.+ ^ 
 
 V 
 
 fL &?. zr "y 1 -r / y-~^ and the fum of v 
 
 / - r - 
 
 v $ for u v 
 
 be divided by v -f 0, and *v by ^ T;, the quotients 
 will be the feries propofed. Therefore, the rule is 
 manifcil. 
 
 EXAMPLES. , 
 
 Given i + T + i +T> &c. ^ infinitum> to find 
 their fum. 
 
 , i a -r- i 4- = 2 the fum required. 
 Given T 6 ^- 4- T!^ 4~ WW> ^c. adinfinitum^ to find 
 their fum. 
 
 , J: 4- S T-k- = 1 tbejum required. 
 
 i ol 
 
 Given 2 J + *. T * T , &c. adinfinitum^ to find 
 their fum, 
 
 , 4 ^- 2 + -J = 1 1 "=.Jum required* 
 
 CHAR XII. 
 
 Of PROPORTION or ANALOGT 
 ALGEBRAICALLY CONSIDERED. 
 
 WHEN quantities are compared together 
 with regard to their differences, or quotients, 
 their relations are exprefled by their ratios. The 
 relation of quantities, arifing from the firft compa- 
 
 rifon 
 
( 37 ) 
 
 nfon, is exprefTed by an arithmetical ratio, that of 
 the fecond, by a georrietri.caFratio ; and the quanti 
 ties themfclves are faid to be in arithmetical, or ge 
 ometrical proportion, as the ratios of their compa 
 nion are arithmetical, or geometrical : Which pro 
 portions, together with fuch others as arife from the 
 alternation, converfion, &e. of thofe- proportions 
 that are of any conQderableufe in Mathematics, will 
 be noticed in the following order. 
 
 SEC T. I. 
 
 Of ARITHMETICAL PROPORTION. 
 
 WHEN* quantities increafc by addition or fubtrac- 
 tion of the fame quantity, thofe quantities are in 
 arithmetical proportion : Thus, a, a -f- d > a 4- id, 
 a + 3<t> &c, or A-, x d, x 2^, x 3 </, &c. 
 are quantities in arithmetical proportion ; wherein 
 the quantity df, which is continually added or fub- 
 tra&ed, is the common difference of the feries ; there 
 fore, when in any four quantities, the difference be 
 tween the firft and fecond, is equal to the difference 
 between the third and fourth, thofe quantities are in 
 arithmetical proportion ; as in thefe, jy, y #, y 
 2 , y 3 n ; where y y n ~ n ; and y -- 2 n 
 y 3 := n. Therefore, &c. 
 
 THEOREM L 
 
 If three quantities be in arithmetical proportion, tie 
 Jum of the two extremes will be double the mean. 
 
 Thus if a> a 4- d, a-\- 2^ are in arithmetical pro- 
 portion, then will a 4- a -i - 2 d n a -\- d -\^a 4- d^ 
 
 T H E O. 
 
( 3Q* ) 
 
 THEOREM II. 
 
 If four quantities be in arithmetical proportion, the 
 film of the two extremes will be equal to thejum of the 
 two means. 
 
 Thus, if /*, a + ^> *~4~ 2 ^ a + 3 ^ are quanti 
 ties ia arithmetical proportion, then will a +0 -f~3*/;z 
 
 THEOREM III. 
 
 / ajeries of arithmetical proportionals, thejum of 
 the two extreme terms, is equal to thejum of any two 
 tsrms equally dift ant from the extremes. 
 
 Let the feries be a, a +d, a+zd, a+$d, a+$d 9 
 &c. to z : Under which write the fame feries with 
 their order inverted; then adding thofe terms toge 
 ther which {land directly oppofite each other, and 
 the fum of any two fuch terms, will be equal to the 
 fum of the firft and laft terms, as plainly appears by 
 the following 
 
 EXAMPLE. 
 
 toz 
 Series inverted, z,z /,a 2 d,z 3 d 3 z ^&$. to a 
 
 thejum of every two terms. 
 
 Now from this example, a rule for finding the 
 fum of all the terms of any arithmetical feries, may 
 be eafily deduced ; for it is plain, that the fum a-j- z 
 ~}-a-}-z~^a-Z) &c.-or *2-fz, taken as many times as 
 there are number of terms, is double the fum of the 
 feries a, a >\-d % a-j-id, &c. Ccnfequently, that fum 
 
 divided 
 
( 309 ) 
 
 divided by a, will be equal to the fum of the fcries ; 
 that is, (puting = number of terms, and j = 
 
 r . r , \a -f z X n na + nz ^ . 
 
 of the lenes) > , - ~ - 1 = s : Or in words, 
 
 2 2 
 
 the fum of the firft and laft terms multiplied with 
 half the number of terms, will give the fum of the 
 feries. 
 
 BUT in any arithmetical feries, the co-efficient of 
 the common difference (^) in any term, is i lefs 
 than the number of terms to that place ; confequent- 
 Jy, its co-efficient in the laft term, is equal to the 
 number of terms lefs i ; and therefore, the laft term 
 z~a-{-n i X ^ # -i- dn d. Confequently, 
 
 a theorem for finding . 
 
 2 
 
 the fum of any arithmetical feries, when the firft term, 
 common difference, and number of terms are given, 
 And univerfally, puting 
 
 a :n firft term of an arithmetical feries, 
 
 d zz common difference) 
 
 I n: laft term, 
 
 n zi number of terms, 
 
 s ~Jum of all tbejeries. 
 
 THEN having given any three of thofe five quan 
 tities, the reft may be found by the following theo 
 rems. 
 
 Theorem i . '4-? = s. Theorem 2. ; _ = 
 2 /-M 
 
 Theorem 7. -Hlf = </. Theorem 4. ? s na = 
 w i 
 
 R r Theorem 
 
( 3*0 ) 
 
 ^ r - j 
 
 Theorems, * s ~~ nl = a . Or, -^Zf + I . /- 
 7* ^ 
 
 ndd+a. a~l-\-d nd. 
 
 SECT. II. 
 Of GEOMETRICAL PROPORTION. 
 
 WHEN of four quantities, the product of the two 
 extremes is equal to the product of the two means ; 
 thofe quantities are in geometrical proportion : As, 
 a, ar y b, br - t where ay,br~ary>b : Alfo, when 
 quantities increafe with a common multiplier, orde- 
 creafe by a common divifor, as, #, ar y ar z > ar 3 , ar 4 , 
 
 &c. and a> ~, , , , &c. thofe quantities 
 r r* r 3 r 4 
 
 arc faid to be in gepmetrical proportion continued, 
 where the common multiplier or divifor r is the com 
 mon ratio. 
 
 T H E O R E M I. 
 
 In any Jenes of quin titles in geometrical proportion 
 continued^ the firft term hath the fame ratio to the Je- 
 cond) as the fecond hath to the third, and as the third 
 to the fourth, &c. 
 
 TiiusJn^r^rV^V^&c. and #, ~, ~, -^. 
 
 r r* r*' 
 
 -a 
 
 ~> &c. a : ar\\ ar \ ar* :: ar % \ ar* :; ar l \ ar* :: 
 
 a a a a a a a 
 
 cue. and a :-:: - : -r ::~T:~T:: : :: 
 r r r % r* r 3 r 3 r+ 
 
 &c. For, ax*r* rr arXar, and aXar 4 arXar 5 5 
 allb, aX~ ~ f x f ? and fo Qn for the reft> 
 
 T H E O. 
 
C 3" ) 
 
 THEOREM II. 
 
 In a feries of geometrical proportionals continued, the 
 produft of the two extremes, is equal to the product cf 
 any two terms equally dift ant from the extremes. 
 
 THUS, in the feries a, ar, ar*, tir 3 , ar*, &c. If x 
 be the laft term, then will 7 be the laft term but 
 
 one, and the laft term but two 5 wherefore, 
 
 ~ ax, the product of the two extremes, and 
 
 sc o.rx 
 
 - = -^- =: ax the product of the fecond and laft term 
 
 M 
 
 but one : That is, ^v^rr ary^~> in like manner, 
 
 r 
 
 X 
 
 tfX# = fff"*X"T> and fo on for the reft. 
 
 THEOREM III. 
 
 tfhefum of any feries of quantities in geometrical pro 
 portion continued, is obtained by multiplying the laft 
 term by the ratio, and dividing the difference between that 
 pro duff and thefirft term, by the ratio lefs i. 
 
 THUS, let the feries whofe fum is required, be 
 a + ar 4- ar* + ar 3 4- ar*, which multiplied with r 3 
 gives rfr4*#r 1 4-flr 3 4-^ 4 +^ r5 > from which fub~ 
 tract the former. 
 
 Thus, I 
 
 Now 
 
Now it is plain, that the difference ar 3 a is equal 
 to thefum of the propofed ferieslriultiplied by r i ; 
 confequently, the fame divided by r 1 3 will give the 
 fum of the feries required: That is, (puting Jzrz 
 ar 5 a 
 
 Or, generally ar-{-ar*+ar' i +ar*,&c. +^+77 
 
 - ar 1 , &c. + +~ 
 
 3G X V 
 
 '+ "7+ T + ~ That is, the fum of any geometrical 
 
 feries wanting the firft term, is equal to the fum of 
 the -fame feries wanting the laft term, multiplied with 
 the ratio. Wherefore, s a~s .vX*" 3 that is, 
 s a--sr rx, and sr s'~:rx a : Hence, s^=z 
 
 r .. x a >. And fmce r is not in the firft term of the 
 r i 
 
 feries, it follows, that in the laft term, its exponent 
 will be i lefs than the number of terms 3 and there 
 
 fore, (puting n ~ number of terms) x ~ ar 
 Confequently,J~{by writing for # its equal ar " " ) 
 
 a . And univerfallv, puting 
 
 r i r 
 
 a-=:frft term of a geometricalferies, 
 
 term, 
 
 s-=Jum of the feries. 
 
 THEN having given any three of the aforefaid 
 quantities, the reft may be readily found by the fol 
 lowing theorems, which are deduced from the above 
 equation, 
 
 Theorem 
 
( 3*3 ) 
 
 rl-a 
 Theorem i. y_ ~ s. 
 
 2. rl+s sr~a. 
 s a 
 
 srs+a 
 4- ' - = / 
 
 THEOREM IV. 
 
 j/7* /b#r quantities are proportional, as a \b \\c\d\ 
 then will any cf the following forms, alfo be proportion^ 
 
 Direftly, a : b:\c\d. 
 
 Alternately, a : c : : b \ d. 
 Inverfely, l\a\\d\ c. _ 
 
 Compoundedly, a-\-b\ b\\ c+d \ d. 
 Dividedly, a \ b a\\c\ d c. 
 
 Mixtly, b+a \ b a : : d+c \ dc. 
 
 SECT. III. 
 
 Of HARMONICAL PROPORTION. 
 
 HARMONICAL proportion arifes from the compari- 
 fon of mufical intervals, or the relation of thofe num 
 bers which affign the lengths of firings founding 
 mufical notes. 
 
 THE moil ufeful part of this proportion in practi 
 cal Mathematics, is contained in the following theo 
 rems. 
 
 T H E O. 
 
THEOREM L 
 
 If three quantities be in harmonica] proportion, the 
 firft will be to the third y as the difference between the 
 firjl andfecondy to the difference between thejeccnd and 
 third. 
 
 THUS, if a y b and r, be in harmonica! proportion, 
 then, as a ' c : : b a : c b : Confequently, ac ab 
 ~cb ca, by multiplying means and extremes: 
 From which equation is deduced the following 
 theorems. 
 
 cb <zac 
 
 Theorem i. 70. Theorem 2. ; ~, 
 
 ic b a-\-c 
 
 Theorem 3. ~ jfe^ 
 
 THEOREM IL 
 
 If four quantities be in harmonica! proportion, the 
 fir ft will be to the fourth , as the difference between the 
 frft and JecQnd> is to the difference, between the third 
 end fourth. 
 
 THUS, if the quantities a> b, c> a 7 , are harmonical 
 proportionals, it will be, a : d : : b a : d - c : 
 Wherefore, ad ac zr db da. From which equa 
 tion, we get the following theorems. 
 db 
 
( -3*5 ) 
 
 CHAP. XIII. 
 
 Of SIMPLE EQUATIONS. 
 
 AN equation is an expreffion, averting the equali-" 
 ty of two quantities, which are compared to 
 gether by writing the quantities with the fign of 
 equality between them. Thus, if #-4-3 is <equal to 
 2# i, the equation is formed thus, x +3 = 2ff n 
 Alfo, 8 3=15 10, 
 
 A SIMPLE equation, is an equation which in 
 volves one unknown quantity, without including 
 its powers. Thus, 3* 1 = 2x^2 is afimpleequa- 
 tion which exprefles the value of the unknown quan 
 tity; when that quantity Hands alone on one fide of 
 the equation, the reft being on the other fide, which 
 if known, we then have a determined value of the 
 unknown quantity in known terms. And the bull- 
 nefs of bringing the unknown quantity to (land 
 alone on one fide of a fimple equation, is called re- 
 duclion of fimple equations : To effect which pur- 
 pofe, we have the following rules. 
 
 RULE I. 
 
 ANY quantity may be taken from one fide of an 
 equation and placed on the other, if you change its 
 fign. Or which is the fame thing, iubtraft the quan 
 tity from both fides. 
 
 FOR, 
 
( 3*6 ) 
 
 FOR, if from thofe quantities which are equal, 
 there be taken the fame quantity, what remains will 
 be equal. 
 
 EXAMPLES. 
 
 Given x 6 20, to find the value of AT. 
 Thus, x~ 20-}- 6, per rule, and x 26 by addition. 
 
 For, 6 taken from x 6, leaves x y and 6 taken 
 from 20, leaves 20 -j- 6, or 26, by the nature offub- 
 traffion. Therefore, &c. 
 
 Given #-1-430 5, to find the value of A?. 
 
 fbuSy x zz 30 5 4 by tranfpofition : 
 
 Or, #*~ 30 9 IT 21 by addition andjultraftion. 
 
 If x 3.4-1 21 : 
 
 fben will x 2 1 -f 3 \ by tranfpofttion : 
 Or, ^"d 23 ^jy addition andjubtraftion. 
 
 RULE IL 
 
 WHEN the unknown quantity is multiplied with 
 any number, it may be taken away by dividing all the 
 reft of the terms in the, equation by it. 
 
 FOR if thofe quantities which are equal, be divid 
 ed by the fame quantity, their quotients will be e- 
 qual. 
 
 EXAMPLES. 
 
 Given 47 1 2 z= 2 y -j- 4 5 to find the value of y. 
 
 Firfty ^y 27= 12 + 4 by tranfpofition : 
 Then, 2y.~ 16 by addition andjubtraffiion : 
 
 *6 
 
 Or, y rrz: 8, per rule. 
 
 If 
 
( 3*7 ) 
 
 If 6y + 3=^4. 18, then will 6y y z: 18 
 tranjpqfition ; <ta*/ 5JXZ= 15 byfubtraftibn. 
 
 Whence, y = =: 5 y dwifion. 
 
 >j 
 
 Let 3* 10 zr 20 #4-6, be given to find #, 
 ^Vr/?, 3* -|- ^ z: 20+6 + 10 by tranffqfition : 
 
 36 
 Or, 4,r =36, ^7rf/ therefore, x = 1 n 9. 
 
 RULE III. 
 
 WHEN any part of the equation is divided by any 
 quantity, that quantity may be taken away by mul 
 tiplying all the reft of the terms by it ; which is the 
 fame as to multiply all the terms in the equation 
 by that quantity. And if thofe quantities which are 
 equal, be multiplied with the fame quantity, their 
 produces will be equal. 
 
 EXAMPLES. 
 
 1} 
 
 Given, r +2 10, to find the value of v. 
 
 ttus, v+ii Soyper rule : 
 And v n 60 12 1148 by tranjfofitionandfultrac^ 
 tion. 
 
 y iy ? 
 Let ^ + +~ ~ 16, be given to 
 
 Men, 12L-2 -16 by addition: 
 
 And $iy + 2411512 by multiffyation , 
 S a 
 
Whence, y^- 15 A 
 
 Alfo, if + 6= 
 
 4jy 3.? = 12 8 by tranffofition. 
 W 'hence, 7 = 4. 
 
 RULE IV. 
 
 IF any quantity be found on both fides of the equa 
 tion, having the fame fign, it may be expunged from 
 both. Alfo, if all the terms of an equation be mul 
 tiplied with the fame quantity, it may be ftruck out 
 of them all. 
 
 EXAMPLES. 
 
 5 then will 2tf= 
 
 per rule : 
 
 And ix A?~2 j or, #=2: 
 
 
 Alfo, if 6x+c ~l+c> then will 6x~b, and -v~* 
 
 ^xa ixa xa da 
 Moreover, if - + - > then will 
 
 C C 
 
 And 4^ ~d by addition andjubtraftion : 
 
 d 
 
 Whence, # = 
 4 
 
 RULE V. 
 
 IF that part of the equation which involves the 
 unknown quantity be a radical expreflion,.it may be 
 
 made 
 
made free from furds by tranfpofing the reft of the 
 terms by the preceding rules, fo that the furd may 
 fiand alone on one fide of the equation : Then take 
 away the radical fign, and involve the other fide of 
 the equation to the pow0rwhofe index is equal to 
 the denominator of the radical fign. 
 
 EXAMPLES. 
 
 20 : 
 
 Then will \/x 4- 3 = 20 4=16 by tranfpofi* 
 tlon : 
 
 Andx+s z: 16 x 16 ~ 256 by involution : 
 
 Or, *= 256-- 3 =253, 
 
 And, if 4+ \/Q.X + 6 9 5 then will \/ix + 6 
 = 94 = 5 by tranfpofition : 
 
 And 2X+6 s: 25 by involution : 
 
 Whence, x ~ z: 9^-. 
 
 In like manner, if 3 \/^+ 3 = 10; then will 
 ? \/^ io 37 s andax^z^b involution - t or, 
 
 RULE VI. 
 
 IF both fides of an equation be a complete power, 
 or can be made fo by the preceding rules, it may be 
 reduced to more fimple terms, by extracting the root 
 of both fides. 
 
 EXAMPLES. 
 
 Given,jy* + y + 9 _ 57 = 87, to find the val 
 ue of>. 
 
 Fitf, 
 
Krft> r + 6y + 9 8 7 + 57 
 
 %'hen, y -f 3 zz 1 2 y extracting the root : 
 Or, y 12, 3 :z: 9 y tranj-pofition. 
 
 Given, p^ z 4- 24^ 4- 16 4^ a +32.7+64, to 
 find the value ofj. 
 
 ^'^> 3^+ 4 iy -f* ^ ^ extraffing the root ; 
 And 3jy 27 1:8 4^x tranffofition ; 
 That is, y ~ 4. 
 
 RULE VII. 
 
 -s. 
 
 ANY analogy may be converted into an equation, 
 by aflerting the produftofthe two extremes equal to 
 the product of the two means, 
 
 EXAMPLES, 
 
 If 6 + AT : 10 :: 4! 6 5 then will 36 + 6x zz 40, 
 ^j? multiplying means and extremes, and 6^=4; 0r, 
 4 
 
 And, if : a :: 10 : 2; then will ~ioa> and 
 
 O y 
 
 30 a 
 4% zz 30 a y or Arm 
 
 4 
 
 And in like manner, if 6 : A; 21:4:5$ then will 
 30 4# 8: 
 
 And 4 A; =30 4- 8 z: 38 s or. A; n 9^. 
 
 COROLLART. / 
 
 HENCE it follows, that an equation may be turned 
 Into 3n analogy, by dividing either fide of it int 
 
 two 
 
( 3" ) 
 
 two fuch parts, which if multiplied together, would 
 produce the fame fide again ; making thofe parts, 
 cither the two means or extremes ; then dividing the 
 other lide in like manner for the other two terms. 
 
 CHAP. XIV. 
 
 CONCERNING the extermination of un 
 known quantities, and reducing thofe equations 
 Which contain them, to a lingle one. 
 
 PROBLEM I. 
 
 To exterminate two unknown quantities, or reduce 
 two equations containing them, to a Jingle one* 
 
 RULE I. 
 
 FIND the value of one of the unknown quantities 
 in each of the given equations, by the rules of the 
 preceding chapter. And puting thefe two values 
 equal to each other, you will have an equation in 
 volving only one unknown quantity > which equa 
 tion if a fimple one, is to be refolved as in the lad 
 chapter, 
 
 EXAMPLES. 
 
 , | -* } to 
 
 .y 
 
 From the firft equation, we have #- 
 
( 3** ) 
 
 And from tlejecond, ,vzz 
 
 ~-? - 3Q+3.? . 
 
 
 ^ / 
 Therefore, 
 
 And 84 6yz:6o4-6y y multiplication : 
 Whence y 84 60 1 2^ : 
 Or, i2yz:24 i 
 
 24 14 y 
 
 And there fore > jn z: 2, and x 
 
 14 2 
 writing if or y its equal)- z:6. 
 
 Given, ] 3^-rJ= ^ 2 ( to ^ nc j v a 
 
 22 J 
 
 From the firft equation > v~ - 3 andtheanalo- 
 
 \j 
 gy turned into an equation, gives $v^zy, or v = 
 
 2y 22 y iy 
 
 - , hnd therefore) - 
 j \j j 
 
 Whence we get y no 57 ~ 6 y ly multiplication : 
 
 And \\y ~ no : 
 
 no 
 Or, j= = 10: 
 
 2^ '2O 
 
 ^ zz z: (^y writing 10 /ir^ ^'/j equaT) -7- 
 
 RULE II. 
 
 FIND the value of one of the unknown quantities 
 in either of the given equations ; and inftead of the 
 unknown quantity in the other equation, fubftitute 
 its value thus found, and there will arife a new e- 
 quation having only one unknown quantity, whofe 
 value is to be found as before. 
 
 EXAMPLES. 
 
( 3*3 ) 
 
 EXAMPLES. 
 
 Give*, \ Z + y ~ T } to find z and y. 
 
 L z y = / 3 
 
 the fir ft equation, we have z~ 10 y, m 
 Juljlituted for z in thejecond equation, 
 
 Gives 10 ^ ~j n 7, 0r 10 2jx n: 7 : 
 
 ./f</ 2JX = 10 - 7=3*. 
 
 Or, 7 = 1.5: 
 
 Whence, z (by writing i.$for y its equal} 10 
 1.5 = 8.5: 
 
 JO -J- 2V 
 
 thefirft equation z z: - - - > an.d this value 
 
 10 + 2y 
 
 Jnlftituted In thefecond equation, gives %y + - ~- 
 
 = 6 5 : 
 
 Or, 6y + 16 + iy zr 130 ; whence, 8^ = 120 : 
 
 120 
 Or ^ = ~"- J 5s andz - 
 
 5 + 15=20. 
 
 RULE III. 
 
 IF the unknown quantity is of lower dimenfion in 
 one of the given equations than in the ofher ; find 
 the value of the unknown quantity in the equation 
 where it is of lead dimenfion, and raife this value to 
 the fame height as the unknown quantity in the 
 other equation ; or on the contrary, Then compare 
 this value with the value of the unknown quantity 
 
 found 
 
C 3*4 ) 
 
 found from the other equation ; and you 'tfill have 
 a new equation, with which proceed as before, 
 
 EXAMPLES. 
 G^en, { J+f,f .1 &, } to find v and y. 
 
 From the fir ft equation vn 10 y ; 
 And therefore, 4 0*z: 10 y\* - 100 
 lTto> 100 ioy+y' t ~6o+y' i ' by rule \ft. 
 Whence, y~z by reduction : 
 Or, 100 2qy-f .>* y* ~ 60 by rule id. 
 V/hence, 4On 
 
 And*u zrio-*-^t:io 
 Given, * - to 
 
 The analogy turned into an equation, gives 
 , which divided by z, gives 32 zs 4 
 
 = 
 
 " 3 ' 
 
 i6y* 
 
 Whence, x* ^r ^--* 
 9 
 
 -^ therefore, ^* +^* -. 25 : 
 Or, 
 
 P R O B. 
 
( 3*5 ) 
 
 PROBLEM II. 
 
 fo exterminate any three unknown quantities, x, y> 
 and z, or to reduce threefimple equations that involve 
 them* to a Jingle one, 
 
 RULE. 
 
 FIND the value of x in the three given equa 
 tions; then compare the firft value of x with the fe- 
 cond, and there will arife a new equation involving 
 onlyjx and z. Again compare the firft, or fecond 
 value of x with the third, and there will arife another 
 equation involving onlyjx and z ; then proceed with, 
 theie two equations as directed in the lad problem 
 
 EXAMPLE. \ 
 
 r 2 .v 4-^4-2? 1 5 1 
 Given, < x 4-67 2 =. 29 > to find x,y and z, 
 
 (. 4-V 1Z -i- 2^ I 2 J 
 
 T m,rwm-Tn- O fX MMM* *U 
 
 the firft equation, we have, x ~ - o 
 
 the fecond, ^^29 4-2 -6.7 : 
 1 2 4- 2% 2y 
 
 4 
 
 I C '"' 2^2 T -- V 
 
 Whence, - -^ - ^294-% 
 
 i -2 4- 22 ajy 
 294-2 ojyn 
 
 From the firfl Qfthefe equations, we get 15 22 - 
 584-22 ioy; ^ 117^:58 154-4.^: 
 
 T t 
 
4?+4 2 
 Whence, y~ \* : 
 
 From the fecond, we have 116+42 24jyr: 
 22 ay : 
 
 104 + 
 That is, 22j 116 12+22; or,yzz - 
 
 104+22 43-|~4 z 
 Consequently, - * 
 
 Whence, 1144+222946 + 882 : 
 And^z 222198 : 
 That is, 662198 : 
 
 ^.. _'98 
 
 , 
 Whence, y = 5, and x 
 
 AND nearly in the fame manner, may be exter 
 minated any number of unknown quantifies ; but 
 there are often much fhorter methods for their exter- 
 mination, which are beft learned by practice j yet 
 fome of them may be thus generally given. 
 
 RULE. 
 
 LET the given equations be multiplied or divid 
 ed by fuch numbers, or quantities, that by addition^ 
 fubtraction, multiplication, divifion, involution or 
 evolution of any two, or more of the equations, one 
 or more of the unknown quantities may vanifh. 
 Then taking the refult and the other equations, and 
 proceed as before, until you have an equation in 
 volving 
 
( 3*7 ) 
 
 volving only one unknown quantity, whofe value 
 may be found by the foregoing rules. 
 
 EXAMPLES. 
 
 Given, j ^gzj? } to find * and> 
 
 Multiply the firft equation with 2, and it will give 
 4,v + 6y 5 8 , andthefecond with 3, gives 9* + 6y n: 93, 
 from which Jubtr aft, 4* + 6y 58 ; and you will have 
 
 35 
 5x^:355 cr, .vzr 7, 
 
 29 2y_29 14_ 
 T" 3 ^ 5< 
 
 r 2^+4^+32=38 i 
 
 Given, < 3^+57+62 = 63 Sto find^,jK, ancl%. 
 
 double thefirft equation Jultraft the Jecond, 
 and from double thejecond^fubtraft the third, and the 
 
 refults will be,\ 
 1 
 
 x + 3^ = 
 
 dg*in, from thefecond of theje equations, Jubtraft 
 thefirft, and the rejult will be x4 ; and from double 
 the firft fubtraft thejecond, and it will give 3y=9 ; or, 
 
 y = - ~ 3. And from the firft of the given equations, 
 
 38 2* 4? 
 2 = 38 2A? 4x5 or>z= - - - 
 
 38 8 12 
 - =6. 
 
 Mifcellaneous 
 
Mifcellaneous Examples. 
 
 Given, 
 3 
 
 y r: 3.2 
 
 tfhejirft equation involved to a fquare, gives v* + 
 *~ 144; tf^ f* ivy +y' L ~ 16 by fub- 
 tracing \vy ( n * 28 ) //^w /* /<^ equation 5 0r ; ^ y 
 
 ~ ^ by evolution : ___ __ _ 
 
 ^f/ therefore, v -\-y-\-v j ~ 1 2 -f 4 : 
 
 16 
 
 Or, 2v ~ 1 6 ; ^;;^'y ~8 : 
 
 Again 3 v-\-y v j = 12 4: 
 
 8 
 =: 8 i or, 
 
 ^ Lto find ^ andj. 
 
 i^y = H4 I 
 ^ L 
 
 J = 9 J 
 
 .F/V/?, *y = 9^ by multiplication : 
 ConJ'equently y vy ~ $y Xj^ = 144 : 
 
 14.4 
 
 is, 97 Z z= 144 ; cr, jy x =: = 16. 
 
 Whence, y~ \/i6 =z 4 ; ^7^ i; 9^ = 36 
 
 -y = $6 j 
 Given, "{ ^ _ o ( to find v and y. 
 
 {vy 
 - 
 y " 
 
 irfti v z: 56 + y ; and therefore, ~ 8 
 
 or, 56 + j = 8j ; whence* y s: r = 8 5 ^^^/ v = 8jr 
 
 = 64. 
 
 Given, 
 
( 3*9 ) 
 
 to find v. 
 
 Given, I tf + /l6"+T r = "^T^ 
 
 That is, vi/ib+v* 4" 164-1^32: 
 ir 16 ^ a : 
 
 involution, v % X i 6-f--y* ~i6 v*]* :n 256 
 
 That is , i6-i; z -4-'y' t iz:256 32 < z; a 
 
 Or, i6v* '2$6 J2i; i ; and iSz; 2 - 4-32^-^ 
 
 , v*= ; or, v= 
 16 i i 
 
 C V a -4- V a "~"~ ^2 7 
 
 Given, \ x -- J~ f to 
 
 Again, x l 2^4"^* a * : 
 7*^^, # 7 i=.\/ aib : _ 
 
 'Therefore^ x+y+x jy ^r */a + 2^ + V^^ 
 , 2AT n \/^ 4- 2^> -f \/ tf 2^ : 
 
 Or, X - 2 
 
 And x -f-y x y 2jr n: -v/^/x 4- 2 ^ -v/^"" 
 
 .... 
 
 Whence, y = 
 
 CHAP, 
 
( 33 } 
 
 CHAP. XV. 
 
 Of the SOLUTION of a variety of QUES 
 TIONS, that produce SIMPLE EQUA 
 TIONS. 
 
 AFTER forming a clear and diftinct idea of the 
 queftion propofed 5 the unknown quantities 
 mult be exprefled by letters, which muft be ordered in 
 fuch a manner, as to exprefs the conditions given in 
 the queftion concerning thofe quantities. Thus, if 
 the fum ( s ) of two quantities (x and jy) are requir 
 ed -, then is x -j-jy s, an exprefiion anfwering that 
 condition. Alfo, if the difference (d) of thofe 
 quantities is required ; that condition muft be ex- 
 prefled thus, x jy~ d (x being the greater) Their 
 product (f) is expreffed thus, xy ~p. Their quo- 
 
 v> 
 
 tient (q) is - q. Alfo, the fum of their fquares 
 
 (a) is expreffed thus, # 4 -\- y* a y and the differ 
 ence of their fquares (b) thus, x z jy* ~b t 
 
 HAVING expreffed the unknown quantities in 
 equations anfwering their relations, or properties, as 
 given in the queftion "; you are next to confider 
 whether your queftion is limited or not -, that is, whe 
 ther the quantities fought, are each of them capable 
 of more known values than one , which may always 
 be difcovered in the following manner. If the 
 equations that arife from expreiling the conditions of 
 the queftion, are in number equal to the quantities 
 fought, then is the queftion truly limited : That is, 
 each df the quantities fought, cannot have more val 
 ues than one in giving the anfwer : But, if the equa 
 tions 
 
ticms exprefllng the conditions of the queftion, are 
 fewer in number than the quantities fought, then the 
 queftion is an unlimited one ; that is, the quantities 
 fought, are each of them of an indeterminate value, 
 and confequently, the queftion propofed, capable of 
 innumerable anfwers. 
 
 AFTER you have discovered that the propofed 
 queftion is limited ; you mud then proceed to exter 
 minate the unknown quantities by the rules already 
 given, or other methods, which you may learn by 
 practice ; to which we now proceed. 
 
 1. What number is that, from which if you take 
 40, the remainder will be 115 ? 
 
 Call the number /ought v : 
 
 Then will v 40 n. 115 by the queftion : 
 
 Or, v n 1 15 + 40 zz 155 the number Jought. 
 
 2. What number is that, from which if you take 
 10, and multiply the remainder with 4, the product 
 will be 30 ? 
 
 Call the number fought v : 
 
 Then will v 10 be the remainder : 
 
 And v 10X4 3 by ths queftion : 
 
 That is> 4& 40 = 30 : 
 
 Or, 4^ = 30 + 40 70 ; or, v n 7 T i?4- 
 
 3. To find two numbers whofe fum is 80, and 
 their difference 16. , 
 
 Let v = the lea ft of the required numbers : 
 
 Then will v + 16 iz the greater by the nature offub- 
 trattion : 
 
 And-v -f v + 16 ~ 80 by the queftion : 
 
 That is y iv ~ 80 16 = 64 : 
 
 Or, V V =132 ; and v+ 16^:32 -f- 16^48, 
 /bf greater number required, 
 
 4- 
 
4. What number is that, which if multiplied with 
 one third of itfelf> will produce the number fought ? 
 
 If you call the number Jo ugh t v : 
 
 i) 
 Then will be one third part of ' <v : 
 
 v 
 
 And v Y>-~v by the queftion : 
 
 v* 
 That is, = v -, or, v* 3^, andv =z 3 the num- 
 
 M 
 
 far fought. 
 
 5. Suppofe the diftance between Boflon and 
 York, to be 150 miles ; and that a traveller fets out 
 from Boflon, and travels at the rate of 5 miles an 
 hour ; another fets out at the fame time from York, 
 and travels at the rate of 8 miles an hour : It is re 
 quired to know how far each will travel before they 
 meet. 
 
 Jfyoufut v for the diflance that muft be travelled 
 by the en e which Jets out from B oft on, and y the diftance 
 travelled -by the other before they meet : 
 
 Then will v -f y rr 1 50, the diftance, travelled by 
 both, and v I y \ : 5 '. 8 by the queftion : 
 
 $y 
 
 That is> 81; z= 5^ 5 or> i>zi 
 
 $y 
 
 *AlfO) v zz 1 50 y 5 conjequently, ^: 1 50 y : 
 
 That is, $y zr 1 200 %y : 
 Whence, y = 1 20013 rz 92-^.. 
 And v = 150 j> = 57 X 9 T - 
 
 6. What fraction is that, if you add r to the nu 
 merator, the value will be 4- ; but if you add I C6 
 the denominator the value will be 
 
 Put -for thefrattionjought : 
 
( 333 ) 
 
 r,> 
 
 Consequently, 
 
 Or, 3^ 6 =: 2jy -f 2 : 
 
 y ~ 8, /^^ denominator : 
 
 y 2 8 2 
 
 rr 3 /^^ numerator : 
 
 \is the fraction required. 
 
 7. What two numbers are thofe whofe fumb6o > and 
 the fum of their fquares 2250 ? 
 
 Call one of the numbers w> and the other y : 
 
 Then -will, i 
 
 'ion. 
 
 rfty iv* + 2 wy -f-JV* rr 60^ zz 3600 ; 
 
 And w* + 2 wy -f- ^ r w 2 " ~f"^ z 
 Therefore, ^wy 2700 : 
 
 * -k 2 wy- -j-^y* - 4 sqy =r 
 
 Whence, w ^'rr v^poo =r 30 : 
 ^w^ 2 T^ = 60 + 30 =z 90, or w = 45 : 
 And y =1 60 wn:6o 45^1 15. 
 
 8. There are three numbers in arithmetical pro- 
 greffion, the firft added to the fecond will make 15., 
 and the fecond added to the third, 21 : What are 
 thofe numbers ? 
 
 Uu Let 
 
( 334 ) 
 
 #, y and z reprefent the three numbers : 
 Then will x +y ~ 15, thejum of the fir ft andje~ 
 d : 
 
 Andy + z 2I > *b*fum of the fetond and third : 
 Alfo, x -f- z zz 2jy j> /<? nature of the proportion. 
 Whence, x 4- jy -f ^ + 2; = 15 -f 21 =36 .- 
 That is, x -f- 2j> -f- z ~ 36 ; c?r, A: -f 36 2j : 
 5^/ ^ 4- z s: 2j s therefore, zy ~ 36 2^ 5 ^r, 
 = 36: 
 
 Whence, y n: 9 j ^t^ ^^115 JK zz 1 5 9 6 : 
 f ~ 21 j^ zz: 21 -9 n 12. 
 
 9. Two merchants traded in partnerfhip ; the 
 fum of their flocks was 600 dollars ; one's flock was 
 in company 8 months, but the other drew out his at 
 the end of 6 months, when they fettled their accounts, 
 and divided the gain equally between them : What 
 was each man's flock ? 
 
 Call one of the flocks x ; then 600 x rr the other : 
 But, x \ 600 -i- x : : 6 : 8 by the queflton : 
 
 8# nj6oo 6 x ; or, 14^ n: 3600 : 
 
 3600 
 Whence, x - = 257^- ; and 600 x = 600 
 
 2574- 342-f- /><? others ftock. 
 
 10. To find three numbers, fuch that if the firfl 
 be added to the fecond, their fum will be 12 ; and 
 the fecond added to the third, their fum will be 20 ; 
 alfo, if the firfl be added to the third, their fum will 
 be 1 6. 
 
 Call the firft number x, thefecondy, and the third z : 
 Then will x -f- y zr 12! 
 
 And y -f- z = 20 V by the queftion. 
 Alfo, x + z~ i6J 
 
 therefore, x +y + x + z ~ 1 2 + 16= 28 : 
 
(335 ) 
 
 fbat is, 1 x +^ + z r: 28 : But y + z ir 20 ; 
 Confequently, ix 4- 20 = 28 ; 0r, 2# = 8 :'' 
 Whence, x = 4, #*/ jy = 12 #=12 4 == 8 : 
 
 2O - T .== 2O 8 :Z 12. 
 
 ii. There are four numbers in arithmetical .pro* 
 greflion ; whereof the product of the two extremes is 
 i r 2, and the product of the two means 130 -, alfo, 
 the fum of the firft aad fecond terms is vj : What 
 are thofe numbers ? 
 
 Put x for the leaf term, and y the common differ 
 ence j then will x, x -\-y, x + 27, x -\- 3^ be thefou^ 
 numbers required : 
 
 ^_ - ^^_ ., 
 
 by the queji 
 
 Whence, **+ 3*y+ 27* ^ x +3^ = 1 30 1 1 2 
 = 18: 
 
 , 2j l ~ 18 j (?r,jy a V 8 ~ 9 
 
 + x 4-j^ r: i? ; ^^ is>'M +3 = 17: 
 
 Or, 2^rri7 3 = 14 : 
 
 Confequently, x V 4 = 7^ $*jtrft term of the $ro~ 
 grejfion ; ^^ therefore, x -f j =: 10, the JeTond term 5 
 ^w^ ^ H- 2jK = 1 3, /#<? third term ; alfo, x -f- 3jy =:' 1 6, 
 the fourth term. 
 
 So thatj, 10, 13, and 16, are the numbers required. 
 
 12. There are three numbers in arithmetical pro- 
 greflion ; the product of the two extremes, is 128, 
 and the product of the lead extreme with the mcan^ 
 is 36; \Yhat are thofe numbers ? 
 
 Call 
 
Call the numbers required, v, y and z j s> and z be 
 ing the extremes y whereof v is the leaft. 
 
 y + z~ iy by the nature of the proportion : 
 
 128 
 zr - from thefirft equation : 
 
 z z: iy - v from the third : 
 
 128. 
 Conjequently y s ^ zc -iy v by equality : 
 
 That is, 128 7= 'iyv v*. But zyv z: 96X2 rr 
 192 : 
 
 tTherefore, 128^:192 ^ a by fubftitution : 
 And v* ~ 192 128645 or, v =z ^64 ^: 8 : 
 
 128 128 
 z ~ ~- = -- zz 16 5 ^^ i; + z 2y ; 
 
 8 + i6 
 
 . ,, ' - z: 1 2 j and therefore the num 
 
 bers fought are 8, 12, 1 6. 
 
 13. To find a fraction, fuch that the fquare of the 
 numerator, added to the denominator, lhall make 
 30 ; and if 2 be added to the denominator, the value 
 of the fraction will be equal to the reciprocal of the 
 numerator. 
 
 Put -for the fraftion fought. 
 
 will v* +y z: 30! 
 
 V i I by the queftm* 
 
 And t l 
 
 jt "T" v j 
 
 Andy* y 4-2X1 ^4" 2 : 
 Confequently, y 4- 2 z: jo y > -that is, 2728*: 
 
 Or, 
 
( 337 ) 
 
 Or, y = V J 4 5 *<* ^ z = 3 y~3<> *4 
 165 or,<v 1/16 : 4. 
 
 4 4 
 
 So that the fraftion fought, is ; for, - 
 
 4=*~- ; therefore, &c. 
 16 4 ^ 
 
 14. To find a number confiding of two places, 
 fuch that the fum of its digits lhall be 5, and if 9 be 
 fubtrafted from it, the digits will be inverted. 
 
 Let v and y refrefent the two digits, v that which 
 ftands in the tenth's 'place. 
 
 Then by the nature of notation, we have lov +y=z 
 ibe number fought. 
 
 therefore, v + y - 5 ? ^ ^ 
 9^ xojy -h ^3 
 
 And lO'y-fjx 9^ xojy 
 
 n y -I- Q 
 
 Whence, $v ~ qy +9 -, or,v=. -- z: ( ly di- 
 
 <vijion) y + i .- 
 
 Alfo, vs y ; and therefore, y + i n 5 y ; tr* 
 
 Andy .-=11 > ^^^7+11124-1=3: St 
 that 32 is the number required. 
 
 15. A certain company at an inn; when they 
 came to pay their reckoning, found that if there had 
 been two perfons lefs in company, they would have 
 paid a dollar a man more ; but if there had been 
 three perfons more in company, they would each of 
 them paid a dollar lefs : What was their reckoning, 
 and the number of perfons to pay it ? 
 
 Put V i= the number of perfons, and y the number of 
 dollars each faid j then will vy ~ the whole recon~ 
 ing. 
 
( 338 ) 
 
 
 5* fy the queftion* 
 
 That is y vy~vy + v 27 2 from the jirft equa~ 
 'tion : 
 
 Or, iy -f- 2 = v : 
 
 And vy~vy v -\- $y 3 from the fecond equa* 
 tion : 
 
 Or, v "=. 3y 3 : 
 
 Confequently, $y 3 ~ iy + 2 ; or, 3y~-iy 
 
 2+ 3 : 
 
 Whence, y =. 5, the number ef dollars each p aid: 
 And v = iy =|- 2 ~ 1 2, w number of -perjjfh : 
 
 Confequently , vy n: 60 dollars^ the whole reckoning. 
 1 6. To find three numbers v, > and w, the produfb 
 
 of each with the fum of the other two being given 
 
 viz. v X? + w ~ = " 93 > J-X ^ + w = 1300, and 
 w X v +J = 1480 : 
 
 r vy j^ <ww p jo rr ^ 
 Or, \ vy -|- ^ n 1300 = ^ 
 t'y^ -f- ryzz 1480 =:C 
 
 Then, vy -|- *vw + ^^ 4- ^J zi^z 4- r, 
 
 rr ^ .* 
 
 therefore, zvw ~a-{- c b >, cr, vw~ 
 
 ^^ ^: But 
 
 ^.a, + c -, or, vy ~ 
 
 Again, vy -{-wy -f- 'yze; + ^7 ~ ^ + ^ ' 
 
 v~a : 
 
 And therefore, we have i wy ir b + c a \ or, wy~ 
 
 c 
 
 '2, 
 
( 339 ) 
 .f. c a 
 
 tmt " ' 
 
 2 
 
 a + b c 
 Buty ~' " i <?#^ y writing this value for y 
 
 ^n the equation, wy 
 
 by redu5lion> 
 
 . 3 
 
 aw +lw cw a -f c b 
 
 vw * - - -- ; or 
 
 a +c b aw+bwcw 
 
 SZ ..... - and therefore by equality > . . 
 
 Q.W b-? c a 
 
 Wkaue, w* = '\ a \^_K 
 
 turned into numbers^ and the root extrafed> w will 
 befeund IZ 37 i whence the other numbers are readily 
 
 a-j-c b a -4- b f 
 
 found ; for v z: ' *" zz i c, and y z^ 
 
 < J Q.W JJ * 2<u 
 
 17. Two women weat to market with 42 eggs, 
 for which they received equal fums of money ; af 
 terwards fays one to the other, if I had fold as many 
 eggs as you, I (hould have received 350 cents ; fays 
 the other, if I had fold no more than you, I ihould 
 have received but 14 cents. Query, the number of 
 ggs each fold, and the particular prices fold at j al~ 
 fo the number of csats each received* 
 
 Let 
 
( 340 ) 
 
 Let 17 =s number of eggs fold by one, and y the num- 
 
 lerfold by the other ; ajfo, u rr price which v eggs were 
 
 fold at per egg, and w the price that y eggs were fold 
 
 'per egg. 
 
 !v+^=4i 
 l*w^&o 
 yu = 14 
 
 w 
 
 14 
 
 3S 
 
 From the third equation we have,, v n - : Prom 
 
 14 
 the fourth equation, uzz > and therefore, vu r^ 
 
 350 14 4900 
 
 - X m - But vu r: yw from the fecond equa- 
 
 4900 
 tion ; wherefore^ - irj; ; or 4900 -^y^w 7 -, and 
 
 : /49o = 7 i ^^^^ ^ = : But y^ 
 
 jo _ 14 
 the fourth equation ; confequently y > or, 
 
 )> writing $u far w 
 
 in thefecond equation, we have vu zr $uy 3 or dividing 
 loth fides by u, we /hall have v ir 5^ : j5/ v ~ 42 .y 
 /r^i thefirft equation ; therefore, $y 31-42 ^ ; cr, 
 
 42 H 
 
 2, ^ w = 5//n: TO. 
 
 1 8. Given the fum (s) and product (f) of two 
 quantities, to find the fum of their fquares> cubes, 
 biquadrates, &c* 
 
 i^/ v and w reprefent the two quantities : 
 tten will\ ^*Jf^7" r l j ^ queftion, 
 
 And 
 
C 
 
 And x -4-jyl* = #* + sry 4- j* =r s* ly involution : 
 Or, 'AT* + sry + jy* 3#y = s 1 if byfubtraff. 
 That is, x* +y*~s* ip Jim tifthefquares. 
 
 Again, x* +y* X x+y * p X s : 
 
 That is, x 3 + xy X ^ +^ 4-^ 3 ~ J 3 
 
 Or, x 3 +J/>-f-^ 3 riJ 3 2^/> ^ writing sp for its 
 
 equal, xy x # +J ; whence, x 
 ~j 3 zspfum of their cubes. 
 
 >/ i, Jf +^y x ^ 
 
 or, (by writing for xyy^x* -\-y* its equal, s^p 
 
 zrj 4 4^ a ^> + 2/* Zl/i of their fourth powers. 
 
 * X* + J s * VP + ^ X s : 
 That is, x* + *j X^ 3 H-j r + y s =* 5 4* 3 P 
 
 &p*s ; and therefore, (by writing for xyXx* + jy 3 //j 
 equal s^p zsp*) we have, x s + s*p jjy* + y* 
 zzJ 5 4J 3 ^> 4- 2J[f a ; ^W ^jy tranfpofition, we get 
 
 4 /?r tbefum of their fifth 
 
 powers -, and Jo en for the reft. 
 
 CHAP. XVI. 
 
 Of QUADRATIC EQUATIONS. 
 
 A QUADRATIC EQUATION, is an e- 
 quation of two dimenfions involving only one 
 unknown quanticy 3 and is either fimple or adfeftcd. 
 
 Xx A 
 
A SIMPLE quadratic, is an equation which invol 
 ves only the fquar'e of the unknown quantity. Thus, 
 v z ~ a* is a fimple quadratic equation. 
 
 BUT when you have an equation which involves 
 the fquare of the unknown quantity, together with 
 its produft with fome known co-efficient, you have 
 what is called an adfefted quadratic equation. Thus, 
 v^+av zi be, is an adfedted quadratic equation. 
 
 ALL adfefted quadratic equations, fall under the 
 three following forms : 
 
 f #* 4- av be 
 viz. < v* - 1 av zr be 
 (. v z av ~ be 
 
 THE folution of adfedled quadratic equations, or 
 finding the value of the unknown quantity in thofe 
 equations, is performed by the following 
 
 R U*L E. 
 
 i. TRANSPOSE all the terms that involve the un 
 known quantity to one fide of the equation, and all 
 the terms that are known to the other fide. 
 
 . I? the fquare of the unknown quantity is mul 
 tiplied with any co-efficient, you muft carl off that 
 co-efficient, by dividing all the terms in the equation 
 by it, that the co-efficient of the higheft dimenfion 
 of the unknown quantity may be unity. 
 
 3. ADD the fquare of half the co-efficient pre 
 fixed to the unknown quantity, to both fides of the 
 equation ; and that fide which involves the un 
 known quantity will then become a complete fquare. 
 
 4. EXTRACT the root from both, fides of the e- 
 qtiation, which will confiftof the unknown quantity 
 connected with half the aforefaid co-efficient ; and 
 therefore by tranfpofing this half, the value of the 
 unknown quantity will be determined. SOL. 
 
( 343 ) 
 
 SOLUTION of the THREE FORMS 
 Of QUADRATICS ILLUSTRATED. 
 
 Let it be required to determine the value ofv, in 
 the fgrm v z + av~bc. 
 
 a* 'a* 
 
 Fir fly v* + av + H be -\ -- by adding thefyr, 
 
 a a 
 
 of - to both fides of the equation : Then v -j- - ^ 
 
 a^ 
 i/bc + by extracting the root of both fides -, or, v == 
 
 a* 
 
 ' -- "ify tranfpofition. But the fquare root of 
 
 any fofitlve quantity, may be either fojitive, or nega 
 tive j that is ', the fquare root of 4- n z may be either -f- 
 -n or n \ for + ;/ X -f ; or, n X n } are re- 
 
 fpeffively equal to + n* . It follows therefore, that all 
 quadratic equations admit of twofolutions, that is, the 
 unknown quantity has two values in the given equation. 
 Thus, in the foregoing example, where v* '+ av -f- 
 
 a* a* a """* 
 
 -~ z: fa + > we may infer, that v -j- - n \/bc+ -. 
 
 ^', for, + </bc + * X + ^tc + - 
 
 4* T" T* 
 
 ~ X V^c + are w f^ equal to be 
 
 a* 
 
 -f- and therefore the two values of v, are v :=: 
 
 ^ a"- a 
 
 " and v n: i/bc + "" : WHOP 
 2 4 ^ 
 
 ambiguity 
 
( 344. ) 
 
 ambiguity is cxprcjfed by writing the uncertain fign Hh 
 
 before </bc + : ThuSyV-k ~ iz v/^ + 
 4 2 4 
 
 4 2 
 exprej/ion for the value of v, viz. 
 
 a . a_ a* 
 
 + ~3 the only negative quantity is~*~v 
 
 
 which is evidently lejs than \/ be -[- > 
 
 quentty, the value ofv is fofitive. : But in thefecond ex- 
 
 a 7 - a 
 
 frejfion, viz. v = \S be 4- --- ? having 
 
 ~ a* a 
 
 \/$f + v $ ~ ^<?/^ negative -, if follows, that 
 
 the value of v. muft alfo be negative. 
 Again, if 2* az bc\ 
 
 tfhen will z* az H ---- ~^-f by adding the 
 
 4 4 ' __ 
 
 Jquare of - to both fides, andz - z: v^^ + "^ 
 2 2 4 
 
 ^ extracting the root > and tier ef ere, z~ 
 
 a a* 
 
 4* t" /iwr ^^ fofitive value cf z, and z ~ \/bc H -- 
 
 4 
 
 * ^ z 
 
 -f~w negative one -, forfince be + is greater than 
 
 ~2 > confluent fy, \/bc + is greater than </ s 
 
( 345 ) 
 
 ~ P a 
 And tfartforc, z~ \Sbc + -f * ** always a ne 
 
 gative quantity. 
 
 And in like manner* the value of z determined in the 
 
 a* 
 third form, viz. z*az be, is z =i .\S fa 
 
 a a* 
 
 4 > where both the values of z will be fejitive, if 
 
 #*""" a 
 
 i$ greater than be > for then z : v/ -- ^ r '^- N ' '**<[ 
 
 vidently a pojitive quantity ; and in thejscond value of 
 
 a* a a z 
 
 z, viz. z \/ ^ + " ^ V pl<tin>that 
 
 /V greater than * bc } fince is greater than Ic ; 
 
 
 and therefore, the \S /V greater than \/ fa ; 
 
 a* a* a . 
 
 confequently, z y be + \S~(~"')tsa 
 
 fofitive quantity. But when be is greater than 
 
 a* 
 
 then be is a negative quantity 5 and fince the 
 
 fquare of any quantity (whether fofitive or negative) 
 
 is always fofitive ; it follows, that ^ fa is im~ 
 
 pojjible y or imaginary j and consequently, z ~ 
 
 a* a 
 
 Y/ fa^~ is imaginary, therefore, in the third 
 
 form, 
 
( 346 ) 
 
 a* 
 form, when be is greater than the Jolution tf the 
 
 equation will be, impojfible. 
 
 EXAMPLES 
 
 Of determining the value of the unknown quan 
 tity in quadratic equations. 
 
 Given, AT* -f- 4*^:32, to find the value of*-. 
 
 Firft, AT -f- 4.x -f- 4 z: 32 + 4, by adding thefquare 
 of half the co-efficient to loth fides : 
 
 Then, \/x* -f-4#-f- 4~ d: V/3^ : 
 
 That is, x + i + 6 ; or, x z: 6 i n 4, cr 
 
 8: Either of which fubftituted for x, will f reduce 
 the given equation. 
 
 Given, 3** 9*= 6, to find x, 
 
 Firft, x* +- -3^ -zby dividing the whole by 3 : 
 
 9 9 
 
 > a: 1 3oc-f--z:"- 2^v completing thefquare : 
 4 4 
 
 3 9 
 
 therefore, x ~ + v^~ 2^y extracting 
 
 the root : 
 
 Given, ay* -~ bv -~ c ~ d, to find v t 
 Firft, av' t l>vzz4 c h tranjpofition ; 
 
 j ' . * J - f 
 
 ^fW ^ - v = 
 a a 
 
( 347 ) 
 
 I b* d~C b"' 
 
 therefore, v*-- v +; = + fy com- 
 
 fie ting thejquare : 
 
 b J^c ~ 
 
 Whence y v ~ ~ v * - -f ? by evokt- 
 
 tlon : 
 
 b ,dc F 
 Or, v=t~ </- + i tranftofaion. 
 
 AI*L equations, wherein there are two terms which 
 involve the unknown quantity, whofe index in one 
 term, is juft double its index in the other, are re 
 duced to equations of lower dimenfions, in the fame 
 manner as quadratics. 
 
 B 
 
 THUS, v 6 -J- bv* IT d ; and v n -f- ^ T = c> are re 
 duced by completing the fquare, and extracting the 
 root, as in quadratics ; and the value of the un 
 known quantity determined by extruding the root 
 of the reiulting equation -, as in the following 
 
 EXAMPLES. 
 
 Given, *y 4 2^ a = 224, to find the value of v. 
 
 Firft, V* 2<y a -f- 1 = 224+ i n 225 by com- 
 fleting the fquare : 
 
 And v* i zz v/22^ ly evolution : 
 Or, v z zz v^^25 4- i by tranfpofition : 
 
 Whence^ <u~ 1/225 -{- ij z ~ 4. 
 
 n 
 
 Given, bv n + cv* d e > to find v. 
 
 Firft y bv n + ctf r= e + d by tranfpofiticK, 
 
 c ^ e 4- d 
 v n +-V~~ ^- ly divifiw : 
 
And V 
 fleting thejquare : 
 
 ( 348 j 
 
 c " c 1 e 
 
 n r \ d c*" 
 
 therefore, <u* -f = v^-* + 
 
 TT71 S ~^ d C * C 
 
 Whence, v = 4- v/-^ -f ~TT T 
 
 r i 
 
 CHAP. XVII. 
 
 f he SOL UriO Nofa Variety of $UE S 7*70 N$, 
 Producing QUADRATIC EQUATIONS. 
 
 i. TT 7 H AT two numbers are thofe, whofe fum 
 \\ is 20, and their product 96 ? 
 
 Call one of the numbers w - 3 then will 20 w be the, 
 ether : 
 
 And w x 20 w z: $6 by the-queftion : 
 That is, low *w* 96 : 
 Or, *y a 2Ow rr 96 ^y tranjfofition : 
 And w z 2oze> + ioo ~ 100 96 ^y complet 
 ing thefquare : 
 
 Therefore, <oo 10 =: Hh \/ioo -96 = v/ 4 
 ^ di 2 ^y evolution : 
 
 Or y w~ + 2 + 10= I a or 8, andia -w rz 20 
 < - 1 2 ~ 8 tbe oiber number* 
 
 i. What two numbers are thofe, \vhofe fum is 36, 
 and the fum of their fquares 720 ? 
 Put w for the greater number : 
 Then will 36 w~tke other : * , 
 
( 349 ) 
 
 And w* /- 36 w]* 7 20 y /* quefiion : 
 Tfttf/ /V, w a + 1 296 7 2W -f- w 2 : 7 20 : 
 Or, 2w* 72-12; = 576 y tranfpofition : 
 dndw* 36 w ~ 188 by dhifion : 
 Wherefore, w* $6w 4- 324 n: 324 288 iz 3^ 
 
 by completing the f quart : 
 
 Consequently, w i8ir \/ 36 z: 6 by evolution: 
 Or, w 6 -f- 1 8 24, ?/^/ 36 w 36 24 
 
 n 12 : 
 
 3. What number being divided by the product 
 of its two digits, the quotient will be 2 j and if 27 
 be added to it, the digits will be inverted ? 
 
 Put w and y for the two digits : 
 Then will 10 w -f y be the number fought, by the 
 nature of notation : 
 
 ~~wj~ >2 
 And low+y + 27 ~ iqy -{- w 
 
 Or, 9 w = 97 27 ^y tranfpofition : 
 
 QV - 27 
 
 1 
 
 \bytb 
 ] 
 
 9 
 
 IO-K; + y -: 207 ; whence, (by writing for 
 
 its equal y 3, in the equation low + y rr 
 <?/ .1 qy 30 + y = 2^ 4 6y : 
 
 2y 2 = 305 or, 2y 4 I7jn 30 by 
 tranfpojition : 
 
 Whence, y* %l.y = 15 ^y dhifion : 
 
 289 289 49 
 
 Andy* ^ y ^-^--j._ ls -^ by cm . 
 
 Dieting thefquare : 
 
 Or, > = \/-7 = - by evolution: 
 4 10 4 
 
 Y y 
 
Confejumtly, J- 
 
 ( 35 ) 
 
 Therefore 36 zV /&* number required. 
 
 4. To find three numbers in geometrical propor 
 tion continued, v/hofe fum is 78 ; and ifthefum 
 of the extremes be multiplied with the-mean, the 
 product will be 1080. 
 
 Putv ~ leaft extreme > andz the greiter ; alfo>y 
 mean : 
 
 *Ihen will v -f- y -f- z zz 7 8 7 r r 
 
 - - J > fa the queftion. 
 
 And v + z X^= 1080 i 
 
 W^/ is, vy -f- zy^z io8Oi andvy + 
 3y wultif lying the fir fl equation ivith y ; 
 
 Whence^y % i^. (by writing for vy-\-zy it 
 787 1080, 
 
 Or 3 j 2 78yn 1080: 
 
 Andy* 78^4*15213:1521 108011441 
 completing thejquare : 
 
 And therefore, y~* 39 rz y / 44 I ~ 
 ticn : 
 
 Or, y~ 39 4; 21 = (becaufe 39 -J- 21 zz 60, /V 
 greater than thejum of the extremes^ which is abjurd) 
 39 21 18 : 
 
 But>vz ~y* = 324 ^j the nature of the proportion : 
 
 ) v^. - > which wrote for <v in the e- 
 
 quation vy + zy zz 1080, JT^J - -- 1- zy rr 1080 : 
 
 , 5932 -f- iSs; 1 zz 10802 : 
 Or,i 8z* 10802; zz 5932 by tranffofition : 
 And z z 602 zz 324 by divifion : 
 Therefore^ z* 602.^-900 900324= 
 completing tkejq 
 
( 35' ) 
 
 Whence, z 30 := i \/ 5 7 6 zr + 24 y evolution : 
 Or, z zz 30 -f- 24=54, and ?; zz 78 2 ^ = 78 
 
 54 1 8 zz 6. Iberefore, 6, 18, 0// 54, are tbe 
 
 numbers required. 
 
 5. There are three numbers in geometrical pro- 
 grefllon, whofe fum is 117, and the fum 01 their 
 fquares 7371 : What are thofe numbers ? 
 
 Call the numbers x } y and v : 
 
 Then will # + y -f* i; zz 1 17 ? 7 
 
 And x* + /+.T- = 7,71 5 ^ 
 
 A'i; jy* ^y /^^ nature of the proportion : 
 A; 4- ^ zz 117 j by tbe firft equation : 
 
 Whence^ x* 4- 2^1? -f* i; z =^= 13689 234^ 
 4y involution : 
 
 Buty ixv zr 2j z , which Julftituted for 2.x<v in the 
 la ft equation, gives #* +iy -f'y 2 13689 234^ -J* 
 /* J? 
 
 Or, ^ a +<y a zr 13689 23/y ^* : 
 
 ^^A? 1 -^ y* =7371 ^* ^y thejecond equations 
 
 Confequently, 737 i jy* zi 13689 234^ > a 5 
 
 Or, 2347 = 136897371 6310: 
 
 6310 
 Whence^y zr 27, and xvy* n 729 : 
 
 OT" 
 
 7 ^9 
 Or, ^ ^ , whichfubftituted in ihs equation x+ 
 
 729 729 
 
 y+v = i IT .gives +27 + i?=n7i cr, ~ 
 
 + T; 117 -27 9 : 
 
 Whence y 729 -j-.i;* ~ 901; ^) f multiplication : 
 
 Or, v* 901; zz 729 by tranJpofit'iQn : 
 
 And therefore,, v* 901; + 2025 zz 2025 729 
 
 -zz 1296 ^ completing tb?f%ua,re : 
 
Conjequently, v 45 -f \/ 1 296 = 36 y evolu 
 tion ; 
 
 729 
 Or, ^ 45 + 36 81, andx - ~ 9. 
 
 And the number 3 required, are 9, 27, 81. 
 
 MISCELLANEOUS QUESflONS, 
 their SOLUTIONS. 
 
 I. Suppofe two cities, A and B, whofe diftance 
 from each other is 216 miles -, and that two cou 
 riers fet out at the fame time, one from A, and the 
 other from B; the firft travels 10 miles a day, and 
 the other 4 miles lefs than the number of days in 
 which they will meet. Query the number of days 
 before they meet ? 
 
 Put x ~ number of days required : 
 
 Then will i o# -J- x 4 X #H 2 1 6 by the queftion : 
 
 That is y IQX + #* ^x zib - t or, x 7 " 4- 6x 
 
 And x* + 6^ + 9z:2i64-9^i: 225 : 
 Whence, A; 4- 3 n ^ v/ 2 25 n: 1 5 s cr, Arn 1 5 ^ 3 
 rr 12, the number of days required. 
 
 2. A traveller fets out from the city A, and tra 
 vels at the rate of 9 miles an hour ; and another at 
 the fame time fets out from the fame city, and fol 
 lows him, travelling the firft hour 4 milts 5 the fe- 
 cond 5 j the third 6, and fo on, in arithmetical pro- 
 greffibn : In what time will he overtake the firil ? 
 
 Put x number of hours in which the firft uill be 
 Overtaken : 
 
 f hen will $x rr the. diftance he travels : 
 
 x i X I +4*4 #+7 
 
 And 
 
( 353 ) 
 
 ~ ^ dtflance tbe other 
 
 travels before he overtakes the firft, by (be nature of the 
 
 x* -f- jx 
 proportion : Confequently, zrcj* by the quef* 
 
 tion : 
 
 Or, x* -4- 7*-^ i8# : Whence, # + 7 = 18 5 0r, * 
 
 ; 1 1 hours, the time required. 
 
 3. There arc four numbers in geometrical pro- 
 greffion, the fum of the extremes is 84, and the furn 
 of the means 36 : What are thofc numbers ? 
 
 Put v andy for the means : 
 
 will and be tbe extremes by tbe nature $/ 
 
 tbe proportion : 
 
 V* JV* I ly tbe queftion : 
 
 H*ti*t p for vy) 
 
 p b. 
 
 But, v 3 -i-jy 3 = (by problem 1 8 
 
 Confidently, pb ~ a* $ap ; cr* p ~ T - - ^r 
 
 ^jy fubjlitution : 
 
 Therefore, V* -}-j 3 r:f; e?r, -j; x s=^f jy j : 
 But, v ~ tf y ; therefore, v* ~ a* ^a*y + 
 
 And tberefwe 3 y* ~~ cy H - - " 
 
 * a And 
 
And-g* * ay rt~ 7- 
 
 ^w^,7 -SAifrr 
 
 , 
 
 - 27, rfv 36- j,r:9; therefore^ 
 
 nnmlers re 
 quired. 
 
 4. Suppofe two cities, A and B, whole diflance 
 from each otiier is 152 miles j and that two men fee 
 out at the fame time from thole cities to meet each 
 dtrkr ; the one which goes from A, travels the firft 
 day i mile, the fecond day 2, the third day 3, and fo 
 on; and the one wnich lets out from B/ goes the 
 firir. day 4 .miles, the fecood day 7, and the third io> 
 a^d fo on. Qjery the' number of days before -they 
 nicer, , and the .number of,milts that each travels? 
 
 Put y ~ number of days before ttiey meet ; 
 
 15? = , 5 3 ij, the gueJHe* : 
 
 Ay ^ -\~ 6 y 
 ! That i i s > ' *=r 152 ; tf>,*4j* -f 6y 304 : 
 
 ^j* +- j-f ~ 6 rr 76 
 
 122 
 
 * y ^ 4 ~^ ^ . 16 
 
 8. , " ' 
 
 % \ 
 
V * ! V 
 
 yl- ; -36, tks number- cf miles 
 
 travelled by tbe ont which Jat -out frcm A> and 
 
 i 
 
 ZT 1 1 6, the dljtancs travelled by the other. 
 
 C H A P. XVIII. 
 
 Oftbs GENESIS, cr 'FG&MATION of E- 
 ^U AT IONS in GENERAL. 
 
 ALL equations of fuperior order, are confider- 
 ed, as produced by the multiplication of equa 
 tions of inferior orders, that involve the fame un 
 known quantity. 
 
 Thus, a quadratic equation may be cdnndered as 
 generated by the multiplication of two fimpte equa 
 tions ; a cubic equation by the multiplication of 
 three flmple equations, or one qiuiciratk and one 
 fimple equation ; and a biquadratic equation by the 
 multiplication of four fi.nple equations, or two quad 
 ratic equations, or one cubic and one fimple equa 
 tion. 
 
 Suppofe w to be the unknown quantity, and a, 4 
 c t d> &c. its feveral values in any fimple equation ^ 
 
 That is, w~a 3 w ~ ^, w n c, w~d, dec. Thta 
 by tranfpoiition, w a ~ o, w b no, w rzrr 
 o, w d ~ o, &c. And the produdl of two of thefe 
 equations as w a x ^ ^zrzO, gives a quadratic 
 .equation, or one of two dimenfions. 
 
 The produft of any three ; as w a X w b X 
 w c r- o, produces a cubic equation, or one of 
 three dimenfions, The 
 
( 356 ) 
 
 The pro&3& of any four of thertt 5 3$ w ^ X 
 se? x '^ c X w d zr o, produces a biquad 
 ratic equation, or one of four dimenfions. 
 
 Hence it appears, that in every equation, the 
 higeft dimenfion of the unknown quantity, is equal 
 to the number of fimple equations; that generate that 
 equation -, and therefore it follows, that every equa 
 tion has as many roots, or values of the unknown 
 quantity, as there are units in the higheft dimenfion 
 of that unknown qnantitv. For fuppoie an equation 
 
 -*. w a X "Jo b X w f o ; and that for iff 
 you fubflitute any of its values (#, b or c) in the giv 
 en equation, then all the terms of an equation will 
 
 vanifh ; for if w ~a,w=.b, and w r, then w a 
 X ^ b X w <: ~ o, becaufe each of the factors 
 are equal to nothing. And after the fame manner, 
 it appears 3 that there are three fuppofitions that give 
 
 ce; a x w ^ X w ~~ c ^ : ^ ut fi nee there are 
 no other quantities befides thefe <*, , r, which fub*-' 
 
 ftituted for w-in the equation w a X w. X 
 w c n: o, will make all the terms vanifh ; it fol- 
 
 lows,that the equations a X w ^ X w c ~ 
 o, can have no more than thefe three roots, or ad 
 mit of more than three folutions. For if you fubfti- 
 tute for is; in the propofed equation, any other quan 
 tity ^, which is neither equal to a, , nor c ; then 
 neither e a, e b> e c y is equal to nothing; 
 arid confequently their product ea X e b X e~c y 
 cannot be equal to nothing, but muft be fome real 
 product : So that no other quantity, btfide's one of 
 thofe before-mentioned, will give a true value of w 
 in the propofed equation. And therefore, no equa 
 tion can have more roots than it contains dimen 
 of the unknown quantity. To 
 
( 351 ) 
 
 To be more plain : Suppofe that AT* icx* + 35** 
 50* + 24 mo, is the equation to be rtfolved ; 
 and that you find it to be the fame as the produdl of 
 
 #~ i X x 2 X # 3 X # 4 : Then you will 
 infer, that the four roots or values of x, are i, 2, 
 3, and 4 j for any of thefe numbers fubftituted for 
 A:, will make that product, and confequentiy, .v 4 - 
 IOAT 3 -f- 35*v* 5O#-{- 24 equal to nothing, accord 
 ing to the propofed equation. 
 
 THE roots of equations are either pofitiye.or ne 
 gative, according as the roots or values of the un 
 known quantity in the fimple equations which prod 
 uce them, are pofitive or negative. Thus, if^z: #, 
 vn b,v=. r, viz d\ then will v + ^n:o, 
 i) -\-b o, *v -\- c ~ Q, and v + d ~ o ; and con-* 
 
 fequently, v + a X.^ + ^.X^ + ^X ^ +- d zi o, 
 will be an equation whofe roots a y ^, r, ^/, 
 are all negative. And after the fame manner, if 
 v ~ ^, ^ zr , -i; f , the equation v a X ^HhJ 
 X ^ f, will have its roots -f *, , -}- r. 
 
 BUT to difcover when the roots of an equation are 
 pofitive, and when negative, and how many there are 
 of each kind, it will be neceflary to confider the figns" 
 and co-efficients of equations, generated from the 
 multiplication of thofe fimple equations that produce 
 them; which will be beft underftood by confidcring 
 the following table, where the fimple equations v ^ 
 V by v f, &c. are multiplied continually with 
 one another, and produce fuccefliv.ely the higher e- 
 quations. 
 
 a 
 
v a 
 X v b 
 
 n:^* avl ,. 
 ^ > + ab o, a quadratic 
 
 v z a 1 
 
 b S X V 1 + ab I 
 
 c J -f ^r > x ^ abc =: o, 4 ^ 
 
 -f ^ J '[equati 
 
 cubic 
 equation 
 
 o, biquad* 
 
 FROM the infpeflion of thefe equations it appears 
 that the co-efficient of the firft term is unity or i. 
 
 THE co-efficient of thefecond term, is the fum of 
 all the roots (a y b> c, d} with contrary figns. 
 
 THE co-efficient of the third term, is the fum of 
 all the produces of thofe roots that can poflibly be 
 made by multiplying any two of them together. 
 
 THE co-efficient of the fourth term, is the fum of 
 all the produces of the roots that can be made by 
 
 combining 
 
( 359 ) 
 
 combining them, three and three : And fo on for 
 any other co-efficient. The lad term is always the 
 product of all the roots, having their figns changed. 
 
 NOTWITHSTANDING thofe fimple equations made 
 ufe of in the foregoing table, in forming the higher e- 
 quations, are fuch as have pofitive roots ; yet the fame 
 reafoning holds,whether the roots are pofitive or neg 
 ative. Whence, if v* pv* 4- qv* rv-{-s~ o, 
 reprefents a biquadratic equation ; then will p be the 
 fuin of all the roots, q the fum of all the products 
 made by multiplying any two of them together, r 
 the fum of all the products made by multiplying any 
 three of them together, and s the product of all four* 
 
 IT likewife appears from infpection, that the figns 
 of the terms in any equation in the foregoing table 4 
 are alternately 4- and : The firft term is always 
 fome pure power of v, and is pofitive : The fecond 
 term is fome power of v, multiplied with the quan 
 tities, a y , r, &c. and fince thefe quanti 
 ties are all negative, it follows, that the fecond term 
 muft alfo be negative. The third term hath for its 
 co-efficient the product of any two of thefe quanti 
 ties, ( #, ^, c , &V.) and fince X gives 
 4- ; it follows, that the third term mud be pofitive. 
 For the fame reafon, the co- efficient of the fourth 
 term, which is formed of the products of any three 
 of thefe negative quantities, muft be negative alfo, 
 and the co-efficient of the fifth term pofitive. But 
 in this cafe, v <?, v = b, v c , v z: d y &c. that 
 is, the roots are all pofitive : Confequently, when the 
 roots of an equation are all pofitive, the figns of the 
 terms are + anc ^ alternately. But, when the 
 roots are all negative \ that is, v a> v ~ b y 
 u rz c y v z= d, &c. then -y-j-^X^-f-^X 
 v 4- c X v-J-^=o, will exprefs the equation pro 
 duced 
 
duced, whofe terms are evidently all pofitive. And 
 therefore when the roots of an equation are all nega 
 tive, there will be no change in the figns of the terms, 
 tonfequently, there will be as many pofitive roots in 
 an equation, as there are changes in the figns of the 
 terms of that equation, and the reft of the roots will 
 be negative. 
 
 HENCE it follows, that the roots of a quadratic e- 
 cjuatipn may be both negative, or both pofitive, or 
 one negative and the other pofitive. Thus, in the 
 
 equations* a 1 7 v - -,\ 
 
 L r X^ 4- ab ~ (v a X v &4 
 
 o, there are two changes of the figns, viz. the firft 
 term is pofitive, the fec<5nd negative, and the third 
 pofitive j confequently, the roots are both pofitive. 
 
 BUT in the equations* -f- al , , 7 
 
 -f- b\ X v + a * = ( v + a 
 
 X v -f ) o, there are no change in the figns, and 
 therefore both the roots are negative. 
 
 AND in like manner, in theequation v* -\- a 
 
 _ _ 
 
 r i) ab IT (v + a X "J ^) o, one of the roots 
 will be pofitive, and the other negative ; for fince 
 the firft term is pofitive, and the lalt negative, it is 
 plain, there can be but one change in the figns, whe 
 ther the fecond term is pofitive or negative. 
 
 HENCE alfo it appears, how that a cubic equation 
 may have all its roots pofitive, or all negative, or 
 two pofitive and one negative $ or two negative and 
 one pofitive. For fuppofe the cubic equation is 
 
 a I 
 
 * \ X v* +"ab ] 
 
 c J + ac \ 
 
 + : lc J 
 
 X v ~ ale ~ (v a X 
 
( 36 1 ) 
 
 ^ b X t> *) o, wherein there are three changes 
 in the figns \ and confequently all three of the roots 
 pofitive. 
 
 AGAIN, fuppofe the cubic equation is of this form, 
 
 v 3 a *} 
 b \ X v 
 
 + * J 
 
 X v + ale = v X 
 
 * I X v j where there are two changes in 
 the figns -, for if -}- b is greater than r, then the fe- 
 
 cond co-efficient a b -J- f muft be negative j if 
 a + b is lefs than r, then the third term will be neg 
 ative 5 for its co-efficient ab ac bc{~'abt 
 X a + b) is, in this cafe negative, becaufe the prod 
 uct a x b is always lefs than the fquare a-\*b X +> 
 and confequently, much lefs than f x * -f- ^ 5 and 
 fince tnere cannot be three changes in the figns, the 
 firft and lad terms having the fame fign ; it follows, 
 that two of the roots of the propofcd equation are 
 pofitive, and the other negative. 
 
 IN like manner, the equation v 3 + a -\- b "ry* 
 H- ab ac -~- bcv ^^r~o,will have two of its roots 
 negative, and the other pofitive j for if a -|- b is lefs 
 tha. c y the feco'nd and third terms muft be negative, 
 by what was proved in the laft example - 9 and if the 
 fecond term is pofitive, that is, a -f- b is greater than 
 f, it is plain there can be but one change in the figns, 
 and confequently but one pofitive root, the other 
 two being negative. 
 
 AND by parity of reaion, the pofitive and negative 
 roots of the other equations may be difcovered ; . 
 
 this 
 
c _f!_ ) 
 
 this method being general, and extends to all kinds 
 of equations whatever. 
 
 CHAP. XIX. 
 
 CONCERNING the TRANSFORMATION 
 cf EQUATIONS, and EXTERMINAT 
 ING their INTERMEDIATE TERMS. 
 
 N Y equation may be transformed into another, 
 ^ ^ whofc roots fhall be greater, or lefs than the 
 roots of the propofed equation by any given differ 
 ence (?) by the following 
 
 A 
 
 RULE. 
 
 ASSUME a new unknown quantity (j) and conneft 
 it with the given difference (e} 9 with the fign + or 
 ~, according as the roots of the propofed equation 
 are to be increafed, or diminifhed ; and make this 
 aggregate equal to the unknown quantity (#) in the 
 propofed equation ; then inilead of the unknown 
 quantity (#) and its powers in the propofed equation, 
 fubilitute this aggregate, (y e) and its powers; 
 and there will arife a new equation, whofe roots will 
 be greater or lefs than the roots of the propofed e- 
 quation, as required, 
 
 EXAMPLES. 
 
 i. Let x 3 px*+qx riro, be an equation 
 to be transformed into another whofe roots fhall be 
 lefs than the roots of the propofrd equation, by the 
 difference e, AJJume 
 
dflume x ~y + e: 
 Then will x* j 3 -f-jyv-f y^ 4. ^ 
 
 r = 
 
 qe 
 
 * 1 ^nation requir. 
 
 2. Let # x 1 1 # -|- 30 n o, be transformed into 
 an equation that fhall have its roots lefs than the roots 
 of the propofed equation by the difference 4. 
 
 AJfume x ~ y -f- 4 : 
 
 16 : 
 
 n^zz iij 44 
 
 + 3 = +3 
 
 J* j^ + 2O, w /^ equation required. 
 
 IN the firft example of the foregoing transforma 
 tions, the co-efficient of the fecond term in the tranf- 
 formed equation, is 3* p ; and if you fuppofe ~ 
 -J./>, and therefore, 3* /? o j then the fecond term 
 of the transformed equation will vanifh. Let the 
 propofed equation be of #dimenfions, and the co-ef 
 ficient of the fecond term f ; and fuppofe # 
 
 P 
 =j + -* then if this value be fubftituted for ^ in 
 
 the propofed equation, there will arife a new equa 
 tion that fhall want the fecond term. For if p = 
 fum of all the roots of the propofed equation, and x 
 
 p 
 ~y -f. - ; it follows, that each value of? in the new 
 
 equation, will be lefs than the value of x in the pro- 
 
 P 
 pofed equation, by -> and fince the number of roots 
 
 is , it follows, that the fum of the values of y> will 
 
 be 
 
be lefs than p, the fum of the values of %, b# n x 
 
 ~ ^ ; that is, the fum of the values of j, is +/> p 
 ^ o ; and fince the co-efficient of the fecorid term 
 in the equation ofy, is the fum of the values ofj, 
 viz. -f- p p 3 which is equal to nothing; it follows, 
 that in the equation of j, arifing from the fuppofi- 
 
 tion of Afjy -{- ~, the fecond term onuft vanifh : 
 
 And therefore the fecond term of any equation may 
 be exterminated by the following 
 
 RULE. 
 
 DIVIDE the co-efficient of the fecond term of the 
 propofed equation by the index of the higheft power 
 of the unknown quantity; and a flu me a new un 
 known quantity ( y ) and annex to it the faid quotient 
 with its fi-gn changed ; then put this aggregate e- 
 qual to the unknown quantity (x} in the propofed 
 equation, and inftead of* and its powers, write thfs 
 aggregate and its powers, and the equation that 
 arifes fhall want the fecond term. 
 
 EXAMPLES. 
 
 Let the .equation x* 8 #-J- i.a ~ o, .be propo- 
 ied to have its fecond term exterminated. 
 8~2~ 4; 
 , x~y-\-4., per rule : 
 Then, x* y* + 8j + 16 
 8^~ 8jx 32 
 
 4 = 
 
 HENCE, 
 
( 36$ ) 
 
 HENCE it appears, that a quadratic equation 
 be refolved without completing the fquare, by ex 
 terminating the fecond term ; for fincejy 1 4 o ; 
 or,y*r:4, and jy ~ v' 4j we fhall have x y + 4:=: 
 
 Let the fecond term of the equation # 3 
 34 o, be exterminated. 
 
 Then, x 3 ~y 3 -f- QJ* + 27^ + 27 
 _9#*n 9?* 547 81 
 + 26% -j-26y-f-78 
 
 J4^ ___ _ 34 
 
 y 3 * jx 10 = 0. 
 
 WHEN the fecond term in any equation is want 
 ing, it is plain, that the equation hath both pofitive 
 and negative roots ; and fmce the co-efficient of the 
 fecond term in any equation, is the difference be 
 tween the fum of the pofitive, and fum of the nega 
 tive^ roots j it follows therefore, that when the pofi 
 tive and negative roots are made equal to each other, 
 that difference vanifhes. Confequently, v/hen an e- 
 quation has the fecond term wanting, the fum of the 
 pofitive roots is equal to the fum of the negative 
 ones. 
 
 HENCE, by the foregoing transformation of equa 
 tions and the exterminating their fecond terms, the 
 pofitive and negative roots are reduced to an equal 
 ity, and the folution of the equation thereby render 
 ed more eafy. 
 
 IF the equation <v* pv* -\-qv mo, be trans 
 
 formed into another, by affuming <v y -}-<?, the 
 
 co-efficient of the third term of the transformed e- 
 
 quation will be y* ipe + q-, now if we fuppofe this 
 
 Aaa co- efficient 
 
( 366 ) 
 
 co-efficient equal to nothmgV aftd refolve the quad- 
 
 ratlc 3** r 2p 4- j 2= o we Ihall have.* g:?...-; ;. ,-. . : 'V ~ 
 
 3 
 
 which fubftttuted for * in the equation v~y-+-* 9 
 the third term of the transformed equation will van- 
 Ifb : Alfo, if the pfopofed equation be of n dimen- 
 fions, the value of e, by which the third term is to 
 TDC exterminated, is found by refolving the quadra- 
 
 IP CLO -\ 
 
 tic equation <?* -\-~~ X e + . 7 z: o, tnat is, 
 
 n n X n i 
 
 by finding the value of e in the co-efficient of the 
 third term of the transformed "equation, when that 
 co-efficient is equal to nothing. And in like man 
 ner, the fourth term of any equation may be, exter 
 minated, by foiving a cubic equation, which is the 
 co-efficient of the fourth term of a transformed equa 
 tion : And after the fame manner, the other terms 
 may be taken away. 
 
 THERE are other transformations which are of ufe 
 in the refolution of equations; of which the moft 
 ufeful, and the only one that we fhall confider, is, 
 when the higheft term of the unknown quantity is 
 multiplied with fome given quantity, to transform 
 the equation into another that ftiall have the co-effi 
 cient of the higheft term unity. 
 
 LET the propofed equation be atf pv* -f- ^i; 
 rzz o ; and fuppofe av ~y y then v ~y -i- a y and 
 this value fubftituted for v in the propofed equation, 
 
 there will arife -^L- -EL- -f S2L r ir cyx>r 
 a 1 a* a a* 
 
 ?2L -|_ zL r n o, and by multiplying the 
 a* 'a 
 
 whole by a*, we fhall have y* y l -\-qay ra* 
 r:O> which gives the following 
 
 RULE. 
 
wfob RULE. - 
 
 
 
 CBANGE the unknown quantity (_v) ii> the propof 
 ed equation, into another (y) r prefix no .co-efficienC 
 to the firft term, pafs the fecond, multiply the third 
 -term with the co- efficient, of the higheft term of the 
 unknown quantity iathe propofed equation, and the 
 fourth term by the fquare of that co-efficient, the 
 fifth by the cube; and fo on, and the higheft terra 
 of the unknown quantity in the refulting equation 
 (hall have its co-efficient unity, as required. 
 
 EXAMPLES. 
 
 Let the equation iv* + 6v 36 zz o, be changed 
 into another that will have unity for the co-efficient 
 of the higheft term of the unknown quantity. 
 
 Tbus,y z + 6y 36 X 2 =i o ; or> y* 4- 6y 72 
 r:o, is the equation required. 
 
 The finding the roots of the propofed equation, 
 and all others of the like kind, will be very eafy when 
 the roots of the transformed equation are found -\ fmce 
 1} = (in this cafe) ~y. 
 
 Transform the equation 5i> 3 101;* H- -i6v 93 
 m o, into another that the higheft term of the im- 
 known quantity may have an unit for its co-efficient. 
 
 Tbusyy* IQJ 4- 8qy 2325 = o, is the. equa 
 tion required. 
 
 CHAP. 
 
( 368 ) 
 
 CH A P. XX. 
 
 Of the RESOLUTION of 
 
 by DIVISORS. 
 
 IF the laft term of an equation is the produft of 
 all its roots ; it follows, that the roots of an e- 
 quation when commenfurable, will be found among 
 the divifors of the laft term ; which gives the fol 
 lowing 
 
 R U L E, 
 
 TRANSPOSE all the terms to one fide of the eqtia- 
 tion. Find all the divifors of the laft term, and 
 fubftitute them fucceflively for the unknown quan 
 tity in the propofed equation $ and that divifor, 
 which fubftituted as aforefaid, gives the refult zi Q, 
 is one of the roots of the equation. But if none of 
 the divifors fucceed, the roots of the equation are 
 for the moft part, either irrational or impoffible.' 
 
 Note. If the laft term cf the fropofed equation is 
 large, and confequently Its divifors numerous 
 they may be diminijhed y by transforming the equa 
 tion into another, by the rules of the laft chapter. 
 
 EXAMPLES. 
 
 Find the roots of the equation x 3 4** -j- io# 
 j 2 = o. 
 
 Here the diyifors of the la/I term y are . i, 2, 3, 4, 6, 
 , i, 2, 3, - 4, 6, 12, which Jub- 
 for.x* ..:........<> ~ 
 
 Gives, 
 
f 1-^44- ioi2 ~ -5 
 j 8 1 6 + 20 1 2 = o 
 j Gives* <{ 27 36 + 30 * 1-2 .= 9 
 | 64 64-4-40 12' 28 
 
 I 2l6 144 + 6O '12 I2O 
 
 Gfc. 
 
 WE omit trying the negative divifors, fince there 
 are three changes in the figns of the propofed equa 
 tion, and therefore none of its roots can be negative : 
 And fince none of the divifors fucceed, except 2 $ 
 it follows, that 2 is the only rational root of the equa 
 tion, the other two being either irrational, or impof- 
 fible. 
 
 Let it be required to find the roots of the equa 
 tion x 3 + ix* 40* +64 0. 
 
 Here the divifors of the laft term y .are i, 2, 4, 8, 
 16,32, which Jubflitutedjucceffi-vely for x in the fro- 
 fofed equation, 
 
 c ! + a 4 o + 64 = 27 
 
 Gives, < 84-8 80 + 640 
 
 (, 64 + 32 160 4- 64 zr o 
 
 WHERE the only divifors that fucceed, are 2, and 
 1 4 ; and fince there are but two changes in the figns 
 of the propofed equation, there mud be one negative 
 root : We are therefore to fubftitute the divifois 
 negatively taken, in order to difcover the other value 
 of#; and on trial, we find that 8 fucceeds. There 
 fore the three roots of the propofed equation, are+2 
 
 BUT when one of the roots of an equation is found, 
 the relt of the roots may be found with lefs trouble, 
 by dividing the propofed equation by the fimple 
 equation, deduced from the root already found, and 
 
 finding 
 
( 37Q ) 
 
 rinding the roots of the quotient, which will be an 
 equation a degree lower than the propofed one. 
 
 THUS, in the laft example the root -f 2 firft found, 
 gives x n: 2 ; or. # 2 ~ o, by which dividing the 
 propofed equation : Thus, 
 # 2) x* -h ix* 40* 4- 64(.v* + 4# 32. 
 
 * 40,*? 
 
 * 8* 
 
 32*4- 64 
 
 32*4- 64 
 
 The quotient will be a quadratic equation x* -(- 
 4# 32 ~ o i which is the product of the other two 
 funple equations, from which the propofed cubic was 
 generated -, and whofe two roots are confequently, 
 two of the roots of that cubic. But the two roots of 
 the quadratic, are +4' an( l. 8. Therefore, the 
 three roots of the cubic equation, are 2, 4, 8, the 
 fame as before. 
 
 THE finding all the divifors of the laft term of an 
 equation, efpecially if that term be large, is much fa 
 cilitated by the following 
 
 R U L . 
 
 t. DIVIDE the laft term by its lead diyifor that ex 
 ceeds .unity, and the quotient by its leaft divifor ; 
 proceeding in this manner, till you have a quotient 
 that is not farther divifible by any number greater 
 than an, unit: And this quotient together with thole 
 rs,,. are. the. Srft dwfprs of the lafl term. 
 
 2, 
 
( 37 1 ) 
 
 . ; 
 
 2. FIND all the produ&s of thofe divifors which 
 arife by combining them two and two, and all the 
 produces which arife by combining them three and 
 three, and fo on, until the continued product of the 
 firft divifors, is equal to the quantity to be divided ; 
 and you will have the divifors required. 
 
 EXAMPLES. 
 
 Thus, fuppofe the laft term of an equation to be 
 60: Then 60-4-2 = 30, 30+2 zz: 15, i$-r-3~:5; 
 therefore, 2X2, 2X3> 2 X 5, and 3 X 5, are the 
 combinations of the twos ; and 2x^X3, 2X2X5? 
 ? X 3 X $, tne combinations of the threes ; alfo, 
 
 2X2X3X51 is the combination of the fours their 
 continued produ6t, equal to the quantity to be di 
 vided. Therefore all the divifors of 60, are 2, 3, 5, 
 
 .4, 6, 10, 15, 12, 20,30, 60. 
 
 And in like manner, the divifors of i cab, are 2, 5, 
 
 a y by 10, 20, ib, $a, $b, ab> ioa> $ab, lab and ioab. 
 
 BUT there is another method for the reduction of 
 
 ^equations by divifors, which is lefs prolix, by re 
 ducing the divifors to more narrow limits, by the fol 
 lowing 
 
 R U L E. 
 
 1. INSTEAD of the unknown quantity in the pro* 
 'pofed equation, fubftitute fuccefllvely the terms of 
 theprogreilion, t, o, -^- 1, &c. and find all the divi- 
 
 ; fors of die fums that refult by fuch fubftitution. 
 
 2. TAKE out all the arithmetical progrefiions that 
 h ca-n 'be found among thofe divifors, whofe terms 
 
 correfjpdncl v?idi the order of the terms, i, o, i. 
 
C 37* ) 
 
 &c. and common difference unity; and the values 
 of A; will be found among the divifors which arife 
 from the fubftirution of x ~o, that belong to thofe 
 progreffions. 
 
 Note. When the arithmetical pregrejjion is increefing 
 according to the order cf the terms i, o, i, the 
 value ofx will be affirmative - } but when the arith 
 metical progrejfion is decreajing, the 'value ofx will 
 be negative. 
 
 EXAMPLES. 
 
 Let x 3 x* 10 x + 6 ~ o, be the propofed 
 equation -, and by fubftituting fucceflively for x> the 
 terms i, o, i, the work will Hand as follows. 
 
 Suppojttions. Refults. Divifors. Ar.P* 
 
 r_ 4 
 x o ># 3 - #* io#-r-6 < 4-6 
 
 l+H 
 
 HERE the progrefTion is decreafing, and 3, that 
 term which (lands againft the fuppofition of x z: o ; 
 therefore, 3, fubftituted for x in the propofed 
 equation, gives, 27 9 + 30 + 6 o ; where 
 all the terms vanifhing, it follows, that 3 is one 
 of the roots of the propofed equation 5 and 2-f-x/2, 
 and 2 v/ 2, the other two roots, found by dividing 
 the propofed equation by #+3, and refolving the 
 quadratic quotient. 
 
 Suppofe it be required to find the roots of the 
 equation v* + 3?> 3 19 z;* 27^ + 90 0. 
 
 Then by fubflituting as before^ the work will Hand 
 as follows. 
 
V ~ I 
 
 <u no 
 
 v i= i 
 
 Refvlt 
 48 
 
 90 
 96 
 
 ( 373 ) 
 
 Divifors. 
 
 1.2,3,4,6, &c 
 1,2, 3,5, 6, &c 
 
 . Progref. 
 
 3, 5 
 
 HERE are five arithmetical progrefTions ; and Tub- 
 ilituting 2, 3, 3, 5, refpeclively for i; in the 
 propofed equation, the whole vanifhes ; the other 
 progreflion being in this cafe ufelefs, fince the num 
 ber of roots are but four. Confcquently,2, 3, 3, 
 5, are the four roots required. 
 
 THERE are many other methods befide thofe which 
 we have here given for the refolution of equations $ 
 which the confined limits of our plan obliges us to 
 omit, and proceed to dif cover the roots of equations 
 by the method of approximation. 
 
 CHAP. XXL 
 
 fbe FINDING the ROOTS of NUMERAL 
 E ^UAriO NS in GENE RAL, ly the ME- 
 ?HOD of APPROXIMATION. 
 
 ALTHOUGH there are other methods for 
 the refolution of equations, than thofe given in 
 the iaft chapter, yet the moft of them are either very 
 prolcx, or confined to particular cafes ; but the fol 
 lowing method of approximation is general, and ex 
 tends to numeral equations of all kinds whatever, 
 and though not accurately true, gives the value of 
 the root to any afligned degree of exactnefs you 
 pleafe, by the following 
 
 Bbb RULE. 
 
374 
 
 RULE. 
 
 t. FIND by trial, a number nearly equal to the 
 root required, and call it r ; and put x for the dif 
 ference between the real root and that already found, 
 then will r x v. 
 
 2. INSTEAD ot'v and its powers in the propofed 
 equation, fubftitute r x and its powers j and there 
 will ariie a new equation involving x and known 
 quantities. 
 
 3. THEN by rejecting all the terms of this new e- 
 quation that involve the powers of A?; and aflum- 
 ing the reft equal to nothing, the value of A? will be 
 determined by means of a fimple equation. 
 
 4. ADD the value of x thus found to r, and you 
 will have a nearer value of the root required ; which 
 if not fufficiently exact, repeat the operation, by fub- 
 ilituting this value for r in the formula exhibiting 
 the value of AT, and it will give a correction of the 
 root; which if not yet exact enough, proceed to a 
 third correction -, and fo on, to any afligned degree 
 of exadtnefs. 
 
 EXAMPLES. 
 
 Given, *v* 4- 6v 31 ~ o, to find v by approx 
 imation. 
 
 ne root found by trial is nearly equal to 3 : 
 Therefore, rn 3, and r + x ~ v : 
 
 = 6r+6x 
 
 6r 31 = o : 
 
( 375 ) 
 Whence, x * * ~ r ~ r (ty writing $forr 
 
 its ^al) 3l ~^~ l = - -3 i and v = 3.3 
 
 yf</ if 3.3 ttjubjtitufedfar r in the equation, x^-i. 
 3i~^-6r mJbM bave x = 31-10.89-19:8 
 2r + 6 6.6 + 6 
 
 .! . ~ .0246, 0r rather x ir .0245, 
 12.6 
 
 - 
 
 Again, if this value bejubftitutedfor r> we Jhall 
 have x r: .000005, and v -=zr -\- x ~3-3 2 455> for a 
 nearer value ofv -, and Jo on, to any ajjtgned degree of 
 exaflnefs. 
 
 Given, v 3 + 2V 73 ^ o, to find v by approx 
 imation. 
 
 <fhe root found by trial, is nearly equal 4 : 
 Therefore, r = 4, and r -{- z n: v : 
 ?hen,v l =: r 3 + 3^ a 2 + 3^2* -i-s 3 - 
 
 73 - 73 
 
 2r + 22 73 rr o ; $r, 
 
 7^ r 3 <2r 73 64 8 
 
 % ,.., - ~ (by writing 4 forz) - 3 -- * 
 
 3^+2 y 4^ + 2 
 
 r= =.02 s ^ therefore, V = r + z 4.02 ; 
 
 ^7^ writing this value for r, in the equation z = 
 73 r 3 2r t 
 
 y* + 2 
 
 64.964808 8.04 
 
 , 
 
 haw z = 
 
( 376 ) 
 
 .004808 
 
 4.019905 nearly. 
 
 And after this manner of reafoning, we may obtain 
 theorems for approximating to the roots of pure 
 powers. 
 
 Thus, if A be a given quantity whofe n root is required^ 
 r the neareft lefs root in the integers, and v the difjer- 
 ence between r and the root required : Then will r n -\- 
 
 2 2 
 
 tj, \ *) 
 
 r J< y 3 , &V. ~ A\ and aJTuming v = 
 
 A r n 
 
 3 
 
 ji 
 
 . cr> more nearly > taking the three frjl terms, 
 
 A- r n 
 
 - 1 nr n ~ l + n X 'lZlr B ~ 2 A-r* 
 
 A-r n nr 
 
 I 
 
 
 
 - - 1 
 
 A-r n 
 
 n i n > and fy writing 
 
 ~^r- X A-r n 
 
( 377 ) 
 
 y for A r , we have v - 
 
 * *+*=.! 
 
 2r 
 
 (by reduction) the theorem for 
 
 n n i 
 
 2 
 
 approximating to the value 'of v y which added to r t 
 will give a correction of the root ; which if not fuf- 
 ficiendy near the truth, the operation muft be re 
 peated,, by fjbftituting the new r in the equation 
 exhibiting the value of v. 
 
 Thus, for example, iuppofe the cube root of 3 is 
 required. 
 
 Here r = i , the near eft lejs root in the integers > and 
 r -f- v root required. 
 
 Therefore, v = ./?_.. = ^-2 = 1 = . 4> ^ 
 
 r + v zr T + .4^: i .4, w^/V^> fubftitutedfor r, and th& 
 
 operation repeated, v will be found ~ .0397 ; there- 
 
 fore, r + v 1.4 + .0397 1-4397 n 
 
 c/*3, very near. 
 
 CHAP. XXII. 
 
 CONCERNING UNLIMITED PROB 
 LEMS. 
 
 HAVING gone through, and explained the 
 methods ufed in arguing limited problems, or 
 fucn as admit of but one folution ; it remains there 
 fore, that we fhew the learner how to reafon about 
 
 thofc 
 
( 378 ) 
 
 thofe problems which are unlimited, or admit of va 
 rious anfwers. 
 
 IT was obferved In Chap, xv, of this Book, that 
 when the equations expreffing the conditions of the 
 queftion, are lefs in number than the quantities 
 fought, the queftion is unlimited, or capable of in 
 numerable anfwers ; yet .all the poflibie anfwers in 
 whole numbers, are for die moft part limited to a 
 determinate number. 
 
 As queftions of this nature admit of fome varia 
 tions as to their general folution ; we lhall therefore 
 confider them in the following problems. 
 
 PROBLEM I. 
 
 * 
 
 Tofind the values of v and y in whole numbers, in 
 tbe equation av jh by c=z o - 3 where a> b and f> 
 are given quantities. 
 
 RULE. 
 
 1. REDUCE the given equation to its lead terms > 
 by dividing it by its greater! common divifor. 
 
 2. FIND the value of v from the given equation ; 
 and reduce the refulting expreffion, by expunging 
 all whole numbers from it, until c be lefs than a y 
 and the co-efficient of y becomes unity. 
 
 3. ASSUME this laft refult equal to fome known 
 whole number, and the expreffion reduced, will give 
 the value of y in known terms - 9 from which the val 
 ue of v may be determined in the given equation. 
 
 Note. If after tbe given equation is divided by its 
 greatefl common divifor, the co-efficients of the 
 unknown quantities, are commensurable to each 
 ether, the qusftion is impojfible. 
 
 EXAM. 
 
( 379 5 
 
 EXAMPLES. 
 
 Given, 109 87 36 ~ o, to find v and y in 
 whole numbers. 
 
 Fir ft, 51; 4y 1 8 = o, ^j dividing the whole by 
 t; or, $v 4-y 18. 
 
 Put W N for any whole number: 
 
 4-V 
 
 * T -*- r- 7 /%* 
 
 ^c? queftion : 
 
 iZ- . 3 + 47 . ,v 3 + 47 
 
 " 3 -j -- > therefore, 
 
 = W ' N,per axiom 9. Alfo> ^- = /^ TV: Confequsnt- 
 
 , $y 3 + 4y __y3l 
 
 fy, --- - -- c~~"^* ^ 3 ferawom 9; 
 
 jy o 
 
 therefore, - rr ; and for the leaft value of 
 
 , ajfume n O, ^;/^/ we fljall havey 3 5^ zz O j 
 
 . 
 = 3, rf^r v n - n 6. 
 
 Given, 26*1; -f- i8y 31140, to find v andj in whole 
 aumbers. 
 
 rfty 13^ -4- 97 = 70 ^y dividing the whole ly 2 : 
 
 j 
 
 ? ~ ny 
 Therefore, - - - ~ W N, per axiom 9 ; #//>, 
 
 5 + 
 _ 
 

 i c -f- 1 1 y 
 X , 7. But, * = 
 
 O 
 
 2 4- i2y 
 therefore, - - TF TV", p er ax. 9. 
 
 -^ , Ar 
 
 whence, -- =: /F7V; 
 
 y 2 
 
 9 : ^^/ = n ; or, y z: i j ^- 2 ; and 
 \j 
 
 a/fuming n zz o, w^ have y ir 2, 
 
 o 
 
 4- 
 
 I OWE my friend a moidore, have nothing about 
 me but crowns, and he has nothing but guineas : How 
 mud we exchange thefe pieces of money, fo that I 
 may acquit myfelf of the debt ? A moidore being 
 valued at 27 fhillings, a crown at 5 Shillings, and a 
 guinea at 21 fhillings. 
 
 Put x ~ number of crowns, andy the number o 
 yeas : 
 
 5^ 2iy zz 27 by the queftion : 
 
 
 5 + 4X + ^-^ ' ^JequentJy, 12 _ 
 2 4-jy 
 
 zz # > or, 2 + y zz 577 ; ^^ affuming n~ 
 
 27 - 
 
 kavey zz 3, //&^ number of guineas, and x~ - 
 
 zr 1 8, /j&tf number of crowns. Therefore, I mujt gros 
 my friend 18 crowns, and he muft give me three gui 
 neas* 
 
 Given, 
 
Given, 4#-f* 17^:1:2900, to find all the poflible 
 values of x and^y in whole numbers. 
 
 Firlt,y - 29 ~-f = W N ; but 
 17 
 
 10 AX 40 1 6x 
 
 . 8. 
 
 iy 
 ax. 9. Alfo* - ^WN\ Confequently, -- -j. 
 
 i therefore, - 
 
 a/ummg this lafl equation =i ff, ^<? gef x =: 17% 6, 
 where > ifnbe taken = i, wejhallhavex 17 6 = 
 
 2900 4* 
 1 1 /0r if/&^ fcj^f ^^/^^ <?/ ^, ^ Jf - " - =z 
 
 1 68 /?r /^^ greateft value of y : Andfince 6 -J- oc^. iy 
 
 z: , /j ^ wi^/f number ; // isflafa, that n -\- 1 /j /^ 
 
 /r^ augment of 6 + x-r-i'l in whole numbers-, and 
 
 therefore, x = 17^ + 17 6, tbefatnd value of x ; 
 
 2QOO A^ 
 
 which fubftituted for x in the equation y ~ - : - 
 
 will give the fecond value ofy : Or, by adding 17 Juc-+ 
 cejfively to the values ofx, and Jubtrafting ^from thofs 
 0fy, we Jhall have all the fojJibU valuss of x and y in 
 whole numbers^ as follows : viz. x 1 1, 28, 45, 
 to 708 j andy zz 168, 164, 160, &fr. /<? 4. 
 
 PROBLEM II- 
 
 To find the leafl whole number AT, /^^/ being divided 
 by the given numbers, a>b, c, d> &c. Jhall leave given, 
 remainders, g, k> I, m> n, G?V t 
 
 C c c RULE, 
 
( 3** ) 
 
 RULE. 
 
 1. SUBTRACT each of the remainders from #, and 
 divide the feveral refults by their refpe&ive divifors, 
 a y b y c, d> &c. and the refuhing quotients will equal 
 whole numbers. 
 
 2. ASSUME the firfl equation equal b, and find the 
 value of x in terms of h. 
 
 3. SUBSTITUTE the value of x in terms of, in the 
 fecond equation j and proceed with the refult as in 
 the lad problem, by expunging all whole numbers, 
 until the co-efficient of h becomes unity, &c. 
 
 4. PUT this expreflion equal />, and find the value 
 of* in terms of/>, by means of the equation of b. 
 
 5. SUBSTITUTE the value of x in terms of/>, in the 
 third equation, with which proceed as before, and fo 
 on, through all tht given equations ; affuming the 
 final refult equal to fome known whole number, and 
 finding the values of the feveral fubflituted letters, , 
 />, &c. from which the value of A; may be. determin 
 ed in known terms. 
 
 EXAMPLES. 
 
 To find the. leaft whole number, that bemg di 
 vided by 7 fhall leave 6 remainder j but being divid 
 ed by 6 fhall leave 4 remainder. 
 
 Put v z: number Jought. 
 
 i) 6 
 djfvnu - ~ b) and we Jh all have v yb + 
 
 which Jubftifuttd for v in thejeccnd equation, gives 
 
( 38,1 ) 
 
 6b 
 But, ~g~ WN: Confequently, 
 
 6h + 2 b+z 
 
 _- _ - w N) md ajfuming ~ 
 
 zr 77, we Jhall have h~ 6n 2 ; wfor* ;/# fo /tfte 
 = i, ^^ ^^77 ^i;<? b ~ 4, ^df y iz 7^ +' 6 z: 34, 
 the number required. 
 
 To find the leaft whole number, that being divid 
 ed by 1 8, fhill leave 14 remainder - y but being di 
 vided by 28, fhall leave 20 remainder. 
 
 Put v ~ number fought. 
 
 v 14. v 20 
 
 Men, ^ =PTN: And, ^ =: W N. 
 
 v \^ 
 A[jume y zi b - 3 and we have v~ 18^4-14, 
 
 which fubftituted for v in the fecond equation y gives 
 
 6 ah i 
 
 -WN-, or, ' l WNby dividing all 
 
 . 
 
 the terms ly 2 ; and 
 
 - X 2 =: = W N. Consequently , ~ -- 
 
 9 h -f- Q ^ "4* 9 
 
 ; - ^ - == ^ jy . ^^^ affuminv - ~ > 
 14 14 14 
 
 w* have h rr i4- 9; /?^ putting n ~ i, 
 ^ zz 14^ 9 5, ^^^/ i; ~ 1 8^ 4- 14= 
 number required. 
 
 Diophantine Problems. 
 
 DIOPHANTINE Problems, fo called, from Diophan* 
 tus their inventor, are fuch as relate to the finding of 
 Pquare and cube numbers, &c. 
 
 THESS 
 
THESE problems are fo exceedingly curious, that 
 Nothing lefs than the moft refined Algebia, applied 
 with the utmoft fkill and judgment, could ever fjr- 
 mount the difficulties which neceffarily attend their 
 folution. The peculiar artince made ufe of in form 
 ing fuch pofitions as ihe nature of the problems re 
 quire, fhews the great ufe of Algebra, or the analy 
 tic arr, in difcovering thofe things that otherwile, 
 would be without the reach of human uoderftandingo 
 
 ALTHO no general rule can be given for the foi-j- 
 tipnofthefe problems; yet the following direction 
 will be very ferviceable on many occations. 
 
 DIRECTION. 
 
 ASSUME one or more letters, for the root of the 
 required fquare, cube, &"c. fuch that when involved 
 tu t./e height of the propofed power, either the given 
 nu nber, or the highelt term of the unknown quan 
 tity mav vanifh. Then if the unknown quantity in 
 the reful tin.-> equation, be of fimple dimenfion, find 
 its value by reducing the equation. But if the un 
 known quantity be ilill a fquare, cube, or other pow 
 er ; affui-ne or her letter or letters, with which pro 
 ceed as before, until the higheft term of the unknown 
 quantity become qf fimple dimenfion in the equa 
 tion. 
 
 EXAMPLES. 
 
 To find a fquare number #% fuch that A?* + i lhall 
 be a fquare number. 
 
 Affume x 2 for the root ofx z -\-i : 
 Then will x 2!* n x* + i ; that is, x* 4* + 4 
 ~ x'* + i ; or, 4* n 4 1~3 i whence ,x r:^, an& 
 
 a 5 . 
 
 w 
 
 9 
 TT 
 
is tie numler required. But if we had aJJumeX 
 
 ar 2 4- i 
 
 /<?r A;'-, wefhould kavehad 
 4r* 
 
 1 
 
 , . 7 . ... 
 + i = - 1 -- * ^wttB /j evidently a fquare 
 
 number ; w*r* r *0y be taken for any number. 
 
 TO 6nd two nu/noers, fuch chat chur product and 
 quotient may be both fquare and cube numbers* 
 
 Ajjume v 9 and v* for the required numbers : 
 
 Then v 9 x ^ 3 ' yl *> ^^ i> 9 -f-i; 3 =^ 6 , ^r<? m- 
 J<?w//> fquare and cube numbers ; where v may be any 
 number taken at plea fare. 
 
 To find four fquare numbers in arithmetical pro- 
 greffion. 
 
 For the fum of the two extremes, ajjiime in* 5 then 
 will thefum of the two means be aijo 2 a by the nature 
 of the proportion : 
 
 For the roots of the two means 9 ajfume n -f 3%, and 
 n 42 : _ _ 
 
 fben will n -f 3zj z + n 42!' = 2/2* : 
 
 That is, n* + 6nz 4- 9^ -h x - *nx + i6z*=i 
 
 - 2WZ -4- 272* = 2W* t 
 
 Or, 25 2 x = 2% ; and by dividing by z, we have 252 
 
 Whence, z~ in~i$ ; andputingn i, we have 
 
 , 6 lit 
 
 And for the roots of the two extremes, ajfums n 
 n 4- z : 
 
 Then will n~^- . 
 Or, w z 
 
 And by reduction, z '~ 2-j-j ~ T ; 
 
Whence, n - -22)% and n 4- zj* zr ~ 
 'two extremes. So that the four fquare numbers in arith 
 metical progrejfion, are ^ T> ^44, |4|, *|. 
 
 To find a number, fuch that being multiplied with 
 one tenth part ofitfelf, and the product increafed by 
 36, fhall produce a fquare number. f 
 
 jP/ 1; for the number fought ; /to v* -MO + 36, *V 
 
 /<? <? 6i fquare number : 
 
 AJTuyne'the root of this fquare ~ v 6, then will 
 
 *v " -' ( | rz: ^"-f-io -f- 36 ; /^7/ /j, 'y 1 i2i;-}-36i^ 
 ^^^- I0 f 36 : 
 
 Or, 1 01; z i zov zr i; 2 j tobence, by redutJion v n 
 '5-% ^ number required. 
 
 To divide a given number 29, confiding of two 
 known fquare numbers 4 and 25, into two other 
 fquare numbers. 
 
 For tbs rcot cf the frU Jquare y aJJ'ume rv 25 and 
 for the root of thefeccnd nv 5 : 
 
 will rv ^ 4- KV J( a n 29 : 
 
 'v* 4^1; 4-4-4- *v a iov 4-25 
 
 Or, r 2 H- a< ^* 4r IO/VT; 4- 29= 295 or, 
 r* 4- /2 2< y* zr 4r4- ioi; ; ^^ ^y dividing by v, we 
 bave r 2 - 4- 2 i; zr 4^ 4- iotf : 
 4r 4. 1 6 
 
 Or, *y ir TT~T" s . ^^^ therefore, rv 2 zz 
 
 2 n - i - a - : and 
 r* 4- a 
 
 cr a 4- c# a 
 
 ^ n 2. wejhall have 
 
 
( 38? ) 
 
 14 
 
 for the root ofthefirfifquare, and 
 
 5 * i * 
 
 zi y /<?r J&* r<?0/ <?/ thefecond. 
 
 To find three iquare numbers in arithmetical pro* 
 grefiion. 
 
 AJfume n* for the mean -, then will in* ~ thefum 
 of the extremes by the nature of the proportion. 
 
 For the root of the greater extreme, ajjume n -j- av, 
 and for the root of the lefs n 3-1; : 
 
 Then will w 3^]*- + + 2-yl* zz 2* : 
 
 FbattSy n* 6 < z; + 91;" + M* + 4^ -f 4^* 2 a : 
 
 Or, 131;* = 2v j ^^^ 4x dividing by v, we have 
 13^ n 2^ 5 whence, v = in~-\2> where n may be 
 any number at pleajure : 
 
 ^fw^ 4x ajjuming n zi i, we Jhall have v zr 2-^-13 : 
 
 dt* 40- 
 
 Whence, i zr r 1 - Tor w ^^/* extreme : 
 *3| 109 
 
 And i + ~ >r the greater : Wherefore, 
 
 49 289 
 
 the numbers required^ are -^*> i, 
 
 THE END OF VOLUME I. 
 
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