The Method of Fluxions and Infinite Series : with Its Application to The

THE METHOD of FLUXIONS AND INFINITE SERIES; WITH ITS of Application to the Geometry CURVE-LINES.

By the INVENTOR Sir I S A A C NEWTON,^ Late Prefident of the Royal Society.

^ranjlated from the AUTHOR'* LATIN ORIGINAL

not yet made publick.

To which is fubjoin'd, A PERPETUAL COMMENT upon the whole Work,

Confiding of ILLU ANN OTATIONS, STRATION s, and SUPPLEMENTS,

In order to make this Treatife the Acomplcat Inftitution for ufe o/' LEARNERS.

By JOHN CO L SON, M. A. andF.R.S. Mafter of Sir free Jofeph fFilliamfon's Mathematical-School at Rochejter. LONDON: Printed by HENRY WOODFALLJ

And Sold at the by JOHN NOURSE, Lamb without Temple-Bar. M.DCC.XXXVI. ' T O

William Jones Efq; F.R S. SIR, [T was a laudable cuftom among the ancient Geometers, and very worthy to be imitated by their SuccefTors, to addrefs their Mathematical labours, not fo much to Men of eminent rank and {ration in the world, as to Perfons of diftinguidi'd in fame Studies. For knew merit and proficience the they fuch could be of very well, that only competent Judges their Works, and would receive them with ''the efteem. So far at leaft I can after thofe they might deferve. copy chufe a Patron for thefe great Originals, as to Speculations, whofe known skill and abilities in fuch matters will enable candor will incline him him to judge, and whofe known of the fhare I have had in the to judge favourably, prefent as to the fundamental of the performance. For part Work, of which I am only the Interpreter, I know it it will need no nor cannot but pleafe you ; protection, to bear ean it receive a greater recommendation, than the name of its illuftrious Author. However, it very naturally itfelf to who had the honour I am fure applies you, (for think it of the Author's and you fo) friendship familiarity had his own confent to nil in his life-time ; who publifli of of his of a nature not elegant edition fome pieces, very this and who have fo an efteem different from ; juft for, as well as knowledge of, his other moft fublime, moil admirable, andjuftly celebrated Works. A 2 But iv DEDICATION. \ But befides thefe motives of a publick nature, I had others that more nearly concern myfelf. The many per- fonal obligations I have received from you, and your ge- nerous manner of conferring them, require all the tefti- of in monies gratitude my power. Among the reft, give me leave to mention one, (tho' it be a privilege I have enjoy 'd in common with many others, who have the hap- of which the free pinefs your acquaintance,) is, accefs you have always allow'd me, to -your copious Collection of whatever is choice and excellent in the Mathernaticks. Your judgment and induftry, .in collecting -thofe. valuable ?tg{^tfcu., are not more conspicuous, than the freedom and readinefs with which you communicate them, to all fuch who you know will apply them to their proper ufe, to the of Science. that is, general improvement Before I take my leave, permit me, good Sir, to join my wiOies to thofe of the publick, that your own ufeful Lu- cubrations fee the with all convenie-nt may light, ipeed ; which, if I rightly conceive of them, will be an excellent methodical Introduction, not only to the mathematical Sciences in general, but alfo to thefe, as well as to the other curious and abftrufe Speculations of our great Author. You all other are very well apprized, as good Judges muft be, that to illuftrate him is to cultivate real Science, and to make his Difcoveries eafy and familiar, will be no fmall improvement in Mathernaticks and Philofophy. That you will receive this addrefs with your ufual candor, and with that favour and friendship I have fo long ind often experienced, is the earneil requeft of, S I R, Your moft obedient humble Servant^

J. C OLSON. (*)

THE PREFACE.

Cannot but very much congratulate with my Mathe- matical Readers, and think it one of the moft for- tunate ciicumftances of my Life, that I have it in my power to prefent the publick with a moft valuable Anecdote, of the greatefl Ma fter in Mathematical and Philofophical Knowledge, that ever appear 'd in the World. And fo much the more, becaufe this Anecdote is of an element ry nature, to fubh'me preparatory and introductory his other moft arduous and for the instruction of Novices Speculations, and intended by himfelf and Learners. I therefore gladly embraced the opportunity that was put into my hands, of publishing this pofthumous Work, becaufe I found it had been compofed with that view and defign. And that my own Country-men might firft enjoy the benefit of I refolved it in an this publication, upon giving Englijh Translation, with fome additional Remarks of my own. I thought it highly and of injurious to the memory reputation the great Author, as well as invidious to the glory of our own Nation, that fo curious and uleful a piece fhould be any longer fupprels'd, and confined to a few private hands, which ought to be communicated to all the learned World for general Inftruction. And more efpecially at a time when the Principles of the Method here taught have been fcrupuloufly fifted and examin'd, have been vigorouily .oppofed and as fome (we may fay) ignominioufly rejected infufficient, by Mathe- matical Gentlemen, who feem not to have derived their knowledge true that cur of them from their only Source, is, from Author's own Treatife wrote exprefsly to explain them. And on the other hand, the Principles of this Method have been zealouily and com- mendably defended by other Mathematical Gentlemen, who yet a feem x lie PREFACE. fern been as little to have acquainted with this Work, (or at leaft to have over-look'd the it,) only genuine and original Fountain of this kind of knowledge. For what has been elfewhere deliver'd by our Author, concerning this Method, was only accidental and oc- and far calional, from that copioufnefs with which he treats of it and illuftrates it here, with a great variety of choice Examples. learned The and ingenious Dr. Pemberton, as he acquaints us in his View of Sir Newton's Tfaac Philofophy, had once a defign of this with the publishing Work, confent" and under the infpectkm of the Author himfelf; which if he had then accomplim'd, he would certainly have deferved and received the thanks of all lovers of Science, The Work would have then appear'd with a double advantage, as the la ft receiving Emendations of its great Author, and likewife in faffing through the hands of fo able an Editor. And among the other effects of this it good publication, poffibly might have prevented all or a of thofe have great part Difputes, which fince been raifed, and which have been fo ftrenuoufly and warmly pnrfued on both the of this fides, concerning validity the Principles of Method. They would doubtlefs in fo a have been placed good light, as would have cleared them from any imputation of being in any wife defective, or not fufficiently demonstrated. But fince the Author's Death, as the Doctor informs us, prevented the execution of that defign, and fince he has not thought fit to refume it hitherto, it became needful that this publication fhould be undertook by another, tho' a much inferior hand. For it was now become highly necefTary, that at laft the great Sir Ijaac himfelf fhould interpofe, fhould produce his genuine Me- thod of it Fluxions, and bring to the teft of all impartial and con- fiderate Mathematicians to its evidence and ; mew Simplicity, to maintain and defend it in his own way, to convince his Opponents, and to teach his Followers Difciples and upon what grounds they mould proceed in vindication of the Truth and Himfelf. And that this be done might the more eafily and readily, I refolved to accom- it with an pany ample Commentary, according to the beft of my and fkill, (I believe) according to the mind and intention of the Au- wherever I it needful and with an thor, thought ; particularly Eye to the fore-mention'd Controverfy. In which I have endeavoui'd to obviate the difficulties that have been to raifed, and explain every in fo full a thing manner, as to remove all the objections of any force, that have been any where made, at leaft fuch as have occtu'd to my obfervation. If what is here advanced, as there is good reafon PREFACE. xi

fon to hope, fhall prove to the fatisfadtion of thofe Gentlemen, who ikfl darted thefe objections, and who (I am willing to fuppofe) had at I fhall be to only the caufe of Truth heart; very glad have con- tributed any thing, towards the removing of their Scruples. But if is here offer'd it fhall happen otherwife, and what fhould not appear and to to be furricient evidence, conviction, demonflration them ; it will be fuch to moil other yet I am perfuaded thinking Readers, who fhall apply themfelves to it with unprejudiced and impartial minds; and then I mall not think my labour ill beflow'd. It fhould however be well confider'd by thofe Gentlemen, that the great number of Examples they will find here, to which the Method of Fluxions the is fuccefsfuUy apply'd, are fo many vouchers for truth of the on that Method is founded. For the Deductions Principles, which are always conformable to what has been derived from other uncon- troverted Principles, and therefore mufl be acknowledg'd us true. This argument mould have its due weight, even with fuch as cannot, as well as with fuch as will not, enter into the proof of the the that has been advanced Principles themfelves. And hypothefn to evade this conclufion, of one error in reafoning being ilill corrected and to and that fo by another equal contrary it, regularly, conftantly, as it mufl be to do here this I and frequently, fiippos'd ; bvpothe/is, not to be becaufe I can think it fay, ought ferioufly refuted, hardly is ferioufly propofed. the is The chief Principle, upon which Method of Fluxions here taken from the Rational built, is this very fimple one, Mechanicks ; which is, That Mathematical Quantity, particularly Extenlion, may be conceived as generated by continued local Motion; and that all Quan- tities whatever, at leaflby analogy and accommodation, may be conceived as generated after a like manner. Confequently there mufl be fuch comparativeVelocitiesofincreafeanddecreafe, during generations, whole Relations are fixt and determinable, and may therefore /pro- blematically) be propofed to be found. This Problem our Author of another not lefs evident here folves by the hjip Principle, ; which is or that it fuppofes that Qnimity infinitely divifible, may (men- fo far as at before it tally at leaft) continually diminifh, lafl, is totally arrive at that be call'd extinguifh'd, to Quantities may vanilhing whk.li are and lefs than Quantities, or infinitely little, any afTign- able Quantity. Or it funnolcs that we may form a Notion, not relative and indeed of abioiute, but of comparative infinity. 'Tis a to the Method of as aifo to the very jufl exception Indivifibles, that have at once to foreign infiniteiimal Method, they rccourfe a 2 infinitely The PREFACE.

little and infinite infinitely Quantities, orders and gradations of thefe, but not relatively absolutely fuch. They affume thefe Quantities finnd & Jewel, without any ceremony, as Quantities that actually and and make with them obvioufly exift, Computations accordingly ; refult of as tlie which muft needs be precarious, as the abfblute exiftence of the Quantities they afiume. And fome late Geometricians 'd thefe have carry Speculations, about real and abfolute Infinity, ftill have much farther, and raifed imaginary Syftems of infinitely great and little and their feveral orders and infinitely Quantities, properties j which, to all fober Inquirers into mathematical Truths, muft certainly appear very notional and vifionary. Thefe will be the inconveniencies that will arife, if we do not rightly diftinguifh between abfolute and relative Infinity. Abfolute as can be Infinity, fuch, hardly the object either of our Conceptions or Calculations, but relative Infinity may, under a proper regulation. Our Author obferves this diftinction very ftrictly, and introduces but little that are fo none infinitely Quantities relatively ; which he arrives at by beginning with finite Quantities, and proceeding by a gradual and neceffary progrefs of diminution. His Computations finite always commence by and intelligible Quantities ; and then at laft he inquires what will be the refult in certain circumftances, when fuch or fuch Quantities are diminim'd in infinitum. This is a conftant practice even in common Algebra and Geometry, and is no more than defcending from a general Propofition, to a particular Cafe is in it. thefe which certainly included And from eafy Principles, managed with a vaft deal of fkill and fagacity, he deduces his Me- if fo thod of Fluxions j which we confider only far as he himfelf with the he has made of has carry'd it, together application it, either or or here or elfewhere, directly indiredly, exprefly tacitely, to the moft curious Difcoveries in Art and Nature, and to the fublimeft Theories : We may defervedly efteem it as the greateft Work of Genius, and as the nobleft Effort that ever was made by the Hun an Mind. Indeed it muft be own'd, that many uftful Improvement?, have been fince and new Applications, made by others, and proba- bly will be ftill made every day. For it is no mean excellence of this Method, that it is doubtlefs ftill capable of a greater degree of and will afford an inexhauftible fund of perfection ; always curious matter, to reward the pains of the ingenious and iuduftrious Analyft. defirous to make this as as As I am fatisfactory poffible, efptcially to the very learned and ingenious Author of the Difcourle call'd The Analyjl, whofe eminent Talents I acknowledge myfelf to have a J great The PREFACE. xlii

veneration for I fhall here endeavour to obviate fome of his great ; to the Method of fuch principal Objections Fluxions, particularly as I have not touch'd upon in my Comment, which is foon to follow. He thinks cur Author has not proceeded in a demonftrative and fcientifical matter, in his Princip. lib. 2. km. 2. where he deduces the Moment of a Rectangle, whole Sides are fuppofed to be variable I fhall the matter Lines. reprefent Analytically thus, agreeably (I think) to the mind of the Author. Let X and Y be two variable Lines, or Quantities, which at different periods of time acquire different values, by flowing or increa- or fing continually, either equably alike inequably. For inflance, let there be three periods of time, at which X becomes A fa, A, a and becomes B and B b and A -+- 7 ; Y f3, B, -+- f fuccefiively where are that be and reflectively ; A, a, B, b, any quantities may aiTumed at pleafure. Then at the fame periods of time the variable Produ

A +- f* x B -+- h, that is, AB T therefore by Sub- traction the whole Increment during that interval of time will be tfB-4-M. QÊ. D. This may eafily be illuftrated by Numbers thus: Make A,rf,B,/, to i other to be af- equal 9, 4, 5, 6, refpeclively; (or any Numbers the be fumed at pleafure.) Then three fucceffive values of X will the three fucceffive values 7, 9, ii, and of Y will be 12, 15, 18, reipcciivcly. xiv The PREFACE. the refpeftively. Alfo three fucceflive values of the Produd XY will be 84, 135, 198. But rtB-f-M = 4xic-f- 6x9= 114 = 19 8_84 . Q.E. O. Thus the Lemma will be true of any conceivable finite Incre- and therefore ments whatever; by way of Corollary, it will be true of infinitely little Increments, which are call'd Moments, and which the the was thing Author principally intended here to demonflrate. 15ut in the cafe of Moments it is to be confider'd, that X, or defi- A and are to nitely ftf, A, A -+- a, be taken indifferently for the fame as alfo Quantity ; Y, and definitely B f/;, B, B -+- ~b. And the want of this Confutation has occafion'd not a few perplexities. Now from hence the reft of our Author's Conclufions, in the fame Lemma, may be thus derived fomething more explicitely. The Moment of the Reclangle AB being found to be Ab -+- ^B, when the contemporary Moments of A and B are reprelented by a and b refpedtively ; make B = A, and therefore b = a, and then the Moment of A x A, .or A*, will be Aa -+- aA, or 2aA. Again, a and therefore b-=. make B = A , zaA, and then the Moment of or will be 2rfA 4 1 or AxA*, A', -f- aA , 3Â*. Again, make B = s 5 and therefore l> and then the of A , = âA -, Moment A xA*, or 4 3 3 3 A will be or . make , 3

And if A" or B" where univerfally, we put =B, A"=. , m and n be affirmative or may any integer Numbers, negative ; then we have A"-* mgA< - mall ma = ;.^B"^' , or b= = aA i, which

is the Moment of B, or of A" . So that the Moment of A" will be The P E E F A C E. xv

1 whether ;;/ be affirmative or or be rtill wtfA"*" , negative, integer fraction. The Moment of AB being M -+- aB, and the Moment of CD B"-'A" -f- maA. B . And fo of any others. Now there is fo near a connexion between the Method of Mo- ments and the Method of Fluxions, that it will be very eafy to pafs from the one to the other. For the Fluxions or Velocities of increafe, are always proportional to the contemporary Moments. Thus if for A, B, C, &c. we write x, y, z, &c. for a, b, c, &c. we may the Fluxion of will be the write x, y, z, &c. Then xy xy -f- xy, m be rnxx*-* whether be or Fluxion of x will , m integer fraction, affiimative or negative; the Fluxion of xyz will be xyz -f- xyz -f- m n m m s xjz, and the Fluxion of x y will be mxx -*y -J- nx yy"~ . And fo of the reft. in Or the former Inquiry may be placed another view, thus : Let A and A-f- a be two fucceflive values of the variable Quantity X, as alfo B and B -+- b be two fucceflive and contemporary values be fucceflive of Y ; then will AB and AB -f- aB-\~ bA+ab two and contemporary values of the variable Product XY. And while X, by increafing perpetually, flows from its value A to A -f- a, or Y flows from B to B -f- b ; XY at the fame time will flow from AB to AB +- aB -+- bA. -f- abt during which time its whole Increment, as appears by Subtraction, will become aB -h bh. -+- ab. Or in thus: Let be to Numbers A, a, B, b, equal 7, 4, 12, 6, refpectively ; fucceflive values of be 1 1 and the then will the two X 7, , two fucceflive values of Y will be 12, 18. Alib the two fucceflive values of the Product XY will be 84, 198. But the Increment aB -+- t>A -J- ah- 48 -f- 42 -+- 24= 1 14= 198 84, as before. And thus it will be as to all finite Increments : But when the In- crements become Moments, that is, when a and b are fo far dirni- lefs at nifh'd, as to become infinitely than A and B ; the fame time ab will become infinitely lefs than either aB or ^A, (for aB. ab :: ab :: A. a and therefore it will vanifh in B. b, and bA. y ) refpect of them. In which cafe the Moment of the Product or Rectangle will be aB -+- bA, as before. This perhaps is the more obvious and direct of in the t relent as there was way proceeding, Inquiry ; but, room for choice, our Author thought fit to chufe the former way,, as xvi The PREFACE. as the more elegant, and in which he was under no neceflity of hav- that ing recourfe to Principle, that quantities arifing in an Equation, which are infinitely lefs than the others, may be neglected or expunged in companion of thofe others. Now to avoid the ufe of this Principle, tho' otherwife a true one, was all the Artifice ufed on this occaiion, which certainly was a very fair and justifiable one. I fhall conclude my Obfervations with confidering and obviating the Objections that have been made, to the ufual Method of finding the Increment, Moment, or Fluxion of any indefinite power x of the variable quantity x, by giving that Inveftigation in fuch a man- as to leave no for it. ner, (I think) room any juft exceptions to And the rather becaufe this is a leading point, and has been ftrangely perverted and mifreprefented. In order to find the Increment of the variable quantity or power rather its relation to the Increment of x confider'd as x, (or } given ; becaufe Increments and Moments can be known only by comparifon with other Increments and Moments, as alfo Fluxions by comparifon Fluxions let us make and let and be with other ;) x"=y, X Y any fynchronous Augments of x and y. Then by the hypothefis we fhall have the for in Equation x-fc-X\* =y -+- Y ; any Equation the variable Quantities may always be increafed by their fynchronous Augments, and yet the Equation will flill hold good. Then by our Author's famous Binomial Theorejn we fhall have y -f- Y = xn * nx"~'X n n *~ 3 &c. or re - -+- -+- x ^=-^*X + x x '-^-V^X ,

n~ l moving the equal Quantities y and x", it will be Y = nx X +- * ny. n x x ^^x'-'^X 3 &c. So that when X deT ^-x"--X -+- ?-^- , notes the of the variable Y will here given Increment quantity A,-, denote Increment of indefinite or x" the fynchronous the power y ; whofe value therefore, in all cafes, may be had from this Series. Now that we may be fure we proceed regularly, we will verify this thus far, by a particular .and familiar instance or two. Suppofe n = 2, l . flows or increafes then Y = 2xX -+- X That is, while x to x +- X, 1 .v* in the fame its Increment 2xX will increafe time, by Y = -+-X , 1 1 to .v 2xX which we otherwife know to be true. 4- -j- X , Again, 1 a 3 fuppofe fl = 3, then Y = 3* X -+- 3*X H- X . Or while x in*. a J creafes to x r+- X, x"> by its Increment Y = 3^ X -h 3^X + X 3 1 1 3 will increafe to x* -f- 3* X -+- ^xX -+- X . And fo in all ,other particular cafes, whereby we may plainly perceive, that this general Conclufion mud be certain and indubitable. This Tie PREFACE. xvii

This Series therefore will be always true, let the Augments X and fo little for the truth docs not at all de- Y be ever fo great, or ever ; of their when are pend on the circumftance magnitude. Nay, they or become it muft be true infinitely little, when they Moments, alfo, Conclufion. But when X and Y are di- by virtue of the general at laft the minifh'd in infinitum, fo as to become infinitely little, of muft needs vanifli as of an greater powers X firft, being relatively vali e than the fmaller So that when are infinitely lefs powers. they obtain the Y=znx*~'X all expunged, we ihall neceflarily Equation ; confe- where the remaining Terms are likewife infinitely little, and there were other Terms in the quently would vanifh, if Equation, than themfelves. But as which were (relatively) infinitely greater as an .there are not, we may fecurely retain this Equation, having and as it us anufeful undoubted right fo to do; efpecially gives piece of information, that X and Y, tho' themfelves infinitely little, or vanifli in to each other as vanifhing quantities, yet they proportion f learn 'd at that the Moment j to nx"~ . We have therefore laft, by which x increafes, or X, is to the contemporary Moment by which s a . their or Velo- x increafes, or Y, as i is to nx"~ And Fluxions, cities of increafe, being in the fame proportion as their fynchronous Moments, we fhall have nx*-'x for the Fluxion of X", when the Fluxion of x is denoted by x. I cannot conceive there can be any pretence to infinuate here, or of that any unfair artifices, any leger-de-main tricks, any Ihifting that have been fo are at all the hypothefis, feverely complain'd of, made ufe of in this Inveftigation. We have legitimately derived in finite that in all cafes the re- this general Conclufion Quantities,

l 1 lation of the Increments will be Y = nx"~ X + x ~~x*'- X*, &c. cafe and are conti- of which one particular is, when X Y fuppofed terminate in But nually to decreafe, till they finally nothing. by nearer and nearer to the thus continually decreafing, they approach inftant of Ratio of i to nx"~\ which they attain to at ihe very the'r is their ultimate vanifhing, and not before. This therefore Ratio, the Ratio of their Moments, Fluxions, or Velocities, by which x and xn continually increafe or decreafe. Now to argue from a to a cafe contain'd under is general Theorem particular it, certainly as the ufual tine of the moft legitimate and logical, well as one of mofl and ufeful ways of arguing, in the whole compafs of the Mathemc- to ftand ticks. To object here, that after we have made X and Y at to or no for fome quantity, we are not liberty make them nothing, is not an the quantity, or vanishing quantities, Objection againft b Method XVlll Tte PREFACE.

Method of Fluxions, but againft the common Analyticks. This this of Method only adopts way arguing, as a conftant practice in the vulgar Algebra, and refers us thither for the proof of it. If we an of have Equation any how compos'd the general Numbers a, b, c, &c. it has been that we always taught, may interpret thefe by any particular Numbers at pleafure, or even by o, provided that the or the Conditions Equation, of the Queftion, do not exprefsly require the contrary. For general Numbers, as fuch, may ftand for definite Numbers in the whole any Numerical Scale ; which Scale be thus (I think) may commodioufly reprefented, &c. 3, 2 > i, o, i, 2, 3,4, &c. where all poffible fractional Numbers, inter- to thefe here are to be mediate exprefs'd, conceived as interpolated. But in this Scale the Term o is as much a Term or Number as any other, and has its analogous properties in common with the refK likewife We are told, that we may not give fuch values to general as could not receive at firft Symbols afterwards, they ; which if admitted I to the is, think, nothing prefent purpofe. It is always moft eafy and natural, as well as moll regular, inftruclive, and ele- to make our as in gant, Inquiries much general Terms as may be, and to defcend to particular cafes by degrees, when the Problem is nearly brought to a conclufion. But this is a point of convenience a only, and not point of neceffity. Thus in the prefent cafe, inflead of defcending from finite Increments to infinitely little Mo- ments, or vanifhing Quantities, we might begin our Computation with thofe Moments themfelves, and yet we mould arrive at the fame Conclufions. As a proof of which we may confult our Au- thor's ownDemonftration of hisMethod, in of this Treatife. x oag. 24. is the In fhort, to require this jufl fame thing as to infift, that a which to Problem, naturally belongs Algebra, mould be folved by Arithmetick which tho' to be common ; poflible done, by purluing all the of the backwards fleps general procefs, yet would be very and and not fo troubkfome operofe, inflrudtive, or according to the true Rules of Art But I to that all our am apt fufpedr, doubts and fcruples about Inferences Mathematical and Argumentations, especially when we are fatisfied that have been and they juftly legitimately conducted, may be refolved into ultimately a fpecies of infidelity and diftruft. Not in refpecl of any implicite faith we ought to repofe on meer human authority, tho' ever fo great, (for that, in Mathematicks, we mould but in utterly difclaim,) refpedl of the Science itfelf. We are hardly to brought believe, that the Science is fo perfectly regular and uniform, 72* PREFACE. xix

form, fo infinitely confident, conftant, and accurate, as we mall re&lly find it to be, when after long experience and reflexion we (hall have overcome this prejudice, and {hall learn to purfue it rightly. We or do not readily admit, eafily comprehend, that Quantities have an infinite number of curious and fubtile properties, fome near and obvious, others remote and abftrufe, which are all link'd together by a neceffary connexion, or by a perpetual chain, and are then only and and difcoverable when regularly clofely purfued ; require our in as well as . truft and confidence the Science, our induftry, application, and obftinate perfeverance, our fagacity and penetration, in order to their being brought into full light. That Nature is ever confiftent with herfelf, and never proceeds in thefe Speculations per faltum, or at random, but is infinitely fcrupulous and felicitous, as in and whenever we may fay, adhering to Rule Analogy. That we make any regular Portions, and purfue them through ever fo great of to the ftricT: Rules of a variety Operations, according Art ; we fhall always proceed through a feries of regular and well- connected tranlmutations, (if we would but attend to 'em,) till at laft we arrive at regular and juft Conclufions. That no properties of Quantity are intirely deftructible, or are totally loft and abolim'd, even tho' to itfelf for if fome be- profecuted infinity j we fuppofe Quantities to come infinitely great, or infinitely little, or nothing, or lefs than nothing, yet other Quantities that have a certain relation to them will only undergo proportional, and often finite alterations, will fym- with and conform to 'em in all their pathize them, changes ; and will always preferve their analogical nature, form, or magnitude, be which will faithfully exhibited and difcover'd by the refult. This we may colledl from a great variety of Mathematical Speculations, and more particularly when we adapt Geometry to Analyticks, and Curve-lines to Algebraical Equations. That when we purfue general Inquiries, Nature is infinitely prolifick in particulars that will refult from them, whether in a direct rubordination, or whether they branch out or even in often collaterally ; particular Problems, we may that thefe are perceive only certain cafes of fomething more general, and may afford good hints and afiiftances to a fagacious Analyft, for afcending gradually to higher and higher Difquilitions, which may be profecuted more univerfally than was at firft expe<5ted or intended. of Thefe are fome thofe Mathematical Principles, of a higher order, which we find a difficulty to admit, and which we {hall never be fully convinced of, or know the whole ufe of, but from much practice and attentive confideration but more a ; efpecially by diligent b 2 peruial, xx The P R E F A C E.

clofe of this and the other peruial, and examination, Works of our illuftrious Author. He abounded in thefe fublime views and in- had an accurate habitual all quiries, acquired and knowledge of thefe, and of many more general Laws, or Mathematical Principles of a not fuperior kind, which may improperly be call'd The Philofophy of and aflifted his Genius and Quantity ; which, by great Sagacity, together with his great natural application, enabled him to become fo compleat a Matter in the higher Geometry, and particularly in the Art of Invention. This Art, which he poflefl in the greateft perfection imaginable, is indeed the fublimeft, as well as the moft difficult of all if it be call'd fuch as not redu- Arts, properly may ; being cible to any certain Rules, nor can be deliver'd by any Precepts, but is wholly owing to a happy fagacity, or rather to a kind of divine Enthufiafm. To improve Inventions already made, to carry them on, when begun, to farther perfection, is certainly a very ufeful and but however is far inferior to the Art of excellent Talent ; Difcovery, as haying a TIV e^u, or certain data to proceed upon, and where juft method, clofe reasoning, ftrict attention, and the Rules of Analogy, to ftrike out to adventure may do very much. But new lights, where ever been fet nullius ante no footfteps had before, trita folo ; this is the nobleft Endowment that a human Mind is capable of, is referved for the chofen few quos Jupiter tequus amavit, and was the peculiar and diftinguifhing Character of our great Mathematical Philofopher. He had acquired a compleat knowledge of the Philofophy of Quan- or its eflential and moft Laws had confider'd tity, of moft general ; it in all views, had purfued it through all its difguifes, and had traced it and Recefles in a it be through all its Labyrinths j word, may faid of him not improperly, that he tortured and tormented Quantities confefs their all poflible ways, to make them Secrets, and difcover their Properties. The Method of Fluxions, as it is here deliver'd in this Treatife, is a very pregnant and remarkable inftance of all thefe particulars. To take a cuifory view of which, we may conveniently enough divide it into thefe three parts. The firft will be the Introduction, or the Method of infinite Series. The fecond is the Method of Fluxions, properly fo culi'd. The third is the application of both thefe Methods to fome very general and curious Speculations, chiefly in the Geometry of Curve-lines. As to the firft, which is the Method of infinite Series, in this the Author opens a new kind of Arithrnetick, (new at leaft at the his time of writing this,) or rather he vaftly improves the old. For he The PREFACE. xxi he extends the received Notation, making it compleatly universal, and fhews, that as our common Arithmetick of Integers received a the introduction of decimal Fractions fo the great Improvement by ; common Algebra or Analyticks, as an univerfal Arithmetick, will receive a like Improvement by the admiffion of his Doctrine of infinite Series, by which the fame analogy will be ftill carry'd on, and farther advanced towards perfection. Then he fhews how all com- be plicate Algebraical Expreffions may reduced to fuch Series, as will true continually converge to the values of thofe complex quantities, or their Roots, and may therefore be ufed in their ftead : whether thofe quantities are Fractions having multinomial Denominators, which are therefore to be refolved into fimple Terms by a perpetual Divi- fion or whether are Roots of or of affected ; they pure Powers, Equa- tions, which are therefore to be refolved by a perpetual Extraction. And by the way, he teaches us a very general and commodious Me- thod for extracting the Roots of affected Equations in Numbers. And this is chiefly the fubftance of his Method of infinite Series. The Method of Fluxions comes next to be deliver'd, which indeed is principally intended, and to which the other is only preparatory and fubfervient. Here the Author difplays his whole fkill, and fhews the great extent of his Genius. The chief difficulties of this he reduces to the Solution of two Problems, belonging to the abftract or Rational Mechanicks. For the direct Method of Fluxions, as it is now call'd, amounts to this Mechanical Problem, tte length of the ibed to the Space defer being continually given, find Velocity of the Mo- tion at any time propofcd. Aifo the inverfe Method of Fluxions has, for a the Reverfe of this which The foundation, Problem, is, Velocity the Motion to the of being continually given, find Space deferibed at any that the or time propofcd. So upon compleat Analytical Geometri- cal Solution of thefe two Problems, in all their varieties, he builds his whole Method. His firft which The relation the Problem, is, 6J fowing Quantities being given, to determine the relation of their Fhixiom, he difpatches does not as is very generally. He propofe this, ufualiy done, Aflow- to its Fluxion for this ing Quantity being given, find ; gives us too lax and an Idea of the and does not vague thing, fufficiently fhew that Comparifon, which is here always to be understood. Fluents and Fluxions are of a relative things n.iture, and fuppofe two at leafr, whofe relation or relations mould always be exprefs'd bv Equations. He therefore that all requires fhould be reduced to Equations, by which the relation of the flowing Quantities will be exhibited, and their comparative xxii f/jg PREFACE. will be more comparative magnitudes eafily eftimated ; as alfo the of their comparative magnitudes Fluxions. And befides, by this means he has an opportunity of refolving the Problem much more than is done. For in the generally commonly ufual way of taking we are confined to. Fluxions,- the Indices of the Powers, which are to be made Coefficients ; whereas the Problem in its full extent will allow us to take any Arithmetical Progreflions whatever. By this means we have an may infinite variety of Solutions, which tho' different in form, will all in the main and yet agree ; we may always chufe the or that which will beft fimpleft, ferve the prefent purpofe. He alfo how the (hews given Equation may comprehend feveral variable and that' means the Quantities, by Fluxional Equation maybe found, notwithstanding any furd quantities that may occur, or even other any quantities that are irreducible, or Geometrically irrational. And all this is and derived demonitrated from the properties of Mo- ments. He does not here proceed to fecond, or higher Orders of for a reafon which will be in Fluxions, affign'd another place. His next Problem An is, Equation being propofed exhibiting the relation the Fluxions to the relation of of Quantities, find of thofe Quan- or to tities, Fluents, one another ; which is the diredt Converfe of the Problem. foregoing This indeed is an operofe and difficult Problem, it in its taking full extent, and, requires all our Author's fkill and ad- dreis which ; yet hefolyes very generally, chiefly by the affiftance of his Method of infinite Series. He firfl teaches how we may return from the Fluxional Equation given, to its correfponding finite Fluential or Algebraical Equation, when that can be done. But when it cannot be or there is .done, when no fuch finiie Algebraical Equation, as is moft the however he finds the commonly cafe, yet Root of that Equation an infinite which anfwers the by converging Series, fame purpofe. And often he mews how to find the or Fluent Root, required, by an infinite number of fuch Series. His for proceffes extracting thefe Roots are peculiar to himfelf, and always contrived with much fub- tilty and ingenuity. The reft of his Problems are an or application an exemplification of the foregoing. As when he determines the Maxima and Minima of in all cafes. When he quantities mews the Method of drawing to whether Tangents Curves, Geometrical or Mechanical ; or however the nature of the Curve be or may defined, refer'd to right Lines or other Curves. Then he {hews how to find the Center or Radius of of Curvature, any Curve whatever, and that in a fimple but manner which he general ; illuftrates by many curious Examples, and fbe PREFACE. xxiii

other that offer themfelves and purfues many ingenious Problems, by he difcufTes another fubtile and the way. After which very intirely to new Problem about Curves, which is, determine the quality of the Curvity of any Curve, or how its Curvature varies in its progrefs in or through the different parts, refpect of equability inequability. to confider the Areas of and fhews He then applies himfelf Curves, as or us how we may find as many Quadrable Curves we pleafe, fuch whole Areas may be compared with thofe of right-lined Figures. Then he teaches us to find as many Curves as we pleafe, whofe Areas may be compared with that of the Circle, or of the Hyper- that (hall be which he extends bola, or of any other Curve affign'd ; to Mechanical as well as Geometrical Curves. He then determines be the Area in general of any Curve that may propofed, chiefly by Series and ufeful Rules for afcer- the help of infinite ; gives many Areas. And the he the taining the Limits of fuch by way fquares the Circle and Hyperbola, and applies Quadrature of this to the con- But he collects ftructing of a Canon of Logarithms. chiefly very- for general and ufeful Tables of Quadratures, readily finding the Areas of Curves, or for comparing them with the Areas of the Conic Sections; which Tables are the fame as. thofe he has publifh'd him- ufe and of thefe felf, in his Treatife of Quadratures. The application he (hews in an ample manner, and derives from them many curious Geometrical Conftructions, with their Demonftrations. Laftly, he applies himfelf to the Rectification of Curves, and mews find as Curves as us how we may many we pleafe,. whofe Curve- Rectification or whofe as to lines are capable of ; Curve-lines, length, the Curve-lines of Curves may be compared with any that fha.ll be concludes in with affign'd. And general, rectifying any Curve-lines either the aflifbncc of his Tables of that may be propofed, by Quadra- tures, when that can be done, or however. by infinite Series. And this is chiefly the fubflance of the prefent Work. As to ,the account be of I in that perhaps" may expected, what have added my Anno- I {hall refer the tations ; inquifitive Reader to the PrefacCj which will go before that part of the Work.

THE -

; THE CONTENTS.

or the CT^HE Introduction, Method of refolding complex Quantities into infinite Series ofJimple Terms. pag. i

i. Fluents to the Prob. From the given find Fluxions. p. 21

2. the Fluxions to the Fluents. Prob. From given find p. 25 To determine the and Prob. 3. Maxima Minima of Quantities, p. 44 Curves. Prob. 4. To draw Tangents to p. 46

To the Curvature in Curve. Prob. 5. find Quantity of any P- 59

6. To the Curvature in Curve. Prob. find Quality cf any p. 75

Prob. 7. To find any number of Quadrable Curves. p. 80

To Curves Areas be to Prob. 8. find whofe may compared thofe of the Conic SecJions. p. 8 1

Prob. 9. Tofind the Quadrature of any Curve ajjigrid. p. 86 number Curves. Prob. 10. To find any of rettifiable p. 124

1. Curves be Prob. 1 Tofind whofe Lines may compared with any Curve- lines ajfigrid. p. 129

12. To Curve-lines Prob. rectify any ajpgn'd. p. 134 THE METHOD of FLUXIONS, AND INFINITE SERIES.

INTRODUCTION : Or, the Refolution of Equations by Infinite Series.

IAVING obferved that moft of our modern Geome-- tricians, neglecting the Synthetical Method of the Ancients; have apply'd themfelves chiefly to the Art the affiftance cultivating of the Analytical ; by of which they have been able to overcome fo many to have all and fo great difficulties, that they feem exhaufted the of and Speculations of Geometry, excepting the Quadrature Curves, Ibme other matters of a like nature, not yet intirely difcufs'd : I thought it not amifs, for the fake of young Students in this Science, to compofe the following Treatife, in which I have endeavour'd to enlarge the Boundaries of Analyticks, and to improve the Doctrine of Curve-lines. 2. Since there is a great conformity between the Operations in the in nor do Species, and fame Operations common Numbers; they are re- feem to differ, except in the Characters by which they B prefented,. 'The Method of FLUXIONS,

firft and the other defi- prefented, the being general indefinite, and nite and particular : I cannot but wonder that no body has thought of accommodating the lately-difcover'd Doctrine of Decimal Frac- tions in like manner to Species, (unlels you will except the Qua- of the fince drature Hyberbola by Mr. Nicolas Mercator ;) efpecially it might have open'd a way to more abftrufe Discoveries. But iince this Doctrine of Species, has the fame relation to Algebra, as the Doctrine of Decimal Numbers has to common Arithme- the of tick ; Operations Addition, Subtraction, Multiplication, Di- of vifion, and Extraction Roots, may eafily be learned from thence,, if the Learner be but fk.ill'd in Decimal Arithmetick, and the Vulgar Algebra, and obferves the correfpondence that obtains between Decimal Fractions and Algebraick Terms infinitely continued. For as in Numbers, the Places towards the right-hand continually in a Decimal or fo it is in decreafe Subdecuple Proportion ; Species are is refpedtively, when the Terms difpofed, (as often enjoin'd in what follows,) in an uniform Progreflion infinitely continued, according to the Order of the Dimenfions of any Numerator or De- nominator. And as the convenience of Decimals is this, that all vulgar Fractions and Radicals, being reduced to them, in fome mea- of fure acquire the nature Integers, and may be managed as fuch ; fo it is a convenience attending infinite Series in Species, that all kinds of fuch as Fractions whofe Denomina- complicate Terms, ( tors are compound Quantities, the Roots of compound Quantities, or of affected Equations, and the like,) may be reduced to the Clafs that to an infinite Series of of fimple Quantities ; is, Fractions, whofe Numerators and Denominators are fimple Terms ; which will no that in the other form longer labour under thofe difficulties, feem'd almoft infuperable. Firft therefore I mail fhew how thefe Re- ductions are to be perform'd, or how any compound Quantities may be reduced to fuch fimple Terms, efpecially when the Methods of computing are not obvious. Then I fhall apply this Analyfis to the Solution of Problems. Divifion and Extraction of 3. Reduction by Roots will be plain from the following Examples, when you compare like Methods of Operation in Decimal and in Specious Arithmetick.

Examples and INFINITE SERIES, 3

. Av ..ift Reduttion '* Examples of by Dhifwn. IjfM/l^^ / The Fraction divide aa b x in the .4. ^ being propofed, by + following manner : 1 faa aax aax a a x* aax* . " aax .

aax O --7 -f-O aax*

o -+-

o - +o flt~* ** Jf* ;.

1 -rr^i_ *-\ " v i r * ^^ tf*^ a* x* a* x* . a* X+ ~ The therefore is _- - - _ . &c. Quotient T JT + T rr +T7-, which Series, being infinitely continued, will be equivalent to Or x the firft Term of the in this j^. making Divifor, manner, - ~ x + toaa + o (the Quotient will be - ?4 n4. 1^ V &c , r~ _ * ** * AV e , % _ found as by the foregoing Procefs. In like manner the 5. Fraction ~- will be reduced to 4 8 I # x ' A:* x or to -{- H- , &c. x-* #-* _f. ^- ^-8 29 *v " 6. And the Fraction r will be reduced to 2x^ 2x i i+x* 3* 1 Ts + yx 13** -j- 34x , &c. Here it will 7. be proper to obferve, that I make ufe of - x-', x-', x-', x-*, &c. for i, &c. of ;r 7,' xs, xi, x^, xl, A4, &c. l for x - *> vx , , &c. and of i v/x, v/*S \/ ^x x'^, x-f.' x * ****&c 1Uifor , i j_^ x >' &c. And this the Rule of as be ^ ^ ? y-^.' by Analogy, may apprehended from fuch Geometrical as Progreflions thefe ; x, x*, x> x, (or i,) a"*,*-',*'*, *, &c. B 2 8. ffie Method of FLUXIONS,

-- 8. In the fame mannerer for 1^ + 1^!, &c. may be wrote

', &c. thus inftead xx be wrote aa > q. And of^/aa may xxl^

.and aa xv|* inftead of the Square of aa xx; and

3 inftead of v/ 10. So that we may not improperly diftinguim Powers into Affir- and Fractional. mative and Negative, Integral

Examples of Reduction by Extraction of Roots. 11. The Quantity aa -+- xx being propofed, you may thus extract its Square-Root.

5 x - J - ' c* - _i_ XXV v (a -4- 4-r 4- aa-+- ^" 2a Sfl3 i6* 1287 2560* aa

xx 4. a*

x* ~* a 4 64

64 a X* " sT* 64^8 z$6a'^ ~ x i; _ 64^ 256 * 5* 6 8 + 64 a I z8rt

_- _ 2^1, &c . 1 7^ i7R/3 n-- /7lt> 7' 1 +

,__i!_lll, &c.'

the Root is found to be 4- Where Jo that a~\--^- ^ ^T,&C. that towards I it may be obferved, the end of the Operation neg- lect all thofe Terms, whofe Dimenfions would exceed the Dimenfions of the laft Term, to which I intend to continue the Root, *' only fuppofe to ,2. and INFINITE SERIES. 5

inverted in this man- iz. Alfo the Order of the Terms may be be found to be ner xx +- aa, in which cafe the Root will a a

10 A* iz A-

the Root of aa xx is -- 13. Thus ^ -Jj ^7 ' of is #'" 8cc. 14. The Root x xx i** 4-.v* Tr**, . . AT A.' A' b*X* g f is -- ## a , Sec. 15. Of -+- -f- ^ '- * * 3 6 . i

thefe due often 17. But Operations, by preparation, may very be abbreviated; as in the foregoing Example to find \/;_***' if the Form of the Numerator and Denominator had not been the fame, I might have multiply'd each by

V t i v Vx \fi xx x i! 2x x as x"> -}- ; _. be reduced to j may * " A- 3 ' ^/axx -\- x-{-xx 2X x.1 infinite Series confifting of iimple Terms.

Of the ReduStion of offered Equations. As to aftedled we mufl be 19. Equations, fomething more particular in explaining how their Roots are to be reduced to fuch Se- as thefe becaufe their ries ; Doctrine in Numbers, as hitherto deliver'd by Mathematicians, is very perplexed, and incumber'd with fo as not to afford fuperfluous Operations, proper Specimens for performing the Work in Species. I fhall therefore firfl (hew how the Refolu- Method of FLUXIONS,

Refolutidn of affected Equations may be compendioufly perform'd and I in Numbers, then fhall apply the fame to Species. 20. Let this l be be Equation _y zy 5 = propofed to refolved, and let 2 be a Number (any how found) which differs from the true Root lefs than by a tenth part of itfelf. Then I make 2 -\-p =y, and fubftitute 2 4-/> for y in the given Equation, by which is l produced a new Equation p> 4- 6p 4- iop i =o, whofe Root is to be fought for, that it may be added to the Quote. 1 Thus becaufe of its rejecting />> 4- 6// fmallnefs, the remaining Equation io/> i = o, or/>=o,i, will approach very near to the truth. Therefore I write this in the Quote, and fuppofe o, i 4- ^ =/>, and fubftitute this fictitious Value of p as before, which produces q* 4- 6,3^ 4- 1 1,23? 4- 0,06 1 =o. And fince 1 1 is near the or 1,23^ 4- 0,06 =o truth, ^= 0,0054 nearly, (that is, dividing 0,06 1 by 11,23, ^ * many Figures arife as there are places between the firft Figures of this, and of the principal QmDte exclufively, as here there are two places between 2 and 0,005) I write 0,0054 in the lower part of the Quote, as being negative; and fuppofing 0,0054 4- r=sg, I fubftitute this as I in before. And thus I continue the Operation as far as pleafe, the manner of the following Diagram :

y~' zy 5 =o and INFINITE SERIES. 7 21. But the Work may be much abbreviated towards the end by this Method, efpecially in Equations of many Dimenfions. Having firft determin'd how far you intend to extract the Root, count fo many places after the firft Figure of the Coefficient of the laft Term but one, of the Equations that refult on the right fide of the Dia- fill'd in gram, as there remain places to be up the Quote, and reject the Decimals that follow. But in the laft Term the Decimals may be neglected, after fo many more places as are the decimal places that are fill'd up in the Quote. And in the antepenultimate Term all that reject are after fo many fewer places. And fo on, by proceeding Arithmetically, according to that Interval of places: Or, which is the fame thing, you may cut off every where fo many Figures as in the penultimate Term, fo that their loweft places may be in Arithmetical Progreffion, according to the Series of the Terms, or are to be fuppos'd to be fupply'd with Cyphers, when it happens otherwife. Thus in the prefent Example, if I defired to continue the Quote no farther than to the eighth place of Decimals, when fubftituted r for four I 0,0054 -f- q, where decimal places are compleated in the Quote, and as many remain to be compleated, I might have omitted the Figures in the five inferior places, which therefore I have mark'd or cancell'd by little Lines drawn through indeed I alfo the firft r J them ; and might have omitted Term , although its Coefficient be 0,99999, Thofe Figures therefore being expunged, for the following Operation there arifes the Sum 0,0005416 -f- 1 1,1 62?% which by Divifion, continued as far as the Term prefcribed, gives 0,00004852 for r, which compleats the the Quote to Period required. Then fubtracting the negative of the from the affirmative arifes part Quote part, there 2,09455148 for the Root of the propofed Equation. likewife be 22. It may obferved, that at the beginning of the if I had doubted whether i Work, o, -f-/> was a fufficient Ap- to the inftead of proximation Root, iof> i = o, I might have that i i fuppos'd o/** -f- op = o, and fo have wrote the firft of its Figure Root in the Quote, as being nearer to nothing. And in this manner it may be convenient to find the fecond, or even the third of the in Figure Quote, when the fecondarjr Equation, about which you are converfant, the Square of the Coefficient of the penultimate Term is not ten times greater than the Product of laft Term the multiply'd into the Coefficient of the antepenultimate Term. And indeed will often fave you fome pains, efpecially in Equations of many Dimensions, if you feek for all the Figures to- 8 Tie Method of FLUXION'S,

to the in this that if extract to be added Quote manner ; is, you the lefier Root out of the three lafl Terms of its fecondary Equation : For thus you will obtain, at every time, as many Figures again in the Quote. 23. And now from the Refolution of numeral Equations, I mall proceed to explain the like Operations in Species; concerning which, it is neceflary to obferve what follows. 24. Firft, that fome one of the fpecious or literal Coefficients, if there are more than one, fliould be diftinguifh'd from the reft, which or be leaft or either is, may fuppos'd to be, much the greateft of or neareft all, to a given Quantity. The reafon of which is, that becaufe of its Dimeniions continually increafing in the Numerators, or the Denominators of the Terms of the Quote, thofe Terms may grow lefs and lefs, and therefore the Qtipte may conftantly approach the as from is faid before of to Root required ; may appear what the Species x, in the Examples of Reduction by Divifion and Ex- traction of Roots. And for this Species, in what follows, I mall ufe of A: or as alfo I fliall ufe generally make z ; y, p, q, r, s, &c. for the Radical Species to be extracted. or 25. Secondly, when any complex Fractions, furd Quantities, happen to occur in the propofed Equation, or to arife afterwards in the Procefs, they ought to be removed by fuch Methods as are fufficiently known to Analyfts. As if we mould have 1 x"= b and from the Pro- y* -+- >' = o,. by x, j 1 multiply duct o extract the Root by* Kyi'-l-fry* bx^ -+ x*-= y. Or we x b and then writing for might fuppofe y x=v, ^~x yt J 6 we mould have i; -+- &*v* fax* -\- 3/5*** ^hx' -+. x = o,. we divide the b whence extracting the Root vr might Quote by x,, if the 3 in order to obtain y. Affo Equation j xy* -f- x$ = o and j were propofed, we might put y?= v, x = z, and fo wri- for and z* for will arife v 6 z=v z* ting vv y, x, there -f- = o ; which Equation being refolved, y and x may be reftored. For the Root will befound^=2-f-s3_|_5~s 5 5cc.andrei1:onngjyandA;, we have x^ J y* = -f- x -+- 6x^ &c. dien fquaring, y =x^-+- 2X ~f- 13*", &c.. 26. After the fame manner if there mould be found negative Di- menfions ofx and be removed the fame jy, they may by multiplying by I x As if we had the I '2.x~ i6y- 3 andjy. Equation x*-}-T> x*-y~ =o, 1 and 3 and there arife -+- 3 5 multiply by x j , would x*y* 3# jy 2_v aa 2ai a 4 A J -r 1 -r-v ~ i 1 O. And U tjie were x = + ?r Equation y \. by; and INFINITE SERIES.

into } there would arife i by multiplying jy xy*-=.a' y* And fo of others. 27. Thirdly, when the Equation is thus prepared, the work be^ the firfr. of the as gins by finding Term Quote ; concerning which, alfo for finding the following Terms, we have this general Rule, when the indefinite or is to be fmall to Species (x 2) fuppofed ; which Caie the other two Cafes are reducible. in the Radical 28. Of all the Terms, which Species (y,/>, q, or r, &c.) is not found, chufe the loweft in refpect of the Dimenlions of the indefinite Species (x or z, &c.) then chufe another Term in the which that Radical Species is found, fuch as that Progreflion of the Dimenfions of each of the fore-mentioned Species, being continued from the Term fir ft afTumed to this Term, may defcend as little as be. And if there much as may be, or afcend as may are any other Terms, whofe Dimenfions may fall in with this muft be taken in 1 ike- Progreflion continued at pleafure, they and wife. Laftly, from thefe Terms thus felected, made equal to nothing, find the Value of the faid Radical Species, and write it in the Quote. fhall 29. But that this Rule may be more clearly apprehended, I the a explain it farther by help of following Diagram. Making into and right Angle BAC, divide its fides AB, AC, equal parts, into raifing Perpendiculars, diftribute the Angular Space equal Squares be from or Parallelograms, which you may conceive to denominated the Dimenfions of the Species x and y, B A 4 as they are here infcribed. Then, when any Equation is propofed, mark fuch of the Parallelograms as correfpond to all its Terms, and let a Ruler be apply'd to two, or perhaps more, of the Paralle- let one lograms fo mark'd, of which be the loweft in the left-hand Column at AB, the other touching the and let all the not the Ruler towards right-hand ; reft, touching the Ruler, lie above it. Then felecl: thofe Terms of the Equation which are reprefented by the Parallelograms that touch the Ruler, and from them find the Quantity to be put in the Quote. out of the 6 s 30. Thus to extract the Root y Equation y 5xy -+- 1 1 i 1 I mark the )'* ja*x y +6a x*-\-& x4=o, Parallelograms belong- C 10 The Method of FLUXIONS, of this ing to the Terms Equation B with the Mark #, as you fee here done. Then I apply the Ruler * DE to the lower of the Parallelo- grams mark'd in the left-hand Column, and I make it turn round towards the right-hand from the C lower to the upper, till it begins A in like manner to touch another, or of the that are perhaps more, Parallelograms mark'd ; and I fee that the fo touch'd to x 3 5 places belong , x*-y* y and_y . Therefore from the Terms 6 z -i-axy-{-aay 2rt =o, I make choice of 2<2 3 and its I write in the y'-i-a^y , Root -\-a Quote. s 1 I a 7 i 7 Alfo c .v I felect vV f 33. having x*y ^c^xy 4- =o, y 4-' 27^5=0 being propofed, whofe

is to the Term I that it be- Root begin by ^ deprefs by s% may 1 come Sz+yt-^azy 2ja !> z~ =o, before I attempt the Refolu- tion. Terms of the are derived the fame 3 5. The fubfequent Quotes by Method, in the Progrefs of the Work, from their feveral fecondary lefs the affair Equations, but commonly with trouble. For whole is perform'd by dividing the loweft of the Terms affected with the fmall x 1 x 3 without the Radical indefinitely Species, (x, , , &c.) Spe- r the with which that radical (/>, q, } &c.) by Quantity Species i of and INFINITE SERIES, n

of one Dimenfion only is affected, without the other indefinite Spe- the cies, and by writing Refult in the Quote. So in the following

the -> ~ } - ~> Example, Terms &c. are produced by dividing a l x, 3 &c. TrW", TTT-v , by ^aa. Thefe 36. things being premifed, it remains now to exhibit the Praxis of z Refolution. Therefore let the Equation y*-{-a y-\-axy za* z x =o be propofed to be refolved. And from its Terms 3 y=-\-a*y 2 =o, being a fictitious Equation, by the third of the foregoing Premifes, I obtain y a=o, and jtherefore I write -{-a in the Quote. Then becaufe -\~a is not the compleat Value ofy, I put and a+p=y, inftead of y, in the Terms of the Equation written in the I the 3 Margin, fubftitute a-\-p, and Terms refulting (/> -{- 1 I from which 3rf/ -f-,?,v/>, &c.) again write in the Margin ; again, according to the third of the Premifes, I felect the Terms -+-^p l -H2 .v=o for a fictitious Equation, which giving p= ^x, I in is write ~x the Quote. Then becaufe ^.v not the accurate Value of I p, put x-\-q=p, and in the marginal Terms for p I fubftitute 3 ^x-t-q, and the refulting Terms (j -^x^+^a^, &c.) I again write in the Margin, out of which, according to the fore- I _I -drx*=o for ficti- going Rule, again feledl the Terms 4^ 3 a tious in the Equation, which giving =^> I write -^ Quote.

fince is not the accurate Value of I make Again, ^ g, -^--{-r=qt and inftead of a I the marginal Terms. And fubftitute ~--\-r' in &4 thus I continue the Procefs at pleafare, as the following Diagram exhibits to view. 12 Method of FLUXIONS,

X3 - X* 2a'

axp T ' a*-x

; 643 *

axq

'31** 509*4

it to continue the 37. If were required Quote only to a certain Period, that x, for inilance, in the laft Term {hould not afcend Dimenfion as I fubftitute the I beyond a given ; Terms, omit fuch as I forefee will be of no ufe. For which this is the Rule, that after the firft Term refulting in the collateral Margin from every Quan- are to be added to the as tity, fo many Terms right-hand, the In- dex of the higheft Power required in the Quote exceeds the Index of that firft refulting Term. 38. As in the prefent Example, if I defired that the Quote, (or .v in the to the Species Quote,) mould afcend no higher than four I all the after Dimenfions, omit Terms after A-*, and put only one x=. Therefore and INFINITE SERIES. 13

Therefore the Terms after the Mark * are to be conceived to be thus the Work continued till at laft we come expunged. And being which or to the Terms -^ -^--H-rfV axr,'m />, q, r, the of the Root to be are reprefenting Supplement extracted, only find fo Terms of one Dimenfion ; we may many by Divifion, \ 131*3 _, 509*4 as we fl^n e want j ng to compleat the Quote. 5121. 16384(13 / _ ... XX 13'SI*'1.*' 509*4kuyAT that at laft we {hall have icc- So y=a 7*-f"6^-t-^l~*- r^I; farther I mail another 39. For the fake of Illustration, propofe to be refolved. From the Example Equation -L_y< .Ly4_f_iy3 iy=. be found to the fifth _^_y z=o, let the Quote only Dimenfion, be after and the fuperfluous Terms rejected the Mark,

_!_ 5j &c.

5 &c. + ^ , -L;S 4 Z'p, &C. 6cc.

s , &c.

% &c.

And thus if we the 40. propofe Equation T 4-rjrJ' '+TT|-T )'' + 7 J 3 to be refolved to the ninth Di- -rT T;' -t-TW' -i-r.)' +y =o, only of the before the the menfion Quote ; Work begins we may reject then as we all the Term -^^y" ; operate we may reject Terms 7 beyond 2', beyond s we may admit but one, and two only after Y4 The Method of FLUXIONS,

f we that the to afcerrd z ; becaufe may obferve, Quote ought always s Interval of two in this .s j z &c. Then by the Units, manner, z, , , fliall s _ _5__ 2; ^_ 3 9 at laft we have J &C. ;'=c fs3_j__|_. T _^_'T ^_. ) 41. And hence an Artifice is difcover'd, by which Equations, tho' affected hi injinitum, and confiding of an infinite number of Terms, may however be refolved. And that is, before the Work begins all the Terms are to be rejected, in which the Dimenfion of the indefinitely fmall Species, not affected by the radical Species, in the or from, exceeds the greateft Dimenfion required Quote ; which, by fubftituting inftead of the radical Species, the firfl Term, of the Quote found by the Parallelogram as before, none but fuch exceeding Terms can arife. Thus in the laft Example I mould have omitted all the Terms beyond y>, though they went on ad injinitum. And fo in this Equation

1 8 4 lS l6 8 &C. -f-3 4S -f-92 ,

1 in z* s 4 z 6 &c. ) j' -}- z*y

that the Cubick Root may be extracted only to four Dimenfions of z, 1 5 in z, 2 I omit all the Terms in infinitum beyond -f-j J.-4_|_.L > 1 1 and all in z a 4 6 and all in .c 2z 4 beyond y- -(-.c , beyond -+-y , tt 4 I aflurr.e this and beyond S-}-;s 42 . And therefore Equation 6 6 1 4 1 be ;> s only to refolved, -^z y* z*y* -{-?* ^ -}-^ ^ z^y* 2z*y ' s i Becaufe?. ~^ { Term of the -i-z'-y 4s4_j_ 8=0. ',(*''- Quote,) of the being fubflituted inflead of y in the reft Equation deprefs'd than four Dimenfions. by z^y gives every where more 42. What I have faid of higher Equations may alib be apply'd to Qi\adraticks. As if I defired the Root of this Equation r 1 .r A* A 4 - h-r-f--; &c.

as f I all in as far the Period x , omit the Terms infinititm., beyond

y in

2" \ 4 -y+ =0. This I refolve either in the ufual manner, by & 4-*-* making and IN FINITE SERIES.

or more j-^; expedition fly by the Method of affected Equations deliver'd before, by which we fhall have where the laft Term or _}'= 3 #> required vanifhes, becomes equal to nothing. 43. Now after that Roots are extracted to a convenient Period, at the they may fometimes be continued pleafure, only by oblerving Analogy of the Series. So you may for ever continue this z-t-i-z* 2; i_2;s &c. is the Root of the infinite ^_^.25_j__'_ 4_{_T j (which Equa- tion i the laft Term thefe 5r==)'-f-^ _j_^5_|_y4 j foe.) by dividing by

' Numbers in order 2, 3, 4, 5, 6, &c. And this, z f^-H-rlo-^' l ' ;9 &c. be continued thefe Num- yjTB.27-f_TrTTTy2 j may by dividing by bers the Series 2x3, 4x5, 6x7, 8x9, &c. Again, &c. be continued at the Terms "-'g , may pleafure, by multiplying thefe &c> And refpectively by Fractions, f} 7, , -, TV, fo of others. 44. But in difcovering the firft Term of the Quote, and fometimes of the fecond or third, there may ftill remain a difficulty to be overcome. For its Value, fought for as before, may happen to be furd, or the inextricable Root of an high affected Equation. Which when it happens, provided it be not alfo impoffible, you may reprefent it by fome Letter, and then proceed as if it were 3 known. As in the Example y*-\-axy-{-ii*-y x 2a>=o : If the 5 Root of this Equation y^^-a'-y 2 =o, had been furd, or un- I mould have for known, put any Letter b it, and then have per- the Refolution form'd as follows, fuppofe the Quote found only to the third Dimenfion. i6 fbe Method of FLUXIONS,

s a 3 ; y -\-aay-\rtxy 2 ft , tf^A- 4jC and INFINITE SERIES, 17

I have 47. Hitherto fuppos'd the indefinite Species to be little. But if it be fuppos'd to approach nearly to a given Quantity, for fmall that indefinitely difference I put fome Species, and that being fubftituted, I folve the Equation as before. Thus in the Equation l x it f}-' ^y* -+- ^y y* -t-y -\-a = o, being known or fuppos'd that x is nearly of the fame Quantity as a, I fuppofe z to be their difference; and then writing a-\-z or a z for x, there will

5 arife y y* -f- jj y* -{-y + z=o, which is to be folved as before. if be to be 48. But that Species fuppos'd indefinitely great, for its Reciprocal, which will therefore be indefinitely little, I put fome Species, which being fubflituted, I proceed in the Refolution as l before. is Thus having y* -+-\ -f-jv x> =o, where x known or fuppos'd to be very great, for the reciprocally little Quantity - I and - for there will arife 1 put z, fobflituting .v, y> -f-.)' + y ~ whofe Root is - z* &c. where =o, .y = ^ ^z + -f- ^2', x reflored. if it will be - being Jyou pleafe, y=:x* H- H 3 9* 8 i**

&c '

49. If it fhould happen that none of thefe Expedients mould fucceed to your defire, you may have recourfe to another. Thus 1 1 in the Equation y* x^y -+- xy* -f- Z) 2y -+- i = o, whereas the firft Term ought to be obtain'd from the Suppofition that 2 1 admits jy-4_j_2yt y + = 0, which yet of no poffible Root; what can be done another you may try way. As you may fuppofe that x is but little different from + 2, or that 2-{-z-=x. Then inftead of there will arife fubftituting 2-{-z A*, y* z'-y* -\zy* 1 and the will 2y -f- = 0, Quote begin from -j- i. Or if you l to be or - 4 fuppole x indefinitely great, = z, you will have ^ 1 >* y --{-- -+-2y* 2y H- i = o, and -f- z for the initial Term of the Quote. , And thus to 50. by proceeding according feveral Suppofitions, and you may extract exprefs Roots after various ways. If mould delire 51. you to find after how many ways this may be done, you mufl try what Quantities, when fubfHtuted for the indefinite Species in the propofed Equation, will make it divifible by_y, -f-or fome Quantity, or by^ alone. Which, for Example l fake, will happen in the Equation y* -}-axy-+-a y x> 20 3 = o, D by 4 1 8 The Method of FLUXIONS,

or or or 2 } T &c. inftead by fubftituting -f-rf, a, za, | , the of .v. And thus you may conveniently fuppofe Quantity x to differ little from -j-tf, or a, or 2a, or za*l^, and thence you may extract the Root of the Equation propofed after fo many ways. And perhaps alfo after fo many other ways, by fup- poling thofe differences to be indefinitely great. Befides, if you take for the indefinite Quantity this or that of the Species which exprefs the Root, you may perhaps obtain your defire after other ways. fictitious Values for the inde- And farther ftill., by fubftituting any finite fuch as az bz 1 &c. and then Species, + , -> ~n^> proceeding as before in the Equations that will refult. 52. But now that the truth of thefe Conclufions may be manifeft that ; is, that the Quotes thus extracted, and produced ad libi-* turn, approach fb near to the Root of the Equation, as at laft to differ from it by lefs than any afilgnable Quantity, and therefore when infinitely continued, do not at all differ from it : You are to confider, that the Quantities in the left-hand Column of the right- hand fide of the Diagrams, are the laft Terms of the Equations that as whofe Roots are p, y, r, s, &c. and they vanifh, the Roots the p, q, r, s, &c. that is, the differences between Quote and the Root fought, vanifh at the fame time. So that the Quote will not then differ from the true Root. Wherefore at the beginning of the Work, if you fee that the Terms in the faid Column will all de- that the fo ftroy one' another, you may conclude^ Quote far extracted is the perfect Root of the Equation. But if it be otherwife, you will fee however, that the Terms in which the indefi- is of few that the ft nitely fhiall Species Dimenfions, is, greate Terms, are continually taken out of that Column, and that at laft none will remain there, unlefs fuch as are lefs than any given Quantity, and therefore not greater than nothing when the Work is continued ad infinitum. So that the Quote, when infinitely extracted, will at laft be the true Root. fake 53. Laftly, altho' the Species, which for the of perfpieuity I have hitherto fuppos'd to be indefinitely little, fhould however be to as as the will ftill fuppos'd be great you pleafe, yet Quotes be true, though they may not converge fo faft to the true Root. This is manifeft from the Anal'ogy of the thing. But here the Limits of the Roots, or the greateft and leaft Quantities, come to be confider'd. For thefe Properties are in common both to finite and infinite Equations. The Root in thefe is then greateft or leaft,. when and INF INITE SERIES. 19 the when there Is the greateft or leaft difference between Sums of and is limited the affirmative Terms, and of the negative Terms ; when the indefinite Quantity, (which therefore not improperly I that the fuppos'd to be fmall,) cannot be taken greater, but Mag- will nitude of the Root will immediately become infinite, that is, become impoffible. 54. To illuftrate this, let AC D be a Semicircle defcribed on the Diameter AD, and BC be an Ordinate. MakeAB = ^,BC=7,AD = ^. Then xx

as before.

Therefore BC, or y, then becomes greateft when iax moft exceeds all the Terms

- that is when * i but " ax > &c > = ** la Sax -f- ga*f- VS^x 4- i6a> VS it will be terminated when x a. For if we take x greater than a the Sum of all the Terms ax ax ax> t ^ S s7 V TbTs *S &c. will be infinite. There is another Limit alfo, when x = o, of the Radical ax to which by reafon of the impoffibility S ; Terms or Limits, the Limits of the Semicircle A, B, and D, are cor^ refpondent.

Tranfttion to the METHOD OF FLUXIONS. of of which 55. And thus much for the Methods Computation, I mall make frequent ufe in what follows. Now it remains, that , for an Illuftration of the Analytick Art, I mould give fome Speci- the nature of mens of Problems, efpecially fuch as Curves will fup- x be that all the difficulties of thefe ply. But firft it may obferved, may be reduced to thefe two Problems only, which I mall propofe ' concerning a Space defcribed by local Motion, any how accelerated or retarded. ~ that -*"* 56. I. The Length of the Space defcribed being continually ( the Motion at ?V, at fill Times) given; to find the Velocity of any ffo^ Tune propofed. / SJLJ tt to JbotA.*** if* 57. II. The Velocity of the Motion being continually given ; find the Length of the Space defcribed at any Time propofed. in the if the of 58. Thus Equation xx=y, y reprefents Length the Space t any time defcribed, which (time) another Space x, and exhibits as by increafing with an uniform Celerity #, mea/ures D 2 defcribed : 20 ?%e Method of FLUXIONS,

defcribed : Then zxx will reprefent the Celerity by which the Space fame moment of to be defcribed y, at the Time, proceeds ; and contrary-wife. And hence it is, that in what follows, I confider Quantities as if they were generated by continual Increafe, after the manner of a Space, which a Body or Thing in Motion defcribes. 59. But whereas we need not confider the Time here, any farther than as it is expounded and meafured by an equable local Motion ; and befides, whereas only Quantities of the fame kind can be compared together, and alfo their Velocities of Increafe and Decreafe : Therefore in what follows I fhall have no regard to Time formally conficter'd,, but I fhall fiippofe fome one of the Quantities propofed, being of the fame kind, to be increafed by an equable to the reft be as it Fluxion, which may referr'd, were to Time j and therefore, by way of Analogy, it may not improperly receive the name of Time. Whenever therefore the word Time occurs in what follows, (which for the fake of perfpicuity and diftindlion I have fometimes ufed,) by that Word I would not have it underftood as if I meant Time in its formal Acceptation, but only that other Quantity, by the equable Increafe or Fluxion whereof, Time is expounded and meafured. '60. Now thofe Quantities which I confider as gradually and I fhall hereafter call or 2 indefinitely increafing, Fluents, Flowing fhall them Quantities, and reprefent by the final Letters of the and z that I them f Alphabet v, x, y, ; may diftinguifh from other Quantities, which in Equations are to be confider'd as known and. and which therefore are the initial U T H > %f& f'df** determinate, reprefented by the Velocities _' .i V*> i*i~- Letters a, b, c, &c. And by which every Fluent is increafed by its generating Motion, (which I may call Fluxions, ( oi V* ffm***4t*'Qr fimply Velocities or Celerities,) I fhall reprefent by the fame Letters thus and z. That for the pointed -y, x, y., is, Celerity of K t4 JO the Quantity v I fhall put v, and fo for the Celerities of the other and I fhall id tti Quantities x, y, z, put x, y, and z refpeftively. '(/ 6 1. Thefe J things being premifed, I mall now forthwith proceed to the matter in hand and } firft I fhall give the Solution of the: two Problems juft now propofed.

PROF, and INFINITE SERIES. 21

P R O B. I.

The Relation of the Flowing Quantities to one another being given, to determine the Relation of their Fluxions. SOLUTION.

1. Difpofe the Equation, by which the given Relation is ex- the prefs'd, according to Dimenftons of fome one of its flowing Quantities, fuppofe x, and multiply its Terms by any Arithmetical - Progreflion, and then by . And perform this Operation feparately for every one of the flowing Quantities. Then make the Sum of the all the Products equal to nothing,, aad you will have Equation required. 2. EXAMPLE i. If the Relation of the flowing Quantities A; and 3 y be X' ax*--{- axy ^ =o; firft difpofe the Terms according then in to x, and according to y, and multiply them the following, manner. Mult. *

makes %xx* zaxx -{- axy * zyy* -f- ayx * * The Sum of the Produdls is -jx** zaxx -k- axy W*-f- ayx=zo, . thei which Equation gives Relation between the Fluxions x and y. x at 3 1 For if you take pleafure, the Equation .v ax -{-axy yt Which it will = o will give y. being determined, be x : y :: 7v* ax : yx^zax -{- ay. Ex. 2. If the Relation of the and z be ex- 3.. Quantities x, y,. r the 3 z'' preis'd by Equation 2j -f- x*y zcyz +- yz* = QJ 22 *The Method of FLUXIONS,

Wherefore the Relation of the Celerities of Flowing, or of the

l Fluxions ,v, v, and z, is tyy* -\- + 2xxy $zz -f- 6zzy zczy

. are and 4. But fince there here three flowing Quantities, .v, y, z, another Equation ought alfo to be given, by which the Relation be among them, as alfo among their Fluxions, may intirely determined. As if it were fuppofed that x -\-y 2 = 0. From whence another Relation among the Fluxions AT-HV z = o would be found by this Rule. Now compare thefe with the foregoing Equa- one of the three and alfo tions, by expunging any Quantities, any one of the Fluxions, and then you will obtain an Equation which the Relation of the reft. will intirely determine whenever there are Frac- 5. In the Equation propos'd, complex I fo Letters for and tions, or furd Quantities, put many each, fup- I work as before. Af- pofing them to reprefent flowing Quantities, exterminate the afTumed as fee terwards I fupprefs and Letters, you done here. If the Relation of the .v and be aa 6. Ex. 3. Quantities y yy x\/aa ## = o; for x

i . as for the o, of which the firfl will give zyy z = o, before, Relation of the Celerities y and z, and the latter will give 2

be - Celerities x and z. Now z being expunged, it will zyy xx for we fhall have = o, and then reftoring x^aa z, zyy -./** g*.>* 4- __ and as was re- 0> for the Relation between x y, ^aa XX quired.

Ex. If .v 3 XX -+- xx = o, 7. 4. ay* 4- j4r \fay expreffes

v : I make and the Relation that is between AT and ^^ = 5;, Lave the three x- ^x \/~ay-+-xx=v, from whence I fhall Equations 6 3 and x 1^=0. ay* + & -u = o, az-\-yz ^ =o, ax*y + the fecond az The firft gives 3**' zayy + z -0=0, gives + & Zy^-yz 3^ = o, and the third gives 4.axx>y-+-6xx'-i-a}>x* and . But 2W= o, for the Relations of the Velocities -y, .v, y, the and INF i NIT E SERIES. 23

the third the Values of & and i', found by fecond and Equations,

? for and iSj z

/. /. v-. . 11 . ,. 7n vz nrft Equation, and there anies %xx* 2a)y-^-~^T.

= o. Then inflead of z and v refloring their Values f and a > . there XX \/ ay -+- xx, will arife the Equation fought ^xx*-2ayy

6*- A- 3 awMf ... . _ . . r , . = o. by which the Relation or the 2 > aa -f- 2^ +yy Velocities x and y will be exprefs'd. 8. After what manner the Operation is to be performed in other is manifefl in the Cafes, I believe from hence j as when Equation are found furd Ra- propos'd there Denominators, Cubick Radicals, within as v 'aa or other com- dicals Radicals, ax -+- \/ xx} any of the like kind. plicate Terms in fhould 9. Furthermore, altho' the Equation propofed there be Quantities involved, which cannot be determined or exprefs'd by any Geometrical Method, fuch as Curvilinear Areas or the Lengths Curve-lines the Relations their be of ; yet of Fluxions may found, as will appear from the following Example.

Preparation for EXAMPLE 5* 10. Suppofe BD to be an Ordinate at right Angles to AB, ancL that ADH be any Curve, which is defined by the Relation between AB and BD exhibited Let be called by an Equation. AB A;, and the Area of the Curve ADB, apply 'd to Unity, be call'd z. Then erect the Perpendicular AC thro' equal to Unity, and C draw CE parallel to AB, and meeting BD in E. Then conceiving thefe two ADB and to Superficies ACEB be generated by the of the Line it is Motion right BED ; manifeft that their Fluxions, the Fluxions of i the x . (that is,, Quantities z t and i x v, or of the s and are to each other as Quantities x,) the generating Lines BD

and BE. Therefore : :: : x BD BE or i, and therefore z = * x BD. 1 1. And hence it that z be is, may involved in any Equation, the Relation between .v and expre fling any other flowing'Quantityjv ; and yet the Relation of the Fluxions x and y may however be dif-

cover'd, 12 . fhe< Method 24

12. Ex. As if the 5. Equation zz -\-axz _y*=r=o were pro- to the Relation between x 1 as alfo pos'd exprefs and; , \/axxx = BD, for determining a Curve, which therefore will be a Circle. The Equation zz-^-axz j^=o, as before, will give 2zz-i- azX axz for the Relation -f- 4_y_y = o, of the Celerities x,y, and z. And therefore fince it is z = x x BD or -x \/ax xx t iubftitute this Value inftead of and there will it, arife the Equation

2xz -t- axx \/ax-r xx 4- axz qyy* = o, which determines the Relation of the Celerities x and y.

DEMONSTRATION of the Solution. of 13. The Moments flowing Quantities, (that is, their indefi- fmall the acceffjon of in nitely Parts, by which, indefinitely fmall are portions of Time, they continually increafed,) are as the Ve- locities of their Flowing or Increafing. if the 14. Wherefore Moment of any one, as x, be reprefented Product of its x into an t>y the Celerity indefinitely fmall Quantity the Moments of the others o (that is, by xo,}

,. , fince the as xo and are the 15. Now Moments, yo, indefinitely p. cttA // natti** tt i e ^cceflions of the .v /fc, u flowing Quantities and y, by which thofe

any therefore the which at all times And Equation, indifferently exprefles of the will as the Relation flowing Quantities, well exprefs the and Relation between x -3- xo y-+-yo, as between x and y: So and be fubftituted in that x -+- xo y -f- yo may the fame Equation for thofe Quantities, inftead of x and y. 1 6. let #' ax* Therefore any Equation -+- axy ^' = be and fubftitute for x and given, x~\-xo } y -j- yo for y, and there will arife

+ $x*oox -f- x*o'' ax 1 2axox ax*oo

axy +- axoy -h ayox -h axyoo y: lyoy- ~ yfooy and INFINITE SERIES. 25

x 3 ax--3 3 =o, which there- 17. Now by Suppofition raxy _}' the divided fore being expunged, and remaining Terms being by o, 1 there will remain ^xx* -f- ^ox -+- x>oo zaxx ax o -f- axy -f- o. But whereas o is ayx _f_ axyo 3_vy* 3y*oy y*oo = fuppofed to be infinitely little, that it may reprefent the Moments of Qiian- that are it will be in relbedl tities ; the Terms multiply'd by nothing of the reft. Therefore I reject them, and there remains $xx* zaxx -f- axy -+- ayx 3_yj*= o, as above in Examp. i. 1 8. Here we may obferve, that the Terms that are not multiply'd by o will always vaniih, as alfo thole Terms that are multiply'd by o of more than one Dimenfion. And that the reft of the Terms will the form that being divided by o, always acquire they ought to have by the foregoing Rule : Which was the thing to be proved. the other 19. And this being now fhewn, things included in the that in Rule will eafily follow. As the propos'd Equation feveral be involved and that the flowing Quantities may ; Terms may be of multiply'd, not only by the Number the Dimenlions of the flow- but alfo other Arithmetical ing Quantities, by any Progreilions ; fo that in the Operation there may be the lame difference of the Terms according to any of the flowing Quantities, and the ProgrefTion be difpos'd according to the fame order of the Dimenlions of each of them. And thele things being allow'd, what is taught belides in and will be Examp. 3, 4, 5, plain enough of itfelf.

P R O B. II.

the An Equation being propofed, including Fluxions of to the Relations O^uantitieS) find of tbofe Quantities to one another.

A PARTICULAR SOLUTION.

i. As this Problem is the Converfe of the foregoing, it muft be folved in a by proceeding contrary manner. That is, the Terms x to the multiply'd by being difpofed according Dimenfions of x ; * muft be divided and then the they by x , by number of their Di- or fome other menfions, perhaps by Arithmetical Progreffion. Then the fame work muft be with the repeated Terms multiply'd by v, y, E or 26 The Method of FLUXIONS, or z, and the Sum refulting muft be made equal to nothing, re- jeding the Terms that are redundant. 2. EXAMPLE. Let the Equation propofed be ^xx* 2axx 4- axy 4- ayx = o. The Operation will be after this manner :

Divide 3 ATA?* 2axx-i-axy Divide * -f- ayx

- 5 by Quot. 3A: 2ax* -\-ayx by Quot. 3 * 4- axy ^. . 2 i. . Divide by 3 Divide by 3 2 i. 1 A; 5 ax 5 * Quote -{-ayx Quote _y 4- axy

3 Therefore the Sum # ax* -f- axy y* = o, will be the required of the is be Relation Quantities x and y. Where it to obferved, that tho' the Term axy occurs twice, yet I do not put it twice in the Sum x'> ax* -+- axy y* =. o, but I rejed the redundant Term. And fo whenever any Term recurs twice, (or oftener when there are feveral flowing Quantities concern'd,) it muft be wrote only once in the Sum of the Terms. 3. There are other Circumftances to be obferved, which I mall/ leave to the of the Artift for it would be needlefs to dwell Sagacity -, too long upon this matter, becaufe the Problem cannot always be folved by this Artifice. I mail add however, that after the Rela- tion of the Fluents is obtain'd by this Method, if we can return, by Prob. i. to the propofed Equation involving the Fluxions, then the work is right, otherwife not. Thus in the Example propofed,

1- after I have found the Equation x> ax -{- axy y* = o, if from thence I feek the Relation of the Fluxions x and y by the firft Problem, I mall arrive at the propofed Equation ^xx* 2axx 4- o. Whence it is that the axy i,yy* -f- ayx= plain, Equation 3 3 o is if the AT -ax*-+-axy _y = rightly found. But Equation xx xy -\- ay = o were propofed, by the prefcribed Method I fhould obtain this ^x* xy + ay = o, for the Relation between Conclufion would be erroneous: Since Prob. i. x and y ; which by the Equation xx xy yx -+- ay = o would be produced, which is different from the former Equation. 4. .Having therefore premiled this in a perfundory manner, I lhall now undertake the general Solution. A and IN FINITE SERIES. 27

A PREPARATION FOR THE GENERAL SOLUTION.

Firft it mufl be that in the 5. obferved, propofed Equation the Symbols of the Fluxions, (fince they are Quantities of a diffe- of which are rent kind from the Quantities they the Fluxions,) in to the ought to afcend every Term fame number of Dimenfions : And when it happens otherwife, another Fluxion of fome flowing be Quantity mufl be underflood to Unity, by which the lower Terms are fo often to be multiply'd, till the Symbols of the Fluxions arife to the fame number of Dimenfions in all the Terms. As if o were the Fluxion the Equation x -+ x'yx axx = propofed, z be be of fome third flowing Quantity z mufl underilood to Unity, by which the firfl Term x mufl be multiply'd once, and the lafl afcend to as axx twice, that the Fluxions in them may many Di- menfions as in the fecond Term xyx : As if the propofed Equation had been derived from this xz -{-xyx- azzx*- = o, by putting z = i. And thus in the Equation yx =}')'-, you ought to imagine x to be Unity, by which the Term yy is multiply'd. 6. Now Equations, in which there are only two flowing Quan- arife to the fame number of tities, which every where Dimenfions, may always be reduced to fuch a form, as that on one fide may be or - or ~ had the Ratio of the Fluxions, 4 , , and on the (as x ,&c.) \ x . y other fide the Value of that Ratio, exprefs'd by fimple Algebraic *

as fee 4- 2 -h 2X And when the Terms ; you may here, = y. foregoing particular Solution will not take place, it is required that you fhould bring the Equations to this form. that Ratio Term is de- 7. Wherefore when in the Value of any is or if that Ratio nominated -by a Compound quantity, or Radical, be the Root of an affected the Reduction mufl be Equation ; perform'd either by Divifion, or by Extraction of Roots, or by the Refolution of an affected Equation, as has been before fhewn. 8. As if the Equation ya yx xa -+- xx xy = o were pro-

firfl or pofed j by Reduction this becomes T-=i-f--^-, -== x ax y

And in the firfl I av+y' Cafe, if reduce the Term ^^., denominated by the compound Quantity a x, to an infinite Series of E 2 fimple 28 The Method of FLUXIONS, Terms - ~ fimple j -f- -f- -+- ^ &c. by dividing the Numerator

the Denominator a I mall have - - y by x, i +- -f- ^ -f.

^ -f- 7; &c. by the help of which the Relation between x and y is to be determined. So 9. the Equation = -j- .XVY.V A: or ^- 4, _y_y xy being given, = A-* x i- and a farther Reduction xx, by 4=4 +V/T -+- A-* : I extract AT the Root out of the fquare Terms -J -f- xr, and obtain the infinite Series 6 f -{-x* x* -f- 2X 5*" -f- 14*', &c. which if I fubftitute for - 6 \/t H- xx, I (hall have = i -f- x* x* -f- 2x X

~ 6 8 &c. or. = x^-ir-x* 2X -+- &c. as 5* , according is either added to -I, or fubtracled from it. 10. And thus if the ~ Equation y* -j- axx*y -f- a'-x^y x*x">

' 1 - 3 3 2x*a>=o were propofed, or -f- ax -f- a >v 2rf = o A:5 A: x I extract the Root of the affected Cubick Equation, and there.

/- X XX ~V ^_111*5_ COQi'4 anfes =a &c . as be feen ^-_|_ a 4. ^ may^ x 4 640 5i2 16384^3 before. 11. But here it may be obferved, that I look upon thofc Terms only as compounded, which are compounded in refpect of flowing Quantities. For I efteem thofe as fimple Quantities which are compounded only in refpect of given Quantities. For they may be reduced to fimple Quantities by luppofing them equal to other givea " Thus I eonfider the -> Quantities.- Quantities "-TT, rr- ^ ^^' c a*4- b' ax-\~bx > 1 4

> v/tfA- H- &c. as becaufe , i bx, Quantities, ~^, L x fimple they may

all be reduced to the i, , \/ex may fimple Quantities ^ -^-, (or x*} &cc. by fuppofing a -f- b =r= e. that the more 12. Moreover, flowing Quantities may the eafily be diflinguifh'd from one another, the Fluxion that is put in the Numerator of the Ratio, or the Antecedent of the Ratio, may not improperly be call'd the Relate Quantify, and the other in the De- nominator, to which it is compared, the Correlate : Alfo the flowing and INFINITE SERIES. 29 be the fame Names flowing Quantities may diftinguifli'd by refpec- for the better of what follows, tively. And understanding you may is or other conceive, that the Correlate Quantity Time, rather any which Time is and Quantity that flows equably, by expounded meafured. And that the other, or the Relate Quantity, is Space, which the moving Thing, or Point, any how accelerated or retarded, defcribes in that Time. And that it is the Intention of the Problem, that from the Velocity of the Motion, being given at every Inftant be of Time, the Space defcribed in the whole Time may determined. in of this Problem be 13. But refpedt Equations may diftinguifli'd into three Orders. Fluxions of and one 14. Firft: In which two Quantities, only of their flowing Quantities are involved. In which the two are involved, 15. Second: flowing Quantities their Fluxions. together with 1 6. Third: In which the Fluxions of more than two Quantities are involved. thefe Premifes I {hall the Solution of the 17. With attempt Problem, according to thefe three Cafes.

SOLUTION OF CASE I.

which alone is contain 'd in 1 8. Suppofe the flowing Quantity, and the accord- the Equation, to be the Correlate, Equation being on one fide to be the ingly difpos'd, (that is, by making only Ratio of the Fluxion of the other to the Fluxion of this, and on the other fide to be the Value of this Ratio in fimple Terms,) mul- Value the Ratio of the Fluxions the Correlate tiply the of by Quan- divide each of its Terms the number of Dimenfions tity, then by with which that Quantity is there afTeded, and what arifes will be other equivalent to the flowing Quantity.

the xxxx ; I x 19. So propofing Equation yy = xy -+- fuppofe to be the Correlate Quantity, and the Equation being accordingly

1 & have - i .v 4 2X &c. Now I mul- reduced, we mall = -f- x -f- ,

' of into and there arifes 3 x f tiply the Value x, .v-f-AT -{- 2X\

&c. which Terms I divide feverally by their number cf Dimenfions, the Refult 1 &c. I And and x +- fv' fv'-f-fv , put =y. by this 30 77je Method ^/"FLUXIONS,

this will be defined the Relation Equation between x and y, as was required.

' '* 3 be -- - 3 20. Let the Equation = a -4- -f- &c. there x 4 6-}

' will arife ax ~ y = -+- j- &c . for the y ' y ZM -^OJ.oi.t~ determiningo Relation between A; and y. 21. thus the x* And Equation _i_ -,, -t- #*, = I I v-J *.!

2 gives y = -f- . + ^ x ** . For the ^ ^ |. *+ multiply - Value of into and it becomes - - A;, -f. ax^ . x* -*- Xv* *; Jf^ X ,

1 or A:- x' 1 ax* which Terms -\- x^-i-x^, being divided by the number of Dimenfions, the Value of y will arife as before.

22. -^ After the fame manner the Equation -. =5-7=== 4- -+- \/ f S7- 1. A

A- - 3 \- cy, gives = -}- H- -i- . For ^_ v/^)' cy~> the

* Value of - there arifes ~ being multiply'd by j, -^ _j_

-{-n'3 or 3 v/^ c 2^^-y* -h -~i ;' + + %y*. And thence -the Value of x refults, by dividing by the number of the Dimen- lions of each Term.

And fo =? = And -1 =- 23. =z\ gives y $z*. , gives ~ * 47 r= , 3 But the . f . = , = For 3f^L Equation ^ ; gives 7 f multiply'd into A: makes a, which being divided by the number of Dimen- ~ fions, which is there arifes an infinite o, , Quantity for the Value

_ 24. Wherefore, whenever a like Term mail occur in the Value

of -. , whofe Denominator involves the Correlate Quantity of one Dimenfion inftead the only ; of Correlate Quantity, fubftitute the Sum or the Difference between the fame and fome other given to be Quantity affumed at pleafure. For there will be the fame Relation of of the Flowing, Fluents in the Equation fo. produced, as of the at firft infinite Equation propofed j and the Relate Quan- tity and INFINITE SERIES. 31

means will be diminifh'd an infinite of tity by this by part itfelf, and will become finite, but yet confifting of Terms infinite in number.

the - if 25. Therefore Equation 4 = being propofed, for x I write ^4- x, affuming the Quantity b at pleafure, there will arife a^ v 11 T^ /* v fl ax^ ax^ c At - : and Divifion 4 T &c - And , by* = rr 4- 77 -rr = & u-^r~X v O b + now the Rule will - 4- ~p &c. for aforegoing give_}'= j ^ 3 ~j^ the Relation between x and y. 26. So if you have the Equation - = - 4-3 xx; becaufe X X ~ write i for there will arife of the Term x -> if you -f- x x, 4

. _f 2X xx. Then the Term into an in- (_ 2 reducing ~-^

2x l 2Ar 3 &c. will have finite Series 4-2 2x4- 4- 2x% you 4 , X 3 4 x* 2x 2x , &c. And then to the Rule ^ 4* _{_ 4- according 1 3 4 s 6cc. for the Relation of y = 4.x ax 4- fx |x 4- ^x , x and y.

1 AT* were 27. And thus if the Equation -.-=x'^-i-x- pro-

becaufe I here obferve the Term x l to be pofed j (or ~j found,

I for and there I tranfmute x, by fubftituting x it, arifes 4

l - - - . _ L __ 1 A;". the Term _' v/ Now x produces

1 3 and the Term \/i x is to i _{_ x _|_ x 4- x , &c. equivalent

1 an(^ therefore or i is j, .i# . a 4-x V^S _ v _ JL;( ^ 1 -i-x 3 that thefe Values the fame as i 4- 4- 4-x 4- |-x , &c. So when

i are fubftituted, I fhall have 4 = i ~f- 2x 4- 4-4-^-x 3 ,6cc. And X 4x 1 4 the Rule =. x x &c - An

As if this were 4 = inflead Equation propofed -^ ^^. c ^_ xi 52 Method of O ^ i/ FLUXIONS,

c and then I mall have or ~i> of .v I write AT, 4= ^ 75

- J and then the Rule = -f,. L. But the ufe of fuch Tranf- by y ^ mutations will appear more plainly in what follows.

SOLUTION OF CASE II.

fo for that involve 29". PREPARATION. And much Equations in the only one Fluent. But when each of them are found Equation, fiift it muft be reduced to the Form prefcribed, by making, that on one fide may be had the Ratio of the Fluxions, equal to an aggregate of fimple Terms on the other fide. fo reduced there be 30. And befides, if in the Equations any Fractions denominated by the flowing Quantity, they muft be freed from thofe Denominators, by the above-mentioned Tranfmutation of the flowing Quantity. o or 31. So the Equation yax xxy aax = being propofed,

i_l _{_ f . becaufe of the Term -, I afiume b at pleafure, and x a x * - if I fhould for x I either write b -+- x, or b x, or x b. As

- . then the Term write b -+- x, it will become 4 = -f- rrr. And

an infinite we mall have being converted byDivifion into Series,

-1-1 , - - < , &C.

72. after the fame manner the = 37 2x J And Equation X + X 2v - .. reafon of the Terms - being propofed; if, by and^.,

i for and i x for there will arife = I write y x, y y X

2 v 2 ^- -^ - '- ^ _ o V -4- 2 x -f- = -4- ~. r . But the Term byJ ZX X* 1 3/ 1 y I -\- y a 3 &c. infinite Divjfion gives i x -+-y xy -f-_y xy* -J-_y xy* t

- a like 2 and the Term _^ by Divifion gives 2_y -i- ^xy t 2~+ xx a 3 5 x 6x . 6x S* 8x &c. There- ^ _f- *-y 4- ^ + iox*y IOAT*,

a 5 fore ' &c. -i- 6x* r-= 3^-i- 3^J -f->' xy* -{- y3 ^y , 6^^ X

33- and INFINITE SERIES. 33

33. RULE. The Equation being thus prepared, when need re- the quires, difpofe Terms according to the Dimenfions of the flowing Quantities, by fetting down fir ft thofe that are not affected by the Relate Quantity, then thofe that are affected by its lead Dimen- fo fion, and on. In like manner alfo diipofe the Terms in each of thefe Clafies according to the Dimenfions of the other Correlate Quantity, and thofe in the firft Clafs, (or fuch as are not affected by the Relate Quantity,) write in a collateral order, proceeding towards the right hand, and the reft in a defcending Series in the left- hand Column, as the following Diagrams indicate. The work being thus prepared, multiply the firft or the loweft of the Terms in the firft Clafs by the Correlate Quantity, and divide by the number of Dimenfions, and put this in the Quote for the initial Term of the Value of the Relate Quantity. Then fubftitute this into the Terms of the Equation that are difpofed in the left-hand Column, inftead of the Relate Quantity, and from the next loweft Terms you will obtain the fecond Term of the Quote, after the fame manner as you obtain'd the firft. And by repeating the Operation you continue the as far as may Quote you pleafe. But this will appear or plainer by an Example two.

EXAMP. i. Let the i 34. Equation 4 = ^x-\-y-\- x*-{-.vy

1 be whofe Terms i T.V A' affected propofed, -+- , which are not the Relate fee in the by Quantity _v, you difpos'd collaterally up- 3-4 The Method of FLUXIONS, beino; divid'-d by the number of Dimenfions 2, gives xx for the fecund Term of the Value of y in the Quote. Then this Term to being likewifc afiumed compleat the Value of the Marginals -{-y there will arife alfo xx and x 5 to be and -+- xv, , added to the Terms -j-x and -{-xx that were before inferted. Which being done, I again a flume the next loweil Terms -f-xx, xx, and -{-xx, which I collect into one Sum xx, and thence I derive (as before) the third Term ; to be in the Value of -|-.ix , put y. Again, taking this Term -i-x 3 into the Values of the marginal Terms, from the 3 and x 3 added I 4 next loweft -f-y# together, obtain ^-x for the of fo the fourth Term of Value y. And on in infinitum. 35. Ex AMP. 2. In like manner if it were required to determine y v - - -- the Relation of x and y in this Equation, -=. I -f- -f- -f- r'-f- < ^ a &* &*

- &c. which Series is to ad I I , fuppofed proceed infinitum ; put in the beginning, and the other Terms in the left-hand Column, and then purfue the work according to the following Diagram. and INFINITE SERIES. 35 In like 37. EXAMP. 3. manner if this Equation were propofed

X 3 3 = 3,v -+- i*y -4-;* j* -t-j .vy -4-;- A^

1 J - 3 6..Y SA- 8.v it is to -f- _V 4- \oxy* IOA-*, &c. and intended Value ot as far as extract the y feven Dimensions of x. I place the Terms in order, according to the following Diagram, and I work as before, only with this exception, that iince in the left-hand Co- lumn y is not only of one, but alfo of two and three Dimensions; (or of more than three, if I intended to produce the Value of y the of x~* I the beyond degree ,) fubjoin fecond and third Powers of the Value of y, fo far gradually produced, that when they are fubftituted by degrees to the right-hand, in the Values of the Marginals 3 6 The Method of FLUXIONS,

Equation required. But whereas this Value is negative, it appears that one of the Quantities x or y decreafes, while the other increafes. And the fame thing is allb to be concluded, when one of the Fluxions is affirmative, and the other negative. 38. EXAMP. 4. You may proceed in like manner to refolve the Equation, when the Relate Quantity is affected with fractional Di- menfions. As if it were propofed to extract the Value of x from this - 1 ; in = iy -+- -J.v -f- -f- , Equation, ^y- zyx* 77* 2_y and INFINITE SERIES. 37 38 tte Method of FLUXIONS,

41. And thus at laft I have compleated this moft troublefo'me all others moft difficult and of Problem, when only two flowing Quantities, together with their Fluxions, are comprehended in an Equation. But befides this general Method, in which I have taken in all the Difficulties, there are others which are generally fhorter, by which the Work may often be eafed; to givefome Specimens of which, ex abundantly perhaps will not be diiagreeable to the Reader. 42. I. If it happen that the Quantity to be refolved has in fome places negative Dimenfions, it is not of ablblute necefllty that therefore the Equation mould be reduced to another form. For thus - the Equation y = xx being propofed, where y is of one negative Dimenfion, I might indeed reduce it to another Form, as i for but the Refolution will be more by writing -f- y y ; expedite as you have it in the following Diagram. and INFINITE SERIES, 3.9 The Method ^FLUXIONS, and INFINITE SERIES, 41

here it be obferved the that as the 50. And may by way, Opera- have inferted between tion proceeded, I might any given Quantity

and - for that is the Terms 4** , the intermediate Term deficient,

and fo the Value of y might have been exhibited an infinite variety of ways. 51. V. If there are befides any fractional Indices of the Dimen- fions of the Relate Quantity, they may be reduced to Integers by fuppofing that Quantity, which is affected by its fractional D- to be to third Fluent and then ftibftitutii menfion, equal any ; by g that Quantity, as alfo its Fluxion, ariling from that fictitious Equation, inftead of the Relate Quantity and its Fluxion. As if the the 52. Equation y= 3*7* -\- y were propofed, where Relate Quantity is affected with the fractional Index .1 of its Dimen- a or fion; Fluent z being afTumed at pleafure, fuppofe y^ = z, z'> i. will be y = ; the Relation of the Fluxions, by Prob. 1 . alfo z* for y = 32Z Therefore fubftituting ^zz* for v, as y, and z* for there will arife 1 z 3 or z x y$, yzz = ^xz*- -+- , = -\-^z, where z performs the office of the Relate Quantity. But after the

is Value of z as z x* -J- , &c. in- extracted, = -f- -f- ^ -^-Q ftead of z reftore y\ and you will have the defired Relation be- 1 3 4 tween x and v that i.v c - an(^ ; is, y? = + -V^ H- T-nr* ; & ^7 6 7 S each '_.v X c - Cubing fide, y =.^x -\- T -+- TYT > ^ 53. In like manner if the Equation y =

1 1 1 -i-v or -i.v ", = Ar-f- , then each fide, v=y>; -+- y' by fquaring -|Jf^ 5 -i- -i-x . But if you mould defire to have the Value of y exhibited an infinite number of make z c x ways, =. -f- -f- -ytf , aiTuming any initial Term c, and it will be ss, that is y, = c* -{- zcx + ^cx*

1 1 * -v 3 + -+- -i-x -t- ^v . But perhaps I may feem too minute, in treating of fuch things as will but feldom come into practice.

SOLUTION OF CASE III.

54. The Refolution of the Problem will foon be difpatch'd, when the Equation involves three or more Fluxions of Quantities. For G between 42 ?$ Method of FLUXIONS, between any two of thofe Quantities any Relation may be afiumed, when it is not determined by the State of the Queftion, and the Re- lation of their Fluxions be found from thence fo either may ; that of them, together with its Fluxion, may be exterminated. For which reafon if there are found the Fluxions of three Quantities, only one need to be two if there be Equation affumedj four, and fo on j that the Equation propos'd may finally be transform'd into another Equation, in which only two Fluxions may be found. And then this Equation being refolved as before, the Relations of the other Quantities may be difcover'd. Let the be zx z o that I 55. Equation propofed -f- yx = ; of may obtain the Relation the Quantities x, y, and z, whofe Fluxions and z are contained in the I at x, y, Equation ; form a Relation pleafure between any two of them, as x and y, fuppofing that x=y, or a or &c. But 2y = -+- z, x=yy, fuppofe at prefent x=yy, and thence x = 2yy. Therefore writing zyy for x, and yy for x, the will be transform'd into this : Equation propofed q.yy z-^-yy* = o. And thence the Relation between y and z will arife, 2yy-{- =.z. In which if x be written for and x* for we mall ^y= yy, y~>, have 2X -f- ~x^ = z. So that among the infinite ways in which be related to each one of is x, y, and z, may other, them here is thefe found, which reprefented by Equations, .v =yy, 2y* +- y* = z, and 2X -+- ^x* = z.

DEMONSTRATION.

thus we have folved the but the 56. And Problem, Demonftra- fo tion is ftill behind. And in great a variety of matters, that we not derive it and with too may fynthetically, great perplexity, from it be fufficient to its genuine foundations, may point it out thus in of That when is fhort, by way Analyfis. is, any Equation propos'd, after you have finifh'd the work, you may try whether from the can return back to the derived Equation you Equation propos'd, by Prob. I. And therefore, the Relation of the Quantities in the derived Equation requires the Relation of the Fluxions in the propofed

: to be Equation, and contrary-wife which was fhewn. were the derived 57. So if the Equation propofed y = x, Equa- l and on the Prob. i. we tion will be y={x ; contrary, by have that becaufe x is And thus y xx, is, y=.x, fuppofed Unity. from and INFINITE SERIES. 4.3

1 from = I 3* -f- xx -+- is derived = tf x* -f- Lx y -+-y xy _y ! x vS &c - thence i. i ^v+ -+- ^o -4T' > And by Prob. y = 2x 1 x! &c. Which two Values of ^-x %x> -+- ^-x* -V ) y agree with each other, as appears by fubftituting x xx+^x> -^x* ->-J-x s <5cc. inftead in the firft Value. , of^ But in the Reduction of I made ufe of an .,8. Equations Opera- tion, of which alfo it will be convenient to give fome account. And its connexion that is, the Tranfmutation of a flowing Quantity by with a given Quantity. Let AE and ae be two Lines indefinitely extended each way, along which two moving Things or Points may pafs from afar, and at the fame time may reach the places A and a, B and A E c p E ' b, C and c, D and d, &c. and let B be the Point, by its diftance from which, -4 : i ^ ? the Motion of the moving thing or in is eftimated fo that point AE ; BA, BC, BD, BE, fucceffively, may be the flowing Quantities, when the moving thing is in the places A, C, D, E. Likewife let b be a like point in the other Line. Then will BA and ba be contemporaneous Fluents, as alfo BC and be, BD andZv/, BE and be, 6cc. Now if inftead of the points B and b, be fubftituted A and c, to which, as at reft, the Motions o are refer'd ; then and ca, AB and cb, AC and o, AD and cd, AE and ce, will be contemporaneous flowing Quantities. There- fore the flowing Quantities are changed by the Addition and Sub- of the AB and ac but are not traclion given Quantities ; they changed as to the Celerity of their Motions, and the mutual refpect of their the Fluxion. For contemporaneous parts AB and ab, BC and be, CD and cd, DE and de, are of the fame length in both cafes. And thus in Equations in which thefe Quantities are reprefented, the contemporaneous parts of Quantities are not therefore changed, not- withftanding their ablblute magnitude maybe increafed or diminimed by fome given Quantity. Hence the thing propofed is manifeft : of this to the For the only Scope Problem is, determine contemporaneous Parts, or the contemporary Differences of the abfolute Quan- tities or defcribed with a Rate of And f, x, _>', z, given Flowing. it is all one of what abfolute magnitude thofe Quantities are, fo that their contemporary or correfpondent Differences may agree with the prcpofed Relation of the Fluxions. this matter Al- 59. The reaibn of may alfo be thus explain'd Let the be and gebraically. Equation y=xxy propofed, fup- G 2 pole 77je Method 44. of FLUXIONS,

i Then Prob. i. x z. So that for =-. pofe x= -+-Z- by = y xxy , =. fince it is that may be wrote y xy -h xzy. Now ,v=s, plain,, though the Quantities x and z be not of the fame length, yet that alike in of that have they flow refpecl: y, and they equal contem- therefore I not the fame poraneous parts. Why may reprefent by Symbols Quantities that agree in their Rate of Flowing,; and to determine, their contemporaneous Differences, why may not I uie v === xy + xxy initead of y = xxy ? 60.. Lartly it appears plainly in what manner the contemporary be from an parts may found, Equation involving flowing Quantities. ~ Thus if -+- x be the when # = 2, then y = Equation, _y = 24. Therefore But when x = 3, then y =. 3.1. while x flows from flow from 2-i to So that the 2 to 3, y will 3.1. parts defcribed in this time are 3 2 = i, and 3-^ 2-i = f . 6 1. This Foundation being thus laid for what follows, I fhall now proceed to more particular Problems. PROB. m.

^ determine the Maxima and Minima H^ A ltijt'1 of

1. When a Quantity is the greateft or the leaft that it can be, at that moment it neither flows backwards or forwards. For if it flows forwards, or increafes, that proves it was lefs, and will pre- it is. the if it flows fently be greater than And contrary backwards, or decreafes. Wherefore find its Fluxion, by Prob. i. and fuppofe it to be nothing. 1 i. If in the x> ax 3 2. Ex AMP. Equation + axy jy = o the x be find the Relation of greatefl Value of, required ; the Fluxions a l of x and y, and you will have 3X.v 2axx -f- axy %yy -i-ayx 1 = o. Then making x = o, there will remain yyy -\- ayx=o, the of this or 3j* = ax. By help you may exterminate either x and the or y out of the primary Equation, by refulting Equation you then of may determine the other, and both them by 3^* -f- ax = o. is the as if had 'd 3. This Operation fame, you multiply the Terms of the propofed Equation by the number of the Dimenfions of the other flowing Quantity.^. From whence we may .derive the famous 2. and INFINITE SERIES. 45 of in famous Rule Huddenius, that, order to obtain the greateft or Relate the leaft Quantity, Equation mufl be difpofed according to of the Dimenfions the Correlate Quantity, and then the Terms are to be multiply 'd by any Arithmetical ProgrelTion. But fince neither nor other I this Rule, any that know yet publiihed, extends to Equa- tions affected with iiird without a Quantities, previous Reduction j I fhall the give following Example for that purpofe. 4. EXAMP. 2. If the greatest Quantity y in the Equation x* a -- xx a xx be to be y~ + 7+ ^ y ~+" = determin'd, feek the of and there will arife the .Fluxions xand^y, Equation 3^^* zayy-{- 1 2^n5 atx 2 ^^v) + Aaxxy-\-6x\* + - A j r \ r r I - fince __ = 0. And by fuppofition y = o, , a 1 2 _xx -\- zay +j* ^ ay -\- omit the Terms multiply'd by y, (which, to fhorten the labour, might have been done before, in the Operation,) and divide the reft "*"-'** by xx, and there will remain %x ^-a"xx = o. When the Re- duction is made, there will arife ^ay-\- %xx = o, by help of which either of the you may exterminate quantities x or y out of the propos'd Equation, and then from the refulting Equation, which will, be Cubical, you may extract the Value of the other. this Problem be had the of 5. From may Solution thefe following. I. In a given .Triangle, or in a Segment of any given Curve, ft> Reft ir.fcribe the greatejl angle. or the II. To draw the greatejl leafl right Line, 'which can lie: a between a given Point, and Curve given in pofition. Or, to draw. a Perpendicular to a Curvefrom a given Point. III. To draw the or the which greatejl leajl right Lines, pajjin?.- can lie bet-ween two through a given Point, others, either right Lines or Curves. IV. From a given Point within a Parabola, to draw a rivbt cut the Parabola Line, which Jhall more obliquely than any other. And to do the fame in other Curves. determine the Vertices their V. To of Curves, greatejl or lealT Breadths, the Points in which revolving parts cut each other, 6cc. VI. To the Points in where hcrce the find Curves, they greatejT or leajl Curvature. VII. To find the Icaft Angle in a given EHi/is, in which the. Ordinates can cut their Diameters.

VIII.. The 4.6 Method of FLUXIONS,

VIII. Of EHipfes that pafs through four given Points, to deter- Circle. mine the greateft, or that which approaches neareft to a IX. 70 determine fuch a part of a Spherical Superficies, which can be illuminated, in its farther part, by Light coming from a great dijlance, and which is refracted by the nearer Hemijphere. more be And many other Problems of a like nature may eafily propofed than refolved, becaufe of the labour of Computation.

P R O B. IV.

To draw Tangents to Curves.

Firft Manner. to the various 1. Tangents may be varioufly drawn, according Relations of Curves to right Lines. And firft let BD be a right Line, or Ordinate, in a given Angle to another right Line AB, as a Bafe or Ab- at fcifs, and terminated the Curve ED. Let this Ordinate move through an inde- to the fo finitely finall Space place bd, that it may be increafed by the Moment is increafed the Moment cd, while AB by ^ A Bb, to which DC is equal and parallel. 1 Let Da be produced till it meets with AB in T, and this Line will in or touch the Curve D d ; and the Triangles dcD, DBT will be fimilar. So that it is TB : BD : : DC (or B) : cd. 2. Since therefore the Relation of BD to AB is exhibited by the

which the nature of the is determined ; feek for Equation, by Curve the Relation of the Fluxions, by Prob. i. Then take TB to BD in the Ratio of the Fluxion of AB to the Fluxion of BD, and TD will touch the Curve in the Point D. Ex. i. AB = x, and BD let their Relation be 3. Calling =jy, 3 -, ax* = o. And the Relation of the Fluxions will x -h axy _y -i be 3xx 2axx-i-axy ^yy* -+- ayx-=. o. So that y : x :: ^xx : ax :: BD : BT. Therefore BT 2ax -4- ay ^ (;-) = ~~ f!X w* ~ ... Therefore the Point D being given, and thence DB and AB, or v and x, the length BT will be given, by which the Tan- gent TD is determined. 4- and INFINITE SERIES. 47 of 4. But this Method Operation may be thusconcinnated. Make the Terms of the propofed Equation equal to nothing : multiply by of the Dimenfions the proper number of the Ordinate, and put the Refult in the Numerator : Then multiply the Terms of the fame Equation by the proper number of the Dimenfions of the Abfcifs, and put the Produdl divided by the Abfcifs, in the Denominator of the Value of BT. Then take BT towards A, if its Value be affirmative, but the contrary way if that Value be negative. o o 13 the 3 ax* multi- 5. Thus Equation* -f- axy y*=o, being 3 z 10 the 3 for the Numerator ply'd by upper Numbers, gives axy 3_y j and multiply 'd by the lower Numbers, and then divided by x, gives 1 for the of 3-x- zax -+- ay Denominator of the Value BT. 6. the 3 bed Thus Equation jy by* cdy -f- -\-dxy = o, (which denotes a Parabola of the fecond kind, by help of which Des Cartes confirufted of fix Dimenfions fee his Equations ; Geometry, p. 42. Ed. An. ^--"fr+'^v Amfterd. 1659.) by Infpeftion gives ^ Qr

r 1 1 And thus a -x* = o, denotes an 7. y (which Ellipfis whofe Center is or ^ BT. fo in others. A,) gives ^ , = And - X 1 8. And take it you may notice, that matters not of what quantity the Angle of Ordination ABD may be. But as this Rule to 9. does not extend Equations afFefted by furd or to Quantities, mechanical Curves ; in thefe Cafes we mufl have recourfe to the fundamental Method.

1 10. Ex. 2. Let A; S xx ay -+- j- \/'ay -+- xx = o be the the Relation between AB and and Equation exprefling BD ; by Prob. i. *"*"* 2 the Relation of the Fluxions will be + V 3*** zayy -f.

*/,.,,, 4 v ~ =0. Therefore it will be

:: : (y x ::) BD : BT. fayT^T-p ^^

II. 48 TJoe Method of FLUXIONS, ii. Ex. Let 3. ED be the Conchoid of Nicomedes, defcribed with the Pole G, the and Afymptote AT, the Diftance LD ; and let

'GA = , LD = c, AB=.v, andBD=;>. And becaufe of fimilar and it will : : : Triangles DEL be LB BD DM : DMG, MG ; that is, v/'cc : : : x : b and therefore yy y -+- y, b-\-y ^/cc yy =yx. Having got this Equation, I fuppofe Vcc yy = z, and thus I fliall have two bz andzz Equations ~\-yz =yx, = cc yy. the of thefe I find the the By help Fluxions of Quantities x, y, and Prob. i. From the firft arifes z, by bz -+-yz -\- yz =y'x -+- xy, and from the fecond 2zz = 2yy, or zz -j- yy = o. Out of thefe if we -^ exterminate z, there will arife -i-yz =yx

-+ which refolved it will be : z - xy, being y x : :

: x : BT. But as is (y ::) BD BD y, therefore BT= .3- - ~ That is, BT = AL -f- -; where the iff BL!- J_l Sign(_J to that the Point prefixt BT denotes, T mufl be taken contrary to the Point A.

12. And hence it SCHOLIUM. appears by the bye, how that of the Conchoid be point may found, which Separates the concave the convex For when is the lea ft from part. AT poffible, D will that Therefore make - be point. AT = v ; and fmce BT z by -\- yv x then v = z -+- 2K -+- Here to morten

for x fubftitute - ^l!5 is the work, > which Value derived from what

- ? - - is before, and it will be -f. z -+- = v. Whence the Fluxions and z found v, y, being by Prob. i. and fuppofing ^=0 and INFINITE SERIES. 49

' .,, ... iy, )K iy-l-zyy Azy-4-zvy --- o. bvProb. -3. there will anfe ~-t-z + -- --=i; = J J y jy z za

in : - for and cc for Laflly, fubftituting this z, yy zz, (which a values of z and zz are had from what goes before,) and making due Reduction, will have -+- -2.be* = o. the Con- you y'- ^by* By ftrudlion of which Equation y or AM, will be given. Then thro' of M drawing MD parallel to AB, it will fall upon the Point D contrary Flexure. if the whofe is to be 13. Now Curve be Mechanical Tangent drawn, the Fluxions of the Quantities are to be found, as in Examp.5. of Prob. i. and then the reft is to be perform'd as before. are cut in 14. Ex. 4. Let AC and AD be two Curves, which the Points C and D by the right Line BCD, apply 'd to the Abfcifs AB in a given Angle. Let AB = x, BD = y, and - = z. Then Prob. i. (by

it be z Preparat. to Examp. 5.) will = x ~T> ^ ^ B~ xBC. let be a or Curve and to deter- 15. Now AC Circle, any known ; mine the other Curve AD, let any Equation be propofed, in which 4 z is involved, as zz +- axz =_y . Then by Prob. i. 2zz +- axz -+- axz = 4X7*. And writing x x BC for z, it will be zxz x BC -+- axx x BC H- axz = 4)7'. Therefore 2z x BC -+- ax x BC -{-

J : : :: : x BD BT. So that if the nature of the az 4jy (y ::) the Ordinate and the Area Curve AC be given, BC, ACB or z ; the Point T will be given, through which the Tangent DT will pafs. 1 6. After the fame manner, if 32 = zy be the Equation to the

'twill be x BC So that : 2 :: Curve AD ; (3.3) 3^ = zy. 3BC

: : BT. And fo in others. (y x ::) BD Ex. Let BD as and let the 17. 5. AB=,v, =y, before, length be z. And a to as 'twill of any Curve AC drawing Tangent it, Cl, x x C/ : or be Bt : Ct :: x z, z = ^- 18. Now for determining the other Curve AD, whofe Tangent is to be drawn, let there be given any Equation in which z is in- '' it will be fo : Bf volved, fuppofe z ==)'. Then z=y, that Ct

: : : : But the Point the Tan- (y x :} BD BT. T being found, gent DT may be drawn. H 19- The Method of FLUXIONS,

19. Thus fuppofmg 'twill be KZ zx and xzsssyy, + = zyj >, for z there will writing arife xz -f- ^-^ = ayy. There- '' y-> ^ O/ fore * -I- : : : f~-' 27 BD : DT. 20. Ex . 6. Let AC be a Circle, or any other known Curve, whofe is and let Tangent Ct, AD be any other Curve whofe Tangent DT is to be drawn, and let it be defin'd by afTuming AB = to the Arch and AC ; (CE, BD Ordinates to AB in a being given Angle,) let the Relation of BD to CE or AE be t. T exprels'd by any Equation. 21. Therefore call AB or AC = x, BD =y, AE=z, and CE = v . And it is that plain v, x, and z, the Fluxions of CE, and each AC, AE, other as CE, Ct, and Et. Therefore are^to *x and C7 = i>, .v x ^ = z. 22. Now let any Equation be given to define the Curve AD, as . = Then z ; and y = therefore Et : y Ct :: (v x "') BD : BT. K Or let the 23. Equation be and it will be . r~>T? I TT- . yz+vx,y-. And therefore CE -4- Et

: Ct :: : : Ct (y x ::) BD BT. Or let the 24. finally, Equation be v* it ayy = y and will be

x . = So that : zayy (3^ =) 3*1;' 31;* x CE 2 ay x Ct :: BD : BT. Ex. Let FC be a 25. 7. Circle, which is touched by CS in C; and let FD be a Curve, which is defined by affuming any Relation of the to Ordinate DB the Arch FC, which is intercepted by DA drawn to the Center. Then letting fall CE, the Ordinate in the Circle, call AC or AF=i, AB

it B T ,S CF = /; and will be tz=(t^=) K . . ^.. and tv v, = (/x -^ =) z. Here I put z negatively, becaufe is AE is dirninifh'd while EC increafed. And befides AE : EC ::

AB : and INFINITE SERIES. 51

AB : BD, fo that zy = vx, and thence by Prob. i. zy -f- yx

and 'tis vx -f- xv. Then exterminating v, z, v, yx ty* tx* = xy. 26. Now let the Curve DF be defined by any Equation, from which the Value of t may be derived, to be fubftituted here. Sup- to the firft and Prob. i. pofe let ^=_y, (an Equation Quadratrix,) by

Whence : it will be / = y, fo that yx yy* yx* = xy. y xx

x :: : _ x : : BT. Therefore BT = x* (y :) BD(;') ADa - x; and AT = xx+yy = ^/. if there will arife 27. After the fame manner, it is // = ly, fo of others. = 6r, and thence AT= - x~ . And z/ / r the Arch the 28. Ex. 8. Now if AD be taken equal to FC, the fame names Curve ADH being then the Spiral of Archimedes ; of the Lines ftill remaining as were put afore : Becaufe of the right Angle ABD 'tis xx and therefore Prob. -{-yy=tf) (by i.) xx +yy = //. Tis alfo AD : AC : :

: fo that znd thence DB CE, tv=y t (by tv vf the Fluxion Prob. i.) -4- =y. Laftly, of the Arch FC is to the Fluxion of the or as right Line CE, as AC to AE, AD

t : : : t : to AB, that is, v x, and thence ix = vf. Compare the Equations now found, and you will fee

and thence xx . And there- 'tis tv -+-ix=y, -\-yy = (tt =) ^^ fore compleating the Parallelogram ABDQ^_, if you make QD : if :: : BT :: : x X : ; that is, QP_ (BD y ::) y ^ you take AP ! > PD will be to the = ; perpendicular Spiral. it will be 29. And from hence (I imagine) fufficiently manifeft, all to be by what methods the Tangents of fcrts of Curves are drawn. However it may not be foreign from the purpofe, if I alfo fliew how the Problem may be perform'd, when the Curves are re-

after other manner whatever : So that hav- fer'd to right Lines, any of feveral the eafieft and moil ing the choice Methods, fimple may always be ufed. H 2 Second $2 The Method of FLUXIONS,

Second Manner.

be a in 30. Let D point the Curve, from which the Subtenfe DG is drawn to a given Point G, and let DB be anOrdinate in any given Angle to the Abfcifs AB. Now let the Point D flow for an infinitely fmall fpace D^/ in the Curve, and in GD let Gk be taken equal to Gd, and let the Parallelo- gram dcBl> be compleated. Then Dk and DC will be the contemporary Mo- --- ments of GD and BD, by which they is transfer'd to are diminifh'd while D d. Now let the right Line ~Dd be produced, till it meets with AB in T, and from the Point T to the Subtenfe GD let fall the perpendicular TF, and then the Trapezia and Dcdk and DBTF will be like; therefore DB : DF :: DC : Dk. the of to 31. Since then Relation BD GD is exhibited by the for the Curve find the Equation determining ; Relation of the Fluxions, and take FD to DB in the Ratio of the Fluxion of GD to the Fluxion of BD. Then from F raife the perpendicular FT, which may meet with AB in T, and DT being drawn will touch the Curve in D. But DT muft be taken towards G, if it be affirmative, and the contrary way if negative. and 32. Ex. i. Call GD = x, BD =_>', and let their Relation 1 be x ~, ax -f- = o. Then the Relation of the Fluxions axy y"= 1 will be ^xx 2axx +- axy -f- ayx ^yy- = o. Therefore ^xx

: ax :: : : : So that zax -h ay ^yy (y x :) DB (y) DF. axy V , .' . in the Curve being ~ 1 Then any Point D given, and thence BD and GD or y and x, the Point F will be given alfo. From whence if the Perpendicular FT be raifed, from its concourfe T with the Abfcifs AB, the Tangent DT may be drawn.

. it that a Rule be derived as well 3 3 And hence appears, might here, as in the former Cafe. For having difpofed all the Terms of the given of Equation on one fide, multiply by the Dimensions the Ordinatejy, in the of a Fraction. and place the refult Numerator Then multiply Subtenfe and its Terms feverally by the Dimenfions of the x, dividing in the the refult by that Subtenfe x, place the Quotient Deno- minator of the Value of DF. And take the fame Line DF towards G if it be affirmative, otherwile the contrary way.. Where you and IN FINITE SERIES, 53

it is no matter how far diftant the Point G you may obferve, that it be at all nor what is the is from the Abfcifs AB, or if diftant, Angle of Ordination ABD. 3 be as before x* ax* -f- = o ; 34. Let the Equation axy J 3 for the and 2ax it gives immediately axy 3>' Numerator, 3** of the Value of DF. -+- ay for the Denominator is to a Conick 35. Let alfo a -+- -x~y=o, (which Equation and for the Denomi- Sedtion,) it gives y for the Numerator, fly nator of the Value of DF, which therefore will be 7 And thus in the thefe will be 36. Conchoid, (wherein thingsGA perform'd more expeditioufly than before,) putting = b,

it will be BD : DL :: = c, GD=x, and BD=^, (;) (c) or : GL Therefore = cb, GA (5) (x <:). xy cy xy cy o. to the Rule ^-^ - that cb = This Equation according gives , to fo is, x

v there will arife v \/ cc \x exterminated, yy 'Jjl U = xy.

1 : .v :: : ,v BD : DF. Therefore y ^/cc yy J2^ (y ::) (ji )

Third Manner.

39. Moreover, if the Curve be refer'd to two Subtenfes AD and BD, which being drawn from two given Points A and B, may meet at the Curve: Conceive that Point to flow D on through an infinitely little Del in Space the Curve ; and in AD and BD take Ak = Ad, and Bc = Bc/; and then kD and cD will be contemporaneous Moments of the Lines AD and - BD. Take therefore DF to BD in the Ratio of the Moment D& to the /r

Moment DC, (that is, in the Ratio of the Fluxion of the Line AD to the Fluxion of the LineBD,) and draw BT, FT perpendicular to BD, AD, meeting in T. Then the Trapezia DFTB and DM: will be fimilar, and therefore the Diagonal DT will touch the Curve. 40. Therefore from the Equation, by which the Relation is defined between AD and BD, find the Relation of the Fluxions by Prob. i. and take FD to BD in the fame Ratio. 41. Ex AMP. Suppofing AD = x, andBD=;', let their Rela- e tion be a o. This is to the of -f- j y = Equation Ellipfes for the fecond Order, whofe Properties Refracting of Light are fhewn by Des Cartes, in the fecond Book of his Geometry. Then the e- Relation of the Fluxions will be y ==o. 'Tis therefore e : d ::(>:# ::) BD : DF. 42. And for the fame reafon if a ^ y = o, 'twill be e : _ d : : BD : DF. In the firft Cafe take DF towards A, and contrary-wife in the other cafe. i. if cafe be- 43. COROL. Hence d-=.e, (in which the Curve comes a Conick Section,) 'twill be DF = DB. And therefore the Tri- angles DFT and DBT being equal, the Angle FDB will be bifected by v -K A the Tangent. 44. and INFINITE SERIES. 55 hence alfo thofe be manifeft of 44. COROL. 2. And things will in themfelves, which are demonstrated, a very prolix manner, by Des Cartes concerning the Refraction of thcfe Curves. For as much are in the in as DF and DB, (which given Ratio of d to e,) refpect of the Radius DT, are the Sines of the Angles DTF and DTB, of the of Incidence the Surface of the that is, Ray AD upon Curve, and of its Reflexion or Refraction DB. And there is a like reafon- of ing concerning the Refractions the Conick Sections, fuppofing that either of the Points A or B be conceived to be at an infinite diftance. of 45. It would be eafy to modify this Rule in the manner the foregoing, and to give more Examples of it : As alfo when Curves are refer'd to Right lines after any other manner, and cannot commodioufly be reduced to the foregoing, it will be very eafy to find out other Methods in imitation of thefe, as occafion mall require.

Fourth Manner.

46. As if the right Line BCD mould revolve about a given Point B, and one of its Points D mould defcribe a Curve, and another Point C fhould be the interfection of the right Line BCD, with another right Line AC given in pofition. Then the Re- lation of BC and BD being exprefs'd by any E- draw quation ; BF parallel to AC, fo as to meet DF, perpendicular to BD, in F. Alfo erect FT perpendicular to DF; and take FT in the fame Ratio to BC, that the Fluxion of BD has to the Fluxion of BC. Then DT being drawn will touch the Curve.

Fifth Manner. But if the Point 47. A being given, the Equation ihould exprefs the Relation between AC and BD } draw CG parallel to DF, and take FT in the fame Ratio to BG, that the Fluxion of BD has to the Fluxion of AC. Sixth Manner. Or if 48. again, the Equation exprefles the Relation between AC and let and in CD; AC FT meet H ; and take HT in the fune Ratio to BG, that the Fluxion of CD has to the Fluxion of AC. A. id the like in others. Seventh *fhe Method of FLUXION

Seventh Manner : For Spirals.

49. The Problem is not otherwise perform'd, when the Curves are refer'd, not to right Lines, but to other Curve-lines, as is ufiial in Mechanick Curves. Let BG be the Circumference of a Circle, in whole Semidiameter AG, while it revolves about the Center A, let the Point D be conceived to move any how, fo as to defcribe the Spiral ADE. And fuppofe ~Dd to be an in- thro' finitely little part of the Curve which D flows, and in AD take Ac = Ad, then cD and Gg will be contemporaneous Moments of the right Line AD and of the Periphery to BG. Therefore draw Af parallel cd, that is, perpendicular to AD, and let the Tangent

DT meet it in T ; then it will be cD : cd : : AD : AT. Alfo let Gt be parallel to the Tangent DT, and it will be : :: : : cd Gg (Ad or AD AG ::) AT At. 50. Therefore any Equation being propofed, by which the Re- lation is find exprefs'd between BG and AD ; the Relation of their Fluxions by Prob. i. and takeAi? in the fame Ratio to AD: And then Gt will be parallel to the Tangent. 51. Ex. i. Calling EG = x, and AD=^, let their Relation be 3 1 5 A: ax -f- = o, and Prob. i. zax-\- : axy jy by 3^* ay 3^*

: : : : : At. The Point / thus ax (y x :) AD being found, draw to will touch Gt, and DT parallel it, which the Curve. is the to the 52. Ex. 2. If 'tis y =y> (which Equation Spiral of 'twill be = y, and therefore a : b : : : x : Archimedes,} j (y :) AD : At. Wherefore by the way, if TA be produced to P,

that it be : :: a : b will be may AP AB y PD perpendicular to the Curve.

Ex. If : 53. 3. xx = by, then 2XX = by, and 2x b :: AD : thus A. And Tangents may be eafily drawn to any Spirals whatever.

Eighth and INFINITE SERIES. 57

Eighth Manner : For Quad ratr ices.

CA. Now if the Curve be fuch, that any Line AGD, being drawn from the Center A, may meet the Circular Arch inG, and the Curve in D; and if the Relation between the Arch BG, and the right Line DH, which is an Ordinate to the Bafe or Abfcifs AH in a given Angle, be determin'd by any Equation to whatever : Conceive the Point D move in the Curve for an infinite- to and the Pa- ly {mail Interval d, rallelogram dhHk being compleat- Jf fo that ed, produce Ad to c, then and D/' will be Moments of Ac = AD ; Gg contemporaneous the Arch BG and of the Ordinate DH. Now produce Dd ftrait on to T, where it may meet with AB, and from thence let fall on and the Perpendicular TF DcF. Then the Trapezia Dkdc DHTF will be fimilar; and therefore D/fc : DC :: DH : DF. And befides if Gf be raifed perpendicular to AG, and meets AF in f; becaufe of the Parallels DF and Gf, it will be DC : Gg :: DF : Gf. There-

: : : : as the or fore ex aquo, 'tis D G^ DH Gf, that is, Moments Fluxions of the Lines DH and BG. the which the Relation of 55. Therefore by Equation exprefies Relation and in- BG to DH, find the of the Fluxions (by Prob. i.) that Ratio take Gf, the Tangent of the Circle BG, to DH. Draw to which in F. at DF parallel Gf, may meet A/* produced And F creel the perpendicular FT, meeting AB in T; and the right Line DT being drawn, will touch the Quadratrix. Ex. i. EG and let it be xx 56. Making = x, DH=;', = fy; then Prob. = Therefore 2.x : b :: : x DH : (by i.)2xx by. (y ::) GJ; and the Pointy being found, the reft will be determin'd as above. But perhaps this Rule may be thus made fomething neater :

:: : Make x :y AB AL. Then AL : AD :: AD : AT, and then DT will touch the Curve. For becaufe of equal Triangles AFD and

'tis x x : : ATD, AD DF= AT DH, and therefore AT AD : (DF or

x : DH or 1 AD : f- AG AL. JB Gf G/::) or)

Let is the 57. Ex.2. x=y, (which Equation to the Quadratrix of the Ancients,) then #=v. Therefore AB : AD :: AD : AT. I 8. 58 *fhe Method ^FLUXIONS, Ex. Let 58. 3. axx=y*, then zaxx=sMy*. Therefore make

: : zax : : : : : : : 3;-* (x y :) AB AL. Then AL AD : AD AT. And thus determine you may expeditioufly the Tangents of any other Quadratrices, howfoever compounded.

Ninth Manner.

59. Laftly, if ABF be any given Curve, which is touch'd by the Line and a right Bt ; part BD of the right Line BC, (being an Or- dinate in any given Angle to the Abfcifs AC,) intercepted between this and another Curve DE, has a Relation to the portion of the Curve AB, which is exprefs'd by any Equation: You may draw a Tangent DT to the other Curve, the of this by taking (in Tangent ^ ^ Contact. and IN FINITE SERIES. 59

Contaft, or a right Angle, or any other given Angle, that is, from to or Lines a given Point, draw 'Tangents, or Perpendiculars^ right that Jhall have any other Inclination to a Curve-line. IV. From any given Point within a Parabola, to draw a right the the or Line, which may make with Perimeter greateji leaft Angle all Curves whatever. poj/ible. And Jb of V. To draw a right Line which may touch two Curves given in or the Curve in two when that can be done. pojition, fame Points, VI. To draw any Curve with given Conditions, which may touch in a another Curve given in pojition, given Point. VII. To determine the RefraSlion of any Ray of Light, that falls upon any Curve Superficies. The Refolution of thefe, or of any other the like Problems, will as not be fo difficult, abating the tedioufnefs of Computation, that there is any occalion to dwell upon them here : And I imagine if may be more agreeable to Geometricians barely to have mention 'd them.

; : P R O B. V.

At any given Point of a given Curve^ to find the Quantity of Curvature.

1. There are few Problems concerning Curves more elegant than c this, or that give a greater Infight into their nature. In order to its Refolution, I mufl: premife thefe following general Confederations. 2. L The fame Circle has every where trie fame Curvature, and in different Circles it is reciprocally proportional to their Diameters. If the Diameter of any Circle is as little again as the Diameter of another, the Curvature of its Periphery will be as great again. If the Diameter be one-third of the other, the Curvature will be thrice as much, &c. II. If a Circle its concave 3. touches any Curve on fide, in any given Point, and if it be of fuch magnitude, that no other tangent can be interleribed in the of Contact near that Point Circle Angles ; that Circle will be of the lame Curvature as the Curve is of, in that Point of Contact. For the Circle that conies between the Curve and another Circle at the Point of Contact, varies lefs from the Curve, and makes a nearer approach to its Curvature, than that other Circle does. And therefore that Circle approaches nea'-eil to its I 2 Curvature, 60 *fbe Method of FLUXIONS,

Curvature, between which and the Curve no other Circle can in- tervene. a 4. III. Therefore the Center of Curvature to any Point of the Curve, is the Center of a Circle equally curved. And thus Ra- dius or Semidiameter of Curvature is part of the Perpendicular to the Curve, which is terminated at that Center. at different Points will 5. IV. And the proportion of Curvature be known from the proportion of Curvature of aequi-curve Circles, or from the reciprocal proportion of the Radii of Curvature. 6. Therefore the Problem is reduced to this, that the Radius, or Center of Curvature may be found. therefore that at three of the Curve

lars ftand ; which is manifeft of itfelf. But there are feveral or 8. Symptoms Properties of this Point C', which may be of ufe to its determination. 9. I. That it is the Concourfe of Perpendiculars that are on each lide at an infinitely little diftance from DC. 10. II. That the Interfeftions of Perpendiculars, at any little finite diftance on each are and divided it fo fide, feparated by ; that thofe which are on the more curved fide D,f fooner meet at H, and thofe are on the other curved fide -Dd which iefs meet more remotely at h. 11. III. If DC be conceived to move, while it infifts perpendicularly on the Curve, that point of it C, (if you except the motion to or of approaching receding from the Point of Influence C,) will be leaft moved, but will be as it were the Center of Motion. 12. IV. If a Circle be defcribed with the Center C, and the diftance DC, no other Circle can be defcribed, that can lie between at the Contact. and INFINITE SERIES. 61

II or Circle n. V. Laftly, if the Center b of any other touching to C the Center of till at la it it co- approaches by degrees this, of the in which that Circle mall incides with 'it ; then any points cut the Curve, will coincide with the point of Contact D. each of thefe the means of 14. And Properties may fupply folving the Problem different ways : But we fliall here make choice of the the moit firlt, as being fimple. Point of the Curve let DT be a DC a 15. At any D Tangent, and the Center of as before. And let Perpendicular, C Curvature, AB be the Abfcifs, to which let DB be apply 'd at right Angles, and which DC meets in P. Draw to and DG parallel AB, CG per- to in which take pendicular it, Cg of any given Magnitude, and to which draw gb perpendicular it, meets DC in

: : : : BD : the Fluxion Cg gf (TB :) of the Ablcifs, to the Fluxion of the Ordinate. Likewife imagine the Point D to move in the Curve an infinitely little diftance Dd, and to drawing de perpendicular DG, and Cd perpendicular to the Curve, let Cd meet DG in F, and $g in/ Then will De be the Momen- tum of the Abfcifs, de the Momentum of the Ordinate, and J/ the Momentum of the contemporaneous right Line g. Therefore DF

. therefore the Ratio's of thefe -De^.^tLJC Having Moments,' or,* which is the fame thing, of their generating Fluxions, you will have to the Line is the the Ratio of CG given C^, (which fame as that of the Point will be DF to Sf,) and thence C determined. 16. Therefore let AB BD and z = x, =y, Cg- = i, g = ; then it will be i : z : : x : y, or z = r- . Now let the Mo- X of z be mentum S-f zxo, (that is, the Product of the Velocity and of an infinitely fmall Quantity o,} and therefore the Momenta and thence Dt'==xxo, de=yx.o, DF = .\o -f- . Therefore X

: :: : zo : xo . 'tisQ-(r) CG (Jf DF ::) + ^ That is, CG= xx \y

J 7- 62 7%e Method of FLUXIONS, whereas we are at to afcribe whatever 17. And liberty Velocity to the Fluxion of the Abfcifs as to we pleafe x, (to which, an the reft be referr'd make x equable Fluxion, may j) = i, and '-^ z-^. then and CG . And thence DG = and yJ = z,' = }'

18. Therefore any Equation being propofed, in which the Rela- to is for the Curve tion of BD AB exprefs'd denning ; firft find Prob. r. and at the the Relation betwixt x and yt by fame time fub- for Then from the ftitute i for ,v, and z y. Equation that arifes, Relation between by the fame Prob. i. find the #, y, and z, and at the fame time fubftitute i for x, and z for y, as before. And thus by the former operation you will obtain the Value of z, and by will have the Value of z which the latter you ; being obtain'd, pro- concave of the duce DB to H, towards the part Curve, that it be - - and draw to and may DH = , HC parallel AB, meet-

in C j then will C be the Center of ing the Perpendicular DC Cur- - vature at the Point D of the Curve. Or fince it is i -|- r.y. 7

PT PT Tk/-> DP TM-T or make DH== ' z 1 i. Thus the 19. Ex. Equation ax^-hx* y =;o being pro- is an to the whofe Latus pofed, (which Equation Hyperbola redtum 2 Tranfverfum ; there will arife Prob. is a, and ) (by i.) a +. zbx

l for and z for in the 2zy o, (writing x, y refulting Equation, which otherwife would have been ax -+ 2&xx = o and zyy ;) there arifes zb 2zz and hence again 2zy = o, (i z being again C the for ,v and y.) By firft we have z an(j b tne wrote = L^L } y i ^^ Therefore Point of the latter z = any D Curve being given, xand from thence and and confequently y, z z will be given, which known, make 7 = GC or and draw HC. being Z DH, 20. As if make definitely you = 3, and b=i, fo that 3#-f- be the condition of xx=yy may the Hyperbola. And if you aliume 9 and x=i, ^11^ = 2, z=, z= TT , DH= gL. raife the li being found, Perpendicular HC meeting the Perpendi- cular and IN FINITE SERIES. 63

drawn which is the make : cular DC before ; or, fame thing, HD

: i : . of HC :: (i z ::) Then draw DC the Radius Curva- ture. the 21. When you think Computation will not be too perplex, you fabfHtute the indefinite Values of z and z into - the may , Value of CG. Thus in the prefent Example, by a due Reduction S * 4 ' r will have DH . Yet the Value of DH you =y -j- ^ by Calculation conies out negative, as may be feen in the numeral Ex- But this that mufl be taken towards B ample. only fhews, DH ; for if it had come out affirmative, it ought to have been drawn the contrary way. be 22. COROL. Hence let the Sign prefixt to the Symbol -\-b changed, that it may be ax -bxx yy=zo, (an Equation to the ilLll^: Ellipfis,) then DH=;--f- . 23. But fuppofing b=. o, that the Equation may become ax -- to the ~ and yy o, (an Equation Parabola,) then DH = y -f- ; thence DG = \a -f- 2X. be 24. From thefe feveral Exprefilons it may eafily concluded, that the Radius of Curvature of any Conick Seftion is always

aa be is the 25. Ex. 2. If x*=ay* xy- propofed, (which Equa- l CiiToid of Prob. i. it will be firft T x tion to the Diodes,") by > =.2azy 6x 2xzz zxzy y-t and then = 2azy-+-2azz -2zy zxzy -4- , T.X a%z -4- *~~ 1 -3*x yy - ^ 2cv+ n-.! 2Z\ : So that z= 3-^. and z= . There- J 2.vy' - zay ay xj .v and fore any Point of the Ciflbid being given, and thence y, - there will be alfo & and z, ; which known, make given being K = CG. _ _ is the 26. Ex. 3. If b-jf-y^/cc yy =.vy were given, (which and Equation to the Conchoid, inpag.48;) make \/cc y\=zv, hi) the firft of there will arife -+- yv = xy. Now thele, (cc _vv for v = vv,) will give (by Prob. i.) 2yz = 2vv, (writing z ;) from thefe and the latter will give l>v -+-yv + zv =y -{- xz. And will be determined. But that z Equations rightly difpofed v and z may alfo be found; out of the laft Equation exterminate the Fluxion and there will arife -I- ~"^ i>, by fubilituting ^ , 7 Method of FLUXIONS,

= y -f- xz, an Equation that comprehends the flowing Quantities, without any of their Fluxions, as the Refolution of the firft Pro- blem requires. Hence therefore by Prob. i. we mall have ^2* byz Ijzv 2)zs )? "\vzv +- ZV = 2Z +- XZ. This Equation being reduced, and difpofed in order, will give z. ' + zz But when z and z are known, make = CG.

27. If we had divided the laft Equation but one by z, then

Prob. i . we - --- by mould have had -f- ^ -f- -f. -i; =

2 ; which ^, would have been a more fimple Equation than the for former, determining z. 28. I have given this Example, that it may appear, how the operation is to be perform'd in furd Equations: But the Curvature of the Conchoid be may thus found a fhorter way. The parts of the b Equation -\-y ^/cc v\' = xy being fquared, and divided by yy, there arifes ~ *" -f. ^ 2by y* = x*, and thence by Prob. i. x or ...

And ^^ 1 hence Prob. i. ~ z m -f- again*^ byJ z, By y4 y/9 zz the firft refult z is determined, and z by the latter. Ex. Let be 29. 4. ADF a Trochoid [or Cycloid] belonging to the Circle whofe Diameter is the Ordinate ALE, AE j and making BD to cut the Circle in L, AB=x, BD and the Arch AL=/, and the Fluxion of the fame Arch

= /. And firfl (drawing the Semidia- meterPL,)the Fluxion ofthe Bafe or Abfcifs AB will be to the Fluxion of the Arch AL, as BL to and INFINITE SERIES. 65

/. that A* or I : / : : v : ~a. And therefore = Then to PL ; is, ^ therefore from the nature of the Circle ax xx = -y-y, and by -~~* Prob. i. a 2X = 2-yy, or = v. nature of the 'tis Arch 30. Moreover from the Trochoid, LD= thence Prob. i v / =z. AL, and therefore -y -M =y. And (by ) -h and / let their Values be lubfti- Laftly, inftead of the Fluxions v a Whence Prob. is de- tuted, and there will arife -^ =z. (by i.) - - make rived -f- = z. And thefe being found, *ut/ w *v z, == DH, and raife the perpendicular HC. and 31. COR. i. Now it follows from hence, that DH = 2BL, CH 2BE, or that EF bifeds the radius of Curvature CO in N. of and And this will appear by fubftituting the values z z now ' found, in the Equation . **= DH, and by a proper reduction of the refult. defcribed the 32. COR, 2. Hence the Curve FCK, indefinitely by Center of Curvature of ADF, is another Trochoid equal to this, whofe Vertices at I and F adjoin to the Cufpids of this. For let be the Circle FA, equal and alike pofited to ALE, defcribed, and

in : let C/3 be drawn parallel to EF, meeting the Circle A Then will Arch FA = (Arch EL= NF =) CA. Line which is at to the 33. COR. 3. The right CD, right Angles Trochoid IAF, will touch the Trochoid IKF in the point C. 34. COR. 4. Hence (in the in verted Trochoids,) if at theCufpid K of the upper Trochoid, a Weight be hung by a Thread at the di- ilance KA or 2EA, and while the Weight vibrates, the Thread be itfelf to the of the Trcchoid fuppos'd to apply parts KF and KI, which refift it on each fide, that it may not be extended into a it it the right Line, but compel (as departs from Perpendicular) to be by degrees inflected above, into the Figure of the Trochoid, while the lower part CD, from the loweft Point of Contact, ftill remains a right Line : The Weight will move in the Perimeter of the lower Trochoid, becaufe the Thread CD will always be perpendicular to it. the 35. COR. 5. Therefore whole Length of the Thread KA is equal to the Perimeter of the Trochoid KCF, and its part CD is equal to the part of the Perimeter CF.

K 36. 66 The Method of FLUXIONS,

Since the Thread its Motion 36. COR. 6. by ofcillating revolves Point as a Center the about the moveable C, ; Superficies through will be the which the whole Line CD continually pafles, to Super- above the Line IF ficies through whichjthe part CN right pafles at as the fame time, as CD* to CN*, that is, 4 to i. Therefore the is the Area and the Area Area CFN a fourth part of CFD ; KCNE is a fourth part of the Area AKCD. fince the fubtenfe is and to 37. COR. 7. Alfo EL equal parallel CN, and is converted about the immoveable Center E, juft as CN the will be moves about the moveable Center C ; Superficies equal in fame that the Area through which they pafs the time, is, CFN, and the Segment of the Circle EL. And thence the Area NFD and the whole area will will be the triple of that Segment, EADF be the triple of the Semicircle. arrives at the the 38. COR. 8. When the Weight D point F, whole Thread will be wound about the Perimeter of the Trochoid KCF, and the Radius of Curvature will there be nothing. Where- fore the Trochoid IAF is more curved, at its Cufpid F, than any an of with the Circle ; and makes Angle Contact, Tangent /3F produ- a Circle can make with a Line. ced, infinitely greater than right of Contact that are 39. But there are Angles infinitely greater than Trochoidal ones, and others infinitely greater than thefe, and and the of them all are fo on in infinitum ; yet greateft infinitely xx x 3 x* 5 lefs than right-lined Angles. Thus = ay, = y, ==ry , &cc. denote a Series of of which x* = dy+, Curves, every fucceeding one makes an Angle of Contact with its Abfciis, which is infinitely can make with the fame Abfcifs. And the greater than the preceding of Contact which the firft xx=ay makes, is of the fame kind Angle z with Circular ones; and that which the fecond x*-=by makes, is of the fame kind with Trochoidals. And tho' the Angles of the fucceed- of the in Curves do always infinitely exceed the Angles preceding, yet of a they can never arrive at the magnitude right-lined Angle. x x*=l> 1 x4 40. After the fame manner ==y, xx=ay, y, = c*y, &c. denote a Series of Lines, of which the Angles of the fubfequents, made with their Abfcifs's at the Vertices, are always infinitely lefs than the Angles of the preceding. Moreover, between the Angles of Contact of any two of thefe kinds, other Angles of Contact may exceed each other. be found ad infwitum, that mall infinitely it of Contact of one kind are in- 41. Now appears, that Angles than thofe of another kind ; fince a Curve of one finitely greater at the of kind, however great it may be, cannot, Point Contact, I he and INFINITE SERIES. 67

lie between the Tangent and a Curve of another kind, however fmall that Curve may be. Or an Angle of Contacl of one kind cannot an of of necefTarily contain Angle Contact another kind, as the whole a Thus the of Contaft of the x* contains part. Angle Curve = cy*, or the Angle which it makes with its Abfcifs, neceflarfly includes the i of Contacl of the Curve x~' and can never be contain'd Angle =^y , by it. For Angles that can mutually exceed each other are of the fame kind, as it happens with the aforefaid Angles of the Trochoid, and of this Curve x> = by*. 42. And hence it appears, that Curves, in fome Points, may be infinitely more ftraight, or infinitely more curved, than any Circle, and yet not, on that account, lofe the form of Curve-lines. But all this by the way only. 43. Ex. 5. Let ED be the Quadratrix to the Circle, defcribed from Center A; and letting fall DB perpendicular to AE, make AB = x, BD =y, and AE = i. Then 'twill be yx yy* yx* =xy, as before. Then writing i for x, and z for y, the l Equation becomes zx zy zx* and = y ; thence, by Prob. i. zx zx* -f- zx zzxx zy* zzyy = y m Then reducing, and i for x and z for again writing y, there arifes z xxxjy ' J, ** and But z & being found, make T = DH, and draw HC as above. 44. If you defire a Conftrudtion of the Problem, you will find it mort. very Thus draw DP perpendicular to DT, meeting AT in P, and make aAP : AE :: PT : CH. For * =r

and zy = g. = -BP; and;ey + x = AP, and -_^_.. z into zy-\-x-=. - into AP=2. Moreover it is i-4-zz = AE x BTy 1 "PT* T> P\ TAT . I nrfr T3T r 1 /i f. U I a \ i r 1 -j- ** r 1 BlJq I j tnereiore : := i-{- rrTT =-T-:T ",) and = BT Bl? BI? 2-

DH. it is : :: : Here = Laftly, BT BD DH CH==^^. be the fame the negative Value only mews, that CH mufl taken way as AB from DH. or other 45. In the fame manner the Curvature of Spirals, of any Curves whatever, may be determined by a very mort Calculation. K 2 46. 68 7&e Method of FLUXIONS, determine the Curvature without 46. Furthermore, to any previous reduction, when the Curves are refer'd to right Lines in any has beer* other manner, this Method might have been apply'd, as done already for drawing Tangents. But as all Geometrical Curves, the conditions are reas alfo Mechanical, (efpecially when defining be reduced to infinite Equations, as I mail mew hereafter,) may I done in this fer'd to rectangular Ordinates, I think have enough it his own in- matter. He that defires more, may eafily fupply by if for a farther illuflration I mall add the Method duftry ; efpecially for Spirals. a Point in 47. Let BK be a Circle, A its Center, and B given its Circumference. Let ADd be a Spiral, DC its Perpen- dicular, and C the Center of Curvature at the Point D. Then drawing the right Line ADK, and CG parallel and equal to AK, as alfo the Per- pendicular GF meeting CD inF: Make AB or AK = i=CG, BK=#, AD==y, and GF = z.. Then con- an little ceive the Point D to move in the Spiral for infinitely Spree Semidiameter and Drf', and then through rfdraw the A/, Cg parallel to fo that G/ cuts and equal to it, draw gf perpendicular gC, gf in to fo that G and draw in/ and GF P; produce GF

: :: : and there- x=i, as above. Alfo CG GF de eD = oyz,

: CF : : de : dD x CF : : dD : fore yz y f Befides CG = oy becaufe = LDAd, d\ = oy x CF?. Moreover, Z_PC

= and raife will eive z ; which being known, make = DH, 1 -f- Z.X.Z. the the Perpendicular HC, meeting the Perpendicular to Spiral DC before drawn in C, and C will be the Center of Curvature. Or which comes to the fame thing, take CH : HD :: z : i, and draw CD. the be will to 50. Ex. i. If Equation ax=y, (which belong at then Prob. or i the Spiral Archimedes,) (by i.) ax=y y (writing 7 ^ for x, and yz for_y,) =yz. And hence again (by Prob i.) o = Wherefore Point D of the and yz+y'z. any Spiral being given,, or there will be - and thence the length AD y, given z = , z= 3 - . ( or) Which being known, make i-t-zz-z :

H-iz :: DA (y) : DH. And i : z :: DH : CH. And hence you will eafily deduce the following Conftrucftion. fo that AB : :: Produce AB to Q, Arch BK Arch BK : BC^, and make AB -+- AQ^: AQj: DA : DH :: a : HC. 2. If ax 1 be the that 51. Ex. =_)" Equation determines the Re- lation between BK and AD; (by Prob. i.) you will have 2axx=. or Thence 2a'x= s -+- 3Jy,-*, 2ax= 3y. again ^zy gsiyy*. 'Tis 'a~ ~ z 9 '- z and z . Thefe therefore = ^7 , = being known, make

i-\-zz K : i-t-zz DA : DH. Or, the work being reduced

1 to a better form, make gxx -f- 10 : gxx -f- 4 :: DA : DH. After the lame if ax* 52. Ex. 3. manner, bxy=yi determines ~ ' to arife the Relation of BK AD ; there will I"* = z,, and bxy -f- $)*. g *~ ; 9 *'- 8 . *7~^;~ = g. From which DH/ and thence the. Point C, is determined as before.

5qi- yo I'he Method of FLUXIONS, thus will the of 53. And you eafily determine Curvature any- or invent Rules for kinds of other Spirals ; any other Curves, in imitation of thefe already given. I have finim'd the ufe 4. And now Problem ; but having made of a Method which is pretty different from the common ways of operation, and as the Problem itfelf is of the number of thofe which are not very frequent among Geometricians : For the illuflration and confirmation of the Solution here given, I mall not think much to give a hint of another, which is more obvious, and has a nearer relation to the ufual Methods of drawing Tangents. Thus if from any Center, and with any Radius, a Circle be conceived to in feveral Points if that be defcribed, which may cut any Curve ; till of Circle be fuppos'd to be contracted, or enlarged, two the Points of interfeclion coincide, it will there touch the Curve. And or recede befides, if its Center be fuppos'd to approach towards, from, the Point of Contadt, till the third Point of interfedtion fhall meet with the former in the Point of Contadt ; then will that Circle

: In like be cequicurved with the Curve in that Point of Contadt manner as I infmuated before, in the laft of the five Properties of the Center of Curvature, by the help of each of which I affirm'd the Problem might be folved in a different manner. a Circle be 55. Therefore with Center C, and Radius CD, let defcribed, that cuts the Curve in the Points d, D, and >-{-/. The fum of the is the Squares of thefe equal to 1 of that -D Square DC ; is, 2VX -+- X* -f- )" -h 2yt -+- / =ss. If you would abbrevi- 1 1 s and it ate this, make v* -f-/ =f, (any Symbol at pleafure,) 1 1 1 o. have found becomes x 2vx -f-jy -f- zfy -+- q = After you will r 1 1* /, y, and q*, you have s-=\/ v -+- q*. for the 56. Now let any Equation be propofed defining Curve, the of the quantity of whofe Curvature is to be found. By help x or this Equation you may exterminate either of the Quantities y, and and INFINITE SERIES. 71

arife an the and there will Equation, Roots of which, (db, DB, , AB, A/3, &c. if you exterminate are "at the Points of interfedtion d, D, J\ &c. Wherefore fince _y,) "three of them become equal, the Circle both touches the Curve, and will alfo be of the fame degree of Curvature as the Curve, in the point of Contact But they will become equal by comparing the Equation with another fictitious Equation of the fame number has three of Dimenfions, which equal Roots ; as Des Cartes has fhew'd. Or more expeditioufly by multiplying its Terms twice by an Arithmetical Progreflion. Let the be is an 57. EXAMPLE. Equation ax =yy, (which to the and fubftitu- Equation Parabola,) exterminating x, (that is, -- ting its Value in the forego- * a ing Equation,) there will arife ^~y*_ -+ zty -f- ? = o. of whofe Roots are to be Three ^ _j_ yi made And for this equal. purpofe 4*2 I o the twice an I multiply Terms by * i o i Arithmetical Progrellion, as you 1 X 2 - fee done here and there arifes + J = j -J that BF 2x Or u = + \a. Whence it is eafily infer'd, = -{- \a, as before. of Parabola draw the 58. Wherefore any Point D the being given, take Perpendicular DP to the Curve, and in the Axis PF = 2AB, and erect FC Perpendicular to FA, meeting DP in C; then will C be the Center of Curvity defired. in the and 59. The fame may be perform'd Ellipfis Hyperbola, but the Calculation will be troublefome enough, and in other Curves generally very tedious.

Of ^uefiions that have fome Affinity to the preceding Problem.

60. From the Refolution of the preceding Problem fome others be fuch may perform'd ; are, I. To find the Point where the Curve has a given degree of Cur- vature. 6 1. Thus in the Parabola, ax=yy, if the Point be required whofe Radius of Curvature is of a given length f: From the Cen- ter of Curvature, found as before, you will determine die Radius 72 7%e Method of FLUXIONS,

to be \/aa which muft be -~^ -+- ^.ax, made equal to f. Then by reduction there arifes x = ^a -f- 1/^aff. II. To find the Point of ReElitude. 62. I call that the Point of ReEiitude, in which the Radius of

Flexure becomes infinite, or its Center at an infinite diftance : Such it is at the Vertex of the Parabola a*x=y*. And this fame Point is commonly the Limit of contrary Flexure, whole Determination I have exhibited before. But another Determination, and that not be derived from this inelegant, may Problem. Which is, the of Flexure fo longer the Radius is, much the lefs the Angle DCJ and alfo (Fig.pag.6i.) becomes, the Moment

- o. i for 3*** 2bxx cdx -4- dxy -f- dxy = Now writing xt a and it zbx whence z for y, becomes 3-v cd-{- dy -f- dxz=.o ; Prob. 6xx zbx -+ dxz +- dxz = o. Here again (by i,) dy + again o for it zb writing i for x, & for y, and z, becomes (>x -+- zdz = o. And exterminating z, by putting b 3* for dz in the dxz there will arife bx Equation 3^,v zbx cd -+- dy -f- = o, o this fubftituted in the room cd-$-dy = ) ory=c-{-^; being of in the of the we fhall have y Equation Curve, x* +- bcd-=z. Q } which will determine the Confine of contrary Flexure. 64. By a like Method you may determine the Points of Rectitude, which do not come the between parts of contrary Flexure. As if Equation x* 4y = o ex- have prefs'd the nature of a Curve ; you firfl, (byProb. i.)4^3 i2ax*-+- i2a*x faz=o, and hence again 12X* 24^7^ -f- 12^' b*z =o. Here fuppofe z = o, and by Reduc- tion there will arife x = a. Wherefore take ABi=fl, and erect the perpendicular BDj this will meet Curve in the Point of Re&itude D, as was required. III. and IN FINITE SERIES. 73 the III. To find Point of infinite Flexure. the Radius 65. Find of Curvature, and fuppofe it to be nothing. Thus to the Parabola of the fecond kind, whole Equation is A;* = 4" q* a that Radius will be CD which be- , = , <7y 6a \/q.ax-\- gxx comes nothing when x = o. or IV. To determine the Point of the greatefl leaft Flexure. 66. At thefe Points the Radius of Curvature becomes either the or leaft. Wherefore the Center of at that mo- greateft Curvature, nor ment of Time, neither moves towards the point of Contact, Fluxion the contrary way, but is intirely at reft. Therefore let the of the Radius CD be found; or more ex- let the of either peditioufly, Fluxion of the Lines BH or AK be found, and let it be made equal to nothing. 67. As if the Queftion were propofed concerning the Parabola of the fecond kind l firft to determine the Center of x = o*y ; aa 9X -> Curvature you will find DH = , ox ?AV and therefore BH = ; make BH 6^'

"- - Hence Prob. t}. But or the (by i.) _j_ ^y == now fuppofe -y, Fluxion of to be BH, nothing ; and belides, lince by Hypothecs 1 1 "' rf and thence fub- A- = .y, Prob. x= i, (by i.) yxx =

AB ==a =<7 and the it will" y'^j- x45| , raifrng perpendicular BD, meet the Curve in the Point of the greateft Curvature. Or, which

is : the fame thing, make AB : BD : 3^/5 : I. 68. After the fame manner the Hyperbola of the lecond kind the l 3 will reprefented by Equation xy = , be moft in the inflected points D and d, which you determine in may by taking the Abfcifs AQ== r, and the erecting Perpendicular QP_=z= v/5, and Q^/> equal to it on the other fide. Then drawing AP and A/>, they will meet the Curve in the points D and d required.

V, 74 The Method of FLUXIONS, V. To determine the Locus of the Center of Curvature, or to defcribe the Curve, in which tbaf* Center is alwaysfound, 69. We have already {hewn, that the Center of Curvature of the Trochoid is always found in another Trochoid. And thus the Cen- ter of Curvature of the Parabola is found in another Parabola of the fecond kind, reprefented by the Equation axx=y*, as will eafily appear from Calculation.

VI. Light falling upon any Curve, to find its Focus, or the Con- that rafted at its courje of the Rays are ref any of Points. 70. Find the Curvature at that Point of the Curve, and defcribe a Circle from the Center, and with-the Radius of Curvature. Then find the Concourfe of the Rays, when they are refracted by a Cir- cle about that Point : For the fame is the Concourfe of the refracted Rays in the propofed Curve. 71. To thefe may be added a particular Invention of the Curva- ture at the Vertices of Curves, where they cut their Abfcifles at right Angles. For the Point in which the Perpendicular to the Curve, meeting with the Abfcifs, cuts it ultimately, is the Center of its Curvature. So that having the relation between the Abfcifs x, Prob. the rela- and the rectangular Ordinate y, and thence (by i.) Value if fubftitute tion between the Fluxions x and y ; the yy, you will be the Radius of r for x into it, and make y = o, Curva- ture. -* " Thus in the ax it is 72. Ellipfis xX=yy, = yy ; of if we x which Value yy, fuppofe^=o, and confequently = />, a of Curvature. fo ^writing i for x, becomes for the Radius And at the Vertices of the Hyperbola and Parabola, the Radius of Cur- vature will be always half of the Latus rectum.

73- and INFINITE SERIES. 75

73. And in like manner for the Conchoid, defined by the Equation zicc cc + zbx the Value of xx = t ~T bb yy, yy (found by

Prob. will be ""* ^ "~~ * Now i.) ^ IT fuppofing y = o, and thence # = c or f, we mail have zb c, or

2(5 -f- f, for the Radius of Curvature. Therefore make AE : EG :: EG : EC, and he : eG :: eG : ec, and you will have the Centers of Curvature C and c, at the Vertices E and e of the Conjugate Conchoids. PROB. VI. To determine the Quality of the Curvature, at a given Point of any Curve.

I. By the Quality of Curvature I mean its Form, as it is more lefs or as it is varied or inequable, more or lefs, in its progrefs thro' different parts of the Curve. So if it were demanded, what is the Quality of the Curvature of the Circle ? it might be anfwer'd, that it is uniform, or invariable. And thus if it were demanded, what is the Quality of the Curvature of the Spiral, which is described by the motion of the point D, proceeding from A in AD with an accelerated velocity, while the right Line AK moves with an uniform rotation about the Cen- ter A ; the acceleration of L 2 which 76 7&? Method of FLUXIONS,

which Velocity is fuch, that the right Line AD has the fame ratio to the Arch BK, defcribed from a given point B, as a Number has

: I if it be to its Logarithm fay, afk'd, What is the Quality of the Curvature of this Spiral 1 It may be anfwer'd, that it is uniformly varied, or that it is equably inequable. And thus other Curves, in their feveral Points, may be denominated inequably inequable, according to the variation of their Curvature. 2. Therefore the Inequability or Variation of Curvature is required at any Point of a Curve. Concerning which it may be obferved, at alike in like 3. I. That Points placed Curves, there is a like Inequability or Variation of Curvature. 4. II. And that the Moments of the Radii of Curvature, at thofe Points, are proportional to the contemporaneous Moments of the Curves, and the Fluxions to the Fluxions. III. that where thofe 5. And therefore, Fluxions are not proportional, the Inequability of the Curvature will be unlike. For there will be a greater Inequability, where the Ratio of the Fluxion of the Radius of Curvature to the Fluxion of the Curve is therefore that ratio of the greater. And Fluxions may not impro- the Index of the of Variation perly be call'd Inequability or the of Curvature. 6. At the points D and d, infinitely near to each other, in the Curve AD^, let there be drawn the

Radii of Curvature DC and dc , and D Curvature. For the Inequability may be call'd fuch and fo great, as the quan-

of that ratio mews it to be : tityj 7^Ja Or the Curvature may be faid to be fo much the more unlike to the uniform II Curvature of a Circle.

fall the Ordinates DB and dbt 7. Now letting perpendicular line in make AB BD to any AB meeting DC P j = #, = y\ and thence B& = xo, it will be Cc = vo; and

-1 T^ x i. = , making = Wherefore and IN FINITE SERIES. 77

Wherefore the relation between x and y being exhibited by any and the Equation, and thence, (according to Prob. 4. 5.) Perpendicu- and the Radius of Curvature i1 and the lar DP or /, being found, , the of the Fluxion

- an<^ - .v there will arife -, / = = > v= = i, y = r^ ) f

v thus and v there will be the Index And y, t, being found, had ^ of the Inequability of Curvature.

> As if in Numbers it were or 2#==n , 9. determin'd, that^=j a and then x= 4 ; y (== + + 7 "= 3v/2. So that

which therefore is the Index of j^= 3, Inequability. 10. But if it were determin'd, that A: =2, then y = 2, ^'=T>

/ 6 will be here = v/5, f =

na -- it is =) byy yy. Alfo (by Prob. 5.) DH =}' -{ where, if for yy byy you fubftitute // aa, there ariies DH =

Tis alfo BD : DP :: DH : DC= - =v. Now (hv Prob.i.) f.ll U 1 and the Equations zaxbxx^yy, aa byy-\-y\-=^t!, give 7 8 77je Method of FLUXIONS,

a bx and and ~ v. give =}')', yy byy = /'/, = And thus v

found, the Index of the being ^ Inequability of Curvature, will aJib be known. Thus in the 13. Ellipfis 2X 3 ATA: where it is a and =}'}', = r, b=-.^ ; if we make then r- " L v ~~ ~" x=-,* * S a 3 x

v and therefore which is the In- ; =|, dex of the Inequability of Curvature. A b P Hence it appears, that the Curvature of this Ellipfis, at the Point D here affign'd, is by two times left inequable, (or 'by two times more like to the Cur- vature of the Circle,) than the Curva- o Jl ture of the Parabola, at that Point of V its from whence an Ordinate let fall the Curve, upon Axis is equaj to half the Latus rectum. 14. If we have a mind to compare the Conclufions derived in > v thefe Examples, in the Parabola 2ax=yy arifes (~= )^ for the V ' s a Index of and in the Inequability j Ellipfis zax bxx=yy, arifes

- - x BP j and fo in the 2ax bxx (^7- =J Hyperbola -+- =yy, the y+3b analogy being obferved, there arifes the Index ("2- ^ \. t J && x BP. Whence it is evident, that at the different Points of any Conic Section conn'der'd apart, the Inequability of Curvature is as the Rectangle BD x BP. And that, at the feveral Points of the Pa- raboh, it is as the Ordinate BD. Now as the is 15. Parabola the moft fimple Figure of thofe that are curved with inequable Curvature, and as the Inequability of its Curvature is fo eafily determined, (for its Index is 6x^ll^i,) there- .. . fore the Curvatures of other Curves may not improperly be compared to the Curvature of this.

1 6. As if it were inquired, what may be the Curvature of the 2X Ellipfis $xx=yy, at that Point of the Perimeter which is determined x = : Becaufe its Index is as by affuming 4., before, it might be anfwer'd, that it is like the Curvature of the Parabola 6.v and IN FINITE SERIES. 79 of the 6.v =)')', at that Point the Curve, between which and Axis is to the perpendicular Odinate equal |. as the Fluxion of the is to the Fluxion 17. Thus, Spiral ADE of the Subtenfe AD, in a certain given Ratio, its concave fide erect fuppofe as d to e; on AP = - x AD perpendicular to AD, y dd ee and P will be the Center of Curvature, and A P t or r=? will be the Index of Inequa- 1J y a.i ee that this has where bility. So Spiral every its Curvature alike inequable, as the Parabola 6x = yy has in that Point of its Curve, from whence to its Abfcifs a perpendicular Ordi- nate is let fall, which is equal to the

1 8. And thus the Index of Inequability at any Point D of the AB

in is found to be . Trochoid, (fee Fig. Art. 29. pag. 64.) Where- fore its Curvature at the fame Point D is as inequable, or as unlike - to that of a Circle, as the Curvature of any Parabola ax yy is at AB the Point where the Ordinate is ^a x -^ thefe Confiderations the Senfe of the 19. And from Problem, as I conceive, mufl be plain enough; which being well underftood, it will not be difficult for any one, who obferves the Series of the to furnifh himfelf things above deliver'd, with more Examples, and to contrive many other Methods of operation, as occafion may re- that he will be able to Problems quire. So manage of a like nature, he is not tedious and (where difcouraged by perplex Calculations,) or no Such are thefe with little difficulty. following ;

I. To find the Point of any Curve, where there is either no Inequabi- or or tie or the lity of Curvature, infinite, grcatej?, leajl. 20. Thus at the Vertices of the Conic Sections, there is no In-

of at the id of the 1 rcchoid it is equability Curvature; Cuf] infi- it is at thofe Points of the nite ; and greatefl Ellif.fis, where the is that where the Rectangle BD x BP greatefl, is, Diagor.al-Lines of the circumfcribed Parallelogram cut the Elliriis, whofe Sides touch it in their principal Vertices. r .n:cr II. 1o determine a Curve of fame definite Species, l'nfp je a C Curvature at Point be and to the Section, liioje any may cqiu:l Jiitiilar Curvature of any other Curve, at a given P./:./ of it. 8 o "The Method of FLUXIONS,

III. To iL-termine a Conk Sctfion, at any Point of which, the Cur- ri?//i7V and Pojition of the tangent, (in refpeSt of the AxisJ) may be like to the Curvature and Pofition of the Tangent, at a Point ajfigrid of any other Curir. 21. The ufe of which Problem is this, that inftead of Ellipfes of fecond the kind, whofe Properties of refradling Light are explain'd by Des Cartes in his Geometry, Conic Sections may be fubftituted, which mall perform the fame thing, very nearly, as to their Re- fractions. And the fame may be underfhood of other Curves.

P R O B. VII. To as as Areas find many Curves you pleafey ivbofe may be exhibited by finite Equations.

I. Let AB be the Abfcifs of a Curve, at whofe Vertex A let the AC = i be raifed, and let CE be perpendicular D drawn parallel to AB. Let alfo DB be a rectangular Ordinate, meeting the right Line CE in E, and the Curve AD in D. And conceive thefe Areas ACEB and ADB to be generated by the right Lines BE and BD, as they move along the Line AB, Then their Increments or Fluxions will be as always the defcribing Lines BE and BD. Wherefore make the Parallelogram ACEB, or AB x i, =.v, and the Area of the Curve ADB call z. And the Fluxions x and z will be as BE and BD; fo that making x = i = BE, then z = BD. a/Turned for 2. Now if any Equation be at pleafure, determining the relation of z and x, from thence, (by Prob. i.) may z be derived. And thus there will be two Equations, the 'latter of which will determine the Curve, and the former its Area. EXAMPLES. Aflume and thence Prob. or 3. ##:=:, (by i.) 2xx=s } 2x=c:, becaufe x=, i.

Aflame and thence will arife =;s an 4. ^=z, ? Equation to the Parabola.

A flume ax* or a'f and there will arife 5. =zz, x*=z, \a^x' =^^, or ^(?x = zz, an Equation again to the Parabola. i 6. and INFINITE SERIES. 81

6 1 f * 6. Affume a x~ =zz,or a*x-' =z, and there arifes a*x = z, the Value of or a'' -j-2xx = o. Here negative z only infinuates, that BD is to be taken the contrary way from BE. c'-a 1 1 if affume -+- c^x* = z , will have zc*x 7. Again you you and there will arife = 2zz ; z being eliminated, 'Z. aa -J-.VA- aa-^-xx - -- 8. Or if affume xx make

= v, and it will be ^ =s,and then (by Prob.i.) ^p ^ Alfo the Equation aa -f- xx = 011; gives 2X = zvv, by the help of which 3 if it exterminate

P R O B. VIII.

To fad as many Curves as you pleafe, -wbofe Areas fiall have a relation to the Area of any given Curve, a/fign- able by finite Equations.

i. Let FDH be a given Curve, and GEI the Curve required, and conceive their Ordinatss DB and EC to move at right Angles upon

A C

.11 G,

their Abfciffes or Bafes AB and AC. Then the Increments or Fluxions drawn of the Areas which they defcribe, will be as thofe Ordinates M into 82 fhe Method of FLUXIONS, that into the into their Velocities of moving, is, Fluxions of their Abfcifles. Therefore make AB = x, BD = v, AC = z, and the the Area let CE =y, Area AFDB = j, and AGEC = /, and the Fluxions of the Areas be s and t : And it will be xv : zy : : s : t. Therefore if we fuppofe x = i, and v=s, as before; it will be and thence - zy = t, =y.

2. Therefore let be affumed one of any two Equations ; which the s the other the may exprefs relation of the Areas and t, and relation of their Prob. let Abfciffes x and z, and thence, (by i.) the - Fluxions t and z be found, and then make =>'.

Ex. i. 3. Let the given Curve FDH be a Circle, exprefs'd by the Equation ax xx = w, and let other Curves be fought, whofe Areas may be equal to that of the Circle. Therefore by the Hy-

- It remains pothefis s=:f, and thence s = f, and y = =^-. to determine z, by afluming fome relation between the Abfciffes x and z.

if then a 4. As you fuppofe ax=zz; (by Prob. i.) =. 2zz: So " it is that for then . But v fubflituting : [ z, y = = =

- ' (\/ax xx \/aa s.s, therefore \/ aa zz = y is the =) a aa * Equation to the Curve, whofe Area is equal to that of the Circle. After if there will 5. the fame manner you fuppofe xx =. z, ariie ~ whence -j and x 2x =s, and thence _)'= (==] ; being

; I exterminated, it will be y=- 7"-' 2Z 2- there arifes o z and 6. Or if you fuppofe cc = xz, = + xz, T-V (5 / thence -- az cc. = y = -- , v 2 2^3

'- \t'isa ax +- = z, Prob. i.) + s=:z, 7. Again, fuppofing (by -^- a mechanical Curve. and thence y ?> which denotes be and let 8. Ex. 2. Let the Circle ax xx =w given again, other aflumed relation Curves be fought, whofe Areas may have any cx s and to the Area of the Circle. As if you afliime + = t, fup- a ZZ. Prob. 'tis c s = t, and = pofe alfo ax = (By i.) + Therefore and INFINITE SERIES. 83

2 ~ Therefore = = ~~; and ax for y fubftituting ^ xx j,

5 and f for x, 'tis;'= 4- ^ v'^ if affume j 9. But you =/, and x = z, you will have s and i z. Therefore - 2!2L O i ^! =/, = y- =j ' K fl

= i; . Now for exterminating v, the Equation ax xx

Prob. therefore 'tis = iJ'u, (by i.) gives ^ 2x= 2vv, and y=.

. if Where you expunge v and A; by fubftituting their values

there \/ax xx and 2;, will arile _)=-" \/tf;s there will arife 10. But if you affume ss = f, and x = zz, Anc and i 2zz; and therefore = V = 4^- 2w=r^, = _y

it will become 5 and x fubftituting \/ax xx and &z, y = \Sa-zz;, which is an Equation to a mechanical Curve. 1 be which 1. Ex. 3. After the fame manner Figures may found, Let have an aflumed relation to any other given Figure. the Hyper- bola cc if affume s and -{- xx = wu be given ; then you = /, will have s f and 2X and thence xx=cz, you = = cz; _)' =

for C-z^ for it .r= -. Then fubftituting v/cc -+- xx j, and x, s; will be y =: - i/cz zz. * 2Z -{- 12. And thus if you affume xv s=zf, and xx = cz, you will have v-^-vx s=t, and 2X = cz. But v=.s, and thence vx = i. Therefore - = ~. But now (by Prob. i.) cc-\-xx y= *** % Then = 1?^ gives x=^-ui;, and 'tis y = ^ fubftituting \/i<,-t-xx ~ for -u, and c*z* for x, it becomes =. , ^ y c ^ if the Ciffoid - 13. Ex. 4. Moreover ^-^^_- =1; were given, to which other related Figures are to be found, and for that purpofe - affume - xx s t - ax xx h, you ^/ax +- = ; fuppofe */ = *** - - and its Fluxion /' therefore h s /. the -, +- = But Equation M 2 =M 84 7%e Method of FLUXIONS,

* A 3 .V. =/j/j gives where if exterminate it will be ^ ==2.^, you /&,

And bcfides fuice it is - s - , 3 3 xx

= t. Now to determine z and z, afTume

aa ax then Prob. i a or z - \/ = z ; (by .) = 2zz, =

' V.. Jaxxx a A;

v/tftf ~. And as this Equation belongs to the Circle, we mall have the relation of the Areas of the Circle and of the Ciflbid.

thus if had ">/ ax ~ s f 14. And you aflumed V xx -h = y there derived and x = z, would have been y-=.\/as> .22-, an Equation again to the Circle. like if 15. In manner any mechanical Curve were given, other mechanical Curves related to it might be found. But to derive geometrical Curves, it will be convenient, that of right Lines de- pending Geometrically on each other, fome one may be taken for the or Abfcifs and that the Area the Bafe ; which compleats Paralle- logram be fought, by fuppofing its Fluxion to be equivalent to the Abfcifs, drawn into the Fluxion of the Ordinate. 1 the I refer it 6. Ex. 5. Thus Trochoid ADF being propofed, to the Abfcifs ABj and the Parallelogram ABDG being compleated, I leek for the complemen- tal Superficies ADG,byfup- pofing it to be defcribed by the Motion of the right Line and therefore GD, its Fluxion to be equivalent to the Line GD drawn into the Velocity of the Motion ; that is, x*v. Now whereas AL is parallel to the Tangent DT, therefore AB will be to BL as the Fluxion of the fame AB to the Fluxion of the Ordinate BD, that and INFINITE SERIES. 85

as r to -j. So that is, that

of the Abfcifs as to : : AB, PL BL, that is, v i :: a \/ ax .v.v,

-y --/ then = v . Then the Fluxion of the Area 2 ax xx vx, ADG, will be 2V7=^=. Wherefore if a right Line equal to --.' <,* xx ly . .x ATV be conceived to be apply 'd as a rectangular Ordinate at B, a point of the Line AB, it will be terminated at a certain geometrical Curve, whole Area, adjoining to the Abfcifs AB, is equal to the Area ADG. 1 8. And thus geometrical Figures may be found equal to other Figures, made by the application (in any Angle) of Arches of a or of other to Circle, of an Hyperbola, any Curve, the Sines right other or verfed of thole Arches, or to any right Lines that may be Geometrically determin'd. 19. As to Spirals, the matter will be very fliort For from the Center of Rotation A, the Arch DG being defcribed, with any Radius the Line in and the in AG, cutting right AF G, Spiral D ; fince that Arch, as a Line moving upon the the Abfcifs AG, delcribes Area of the Spiral AHDG, fo that the Fluxion of that Area is to the Fluxion of the Rectangle i x AG, as to i if raife the the Arch GD ; you perpendicular right Line GL equal to that Arch, by moving in like manner upon the fame Line AC, it will defcribe the Area A/LG equal to the Area of the Spiral AHDG : The Curve A/L being a geometrical Curve. And fartlirr, if the Subtenfe AL be drawn, then A ALG = | GD Sector therefore xGL = |AGx = AGDj the complernental and will alfo be Segments AL/ ADH equal. And this not only agrees to the Spiral of Archimedes^ (in which cafe A/L becomes the Parabola of Apoliomus,) but to any other whatever; fo that all of them may be converted into equal geometrical Curves with the fame eale. 20. 86 tte Method of FLUXIONS,

20. I might have produced more Specimens of the Conftruction of this Problem, but thefe may fuffice; as being fo general, that whatever as yet has been found out concerning the Areas of Curves, is in or (I believe) can be found out, fome manner contain'd herein, and is here determined for the moil part with lefs trouble, and without the ufual perplexities. that 21. But the chief ufe of this and the foregoing Problem is, nffuming the Conic Sections, or any other Curves of a known magnitude, other Curves may be found out that may be compared with thefe, and that their defining Equations may be difpofed orderly in a Catalogue or Table. And when fuch a Table is contracted, when the Area of any Curve is to be found, if its defining Equation be either immediately found in the Table, or may be transformed into another that is contain'd in the Table, then its Area may be known. Moreover fuch a Catalogue or Table may be apply'd to the determining of the Lengths of Curves, to the finding of their the Su- Centers of Gravity, their Solids generated by their rotation, to the of other perficies of thofe Solids, and finding any flowing quantity produced by a Fluxion analogous to it.

P R O B. IX.

To determine the Area of any Curve propofed.

1. The refolution of the Problem depends upon this, that from the relation of the Fluxions being given, the relation of the Fluents if the Line may be found, (as in Prob. 2.) And firft, right BD, by the motion of which the Area required AFDB is defcribed, move upright upon an Abfcifs AB the Paral- given in pofition, conceive (as before) in mean time lelogram ABEC to be defcribed the on the other fide AB, by a line equal to unity. of the Pa- And BE being fuppos'd the Fluxion of the Area rallelogram, BD will be the Fluxion required. 2. Therefore make AB = x, and then alfo ABEC=i \x=x, it be and BE = x. Call alfo the Area AFDB = z, and will BD=z, as alfo becaufe x=i. Therefore by the Equa- =~,X

the the ratio of the Fluions - tion expreffing BD, at fame time

IS and INFINITE SERIES. 87

is and thence Prob. 2. be found the exprefs'd, (by Cafe i.) may relation of the flowing quantities x and z. Ex. i. or is 3. When BD, z, equal to fome fimple quantity.

Let there be ~ = z, r to the 4. given , (the Equation Pa- and - rabola,) (Prob. 2.) there will arife = z. Therefore a ^> or -L AB x BD, = Area of the Parabola AFDB.

c. Let there be eiven fan to a Parabola of J ^ aa = z, Equation*

the fecond and will that ~ AB x BD kind,) there arife -^ = z, is, = Area AFDB.

6. Let there be given ~ z XX ' or a^x 1 = x-:, (an Equation to an of the Hyperbola fecond kind,) and there will arife a 3 x 1 z or z. 7 = That is, AB x BD Area of an infinite on the other fide of = HDBH, length, lying the Ordinate BD, as its negative value insinuates. there would arife j. And thus if there were given ^ = z, Z. 2XX

8. Moreover, let ax = zz, or a*x* = z, (an Equation again that i-AB to the Parabola,) and there will arife ~a^x^ = z,, is, x BD = Area AFDB.

. then or 2 x AFDH. 9 Let ~=zz-t za*x = s, AB BD =

f or 2 x BD 10. Let =zz', then ^ = s, AB = HDBH.

1 1. Let ax* z~> then or AB xBD = AFDH. = ; fV = z, i And fo in others. 12. Ex. 2. Where of fuch z is equ.il to an Aggregate Quantities. 13. LetAT-H^ij then^-h *- = z> , J

~ . 14. Let ^ \

Let z then 2x^ x 4** = 2r - 15. 3*i ^ ; +- a Divifion is 1 6. Ex. 3. Where previous reduction by required. Let there be to the 17. given j~, =.& (an Equation Apollonian it will be Hyperbola,) and the divifion being performed in injinittun, s l%e Method of FLUXION ,

5 ?f ^l &c. And Prob. as x __ _ ^ 4. , thence, (by 2.) * 11 1. a x ^^ will obtain in the fecond Set of Examples, you z= -y --^

a /. "x U~A ^

* 5^/3 A/4

and divifion it will be 1 8. Let there be given ^ ==*, by ^ 1 ~J XX

l x 6 &c. or elfe - &c. And " , s= -^ -f- -.,' ~=i x*-{-x* 1 X X4 A.

3 r 7 Prob. 2 x , &c. thence (by 2.) = ^ -f-^ 1# =AFDBi

-i ^- - or 2 , &c. =HDBH. = H- 5 X 3X* SA.'' L A

Let there be and divifion it will 19. given , ~!Li X =z, by

1 be z = 2x^ 2X + 7^ I3AT -f- 34*% 6cc. And thence (by

' 8 Prob. x' 3x3 A< &c - 2.) z = $x* -f- yx* T + V ^ 20. Ex. 4. Where a previous reduction is required by Extraction of Roots.

21. Let there be given z = \/ aa -\- xx, (an Equation to the an Hyperbola,) and the Root being extracted to infinite multitude * be a - of * it will z=i 4- ~\ r, &c. whence terms, ft a Q f,9.*if.7-f,l I I -7 fit* . r . X~ X x ^x as in the ss = ax+ -h &c - foregoing 6 , 77^ TT^ 22. In the fame manner if the Equation z = \/aa xx were given, (which is to the Circle,) there would be produced z=ax

s b ^Oi.J ii2a 23. And fo if there were given z-=\/x xx, (an Equation alfo to the Circle,) by extracting the Root there would arife x'f therefore z = x* 4-** -r'-g-x^, 6cc. And z = .ix* 1 V z TT . ___ _ 24. Thus s === v//z<* -^- AV xx, (an Equation again to the Cir- bxx - - .\ extraction of the Root it j occ. cle,) by gives z=a-\- gjs - , ^* Jf3 /'*v3 whence 2; <7Ar -- -,--- &c. = -f- , 4 6< 24^ I

25. And thus v^~ZT7~ = ^, by a due reduction gives

z=i-+- T^-V* -h &c. then 2 AT 3 43^4, = -f- ^ -f- TV^ S &c. H-irf -f^ _l_^ + TV^ T '- Vo^ 26. and INFINITE SERIES. 89

26. Thus 5 the of finally z=l/a* -t-A' , by extraction the Cubic ~ ~ &c. then Root, gives z=a -+- -+- w and (by Prob. 2.)

*=** &c. AFDB. elfe + -~, gfr + T> = Or *= C '' ** A 1 1 And thence * = 7 &c. = HDBH. 567* Ex. a 27. 5. Where previous reduction is required, by the refolution of an affected Equation. 28. If a Curve be defined by this Equation z> + a*z x 3 2a"' x = o, extradl the Root, and there will arife z = a .

: : j_ _j_ !4^-. &c. whence will be obtain'd as before z-=ax 64.2 5 i zaa

But if ? 29. z~' cz* 2x*z c *z -f- 2x -+- c* = o were the Equation to the Curve, the refolution will afford a three-fold Root; f? either z = c + .v &c. or S c !i' -, + Jl,1 = .v-f- V 32' <- A or - f - s = c --(_ &c. And hence will arife the 2" 2rc At T values of tb,e three correfponding Areas, z = ex + x*

1 &c. 2; -f- T^t, = r^ i.v + ^ ^0, &e. and x = ex ~A 5 X 4 .,S - --! - flrr 1 CCC> 6c 8. 24^' 30. I add nothing here concerning mechanical Carves, becaufe their reduction to the form of geometrical Curves will be taught afterwards. 31. But whereas the values of z thus found belong to Areas which are fometimes to a finite fituate, part AB of the Abfcifs, to a fometimes part BH produced infinitely towards H, and fometimes to both parts, according to their different terms: That the due value of the Area be may alTign'd, adjacent to any portion of the Abfcifs, that Area is always to be made equal to the difference of the values of which to the z, belong parts of the Abfci/s, that are terminated at the beginning and end of the Area.

32. For Inflance ; to the Curve exnrefs'd bv the Equation i-^-'xx

' m^^ JTC- fhe Method of FLUXIONS,

} ~, it is found that z=x ^x &c. that I de- _l_ 4_,vS Now may termine the quantity of the Area the MDll, adjacent to the part of Abfcifs /'B; from the value of z, which arifes by putting AB = x, and there I take the value of z, which arifes by putting Ab=x, &c. x x> &c. the value of remains x -Lx* + ^-x', + -J-x', be to that Area WDB. Whence if A*, or x, put equal nothing, be had the whole Area AFDB = x x' -+- -^x', &c. jqere will - fame Curve there is alfo found z, ==. -+ 33. To the

what is the Area L, &c. Whence again, according to before, 5** * I 1 1 ^ I I &/~f* *"T ppr^TOt'f1 1 I -- " ' ' ^""1 J. ijCl CIUI C - OCC. __ t _. ,_ '- T ]V\T\ oCC 1 -) Area bdH toward if AB, or x, be fuppofed infinite, the adjoining be to - H, which is alfo infinitely long, will equivalent ^ - &c. will -f- . &c. For the latter Series -f- ~, CA ^-35 vanifh, becaufe of its infinite denominators.

it 34. To the Curve reprefented by the Equation a-\- = Z,^ - :s found, that z=.ax -. Whence it is that x ax i X X - -4- = Area &/DB. But this becomes infinite, whether x be fup- or x infinite and therefore pofed nothing, ; each Area AFDB and &/H is infinitely great, and the intermediate parts alone, fuch as &/DB, can be exhibited. And this always happens when the Ab- fcifs x is found as well in the numerators of fome of the terms, as in the denominators of others, of the value of z. But when x is only found in the numerators, as in the firft Example, the value of z, belongs to the Area fituate at AB, on this fide the Ordinate. And when it is only in the denominators, as in the fecond Example, that the all value, when figns of the terms are changed, belongs to the whole Area infinitely produced beyond the Ordinate. 35. If at any time the Curve-line cuts the Abfcifs, between the b and in points B, fuppofe E, inftead of the Area will be had the difference &/E* BDE of the Areas at the diffe- of the Abfcifs to if rent parts ; which here be added the Rectangle t G he Area dEDG will be obtain'd. t and INFINITE SERIES.

it is to be that in the value 36. But chiefly regarded, when of & is divided x of one dimension the corre- any term by only ; Area to that term to the Conical and there- fponding belongs Hyperbola ; it in fore is to be exhibited by felf, an infinite Series : As is done in what follows. fl3 ~ glA Let z, be an to a Curve and divifion J77. '= Equation ; byJ ax -f- xx it becomes - z = 2a +- 2X _ h^ &c. and thence aa y 2X> X* l 1 ' Z = 2ax -f- x To* &c. And the Area &/DB l^ j ^T

aa 2*5

,&*. zax xx ,

Where by the Marks and I denote the little Areas belonging 1-1 1 - aa aa to the Terms and

Now that and be I make or x to 38. |^ |j| may found, Kb, y be and bE definite, indefinite, or a flowing Line, which therefore I call ; fo that it will be to that Area ;' -^; = Hyperbolical adjoin-

to - - - - B, that is, But Divifion it will be = ing byJ ~ j x \ y x

or - A4 therefore, x

. and therefore the whole Area -* ' ' required WDB = X 2A321 3 . xx H -, &c. the fame or have been ufed for 39. After manner, AB, x, might a definite Line, and then it would have been

if in and be affumed to be 40. Moreover, Z>B be bifefted C, AC and indefinite then AC of a definite length, and Cb CB ; making ' ~ and C or CB 'twill be bd= s=-\- -)- = i>, =_)', -^

therefore the Area -{- _i- ^-^-' &c. and Hyperbolical adjacent N 2 to 'A Mt&od of FLUXIONS,

a V to the Part of the Abfcifs &C will be r I -~ - ~ - .&c. Twill be alfo DB = = ? + ~ ^ + &c. And therefore the Area adjacent to the other part of the Abfcifs CB 1 11 7* st^ 4 Si"*- 1 " ' ' - -f- , , &c. And the of thefe = l + 1 * 1 Sum f 2f Jf 4' 5' Areas 7- -\~ ~r ~r, &c. will be equivalent to -|

a 3 41. Thus in the Equation -f- z,* ~$- z x~= =o, denoting the nature of a Curve, its Root will be z = ,v y 8 J A X 8 I A i _ _1 _ &c. Whence there arifes z, =-. Lxx -x _!_

' Area Y /^r^ 6cc. And the T A ' ox Six' KC

7 &c. that .'XA ^' T, is,=:|.v T Six

&c. _ - ^ &c. - i - - for the moft 42. But this Hyperbolical term, pnrt, may be very the commodioufly avoided, by altering beginning of the Abfcifs, or it fome en that is, by increafing diminiihing by gi\ quantity. As * ? in the former where = z was to Example, v + v v the Equation b to be the the Curve, if I fhould make beginning of the Ablcifs* to be of determinate and fuppofmg Al> any length 4/7, for the re- I fliall now write x : mainder of the Abfcifs B, Thst is, if I dimi- x a nifti the Abfcifs by a, by writing -f- inftead of x, it will -become ^~^,. = ~> and ' 4 z -' _!! &Ci whence arifes s ^ &c. = = \ax 4--^'bia 273 j

Area . another and another for the be- 43. And thus by affuming point Area of Curve be exr-ivib'd an ginning of the Abfcifs, the any may infinite variety of ways. = z have been refolved 44. Alfo the Equation rj-p might " into the two infinite Series z, --- -+- &c. a -f .v 2 1 "-^ .V X X }

**--}-; &c. where there is found no Term divided b} the fir ft 2 Power and INFINITE SERIES. 93

Power of x. But fuch kind of Series, where the Powers of A* afcend in the numerators of the in the denominators infinitely one, and of the other, are not fo proper to derive the value of z from, by Arithmetical computation, when the Species are to be changed in- to Numbers. difficult can occur is to un- 45. Hardly any thing to any one, who dertake fuch a computation in Numbers, after the value of the Area in Yet for the the is obtain'd Species. more compleat illufhation of I mall add an foregoing Doctrine, Example or two. be 46. Let the Hyperbola AD propofed, whofe Equation is \/x-+-xx=z; its Vertex being at A, and each of its Axes is equal to Unity. From what goes before, its Area ADB=-i.v> " 1 ? -+- A*' -+- &c ' that j'^ T'T* T^P*'"' ' * ' 4 s is x* into Lx x* .v .v -V -+- T T + y T T4T > &c. Series be which may infinitely produced by thelaft of multiplying term continually by the fucceeding terms this i-J 5 ~~'" q Proereffion #. .v &c. That is 2 ^^rA - Xix ^^'v S 47 6-9 8.-n 10.15*. 1 3 the firft term I_ fecond term -L.v* : Which ^..v x x' makes the 2 -5

" l 'd -~ multiply by x makes the third term TV-vl : Which mul-

' ? makes .v and fo ad tiply'd by ^ x T T the fourth term; hifini- tuin. Now let AB be affumed of any length, fuppofe ^, and writing this Number for and its for and firft term .v, Root 4 x*, the ^x^ or y x T> being reduced to a decimal Fraction, it becomes c into '- ^ the fecond -^3333333> & - This makeso.oo625 term.

~ ' v This into makes 0.0002790178, &c. the third term. And 4-7 4 fo on for ever. But the term?, which I thus deduce by degrees, I in difpole in two Tables; the affirmative terms one, and the negative in another, and I add them up as you fee here.

-i-o. 94 "The Method of FLUXIONS, + 0.0833333333333333 00002790178571429 62500000000000 34679066051 271267361111 834^65027 5135169396 26285354 144628917 961296 86 6 4954581 3 7 190948 1663 7963 75 35 2 _ 1

1 0.0002825719389575 -f- 0.0896109885646518 4- 0.0896109885640518 "0^3284166257043

Then from the fum of the Affirmatives I take the fum of the ne- and there remains for the gatives, 0.0893284166257043 quantity Area which was to be found. of the Hyperbolic ADB ; let the Circle be 47. Now AdF propofed, which isexpreffed by the equation \/x xx = z > is and from what that is, whofe Diameter unity, will be -!#* goes before its Area AdB ..#* ' T Txi -fT^i &c> I n which Series, fince the terms do not differ from the terms of the Se- ries, which above exprefs'd the Hyperbolical Area, unlefs in the and elfe remains to be to Signs -4- ; nothing done, than terms with other conned: the fame numeral fignsj that is, by fubtracting the connected fums- of both the afore -mention'd tables, 6 1 from the firft term doubled 0.08989 3 560 503 93 0.1666666666666, &c. and the remainder 0.0767731061630473 will be the portion A^B of the ciicular Area, fuppoiing AB to be a fourth part of the diameter. And hence we may obferve, that tho' the Areas of the Circle and Hyperbola are not compared in a Geometrical confidera- is dilcover'd the tion, yet each of them by fame Arithmetical computation. 48. The portion of the circle A^/B being found, from thence the whole Area may be derived. For the Radius dC being drawn, or of multiply Ed, or -^v/S? Ulto -^C, i, and half the product or 12 8 2 '" ^e e va ^ ue f the s-Vs/3' -54 ^5 773^5 75 w ^ Triangle cWB; which added to the Area AdB, there will be had the Sector ACd = 0.1308996938995747, the fextuple of which whole Area. 49. And and INFINITE SERIES. 95

49- And hence by the way the length of the Circumference will be 3.1415926535897928, by dividing the Area by a fourth part of the Diameter. the 50. To thefe we mail add calculation of the Area comprehended between the Hyperbola dfD and its Afymptote CA. Let be the and C Center of the Hyperbola, putting

'twill be -^ -^ a+x =BD, and -=.bd; whence " the Area - - AFDB = bx 4- 4 -*, &c. and the Area

4- , &c. and the fum 0aL>&=. 2ox-\ ? c AB /'/.<

? ~ and 4- 4- let us CA Kb ^ , &c. Now fuppofe = AF=i, or AB i.i and thefe = TL., Cb being 0.9, and CB = ; fubftituting for Series numbers a, b, and x, the firft term of the becomes 0.2, the fecond and fo as 0.0006666666, &c. the third 0.000004 ; on, you fee in this Table. O.2OOOOOOOOOOOOOOO 6666666666666 40000000000 285714286 2222222 l8l82

The fum 0.200670695462151 1= Area bdDB. defired 51. If the parts of this Area Ad and AD be feparately, fubtract the lefler BA from the greater dA, and there will remain - -- and 3-+ -^4- h &c. Where if i be wrote for a b, and reduced to decimals will iland -jig. for x, the terms being thus; O.O IOOOOOOOOOOOOOO 500000000000 3333333333 25000000 2OOOOO 1667

The fum o. = A^ AD, 5 2 - The Method 96 of FLUXIO N s,

if this difference of 52. Now the Areas be added to, and fubtracted from,their fum before found, half the aggregate o. 1053605156578263 will be the greater Area hd, and half or the remainder 0.0953101798043248 will be the lefler Area AD. 53. By the fame tables thofe Areas AD and hd will be obtain'd alfo, when AB and Ab are fuppos'd T ~, or CB=i.oi, and if d> = o.gg, the numbers are but duly transferr'd to lower places, as may be here feen. O O2OOOOOOOOOOOOOC0 O.O30ICOOOOOOOO3OO 66666666666 50020000 4000000 3^ 28 Sum 0.0001000050003333 AJ AD. Sum o 020000(5667066(195 = ==AD. 54. And fo putting AB andA=-~o-> orCB=i.oor, and' 0^ = 0.999, there will be obtain'd Ad= 0.0010005003335835, and AD = o. 0009995003330835. and 55. In the fame manner (if CA and AF= i) putting AB A = o.2, or 0.02, or 0.002, the fe Areas will arife, A^=o.223 1435513 142097, and ADz=o. 1823215567939546, or A

Area's to and will be the Area belonging ^-| 2, AFcT/3, C/3 being 3..

as it is ~ and 2 a addition of Again, = 5, x5= 10, by due Areas will be obtain'd 1.6093379124341004 = AF^/3, when c and 10. /3=5; =AF < when = 2.3025850929940457 T/3, C/3 And thus, fince 10x10=100, and 10x100=1000, and ^5

x ' ' x 10 and i.i .' I and xo.98 = 7, lox = n, and ^ )

- it is =499 ; plain, that the Area AF^/3 may be found by the of the Areas found i i compofition before, when C/3 = oo j ooo i 7> and IN FIN ITE SERIES, 97

7; or any other of the above-mention'd numbers, AB = BF being llill unity. This I was willing to infinuate, that a method might be derived from hence, very proper for the conftrudtion of a Canon of Logarithms, which determines the Hyperbolical Areas, (from be fo which the Logarithms may ealily derived,) correfponding to many Prime numbers, as it were by two operations only, which are not very troublefome. But whereas that Canon feems to be deriva- ble from this fountain more commodioufly than from any other, what if I mould point out its contraction here, to compleat the whole ? of the 57. Firfl therefore having affumed o for the Logarithm as is number i, and i for the Logarithm of the number 10, gene- 1 rally done, the Logarithms of the Prime numbers 2, 3, 5, 7, 1, the 13, 17, 37, are to be inveftigated, by dividing Hyperbolical Areas now found by 2.3025850929940457, which is the Area correfponding to the number 10: Or which is the fame thing, by mul- for tiplying by its reciprocal 0.4342944819032518. Thus Inftance, if 0.69314718, &c. the Area correfponding to the number 2, were multiply'd by 0.43429, &c. it makes 0.3010299956639812 the Lo- garithm of the number 2. in the 58. Then the Logarithms of all the numbers Canon, which are made by the multiplication of thefe, are to be found by the addition of their Logarithms, as is ufual. And the void of this places are to be interpolated afterwards, by the help Theorem.

is be A- 59. Let be a Number to which a Logarithm to adapted, the difference between that and the two neareft numbers equally diflant on each fide, whofe Logarithms are already found, and let d be half the difference of the Logarithms. Then the required Loga- of the Number n will be obtain'd +- rithm by adding d-\- gr^, &c. to the Logarithm of the leffer number. For if the numbers or are expounded by C/>, C/3, and CP, the rectangle CBD C,&T=i, as and the Ordinates and raifed if n be wrote before, pq PQ^being ; ~ for C/3, and x for or /3P, the Area or -+- ~, p pgQP ^ } + &c. will be to the Area or *- pq}$ +- ^ -f- ^, &c. as the difference between the of the extream Logarithms numbers or 2(i, to the difference between the Logarithms of the leffer and of the middle O one; g 8 Tie Method of FLUXIONS, dx dx* dx* - -+- -f- &C. that the one: which therefore will be . , is, when x 3 - A' A"* a +- -4- &c.

- is -4- &c. divifion perform'd, d-\- s * 2n i Zfj

60. The two firft terms of this Series d-\- I think to be accu- 2n rate enough for the construction of a Canon of Logarithms, even were to be to fourteen or tho' they produced fifteen figures; provided the number, whofe Logarithm is to be found, be not lefs than 1000. And this can give little trouble in the calculation, becaufe x is generally an unit, or the number 2. Yet it is not neceffary to interpolate all the places by the help of this Rule. For the Logarithms of numbers which are produced by the multiplication or divifion of the number laft found, may be obtain'd by the numbers whofe Logarithms were had before, by the addition or fubtraction of their Logarithms. Moreover by the differences of the Loga- rithms, and by their fecond and third differences, if there be occa- the void be more the lion, places may expeditioufly fupply'd ; foregoing Rule being to be apply'd only, when the continuation of fome is in to thofe full places wanted, order obtain differences. 6 1. By the fame method rules may be found for the intercalation of Logarithms, when of three numbers the Logarithms of the leffer and of the middle number are given, or of the middle number and of the greater; and this although the numbers mould not be in Arithmetical progreffion. 62. Alfo by purfuing the fteps of this method, rules might be for the conftruction of the tables eafily difcover'd, of artificial Sines and Tangents, without the affiftance of the natural Tables. But of thefe things only by the bye. 63. Hitherto we have treated of the Quadrature of Curves, which of terms are exprefs'd by Equations confirming complicate ; and that by means of their reduction to Equations, which confift of an infinite number of fimple terms. But whereas fuch Curves may fometimes be fquared by finite Equations alfo, or however may be compared with other Curves, whofe Areas in a manner may be confider'd as known ; of which kind are the Conic Sections : For this reafon I thought fit to adjoin the two following catalogues or tables to of Theorems, according my promife, conflructed by the help of the jtb and Bth aforegoing Propofitions. 64. and IN FINITE SERIES. 99

64. The firft of thefe exhibits the Areas of fuch Curves as can be and the fecond contains fuch whole Areas be fquared ; Curves, may compared with the Areas of the Conic Sections. In each of thefe, the letters d, e, f, g, and h, denote any given quantities, x and z the Abfcifles of Curves, v and y parallel Ordinares, and s and t Areas, as before. The letters and 6, annex'd to the quantity z, denote the number of the dimenfions of the fame z, whether it be or affirmative or if integer fractional, negative. As =3, then JZ1ZZZ2 3 z l s z-=z-~> or-' and z*-' , "=z , , &+' = z*, =z*. 3 in the values for the fake of 65. Moreover of the Areas, brevity, inftead of this Radical or is written R \Se-{-f& t , inflead of which the value of the is and/

t, CO rt

I ii n '5 en e* I

i UCO 3 H N s

I *

u 1 v, N 1 S 1 a

Curve +

' *~ T1 *J- V ^~ - 1 N CO and INFINITE SERIES. 101

T- * T*

II II II II II II

s o bo G

* ** c

t X

II T U 1 OJ 01 a f ^ o ^ \ X" u O ol \O i iM ""* I I X X v * s CO s: 1 I t H-

cno *?> oa'j j? N cr> M M

-f' f i ?r

N M

x x" e IO2 jff>e Method -o^ FLUXIONS, Other of the fame kind been added but 67. things might have ; I fhall now on pafs to another fort .of Curves, which may be com- with the pared Conic Sections. And in this Table or Catalogue have the you propofed Curve reprefented by the Line QE^R, the of beginning whole Abfcifs is A, the Abfcifs AC, the Ordinate CE, the beginning of the Area a^, and the Area defcribed a^EC. But the beginning of this Area, or the initial term, (which com- either monly commences at the beginning of the Abfcifs A, or recedes to an infinite diftance,) is found by feeking the length of the Abfcifs Aa, when the value of the Area is nothing, and by eredling the perpendicular a^/. 68. After the fame manner have you the Conic Sedlion reprefented the Line whofe is by PDG, Center A, Vertex a, rectangular

Semidiameters Aa and AP, the beginning of the Abfcifs A, or a, or a, the Abfcifs AB, or aB, or aB, the Ordinate BD, the Tangent DT meeting AB in T, the Subtenfe aD, and the Re&angle infcribed or adfcribed ABDO. the letters before it 69. Therefore retaining defined, will be AB or aB and AC = z, CE=y, a.%EC = t, = x, BD = i;, ABDP or aGDB=j. And befides, when two Conic Sections are of the required, for the determination any Area, Area of the latter mall be call'd Abfcifs and the Ordinate for

Tl .V3 . >*, + i? Q BL, O rt O 2 S S Q M ea O OH u Q Q Q O O pa 14 rt o o >s| = a c CO en O O Q _2 Q eg O 3 Q

-y CO

** V V

4- o U. --V- 104 Method of FLUXIONS,

a Q o O Q a c O c O rt o h O O .5 cj OJ Q o o O o Pi Q Q n O O rt C O m 2 B (LI3 ^1^

X 15 ^

"T I dina K fr t + SJ 3| ed 4 I s %

C II 3 II

+ v

a - ji 3 U - V +

^ 4- I 4- o H fe -V" and INFINITE SERIES. < Q

CO Q O

u + 1 5 13 +

) $ N + 1 I.Hf 4- 1 + H Mj Tf-

P s-

i U ^ ^ V S) u O

*-It tt

<5 + b

4 I?

U '

o .

X io6 Method of FLUXIONS,

J s to _u o

I 3 U + +

t/>

o fa

X x and INFINITE SERIES, 107

on to illuftrate the Theorems that 71. Before I go by Examples are deliver'd in thefe claffes of Curves, I think it proper to obferve, whereas in the I have 72. I. That Equations reprefenting Curves, all the of the and i all along fuppofed figns quantities d, e, f\ g, />, it fhall that are to be affirmative ; whenever happen they negative, Abfcifs and Or- they muft be changed in the fubfequent values of the ninate of the Conic Section, and alfo of the Area required. II. numeral and when 73. Alfo the figns of the Symbols 0, they Areas. are negative, muft be changed in the values of the More- over their Signs being changed, the Theorems themfclvcs may ac- the ot quire a new form. Thus in the 4th Form of Table 2, Sign d ' ~~~' the Theorem becomes I being changed, 3d -;_ iv,.-j- ,-^ -}> ~~^

that x , &c. is, =}', *"==*, 7=^=' cz -f-/a is be in into 2.w 3^===^. And the fame to obferved others. feries of each the 2d of the ift Ta- 74. III. The order, excepting continued each ad For in the Series of ble, may be way infinitum. the numeral co-efficients the -;d and 4th Order cf Table i, of the are fonn'd multi- initial terms, (2, 4, 16, 96, 768, Sec.) by &c. plying the numbers 2, 4, 6, 8, ro, continually the co-efficients of the terms are de- into each other ; and fubfequcnt rived from the initials in the 3d Order, by multiplying gradually by &rc. or in the Order multi- , -Li, 1 > A,* , 4th by -rV. &C. But the co-efficients plying by i, 4-, f, T> &c. a rife of' the denominators i, 3, 15, 105, by multiplying the &c. into each other. numbers i, 3, 5, 7, 9, gradually ft d h Series of the i 1 But in the ad the , 2 , , and 75. Table, 3'', 4 c; ", io th Orders are produced in infinitum by diviiion alone. Thus having .4--1-' t/x ift if the diviiion to a = v, in the Order, you perform convenient there will arifo ~z ^ 7 ^ period, j~ 'j

to ==.)'. The firft three terms belong the ift Order of

to the ift cf Table i, and the fourth term belongs Species this Order. d ~ 3n Jc --4 <: n ~ - z Whence it that the Area is + ~ appears, 7^- 1^f r?r

__ _il s . s for the Area of the Conic Section, whofe Abfcifi } putting *' d - is and Ordinate v = r--. x=r , g : P 2 io8 7&e Method of FLUXIONS, the Series of the 76. But ^th and 6th Orders may be infinitely continued, by the help of the two Theorems in the 5th Order of Table i. by a due addition or fubtraction : As alib the 7th and 8th Scries, by means of the Theorems in the 6th Order of Table i. and the Series of the nth, by the Theorem in the roth Order of Table i. For inftance, if the Series of the 3d Order of Table 2. beto be farther continued, 6 = and the ift Theorem of the fuppofe 4>j, ~~1 ~ 4l| 3>1 1 jth Order of Table i. wll become 8fts~ . 5/b~ into =. -^-=^f. But according to the 4th Theorem of ~ to be for it is this Series produced, writing ^ 'Qfr'-'S/*' and __ t

and there will So that fubtrafting the former values of / /, remain S ' a 4J J- 1 Ii;/?} R _,, - , . / 2 10/1/3 Thefe qnez v/^-h/ =/> I2e ft being mul- -~ij j - for xv*, there will arife tiplied by ; and, (if you pleafe) writing

of the Series to be a 5th Theorem produced,'

- , 1 ! s = v, and -r- = f. of thefe Orders alfo be otherwife 77. IV. Some may derived from others. As in the 2d Table, the 5th, 6th, 7th, and nth, from the 8th; and the 9th from the loth : So that I might have omitted them, be of fome tho' but that they may ufe, not altogether necefftry. Yet I I have omitted fome Orders, which might have derived from the ifr, and 2d, as alfo from the 9th and loth, becaufe they were affected by Denominators that were more complicate, and therefore can hardly be of any ufe. the of is 78. V. If defining Equation any Curve compounded of feveral Equations of different Orders, or of different Species of the be the fame Order, its Area mufl compounded of correlponding A- care that be connected with reas ; taking however, they may rightly not or their proper Signs. For we mufl always add fubtra

d their the feveral inconveniency, I have prefix' proper Signs to Va- lues of the Areas, tho' ibmetimes negative, as is done in the jth of Table 2. and yth Order is farther to be about the of the 70. VI. It obferved, Signs Areas, either that the that -f- * denotes, Area of the Conic Section, adjoin- is to be added to the other in the value ing to the Abfcifs, quantities fee the ifl or that the Area on the other of t , ( Example following ;) fide of the Ordinate is to be fubtracled. And on the contrary, s denotes ambiguoufly, either that the Area adjacent to the Abfcifs is to be fubtradled, or that the Area on the other fide of the Ordinate convenient. the of if is to be added, as it may feem Alfo Value f, it comes out affirmative, denotes the Area of the Curve propoled ad-

: if it be it joining to its Abfcifs And contrariwife, negative, reprefents the Area on the other fide of the Ordinate. 80. VII. But that this Area may be more certainly defined, we its as to its Limit at the at mull enquire after Limits. And Abfcifs, the Ordinate, and at the Perimeter of the Curve, there can be no un- initial or the from whence its de- certainty: But its Limit, beginning obtain various In the fcription commences, may pofitions. following Examples it is either at the beginning of the Abfcifs, or at an infinite diftance, or in the concourfe of the Curve with its Abfcifs. But it elfewhere. And wherever it it be may be placed is, may found, by f becomes ieeking that length of the Abfcifs, at which the value of an the fo raifed nothing, and there erecting Ordinate. For Ordinate will be the Limit required. the 8 1. VIII. If any part of the Area is pofited below Abfcifs, / will denote the difference of that, and of the part above the Ab- fcifs. 82. IX. Whenever the dimenfions of the terms in the values of (hall afcend too or defcend too be .v, i;, and /, high, low, they may or fo often .reduced to a juft degree, by dividing multiplying by any be to the office of Uni- given quantity, which may fuppos'd perform thole dimenfions mail be either too or too low. ty, as often as high 83. X. Befides the foregoing Catalogues, or Tables, we might allb of Curves other be the conftrucT: Tables related_tp_ Curves, which may moftfimple intheirkind; as to

84. EXAMPLE I. Let QER be a Conchoidal of fuch a kind, that the Q Semicircle QHA being defcribed, and AC being creeled perpendicular to R the Diameter A Q^_ if the Parallelo- gram QACI be compleated, the Dia- gonal AI be drawn, meeting the Se- micircle in H, and from H the'per- be let fall to pendicular HE 1C ; then the Point E will defcribe a Curve, whole Area ACEQJs fought. make ^.Therefore AQ^==a, AC=z, CE=y, and becaufe of the continual be Proportionals AI, AQ^, AH, EC, 'twill ECor_>'= --^ 86. Now that this may acquire the Form of the Equations in the Tables, make =2, and for z~- in the denominator write z*, and * a*z~-* for or in ;]-' the numerator, and there will arife_y = flf an of the ift of the ad Order of Table 2, > Equation Species a -\-x,

3 and the Terms it will rf e and being compared, be^ = , = a*,

3 1 1 I j .fo that 4/ .J'' i tf .*; and xv 2s v ii x, = -u, /f= T-<

t. the values found to a 87. Now that of x and v may be reduced number of dimen lions, choofe any given quantity, as a, by which, as unity, a* may be multiplied once in the value of x, and 1 in the value of v, a> may be divided once, and ^x twice. And by l 1 .v and xv this means you will obtain s/"^niTr =^,^/a =1', is thus. 2s, t: of which the conllradion Arch 88. Center A, and Radius AQ^_ defcribe the Qigadrahtal the in take AH ; raiie BD QDP ; AC AB = perpendicular meeting that Arch in D, and draw AD. Then the double of the Scclof For ADP will be equal to the Area fought ACEQ^

' or-y and .vj 2s= 2 2 AB.?=) BD, ; A ADB that or = 2*A ADB'-f- aBDP, is, either = aOAD, or=2DAP: Of which values the affirmative aDAP belongs to the Area ACEQ, on this fide EC, and the negative aC^AD belongs to the Area RE R extended ad infi.ritum beyond EC. 'of found fometimes be 89. The folutions Problems thus may the le- made more elegant. Thus in the prefent cafe, drawing RH midiameter and INFINITE SERIES. in

midiameter of the Circle QH A, becaufe of equal Arches QH and DP, is Sector and therefore a the Sector QRH half the DAP, fourth part of the Surface ACEQ^ be a is 90. EXAMPLE II. Let AGE Curve, which defcribed by the Angular point E of the Norma AEF, whilft one of the Legs AE, being interminate, paffes continually through the given point A, and the other CE, of a given length, r, flides upon the right Line AF given in pofition. Let fall EH perpendicular to AF, and compleat the and Parallelogram AHEC ; calling AC = z, CE =_y, and

EF = rf, becaufe of HF, HE, HA it be continual Proportionals, will HAor y= be 91. Now that the Area AGEC may known, fuppofe = *, tr-i fl in the or 2 will be =}' Here "ce z = , and thence it 'j==a ~z^1 the value di- numerator is of a fraded dimenfion, deprefs of/ by ~ I ~ V) an of the z&, and it will be = S> Equation viding by 7=7=* y a ~ i of Table 2. And the terms com- ad Species of the ;th Order being and a*. So that z 1 pared, it is

92. Ex- 112 The Method of FLUXIONS,

III. Let AGE be the Ciflbid to the 92. EXAMPLE belonging Circle ADQj defcribed with the diameter AQ.. Let DCE the be drawn perpendicular to diameter, and meeting the Curves in D and E. And na- z and ming AC = t CE =.y, a becaufe of AQj== ; CD, CA, CE continual Proportio- or nals, it will be CE y =

:, and dividing by z, 'tis X ~~ l = / Therefore zr~ y ' az I

== ^, or i = ,and thence y = an Equation or V aai-i of the Order of Table 2. Terms therefore the 3d Species 4th The 'tis d-=. e Therefore being compared, I, = i, and f=a.

% = x,

s fo that it is *AC = x, CD = v, and thence ACDH = ; of the Ciflbid 3ACDH 4AADC = 3* 2xv = t = Area is the Area ACEGA. Or, which fame thing, 3 Segments ADHA = ADEGA, or 4 Segments ADHA = Area AHDEGA. IV. Let 93. EXAMPLE PE be the firft Conchoid of the Ancients, defcribed from Center G, with the Afymptote AL,. and diftance LE. Draw its Axis GAP, and let fall the Or- dinate EC. Then calling AC =: z, CE =.y, GA = a, and . becaufe of the Pro- Ap c ; A C : CE AL : : portionals it will be CE or GC : CE, y

* be found from hence, the 04. Now that its Area PEC may And if Ordinate CE are to be confider'd feparately. paits'of the it is CD , Ordinate CE is fo divided in D, that = v/^ the and and INFINITE SERIES.

will a de- and DE = *\/V ^ ; CD be the Ordinate of Circle fcribcd from Center A, and with the Radius AP. Therefore the Area is and there will remain the other part of the PDC known, to be found. Therefore fince the of the Or- part DPED DE, part it is is to z* dinate by which defcribed, equivalent -\/e* ; fup-

and it becomes z* an of pofe 2 = w, -^/e* = DE, Equation of the of Table 2. The terms therefore the ift Species 3d Order f c t and therefore being compared, itisd=t>, = , and/= i; 1 l - 1 . = x, \/ i -+- c* x = v, and zbc s -- = t. j 1 Z Z. reduce them to a number of 95. Thefe things being found, juft too and dimenfions, by multiplying the terms that are deprefs'd, fome If this dividing thofe that are too high, by given Quantity. ~ c* % and be done by c, there will arife = x,

-- = t : The Conflruclion of which is in this manner. c ex and Parameter 96. With the Center A, principal Vertex P, aAP, defcnbe the Hyperbola PK. Then from the point C draw the right Line CK, that may touch the Parabola in K : And it will be, as AP to 2AG, fo is the Area CKPC to the Area required DPED. fo revolve about the Pole 97. EXAMPLE 5. Let the Norma GFE G, as that its angular point F may continually flide upon the right in then conceive the Curve PE to be de- Line AF given pofition ; fcribed by any Point E in the other Leg EF. Now that the Area of this Curve may be found, let fall GA and EH perpendicular to the right Line AF, and compleating the Pa- rallelogram AHEC, call AC

= 2, CE=j, AG = , and EF=; and becaufe of the Proportionals HF : EH : : AG : AF, we mall have AF = bz . Therefore CE or , y V a zz b But whereas

of a Circle defcribed with the Semidiameter c ; about the Center A let *fhe Method of FLUXIONS, meets ia let fuch a Circle PDQ_be defcribed, which CE produced the hel of which ~ D then it will be DE = : B? P EqUa ; ^=rS tion there remains the Area PDEP or DERQ^to be determin'd.

and and it will be i^~ > Suppofe therefore =:2, G=^ 3 DE= V ft ^ of the Order of Table i. sn Equation of the ift Species 4th And be cc the Terms being compared, it will b-= d, =e, and j ==/; fo that bV cc zz = l>R=f. as value of t is and therefore the 98. Now the negative, Area / lies the Line that its initial Limit reprefented by beyond DE ; may be found, feek for that length of z, at which t becomes no- c. continue thing, and you will find it to be Therefore AC to Q^> that it may be AQ==c, and erect the Ordinate QR.; and DQRED will be the Area whofe value now found is b\/cc zz. fhould the 99. If you define to know quantity of the Area at the Abfcifs co-extended with without PDE, pofited AC, and it, knowing the Limit QR, you may thus determine it. 100. From the Value which / obtains at the length of the Ab- fubtract its value at the of the Abfcifs fcifs AC, beginning ; that is, from b\/ cc zz fubtract &, and there will arife the defired quantity A: b\/ LC zz. Therefore compleat the Parallelogram PAGK, and let fall DM perpendicular to AP, which meets GK the will be to the in M ; and Parallelogram PKML equal Area PDE. 101. Whenever the Equation defining the nature of the Curve cannot be found in the Tables, nor can be reduced to limpler terms nor other means it muft be transform'd into by divifion, by any ; of to in the manner other Equations Curves related it, fhewn in Prob. 8. till at laft one is produced, whofe Area may be known by the Tables. And when all endeavours are ufed, and yet no fuch can be found, it may be certainly concluded, that the Curve pro- be pofed cannot compared, either with rectilinear Figures, or with the Conic Sedions. 102. In the fame manner when mechanical Curves are concern'd, they muft fir ft be transform'd into equal Geometrical Figures, as is fhewn in the fame Prob. 8. and then the Areas of fuch Geometri- cal Curves are to be found from the Tables. Of this matter take the following Example. 103. and IN FINITE SERIES. 115

6. it be determine the of 103. EXAMPLE Let propofed to Area Arches of aie the Figure of the any Conic Section, when they made Ordinates on their Right Sines. As let A be the Center of the Conic Section, " AQ_and AR the ^ V .' ^\ Semiaxes, CD the Ordinate to the Axis AR, and PD a Per- pendicular at the point D. Alfo let AE be the fa id mechanical Curve meeting CD in E; and from its nature before defined, CE will be equal to the Arch QD. There- fore the Area AEC is fought, or com- the pleating parallelogram ACEF, the excefs AEF is required. To which purpole let a be the Latus rectum of the Conic Section, and b its Latus tranfverfum, or 2AQ^_ Alfo let AC=z, and CD=_>';

it be then will V^bb -f- -zz =y, an Equation to a Conic Section, ~- as is known. Alfo PC= -z, and thence PD = v/^H zz. 104. Now fince the fluxion of the Arch QD is to the fluxion of the Abfcifs as to if the fluxion of the AC, PD CD ; Abfcifs be fup- of the pos'd i, the Fluxion Arch QD, or of the Ordinate CE, **+"-*~~ 4 will be i/ . Draw this into FE, or z, and there

will arife z / for the fluxion of the Area AEF.

If therefore in the Ordinate CD you take CG - -zz the Area which is defcribed V , AGC, by CG -zz

moving upon AC, will be equal to the Area AEF, and the Curve AG n6 77je Method of FLUXIONS,

AG will be a Geometrical Curve. Therefore the Area AGC is fought. To this purpofe let z* be fubflituted for z* in the laft

< Equation, and it becomes &*-* = CG, an i; \/^-j-, j-^ Equa.- M tion of the ad Species of the i ith Order of Table 2. And from a - comparifon of terms it is d = i, e-=.i-bb ~ and =,/= ,

$= : fo that ~] xx r, i>. * zz=x. \/ -f- and a \/^bb* a ' Afl a /

~s t. DP and /. = That is, CD = x, = v, Jj = And this is the Conftruction of what is now found. 105. At Q^ erect QK perpendicular and equal to QA, and thro* the point D draw HI parallel to it, but equal to DP. And the Line KI, at which HI is terminated, will be a Conic Section, and Area be to the comprehended HIKQ^will the Area fought AEF, as b to a, or as PC to AC. if 106. Here obferve, that you change the fign of b, the Conic Section, to whofe Arch the right Line CE is equal, will become an if make b the Ellipfis; and befides, you = , Ellipfis becomes- And in this cafe the line KI becomes a a Circle. right line parallel

the has 107. After Area of any Curve been thus found and conftrucled, we fhould confider about the demonftration of the con- that afide all ftruction ; laying Algebraical calculation, as much as be and may be, the Theorem may adorn'd, made elegant,, fo as to become fit for publick view. And there is a general method of de-- monftrating, which I mail endeavour to iiluftrate by the following Examples.

the in Demonftration of Conjlruflion Example 5. the Arch a 1 08. In PQ^take point d indefinitely near to D, and draw de and dm to (Figure p. 113.) parallel DE and DM, meeting DM and AP in p and /. Then will DE^/ be the moment of the Area PDEP, and LM// will be the moment of the Area LMKP. Draw the femidiameter AD, and conceive the inde- fmall arch ~Dd to be as it a finitely were right line, and the tri-

and will be and therefore : : angles -D/^/ ALD like, D/> pd:: AL LD.

: :: : : :: : But it is HF EH AG AF ; that is, AL LD ML DE; and

: : : : x therefore Dj> pd ML DE. Wherefore Dp DE = pd x ML That and IN FINITE SERIES, 117

is to the LM;;//. That is, the moment DEed equal moment And fince this is demonflrated indeterminately of any contemporaneous moments whatever, it is plain, that all the moments of the Area PDEP are equal to all the contemporaneous moments of the Area PLMK, and therefore the whole Areas compofed of thofe moments each other. D. are equal to C^JE.

the in Demonftration of ConftruSfion Example 3. be the and 109. Let DEed momentum of the fuperficies AHDE, A

Befides it is DC : QA (aDK) : : AC : DE. And therefore

: :: Cc : 2Dd :: DC aDK

AC : DE, and Cc x DE = zDd-x. AC. Now to the moment of the periphery Dd to the tan- produced, that is, let fall the gent of the Circle, and AI will perpendicular AI, So that be equal to AC. zDd x AC = zDd x AI = 4 So that moment Triangles AD

Bemvnftratwn iiS "The Method of FLUXIONS,

the Demonftration of ConftruRion in Example 4. no. Draw ce to parallel CE, and at an indefinitely fmall diflance from it, and the tangent of the ck Hyperbola t and let fall KM perpendicular to AP. Now from the nature of the Hyper- bola it will be AC : A? :: AP : AM, and therefore AC? : :: APV :: GLq AC?: LE? (or ')

: and : AP? AM? ; divlfim* AG/ AL? (DE?) ::.AP?: AM? AP?(MK?) ; And invent, AG:

AP :: DE : MK. But the little Area DEed is to the Tri- as angle CKr, the altitude DE is to half the altitude KM ; that is, as AG to -LAP. Wherefore all the moments of the Space PDE are to all the contemporaneous moments of the Space PKC, as AG to 4-AP. And therefore thofe whole Spaces are in the fame ratio.

Demonjlration of the Conjlruftion in Example 6.

in. Draw c*/ and parallel infinitely near to CD, (Fig. in p. 115-) the Curve in and draw meeting AE e, hi and fe meeting DCJ in p and Then the fimi- q. by Hypothefis ~Dd= Eg, and from the litude of the and it : Triangles Ddp DCP, will be D/> (Dd) ( Eq :: P : (PD) HI, fo that Dp x HI = Eg xCPj and thence x HI :: Dp (the moment HI/'/.)): Eg x AC (the moment EF/e)

E : :: : fince ? xCP EyxAC CP AC. Wherefore PC and AC are in the given ratio of the latus tranlverfum to the Jatus rectum of the Conic Section QD, and fince the moments HI//) and EFfe of the Areas HIKQ^and AEF are in that ratio, the Areas themfelves will be in the fame ratio. Q-^E. D. 112. In this kind of demonilrations it is to be obferved, that I affume fuch for quantities equal, whofe ratio is that of equality : And that is to be efteem'd a ratio of equality, which differs lefs from than ratio equality by any unequal that can be affign'd. Thus in the laft demon ftration I fuppos'd the rectangle E^xAC, or FE?/, to be to the equal fpace FEt/j becaufe (by realon of the difference lefs than Eqe infinitely them, or nothing in comparifon of them,) they and INFINITE SERIES. 119 ratio of for they have not a inequality. And the fame reafon I HI//6 and fo in others. made DP x HI = ; here made ufe of this method the 1 13. I have of proving Areas or to a of Curves to be equal, have given ratio, by the equality, or of their moments becaufe it has an to by the given ratio, ; affinity the ufual methods in thefe matters. But that feems more natural of which depends upon the generation Superficies, by Motion or Fluxion. Thus if the Confbuclion in Example 2. was to be de- the monftrated : From the nature of Circle, the fluxion of the right

1 is to the fluxion of the line as line ID (Fig. p.i 1.) right IP, AI to

: : : : the ID ; and it is AI ID ID CE, from nature of the Curve

x to AGE ; and therefore CE x ID = ID x IP. But CE ID = the fluxion of the Area PDI. And therefore thofe Areas, being generated by equal fluxion, muft be equal. Q^E. D. 1 14. For the fake of farther illustration, I fliall add the demonflration of the Confrruc~r.ion, by which the Area of the Ciffoid is the determin'd, in Example 3. Let lines mark'd with points in the fcheme be expunged; draw the Chord DQ^ and the Afymptote QR of the Ciffoid. Then, from the nature of the Circle, it Is DQj- = AQ_x CQ^, and thence (by Prob. i.) Fluxion of DQj= AQjcCQ. And therefore AQ_: 2DQj CX^ Alfo from the nature of the Ciffoid it is ED : AD :: AQ^: DQ^ There- fore ED : AD : : and EDxCC^=ADx2DQ^, or 4xiADxDQ^ Nowfmce is at the DQ__ perpendicular end of AD, revolving about

and i AD x to the fluxion A ; QD = generating the Area alfo x its quadruple ED CQ^== fluxion generating the Ciffoidal Area Wherefore that Area QREDO. QREDO infinitely long, is generated quadruple of the other ADOQ^ Q^E. D. SCHOLIUM. 120 The Method of FLUXIONS,

SCHOLIUM.

115. By the foregoing Tables not only the Areas of Curves, but of other quantities any kind, that are generated by an analogous way of flowing, may be derived from their Fluxions, and that by the affiftance of this Theorem : That a quantity of any kind is to an unit of the lame kind, as the Area of a Curve is to a fuperficial if fo be that the fluxion that be to an unity ; generating quantity unit of its kind, as the fluxion generating the Area is to an unit of its kind alfo that as the Line ; is, right moving perpendicularly upon the Abfcifs (or the Ordinate) by which the Area is defcribed, to a linear Unit. Wherefore if any fluxion whatever is expounded by fuch a moving Ordinate, the quantity generated by that fluxion will be the Area defcribed fuch or if the expounded by by Ordinate ; Fluxion be expounded by the fame Algebraic terms as the Ordinate, de- the generated quantity will be expounded by the fame as the fcribed Area. Therefore the Equation, which exhibits a Fluxion of any kind, is to be fought for in the firft Column of the Tables, and the value of t in the laft Column will mow the generated Quan- tity. _ 1 1 6. As if \/ 1 -h exhibited a Fluxion of any kind, make it and that it be reduced to the form of the equal to y, may Equations

' in the Tables, fubftitute z* for z, and it will be z~ 8a i8z -id _ + ,~ gz -p. _, - i - R> and thence \S +- a == =/. Therefore it is the Z which is quantity ^~ 1/1 -4- generated by the Fluxion

J^l- 3 17, And thus if v'l -f- reprefents a Fluxion, a due re- 7 by 9 out of the duftion, (or by extracting & radical, and writing _

for 2~^) there will be had -or, */s&--! =7, an Equation of z ga* the ad Species of the 5th Order of Table 2. Then comparing the terms, and INFINITE SERIES. 121

7 - e i. So that x terms, it is d=. i, = , and/= = = **,

' and J =-* = A Which found, the _j_ ^ = -u, 4 being 7

the fluxion v/ j L^Z w ill be quantity generated by + known, by

its own as the is to making it to be to an Unit of kind, Area j* or which comes to the the fuperficial unity ; fame, by fuppofing a but a of an- quantity t no longer to reprefent Superficies, quantity other kind, which is to an unit of its own kind, as that fuperficies k to fuperficial unity.

_l a linear I 1 1 8. Thus \/i 4- ~ to Fluxion, reprefent fuppofing T 9 but a Line that imagine t no longer to fignify a Superficies, ; Line, Area: for inftance, which is to a linear unit, as the which (accord- is to a or ing to the Tables) is reprefented by t, fuperficial unit, that Area to a linear that which is produced by applying unit. On the which account, if that linear unit be made e, length generated

~ . this foundation by the foregoing fluxion will be And upon thofe Tables may be apply'd to the determining the Lengths of Curve-lines, the Contents of their Solids, and any other quantities whatever, as well as the Areas of Curves.

Of ^uejlions that are related hereto.

I. To approximate to the Areas of Curves mechanically,

is of 119. The method this, that the values two or more right- be fo lined Figures may compounded together, that they may very the value nearly conftitute of the Curvilinear Area required. 120. Thus for the Circle AFD which is denoted by the Equa- tion .v xx zz found the value of = r } having the Area AFDB, viz. ** #* /,** J-x*, &c. the values of fome Rectangles are to be fought, fuch is the value x\/x xx, or x*

z* T#* TV#% &c - of the rectangle BD x AB, and x^/x, or #', the value of AD x AB. Then thefe values are to be multiply'd by for and then any different letters, that ftand numbers indefinitely, R to 122 2^2 Method of FLUXIONS, to be added together, and the terms of the fum are to be compared with the correfponding terms of the value of the Area AFDB, that as far as is poffible they may become equal. As if thofe Parallelo- grams were multiply'd by e and f, the fum would be ex* \ex^

{$$, &c. the terms of which being compared with thefe terms x * x * &c. there arifes and ^ ,^ TV*% +/=-!, i^= 4., * 4 or e and e So that x AB x = , /= % = Tr ^-BD -f- TTAD * AB = Area AFDB very nearly. For ^-BD x AB -f. TTAD x AB is - equivalent to .!#* 4.** _^.v* _L.,v*, &c. which being fubtracted from the Area AFDB, leaves the error only T'-# -j- TV#*, &c. 121. Thus if AB were bifected in E, the value of the rectangle AB x DE will be x\/x %xx, or x* -^x* -2-#* -- x*, &c. And this compared with 128 r 1024 8DE + zAD the rectangle AD x AB, gives into AB = Area AFDB, the error being only

J-x* -\-- x* &c. which is always lefs than 560 5760 of the whole even tho' AFDB TJ^JTJ. part Area, But this were a quadrant of a Circle. Theorem may be thus pro- fo is the AB pounded. As 3 to 2, rectangle into DE, added to a difference between and fifth part of the AD DE, to the Area AFDB, very nearly. 122. And thus by compounding two rectangles ABxED and AB x BD, or all the three rectangles together, or by taking in ftill Rules be more rectangles, other may invented, which will be fo much the more exacT:, as there are more Rectangles made ufe of. of And the fame is to be understood the Area of the Hyperbola, or one of any other Curves. Nay, by only rectangle the Area may as often be very commodioufly exhibited, in the foregoing Circle, to the by taking BE to AB as v/io 5, rectangle AB x ED will be Area as to the error to the AFDB, 3 2, being only TfTAT* -fr-

to determine the II. The Area being ghen, Abfcifs and Ordinate. 123. When the Area is exprefs'd by a finite Equation, there can be no difficulty : But when it is exprefs'd by an infinite Series, the affected root is to be extracted, which denotes the Abfcifs. So for the W^^ ** w and INFINITE SERIES. 123 the Hyperbola, defined by the Equation ^ = z, after we have * found * bx - &c. that from the = -^ -+- , given Area the Abfcifs x may be known, extract the affedled Root, and there will arife x - = + ^ + 4- -JjjL , &c. And

if the Ordinate .5 divide ab /z moreover, were required, by 4- AT, that is, a -f- -+ ~ &c. and there will arife -f- , z=l>> by } -^ s

as to the 124. Thus Ellipfis which is exprefs'd by the Equation a%x * ax -xx = zz, after the Area is found z = ^a?x* i 1 I , x , Hf_, &c. write i;' for and / for and it becomes ^!^ , x*,

= t* &c. and the root ^ -i-j, extracting /=

&c. is to equal x. And this value being fubflituted inftead of x in a the Equation ax -xx = zz, and the root being extracted, there arifes * = **. 3 38*' __ 4Q7^7 ^L 5cc> So that from 5<: '7Sf* 225018 z, the given Area, and thence v or ./"I, the Abfcifs # will be f za* and the Ordinate z. given, All which things may be accommodated to the if the of Hyperbola, only flgn the quantity c be changed, wherever it is found of odd dimenfions.

R O B. 124- *The Method of FLUXIONS,

P R O B. X.

1o find as many Curves as we pleafe, vohofe Lengths may be exprcfsd by finite Equations.

1. the foltirion of The following pofitions prepare the way for this Problem.

2. I. If the right Line DC, ftanding perpendicularly upon any. Curve AD, be conceived thus to move, all its points G, g, r, &c. will defcribe other Curves, which are equidiftant, and perpendicular to that line : As GK, gk, rs, &c. II. If 3. that right Line is continued indefinitely each way, its extremities will move contrary ways, and therefore there will be a Point between, which will have no motion, but may therefore be call'd the Center of Motion. This Point will be the fame as the Center of Curvature, which the Curve AD hath at the point D, as is mention'd before. Let that point beC. 4. III. If we fuppofe the line AD not to be circular, but unequably curved, fup- lefs will pofe more curved towards r any points, and INFINITE SERIES. 125

that the whole Curve lies on the fame fide of the points, it is plain and therefore is not but touch'd it. right line DC, cut, only by 6. Here it is fuppos'd, that the line

10. 126 Method of FLUXIONS, 10. EXAMPLE. Let ax =yy be the Equation to the Curve, Prob. which therefore will be the Apollonian Parabola. And, by 5. will be found AL=|

^ 2if -+. ax. ,* and DC = a Which being obtain'd, the Curve KC is determin'd by AL and LC, and its Length by DC. For as we are at and liberty to aflume the points K C anf where in the Curve KC, let us of fuppofe K to be the Center Cur- vature of the Parabola at its Vertex ; or and putting therefore AB and BD, it will be x and y, to be nothing, DC = -irf. And this is the Length AK, or DG, which being fubtracted from the former indefinite value of DC, leaves GC or KC = -^- V aa +. ax \a. what is 11. Now if you defire to know what Curve this is, and its without relation to the call KL z Length, any Parabola ; = t or and and LC = v, and it will be &==. AL \a = 3 x, ^z = AT, ' - ax Therefore S! CL or == = =yy.''' 4v/- = = = v, 2 27 t aa 2 7 # kind. u* j which fhews the Curve KC to be a Parabola of the fecond And for its Length there arifes llil ^/^aa -f- az a, by writing ~z for >r in the value of CG. 12. The Problem alfo may be refolved by taking an Equation, which fhall exprefs the relation between AP and PD, fuppofing P to be the interfeclion of the Abfcifs and Perpendicular. For calling AP=,v, and PD =/, conceive CPD to move an infinitely fmall fpace, fuppofe to and in and ta- the place Cpd} CD Cd king CA and CeT both of the fame given length, fuppofe = r, and to CL let fall the perpendiculars A^ and of which call fy y Ag, (which =z) may meet Cd inf. Then compleat the Parallelogram gyfe, and making and the fluxions of the x,y, z quantities ,v, y, and x, as before it and IN FINITE SERIES. 127

' " 1 it will be Ae : A/ :t A?P All* Q"P : CA] :: TT '

? :: : C P. Then t :: : CP. And A/: P/> CA a>quo Ae:Pp ^11 is the of the Abfcifs the acceiTion But P/> moment AP, by of which and is the it becomes Ap ; Ae contemporaneous moment of the per- the decreafe of which it becomes pendicular Ag-, by fy. There- the fluxions of the lines fore Ae and Pp are as Ag (z) and AP (x), ~- 2, : x :: : that is, as z and x. Wherefore CP. And fmce it

* a i and i it will be is Cgl = CAI AgT = &&, CA = ; * ~*z fmce of the CP_= . Moreover we may aflume any one

to the reft are to be three x,y, and z for an uniform fluxion, which if x be that fluxion, and its value is unity, then CP = referr'd,

it : :: CP : alfo : 13. Befides is CA (i) Ag (z} PL; CA (i) Cg is 2Z : : : therefore it zz CP CL ; PL = , and CL = )

~ z j Zz. Laftly, drawing /^parallel to the infinitely fmall X to will be the Arch D. Therefore Pp Pg fluxions of AP as i and Therefore becaufe of fimilar (x) and PD (;'), that is, y. fmce and or i and triangles Ppq and CAg, CA Ag, z, are in the

. have this fame ratio, it will be y = Whence we folution of the Problem. which the 14. From the propofed Equation, exprefles relation relation of the fluxions x and between x and^x, find the y, (by Prob. i.) and there will be had the value of to which z putting x = i, _)-, the lafl is equal. Then fubftituting z for/, by help of the Equa- find relation of the and tion the Fluxions x,y, z, (by Prob. i.) and

i the value again fubftituting for x, there will be had of z. Thefe found make ^21= CP, z x CP = PL, and CP x v/ 1 yy being Z = CL; and C will be a Point in the Curve, any part of which KG is equal to the right Line CG, which is the difference of the to the from the tangents, drawn perpendicularly Curve \)d points C and K, I2 8 7%e Method of FLUXIONS, Let be the which the rela- 15. Ex. ax=yy Equation exprefles and tion between AP PD ; and (by

it will be firft or Trob. i.) ax= 2yy, a = 2yz. Then zyz -f- zyz = o, or = z. Thence it is CP = y

I yy l_J

4-vv aa.

And from CP and PL taking away y

and x. there remains CD = , aa ~ and AL = ?a . Now I take c away y and x, becaufe when CP and PL have affirmative values, they fall on the fide of the point P towards D and A, and they ought to be diminiihed, by taking away the affirmative quantities PD and AP. But when they have negative values, they will fall on the contrary fide of the point P, and then they muft be encreafed, which is alfo done by taking away the affirmative quantities PD and AP. 1 6. Now to know the Length of the Curve, in which the point is of its rauft ieek C found, between any two points K and C ; we the length of the Tangent at the point K, and fubtradt it from CD. As if K were the point, at which the Tangent is terminated, when CA and Ag, or i and z, are made equal, which therefore is fituate itfelf i for in in the Abicifs AP ; write z the Equation a= 2yz, whence a=2y. Therefore for y write ^a in the value of CD, that is in and it comes out a. And this is the , length

of the at the K, or of DG ; the difference between Tangent point -- which and the foregoing indefinite value of CD, is -i#> that

is GC, to which the part of the Curve KC is equal. that it 17. Now may appear what Curve this is, from AL (having firft changed its fign, that it may become affirmative,) take AK, which will be ^a, and there will remain KL = %a, which

call /, and in the value of the line CL, which call v, write for

aa anc^ l ^ere l a"fe or -> v \/^at = ; = vv, which is an Equation to a Parabola of the fecond kind, as was found before. i a, and INFINITE SERIES. 129

1 8. When the relation between t and v cannot conveniently be reduced to an Equation, it may be fufficient only to find the lengths PC and PL. As if for the relation between AP and PD the Equa- 3 tion =o were affumed; from hence Prob. i.) ^x-^-^y _}' (by 1 firft there arifes a 4-^*2 y*z = o, then aaz zyyz y*z=o,

therefore it is and are and z = ,' z = . Whence yy aa aa yy

""'- and the is given PC = , PL = 2rxPC, by which point C determined, which is in the Curve. And the length of the Curve, between two fuch points, will be known by the difference of the or two correfponding Tangents, DC PC y. 19. For Example, if we make a= i, and in order to determine of the take 2 then fome point C Curve, we y = ; AP or x becomes .y 3"'.v_ _ . -_. - * "- 2 and PI - z z T> 1VPC > ana rLl Zaa T' T> ? to determine another if we take it Then point, ^' = 3, will be AP=6, =i, z = >ir, PC= 84, andPL= ioi. if from Which being had, y be taken PC, there will remain 4 in firil and in the for the the cafe, 87 fecond, lengths DC j the difference of which 83 is the length of the Curve, between the two points found C and c. 20. Thefe are to be thus underftood, when the Curve is continued between the two points C and c, or between K and C, without that Term or Limit, which we call'd its Cufpid. For when one or more fuch terms come between thofe points, (which terms of the are found by the determination greateft or leaft PC or DC,) the lengths of each of the parts of the Curve, between them and the or muft be and then points C K, feparately found, added together. PROB. XI.

To find as many Curves as you pieofe, whofe Lengths may with the be compared Length of any Curve propofed, or with its Area to a Line the applied given y by help of finite Equations.

i. It is performed by involving the Length, or the Area of the in the is propofed Curve, Equation which affumed in the foregoing Problem, to determine the relation between AP and PD (Figure Art. 12. pjg. 126.) Eut that z, and z may be thence derived, (by S Prob. 130 7%4 Method of FtuxioNS, the fluxion of the or of Prob. i.) Length, the Area, muft be firft difcowr'd. 2. The fluxion of the Length is determin'd by putting it" equal to the fquare-root of the fum of the fquares of the fluxion of the Ab- fcifs and of the Ordinate. For let RN be the perpendicular Ordi- nate, moving upon the Abfcifs MN, and let QR be the propofed Curve, at which RN is terminated. Then calling MN and = s, NR=/, and QR='i>, their

/R/ -f-Tr*

= Rr, or \/V -f- f- =

third that is, z and hence Which u=y, ^ = } ^ ^'=2;. being and INFINITE SERIES. 131

1 muft take PC -^. and DC being found, you = , PL=/x PC,

it that the of == PC y, or PC QR- Where appears, length cannot be but at the fame time 'the the given Curve QR found, be and from thence the length of the right Line DC muft known, in the is found and fo on the length of the Curve, which point C ; contrary. make 6. Ex.2. The Equation as ss = ff remaining, # = j,

d irv And the firft there will be found an ^ax-=.^ay. by ^ = -y,

as But the fecond i and therefore = v. And above. by = s, ^ 2iw or i = z. by the third 4^ = 407, (eliminating -y) ^

" 3L Then from hence == z ,

there be three aa = st, a + j. Ex. 3. Let fuppos'd Equations, the which denotes an *s = x, and A: -f- v =}' Then by firft, and therefore 4- " Hyperbola, it is o=rf+/i, or 7 = ', '-V"

is and therefore V/M -f- tf = v. By the fecond it 3* = i,

- it is i -u == or i v/w -+- = v. And by the third + y t + 3' that

~ there will arife from thence equal to iv, or | -f- = 7C'i;, ^

~ 1. 2W7i;. And firft for then for s, and ^ = fubftituting /', arife iv z. Now and z there will = = _>' dividing by aw, P^3 being found, the reft is perform'd as in the fivft Example. 8. Now if from any point Q_of a Curve, a perpendicular QV is be let fall on MN, and a Curve is to be found whofe length may known from the length which arifes by applying the Area QRNV

let Line be call'd the to any given Line ; that given E, length

is be call'd its v. which produced by fuch application

and the fluxions lines the fame time ; v and } of the v and MN, or of the which thofe to (or s,) lengths arife by applying Areas the s

Line are in the fame it be ~ . given E, ratio ; will v= Therefore

by this Rule the value of v is to be inquired, and the reft to be perform'd as in the Examples aforegoing. be 9. Ex. 4. Let QR an Hyperbola which is defined by this

aa -+ = // ; and thence arifes Prob. Equation, (by I.) =tf,

or = t. Then if for the other two Equations are aflumed x=s

v the firft will i whence v } and and y = ; give = j, = ^ =

latter will hence the give = v, or z = then from z= , y -g, ^

and fubftituting or for t, it becomes z = ~ . Now y and z ct ft hit

being found, make -r~ === CP, and_y x CP =n PL, as beforehand

thence the Point C will be determin'd, and the Curve in which all fuch points are fituated : The length of which Curve will be known from the length DC, which is equivalent to CP v, as is fufficiently fliewn before. 10. There is alfo another method, by which the Problem may be refolved and that is fluxions are either ; by finding Curves whofe equal to the fluxion of the propofed Curve, or are compounded of the fluxion of that, and of other Lines. And this may fometimes be of ufe, in converting mechanical Curves into equable Geometri- cal of in Curves ; which thing there is a remarkable Example fpiral lines.

1 1. be a Let AB right Line given in pofition, BD an Arch mov-< ing upon AB as an Abfcifs, and yet reas its a taining A Center, AD^ Spiral, at arch is which that continually terminated, bd an arch near or indefinitely it, the place into which the arch BD by its motion next a to the arrives, DC perpendicular arch bdt dG the difference of the arches, AH another Curve equal to the Spiral AD, BH a Line right moving perpendicularly upon AB, and terminated at the Curve AH, bh the ^ ~B~

little and lince bb. And in the infinitely triangles DG/ HK, DC to fame third Line and therefore and HK are equal the Bb, equal are to each other, and Dd and Hh (by hypothecs) correfpondent and therefore as alfo the at G parts of equal Curves, equal, angles the third fides dC and hK will be and K are right angles ; equal alfb. Moreover fince it is AB : BD :: Ab : bC :: hb AB (Qb) : * - therefore - = CG. If this be taken bC BD (CG) j A B away * dC /6K. Call from dG, there will remain dG & = = and their fluxions therefore AB=*, BD=-y, andBH=>', fince dG, and /jK are the z, v, and y refpedtively, B, contemporaneous moments of the fame, by the acceflion pf which they become bd and and therefore are to each other as the fluxions. A, t bb, let the fluxions be Therefore for the moments in the lafl Equation the letters for the Lines, and there will arife-y . fubftituted, as alfo if z be or the ^ ==" -.y. Now of thefe fluxions, fuppos'd equable, ~ *'

the will be i; unit to which the reft are refer'd, Equation ^=)'-

12. Wherefore the relation between AB and BD, (or between z which the is defined, and v,) being given by any Equation, by Spiral and thence alfo the fluxion the fluxion v will be given, (by Prob. i.) this will it to v. . And (by Prob. 2.) give ;', by putting equal ^ of it is the fluxion. the line y, or BH, which

the were which is to the i?. Ex. i. If Equation jzrzr-u given, 2 of Archimedes, thence Prob. - = v. From hence Spiral (by i.) -^

- or - and there will remain - and thence Prob. take , , =y, (by 2.)

2?_-r. Which fhews the Curve AH, to which the Spiral AD isi 2U to the Parabola of whofe Latus reclum is equal, be Apollonius, 2??; or whole Ordinate BH is always equal to half the Arch BD. If is defined the 14. Ex. 2. the Spiral be propofed which by 5 }_

3 1 a =a'v , or v there arifes Prob. Equation =;^ , (by i.) =-r, T 2^T I _l_ or ~- which if take , there will remain v, ano from you ^, , = x T fl 2i i

; Prob. will be ^l v. i. nrr thence (by 2.) produced = That ; -BD 3^ EU, AH being a Parabola of the fecond kind, t > 134 tte Method of FLUXIONS,

If the to the be thence 15. Ex. 3. Equation Spiral z

there will remain ~.. - the ^/- ?, , = y. Now fince quantity generated by this fluxion y cannot be found by Prob. 2. unlefs it be refolved into an infinite Series; according to the tenor of the Scho- it to firft lium to Prob. 9. I reduce the form of the Equations in the of the for z, it column Tables, by fubftituting z* ; then becomes

=.y, which Equation belongs to the 26. Species of the 4th

Orderof Table i. And by comparing the terms, it is d=,e=:ac, 2 -~ - andf=c, fo that ^ ac -f- cz == f=y. Which Equation belongs to a Geometrical Curve AH, which is equal in length to the Spiral AD.

PROB. XII.

To determine the Lengths of Curves.

1. In the foregoing Problem we have fhewn, that the Fluxion of a Curve-line is equal to the fquare-root of the fum of the fquares of the Fluxions of the Abfcifs and of the perpendicular Ordinate. Wherefore if we take the Fluxion of the Abfcifs for an uniform and determinate meafure, or for an Unit to which the other Fluxions are to be refer'd, and alfo if from the Equation which defines the Curve, we find the Fluxion of the Ordinate, we mall have the Fluxion of the Curve-line, from whence (by Problem 2.) its Length may be deduced. 2. Ex. i. Let the Curve FDH be propofed, which is defined by

-- - '- the Abfcifs AB and the the Equation -f- =_y ; making = s, Ordinate DB Then moving =y. Jr from the Equation will be had, 3 J v- the Prob. = y,/ ' ' - ^ \(bys i.) aa 12Z.S. of fluxion z being i, and y being the fluxion of y. Then adding the X~ of the the fquares fluxions, fum be == and v/ill -h |-f- -^ it, extracting the root, and INFINITE SERIES. 135

and thence Prob. = : t . Here / ftands for the = t, (by 2.) ^ ^ fluxion of the Curve, and / for its Length. if the

AB be look'd upon as indefinite, there will remain - _i_ -1 aa 1 2ft 24 for the value of

If know the of the Curve which is 4. you would portion repre- value of / to be to fented by /, fuppofe the equal nothing, and there -- or .. Therefore if take > arifes z* = , z= you 12 AB=-^- V*z y, 2 bd the and eredT: the perpendicular t length of the Arch ^D will be t or And the fame is to be underflood of all Curves aa 11% in general. After the fame manner which we have determin'd the 5. by if the -^L be length of this Curve, Equation ^ -f- =y propofed, nature of another Curve there will for defining the ; be deduced

^ . _lL -=.t\ or if this Equation be propofed, La*y?~*. 3" 1 * "* 2_ 5 '= t. Or in there will arife ^ -f-i^ general, if it is cz* -{-

* .- is u fed _ =_>', where 6 for reprefenting any number, either ,-8" Integer or Fraction, we (hall have cz* = /. o 4&Qi od

1 is z* or ~ and thence Prob. i. is derived Equation = ay , =_y, .by

~ < Therefore 1 -+- 2f: = i -+- ss . Now fmce the ar==y. yy 2a 4<* length of the Curve generated by the Fluxion / cannot be found by Prob. 2. without a reduction to an infinite Series of fimple Terms, I confult the Tables in Prob. 9. and according to the Scholium belong-

to I 1 . thus ing it, have / = v/ -t- And you may find

1 the lengths of thefe Parabolas Z = ay*, 2? r= ay*, z> = ay*, &c. * 8. Ex. 4. Let the Parabola be propofed, whofe Equation is 4 * 3 or Prob. will arife , and thence = = rfy (by i.) 1^ _y. ^=:^;"

v/ Therefore 1 -f- i^ =

1 sF = x, v/i -f- ^ = v, and |j=?. Where x denotes the Ab- 7 9 the and s the Area of the and / the fcifs, y Ordinate, Hyperbola, the Area to linear length which arifes by applying %s unity. the fame manner the of the Parabolas z 6 9. After lengths =ay', 7 z' &c. alfo be reduced to the Area of the z* :z=y , =ay', may Hyperbola. the CuToid of the Ancients be jo. Ex. 5. Let propofed, whole

. is __ and thence Prob. Equation ^T^jL" ; (by i.)' V az. 2.Z. 22,*

/ and therefore ^ i t v az zz=y, -^ ^/"^ = yy -f- = ;

' 2? for or becomes v/ az" which by writing ^ z~\ ^ -f- 3 = /, of the Order of Table 2 an Equation of the ift Species 3d ; then fo the it is = d, =. e, and ^ that comparing Terms, ^ 3 =^5 3 i /: 20;' 4

of which if i =y, to the fquare be added, there ariies zy az aa 4 of the fluxion of the arch A.J. , the fquare To which 02 Szz there will arife whofe if i be added again, -^ ^ ^- , fquare-root

=.* __ is the fluxion of the Curve-line AD. Where if z be ex- 2y/ az 2ZZ ~ tracted out of the radical, and for z be written c", there will be " -- a Fluxion of the ift of the Order of ' , Species 4th

Table 2. Therefore the terms being collated, there will arife d=.^a,

fo that ux _.v.v < e = 2, =rt; z= = x, \/ =

and -1 + ,= into, Method FLUXION J3 8 of s. the line

is its Abfcifs , and per-

is the pendicular Ordinate length BD, which arifes by applying the Area a^To. to linear unity. Now that the length of this Curve VD may be determin'd, I feek the fluxion of the Areaa

and I find it to be -^

' v/ b ax, putting AB =, and its fluxion unity. For

'tis AT its fluxion is whofe half drawn = = and , o p za v z

into the altitude or v/ - is the fluxion of the Area /3<^, a + , defcribed the that fluxion is , by Tangent

Ordinate BD. To the of this add the of fquare ~^~ i, fquare fl l6^ ta ^~ root the fluxion BD, and there arifes , whofe ^+a -^ z-\- ibfrz*- is the fluxion of the Curve VD. But this

is fluxion of the ift Table 2 : and a Species of the 7th Order of terms there will - the being collated, be =

l =g, and therefore z = x, and \/a b a*x -f- to one Conic whofe (an" Equation Section, fuppofe HG, (Fig. 2.) is v alfo *- Area EFGH j, where EF = #, and FG = ;) ==%,

i and - to another Conic */i6bb a*% + a&t- = Y) (an Equation Section, and INFINITE SERIES. 139

whofe Area is Section, (hppofe ML (Fig. 3.) IKLM

and thence, Prob. will be - O r =)', (by i.) had^ = (

To the fquare of this add i, and the root of the fum \/aa 4- tzz aa -\- bz.z. + bits. /. as this fluxion is not to be will be ^/ = Now found to infinite Series and firft in the Tables, I 'reduce it an ; by divifion " y i / 3 y 4 / 1 1 '5 * t 1 2:4 2 z &c - a d it becomes ;= extracting

, & &c.c AndA the root, t == a- z ,* be had the of Arch hence (by Prob. 2.) may length the Hyperbolical

the of 17. If the Ellipfis \/aa bz,z=.y were propofed, Sign be had z b ought to be every where changed, and there will 4- 7 - ^ *-z' 1 i &c. for the of its & _f- -^ ^t_s , length

it be z Arch. And likewife putting Unity for b, will -+- -^ -f-

Jil &c. for the of the Circular Arch. Now the 3ii_4_ >, lengthO 104 I I 2V.'' mul- numeral coefficients of this feries may be found adinfinitumt by

> the terms of this , tiplying continually Progreflion j , ^-

' ' S x 9 10 x i i be whole 18. Ex. 9. Laftly, let the Quadratrix VDE propofed, Vertex is V, A being the Center, and AV the femidiameter of the interior Circle, to which it is adapted, and the Angle VAE being a right Angle. Now any right Line AKD being drawn through A, cutting the Circle in K, and the Quadratrix in D, and the perpendiculars KG, DB being let fall it to AE } call AV AG = c;, VK x, and BD and =.a, = = y, T 2 will The Method of FLUXIONS.

- will be as in the foregoing Example, x =.z 4- -r~ 4- j; 4-

Extract the root and there will arife x &c. js, z= ^ 4- 7 * l , fubtract or a and the , &c. whofe Square from AKq. s s 4 " a 4 of the remainder # &c. will be GK. root 4- ,; *, 2^j -^9Aa9 *7?r\/j9 Now whereas by the nature of the Quadratrix 'tis AB = VR = x, and fince it is AG : GK :: AB : BD (y), divide AB x GK by AG, And and there will arife = a , &c. thence, y ^ ^ -^--, - &c. to the of Prob. = , fquare (by i.) y ^ ^. ^ will be i which add i , and the root of the fum 4- -f- ^'-J il il

_6o4^ __ Prob. / be &c ^ \vhence (by 2.) may obtain'd, 1Z /S~S U or the Arch of x the Quadratrix ; viz. YD = 4- ^j -f

6 4 'v7 &c. 895025

THE THE METHOD of FLUXIONS AND INFINITE SERIES;

O R,

A PERPETUAL COMMENT upon

the foregoing TREATISE, u

; THE METHOD of FLUXIONS AND INFINITE SERIES.

ANNOTATIONS on the Introduction :

OR,

The Refolution of Equations by INFINITE SERIES.

S E c T. I. Of the Nature and ConftruElion of Infinite or Converging Series.

great Author of the foregoing Work begins it with a fhort Preface, in which he lays down his main defign very concifely. He is not to be here underftood, as if he would reproach the modern Geometricians with deferting the Ancients, or with abandoning their Synthetical Method of Demonftration, much lefs that he intended to difparage the Analy- tical Art for on the he has nauch ; contrary very improved both Methods, and particularly in this Treatife he wholly applies himfelf to cultivate Analyticks, in which he has fucceeded to univerial ap- plaufe and admiration. Not but that we mail find here fome examples of the Synthetical Method likewife, which are very mafterly all remains and elegant. Almoft that of the ancient Geometry is indeed Synthetical, and proceeds by way of demonftrating truths already known, by mewing their dependence upon the Axioms, and other : 144 -tb e Method of FLUXIONS,

other fir ft Principles, either mediately or immediately. But the hiiinefs of Analyticks is to invcftiga'te fuch Mathematical Truths as really are, or may be fuppos'd at leaft to be unknown. It afiumes thofe Truths as granted, and argues from them in a general man- till after a .fcries of in ner, argumentation, which the -feveral fteps have a. neceftary. connexion wjth each other, it arrives at the knowledge of the propofition required, by comparing it with fomething really known or given. This therefore being the Art of Invention, it deferves to be cultivated with the utmoft certainly induftry. Many of our have been modern Geometricians perfuaded, by confidering intricate labour'd Demonftrations of the the and Ancients, that they .were Mailers of an Analyfis purely Geometrical, which they ftudi- ouily conceal'd, and by the help of which they deduced, in a direct and fcientifical manner, thofe abftrufe Proportions we fo much ad- mire in tome of their writings, and which they afterwards demonftrated Synthetically. But however this may be, the lofs of that Analyfis, if any fuch there were, is amply compenfated, I think, our Arithmetical or by prefent Algebraical Analyfis, especially as it is now improved, I might fay perfected, by our fagacious Author in the Method before us. It is not only render 'd vaftly more univerfal, and exterriive than that other in all probability could ever be, but is likewife a moft compendious Analyiis for the more abftrufe Geome- trical Speculations, and for deriving Conftructions and Synthetical Demonftrations as from thence ; may abundantly appear from the enfuing Treatife. 2. The conformity or correfpondence, which our Author takes notice of here, between his new-invented Doctrine of infinite Series, and the commonly received Decimal Arithmetick, is a matter of confiderable importance, and well deferves, I think, to be let in 3. fuller Light, for the mutual illuftration of both ; which therefore I fhall here attempt to perform. For Novices in .this Doctrine, tJho' they inay already be well acquainted with the Vulgar Arithmetick, and with the Rudiments of the common Algebra, yet are apt to appre- abftrufe and difficult in infinite Series hend fomething ; whereas indeed they have the fame general foundation as Decimal Arithmetick, efpecially Decimal Fractions, and the fame Notion or Notation is only tarry'd ftill farther, and rendered more univerfal. But to mew this in fome kind of order, I muft inquire into thefe following particulars. Firft I muft (hew what is the true Nature, and what are the genuine of our Scale of Arithmetick. Principles, common Decimal Secondly what is the nature of other particular Scales, which have been, or may and INFINITE SERIES. 145

is of may be, occasionally introduced. Thirdly, what the nature a which the foundation for the Doctrine of infinite general Scale, lays Scale ot Series. Laftly, I ihall add a word or two concerning that Arithmetick in which the Root is unknown, and thcrefoi-e propofcd occafion to the of Affected to be found ; which gives Doctrine Equa- tions. Firft then as to the common Scale of Decimal Arithmetick, it is that ingenious Artifice of expreffing, in a regular manner, all conceivable Numbers, whether Integers or Fractions, Rational or Surd, of by the feveral Powers the number Ttv/, and their Reciprocals; with the affiftance of other fmall Integer Numbers, not exceeding Nine, which are the Coefficients of thofe Powers. So that Ten is here the Root of the Scale, which if we denote by the Character X, as in the Roman Notation and its feveral Powers by the help of this 1 1 3 Root and Numeral Indexes, (X = 10, X = ico, X = 1000, 4 is ufual the Coefficients X = 10000, &c.) as ; then by ailuming as (hall we form or o, i, 2, 3, 4, 5, 6, 7, 8, 9, occafion require, may inflance 4 3 exprefs any Number in this Scale. Thus for 5X -f- jX -f- 1 1 will be a this 4X + 8X -rf- 3X particular Number exprefs'd by Scale, and is the fame as 57483 in the common way of Notation. Where we may obferve, that this laft differs from the other way of of Notation only in this, that here the feveral Powers X (or Ten) and are left are fupprefs'd, together with the Sign of Addition -f-, to be fupply'd by the Underftanding. For as thofe Powers afcend is or regularly from the place of Units, (in which always X, i, here is muhiply'd by its Coefficient, which 3,) the feveral Powers therefore be the will ealily be understood, and may omitted, and Coefficients only need to be fet down in their proper order. Thus 6 5 3 the Number 7906538 will (land for yX -+- gX -f- oX* -+-6X -f- ^X* -f-3X' -f-3X, when you fupply all that is underftood. And be the Number 1736 (by fuppreffing what may ealiiy -underftood,) 1 will be to 3 6 and the like of all other equivalent X -+- 7X -f- 3X -f- ; Integer Numbers whatever, exprefs'd by this Scale, or with this Root X, or Ten. The fame Artifice is uniformly carry'd on, for the expreffing of all Decimal Fractions, by means of the Reciprocals of the ll-vcral

of fuch as i 1 c.:c. Powers Ten, = o, ; = 0,0 ; = 0,001 ; ^ 5^1 ^ which Reciprocals may be intimated by negative Indices. Thus the : 4 Decimal Fraction 0,3172 (lands for 3X~'-j- iX~~ -f-7X -{- 2\~~ i is and the mixt Number 526,384 (by {applying what underfl ; U becomes Method

4 2X> X~' 1 and becomes 5 X + -f- 6X -f- 3 -f- 8X" -f- 4X- ; the infinite or interminate Decimal Fraction 0,9999999, &c. ftands for 1 3 4 5 & X~ &c. which infi- 9 X^' -f- gX- -4- 9X~ H- 9X~ -f- 9 -+- yX~ , nite Series is equivalent to Unity. So that by this Decimal Scale, (or by the feveral Powers of Ten and their Reciprocals, together with their Coefficients, which are all the whole Numbers below Ten,) all conceivable Numbers may be exprefs'd, whether they are integer or rational or irrational at leaft of a fracled, ; by admitting continual progrefs or approximation ad infinitum, And the like may be done by any other Scale, as well as the Deci- mal Scale, or by admitting any other Number, befides Ten, to be the Root of our Arithmetick. For the Root Ten was an arbitrary Number, and was at firft aflumed by chance, without any previous confideration of the nature of the thing. Other Numbers perhaps may be affign'd, which would have been more convenient, and which have a better elaim for being the Root of the Vulgar Scale of Arith- metick. But however this may prevail in common affairs, Mathe- maticians life other Scales and therefore in make frequent of ; the fecond place I (hall mention fome other particular Scales, which have been occafionally introduced into Computations. The moft remarkable of thefe is the Sexagenary or Sexagefimal Scale of Arithmetick, of frequent ufe among Aflronomers, which expreffes all poffible Numbers, Integers or Fractions, Rational or Surd, by the Powers of and certain numeral Coefficients not Sixty, exceeding fifty- nine. Thefe Coefficients, for want of peculiar Characters to reprefent them, muit be exprefs'd in the ordinary Decimal Scale. Thus if ftands for 60, as in the Greek Notation, then one of the/e Num- bers will be or in the Scale 53^ -f- 9^' -+- 34!, Sexagenary 53", 9*, 34, which is equivalent to 191374 in the Decimal Scale. Again, will be the fame as the Sexagefimal Fraclion 53, 9', 34", 53^= -f- z which in Decimal Numbers will be 9|f+ 34~ , 53,159444, &c. aa infinitum. Whence it appears by the way, that fome Numbers may be exprefs'd by a finite number of Terms in one Scale, which in another cannot be exprefs'd but by approximation, or by a progreffion of Terms in infinitum. Another particular Scale that has been confider'd, and in fome meafure has been admitted into practice, is the Duodecimal Scale, which exprefles all Numbers by the Powers of Twelve. So in common affairs we fay a Dozen, a Dozen of Dozens or a Grofs, a Dozen of GrofTes or a great Grofs, Off. And this perhaps would have been the mod convenient Root of all otherSj by the Powers of which to and IN FINITE SERIES. 147 to conftruct the Scale of Arithmetick as not fo popular ; being lig its and all below but that Multiples, it, might be eafily committed and it admits of a of Divifors to memory ; greater variety than any Number not much greater than itfelf. Befides, it is not fo fmall, 'but that Numbers exprefs'd hereby would fufficiently converge, or arrive near by a few figures would enough to the Number required; of is an the contrary which inconvenience, that muft neceflarily attend the taking too fmall a Number for the Root. And to admit this Scale into practice, only two fingle Characters would be wanting, to denote the Coefficients Ten and Eleven. Some have confider'd the Binary Arithmetick, or that Scale in which TIDO is the Root, and have pretended to make Computations to find considerable in it. this can never by it, and advantages But be a convenient Scale to manage and exprefs large Numbers by, becaufe the Root, and confequently its Powers, are fo very fmall, that they make no difpatch in Computations, or converge exceeding flowly. The only Coefficients that are here necelTary are o and i. Thus i x 2 5 -f- i x 2* -h o x2 3 + i x2* -f- i x 2' -f- 0x2 is one of thefe Numbers, (or compendioufly 110110,) which in the common No- tation is no more than 54. Mr. Leibnits imngin'd he had found great Myfteries in this Scale. See the Memoirs of the Royal Academy of Paris, Anno 1703. In affairs have common we frequent recourfe, though tacitly, to Millenary Arithmetick, and other Scales, whofe Roots are certain Powers of Ten. As when a large Number, for the convenience of read- ing, is diftinguifli'd into Periods of three figures: As 382,735,628,490. Here 382, and 735, &c. may be confider'd as Coefficients, and the the Scale is Root of 1000. So when we reckon by Millions, Billions, Trillions, &c. a Million may be conceived as the Root of our Arith- Alfo when divide a into metick. we Number pairs of figures, for the Extraction of the into ternaries Square-root ; of figures for the of the Cube-root Extraction ; &c. we take new Scales in effect, whofe Roots are 100, 1000, &c. Any Number whatever, whether Integer or Fraction, may be made the Root of a particular Scale, and all conceivable Numbers may be or that of exprefs'd computed by Scale, admitting only integral and affirmative whofe number the Coefficients, (including Cypher c) not be than the Root. need greater Thus in (Quinary Arithmetick, in which the Scale is of the Powers of the compofed Root 5, the Coefficients need be the five Numbers only o, i, 2, 3, 4, and yet all whatever are Numbers expreffible by this Scale, at leaft by approxi- U 2 mation, j^B 77oe Method of FLUXIONS, mation, to v/hat accu-racy we pleafe. Thus the common Number this Arithmetick would be x 4 2 x 2827,92 in 4 5 -+- 5' -|- 3 x 5* -\~ s I I or if we the feveral ox5 H-2x5-f-4x5~ H-3x 5~ ; may fupply as do thofe of Powers of 5 by the Imagination only, we Ten in the common Scale, this Number will be 42302,43 in Quinary Arithme- tick. All vulgar Fractions and mixt Numbers are, in fome meafure, the or the expreffing of Numbers by a particular Scale, making Deno- minator of the Fraction to be the Root of a new Scale. Thus is 1 is in effect o x 2 ; and the fame as 8 x '-f- x 3 + x^" 8-f- 5 3 j-'j reduced to this Notation will be or ra- and 25-5- 25x9 + 4x 9' , 1 . fo of all ther 2x9' -4- 7x9 -4-4X9"" And other Fractions and mixt Numbers. A Number computed by any one of thefe Scales is eafily reduced inftead of the in to any other Scale affign'd, by fubftituting Root one Scale, what is equivalent to it exprefs'd by the Root of the other Scale. Thus to reduce Sexagenary Numbers to Decimals, becaufe and therefore s 6X 1 3 60 = 6x10, or|=6X, | = 3 , ^=2i6X , &c. by the fubilitution of thefe you will eafily find the equivalent Decimal Number. And the like in all other Scales. The Coefficients in thefe Scales are not neceflarily confin'd to be lefs than the affirmative integer Numbers Root, (tho' they mould be fuch if we would have the Scale to be regular,) but as occafion may be affirmative or require they may any Numbers whatever, negative, indeed come out integers or fractions. And they generally promif- cuoully in the Solution of Problems. Nor is it neceflary that the Indices of the Powers mould be always integral Numbers, but may and the be any regular Arithmetical Progreffion whatever, Powers themielves either rational or irrational. And thus (thirdly) we are of what is call'd an univerfal come by degrees to the Notion Series, or an indefinite or infinite Series. For fuppofing the Root of the Scale to be indefinite, or a general Number, which may therefore or &c. and the be reprefcnted by x, y, affuming general Coefficients are or affirmative or a, b, c, d, &c. which Integers Fractions, nega- it we form fuch a Series as tive, as may happen ; may this, ax* -f- l lx* _j_ ex* -f- dx -f- ex, which will reprefent fome certain Number, the Scale whofe Root is x. If fuch a Number exprefs'd by pro- is call'd an ceeds in hfif.itum, then it truly and properly Infinite x Series, or a Converging Series, being then fuppos'd greater than Such for is x 3 &c. where Unity. example + \x~'-\-^.x'--+ ^*~ , the reft of the Terms are underftood ad in/initum, and are iniinuated and INFINITE SERIES. 149 bv, oV. And it may have any dcfcending Arithmetical Progreffion m m~ l 1 for its Indices, as x \x -+- ^v* -+-"*..\s, Gfc. led And thus we have been by proper gradations, (that is, by arguing from what is well known and commonly received, to what to difficult and to of before appear'd be obfcure,) the knowledge infinite Series, of which the Learner will find frequent Examples in the lequel of this Treatife. And from hence it will be eafy to make the following general Inferences, and others of a like nature, which will be of good ufe in the farther knowledge and practice of viz. the firft of Series is al- t-hefe Series ; That Term every regular ways the mo ft coniiderable, or that which approaches nearer to the Number intended, (denoted by the Aggregate of the Series,) than

: is and fo on : any other lingle Term That the fecond next in value, That therefore the Terms of the Series ought always to be difpoled

our : in this regular defcending order, as is often inculcated by Author That when there is a Progreflion of fuch Terms-/;? infinitum, a few of the firft Terms, or thofe at the beginning of the Series, are or fhould fufficient to the whole and that thefe be a Approximation ; may come as near to the truth as you pleafe, by taking in ftill more Terms : That the fame Number in which one Scale may be exprefs'd by a finite number of Terms, in another cannot be exprefs'd but by an infinite Series, or by approximation only, and vice versei : That the fo the cafen'.i the bigger Root of the Scale is, by much fafter, Series for the of the paribus, the will converge ; then Reciprocals Powers will be fo much the lefs, and therefore may the more fafely T be neglected : That if a Series coir e os by increafing Powers, fuch 4 the Scale as ax -^ bx* -+- ex* -|-

I took notice in the fourth place, that this Doctrine of Scales, and Series, gives us an eafy notion of the nature of affected Equations, or fhews us how they ftand related to fuch Scales of Numbers. In Inflances the other of particular Scales, and even of general ones, the Root of the the and the are all Scale, Coefficients, Indices, fiippos'd to be given, or known, in order to find the Aggregate of the Series, which is here the thing required. But in affected Equations, on the contrary, the Aggregate and the reft are known, and the Re ot of the Scale, by which the Number is computed, is unknown and re- Thus in the affected quired. Equation $x* -j- 3*2 -f- ox* -+- 7*- 53070, the Aggregate of the Series is given, viz. the Number to find the is 53070, x Root of the Scale. This eafily difcern'd to be 10, or to be a Number exprefs'd by the common Decimal Scale, efpecially if we fupply the feveral Powers of 10, where they are un- 4 3 1 derftood in the Aggregate, thus 5X -+- 3X -f-oX +7X' -4-oX = 53070. Whence by companion 'tis x = X=io. But this will not be fo eafily perceived in other instances. As if I had the f 1 Equation 4^+4- ax 3 -f- 3** -f-ox" -f- 2x -f- ^x~ -f- ^x~ = 2827,92 Ihould not I fo eafily perceive that the Root x was 5, or that this is a Number exprefs'd by Quinary Arithmetick, except I could reduce it to this 2x 3 2 x * form, 4x5* -+- $ + 3*5* + 0x5' -f- 5 H- 4x5 ; -+- 3 x 5~~ = 2827,92, when by comparifon it would preiently ap- that the be So that the pear, Root fought muft 5. finding Root of an affected Equation is nothing elfe, but finding what Scale in Arith- jnetick that Number is computed by, whofe Refult or Aggregate is in the common Scale which is a Problem of ufe given ; great and extent in all parts of the Mathematicks. How this is to be done, either in Numeral, Algebraical, or Fluxional Equations, our Author will inflruct us in its due place. Before I difmiis this copious and ufeful Subject of Arithmetical

Scales, I fhall here make this farther Observation ; that as all conceivable Numbers whatever may be exprefs'd by any one of theie Scales, or by help of an Aggregate or Scries of Powers derived frcm Root fo likewife Number whatever be any ; any may exprefs'd by fome fingle Power of the fame Root, by affuming a proper Index, integer or fracted, affirmative or negative, as occafion fhall require. Thus in the Decimal Scale, the Root of which is 10, or X, not only the Numbers i, 10, 100, 1000, &c. or i, o.i, o.oi, o.ooi, &c. the feveral of and their that is, integral Powers 10 Reciprocals, may be the Powers of X or viz. X X' X 1 s , , X exprefs'd by fingle 10, , , or X- 1 &c. but alfo all the inter- X, , X~% X--% refpectively, mediate and INFINITE SERIES. 151

mediate Numbers, as 2, 3, 4, Gff. u, 12, 13, Gfr. may be exprefs'd Powers of X or if we aflame Indices. by fuch fingle 10, proper JOI03> &C- X '477",&c. =__ Xo/o-.o, &e. Qr jj Thus 2 = X' , 3 = 4 g^ 8 &C - ' &e> _.X''4'3!>. i 2=== X'>7i" 456 = X*.s89s,&c. And the like of are all other Numbers. Thefe Indices ufually call'd the Logarithms of to which and are the Numbers (or Powers) they belong, fo many Ordinal Numbers, declaring what Power (in order or fucceflion) any of Root : And different Scales of Lo- given Number is, any aflign'd different Roots garithms will be form'd, by afluming of thofe Scales. But how thefe Indices, Logarithms, or Ordinal Numbers may be conveniently found, our Author will likewife inform us hereafter. All that I intended here was to give a general Notion of them, and to mew their dependance on, and connexion with, the feveral Arith- metical Scales before defcribed. of It is eafy to obferve from the Arenariiu Archimedes, that he difcufs'd this of Arithmetical had fully confider'd and Subject Scales, he there the in a particular Treatife which quotes, by name of his or in which it there he had laid the a'^^tl, Principles ; (as appears) foundation of an Arithmetick of a like nature, and of as large an extent, as any of the Scales now in ufe, even the moft univerlal. It a appears likewife, that he had acquired very general notion of the Dodtrine and Ufe of Indices alfo. But how far he had accommodated an Algorithm, or Method of Operation, to thofe his Princi- uncertain till that Book can be ples, muft remain recover'd, which than it is a thing more to be wim'd expedled. However may be his Genius and that fairly concluded from great Capacity, fince he thought fit to treat on this Subject, the progrefs he had made in it was very confiderable. But before we proceed to explain cur Author's methods of Ope- it be ration with infinite Series, may expedient to enlarge a little and to farther upon their nature and formation, make fome general Reflexions on their Convergency, and other circumftances. Now their formation will be beft explain'd by continual Multiplication after the following manner. the a bx 1

- . If the Root x= thefe two are multiply'd together, they will 3 2Xf?* *i &c. produce* + 2f_V + "1^5^ + V, a + a a n 152 The Method of FLUXIONS,

. if inftead of here its - o ; and x we fubflitute value , the Series

' ap fy+" q q q if 11* 9 j* "*" aq &c. o or if we divide and it will . = ; by -, tranfpofe, be t* tp + bg p dj> + eg /* ep + Jq - .... x x x ,7 : + , &c. = which Series, y j ^ thus derived, may give us a good infight into the nature of infinite Series in general. For it is plain that this Series, (even though it to were continued infinity,) mufl always be equal to a, whatever be to be the values of b &c. may fuppofed p, q, a, y c, d} For - the firft of the will be removed or , part firflTerm, always deflroy'd by its equal with a contrary Sign, in the fecond part of the feeond

firfl Term. And x- , the part of the fecond Term, will be re- i i moved by its equal with a contrary Sign, in the fecond part of -the third and fo on : So as to leave -- or for Term, finally , a, the

Aggregate of the whole Series. And here it is likewile to be obferv'd, that we may flop whenever we pleafe, and yet the Equation will be good, provided we take in the Supplement, or a due part of the next Term. And this will always obtain, whatever the nature of the Series may be, or whether it be converging or diverging. If the Series be diverging, or if the Terms continually increafe in value, then there is a neceflity of taking in that Supplement, to preferve if the .the integrity of the Equation. But Series be converging, or if the Terms continually decreafe in any compound Ratio, and there- vanifh or to the fore finally approach nothing ; Supplement may be as of fafely neglected, vanishing alfb, and any number Terms maybe taken, the more the better, as an Approximation to the Qium- a. thus from a due of this fictitious tity And confederation Series, of all or Series the nature converging diverging may eafily be apprehended. Diverging Series indeed, unlefs when the afore-mention'd increafing Supplement can be affign'd and taken in, will be of no feivice. And this Supplement, in Series that commonly occur, will be generally fo entangled and complicated with the Coefficients of the Terms of the Scries, that altho* it is always to be understood., neverthelef?, ii is often impoffible to be extricated and affign'd. But however, converging Series will always be of excellent ufe, as Affording a convenient Approximation to the quantity required, when it cannot be othei wile exhibited. In thefe the Supplement aforefaid, tho' and INFINIT E SERIES. 153

tho' generally inextricable and unnflignable, yet continually decreafes with the Terms of the and lefs along Series, finally becomes than any aflignable Quantity. lame The. Quantity may often be exhibited or exprefs'd by feveral Scries that Series is to be converging ; but mod edeem'd that has the Rate of greateft Convergency. The foregoing Series will converge fo much the cteteris as is lefs than or as the fader, paribus, p q y - Fraction is lefs than Unity. For if it be equal to, or greater than

Unity, it may become a diverging Series, and will diverge fo much as is the fader, p greater than q. The Coefficients will contribute little or nothing to this Convergency or Divergency, if they are fuppos'd to increafe or decreafe (as is generally the cafe) rather in a fimple and Arithmetical, than a compound and Geometrical Propor- tion. To make fome Edimate of the Rate of Convergency in this Series, and by analogy in any other of this kind, let k and / reprefent two Terms indefinitely, which immediately fucceed each other in the progrefTion of the Coefficients of the Multiplier a -+- bx ex* 3 &c. and let the number n the order -if -f-^x , reprefent or of k. the Series place Then any Term of indefinitely may be reprefented where the be or by -f- l'-Jf-~*- Sign mud -+- , accor- ?" is if ding as n an odd or an even Number. Thus == i, then *_LlL^Z k 1 and the firft will be . = a, = 1', Term -f- ]f ==2j

c then & l and the fecond Term will be = />, = c, ^~p. And fo of the red. if the next in the Alib m be Teim aforefaid pro- 7 after then l will be two grefTion /, -f- -^~lp"~ -f- ^ /." any fuc- ?" ?" cefiive Terms in the fame Series. Now in order to a due Conver- that is gency, the former Term abfolutely confider'd, fetting afide the the Signs, mould be as much greater than fucceeding Term, as therefore that conveniently may be. Let us fuppoie JL^Jp-i j s i"

' all the common factor \ than or ( dividing by c" greater " ^p",' } ~^ ' r t" + /f? - ^ is than or both that greater ^ , ( multiplying by pq, )

is than or the com- that Ipq -f- krf greater nip* +- Ipq, (taking away 1 that is than or a farther mon IpqJ kf greater //.y, , (by Diviiion,) - and as as be. that x is greater than unity ; much greater may fl X This 7%e Method of FLUXIONS,

effeft a double the is This will take on account ; firft, greater k in of and the is in of refpecl: ;;;, fecondly, greater 5* refpect p\ Now ex* if the in the Multiplier a -\-bx -f- -\-dx>, &c. Coefficients a, b, are in then k will be r, &c. any decreafing ProgreiTion, greater than is than fo that a k will be /, which greater m ; fortiori greater than therefore a m. Alfo if q be greater than p, and (in duplicate ratio) So that is the of j* will be greater than /*. (cater faribus) degree Convergency is here to be eftimated, from, the Rate according to w hich the Coefficients a, b, c, &c. continually decreafe, compounded with the Ratio, (or rather its duplicate,) according to which q fhall to be than be fuppos'd greater />. n / as the Term A will be The fame things obtaining before, .j_ i what was call'd the Supplement of the Series. For if the Series be continued to a number of Terms denominated by n, then inftead of this all the reft of the Terms in itifinitutn, we may introduce Sup- value inftead plement, and then we fhall have the accurate of a, of an approximation to that value. Here the firft Sign is to be taken if n is an odd number, and the other when it is even. Thus if and we fhall n= i, and confequently k=a, /= <, have * b etll t == a. Or if == 2, and /= c, then lX x + q

. illf-a -4- c\i f 7 j .i bb-^-a-j ff->r p <{$ cq J-If _L_L_I - - L---a. Or if n = 3, /= a, then x -4- f i 1 i q x ^ =.$. And fo on. Here the taking in of the Supple- the value of it ment always compleats a, and makes perfect, or which will whether the Series be converging diverging ; always that can be the beft way of proceeding, when Supplement readily be known. But as this rarely happens, in fuch infinite Series as ge- muft have recourfe to infinite nerally occur, we converging Series, wherein this Supplement, as well as the Terms of the Series, are and therefore after a number of infinitely diminifh'd ; competent them are collected, the reft may be all neglected in infinitum. the better to aflift the From this general Series, Imagination, we will defcend to a few particular Inftances of converging Series in Let the Coefficients &c. be Numbers. a, />, c-, d, expounded by pure * < then ** _ x ^ to, ; x , + ,, , | ; refpectively ^ ^ _-2^ x / ^ (XC< J or L^_fl^ x H 4, &C. ' ^c! ' x ^_^-+5ix 3' r.x; 7 3 4? 4x55. 5 5 27 ? f

'. lefs r . That the Series hence arifmg may converge, make/ than a?:d IN FINITE SERIES. 155

- than in given ratio, = ~, or = i, q = 2, then q any fuppofe />

A x &c. i. That this Series of |.x|H-4^x^ TV -J., = is, Fractions, which is computed by Binary Arithmetick, or by the Reciprocals of the Powers of Two, if infinitely continued will if finally be equal to Unity. Or we defire to flop at thefe four Terms, and inftead of the reft ad infinitum if we would introduce the Supplement which is equivalent to them, and which is here known to be x or we Hull have - j Ty, TV, 4 | -+- T^ -f- as is to let the be T'o- = i, eafy prove. Or fame Coemdents ex-

&c. then it will be - - pounded by i, |, -i, i, -f, -+-

f . 4iz^ 1 f=4f / & Thu Series m ehhei X X 3 1 3 47 J* 4 5? i be continued or after number of Terms infinitely, may be fum'd any i, _ n ~ infteadof all ;?, the exprefs'd by by introducing Supplement ; H-IXJ* the then -2 reft. Or more particularly, if we make (jr= $p, _f.

-- - ^ 7-^-. . v/hich is a -+- -f ( &c. = i, Number 6x5! liXjS 20X^4 30X;;!' exprefs'd by Quinary Arithmetick. And this is eafily reduced to the Decimal ~ for and the Coefficients Scale, by writing -f, reducing ; for then it will become 0,99999, &c. = i. Now if we take thefe five mall have Terms, together with the Supplement, we exadly 11- -12 - -f- r - -f- -~ 4- ^-, = i. Again, if 2x5 6x,i + 12x5} 20x54 + 30x5' 6x;

we make here ioo/^, we fhall have the Series JJ77= " 6 40 ~ 9 9 < co 27 ^^- >c -i- -': x x x -f- - + X i X 3 iccoo 3 4 oooooo 4 5 locoocooo which converges very fa ft. And if we would reduce this to the regular Decimal Scale of Arithmetick, (which is always fuppos'd to be to done, before any particular Problem can be faid be coinplcatly folved,) we muit let the Terms, when decimally reduced, orderly under one another, that their Amount or Aggregate may be tlifco- ver'd and then will in the the ; they ftand as Margin. Here Ag- gregate of the firfc five Terms is 0,99999999595, 0,985 which is a near Approximation to the Amount of the whole infinite or to Series, Unity. And if, for proof-

' to +/ 1L lake, we add this the _ = ,- Supplement 5 '" ' + , ,/' |OJ

the . be = 0,00000000405, wh< Unity exaclly. X 2 There T f6 The Method 3 of*f FLUXIONS,

There are alfo other Methods of forming converging Series, whether general or particular, which fhall approximate to a known quan- and therefore will be to the nature of tity, very proper explain Con- vergency, and to mew how the Supplement is to be introduced, when in order it can be done, to make the Series finite ; which of late has been call'd the Summing of a Series. Let A, B, C, D, E, &c. &c. be of and a, />, c, d, e, any two Progrcffions of Terms, which A is to be exprefs'd by a Series, either finite or infinite, compos'd of itfelf and the other Terms. Suppofe therefore the firft Term of the Series to be a, and that p is the fupplement to the value of a. ~ a Then is A a or = . As this is the whole = -}-/>, p Supple- ment, in order to form a Series, I fhall only take fuch a part of it - as is the Fraction and for the fecond denominated by , put q Sup-

That I will afiimie - - plement. is, = (p=) -XTJ -\-q, or /A a b \ A a E b ...... , f xi x ' as 1S 1S the whole q R=7 ~~B~ Again, of the I fhall aflume fuch a of it as is value Supplement q> only part de- the and for the next r. nominated by Fradion > Supplement put

. . /A a = (?=) -g- x orr = (-- x

x rr- x Now as this is the whole value of the Supplement r, I only afTume fuch a part of it as is denominated

the Fraction - and for the next s. That ~ by , Supplement put is,

Bl> Cc A a B /; Cc, A a r - x - or s x x (^ ) x -rr-a -+- s, = -77- x -7 = = ' ^ I D ^ U Ij

B /; Cc 7 A a B '> Cc T>d A j /- c i x r- 77 x . And lo on as far x x TJ r- x

that at lafr. the value of as we pleafe. So we have A.'=a-\-p, - where the Supplement p = ~l)-\-q, where the fecond Supple-

A a b a l> B A E Cc , inent == x TT-C where r x x q g -}- r, = g ^~ -]y 4- s, A B b C c D d where s = '^- x -7- x -rr x r-e-\- 1. And fo on ad tnfinitum. D (*. U H,

a. B /; A c _,. r 11 A A A a a Eb C , That is A a b x x x finally = -+- .+- ^-c -\ -7 -jj- A a 7, b Cc D d c \ -a r^ TT\ -O Of* x TV- x x Kc. where ere. and a -\- -jj- -J7-e, A, B, C, D, E, y 6cc. be b, r, d, e, may any two Progreffions of Numbers whatever, or whether regular defultory, afcending or defcending. And when it and INFINITE SERIES. 157

in thefe that either A = a, or or it happens Progreffions, B=^, ___ 5cc. then the Series terminates of itfelf, and exhibits the vilue of A in a finite number of Terms : But in other cafes it ap- to the value of A. But in the cafe of an proximates indefinitely the faid to re- infinite Approximation, Progreffions ought proceed to feme Hated Law. Here it will be to ob- "ularlv, according eafy if 1C and k are to two Terms indefi- fcrvc," that put reprefent any aforefaid whofe are denoted the nitely in the Progreffions, places by if L and / are the Terms number ;/, and immediately following ; then the Term in the Series denoted by n -f- i will be form'd from (v /- it /. As if n = i, the preceding Term, by multiplying by -^ fecond Term will K = A, k = a, L = B, l=b, and the

-L " j A T L - - -- i /t ', f-* 17 t"1~ipn TC "R k u ** " * ' DC 1 r> H A a, B A a B f> the third Term will be r == x r; I z c, and jr-^* ~7cT ~tT ~Tr~ then the and fo of the reft. And whenever it fhall happen that L =/, and no farther. And the Series will ftop at this Term, proceed fo much the catcris as the Series approximates fafter, paribus, nearer to Numbers A, B, C, D, &c. and a, b, c, d, &c. approach each other refpedively. Let Now to give fome Examples in pure Numbers. A, B,C, D, &c- then we &c. = 2, 2, 2, 2, &c. and a, b, c, d, &c. = i, i, i, i, fo fhall have 2 = i -h 1 H- + T +* -V> &c - And always, when are of the Series will be a the given Progreffions Ranks equals, G<~ metrical Progrefnon. If we would have this Progieffion ftop at the next Term, we may either fuppofe the firft given Progreilion or fecond to be 'tis all to be 2, 2, 2, 2, 2, i, the i, i, i, i, i, 2, that one which. For in either cafe we mall have L= /, is F ==/, TC ^ p* laft muft be - or i. and therefore the Term multiply'dby , =

Then the Progreffion or Series becomes 2 = I +T-r-ir~+"T + Tci 'if &c. &c. and +-TT- Again, A, B, C, D, = 5, 5, 5, 5, a, b, c, d, + - c - &c. =4, 4> 4, 4, &c then 5 = 4 H- T + T T + TTT -H *-TT, & ' or = H- T T -i- -4 T + T!T> &c Or if A, B, C, D, &c. = 4, ^. 6cc. and f &c. &c. then 4, 4, 4,

~ - mu^ be the la ft and therefore = ""7^ ^ T niultiply'd by the Series becomes 6 .1. x - - .. Term. So that 5 = 1.7 -f- -^-S f x

* ' '* ' I * ' 4 3 * ^ 3 v ^ v TO f Tf for --> X n X X X 10 vX vX> vx 1 r A R CD (XU 2 T9 T T T T T T T TJ **> ^ ^> ^> > &c. 2, &c. then 2 3, &c. and <--, b, c, d, = i, 3, 4, = 1-4- 4, 5, &c. If T-+^x^ 3 +|x^xi4-|-|x^xix^5, A,B,C,D,6cc. and &c. = &c. then i =^ i, 2, 3, 4, &c. ^, b, c, d, 2, 3,4, 5, =

- X X 6 &c - And from 13 + T x|4 T XT; infinite other Series be this general Series may particular eafily de- fliali to the rived, which perpetually converge given Quantities ; chief ufe of which Speculation, I think, will be, to iliew us the nature of Convergency in general. that be There are many other fuch like general Series may readily form'd, which mall converge to a given Number. As if I would confliucl a Series that flrali converge to Unity, I fet down i, together with a Rank of Fractions, both negative and affirmative, as here follows.

'* - - - &c I --"""""'"'

-h and INFINITE SERIES, 159

affirmative and iiich as are feen here below ; with it, both negative, will the which being added together obliquely as before, produce following Series.

i 4- j6o *fi>e Method of FLUXIONS,

tation of the ufual praxis of Divifion in Numbers. In order to obtain the Quotient of aa divided by b -f- x, or to relblve the com- the pound Fraction T|T- into a Series of fimple Terms, firft find

of aa divided l> the firft of the Divifor. This Quotient by } Term which write in the Then the Divifor is ^ , Quote. multiply by this Term, and fet the Product aa -h ^ under the Dividend, from

whence it muft be fubtracted, and will leave the Remainder ~ . divide Then to find the next Term (or Figure) of the Quotient, the Remainder by the firft Term of the Divifor, or by b, and put the Quotient "~ for the fecond Term of the Quote. Multiply the Divifor by this fecond Term, and the Product ^ ^r the from it muft be fub- fet orderly under laft Remainder ; whence find to find the new Remainder -h . Then to the tracted, "-^-bo next Term of the Quotient, you are to proceed with th-is new and fo on in The Remainder as with the former ; infimtum. Qup- * - r . a* K* c c*x* a*x3 c , tient therefore is &c. into i j -+- ^- , (or -j ? So that this the Number or .+- ^ ^ , 6cc.) by Operation

1 1 a is reduced from that Scale in Quantity ^ , (or x^-t-*!" ) Arithmetick whofe Root is b -+ x, to an equivalent Number, the

is . And Root of whofe Scale, (or whofe converging quantity) this Number, or infinite Series thus found, will converge fo much the fafter to the truth, as b is greater than x. an inftance or two in To- apply this, by way of illustration, to common Numbers. Suppofe we had the Fraction |, and would jeduce it from the feptenary Scale, in which it now appears, to an mall the Powers of 6. Then equivalent Series, that converge by in the we (hall have = ; and therefore , j foregoing general ^ ^ \ b and and the Series Fraction , make a-=. i, = 6, #==1, -^-x b -"j~ ~ be to will become f + ^ ^, &c. which will equivalent if it to a Series the Powers Y. Or we would reduce converging by

of make and .v = i, becaufe ~ , a= i, ^=8, 8, f= then and IN FINITE SERIES. j6r

~ &-c - which Series will fafter then = T + ~* -+- & -+- ^ > converge than the former. Or if we would reduce it to the common Denary (or

becaufe niake l> and Decimal) Scale, f -~r- , a= i, = 10,

then <*c ' x= 3 ; 7 = -rV -4- -4-0- -+- Wo-o- -f- -o-Vo-o- + TO-O-^O-S-J as be collected. hence = 0,1428, &c. may eafily And we may va- obferve, that this or any other Fraction maybe reduced a great to infinite Series but that Series will iafteft riety of ways ; converge to the truth, in which b mall be greateft in refpect of x. But that Series will be mod eafily reduced to the common Arithmetic^, which converges by the Powers of 10, or its Multiples. If we or i mould here refolve 7 into the parts 3 -f- 4, or 2+5, -f- 6,

or : fuch &c. inftead of converging we mould have diverging Series, to be taken in. as require a Supplement And we may here farther obferve, that as in .Divifion of com- of Divifion whenever we mon Numbers, we may flop the procefs and inftead of all the reft of the ad in- pleafe, Figures (or Terms) write the Remainder as a and the finituniy we may Numerator, 'Divifor as the Denominator of a Fraction, which Fraction will be

: in the the Supplement to the Quotient fo the fame will obtain Divifion of Species. Thus in the prefent Example, if we will flop a "~ the firft Term of the we mall have . at Quotient,' = ^ -^- o ^L. b + X b X /; |- x

ft the fecond then "-~r Or if we will op at Term, --r\. = j -f- then Or if we will flop at the third Term, ^- = ^

in which _ . And fo in the Terms, ^-x fucceeding be to make the thefe Supplements may always introduced, Quotient Obfervation will be found of ufe in fome of compleat. This good Fraction is not to the following Speculations, when a complicate but to be or to be reduced to a be intirely refolved, only deprefs'd, form. fimpler and more commodious For hav- Or we may hence change Divifion into Multiplication. firft of the and its or ing found the Term Quotient, Supplement, ta aax *' K aa - i i -i /lit it , we fhall the = -, multiplying by Equation ^ -^x ? -^- fo that this value of haveIldVC = * 3- a ,' fubftituting i T~^- ant Ml aa aa aaX _ffL_ in firft it will become = -f- the Equation, ^ y -^ a>A'*- . where the two firft Terms of the Quotient are now known. Y Multiply 162 The. Method of FLUXIONS*

this and it will become Multiply by ^ , *- , which fubfthuted in the laft it will become L being Equation, ra 1 i aa aav fi^.v a*** a' x*' i .1 c r- n - - r =. ---- -4 -- --1- -. . where the four nrlt r- ' t-^-x b b* b* I* iS+i*X Terms of the Quotient are now known. this Again, multiply ' 6 ,-, . , A.4 rf 5 .v4 fl-.v4 a*x* a*x it --- ~ , and will become = Equation by ^ ^7^x * JT+

- laft r- r which fubftituted in the s -p -, 8 , being Equation z 7 1 ! ... aa a* a .v a*x* .<3 a**4 a x , - - - it will become 1 4- = 4 f p- i 6 17 i V 8 -4 of firft are fyi- ^- 5T- , where eight the Terms now hi b t9-^-6x the known. And fo every fucceeding Operation will double number of Terms, that were before found in the Quotient. This method of Reduction may be thus very conveniently imitated in Numbers, or we may thus change Divifion into Multipli- I find the of the cation. Suppofe (for inftance). would Reciprocal the value of the Fraction ' Decimal Prime Number 29, or T T. m Numbers. I divide 1,0000, Gfc. by 29, in the common way, fo far as to find two or three of the firft Figures, or till the Remainder be- afliime to comes a fingle Figure, and then I the Supplement compleat the Quotient. Thus I mail have T ~ =. 0,03448^ for the compleat Quotient, which Equation if I multiply by the Numerator 8, it will I fubftitute give ^ = 0,275844^., or rather ^.==0,27586^. this initead of the Fraction in the firft Equation, and I (hall have ^=1:0,0344827586^. Again, I multiply this Equation by 6, * and then Subftitution ' == and it will give T 7 = o, 2068965517^, by T 7 0,03448275862068965517^. Again, I multiply this Equation by 7, 7 anditbecomes T ?=o,24i3793io3448275862oi|-,andthenbySubfti-

number of where every Operation will at leaft double the Figures be an found by the preceding Operation. And this will eafy Expe- dient for converting Divifion into Multiplication in all Cafes. For it be multi- the Reciprocal of the Divifor being thus found, may into the Dividend to the ply'd produce Quotient. . . , aa aa n*x -** **S , c , here Now as it is found, that j =7 77 -+ ~jr Z7~> will b is than A* fo when &c. which Series converge when greater ; is than that the it happens to be otherwife, or when x greater b, Powers of x may be in the Denominators we muft have recourfe to the and INFINITE SERIES, 163

in which we fhall find the other Cafe of Divifion, -^-^ = ^ a i &c. and where the Divifion is 'd as _j_ ^- "^ , perform before. 6. In thefe of our the Procefs of Divifion 5, Examples Author, exercife of be thus exhibited : (for the the Learner) may

o x i +o 1 AT .V4

+7*-

Now in order to a due Convergency, in each of thefe Examples, we muft fuppofe x to be lefs than Unity; and if x be greater than l Unity, we muft invert the Terms, and then we fhall have XX "^ 1 i i * 1 I I I c &c - = ^ ^ + 7*

*/* io. This Notation of Powers and Roots 7, 8, 9, by integral and fractional, affirmative and negative, general and particular Indices, was certainly a .very happy Thought, and an admirable Improve- ment of Analyticks, by which the practice is render'd eafy, regular, and univeifal. It was chiefly owing to our Author, at leaft he car- .ried on the Analogy, and made it more general. A Learner fhould be well acquainted with this Notation, and the Rules of its feveral Operations fhould be very familiar to him, or otherwife he will often find himfelf involved in difficulties. I fhall not enter into any far- difcuffion of it as not ther here, properly belonging to this place, or fubject, but rather to the vulgar Algebra. 1. Author to 1 The proceeds the Extraction of the Roots of pure Equations, which he thus performs, in imitation of the ufual Pro^ cefs in Numbers. To extract the of Square-root aa +- xx ; firft the is be Root of aa a, which muft put in the Quote. Then the Square of this, or aa, being fubtradted from the given Power, leaves -+-xx for a Refolvend. Divide this by twice the Root, or 2a, which is Y 2 th 164 ?$ Method' of FL u X r or N s, the firft part of the Divifor, and the Quotient muft be made the fecond Term of the Root, as alfo the fecond Term of the Divifor. x the Divifor thus or -za ~ the fecond Multiply compleated, -J- , by Term of the Root, and the Produft xx + muft be fubtrafted from the Refolvend. This will leave , for a new Refolvend, 4-" which divided firft being by the Term of the double Root, or 2tf, . A will for the third Term of the Root. Twice the Root give j before with this Term added to or 2a befound, it, -+- ^ -^ , ^- ins multiply 'd by this Term, the Product 1- muft 4a* 8^4 640'' be fubtrafted from the laft Refolvend, and the Remainder -f-

. B will be a new Refolvend, to be proceeded with as before, the next of the Root and fo as far as for finding Term ; on you ' '- -' pleafe. So that we (hall have \/ aa -+-xx = a+ _ _i_ ~ 1 T.a oa* io*

obferve from that in It is eafy to hence, the Operation every new Column will give a new Term in the Quote or Root; and therefore no more Columns need be form'd than it is intended there mall be Terms in the Root. Or when any number of Terms are thus extraded, as many more may be found by Divifion only. Thus hav-

firft ing; found the three Terms of the Root a -f- , by 2a fcu3 " J v^ v4 -za the third Remainder their double -\ , dividing or Re- folvend the three firft Terms of the Quotient -\- 4 7^: , ^* in. 04*. l6&^ c* 8 7x l ; ',- will be the three Terms of the Root. ' H fucceedingfj 1 2 Oil 2 COrt*

The Series a -f- ^c< t^us f untl f r the ^i H TT* > fquare- root of the irrational quantity aa -f- xx, is to be understood in the a is to following manner. In order to due convergency a be (iippos'd - greater than x, that the Root or converging quantity may be leis than Unity, and that a may be a near approximation to the fquare- the root required. But as this is too little, it is enereafed by fmall

which makes it too Then the next quantity , now big. by Operation and INFINITE SERIES. 165

the ftill Operation it is diminim'd by fmaller quantity ^; which diminution being too much, it is again encreas'd by the very fmall which makes it too in order to be farther di- quantity -7--r , great,

thus it minifli'd by the next Term. And proceeds in infinitum, the Augmentations and Diminutions continually correcting one another, and till the till at lalt ihey become inconfiderable, Series (fo far con- is near to the tinued) a lufficiemly Approximation Root required. 12. Wh-ii a is Ids than x, the order of the Terms muft be inverted, 01 ihe fquare-root of xx -+- aa muft be extracted as before; in which cafe it will be x -+- ', &c. And in this Series 2X -f-.5 the converging quantity, or the Root of the Scale, will be -. Thefe two Scries are by no means to be understood as the two different Roots the aa for each of the two Series will exhibit thofe of quantity -+- xx -, are two Roots, by only changing the Signs. But they accommodated to the two Caf s of Convergency, according as a or x may happen to be the greater quantity. I (halt here refclve the foregoing Quantity after another manner, the better to prepare the way lor what is to follow. Suppofe then fi'-d the value of yv=.cni-\- xx, where we may the Root y by the f 11 ,wir Proccfy ; aa -+- XX= ;.j = (\f)' = yy rf-f-/) aa^-zap~ -\-pp-, or zap -+- pp = xx = (If p = + q} xx + zaq -{-

or - --qq; 2rf?-J- ^ -H^= = (if ?=== _^ r or + >

'-, - r rr . -- &c. which Procefs = j) = 6 (if + 1 may oi,' O-|t. VU* J wo-ds. be thus explain 'd in In order to find V ua --xx, or the Root y of this Equation 1 wheie a is to be yy-=aa-\-xx, iuppofc y = ^-f-/', undeiftood as near arion to the value of nearer the bet- a pretty Approxii: _y, (the is the lnv.,11 to or the ter,) and p Supplement that, quantity which makes it compleat. Then by Subftitution is deiivcd the fir It Sun- whole is to plementiil Lqu^i'oa zap -+-//; = xx, Root/; bt fou:,d. is n:iich than is INOW as 2uJ> bigger ff, (lor za bigger than the Sup- - v;c fh;.!l have or at leaft ve (hall have plement/,) nearly p ,

: to = -f- -', the fecoiid exactly ;- ; fuppofmg q reprefent Supple- ment j66 *ft>e Method of FLUXIONS, ment of the Root. Then by Subftitution zaq -+- ^q -4-^= =

^1 will be the fecond Supplemental Equation, whofe Root q is the

will be a little fecond Supplement. Therefore q quantity, and qq

fo -that much lefs, we mall have , or nearly q= g--3 accurately

if r be made the third to the Root. q =. ^ -f- r, Supplement

And therefore zar -f- r r -f- r* = f- will be the U 4^ ou*r L^," third Supplemental Equation, whofe Root is r. And thus we may go on as far as we pleafe, to form Refidual or Supplemental Equa- tions, whofe Roots will continually grow lefs and lefs, and therefore will make nearer and nearer Approaches to the Root y, to which is the Root of this they always converge. For y =5= a -{-/>, where p Equation zap-- pp-=xx. Or y =: a~\- -+-g, where q is the

of this -- . Or ; a Root Equation zaq -\ -q-\-qq-=z ^ y -f- * -. -f- r. where r is the Root of this Equation zar -f- r Ztt oa> a ~ rr-=. -~ ~. And fo on. The "Refolution of one I any in the of thefe Quadratick Equations, ordinary way, will give the will the value refpeclive Supplement, which compleat of y. I took notice before, upon the Article of Divifion, of what may be call'd a Comparifon of Quotients; or that one Quotient may be exhibited by the help .of another, together v/ith a Series of known or iimple Terms. Here we have an Inftance of a like 'Comparifon of Roots; or that the Root of one Equation may be exprels'd by the Root of another, together with a Series of known or fimple Terms, which will hold good in all Equations whatever. And to mall hereafter find a like carry on the Analogy, we Comparifon of one for a Fluents ; where Fluent, (fuppofe, inftance, Curvilinear Area,) will be exprefs'd by another Fluent, together with a Series of fimple Terms. This I thought fit to infinuate here, by way of that I the conftant anticipation, might mew uniformity and har- mony of Nature, in thefe Speculations, when they are duly and regularly purfued. for But I mall here give, ex abundanti, another Method this, and fuch kind of Extractions, tho' perhaps it may more properly be- of which is foon to long to the Refolution Affected Equations, fol- however it ferve as an Introduction to their Solution. low ; may j The and INFINITE SERIES. 167

The firft or Refidual Supplemental Equation in the foregoing Pro- cefs was 2ap -\-pp-=. xx, which may be refolved in this manner.

Bccaufe it will be Divilion " - ' y by3 p = -{ -f- />= -^-,za za Aa* ^ + t ** ! x*tA

, &c. Divide all the Terms of this Series the ,-^7 + -^ (except fir and then ft) by p, multiply them by the whole Series, or by the value of and - ' 3-^ />, you will have p = -f- ia 8*' + 8*4 3Z 6cc. , where the two firft are clear'd of Divide all the ^ -g Terms />. Terms of this the and them Series, except two firft, by />, multiply the value of or will have a Series by />, by the firft Series, and you for in p which the three firft Terms are clear'd of p. And by repeating the Operation, you may clear as many Terms of p as you So that at laft will have ~ -+- pleafe. you p = , 7^ value of as before. -+- ^~, &c. which will give the fame y of thefe and 13, 14, 15, 16, 17, 18. The feveral Roots Examples, of all other pure Powers, whether they are Binomials, Trinomials, or any other Multinomials, may be extracted by purfuing the Me- thod of the foregoing Procefs, or by imitating the like Praxes in Numbers. But they may be perform'd much more readily by gene- there will ral Theorems computed for that purpofe. And as be frequent occalion, in the enfuing Treatiie, for certain general Opera- tions to be perform'd with infinite Series, fuch as Multiplication, of Roots 1 Divilion, railing of Powers, and extracting ; mall here derive fomc Theorems for thofe purpofes. I. Let A H- B 4- C + D -+- E, &c. P-f-Q^-R-f-S-t-T, &c. and the of three feveral Series a _l__j_^_{_j\.4_ g) &c. reprefent Terms let &c. into refpedlively, and A-|-B-{-C-f-D-|-E, P+Q-t-R-f-S+T, a, the known Rules of &c. = -\- /B -{- y -i-

. x 1- 4- "^.-t-K -f- o -t- i7ov. = AP +BP -i-Cf'+DP-f-E,-, <3c.

And 1 68 'The Method of FLUXIONS,

And this will be a ready Theorem for the Multiplication of any Series into each other as in the infinite 5 following Example.

(A) (B) (C) (D) (E) (P) (QJ (R) (S) (T) 1 X* *J A'4 ,, . -x &c - afr*+ + & + &> mto*-fx-f- 1 &>+X^+i*?,rb +~, $cc, =^+t^+tf^ X* A 4 a JL/rv v* -P. '- = _ 1 1 , t* A ^^TT ** V- _ 9# \2a" +**'+ +7|? * 4 i! _ '* 7a i \a

-*-.3* 9 ^ And fo in all other cafes.

II. From the fame Equations above we fhall have A = -.

.-DQ.-CR-BS-AT ^ ^ And then by Subftitution i ^(A+ B-J-C+D + E, &c. =) S + a

p p of one will ferve commodioufly for the Divifion infinite Series by another. Here for conveniency-fake the Capitals A, B, C, D, &c. in to the are retained the Theorem, denote firft, fecond, third, fourth, &c. Terms of the Series refpedively. M (0) divide the Series Thus, for Example, if we would #* _f. .ax -+. p (>) (/) (t) ( J (QJ W (S) (ij 2 ' " ii * . x z , &c. the Series -f- _}_ ^-ii^-_{_ by a+^x-i- ~^. , &c. a*"~ T* will - the Quotient . be a -f- -f- fx*

&c. Or , reftoring the Values of &c. which the A, B, C, D, reprefent feveral Terms as they /land in order, the will become a _i_ .11 Quotient f# z &Ut r + *^ ' ^' 5 7 ga 3 And after the fame manner in all other Examples.

HI. and INFINITE SERIES. 169

III. In the laft Theorem make .r=r, /3 = o, o>=o, ^ = 0,60:. . then ^_ V. l' p ~~F~ ~~p DQ+CR+BS+AT &(, whkh Theorcm ^ win readl]y find th cal of any infinite Series. Here A, B, C, D, &c. denote the feveral Terms of the Series in order, as before. p ( ) (QJ the Thus if we would know Reciprocal of the Series a-\- f.v-{-

*- &c. And reftoring the Values of A, B, C, D, &c. it will be

------^~- &c. for the > 1- h Reciprocal required. la 12^ 84 720*' ^- 2. = i + f.v + &c. And . i*+i<, ,. i f f l x + f..- A< ..{ fa of others. IV. In the firft Theorem if we make P=A, Q^==B, R = C, that if we make both to be the fame Series mail have S=D, &c. is, ; we * A+B+C+D+E+F+G7&^ I tf= A+ zAB + zAC+ zAD + 2 AE + zAF + zAG.tff. + B 1 + zBC + 2 BD + zBE + aBF + L* + zCD+ zCE + D* which will be a Theorem for finding the Square of any infinite Series.

Fv i -_-'_ - . " 5 7 1 aa Sa'^lba iz8a 256^ 4^* ga^iea* I zSafl" 25!., .'<>

8 s 64 S i 2a

1 L - *^c x bx* o i t 1x' txl A-4 Ex. --- -- &c. J3. 2 ^a H ^3 '

u i TTTI

.7/4 4 64*4 Ii

Ex. . _l _H_l^- ^-, I *_ ii._fil , 30." 4 2 JH 8 - > 2434" 9

if V. In this laft Theorem, we make A*= P, aAB = , 2 AC Qv 1 B 1 2AD -+- 2BC = S, 2 AE -+- 2BD -f- C = T, &c. we _f- =R, * O R .- R S "* BC

B , D fhallhave A = P^ = ^-, C==-^- = -^- , E== T ~ 2BD ~ C - &c. Or p &c. ^ iA , + Q + K-hS+TH-U, | = pi z Q R B S 2RC T 2BD C^ U 2 BE ^CD r - -4- -4~ 1 -4- -4- ^^ '

Ex 2^1- "'~~o

VI. Becaufeit is by the fourth Theorem a -{-@-i--y-\-

1 &c. write a j8S 2a^ -f- P, Q^ R, S, T, , 2a/3, 2> + 2/3y, 2ag- i / &c. , refpedively. Then X A

And this will be a Theorem for finding the Reciprocal of the Square of any infinite Series. Here A, B, C, D, &c. ftill denote the Terms of the Series in their order. VII. If in the firft Theorem for P, Q^ R, S, &c. we write 4 &c. is A*, 2AB, 2AC + B , 2AD -H 2BC, refpedively, (that 1 5 . A+B+C+D,&c.| 3 byTheor.4.)wemallhaveA+B+C+D+E+F 3 6cc.| s 1 1 &c. = A -i- 3A*B + sAB -h sA*D -j- 3AC -f- 360, 6ABC+ 36^0 + 3BD B' + 6ABD-f- 6ACD -- 6ABE

t Series. which will readily "give the Cube of any infinite 11 v9' A- r. 13 X ^

*' *'* ^ yjf^ *" " " " ^^ * "T~ 2* 11 15 3 Ex. and INFINITE SERIES, 171 Ex.2. t*1 -i~ VIII. In the laft if we make A 3 Theorem, =P, 3A*B O , 1 '+.3A'C = R, B'-f-6ABC-|-3A D = S, &c. then A=PT, Q_ R 3 AB* _ S-6ABC Bi B = , C = x , U = - - , fisc. that is p: ?A j^ ^ Sec. , I i^K+ +l+ root of any infinite Series may be extracted. Here alfo A, B, C, D, in order. &c. will reprefent the Terms as they ftand 1 15 s T? x' 8* 1 x IPX'* 7*"* 7 _ _* ;** ~ - ~ "'^ I ^+ 8i 8 -~"I" z' I 243a 4 ' 7 8 ' 4 Ex. 2. A; x 6cc. l^ ** , &c. f* -h T 7 H- T | T , =t**-t-rT H-Trr^ J IX. Becaufe it is the feventh a -f- -\- , &c. by Theorem + y j a* i 1 5 &c. in the third Theorem for + 3a /3 -f- 3 a/3 -f- /3 , P,

i 1 } R, S, T, &c. write ', 3j8, 3/S -f-Sa ^ /3 -f- 6a/3>-f- &c. then 3'fr refpeflively ;

This Theorem will give the Reciprocal of the Cube of any infinite for in Series ; where A, B, C, D, &c. ftand the Terms order. 1 firft if ; X. in the Theorem we make P=A , Laftly, Q4 ==3A B,

> &c. we {hall have

4 s 1 1 A+B-f-C-i-D, &c. I =A^H-4A B-{-6A B -|-4ABs&c. which

will be a Theorem for finding the Biquadrate of any infinite Series. And thus we might proceed to find particular Theorems for any other Powers or Roots of any infinite Series, or for their Recipro- or fractional of thefe all will cals, any Powers compounded ; which be found very convenient to have at hand, continued to a competent number of Terms, in order to facilitate the following Operations. Or it may be fufticient to lay before you the elegant and general Theorem, contrived for this purpofe, by that fkilful Mathematician, and my good Friend, the ingenious Mr. A. De Mo'rore, which was firft publifh'd in the Philofophical Tranfa&ions, N 230, and which will readily perform all thefe Operations.

Z 2 Or 172 The Method of FLUXIONS, have recourfe to Or we may a kind of Mechanical Artifice, by the which all foregoing Operations may be perform'd in a very eafy and general manner, as here follows. When two infinite Series are to be multiply 'd together, in order to find a third which is to be their Product, call one of them the and the Multiplicand, other the Multiplier. Write dawn upon your the of the Paper Terms Multiplicand, with their Signs, in a defcend- fo that ing order, the Terms may be at equal diftances, and juft under one another. This you may call your fixt or right-tand Paper. Prepare another Paper, at the right-hand Edge of which write down of the Terms the Multiplier, with their proper Signs, in an afcending Order, fo that the Terms may be at the fame equal diftances from each other as in the Multiplicand, and juft over one another. This you may call your moveable or left-hand Paper. Apply your movenble Paper to your fixt Paper, fo that the firft. Term of your Multiplier may ftand over-againft the firft Term of your Multipli- cand. Multiply thefe together, and write down the Product in its for the firft of the place, Term Product required. Move your move- abie Paper a ftep lower, fo that two of the firft Terms of the Mul- ftand two of the firft tiplier may over-againft Terms of the Multi- plicand. Find the two Produces, by multiplying each pair of the that ftand one Terms together, over-againft another ; abbreviate them if it may be done, and- fet down the Refult for the fecond of the Term Product required. Move your moveable Paper a ftep the firft lower, fo that three of Terms of the Multiplier may ftand the firft over-againft three of Terms of the Multiplicand. Find the three Products, by multiplying each pair of the Terms together that another fet ftand over-againft one j abbreviate them, and down the Refult for the third Term of the Product. And proceed in the lame manner to find the fourth, ana all the following Terms. I ihall iiluftrate this Method by an Example of two Series, taken Scale of 01 Decimal from the common Denary Arithmetick ; which the in all will equally explain Procefs other infinite Series whatever. Let the Numbers to be multiply 'd be 37,528936, &c. and or 10 where it is 528,73041, &c. which, by fupplying X underftood, will become the Series 3X -\- jX -+- jX-'-f- aX-'-f- 8X-* &c. _j_ 9 X-44- 3X-5H- 6X- and 5 X* -f- aX 4- 8X -j- 7X- + l J 3X~ -t- oX- -+-4X-4-f- iX-s, &c. and call the firft the Multipli- cand, and the fecohd the Multiplier. Thefe being difpofed as is will ftand as follows. prefcribed, Multiplier, and INFINITE SERIES.

Multiplier, Multiplicand ?X

-H4X-+ -f-oX-'

8X t 174 ^}e Method of FLUXIONS, der one another. This is your fixt or right-hand Paper. Prepare another Paper, at the right-hand Edge of which write down the Terms of the Divifor in an afcending order, with all their Signs the fo the changed except firft, that Terms may be at the fame equal distances as before, and jufl over one another. This will be your moveable or left-hand Paper. Apply your moveable Paper to your fixt Paper, fo that the firft Term of the Divifor may be over-againft the firft Term of the Dividend. Divide the firft Term of the Di- vidend by the firft Term of the Divifor, and fet down the Quotient over-againft them to the right-hand, for the firft Term of the Quo- tient required. Move your moveable Paper a ftep lower, fo that two of the firft Terms of the Divifor may be over-againft two of the firft Terms of the Dividend. Colleft the fecond Term of the the Dividend, together with the Product of the firft Term of Quo- tient now found, multiply'd by the Terms over-againft it in the left- hand thefe divided the firft the Divifor will be Paper ; by Term of the fecond Term of the Quotient required. Move your moveable the Paper a ftep lower, fo that three of the firft Terms of Divifor may ftand over-againft three of the firft Terms of the Dividend. Collecl the third Term of the Dividend, together with the two Pro- duds of the two firft Terms of the Quotient now found, each be- in the left-hand ing multiply'd into the Term over-againft it, Paper. Thefe divided by the firft Term of the Divifor will be the third Term of the Quotient required. Move your moveable Paper a ftep lower, fo that four of the firft Terms of the Divifor may ftand over- againft four of the firft Terms of the Dividend. Collecl: the fourth Term of the Dividend, together with the three Products of the three firft Terms of the Quotient now found, each being multiply'd by the Term over-againft it in the left-hand Paper. Thefe divided by the firft Term of the Divifor will be the fourth Term of the Quo- to find tient required. And fo on the fifth, and the fucceeding Terms. For an Example let it be propofed to divide the infinite Series

I2IA-5 28|X4 ., , 1 x 1 &c. the1C'Series a x .a* + tax 4- H- ^ + 7^1 , by 4- f &c. Thefe as is _]_ -j_ L -+- -^ , being difpofed prefcribed, will ftand as here follows.

Divifor, and INFINITE SERIES. 175

Divifor, ; fr 7^^ Method of FLUXIONS,

2 ^ c - on tne otner when meet will > ij ' 3-4> Paper, they together, compkat the numeral Coefficients. Apply therefore the fecond Term .of the move-able Paper to the uppertnoft Term of the fixt Paper, of ;ind the Product made by the continual Multiplication the three Factors thatftand in a lin-e over-againft one another, [which are the fecond Term of the given Series, the numeral Coefficient, (here the given Index,) and the firft Term of the Series already found,] divided by the firft Term of the given Series, will be the fecond Term of the Series required, which is to be let down in its place over- againft I. Move the moveable Paper a ftep lower, and the two Produces made by the multiplication of the Factors that ftand over- care be -againft one another, (in which, and elfewhere, muft had to take the numeral Coefficients compleat,) divided by twice the firft Term of the given Series, v/ill be the third Term of the Series required, which is to be fet down in its place over-againft 2. Move the moveable, Paper a ftep lower, and the three Products made by the multiplication of the Factors that ftand over-againft one another, divided by thrice the firft Term of the given Series, will be the fourth Term of the Series required. And fo you may proceed to find the next, and the fubfcquent Terms. It may not be amifs to give one general Example of this Reduc- tion, which will comprehend all particular Cafes. If the Series az _l_ b^ _j_ c&' -+-dz*, ,&c. be given, of which we are to find any Power, or to extract any Root; let the Index of this Pov>er or Root be m. Then prepare the moveable or left-hand Paper as you fee below, where the Terms of the given Scries are fet over one another in order, at the edge of the Paper, and at equal diftances. Alfo after every Term is put a full point, as a Mark of Multiplication, and after every one, (except the firft or loweft) are put the feveral as &c. with the Multiples of the Index, m, zm, pn, 40;, negative Likewife a be undei Sign after them. vinculum may flood to to with the other be placed over them, connect them parts of the numeral Coefficients, which are on the other Paper, and which make them compleat. Alfo the firft Term of the given Series is the reft a to denote its a or feparated from by line, being Divifor, the Denominator of a Fraction. And thus is the moveable Paper prepared. fixt or write down the natu- To prepare the right-hand Paper, at the fame ral Numbers o, i, 2, 3, 4, &c. under one another, equal diftances as the Terms in the other Paper, with a Point after them the firft 1 erm o as a Mark of Multiplication ; and over-againft write and INFINITE SERIES.

m firft reft ot write a*"z for the Term of the Series required. The to be wrote as (hall be the Terms are down orderly under this, they found, which will be in this manner. To the firft Term o in the fixt Paper apply the fecond Term of the moveable Paper, and they ~~ exhibit this Fraction *-* m - " z reduced will then , which being as,. I

<~ I t muft be fet in its for the fecond to this aw* &s*+ , down place, Term of the Series required. Move the moveable Paper a ftep lower, a m and you will have this Fraction exhibited + cz*. 2m o. a z

az. 2 m m- l "LL m--b* which being reduced will become mu c-{- mx a xz +'~, of the Series to be put down for the third Term required. Bring have the down the moveable Paper a ftep lower, and you will m n Fraction -f- dz,*. yn o. a z .+- cz*.

L m- l 3- bz?. m ma *c -+- m x a b

az. 3

for the fourth Term of the Series required. And in the fame manner are all the reft of the Terms to be found.

Moveable Fixt Paper Paper, &c. o. i. m 2. - a^i-o* -f- ma ~*c x z"

' - m m 1 m l *. m -3 . m x -x- -a *l>>+mx.'- a 6c+Ma dxz" J 7. 1 T az.

N. B. This Operation will produce Mr. De Moivre's Theorem mentioned before, the Inveftigation of which may be feen in the place there quoted, and fhall be exhibited here in due time and this therefore will of the place. And fufficiently prove the truth prefent Procefs. In particular Examples this Method will be found very eafy and practicable. A a But 178 The Method of FLUXIONS, But now to mew fomething of the ufe of thefe Theorems, and fame time to the for jit the prepare way the Solution of Affected and we will here Fluxional Equations; make a kind of retrofpect, and refume our Author's of Examples fimple Extractions, beginning Divifion with itfelf, which we fhall perform after a different and an eafier manner.

Thus to divide aa by b -f- x, or to refolve the Fraction

into a Series of Terms ; make or - fimple r^=y, by -f- xy , Now to find the the quantity y difpofe Terms of this Equation after a 1 and in this manner = , proceed the Refolution as fee + *-J J you is done here. 1 a*x a*.* *x t I =** a^J a -T-*--T* 7T + -77- > &C..

+ xy\ h-r TT + -75 ,

C IT + -JT- > OCC.

of a*- Here by the difpofition the Terms is made the firft Term of the Series to and belonging (or equivalent) by, therefore dividing will be the firfl of the Series by b, Term equivalent to y, as is fet a^x down below. Then will + be the firft Term of the Series

-4- xy, which is therefore fet down over-againft it; as alfo it is fet but a down over-againft by, with contrary Sign, to be the fecond a Term of that Series. Then will ~ be the fecond Term of yt a to be fet down in its place, which will give -^- for the fecond Term of and this with a fet -f- xy ; contrary Sign muit be down for third ~- the Term of by. Then will + be the third Term of

and therefore ~ will be the third of y, + Term 4- xy, which with a mufl be made the fourth contrary Sign Term of by, and therefore '~ will be the fourth Term of y. And fo on for ever.

Now the Rationale of this Procefs, and of all that will here follow of the fame kind, may be manifeft from thefe Confiderations. The unknown Terms of the Equation, or thofe wherein y is found, are (by the Hypothecs) equal to the known Term aa. And each of thofe a?id IN FINITE SERIES. 170 thofe unknown Terms is refolved into its equivalent Series, the Ag- of which muft (till be to fame gregate equal the known Term aa ; (or perhaps Terms.) Therefore all the fubfidiary and adventitious Terms, which are introduced into the Equation to aflift the Solution, the (or Supplemental Terms,) muft mutually deftroy one another. refolve Or we may the fame Equation in the following manner :

a* la* k*a* . ------Ha* = -4 -- , &c. y^ A" .V X A;4

1 Here a is made the firft Term of -+- and therefore muft xv," x

firft of be put down for the Term y. This will give + for the a firft Term of by, which with contrary Sign muft be the fecond --~ muft Term of -+- xy, and therefore be put down for the fecond of Then will be the fecond Term of Term y. ^ byy will be which with a contrary Sign the third Term of -|- xy, and - - therefore + will be the third Term of y. And fo on. There- fore the Fraction propofed is refolved into the fame two Series as were found above.

- If the Fraction : were given to be refolved, make 1 + * ' + -V" l v or x the Refolution of which is little t y -+- y=. i, Equation

the in the manner : rrxpre than writing down Terms, following

7 ---- 1 y = i x+x xx,&cc. y (-x- *-4-|_x- & ccc.. i x~ &c. , +x*y 3 = x-'-+x- ,

as i is the Here in the firft Paradigm, made firft Term of y, fo 1 will x be the firft Term of x*y, and therefore x*- will be the be fecond Term of y, and therefore x* will the fecond Term of therefore x* will be third Term of &c. in x*y, and -+- y ; Alfo the fecond Paradigm, as i is made the firft Term of x*y, fb will -f- x~'- be the firft Term of y, and therefore x~- will be the fecond or will be the fecond of Term of x*y, x~* Term y ; &c. A a 2 To 180 tte Method of FLUXIONS,

i 3. zx ~*

. - To refolve the compound Fraction into fimple Terms, i i a * 2.y 13 i v make 7 =y, or 2** # = y 4- AT^ ^xy, which E- I+* 1 3* quation may be thus refolved :

= 2A^ * X^

1 3** + 34^* 73**, &c. 3 &c. 34* } 39**, &c.

Place the Terms of the Equation, in which the unknown quantity y is found, in a regular defcending order, and the known Terms above, as you fee is done here. Then bring down zx^ to be the firfl Term of y, which will give -f- 2x for the firfl Term of the Series a 4- x*y, which mufl be wrote with contrary Sign for the fecond will the fecond Term of y. Then Term of 4- x^y be 2x%, and the firfl Term of the Series 3*7 will be 6x^, which together make SAT*. And this with a contrary Sign would have been wrote for the third Term ofy, had not the Term x* been above, which J * it for the third of reduces to 4- jx Term y. Then will 4- yx* be the third Term of 4- x*y, and 4- 6x* will be the fecond Term collected with a will of 3fly, which being contrary Sign, make for the of and fo as in 1 3** fourth Term y ; on, the Paradigm. If we would refolve this Fraction, or this Equation, fo as to accommodate it to the other cafe of convergency, we may invert the Terms, and proceed thus :

1 O W f xV *- -i- 3-v /

1, &c. ' 1 y = fX' -f- 7 4f* ft*- , 6cc.

down AT* to be the firfl Term of whence Bring 3-vy, ~\- ^ in its will be the firfl Term of y, to be fet down place. Then the firfl and INFINITE SERIES. 181

with a firft Term of +- x^y will be -f- fx, which contrary Sign will be the will be the fecond Term of 3*?, and therefore -+- f the fecond of will be -f- fecond Term of y. Then Term -+- x^y f#s with a and the firft Term of y being -+- fx*, thefe two collected have made *.#* for the third of contrary Sign would Term 3*}', z had not the Term +- zx' been prefent above. Therefore uniting which thefe, we fhall have -f- x* for the third Term of 3*7,

r the third of Then will the third will g lve ?j-x~* f Term y. of Term of -+- xh be if, and the fecond Term y being -+- -%, the thefe two collected with a contrary Sign will make -f- if for 1 and therefore will be l ^e fourth fourth Term of T,xy, TT*"" of and fo on. Term y -, for to on to the Author's And thus much Divifion ; now go pure or fimple Extractions. of To find the Square-root of aa -f- xx, or to extract the Root y xx a then we fhall have this Equation yy = aa-{- ; make y = -+-/>, by Subftitution zap -f- pp = xx, of which affected Quadratick Equa- this tion we may thus extract the Root p. Difpofe the Terms in oa manner zap-^= xx, the unknown Terms in a defcending order H-/AJ fide of the one fide, and the known Term or Terms on the other Equation, and proceed in the Extraction as is here directed.

*4 * *' -) - 5x8 7 **/==* - + s74 5i + H53. _^i + -\.__ + + 8 ' "J J ^4* 8*4 640*^!_^:,12Sa 6 9 10 x* A4 * CA 7* ., * -I- , h -t-f-t &C. f za 8a l \6a ! izSa 1 25O'

1 is the firft this Difpofition of the Terms, x made Term of By x the Series to then fhall have for the firft belonging zap ; we Term of the Series p, as here fet down underneath. Therefore

will be the firft Term of the Series *, to be put down in its 4474 1 place over-againft p . Then, by what is obferved before, it muft be put down with a contrary Sign as the fecond Term of zap,

- which will make the fecond Term to be . there- of/> ^ Having fore v i 77jt2 Method of F L n o !-; / s,-

two firft Terms of *- fore the P = ~, we fhall liave, (by any the Methods for of foregoing finding the Square of an infinite Se- 1 ries,) the two firft Terms of p = ~ . which la ft Term 3- ' AfCi #4 4 irmft be wrote with a contrary Sign, as the third Term of zap. Therefore the third Term of * is ^ and , the third Term of * ' Lp*

the aforefaid will be -~ is to (by Methods) } which be wrote with a contrary Sign, as the fourth Term of zap. Then the fourth 8 Term of will be and therefore the fourth is p -||_, Term of/* """ which is to be wrote with a for the fifth 7IsI ' contrary Sign

Term of zap. This will give 2^- for the fifth Term of p . and fo 2^O' in the we may proceed Extraction as far as we pleafe. Or we may difpofe the Terms of the Supplemental Equation thus :

J > a* ---- ~ c -f- zax za* -+- * &c ' zap x ^ ' &c. A X 3 ,y

- * * ,&c.

a 4 * is made the firft Term of the and therefore Here Series/ , x, be firft (or elfe x,) will the Term of p. Then zax will be the firft Term of zap> and therefore zax will be the fecond Term of 1 # a 6cc. the . So that becaufe = 2rfx, by extracting Square-root p"- /> of this Series by any of the foregoiug Methods, it will be found &c. or a will be the fecond Term of the Root / x a, />. Therefore the fecond Term of zap will be 2<2% which muft be with a for the third Term 1 and thence wrote contrary Sign of/ , (by of will be -- Extraction) the third Term / . This will make the

of to be which makes the fourth 4 .third Term zap , Term of/

- and therefore o will be the fourth Term to be , (by Extraction) z of/. This makes the fourth Term of zap to be o, as alfo of / . Then ^ will be the fifth Term of/. Then the fifth Term of I zap and INFINITE SERIES, 183

be . which will make the fixth Term of to be zap will 5 , />* f 4*

.ll therefore o will be the fixth Term of &c. ; and p, will be deficient fo that in the Here the Terms alternately ; given aa the Root will be a x a Equation yy = -\- xx, y = -f- -f- "->

is x &c. which is the as &c. that = -f- -h s , fame y ^-} -^- if we fhould change the order of the Terms, or if we fhould change a into x, and x into a. If we would extradl the Square-root of aa xx, or find the aa xx make a as be- Root y of the Equation yy = ; y = -f- p, x fore then which be refolved as in the fol- ; zap -f-/* = x*-, may lowing Paradigm :

.v4 X 6 8 J _ ^.V z f I" 4fl 8a4 64,1 6 8 f *4 X' <;* 1 - t> \ 1- ; H- ^~; -f- f -{- + r J 4 84 6^.6 ^ "4 JC CX "7 X c - . . __ k~^_ i . J* ,^_ ^^^^ ^^^ J, cSCC*

1 if we mould to make x the firft Term of 1 Here attempt -J-/ , 1 have x or for the fi rfl Term we mould ^/ , x^/ i, of/ ; which no Series can be form'd being rnpoflible, fliews from that Suppofi- tion. # or the To find the Square-root of xx, Root y in this Equa- make x^ then 1 tion yy = x xx, y = + p, x -+- zx^p -f- p = x - xx, or zx^p -+- /* = x*, which may be refolved after this manner :

The Terms being rightly difpofed, make x* the firft of zx^p; then will x* be the firft Term of p. Therefore a 3 will be the firft Term of which is alfo to be with ~\- px / , wrote for the fecoiid - * a contrary Sign Term of 2x'-p, which will give fA for the lecond Term of p. Then (by fquaring) the fecond Term of will be 4 which will - 4 for the fecond Term of ^ i^ , give i* 184 ffi? Method of FLUXIONS,

and therefore for the third Term of and zx^p, -V^ p ; fq op.

Therefore in this it will be x'* A-'" x* Equation y=z f f rV*''"* &c. So to extract the Root of this y Equation yy =.aa-\-bx xxt make then bx y = a-{-p } zap -+- p* = xx, which may be thus refolved.

* =fa X + *fi, &C . 1 L'-x'

tx x* Ixl

l Make bx the firft Term of zap; then will ~- be the firft Term 20.

1 b of Therefore the firft Term of will be is p. p -+- ^ , which alfo to be wrote with a contrary Sign, fo that the fecond Term of

be -< x* which will make the fecond zap will ^- , Term of * to be ' Then the fecond p gjr by fquaring, Term of/* will be *-* which muft be wrote with a 7 -g^ 5 contrary Sign for the third Term of zap. This will give the third Term of p as in the Example; and fo on. Therefore the Square-root of the bx will be a ~ Quantity a^ -f- xx -+ ^ -f. -^ _,

a* Alfo if we would extract the of < we Square-root _ , may extract the Roots of the Numerator, and likewife of the Denomi- as nator, and then divide one Series by the other, before ; but more thus. or 1 dire.ctly Make _**! = yy, i -{- ax = yy b*x*y*. z Suppofe = i -f- p, then ax* = bx*- y zp -+-p zbx*-p bx*-p*, which Suppplemental Equation may be thus refolved.

zp and INFINITE SERIES, 185

+TT*\ 1 &c. ab ^ab ,

~a^b t &c.

&c. bx^p* _ ,

1 of will l Make ax -f- bx* the firfl Term 2/>, then frf.v -f- f&v will the firfl "be the firfl Term of /. Therefore abx* b*x* be the firfl Term of 2bx*p, and â*x* -f- -âbx* -f- -^bx* will be and their muil Term of/*. Thefe being collected, Signs changed, be made the fecond Term of 2/, which will give abx* -f- |J*A %a*x* for the fecond Term of/. Then the fecond Term of 2bx*p 6 l>*x 6 i bx 6 and the fecond Term of will be -âb^x -f- â > p* will be found a l bx 6 6 6 6 and (by fquaring) f +- \ab*x j-a''X -{- $frx , l 6 6 4 the firfl Term of bx*p will be â^bx -âb'-x f^'AT ; which being collected and the Signs changed, will make the third half will be the third of and fo on as Term of 2p, which Term p ; far as you pleafe. And thus if we were to extract the Cube-root of a* 4- x*, or the a 3 tf 3 make a then Root y of this Equation 7' = 4- j y = -f-/, by 3 1 1 or i Subflitution a -f- 3d / -f- âp -+- p> = & -+- x*, 3 / -f S be thus refolved. = A: , which fupplemental Equation may

l 243 B b The Method of FLUXIONS, of the The Terms being difpos'd in order, the firft Term Series will be which will make the firft Term to be *. Thiss

1 . And this will make the will make the firft Term of/ to be ^ firfl Term of to be which with a muft be ^ap* ^ , contrary Sign of therefore the fecond Term will the fecond Term 3/z*/>, and ofp 1 be . Then the fecond Term of will be r (by fquaring) ^ap

*"= . ff! the firft Term of win be . Thefe and (by cubing) 6fi Oa* 27<:< r y9 make which with a muft be being collected , contrary Sign the third Term of ^a^p, and therefore the third Term of p will be - ill . third of will be _j___ Then by fquaring, the Term ^ap* and by cubing, the fecond Term of/ 3 will be ^^, which being

will make anc^ therefore the fourth Term collected y^-j > of-^^p * -- - will be and the fourth Term of will be 11 . And ^T,' p 8j

SECT. II L The Refolution of Nttmeral AffeSted Equations*

W as to the Refolution of affected Equations, and firft in be- Numbers ; our Author very juftly complains, that fore his time the exegefa numcroja, or the Doctrine of the Solution of affected Equations in Numbers, was very intricate, defective, and inartificial. What had been done by Vieta, Harriot, and Oughtred in this" matter, tho' very laudable Attempts for the time, yet how- had rea- ever was extremely perplex'd and operofe. So that he good their fubftituted a fon to reject Methods, efpecially as he has much better in their room. They -affected too great accuracy in purfuing exact and INFINITE SERIES. 187 which led them into exact Roots, tedious perplexities ; but he knew very well, that legitimate Approximations would proceed much more and would anfwer regularly and expeditioufly, the fame intention much better. be 20, 21, 22. His Method may eafily apprehended from this one Inftance, as it is contain'd in his Diagram, and the Explanation of it. Yet for farther Illuftration Lfhall venture to give a fhort rationale of it. When a Numeral Equation is propos'd to be refolved, he to the as takes as near an Approximation Root can be readily and conveniently obtain'd. And this may always be had, either by the known Method of Limits, or by a Linear or Mechanical Conitruc- tion, or by a few eafy trials and fuppofitions. If this be greater or lefs than the Root, the Excefs or Defect, indifferently call'd the Sup- plement, may be reprefented by p, and the affumed Approximation, are to be in together with this Supplement, fubftituted the given. Equation inftead of the Root. By this means, (expunging what will will be be fuperfluous,) a Supplemental Equation form'd, whole Root of the of the is now p, which will confift Powers affumed Approxima- tion orderly defcending, involved with the Powers of the Supplement regularly afcending, on both which accounts the Terms will be con- ratio or if tinually decreafmg, in a decuple falter, the affumed Ap- proximation be -fuppos'd to be at leaft ten times greater than the Supplement. Therefore to find a new Approximation, which fhall it will be fufficient to nearly exhauft the Supplement p, retain only the two firft Terms of this Equation, and to feek the Value ofp from the the refulting fimple Equation. [Or fometimes three firft Terms may be retain'd, and the Value of p may be more accurately found from the refulting Quadratick Equation; Sec.] This new Approxi- be mation, together with a new Supplement g, muft fuhftituted initead of p in this laft fupplemental Equation, in order to form a the fame be fecond, whofe Root will be q. And things may obferved of its this fecond fupplemental Equation as of the firft; and Root, or an to be difcover'd after the fame manner. And Approximation it, may thus the Root of the given Equation may be profecuted as far as we pleafe, by finding new iiipplemental Equations, the Root of every one of which will be a correction to the preceding Supplement. So in the 3 = o, 'tis to prefent Example jy 2y 5 eaiy perceive, that 2 = ; for 2x2x2 2x2 = 4, which mould make y fere 5. Therefore let p be the Supplement of the Root, and it will be y = 2. -{-/>, and therefore by fubftitution i -f- lop -+- 6p* -\-p= = Q. As p is here fuppos'd to be much lefs than the Approximation 2, B b 2 ty i88 The Method of FLUXIONS, by this fubftitution an Equation will be form'd, in which the Terma will gradually decreafe, and Ib much the fafter, cateris parities, as 2 is greater than p. So taking the two firft Terms, i -f- io/>=o, x or =. _. or a fecond 'tis fere, p T fere ; affuming Supplement q, This fubftituted for in the laft p = T'o- -h ? accurately. being p Equation, it becomes o, 6 1 -+- 1 4- = o, which is 1,237 + 6>3?* . and extrac- what is (hewn before, concerning the raifing of Powers m m m I P* wP &c. ting Roots, it will follow that y = P -h/> = -f- -'/>, m all the , or that thefe will be the two firft Terms of y ; and reft, into the of be And for being multiply'd Powers />, may rejected. ~ m~ l m~ I m l m~l m~- the fame reafon y = P -h m iP p, &c. y = P -+ fub- m 2 P"-=p, &c. and fo of all the reft. Therefore thefe being ftituted into the Equation, it will be n- l a ]>> .4- niaP , &c."l ~ p -, &c. n o Or P" m 2c ~*p, &c. >= ; dividing by ,

m 7 JP "-", &c. &c.

'- -j-^/P-s , &c. -\-m &c. = o. From whence the Value of ^dP~*p, taking />, *P-' cP- rfp-* ar,. we mail = + + + . and have/ --1 m . ma?- lbV~' z^~3 _, Jjff + -{.m + J^P-4

confequently^ J r=

,. To reduce this to a more commodious make - form, Pi= , whence

I i P=A-'B, P- =A- B% &c. which being fubftituted, and alfo the Numerator and Denominator A"7 it will be multiplying~ ~ ~ by , 'B +" A."-"-B*+ =4rfA"-?Bi. ^c-. will

be a nearer to the or fo much Approach Rootjy, than jp P, and the 2 77je Method of FLUXIONS,

' the nearer as is near the Root. And hence we may derive a very convenient and general Theorem for the Extraction of the Roots of Numeral Equations, whether pure or affected, which will be this. m 1 1 m~~- m~ Let th,e s &c. general Equation ay -^- by" -+ cy -f- d) , o be to be folved if the Fraction - be =: propofed ; affumed as near the Root y as conveniently may be, the Fraction

1 m 1 m -S 3 z 3 zcA F f . . iAA +7 4. 3n/A 4B4,'fef nearer Approximation to the .Root. And this Fraction, when com- inflead of the Fraction - which means a puted, may,be,ufed , by

-be had and fo till we Bearer Approximation may again ; on, approach as near the true Root as we pleafe. This general Theorem may be conveniently refolved into as many particular Theorems as we pleafe. Thus in the Quadratick Equa- 1 z A -4- rB 1 it will be In the Cubick tion y by === c, = , ,"7D p , fere. + if y if * 2t\ X f J DO D

...... 2 A it will be + ty + cy = d, y == . Equation y* 3/^' And the llke of hlSher Equations. For an of the Solution of a let ; 'Example Quadratick Equation, it.be propofed to extract the Square-root of 12, or let us find the 1 with value of_y in this Equation y #= 1.2. Then by comparing

<: 12. And the general formula, we fliall have b =. o, and = /I or taking 3 for the firft approach to the Root, making g =T> that and we fliall have Substitution ^==. is, Az=3 B;= i, by y ^~ =4-, fora nearer Approximation. Again, making A = 7 12l and B = 2, we fliall have y = == || for a nearer Approxi- 14 X 2 mation. Again, making A = 97 and B = 28, we fliall have _y= + i:! x 28i 97j . __ lil7 for a nearer Approximation. Aeain, '94* 2 S45 2 _ __ ' A= and B 2 we ^a11 have making 18817 =543 > y=

7o8ic8o77 /- A i -r i == ior a nearer And if we on in the 1|^ Approximation. go fame method, we may find as near an Approximation to the Root as y/e pleafe, This and INFINITE SERIES. 191

This Approximation will be exhibited in a vulgar Fraction, which, if it be always kept to its loweft Terms, will give the Root of the in the fhorteft and manner. it will al- Equation fimpleft That is, ways be nearer the true Root than any other Fraction whatever^ whofe Numerator and Denominator are not much larger Numbers than its own. If by Divifion we reduce this laft Fraction to a De- cimal, we mall have 3,46410161513775459 for the Square-root of 12, which exceeds the truth by lefs than an Unit in the lall place.- For an Example of a Cubick Equation, we will take that of our Author j * 2? = and therefore b = o, _y 5, by Companion for the firft to the =. 2, and d==. 5. And taking 2 Approach we mall or = 4., that is, A = 2 and B=i, Root, making ^- have by Subftitution y ==- = 44 f r a nearer Approach to the Root. Again, make A = 21 and B = 10, and then we 9- 1 + 2500 __ Hj-L mall have y = for a nearer Approximation. 6615 1000 5615 Again, make A= 11761 and 6 = 5615, and we mall have ~ . y = = 3 J 16 1 9759573 495 3x11761 1 ^561 5 2x5615 proximation. And fo we might proceed to find as near an Approxi' mation as we think fit. And when we have computed the Root in a near enough Vulgar Fraction, we may then (if we pleafe) reduce it to a Decimal by Divifion. Thus in the prefent Example we fhall have ^ = 2,094551481701, &c. And after the fame manner we may find the Roots of all other numeral affected Equations, of whatever degree they may be.

SECT. IV. The Refolution of Specious Equations by infinite Series and the the ; firft for determining forms of Series^ and their initial Approximations.

the of 23, 24. TTT^ROM Refolution numeral affected Equations, our Author to find the J/ proceeds Roots of Literal, Spe- cious, or Algebraical Equations alfo, which Roots are to be exhibited by an infinite converging Series, confiding of fimple Terms. Or are to be they exprefs'd by Numbers belonging to a general Arithme- tical Scale, as has been explain'd before, of which the Root is denoted by .v or z. The affigning or chufing this Root is what he means here, by diftinguiming one of the literal Coefficients from the if are is reft, there feverul. And this done by ordering or difpofing the Method of FLUXIONS, the Terms of the given Equation, according to the Dimenfions of that Letter or Coefficient. It is therefore convenient to chufe fuch a Root of the Scale, (when choice is allow'd,) as that the Series may Fraction lefs converge as faft as may be. If it be the leaft, or a than Unity, its afcending Powers muft be in the Numerators of the Powers Terms. If it be the greateft quantity, then its afcending muft be in the Denominators, to make the Series duly converge. be If it be very near a given quantity, then that quantity may conveniently made the firft Approximation, and that fmall difference, or Supplement, may be made the Root of the Scale, or the converging quantity. The Examples will make this plain. 25, 26. The Equation to be refolved, for conveniency-fake, iliould before its Refo- always be reduced to the fimpleft form it can be, Jution be for this will the leaft trouble. But attempted ; always give all the Reductions mention'd by the Author, and of which he gives con- us Examples, are not always neceflary, tho' they may be often Roots of venient. The Method is general, and will find the Equa- tions involving fractional or negative Powers, as well as cf other Equations, as will plainly appear hereafter. 27, 28. When a literal Equation is given to be refolved, in diftin- its is to guifhing or affigning a proper quantity, by which Root con- three cafes or varieties all verge, the Author before has made ; which, for the fake of uniformity, he here reduces to one. For becaufe be as fmall the Series mull neceffarily converge, that quantity muft other that its -,as poffible, in refpect of the -quantities, afcending chufe Powers may continually diminim. If it be thought proper to its muft be intro- the greateil quantity, inftead of that Reciprocal if it duced, which will bring it to the foregoing cafe. And approach fmall difference be near to a given quantity, then their may introduced into the Equation, which again will bring it to the firft cafe. So that we need only purfue that cale, becaufe the Equation is always fuppos'd to be reduced to it. But before we can conveniently explain our Author's Rule, for finding the firft Term of the Series in any Equation, we muft confider the .nature of thofe Numbers, or Expreffions, to which thefe are and in this literal Equations are reduced, whofe Roots required ; Inquiry we ihall be much aiTifted by what has been already difcourfed of Arithmetical Scales. In affected Equations that were purely nume- the feveral Powers ral, the Solution of which was juft now taught, or of the Root were orderly difpoied, according to a fingle limple and there Arithmetical Scale, which proceeded only in longum, was fufficient #nd INFINITE SERIES.

for their Solution. fafficient But we muft enlarge our views in thefe literal affected Equations, in which are found, not only the Powers of the Root to be extracted, but alfo the Powers of the Root of the of the Scale, or converging quantity, by which the Series for the the is to Root of Equation be form'd ; on account of each of which circumftances the Terms of the are Equation to be regularly difpofed, and therefore are to conftitute a double or combined Arithmetical which muft both Scale, proceed ways, in latum as well as in longum, as it were in a Table. For the Powers of the Root to be extraded, are be fuppofe y, to difpofed in longum, fo as that their Indices may conftitute an Arithmetical Progreffion, and the vacancies, if any, may be fupply'd by the Mark #. Alfo the Indices of the Powers the pf Root, by which the Series is to converge, fuppofe x, are to be in difpofed latum, fo as to conftitute an Arithmetical Progreffion, and the vacancies may likewife be fill'd up by the fame Mark *, it hall be when thought neceffary. And both thefe together will make a combined or double Arithmetical Scale. Thus if the Equa- tion s -6a* x* fax* y $xy 4- i!y4 7* #/ 4- 4- =a=-o, were given, to find the Root the Terms y, may be thus difpofed : 6 V4 y y* yS y* yl yo x

= 0;

the f Alfo Equation v by* 4- gbx\ x* =o fhould be thus dif-

in order to its Solution : jpofed, y' * * by* * Method of FLUXIONS,

When the Terms of the Equation are thus regularly difpos'd, ft for Solution to the is then ready ; which following Speculation will be a farther preparation. 29. This ingenious contrivance of out' Author, (which we may call Tabulating the Equation,) for finding the firft Term of the Root, (which may indeed be extended to the finding all the Terms, or the form of the Series, or of all the Series that may be derived from the given Equation,) cannot be too much admired, or too care-

: reafon and foundation of fully inquired into The which may be thus generally explained from the following Table, of which the Construction is thus.

a^-bt, ba+bb ja-\-bb

2-+J*

+4* 40+4^

711+3^

zJ za-\-zb \a-\-zb 70+

za+b

b b za b b a b b 3 a 50 bab

za zb zl azb 3 a zb bazb

; 3 73*

and In a Pfor.e draw any number of Lines, parallel equidiftant, and cthers_at; right Angles to them, fo as to divide the whole Space, as far as is neceffary, into little equal Parallelograms. Aflume any one of thefe,- in which write the Term o, and the Terms a, za, 30, 4.a, &c. in-the fuceeeding Parallelograms to the right hand, as alfo the Terms -*-^ 2a, 3^7, &c. to the left hand. Over the Term o, in. the fame Column, write the Terms ^, zb, 3^, 4^, &c. fuc- and the &c. underneath. And ceffively",' Terms b, zb, 3^, thefe Ave ma^f call primary Terms. Now to infert its proper. Term the two 'Terms in any other afitgiVd. Parallelogram, add primary that' ftand if each- and write the Sum together.., over-againft way, all the be- in the given Parallelogram. And-* thus Parallelograms as is oecafion the whole. ing fill's, as-far there every way, Space will and INFINITE SERIES. 195

-will become a Table, which may be called a combined Arithmetical of the Numbers a and ProgreJJion in piano, compofed two general t\ of which thefe following will be the chief properties. to the Series Any Row of Terms, parallel primary o, a, za, ^a, &c. will be an Arithmetical Progreflion, whofe common Difference and it be fuch at Row or is a ; may any Progreflion pleafure. Any to the Series &c. will be an Column parallel primary o, , zb, 3^, Arithmetical Progreflion, whofe common difference is ^j and it may be any fuch Progreflion. If a ftr-ait Ruler be laid on the Table, the Edge of which mall pafs thro' the Centers of any two Parallelo- whatever all the Terms of the whofe Cen- grams ; Parallelograms, ters mail at the fame time touch the Edge of the Ruler, will conftitute will coniiit of an Arithmetical Progreflion, whofe common difference firfl will be fome of and the other two parts, the of which Multiple a, a Multiple of b. If this Progreflion be fuppos'd to proceed injeriora. or towards the lower verjus, or from the upper Term Parallelogram ; difference be fub- each part of the common may feparately found, by to the from the tracling the primary Term belonging lower, primary Term belonging to the upper Parallelogram. If this common difference, when found, be made equal to nothing, and thereby the Re- into a lation of a and b be determined ; the Progreflion degenerates it an Arithmetical Hank of Equals, or (if you pleafe) becomes Progref- fion, whofe common difference is infinitely little. In which cafe, if all the of the Parallelo- the Ruler be moved by a parallel motion, Terms mall at the be found to touch the grams, whofe Centers fame time Edge of the Ruler, fhall be equal to each other. And if the motion of the Ruler be continued, fuch Terms as at equal diftances from the firfl: fituation are fuccerTively found to touch the Ruler, fliall form an Arithmetical Progreflion. Laftly, to come nearer to the cafe in hand, if any number of thefe Parallelograms be mark'd out and di- the or and at flinguifh'd from. reft, aflign'd promifcuoufly pleafure, through whofe Centers, as before, the Edge of the Ruler ihall fuc- in its from ceflively pafs parallel motion, beginning any two (or more) are initial or external Parallelograms, :whofe Terms made equal ; an Arithmetical Progreflion may be found, which ihall comprehend and take in all thofe promifcuous Terms, without any regard had to the Terms that are to be omitted. Thefe are fome of the properties of this or of a combined Arithmetical in Table, Progreflion piano>, by . which we may eafily underfland our Author's expedient, of Tabu- lating the given Equation, and may derive the neceflary Confequen- ~es from it. C c 2 For 196 The Method of FLUXIONS,

For when the Root y is to be extracted out of a given Equation, confifting of the Powers of y and x any how combined together- other of which is to be promifcuoufly, with known quantities, x as before fuch a value the Root of the Scale, (or Series,) explain'd ; of y is to be found, as when fubftituted in the Equation inftead of to y, the whole (hall be deftroy'd, and become equal nothing. And firft the initial Term of the Series^or the firfl Approximation, is to be be 1 found, wtyich in all cafes may Analytically reprefented by Ax* ; or m &c. So that we mail have 1 we may always put y = Ax , y = m &c. 3 A*x* 6cc. 4 A 4 *4 &c. And fo of other , _}> = *, A*x, _>' = Powers or Roots. Thefe when fubftituted in the Equation, and by feveral of that means compounded with the Powers x (or z} already found there, will form fuch a combined Arithmetical Progreffion in flano as is above defcribed, or which may be reduced to fuch, by making a=m and = I. Thefe Terms therefore, according to the nature of the Equation, will be promifcuoufly difperfed in the the vacancies be conceived to be Table j but may always fupply'd, before mention'd. That the and then it will have the properties is, initial or Ruler being apply'd to two (or perhaps more) external could not be at the be- Terms, (for if they were not external, they ginning of an Arithmetical Progreffion, as is neceflarily required,) and thofe Terms being made equal, the general Index m will thereby be determined, and the general Coefficient A will alfo be known. If the external Terms made choice of are the loweft in the Table, which is the cafe our Author purfues, the Powers of x will proceed be and then a Series by increafing. But the higheft may chofen, will be found, in which the Powers of x will proceed by decreafing. And there may be other cafes of external Terms, each of which will eommonly afford a Series. The initial Index being thus found, the other compound Indices belonging to the Equation will be known be in alfo, and an Arithmetical Progreffion may found', which they the form are all comprehended, and confequently of the Series wifll be known. Or inftead of Tabulating the Indices of the Equation, as above, it will be the fame thing in effedt, if we reduce the Terms themfelves to the form of a combined Arithmetical Progreffion, as was fhewn before. But then due care mufl be taken, that the Terms may be at diftances otherwife the Ruler cannot be rightly placed equal j ac- to difcover of the as tually apply'd, the Progreffions Indices, may be done in the Parallelogram. For and INFINITE SERIES, 197

For the fake of greater perfpicuity, we will reduce our general Table or combined Arithmetical Progreffion in piano, to the parti- in which a-=.m and b=. i which will th.n cular cafe, -, appear thus :

- 2M+0 m+6 JOT+6 5 M-6

_,, +5 + 5 "-+5 "'"+5 m w+4 + 4 2W+4 4'"+ 4 7 +4

zm-if 3 + 3

2 - AO-f- 2 CT+2 2OT-4-2 4W+2

m-\- 1 OT+I 2OT+ 1

5^2

< > 601 2m 3 m 3 3 3 3 3 3 3

fubfervient to the Now the chief properties of this Table, prefent will be thefe. If be feledted, and an- purpofe, any Parallelogram and if their other any how below it towards the right hand, included the Numbers be made equal, by determining general Number m, affirmative alfo if the of the which in this cafe will always be ; Edge of thefe two all the Ruler be apply 'd to the Centers Parallelograms ; Numbers of the other Parallelograms, whofe Centers at the fame time touch the Ruler, will likewife be equal to each other. Thus if the be as alfo the Parallelo- Parallelogram denoted by m -+- 4 feleded, if make m we mall have 2 ; and we -t- = -H 2, gram 377* -f- 4 ^m m=i. Alfo the Parallelograms ;;; -h 6, m -f- 4, 3^7 -|- 2, $m, m 2 &c. will at the fame time be found to touch the of j , Edge the Ruler, every one of which will make 5, when m= i. And the fame things will obtain if any Parallelogram be felecled, and another any how below it towards the left-hand, if their included Numbers be made equal, by determining the general Number m, which in this cafe will be always negative. Thus if the Parallelo- gram denoted by 5/w-i-4be felecled, as alfo the Parallelogram 402 -f- 2; and if we make ^m-\-^.-=^.m -t-2, we fliall have>= 2. Alib 6cc. the Parallelograms 6w+6, 5^4-4, 4^ + 2, 3?;;, zm 2, will 198 7%e Method of FLUXIONS, found at the fame time to will be touch the Ruler, every one of which will make 6, when m = 2. The fame things remaining as before, if from the firft fituation of the Ruler it (hall move towards the right-hand by a parallel motion, it will continually arrive at greater and greater Numbers, which at diftances will form an equal afcending Arithmetical Progeffion. Thus if the firft two felected Parallelograms be zm 1 = whence 5;;; 3, the Numbers in all the m=., correfponding Parallelograms will be Then if the Ruler moves towards the -j. right-hand, into the parallel fituation %m-\- i, 6m I, &c. thefe Numbers will each be 3. If it moves forwards to the fame diftance, it will arrive at &c. which will each be 4/7; -{-3, 7/ +- i, 5^. If it moves forward to the fame it will arrive at again diftance, yn -f- 5, %m -f- 3, &c. which will each be And fo on. But the Numbers 8f. f, 3, 52., 8y, &c. are in an Arithmetical Progreffion whofe common difference is 2-i. And the like, mutatis mutandis, in other circum- fiances.

And hence it will follow

Arithmetical whofe common difference is or creafing Progreffion, , 3. And hence it will follow alfo, if in this laft fituation of the Ruler it moves the contrary way, or towards the left-hand, it will continually arrive at lefler and lefler Numbers, which at equal diftances will form a decreafing Arithmetical Progreflion. Now if out of this Table we fhould take promifcuoufly any number of in Parallelograms, their proper places, with their refpeclive Num- and INFINITE SERIES, 199

all the reft Numbers included, neglecYmg ; we mould form fome certain fuch as of which thefe would be the Figure, this, properties.

"M-3

2OT-J-I 5;;;+ 1

The Ruler being apply'd to any two (or perhaps more) of the Parallelograms which are in the Ambit or Perimeter of the Figure, and their that is, to two of the external Parallelograms, Numbers the if being made equal, by determining general Number m ; the Ruler paffes over all the reft of the Parallelograms by a parallel motion, thofe Numbers which at the fame time come to the Edge of the Ruler will be equal, and thofe that come to it fuccefllvely will form an Arithmetical Progreffion, if the Terms mould lie at equal diftances or be reduced to Terms ; atleaft-they may fuch, by fupplyingany that may happen to be wanting. Thus if the Ruler fhould be apply'd to the two uppermoft and external Parallelograms, which include the Numbers 3/w-f-^ and and if be made we mall have fo that ^m ~}_ 5, they equal, m = o, will each of thefe Numbers be 5. The next Numbers that the Ruler will be 6/ will arrive at m -f- 3, 4;;; +3, -f- 3, of which each will la ft are zm of which each is i. So that be 3. The -f- i, 5>-f- i, are here #2 = 0, and the Numbers arifing 5, 3, i, which form a decreafing Arithmetical Progreffion, the common difference of which is 2. And if there had been more Parallelograms, any how difpofed, their Numbers would have been comprehended by this Arithmetical or at leaft it been Progreffion, might have interpolated with other Terms, fo as to comprehend them all, however promifcuoufly and have been taken. irregularly they might Thus fecondly, if the Ruler be apply'd to the two external Pa- and if thefe rallelograms 5/72+ 5 and 6m-}- 3, Numbers be made mail have and the Numbers themfelves will be equal, we m = 2, each ic. The three next Numbers which the Ruler .will arrive at will 20O The Method of FLUXIONS,

each and the two laft will be will be 11, ^ach 5. But the Num- will be bers 15, n> 5. comprehended in the decreafing Arithmetical Progreffion 15, 13, 1 1, 9, 7, 5, whofe common difference is 2. if the Ruler Thirdly, be apply'd to the two external Parallelograms and and if 6m -f- 3 5*0-4-1, thefe Numbers be made equal, we fhall have tn = 2, and the Numbers will be each The two next 9. Numbers that the Ruler will arrive at will be each 5, the next will be 3, the next i, and the laft -+- i. All which will be in the Arithmetical comprehended afcending Progreffion 9, 7, 5, 3, i, -+- i, whofe common difference is 2. Fourthly, if the Ruler be apply'd to the two loweft and external and and if Parallelograms 2m-\-i 5/77 -+- i, they be made equal, fhall have fo that each we again m = o, of thefe Numbers will be i . next three The Numbers that the Ruler will approach to, will each be and the laft But the Numbers will be 3, 5. i, 3, 5, compre- in hended an afcending Arithmetical Progreffion, whofe common difference is 2. if the Ruler Fifthly, be apply'd to the two external Parallelograms in -f- 3 and 2m +- i, and if thefe Numbers be made equal, we fhall have m = 2, and the Numbers themfelves will be each The 5. next that the will three Numbers Ruler approach to will each be 1 1, and the two next will be each the 1 15. But Numbers 5, 1, 15, will be in the Arithmetical comprehended afcending Progreffion 5, 7, 9, II, 13, 15, of which the common difference is 2. if the Ruler be Laftly, apply'd to the two external Parallelograms if thefe pn -f- 5 and m-\- 3, and Numbers be made equal, we fhali have m=. I, and the Numbers themfelves will each be 2. The next Number to which the Ruler approaches will be o, the two next are each i, the next 3, the laft 4. All which Numbers will be found in the Arithmetical defcending Progreffion 2, I, p, i, 2, 3, 4, whofe common difference is i. And thefe fix are all the poffible cafes of external Terms. Now to find the Arithmetical Progreffion, in which all thefe re- fhall be find fulting Terms comprehended ; their differences, and the greateft common Divifor of thofe differences fhall be the common difference of the Thus in the fifth cafe Progreffion. before, the refulting Numbers were 1 whofe are 5, 1,15, differences 6, 4, and their greateft common Divifor is 2. Therefore 2 will be the common difference of the Arithmetical which will include all the Progreffion, refulting without Terms. But the Numbers 5, n, 15, any fuperfluous .ap- of all plication this will be beft apprehended from the Examples that are to follow. 30 and INFINITE SERIES. 201

before the form of this 30. We have given Equation, y< $xy* 1 6

6 6m s m l a tIB have .\- x$ + .v + tion, we fhall A $A -+- -^xv+s 7*A -f.

3 J felected 6fl x -f-^.x'4, &c. = o. Thefe Indices of AT, when from with their will ftand the general Table, refpective Parallelograms, thus:

4 2 02 Tfo Method of FLUXIONS,

6 5 This here will give the Equation A 7rt*A* -j- 6 and hence each of the Numbers 9. The Ruler in or its motion will next arrive at $m-\- i, 8f. Then at zm -f- 2, or at And at But thefe Numbers 5. Then 4. laftly 3. 9, 8f, 5, 4, in an Arithmetical of 3, will be comprehended Progreffion, which the common difference is i. So that the form of the Series here &c. if will be y =A.v* -f- Ex -+- Cx^ -f- D^, But we put the two in order to obtain external Terms equal to nothing, the firft Ap- A4 I mail 6 or A 1 - which proximation, we have A + =o, -f- = o, will afford none but impoffible Roots. So that we can have no initial Approximation from this fuppofition, and confequently no Series. to the third laft cafe of external But laftly, try and Parallelograms, to and which we may apply the Ruler 4 4^2-4-3, being made equal, of the Numbers will be will give m = -, and each 4. The next or the next or Number will be 3 ; the next 2m -\- 2, 2| ; 50* -{- i, 27; the laft will be 6m, or if. But the Numbers 4, 3, af, 27, if, will all be found in a decreafing Arithmetical Progreffion, whofe

common difference will be f . So that Ax* + Bx H- Cx~* -+- Dx~s if the 6cc. may reprefent the form of this Series, circumftances of the and INFINITE SERIES. 203 the Coefficients will allow of an Approximation from hence. But if we make the initial Terms equal to nothing, we mall have a,

-\- b* o, which will give none but impoflible Roots. So that we can have no initial Approximation from hence, and confequemly no Series for the Root in this form. s 1 # ; when the 3 i. The Equation y by -+- qbx* =o, Terms to a double Arithmetical will have are difpofed according Scale, the before from it be form as was (hewn ; whence may known, what cafes of external Terms there are to be try'd, and what will be the circumftances of the feveral Series for the Root y, which may be derived from hence. Or otherwiie more explicitely thus. Putting 1" this Ax for the firft Term of the Series y, Equation will become 1 " 1 6cc. o. So by Subftitution A'A.-?" M * -f- gbx* x*, = that if we take thefe Indices of x out of the general Table, they will ftand as in the following Diagram. Now in order to have an afcending the Ruler to Series for y, we may apply the two external Parallelograms 2 and 2W, which therefore being made equal, will - and each of the Numbers give m i, will be 2. The Ruler then in its parallel will firft come to and then to or progreis 3, yn, 5. But the Num- are all contain'd in an bers 2, 3, 5, afcending Arithmetical Progrefiion, whofe common difference is i . Therefore the form of the Series 1 will here be = AA; -f- B* -f-(*', &c. And to determine the ;' we fhall have the 1 -- firft Coefficient A, Equation bfcx -f- qbx* o, a that is A + So that either or A = 9, = 3. 4-3*, or 3^ may be the initial Approximation, according as we intend to extract the affirmative or the negative Root. mall have another cafe of external We Terms, and perhaps another afcending Series for_y, by applying the Ruler to the Parallelo- which grams 2; and 5;^, Numbers being made equal, will g;ive =zo. the when we m (For by way, put 2;= 5/77, we are not at to that becaufe liberty argue by Diviiion, 2=5, this would bring us to an And the laws of absurdity. Argumentation require, that no muft be but Abfurdities admitted, when they are inevitable, and are of ufe to mew the of when they falfity fome Supposition. We fliould therefore here Subtraction, thus: Becanfe cm ^t>i argue by t then 2:>i = o, or = o, and therefore m o. This Cau- 5//f pn = tion I thought the more necellary, becaufe I have obferved f >mc, D d 2 who 204 "The Method of FLUXIONS, who would lay the blame of their own Abfurdities upon the Analy- tical Art. But thefe Abfurdities are not to be imputed to the Art, but rather to the unikilfulnef of the Artift, who thus abfurdly ap-

the of his 777. we {hall plies Principles Art.) Having therefore = o, alfo have the Numbers 2/77. o. The Ruler in its = 577*' = parallel motion will next arrive at 2 then at the Numbers ; and 3. But o, will 2, 3, be comprehended in the Arithmetical Progreffion o, i, 2, 3, whofe common difference is i. Therefore y = A -+- Ex -+- CAT*, &c. will be the form of this Series. Now from the exterior Terms A*

3 bA* = o, or A = by or A = fi, we {hall have the firft Term of the Series.

There is another cafe of external Terms to be try'd, which poffi- for the Ruler to bly may afford a defcending Series y. For applying the and and thefe we (hall have Parallelograms 3 5777, making equal, the Ruler 7/7=4, an d each of thefe Numbers will be 3. Then will come to 2 or -- the Numbers ; and laftly 2777, But 3, 2, if, will be comprehended in a defcending Progreffion, whofe common difference is f. Therefore the form of the Series will be y = Ax^ T r 3 3 _f. BA" -|- CA^ -f- D, &c. And the external Terms A .v A: = o will give A= i for the firft Coefficient. Now as the two former' cafes will each give a converging Series for y in this Equation, when .v is lefs fo this cafe will afford us a Series when x is than Unity ; than which will fo much the the greater Unity ; converge fafter, greater x is fuppofed to be. 32. We have already feen the form of this Equation y> -\-axy -f- aay A? 3 2# 3 =o, when the Terms are difpofed according to a double Arithmetical Scale. And if we take the fictitious quantity to the Root we {hall Ax* to reprefent the firft Approximation ;', m l 3 3 have m + a'-Ax" A' 2^ Sec, by fubftitution A'X= -f- aAx -+- , = o. Thefe Terms, or at leaft thefe Indices of x, being felecled out of the general Table, will appear thus. Now to obtain an afcending Series for the Root y, we may apply the Ruler to the three external Terms o, 777, 3777, which being made equal, will give m = o. Therefore thefe Numbers are each o. In the next place the will i Ruler come to 777.4- i, or ; and laftly the are contain'd in the to 3. But Numbers o, i, 3, Arithmetical Progreffion o, i, 2, 3, whofe common difference is i. Therefore 1 the form of the Root is y=. A -+- Ex -{-Cx -+- Dx>, 6cc. Now 1 if the Equation A 3 + a A <2a' =o, (which is derived from the initial and INFINITE SERIES. 205

1 initial Term?,) is divided by the factor A -f- ah. ~t- 2a*, it will give the Quotient A a = o, or A=.a for the initial Term of the Root^y. If we would alfo derive a defcending Series for this Equation, we Ruler to the external which may apply the Parallelograms 3, yn, to each will i alio thefe Num- being made equal other, give m = ; to or 2 bers will each be 3. Then the Ruler will approach m-\- i, ; or i to o. But the Numbers are a de- then to //;, ; laftly 3, 2, i, o, difference creafing Arithmetical Progreflion, of which the common is i. So that the form of the Series will here be y=Ax -+- B -+- CA, ' Dx~- &c. And the form'd the external Terms -f- , Equation by will be A 3 x 3 .v 3 = o, or A= i. s 1 c'x 3- c 7 33. The form of the Equation x*)' y+X} -f- = o, Arithmetical we have feen, as exprefs'd by a combined Scale, already varieties of external with which will eafily mew us all the Terms, ra their other Circumftances. But for farther illuftration, putting A,v for m t x^ + l the firft Term of the Root y, we ("hall have by fubftitution A I i "I + I c'x* Thefe Indices of x 36--A ,v -+- c\ &c. =o. being tabulated, will ftand thus. have Now to an afcending 2 Series, we mufl apply the Ruler to the two external Terms o and yn -\- 2, which being made equal, will give the two will be each o. The next m .*-, and Numbers anting the la ft is 2. the Ruler arrives at is zm or ; and Number that + r, .J. Arithmeti- But the Numbers o, i, 2, will be found in an afcending

is -i-. Ax~ cal Progreffion, whofe common difference Therefore y =.

'> the form of the Root. deter- _l_ B.v -f- C -f- D.x^, &c. will be To mine the firft Coefficient A, we fhall have from the exterior Terms 7 7 c'\ Therefore A'-f-6- = o, which will give A = y^c =

to the Root will be ==. , the firft Term or Approximation y J/-^ &c. if can obtain a We may try we defcending Series, by applying the Ruler to the two external Parallelograms, whofe Numbers are 2 made will ;;; and thefe and 5;-f-2, which being equal, give = o, next arrive at Numbers will each be 2. The Ruler will 2///-J- i, or at o. But the Numbers 2, i, o, form a de Icon i ; and laftly cling difference is i. So that die form of the ProgreiTion, whofe common be B,v Cv- J &c, And the Series will here y = A -f. + , putting initial 206 The Method of FLUXIONS, initial Terms equal to nothing, as they ftand in the Equation, we 1 c*x* or for the firft ihall have A'* = o, A = -\- a^y* 27^ = o, it may be thus refolved without any preparation. When reduced to 6 3 6 our it will ftand thus, 8z -}-az \* * * ,, form, y/ J 1 and f=o; by* * 27^9 3 B ! +6 1 m+' s putting_y=A ',&:c.it willbecome 8A*z*" +aA z*' * * &c.7 * * 27^3 m firft of external Terms will +s The cafe give $A*z* 27.^' = o, whence 3/^-1-6 = 0, or m=s 2. Thefe Indices or Numbers other therefore will be each o ; and the 2/-f- 6 will be 2. But 0,2, will be in an afcending Arithmetical Progreffion, of which the common difference is 2. So that the form of the Series will be y=. Az~~- 1 -|- B -h Cs. -+- Dz*, &c. And bccaufe 8A' = 27^9, or 2A=3^3, 3 it will be A = J-0 . Therefore the firft Term or Approximation to 3 will be -^- the Root * 2 2. mJ 6 But another cafe of external Terms will give aA*-z~- c or /;; = o, whence 2w-f-6 = o, = 3. Thefe Indices or Num- bers therefore will be each o and the other 6 will be j yn -+- 3. will found in a But o, 3, be defcending Arithmetical P/ogrefiion,

whofe common difference is 3 . So that the form of the Series will be Az~* 6 Cs-' ccc. And becaufe ^A 1 y = -f- Ez~ -f- , = 27^', J tis A + x^ 4 f r ^''^ Coefficient. = 3v/3 > ^ is cafe of external Laftly, there another Terms, which may pom"- a + 6 m^~ 6 bly afford us a defcending Series, by making SA*z3 -f- aA*z"- o. And the Numbers will be each to 6 =: o ; whence m = equal ; the other Number, or Index of z, is o. But 6, o, will be in a of defcending Arithmetical Progreffion, which the common difference the of the Series will 6 is 6. Therefore form be _y= A -f- Ez~ -f- 1 11 &c. Alib becaufe 8A -+- aA it is A for Oc- , = o, = {a the firft Coefficient. I fhall one in order to fhew produce Example more, what variety of Series may be derived from the Root in fome Equations; as alib to fhew all the cafes, and all the varieties that can be derived, in the Let us prefent ftate of the Equation. therefore affume this Equation, a 6 1,vl 3^ /. ClI I fl\* --_ z _ - _ _ _ y* + x -- + _ _j_ _ .+. ^ = o, or

3 1 i l 3- 3 z 1 6 rather y a~ x -\- x> x -+- a A. a x~ s ~ y a>y~- a\) \y~ -}- i a= o. if m -+- = Which we make A.\ , &c. and }' = difpofe and INFINITE SERIES. 207 the Terms to difpofe according a combined Arithmetical Progref- fion, will appear thus :

**** * m .*x" +** *

Now here it is plain by the difpofition of the Terms, that the Ruler can be apply'd eight times, and no oftner, or that there are cafes of eight external Terms to be try'd, each of which may give a Series for the Root, if the Coefficients will allow it, of which four will be afcending, and four defcending. And firft for the four cafes of afcending Series, in which the Root will converge by the afcend- Powers ing of x ; and afterwards for the other four cafes, when the Series converges by the defcending Powers of x. I. the Apply Ruler, or, (which is the fame thing,) afTume the s 1 1 "1 -* Equation a A~=x~^ a"' A- *- = o, which will give 3/77 = 2in or 7/7= alfo The Number 2, 2; A=^. refulting from thefe is Pailer Indices 6. But the in its parallel motion will next come to the Index . then to or 2 3 zm-{- 2, ; then to o ; then to zm or 2 then to and to 2, ; 3 ; laftly 3/7; and 2/774- 2, or 6. But the in Numbers 6, 3, 2, o, 2, 3, 6, are an afcending Arithmetical Progrellion, of which the common difference is i and therefore the 1- ; form of the Series will be y = Ax --Bx* - -f- C.v, &c. and its firft Term will be . a 6 l t i II. Affume the Equation a x~ a''A x-- ==z o, which will or alfo give 3 = zm 2, m = f } A = a*. The Num- ber hence is ; the next will be or iJL refulting 3 37/7, ; the 2/72 next 2, or i ; the next o the next j ; 2/>-f- 2, or ; the next or i ; the two laft zm 2 and are 3/7;, 4- 3, each 3. But the Numbers 3, i, j, o, i, i|, 3, will be found in an afcending Arithmetical Progreffion, of which the common difference is and therefore f ; the form of the Series will be y = Ax^ +- Bx +- + Dx% &c. and its firft Term will be + ^/ax. III. 208 7?je Method of FLUXIONS,

6 1 11""- III. Aflame the Equation a x~* a* A. .* = o, which will give 3 = 2?/7 2, or;;;= f; alfo A = + a*. The Num- ber is the next or next refulting 3 ; 3;?;, if ; the 2m 2, or i the next o the next or i the next or ; ; 2m -+- 2, ; 3z, the laft if; two 3 and 2m -f- 2, which are each 3. But the Numbers all 3, if, i, o, i, if, 3, will be comprehended in an afcending Arithmetical Progreiiion, of which the common difference is and therefore the form of the Series will be - f ; y A.y~ B &c. and the firft will be f ~h -f- Cx* -f- Dx, Term a*x~' , or "v/;- 3 1 2 IV. Affume the Equation A A: 3 ^'A *-'*- = o, which will give 3 = 2; 2, or ;/z = 2; alfo A = a*. The Number is 6 the next will be the next refulting ; 3 ; 2m -{-2, or 2 ; the next the next or 2 the next laft o; 2m 2, ; 3 ; the two and is 3/tf 2#?4-2, each of which 6. But the Numbers 6, 3, 2, o, 2, 3, 6, belong to an afcending Arithmetical Progref- fion, of which the common difference is i. Therefore the form of the Series will be y = Ax~- +- Bx~' -+- C -f- Dx, &c. and its firft Term will be ^ The four defending Series are thus derived. 1 m l I. Afllune the Equation Au 3 a-'A x"- + o, which will - = 2/w -4- 2, or #2 = 2; alfo A = . The Number re- give 3;;z

is 6 the next will be the next 2m or fulting ; 3 ; 2, 2 ; the next O; the next 2;;z-f-2, or < 25 the next 3; the two laft each of is 3/72 and 2m 2, which 6. But the Numbers to a 6, 3, 2, o, 2, 3, 6, belong defcending Arithmetical Pro- difference is i. greflion, of which the common Therefore the form 1 of the Series will Ax* Ex C D.*- &c. firft be/ = -i- ~f- -f- , and the X S, Term will be . a l i im ~Jri II. Affume the Equation x* a~ A x = o, which will give 2 or ;;:= alfo A + a*. The Number 2m -+- = 3, f ; = refulting

is ; the next wi: be or the next or i 3 3;^, if; 2;/z-f-2, ; the

next o the next ,/ or i the next ; 2, ; yn, or if; the laft two 3 and 2m 2 are each 3. But the Numbers 3, - if, i, o, i, if, 3, belong to a defcending Arithmetical ProgreiTon, of which the common difference is i. Therefore the form of the Series will = Ax^-i-Ex+Cx~^-{- DAT*', &c. and be_)' the firft Term will be + ^/ax. III. and INFINITE SERIES. 209 III. Aflume the Equation x 5 <7A-** *+ = o, which will' 2 or TW alfo A + a*'. gve 3 = w H- 2, = f ; = The Num- ber from hence is the next will be or refulting 3 ; 3;;?, if ; the next or i the next the i 2m -+-2, ; o ; next 2m 2, or ; the next or the laft 3777, if ; two 3 and 2m 2, each of whichare But the 3. Numbers 3, if, i, o, i, if, 3, are comprehended in a defcending Arithmetical Progreflion, of which the common difference is f . Therefore the form of the Series will l bcy=Ax~*-t-Bx~'-i-Cx~~ -l-Dx-% &c - and the firft Term will be + a*x~* or + a - .

6 m I IV. Laftly, aflume the Equation a A-ix~i rfA.- which will = 2m -f- 2, or m 2 ; alfo A ==='#. give 3;;; = The Number is 6 be the next refulting ; the next will 3 ; zm 2, or 2 the next or the next ; o ; the next 2m -f- 2, 2 ; 3 ; the two next 3#; and 2m 2, are each 6. But the Numbers 6, 3, 2, o, 2, 3,' 6, belong to a defcending Arithmetical Progref- iion, of which the -common difference is r. Therefore the form of 1 3 5 the Series will

And this may fuffice in all Equations of this kind, for finding the farms of the feveral Series, and their firft Approximations. Now we muft proceed to their farther Refolution, or to the Method of finding all the reft of the Terms fucceffively, no

.SECT. V. The Refolution of Affe&ed Specious Equations, firofecuted by various Methods of Analyfis.

35. TTT ITHERTO it has been fhewn, when an Equation is ~J_ propofed, in order to find its Root, how the Terms of the Equation are to be difpoied in a two-fold regular fucceffion/fo as thereby to find the initial Approximations, and the feveral forms of the Scries in all their various circumftances. Now the Author proceeds in like manner to difcover the fubfequent Terms of the Series, which may be done with much eafe and certainty, when the form of the Series is known. For this end he finds Refidual or Supple- a mental Equations, in regular fuccefTion alfo, the Roots of which are a Series of continued Supplements to the Root required. In every one of which Supplemental Equations the Approximation is E e found, 2io The Method of FLUXIONS, found, by rejecting the more remote or lefs confiderable Terms, and- fo reducing it to a fimple Equation, which will give a near Value of the Root. And thus the whole affair is reduced to a kind of Comparifon of the Roots of Equations, as has been hinted already. The Root of an Equation is nearly found, and its Supplement, which, ihculd make it is the Root of an inferior the compleat, Equation > Sup- of which is the Root of an inferior fo on plement again Equation ; and ever. that for Or retaining Supplement, we may flop where we pleafe. 36. The Author's Diagram, or his Procefs of Refolution, is very to be underflood however it be thus farther eafy ; yet may explain'd. Having inferted the Terms of the given Equation in the left-hand Column, (which therefore are equal to nothing, as are alfo all the fubfequent Columns,) and having already found the firft Approxi- to the Root to be a inflead of the he fubflitutes its mation ; Root y equivalent a-\-p in the feveral Terms of the Equation, and writes the Refult over-againfl them refpedtively, in the rightrhand Margin. Thefe he collects and abbreviates, writing the Refult below, . in the of all the too a left-hand Column ; which rejecting Terms of high the i i compofition, he retains only two loweft Terms ^.a p-\-a x=.o^ which give p = x for the fecond Term of the Root. Then afluming/> = -%x-}-q, he fubflitutes this in the defcending Terms to the left-hand, and. writes the Refult in the Column to the right- hand. Thefe he collects and abbreviates, writing the Refult below in the left- hand- Column. Of which rejecting again all the higher the two loweft I -cx i o Terms, he retains only, ^a*q TT = i which for the third Term of the Root. And fo on. give a Or in imitation of a former Procefs, (which may be feen-, ,pag; of and all fuch like be 165.) the Refolution this, Equations, may thus perform'd. 2fl'= i a* !L Or i)3_|_tf^y= (if y=-a-rp} + '$a*'p-\-T> ap -}-p* } collecting 3 a * -+.axy-h A? + +ay > andexpung- J ing,

I r collecting and expunging,

X which Procefs the Root will be found =

Or in imitation of the Method before taught, (pag. 178, &c.) we refolve the firft of this may thus Supplemental Equation Example ; >a 3 a'-x x* where the w>. W-p -f- axp -f- 3^/ -4-/ = -+ -, Terms muft be difpos'd in the following manner. But to avoid a great deal it be here that of unneceffary prolixity, may obferved, y = a, &c. briefly denotes, that a is the firft Term of the Series, to be derived for the Value of y. Alfo^=* f#, &c. infinuates, that fx is the fecond Term of the fame Series y. Alfo y = * * -f- 644 &c. that -4- is the third of the Series with- infinuates, 1 r Term y, 643 ** out any regard to the other Terms. And fo for all the fucceeding

Terms ; and the like is to be underftood of all other Series whatever.

4*'/l == a*x * +*

, &c. 40963^ 13 1x4 c 7T7T &c -

this it be To explain Procefs, may obferved, that here a*x is l made the firft Term of the Series, into which ^a p is to be refolved or 4 a*x, &c. and therefore x, &c. which ; t.a*p = p = below, Then is -f- 1 &c. is fet down axp = ^ax , and (by fquaring) l a &c. each of which are let in their _f-3^) = -t-Tz-ax , down pro- Thefe Terms will per Places. being collecled, make -V^S which with a contrary Sign muft be fet down for the fecond Term

a l * &c. and therefore -?- > of ; or = -r'^ax } = * -f- ^a*p 4d /> + p

a &c. and * il. > &c. Then , = axp=.*-^-^ (by fquaring) 3

5 3 and c - T^efe collected &c. (by cubing) /> = W^" ' ^ being to be wrote down with a will make ^, contrary Sign; and with A: 3 one of the Terms of the this, together , given Equation, l * * --'x* -^- will make = -f- } &c. and therefore />= * * -f- ? a*p '*" a s i 1*4 * * ~~ &c. and * * &c. Then axp= -J- , (by fquaring) 3^/1* = E e 2 212 Ih* Method of FLUXIONSJ

-1^1 _, 1&22 &c . and * * -f- &c. all which ' N(by cubing) = ', ] 4096.1 ft! 1024* a will 1 collected with Sign, make 4tf = ***-f- being contrary />

59i_* &c. and therefore -f- &c. And the , ,'' by /=*** 163841^ 40961* continue the Extraction as far fame Method we may as we pleafe. The Rationale of this Procefs has been already deliver'd, but as it will be of frequent ufe, I fhaM here mention it again, in feme- of what a .different manner. The Terms the Equation being duly order'd, fo as that the Terms involving the Root, (which are to be refolved into their refpecttve Series,) being all in a Column on one on the other fide fide, and t,he known Terms ; any adventitious Terms may be introduced, fuch as will be neceffary for forming the feveral Series, provided they are made mutually to deftroy one another, that the integrity of the Equation may be thereby preferved. a Thefe adventitious Terms will be fupply'd by kind of Circulation, , which w^ill make the work eafy and pleafant enough ; and the ne- the Powers or of fuch Series as com- ceffary Terms of fimple Roots, muft be derived one of the pofe the Equation, by one, by any Theorems. - - foregoing , Or if we are willing to avoid too many, and to0 high Powers - in in thefe Extraction's, we may proceed' the following manner. The Example mall be the fame Supplemental Equation as before, , be reduced to this ax -4- which may form, 4a* -f- -+- ^ap pp -x.p =s 3 the of which Refolution be thus : , * " a*x 4-# j ' may .'

' ' a -'! 4rf H- ax - . .- . __, 3 . $

5 i 2a a - &c: -4-/ h TV** 77^ ,

X* I 3 I*S 64 5i2i 16384^3^

The Terms 4^* -f- ax-\- ^p-^-fp I call the aggregate Factor, of or which I place the known part parts 4<2* -{- ax .above, and the in a to the fa unknown, parts ^ap -f- pp Column left-hand, as that to be be their refpeclive Series, as they come known, may placed them. Under thefe a Line is to receive regularly over-againft drawn, 2, the and INFINITE SERIES. 2*3

the Series beneath is aggregate it, which -form'd by the Terms of the aggregate Factor, as they become known. Under this aggregate Se- ries comes the Factor or the fimple />, fymbol of the Root to be extracted, as its Terms become known alfo. Laftly, under all are the of the in their known Terms Equation proper places. Now as thefe laft Terms (becaufe of the Equation) are equivalent to the Pro- duct of the two Species above them ; from this confideration the Terms of the Series p are gradually derived, as follows. the initial the Firft, Term 4^ (of aggregate Series) is brought down into its place, as having no other Term to be collected with it. Then becaufe this 'd the firft of Term, multiply by Term />, is to the firft of the fuppofe q, equal Term Product, that is, ^.a'-q it will cr = a*x, be q = ~x, p = -L.v, &c. to be put down in its Thence we (hall have T &c. which to- place. > ap-=. %ax, gether with -}-ax above, will make -^^'ax for the fecond Term ; of Series. if the aggregate Now we fuppofe r to reprefent the le-- of and to be in its cond Term p, wrote place accordingly ; by crofs- multiplication we lhall have ^.a^r -'y-ax^ =^ o, becaufe the fecond _v^

Term of the Product is abfent, or=ro. Therefore r-=. , which 64*' be fet down in its And hence * 3 may now place. yap = -f- -^x -, &c. and &c. which collected will l p* = ^x*, being make ^~x , for the third Term of the aggregate Factor. Now if we fuppofe s to reprefent the third Term of p, then by crofs-multiplication, (or 1 for of infinite ; by our Theorem Multiplication Series,) q.a 4-

* 3 x 3 is the third Term of the There- ^ = ; (for Product.) 256 256 l s - to be fet down in its * -4- fore = , place. Then -lap = * 512^* 1 * &c. and i = A- , &c. which together will make zea 5 I 2a 3 ^2l of the Series. _j_ for the fourth Term aggregate Then putting

fourth of fliall have /.to reprefent the Term p, by multiplication we whence / to be ^ = o, = ' 2048^ 4096* -^L fet down in its place. If we would proceed any farther in the Ex- traction, we mufl find in like manner the fourth Term of the Se- the third Term of in order to find the fifth ries 3/, and p*-, Term of the aggregate Series. And thus we may eafily and furely carry of on the Root to what degree accuracy we pleafe, without any of danger computing any fuperfiuous Terms ; which will be no mean advantage of thefe Methods. Or Method of FLUXIONS,

Or we may proceed in the following manner, by which we fliati avoid the trouble of railing any fubfidiary Powers at all. The Sup- of the fame plemental Equation Example, ^cfp +- axp -f- ^ap* -f- be p= = a*x-{-x*, (and all others in imitation of this,) may reduced to this a*x form, /\.a*- -+- ax-+- ^a -{- p x/> x/> = + x>, which may be thus refolved.

4#* -f- ax

The Terms being difpofed as in this Paradigm, bring down 4^* the firft of the ftill for Term aggregate Series, as it may be call'd, to firft and fuppofe q reprefent the Term of the Series p. Then will 4^*5' = a*x, or^= ~x, which is to be wrote every where for the firft of firft Term p. Multiply +- 3*2 by x for the Term of 3#-f-/>x/>, with which product ax collect the Term above, or the Refult ~#x will be the fecond of the -+- ax ; Term aggregate Series. let r the fecond of and Then reprefent Term />, we fhal'l have 1 or r to be by Multiplication q.a'-r -r-s-^x = o, = ^ , as wrote every where for the fecond Term of^>. Then above, by crofs- ~~ 1 1 we fhall have x -I- -rV-v = ^or tne third multiplication 3^ a ^V^ Term of the aggregate Series. Again, fuppofing s to reprefent the third Term of p, we ftiall have by Multiplication, (fee the Theorem,

1 i for that A.a s . that . to purpofe,) +' , ', =x is, s=^^a ' 256 256 5iz be wrote every where for the third Term of p. And by the lame way of Multiplication the fourth Term of the aggregate Series will be found to be \-2L. which will make the fourth -of to be } Term p

And fo on.

Among all this variety of Methods for thefe Extractions, we is muft not omit to ftipply the Learner with one more, which common and INFINITE SERIES/ mon and obvious enough, but which fuppofes the form of the Se- ries required to be already known, and only the Coefficients to be unknown. This we may the better do here, becaufe we have al- determine ready fhewn how to the form and number of fuch Se- cafe This Method confifts in the ries, in any propofed. aflumption of a general Series for the Root, fuch as may conveniently repre- the fubftitution of which in the fent it, by given Equation, the general Coefficients may be determined. Thus in the prefent Equa- 5 3 tion y= 4- axy 4- aay A' 2a = o, having already found (pag. 204.) the form of the Root or Series to be y = A 4- Bx -+- Cx*, &c. by the help of any of the Methods for Cubing an infinite Series, we may eafily fubrtitute this Series inftead of y in this Equation, which will then become

3 l l 1 3 A 4~ 3 A ijX 4~ ^n.D x 4~ -D'AJ" 4~ 3 ^"*> *^c* 4^ 3A*C 4- 6ABC4- 36^ } 4- 6ABD o.

1- aA.x 4- aBx 4- aCx* -f- aDx*, &c.

Now becaufe x is an indeterminate quantity, and muft continue' fo to be, every Term of this Equation may be feparately put equal to- nothing, by which the general Coefficients A,B, C, D, &c. will be de- to Values and this means the will be termined congruous ; by Root^ 1 known. A 3 4- a A 2a~> = o, which will Thus, ( i.) give A=/r, a or as before. (2.) 3A B -+- aA -+- a*-B = o, B==

a B 3 D i = o, (4.) -4-6ABC-j-3A*D4-rfC-H orD'=^_> AO 6ABD 4- A*E 4- aD 4- ^E = or = (5.) 3 + 3B-C + 3 o, &c. _^2_ . And. fo on, to determine F, G, H, Then by fubfti- 163^4^^ Values of &c. in the aflumed tuting thefe A, B, C, D, Root, we (hall have the former Series y =ax+ ^4- '+ ;%^> &c. refolve this Or laftly, we may conveniently enough Equation, or fame it to the any other of the kind, by applying general Theorem, CT for the Roots of affedted in Num- ra . 1 90. extracting any Equations to this form i3 * a 1 bers. For this Equation being reduced ; 4- 4-^x 2. x/ 2l b The Method of FLUXIONS,

3 c inftead x 2rt -+-A:' x.y = o, we fliall have there #2 = 3. And of the firft, fecond, third, fourth, fifth,- &c. Coefficients of the Powers if write 2# 5 x ? of y in the Theorem, we 1,0, aa -f- ax, , o, - - and if the firft > .&c. refpectively ; we make Approximation = 4 " A * or a and B i ; we. fliall have , for a nearer A= = a , Approxi- 4 -f- * mation to the Root. Again, if we make A= 4^' -f- x*, and fliall have B 4^ -f- ax, by Subftitution we the Fraction ,* ,. .t5 -(- 48a*.v4 -f- i Zfi4 -f- zqSjt .* * +1*9

nearer Approximation to the Root. And taking this Numerator for A, and the Denominator for B, we fliall approach nearer ftill. But this laft Approximation is fo near, that if we only take the firft five Terms of the Numerator, and divide them by the firft five Terms of the Denominator, (which, if rightly managed, will be no troublefome Operation,) we fliall have the fame five Terms of the Series, fo often found already. And the Theorem will converge fo faft on this, and fuch like oc- -cafions, that if we here take the firft Approximation A = a, (ma- ** fliall have B i we = , &c. a &cc. king = ,) y -^ ^ = ~x, A .And if again we make this the fecond Approximation, or a we fliall have t*, (making B = i,) y =

a 4 ax * 1 z 4 ~T -i 4 5 if again we make this the third Approximation, or A=:a &c. 'B== we fliall have the of _ (making J,) Value the D ti& ^ * *-** true Root to eight Terms at this Operation. For every new Oper- ation will double the number of Terms., that were found true by the laft Operation. ftill with the fame we To proceed Equation ; have found before, that we likewife have a pag. 205, might defcending Series in this 1 &c. for the form, v = AA'H-B -j-Cx- , Root y, which we fliall or three for the extract two ways, more abundant exemplification of It has been this Doctrine. already found, that A= i, or that x is the firft Approximation to the Root. Make therefore y =. x and fubftitute this in the 3 x> given Equation jy -f- axy -f- any *a= = o, which will then become ^p -f- axp -f- a?p -\- ^xf 1 a^x 2 3 -4- ax -f- = o. This may be reduced to this form ax a* ax"- a*x -^ _t_ -+- -f- 3.v/> -{-/* x/> = + 2a*, and may be refolved as follows. OAT* and INFINITE SERIES. 217

1 3A' -f- aX -f- rt* ax _ a * IE1 _+_ t Sec.

. _ 4_ _+_ &c> p* + ^ _; s

* -f- rfl +- &c. 3-v* t ^ , " 4 *""* 64 c -- /,..' _i_^ " ~~-U_ _i-~ y 3 3-v 81^^

The Terms of the aggregate Factor, as alib the known Terms of the l Equation, being difpofed as in the Paradigm, bring down ^x for lire firft of the Series and to Term aggregate ; fuppofing q repre- the lent firft Term of the Series p, it will be 3^^ = ax*, or q=- y, for the firft Term of p. Therefore ax will be the firft of to be in its Term 3^ put down place. This will make the fecond Term of the Series to be fo if ; re- aggregate nothing ; that prefent the fecond Term of p, we fliall have by multiplication 3vV 1 or / of to be = a *;, = "_ for the fecond Term p, put down in 1 its place. Then will a be the fecond Term of $xp, as alfo 1 will be the firft to be fet down each in their ^d"~ Term of/ , places. 1 of this will be which is to be the The Refult Column -^z , made third Term of the aggregate Series. Then putting s for the third Term we mall have rt3 == 2(l ' of/, by Multiplication ^x^s -V > or $52- fhall s= . And thus byJ the next Operation we have / = 1

and fo on. "Or if we would refolve this reildual Equation by one of the fore- the of and going; Methods, by which railing Powers was avoided, alone we wherein the whole was performed by Multiplication ; may

1 1 1 reduce it to this form, 3* + fix -f- a -f- 3* -j-/ x/> x/ = ^.v

of will be thus : d^X _j_ 2n* , the Refolution which

F f j.v- 2I 8 tte Method of FLUXIONS,

3** -f- ax -h a* * --+ 3* T* ~

a &c. fa* + , --- . 3*- 8ix* 243*3'

The as in Terms being difpos'd the Example, bring down 3*'* for firft of the the Term aggregate Series, and fuppofing q to reprefent the firft of the Term Series p, it will be yx^q = ax*, or q = La. Put in its down -+- 3* proper place, and under it (as alfo after the firft it) put down Term of/, or La, which being multiply'd, and collected with -j- ax above, will make o for the fecond Term the of aggregate Series. If the fecond Term ofp is now reprefented we fhall have * or r to by r, ix^r = a'-x, = , be put 3^* down in its feveral places. Then by multiplying and collecting we mail have -f- a* for the third Term of the aggregate Series. And s for the third of fhall have putting Term p, we by Multiplication

' rf3 3 T =2d or j= . And fo on as far as we 3Ar*j T , |^ pleafe. Laftly, inftead of the Supplemental Equation, we may refolve the given Equation itfelf in the following manner :

* 28*4 1 - ax Sec. %a*x -f- \a* , ----- 1 f- ax La*x La* +- , &c.

y= x 'a 243^5

becaufe it is &c. it will be Here y~> =x*, y = x, &c. and therefore -I A &c. which muft be fet down in its _t_ xy =-f- , place. Then be wrote with a that it it muft again contrary fign, may be y= == * &c. and therefore the rfx*, (extracting cube-root,) /= * a &c. Then -+- a*y = 4- a*x, &c. and + axy = * , j-^^, &c. which and INFINITE SERIES. 219 collected with a which being contrary fign, will makers = * * l &c. and * * &c. Hence -f- a JLa*x, (by Extraction) y = , y 3 &c. and * * = # frt , -f- ^v>'= a*, &c. which being collected with a contrary fign, and united with -f- 20 J above, will * * * 5 make = , &c. whence * * * > y"' f^ (by Extraction) y = 4-^

&c. Then -+- a = * * &c. and -f- = * * * 7 j*, axy + ^7' collected a &c. which being with contrary fign, will make y* =

&c ' and l ^en * * * * 177 ' (by Extraction)^' ==* * * * + &c. And fo on. 37, 38. I think I need not trouble the Learner, or myfelf, with of the giving any particular Explication (or Application) Author's Rules, for continuing the Quote only to fuch a certain period as {hall be before determined, and for preventing the computation of fuperfluous Terms ; becaufe mod of the Methods of Analyfis here deli- no Rules at nor is leaft ver'd require all, there the danger of making any unneceflary Computations. are to find the of an 39. When we Root y fuch Equation as 1 3 -+- v4 &c - = *> tllis is call'd this, y t)' fj t + t>"> ufually a Series. For as here the is the Reverfion of Aggregate z exprefs'd by the Powers of y; fo when the Series is reverted, the Aggregate y will be exprefs'd by the Powers of z. This Equation, as now it the to (lands, fuppofes z (or Aggregate of the Series) be unknown, and that we are to approximate to it indefinitely, by means of the and its Or otherwife the known Number y Powers. ; unknown Number z is equivalent to an infinite Series of decreafing Terms, an Arithmetical of the exprefs'd by Scale, which known Number y is the Root. This Root therefore muft be fuppofed to be lefs than Unity, that the Series may duly converge. And thence it will follow, that z, alfo will be much lefs than Unity. This is ufually called a Logarithmick Series, becaufe in certain circumftanp.es it ex- the preffes the Relation between Logarithms and their Numbers, as will appear hereafter. If we look upon z, as known, and therefore y as unknown, the Series mull be reverted; or the Value of y muft a Series of be exprefs'd by Terms compos'd of the known Num- ber z and its Powers. The Author's Method for reverting this Se- ries will be obvious from the confideration of his very Diagram ; meet with another and we mall Method hereafter, in another part of be therefore in it his Works. It will fuffiqient this place, to perform after the manner of fome of the foregoing Extractions. F f 2 y Method of FLUXIONS,

a 3 4 } a &c - y 1 = + |~ -+- f:i + TV- 4- T5o-3 >

5 4 fV- > h f* -4- 'f^ -H AS',' &c. 4 _. fS fa', &c. M L. AX* &C. A./ -s J * Sec.

In this Paradigm the unknown parts of the Equation are fet down in a defcending order to the left-hand, and the known Number z is fet down over-againft y to the right-hand. Then is y = z, Sec. 1 and therefore Sec. which is to be fet down in its fj* = fa ,

c - place, and alfo with a contrary fign, fo that _}'= * -f- f % & 1 And therefore (fquaring) f^ = * f 2', Sec. and (cubing) 3 3 Sec. which collected with a -h fy =4- fa , Terms contrary fign, make y= * * -f- -.^s, Sec. And therefore (fquaring) f_y* = 3 4 4 * 24 c - an d * Sec. and * & -4- = -|- , f/ rV (cubing) f_)' fa 4 Sec. which Terms collected with a make =: f.?. , contrary fign, 1 a f c - Therefore * # * &c. = ***-{- -j 54 ' ^ = f.s , y -?- y_y

5 H- f^ = -{- f^% &c. which Terms collected with a contrary fign, s #- Sec. And fo of the reft. make y= ***-{- -4-a , 3 5 40. Thus if we were to revert the Series y -f- f/ -f- ^V>' + TT-T^

c - of -f- T .fy T y' -h TTTS->''S ^ = ^, (where the Aggregate the Se- ries, or the unknown Number a, will reprefent the Arch of a Circle, is its is the whole Radius i, if right Sine reprefented by known confider'd as un- Number y,) or if we were to find the value of r,

confider'd as ; known, to be exprefsd by the Powers of a,, now known we may proceed thus :

o* 5 1 Lo*3 ,^1^ f ]_ ____'__ *?" -.{ ^9 o^C

3 5 i>>9 vVvATr* + T"T"3""a" 3 Sec. j

The Terms being difpofed as you fee here, we mall have jy==a, Sec. and therefore 5 3 Sec. which makes (cubing) fj = fa , y = * 3 Sec. fo that we lhall have 3 * fs , (cubing) + f_>' = -rV^'j Sec. and alfo 1 ? -a !r Sec. and with a -^y =-5 5 , collecting contrary fign, y and INFINITE SERIES, 221

' 7 - Hence \<-> * * &c. and r==* * -+-TT.T-'. &c = T TV > ' ^y 7 7 7 &<--. * &c- and = > and with a TV~ , TTT.v TTT~ collecting v = * * * &c< Anct lo on ' contrary tign, WT^"' this Extraction another of It" we fhould defire to perform by the the to be reduced foregoing Methods, that is, by fuppoiing; Equation 4 6 to this- i +- &<' x it form -+- j-_v* 4- -rV TTT.' + TTTT^'^ ;==;, may be fufHcient to let down the Praxis, as here follows.

I The 222 Method of FLUXIONS. moft but perhaps readily by fubftituting the Value of y now found in the given Equation, and thence determining the general Coeffi- cients as before. which the Root will be found to be By _)' =

* i i J 3 9 9 I 6 i ~ f, _ - I- I '_ i I fy7 3 Z.Oor Z^ *;7 *_2-~ 6 **-*-. 3 gt** TT T i TTTT' T^ T JTT^rT > 42. To refolve this affected Quadratick Equation, in which one m of the Coefficients is an infinite Series if we =. Ax &c. ; fuppofe y , we (hall have (by Subftitution) the Equation as it ftands here below. m the we {hall have aAx Then by applying Ruler, -+- 4 =o, whence m = 4, and A = ~ . The next Index, that the Ruler in its motion will arrive or parallel at, is m -+- I, 5; the next is m-\-2, or 6; &c. fo that the common difference of the Progrel- fion is and the i, Root may be reprefented by y = Ax* -{-Ex* -f- Cx 6 &c. which be extracted as , may here follows.

x" and INFINITE SERIES. 223

Methods it will be taught before, found y = a + x -f- _,_ g, &c. in the Now given Equation, becaufe the infinite Series a -\- x -+- * 1 vj v4 &c. is a Geometrical -4- ji 4- ^r y ProgrcfTion, and therefore is a to - as be Divifion if equal , may proved by ; we fubftitute this,

the will become - - Equation _>* y -f- ^ o> And if we extract the fquare-root in the ordinary way, it will give y r= a a ...4+ -x * -~^ or ^ exa Rootj And if this Radical ia~ lax be refolved, and then divided by this Denominator, the fame two Series will arife as before, for the two Roots of this Equation. And this fufficiently verifies the whole Procefs. 43. In Series that are very remarkable, and of general ufe, the Law of Continuation (if not obvious) mould be always affign'd, when that can be conveniently done; which renders a Series ftill more ufeful and elegant. This may commonly be difcover'd in the Compu- to the tation, by attending formation of the Coefficients, efpecially if we put Letters to reprefent them, and thereby keep them as general as may be, defcending to particulars by degrees. In the Logarithmic for 1 the Series, instance, z=y {y -f- .lys y4, &c. Law of Confecution is very obvious, fo that any Term, tho' ever fo remote, be may eafily aflign'd at pleafure. For if we put T to reprefent any Term indefinitely, whcfe order in the Series is exprefs'd by the natural Number then will + where the muft be , T = -j", Sign 4- or according as m is an odd or an even Number. So that the l 101 hundredth Term is is &c. In the L-y , the next -j-_J_^ , Reverie this 3 4 of Series, or y = z,~\- f-s* +- fa; -+- -y^a -4- TTo-^S &c. the Law of Continuation is thus. Let T reprefent any Term whofe order in is then is indefinitely, the Series exprefs'd by m ; m which Series in the Denominator mud: be con- T= ^p- , tinued to as many Terms as there are Units in m. Or if c ftands for the Coefficient of the Term immediately preceding, then is T=

_ -+- TTO-S' ToV^ 27 + I _r' &c. which the Relation between the Circular Arch _rT TT > (by be thus. a*nd its light Sine is exprefs'd,) the Law of Continuation will 224 *?%* Method of FLUXIONS,

If T be any Term of the Series, whofe order is exprefs'd by w, and _im I _ f ". before then , if c be the Coefficient immediately ; T = zm I x zm 2 3 A And in the Reverie of this Series, or z. = y -f- ^v -f- -?)' -f- T T y~ r 9 cc. the Law of Confecution will be thus. If T _f-_|^-T , repre- the of whole in the Series is and if fents any Term, Index place ;//, ""

Coefficient then . . ,,i-i_ c be the preceding j T = And 2/11 I X 2 2 the like of others. 44, 45, 46. If we would perform thefe Extractions after a more Indefinite and general manner, we may proceed thus. Let the given be -}- -\- rf.vv 2^x-=o, Equation a\v _ z -}- p, where Margin. Suppofe y _ ; y /; is to be conceived as a near Approximation to the Root y, and p as its fmall Supplement. When this is fubftituted, the Equation will ftand as it .v ? a ~P I ^'f" does here. Now becaufe and f> + + + f=~\ -^ " f * are both fmall the moil _i_ 3 quantities, '/ ^ = confiderable are at the be- /* a s quantities + + */. ^ * of the from ' ginning Equation, _ x whence they proceed gradually di- both downwards and towards the as minifliing, right-hand ; oug;ht to be when the Terms of an always fuppos'd, Equation are dilpos'd according to a double Arithmetical Scale. And becaufe inftead of

one unknown we have here introduced If and quantity _v, two, />, determine one of them as the we may b, neceffity of the Relblution iliall remove therefore the require. To moft confiderable Quantities of the and to leave out Equation, only a Supplemental Equation, whofe Root we 6* a*b is/>; may put -+- 2^ = o, which Equa- tion will determine b, and which therefore henceforward we are to look as known. And for 1 upon brevity fake, if we put a -+- 3^* c, we mail have the Equation in the Margin. Now here, becaufe the two initialTerms abx are the moft con fiderable of -f- cp -+- \. * ? aox+axp = the Equation, which might be removed, if * r for the nrft Approximation to^ we fiiould afiume and the ^ , refulting Supplemental Equation would be de-

lower ; therefore make = _f- and fubftitution we prefs'd p q, by

-flwll have this Equation following. Or and INFINITE SERIES. 225

if + 3*?* Or in this Equation, J7 . ^+ 3 * e make ^ = ^,

* i itwillaffume a > + **? ' ^ -+- =/; .e.1 V this form.

< 3 x* the to be Here becaufe Terms next removed are-f- cq -j-^-x^we may l and Sub- 3 put y = -x +- r, by * +

Or by another Method of Solution, if in this Equation we affumc A -}- Cx* Dx 3 &c. and fubflitute this in (as before) y = -\-Bx +- , the Equation, to determine the general Coefficients, we fhall have e . a\ c -ja' a c , . , A ---x x*-\-- - x"' &c. wherein A is the y = -t- ,.-- t

5 1 Root of the Equation AS -f. a* A art = o, and c = 3 A* + a*. 47. All Equations cannot be thus immediately refolved, or their Roots cannot always be exhibited by an Arithmetical Scale, whofe is one Root of the Quantities in the given Equation. But to perform the Analyfis it is fometimes required, that a new Symbol or Quantity fhould be introduced into the Equation, by the Powers of which the Root to be extracted may be exprefs'd in a converging Series. And the Relation between this new Symbol, and the Quantities of the Equation, mu ft be exhibited by another Equation. Thus if it were propofed to extradl the Root y of this Equation, 1 3 4 &c. it be in vain to x = fi-\-y , would 4/ -Hy}' ^_}' expedt, that it might be exprefs'd by the fimple Powers of either x or a. For the Series itfelf fuppofes, in order to its converging, that y is fmall Number lefs than but a are fome Unity ; x and under no fuch limitations. And therefore a Series, compofed of the afcendiag Powers of x, may be a diverging Series. It is therefore neceflary to introduce a new Symbol, which mall alfo be fmall, that a Series G g form'd

4- Method 226 Ibe of FLUXIONS, form'd of its Powers may converge to y. Now it is plain, that x tho' ever fo muft be each becaufe ami rf, great, always near other, difference &.c. is a (mall therefore their y y* } quantity. Aflame the Equation x a = z, and z will be a fmall quantity as required; and being introduced inftead of x a, will give z-=y y* -f- 4 &c. whofe Root extracted will be ^y* -ly , being y = z->t-^^ 3 s4 ^cc> as before. -j-y.2 4-TV > Thus if we had the i3 x 3 to find 48. Equation _> -f-j* -h_y = o, we have a Series for of the the Root y ; might y compofed afcending Powers of x, which would converge if x were a fmall quantity, lels than Unity, but would diverge in contrary Circumftances. Suppo- that x was to be a in this cafe the fing then known large Quantity ; Author's Expedient is this. Making & the Reciprocal of x, or fup-

l the - inftead of x he introduces z into the pofing Equation x= , Equation, by which means he obtains a converging Series, confining of the Powers of z afcending in the Numerators, that is in reality, of the Powers of x afcending in the Denominators. This he does, the Cafe he to himfelf but in the Method to keep within propofed ; here purfued, there is no occafion to have recourfe to this Expedient, it being an indifferent matter, whether the Powers of the converg- or ing quantity afcend in the Numerators the Denominators. in the 5- * Thus given Equation y> 4-j 4- jy ? '

m 1 m Ax A**'" A *"" Ax * &c.? king y = , 6cc.) 4- 4- , j v* .3

3 5 by applying the Ruler we mall have the exterior Terms A A* A' or = o, or m=. i, and A = i. Alfo the refulting Number Index next Term to which the Ruler will is 3. The approaches give 2/11, or 2; the laft m, or i. But 3, 2, i, make a defcending Progreffion, of which the common difference is i. Therefore the form of the Root will be Ax B Cx" 1 DAT"* &c. which we y = 4- 4- 4- , may thus extract.

+y '

1 3 l} therefore 1 Becaufe =A will be =A- , &c. >)' ,&c.it _y=x,&c.and _y which will make y* = * #*, &c. and (by Extraction)/ = * -^ * ~x &c. which with A* &c. Then (by fquaring)^* = } below, the 3 * * ~x &c. and therefore and changing Sign, makes j = } y and INFINITE SERIES. 227

* &c. Then * * &c. and * v = * }*~", ;* = .1, y = , the make * * * &c. which together, changing Sign, y> = -4- , &c. &c. and ;'=*** + TV*~S Then y- * * -f- 44*-', * and therefore _ Sec. and = * *""'> &c. ^ # ,. ^ _^_ _>' 75 ^.^-^ ~3 * * * * x c - &c. and = -f- -pT > & _v as this is accommodated to the cafe Now Series of convergency fo derive when x is a large Quantity, we may another Series from hence, which will be accommodated to the cafe when .v is a fmall direct us the quantity. For the Ruler will to external Terms Ax* 5 whence and i and the x = o, m= 3, A= ; refulting Num-

ber is The next Term will or 6 ; and the lair, is 3. give 2m, 3*77, will or 9. But 3, 6, 9 form an afcending Progremon, of which the 6 difference is Therefore v =Ax'' Ex 9 common 3. -+- -t-Cy , &c. will be the form of the Series in this cafe, which may be thus derived.

x> 6 # * 8 y = x -+- 4*" -f- , &c. -} 14*' x s 2X> 11 2x" 8 h +- 3AT 7A- , &c.

6 Ar 3 &c. it will &c. and therefore Here becaufe_)' = , bej>*=x , * x 6 &c. Then * &c. and v = , y- = 2x, ^5=^9, &c. and therefore y = * * H- .V, &c. Then y- = * * -j- 3^'*, 6cc. and 11 3 * &c. and * * * &c. y = S-^ ) therefore7= -f. o, The Expedient of the Ruler will indicate a third cafe of external m K Terms, which may be try'd alfo. For we may put A=x= -{- A*x*' m -f- Ax = o, whence m = o, and the Number refulting from the other Term is 3. Therefore 3 will be the common difference of the and the form of the be Progrelfion, Root will _y= A -{- Bx' -{- Cx 6 &c. But the A 5 A a A will , Equation -f- + = o, give A = o, which will reduce this to the former Series. And the other two Roots of the Equation will be impofftble. 3 If the of this r x"' Equation Example jy -f- y* -{- o be the factor mall have multiply'd by y i, we the Equation y* y x' or r+ * # * ... X~'y -f- = o, - y ) which when re- - t -- AT C=o, A'_) J 'folved, will only afford the fame Series for the Root y as before. l 1 49. This Equation \* x\y -h xy + 2f' zy -+- i = o, when reduced to the form of a double Arithmetical Scale, will ftand as in the Margin. C g 2 Now 2 2 8 The Method of FLUXIONS,

* 2>-f i Now the finl Cafe of external _> +2."- *- * Terms, fhewn by the Ruler, in _ L order for an Series, will _ afcending Or making y Axm> fc 1 i " make A'.**" _|- 2 A^ 2 A*" m M^ # zAljc or ;;/ where the j -+- i = o, = o ; tm ri Number is alfo o. The \4-s refulting; A 1* ^ i 11-1 fecond is zm -h i, or i ; the third zm-}- 2, or 2. Therefore the Arithmetical Progreffion will be o, difference is i and it will be i, 2, whofe common ; confequently 1 the v == A -f- Bx -+ Cx -+- Dx*, &c. But Equation A +- zA* of the firft "_ zA -H i o, which mould give the Value Coeffi- but Roots fo that cient, will fupply us with none impoffible ; y, an the Root of this Equation, cannot be exprefs'd by Arithmetical Scale whofe Root is x, or by an afcending Series that converges by the Powers of x, when x is a fmall quantity. to be firft the As for defcending Series, there are two cafes try'd ; 1 im l AT + whence zm or Ruler will give us A**** A = o, ^m = -f- 2, is the next will fff i and A=+ i. The Number arifing 4; }

next or 2 the next or i ; the laft o. be zm -f- i, or 3 ; the 2w, ; m, i for its common But the Arithmetical Progreffion 4, 3, 2, i, o, has^ Series will be difference, and therefore the form of the y = Ax -+ our ufual B 4- CAT"', 8cc. But to extract this Series by Method, it will be beft to reduce the Equation to this form, / x* 4- x -+- z

l thus : __ 2y~ 4- y~* = o, and then to proceed

~ 1 -_ i A: x x 2 -f- ZX~* | > &c-

I h A;~ , &C. 97' 77 c

Becaufe = x* x 2, &c. 'tis therefore (by Extradlion) jy l _ JL &c. Then = y x %x~ , (by Divifion) zy~* zx-*, * 1 &c. and * * * &c. fo that y = * * -f- 2*- , (by Extradion) y = l 1 -i-^" &c - and == &c. Then = * -f- ? y~* *"*, _j_ -VAT-% zy~ 1 united with a =* * * * &c.' which being contrary fign, make^ ~ '~I 3 c> ant^ therefore Extraction = ****- 4-i-s-v T A > & by y

. In the other cafe of a defcending Series we mall have the Equation 1 z or m i, and , i whence zm o, = A *""^ -f- =o, +- = will be hence is o ; the next zm r, A i . The Number arifing + Cjf and INFINITE SERIES. 229 or i the next or 2 and the laft or But ; 2//v, -, 4w, 4. the Numbers o, I, 2, 4, will be found in a defcending Arith- metical Progrelfion, the common difference of which is i. There- fore the form of the Root is y = A.x~' -+- Bx~'- -+- Cx~*, &c. and the Terms of the Equation mufl be thus difpofed for Refolution.

--- - 2X - I-f- A*"" 1

1 Here becaufe it is x &c. it will be Extraction of the y~- = , by l Square-root y~ =x, &c. and by finding the Reciprocal, y = x~', l &c. Then becaufe zy~ = 2X, &c. this with a contrary Sign, 1 and collected with x above, will make_y = * -{- x, &c. which I (by Extraction) makes y~ = * -+ i, &c. and by taking the Reci- * i 6cc. Then becaufe * &c. procal, /== ^^~ , zy~* = i, this will with a contrary fign, and collected with 2 above, make l y~* = * * i, 8ec. and therefore (by Extraction) y~ = * * 3 , &c. and = * * -f- y &cc. Then becaufe 4*"" (by Divifion) jy ^x~ l 2y~' = * * -}- -^"S ^ wiH be y = # * * %x~*y &c. and 1 1 " 1 J * * &c - and c - Then j- = * 4* , >'=*** -V*~ > & becaufe * * 1 &c. and &c. thefe 2y~' = * -f- -fA;- , /* = x~~-, z collected with a contrary fign will make y~ = * * * * V-v~% 4 &cc. and * * * * c - an^ ** ** y~' = -V*~S & 7 = -f- rlT*" * &c. Thefe are the two defcending Series, which may be derived for the Root of this Equation, and which will converge by the Powers of x, when it is a large quantity. But if x mould happen to be fmall, then in order to obtain a converging Series, we much change the Root of the Scale. As if it were known that x differs but little from 2, we may conveniently put z for that fmall difference, or aflame .v 2 &. we rmy the Equation = That is, irulead of x in this Equation fubftitute 2 + 2, and we mall have a new Equa- -- tion y* zy* ^zy* 2y + I = o, which will appear as in the Margin. Here 230 The Method of FLUXIONS,

Here to have an afcending Se- .'* * * 2; 4- ' ~ = l> zAz'" > ries, we muft put A+z*? ?*'

whence m and . + 1=0, = o, Or k * _ T-l__ -KT 1 1 i. Number hence , A = The A 4 A4., arifing is o ; the next is 2/H-i,

i laft or or ; and the 2m -f- 2, 2. But o, 1,2, are in an afcending Progreffion, whole common difference is i. Therefore the form of a the Series is A B;s -f- Cs -+- D;s 3 6cc. And if the Root y = -f- , y be extracted by any of the foregoing Methods, it will be found y =. 1 i 6cc. Alfo we hence find two Se- -+--iz -^s , may defcending ries, which would converge by the Root of the Scale z, if it were a large quantity. field for the 50, 51. Our Author has here opened a large Solution of thefe Equations, by Shewing, that the indeterminate quantity, or what we call the Root of the Scale, or the converging quantity, a and thence new Series may be changed great variety of ways, will be derived for the Root of die Equation, which in different circum- /tances will converge differently, fo that the moft commodious for the preSent occafion may always be chofe. And when one Series does not fufEciently converge, we may be able to change it for another that (hall converge falter. But that we may not be left to or uncertain interpretations of the indeterminate quantity, be obliged to make at random he us this Rule for Suppositions j gives finding initial Approximations, that may come at once pretty near the Root required, and therefore the Series will converge apace to it. Which Rule amounts to this: We are to find what quantities, when fubftituted for the indefinite Species in the propofed Equation, will make it divifibk by the radical Species, increaSed or diminished by another quantity, or by the radical Species alone. The fmall difference that will be found between any one of thofe quantities, and the indeterminate quantity of the Equation, may be introduced inftead of that indeterminate quantity, as a convenient Root of the Scale, by which the Series is to converge.

; Thus f the Equation propofed be y= -f- axy -f- cSy x* 2# = o, and if for x we here Substitute #, we Shall have the Terms 3 2a i which are divisible the be- _y -f- y 3^*, by y a, Quotient Therefore the ing y* -h ay -f- 3*2*. we may fuppofe, by foregoing Rule, that a x = & is but a fmall quantity, or inftead of x we may Substitute a z in the propoied Equation, which will then 5 become y* -f- 2a*y azy -\- y-z 3"* -t- z= 2# = o. A Series and INF j NITE SERIES, 231 Series derived from of the Powers hence, compofcd afcending of z9 mull converge faft, crtfcris parifats, becaule the Root of the Scale z is a (mail quantity. in the fame if Or Equation, for x we fubftitute a, we fliall have the Terms \* a 3 which are divifible , by y a, the Quo- tient being y* -4- ay -f- a*. Therefore we may fuppofe the difference between a and .v to be but little, or that -a x = z is a fmall and in quantity, therefore (lead of .v we may fubftitute its- equal a z in the given Equation. This will then become r 3 l z a* azy -f- T,a -f- 303* = o, where the Root y will con- the verge by Powers of the fmall quantity z. Or if for x we fubftitute we lhall have the Terms 3 za, _>' 6^ 3 which are divilible the a*? -4- , by y-\- za, Quotient being _)* x . Wherefore is but a fmall zay-i- 3rt we may fuppofe there difference between za and x, or that za x =z is a fmall and therefore infread of introduce its quantity ; x we may equal za into the will then z Equation, which become jv* a'-y azy -4- 6a> -f- iza*z -f- 6az* -f-s } = o. if for Laftly, x we fubftitute z~*a, we fliall have the Terms 3 which are divifible the Radical alone. jy z^a'-y -4-tf*y, by y, Species Wherefore we may fuppofe there is but a fmall difference between

or that - z^'a and x, z^a x = z is a fmall quantity ; and therefore inflead of x we may fubfthute its equal 2?a z, which will reduce the Equation to y* -f- i ^z x a"y azy -+- 3^4 x a^z 1 wherein the Series for -\- 3^2 x az -f- Z' = o, the Root y may converge by the Powers of the fmall quantity z. But the reafon of this Operation ftill remains to be inquired into, which I mall endeavour to explain from the prefent Example. In x 3 za* the indeterminate the Equation y~> -\- axy -f- a*y =o, its muft be of all quantity x, of own nature, fufceptible poffible at if it had would be fhew'd Values ; leaft, any limitations, they by impoflible Roots. Among other values, it will receive thefe, a, a, - - r za, z~*a, 6cc. in which cafes the Equation would become y* 1 +- o, ; a 1 = o, a -4- 6a* = o, 3 za*y 30* = 7 y* y _y &cc. as thefe admit 2^a*y -f- a'-y =. o, refpedtively. Now Equations as their divifible or of jull Roots, appears by being by y -f- another quantity, and the laft by y alone; fo that in the Refolution, the whole Equation (in thofe cafes), would be immediately exhaufted : And in other cafes, when x does not much recede from one of thofe Values, 232 The Method of FLUXIONS,

Values, the Equation would be nearly exhaufted. Therefore the introducing of z, which is the fmall difference between x and any thofe muft the and z itfelf mull pne of Values, deprefs Equation ; be a convenient quantity to be made the Root of the Scale, or the converging Quantity. I (hall give the Solution of one of the Equations of thefe Exam- 3 which mall be this, -f- -4- a* = o, or ples, _y azy y-z ^az*

5 ; it Here becaufe # , &c. will be &c. Then _>' = y = a, azy 6cc. which muft be wrote fl2r, again with a contrary fign, and united with 3^*2 above, to make y* = * 2a*z, &c. and 1 therefore ==. * &c - Then * 6cc. and y -f-*' s>' = -f- ^az ,

V = * * , &c. Then * " 3 a 3 ssz # * -f- .Is , &c. and = *** &c. and * * * _)'* J-.ss, y = 5 217Z <> OCC. , Author hints at other of a The many ways deriving variety of the fame as when Series from Equation ; we fuppofe the afore-men- difference z to be and from tion'd indefinitely great, that Suppofition we find Series, in which the Powers of z (hall afcend in the Deno- Cafe we have all minators. This along purfued indifcriminately with other in which the Powers of the the Cafe, converging quantity in the and therefore we afcend Numerators, need add nothing here about it. Another is, to affume for the Expedient converging quantity fome other quantity of the Equation, which then may be confider'd as indeterminate. So for we here, inftance, may change a into x, and x into a. Or to affume Relation at laftly, any pleafure, (fup- x ~- x az -f- bz\ = , x 3 between pofe = J^ 5 &c.) the indeterminate of the and quantity Equation x, the quantity z we would introduce into its which room, by new equivalent Equations be and then may form'd, their Roots may be extracted. And afterwards the value of z be x may exprefs'd by } by means of the affumed Equation.

52. The and INFINITE SERIES. 233 Author in a us a of 52, The here, fummary way, gives Rationale of his whole Method Extractions, proving a priori, that the Series continued in will thus form'd, and infinitum, then be the juft Roots of the propofed Equation. And if they are only continued to a of competent number Terms, (the more the better,) yet then will they be a very near Approximation to the juft and compleat Roots. For, when an Equation is propofed to be refolved, as near an Ap- is to the proach made Root, iuppofe y, as can be had in a lingle of the the and be*. Term, compofed quantities given by Equation ; caufe there is a Remainder, a Relidual or Secondary Equation is thence form'd, whole Root p is the Supplement to the Root of the given Equation, whatever that may be. Then as near an approach as a is made to /, can be done by a lingle Term, and new Relidual Equation is form'd from the Remainder, wherein the Root q is the Supplement to p. And by proceeding thus, the Relidual Equations are continually deprefs'd, and the Supplements grow perpetually lels and lefs, till the Terms at laft are lefs than any affignable quantities. We may illuftrate this by a familiar Example, taken from the ufual Method of Divifion of Decimal Fractions. At every Operation we put as large a Figure in the Quotient, as the Dividend and Divifor as leave the leaft will permit, fo to Remainder poflible. Then this Remainder (applies the place of a new Dividend, which we are to exhauft as far as can be done by one Figure, and therefore we put the greateft number we can for the next Figure of the Quotient, and thereby leave the leaft Remainder we can. And fo we go on, either till the whole Dividend is exhaufted, if that can bz done, or till we have obtain'd a fufficient Approximation in decimal places or the fame of that figures. And way Argumentation, proves our Au- thor's Method of Extraction, may ealily be apply'd to the other ways of Analylis that are here found. it is that tho' the 53, 54. Here feafonably obferved, indefinite Quantity fhould not be taken fo fmall, as to make the Series converge very faft, yet it would however converge to the true Root, tho' by more fteps and flower degrees. And this would obtain in even if it never fo proportion, were taken large, provided we do due Limits of not exceed the the Roots, which may be difcover'd, either from the given Equation, or from the Root when exhibited by a Series, or may be farther deduced and illuftrated by fome Geometrical Figure, to which the Equation is accommodated. So if the given Equation were yy =. ax xx, it is eafy to obferve, that neither^ nor x can be infinite, but they are both liable to H h flv.rul The Method of FLUXIONS, ieveral Limitations. For if x be fuppos'd infinite, the Term ax the would vaniih in refpedt of xx, which would give Value ofjyy on this Nor can x be for then the impoffible Supposition. negative; Value of yy would be negative, and therefore the Value of_y would If x then o allb is again become impoffible. = o, is^ = ; which one Limitation of both quantities. As yy is the difference between ax and xx, when that difference is greateft, then will yy, and con- this fequently^, be greateft alfo. But happens when x = a, as as from the Prob. alfo y = ftf, may appear following 3. And in is number of whether general, when y exprefs'd by any Terms, finite or infinite, it will then come to its Limit when the difference the affirmative and Terms as is greateft between negative j may ap- from the fame Problem. This laft will be a Limitation for pear yt but not for x. Laftly, when x = a, then_y = o; which will limit to be than both x and y. For if we fuppofe x greater a, the ne- over the and the gative Term will prevail affirmative, give Value of will make the Value of So that yy negative, which y impoffible. upon the whole, the Limitations of x in this Equation will be thefe, that it cannot be lefs than o, nor greater than a, but may be of any intermediate magnitude between thofe Limits. Now if we refolve this Equation, and find the Value of y in an ftill difcover the fame infinite Series, we may Limitations from thence. For from the Equation yy = ax xx, by extracting the 3. _5. fhall have ~ fquare-root, as before, we y =. a^ L za* Sa 1

1 ' X ** * 3 X - i - O TT c. that == d*x* into i , &c. Here , is, y i6a z for then x? would be an x cannot be negative ; impoffible quantity. for then the ~ Nor can x be greater than a ; converging quantity or the Root of the Scale by which the Series is exprefs'd, would be and the Series would and greater than Unity, confequently diverge,, between not converge as it ought to do. The Limit converging and and therefore o diverging will be found, by putting x=a, y = ; i in which cafe we fhall have the identical Numeral Series i = _' &c. of the fame nature with fome of thofe, which we _l_ ^ -if. r , have elfewhere taken notice of. So that we may take x of any intermediate Value between o and a, in order to have a converging fo much fafter Series. But the nearer it is taken to the Limit o, true Root and the nearer it is taken the Series will converge to the ; the flower. But it will to the Limit a, it will converge fo much however 4 and INFINITE SERIES. 235

'however converge, if A: be taken never fo little lefs than a. And by Analogy, a like Judgment is to be made in all other cafes. The Limits and other affe&ions of y are likewife difcoverable from this Series. When x = o, then y = o. When x is a nafcent quan- to be all but the rirft tity, or but juft beginning pofitive, the Terms will a and x. may be negledted, and y be a mean proportional between Alfo y = o, when the affirmative Term is equal to all the negative

- - u- -? x a. Terms.or when i= -f- , &c. that is, when = z 8a* ib3 ' f For then i = -f. _f_ _ &c. as above. will be a 4. rj Laftly, y Maximum when the difference between the affirmative Term and all the will be found negative Terms is greateft, which by Prob. 3. when x = ^a. Now the Figure or Curve that may be adapted to this Equation, and to this Series, and which will have the fame Limitations that they have, is the Circle ACD, whofe Diameter is AD = a, its Ab- /cifs AB = x, and its perpendicular Ordinate BC =.}' For as the Ordinate BC=^ is a mean proportional between the Segments of the Diameter AB rrn x and BD = a x, it will be yy ==. ax xx. And therefore the Ordi- nate BC will be the fore- = _y exprefs'd by going Series. But it is plain from the nature of the Circle, that the Abfcifs AB cannot be extended back- fo as to neither can it be forwards, become negative ; continued wards beyond the end of the Diameter D. And that at A and D, where the Diameter begins and ends, the Ordinate is nothing. And the greateft Ordinate is at the Center, or when AB = ^

SECT. VI. fo the 'Trqnfitton Method of Fluxions.

' "^HE learned and Author thus accom- 55. | fagacious having plifh'd one part of his deiign, which was, to teach the Method of converting all kinds of Algebraic Quantities into fimplc Terms, by reducing them to infinite Series : He now goes on to fhew the ufe and application of this Reduction, or of thefe Series, in the of is Method Fluxions, which indeed the principal defign of this Treadle. For this Method has fo near a connexion with, and dependence upon the foregoing, that it would be very lame and defective without it. He lays down the fundamental Principles of H h 2 this The Method of FLUXIONS, this Method in a very general and fcientiflck manner, deducing them from the received and known laws of local Motion. Nor is this inverting the natural order of Science, as Ibme have pretended, by introducing the Doctrine of Motion into pure Geometrical Spe- culations. For Geometrical and. Analytical Quantities are belt conceived as generated by local Motion; and their properties may as well be derived from them while they are generating, as when their generation is fuppos'd to be already accomplifh'd, in any other way. or curve is defcribed A right line, a line, by the motion of a point, a fmface by the motion of a line, a folid by the motion of a fur- an the rotation of a radius all face, angle by ; which motions we may conceive to be performed according to any ftated law, as occafion (hall require. Thefe generations of quantities we daily fee to obtain in rerum naturd, and is the manner the ancient Geometricians had often recourfe to, in confidering their production, and then de- their ducing properties from fuch adhial defcriptions. And by analogy, all other quantities, as well as thefe continued geometrical be as a quantities, may conceived generated by kind of motion or progrefs of the Mind. The Method of Fluxions then fuppofes quantities to be generated by local Motion, or fomething analogous thereto, tho' fuch generations indeed may not be eflentially neceflary to the nature of the thing fo generated. They might have an exiftence independent of thefe motions, and may be conceived as produced many other ways, and yet will be endued with the fame properties. But this conception, of their being now generated by local Motion, is a very fertile notion, and an exceeding ufeful artifice for discovering their pro- a to the for a and me- perties, and great help Mind clear, diftincl:, thodical perception of them. For local Motion fuppofes a notion of time, and time implies a fucceffion of Ideas. We eafily diflin- what and will in thefe guifh it into what was, is, what be, ge- fo confider nerations of quantities ; and we commodioufly thofe be too for our and things by parts, which would much faculties, extream difficult for the Mind to take in the whole together, without distributions. fuch artificial partitions and Our Author therefore makes this eafy fuppofition, that a Line may be conceived as now defcribing by a Point, which moves either or equably or inequably, either with an uniform motion, elfe according to any rate of continual Acceleration or Retardation. Velocity is a Mathematical Quantity, and like all fuch, it is fufceptible of be infinite gradations, may be intended or remitted, may increafed or and INFIN ITE SERIES. 237

or dlminifhfd in different parts of the fpace delcribed, according to an infinite variety of fluted Laws. Now it is plain, that the fpace thus defcribed, and the law of acceleration or retardation, (that is, the velocity at every point of time,) mufl have a mutual relation and muft determine each to each other, mutually other ; fo that the other one of them being affign'd, by neceflary inference may be it. this is derived from And therefore ftrictly a Geometrical Pro- blem, and capable of a full Determination. And all Geometrical 1 or Propoluions once demonftrated, duly investigated, may be fafely made ufe of, to derive other Proportions from them. This will divide the prefent Problem into two Cafes, according as either the is in to find the Space or Velocity affign'd, at any given time, order other. Arid this has given occasion to that diftin<5lion which has each lince obtain'd, of the dirctt and irrcerje Method of Fluxions, of which we fhall now confider apart. 56. In the direct Method the Problem is thus abftractedly pro- i or pofed. From the Space defer bed, being continually given, affumed, at to the the or being known any point of Time ajjigrid ; find Velocity of Motion at that Time. Now in equable Motions it is well known, that the Space defcribed is always as the Velocity and the Time of or the is as the de- defcription conjunclly ; Velocity directly Spice fcribed, and reciprocally as the Time of defcription. And even in inequable Motions, or fuch as are continually accelerated or retarded, according to fome ftated Law, if we take the Spaces and Times very fmall, they will make a near approach to the nature of equable Mo- the fmaller thole are taken. if tions ; and flill the nearer, But we be may fuppofe the Times and Spaces to indefinitely fmall, or if they are nafcent or evanefcent quantities, then we fhall have the Ve- little as that and locity in any infinitely Space, Space directly, as the therefore of all tempufculum inverlely. This property inequable Mo- us tions being thus deduced, will afford a medium for folving the as will be fhewn afterwards. So that the prefent Problem, Space defcribed being thus continually given, and the whole time of its at the end of that time will be thence de- defcription, the Velocity terminable. abflract Mechanical which 57. The general Problem, amounts to the lame as what is call'd the inverfe Method of Fluxions, will be this. From the Velocity of the Motion being continually given, to de- at Time termine the Space defcribed, any point of affign'd. For the Solution of which we fhall have the afTiflance of this Mechanical Theorem, that in inequable Motions, or when a Point defcribes a Line 2 <>8 *!} Method of FLUXIONS,

Line according to any rate of acceleration or retardation, the indefi- defcribed in little will be in nitely little Spare any indefinitely Time, ratio of the and the or will a compound Time Velocity ; thejpafiolum be as the velocity and the tempiijculum conjunctly. This being the Law of all equable Motions, when the Space and Time are any finite quantities, it will obtain allb in all inequable Motions, when the Space and Time are diminiih'd in infinitum. For by this means all inequable Motions are reduced, as it were, to equability. Hence the Time and the Velocity being continually known, the Space delcribed be alfo as will more from what follows. may known ; fully appear ThisTroblem, in all its cafes, will be capable of a juft determina- it in its full we mult it tion ; tho' taking extent, acknowledge to be a very difficult and operofe Problem. So that our Author had good reafon for calling it moleftijfimum & omnium difficilltmum pro- blema. fix his our Author illuftrates 58. To the Ideas of Reader, his x general Problems by a particular Example. If two Spaces and y are defcribed by two points in fuch manner, that the Space x being uniformly increafed, in the nature of Time, and its equable velocity the x and if the increafes in- being reprefented by Symbol ; Space y equably, but after fuch a rate, as that the Equation y =. xx ihall relation between thofe x always determine the Spaces j (or being will thence then the continually given, y be known ;) velocity of 2xx. the increafe of y fhall always be reprefented by That is, if the the of the increafe of fymbol y be put to reprefent velocity y, then will the Equation y =. zxx always obtain, as will be (hewn hereafter. Now from the given Equation y = xx, or from the relation of the the and or its Spaces y and x, (that is, Space Time, representative,) being continually given, the relation of the Velocities y=.2xx is found, or the relation of the Velocity y, by which the Space increafes, to the Velocity x, by which the reprefentative of the Time increales. And this is an inftance of the Solution of the firft general Problem, or of a particular Queftion in the direct Method of Fluxions. But -.vice versa, if the kit Equation y = 2xx were given, or if the Ve- is were locity y, by which the Space y defcribed, continually known if from the Time x being given, and its Velocity x , and from thence. we ihould derive the Equation y = xx, or the relation of the Space and Time : This would be an inftance of the Solution of the fecond or of a of the inverfe .general Problem, particular Queftion Method of Fluxions. And in analogy to this defcription of Spaces by moving points, our Author confiders all other quantities whatever as generated and INFINITE SERIES. 239 continual nerated and produced by augmentation, or by the perpetual acceffion and accretion of new particles of the fame kind. the 59. In fettling Laws of his Calculus of Fluxions, our Author very fkilfully and judicioufly difengages himfelf from all confideration of Time, as being a thing of too Phyfkal or Metaphyfical a nature to be admitted here, efpecially when there was no abfolute neceffity for it. For tho' all Motions, and Velocities of Motion, come to be or when they compared meafured, may feem neceflarily a notion of like all to include Time; yet Time, other quantities, may be reprefented by Lines and Symbols, as in the foregoing example, efpecially when we conceive them to increafe uniformly. And thefe reprefentatives or proxies of Time, which in fomc mea- fiire may be made the objects of Senfe, will anfwer the prefent purpofe as well as the thing itfclf. So that Time, in fome fenle, may be laid to be eliminated and excluded out of the inquiry. By this means the Problem is no longer Phyfical, but becomes much more fimple and Geometrical, as being wholly confined to the defcription of Lines and Spaces, with their comparative Velocities of increafe and decreafe. Now from the equable Flux of Time, which we to be the conceive generated by continual acceflion of new particles, or Moments, our Author has thought fit to call his Calculus the Method of Fluxions. 60, 6 1. Here the Author premifes fome Definitions, and other neceflary preliminaries to his Method. Thus Quantities, which in or any Problem Equation are fuppos'd to be fufceptible of continual increafe or he calls or decreafe, Fluents, flowing Quantities ; which call'd variable or are fometimes indeterminate quantities, becaufe they are capable of receiving an infinite number of particular values, in a regular order of fucceilion. The Velocities of the increafe or de- fuch are call'd creafe of quantities their Fluxions ; and quantities in the fame Problem, not liable to increafe or decreafe, or whofe Fluxions are nothing, are call'd conftant, given, invariable, and determinate diftindlion of is quantities. This quantities, when once made, care- obferved the fully through whole Problem, and infinuated by proper the firft Letters of Symbols. For the Alphabet are generally appro- priated for denoting conftant quantities, and the laffc Letters commonly lignify variable quantities, and the fame Letters, being pointed, the Fluxions of thofe variable repreient quantities or Fluents refpec- diftinction between is tivcly. This thefe quantities not altogether arbitrary, but has fome foundation in the nature of the thing, at leafl during the Solution of the prefent Problem. For the flowing or 7#* Method 24-O of FLUXIONS. be or variable quantities may conceived as now generating by Motion, or and the conftant invariable quantities as fome how o other al- in .ready generated. Thus any given Circle or Parabola, the Diame- or Parameter are conftant or but the ter lines, already generated ; Abfcifs, Ordinate, Area, Curve-line, &c. are flowing and variable quantities, becaufe they are to be underftood as now defcribing by diftinc- local Motion, while their properties are derived. Another tion of thefe quantities may be this. A conftant or given Irne in any Problem is tinea qtitzdam^ but an indeterminate line is linea qua-vis vel qutzcunque, becaufe it may admit of infinite values. Or laftiy, conftant quantities in a Problem are thofe, whole ratio to a common their is be but in variable Unit, of own kind, fuppos'd to known ; quantities that ratio cannot be known, becaufe it is varying perpetually. This diftinction of quantities however, into determinate and indeterminate, fubfifts no longer than the prefent Calculation requires; for as it is a diftinftion form'd by the Imagination only, for its own de- conveniency, it has a power of abolifhing it, and of converting terminate quantities into indeterminate, and vice versa, as occaiion of which fhall fee Inftances in what follows. In may require ; we a Problem, or Equation, theie may be any number of conftant quantities, but there muft be at leaft two that are flowing and indeter- all the reft minate ; for one cannot increafe or diminifh, while continue the fame. If there are more than two variable quantities in a Problem, their relation ought to be exhibited by more than one Equation.

ANNO- ( 241 )

ANNOTATIONS on Prob.i,

O R,

The relation ofthe flowing Quantities being given, to determine the relation of their Fluxions.

I. Fluxions t(f SECT. Concerning of the firft orcler^ and Jlnd their Equations.

HE Author having thus propofed his fundamental Pro-' blems in an abftra

: But that is to their of pofe owing great degree univerfality. We are to form, as it were, fo many different Tables for the Equation, as there are in flowing quantities it, by difpofing the Terms according to the Powers of each quantity, fo as that their Indices may' form an Arithmetical Progreflion. Then the Terms are to be mul- in the ' tiply'd each cafe, either by Progreflion of the Indices, or by the Terms of other Arithmetical any Progreflion, (which yet mould ' .have the fame common difference with the the Progreffion of Indices ;) I i as Tfo Method 242 of FLUXIONS. the Fluxion of that as alfo by Fluent, and then to be divided by the Fluent itfelf. La ft of all, thefe Terms are to be collected, according to their proper Signs, and to be made equal to nothing; which will be a new Equation, exhibiting the relation of the Fluxions. indeed is not fo This procefs fhort as the Method for taking Fluxions, be relent which he el fe where (to given p lyv) delivers, and which is follow'd but it makes fufficient commonly ; amends by the univer- of lality it, and by the great variety of Solutions which it will afford. For we may derive as many different Fluxional Equations from the lame given Equation, as we .(hall think fit to affume different Arith- metical Progreffions. .Yet all thefe Equations will agree in the main, and tho' differing in form, yet each will truly give the relation of the Fluxions, as will appear from the following Examples. 2. In the firft Example we are to take the Fluxions of the Equa- tion x> ax 1 where the -{- axy y"> = o, Terms are always over to one brought fide. Thefe Terms being difpofed according to the powers of the Fluent x, or being conlider'd as a Number ex- 1 is will - prefs'd by the Scale whofe Root x, iland thus x> ax -f- the Arithmetical ayx* y>x = o; and affuming Progrefiion 3, 2, is of the ], o, which here that Indices of x, and multiplying each Term by each refpedlively, we fhall have the Terms jx 3 zax- * which i or xx~ l H- ayx j again multiply'd by , , according to the will make 1 2axx Rule, ^xx -f- ayx. Then in the fame Equa- tion making the other Fluent/ the Root of the Scale, it will ftand 5 1 o and thus, _y -f- oy*-i- axy ax*y = ; affuming the Arith- - > metical Progreffion 3, 2, I, o, which alfo is the Progreffion of the Indices of y, and multiplying as before, we fhall have the Terms

; * -+- *, which -- or will 3_)' axy multiply'd by , yy~*, make a ax Tlien colle(^i n the the 1 3i>' -+ J- g Terms, Equation yxx o will zaxx + ayx tyy* -f- axj = give the required relation of the For if we refolve this Fluxions. Equation into an Analogy, we fhall 2 1 have x : : : ax zax in all y 3>' i^x -h ay -, which, the values that can x and y affume, will give the ratio of their Fluxions, or the of their increafe or comparative velocity decreafe, when they flow according to the given Equation. Or to find this ratio of the Fluxions more immediately, or the value of the Fraction 4' fewer by fteps, we may proceed thus. Write down the Fraction ? with the note of equality after it, and in the Numerator and INFINITE SERIES. 243 Numerator of the equivalent Fraction write the Terms of the Equa- to with their each tion, difpos'd according x, refpective figns ; be- the Index of ing multiply'd by x in that Term, (increafed or diminifh'd, if you pleafe, by any common Number,) as alib divided by .v. In the Denominator do the fame by the Terms, when dii- to the pofed according y, only changing figns. Thus in the pre- 1 fent x"' ax 3 (lull Equation -f- axy ;' = o, we have at once y i,x*2ax-\-av + ~ * J>* * ax * Let us now apply the Solution another way. The Equation x ; ax* -f- axy y* = o being order'd according to x as before, will be x 1 ax* 1 o and the Indices -(- ayx y*x =. ; fuppofing of x to be increas'd by an unit, or aifuming the Arithmetical Pro- -~~ ~ the greffion , t , , and Terms -j- ^ multiplying refpectively, we fhall have thefe 1 Terms ^.xx* ^axx -}- zayx y-xx- . Then the Terms to will become 3 1 ordering according /, they _)' -f- oy f -\-axy -i- x*y =.0; and fuppofing the Indices ofy to be diminifli'd ax*

an unit, or the Arithmetical Progreffion ^ L Si iJ, by afluming , ' ' .> y y y and the multiplying Terms refpecYively, we mall have thefe Terms 1 2yy* * * x*yy- + ax*yy~*. So that collecting the Terms, we lhall have 2 l 4.v.v* -^axx +- ayx y>xx~ zyy* x'>yy-' -+- ax*yy~* = o, for the Fluxional Equation required. Or the ratio * * . c ^1 T>1 -11 i y 4X T a -f-f- 2ay v'v . , . of the Fluxions will be - -. 3_J : _ = , which ratio x 2 J l l Z) * * -f-As, ax \< be found may immediately by applying the foregoing Rule. Or contrary-wife, if we multiply the Equation in the fir ft form v the ~ ? ~ flinll 1 by Progreffion , , , we have the Terms zxx } ^ axx * 1 -\-ytx\- . And if we multiply the Equation in the fc- y y cond form - ^ l - we fiiall have the , Terms by , 5 , 4^* *, ! 1 H- zcixy -+- x=j}~ cx-yy~'. Therefore collecting 'tis a.v.v ^v.v > 1 + rxx~> ^v}*+ 2axy-i-x>j}- ~fix y}-'~o. Or the ratio of the Fluxions will be which | = ^ ^ ~^:^^.,-r , might l.avc been found at once by the foregoing Rule. And in if the x"> - % -- general, Equation -ax axy y* o, in the form x- a\- ; ,v -f- = o, be multiply'd by the Terms " of this Arithmetical ;+ 3 "L+J. v .v ; ProgreffionO ) -v, JLr w\> 11;n the Terms m produce -\-y.\-~ m-+-2n>:x-{- m -\- icxt mj'xx-'-, I i 2 and e 244 ^)e Method of FLUXIONS, fame reduced to and if the Equation, the form y*-\- b; 'd _f- K\y= o, multiply by the Terms of this Arithmetical Pro- ax 1 " ^ w*^ ro(^ uce t^ie Terms grerTion Mjs 7, ~7~7' "^' P

l l * H- H- iaxy-\-nx~>yy-* nax yy~ . Then collecting the Terms, 1 i ilia 11 have m vx we -\- 3. ; -i- arfxv H- m-\-.iaxy my"'.\x~ w 1 * * l for the ~f- 3.X) 4-^-t- irftfy -f- nx*yy~~ nax*yy* = o, Fluxional Equation required. Or the ratio of the. Fluxions will be * -4- js I m 3*" -(- z* -j- m -j- ay m)$x . . 1 . . . - - : - ^- - : which have been = - -- ; might * r l n -j- D * ;? -j- I ax nx'j -f- nax^y found immediately from the given Equation, by the foregoing Rule. be determined Here the general Numbers m and n may pro lubitu, by which means we may obtain as many .Fluxional Equations as we And thus pleafe, which will all belong to the given Equation. we that is beft ac- may always find the fimpleft Expreffion, or which commodated to the prefent exigence. Thus if we make m = o, *'*-*"* + "> ;; mall , as found before. Or if and == o, we have 4 = a X 3j ax a n 11 l y 4* - : ; we make and n= i , we ihall have ^ = i, x = *- as before. Or if we make m=- i, and n = i, we fhall have ax A - +_> =. _ before. Or if we make m ?, - ' - J = J x fy- zax x*j '-\~axij

i n 11 l. V and n -r- ihall have - =. 7,' we =

Qthers _ th j s var i of Now etyJ of Solutions y -(- 3^4 ^axi will beget no ambiguity in the Conclusion, as poffibly might have been fufpected; for it is no other than what ought neceffarily to arife, from the different forms the given Equation may acquire, as will appear afterwards. If we confine ourfelves to the Progremon of the Indices, it will bring the Solution to the common Method of taking Fluxions, which our Author has taught elfewhere, and which, becaufe it is eafy and expeditious, and requires no certain order of the Terms, I mall here fubjoin. For every Term of the given Equation, fo many Terms mufr. be form'd in the Fluxional Equation, as there are flowing Quantities in that Term. And this muft be done, (i.) by multiplying the Term the Index of each contain'd in it. by flowing Quantity (2.) By it the itfelf dividing by quantity j and, (3.) by multiplying by its Fluxion. Thus in the foregoing Equation x> ax* -f- ayx y 3 3 the Fluxion to the Term .v 3 is or = o, belonging , ^x^x. The and INFINITE SERIES. 245

1 to ax is - - or The Fluxion belonging , zaxx. The

. avxv ayxx to is or Fluxion belonging ayx 1- , axy -f- ayx. And the

Fluxion to 3 is or that belonging / , y-y. So the of the whole or Fluxion Equation, the whole Fluxional Equation, a 1 is A- zaxx -f- -f- Thus the 3-v ayx ayx 3_>' _y=o. m Equation x will mxxm-* and the m =}', give =.y ; Equation x z," y, will m l t z" nxm for give mxx -f- zz"~ = y its Fluxional Equation. And the like of other Examples. If we take the Author's funple Example, in pag. 19, or the Equa- tion or rather x* is l y = xx, ay = o, that ayx x y = o, in order to find its moft Fluxional it be general Equation ; may perform'd by the Rule before given, fuppofing the Index of x to be encreas'd by m, and the Index of y by ;;. For then we {hall have

? "-'-g+-'* firft = _ For the Term of the diredtly x z ' n given nx y -|- \a the Index of Equation being ayx, this multiply'd by x increas'd by l that is and divided will for the firlt 7/7, by ;;z, by x, give mayx~ Term of the Numerator. Alfo the fecond Term being x*y, this multiply'd by the Index of A- increas'd by m, that is by w-f- 2, and 2X for the fecond Term of the divided by ,v, will give m -h Nu- merator. Again, the firft Term of the given Equation may be now Index of increas'd ,Y*J, which multiply'd by the y by n, that l l is divided will the nx for by ;;, and by r, give (changing fign) y~ the firft Term of the Denominator. Alib the fecond Term will then be which the Index increas'd cyx, multiply'd by of/ by ;/, that is divided will by n -f- i, and by y, give (changing the Sign) n -|- \a for die fecond Term of the Denominator, as found above. the Now from this general relation of Fluxions, we may deduce as ones as Thus if we make many particular we pleaie. ///= o, and fhall -- -- or 7/r=o, we have r , ay = 2xx, agreeable to our Solution in cited. Or if Author's the place before we make;= 2, 1 2TA 2tfl> . ., 1 n II 1 - aiid ;z= r, we lhall have = Or if we make = -7^7 -77

;/ - - - //v and we (hall have . = o, = i, = ^' = Or if V X j A" - -^-^ we make n o. and m-=. we fhall have . -1-, = 2, = m v a ,v before. All and other as which, innumerable cafes, may be eafilv* a fubftitution of Or we proved by equivalents. may prove it c: rally e 246 tf>e Method of FLUXIONS,

i I the it is a~ in the thus. Becaufe by given Equation , rally ~ y=x ,,. . /-i V mayx m-\-zx c r , , , of the ratio 4 f r mbmtute its and value = gA& -. _7^ri7 7 value, OT* - 2X /, 11 V V + 2X have ~- -= as above. we fhall = = x na n -+- i a a of the fecond is 3 3. The Equation Example 2j -f- x*y 2cysi -4- ^z* Z' = o, in which there are three flowing quantities y, x, and z, and therefore there muft be three operations, or three Tables form'd. Firft the to mufl be difpofe Terms according y, thus ; and the Terms of the Pro- 2j3 _j_ oja _{_ x*y z~>y= o, multiply by - 2CZ

1 1 1 2 1 i greffion xjj"" , ixj/y" , oxj/y"" , xj//- , relpeclively, (where the Coefficients are form'd by diminishing the Indices ofy by the com- will be * mon Number :,) and the refulting Terms qyy* * -f- &yy*.

theTerms to . Secondly ; difpofe according x, thus-> yx*--}-ox-t-2y">x=o 3 2cz

i r the Terms of the xxv~ . and multiply by ProgreiTion 2xxx~\ , l the Coefficients are the fame as the Indices of oy.xx~ , (\vhere x,) here is *. and the only refulting Term -+- 2yxx * Laftly, difpofe

the to thus ; z= Terms according z, -+-^y^ 2cyz--x*yz=o J -4- 2}" the I 2xzz~'f fx.zz~ ! and multiply by Progreffion 3xs~ , , , oxzz*, (where the Coefficients are alfo the fame as the Indices of z,} and the Terms will be ^zz* -h 6yzz-~-2cyz * . Then collecting all fhall have the Fluxional 1 thefe Terms together, we Equation fyrj + i ~3yy _|_ av,v.v yzz* -+- 6yzz 2cyz =. o. Here we have a notable inftance of our Author's dexterity, at for For in one of thefe finding expedients abbreviating. every Ope- rations fuch a Progreffion is chofe, as by multiplication will make deftrudtion of the Terms. which means he the greateft By arrives at the fhorteft Expreffion, that the nature of the Problem will allow. It we mould feck the Fluxions of this Equation by the ufaal me- if a flu thod, which is taught above, that is, we always me the Pro- fhall have oreffions of the Indices, we 6yy* -+ 2xxy -\- xy 2cyz o which has two Terms zcyz -+- ~}yz* ~r- dyzz 3'zz* = ; more' than the other form. And if the Progreffions of the Indices in each t (-j increas'd, cafe, by any common general Numbers, we may form the moil: general Expreilion for the Fluxional Equation, that the Problem will admit of. 3 4- and INFINITE SERIES. 247

4. On occafion of the laft Example, in which are three Fluents and their Fluxions, our Author makes an ufeful Obfervation, for the Reduction and compleat Determination of fuJi Equations, tho' it be derived from the Rules of the vulgar Algebra ; which matter may be confider'd thus. Every Equation, conlilling of two flowing or variable Quantities, is what correfponds to an indetcrmin'd Pro- blem, admitting of an infinite number of Anfwcrs. Therefore one of thofe quantities being afiumed at pleafure, or a particular value to the other will alfb being affign'd it, be compleatly determined. in the derived And Fluxional Equation from thence, thofe particular values being fubftituted, the Ratio of the Fluxions will be given in Numbers, in any particular cafe. And one of the Fluxions being taken for Unity, or of any determinate value, the value of the other may be exhibited by a Number, which will be a compleat Determi- nation. But if the given Equation involve three flowing or indeterminate of them muft be a/Turned to determine the third Quantities, two ; or, which is the fame thing, fome other Equation muft be either given or aflumed, involving fome or all the Fluents, in order to a compleat Determination. For then, by means of the two Equa- tions, one of the Fluents may be eliminated, which will bring this to the former cafe. Alfo two Fluxional Equations may be derived, involving the three Fluxions, by means of which one of them may be eliminated. And fo if the given Equation mould involve four Fluents, two other Equations fliould be either given or afTumed, in order to a compleat Determination. This will be fufficiently explain 'd by the two following Examples, which will alfo teach us how complicate Terms, fuch as compound Fractions and Surds, are to be managed in this Method. 6. Let the be a* x*/ a* x- = o, 5, given Equation y* of which we are to take the Fluxions. To the two Fluents y and third if x we may introduce a ;c, we aflume another Equation. Let that be z = x\/a*- x~, and we mall have the two Equations 1 y- a- & = o, and a'-x x* z* r= o. Then by the fore- their going Solution Fluxional Equations (at leaft in one cafe) will be 2jy z = o, and a*xx zxx> zz = o. Thefe two Fluen- tial Equations, and their Fluxional Equations, may be reduced to one Fluential and one Fluxional Equation, by the ufual methods

: that of Reduction is, we may eliminate z and z by fubftituting 1 1 1 1 their values yy aa and zyy. Then we fhall havej a x\/ a .v 248 fix Method of FLUXIONS,

" "

- "__ == o. , ! Q and = Or the J 2yy by taking away furds,

z 4 l z 4 'tis a"x ^ y* 4- 2a y rt = o, and then a*xx 2xx=. -- za* = o.

if the be x 5 - 7. Or given Equation ay* -f- x^^/ay -\-x*-

1 find its to the two = o, to corresponding Fluxional Equation ; ,v and others .z and flowing quantities y we may introduce two i', and thereby remove the Fraction and the Radical, if we affume the two Equations -~ = z, and x*~i/ay-t-xx=zv. Then we (hall T. ^ +_>' 1 have the three Equations x= ay -\- z i;=o, az-\-yz 6 x i<-~ which will the three by* r o, and ayx* -f- = o, give Fluxional Equations ^xx* zayy -+- z V = o, az +- yz -+- yz s = o, and -+- -f- 6xx 2vv= o. Thefeby,- "^byy* ay'x* ^.ayxx' known Methods of the common Algebra may be reduced to on& one Fluxional x and and their Fluential and Equation, iavolving y y Fluxions, as is required. 8. And by the fame Method we may take the Fluxions of Bino- mial or other Radicals, of any kind, any how involved or complicated with one another. As for inflance, if we were to find the

Fluxion oF-Vwf -\-*/aa xx, put it equal to y, or make ax-i~ xx=yy. Alfo make

tute its value vax-+-\/aa xx, we mall have the Fluxion re-

-J an A Jf , ax xx , 1 T^ ------: ..., - . And other quired = Exam-- y x many 7.1/fta xx yax + y aa xx a like- kind will be found in the of this Work. pies of fequel 1 12. In the is 9, 10, 1, Examp. 5. propofed Equation zz -{- axz 4 m which there are three variable and .)' => quantities x, y, z, and therefore the relation of the Fluxions will be 2zz -|- axz ~ === o. But as there wants another Fluential _j_ ax 4j/j-3 Equa- a tion, and thence another Fluxional Equation, to make compleat

if Fluxional were or < determination ; only another Equation given x and afTurned, we mould have the required relation of the Fluxions y,.. Suppofe and INFINITE SERIES. 249

this Fluxional were xx then Suppofe Equation i=.vv/^-v ; by fubftitution we mould have the Equation zz -f- ax x x^/ax xx

5 3 -f- axz = o, or the x :: : 2Z -4- ax x 4)7 Analogy :y 4_>' v/rftf .vx -f- rf;s, which can be reduced no farther, (or & cannot be eliminated,) till we have the Fluential Equation, from which the Fluxional Equation z=x\/ax xx is fuppos'd to be derived. And thus we may have the relation of the Fluxions, even in fuch cafes as \re have not, or perhaps cannot have, the relation of the Fluents. But tho' this Reduction may not perhaps be conveniently per- be forni'd Analytically, or by Calculation, yet it may poffibly per- of Curves form'd Geometrically, as it were, and by the Quadrature ; and as we may learn from our Author's preparatory Proportion, from the following general Conliderations. Let the right Line AC, perpendicular to the right Line AB, be conceived to move always its defcribe the line AB. parallel to itfelf, fo as that extremity A may and Let the point C be fixt, or always at the fame diftance from A, let another point move from A towards C, with a velocity any how does accelerated or retarded. The parallel motion of the line AC from not at all affect the progreffive motion of the point moving A towards C, but from a combination of thefe two independent defcribe the morions, it will Curve ADH ; while at the fame time the fixt point C will defcribe the right line CE, parallel to AB. Let the line AC be conceived to move thus, till it comes into the place BE, or BD. Then the line AC is conftant, and remains the fame, while the indefinite or flowing line becomes BD. Alfo the Areas defcribed at the fame time, ACEB and ADB, are likewife and their flowing quantities, velocities of defcription, or their muft be as their Fluxions, neceflarily refpeclive defcribing or and lines, Ordinates, BE BD. Let AC or BE be Linear Unity, or a conftant known right line, to which all the other lines are to or refer'd in be compared ; juft as Numbers, r.M other Numbers refer'd to or are tacitely i, to Numeral Unity, as being the fim- of all Numbers. And let the Area pleft ADB be fuppos'd to be apply 'd to BE, or Linear Unity, by which it will be reduced from the order of Surfaces to that of Lines ami let j the refulting line be call'd z. That make the Area if is, ADB = z x BE ; and AB be call'd x, then is the Area ACEB = x x BE. Therefore the K k Fluxions 1 Ibe Method 25 o" of FLUXIONS, Fluxions of thefe Areas will be z x BE and x x BE, which are as z and x. But the Fluxions of the Areas were found before to be as BD to BE. So that it is z : x : : ED : BE = i, or z = x x BD. the Fluxion of Area will be as the Consequently in any Curve, the Ordinate of the Curve, drawn into the Fluxion of the Abfcifs. Now to apply this to the prefent cafe. In the Fluxional Equa- tion before affumed z=x

of no finite know they will admit common meafure, or that their pro- be exhibited in rational tho' ever fo portion cannot Numbers, fmall, a feries of decimal or other but may by parts continued ad infinitum. In common Arithmetick we know, that the vulgar Fraction the decimal Fraction 1., and 0,666666, &c. continued ad infinitum^ are one and the fame thing j and therefore if we have a fcientifick notion of the one, we have likewife of the other. When I de- line icribe a right with my Pen, fuppofe of an Inch long, I defcribe firft one half of the line, then one half of the remainder, then one half of the next and fo on. I remainder, That is, actually run over all thofe infinite divifions and fubdivifions, before I have compleated the Line, tho' I do not attend to them, or cannot diftinguifh them. And by this I am indubitably certain, that this Series i JL _' of Fractions -f- _j_ -.-}- r> &c. continued ad infinitum, is pre-

to , cifely equal Unity. Euclid has demonflrated in his Elements, that the Circular Angle of Contact is lefs than any aflignable right- lined Angle, or, which is the fame thing, is an infinitely little Angle in comparifon with any finite Angle : And our Author fhews us fHll greater My fteries, about the infinite gradations of Angles of Con- tact. In Geometry we know, that Curves may continually approach towards their Arymptotes, and yet will not a&ually meet with them; till both are continued to an infinite diftance. We know likewife, that many of their included Areas, or Solids, will be but of a finite and determinable magnitude, even tho' their lengths mould be actually continued ad infinitum. We know that fome Spirals make infinite Circumvolutions about a Pole, or Center, and yet the whole Line, thus infinitely involved, is but of a finite, determinable, and aflignable length. The Methods of computing Logarithms fuppofe, that between any two given Numbers, an infinite number of mean Pro- portionals maybe interpofedj and without fome Notion of Infinity are their nature and properties hardly intelligible or difcoverable. in the And general, many of moft fublime and ufeful parts of knowledge muft be banifh'd out of the Mathematicks, if we are fo fcrupulous as to admit of no Speculations, in which a Notion will be included. as of Infinity neeeflarily We may therefore fafely admit of Moments, and the Principles upon which the Method

of Fluxions is here built, . as any of the fore-mention'd Specula- tions.

The nature and notion of Moments being thus eftablifli'd, we may pafs on to the afore -mcnticn'd Theorem, which is this. and INFINITE SERIES. 253

(contemporary) Moments offairing quantities are as the Velocities of or that as their if this be flowing increafing ; is, Fluxions. Now proved of Lines, it will equally obtain in all flowing quantities whatever, which may always be adequately rcprefented and expounded by Lines. But in equable Motions, the Times being given, the Spaces defcribed will be as the Velocities of Defcription, as is known in Mechanicks. And if this be true of any finite Spaces whatever, or of all Spaces in general, it muft alfo obtain in infinitely little Spaces, which we call Moments. And even in Mo- tions continually accelerated or retarded, the Motions in infinitely little Spaces, or Moments, muft degenerate into equability. So that the Velocities of increafe or decreafe, or the Fluxions, will be always as the contemporary Moments. Therefore the Ratio of the Fluxions of Quantities, and the Ratio of their contemporary Moments, will always be the fame, and may be ufed promifcuoufly for each other. 14. The next thing to be fettled is a convenient Notation for thefe Moments, by which they may be diftinguifh'd, reprefented, compared, and readily fuggefted to the Imagination. It has been for appointed already, that when x, y, z, v, &c. ftand variable or flowing quantities, then their Velocities of increafe, or their Fluxions, fhall be therefore will be reprefented by x, y, z, -j, &c. which proportional to the contemporary Moments. But as thefe are only or Velocities, magnitudes of another Species, they cannot be the Mo- ments themfelves, which we conceive as indefinitely little Spaces, or other here analogous quantities. We may therefore aptly introduce the Symbol o, not to ftand for abfolute nothing, as in Arith- but a rnetick, vanifhing Space or Qtiantity, which was juft now but finite, by continually decrealing, in order prefently to terminate in mere is become lefs than nothing, now any affignable Qinintify. And we have certainly a right fo to do. For if the notion is in- and no contradiction telligible, implies as was argued before, it may be furely infinuated by a Character appropriate to it. This is not the which would be to the aligning quantity, contrary hypothefis, but is a mark to only appointing reprefent it.- Then multiplying the' Fluxions the fhall have by vanishing quantity

arc &c. as o (land Spaces, reprefented by A-, y, z, -v, may for a. of the fame -vanishing quantity kind, and as x, y, z, v, &c. may for their Velocities of increafe or if ftand decreafe, (or, you pleafe, Numbers to thofe fpr proportional Velocities,) then may xo, yo, &c. zo, i-o, always denote their refpedive fynchronal Moments, .or momentary accefiions, and may be admitted into Computations this .accordingly. And we corne now to apply. muft now recourfe 15. We have to a very notable, ufeful, and extenfive property, belonging to. all Equations that involve flowing Which that in the Quantities. property is, progrefs of flowing, the Fluents will continually acquire new values, .by the accefilon of contemporary parts of thofe Fluents, and yet the Equation will be equally true in all thcfe, cafes. This is a neceffary refult from the Na- ture and Definition of variable Quantities. Confequently thefe Fluents be .rnay any .how increafed or diminifh'd by their contemporary or Increments Decrements ; which Fluents, fo increafed or diminiihed, may be fubflituted for the others in the Equation. As if an mould involve the Fluents x and with Equation _y, together any given quantities, and X and Y are fuppofed to be any of their contemporary Augments reflectively. Then in the given Equation we fubflitute x -f- X for and Y for and the may x, y -+- -y, yet Equa- tion will be or .the of the .good, equality Terms will be prefer ved. .So if X and Y were contemporary Decrements, inflead of x and fubflitute x y we might X and y Y reflectively. And as this inuft hold good of all contemporary Increments or Decrements whatever, whether finitely great or infinitely little, it will be true likewife of in flea d contemporary Moments. That is, of .r and y in fubflitute xo any Equation, we may .v-f- and y -t-jo, and yet we ihall flill have a good Equation. The tendency of this will appear from what immediately follows. 16. The Author's fingle Example is a kind of Induction, and the proof of this may ferve for all cafes. Let the Equation x s a.\* a 5 =o be as before, the variable + xy _>' given including quantities x and inftead of which we r, may fubflitute thefe quantities increas'd by their contemporary Moments, or x -- xo and Tlien we ihall have the x xo 3 y -i-yo respectively. Equation -+- | a a x x + AO i -f- a x x -|- xo x y -Jo~ T+"}f > = o. Thefe Terms .being expanded, and reduced to three orders or columns, according as the vanifhing quantity o is of none, one, or of more /limenfions, will ftand as in the Margin. and INFINITE SERIES; 255

3 * A *"* the Terms of the fir ft * ?*w +3 r 1 17, 18. Here + or remove or one column, deftroy _ fl .v i_ 2f,^ox _ all'l order, \ another, as being absolutely equal to no- +a.rj> + a.\iy -\-axjs- )>=o, thing by the given Equation. They be- _ )3 ^ _,;>,_ | ing therefore expunged, the remaining "j. j Terms may all be divided by the com- whatever it is. This all the Terms mon Multiplier - 3 o then fubftitution we flwll ax* 4- axy _y = ; by a have x X 3 a x .v X V 3 | 4- 4- | 4- 4-#x.v4-Xx/4-Y y | 1 z 1 o or in termini* .v 5 X -+- 3 ax = ; expanfis -f- 3X -f- 3xX X 3 a 4 2rfxX aX* -t- axy -\-

~^- which is now the ratio of the Moments. P = ax , And - y this is the fame ratio as that of the Fluxions, or it will be

1 - .V 2f>x--ai . . or a = zaxx 4- as wss 3_)' axy $x-x ayx, found before. In this way of arguing there is no aflumption made, but what is both of the iuflifiable by the received Methods ancient and modern Geometricians. We only defend from a general Proportion, which is in is undeniable, to a particular cafe which certainly included ir. That 256 The Method of FLUXIONS, the relation of the variable we thence That is, having Quantities, da-eddy deduce the relation or ratio of their contemporary Aug- and we deduce the relation or ratio of ments ; having this, directly thofc contemporary Augments when they are nafcent or evanefcent, or to be in a are juft beginning juft ceafing ; word, when they Mo- ments, or vanilliing Quantities. To evade this realbning, it ought to be proved, that no Quantities can be conceived lefs than afiign- able Quantities; that the Mind has not the privilege of conceiving as that the of Quantity perpetually diminiiLingy/w^w ; Conception an includes a a .vanishing Quantity, a Moment, Infinitefimal, &c. contradiction : In fhort, that Quantity is not (even mentally) divifible for that the mufb be reduced at ad infinitum ; to Controverfy laft. But I believe it will be a very difficult matter to extort this Principle from the Mathematicians of our days, who have been fo in are convinced of the long quiet poiTefTion of it, who indubitably evidence and. certainty of it, who continually and fuccefslully ap- to and ply it, arid who- are ready acknowledge the extreme fertility fo occalions. ufefulnefs of it, upon many important I but to account for thefe two cir- 19. Nothing remains, think, .cumilances, belonging to the Method of Fluxions, which our Au- here. Firft that the whofe thor briefly mentions given Equation, Fluxional Equation is to be found, may involve any number of has been and flowing quantities. This fufficiently proved already, we have feen feveral Examples of it. Secondly, that in taking Fluxions we need not always confine ourfelves to the progreffion of Arithmetical the Indices, but may affume infinite other Progreflions, deferve a little farther illu- as conveniency may require. This will refult ftration, tho' it is no other than what muft neceiTarily from the different forms, which any given Equation may afTume, in an 1 infinite Thus the x 3 ax -4- 3 = variety. Equation axy j o, m will become being multiply'd by the general quantity x y", #"+*>' m 1 m+l 1 m -r- ax -$- -h ax x = o, which is virtually the fame y" y"'t' y"~^^ Equation as it was before, tho' it may aiTume infinite forms, accor- and n. And if we take the ding as we pleafe to interpret m Fluxions of this Equation, in the ufual way, we mall have m m n~^ m i 1 nax + y* -j- nx^rty}*- m -+- zaxx ^y" ^yy -f- I a l ! B m~ ''* n irf.Y" mxx n -f- + 'j/)- y we mail have m .5= o. Now if we divide this again by x"}", l 4- nx*j>y~* m -f- 2axx nax*yy~~ -+- m -+- laxy 4- n-\- \axy n a which is the fame as /xx~*y* -f- 3j/y ?= o, general Equation was derived before. And the like may be underftood of all other SECT. Examples. and INFINITE SERIES. 257

SECT. II. Concerning Fluxions of fuperior orders^ and the method of deriving their Equations.

this our Treatifc Author confiders only fir ft Fluxions, and has not fit to extend his IN thought Method to fuperior orders, as not di- within his rectly foiling prefent purpofe. For tho' he here purfues which Speculations require the ufe of fecond Fluxions, or higher orders, yet he has very artfully contrived to reduce them to firft and to avoid the Fluxions, necefTity of introducing Fluxions of fuperior orders. In his other excellent Works of this kind, which have been publifh'd by himfelf, he makes exprefs mention of them, their nature he difcovers and properties, and gives Rules for deriving their Equations. Therefore that this Work may be the more fer- viceable to Learners, and may fulfil the defign of being an Inftitu- I mall here make fome the tion, inquiry into nature of fuperior Fluxions, and give fome Rules for finding their Equations. And afterwards, in its proper place, I mail endeavour to (hew fomething of their application and ufe. Now as the Fluxions of quantities which have been hitherto con- or their fider'd, comparative Velocities of increafe and decreafe, are and of their own themfelves, nature, variable and flowing quantities and as fuch are themfelves alfo, capable of perpetual increafe and de- or of acceleration and retardation crea&, perpetual ; they may be treated as other flowing quantities, and the relation of their Fluxions be and may inquired difcover'd. In order to which we will adopt our Author's Notation already publifh'd, in which we are to con- that as &c. have their Fluxions fo ceive, x, y, z, #, j, z., &c. thefe likewife have their Fluxions x, /, z,&c.which are the fecond Fluxions of &c. And thefe x, v, z, again, being ftill variable quantities, have j. their Fluxions denoted by x, y, z, &c. which are the third Fluxions of &c. And thefe x, y, z, again, being ftill flowing quantities, have their Fluxions x, /, z, &c. which are the fourth Fluxions of &c. And fo we x, y, z, may proceed to fuperior orders, as far as there mall be occafion. Then, when any Equation is propofed, con- of futing variable quantities, as the relation of its Fluxions may be found what has been before fo by taught ; by repeating only the fame and operation, confidering the Fluxions as flowing Quantities^ the L 1 relation Method 258 The of FLUXIONS, relation of the fecond Fluxions may be found. And the like for all of Fluxions. higher orders Thus if we have the Equation y* ax = o, in which are the two Fluents y and x, we fhall have the firft Fluxional Equation zyy - as ax o. And here, we have the three Fluents j>, y, and x, if we take the Fluxions again, we fhall have the fecond Fluxional o. And as there are four Equation zyy -+- zy* ax= here, Fluents if we take the Fluxions we fhall have the y, y, y, and x, again, .. . third Fluxional -f- ax = or Equation zyy + zyy ^.yy o, zyy 4-

as there are five Fluents bjy ax = o. And here, y, y, y, y, and x, if we take the Fluxions again, we fhall have the fourth Fluxional

l or Equation zyy + zyy -f- 6yy -+- 6y ax = o, zyy -+- Syy -f- 6y*

there fix ax = o. And here, as are Fluents y, y, y, y, y, and xy

if we take the Fluxions again, we fhall have zyy + zyy -f- 8yy -{-

i ax for the fyy _j_ i zyy ax = o, or zyy +- oyy -f- zoyy = o, to the fifth Fluxional Equation. And fo on fixth, feventh, 6cc. Now the Demonftration of this will proceed much after the manner as our Author's Demonftration of firft Fluxions, and is indeed included in it. For in the ax o virtually given Equation^* = } at the fame time and if we fuppofe y and x to become y -f- yo x-)- xo, if and xo to denote the Moments (that is, we fuppofe yo fynchronal Fluents and then fubftitution we fhall have * of the y x,) by ~y +yo\ 1 a x x -f- xo = o, or in termini* expanjis, y -f- zyyo -+-y*o* ax 1 1 1 o. ax ^ and divi- axo = Where expunging y = o, andj/ , it will be ax o for the firft fluxional ding the reft by o, zyy = in this if we the Equation. Now Equation, fuppofe fynchronal and to and xo Moments of the Fluents y, y, x, beyo} yot refpedively ; we fubftitute and xo in for thofe Fluents may y -f-jj/o, y -+-yo, x+ the kft Equation, and it will become zy-t-zyoxy-l-yo axx + xo

r. ax axo o, or expanding, zyy -f- zyyo +- zyyo -+- zy'yoo = o. Here becaufe zyy ax= o by the given Equation, and becaufe vanishes divide the reft and we fhall have + zy'yoo ; by o, zy* zyy ax = o for the fecond fluxional Equation. Again in this Equa- the of the Fluents tion, if we fuppofe Synchronal Moments y, y t

and to be and xo ; for thofe Fluents y, xt yo, yo, yo, refpedively we and INFINITE SERIES. 259

a^id x xo in the lad we may fubftitute y+yo, y + yo, y-t-yo, +-

.. a j become \ Equation, and it will 2x7 -\-yo +- zy -+- 2yo x y -f- yo

l a x x _j_ xo o, or expanding and collecting, 2j* + 6yyo -t- 2y*o

1 2 ax axo = o. But here becaufe s _}_ yy -+- 2yyo -t- s;^ 2j' the laft the reft _l_ rfx = o ; by o, and 2/_y by Equation dividing expunging all the Terms in which o will ftill be found, we fliall o for the third fluxional have 6yy -+- 2yy ax = Equation. And in like manner for all other orders of Fluxions, and for all other Examples. Q^ E. D. To illuftrate the method of rinding fuperior Fluxions by another 5 3- Example, let us take our Author's Equation # ax -{-axy y> = o, in which he has found the fimpleft relation of the Fluxions a o. Here to be 3x^ zaxx -h axy +- axy 3^/7* = we have the and the fame Rules the flowing quantities x, y, x, y ; by Fluxion of i this Equation, when contracled, will be 3#w + 6x*x 2axx

s !L zax* H- -+- -\- = o. And in this axy 2axy axy 3vy 6jf y Equa- tion we have the flowing quantities x, y, x,y, x, y, fo that taking the Fluxions again by the fame Rules, we fhall have the Equation,

l 6x 3 6axx when contracted, ^xx -f- iSxxx -{- 2axx -f- axy -f-

s i o. %axy -+- T,a.\y -f. axy 3yy* fyyy 6y = And as in this there are found the Equation flowing quantities x, y, x, yy x, y,

x, y, we might proceed in like manner to find the relations of the fourth Fluxions to this all belonging Equation, and the following orders of Fluxions. And here it may not be amifs to obferve, that as the propofed Equation expreffes the conflant -elation of the variable quantities x and and as the firft fluxional the y -, Equation exprefles conftant re- variable finite i.nd lation of the (but alTignable) quantities x and y, which denote the comparative Velocity of increafe or decreale of x and y in the propcfed Equation : So the fecond fluxional Equation will the conftant relation of the variable finite and exprefs (but aflig- x and which denote the of nable) quantities y y comparative Velocity the increafe or decreafe ot .v and_y in the foregoing Equation. And in the third fluxional Equation we have the conftant relation ot the variable

(but finite and aflignable) quantities .v and r, which will denote the L 1 2 com- 260 The Method of FLUXIONS,

of the increafe or decreafe of "x and in comparative Velocity "y the foregoing Equation. And fo on for ever. Here the Velocity of a Velocity, however uncouth it may found, will be no abfurd Idea but on the when rightly conceived, contrary will be a very rational and Notion. If there be fuch intelligible a thing as Motion any how continually accelerated, that continual Acceleration will be the Ve-

of a ; and as that variation be locity Velocity may continually va- that accelerated or ried, is, retarded, there will 'be in nature, or at leafl in the the of a of a Understanding, Velocity Velocity Velocity. Or in other the Notion words, offecond, third, and higher Fluxions, muft be admitted as found and to genuine. But proceed : We much abbreviate the may Equations now derived, by the known Laws of From the 1 Analyticks. given Equation x* ax -+- ax = ^ere is found a new y y"' Equation, wherein, becaufe of two new x and are at Symbols y introduced, we liberty to aflume another belides this now in Equation, found, order to a jufl De- termination. For we fimplicity-fake may make x Unity, or any other conftant ; that we x to quantity is, may fuppofe flow equably, and therefore its is uniform. Make therefore x i Velocity = } and the firft fluxional will become Equation 3^* 2ax -+- ay + axy 1 o. So in the a 3j)/)' = -f- 6x*x 2axx Equation 3x.v 2ax* -+. axy -i- zaxy -h axy 3 vj* 6y\y = o there are four new Sym- bols and and introduced, x, y, x, r, therefore we may afiume two other which the congruous Equations, together with two now found, will amount to a compleat Determination. Thus if for the fake of make one to fimplicity we be x = i, the other will' be neceflarily .v and thefe =o ; being fubftituted, will reduce the fecond fluxionaj

to 6x 2.0. Equation this, -f- iay -f- axy ^yy- 6y*y o. And thus in the next wherein there Equation, are fix new Symbols x, x, x, ; befides the three now }', y, y Equations found, we may take and thence x= i, x=o, x= o, which will reduce it to 6 -f- $ay -+- axy yy* i $yyy 6f> == o. And the like of Equations of fucceeding orders. But all thefe Reductions and Abbreviations will be beft made as the are derived. Thus the Equations propofed Equation being x~> ax* + axy y= = o, taking the Fluxions, and at the fame time making x= i, (and confequently x, x, &c. =o,) we (hall have zax 3** + ay + axy zyy* = o. And taking the Fluxions again. and INFINITE SERIES. 261

will be it 6x 20. i again, -f- zay +- axy 3yy* 6y*y = o.

And the Fluxions it will be taking again, 6 -f- $d'y -+- axy %yy* = o. And the Fluxions it will be 6y*> taking again,

4 2 1 = o. And fo as axy 3^ 4-yy'y i%y y 3677* on, far as there is occafion. But now for the clearer apprehenfion of thefe feveral orders of Fluxions, I (hall endeavour to illuftrate them by a Geometrical to a Figure, adapted iimple and a particular cafe. Let us allume 1 the Equation y r=ax, otyzs=ia*x*, which will therefore belong to whole is Abfcifs the Parabola ABC, Parameter AP = tf, AD = x, Ordinate LD where is a at the Vertex A. and =y ; AP Tangent Then taking the Fluxions, we fhall have y = yaPsve~~*. And fuppofing the Parabola to be defcribed by the equable motion of the Ordinate upon the Abfcifs, that equable Velocity may be expounded the Line or x a. by given Parameter a, that is, we may put = Then

~ "? -?- will us \t\v]\ibey=(a*x *= = = ) ', which give 2X ' zxk 2X this Conftrudtion. Make x (AD) : y (BD) :: a (|AP) : DG =

= y, and the Line DG will therefore zx J reprefent the Fluxion of y or BD. And if this be done every where upon AE, (or if the Ordinate DG be fuppos'd to move upon AE with a parallel motion,) a Curve GH will be conftiucted or delcribed, whofe Ordi- nates will every where expound the Fluxions of the correfponding Ordinates of the Pa- rabola ABC. This Curve will be one of the Hyperbola's between the Afymptotes 3. and AP for its -11 AE ; Equation isjx= ,

Or yy = . " from the Again, = , or 2 Equation y *y = ay, by taking the Fluxions and x again, putting =a as before, we fhall have

zay -{- = j where the 2xy=aj,ory J negative fign {hews only, is to be confider'd rather as a that_y retardation than an acceleration, or an acceleration the contrary way. Now this will give us the following 202 ?2* Method of FLUXIONS,

following Conftruaion. Make x (AD) : y (DG) : : \a (iAP) ; the Line will DI = y, and DI therefore reprefent the Fluxion of DG, or of j, and therefore the fecond Fluxion of BD, or of/. And if this be done every where upon AE, a Curve IK will be comlructed, whofe Ordinates will always expound the fecond Fluxions of the correfponding Ordinates of the Parabola ABC. This Curve likewife will be one of the Hyperbola's, for its Equation is y = * /Jy fl* G. a* ^ 1 6*5 v from the ^- Again, Equation = , or = t y 2xy ay'

the mail ^by taking Fluxions we have 2ay zxy =: ay., or

which will us this Conftrudlion. : ~y=~ , give Make x (AD) y (DI) :: \a (|AP) : DL=y, and the Line DL will therefore fecond reprefent the Fluxion of DI, or of y, the Fluxion of DG, of or if this be or of y, and the third Fluxion BD, of^. And done every where upon AE, a Curve LM will be conflructed, whofe Ordinates will always expound the third Fluxions of the correfponding Ordinates of the Parabola ABC. This Curve will be an Hyper-

" and its will be '=-1 or bola, Equation y=. ; , yy= 64*"*

And fo we might proceed to conftrucl Curves, the Ordinates of which (in the prefent Example) would expound or reprefent the fourth, fifth, and other orders of Fluxions. We might likewife proceed in a retrograde order, to find the. Curves whofe Ordinates mall reprefent the Fluents of any of thefe 3. when As if we had Fluxions, given. = , La*xx~* or if y = }

the Curve GH were given ; by taking the Fluents, (as will be

taught in the next Problem,) it would be y = ^-r = (a^x*= )

- which will us this Conftruction. Make : , give \a (|AP) 2 .v :: : DB and the Line will (AD) y (DG) = -J , DB reprefent the Fluent of or if this DG, of y. And be done every where upon the Line AE, a Curve AB will be con ftru died, whofe Ordinates will always expound the Fluents of the correfponding Ordinates of the Curve GH. This Curve will be the common Parabola, whofe i Parameter and INFINITE SERIES. 263

Line is Parameter is the AP = a. For its Equation y = a*x'* t or yy=ax. So if we had the Parabola ABC, we might conceive its Ordinates of which the to reprefent Fluxions, correfponding Ordinates of fome other Curve, fiippofe QR, would reprefent the Fluents.

To find which Curve, put y for the Fluent of y, y for the Fluent Iff / n I .. .: &c. &c. &c. be a Series of of y, (That is, let, _/, y, y, /, j/, y, y, Terms proceeding both ways indefinitely, of which every fucceed- the Fluxion of the and vice versa ing Term reprefents preceding, ; according to a Notation of our Author's, deliver'd elfewhere.) Then becaufe it the Fluents it is_y = (div*=

' ' .x % \ Z 2f!i! will be J2. ; which will us this Con- y = [, = = ) give W 3 y 3* ftrudion. Make $a (|AP) : ft (AD) :: y (BD) : -^ =y = DQ^ Line the Fluent of and the DQ^will reprefent DB, or ofy. And if the fame be done at every point of the Line AE, a Curve QR will be form'd, the Ordinates of which will always expound the Fluents of the correfponding Ordinates of the Parabola ABC. This Curve alfo will be a Parabola, but of a higher order, the Equation 3. I I * which or . of is^= , yy = 3^

zx 2 3ilJL \ ^fl . Again, becaufe y = f ~ == = taking the Flu- l \ $a? $a v. a J ->.a.'- " 7 / "* i ents it will be JfL.sff! which will us y=( |x"= W , give this

Conftruaion. Make : x : : : | (|AP) (AD) y ( DQ^J = _y / = DS, and the Line DS will the Fluent of or reprefent DQ^, of_y. And if the fame be done at every point of the Line AE, a Curve ST will thereby be form'd, the Ordinates of which will expound the Fluents of the correfponding Ordinates of the Curve QR. This // i //// Curve will be a Parabola, whofe is 1^1 or Equation jy= , yy =

. And fo we on as far as we ^-. might go pleafe, Laftlv, 264 The Method of FLUXIONS, the Laftly, if we conceive DB, common Ordinate of all thefe to be Curves, any where thus conftrucled upon AD, that is, to be divided in the 6cc. from whence to thus points S, Q^ B, G, I, L, are drawn 6cc. to AP Ss, Qtf, B^, Gg, I/, L/, parallel AE ; and if this Ordinate be farther conceived to move either backwards or forwards upon AE, with an equable Velocity, (reprefented by AP = tf = x,) and as it defcribes thefe Curves, to carry the aforefaid Parallels with it in its motion : Then the along points s, q, b,g, &c. will in fuch a in the i, /, likewife move manner, Line AP, as that the Velocity of each point will be reprefented by the diflance of the next from the point A. Thus the Velocity of s will be re- of b prefented by Aq, the Velocity of q by A, by Ag, of g by A/, of / &c. Or in other will be the Fluxion of A.S by A/, words, Aq ; Al> will be the Fluxion of or the fecond Fluxion of As Ag, ; Ag will be the Fluxion of Ab, or the fecond Fluxion of Aq, or the third. Fluxion will Fluxion of or the of As ; Ai be the Ag, fecond Fluxion of or the of or the fourth Ah, third Fluxion Aq, Fluxion of As ; and fo on. Now in this inftance the feveral orders of Fluxions, or Velocities, are not only expounded by their Proxies and Reprefen- tatives, but alfo are themfelves actually exhibited, as far as may be done by Geometrical Figures. And the like obtains wherever elfe we make a mews the relative beginning ; which fufficiently nature of all thefe orders of Fluxions and Fluents, and that they differ from each other and in the manner of by mere relation only, conceiving. And in has obferved from this general, what been Example, may be eafily accommodated to any other cafes whatfoever. Or thefe different orders of Fluents and Fluxions may be thus explain'd abftractedly and Analytically, without the afliftance of Curve- lines, by the following general Example. Let any conflant and known quantity be denoted by a, and let a" be any given Power n or Root of the lame. And let x be the like Power or Root of m m the variable x. a : : : : and indefinite quantity Make x a y, or m

l ~m m a . alfo will be an indefinite y = = x Here y quantity, ^a

which will as foon as the value of is become known x affign'd. l ~m m~ 1 Then the it will be ma xx and taking Fluxions, y = ; fuppofing x to flow or increafe uniformly, and making its constant m m Velocity or Fluxion x = a, it will be y = ma* x -*. Here if

1 n m for a x we write its value it will be = that is, x : y, y ,

ma : : : So a y y. that y will be alfo known and affignable Quan- tity, and I N FINITE SERIES. 265

x therefore is the tity, whenever (and y) affign'd. Then taking we mall 1- Fluxions again, have^=wxw ia* "xx"- = ;;; x 1 m m~ l ; irtS"""^*"" ; or for ma"-~ x its it writing value y, will be ta - v ~~ : : : , that x m la : will y = x is, y y. So that y become a known when x therefore is quantity, (and y and y) affign'd.

Then the Fluxions we fhall taking again, have y = m x m i x m m ?.a*-m\ or -*, that x : m za :: : y=.-^~ , is, y y where alfo will be is y known, when x given. And taking the Fluxions again, we fhall have y = rnx m i x m 2 x m

- - ; that : :: : So that will = is, x m 30 y /. y alfo be whenever x is known, given. And from this Inductipn we may in conclude general, that if the order of Fluxions be denoted by any number or if n integer ?/, be put for the number of points over the n ^_____ ll-l-i it Letter t will be x : na : : : or y always m y y ; from the Fluxion of order any being given, the Fluxion of the next immediate order may be hence found. ______"+t n

Or we thus invert the : may m na x : : : proportion y y} and then from the Fluxion given, we fhall find its next immediate Fluent. As if = 'tis m za : A; : : : - 2, y y. If n i

: 'tis m \a x :: : If 72 'tis : = ma : : : y y. 0, x y y. And the if fame n== 'tis ia : obferving analogy, i, m-- x :: y :

is 1 ; where for the Fluent or for with a y y put of; , y negative point. l -m m 1 1 And here becaufe x it will be : :: -" y=.a , m 4- la x a *" : ' I l ~V+ v"1-*- 1 or = : which alfo y, y _ = ^ may thus appear. Be- m-\-\a m-\-\a

caufe = {a*-<"x*> __Zj__T il y = , the =) taking Fluents, (fee the

/ m-f,, next it will be - Problem,) = . if y ^ m Again, we make = 2,

I I II . -- v * +... : x ; .. tism-{-2a :: or .. y y} y = = - . For a M m becaufe 266 The Method of FLUXIONS,

.f - x - == the Fluents becaufe y = m , takingw it will be

. if make -. we _ m+ i Again, = 3, 'tis m -|-

And fo for l"~'~* m l X x -t-3 + m -\- 2 j+3a all other fuperior orders of Fluents. this fuffice in And may general, to mew the comparative nature of and properties thefe feveral orders of Fluxions and Fluents, and the to teach operations by which they are produced, or to find their refpeftive fluxional Equations. As to the ufes they may be apply 'd when that will to, found, come more properly to be confider'd in another place.

SECT. III. Tfte Geometrical and Mechanical Elements of Fluxions,

foregoing- Principles of the Doftrine of Fluxions being THEchiefly abftradted and Analytical I mail here endeavour, after a general manner, to (hew fomething analogous to them in Geo- a.nd Mechanicks which metry ; by they may become, not only the of the objeft Underftanding, and of the Imagination, (which will only prove their poffible exiftence,) but even of Senfe too, by making them adually to exift in a vifible and fenfible form. For jt is now become neceffary to exhibit them all manner of ways, in order to give a fatisfaclpry proof, thai they have indeed any real exiftence at all.

And fir ft, by way of prepara- ~ tion, it will be convenient to confider Uniform and equable motions, as alfo fuch as are alike inequable. Let the right Line AB be defcribed the by equable motion of a point, which is now at E, and will prefently be at G. Alfo let the Line CD, parallel to the former, be defcribed the by equable motion of a point, which is in H and K, at the farne times as the former is in E and G. Then will EG and be HK contemporaneous Lines, and therefore will be proportional to the and INFINITE SERIES. 267 the indefi- the Velocity of each moving point refpedlively. Draw in then becaufe of like nite Lines EH and GK, meeting L ; Tri-^ the Velocities of the E and which angles ELG and HLK, points H, were before as EG and HK, will be now as EL and HL. Let the defcribing points G and K be conceived to move back, again, with the fame Velocities, towards A and C, and before they ap- be in at fmall proach to E and H let them found g and ^, any diftance from E and H, and draw gk, which will pafs through L ; then ftill their Velocities will be in the ratio of Eg and H/, be thofe Lines ever fo little, that is, in the ratio of EL and HL. Let with the moving points g and k continue to move till they coincide in cafe the Lines and will E and H ; which decreeing Eg H pafs lefs and will through all polYible magnitudes that are and lefs, finally at the fame become vanishing Lines. For they muft intirely vanifh with and moment, when the points g and k mall coincide E H. In all which ftates and circumftances they will ftill retain the ratio of EL to HL, with which at laft they will finally vaniih. Let thofe points ftill continue to move, after they have coincided with E and H, and let them be found again at the fame time in y and are K, at any diftance beyond E and H, Still the Velocities, which now as Ey and H*, and may be efteemed negative, will be as EL and HL, whether thofe Lines Ey and Hx are of any finite magni- or are nafcent Lines that if the Line its tude, only ; is, yx.L, by angular motion, be but juft beginning to emerge and divaricate from EHL. And thus it will be when both thefe motions are equable as alfo are alike in both motions, when they inequable ; which cafes the common interfedlion of all the Lines EHL, GKL, gkL, &c. -will be the fixt point L. But when either or both thefe motions are fappos'd to be inequable motions, or to be any how continually accelerated or retarded, thefe Symptoms will be fomething different ; for then the point L, which will ftill be the common interfeclion of thofe Lines when they firft begin to coincide, or to divaricate, will no longer be a fixt but a moveable point, and an account muft be had of its motion. For this purpofe we may have recourfe to the following Lemma. an indefinite fixt Let AB be and right Line, along which anothe: indefinite but moveable right Line DE may be conceived to move or roll in fuch a manner, as to have both a progreflive motion, as alfo aa motion about a moveable angular Center C. That is, the common of interfection C the two Lines AB and DE may be fuppofed to move with any progreffive motion from A towards B, while at the M m 2 fame 26S The Method of FLUXIONS,

tame time the moveable Line DE revolves about the lame point C, with any angular motion. Then as the Angle ACD continually decreafes, and at laft vanifhes when the two Lines ACB and DCE coincide even then the ; yet point of interfection it be ftill C, (as may call'd,) will not be loft and annihilated, but will appear again, as foon as the to or Lines begin divaricate, to feparate from each if be other. That is, C the point of interfeclion before the coincidence, and c the point of interfection after the coincidence, when the Line dee {hall out of there again emerge AB ; will be fome intermediate point L, in which C and c were united in the fame point, at the moment of coincidence. This point, for diftin&ion-fake, may be call'd the Node, or the point of no divarication. Now to apply this to inequable Motions : be defcribed Let the Line AB by the continually accelerated mo- is in tion of a point, which now E, and will be prefently found inr Line G. Alfo let the CD, parallel to the former, be defcribed by the equable motion of a point, which is found in H and K, at the fame times as the other point is in E and G. Then willEG and HK be contem- Lines poraneous ; and producing EH and GK till they meet in I, thofe contemporaneous Lines will be as El and HI Let the refpedlively. defcribing and be conceived points G K to move back again towards A and C, each with the fame of degrees Velocity, in every point of their mo- as had before tion, they acquired ; and let them arrive at the fame time at g and k, at fome fmall diftance from E and H, and draw gki meeting EH in /. Then Eg and Hk, being contemporary Lines and little alfo, very by fuppofuion, they will be nearly as the Ve- locities and INFINITE SERIES, 269 and that at E and which locities at g k, is, H ; contemporary Lines will be now as E/' and H/'. Let the points g and k continue their motion till they coincide with E and H, or let the Line GKI its and motion in or gki continue progremve angular this manner, and let till it coincides with EHL, L be the Node, or point of no divarication, as in the foregoing Lemma. Then will the laft ratio of the vanifhing Lines Eg- and lik, which is the ratio of the Velo- ' cities at E and H, be as EL and HL refpe&ively. If the Hence we have this Corollary. point E (in the foregoing be to move from towards with a figure,) fuppos'd A B, Velocity any how accelerated, and at the fame time the point H moves from with an if C towards D equable Velocity, (or inequable, you pleafe ; ) thofe Velocities in E and H will be refpectively as the Lines EL and HL, which point L is to be found, by fuppofmg the contemporary Lines EG and HK continually to dkninim, and finally to vanim. Or by fuppofmg the moveable indefinite Line GKI to move with a in fuch as that progreffive and angular motion, manner, EG and HK fhall always be contemporary Lines, till at laft GKI mall coincide with the Line EHL, at which time it will determine the Node L, or the point of no divarication. So that if the Lines AE and their Velocities of de- CH reprefent two Fluents, any how related, at or their will be in the fcription E and H, refpe&ive Fluxions, ratio of EL and HL. And hence it will fol- ^ low alfo, that the Lo- cus of the moveable or Node that point L-, IH all the is, of points of C T> no divarication, will be fome Curve-line L/, to which the Lines EHL and GK/ will always be Tangents in L and /. And the nature of this Curve L/ may be determined by the given relation Fluents or of the Lines AE and CH ; and vice versa. Or however the relation of its intercepted Tangents EL and HL may be determined in all cafes ; that is, the ratio of the Fluxions of the given Fluents. For 270 tte Method of FLUXLONS, For illuftration-fake, let us apply this to an Example. Make the Fluents AE= y and CH ;= x, and let the relation of thefe be always this x". Make the Lines exprefs'd by Equation y = contemporary EG = Y and HKs=X.; and becaufe AE and CH are contempo- fhall have the Lines and rary by fuppofition, we whole AG CK

x . contemporary alfo, and thence the Equation y -f-Y= -j-X | This by our Author's Binomial Theorem will produce y -+- Y = x" + 1 x" will be- -nx"~ X -+- n x"-^-x*~*X* , &c. which becaufe = ( y )

I 1 x &c. or in an : come Y= x"- X-J- ^-^Ar'-^X , Analogy, X

n~ l y :: i : nx + x ^-^""^X, &c. which will be the general relation of the contemporary Lines or Increments EG and HK. Now let us fuppofe the indefinite Line GKI, which limits thefe contemporary Lines, to return back by a progrefiive and angular motion, fo as always to intercept contemporary Lines EG and HK, and that to determine the finally to coincide with EHL, and by means and to di- Node L; that is, we may fuppofe EG = Y HK = X, minifli hi i-nfinitum, and to become vanifhing Lines, in which cafe l we fhall have X : Y : : i : nx"~ . But then it will be likewife X :

: :: : : : : or i : nx"~' : : Y : : HK EG HL EL x y, x :y, ory=nxx'. And hence we may have an expedient for exhibiting Fluxions and Fluents Geometrically and Mechanically, in all circumftances, fo as to make them the objects of Senfe and ocular Demonftration. Thus in the laft figure, let the two parallel lines AB and CD be defcribed by the motion of two points E and H, of which E moves and be to any how inequably, (if you pleafe) H may fuppos'd move and let the and equably and uniformly ; points H K correfpond to E and G. Alfo let the relation of the Fluents AE =r y and CH = x be defined by any Equation whatever. Suppofe now the defcribing points E and H to carry along with them the indefinite Line EHL, in all their motion, by which means the point or Node L will defcribe fome Curve L/, to which EL will always be a Tan- be the gent in L. Or fuppofe EHL to Edge of a Ruler, of an indefinite length, which moves with a progreffive and angular motion thus combined together ; the moveable point or Node L in this have leaft Line, which will the angular motion, and which is always the point of no divarication, will defcribe the Curve, and the Line itfelf will a to it in or Edge be Tangent L. Then will the feg- ments EL and HL be proportional to the Velocity of the points or will exhibit the ratio of the E and H refpeclively ; Fluxions y a-nd x, belonging to the Fluents AE=y and CF = x. i Or and INFINITE SERIES. 271

Or if we fuppofe the Curve L/to be given, or already conftmcled, we may conceive the indefinite Line EHIL to revolve or roll about itfelf to as a to move it, and by continually applying it, Tangent, from the fituation EHIL to GK.ll. Then will AE and CH be the Fluents, the fenfible velocities of the defcribing points E and H will be their Fluxions, and the intercepted Tangents EL and HL will be the redlilinear meafures of thofe Fluxions or Velocities. Or it

: obftacle in form of a may be reprefented thus If L/ be any rigid Curve, about which a flexible Line, or Thread, is conceived to be a Line which wound, part of which is ftretch'd out into right LE, will therefore touch the Curve in L ; if the Thread be conceived to be farther wound about the Curve, till it comes into the fituation this motion it will even to the the fame L/KG ; by exhibit, Eye, of or their increafing Fluents as before, their Velocities increafe, of thofe Fluxions, as alfo the Tangents or rectilinear reprefentatives Fluxions. And the fame may be done by unwinding the Thread, in the manner of an Evolute. Or inftead of the Thread we may the make ufe of a Ruler, by applying its Edge continually to curved Obftacle L/, and making it any how revolve about the move- all manners the able point of Contadl L or /. In which Fluents, Fluxions, and their rectilinear meafures, will be fenfibly and mecha- be allowed to have a nically exhibited, and therefore they muft place but in rernm naturd. And if they are in nature, even tho' they were more if are fenfible barely pofiible and conceiveable, much they the fome me- and vifible, it is the province of Mathematicks, by and thod or other, to investigate and determine their properties proportions. the curved Or as by one Thread EHL, perpetually winding about fee and obftacle L/, of a due figure, we mall the Fluents AE CH at rate the mo- continually to increafe or decreafe, any aflign'd, by tion of the Thread EHL either backwards or forwards ; and as we of E and (hall thereby fee the comparative Velocities the points H, and and alfo the Lines that is, the Fluxions of the Fluents AE CH, EL and HL, whofe variable ratio is always the rectilinear meafure of wind- thofe Fluxions : So by the help of another Thread GK/L, in its and then out into a ing about the obftacle part /L, ftretching or and made to move backwards or for- right Line Tangent /KG, if the firft be at reft in fitua- wards, as before ; Thread any given tion EHL, we may fee the fecond Thread defcribe the contempo- Increments and which the Fluents porary Lines or EG HK, by if is made to AE and CH are continually increafed ; and GK/ approach e Method 272 ^ of FLUXIONS, we fee thofe Lines conti- proach towards EHL, may contemporary imallv to diminim, and their ratio continually approaching towards to and the the ratio of EL HL ; continuing motion, we may pre- Lines to unite as fently fee thofe two actually coincide, or to one Line, and then we may fee the contemporary Lines actually to va- ntfh at the fame time, and their ultimate ratio actually to become that of EL to HL. And if the motion be ftill continued, we mall fee the Line GK/ to emerge again out of EHL, and begin to defcribe other contemporary Lines, whofe nafcent proportion will be that of EL to HL. And fo we may go on till the Fluents are exhaufted. All thefe particulars may be thus eafily made the objects of fight, or of Ocular Demonftration. This may ftill be added, that as we have here exhibited and re- firft do prefented Fluxions geometrically and mechanically, we may the fame thing, mutatis mutandis, by any higher orders of Fluxions. Thus if we conceive a fecond figure, in which the Fluential Lines fhall increafe after the rate of the ratio of the intercepted Tangents (or the the firft then will ex- Fluxions) of figure ; its intercepted Tangents pound the ratio of the fecond Fluxions of the Fluents in the firft figure. Alfo if we conceive a third figure, in which the Fluential Lines fhall increafe after the rate of the intercepted Tangents of its will the the fecond figure ; then intercepted Tangents expound third Fluxions of the Fluents in the firft figure. And fo on as far as we pleafe. This is a neceflary confequence from the relative nature of thefe feveral orders of Fluxions, which has been fhewn before.

And farther to mew the univerfality of this Speculation, and how well it is accommodated to explain and reprefent all the circumftances of Fluxions and Fluents; we may here take notice, that it may be alfo adapted to thofe cafes, in which there are more than two Fluents, which have a mutual relation to each other, exprefs'd by one or more Equations. For we need but introduce a third parallel Line, and fuppofc it to be defcribed by a third point any how moving, and that any two of thefe defcribing points carry an indefinite Line along with them, which by revolving as a Tangent, defcribes the Curve whofe Tangents every where determine the Fluxions. As alfo that any other two of thofe three points are connected by another indefinite Line, which by revolving in like manner defcribes fuch fo there or another Curve. And may be four more parallel Lines. All but one of thefe Curves may be affumed at pleafure, when they are not given by the ftate of the Queftion. Or Analy- tically,

' / ///. j //'//<'. i , i( //. uvuutm v /(v Yfa/?//// 2-3. and INFINITE SERIES. 273

fo be not tically, many Equations may aflumed, except one, (if as is of concern'd. given by the Problem,) the number the Fluents I believe it But laftly, may not be difficult to give a pretty good notion of Fluents and Fluxions, even to fuch Perlbns as are not much verfed in Mathematical Speculations, if they are willing to be iniorm'd, and have but a tolerable readinefs of apprehenfion. This I {hall here attempt to perform, in a familiar way, by the inftance of a Fowler, who is aiming to (lioot two Birds at once, as is reprefented in the Frontifpiece. Let us fuppofe the right Line AB to or to be parallel the Horizon, level with the Ground, in which is be- a Bird now flying at G, which was lately at F, and a little fore at let be E. And this Bird conceived to fly, not with an equable or uniform fwiftnefs, but with a fwiftnefs that always increafes, (or with a Velocity that is continually accelerated,) according to fome known rate. Let there alfo be another right Line CD, parallel to the former, at the fame or any other convenient diftance from the Ground, in which another Bird is now flying at K, which was lately at and a little before at at the fame of time as I, H ; juft points the firft Bird was at G, F, E, refpectively. But to fix our Ideas, and to make our Conceptions the more fimple and eafy, let us imagine this fecond Bird to defcribe fly equably, or always to equal parts of the Line CD in equal times. Then may the equable Velocity of this Bird be ufed as a known meafure, or ftandard, to which we may always compare the inequable Velocity of the firft Bird. Let us now fuppofe the right Line EH to be drawn, and continued to the fo that the point L, proportion (or ratio) of the two Lines EL and HL may be the fame as that of the Velocities of the two at Birds, when they were E and H refpeclively. And let us farther fuppole, that the Eye of a Fowler was at the fame time at the and that he point L, directed his Gun, or Fowling-piece, according to the right Line LHE, in hopes to moot both the Birds at once. But not himfelf thinking then to be fufficiently near, he forbears to difcharge his Piece, but ftill pointing it at the two Birds, he continually advances towards them according to the direction of his Piece, till his Eye is prefently at M, and the Birds at the fame time in F and in the fame Line I, right FIM. And not being yet near enough, we may fuppofe him to advance farther in the fame manner, his Piece being always directed or level'd at the two Birds, while he himfelf walks forward according to the direction of his Piece, till his is now at and the in Eye N, Birds the fame right Line with his Eye, at K and G. The Path of his Eye, delcribed by this N a double Method 274 The of FLUXIONS, double motion, (or compounded of a progreffive and angular mo- be ibme Curve-line Plain as tion,) will LMN, in the fame the reft of the figure, which will have this property, that the proportion of of the diftances his Eye from each Bird, will be the fame every as that of their Velocities. his where refpeftive That is, when Eye was at L, and the Birds at E and H, their Velocities were then as EL and HL, by the Conftruftion. And when his Eye was at M, and the Birds at F and I, their Velocities were in the fame proportion as the Lines FM and IM, by the nature of the Curve LMN". And when his Eye is at N, and the Birds at G and K, their Velo- cities are in the proportion of GN to KN, by the nature of the fame Curve. And fo univerfally, of all other fituations. So that the Ratio of thofe two Lines will always be the fenfible meafure of the ratio of thofe two fenfible Velocities. Now if thefe Velocities, or the fwiftneffes of the flight of the two Birds in this inflance, are call'd Fluxions; then the Lines defcribed by the Birds in the fame time, may be call'd their contemporaneous Fluents; and all inftances whatever of Fluents and Fluxions, may be reduced to this Example, and may be illuflrated by it. And thus I would endeavour to give fome notion of Fluents and

Fluxions, to Perfons not much converfant in the Mathematicks j but fuch as had acquired fome fkill in thefe Sciences, I would thus to and to proceed farther inflrudl, apply what has been now deliver'd. The contemporaneous Fluents being EF=_y, and Hl=.v, and their rate of flowing or increafing. being fuppos'd to be given or their relation be an known ; may always exprefs'd by Equation, which will be compos'd of the variable quantities x andjy, together with any known quantities. And that Equation will have this proof thofe variable that as and perty, becaufe quantities, FG IK, EG and HK, and infinite others, are alfo contemporaneous Fluents; it will indifferently exhibit the relation of thofe Lines alfo, as well as or fubflituted in the of EF and HI ; they may be Equation, inftead of x and y. And hence we may derive a Method for determining the Velocities themfelves, or for finding Lines proportional to them. IK in the I For making FG =Y,.and = X ; given Equation may inftead of and x inftead of which fubftitute y -}- Y ^y, -f- X x, by I fhall obtain an Equation, which in all circumftances will exhibit the relation of thofe Quantities or Increments. Now it may be plainly if is to perceived, that the Line MIF conceived continually approach in nearer and nearer to the Line NKG, (as jufl now, the inftance of the Fowler,) till it finally coincides with it; the Lines FG = Y, and and INFINITE SERIES. 275

and IK = X, will continually decreafe, and by decreafing will approach nearer and nearer to the Ratio of the Velocities at G and K, and will finally vanifh at the fame time, and in the proportion of in the thofc Velocities, that is, Ratio of GN to KN. Confequently in the Equation now form'd, if we fuppofe Y and X to decreafe continually, and at laft to vanifh, that we may obtain their ultimate Ratio mail obtain the of to ; we thereby Ratio GN KN. But when Y and X vanifh, or when the point F coincides with G, and I with H, then it will be EG = y, and HK = .x'; fo that we fhall have y : x :: GM : KN. And hence we mall obtain a Fluxional Equa- tion, which will always exhibit the relation of the Fluxions, or Ve- locities, belonging to the given Algebraical or Fluential Equation. Thus, for Example, if EF=j', and HI = x, and the indefinite Lines y and A: are fuppofed to increafe at fuch a rate, as that their x 1 ax* relation may always be exprefs'd by this Equation + axy o then y* = ; making FG=Y, and IK = X, by fubftituting y -f- Y for j, and x -+- X for x, and reducing the Equation that will Z 3 - arife, (fee before, pag, 255.) we fhall have ^x"-X -f- 3#X -f- X 1 ! zaxX aX -f- axY -\- aXj -+- rfXY 3y*Y 3jyY* Y = o, which may be thus exprefs'd in an Analogy, Y : X :: 3** 2 ax 1 .+. ay +- 3tfX -h X aX : ^ ax aX -+- 37Y -+- Y*. This Analogy, when Y and X are vanishing quantities, or their ultimate

will : : : Ratio, become Y X : 3** ^ax -f- ay 3^* ax. And

is : v : it : :: becaufe it then Y X :: GN : KN :: x, will be y x 1 : the 3X zax -+- ay 3^* ax. Which gives the proportion of Fluxions. And the like in all other cafes. Q^. E. I. We might alfo lay a foundation for thefe Speculations in the following manner. Let ABCDEF, 6cc. be the Periphery of a Polygon, of or any part it, and let the Sides AB, BC, CD, DE, &c. be of any magnitude whatever. In the fame Plane, and at any diftance, draw the two parallel Lines /6, and bf\ to which continue the right Lines AB4/3, BCcy, &c. the as in the DEes, meeting parallels figure, Now if we fup- n N 2 pofe 276 7%e Method of FLUXIONS, or to be pofe two moving points, bodies, at $ and b, and to move time to and with in the fame y c, any equable Velocities ; thofe be to each other as Velocities will @y and be, that is, becaufe of the as and bE. Let them fet out from and parallels, /3B again y c, arrive at the fame time at and with Velocities and ^ d, any equable ; thole Velocities will be as and that as yfr cd, is, yC and cC. Let them depart again from and d, and arrive in the fame time at g Velocities thofe Velocities will be as and e, with any equable ; S-t as it will be and de, that is, J^D and dD. And the fame thing every where, how many foever, and how fmall foever, the Sides of the Polygon may be. Let their number be increafed, and their magnitude be diminim'd in infinitum, and then the Periphery of the Poly- gon will continually approach towards a Curve-line, to which the &c. will become as alfo the Lines AB^/3, ECcy, CDd, Tangents -, Motions may be conceived to degenerate into fuch as are accelerated or retarded continually. Then in any two points, fuppofe and d, where the defcribing points are found at the fame time, their Velo- cities (or Fluxions) will be as the Segments of the refpeclive Tan- and the Lines and gents cTD and dD ; /3^ bd, intercepted by any be or two Tangents J>D and /SB, will the contemporaneous Lines, Fluents. Now from the nature of the Curve being given, or from its Lines be the property of Tangents, the contemporaneous may found, or the relation of the Fluents. And vice versa, from the be Rate of flowing being given, the correfponding Curve may found.

ANNO- and INFINITE SERIES. 277

ANNOTATIONS on Prob.i-

O R,

The Relation of the Fluxions to beingo given,& 7 find the Relation of the Fluents.

I. Solution SECT. A particular ; with a preparation for the general Solution, by 'which it is diftribitted into- three Cafes.

E are now come to the Solution of the Author's fecond fundamental Problem, borrow'd from the Science

of Rational Mechanicks : Which is, from the Velo- cities of the Motion at all times given, to find the defcribed or to find the quantities of the Spaces ; Fluents from the given Fluxions. In difcuffing which important Problem, there will fome at be occafion to expatiate thing more large. And firft it may not be amifs to take notice, that in the Science of Computation all the Operations are of two kinds, either Compolitive or Refolutative. The Compolitive or Synthetic Operations proceed neceffarily and di- in their feveral and not rectly, computing qit(?fita> tentatively or by Such are way of tryal. Addition, Multiplication, Railing of Powers, But and taking of Fluxions. the Refolutative or Analytical Opera- Extraction of tions, as Subtraction, Divifion, Roots, and finding of forced to and Fluents, are proceed indirectly tentatively, by long to arrive at their feveral and deductions, qutefita ; fuppofe or require to and confirm the contrary Synthetic Operations, prove every llep of the Procefs. The Compofitive Operations, always when the data are finite and terminated, and often when they are interminate i or The Method of FLUXIONS,

finite conclufions whereas often in or infinite, will produce ; very the Refolutative Operations, tho' the data are in finite Terms, yet cannot be obtain'd without an infinite Series of Terms. the quafita Of this we mall fee frequent Inftances in the fubfequent Operation, of returning to the Fluents from the Fluxions given. fuch The Author's particular Solution of this Problem extends to * -+- ayx. The aggregate of thefe, neglecting the Term is x* ax* the redundant ayx, -\- axy _}" = o, Equation it be required. Where muft noted, that every Term, which occurs more than once, mult be accounted a redundant Term. So and INFINITE SERIES. 279

1 So if the propoied Equation were m -f- ^yxx* m-\- 2(jyxx -f- 4 ;// -+- i ay* xx m} -v n-\- $xyy> -\-n-\- lax^yy -+- nx+y nax>y =. o, whatever values the general Numbers m and n may acquire ; if thofe Terms in which x is found are reduced to the Scale whofe 1 Root is x, they will ftand thus : m -+- yyxx' m -+- zayx*. -+ m-\-\ay*xx my**-, or expunging the points they will become 1 m -+- %yx+ m -f- Ziivx* +- m -+- \ay-x m\*x. Thefe being divided refpedtively by the Arithmetical Progreffion m -f- 3, m-\-2,. will 1 1 Alio m-\- i, m, give the Terms yx+ ayx -f- ay'-x y+x. the Terms in which is found reduced to the Scale whofc y ; being Root isy, will ftand thus : n -4- ^xyy* * +- n -+ iax*jy-{- nx*y; nax"=y or the n iaxi expunging points they will become n -\~^ x^ * -^~ ~^~ )" Pro- +- nx+y. Thefe being divided reipeclively by the Arithmetical

will grefTion ^-{-3, ?i-{- 2, ?z-|-i, ;;, give the Terms xy* -\- 1 ax*}' -)- x+y ax*y. But thefe Terms, being the fame as the former, all be confider'd as mull redundant, and therefore are to be rejected. i i So that -f- x or the yx* ayx* ay y^x=o ) dividing by yx, 1 Equation x* ax -\-ayx y* = o will arife as before.

l Thus if we had this Fluxional Equation mayxx~ m -+- 2xx nx*yy~* -+- ;z-f- \ay =. o, to find the Fluential Equation to which I it the * the belongs ; Terms mayxx~ m -f- 2xx, by expunging points, and dividing by the Terms of the Progreffion m, m-\- 1, w-t-2, the 1 will give Terms ay x*. Alfo the Terms nx^yf -+- n-\-iay, the by expunging points, and dividing by n, n-\- i, will give the x 1 Terms -f- ay. Now as thefe are the fame as the former, they are to be efteem'd as redundant, and the Equation required will be 1 ay x = o. And when the given Fluxional Equation is a general one, and adapted to all the forms of the Fluential Equation, as is the cafe of the two laft then all the Examples ; Terms ariling from the fecond Operation will be always redundant, fo that it will be fufficient to make only one Operation. Thus if the were 1 3 J given Equation ^.yy -f- z yy~ -f- 2yxx 3:32* H- 6}'z.z 2cyz = o, in which there are found three flowing quantities the in which x is is j only Term found 2yxx, in which expunging the point, and then dividing by the Index 2, it will be- 1 l . Then the Terms in which is found are come^* y 4^*4- z*yy~~ t which the become # * 3 and expunging points ^ 4-s , dividing by, 280 72k Method of FLUXIONS,

5 ; by the Progreffion 2, i, o, i, give the Terms aj s . Laftly the Terms in which z is found are yzz* -J- 6yzz zcyz, which the become and expunging points 32;"' -f- 6yz* 29-2, dividing 1 by the Progreffion 3, 2, i, give the Terms & + T.yz zcyz. Now if we collect thefe Terms, and omit the redundant Term z*, z 1 mall have z"' we yx -+- 2y> -f- yz 2cyz = o for the Equa- tion required. But thefe deductions are not 3, 4. to be too much rely'd upon, till they are verify 'd by a proof; and we have here a fure method of proof, whether we have proceeded rightly or not, in returning from the relation of the Fluxions to the relation of the Fluents. For every refolutative Operation mould be proved by its contrary com- the pofitive Operation. So if Fluxional Equation xx vice versa. And as this Equation virtually involves three variable quantities, and INFINITE SERIES. 281

either Fluential or quantities, it will require another Equation, Fluxionai, for a compleat determination, as has been already ob- i becomes ferved. So as the Equation yx = xyy, by putting x = in like manner this and the yx=yy; Equation requires fuppofes other. II. Here we are fome ufeful in 6, 7, 8, 9, 10, taught Reductions, order to prepare the Equation for Solution. As when the Equation contains only two flowing Quantities with their Fluxions, the ratio of the Fluxions may always be reduced to fimple Algebraic Terms. The Antecedent of the Ratio, or its Fluent, will be the quantity to the for the be extracted ; and Confequent, greater fimplicity, may o is be made Unity. Thus the Equation zx +- 2xx yx y = y to - 2 2X or 'tis = 2 reduced this, = -+ y, making x=i, y

So the xa -f- xx = o, ma- _^_ 2 x y. Equation ya yx xy a y i i, will become -f- = i = = ) king x= y ( ~^+ jdb + 2L f!? ^ &c. Divilion. But we the _!_ _j_ _|_ , by may apply par- 1 ticular Solution to this Example, by which we mail have {x xy

and thence =."- . the __ ## _{_ tfy = o, y ^~- Thus Equation = making x=i, becomes yy -+- xx, and ex- yjr xy-{-xxxx, =y 'tis -i ~ the Series tracting the fquare-root, y = \/ -j- xx = 1 6 8 2.-4-X x*-{-2X 5X -f- I4x', &c. that is, either y = i -j- * 4 6 8 10 &c. or x 1 x 4 6 x x -{-2x jx -f- I4* , y = -f- zx 8 10 &c. the i 1 x 1 _j_ rx I4-.V , Again, Equation y> -+-axx*y-{-a y 3 3x 3 2x'tf ? x= i, becomes x X =o, putting _y -\-axy -\-ay - 2<7 3 o. Now an affected Cubic Equation of this form has 1 mail been refolved before, (pag. 2.) by which we have y = a ^x + ? 4 xx ^9^ c, iji* " ' ~*~ ' 6^ uz" 1 16384^5 12. For the fake of perfpicuity, and to fix the Imagination, our Author here introduces a diftinction of Fluents and Fluxions into Relate and Correlate. The Correlate is that flowing Quantity which is or he fuppofes to flow equably, which given, may be arTumed, at any point of time, as the known meafure or ftandard, to which the Relate Quantity may be always compared. It may therefore denote Time and its or an very properly ; Velocity Fluxion, being uniform and conftant quantity, may be made the Fluxionai Unit, or the known meafure of the Fluxion (or of the rate of flowing) of the Relate Quantity. The Relate Quantity, (or Quantities if ieve- O o ral 2 8 2 The Method of FLUXIONS,

is that which is ral are concern'd,) fuppos'd to flow inequably, with; of acceleration or retardation and ts any degrees ; inequability may or reduced as it were be meafured, to equability, by conihntly com- it with its paring correfponding Correlate or equable Quantity. This therefore is the Quantity to be found by the Proble'm, or whofe is to be extracted from the Root given Equation. And it may be as a defcribed the conceived Space by inequable Velocity of a Body in while the or Point motion, equable Quantity, or the Correlate, or meaiures the time of reprefents defcription. This may be illuftrated our common Mathematical by Tables, of Logarithms, Sines, &c. In the Tangents, Secants, Table of Logarithms, for inflance, the Numbers are the Correlate as Quantity, proceeding equably, or while their by equal differences, Logarithms, as a Relate Quantity, proceed inequably and by unequal differences. And this refemblance r would more nearly obtain, if w e mould fuppofe infinite other Num- their to be bers and Logarithms interpolated, (if that infinite Num- where the ber be every fame,) fo as that in a manner they may become continuous. So the Arches or Angles may be confider'd as becaule the Correlate Quantity, they proceed by equal differences, while the Sines, Tangents, Secants, &c. are as fo many Relate Quan- of is tities, whofe rate increafe exhibited by the Tables. 13, 14, 15, 16, 17. This Diflribution of Equations into Orders, or Gaffes, according to the number of the flowing Quantities and tho' it be not of abfolute their Fluxions, neceflity for the Solution, ferve to make it more may yet expedite and methodical, and may fupply us with convenient places to reft at.

SECT. II . Solution of the Jirft Cafe of Equations. r 20, 21, 22, firft Cafe of wherr 18, 19, 23. ~|~^HE Equations is, -i. the ? or what Quantity , fupplies can te its place, always found in Terms compofed of the Powers of x, and known Quantities or Numbers.. Thefc Terms are to be to be multiply'd by x, and divided by the Index of .v in each Term,, which will then exhibit the Value of Thus in the a jr. Lquationj/ = .xi/ l x it has been found ~ i s -+- x*, that = -t-x* x* -f- 2x $x* -f-

10 - &cc. Therefore x* 2,v 7 I4* , =^-4- x'-f- 5^' -t- 14*'*,

1 1 9 3 &c. and = x -f- , confequently y + jX* -fx ^x" J-x -j-l^A" be &c. as may ealily proved by the direct Method. But and INFINITE SERIES. 283 be refolved more But this, and the like Equations, may readily by a Method form'd in imitation of fome of the foregoing Analyfes, after this manner. In the given Equation make x = i ; then it

is thus refolved : will bej)* =j/-l-.v*, which

S 9 X* -+- X* 2X &C. = -f- pC , H4 & 9 J .V 4 2X &C. y*-y $ -f- 5X ,

of 4 be the firft Make AT* the firft Term y ; then will x Term 1 of which is to be with a for the fecond j/ , put contrary Sign 6 fecond Term of y. Then by fquaring, -f- 2X will be the Term Therefore of j/, and 2x* will be the third Term of y. 8 of will be the 5# will be the third Term j/, and -f- 5*" fourth of and fo on. Therefore the Term y ; taking Fluents, y = 5 7 one Root of the I..V -+- -fx* ix -f-4-x, &c. which will be Equation. And if we fubtradt this from x, we (hall have y = x -+- 3 7 -i-A; for x ^.v* -f- AX', &e. the other Root. -v -^ ^ if c - that if ax So = a 4-r -4- r h > & is, = f 04^ 5 I 2* A' . c , # I?IA.'4 &c " then &c. ' v=^->^4- + 6? li ^,

* A jj-T ^

i s -^yii: yx ya &c then Y j. v .

or c *. If 4 = -, = or ^--**. ==f * , ex? x c

f then _y=^ .

v ' if '- or - ^ ^v Laftly, = ~, = = ; dividing by the In- a it will be - or dex o, y = , y is infinite. That this Expreffion, be or value of y, mufl infinite, is very plain. For as o is a vanim- or lefs than - ing quantity, any affignable quantity, its Reciprocal muft be than or bigger any affignable quantity, that is, in-

finite. O o 2 284 The Method of FLUXIONS, this to be Now that quantity ought infinite, may be thus proved. the 4 - let AB In Equation = x , reprefent the conftant quantity a, and in CE let a point move equably from C towards E, and defcribe the Line of Avhich CDE, let any indefinite part CD be x, and its in where is equable Velocity D, (and every elfe,) reprefented

A o, E

c and INFINITE SERIES, 285 x write b and we fhall have ~ -- and then we may +- x, = , by 1 r i i -r-v r vx f ax \ ax "X and Divifion it is -4- :. J -r Multiplication = ( = l -f. x V*-}-* / b t

~ &c. and therefore "- ~ 6cc. -77 , }'= ^- \ -f- ~,

- -' 2.6. So if ~ ' becaufe of the Term which = -J- 3 xx, , would give an infinite value for ^, we may write j -f- x inftead of - ~ y X, and we fhall then have = -4-2 zx xx, or = 1 X X | X x 1 y 2X 2X or Divifion - - x j -^ 1- x"', by = 4x 4x* -f- zx s &c. and therefore 2x* 3 4 -f- , y=.^x -+- ^.x |x -f- r &c. ^x , y 1 Or the ~ z zx x that is Equation = -^^ +- , y -f- xy 1 j x be thus refolved : = 4 jx , may

11 y^ = 4 * 3* A: J 1 AT 3 2X 4 ^" 4X -J- 4X -j- , &C, x 3 2x 4 6cc. H- xyj h 4^ 4x* 4- , AT* 2x 3 2x 4 y = 4 4.x -f- -f- , &c. 4 * T 2X"' _ i X x c . = 4.V -{- ^X3 _|_ t &

firft of Make 4 the Term j, then 4x will be the firfl Term of will be the fecond xy, and confequently 4* Term of j. Then a will be the fecond Term of and therefore fc 4x .vy, -(- 4x* 3x , be the third fo or x*, will Term of_y ; and on.

J if -. 27. So = AT~^ -f- x~ x'~, becaufe of the Term x~' change x into i x, then == -.' -+- s/ i x. But X I X yi ,v by the foregoing; Methods of Reduction 'tis = i -f- x -+- x* * I X

4 x 5 and x I -+- , 6cc. v/i = 4-^ r-^ -rrx^ &c. a"d

Therefore collecting thefe according to their Signs, 'tis 4- i 4-

1 3 &c. that 3 4 2.v-|- ix -t- T^-x , -f- &c. is-^ =x4-2x* |x + ^x , a and therefore x 4- x -f- ix s 4 &c. y = 4- ^x , 28. So if the were ~ given Equation == " * ~^ ~ ~" X i. ii^^C -!- 3t"A* X% - . '^ ; the of x. that is. inftead of x write ' change beginning t A | x, 286 7$ Method of FLUXIONS,

- y f3 c*-X yx -- l 1 c then c">x~* c x- or c*x~* . x, = Al = , ^ =

1 l and therefore . c*x- , _>' = ^c=x-~ -\-c*x~

SECT. III. Solution the of fecond Cafe of Equations.

2 9> 3- TT^Quations belonging to this fecond cafe are thofe, wherein the two Fluents and their M^ Fluxions, fuppofe x and or x and y, j, any Powers of them, are promifcuoufly involved. As our Author's are Analyfes very intelligible, and fee'm to want but little explication, I mall endeavour to refolve his Examples in an eafier and than is done here fomething fimpler manner, ; by applying to them his own artifice of the Parallelogram, when need- or the of a combined Arithmetical ful, properties Progreffion in piano, before : alfo as explain'd As the Methods before made ufe of, in the Solution of afTeclcd Equations. 31. The Equation yax xxy aax = o by a due Reduction "- ~ ~ in - .becomes = -+- , which, becaufe of the Term there is occafion for a Tranfmutation, or to change the beginning of the Correlate x. therefore the Quantity ArTurning conftant quantity b, we 4- -f- whence Divifion will be may put = ^ -^ , by had z y v a ax ax ax* e i i -.-, . , == -I- -+- > &c ' which is then -j -^ y -ji 77 Equation prepared for the Author's Method of Solution. to But without this previous Reduction an infinite Series, and the Reiblution of an infinite Equation confequent thereon, we may perform the Solution thus, in a general manner. The given Equa- - or x tion is now 4 = -|- , = i, it is -f- j- putting aby axj 1 a which be thus refolved : /y ~+- yx -f- , may

\ aby =

xy and INFINITE SERIES. 287

1- is done make a the firft Difpoiing the Terms as you fee here, then ~ will be the firft Term of and thence Term of aby, j, -|x a will be the firft Term of y. So that x will be the firft Term of * b ax will be the firfl of two to- axv,* and Term by. Thefe / i i / or ax - with a be gether, -,x = x, contrary Sign, mud put for the fecond down Term of aby. Therefore the fecond Term of y will be '-~-x, and the like Term of y will be A*. Then the ' t>- -' ^b- a '-'-~ fecond Term of will be A*, and the fecond Term of ab..\y b" x and the be ly will be -^~ *, firft Term of xy will yA*. ~ 2 * ~ three make - which with a Thefe together _ A*, contrary Therefore the Sign muft be made the third Term of aby. third Term of y will be ~-^-'A* and the third Term of y will be r ZaL ' * t , / * _^_ 7 - 3 A' . fo in' cafe if And on. Here a particular we make.

we mall have the Series x * , &c. b =. a, fimple y =. -+- ^ 7 Or if we would have a defcending Series for the Root y of this Equation, we may proceed as follows :

1 i b zab -+- x &c. xy~\ =

'-, &c. 3 6cc.

a ; ^ 2rt*A-~ 3 &C. JJ/=: rt A~ - -4-/>X ,

and make a* the firft Term of the Difpofe the Terms as you fee, - of will Series xy, then will be the firft Term y, and a*x~"- l of be the firft Term of y. Then will -f- a"-bx~ be the firft Term I and a"'X~ will be the firft of which by, Term axy y together b x a l t this be make a-\- x~ ; therefore with a contrary Sign muft the fecond Term of xy. Then the fecond Term of y will

3i 1 3 be x~~ and the fecond will be l>-x.2a x~ , a-\-by.a ) Term ofj/ a-\- Therefore the fecond Term of by will be a -f- b x tf*Av~*, and 2 88 TZe Method of FLUXIONS, and the fecond Term of axy will be a-\-b* 2a*x~*, and tlie of will be a^bx~- firft Term aby ; which three together make 1 1 ^za -I- zab -+- b* xa *.*. ThiswithacontrarvSisnmuftbethethird % 2 3 Term of xy, which will give 2a* -+- zab -+- b x a x~ for the of and fo on. if third Term y ; Here we make b=.a, thenj= 1 a za* ca4 .

-- r 3' &C. x +- *' ^Tx* are And thefe all the Series, by which the value of y can be exhibited in this Equation, as may be proved by the Parallelogram. For that Method may be extended to thefe Fluxional Equations, as well as to Algebraical or Fluential Equations. To reduce thefe Equations within the Limits of that Rule, we are to confider, that m as the initial the in both thefe Ax may reprefent Term of Root jr, m kinds of or becaufe it be Ax &c. fo in Equations, may y = , ~ Fluxional mall have m I Equations (making #=1, we a\foy=mAx ) m t 6cc. or for Ax 6cc. 'tis &c. So that in writing y , y = myx~* , every Term of the given Equation, in which y occurs, or the Fluxion of the Relate Quantity, we may conceive it to take away one Di- menfion from the Correlate Quantity, fuppofe x, and to add it to the Relate to Reduction Quantity, fuppofe y ; according which we may inlert the Terms in the Parallelogram. And we are to make a like Reduction for all the Powers of the Fluxion of the Relate Quantity. This will bring all Fluxional Equations to the Cafe of Algebraic Equations, the Refolution of which has been fo amply treated of before. the the Thus in prefent Equation aby -+- axy = by -f- yx +- aa, be inferted ' were fub- Terms mufl in the Parallelogram, as if yx~ inftead of fo in the ftituted y ; that the Indices will ftand as Margin, and the Ruler will give only two Cafes of external Terms. Or rather, if we would reduce this Equation to the form of a double Arithmetical Scale, as explain'ci before, we mould have it in this form. Here in the firft Column are contain'd thofe which have of one or what _ Terms y Dimenfion, y i_ 2 1 ax is to it. In the fecond is a + i J C~ equivalent Column , ; or y of no Dimenfions. Alfo in the firft Line is

. or fuch in is xy, Terms which x of one Dimenfion. In the,

Line are the Terms . , fecond by~l 1 ,

of becaufe is as if it {iqiis .v, -j- axy regarded were ay. Laftly, in the third line is or the Term in which x is aby t of one negative Dimenlion and INFINITE SERIES. 289

J Dimension, becaufe -\-aly is confider'd as if it were -f- abx~~ )\ And thefe Terms being thus dilpos'd, it is plain there can be but two Cafes of external Terms, which we have already difcufs'd. ?2. If the be -TV 2 x -4- or oropofed Equation = jy > O y xx 'tis -f- 2.v l 1 o the making x= i, y 3_v -f- xy~ 2yx~ = ; mall Solution of which we attempt without any preparation, or without any new interpretation of the Quantities. Firft, the Terms to a are to be difpos'd according double Arithmetical Scale, the Roots and then will Itand of which are y and .Y, they as in the Margin. The Method of doing this with certainty in all cafes is as follows. I obferve in p- 2.V + Xj~ the Equation there are three powers of 1 1 * which are and ; there- y, y , y, 7-' - 2JX- in order at the fore I place thefe top of the Table. I obferve likewife that there are four Powers of x,

1 I l are .v A and x~ which I in order in a Column which , x, , , place hand or it will be to conceive this to be done. at the right ; enough in Then I infert every Term of the Equation its proper place, ac- to its Dimenfions of and x in that Term the cording y ; filling up vacancies with Aflerifms, to denote the abfence of the Terms be- to them. The Term I infert as if it were longing y _}'*""', before. Then that if the as is explain'd we may perceive, we apply Rukr to the exterior Terms, we mail have three cafes that may pro- fourth is that of or duce Series ; for the cafe, which direft afcent to be as never defcent, is always omitted, affording any Series. To which will arife from two begin with the defcending Series, the s and . external Terms 2x -f- xy~ The Terms are to bsdifpos'd, and the Analyfis to be performed, as here follows :

-J44*-*, &c.

l then 1 Make xy~ = 2X, 6cc. y- = 2, &c. and by Divifion &c. Therefore &c. and "T=4, 3>'=T, confequently A-> '= or 1 1 &c. and Divifion 4, &c. y- = # -I*" , by y = * -f- 1 ~ 1 l &c. Therefore x &c - an d x\~ , 2)' = *^ y -f.*- confequently 1 l &c. So that * * c ' anc^ ^' v ^" r^ % * T*" ) y~ = T*"~% ^ ^7 v - c - anc^ fion y = * * -f- r fx~'^c . Then 3v = * # -j- 4rA ~% ^ P p y Method of FLUXIONS, &c. and A 6cc. Thefe three- y == * -f- i*-"-, zyx~* = % i l make , and therefore * * * together 4- r^x- xy~ = 44*"% J &c. fo that y * * * -f- V|T*~~ &c - A "d fo on. Another defcending Series will arife from the two external Terms

be thus extracted : -4- -y and 2X, which may

3 &c. zx f -f- 41*-' i|*- , ' _ ^4x-*, &c. +^X-, &C. i-x-% &c.

* Make 3/ = 2X, &c. then y = ^x, &c. and (by Divifion) y 1 and c - There- = x~*, &c. x>'~ =|,&c. and y=- T> & &c. and S and fore * , = * T) &c. = _y T (by Divifion) 3_y 1 &c. and * &c. and xy~* = * -g-*" , _y= o, zyx~* = 1 &c. Therefore * 4- , &c. and jy = * * 4- ~x~\ 3_y=* i-j..*- J 6cc. &c. ^.i^ , The afcending Series in this Equation will arife from the two ex- l or the ternal Terms 2yx~* and xy~ ; multiplying whole Equa- one of the external be clear'd from tion by y, (that Terms may i 1 we mall have 4- x 4- x~ = o, of which y,) yy 3^* zxy 2y the Refolution is thus :

- \ S 9 - v* -1 vl- _I_. - v3 M^~ t^Vv # ^S~" ^^* ^ "T"*\ LA/V&r * Jlr / ^ *r ' and INFINITE SERIES. 291 and a 1 of Then = **> &c. 2v*x~ * * A* , y. y'y confequently = 1 * * &c. and 8cc. and y = 4-v*, by extracting the fquare-roor,

Then yy = -f- o, &c. and 2.vy = -4--V-'

l t &c. and therefore x~ = * * &c. and * * * 2)' -^^S _>' = 5 &c. |.v , &c. 33, 34. The Author's Procefs of Refolution, in this and the following Examples, is very natural, fimple, and intelligible; it proceeds Jeriatim & terminatim, by p'afling from Series to Series, and by gathering Term after Term, in a kind of circulating manner, of which Method we have had frequent inftances before. By this means he collects into a Series what he calls the Sum, which Sum is the value of - or of the Ratio of the Fluxions of the Relate ,v and Correlate in the and then the former given Equation ; by Pro- blem he obtains the value of y. When I firft obferved this Method of Solution, in this Treadle of our Author's, I confefs I was not u little pleafed ; it being nearly the fame, and differing only in a few circumftances that are not material, from the Method I had happen'd to fall into feveral years before, for the Solution of Algebraical and Fluxional Equations. This Method I have generally purfued in the courfe of this work, and fliall continue to explain it farther by the following Examples. l The Equation of this Example i 3^ -f- y + x -+- xj y o = being reduced to the form of a double Arithmetical Scale, will (land as here in the and the , Margin ; ~ v v<) Ruler will difcover two cafes to be try'd, of xI which one may give us an afcending, and the other a defcending Series for the Root y. And firft for the afcending Series.

Terms as The being difpofed you fee, makej/=i, &c. then &c. Therefore y=x, y = x, &c. the Sign of which Term it will * x being changed, bej/= -{- 3 AT, &c. = * 2.v, &c. P p 2 and 292 77->e Method of FLUXIONS, * x x &c. Then and therefore y = , y = * -+- #% &c. and thefe each 'tis .vy = *% &c. deftroying other, y = * * -+**, therefore * 3 &c. and = , &c. Then _y *-t-7.* _y=** ^x*, and &c. it will be 5 &c. = * -f- x', = * * * .I* , &c. xy j- therefore * und y = * * ^x* y &c. &c. in The Analyfis the fecond cafe will be thus :

* 3 -+ I2X~ , &C. 1 x 6*- . h 4 ~t- * , &c 2 f~ 1 2X~1, &C. 1 Z V = AT 4 6*-" +- 6.V~ * I2AT~*, 6CC.

x l Make = , &c. then = x, &c. Therefore xy ;' _y &c. and = x, changing the Sign, 'tis xy-=. % x 3*, &c. * &c. and = 4*, therefore y = * -h 4, &c. Then jy= * &c. 4, andj = i, &c. and changing the Signs, 'tis xy = * * H- 5 -f- i, 6cc. = * # -l- 6, &c. and y = * * 6x~*, &cc. &c.

y 35, 36. If the given Equation were ^ ==:i-f-^-f.^._f_^-

H &c. its Refolution be thus : ^ , may perform'd

zx'i &c- a X &c - 4 > XV A*

, &c. A 1 * 1, &c. - a* , &c.

A* A4 *" f * 2^2 *^ ^^4 "T~ i/)3 "I" o>,4

Make y i, &.c. then x, &c. Therefore 2 f y= o a &c. and * &c. and therefore y + ;, _y =*+-, &c. Then

~i = J7 j therefore

&c. and * * * * = , &c. And fo on. y = , j + Now and INFINITE SERIES. 293

Now in this Example, becaufe the Series | -+- ^ 4- ^ -+.

=? . &c. is to it will be or . ' equal ' h I, ay' xy 4 a x /y= A; J tf that A o which =jy + *", is, jx -f- rfx xx ay -+- j = ; the Equation, by the particular Solution before deliver'd, will give relation of the Fluents yx ay -{-ax I** = o. Hence y = a* -_xx , * an(j Divifion x -f- -, -f- r ', &c. as found a x Jy= za h za~ 2a* above. 37. The Equation x-' of this Example being tabulated, or reduced to a double Arithmeti- cal Scale, will ftand as here in the Margin. Where it may be obferved, that becaufe of can be but one the Series proceeding both ways ad injinitum, there cafe of exterior Terms, of which the Solution here follows: 294 Th* Method of FLUXIONS, be an affirmative was fuppos'J to Velocity. This Remark mull take as often as there is occafion for it. place hereafter, this the Author 38. In Example puts x to reprefent the Relate or the be Quantity, Root to extracted, and y to reprefent the Cor- relate. Bat to prevent the confufion of Ideas, we mall here change into fo .v into y, and / A", that y (hall denote the Relate, and x the Correlate Quantity, as ufual. Let the given Equation there-fare be i 1 - ~x zx' whofe is to = 4** -\- -h -+- } Root 2xy' -f^ 7#* y be extracted. Thefe Terms being difpoled in a Table, will ftand thus: And the Refolution will be as follows, taking y and -t- x for the two external Terms.

5 I * * * ** 1 + J _j_ ^i 2J34-4* X 1 * * '* i: Z- _|- 7A; If -t X 1 a x * 1 - - J = I*' * A'H-ZX + I*** X 1

* * * 4..1 x/ # * * * * * *

1 At. * * y

a &c. -l-x &c. becaufe it is j/=|A;, then^ = , Now jx=s * &c. it will be alfo * &c. And whereas it o, y= o, is^ = |-x, &c. it will be zxy^ = x*, &c. and therefore y = * # -f- x* &c. * &c. then * * >r 3 &c. be- 4**, = 3**, _y= , Now caufe it is y = * -f- o, &c. it will be alfo y^ == * -\- o, &c. and 5 2Ay = * -f- o, &c. and confequently y E= ***-{- 7^^, &c. and therefore y = ***-{_ 2**, &c. And fo on. There are two other cafes of external which Terms, will fupply us with two other Series for the Root y, but they will run too much into Surds. This fufficient to (hew the may be univerfality of the Method, and how we are to proceed in like cafes. 39. The Author mews here, that the fame Fluxional Equation often afford a of Series for may great variety the Root, according as fhall introduce we any conftant quantity at pleafure. Thus the of Art. or Equation .34. j/=i 3* -\-y -j- # -f- xy, may be refolved after the following general manner: and INFINITE SERIES. 295

l r^3*+ ** y = + * * 1 1 a 4. x -fza* + i

1 ax ax ^ Ji ax x* axt,, isV.

Here inftead of 6cc. we make &c. making ji/ = i, may _y=o, becaufe and therefore y = a, &c. then y = o, &c. then y a, 6cc. and confequently y = * -f- tisA' 1 &c. and therefore we have fourth Example = i}' , may eafily x^ c> it muft be when reafon of the = -i->'> & Secondly, done, by fquare-root of a negative Quantity, we fhould otherwife fall upon aflame a impoflible Numbers. Laftly, we muft fuch Number, when otherwife there would be no initial Quantity, from whence to begin the of the Root that when the Relate computation ; is, Quantity, or its Fluxion, affects all the Terms of the Equation. 41,42,43. The Author's Compendiums of Extraction- are very curious, and fhevv the univerfality of his Method. As his feveral ProcciTes want no explanation, I lhall proceed to refolve his Exam- if the ples by the. foregoing general Method. As given Equation x 1 or x 4 the Refolu-tion werej=:- , y /-' = , might be thus : y 296 The Method of FLUXIONS,

3 T O * * < . X* l<3-7x = , &c. 'y

1 I f f ' a.~ -f- a~*x \a~$x*- -\- a~ 7 x*, &c. J 4-^~*A" ; * ** /7 _1_ - _I_ 'IL 1^ AT -t- -4- see. j ,77 ga , ,

Make = &c. then conftant a, it _y o, afluming any quantity may l l be &c. 'Then Divifion > a~ &c. and y= a, by y~ = , 1 a~ l 6cc. therefore = * -f- a* , &c. and = * -4- x, _y confequently _y l Then by Divifion y~ = * -{- -3x, &c. and therefore y = 3 * * &c. and * * &c - Then a~ix, confequently _y = 4~ ^S 1 1 Divifion s'x &c. and therefore = again by y-' = 4^ , y 1 A; a-*x* * * *H-|d~5 .vv&c.and confequently/ = * * * ^x', ihall &c. And fo of the reft. Here if we make a = i, we have i x IK* .x* 4 = -f- -f- , &c. y |-|-Ar Or the fame Equation may be thus refolved :

' 1 3 8 J - y~ == - A" -f- 2 AT" -f- I4AT- -+- 2 l6x- 3, &C. J ' 8 I 2i6x'~~ 3, &c. AT" 2 7 JZ 28ox- I 7 = -f- 2AT~ + l8x- + ) &C,

Make z &c. 2Ar~3 y~'= A-S&c. or_y=^~ , Thenj/= , therefore l 3 &cc. and Divifion 2fc.and y =z* -\-2x~ , confequently by 1 -f-2 .v~ 7 &c. 8 and therefore r=* , Then j/=* i4x~ ,&c. y- 8 12 * &c. Divifion &c. Then = *4-i4.v~ , and by _)'= * *+i8^^ , * J 3 l * 2 l ^ * 21 6jf 3 &c. and therefore = * * -f- l6x~ y = y~ } and Divifion 17 &c. And fo on. &c. by y = * * * + 28o^~ , Another afcending Series may be had from this Equation, viz. ** X' -f- -f- &c. it and y=^/2x \ ^ , by multipying by y, i the firft of then making Term yj. J 44. The Equation y = 3 -+- 2y x~ y- may be thus refolved :

1 -4- ojc- , &c. IN FINITE SERIES. 297 ke then Ma y = 3 ,Scc. y = ^x, 6cc. Therefore zy = 6x, &c. I i &c. and x~ y = c)x, and confequently_>' = * 3*, 6cc. Therefore y == * &c. Then * 1 6cc. and x~ I l = IA*, a_>' = -f- 3* , y == 4 * 6cc. * 6# &c. and 3 9**, = * -f- , = * * -f- 2AT Therefore^ / , &c. &c. the Refolution be Or may perform'd after thefe two following manners :

zy 1=3 *-' -f- IA *,&c. ;'*-'\= *

l -* ' 2 _? v~~ 1 v~~ '4 v~"~ &"f*A'^* _j _/ ^ ^^^r.-v 4 "F^ j T = 2A--j-i 3 A-~' &C.

Make 6cc. then &c. and zy-=. 3, &c. or_y =r ?, j= o, 1 l or x -iy-- =-j- , &c. Therefore = * , &c. / %x- 2_y %x~ 1 ~ z * -f- T*"", } &c. and * x c - ancl = y = % > & by fquaring x~'j* 1 2 2 -'''" * , &c. and therefore * 6cc. and = I 2_>'=r* -f- ^A*~ , 2 * * &c. fo on. y = I-*" } And divide J Again, the whole Equation by y, and make x y = 2, &c.

1 " &c. And becaufe &c. and 1 &c. thenj' = 2A;, j/=2, j'^ni^ , "" 1 1 1 1 1 "" 1 Aj &c. c - = , and = I-* } ^ therefore 'tis^" 3>'~~ yx~* 1 1 * ^C - an(l * c - Then becaufe = H- T-^" ) y = H- T> & j^y" == I 2 l ~4 &c. and * c - 'tis * o, = -+- -I-*" , & = * * v + y jx~ T" > &c. and y * * TX~*> ^cc> ^c - 45, 46. If the propofed Equation be_y = y -\- x~' X~-, its Solution may be thus :

77 = x-*+x~ /; s 4-jrJ _!_*- 298 The Method of FLUXIONS, of which the Fluents as we pleafe, may be exprefs'd in finite Terms; return to thefe ibmetimes but to again may require particular Expe- dients. Thus if we aflume the 2x x* } Equation y = -f- ^ , the and taking Fluxions, putting x= i, we {hall have = 2 jr 1 as alfo ~ i |x -f- -for , = ^x -+- ~x*. Subtract this laft from

t the and we fhall have i foregoing Equation, j/ = 2#-f-l* , ZX the Solution of which here follows.

Let the be , 47. propos'd Equation y= -f- i 2#-{- !#*, of which the Solution may be thus :

1 \ e ex' ' fx gx* __ fx = o.

By tabulating the Terms of this Equation, as ufual, it may be l obferved, that one of the external Terms y -+- ^yx~ is a double Term, to which the other external Term i belongs in common.

Therefore to feparate thefe, afllime y = zex, &c. then . i-

&c. and i e &c. and = e, confequently y = -f- , therefore y ==x &c. or -f- ex, That is, becaule 2ex = x -f- ex, ze=. i -j-e, 'tis ez= or &c. So if we make z i, _y= 2x, _y=* -f- 2/x , 5cc. then i- = * y^j &c. therefore y = * ~\~fx 2x, &c. and 1 v * &c. that or -i. = -+- l/x A-*, is, 2/= |/ i, /= So that == * &c. So if we = * * +- &c. then _y -f-A'% makejv' zgx*, 1 &c. and therefore 1 s== * * , = * * -4- ix ,

* * -f- -f- &c. or or -_ __. fo and/ = |*-> ^s 2g = -jg+, ^ > 3 4 that * &c. So if we make *. 2/^x j =# ^-.v , / = * * , &c. then

3 /w &c. and therefore * * * -f- /JA: 3 -=*#*- , _y= , &c. and * * * 4 &c. Butbecaufe here /= -f- ^r^ , 2^=^^, this Equa- tion would be ubfurd except = o. And fo all the fubfequent Terms will this vanifh in infinitum, and will be the exact value of y. And the fame may be done from the other cafe of external Terms, as will appear from the Paradigm. 48. Nothing can be added to illuflrate this Investigation, unlefs

would demonftrate it as is we fynthctically. Becaufe^ =ex* } here found, and INFINITE SERIES. 299

( found, therefore y = e:v+~', or y = Iff! . Here in (lead of ex'

~ as and we fhall = , at firft. fubftitute y, have_y x given x~- 2.V 49, 50. The given Equation y =yx~- 4- -f- 3 4- 4*-' may be thus reiblved after a general manner.

n 2x 1 4- x- 2 .v-* &c. y = 4- 3 4A- 4- i-*-* , ' I 3 / -f- i -f- 4-v 4-rf.v~ rtx~~ 4- ax~* ~ z "~' 3 -~ x ----- * A a*~* + *~ &c. )\ 4 T*~* , J -\-ax~* ax~* x l #-' z .x~* y= -f- 4.v + a +- fx~ ,.&c. z 3 ^A-"~' -j- ax~ fax" .

1 1 1 z x 6cc. Therefore i 2* &c. = , x~ = , Make_y = , then; y ficc. conlequently y = * -f- i -+- 3, &c. = * 4, 6cc. and therefore z * &c. and confe- j = * -+- 4.x, &c. Then x~ y = 4-v~", quently y = * * -f- o, 6cc. and therefore afluming any conftant quantity a, it may be y = * * -+- a, &c. Then #-*_y = * * 1 z ^A,-" &c. and therefore , * * * ax~* &c. and , j = -+- +- x~ , 1 * * fix' 1 x" &c. And fo on. Here if we make y=. * , 1 a 'tis * * H- , &c. = o, ^ = + 4* ; ^ ^ 3 of this is which 51, 52. The Equation Example y=. ^xy* -\-y, we fliall refolve by our ufual Method, without any other prepara- be tion than dividing the whole by j*, that one of the Terms may which will reduce it clear'd from the Relate Quantity ; yy~^ ^ thus : c= 3r, of which the Refolution may be

' s X-- * 3 -~-x &c. 3x -f- f -f- TT -|- ^rx* + , * 3 4 X -V^ TTTT^ TT4 &C ' 6 7 y = fx -+ T'^ -4- TTY*"> &c -

Make = .&c. or the Fluents, = |jc% 6co. jJ;y"~^ 3#, taking %y~' or 1 &c. or 6 &c. And becaufe y^ = fx , y = fAr , y$ = fx*, 1 it will be &c. and therefore ^ 4- &c. jj)T~7 ==*_{- f^ , ^= 5 l x7 &c. and * -f- &c. and * 4- T > ^.x , y^ = ^x , by cubing y= V 'tis v? &c. Then becaufe * &c - = * * -f- -V > y"' = rV^'S ji/y""^ ' 4 ' 4 * &c. and * * v > &c.and therefore * # -+- T , = T Tr- 37' = T ;-j + * * v8 &c - And * on - &c. and by cubing ;= 4- T TT- ) Qjl 2 53- -oo 7%e Method of FLUXIONS,

in the x i== zx 53. Laftly, Equation y = zy^ -+- ty*, orjj/y -4- xx*, afTuming c for a conftant quantity, whofe Fluxion therefore is o, and taking the Fluents, it will be 2V*= 2c -f- 2x -f- ~x'^, or 1 -+- x -f- -i-x'. Then c -+- 2cx -f- A-* -f. y*=c by fquaring, _> = 3 -icx* - Here the Root receive as diffe- + !** -+ f^ _y may many rent values, while x remains the fame, as c can be interpreted diffe- 1 rent ways. Make c = o, then y = x -+- -ix* -+- loc*. is here The Author pleas'd to make an Excufe for his being fo in will minute and particular, dilcuffing matters which, as he fays, into but I think this but feldom come practice ; any Apology of kind is needlefs, and we cannot be too minute, when the perfection of a Method is concern'd. We are rather much obliged to him for giving us his whole Method, for applying it to all the cafes that and for arife. may happen, obviating every difficulty that may The of thefe Extractions is there ufe certainly very exteniive ; for are no the inverfe Problems in Method of Fluxions, and efpecially fuch as are to be anfwer'd by infinite Series, but what may be reduced to fuch Fluxional Equations, and may therefore receive their Solutions from hence. But this will appear more fully hereafter.

SECT. IV. Solution of the third Cafe of Equations, with fame neceffary Demonftrations.

the more methodical Solution 54. TT* O R of what our Author calls a moft troublejbme and difficult Problem, (and furely the Inverfe Method of Fluxions, in its full extent, deferves to be call'd fuch a Problem,) he has before diftributed it into three Cafes. The firft Cafe, in which two Fluxions and only one flowing Quan- in the he has tity occur given Equation, difpatch'd without much the affiftance difficulty, by of his Method of infinite Series. The fecond Cafe, in which two flowing Quantities and their Fluxions are any how involved in the given Equation, even with the fame affiftance is flill an operofe Problem, but yet is difculs'd in all its varieties, by a fufficient number of appofite Examples. The third Cafe, in which occur more than two Fluxions with their Fluents, is here very artfully managed, and all the difficulties of it are reduced to the other two Cafes. For if the Equation involves (for inftance) three Fluxions, with fome or all of their Fluents, another Equation ought to be given by the Queftion, in order to a full De- terminationj and INFINITE SERIES. 301 as has been termination, already argued in another place; or if not, the Queftion is left indetermined, and then another Equation may be affumed ad libitum, fuch as will afford a proper Solution to the And the reft Queftion. of the work will only require the two with former Cafes, fome common Algebraic Reductions, as we fhall fee in the Author's Example. to confider 55. Now the Author's Example, belonging to this third Cafe of finding Fluents from their Fluxions given, or when are there more than two variable Quantities, and their Fluxions, either or underftood in exprefs'd the given Equation. This Example is z in zx 4- yx = o, which becaufe there are three Fluxions A-, and therefore y, z, (and virtually three Fluents x, y, and z,) and but one I Equation given ; may affume (for inftance) x=y, whence x and fubftitution =JK, by zy z -\-yy = o, and therefore zy as & + T)'* = Now here are only two Equations x y== o l and zy z-\-^y =o, the Quantities x, y, and z are ftill variable and Quantities, fufceptible of infinite values, as they ought to be. Indeed a third Equation may be had, as zx z-\-x* = o; but as this is derived only from the other two, it brings no new limitation with but leaves it, the quantities ftill flowing and indeterminate quantities. Thus if I mould affume zy=a-\-z for the fc- cond then Equation, zy=z, and by fubftitution zx zjr-k-yx=;o, or - x .Ixv 1 y = j^ = -f- -f- ^x'-x, &c. and therefore y = x -+- ix

S &c - which two are a H-TT# J Equations compleat Determination. if we affume with the Author s and thence Again, x=j , x=Z)y t 1 1 we mall have fubftitution z = o, and thence by <\.yy -^-yy zy z -+- ^ = o, which two Equations are a fufficient Determina- tion. indeed have a zx o but as We may third, z -+- ^x^ = ; this is included in the other two, and introduces no new limitation, will ftill the quantities remain fluent. And thus an infinite variety 1 of fecond Equations may be aflumed, tho it is always convenient, that the affumed Equation fliould be as fimple as may be. Yet fome caution muft be ufed in the choice, that it may not introduce fuch a limitation, as fhall be inconfiftent with the Solution. Thus if I fhould affume zx z= o for the fecond Equation, I mould have o to be zx z = fubftituted, which would make yx = o, and therefore would afford no Solution of the Equation. 'Tis eafy to extend this reafoning to Equations, that involve four or more and their but it would be Fluxions, flowing Quantities , needlefs here to multiply Examples. And thus our Author has com- for- pleatly folved this Cafe alfo, which at firft view might appear midable Method 302 7%e of FLUXIONS, midable enough, by reducing all its difficulties to the two former Cafes. Author's of 56, 57. The way demonstrating the Inverfe Method Fluxions is but of Short, fatisfactory enough. We have argued elfe- that from the where, Fluents given to find the Fluxions, is a direct and and fynthetical Operation ; on the contrary, from the Fluxions to find the is given Fluents, indirect and analytical. And in the order of nature mould Synthefis always precede Analyfis, or Com- mould before pofidon go Refolution. But the Terms Synthefis and are often ufed in a and taken Analyfis vague fenfe, only relatively, as in this For the direct of place. Method Fluxions being already demonftrated the fynthetically, Author declines (for the reafons he to demonstrate the Inverfe gives) Method fynthetically alfo, that is, primarily, and independently of the direct Method. He contents himfelf to it that prove analytically, is, by fuppofing the direct Me- as demonstrated thod, fufficiently already, and Shewing the neceSTary connexion between this and the inverSe Method. And this will al- be a full of the ways proof truth of the conclufions, as Multiplica- tion is a good proof of Division. Thus in the firlt Example we that if the is 1 found, given Equation y -f- xy y=^x x I, we Shall have the Root x 1 x 3 -* 6 y=x -j- f -f- ^.x* -^r* , cc. To the truth of which prove conclufion, we may hence find, the direct i 2x 1 3 by Method, _y = -{-x .i* -f-f#* TTX', &c. and then fubStitute theSe two Series in the given Equation, as follows;

------jf 1 _. < _>_ , __ 6 y f_ X _ 4- ]_ X X _J_ X _^X - 6 _{_ X ------1_ X __. #3 3.4 __ _?_ y _j_ ^f _{_ X 5 -- r A-* 6 y -f. 2 X -{- ^ ^ + -rX' ^X x 1

thefe mall find Now by collecting Series, we the refult to produce the and given Equation, therefore the preceding Operation will be fufticiently proved. In this and the 58. fubfequent paragraphs, our Author comes to and fome of the chief open explain My Steries of Fluxions and Fluents, and to give us a Key for the clearer apprehenfion of their nature and properties. Therefore for the Learners better instruction, I Shall not think much to inquire fomething more circumstantially into this matter. In order to which let us conceive any number of right ae &c. extended both Lines, AE, t as, indefinitely ways, along which or a Body, a defcribing Point, may be fuppofed to move in each Line, and INFINITE SERIES. 303

Line, from the left-hand towards the right, according to any Law or Rate of Acceleration or Retardation whatever. Now the Motion of every one of thefe Points, at all times, is to be eftimated by its diftance from fome fixt in the fuch point fame Line ; and any Points be chofen for this in each in may purpole, Line, fuppoie B, I), /3, which all the Bodies have been, are, or will be, in the fame Mo- ment of Time, from whence to compute their contemporaneous Augments, Differences, or flowing Quantities. Thefe Fluents may be conceived as negative before the Body arrives at that point, as in nothing when it, and as affirmative when they are got beyond it. In the rlrft Line AE, whole Fluent we denominate by x, we may the to move or with luppofe Body uniformly, any equable Velocity ; then may the Fluent x, or the Line which is continually defcribed,

A B C J> :E

c/^ , * /? 9" 8 2 ! 1-1 II reprefent Time, or {land for the Correlate Quantity, to which the feveral Relate Quantities are to be constantly refer'd and compared. For in the fecond Line ae, whofe Fluent we call y, if we fuppofe the Body to move with a Motion continually accelerated or retarded, according to any conftant Rate or Law, (which Law is exprefs'd by of x and and known then any Equation compos'd y quantities j) will there always be contemporaneous parts or augments, defcribed in the two Lines, which parts will make the whole Fluents to be contemporaneous alfo, and accommodate themfelves to the Equation in all its Circumftances. So that whatever value is afiumed for the value of Correlate x, the correfponding or contemporaneous the Re- late y may be known from the Equation, and vice versa. Or from the Time being given, here represented by x, the Space represented be known. The we call of the by y may always Origin (as may it) Fluent x is mark'd by the point B, and the Origin of the Fluent y the at the fame time are in by the point b. If Bodies found A and then will the Fluents be BA and ba. If , contemporaneous at the fame time, as was fuppofed, they are found in their refpec- and then will each Fluent be If at the tive Origins B , nothing. fame time they are found in ^ and c, then will their Fluents be -1- BC and -\-bc. And the like of all other points, in which the i moving 304 The Method of FLUXIONS, moving Bodies either have been, or fliall be found, at the fame time. As to the Origins of thefe Fluents, or the points from whence we to tho' muft be conceived begin compute them, (for they to be variable and indetermined in refpedt of one of their Limits, where the de- are fcribing points are at prefent, yet they fixt and determined as to their other Limit, which is their Origin,) tho' before w appointed the Origin of each Fluent to be in B and b, yet it is not of abfolute mould or at the fame neceffity that they begin together, Moment of Time. All that is neceflary is this, that the Motions may continue obferve the fame rate as before, or that they may of flowing, and have the fame contemporaneous Increments or Decrements, which will not be at all affected by changing the beginnings of the Fluents. The Origins of the Fluents are intirely arbitrary things, and we other we If may remove them to what points pleafe. we remove them from B and b to A and c, for inftance, the contemporaneous Lines will ftill be AB and ab, BC and be, &c. tho' they will change their names. Inftead of AB we fhall have o, inftead of B or o inftead of we fliall have we fliall have -+- AB, -+- BC -f- AC ; &c. So inftead of ab we fliall have ac -{-be, inftead of b or o we fliall have be, inftead of-f- /Wwe fliall have -+- bc + cd, &c. That which determines the Law of is, in the Equation general flowing increafe or diminifh or or or increafing, we may always x, y y both, occafion and the by any given quantity, as may require, yet Equa- ftill the rate of which is all tion that arifes will exprefs flowing ; that the ufe and of which is neceffary here. Of conveniency Reduction we have feen feveral in fiances before.

If there be a third Line a.e, defcribed in like manner, whofe its with Fluent may be z, having parts correfponding the others, as

c - there muft be another either or a/3, &y, y, & Equation, given aflumed, to afcertain the rate of flowing, or the relation of z to the Correlate x. Or it will be the fame thing, if in the two Equations the Fluents x, y, Z, are any how promifcuoufly involved. For thefe and determine the of two Equations will limit Law flowing in each Line. And we may likewife remove the Origin of the Fluent z of the Line And fo if there were to what point we pleafe a. more Lines, or more Fluents. has been faid an inftance. 59. To exemplify what by eafy Thus inftead of the Equation y=xxy, we may aflume y = xy -+- xxy, is or x is diminifli'd for where the Origin of x changed, by Unity ; inftead of x, lawfulnefs of which j -J-- x is fubftituted The Re- duction and INFINITE SERIES. 305 be thus duftion may proved from the Principles of Analyticks. Make x i -\-z, whence x=z, which (hews, that xand2 flow or increale alike. Subftitute thefe infteadofx and x in the Equation^':=xxy, and it will become y = zy -+- zzy. This differs in nothing elle from the afTumed Equation y = xy -f- xxy, only that the Symbol x is changed into the Symbol z, which can make no real change in the argumentation. So that we may as well retain the dime Symbols as were at - given firft, and, becaufe z-=x- i, we may as well fuppofe x to be diminiih'd by Unity. 60, 6 1. The Equation expreffing the Relation of the Fluents will at all times of their for give any contemporaneous parts ; afluming different values of the Correlate Quantity, we ma'!, thence have the correfponding different values of the Relate, and then by fubtradion we fhall obtain the contemporary differences of each. Thus if the - were x where is to be a given Equation y = -{- , x fuppos'd quan- or tity equably increafmg decreafing ; make x = o, i, 2, 3, 4, 5, &c. then = infinite, 2, &c. fucceifively, y 2|, 3.1, 4^, 5-^-, refpectively. And taking their differences, while x flows from o to i, i to from to 2, from 2 3, &c. y will flow from infinite to 2, from from to &cc. that 2 to 2-i-, 2| 3.1, is, their contemporaneous parts will be i, &c. and &c. i, i, i, infinite, i, , -{.I, refpeclively. Likewife, if we go backwards, or if we make x negative, we mall &c. have x = o, i, 2, which will make _y= infinite, 2, &c. fo that the differences will be as be- 2-i-, contemporaneous fore. Perhaps it may make a ftronger impreffion upon the Imagina- tion, to reprefent this by a Figure. To the rectangular Afymptotes GOH and KOL let ABC and DEF the be oppofite Hyperbola's ; bifed An- the indefinite gle GOK by right Line yOR, perpendicular to which draw the Diameter BOE, meeting the Hyperbola's in B and E, from whence draw BQP and EST, as alfo CLR and DKU parallel to GOH. Now if OL is made to reprefent the indefinite and equable -' quantity x in the Equation y = x -f-

l then CR For CL - may reprefent y. = ^ = , (fuppofing

= OL = x , therefore CR =^ LR -4- R r or 306 *The Method of FLUXIONS, or x -f- Now the of or in if y = ^ Origin OL, x, being O j then or x = o, CR, y, will coincide with the Afymptote OG, and therefore will be infinite. If x= i = then = BP=2. OQ^ _y If x = 2 = OL, then y = CR = 2 i. And fo of the reft. Alfo the if proceeding contrary way, x = o, then y may be fuppofed to coincide with the Afymptote OH, and therefore will be negative and infinite. If x = OS = i, then y = ET = 2. If x = OK 2, then = Dv = 2~, &c. And thus we = _y may purfue, at leaft by Imagination, the correfpondent values of the flowing quantities x and_y, as alfo their contemporary differences, through all their to their poiTible varieties ; according relation to each other, - as exhibited the x . by Equation y = +- .

The Transition from hence to Fluxions is fo very eafy, that it may be worth while to proceed a little farther. As the Equation expreffing the relation of the Fluents will give (as now obferved) of their or differences fo if thefe differences any contemporary parts ; are taken very fmall, they will be nearly as the Velocities of the are defcribed. moving Bodies, or points, by which they For Mo- tions continually accelerated or retarded, when perform'd in very if fmall fpaces, become nearly equable Motions. But thofe differences are conceived to be dirninifhed in infiriitum, fo as from finite differences to become Moments, or vanifhing Quantities, the Mo- therefore the tions in them will be perfectly equable, and Velocities of their Defcription, or the Fluxions of the Fluents, will be accurately as thofe Moments. Suppofe then x, y, z, &c. to reprefent Fluents in any Equation, or Equations, and their Fluxions, or Ve- locities of increafe or decreafe, to be reprefented by x, y, z, &c. to be and their refpedlive contemporary Moments op, oq, or, &c. will be the of the of where p, q, r, &c. Exponents Proportions the Moments, and o denotes a vanifhing quantity, as the nature of will as Moments requires. Then x, y, z, Sec. be op, oq, or, &c. as So that &c. be ufed inflead that is, p, g, r, &c. ,v, y, z, may

r ^c - ni the of the Moments. That is, the of/>, ?> -> designation fyn- of be chronous Moments x, y, z, &c. may reprefented by ox, oy, be oz, &c. Therefore in any Equation the Fluent x may fuppofed to be increafed by its Moment ox, and the Fluent y by its Moment &cc. or x &c. be fubftitnted in the oy, -+- ox, y -{- oy, may Equation flill be inflead of x, y, &c. and yet the Equation will true, becaufe the Moments are fuppofed to be fynchronous. From which Ope- ration and INFINITE SERIES, 307 ration an Equation will be form'd, which, by due Redu&ion, muft the relation the Fluxions. neceflarily exhibit of Thus, for example, if the Equation y = x -+- z be given, by Subftitution we fliall have y -f- oy =. x -f- ox -+- z -+- oz, which, be- will become ox or x caufe y = x -+- z, oy = -f- oz, y = -f- z, winch - is the relation of the Fluxions. Here again, if we afllime z = > the or zx = i, by increafing Fluents by their contemporary Mo- ments, we fliall have z -+- oz x A- -f- ox = i, or zx + ozx -f- oxz -f- oozx = i. Here becaufe zx = i, 'tis ozx -f- o.\z -+- oozx = o, or ~x-f- A- -+- 02X = o. But becaufe ozx is a vanifliing Term in

the 'tis zx A z or refpect of others, -f- = o, z === -f =

Now as the Fluxion of z conies out negative, 'tis an indication that in as A- increafes z will decreafe, and the contrary. Therefore the x if z - or if the relation of the Fluents Equation y = -+- z, = ,

- then the relation of the Fluxions will be x be y = x -+- , y =

- And as before, from the Equation y =: x -+- we derived the or the fo from the contemporaneous parts, differences of Fluents ; Fluxional Equation y = x ^ now found, we may obferve the rate of flowing, or the proportion of the Fluxions at different values of the Fluents.

1 1 becaufe it is x : : : I : i : : x : x i For y \ ; when # = o, or when the Fluent is but beginning to flow, (confequently

is it will be x : :: o : i. That the Ve- when y infinite,) y is, locity wherewith x is defcribed is infinitely little in comparifon of the velocity wherewith^ is defcribed; and moreover it is infinuated, (becaufe increafes finite tho' never of i,) that while x by any quantity, fo will an infinite at the little, y decreafe by quantity fame time. This will appear from the infpeclion of the foregoing Figure. When

then : : : i : x= i, (and confequently _)'= 2,) x y o. That is, x will then flow infinitely faflrer than y. The reafon of which is, its or that y is then at Limit, the leaft that it can poflibly be, and therefore in that place it is ftationary for a moment, or its Fluxion is nothing in comparifon of that of x. So in the foregoing Figure, BP is the lea ft of all fuch Lines as are reprcfented by CR. When

it will A- : :: : Or x=2, (and therefore y = 27,) be y 4 3. R r 2 the Method of FLUXIONS,

is there than that in the the Velocity of x greater of^, ratio of 4 then x : :: : 8. And fo on. So that to 3. When x=3, y 9 the Velocities or Fluxions conftantly tend towards equality, which - they do not attain till (or CL) finally vanishing, x and y become the like be obferved the values of equal. And may of negative x and y.

SECT. V. 77je Refolution of Equations^ whether Algebrai- cal or Fluxionalt by the ajfiftance of fuperior orders of Fluxions.

the foregoing Extractions (according to a hint of our Au- be more and ALLthor's,) may perform'd fomething expeditioufly, without the help of fubfidiary Operations, if we have recourfe to orders (hew this firft an Inftance. fuperior of Fluxions. To by eafy Let it be required to extradl the Cube-root of the Binomial 3 1 3 a> x or to find the Root of this a -\- x"' ; -f- , y Equation y = 3 or rather, for fimplicity-fake, let it be_) =a* -+- z. Then y=.a, &c. or the initial Term of y will be a. Taking the Fluxions of l fhall have or . this Equation, we -$yy- = z = i, y = -^y~ But 1 is &c. fubftitution it will be 6cc. and as it y = a, by y = jtf" , 1 'tis &c. a is taking the Fluents, y= * -f- \a~ z^ Here vacancy left for the firft Term of y, which we already know to be a. For another Operation take the Fluxions of the Equation jj/=: j/~* ;

5 whence y = %yy~* = jy~ - Then becaufe y = a, &c. 'tis y = ?&"*, &c. and taking the Fluents, 'tis y = * 3 l z, &c. and the Fluents 'tis = * * , $a~ taking again, jy- ^a-^z 6cc. Here two vacancies are to be left for the two firfl Terms of the next y, which are already known. For Operation take the

of the s that - 6 Fluxions , is, = -f- =. Equation y=. ^y~ y 9-yy~

8 8 . Or becaufe 'tis &c. _l_^.ji- _7=c,&c. _y=if(S- , Then taking the Fluents, 'tis * a-*z, &c. = * * 1 6cc. and y= L^. y -^a-^z , * * * s &c. for another the y = -v Ta~*z*, Again, Operation take

of the 9 Fluxions Equation y =: ~~y^ ; whence y = -*f^~

11 - Or becaufe &c. 'tis = Iry" y = a, "y = IT^"", &c.

1 Then taking the Fluents, = * ^, &c. = * * y ^a- jy and INFINITE SERIES. 309

1 &c. * * * _+._ _.rt "z 5 &c. and * * * * H-a-^z , y = , _)' = '^-?-a-"z+, &c. And fo we may go on as far as we pleafe. We have therefore found at that v a -. laft, = -\ 1 , -f- ~gB y. 9 8i

or for z 'tis x- ~ , &c. x*. = -+- ( ^ writing fa + ^ C* 1* - - " ~ I -n l\f\^ ' I ' biXI ,3 ^1 m if would refolve a x into an Or univerfally, we -f- \ equivalent ra m a x and we fhall have a for the infinite make \ Series, y = -f- , m firfl of the Series or it will be a &c. be- Term y, y = , Then j_ i m j the Fluxions we fhall have caufe y = a +- x, taking ~yy = m m it will x or . But becaufe it is a , &c. be = i, y = my y = ~ m~ 1 m t t/m &c. and now the 'tis * ma x y y = , taking Fluents, y = m 6cc. becaufe it is the Fluxions it will Again, y = my , taking m x and becaufe am Sec. be y = m lyv *= m ly ; y = , m~'i ia &c. the 'tis * 'tis/ = wxw , And taking Fluents, y = I ~ ^=- m 2 ? x / i~% 6cc. and therefore y = * * m x d x% &c. 2 _ j becaufe it is m x //; the Fluxions it Again, y = 17 "', taking m - - i x ;// m x / i x will be y = m 2j/y = m a &c. 'tis m x becaufe := , = and 7 _y i x w/ And taking the Fluents, 'tis /== * m x w x m 2am &c. and v = * y =**/;/ x '^^- -ix*, CT fo as far as we x ~y-3X 3, 6cc. And we might proceed pleafe, had not been ma- if the Law of Continuation already fufficiently~ a* mam l x nifefl. So that we fhall have here a -+- x I = -f- -+- * ~~" ftt __ r m ^ 2 ni * m ^ l - - - m m x -- x - m x !LU fl x 4- w x x a ~^ -+- 2 2 3 ?^-V-4;c<, &c. tho' difcover'd Vhis is a famous Theorem of our Author's, by or rather him after a very different manner of Investigation, by .,- lldufr*>v of his Binomial >'$ vntrrrU*t> Indudion. It is commonly known by the name a Theorem, becaufe by its amftance any Binomial, as + x; may JujJf&rcL f Root of it be ex- be railed to any Power at pleafure, or any may & -^fajfifa1 m- ' . it is that m is tradted. And obvious, when interpreted by any Cff **'* r^ t r .3

i and INFINITE SERIES. 311

*'"*= * "'+' ~ , ra a" m x w x /lull have a + x I = -f- ^. -f- -7- ^7 =^*=fX;0i.^ Now this Series will Hop of its own accord, at a finite number is and that of Terms, when m any integer negative Number ; is, of is found. when the Reciprocal of any Power a Binomial to be But in all other cafes we mall have an infinite converging Series for the Power or Root required, which will always converge when a becaufe the the or the and x have the fame Sign ; Root of Scale,

is which is lefs than converging quantity, ^ , always Unity. Series from By comparing thefe two together, or by collecting

, : m i _ a - - we fliall have the two each the common quantity ^ ,

_ . TO x Series - x - x - &c. equivalent -+- -j- -f- -j-x , = x 2 , 11 "L_ 11 x + x x ==^=- , &c. from whence we 2 5 a + x |* 3 a-i-x I might derive an infinite number of Numeral Converging Series, not inelegant, which would be proper to explain and illuftrate the nature of Convergency in general, as has been attempted in the former part of this work. For if we aflume fuch a value of i as will make either of the Series become finite, the other Series will exhibit the quantity that arifes by an Approximation ad infinitum. A; be And then a and may afterwards determined at pleafure. As another Example of this Method, we fliall fhew (according to derive de Moivre's Theorem for to promife) how Mr. elegant ; to indeterminate or for raifing an Infinitinomial any Power, extracting any Root of the fame. The way how it was derived from the abftract coniideration of the nature and genefis of Powers, (which indeed is the only legitimate method of Inveftigation in the prefent cafe,) and the Law of Continuation, have been long ago communicated and demonftrated by the Author, in the Philofbphicai Tranf- actions, N 230. Yet for the dignity of the Problem, and the better to illuftrate the prefent Method of Extraction of Roots, I lhall deduce it here as follows.

the a cz* itz* * Let us aflume -+- b~ -+- -f- -4- ez* t &.c. Equation | where the value of is to be found an infinite = )', y by Series, of m the firfl: is known to be a or it is which Term already , y a", 1 &c. Make v = a -f- l>z -f- cz, -J- dz> -f- ez*, &c. and putting -y b z = i, and taking the Fluxions, we mail have r^zi -+- 2cz -+- , I2 Tie Method of FLUXIONS,

it is where &c - Then becaufe = u", = mviT-', 7, &c. we = -y = _y if we make = <*, the it will be = * maF*ks ma'-'b, &c. and taking Fluents, _> t

1 it is becaufe mvv"- , = another = _y For Operation, _y z i;"'~ . And becaufe = -f- + 12^2;*, _l_ their values a, &c. ^, &c. and 2c, 6cc. y, v, and -yfubftituting refpec- 1 - 1 m- z x OT i6*a &c. and we fliall have = zmca" -+- m , tively, jy m- J * 2fnca z -\-m-x. m i*tf*-**, 6cc. and taking the Fluents 7 = m- l m- 2 1 * * mca z* -\-m~x. z ) taking the Fluents again, y = ^^^a &c. .. - 1 ~ m iv 1 vm 1 For another Operation, becaufe/ = mw -+- mxm t

m~ l m~ z t m i / iv 2 v *v">. 'tis y m v"j + yn x -vv-{-m\m ixw

And becaufe -u = 6^+24^2;, &c. for v, v, v, v, fubftituting a,

m~ l fliall have 6mda 6cc. b, &c. 2c, &c. 6^/, &c. we )"=, -f- 6m x

m~'i 3 m w \bca -\- m x. m i x m 2l> a ~*, &c. And taking the m~ l m Fluents it will be y = * 6mda z -+- 6m x m \bca -*z -+- m x m 8cc. l/ * * w i Km 2foa ~*z, =

m 1 - l -iz &c. * * * mda m z* x ^^ x w 2foa , and/ = -+ m x

m x . And fo on z' /-

fliall therefore have tf finitum. We

And if the whole be multiply'd by s", and continued to a due it have the form of Mr. de Moivres Theorem. length, will The Roots of all Algebraical or Fluential Equations may be ex- an tradled by this Method. For Example let us take the Cubick 3 Equation y* -\~axy -\-a*y x"' 2 =o, fo often before refolved, in which y = a, &c. Then taking the Fluxions, and making z x we fliall have -+- +- -f- = o. Here = i, ^yy- ay axy a*-y 3x 1 if for y we fubftitute a, &c. we fliall have ^y -\- a 4- axy 3**, &c. =o, or>= -j:+^" = ^' &c - =.-- &C' A"d v * &c. Then the Fluxions taking the Fluents, = -i-x, taking j again and INFINITE SERIES. 313

1 again of the lafl Equation, we fliall have 37}* -f- 6y y 4- 2 ay 4- axy l 4- a 6x= o. Where if we make = a, &c. and = y 7 7 ^., we fliall have __ _!_ &c. and therefore &c. 'y= ZZif+j''-^: ' 43% CSV. 32a

1 '"" AT ' &c. and * * c - *y = * 4 , y= -h z & Again, ?yy* -{- iSyy'y "\2.ii (3 -I.'? ^

6 _j_ 675 -f- 3^7 -f- axy -\- a*-j = 0. Make 7 = a, &c. 7 =

" ^ + ^~ ^-4 6 6cc. then 5j; C- __ &c. and = , ^ '7 7 3 2a 4a

* = , &c. and therefore = , &c. * * 6cc. ^|j, 7 ^^ 7 ^ ^ ,

and * * * - &c. 7 = ^ 5 Again, 377* -1-24777-1- 18^*74- 3677*

o. Make 4- 4^7 4- ^7x7 4- a*y = 7 = a, &c. 7 = ^, &c./ =

- &c. and - &c. then , 7 = |^ , 7 =

'- &c. , &c. * :, and7=# ^-. 7 = *

&c. and * * * * ' ^C ' ^nc^ ^ on as ^ar as we , y = r68a

Therefore the Root is y x &c. =ai + H-H^ ' -|2|fl J 643f- 12^- + i68' The Series for the Root, when found by this Method, muft al- have its Powers but if defire likewife to find ways afcending ; we a Series with defcending Powers, it may be done by this eafy artifice. As in the prefent Equation y* -\- axy -f- a*y x"' 2a* = o, we may conceive x to be a conftant quantity, and a to be a flowing or to a confufion of quantity ; rather, prevent Ideas, we may change a into and x into and then the will be 5 x, a, Equation _y -f- axy -+ x*y rt 3 ax 3 = o. In this we mall have y = a, &c. and ta- the Fluxions, 'tis -4- -f- -f- +- 6x* = o, king 3j/y* ay axy 2xy x*y z "~" ay 2X],"-r- 6-V n . i r o > #& e> . tis or 7-1 But becauie &c. - , ficc. y= H r y=a, ^y= y +<**+* r 3 == i, &c. and therefore/ = * -i-.v,&c. Again taking the Fluxions 4 1 'tis -f- 4- +- -+- I2x = o, or 3_)^ 6)/ / zay axy -\-2y-\- ^.xy x*y - -6^-2^-2, -yj>-^-'j-4y+"*1 Qr / yJ+flX-f-* y* ~^ + 2a = a, &c. and = &c. 'tis == &c. king _y _y |, f~ , y 3^ &c. and =s , = , &c. and , &c. _y yrt ^ ;'=** ^

5 Again it is 3vy* -t- i8;7/ -f- 6y -f- 3^} 4- axy -+. 67 H- 6x/ H- x*_y S f 12 The Method of FLUXIONS, -rv- ? -- _ 12 = O, Or y = 4 + v -T 2 + 2 + 12 a &c. y = 4-' &c - and y , v t = = -, &c.)~/ / y 3> 3** &c. Then the * &c. = -^ , taking Fluents, 7 = , &c.

* &c. and * * c. And fo on. There- y = * ^T , y = , 2"

mall have

have -f- or . But tion, we (hall ay 2xy -+- xy = o, y= -^T 1

to have a conftant for firft becaufe we are quandty the Term of y, the i Then we may fuppofe y='~^^ = o, &c. taking Fluents and * * 'tisj/= * o, &c. y =. o, &c. Then taking the Fluxions

'tis -f- -{- = o, or Here again, a'y 27 + 4*7 xy y = "J'^.'T* their values and mall if for y and_y we write i, &c. o, &c. we have

il &c. whence * &c . * }'= ^ , y , y = ^ , c .

* * * > ^c - the Fluxions and 7 == ,71 Taking again, 'tis

J ay +6y + 6xy +xy = o, 01-7 ~*?~^ = o, <5cc. There-

&c. * * &c. fore y = * o, _y= o., j.'= * * * o, &c. y =. **#

~ 2 4 1 &-C. A' or - Q, Again, ay -+- I2y 4- 8^,7 -f- j = o, y = l ^!.'' and INFINITE SERIES.

&c. , &c. Then = **4- , &c. = + Jr^*^^, ;< ^ &c. and , &c. = * * * * 4- , _>>=:***-{- jy 45 _}=***** - - - &c. -f- loxy -f- = o, whence "' } Again, a*y-\- zoy xy y=s 5 r56 7? &c. or ***** o, Again, a*j+ 307 -f- i2*y -+- A,"^ = o, y=.

9 x 2 rf- &c. Then =- . _ . , &c y~^=:- 30 4 > !^2f ,

12x30** - 4x301-3 ==- * * -- &c. * * * , &c. *** * j ^ , y = ^ ; = 30*4 6*5 o * 6 c^c. -- -, &c. = # * * * * , _y -^ j)/=******

A' 7 fo on. So that c. and _/ = ******* , &c. And we

ox- 1 ' 4 have here * # -H OA > &c - that = y = ^ + + ^; is, _y *! ' fl , . c * ' & P + 5^ /' This Example is only to {hew the universality of this Method, we are to in other like cafes for as to the and how proceed ; Equa- have refolved tion itfelf, it might been much more fimply and ex-

in the manner. Becaufe - peditioufly, following y = -^- } by

i ~ *- *L Divifion it will be , &c . y = + ^ 6 -f- And ta- x - the Fluents, = -+- 6 4- > &c. king y ^ ; ^ 1 In the fame a*x x if it were Equation ^ 4- ^ = o, requir'd or if to exprefs x by y, (the Tangent by the Arch,) x were made the Relate, and y the Correlate, we might proceed thus. Make x jc or si - y = i, then a* a?x +- x* = o, = 14 = i 5cc J G,

&e. And the 'tis x Then x = * y, taking Fluxions, = ^ =s

whence A- &c. and , &c. = &c. = * o, x * * ~-^ 04-^, = o, that the this Series will be &c. So Terms of alternately deficient, and therefore we need not compute them. Taking the Fluxions

. , * Zxx 2 n i-i->i r 2V tis x = -j- =: , &c. Therefore x * again, = -^ , &c.

&c. and # > A- = * * , ^== * * , &c. Again, x=z j-, ^ -^ -l-^r S f 2 and Method of FLUXIONS,

>X xx 2XXXX c ,n_. tf ,2 A- . and >- and again, = -f- -^ h Subitituting i, &c.

for # and x and alfo 6cc. for it x &c. } o, # and #, will be =

16 c t. i6y c gy* - , &c. whence x =. * , occ. * ** , &c. x * # * = ' = ' C J 21* &c. A- * * * * &c. and . x &c.. _, , = ^ , ==-***#, ' " J ' 6 6 :: 5 ' 7 I . 20xx+ iQAr- 4" 2A ;xr J 2n*- -4--;o.;x4-i2.v.v-(-2.vA- and x -- Again, x = - - , again, = - -

l for and x &c. &c. and &c. re- Here x, x, writing i, ^ , ,

8 6 x &c . Sec. Then x fpeaively, 'tis = ^U. , ^ , =

J v ^i &c. J &c. , &c. 2==..... , = ..i4\ x=***i^- fi 3^ 4 * * * * * &c. x= &c. and &c. , ****** ig , ^= 7^ JJpr, That X *, &c. is, =y+.^ + -l. +

6 7.1 Cfl l let us take the , For another Example,, Equation ^*jj* .y j> if the Radius of a Circle be denoted /z*x* o, (in which, by a, of the then the and if y be any Arch fame,, correfponding right Sine x from which we are to extract the will be denoted by j) Root y. 1 then it will be or Make x=i, a*y*~ x^y = a*, ;}* = -^^ M or &c. and therefore * &c. J} &c. j/= i, y=. x, Taking 1 fhall have 2X o or the Fluxions we za*yy zxy* yy = ) a*y> nc^ ta __ or ' ^^ ^ the Fluxions Xy xy = 0, "y = ^r~7i = ^ing.

^ a or __. . 'tis o, = y again, a*y j/ 3* /= ^

* * &c. and * * 1- , Therefore = * &c. = , _y = * } J } ^

a a <7 = o, and , &c. Then ;y 4^ 5^' x*y again j/ 9_y

x^=o, orj = i^=,&c. = J,&C . There-

' * * * * &c. fore y := * &c, * > &c. = , , y ^ ^ 4 ^5^ * * * INFINITE SERIES, 317

. 6cc. r -IL * * , , and = , &c. the j^-4 ***#* Taking 6 :: S 6 7 l 'tis n Fluxions i6y = o, and again, y g.y x*y again, a*jr

2^Y + i i xv c 1 or ~ &c. z$y 1*7 x*y = o, ;< = -^ = -/, =

&c. &c. v * * &c. ,' Therefore * = = 6 ^ jJ ^.v,a* J ^V,2a

* * * -2-3 r * x4 c> === * * * = x t &c. * * * > ^ * * y _)' = gJT y

r &c. r , &c. and *****#* ^-|x , y = ****** 6 y = ~ -^- -^ -^-. . &c. Or v x &c. J * = -i- l 4- 4- , 6 TI2a t>

Fluxions 'tis axx 2a*xx = o, or x= =: o, &c,- Therefore x = * o, &c. x =: * * o, &c. Taking the Fluxions

'tis x &c. Thence * &c. again, = ^ = ^ , x= ^,

&c. * * - &c. x x = , andx=* , Again, = ^> 2^1 Oi fi ~ and x -+- &c. Therefore x , &c. x = = ^ , = ^ = 1 J 4 t ) j -^ occ. x * * * x * * * - occ. and x , = 7 , &c. = ,

v* x x * * - &c. x and x === *, Again,* = , = i ?r/24 O a fl 6 ? 1 i v y - - x * ~ * , 6cc. Therefore x , = 6 = ^j

* * * -1 &c. x * * * &c. x , 6cc. x , = -^-j g

j.J V* J -^- &c. x - f &c. and ** _ t , = , x= I 2Ofl* .. 72Ofl -^ *__ - - 6cc. And therefore x= -f- , y> 6 < ^ &c. out this If it were required to extraft the Root y of Equation, 1 we *y . .x*y* + m*y* w ^ = o, (where x =s i,) might proceed 3i 8 The Method of FLUXIONS,

'" v - - ceed thus. Becaufe ==. = &c. 'tis See. y~ ;*, = ;;;, ^ ~^ ;/ &c. the and_y= -# w*> Taking Fluxions, we fhall have 2a*yy or 2xy* 2x*yy -bJzr$*yy = o, ay xy xy -+- my o, or

^c - Therefore the 'tis r= p ^r= j taking Fluxions again,

1 v 3xy x / = o, or y = -I ^^_ jf^l; and making y = m, &c.

~ OT ' "~ OTX ' '" x ; 'tis &c. and therefore Xy * A "'x, &c. y = y= = -^ ^ /2

OT x ' &c. and y = * * * &c. the i^LV, * ~^\-', Taking za'' zx. ya

1 Fluxions 'tis -+- m o and again, a*y 4 x_y $xy xy = j again,

" 2 "'' x ~t~ vv x i/ Q ->' cfy {- in* 9 x y 7xy o, or y ___

x ,--x 9 -^ ;" ' ~ g' lx 9 ~ CT &c> Therefore * y= \y &c ' v 4 4 ^

Series is to a Theorem of our This equivalent Author's, which (in for another place) he gives us Angular Sections, For if A- be the to Radius a then will Sine of any given Arch, ; y be the Sine of another Arch, which is to the firft Arch in the given Ratio of m to i. Here if be odd the Series will m any Number, become finite j and in other cafes it will be a converging Series. And thefe Examples may be lufficient to explain this Method of it carries Extraction of Roots ; which, tho' its own Demonftration for evidence be along with it, yet greater may thus farther illustrated. Roots In Equations whofe (for example) may be reprefented by the 4 Series A -+- Ex Cx Dx 3 6cc. due general y = -f- + , (which by Re- duction may be all Equations whatever,) the firfc Term A of the Root will be a given quantity, or perhaps = o, which is to be known from the circumftances of the Queilion, or from the given Equation, and INFINITE SERIES. 319 have been abi Equation, by Methods that ^antly explain'd already. flrall have have Then making x= i, we y = B -f- 2C.v -+- 30**, &c. where B likewife is a conftant quantity, or perhaps = o, and firft Term of the Series This therefore is reprefents the y. to be derived from the firft Fluxional Equation, either given or elfe to and becaufe it is &c. the be found ; then, y = B, by taking Fluents it will be y = * Ex, ccc. whence the fecond Term of the Root will be known. Then becaufe it is_y= zC -f- 6D.v, &c. or becaufe conftant zC will the firft of this is the quantity reprefent Term y ; to be derived from the fecond Fluxional Equation, either given or to be found. And then, becaufe it is y = zC, &c. by taking the 1 Fluents it will be * &c. and * * Cx cc. y = zCx, again y = , by which the third Term of the Root will be known. Then becaufe

it is_y=6D, &c. or becaufe the conftant quantity 6D will repre-

fent the firft of the Series this is to be derived Term y ; from the

third Fluxional Equation. And then, becaufe it is y = 6D, &c. the Fluents it will be by taking v = * 6Dx, &c. y = * * 3D*'-, See. and * 3 &c. the _)'==.* * D* , by which fourth Term of the Root will be known. And fo for all the fubfequent Terms. And hence it will not be difficult to obferve the compofition of the Co- efficients in moft cafes, and thereby difcover the Law of Continua- tion, in fuch Series as are notable and of general ufe. If ihould defire you to know how the foregoing Trigonometri- cal are derived from the Equations Circle, it may be fhewn thus : on the Center with Radius a let A, AB = t the Quadrantal Arch BC be defcribed, and draw the Radius AC. Draw the Tangent BK, and through any point of the Circum- c ference D, draw the Secant ADK, meeting the Tangent in K. At any other point d of the Circumference, but as near to D as may be, draw the Secant A.tte, meeting BKin/ ; on Center A, with Radius AK, defcribe the Arch K/, meeting A in /, Then fuppofing the point d continually to approach towards D, till : it c<- .ncides finally with it, theTri- lineum K//6 will continually approach to a right-lined Triangle, and to funilitude w/ith the Triangle ABK : So that when Dd is a Moment 320 The Method of FLUXIONS,

Moment of the Circumference, it will be K-! = ^4 x =~ Da &.1 L)J AB

. Make AB the x ^ = a, Tangent BK = x, and the Arch and inflead of the BD=y ; Moments Kk and Dd, fubftitute the " + 2 Fluxions x and and it will be - - *- or v proportional Jy, = , a y a* J + x*y a*x = o. From D to AB and de let fall the Perpendiculars DE and Dg-, which in Dg meets de, parallel to DE, g. Then the ultimate form of the Trilineum Ddg will be that of a right-lined Triangle fimi- / lar to : : F DAE. Whence "Dd dg :: AD AE = v /iJJ$ D&q. Make AD = a, BD=_>', and DE=x; and for the Moments Dd, dg, fubftitute their proportional Fluxions y and x, and it will

1 1 1 3- 1 1 be : x : : a : a AT*. : x :: a* : a x or y \/ Or^ , a^y 1- - x^y* a'-x = o. the Hence Fluxion of an Arch, whofe right Sine is x, being and likewife the Fluxion of an whofe exprefs'd by f^_^ ; Arch,

Sine is if thefe Arches are to right y, being exprefs'd by i?_ ,. ; each other as i to m, their Fluxions will be in the fame proportion, "x v x and vice versa. Therefore , : x : : i : ;, or . . J a x */ a y da x or l = -T37i } putting #= i, 'tis a*y

== o ; the fame as before refolved. ' Equation We might derive other Fluxional Equations, of a like nature with thefe, which would be accommodated to Trigonometrical ufes. As if/ were the Circular Arch, and x its verfed Sine, we mould have the a^x* = o. Or if were the Equation zaxy* x'-y'- y Arch, it would be l 4 and x the correfponding Secant, x^y* a*-x*y # x* = o. Or inftead of the natural, we might derive Equations for the artificial Sines, Tangents, Secants, &c. But I fhall leave thefe others that Difquifitions, and many fuch might be propofed, to ex- ercile the Induftry and Sagacity of the Learner.

SECT, and INFINITE SERIES. 321

VI. An 'SECT. Analytical Appendix', explaining fome Terms and Expreffions in the foregoing work.

mention has been frequently made of given Equations, and others a ad and the like I mall take oc- BEcaufe framed libitum, ; calion from hence, by way of Appendix, to attempt fome kind of or explanation of this Mathematical Language, of the Terms giver/, and or which afligjfd, affiimed, required Quantities Equations, may give light to fome things that may otherwife feem obfcure, and doubts and are to arife in may remove fome fcruples, which apt the Mind of a Learner. Now the origin of fuch kind of ExpreiTions of in all probability feems to be this. The whole affair purfuing Mathematical Inquiries, or of refolving Problems, is fuppofed (tho' or the Pro- tacitely) to be tranfacled between two Perfons, Parties, or the pofer and the Refolver of the Problem, (if you pleafe) between like Mafter (or Inftruclor) and his Scholar. Hence this, and fuch Phrafes, datam reffam, vel datum angtthim, in iniperata rations Je- care. As Examples inftrudl better than Precepts, or perhaps when both are join'd together they inftrucl beft, the Mafter is fuppos'd to propofe a Queftion or Problem to his Scholar, and to chufe fuch as thinks fit the Scholar is Terms and Conditions he ; and obliged to folve the Problem with thofe limitations and reftriclions, with thofe Terms and Conditions, and no other. Indeed it is required on the part of the Mafter, that the Conditions he propofes may be confident with one another for if involve ; they any inconfiftency or contradiction, the Problem will be unfair, or will become abfurd and impomble, as the Solution will afterwards difcover. Now thefe Conditions, thefe Points, Lines, Angles, Numbers, Equations, Gfr. that at firft enter the ftate of the Queftion, or are fuppofed to be chofen or given by the Mafter, are the data of the Problem, and the Anfwers he to receive are the it fometimes expects qii(?/ita. As may happen, that the data may be more than are neceffary for determining ft'.on and lo interfere with one and the the^Qiu , perhaps may another, now become fo be Problem (as propofed) may impolTible ; they may fewer than are neceffary, and the Problem thence will be indetermin'd, and may require other Conditions to be given, in order to a compleat De- termination, or perfectly to fulfil the quafita. In this cafe the Scholar is to what is and at his difcretioa a me fuch and fo fupply wanting, miy (Jit many otherTerms and Conditions, Equations and Limitations, as he finds T t will 322 7&? Method of FLUXIONS,

will be neceffary to his purpofe, and will befl conduce to the fim- the and neateft Solution that be pleft, eafieft, may had, and yet in the moft general manner. For it is convenient the Problem fhouM as as the be propofed particular may be, better to fix the Imagina- and the Solution mould be made as tion ; .yet general as poffible, that it may be the more inftrucHve, and extend to all cafes of a like nature.

Indeed the word datum is often ufed in a fenfe which is fome- different from but which thing this, ultimately centers in it. As is call'd a when one is not that datum, quantity immediately given, however is infer'd from which but neceffirily another,, other perhaps infer'd from a and fo on in a is neceffarily third, continued Series, it is infer'd from a which is till neceffarily quantity, known or given the fenfe before This is the Notion in explained. of Euclid'?, data, other of that and Analytical Argumentations kind. Again, that is often call'd a given quantity, which always remains conftant and in- while other or circumftances variable, quantities vary ; becaufe fuch as thefe only can be the given quantities in, a Problem, when taken in the foregoing fenfe. all this the fenfible and To make more intelligible, I /hall have, a few of recourfe to pradlical inftances, by way Dialogue, (which, was the old didadlic method,) between Mafter and Scholar; and or this only in the common Algebra Analyticks, in which I fhall borrow my Examples from our Author's admirable Treatife of Univerfal Arithmetick. The chief artifice of this manner of Solu- in that as faft as tion will confift this, the Mafter propofes the Con- ditions of his Queftion, the Scholar applies thole Conditions to from them makes all the ufe, argues Analytically, aeceffary deductions, and derives fuch confequences from them, in the fame order as he will they are propofed, apprehends be mcft fubfervient to the he that can do this, in all Solution. And cafes, after the fureft, fim- readieft will be the beft pleft, and manner, ex-tempore Mathemati- cian. But this method will be beft from the explain'd following Examples. to I. M. A Gentleman being 'willing diftribiite Abns S. Let the Sum he intended to diftribute be reprefented by x. M. Among S. Let the number of be then - fbme poor people. poor _}>, would been the fhare of each. have M. He wanted 3 fiillings S. Make for the lake of and let 3 = rf, univerfality, the pecuniary Unit be then the to be distributed one Shilling ; Sum would have been x-{-a, and" and INFINITE SERIES. 32"

and the fhnre of each would have been ^^- . M. So that each y S. ^ might receive $ fallings. Make 5=^, then = b, whence he x = by a. M. "Therefore gave every ot.e 4 fallings. S. Make the diftributed will be 4=f, then Money cy. M. And he has 10 S. Make io fallings remaining. = d, then cy -f- d was the Money he intended at firft to or d diftribute; cy -+- = (x =) by- a, or

=5 . M. J^rf* w<2J the number ? S. The y ^jt-f of poor people

3 "*"' number was y = 7 = = 1-2. M. ^W /60w much Alms '' ? 4 tf/tf Of tff ^ry? intend to diftribute ? S. He had at firft x = by a = 5x13 3 = 62 fhillings. M. How do you prove your Solution? was firft S. His Money at 62 fhillings, and the number of poor 1 But if his had been === r people was 3. Money 62-4-3 ^5 ^ 3 x 5 each have fhillings, then poor perfon might received 5 millings. But as he to each that will be gives 4 fhillings, 13x4=52 fhillings diftributed in all, which will leave him a Remainder of 62 52 c= 10 fhillings. at his II. M. A young Merchant, firjl entrance npon bufmefs, began the World with a certain Sum of Money. S. Let that Sum be x, the pecuniary Unit being one Pound. M. Out of which, to maintain he 100 S. the himfclf the Jirjl year, expended pounds. Make given number ioo = tf; then he had to trade with x a. M. He at the end the it traded with the reft, and of year had improved by a I will aflume third part. .S. For univerfality-fake the general number and will make = n i, n = then the n, ^ (or ;) Improve- ment was n i xx a = nx na x -f- a, and the Trading- fiock and Improvement together, at the end of the firft year, was did the the S. MX na. M. He fame thing fecond year. That is, his whole Stock being now nx na, deducting a, his Expences for this year, he would have nx na a for a Trading-ftock, and nx na or n ix a, n'-x n*a nx -f- a for this year's Im- which provement, together make n'-x n*a na for his Eftate at of the fecond the die end year. M. As aljo third year. S. His a x whole Stock being now ;< x n a na, taking out his Expences for the third year, his Trading-ftock will be n*x n'-a na a, ^~ and the Improvement this year will be n i X*A- n'-a na a, J or x n=a n'-x -f- a, and the Stock and Improvement together, or his whole Eftate at the end of the third year will be n*x n*a

1 _ n a na, or in a better form n"'x -+ "-^na. In like manner T 2 if 324 Th* Method of FLUXIONS, the fourth his Eftate now n*x if he proceeded thus year, being n i a _ rf a na, taking out this year's Expence, his Trading-flock will be n>x n>a if a na a, and this year's Improvement is n Fx n=x is a ifa na a, or n*x n^a n*x -f- a, which added to his Trading-ftock will be n*x n*a tfa 72-^2 * na, or 11* x -f- na, for his Eftate at the end of the fourth

his s - year. And fo, by Induction, Eftate will be found n x -4- na at the of fifth And end the year. univerfally, if I aflume the ge- m _~ l neral Number m. his Eftate will be nx -f- -na at the end of i any number of years denoted by m. M. But he made his Eftate m double to -what it was at S. 2 then n firft. Make = , x -t- m m - or - = bx, x = ==-#. M. At the end l _^na of 3 years. n i x n" b

S. Then #2=3, a-=. 100, b = 2, n = %, and therefore x=s..

400 = 1480. M. j%!2

caufe it is c : : : x : - will move his whole x in the f , A fpace

time Alfo I will aflume the Y Velocity of B to be fuch, that it will move 8 = d Miles in 3 ==:g Hours. Then becaufe it is d :

:: e x : will move his g j-g> B whole fpace e x in the time

But moves a - - - '-7%. M. A given time S. Let that time be i b Hour. M. = Before B begins to move. S. Then A"s time is to U's time added to the time h or equal y ^ = '^g -f- h. M,' and INFINITE SERIES, 325 will be the that each M. Where will they meet., or what fpace ? S. From this we fhall have x = have defcribed Equation * s*i 1^ r x = Miles, which will X7=='/ X7==' J 7' 35 oxz-r7x2 x 3' 37 e be the whole fpace defcribed by A. Then x= 59 35 = 24 Miles will be the whole fpace defcribed by B. Acres . IV. M. Jf 12 Oxen can be maintained by the Pafture 0/37 S. Make 12 of Meadow-groundfor 4 weeks, = #, 3! = ^, 4=cj to be determin'd after- then aiTuming the general Numbers e, f, h, wards as occafion Ihall require, we mall have by analogy

Oxen If 3 26 7%e Method ^FLUXIONS,

- ~* 1

Time Oxen Time Oxen

,, ace h c . ace df *' , df - c - " - c - tO ' t r Ti 7 m T TZ ' * bh f c h bh which will be the number of Oxen that may be maintain'd by the of it growth only the pafture e, during the whole time h. But was found before, that without this growth of the Grafs, the Oxen ^ might be maintain'd by the pafture e for the time h. There- ~ "" fore thefe two or x - will be the together, ^ _f_ -^-^ , ber of Oxen that may be maintain'd by the pafture e, and its growth together, during the time b. M. How many Oxen may be main- 1 8 week s ? S. x to tain d by 24 Acres offitch pafture for Suppofe be that number of Oxen, and make 24=^, and h= 18. Then by analogy Oxen Pafture If Then ex J requ ij. e

. , r i ex ace And conlequentlyT J = j-r g bh dft ~ a^j ac /> r x f jf = T +

1 2 x i 21*9 4 14 fi

I O J-j V. M. If I have an Annuity S. Let x be the prefent value i then x* will of i pound to be received year hence, (by analogy) of I to be received 2 &c. be the prefent value pound years hence, will be the value of i to be re- and in general, x" prefent pound in the cafe of an the ceived m years hence. Therefore, Annuity, Series x -f x* -+- x"' ~+- x*, &c. to be continued to fo many Terms as there are Units in m, will be the prefent value of the whole be for Annuity of i pound, to continued m years. But becaufe

+ . 1 r x'' 4 &c. continued to fo -- =x-{- x -f- H-A' , many Terms I X Units in Divifion therefore as there are in, (as may appear by 3)

y *~ of i will reprefent the Amount of an Annuity pound,

to be continued for ;;/ years. M. Of Pounds. S. Make = a* and INFINITE SERIES. 327 then the = a, Amount of this Annuity for m years will be

a. M. To be continued ^ for 5 years fuccejji'vely. S. Then Which I S. m = 5. M. Jellfor pounds in ready Money. Make

then 1 = c, -a = c, or x"" -- I x

In any particular cafe the value of x may be found by the Refolu- tion of this affected Equation. M. What Interefl am I alfav'd per centum per annum? S. Make 100 = ^; then becaufe x is the prefent value of i pound to be received i year hence, or (which is the fame thing) becaufe the prefent Money x, if put out to ufe, in i year will produce i pound; the Intereft alone of i pound for i will be i year x, and therefore the Intereft of 100 (or K) for i will be pounds year be b bx y which will known when x is known. And this might be fufficknt to (hew the conveniency of this Me- thod but I mall farther illuftrate it ; by one Geometrical Problem, which mall be our Author's LVII. VI. In the M. right Line AB I give you the ftuo points A and B. their S. Then diftance AB = m is given alfo. M. As likenaife the two points C and D out of the Line AB, S. Then conlequently the figure ACBD is given in magnitude and fpe- cie ; and producing CA and CB towards d

and

:: = = G/. Subtraft this from Ge-=&, and there Bf(c) :/A ; - will remain le=.b . Then becaufe of the fimilar Triangles chk

it and will be : : : ck : ch= . And CHK, CK (d) CH (e) (y) 'j

: :: ck : . A/J . CK (/O HK (/) ( hk = Therefore = AB y] -\

hk m X But it is A/6 x : cb Bk = .. (m 'f) 2) ::

c -v b : le . Therefore m x x = or (a) (b ) f J ^,

dc ae bft. xv demy bdx* + bdmx = o. In which the indeterminate x and arife Equation, becaufe quantities y only to two Dimenfions, it mews that the Curve defcribed by the point C is a Conic Section. in M. Ton have therefore folved the Problem general, but you fionld Conic Sections in now apply your Solution to the feveral fpecies of par-

done in the manner : ticular. S. That may eafily be following e + l'f ctl __ 2 an(j then the will be- p t foregoing Equation - bdmx and ex- come fcf zpcxj> demy bdx'- 4- = o, by trading and INFINITE SERIES. 329

the - trading Square root it will be y = -.x -f. -f-

ft 1*"1 ''d " . . . . V I'P I

~ L that if the Term 4- x X were abfent, or if -4- = o, or j jj ^

that if -- its r = ; is, the quantity (changing fign) fhould

to then the be a Parabola. But if the be equal ^ , Curve would fame Term were prefent, and equal to fome affirmative quantity, ? - that is, if -f- be affirmative, (which will always be when

is or >} affirmative, if it be negative and lefs than -. the Curve

will be an Hyperbola. Laftly, if the fame Term were prefent and

negative, (which can only be when - is negative, and greater than

the Curve will be an or a Circle. y> Ellipfls I mould make an apology to the Reader, for this Digreffion from the Method of Fluxions, if I did not hope it might contribute to his entertainment at leaft, if not to his improvement. And I am whoever fhall fully convinced by experience, that go through the reft of our Author's curious Problems, in the fame manner, (where- he left in, according to his ufual brevity, has many things to be or fupply'd by the fagacity of his Reader,) fuch other Queflions and Mathematical Diiquifitions, whether Arithmetical, Algebraical, collected Geometrical, &c. as may eafily be from Books treating I fhall do this after the on theie Subjects ; fay, whoever foregoing manner, will find it a very agreeable as well as profitable exercife : of As being the proper means to acquire a habit Investigation, or of arguing furely, methodically, and Analytically, even in other as well as as Sciences fuch are purely Mathematical ; which is the great end to be aim'd at by thefe Studies.

U u SECT, 330 7%e Method of FLUXIONS,

VII. The a SECT. Conclufion ; containing Jhort recapitulation or review of the whole.

E are now arrived at a period, which may properly enough be call'd the tie conclujion of Method of Fluxions ami Infinite for the of is Series ; defign this Method to teach the nature of Series in and of Fluxions and general, Fluents, what they are, how they are and what derived, Operations they may undergo ; which defign be faid be (I think) may now to accompliili'd. As to the application of this Method, and the ufes of thefe Operations, which is all that now remains, we mall find them infilled on at large by the Author in the curious Geometrical Problems that follow. For the whole that can be either Series or done, by by Fluxions, may eafily be to the reduced Refolution of Equations, either Algebraical or Fluxional, as it has been already deliver'd, and will be farther apply'd and purfued in the fequel. I have continued my Annotations in a like manner upon that part of the Work, and intended to have added them here but the matter to fo faft under ; finding grow my hands, and feeing how impoffible it was to do it juftice within fuch narrow limits, and alfo perceiving this work was already grown to a competent fize; I refolved to lay it before the Mathematical Reader unfinifh'd as it is, referving the completion of it to a future if I mall find opportunity, my prefent attempts to prove acceptable. Therefore all that remains to be done here is this, to make a kind of review of what has been hitherto deliver'd, and to give a fum- in mary account of it, order to acquit myfelf of a Promiie I made in the Preface. And having there done this already, as to the Au- thor's part of the work, I (hall now only make a fhort recapitulation of what is contain'd in my own Comment upon it. And firft in my Annotations upon what I call the Introduction, or the Refolution of Equations by infinite Series, I have amply purfued a ufeful hint given us by the Author, that Arithmetick and Algebra are but one and the fame Science, and bear a ftridl analogy to each both in their Notation and the firft other, Operations ; computing after a definite and particular manner, the latter after a general and indefinite manner : So that both together compofe but one uniform Science of Computation. For as in common Arith- metick we reckon by the Root Ten, and the feveral Powers of that fo in or the Root ; Algebra, Analyticks, when Terms are orderly dilpos'd and INFINITE SERIES. 331

is we reckon other Root and its difpos'd as prefcribed, by any Powers, or we may take any general Number for the Root of our Arithmetical Scale, by which to exprefs and compute any Numbers as in common Arithmetick we required. And approximate continually to the truth, by admitting Decimal Parts /;; infnititm, or by the ufe of Decimal Fractions, which are compofed of the reciprocal Ten fo in our Author's or in Powers of the Root ; improved Algebra, the Method of infinite converging Series, we may continually ap- or an fuc- proximate to the Number Quantity required, by orderly cefiion of Fractions, which are compofed of the reciprocal Powers the in of any Root in general. And known Operations common Arithmetick, having a due regard to Analogy, will generally afford for the like us proper patterns and fpecimens, performing Operations in this Univerfal Arithmetick. Hence I proceed to make fome Inquiries into the nature and formation of infinite Series in general, and particularly into their of and where- two principal circumftances Convergency Divergency; in I attempt to (hew, that in all fuch Series, whether converging is a if not is or diverging, there always Supplement, which exprefs'd which when it can be ai- however to be underftood ; Supplement, certained and admitted, will render the Series finite, perfect, and accurate. That in diverging Series this Supplement muft indifpen- fablv be admitted and exhibited, or otherwise the Conclufion will be imperfect and erroneous. But in converging Series this Supplement it diminifhes the may be neglected, becaufe continually with Terms of the Series, and finally becomes lefs than any affignable quantity. And hence arifes the benefit and conveniency of infinite converging whereas that is fo and Scries ; that Supplement commonly implicated of entangled with the Terms the Series, as often to be impoiliblc to in Series it be be extricated and exhibited ; converging may fafely neg- the re- lected, and yet we mall continually approximate to quantity a in quired, And of this I produce variety of Inftances, numerical and other Series. I then go on to mew the Operations, by which infinite Scries are either produced, or which, when produced, they may occasionally undergo. As firft when fimple fpccious Equations, or purs Powers, are to be refolved into fuch Series, whether by Divifion, or by Ex- traction of Roots ; where I take notice of the ufe of the afore-men- Scries be render'd that tion'd Supplement, by which may finite, is, may be compared with other quantities, which are confider'd as ufeful given. I then deduce feveral Theorems, or other Artifices, Una for 332 tte Method of FLUXION s, for the more expeditious Multiplication, Divifion, Involution, and of infinite Evolution Series, by which they may be eafily and rea- in all cafes. I the dily managed Then fhew ufe of thefe in pure Equations, or Extractions; from whence I take occasion to introduce a new praxis of Refolution, which I believe will be found to be very eafy, natural, and general, and which is afterwards ap- of ply'd to all fpecies Equations. I on with our Then go Author to the Exegefis numerofa, or to Solution of affefted in the Equations Numbers ; where we mall find his Method to be the fame that has been publifh'd more than once in his be other of pieces, to very {hort, neat, and elegant, and was a great the its firft Improvement at time of publication. This Method is here farther explain'd, and upon the fame Principles a general Theo- rem is form'd, and diftributed into feveral fubordinate Cafes, by which the Root of any Numerical Equation, whether pure or af- be with fected, may computed great exactnefs and facility. From Numeral we pafs on to the Refolution of Literal or Speci- affected infinite Series in the firfl ous Equations by ; which and chief difficulty to be overcome, confifts in determining the forms of the feveral Series that will arife, and in finding their initial Approxima- tions. Thefe circumftances will depend upon fuch Powers of the Relate and Correlate Quantities, with their Coefficients, as may happen to be found promifcuoufly in the given Equation. Therefore the Terms of this Equation are to be difpofed in longum & in latum, or at lenft the Indices of thofe Powers, according to a combined in as is there or Arithmetical Progreffion p/ano, explain'd ; according to our Author's ingenious Artifice of the Parallelogram and Ruler, the reafon and foundation of which are here fully laid open. This will determine all the cafes of exterior Terms, together with the of the Indices and therefore all the -forms Progreffions ; of the feveral Series that may be derived for the Root, as alfo their initial Coefficients, Terms, or Approximations. of We then farther profecute the Refolution Specious Equations, of or a by diverfe Methods Analyfis -, we give great variety of Pro- which the Series for the are cefTes, by Roots eafily produced to any of are number Terms required. Thefe ProcefTes generally very lim- and the Theorems before ple, depend chiefly upon deliver'd, for finding the Terms of any Power or Root of an infinite Series. And the whole is illustrated and exemplify 'd by a great variety of In- ftances, which are chiefly thofe of our Author. The and INFINITE SERIES. 333

The Method of infinite Series being thus fufficiently dilcufs'd, we make a Tranfition to the Method of Fluxions, wherein the nature and foundation of that Method is explain'd at large. And fome general Observations are made, chiefly from the Science of Rational the is Mechanicks, by which whole Method divided and diftinguiih'd into its two grand Branches or Problems, which are the Diredt and Inverfe Methods of Fluxions. And fome preparatory Nota- tions are deliver'd and explain'd, which equally concern both thefe Me- thods. I then proceed with my Annotations upon the Author's firft Pro- blem, or the Relation of the flowing Quantities being given, to determine the Relation of their Fluxions. I treat here concerning Fluxions of the firft order, and the method of deducing their Equa- tions in all cafes. I explain our Author's way of taking the Fluxions of any given Equation, which is much more general and fcientifick than that which is ufually follow'd, and extends to all the varieties of Solutions. This is alfo apply'd to Equations involving feveral flowing Quantities, by which means it likewife comprehends thofe cafes, in which either compound, irrational, or mechanical Quan- tities may be included. But the Demonftration of Fluxions, and of the Method of taking them, is the chief thing to be confider'd here; which I have endeavour'd to make as clear, explicite, and fatisfactory as I was able, and to remove the difficulties and objections that have been raifed againft it : But with what fuccefs I muft leave to the judgment of others. I then treat concerning Fluxions of fuperior orders, and give the Method of deriving their Equations, with its Demonftration. For tho' our Author, in this Treatife, does not expreffly mention thefe orders of Fluxions, yet he has fometimes recourfe to them, tho' ta- I that are a citely and indirectly. have here ("hewn, they necelTary

refult from the nature and notion of nrft Fluxions ; and that all thefe feveral orders differ from each other, not abfolutely of and effentially, but only relatively and by way companion. as as And this I prove well from Geometry from Anaiyticks ; ot and I actually exhibit and make fenfible thefe feveral orders Fluxions.

But more efpecially in what I call the Geometrical and Mechani- the cal Elements of Fluxions, I lay open a general Method, by help of Curve-lines and their Tangents, to reprefent and exhibit Fluxions and Fluents in all cafes, with all their concomitant Symptoms and AffecYions, f 3e Method 334 ^ of FLUXIONS,

Aiic&ions, after a plain and familiar manner, and that even to ocular view and infpedlion. And thus I make them the Objects of Senfe, not their exiitence is by which only proved beyond all poflible con- the tradiftion, but alfo Method of deriving them is at the fame time fully evinced, verified, and illuftrated. Then follow my Annotations upon our Author's fecond Problem, or the Relation of the Fluxions being given, to determine the Re- lation of the or Fluents which is flowing Quantities ; the fame thing of as the Inverfe Method Fluxions. And firft I explain (what out- a Solution Author calls) particular of this Problem, becaufe it cannot be generally apply 'd, but takes place only in fuch Fluxional Equa- tions as have been, or at leaft might have been, previoufly derived from fome finite Algebraical or Fluential Equations. Whereas the Fluxional Equations that ufually occur, and whofe Fluents or Roots are required, are commonly fuch as, by reafon of Terms either re- or cannot be refolved this dundant deficient, by particular Solution ; but muft be refer'd to the following general Solution, which is here distributed into thefe three Cafes of Equations. The firft Cafe of Equations is, when the Ratio of the Fluxions of the Relate and Correlate Quantities, (which Terms are here ex- be the Terms of the plain'd,) can exprefs'd by Correlate Quantity alone in which Cafe the Root will be obtain'd an ; by eafy procefs : In finite Terms, when it may be done, or at leaft by an here a ufeful is infinite Series. And Rule explain'd, by which an infinite Expreffion may be always avoided in the Conclufion, often which otherwife would occur, and render the Solution inexpli- cable. The fecond Cafe of Equations comprehends fuch Fluxional Equa- tions, wherein the Powers of the Relate and Correlate Quantities, with their Fluxions, are any how involved. Tho' this Cale is much the it is a more operofe than former, yet folved by variety of eafy and fimple Analyfts, (more fimple and expeditious, I think, than thofe of our Author,) and is illuftrated by a numerous collection of Examples. laft Cafe of The third and Fluxional Equations is, when there are than two Fluents and their Fluxions involved more j which Cafe, without much trouble, is reduced to the two former. But here are alfo explain'd fome other matters, farther to illuftrate this Dodlrinej as the Author's Demonftration of the Inverfe Method of Fluxions, the Rationale of the Tranfmutation of the Origin of Fluents to other i places and INFINITE SERIES. 335 the of the Incre- places at pleafure, way finding contemporaneous ments of Fluents, and fuch like. Then to conclude the Method of Fluxions, a very convenient and is and for the Refolution general Method propofed explain'd, of all kinds of Equations, Algebraical or Fluxional, by having recourfe of Fluxions. This is to fuperior orders Method indeed not contain'd in our Author's prelent Work, but is contrived in purfu- ance of a notable hint he gives us, in another part of his Writings. And this Method is exemplify 'd by feveral curious and ufeful Pro- blems. Laftly, by way of Supplement or Appendix, fome Terms in the Mathematical Language arc farther explain'd, which frequently occur in the foregoing work, and which it is very neceflary to appre- a fort of Praxis is to this hend rightly. And Analytical adjoin'd to it the more and in which is Explanation, make plain intelligible ; exhibited a more direct and methodical way of refolving fuch Alge- as are or at- braical or Geometrical Problems ufually propofed ; an tempt is made, to teach us to argue more cloiely, dhtinctly, and Ana- lytically. And this is chiefly the fubftance of my Comment upon this part of our Author's work, in which my conduct has always been, to endeavour to digeft and explain every thing in the moft direct and natural order, and to derive it from the moft immediate and genuine I in the of a Principles. have always put myfelf place Learner, and have endeavour'd to make fuch Explanations, or to form this into fuch an Inftitution of Fluxions and infinite Series, as I imagined would have been ufeful and acceptable to myfelf, at the time when I fidl enter'd upon thefe Speculations. Matters of a trite and eafy nature

over a animadverfion : But in of I have pafs'd with flight things more or I have novelty, greater difficulty, always thought myfelf obliged and conlcious to to be more copious and explicite ; am myfelf, that I have every where proceeded cumjincero ammo docendi. Wherever I have fallen fhort of this defign, it fliould not be imputed to any want of care or good intentions, but rather to the want of fkill, or to the abftrufe nature of the lubject. I (hall be glad to fee my de- and (hall be fects fupply'd by abler hands, always willing and thank- ful to be better instructed. will and What perhaps give the greateft difficulty, may furnifli mod matter of objection, as I apprehend, will be the Explanations before given, of Moments, -vanifiing quantities, infinitely little quan- titles, 236 The Method of FLUXIONS, which our Author makes ufe of in this fjfies, and the like, Treatife, and elfe where, for deducing and demonftrating hisMethod of Fluxions. I fhall therefore here add a word or two to my foregoing Explana- this this feems to tions, in hopes farther to clear up matter. And be the more necefTary, becaufe many difficulties have been already ftarted about the abftracl nature of theie quantities, and by what name they ought to be call'd. It has even been pretended, that they and and it are utterly impoffible, inconceiveable, unintelligible, may therefore be thought to follow, that the Conclu lions derived by their at if not means muft be precarious leaft, erroneous and impoflible. Now to remove this difficulty it mould be obferved, that the only Author to denote thefe is the Symbol made ufe of by our quantities, or affected fome Coefficient. But this letter o, either by itfelf, by firft a finite and which Symbol o at reprefents ordinary quantity, mu ft be understood to diminim continually, and as it were by local till fome certain time it is and termi- Motion ; after quite exhaufted, nates in mere nothing. This is furely a very intelligible Notion. But to go on. In its approach towards nothing, and juft before it becomes abfolute nothing, or is quite exhaufted, it muft neceflarily of all it cannot pafs through vanifhing quantities proportions. For an to at once that were pafs from being affignable quantity nothing ; and not which is to to proceed per fa/turn, continually, contrary the it is an tho' ever fo Suppofition. While affignable quantity, little, it is not yet the exact truth, in geometrical rigor, but only an Ap- to it and to be it muft be lefs than proximation ; accurately true, that it muft be any affignable quantity whatfoever, is, a vanifhing the of a or quantity. Therefore Conception Moment, vanishing be admitted as a rational Notion. quantity, muft But it has been pretended, that the Mind cannot conceive quan- be fo far and fuch as thefe are tity to diminifh'd, quantities repre- I cannot fented as impoffible. Now perceive, even if this impofli- that the bility were granted, Argumentation would be at all affected or that the Concluiions would be the lefs certain. by it, The im- of arife from the narrownefs and poffibility Conception may imper- not fection of our Faculties, and from any inconfiftency in the nature of the thing. So that we need not be very folicitious about thefe the pofitive nature of quantities, which are fo volatile, fub- as to our tile, and fugitive, efcape Imagination ; nor need we be in what name are to be call'd but much pain, by they j we may confine ourfelves wholly to the ufe of them, and to difcover their properties, and INFINITE SERIES. 337 are not introduced for their own but properties. They fakes, only to us to the of as fo many intermediate fteps, bring knowledge other are to quantities, which real, intelligible, and required be known. It is fufficient that we arrive at them by a regular progrefs of di- and a and of and that minution, by juft neceflary way reafoning ; and leave us they are afterwards duly eliminated, intelligible and indubitable Conclusions. For this will always be the confequence, let the media of ratiocination be what they will, when we argue according to the ftriet Rules of Art. And it is a very common thing in Geometry, to make impoffib'e and nbfurd Suppofitions, which is the fame thing as to introduce iinpoffible quantities, and by their means to difcover truth. We have an inftance fimilar to this, in another fpecies of Quan- tities, which, though as inconceiveable and as impofTible as thefe can be, yet when they arife in Computations, they do not affect the Conclufion with their impoffibility, except when they ought fo to do; but when they are duly eliminated, by juft Methods of Reduction, the Conclufion always remains found and good. Thefe. Quantities are thofe Quadratick Surds, which are diftinguifh'd by and as the name of impoffible imaginary Quantities ; fuch ^/ i, &c. For ^/a, v/ 3, v/ 4, they import, that a quantity or number is to be found, which multiply'd by itfelf mall produce a - which is negative quantity ; manifeftly impoffible. And yet thefe have all varieties of to one quantities proportion another, as thofe to the aforegoing are proportional poffible and intelligible numbers 8cc. and when arife in I, ^/2, v/3, 2, respectively -,. they Compu- tations, and are regularly eliminated and excluded, they always leave a juft and good Conclufion. Thus, for Example, if we had the Cubick Equation x~> lax" from whence we were to extract the -J-4IX 42 =o, Root x ; by proceeding according to Rule, we mould have this fiird Ex- for the x - preffion Root, = 4 -f- y'3 4- v/ -"fr-f- ^J^ ,/ -^, in which the -~ is involved impoffible quantity ^/ ; and yet this Expreffion ought not to be rejected as abfurd and ufelefs, becaufe, by a due Reduction, we may derive the true Roots of the Equation from it. For when the Cubick Root of the firft inn- culum is rightly extracted, it will be found to be the impoffible Number i as and when the -+- ^/ , may appear by cubing ; Cubick Root of the fecond vinculum is extracted, it will be found

to be i thefe \/- j- Then by collecting Numbers, the X.x im- 338 77je Method of FLUXIONS, will be and the Root of impoffibie Number

THE

CONTENTS of the following Comment.

I on the Introduction or the ; Refolution of *-*/JNnotations Equations by Infinite Series. pag. 143 Sedt. I. the nature and Of conjlruttion of infinite or converging Series. P-H3 Sedt. II. The Refolution offimple Equations, or ofpure Powers, Series. = ~ by infinite p. 1 59

Sedt. III. The Numeral 1 Refolution of AffectedEquations, p. 8 6 Sedt. IV. The Refolution of Specious Equations by infinite Se- ries the the ; andfirjifor determining forms of Series, ami their initial Approximations. P- 1 9 1

Sedt. V. The Refolution ofAJfe&ed fpecious Equations proje-

various Methods . . cuted by ofAnalyjis. p. 209 Sedt. VI. Tranfition to the Method of Fluxions. P-235

II. Annotations on Prob. i. or, the Relation of the flow- to determine ing Quantities being given-t the Relation Fluxions. of their p. 241

I. Fluxions the to Sedt. Concerning of firft Order , and find their Equations. p.24i

Sedt. II. Concerning Fluxions of fuperior Orders, and the

method their > of deriving Equations. ] Seft. III. The Geometrical and Mechanical Elements of Fluxions, i p.266 [T] III. CGNTENTa

on Prob. 2. or the Relation the ,111. Annotations y of Fluxions being given, to determine the Relation of the Fluents. p.2 77 I. 'with a Se. A particular Solution ; preparation to the general Solution, by which it is dijlributed into three Cafes. p.a// Sedl. II. Solution the - 282 of firft Cafe of Equations. p. Sedt. III. Solution of the fecond Caje of Equations. -p.286 Seft. IV. Solution of the third Caje of Equations, with fome neceffary Demonftrations. - . P-3OQ

Sedt V. The Refolution of Equations, whether Algebraical or the Fluxional, by> afliftancediJ ofj j[uperiorr Orders ofj . - r luxions. - - p-3o Sedt. VI. An Analytical Appendix, explaining Jome Terms and ExpreJJiom in the foregoing Work. P-3 2 I VII. The a ,Se<5t. Conclufwn , containing j}:ort recapitulation or review of the whole. - - P-33 Reader is defired to correfl the following Errors, which I hope will be thought THEbut few, and fuch as in works of this kind are hardly to be avoided. 'Tis here neceflary to take notice of even literal Miftakes, which in fubjefts of this nature are often very material. That the Errors are fo few, is owing to the kind affillance of a flcilful Friend or

the IVefs ; as of a Printer. two,_ who fupplyfd my frequent abfence from, alfo to the care diligent ERRATA.

In tie Preface, pag. xiii< lirt, 3. read which Ibid. 1. read P. 1 I. 1 2. read CE x \Q is here fubjoin'd. 5. for matter read Hyperbola. 19. to the Fluxion of the' Area, and manner. Pag. xxiii. 1. alt. far Preface, &e. = ACEG ); nWConclufton of this Work. P- 7. \.T,i.for - =.. lDxIP = . Ibid. ~{- read P. 15. 1.9. ready !>**+ -&*' P.I3J.1.8. readJf

. -',4, &c. P'. 1. read P. \. 17. 17. 32. 19, read. ! . P. 135. 1. 15. read P. 9* 9" I. ^WAb&Jifs^AB. P. l,'2j. read - . P. 35, 1. 3: for lOtfjr* read 13.8. 9. 145. \.fenult. 3 read 7\~~ . P. 149. 1. 2O. read whkh irt; P. ~~r'~ 157. I.i3./-f^ ax. P.i68. l.j. retd^ax. loxty. P. 62. 1.4. read . P. 63.!. 31. P. 1 P-I7I. \.\j.fir Reread $*. 77.' \.l$.rcait

y- y~- . .... firyreatl-y. Ibld:\.ult.for read P. 1. 1 6. to P. 204. read 2m, 213^ [.-j. far- read Series. P. 1. 5 Species 229, 21* for x retu( P. 64. \.q. for 2 read z. Ibid. \. 30. read t. x 4. /i/V. 1. x 4'readx *. P. read 24. for 234. P. 82. 1. 17. read zzz. P. 87. 1. 22. 1. 2. ^or yy ready. P. 236. 1. 26. ;vW genera- + 2A>. Ibid. 1.22,24. reaJAVDK. P.t ting. P. 243. 1. 29. read. ax*yi*. P. 284.

1. uit. read i . P. 1. .7 -,', read \,2-[. read. * P.-gz. 1.5. read-\- . Ibid. 289. 17. fur right j^

1 left. P. 1. 2.- x 4 z read x. P. 1.8. . i, read' \.z\.for 104. read 6;jt 295. ', --^-J-ax*. ' P. read '. P. P. 109. 1. 33. dele as ofen. P. I 10. 1. 29. read 297. \.ig.forjx y* 298. 1.14.

read . y. 1\3O4. 1. 20, 21 dil: -(- be. P^og. m~t> 1 1. 1 8. a . 1 1. . read P. 3 7. tilt, read a'-j / 1 l ^ x P. 1 and v '=. nj, 1 7. for Parabola ADVERTISEMENT.

Lately publijtid by the Author,

BRITISH HEMISPHERE, or a Map of a new contrivance, for Minds in the firft THEproper initiating young Rudiments of Geography, and the ufe of the Globes. It is in the form of a Half- Globe, of about 15 Inches Diameter, but comprehends the whole known Surface of our habitable Earth ; and mews the iituation of all the remarkable Places, as to their Longitude, Latitude, Bearing and Diftance from London, which is made the Center or Vertex of the Map. It is neatly fitted up, fo as to ferve as well for ornament fufficient Inftructions are to make it as ufe j and annex'd, intelligible to every Capacity.

Sold by W. REDKNAP, at the Leg and Dial near the Sun Tavern in at near St. Fleet-jlreet ; and by ]. SEN EX, the Globe Dunftan's Church. Price, Haifa Guinea. and INFINITE SERIES. 339 contribute to the Advancement of true Science. In fhort, it will than enable us to make a much greater progrefs and profkience, we othervvife can do, in cultivating and improving what I have elfewhere call'd The Philofopby of Quantity.

FINIS. 3T .

.. I